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Motion-assisted vat photopolymerization: an approach to high-resolution additive manufacturing

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MOTION-ASSISTED VAT PHOTOPOLYMERIZATION:
AN APPROACH TO HIGH-RESOLUTION ADDITIVE MANUFACTURING

By

Han Xu

A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(INDUSTRIAL AND SYSTEMS ENEGINEERING)

May 2023

Copyright 2023 Han Xu

i

Acknowledgments
The journey of my Ph.D. has been a unique and transformative experience, and it has been
shaped and enriched by the contributions of numerous individuals to whom I owe my deepest
gratitude.
At the forefront of these is my Ph.D. advisor, Yong Chen. His sagacious guidance,
unwavering support, and profound wisdom have been the cornerstone of my research work. His
mentorship has been an invaluable resource, and for this, I am profoundly indebted.
My heartfelt thanks are extended to my parents, Guohua Xu and Bin Yu. Their unwavering
faith in my abilities, constant encouragement, and boundless love have been my beacon of hope
and motivation throughout this journey.
My wife, Mingyang Li, deserves special mention. Her patience, understanding, and
consistent financial support have been my bedrock. Her faith in me has provided the strength I
needed to surmount the challenges of this journey.
I am deeply appreciative of my esteemed dissertation committee members, Qiang Huang,
Huangbo Zhao, and Satyandra Gupta. Their discerning feedback and invaluable critiques have
significantly shaped my research and broadened my perspectives.
My sincere gratitude goes out to the professors who have contributed immensely to my
research: Huachao Mao from Purdue University, Laiming Jiang from Sichuan University, Chi
Zhou from the University at Buffalo, Xiangjia Li from Arizona State University, and Yang Yang
from San Diego State University.
ii

To my colleagues in our lab - Jie Jing, Yang Xu, Yeowon Yoon, Zhonghao, Manqi Li, and
Yichen Cui, your intellectual companionship and collaborative efforts have added great value to
my research experience.
I am also indebted to the dedicated lab assistants who have played an integral role in our
work: Shuai Chen, Renzhi Hu, Junhong Zhu, Jiaqi Nie, Yiyu Chen, Jiahui Ren, Hanyu Zhao, and
Muqun Hu. Your diligence and assistance have significantly contributed to the success of our
research.
My friends - Siyuan Yao, Yiqi Li, Zifan Zhu, and Jinwen Ren, your unwavering support
and enduring friendship have been a source of comfort and strength during this journey.
Finally, I wish to recognize the University of Southern California and the Viterbi School
of Engineering for providing an enabling academic environment and the necessary resources that
facilitated my research.

iii

T able of Contents
Acknowledgments............................................................................................................................ i
List of Tables ................................................................................................................................. vi
List of Figures ............................................................................................................................... vii
Abstract ......................................................................................................................................... xii
Chapter 1: Introduction ................................................................................................................... 1
Chapter 2: Literature Review .......................................................................................................... 8
2.1 Lateral Resolution Challenges for Vat Photopolymerization ................................... 8
2.2 Vertical Resolution Challenges for Vat Photopolymerization ................................ 15
2.3 Surface Roughness Challenges for Vat Photopolymerization ................................ 19
2.4 Mask Shifting Technologies in Photolithography and Stereolithography .............. 29
Chapter 3: Research Questions and Hypotheses ........................................................................... 32
3.1 Statement of Problems ............................................................................................ 32
3.1.1 Limitations of Vertical Resolution in MIP-VPP .............................................. 33
3.1.2 Limitations of Lateral Resolution in MIP-VPP ............................................... 34
3.1.3 Limitations of Surface Roughness in MIP-VPP .............................................. 35
3.2 Hypothesis............................................................................................................... 37
Chapter 4: Results and Evaluations .............................................................................................. 40
4.1 Improving Vertical Resolution through Zooming-Focused MIP-VPP ................... 40
4.1.1 High Resolution Model Reconstruction via Frustum Layer Stacking ............. 41
iv

4.1.2 Overview of the Zooming-Focused MIP-VPP Process ................................... 49
4.1.3 Hardware Implementation of Zooming-Focused MIP-VPP ............................ 52
4.1.4 Parameter Optimization for Zooming-Focused MIP-VPP .............................. 53
4.1.5 Light Intensity Control of the Zooming Focused MIP-VPP ............................ 63
4.1.6 Printing Results and Discussion....................................................................... 77
4.1.7 Conclusion ....................................................................................................... 90
4.2 Improving Lateral Resolution through Spatiotemporal MIP-VPP ......................... 92
4.2.1 Modelling Non-Uniform Curing Behavior ...................................................... 93
4.2.2 Overview of the Spatiotemporal MIP-VPP Process ........................................ 95
4.2.3 Photopolymerization Simulation of Spatiotemporal MIP-VPP ....................... 97
4.2.4 Optimizing Mask Image Planning for Spatiotemporal MIP-VPP ................. 101
4.2.5 Generation of Mask Images using Error Diffusion Algorithm ...................... 104
4.2.6 Computational Results and Simulation Verification ..................................... 115
4.2.7 Experimental Validation ................................................................................ 121
4.2.8 Printing Results Discussion ........................................................................... 125
4.2.9 Conclusion ..................................................................................................... 129
4.3 Improving Surface Roughness through Vibration-Assisted MIP-VPP ................ 130
4.3.1 Issue of Contour Aliasing and 'Dark Zone' Aliasing ..................................... 134
4.3.2 Overview of the Vibration-Assisted MIP-VPP Process ................................ 142
4.3.3 Materials and Methods ................................................................................... 152
v

4.3.4 Results of Vibration-Assisted MIP-VPP ....................................................... 157
4.3.5 Discussion ...................................................................................................... 170
4.3.6 Conclusion ..................................................................................................... 172
Chapter 5: Conclusion and Recommendation ............................................................................. 174
5.1 Research Question and Hypothesis Testing Results ............................................. 174
5.2 Contribution .......................................................................................................... 177
5.3 Future Work .......................................................................................................... 178
Bibliography ............................................................................................................................... 182

vi

List of T ables
Table 1: Design Parameters for Zooming-Focused Printer
Table 2: Study of Computational Costs
Table 3: The X-Direction Resolution of the Piezo Actuator
Table 4: The Y-Direction Resolution of the Piezo Actuator

vii

List of Figures
Figure 1: The schematic plot of vat photopolymerization
Figure 2: Three main variants of vat photopolymerization
Figure 3: The structure and principle of DMD
Figure 4: The structure and principle of LCD
Figure 5: Industrial application of MIP-VPP with high resolution requirements
Figure 6: Projection-based microscale VPP technologies
Figure 7: Projection-based multi-scale VPP technologies
Figure 8: Technology invocation of the MIP-VPP to improve Z resolution
Figure 9: The lateral surface finish improvements in VPP
Figure 10: Layerless VPP Processes to improve the vertical surface finish
Figure 11: Post-coating processes to improve the vertical surface finish
Figure 12: Vertical resolution limited by staircase effect
Figure 13: Lateral resolution limited by pixel size and UV energy distribution
Figure 14: 3D printing of optical lenses using zooming-focused MIP-VPP
Figure 15: A schematic illustration of zooming-focused MIP-VPP
Figure 16: The generation of the frustum layer stacking
Figure 17: The profile of the slant angle in the Z direction
Figure 18: The profile of a frustum layer sliced from a hemispherical model
Figure 19: The schema of zooming-focused MIP-VPP
Figure 20: The hardware design and implement of zooming-focused MIP-VPP
Figure 21: The effect of the position of the optical system of zooming-focused MIP-VPP
on the fabrication scale
viii

Figure 22: The parameters affects the resolution of zooming-focused MIP-VPP
Figure 23: The parameters affect the fabrication speed of zooming-focus MIP-VPP
Figure 24: Manual focusing the mask images to building platform at different
magnification rate
Figure 25: Dynamic focusing the mask images to building platform at different
magnification rate
Figure 26: Continuous mask image change of zooming-focused mask image projection
Figure 27: Light intensity distribution model of zooming-focused MIP-VPP
Figure 28: Light cone captured by the objective lens
Figure 29: Simulation of the light intensity distribution on the building platform over
different magnification rate
Figure 30: Two steps compensation of relative light intensity
Figure 31: Two steps compensation of relative light intensity
Figure 32: Curing depth study of after two steps light intensity compensation
Figure 33: Comparison of the photopolymerization behavior between original and
compensated light intensity
Figure 34: Design of the compound parabolic concentrator
Figure 35: The testcase of the compound parabolic concentrator printed by zooming-
focused MIP-VPP
Figure 36: Dimensional accuracy of the 3D printed CPC elements
Figure 37: SEM images of the compound parabolic concentrator printed by zooming-
focused MIP-VPP
Figure 38: The AFM images of the surface roughness of the CPC elements
ix

Figure 39: Performance of the 3D printed compound parabolic concentrator
Figure 40: Fisheye lenses and laser beam expanders fabricated via zooming-focused MIP-
VPP
Figure 41: Optical microscope image of zooming-focused MIP-VPP printed fisheye lens
over different curvature
Figure 42: Optical performance of the 3D printed fisheye lenses
Figure 43: Experimentally measured MTF for zooming-focused MIP-VPP printed lens
Figure 44: 3D printed beam expander testcases
Figure 45: Beam expanding performance of the 3D-printed laser beam expander
Figure 46: Customized geometry printed by zooming-focused MIP-VPP
Figure 47: Simulation of the non-uniform polymerization behavior
Figure 48: Principle of subpixel shifting for n=2
Figure 49: Comparison of the mask image and simulated curing profile between static
process and sps-MIP-VPP process
Figure 50: Control of the dimensional tolerance and geometrical tolerance of the sps-MIP-
VPP
Figure 51: Light intensity model of each calibrated pixel
Figure 52: Illustration of boundary erosion method
Figure 53: An open-loop pixel blending process
Figure 54: Error diffusion algorithm based on closed-loop pixel blending process
Figure 55: Iteration process controlled by the optimization parameters
Figure 56: An example to show the error diffusion process
Figure 57: Adaptive error diffusion with improved efficiency
x

Figure 58: Simulation of the Subpixel shifting process
Figure 59: Simulation of the Lateral aliasing elimination via subpixel shifting process
Figure 60: Dimensional accuracy and positional accuracy simulation for multiple squares
Figure 61: The hardware of piezo-actuated subpixel shifting MIP-VPP system
Figure 62: Subpixel shifting implementation
Figure 63: Experimental results and analysis of the dimensional accuracy
Figure 64: Experimental results and analysis of the application
Figure 65: The spatial and temporal domain control of various vat photopolymerization
Figure 66: Surface roughness requirements for various industrial products
Figure 67: The contour aliasing and ‘dark zone’ aliasing on LCD mask images
Figure 68: A cylindrical testcase printed by commercial VPP 3D printer
Figure 69: The relative light intensity of a single pixel on the elegoo2.0 LCD photo mask
Figure 70: The convoluted light intensity distribution simulated from the single pixelution
in X direction
Figure 71: The practical measured convoluted light intensity of the mask image
Figure 72: The schema of the vibration assisted MIP-VPP to improve the surface
roughness
Figure 73: Vibration pattern of the LCD mask images
Figure 74: The simulation of the convoluted light intensity in X direction
Figure 75: The simulation of the convoluted light intensity in Y direction
Figure 76: The relationship between the light intensity variation and the numbers of steps
of vibration
Figure 77: The vibration pattern designed based on the light intensity simulation
xi

Figure 78: The piezo actuator and its controller
Figure 79: The design of piezo stage amplifier
Figure 80: The mask image shifting controlled by input signal
Figure 81: Schematic of the Vibration-Assisted MIP-VPP experimental hardware
Figure 82: Example of the cylinder testcases to measure the surface roughness
Figure 83: Surface roughness of the printed cylinder testcase with different vibration
pattern
Figure 84: Contour smoothness of vibration-assisted VPP
Figure 85: The contour smoothness of the bottom line of the triangle testcase
Figure 86: The contour smoothness of the slope line of the triangle testcase
Figure 87: The dimensional accuracy of the vibration-assisted VPP
Figure 88: The quantification comparison of the dimensional accuracy of the vibration
assisted VPP
Figure 89: The transparency of the parts printed by the vibration assisted VPP
Figure 90: Modulate transfer function of the USAF 1951 calibration target

xii

Abstract
Additive manufacturing (AM), which enables the direct fabrication of products with
complex geometries from computer-aided design (CAD) models in a single step, has gained
widespread adoption across various industries. Among the different AM techniques, mask image
projection-based vat photopolymerization (MIP-VPP) transforms liquid monomers into solid
polymers using digitally controlled ultraviolet (UV) mask images, pushing the resolution limits of
the AM process to sub-millimeter scales. The advent of MIP-VPP has paved the way for creating
products with industrial-grade precision, opening up opportunities for applications in dental,
surgical, biomedical, and even optical fields.
Most existing AM processes employ layer stacking, using thin sheets to create three-
dimensional products. Hundreds of laminar sheets, generated by slicing a CAD model, are
converted into pixelized mask images and projected onto the polymerization interface in a layer-
by-layer fashion. The intrinsic discontinuities in pixels and layers constrain the resolution in both
the lateral plane and the vertical direction, as well as the surface roughness of the printed parts.
Conventional methods attempt to increase resolution by reducing the pixel size of the mask image,
which in turn decreases fabrication speed. Owing to the additive nature of MIP-VPP, a trade-off
exists between process resolution and fabrication speed. This trade-off hampers the production
efficiency of MIP-VPP when fabricating high-resolution components, such as microfluidic devices,
optical components, and microneedles. In this dissertation, we aim to address the limitations of
feature resolution and surface roughness in MIP-VPP to expand its applications and enhance its
performance.

xiii

Traditional MIP-VPP processes use mask images with resolutions limited to tens of
micrometers to define geometry and cure one layer of resin in a few seconds. However, the
frequency of mask image changes can be thousands of hertz, leading to a mismatch between
exposure time and mask image frequency. Inspired by this, we developed motion-assisted MIP-
VPP, applying multiple mask images with adjusted positions, pixel sizes, or light intensities during
a single layer fabrication in split exposure times. This approach involves superimposing sub-mask
images to determine the curing result of one layer, improving resolution and surface quality
without sacrificing fabrication speed. Building upon this foundation, we investigated three
different motion-assisted methods to enhance resolution and surface quality.
This research addresses lateral resolution, vertical resolution, and surface roughness
limitations separately. Lateral resolution is constrained by mask image aliasing, resulting in
pixelated contours in digital masks. Vertical resolution is hindered by staircase aliasing generated
from the standard thin sheet layer stacking. Surface roughness is affected by both contour aliasing
and 'dark zone' aliasing, arising from pixels and light intensity distribution in the mask image. We
developed three mask-shifting MIP-VPP technologies to improve lateral and vertical resolution,
as well as surface roughness. Shifting the image mask laterally during one layer fabrication
enhances lateral resolution without reducing pixel size. Adjusting the magnification rate during
continuous printing creates smoother layer stacking without affecting fabrication speed. Applying
XY-plane vibration to the mask image improves surface smoothness without compromising
fabrication speed or dimensional accuracy. The methods, hardware, and software of the motion-
assisted MIP-VPP processes are thoroughly demonstrated in this dissertation.
We developed a zooming-focused mask image projection vat photopolymerization
(zooming-focused MIP-VPP) process to increase vertical resolution in additive manufacturing.
xiv

Zooming-focused MIP-VPP is a continuously printing AM system with a dynamically controlled
focus distance, enabling nano scale increments for mask image change during continuous printing.
This method offers two vertical resolution improvements: 1) reduced slicing thickness of the CAD
model by minimizing mask image increment, and 2) stacking 3D frustum sheets with controlled
slope angles instead of normal thin sheets, providing a closer approximation to the ideal model.
To support zooming-focused MIP-VPP fabrication, we designed a mask-shifting zooming focus
system that dynamically controls the positions of the imaging lens and mask image LCD screen.
The process planning method generates a shifting path for the zooming focus system, while light
intensity variation and photopolymerization behavior are modeled and compensated. This
approach allows high-resolution approximation of the vertical profile without sacrificing
fabrication speed.
We developed a spatiotemporal mask image projection vat photopolymerization (sps-MIP-
VPP) process to increase lateral resolution by implementing a mask shifting imaging projection
system and a corresponding data-driven mask image planning method. The sps-MIP-VPP exposes
a set of shifted sub-mask images in sub-pixel distances, yielding a photopolymerization result
equivalent to a higher-resolution mask image while maintaining fabrication speed. A digital light
processing (DLP) UV projector, mounted on piezo-actuated linear stages, facilitates sub-mask
image shifting. A rhomboid-based flexible hinge stroke amplifier amplifies the piezo actuator's
travel distance, enabling a 100Hz shifting frequency with minimal repetitive error. Traditional 2D
layer slicing software is inadequate for achieving sub-pixel resolution mask images, so we
proposed a data-driven mask image planning method that optimizes pixel blending, synchronizes
processes, characterizes light and material properties, and calibrates the system. This generic
xv

approach can be employed in various MIP-VPP systems, including continuous printing AM
systems like continuous liquid interface production (CLIP) and zooming focused MIP-VPP.
We developed a vibration-assisted mask image projection vat photopolymerization
(vibration-assisted MIP-VPP) process to enhance surface quality in additive manufacturing. This
innovative approach applies an XY-plane vibration to the mask image during fabrication. By
vibrating the mask images at high frequency during a single layer of printing, the light intensity
distribution between the pixels are uniformed, and pixelated aliasing are blurred. The vibration-
assisted MIP-VPP addresses both contour aliasing and 'dark zone' aliasing, which stem from pixel
discontinuities and uneven light intensity distribution in the mask image. To support this process,
we designed a mask-shifting system capable of rapidly and accurately applying vibrations to the
mask image during fabrication. This novel approach demonstrates the potential of vibration-
assisted MIP-VPP to push the boundaries of surface quality in MIP-VPP without sacrificing the
dimensional accuracy.
As a proof of concept, several industrial products were fabricated to verify the effectiveness
and efficiency of the proposed motion-assisted MIP-VPP. We demonstrated two optical
applications benefiting from the zooming-focused MIP-VPP: a compound parabolic concentrator,
exhibiting excellent solar energy gathering properties, and customized fisheye lenses with various
curvatures. Additionally, a dental model was fabricated using spatiotemporal MIP-VPP to
showcase the surface finishing and potential application of the process. These test cases
demonstrated the potential of motion-assisted MIP-VPP in broadening the application of 3D
printing for high-resolution industrial products.

1

Chapter 1: Introduction
Additive manufacturing (AM)
[1-3]
, also known as three-dimensional (3D) printing,
fabricates 3D objects by accumulating solid material
[4]
at designated locations, converting
computer-aided design (CAD) models
[5, 6]
into physical products. Compared to conventional
subtractive and reshaping manufacturing, AM sequentially adds materials to form the target
products
[7]
, significantly improving time efficiency, material efficiency, environmental
friendliness, human interaction, customizability, and shortening product development cycles
[8-13]
.
There are three main categories of AM processes: ceramic-based AM
[2]
, metallic-based AM
[14]
,
and polymer-based AM
[2]
. Our focus is on liquid-based AM, thanks to its broad application and
low-cost properties. Liquid-based AM can be further classified into vat-photopolymerization
[15]
,
liquid thermal polymerization
[16]
, direct ink writing
[17]
, and fused deposition
[18]
.
Among the various AM processes, vat-photopolymerization (VPP), also known as
stereolithography, stands out due to its rapid building speed, high dimensional accuracy, and
material diversity. It is widely used in prototyping, model presentation, biomedical devices
[19-21]
,
microfluidics
[22, 23]
, energy recycling
[24-26]
, metamaterials
[27-29]
, and the optical industry
[30-33]
. The
first VPP-based AM machine was developed by CW Hull
[34]
. In his design, 3D models in STL
format are sliced into thin sheets, each representing a cross-sectional image of the model with a
designated thickness. These sheets are deposited in a layer-by-layer sequence, controlled by a
computer system. The 3D object is physically formed through photopolymerization of liquid resin
triggered by ultraviolet (UV) radiation. In the original design (Figure 1), a scanning laser diode
focuses a laser beam top-down onto the surface of photocurable resin in a vat. Mounted on an X-
Y translation stage, the laser diode controls the laser beam's speed and direction. After scanning
2

one layer, the Z-stage moves down to recoat liquid resin on the newly fabricated part. This process
repeats until all layers are completed.

Figure 1: The schematic plot of vat photopolymerization
[35]

Compared to other AM technologies, VPP offers several advantages in terms of geometric
resolution, surface finishing, scalability, and material options
[36, 37]
. For instance, the ELEGOO
Mars Pro printer boasts a resolution of 50 μm. Surface roughness for 3D printed parts varies from
a few micrometers on flat surfaces to 100 μm on 45° slanted surfaces. Commercial mask
projection-based VPP fabrication speeds can reach 20-36mm per hour, and the building scale
ranges from the micro-scale (less than 1mm) to the centimeter-scale (around 50cm). Furthermore,
VPP supports a wide variety of printable materials, including plastics, ceramics, and biomedical
materials, catering to different functional requirements.
3

Figure 2: Three main variants of vat photopolymerization

VPP variants, classified by the dimension of the unit operation in the photopolymerization
of 3D objects, have evolved from Hull's prototype. They include line scanning-based VPP
[38]
, mask
image projection-based VPP (MIP-VPP)
[39]
, and volumetric-based VPP
[40]
, shown in Figure 2. As
process resolution increases, fabrication speed and scale decrease
[41]
. MIP-VPP has gained
popularity due to its rapid fabrication speed and reasonable resolution. MIP-VPP projects 2D UV
mask images onto the building platform, generating each layer in one shot. Mask images are
generated by a dynamic photomask driven by a microcontroller
[42]
. MIP-VPP offers a balance
between fabrication speed and resolution. Commercial MIP-VPP printers can achieve 50 μm
resolution
[43]
, while volumetric-based VPP can only reach 500 μm resolution
[40]
. MIP-VPP is also
ten times faster than laser scanning-based VPP
[43]
.
4

Figure 3: The structure and principle of DMD
[44]

MIP-VPP performance in fabrication speed, scale, and process resolution is largely
dependent on mask images. These images are generated by photomasks, which, unlike
conventional photolithography photomasks, can be dynamically controlled to create designated
patterns for each layer. Digital micromirror devices (DMD) or liquid crystal displays (LCD) are
typically used as dynamic photomasks. DMD is a micromirror array consisting of millions of
individually controlled micromirrors (Figure 3)
[39]
. Driven by an actuator, a micromirror can
switch between "on" and "off" states. When in the "on" state, light reflects off the mirror, passes
through an imaging lens, and focuses on the building platform. In the "off" state, light is reflected
into a black light absorber to prevent projection. UV light reflected from the DMD forms mask
images on the building platform.
Alternatively, an LCD screen can be used as a dynamic mask (Figure 4). Each pixel's
liquid crystal can be individually set to a "transparent" or "opaque" state by controlling the
voltage
[45]
. Contracted UV radiation from the light source passes through transparent pixels and is
blocked by opaque pixels. Unlike DMD-based VPP, LCD-based VPP is smaller, does not require
a lens for imaging, and is more compact. However, its large pixel size and low UV light
transmission rate limit fabrication speed and resolution
[35]
.
5

Figure 4: The structure and principle of LCD
[45]

In the MIP-VPP process, three primary issues arise: lateral resolution, vertical resolution,
and surface roughness. The lateral resolution is determined by the pixel size of the mask images
[46]
.
And there is a trade-off between lateral resolution and fabrication speed. Smaller pixel sizes result
in higher resolution, while fabrication speed is influenced by the total area of the mask image
[47]
.
A larger mask image also allows for faster fabrication of the same volume. To achieve both high
speed and high-resolution fabrication, it is necessary to simultaneously reduce pixel size and
increase the total area of the mask image. However, traditional methods that shrink the DMD
array's element size to a few micrometers are not only costly but also physically unscalable
[48]
.
The vertical resolution in MIP-VPP is limited by the staircase stepping effect. This process
inherently experiences staircase stepping aliasing when printing sloped surfaces
[49]
. Layer stacking
employs a series of right prisms to approximate these sloped surfaces, inevitably resulting in a
staircase-like appearance. To minimize staircase aliasing, it is necessary to use thinner layers, but
this approach simultaneously reduces fabrication speed and necessitates a more precise mechanical
system. The challenge of increasing process resolution without sacrificing fabrication speed and
scale has persisted as a problem in the MIP-VPP field for decades.
6

The surface roughness in MIP-VPP is limited by both the staircase stepping aliasing and
the lateral aliasing. The staircase stepping effect contributes to surface roughness on sloped
surfaces, while lateral aliasing stems from the pixelation of the mask images. The combination of
these two factors results in a less-than-ideal surface finish, especially when the object requires
intricate details or smooth contours. Addressing these limitations to achieve better surface quality
remains a challenge for MIP-VPP technology.
In the meantime, there exist considerable demands for fabricating objects with high-
resolution micro-features or smooth surface finishing. Examples are abundantly observed in
commercial products such as airtight connectors, ball bearing, micro lens array, artworks. Besides,
the recent research in metamaterials, customized optical components, 4D printing, and biomedical
brings a series of high-resolution applications for MIP-VPP, shown in Figure.5
[50-54]
. However,
the trade-off between the resolution and the speed limits the production efficiency of these
applications, preventing them from making use of the customizability and design freedom of MIP-
VPP.

Figure 5 ： Industrial application of MIP-VPP with high resolution requirements
[50-54]

7

There is a significant research gap between the growing demand for high-resolution
structures and the fabrication processes that enable them. As a result, it is both urgent and desirable
to develop a fabrication process capable of producing high-resolution products without sacrificing
fabrication speed. This proposal aims to investigate the aliasing of the MIP-VPP process and
develop both hardware and software approaches to improve its resolution.
Traditional MIP-VPP processes use mask images with resolutions limited to tens of
micrometers to define geometry and cure one layer of resin in just a few seconds. However, the
frequency of mask image changes can reach thousands of hertz, resulting in a mismatch between
exposure time and mask image frequency. Inspired by this, we developed motion-assisted MIP-
VPP, which applies multiple mask images with adjusted positions, pixel sizes, or light intensities
during a single layer fabrication in split exposure times. In this case, the curing result of one layer
is determined by the superimposed sub mask images, achieving high resolution. Moreover, we
found that shifting the photomask laterally increases the lateral resolution of MIP-VPP, while
shifting it vertically reduces stair-stepping. Lastly, we control the vibration frequency of MIP-VPP
to improve the surface smoothness of the printed object by eliminating contour aliasing and light
intensity variation.
The following sections are organized as follows: Section 2 presents a literature review on
related work aimed at improving the resolution and surface roughness of MIP-VPP; Section 3
outlines the key problems of resolution and surface roughness in MIP-VPP; Section 4 provides the
main results of the proposed techniques for improving resolution and surface roughness; and
Section 5 concludes with a discussion of future work.
8

Chapter 2: Literature Review
This section briefly summarizes the state of the art in VPP processes, with an emphasis on
methods for improving resolution and surface roughness. Firstly, we will review the methods for
improving lateral and vertical resolution, along with the trade-off between process resolution and
fabrication speed. Secondly, we will also review techniques for improving surface roughness,
including in-situ techniques and post-processing techniques. Finally, we will review mask shifting
techniques used in photolithography and stereolithography.
2.1 Lateral Resolution Challenges for Vat Photopolymerization
Feature resolution generally determines the minimal printable features of an AM process.
The resolution requirement varies depending on the specific applications of manufactured products.
Although hundreds of microns resolution are enough for consumer products such as phone shells,
pen holders, and home accessories, other products, like microfluidic chips, microneedle arrays,
and compound lenses, require resolution at tens of micrometers or smaller
[36]
. Hence, feature
resolution has been an essential part of any AM process. Since VPP can fabricate parts in three
dimensions, the resolutions of a VPP process should be measured and evaluated in the XY plane
and in the Z direction (the layer-building direction). The two resolutions differentiate in working
principles and affect the parts’ printability in different scenarios
[55]
.
The XY resolution is rated by the spot size of a laser or the pixel size of an LCD or DLP
projection systems, and depends on the photocurable materials used in VPP. It can be used to
estimate the minimum feature size, namely the cross-section area that can be fabricated in the XY
plane. The minimum printable feature size within an XY plane, such as discrete pillars, is usually
many times its laser spot size or pixel size. Many factors, including light sources, optics, and
9

material properties, jointly determine the final cured shape, which is usually larger than the
incident beam itself
[56, 57]
.
There exists an inherent trade-off within projection-based VPP between the XY resolution
and the print area
[58]
. The resolution of the projection-based VPP depends on the kind of projection
apparatus and its matching lens system and is coupled with the print area. A LCD is cheap but has
a larger pixel size and more importantly, lower optical contrast than a DLP. A DLP system has a
better resolution because of its high contrast and transmittance ratios
[35]
. Both DMD chips and
LCD screens have a limited quantity of pixels (e.g., 1920 1080), which constrains the print area
to maintain an acceptable resolution. The printing resolution will decrease with the increase of
pixel size and the simultaneous scaleup of the print area. For example, for microscale VPP
[59-63]
,
if the light source is mapped to a print area of 19.2 mm  10.8 mm, each pixel size is 10 µm.
However, for mesoscale VPP
[64-67]
, each pixel is typically around 100 µm if the print area is 192
mm  108 mm. This dilemma of the part-to-pixel (PTP) size ratio hinders projection-based VPP
for large-area and high-resolution fabrication.
A main approach to increase XY resolution is to design light illumination apparatus and
related optical systems with higher resolution. Researchers have proposed various methods to
reduce the pixel size and applied them in the projection-based
[59-63]
, laser-based
[68-71]
, and
volumetric
[40, 72-75]
VPP systems. The early version of the projection-based microscale VPP
apparatus was developed by Sun and his coworkers in 2005 using a DMD as the dynamic mask
[62]
. As shown in Figure 6a, a CAD model is first sliced into a sequence of 2D patterns. Then,
these 2D patterns are transferred to the DMD chip, which modulates the UV light illumination.
The mask image projection passes through a reduction lens and focuses on the resin surface with
a reduced pixel size as small as 1.1 μm. The patterned UV irradiation triggers localized
10

photopolymerization, forming one layer of the part. The Z stage lowers the printing platform to
refill fresh resins to fabricate the subsequent layers. The first developed projection-based micro
VPP system has demonstrated 3D microstructures, such as a micropillar array (Figure 6b) and
micro spring array (Figure 6c) with feature sizes as small as 25 µm. Later Zheng et al.
demonstrated the printing of ultralight and ultrastiff mechanical metamaterials using their
developed projection-based microscale VPP system (Figure 6d)
[63]
. Micro lattice structures with
a resolution of ~5 µm are successfully printed using photopolymers (Figure 6e). With further
postprocessing, such as electroless plating and atomic layer deposition, hollow-tube metallic
lattices (Figure 6f) and ceramic lattices (Figure 6g and 6h) were also fabricated. Inspired by the
computer numerically controlled (CNC) accumulation process, Yang et al. developed a microscale
3D printing process – immersed surface accumulation (ISA) 3D printing system in which a light
guide tool consisted of optical fibers and an objective lens for 2D light beam projection (Figure
6i)
[59]
. The light guide tool is merged inside a tank filled with photocurable liquid resin. By
combining dynamically controlled light beam projection and the 5-axis movement, the ISA-3D
printing method can selectively cure the liquid resin. It can project UV patterns on a 3.67 mm ×
2.75 mm with a resolution of 2.5 µm per pixel. Using the ISA-3D system, the authors successfully
printed bioinspired super-hydrophobic structures with a 35 µm feature size. Recently, Toombs et
al. developed a microscale computed axial lithography (micro-CAL) technique, one of microscale
volumetric VPP systems (Figure 6j)
[73]
. The printer coupled a laser light source into an optical
fiber and demagnified light pattern defined by the DMD. With gradient descent digital mask
optimization, the micro-CAL system enabled rapid printing of complex 3D microstructures with
minimum feature sizes of 20 µm and 50 µm in polymer and fused silica glass, respectively.
11

Figure 6: Projection-based microscale VPP technologies. (a)-(c) The first proposed microscale
projection-based microscale VPP. Reprinted from
[62]
, Copyright (2005), with permission from Elsevier. (a)
Schematic of the system design. (b) high aspect-ratio micro rod array consists of 21 × 11 rods with an overall
size of 2 mm × 1 mm. The rod diameter and height are 30 μm and 1 mm, respectively. (c) Micro coil array with a
coil diameter of 100 μm and a wire diameter of 25 μm. (d)-(h) Fabrication of ultralight, ultrahigh-stiffness
stretch-dominated micro lattices. Reprinted from
[63]
, Copyright (2014), with permission from AAAS. (d)
Projection-based microscale VPP capable of fabricating arbitrary microscale 3D structures with a resolution of
~5 μm. Octet-truss micro lattices with varied constituent materials and configurations: (e) solid polymer HDDA;
(f) hollow-tube metallic Ni-P; (g) hollow-tube ceramic (alumina); (h) solid ceramic (alumina). (i) Schematic
illustrations of the ISA-3D printing system. Inserts show the magnification of the light guide tool, optical fiber
with projected 2D micro patterns, models, and SEM images of the 3D-printed eggbeater arrays. (j) and (k)
Volumetric VPP for microscale structures. Reprinted from
[73]
, Copyright (2022), with permission from AAAS. (j)
Schematic of microscale volumetric VPP setup. (k) A cubic lattice printed in monomeric photopolymer with a
minimum positive feature size of 20 μm.

Intensive research has been done to address the trade-off between part size and XY
resolution in the VPP technology. The first category of the strategies is to combine n×m projects
to cover an enlarged projection area or to combine two light sources with fabrication capabilities
12

at different scales
[76-79]
. Mao and his coworkers presented a multi-scale VPP system by using
dynamic apertures to shape laser beams for different size scales (Figure 7a)
[78]
. For example, a
laser with a large aperture size (0.5 mm) is utilized to fabricate the interior of a part or macroscale
features with a larger layer thickness, while a beam shaped by a tiny aperture size (50 μm) is used
to cure microscale features with a smaller layer thickness (Figure 7b). To further increase the
fabrication speed, Jia et al. utilized a DLP projector to fabricate the macroscale body of the part
and a focused laser beam to print the micro-textures (Figure 7c)
[77]
. The software system enables
designing the distribution and density of specific microscale textures on a macro-object by
generating projection images and a laser toolpath for the two integrated light sources. However,
this method still has a limited printable area due to the fixed light sources used in the VPP system.
The second type is to split a projection area into many sections such that multiple
projections at various places can be used to complete the layer-by-layer fabrication. A
straightforward method to break the print area limit is the well-known step-and-repeat process
[80,
81]
. A projector or the build platform moves and stops at a specified position by motorized stages
and then makes a projection. Each subarea is printed using a consistently higher resolution than
static projection. Figure 7d shows a large-area VPP 3D printer based on the step-and-repeat
principle developed by Lee et al.
[80]
. The projector stepped in the X direction after each exposure
until the end of the row. Then the projector shifts in the Y direction to repeat the previous process
until the end of the layer. Hence, macroscale objects can be fabricated without losing detailed
micro-features. However, the fabrication time will dramatically increase when manufacturing
large objects due to the frequent small displacement and on-off switch of the projector.
Later, researchers proposed a moving light strategy
[82-86]
to eliminate the extra transition
time in the step-and-repeat method. Unlike the step-and-repeat approach, the projector
13

continuously moves while simultaneously projecting mask images to finish the fabrication of one
row. The projector then shifts to the next row until the whole layer is fabricated. However, the
projector needs to constantly refresh the mask image with the movement of a one-pixel distance.
Otherwise, the motion blur effect will lead to incorrect curing of geometric features. Figure 7e
shows a DMD projection-based VPP setup based on the moving light method developed by Emami
et al.
[82]
. The process needs additional segments for an accelerated and decelerated motion for
each continuous movement. The image projection happens during uniform motion at a constant
scanning speed (0.64 mm/s). Given a layer thickness, a larger projector scanning speed is always
desired for fabrication efficiency. The high image refresh rate required for the projector limits the
achievable moving speed and printing efficiency.
Zheng et al. expanded the print area by utilizing two-axes galvo mirrors to rapidly move
the projection image’s position along the resin surface, similar to the laser-based VPP (Figure 7f)
[50]
. The method is fast (3.333 mm
3
/s) with high resolution (5 μm). However, it requires a
customized f-theta lens to compensate for the varying focal length so the reflected 2D images can
be focused on the building plane. Consequently, the limited field-of-view of the f-theta lens
constrains the achievable printing area (100 mm ×100 mm). Recently Yang et al. developed a
hopping light VPP (HL-VPP) method by synchronizing linear scanning projection with a galvo
mirror’s rotation to address the printing area limit and the printing efficiency due to limited
projector refresh rate (Figure 7g)
[87]
. The projector moves continuously at a constant speed while
the one-axis galvo mirror rotates periodically to compensate for the projector’s linear movement
so synchronized hopping motion can be achieved. By this means, HL-VPP can simultaneously
achieve large-area (over 200 mm), fast-speed (scanning speed of 13.5 mm/s), and high-resolution
14

(10 μm pixel size) fabrication. The distinguishing characteristic of HL-VPP is that it allows for
hundreds of times lower projector refresh rate (10 Hz) without motion blur.
Figure 7: Projection-based multi-scale VPP technologies. (a) and (b) Multi-scale VPP using shaped
beams. (a) Schematic of shaped beam optics. (b) Schematic diagram of the multi-scale fabrication process. The
multi-scale features in the XY plane are achieved by the large and small beams, and the multi-scale resolution
along the Z-axis is realized by the large and small layer thickness. (c) Multi-scale VPP printing of macroscale
objects with micro textures achieved by hybrid light sources. The projection-based VPP fabricates the
macroscale solid body, and microscale structures on its surfaces are manufactured by the laser-based VPP
method. (d) Fabrication procedure of Step-and-repeat approach. Reprinted from
[80]
, Copyright (2015), with
permission from Springer Nature. (e) Multi-scale VPP process design based on Moving Light strategy. Reprinted
from
[82]
, Copyright (2014), with permission from Elsevier. (f) Multi-scale VPP printing of hierarchical
mechanical metamaterial using two-axes galvo mirrors. Reprinted from
[50]
, Copyright (2016), with permission
15

from Springer Nature. (g) Structure of HL-VPP prototype and a large-area print with both microscale features
and macroscale features.

2.2 Vertical Resolution Challenges for Vat Photopolymerization
The resolution of VPP in the Z direction is rated by layer thickness. VPP technology can
be divided into top-down and bottom-up configurations based on the light source position. The
build platform is either immersed in the resin vat in the top-down configuration or dipped into the
resin vat in the bottom-up configuration. One layer of liquid resin is either constrained between
the resin vat and the build platform or left on the build platform freely. In comparison, the bottom-
up VPP configuration allows for much thinner layer thickness than the top-down VPP
configuration, especially for viscous resins
[88]
. A high layer resolution (i.e., small layer thickness)
allows for thin features in the Z direction and a good surface finish but will significantly increase
the total print time. For this reason, the layer thickness for commercial VPP machines is typically
25 ~ 100 μm. In addition to layer thickness, photocurable material’s optical properties play an
essential part in the curing process, determining the minimum features in the Z direction such as
overhang features or thin gaps. Jacob’s working curve describes the Z dimension for overhang
features without a constrained surface on the top. For internal void features like channels and gaps,
light irradiation to build the roof layer (i.e., the layer encloses the void) and subsequent layers will
potentially cure the resin inside the channels. This is called over-curing effect, which is common
in 3D printing microfluidic chips containing microchannel networks within transparent material
blocks.
VPP technologies have a severe limitation in the Z direction when fabricating features like
microchannels. Most VPP-based 3D printers can only fabricate transparent channels with more
16

than 200 µm height for the channel height dimension. This is a common issue in microfluidic
device fabrication, where transparency is desired. The practical limit of minimum channel height
results from resin’s over-curing. That is, the light irradiation required to build the channel roof can
potentially photopolymerize the resin residing inside the channel. In addition, additional light will
penetrate the previously built layers when curing the subsequent layers, leading to channels’
blockage.
Researchers have investigated different strategies to address this issue. One type of the
strategies is to decrease the light penetration depth of the used liquid resin by adding
photosensitizing additives for the visible blue light source (405 nm)
[89, 90]
or shifting the light
 source from visible blue light to UV light ( 385 nm)
[91, 92]
and adding UV-absorbing dyes
[93-96]
.
Zhang et al. employed a 405 nm light source and quinoline yellow as a photo absorber for VPP
resin and successfully fabricated microchannels with a cross-section as small as 100 μm × 100 μm
(Figure 8a). Gong et al. investigated different photo absorbers and process parameters
experimentally and finally chose Sudan I for their resin formulation (Figure 8b). With custom
resin formulation and optimized process parameters, they successfully fabricated 50-μm-height
microchannels on a commercially available 3D printer equipped with a 405 nm light source
(Figure 8c-8e). However, photo-absorbing dyes quinoline yellow and Sudan I render the prints
colored
[96]
. Kuo et al. enhanced the resolution of VPP-printed microchannels without
compromising transparency by using photosensitizer isopropyl thioxanthone (ITX)
[96]
. ITX
absorbs strongly at 385 nm (Figure 8f). This absorbance increase allows for VPP printing of
transparent microchannels of 400 μm height with commercially available 3D printers using a 385
nm light source, which is still quite big (Figure 8g). Based on the previous study
[89]
, Gong et al.
utilized a 385 nm UV light engine (Figure 8h), yellowish UV absorber – 2-nitrophenyl phenyl
17

sulfide (NPS), and optimized process parameters to achieve a flow channel with a height as small
as 18 μm (Figure 8i)
[94]
.
Yang et al. proposed a new approach called In-situ Transfer VPP (IsT-VPP) for
microchannels fabrication (Figure 8j)
[97]
. The key idea of IsT-VPP is to print the channel-roof
layer (i.e., the top layer portion that encloses the channel) separately via double exposure on the
resin vat surface using an additional build platform called an auxiliary platform. When printing
the channel roof, the auxiliary platform is utilized to prevent light penetration into the residual
liquid resin inside the channel. The channel roof is then in-situ transferred to the built part with the
second exposure of a planned mask image. All the other layers are printed using the normal VPP
process without the auxiliary platform. The integrated auxiliary platform dramatically cut down
the energy penetration to the liquid resin in the channels. With 405 nm light sources and
commercial photopolymers, the authors successfully demonstrate microfluidic channels with high
Z-resolution (within 10 μm) and accuracy (±1 μm) with no requirements on liquid resins such as
reduced transparency (Figure 8k).
18

Figure 8. Technology invocation of the MIP-VPP to improve Z resolution. (a) Cross-section and top
views of stepwise narrowing channel printed using optimized resin formulation with quinoline yellow and
perfused with dyed fluid. The insert shows the design of the narrowing channel. Reprinted from
[93]
, Copyright
(2017), with permission from The Royal Society of Chemistry. (b)-(e) Optical approach and application of Sudan
I in resin formulation for 3D-printed microfluidics. Reprinted from
[89]
, Copyright (2015), with permission from
The Royal Society of Chemistry. (b) Measured resin absorbance compared with 3D printer’s LED emission
spectrum for several commercial and custom resins with Sudan I. Channels with the height of (c) 50 μm (d) 70 μm
(e) 90 μm printed by optimized process parameters and resin formulation. (f) and (g) Application of ITX and 285
nm UV light source for 3D printing microfluidics. Reprinted from
[96]
, Copyright (2019), with permission from
John Wiley and Sons. (f) Absorbance spectra of resin compositions and the power spectrum of the 385 nm UV-
LED DLP projector. (g) Cross-section view of 3D-printed stacked channels using optimized resin formulation
containing ITX. (h) and (i) Custom 385 nm light engine-based 3D printer and resin containing NPS for 3D
printing microfluidic channels. Reprinted from
[94]
, Copyright (2017), with permission from The Royal Society of
Chemistry. (h) CAD model of the custom 3D printer using UV light engine. (i) SEM images showing cross
sections of flow channel printed with optimized resin formulation containing NPS. (j) and (k) IsT-VPP strategy
19

for transparent microfluidic device fabrication. (j) The detailed structure of the IsT-VPP apparatus. (k) SEM
image showing cross-sections of multi-layer microfluidic channel printed by IsT-VPP.

2.3 Surface Roughness Challenges for Vat Photopolymerization
Besides dimensional accuracy and feature resolution, surface quality is an essential end-
product property. The surface quality of an engineering part is an evaluation of its surface
imperfections, such as scratches, pits, digs, and textures
[98]
. Good surface quality can reduce
friction and improve wear resistance, resist corrosion, fatigue, and optical performance
[99-104]
.
Both surface roughness and surface defects are chosen as the indicator of the surface quality in
this review to cover the surface imperfections in different size scales. Depending on the specific
application of the products, the surface roughness requirements vary from millimeter scale for
regular structures and micrometer scale for surface appearance, to sub micrometer scale for
products’ mating requirements and nanometer scale for optical properties. The surface defects of
3D-printed parts such as digs or pits are millimeter scale, which are usually related to supporting
structures. Although VPP makes breakthrough in its achievable speed and resolution, the poor
surface quality has been regarded as a barrier for applying VPP to high precision application
[105-
108]
. For example, the conflicts between building size, printing speed, and surface roughness
challenge researchers in developing pragmatic VPP processes
[109, 110]
. VPP’s surface quality
highly depends on the adopted printing process. A 3D CAD model is first sliced into numbers of
thin layers, which are then used to sequentially fabricate each layer by projecting UV mask images
on the photocurable resin. Two types of common aliasing exist when building 3D objects. First,
mask image projection of each layer generates the lateral aliasing between the cured layer's
footprint and the sliced cross-sectional images
[111]
. Second, the layer-by-layer stacking of the thin
sheets generates the vertical aliasing known as stair-case stepping
[112, 113]
. To improve VPP’s
20

surface roughness, both the lateral and vertical aliasing need to be addressed. Besides, the removal
of the supporting structure are usually carried out by expensive manual polishing, bringing
surfaces defects due to such supports
[114]
. Research on new supporting strategy for VPP to reduce
surface defects is needed.
An UV mask image projected to the surface of a resin vat is used in the vat
photopolymerization to solidify a cross-section of 3D-printed parts. The lateral surface finish of
VPP is determined by the mask image resolution and the light intensity distribution
[54, 111]
.
Although the use of an ultra high-resolution LCD or DMD is a straightforward solution to tackle
the lateral aliasing, the cost of a photomask increases as the pixel size reduces. The typical size of
a commercial photomask limits the lateral surface finish to 10 - 50
[115, 116]
. Moreover, the
light intensity from a photomask pixel follows Gaussian distribution
[116, 117]
. The light intensity of
the mask image will convolute from all the mask pixels, inherently causing minute ‘dark’ spaces
between micro-mirrors or LCD elements
[118, 119]
. Hence, the development of VPP process for
higher mask image resolution and uniform light intensity distribution has been the target of many
researchers.
Researchers have developed many processes including grayscale pattern, focus deviation,
and mask image oscillation to get uniform light intensity distribution of a mask image. Thanks to
the high frequency control of a DLP projector, the grayscale values of a pixel in a mask images
can be easily set, enabling sub-voxel level light distribution control
[120-122]
. Luongo et al.
investigated the grayscale pattern of a single layer to print smooth microfeatures and thereby
modify the reflectance properties of the surface
[123]
. The bunny testcases were printed by the mask
images with designed grayscale patterns (Figure 9a). From the left to right, the bunny was printed
without any grayscale pattern applied, with an isotropic 2D sinusoid function applied, and using
21

sparse convolution noise with low and high amplitudes. Both the sinusoid pattern and the noise
function reduced surface roughness. Further study needs to be done to optimize the grayscale
patterns for smoother surface roughness. Besides grayscale patterns, focus control is widely used
in manipulating the mask images. Shan et al. proposed a defocused VPP by increasing the gap
between the LCD screen and the resin vat
[111]
. The defocused mask image flattens Gaussian
distribution of the light intensity of pixels
[124]
. The pixel convoluted light intensity results are
much more uniformed. The SEM images in Figure 9b present the top and side views of the straight
spoke testcases. Surfaces with similar roughness were printed by the conventional VPP with and
without grayscale patterns. In contrast, the defocused VPP process significantly smoothened both
top and side surfaces. A 1 μm surface roughness was achieved by the defocused VPP, compared
to 7 μm of the same surface without defocusing in printing. A similar idea to flatten the Gaussian
distribution of the UV energy is to oscillate the mask image
[54, 117, 118, 125]
. Chao et al. presented an
oscillation-assisted DLP-based 3D printing approach for fabricating micro lens arrays with
optically smooth surface
[54]
. Mechanical oscillation was applied to the projection lens of a bottom-
up micro-DLP projection system (Figure 9c). The 3D-printed micro lenses has about 1 nm surface
roughness, achieved by combining mechanical oscillation and grayscale UV exposure
[54]
. The
sinusoidal oscillation is performed along the diagonal direction of the projection area. The
oscillation of the projection lens smears the discrete pixels and eliminates the dark grids between
pixels. It was observed that the oscillation eliminated the light intensity fluctuation and unified the
light intensity within the mask projection image. The image resolution of the fabricated micro lens
arrays is further evaluated using the negative USAF1951 resolution test chart with six groups of
patterns from +2 to +7 and each group of the pattern consists of six elements. The highest
resolution achieved was 456.6 line/mm.
22

The formulation of photocurable resin is essential for VPP’s surface finish. Several studies
improved lateral printing resolution by using formulated resin to control the light energy scattering
and the penetration depth
[71, 94, 126-128]
. The established relationship between the prepolymer
composition and the curing depth has been characterized by analytical models
[126]
. Based on the
established models, Kowsari et al. investigated the comparative effects of resin formulation on
printing characteristics
[119]
. The effect of custom formulated mono-, di-, and tri-functional (meth)
acrylate-based photopolymer constituent types and concentrations on the shape evolution, size,
resolution, and surface roughness of features was investigated. The single-layer feature printed by
the bottom-up VPP platform (Figure 9d) shows that the di(meth)acrylate-based solution that
produced rounded single-pixel features and optically clear surfaces. These differences can be
explained by changes in refractive index during curing, causing a self-focusing effect on the
investigated resins. Mono-, di-, and tri-functional monomers and varying molecular weights would
lead to different shrinkages and, therefore, different change of refractive indices while curing. Less
change of refractive indices gives less self-focusing effect, providing a smoother surface. This
finding enables the printing of a thin layer with microscale surface roughness. Namgung et al.
further studied the key printing parameters, including resin formulation, building orientation, layer
thickness, and layer offset, leading to a thin layer of smooth surface
[129]
. The surface finish of the
optimized parameter is characterized as 645.1 line/mm, compared to 80.6 line/mm without
optimization.
Traditional polishing techniques can be applied in-situ or after VPP to improve parts’
surface finish. Instead of physically polishing the 3D-printing parts, resin coating can be applied
after the printing process. The surface roughness was largely reduced by simply coating a thin
layer of the monomer liquid resin to the part after its removal from the print bed
[130]
. Figure 9e
23

compares the transparency of the 3D-printed thin discs. The thin discs with resin coating (Figure
9e-right) is 60% better in transmittance. The mechanical polishing is also used to fabricate optic
elements to achieve nanoscale surface roughness. Figure 9f shows a six-axis robot moves the
polishing tool on the optical surface in accordance with a pre-programmed path. The layer structure
shown in the 3D-printed spherical lens was eliminated after polishing. A surface roughness of 10
nm can be achieved by machining
[131]
. The building platform and the separation film affects the
surface roughness of the fabrication results since the textures from the separation film and the
building platform can be imprinted to the printed parts. To minimize the roughness of both surfaces,
researchers used smooth glass slides for the vat and building plate surfaces
[92, 129]
. A passivating
silane was derivatized to the bottom vat glass slide to ensure the separation after one layer
fabrication. The surface roughness difference between the glass slides vat and the conventional
PTFE vat are demonstrated in Figure 9g.
24

Figure 9: The lateral surface finish improvements in VPP. (a) Reducing the micro surface aliasing by
using mask images with sinusoid grayscale patterns
[123]
; (b) Reducing the pixelated surface aliasing by
defocusing the LCD mask images. From left to right, standard, 0.1mm defocusing, and 1mm defocusing mask
images were used in printing the results
[111]
; (c) Vibration-assisted micro-stereolithography to fabricate optical
lenses. A sinusoid vibration was applied to the mask images in printing lenses, with their surface roughness less
than 10 microns
[54]
; (d) One shot printing of a thin optically clear surface. Chemical composition of the resin
determines the surface roughness
[119]
; e) Liquid resin coating to generate transparent surface
[130]
; f) Post
processing to improve the surface quality of a 3D-printed fisheye lens. Scale bar: 1 cm
[131]
; g) Disc samples
printed on silane coated glass plates and PTFE film. The silane coated glass plates leads to a smoother result
[129]
.

The surface finish of a 3D-printed object is poor due to the stair-stepping effect of the layer-
based VPP process
[132]
. Conventional MIP-VPP follows the schema of layer-by-layer stacking of
25

2.5D thin sheets in fabricating 3D objects. Laminar sheets generated from the sliced CAD model
with specific layer thickness are converted into pixelated mask images and projected to the
photocurable resin. The intrinsic discontinuities in pixels and layers lead to staircase stepping on
part surfaces, which deteriorates their surface quality. A typical layer thickness used in VPP is
between 15 μm - 150 μm, which leads to similar size of staircase stepping
[133, 134]
. A logical
strategy for improving surface finish requires reducing pixel size and layer thickness. Instead,
researchers also investigated layerless fabrication to eliminate the stair-stepping effect. And
meniscus coating after the layer-based or layerless fabrication can be used to further improve parts’
surface quality.
process planning by converting 3D geometry into 2D contours is dramatically simplified
[135]
. However, the stair-stepping effect comes with the layer-based fabrication approach,
deteriorating the vertical surface finish. Attempts of developing layerless VPP processes to tackle
the vertical surface finish problem have been made. For example, computer numerically controlled
(CNC) accumulation has been developed that make an analogy of the 5-axes Most of the VPP
processes are based on layers. A significant benefit of the layer-based approach is the CNC
machining in AM
[116, 136]
. A fiber optic cable connected with an ultraviolet (UV) LED and related
lens is served as an accumulation tool. By controlling the on/off state of the UV-LED and the
multi-axis motion of the cable, a physical model can be built in the shape of the accumulation tool
path (Figure 10a). This technique has shown it can significantly reduce the staircase stepping. Pan
et al. investigated the use of CNC accumulation in fabricating conformal features on curved
surfaces
[137]
. Highly flexible tool motions have been shown to be critical for building conformal
features on curved surfaces. The CNC accumulation process achieved less than 10 μm of the
surface defect, compared to 50 μm in the layer-based fabrication approach (Figure 10b).
26

Another way to fabricate layerless parts is using the continuous liquid interface production
(CLIP). The CLIP process demonstrates the continuous generation of monolithic polymeric parts
[66]
. It utilizes an oxygen-permeable, UV transparent window below a liquid resin bath, maintaining
a liquid interface below the advancing part. Suction force regenerated from the advancing part
constantly renews liquid resin. Less than 1μm layer thickness can be achieved by the CLIP process
(Figure 10c). Much research has been done based on CLIP to further extend its applications.
Chong et al. investigated a large-scale CLIP process for SIOC ceramic components
[138]
. It changed
original CLIP process to a bottom-up CLIP processes, resulting in an abundance of oxygen supply
and a considerably fast printing speed. Moreover, the CLIP process was combined with other
surface improving techniques. Shao et al. synergizing grayscale photopolymerization and micro-
CLIP processes to rapid 3D printing optical lenses
[33]
. The use of polydimethylsiloxane thin film
possessing much refined nanoscopic porosities as the functional substitute of Teflon membrane
was reported to significantly reduce the surface roughness to 13.7 nm (Figure 10d). For some
circumstances where continuous fabrication is not applicable, such as large scale, high resolution,
and multi-material fabrication, adaptive slicing is an idea approach to blur the layer stepping
[139-
141]
. Mao et al. introduced a method to efficiently generate slicing plans by a new metric profile
that can characterize the distribution of deviation errors along the building direction (Figure 10e).
It uses smaller layer thicknesses at the places with higher surface finish requirements while using
larger layer thicknesses at other places with lower requirements to achieve high fabrication
efficiency.
27

Figure 10: Layerless VPP Processes to improve the vertical surface finish. (a) The schematic plot of the
CNC accumulation, an optical fiber inserted into the resin vat cures liquid resin at the tip of the fiber, while the
optical fiber moving continuously
[136]
; (b) Layerless VPP achieved by CNC accumulation compared to the
traditional VPP process, smooth surface can be observed from CNC accumulation
[137]
; (c) Continuous liquid
interface production process where1 µm layer thickness can be achieved by a ‘dead zone’
[66]
; (d) Continuous
liquid interface process combined with grayscale mask images, Scalebar: 20μm
[33]
;(e) Adaptive slicing processes
vary the layer thickness based on the fabrication requirements
[139]
.

Liquid meniscus coating is a post processing method that utilizes the meniscus equilibrium
of the fluid interfaces at the corners of intersection layers to smooth curved surfaces
[142]
. Figure
11a shows meniscuses will be formed after raising the built layers out of the liquid resin.
Accordingly, another mask image can solidify the liquid resin in the meniscus areas. Much effort
has been made to closely match the meniscus equilibrium to the designed pattern
[92, 117, 143]
. Based
28

on the developed meniscus models in various scenarios, a process planning method has been
developed
[92]
. Compared to the use of ultra-thin layers that significantly elongates the building
time, the developed technique enables using a much larger layer thickness to achieve desired
surface finish. Meniscus coating can be combined with thermally post-curing resin to achieve
uniform conversion
[144]
. Post-processing was employed to raise conversion and improve the
mechanical properties of 3D-printed parts, commonly via UV exposure as well as thermal curing
(Figure 11b). The thermal initiator performs a post-cure to drive the material to be fully cured.
The authors show that this approach enables fast printing times (5 μm/s) precise control over
vertical resolution and decreased surface corrugations on a 10’s of microns scale. Another attempt
was to combine the liquid meniscus coating with the grayscale mask images
[32]
. Applying it to the
VPP results in the grayscale transition region with better surface coverage extended further along
the horizontal surface rather than a sheer transition associated with binary polymerization (Figure
11c). However, due to the complications with the underlying processes, it remains difficult to
precisely control the geometry of the grayscale transition region. The 3D-printed results
demonstrate a maximal imaging resolution of 373.2 line/mm with low field distortion less than
0.13% across a 2 mm field of view. Lastly, a rapid and smooth printing process can be achieved
by combing liquid meniscus coating and the continuously liquid interface. Zhang et al. controlled
the confinement of the liquid-solid interface and the continuous printing mode, and used excess
resin adhering to the cured structure to generate a liquid film when scraping off
[145]
(Figure 11d).
A centimeter-scale contact lens structure with extreme smoothness (sub 1.3 nm) was printed by
this process.
29

Figure 11: Post-coating processes to improve the vertical surface finish. (a) The meniscus coating
process dips the 3D-printed parts into liquid resin to form a thin layer of meniscus equilibrium to reduce the
vertical aliasing
[143]
; (b) A post processing by raising the temperature to thermally post-cure the polymers. The
interlayer variation can be homogenized after thermal curing
[144]
; (c) The meniscus coating process is combined
with grayscale mask images to generated sub-micron surface roughness, and the grayscale mask images generate
partially cured boundary layers to file the residual gap from the meniscus coating, Scalebar: 200 μm
[32]
; (d) The
meniscus coating process was combined with continuous liquid interface to generate optical smooth surfaces, a
liquid meniscus coating was automatically formed during the continuous pulling up, Scalebar: 5 mm
[145]
.

2.4 Mask Shifting Technologies in Photolithography and Stereolithography
Mask shifting technologies change the position or angle of the photomask during the mask
image projection, giving an additional degree of freedom in controlling the mask image. Mask
30

shifting technologies have been used in photolithography and stereolithography for a long time.
Here we review several typical usages.
In photolithography the mask shifting is used to fabricate tiny features which is smaller
than the photomask. K. Indykiewicz et al. apply the mask translation and mask rotation techniques
to double patterning lithography to fabricate narrow parallel submicrometric stripes. Thin stripes
with 150 nm in width can be fabricated by photomask with 3000 nm intervals
[146]
. A. Ashok et al.
developed a pseudorandom phase mask from subpixel shifting to super-resolution imaging,
overcoming the pixel-limited resolution of digital imagers. As a result, 50% resolution
improvement can be achieved over a conventional multi-frame imager
[147]
.S. Tetsuno et al. use
two subpixel-shift imaging methods, which use a series of mask illuminations and a single channel
detector that enhances lateral spatial resolution in proportion to the number of shifts
[148]
. However,
there is an essential limitation of the mask shifting process. Mask shifting cannot increase the
density of the tiny features. Though the size of features on a silicon wafer is reduced, the number
of features is not increased.
In stereolithography, mask shifting is used to enlarge the fabrication scale or switch the
mask image pattern. The scanning-projection-based stereolithography developed by M. Emami
takes advantage of a high-resolution DMD device and a moving stage to combine the high
resolution and large-scale fabrication
[82]
. M. Hedge developed large-area mask-projection
scanning stereolithography, demonstrates the scalability of 3D structures to fabricate thermoplastic
polymers
[149]
. H. Mao et al. uses mask shifting techniques to switch the shaped beams for
scanning-based stereolithography. Micropatterns ranging from 400 μm to 20 μm can be achieved
by switching different apertures patterns on the photomask
[78]
. None of the previous work
successfully uses mask shifting to increase the process resolution.
31

In the past decade, tremendous effort has been devoted to various process planning for the
VPP, either in the temporal domain or the spatial domain, to improve the process resolution.
However, little work has been done on the process planning in both spatial and temporal domains,
resulting in a loss of performance in process resolution and fabrication speed. Though lots of the
research has been down to eliminate the stair-stepping and to improve the vertical resolution, none
of the research before can balance the fabrication speed, printable scale, model fidelity,
computational and mechanical limitation of the set-up.

32

Chapter 3: Research Questions and Hypotheses
3.1 Statement of Problems
MIP-VPP has made significant breakthroughs in balancing design freedom, fabrication
speed, cost-efficiency, dimensional accuracy, and material efficiency. However, it is challenging
to achieve high-resolution features or smooth surfaces at a rapid fabrication speed. As stated in the
introduction, the MIP-VPP process slices the 3D CAD model into thousands of thin layers and
sequentially fabricates each layer by projecting UV mask images onto the photocurable resin. Two
types of aliasing are generated in physically constructing the 3D parts. Firstly, mask image
projection of each layer generates lateral aliasing between the cured layer's footprint and the sliced
cross-sectional images. Secondly, the layer-by-layer stacking of thin sheets generates vertical
aliasing, known as stair-stepping. To improve the process resolution and surface roughness of
MIP-VPP, both lateral and vertical aliasing must be addressed.
In conventional MIP-VPP, achieving higher process resolution precision and lower surface
roughness requires reducing the size of the pixels in mask images. However, smaller pixel sizes
lead to slower fabrication speeds and reduced product throughput. The tradeoff between product
throughput and product accuracy poses challenges for MIP-VPP. To commercialize MIP-VPP for
high-resolution applications, the main challenge lies in designing an MIP-VPP-based AM process
that maintains high product resolution without sacrificing throughput.
The process resolution and surface smoothness are limited by both lateral and layer
stepping aliasing. Due to the layer-based characteristics, material deposition in a layered manner
is an anisotropic process, resulting in different aliasing between the lateral and vertical directions.
33

To simplify the problem, we have investigated the principles of lateral resolution, vertical
resolution, and surface roughness separately.
3.1.1 Limitations of Vertical Resolution in MIP-VPP
The vertical resolution is mainly determined by the layer thickness, which is the gap
between the slicing planes, and the non-uniformity of photocuring behavior within each layer.
MIP-VPP stacks layers, which are in 2.5D prism shapes to form the 3D parts. The shape of a prism
layer is defined by the geometry of the corresponding sliced images, resulting in layer stepping
when there is complex geometry between neighboring slices
[141, 150]
. It has been proved that the
layer stepping can be significantly reduced when a finer layer thickness is used
[151]
. However,
using finer layer thickness can be computationally expensive and more time-consuming since the
on-and-off switch of the light source and the up-and-down recoating of the building platform
extend the fabrication time. Figure12 shows the layer stepping generated by different layer
thicknesses for slicing, and the aliasing volume reduces as the layer thickness decreases.
[11]
Figure 12: Vertical resolution limited by staircase effect

34

Additionally, the photocuring behavior is non-uniform within a single layer. Based on
Beer-Lambert absorption (, , ) = (, , 0)

, where is the irradiance at any arbitrary
point (, , 0) in the building area, is the penetration depth of the resin at specified light source,
the light intensity falls of exponentially from the penetration surface
[152]
. The curing behavior
without vertical constrain can be derived as a paraboloidal shape instead of a cubic or cylindrical
shape for a single pixel light source with Gaussian energy distribution. The paraboloidal curing
result leads to a widely bottom and narrower top in a single layer. To increase the vertical
resolution of the process, both the layer stepping and non-uniformity with layer need to be
considered.
3.1.2 Limitations of Lateral Resolution in MIP-VPP
The lateral resolution is mainly limited by the mask images, which is sliced from the input
three-dimensional CAD models, and the nonuniformity of the polymerization behavior. Firstly,
almost all of the MIP-VPP processes, either DMD-based or LCD-based, use discrete mask images.
The size of the mask image element limits the smallest printable feature and brings discrete aliasing
on the continuous boundary. For a typical MIP-VPP, the lateral resolution is around 20~50
μm/pixel. To achieve a higher resolution, we have to sacrifice the fabrication speed or fabrication
scale.
Secondly, unlike the CNC machining, the material’s photocuring behavior in VPP is highly
nonuniform, including the non-uniformity within every pixel and non-uniformity between
pixels
[153]
. 1) nonuniform in a pixel: the pixel of a projection image is Gaussian distributed rather
than uniformly distributed
[64]
. Hence, the curing depth from a single pixel varies according to the
Beer-Lambert law of absorption
[39]
. In particular, a single-pixel generates a paraboloidal shape
instead of a cubic or cylindrical shape. 2) Nonuniform across the whole layer: each pixel on the
35

photomask has different geometries and intensities. The non-uniformity of the photocuring
behavior distorted the photopolymerization result from the mask images, contributing to the
aliasing in the lateral plane as well. Figure 13a shows the lateral resolution limited by the size of
the pixels. For the same CAD model, the mask image resolution decreases as the size of the pixel
increases. Figure 13b shows the difference between the photopolymerization result and the
projected mask image. Aliasing generated by non-unformal polymerization behavior blurs the
sharp corner in the original mask image. To improve the lateral resolution, both the aliasing of
mask images and the aliasing of polymerization behavior need to be considered.

Figure 13: Lateral resolution limited by pixel size and UV energy distribution

3.1.3 Limitations of Surface Roughness in MIP-VPP
Surface roughness, referring to fine-scale irregularities in a component's surface texture, is
a crucial factor in MIP-VPP alongside process resolution. Factors affecting surface roughness
36

include mask image aliasing, light intensity uniformity, linear stage accuracy, surface quality of
the separation film and building platform, and resin composition. Some applications require
surface roughness as high as 10nm, posing higher demands on the fabrication process compared
to dimensional accuracy or printing resolution.
As addressed in section 3.1.2, pixelated aliasing results from the discrete nature of mask
images, causing jagged boundaries in fabricated parts. And Non-uniform photocuring behavior
within each layer and across the entire build area also contributes to surface roughness in MIP-
VPP parts. While grayscale mask images can partially reduce pixelated aliasing, achieving nano-
scale surface roughness is still challenging. Conventional resolution-improving processes
primarily focus on overall process resolution and may not adequately address such surface
roughness.
Various attempts have been made to improve surface roughness in MIP-VPP, including
grayscale mask images, linear vibration of mask images, mask image defocusing, post-machining,
and post-material coating. Although progress has been made, challenges persist, particularly as
previous processes improve surface roughness at the cost of fabrication speed or dimensional
accuracy. A targeted approach to reduce nano-scale surface roughness without sacrificing speed
and accuracy is required.
The proposal's main question revolves around designing an MIP-VPP-based AM process
that achieves high process resolution and surface roughness without compromising each other or
the product's throughput. The main question comprises the following sub-questions: 1) How can
motion-assisted MIP-VPP enhance vertical resolution without sacrificing fabrication speed or
building volume? 2) How can motion-assisted MIP-VPP improve lateral resolution without
affecting fabrication speed or building volume? 3) How can motion-assisted MIP-VPP reduce
37

surface roughness in MIP-VPP-fabricated products without compromising process resolution,
fabrication speed, or building volume?
3.2 Hypothesis
We need to address the proposed questions above to design a MIP-VPP process with high
process resolution and high surface roughness while maintaining reasonable product throughput.
Our research methodology employs motion-assisted MIP-VPP to reduce lateral and vertical
aliasing, as well as surface roughness.
For vertical resolution, we discovered that layer stepping is generated by the normal layer
stacking. The normal layer stacking, which is in a prism shape, contains only the information of
the sliced image, and the slope angle information of the sidewall is lost. By changing the normal
prism stacking model to a frustum stacking model, which contains the slope angle information for
each layer, stair-stepping can be significantly reduced. Following this concept, we found that
shifting the photomask vertically during the printing process changes the focal length of the
imaging system, allowing the zooming focal length to generate a frustum stacking model.
For lateral resolution, we found that MIP-VPP fabricates one layer in a few seconds,
whereas the frequency of mask image changes can be as high as hundreds of hertz. The
considerable mismatch between exposure time and mask image frequency inspired us to apply
multiple mask images during a single layer fabrication in split exposure times. In this case, the
curing result of one layer is determined by the superimposed sub-mask images. If we shift each
sub-mask image by a sub-pixel distance, the superimposed results can achieve higher resolution
in the lateral direction.
38

For surface roughness, we implemented an XY plane vibration to the photomask during
exposure. The adaptive vibrations help in mitigating pixelated aliasing and non-uniform
photocuring behavior, leading to a smoother surface finish. Additionally, by carefully controlling
the vibration frequency and amplitude based on the part's geometry, we can selectively target
regions that require higher surface quality without affecting the overall fabrication speed and
dimensional accuracy.
If we can address the lateral aliasing by laterally shifting the mask images and resolve the
layer stepping by shifting the focus distance, both the lateral and vertical resolution can be
improved without sacrificing the fabrication speed. Then, we can reduce the surface roughness by
applying a vibrated mask image. Based on this idea, the underlying hypotheses include:
Q1: How can motion-assisted MIP-VPP enhance vertical resolution without sacrificing
fabrication speed or building volume?
To answer this question, the following hypotheses are investigated:
Hypothesis 1.1: A frustum layer stacking compared to normal layer stacking can
effectively reduce the vertical staircase.
Hypothesis 1.2: A smooth change of mask images can be achieved by zooming the focal
distance of the MIP-VPP hardware.
Hypothesis 1.3: The combination of the continuous liquid interface production and the
continuous change of the mask images enable the frustum layer stacking

Q2: How can motion-assisted MIP-VPP improve lateral resolution without affecting
fabrication speed or building volume?
39

To answer this question, the following hypotheses are investigated:
Hypothesis 2.1: Higher resolution mask images can be achieved by replacing the
original mask image to a series of superimposed sub images with a shifted position.
Hypothesis 2.2: The sub-images can be generated computationally efficiently from the
original mask image by error diffusion.
Hypothesis 2.3: The high-resolution mask image shifting can be achieved by A piezo
actuated pixel shifting mechanism.

Q3: How can motion-assisted MIP-VPP reduce surface roughness in MIP-VPP-
fabricated products without compromising process resolution, fabrication speed, or building
volume?
To answer this question, the following hypothesis are investigated:
Hypothesis 3.1: The nonuniform light intensity from the LCD mask image can be
uniformed by vibrating the mask image at certain frequency and amplitude.
Hypothesis 3.2: The contour aliasing from the LCD mask image can be reduced by
vibrating the mask image at certain frequency and amplitude.
Hypothesis 3.3: The vibration amplitude applied in the printing is so small that
dimensional accuracy of the products will not be affect.

40

Chapter 4: Results and Evaluations
In this chapter, we will demonstrate some preliminary results that have partially validated
our hypotheses. The chapter presents the study of motion-assisted MIP-VPP including zooming-
focused MIP-VPP, spatiotemporal MIP-VPP, and vibration assisted MIP-VPP
4.1 Improving Vertical Resolution through Zooming-Focused MIP-VPP
The vertical resolution of the MIP-VPP is limited by staircase aliasing. A new continuous
mask image projection-based vat photopolymerization process is reported that can directly shape
polymer materials into optical lenses with microscale dimensional accuracy (14.7 μm) without
additional post-processing. The main idea is to utilize frustum layer stacking instead of the
conventional normal layer stacking to eliminate staircase aliasing and enable nanoscale surface
roughness. A continuous change of ultraviolet (UV) mask images is achieved using a zooming-
focused projection system to generate the frustum layer stacking with controlled slant angles. The
dynamic control of image size, objective and imaging distance, and light intensity involved in the
zooming-focused mask image projection-based vat photopolymerization (zooming-focused MIP-
VPP) are systematically investigated. The experimental results reveal the effectiveness of the
proposed processes. The 3D-printed optical lenses with various designs, including parabolic lenses,
fisheye lenses, and a laser beam expander, are fabricated with surface roughness (

) of 3.4 nm
without post-processing. The dimensional accuracy and optical performance of the 3D-printed
micro compound parabolic concentrators and micro fisheye lenses are studied, demonstrating this
new manufacturing process's fast and precise nature. The use of frustum layer stacking in 3D
printing may open a new avenue for future optical component and device fabrication.
41

4.1.1 High Resolution Model Reconstruction via Frustum Layer Stacking
Additive manufacturing, also known as 3D printing, fabricates physical objects from digital
CAD models by accumulating material at designated locations.
[154]
Benefiting from rapid
turnaround and high customizability, additive manufacturing has been widely used in dental,
surgical, biomedical, and other applications.
[107, 155-162]
Among all the AM processes, MIP-VPP
converts liquid monomers into solid polymers in a layer-by-layer manner using controlled
ultraviolet (UV) light, achieving the micro-scale resolution and centimeter scale fabrication size.
It has been a widely used AM process to fabricate parts with high accuracy and surface finish,
which sheds light on direct printing optical lenses.
[59, 163-165]
Many attempts have been made to 3D
print optical components using MIP-VPP apparatuses.
[166, 167]
However, most optical lenses are
still not 3D printable by layer based printing due to the extreme high resolution limits (10 nm ~
500 nm).
[168-170]
MIP-VPP follows the schema of stacking 2.5D thin sheets layer-by-layer in
fabricating 3D products. Laminar sheets generated from the sliced computer-aided design (CAD)
model with specific layer thickness are converted into pixelated mask images and projected to the
photocurable resin, as shown in Figure 14a-left. The curing results from a sliced layer is a 2.5D
extrusion of the mask images from that layer, as shown in Figure 14a-right. The red curve in the
magnified figure is the profile of designed CAD models, whereas the green area is the cured layers.
The Intrinsic discontinuities in pixels and layers lead to the staircase stepping on the printed surface,
which deteriorates 3D objects’ surface quality. A critical barrier for AM to overcome is how to
reduce pixel size and layer thickness to achieve the high resolution and surface finish that are
required by optical lenses.
Continuous liquid interface production (CLIP) developed by Carbon 3D offers a strategy
to reduce the layer thickness to a submicron scale.
[171]
As Figure 14b-left shows, the CLIP process
42

projects a high-framerate UV mask video onto the building platform while the building platform
continuously elevates from the resin vat. The high framerate mask video enables the super slicing
of the CAD model to less than 1 µm thin sheets, significantly reducing the staircase stepping.
Based on this process, Chen et al. introduced a CLIP process combining grayscale exposure and
meniscus coating that fabricates aspherical lenses with surface roughness down to 7 nm after
meniscus coating.
[33, 172]
Zhang et al. introduced a single droplet continuous MIP-VPP that uses
the liquid film covering to fabricate contact eye lenses.
[145]
However, while the layer thickness is
significantly reduced in the CLIP processes, the change of mask images is still not smooth, mainly
because the pixel size of a mask image defines the smallest projection light change
[54, 119]
, as
Figure 14b-right shows. As illustrated in yellow lines, the staircase stepping caused by the
discrete change of mask images is at a one-pixel scale (at 10 µm level), which is still not smooth
for optical lens fabrication. Thus, in the previous studies, the 3D-printed optical lenses need to be
post-processed using meniscus coating or polishing to reach a good surface finish. In addition to
reducing the manufacturing speed, the post-processing often makes dimensional accuracy
uncontrollable.
[132, 142, 173]
A zooming-focused MIP-VPP is developed as a dimensionally accurate and time-efficient
strategy to circumvent the critical issues for optical lens fabrication. The proposed process utilizes
a moving objective lens to dynamically change the projection images while continuously elevating
the building platform (Figure 14c-left). This configuration enables the frustum layer stacking with
controlled layer tilting angles, as shown in Figure 14c-right. Unlike 2.5D layer stacking, each
layer of the frustum layer stacking has a slope for its side surface (defined as a slant angle). Hence,
the frustum layer stacking requires not only the information on the cross-sectional image but also
the slant angle of the 3D model. As a result, the 3D-printed surfaces by the frustum layer stacking
43

have three orders of magnitude smaller staircase stepping errors than those by the 2.5D layer
stacking under the same layer thickness. By combining frustum layer stacking and the super slicing
technique from CLIP, the A zooming-focused MIP-VPP is developed as a dimensionally accurate
and time-efficient strategy to circumvent the critical issues for optical lens fabrication. improves
the surface smoothness to less than ten nanometers, enabling the direct printing of optical lenses
with a smooth surface finish (3.4 nm) and accurate dimension (14.7 μm).
Here, we report a new strategy of 3D printing super-sliced frustum layers by continuously
changing the mask images on the continuously elevating building platform for optical lens
fabrication. To develop the zooming-focused MIP-VPP process, we first present essential
parameters in terms of the fabrication scale, speed, and resolution, satisfying the requirements of
optical lens fabrication. Next, the model of the light intensity distribution of the zooming-focused
imaging system is systematically evaluated. This is followed by a model-based light intensity
compensation to achieve uniform light intensity during fabrication as the focus changes. Finally,
the test cases of compound parabolic concentrators, fisheye lenses, and a laser beam expander with
customized sizes and shapes have been designed to demonstrate the dimensional accuracy, surface
smoothness, and optical properties of this new AM process.
44

Figure 14: 3D printing of optical lenses using zooming-focused MIP-VPP. a-c) The comparison of the
MIP-VPP processes and the resulting surface finish. a) Optical lenses printed by layer-based MIP-VPP. The
staircase stepping is formed between layers. b) Optical lenses printed by continuous liquid interface production
(CLIP). The pixel size limits the surface finish. c) Optical lenses printed by zooming-focused MIP-VPP and the
frustum layer stacking.

Zooming-focused MIP-VPP eliminates staircase stepping by employing a continuously
changed mask image to build a frustum layer with a super-sliced layer thickness. Figure 15 shows
the overview of zooming-focused MIP-VPP. 1) A 3D CAD model is designed based on the
geometry of an optical lens. 2) The designed 3D CAD model is super sliced into a set of thin
frustum layers whose slant angle and cross-sectional images are correspondingly characterized
over the Z direction of the CAD model. 3) The motion planner converts the data of the slant angle
45

into the moving velocity of the objective lens, the liquid crystal display (LCD) photomask, and
the building platform. 4) A zooming-focused MIP-VPP apparatus builds the frustum layers in
continuous printing. 5) After bathing and nitrogen flow cleaning, the printed optical lenses can
achieve less than 3.4 nm surface roughness and micron-scale dimensional accuracy.

Figure15: A schematic illustration of zooming-focused MIP-VPP. The process starts with CAD design,
followed by profile characterization, velocity calculation, zooming-focused mask image projection, and final
cleaning.

The essence of this process is that the frustum layer stacking is applied to provide a close
approximation in regenerating 3D models and to reduce the staircase stepping (Figure 16a). Note,
in our method, the side surface of each frustum layer has the same slant angle . Hence it only
applies to the symmetric shapes along the Z-axis. Fortunately, most optical lenses are symmetric
46

along the optical axis. A frustum layer of an optical lens is generated by following three steps: 1)
the CAD model is sliced into a series of cross-sectional images under a certain layer thickness

.
2) The difference between a cross-sectional image and its neighboring cross-sectional images is
given by

() =
(

)

, where

is the characteristic length of the

layer cross-section
image. 3) Based on the

and the ( − 1)

cross-sectional images, a frustum layer can be
constructed as a linear interpolation of its neighbors, whose slope angle of the side surface is
defined as =

/

. To evaluate different layer stackings’ performance, we use an
intermediate metric profile in the previous work to quantify the approximation error of a CAD
model.
[141]
The layer stacking approximation errors by the 2.5D and frustum layer stackings under
the same layer thickness

are demonstrated in the Figure 16, shown as the red color for both
layer stacking methods. In Figure 16a, the 2.5D layer stackings marked in green have vertical
staircase stepping for each layer. The 2.5D layer stacking omits the geometric information between
two cross-sectional images in the approximation, leading to a noticeable missing error for optical
lenses. In comparison, the frustum layer stacking (Figure 16b) considers both the cross-sectional
images and the corresponding slant angle of the side surface of each layer, leading to a volumetric
error

that is three orders smaller.
47

Figure 16: The generation of the frustum layer stacking. The printed part is marked in green. The metric
error of the layer stacking is marked in red. a) the metric error of 2.5D layer stacking b) the metric error of the
frustum layer stacking.

To physically enable the frustum layer stacking in MIP-VPP, we continuously adjust the
size of the mask images during the printing process. That is, the printing of a frustum layer with a
certain slant angle can be achieved by controlling

and

, where

is the changing speed of
the projection images and

is the part printing speed in the Z direction (Figure 16b).

is further
determined by the velocity of the LCD

, focal length, the diameter of the mask images on the
LCD screen, and the distance between the objective lens and the building platform (refer to Method
S1, supporting information). Thus, the slant angle is governed by the following equation:

() =

= −
2

(

− )

(1)

48

where is the focal length of the objective lens, is the diameter of the mask images, and

is the distance between the objective lens and the building platform. In this case, the designated
slant angle of the frustum layer can be achieved by manipulating the velocity of the LCD
photomask

and the printing speed

. An example of motion planning to generate the desired
frustum layer stacking of different optical lenses is demonstrated in Figure 17. The CAD models
of three different parabolic lenses are converted into slant angles over the Z height shown in the
blue curve. Based on the slant angle , the velocity ratio

/

over the Z height calculated using
Equation (1) are plotted in Figure 17 in orange. The value of

/

increases as the Z height
increases from 0 mm to 3 mm since the profile slant angle decreases. At the very tip of the
parabolic lens,

/

will reach its maximum to fabricate a nearly flat frustum layer. The printing
speed

is related to light intensity and the curing depth of photocurable resin, which will be
discussed in later sections.

Figure 17: The profile of the slant angle in the Z direction. The slant angle (blue curves) is determined
by the moving velocity (orange curves) of the LCD mask and the building platform
49

4.1.2 Overview of the Zooming-Focused MIP-VPP Process
To verify our idea, an in-house developed zooming-focused MIP-VPP apparatus was built,
comprising a computer for the process planning, a microcontroller for the motion control, a 405
nm UV light emitting diode (LED), an objective lens movable in Z, a LCD mask movable in Z, a
glass building platform, and a resin vat coated with Polydimethylsiloxane (PDMS). (Figure S1,
supporting information) The apparatus is configured in the bottom-up configuration.
[174, 175]
The
UV radiation emitted from the LED array shoots through the LCD photomask and is projected by
the objective lens to the building platform. A light homogenizer is attached to the bottom of the
LCD photomask to convert parallel UV light into diffuse light to generate uniform light intensity
distribution. The objective lens and the LCD photomask are mounted on high-resolution Z linear
stages that are separately controlled to adjust the objective and imaging distances during printing.
The oxygen embedded in the PDMS membrane inhibits the free radical polymerization reactions,
creating a non-photocurable dead zone near the membrane to prevent adhesion while
simultaneously facilitating a sustained supply of fresh resin as the 3D-printed part is pulled up
vertically in a continuous motion.
[64, 176]
Figure 18-19 demonstrate the printing of a frustum layer
via zooming-focused MIP-VPP.
50

Figure 18: The profile of a frustum layer sliced from a Hemispherical model. b) The components of
zooming-focused MIP-VPP system.

Figure 18 shows the magnified profile of a designed optical lens sliced into a frustum layer.
Points 1 and 2, marked in yellow, are the bottom and the top of the frustum layer. The thickness
of the frustum layer

is 1 µm, and the slant angle is . The distance of the micro steps is the
theoretical resolution which determined by the linear stage and focal length. Figure 19 shows the
motions of the LCD photomask, the objective lens, and the building platform from point 1 to point
2. The objective lens starts at the position

and the LCD photomask starts at the position

.
During the continuous printing, the objective lens moves up at a velocity

and the LCD
photomask moves down at a velocity

to form the slant angle of the frustum layer at each
time moment. The objective lens ends at position

and the LCD photomask ends at the position

to finish this frustum layer. As Figure 18 shows, a series of mask images with micro-step
changes can be achieved on the building platform (marked in black). The size of projected mask
images for micro-steps can be controlled by

and

, governed by the equation:
51

1

=
1

+
1

(2)

is the focal length of the objective lens. The magnification rate of the mask image,
namely ratio between projected images on the building platform and the mask image on the LCD
is governed by the equation:
=

(3)

By changing the ratio of the

and

, we can change the projected mask image size
without changing the pixels on the imaging mask. By controlling the changing rate of the
magnification rate, the slant angle of the frustum stacking can be controlled. The relationship is
given in the function:

() =

()
0. 5 ∗

∗ ()
(4)

() is the slant angle of the frustum stacking for pixel at layer ,

is the moving
up speed of the building platform.

is the distance from the pixel , to the center of the LCD
mask. () is the changing rate over time of magnification at layer . We can easily find that
the different slant angle can be applied to each layer independently, which enable the frustum
stacking approximation proposed in the section 4.1.1.
52

Figure 19: The schema of zooming-focused MIP-VPP

4.1.3 Hardware Implementation of Zooming-Focused MIP-VPP
The prototype of zooming-focused MIP-VPP is demonstrated in the Figure 20a. A
customized zooming focus 3D printer built in this study comprises a computer for process control,
a microcontroller for the precise control of the simultaneous movement of the zooming focus
module, 1280x800 resolution LCD mask image screen featuring 405nm UV light LED, An UV
projection lens (Olympic Plan C), three motorized linear stage, and a resin vat. As illustrated in
Figure 20b, an PDMS film is coated at the top of surface of the resin vat. As the UV light being
projected into the liquid resin from below, the oxygen diffused through the membrane effectively
inhibit the free radical polymerization reactions. It creates a nonpolymerizing dead zone near the
membrane to prevent adhesion while simultaneously to facilitate sustained supply of fresh resin as
the 3D printed part being pulling up vertically in a continuous motion. Another detailed parameter
of the physical set-up is denoted in the Table 1.
53

Table 1: Design Parameters for Zooming-Focused Printer
component parameter description
Objective lens F = 7.8mm, 18mm,
28.9mm
Effective focal length
NA = 0.1, 0.25, 0.4 Numerical aperture
DOF = 55um, 8.8um,
3.4um
Depth of focus
LCD mask Res = 50um Pixel size
a = 10mm Dimension
Linear stage ΔX = 3um Minimum step
L = 350mm Travel
Vmax = .9in/sec rated speed
LED 1280*800 resolution

Figure 20: The hardware design and implement of zooming-focused MIP-VPP. a) CAD design of the
Zooming-Focused MIP-VPP machine. b) physical set-up of the Zooming-Focused MIP-VPP machine.

4.1.4 Parameter Optimization for Zooming-Focused MIP-VPP
The essential performance of the Zooming-Focused MIP-VPP is characterized in terms of
the fabrication scale, speed, and resolution are characterized. The fabrication scale is the size of
54

the mask images, which is determined by the magnification rate of the projecting system and the
mask image size. The quantitated characterization of the fabrication scale is given in the equation:

=

∗ (5)

is the fabrication area.

is the mask image size. is the magnification rate given by
eqn.(8) and eqn.(9). The relationship between the working fabrication area and the position of the
optical systems is determined.
Figure 21 shows the relationship between M and

at three different focal lengths f =
7.8mm, 18mm and 28.9mm. As the magnification rate increasing the,

increases whereas

decreases. The maximum value of

is 350mm, limited by the length of the linear stage. The
maximum value of

is 25mm, limited by the setup configuration. The range of magnification
rate calculated Based on the setup capability are M = 0.025~2.2 when f = 7.8mm, M = 0.06~0.35
for f = 18mm. It is not capable to focus image on building platform for f = 28.9mm. The

is
given by the objective lens diameter. When the diameter of the mask image is larger than the
diameter of the objective lens, the boundary light intensity will become too dim to solidify the
resin. The detail of light intensity control will be discussed in the next section. Here the

is set
to 100

for the 10mm diameter objective lens in our experiment. Thus,

is 220

for f =
7.8mm and 35

for f = 18mm. Using an objective lens with a smaller focus lens reduces the
value of the same magnification rate, miniaturing the volume of the 3D printer, and increasing the
fabrication scale.
55

Figure 21: The effect of the position of the optical system of zooming-focused MIP-VPP on the
fabrication scale

Besides the fabrication scale, the process resolution is another essential performance of the
Zooming focus MIP-VPP. Similar to all of the VPP techniques, the resolution of the projected
mask image, which is defined by the slightest controllable difference between two projected mask
images, gets worse as the focal length of the objective lens reduces. The following equation
governs the relationship between the focal length and the resolution:

= 0. 5 ∗

∗

∗

= 0. 5 ∗

∗

∗

= (

,

)
(6)

56

where is resolution of the projected mask image.

is the resolution limited by the
linear stage that controls the objective lens.

is the resolution limited by the linear stage that
contro.ls the LCD mask.

is the distance of unit steps of the linear stage, which is 3um for our
setup.

is varying from 0 to 5mm, depending on the position of the pixels. Considering the
eqn.(10) for a given objective lens, the

and

are injective function of M, governed by

=
(1 + ) and

= 1 +

.

can be written as the function of focal length:

= 0. 5 ∗ −

(

− )

∗

∗

= 0. 5 ∗
1

∗

∗

= (

,

)
(7)

The

is a fixed number for a zooming focus printer with specified objective length and
linear stage. The

is a varying number increasing as the magnification rate increases. The
relationship between the resolution and the magnification rate is plotted in the Figure 22.
57

Figure 22: The parameters affect the resolution of the zooming-focused MIP-VPP. a) process resolution
over imaging distance. b) process resolution over objective distance

There are several features of zooming focused MIP-VPP disclosed by Figure 22. First, the
equation shows that the resolution of the zooming focus increases as the focal length increase.
Their focal lengths vary from 5mm to 30mm for the commonly used objective lens. Finer
resolution can be achieved by using a larger focal length. However, larger focal lengths lead to a
smaller fabrication scale, discussed previously. To balance the fabrication scale and the process
resolution, we select the objective lens with focal length f = 18mm for our setup. The
corresponding resolution for the mask image is 0.83um, and the corresponding fabrication scale is
35

The last essential parameter of the process performance is the fabrication speed. The
fabrication speed is limited by the hardware and the photopolymerization speed of the resin. The
photopolymerization speed

= 5~ 100 /, which is similar to other continuous mask image
58

projection process. The hardware limitation of the linear stage has to be considered as well. The
theoretical limitation of the linear stage 0.9inch/sec(22.8mm/s). Given in the frustum
approximation, =

/

=

/

, Where:

= 0. 5 ∗

∗

∗

= 0. 5 ∗

∗

∗

= (

,

)
(8)

The frustum approximation uses a slant angle from 0 to 90 degrees to approximate the
surface curvature. However due to the hardware limitation of the

, the slant angle cannot go to
0. Here we chose = 10 degrees as the smallest slant angle. Assuming the linear stage can reach
its maximum rated speed(

,

), the maximum printing speed can be calculated from

=

. Figure 23 shows the relationship between the

and the magnification rate speed. The
dash line is the working range of the magnification rate. Because of the nonlinear relation between
the magnification rate and the image distance, the maximum printing velocity presents a nonlinear
increment as the magnification rate increase. Besides, as the slant angle increases, the maximum
printing speed increases. In the usual working range, the magnification rate is within 0.07~0.35.
The slowest printing speed for = 10 is 5.4um/s, which is acceptable for high-resolution tiny
feature fabrication. In the large magnification rate printing, the velocity limitation from the linear
stage is larger than 500 um/s. So at large magnification rate printing speed is governed by the
photopolymerization speed, which is 100um/s.
59

Figure 23: The parameters affect the fabrication speed of zooming-focus MIP-VPP

The purpose of dynamic image focusing is to maintain the focusing plane on the surface of
the building platform during the focus zooming. Unlike the conventional process, focus zooming
changes the objective distance

and the mask distance

. A position planning method is
developed here to determine the

and

based on given magnification rate.
According to eqn.(9) the position of the objective lens and the LCD mask can be mapped
to the magnification rate by =

. However, the travel distance of the linear stage is related to
their initial position as well. Shown in the Figure 24, the schematic plot explains how the initial
position affects the focusing plan. The

,

are the initial position of the linear stage which is
defined during the assembly of hardware. The , are the travel distance command sent by micron
controller for corresponding magnification rate. We have

= (

−

) + ( − ), and

=
60

(

+ ). Plug in eqn.(8) and eqn.(9) we get the position planning function considering the initial
position of the linear stage:
1

+
+
1
(

−

) + ( − )
=
1

=

+
(

−

) + ( − )

(9)
and are the travel distance of the linear stages,

are the mounting position of the
linear stages, is the focal length, is the magnification rate.

Figure 24: Manual focusing the mask images to building platform at different magnification rate

To get the travel distance of the objective lens = ( ) and the LCD mask = ( ),
the

,

in the proposed function need to be manually calibrated. To calibrate

,

, we manual
focus the calibration mask images sharply onto the building platform, shown in the Figure 23. We

61

measure the travel distance of the objective lens

and the imaging mask

corresponding the
magnification rate

Tens sample points with the magnification rate from 0.07 to 0.35 are
manually focused. The calibrated (

, = 1,2,3 … 10) , are fitted in the proposed dynamic
focusing eqn(13). The fitting result is given as follow:

can be calculated based on the fitting points. The more sample points are used the
high accuracy the initial position it will be. By using ten sample points, we can get

=
92. 42 (91. 11,93. 7) and

= 17. 96 (17. 86,18. 05) . Then travel distance of the linear
stage corresponding to the magnification rate can be calculated from the dynamic focusing model.
After manual calibration, the initial position needs to be verified by testcase printing. The
calibration points fitting model and the printed result is demonstrated in the Figure 25. The
calibration chessboard patterns are printed at different magnification rate. The testcase
demonstrate a well-focused mask image by sharp boundary of the chess board from M = 0.34 to
M =0.10. Fig.38(f) shows the calculated x, y curve for given

,

. A mapping plot between the
magnification rate to the x, y can be generated, which is used for printing process planning.
62

Figure 25: Dynamic focusing the mask images to building platform at different magnification rate a-e)
chessboard calibration pattern printed with magnification rate from M=0.34 to M =0.10.f) fitted model based on
manual calibration points.

We demonstrate the projected mask images on the building platform generated by the
zooming focus imaging system. By using dynamic focused mask image on the building platform,
the submicron resolution mask images can be sharply projected onto the building platform, shown
in Figure 26 projected mask images are demonstrated at different magnification. From left to right,
the diameter of the mask images reduced from 2040um to 540um. The image at the bottom shows
a zoom-in view of the circle patterns with diameters 1449um, 1500um, 1501um. The accurate
control of the magnification rate achieves a 1um resolution on the mask images. The sub-micron
scale dimensional control solves the smooth change of the mask images, enabling the smoothing
printing of the slant surface of the frustum stacks. Moreover, the sub-micron resolution control
allows a finer slicing thickness for a CAD model. Unlike the fixed focused mask image, which is
drawn by pixels, sub-micron increment of the mask images sizes in Zooming-Focused MIP-VPP
is more accurate to draw the cross-sectional images for each layer.
63

Figure 26: Continuous mask image change of zooming-focused mask image projection

The noticeable finding here is that the zooming focus system can provide a sub-micron
resolution using a low-cost imaging system. This resolution is a breakthrough for mask image-
based vat photopolymerization (VPP) process with millimeter-scale printing area.
4.1.5 Light Intensity Control of the Zooming Focused MIP-VPP
The previous section demonstrates precise control on the geometry of the mask image.
Besides the geometric accuracy of the mask images, the photopolymerization behavior of the
liquid resin also determines the final part accuracy. The photopolymerization behavior can be
modelized by the received UV dose and the curing threshold of the resin, governed by the equation:

=

ln

(10)

For fixed light intensity imaging system = ∙ , the curing threshold can be written as
64

=

ln (
∙

) (11)

is the curing thickness of the resin.

is the penetration depth varying from materials.
is the light intensity. is the exposure time. E is the absorbed energy dosage.

represents a
critical energy dosage. In our experiment, Formlabs Clear Resin with the curing depth

= 276um
is used. To get consistent curing behavior, a uniform curing depth is required.
For the specific resin used in this experiment, uniform curing depth can be achieved by
uniformly distributing UV dose on the mask image. However, due to the incident angle and the
magnification rate, the uniform light intensity images on the LCD mask will lead to an ununiform
light intensity distribution on the building platform. This section investigated the light intensity
distribution model and developed a light energy control method based on the physical model to
get uniform light intensity on the building platform.
Several assumptions are made to modulate of light intensity distribution: 1) the LED
collimated by fisheye array shoots parallel light to the LCD mask. 2) There is no spatial variation
of the LED light intensity in the fabrication area. 3) The scattering distribution of the light passing
through the light diffuser follows the Gaussian distribution over the output angle, as Figure 26
shows. 4) There is no energy loss while light passes through the diffuser. 5) The spherical
aberration of the objective lens is negligible. These assumptions are made based on preliminary
experiments which justify the rigorousness. Some inference can be obtained from the assumptions.
The incident light on the diffuser with a constant light intensity(

) can be inferred from
assumptions 1 and 2. The

is independent from the LCD locations and the distance between
LED and LCD. The total light intensity of the scattered light(

) is identical to

.
65

Figure 27 shows the light path of Zooming-Focused MIP-VPP process, The incident light
on the diffuser is scattered into a hemisphere with the light intensity in gaussian distribution. Only
a portion of the light shoots in the objective lens with the finite diameter(d). The diameter(d) of
the objective lens and the objective distance(

) determine the acceptance angle of the scattered
light. An acceptance cone with the spherical angle derived from integrating the acceptance angle
over the from [ 0,2] , is the light collected by the objective lens. All light collected by the objective
lens will be projected to the building platform.

Figure 27: Light intensity distribution model of zooming-focused MIP-VPP

Based on the schematic plot, we obtain the light intensity distribution model in the
following steps. The energy of the diffused light over the scattering angle() follows the Gaussian
distribution.
~ (0,

) (12)
At any point of the diffuser, the emitting light intensity at certain angle can be written as:
66

(,

) =

√2
exp (−

2

) ， (13)
Where

is the emitting light intensity in the normal direction , is the diffused angle,
is the spherical angle of the acceptance cone, A is the area of diffuser. The identical assumption
defines the relationship between the

and (,

). The following equation helps to calculate the
diffused light intensity (,

):

=

= (,

)
.

(14)
The light flux on the building platform which is what we need to calculated is sum of light
flux that collected by the objective lens. The light flux from a unit area of any points on the
building platform can be written as the integration of the light intensity along the spherical angle
of the acceptance cone whose vertex is the corresponding points on the image mask.

= (,

)

= (,

)

( )

(15)
Where

( ) is the angle between the generatrix of the acceptance cone and the height of
the acceptance cone. is the directional angle indicating the component of a generatrix on the
objective lens surface. is the angle between an arbitrary beam of light to the height of the
acceptance cone. is the area of diffuser. The height of the acceptance cone is identical to the
objective distance

.
67

Figure 28: Light cone captured by the objective lens

The

( ) , shown in the Figure 28 can be obtained by some simple mathematical
manipulation in following steps:

( ) = arctan

( )

;
ℎ :

( ) =
2

+ √∆
2
;
ℎ : ∆= (2

)

− 4(

−

);
=

( )

+

−

2

( )

;
(16)

Where is the radius of the objective lens.

is the central distance which is the relative
position of the incident point on the diffuser referred the center of the objective lens. From the
simultaneous functions given above, we obtain that

( ) = Θ( ,

,

, ). After plugging in the
light flux equation, we obtain the light flux on the building platform.

= (,

)

= (,

)
( ,

,

,)

(17)
68

From the output light flux function, we found that the

= Φ (

,

, ), where the
output light flux is the function of objective distance

, central distance of pixel

, and the radius
of objective lens .
The light intensity on the building platform is given as well. The output light intensity is
output light flux divided by the projecting area . is the focal length of the objective lens.

=

=

(,

)
( ,

,

,)

ℎ

= =

−

=

( ,

, , ,

)
(18)

The output light intensity function discloses that the output light intensity is the function
of magnification rate , central distance

, objective lens radius , the objective lens focal
length and the unit light intensity

. For a specific Zooming-Focused MIP-VPP printer, the
,

are constant numbers. The output light intensity is only dependent on the
magnification rate and the central distance.

=

( ,

) (19)

The function above is the final model to describe the light intensity distribution on the
building platform. We can predict the light intensity of Zooming-Focused MIP-VPP process for a
given magnification rate and pixel position. Later a simulation result of a specific printer shows
the trend of the light intensity distribution. And we developed a compensation method based on
this model to get uniform light intensity distribution.
69

What we derived above is a generic model for all projection based VPP process. For a
conventional Projection based VPP, the magnification rate is a constant determined by the
hardware. The

is determined by the central distance

. This model is applicable for light
intensity modeling for generic projection based VPP process as well.
1) Light Intensity Simulation
Based on the light intensity distribution model developed above, a simulation of the light
intensity distribution of our Zooming-Focused MIP-VPP prototype is demonstrated in this section.
We plug the physical parameter into the model

=

( ,

, , ,

) and

=
∫ ∫ (,

)
.

, where R = 8mm, f =18mm. Since the energy dosage of the UV radiation
affects the photopolymerization behavior directly, we present the relative energy dosage /

. in
simulation.

=

( ,

, , ,

) ∙t

(

) ∙
(20)

Given the eqn.(15) = ∙, /

is identical

/

. Given

is a constant in our
setup, the relative magnitude of

can be calculated from the /

. This transformation gets rid
of the

term in the model, making the math simpler. The simulation result of /

is
demonstrated in the figure below.
70

Figure 29: Simulation of the light intensity distribution on the building platform over different
magnification rate.

As shown in the simulation result Figure 29, the simulation shows the relative UV energy
dosage /

over the objective distance

and the pixel position

. The objective distance

is another name of the magnification rate, given the equation =

. The /

is stronger as
the pixel locating closer to optical center, shown by in the pixel position axis in the simulation.
Besides, the /

is stronger when the

is larger, because the larger

gives smaller
magnification rate M which projects similar amount of light flux to a smaller region.
In the working domain of our prototype, the /

is varying from 1 to 1.9. Since

is a
constant for our LED light source, the is varying from 1 to 1.9 because of different

and

.
Nearly two times the UV dose variance leads to undesired curing results. If we only consider single
layer photopolymerization with a fixed magnification rate, the resin in the center of the LCD mask
will be overcured while resin on the boundary will be unsecured. If we only consider the
photopolymerization result from a single pixel, the resin is overcured in small magnification rate
71

printing and under-cured in large magnification rate curing. To get uniform photopolymerization
result, a uniform light intensity must be implemented by certain compensation.
2) Light Intensity Compensation
The non-uniformity over the pixel positions and the magnification rates must be solved
together to get controlled uniform light intensity. The Figure 30 shows the relative light intensity
over the pixel position and the magnification rate respectively. We found that the relative light
intensity over the pixel position following a similar pattern for all of the magnification rate, which
indicates a possibility to decouple the two related parameters.

Figure 30: Two steps compensation of relative light intensity

Here we assume for any objective distance

(magnification rate), /

over the pixel
position

can be written in the following form:

(

,

) = (

) + (

) (21)

If this assumption stands, we can separate the parameters

and of

during the
compensation we can design a grayscale pattern to with energy dosage change ′ (

) to adjust
the UV radiation dose over the pixel position. Then we can design a dynamic the exposure time
72

with energy dosage change ′ (

) to adjust the UV radiation dose over the objective distance.
Over goal is that a compensated relative energy dosage

has a constant energy dosage, where
the grayscale pattern and dynamic exposure time are applied to original

(

,

).

(

,

) = (

) +

(

) + (

) +

(

) = (22)

The assumption is justified in the working range of the

,

by quantitative method. We
construct a function (

,

) = (

) + (

). And we let (0,70) =

(0,70) = 1. 4028,
(0, 270) =

(0,270) = 1. 8867 . The (270) − (70) = (0, 270) − (0, 70) = 0. 4839 .
Then we let (8,70) =

(0,70) = 0. 8048 . We can obtain that (8,270) = (8,70) +
(0, 270) − (0, 70) = 1. 2887. The actual value of the

(8,270) = 1. 1911. The maximum
error between the constructed function (

,

) and the

(

,

) is 7.5%, which is negligible
in practical printing. It is safe to proximate the

(

,

) by (

,

). In this case we can treat
the light intensity variation over the pixel position and the over the objective distance
(magnification rate) separately. We let (

) =

(

, 70) and (

) =

(8,

) in the
following compensation steps.
The Figure 30a and 30b the two steps compensation of the relative light intensity. First a
grayscale mask is applied to the LCD mask image based on the model. In the grayscale mask
image applies a grayscale value

(

) from 0/255 to 255/255 to each individual pixel on the LCD
mask. the grayscale pattern is constructed as following function:
73

(

) = (

)/(

)

(

) = (

) −
(

)

(

)

(23)

Where

is the relative energy dosage of the pixels at the boundary

= 8 of the
LCD. The light intensity after applying grayscale equals to

= (

) + (

) +

(

) (24)

From the equation above, we can find that the grayscale mask

(

) eliminates the energy
dosage variation over pixels. The relative energy dosage is only determined by the objective
distance

, shown in the Figure 30b
Similarly, we can construct exposure time function

(

), letting the following function
stands:

(

) = (

)/

(

)
(

) +

(

) =
(25)

Where is a constant energy dosage. The dynamic exposure time

(

) changes along
the objective distance, eliminating the energy dosage variation over the magnification rate. The
final energy dosage after compensation is a constant over the pixels and magnification rate, which
justify the fidelity of two steps compensation process.

= (

) + = (26)

74

3) Light Intensity Compensation Result.
In the first step, grayscale mask image compensation is applied to LCD screen. The Figure
31 shows the simulation with the grayscale mask

(

) calculated from the equation discussed.
The two most left figures show the mask images applied to LCD. The top left is the binary mask
image without any compensation. The bottom left is the grayscale mask image. The figures on the
right show the relative energy dosage

with grayscale and without grayscale over the
magnification rate from 0.35 to 0.07. The

is larger in the center of the mask and smaller on the
boundary for binary mask images. But the

is uniform in the whole LCD screen by using
grayscale mask. Also, we can see that the uniformity of

over pixels maintain well for mask
image at all of magnification rate. This also justifies our two step compensation assumptions.

Figure 31: Two steps compensation of relative light intensity

Though the energy dosage is uniformed over the pixels after the first step compensation,
the energy dosage increases as the magnification rate decreases. In the second step, the light
intensity variation over the magnification rate change is compensated by applying different
exposure time based on the model previous model and calibration result. Figure 32 shows the
exposure time to curing 76um layer thickness Formlabs Clear resin at the magnification rate from
75

0.07 to 0.34. The black line is the predicted exposure time based on the compensation model. The
red dots are the practical exposure time based on calibration. The practical exposure times align
well with the theoretical model, which proves the feasibility of our light distribution model. In the
continuous printing process, the exposure time can be converted into the pulling-up speed of the
building platform. The light intensity uniformity can be achieved by controlling the grayscale
pattern and the controlled pulling up speed based on the previous model.

Figure 32: Curing depth study of after two steps light intensity compensation

The practical photopolymerization test case of a chessboard after compensation is
demonstrated in Figure 33. Figure 33a shows a cone shape testcase printed by Zooming-Focused
MIP-VPP prototype. The magnification rate of the mask image, which is a circle, changes during
the printing .The left picture of Figure 33a shows the original result without exposure time
76

compensation. The printed part has a rough surface and a twisted profile, resulting from the
overcure of and under-cure of the resin at different magnification rate. The right picture of Figure
33a shows the printing result with compensated exposure time. A smooth surface with a straight
profile contour can be seen in the picture.
Figure 33b shows a chess board testcase printed with and without grayscale compensation.
The left pictures of Figure 33b show the mask image with and without grayscale. The right
pictures of Figure 33b show the photopolymerization result of the corresponding mask image.
From the result, we can find that the resin in the center of the chessboard is overcured, whereas
the resin on the boundary is under-cured. After compensation, uniform sample dots are printed
for all of the magnification rates.

Figure 33: Comparison of the photopolymerization behavior between original and compensated light
intensity.

This section studied the light intensity distribution of various parameters in terms of
fabrication scale, process resolution, and fabrication speed. Uniform light intensity is achieved by
two steps of light intensity compensation. With uniform light intensity, we can control the
77

photopolymerization behavior of the resin alongside the geometry of the mask images. These give
us the capability for smooth and accurate surface fabrication.
4.1.6 Printing Results and Discussion
Benefiting from the Zooming-Focused MIP-VPP process, the layer stepping on the surface
can be eliminated without sacrificing dimensional accuracy. This enables the outstanding smooth
surface quality with accurate dimension for VPP based AM. Lots of new applications of the VPP
process can be developed by using Zooming-Focused MIP-VPP. Here we use customized optical
devices as an example to demonstrate the performance of the Zooming-Focused MIP-VPP. Two
test cases for different optical purposes are demonstrated in this section. The printed test case is
evaluated in terms of dimensional accuracy, surface roughness, and optical performance.
The compound parabolic contractor is the critical component of solar energy gathering
devices. A compound parabolic contractor lens can contract the light flux from its inlet surface to
its outlet surface with little energy lost with a specifically designated parameter. Figure 34a shows
the principle of these devices. The input light within the acceptance angle will be reflected by the
glossy sidewall surface to the outlet of the device. The acceptance angle is determined by the
geometric shape of the sidewall, which enables the total internal reflection(TIR). This device
requires a fabrication process with high surface smoothness and dimensional accuracy, which is
difficult for conventional VPP to fabricate.
78

Figure 34: Design of the compound parabolic concentrator. a) principle of the compound parabolic
concentrator. b) CAD design of the compound parabolic concentrator. c) sliced profile image and cross-sectional
images of the CAD model.

Compound parabolic concentrator (1.0mm diameter of the output area) with acceptance
angles 20, 30, and 40 degrees are designed and printed to prove the successful preparation of
optical devices with eliminated step effect. CAD models are designed according to the TIR
requirements to make all of the incident light within the acceptance angle can be shot to the outlet
of the contractor. The CAD model, in Figure 34b is converted to a cross-sectional image and a
profile lens indicating the cross-sectional area, shown in Figure 34c. The cross-section area will
be converted to the magnification rate of the LCD mask for zooming focus printing. The printing
result of the compound parabolic contractor is presented in Fig.47. The printed parts show a
smooth surface and good transparency because of the continuous printing and submicron
resolution of the mask images.
The dimensional accuracy and the surface quality are evaluated in Fig.47. The compound
parabolic contractor with a different designed acceptance angle of 20 degrees, 30 degrees, 40
degrees are compared with the designed dimension. The blue curves show the designed profile
79

curve of the compound parabolic contractor, fitting well with the printed sample, indicating an
excellent dimensional accuracy control.

Figure 35: The testcase of the compound parabolic concentrator printed by zooming-focused MIP-VPP

The dimensional accuracy of the CPC elements with acceptance angles of 30° is measured
from optical microscopy images, shown in Figure 36. The radius of the designed CPC over the Z
height is plotted as the black curve. And the mean radius of the printed CPC over the Z height is
plotted in orange. The mean value of the surface profile matches well with the design value. The
red dash line in Figure 36a shows the standard deviation of the surface profile. The standard
deviation of less than 8 µm confirms the good repeatability. Figure 36b shows the absolute error
between the designed and printed radius over the Z height. The absolute deviation is well controlled
within the range from -6.8 µm to 7.9 µm plotted in the functional region for light reflection. The
deviation is mainly due to the light intensity distribution. The absolute deviation can be further
reduced by better light intensity calibration. Beyond the functional region, the absolute deviation
increases due to the material overcure at the base region.
80

Figure 36: Dimensional accuracy of the 3D printed CPC elements a) The surface profile of printed CPC
elements. The black curve is the designed profile, and the orange curve is the printed profile. The standard deviation
between samples is demonstrated in the red dash curve. b) The absolute deviation of radius over Z height between
the designed and printed CPC elements.

The scanning electron microscopy (SEM) image of the samples (Figure 37) demonstrates
a smooth surface quality. The left figures show the images with a scale bar of 100um. The layer
stepping aliasing, which is commonly seen in the conventional VPP process, is largely reduced in
Zooming-Focused MIP-VPP. Flat top surfaces and sharp corners, which are hard to fabricate via
coating post process, can be seen on the printed sample. The right figures show the zoom-in images
of the left ones with the scale bar 10um. Even under the 10um scale SEM images, there is no
significant layer steps on the surface, presenting outstanding surface quality control compared to
other VPP processes.
81

Figure 37: SEM images of the compound parabolic concentrator printed by zooming-focused MIP-VPP.
The left scale bars: 100 µm. The right scale bars: 10 µm.

Furthermore, the surface roughness of the TIR surface measured via atomic force
microscopy (AFM) is 3.4 nm within a 2 × 2 sampling region, quantified by the root mean
square (

) (Figure 38). The maximum roughness depth

in the measured region is 8.0 nm.

Figure 38: The AFM images of the surface roughness of the CPC elements

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The surface roughness result confirms zooming-focused MIP-VPP is of high printing
precision with the stair-stepping defect eliminated. Unlike previous studies relying on the meniscus
coating process, the dimensional accuracy of the fabricated sample is explicitly defined by the
zooming-focused MIP-VPP process, providing a controlled dimensional accuracy. We used light
flux at the receiver to characterize the performance of the 3D-printed CPC samples concentrated
on their outlets. Figure 39 shows the illumination performance over the varying incident angle.
The lenses with acceptance angles equal to 20 °, 30 °, and 40 ° are noted in different colors. For the
CPC elements with a = 20° , they have a light flux of 96k lux at the receiver of the devices when
light is shooting at 0° incident angle. As a comparison, the light flux is 15k lux without CPC
elements for the same area and incident angle. The CPC elements with = 20° have a huge light
flux drop as the incident light angle is larger than 20° . In comparison, the devices with a = 40°
have a wider acceptance angle, showing a slower light flux change over the incident angle of the
laser beam. The surface quality and dimensional accuracy of the 3D-printed samples are
comparable to those made by conventional optics fabrication methods
[177]
. Therefore, zooming-
focused MIP-VPP makes it possible to fabricate high-quality optical lenses rapidly.
83

Figure 39: Performance of the 3D printed compound parabolic concentrator.

To present the advantages of fabricating complex optical structures, we further demonstrate
the capability of zooming-focused MIP-VPP in 3D-printing a fisheye lens and a laser beam
expander consisting of multiple optical lenses.
[105, 178, 179]
Figure 5a shows the surface profile of a
fisheye lens, in which the spherical lens is placed on a cylindrical base. The diameter of the fisheye
lens is = 1. 5 ; the height of cylinder base is ℎ

= 0. 5 ; the curvature of the fisheye
lenses varies from 0. 63

to 0. 4

, respectively. Figure 5b shows the optical
microscopy images of the 3D-printed fisheye lens. The SEM image of its top surface shown in
Figure 5c reveals smooth surfaces without visible stepping or pixelized aliasing under 230X
magnification.
84

Figure 40: Fisheye lenses and laser beam expanders fabricated via zooming-focused MIP-VPP. a) Schematic
design of the fisheye lens. b) Optical microscope image of the zooming-focused MIP-VPP printed fisheye lens,
scale bar: 1 mm. c) SEM images of the zooming-focused MIP-VPP printed fisheye lens from the side view, scale
bars: 100 µm.

Besides the nanoscale surface roughness, the dimensional accuracy of the fisheye lenses is
studied. The fisheyes lenses structured with designed curvature of 0. 4

0. 5

0. 55

, and 0. 63

were printed. Their fabrication time is around 2 minutes . The printed
curvatures measured from optical images are 0. 413

0. 504

0. 549

, and
0. 632

, ensuring the dimensional fidelity, as displayed in Figure 41. The lens with a smaller
curvature has a larger error because, when the curvature is small, zooming-focused MIP-VPP
requires a fast motion of the LCD and the objective lens, leading to larger errors.
85

Figure 41: Optical microscope image of zooming-focused MIP-VPP printed fisheye lens over different
curvature, scale bar: 1 mm.

As shown in Figure 42a, the lateral resolution of the 3D-printed fisheye lens is
characterized by imaging a USAF1951 resolution target at multiple illumination wavelengths. The
Multiple bandpass filters with center wavelengths of 441 nm, 635 nm, and white light were used
for illumination. The aperture size of the imaging system is 3 mm. By imaging the resolution target
using white light, the 3D-printed fisheye lenses can resolve the most minor feature in element 6 of
group 7 (Figure 42b).
86

Figure 42: Optical performance of the 3D printed fisheye lenses a) Schematic illustration of the
experimental setup to characterize the imaging resolution of the 3D-printed fisheye lens. b) Imaging resolution of
the printed fisheye lens measured by USAF 1951 resolution target, scale bar: 500 µm (left), scale bar: 50 µm
(right).

Subsequently, the experimentally measured image modulation as a function of spatial
frequency is calculated to define the modulation transfer function (MTF), Shown in Figure 43. A
10% modulation of the MTF is used as a threshold to determine the imaging resolution. The MTF
curves indicate the maximum imaging resolution at the spatial frequency of 244.8, 265.5, and
306.8 lp/mm under the illumination of blue, red, and white light, respectively.
87

Figure 43: Experimentally measured MTF for zooming-focused MIP-VPP printed lens at wavelength
441 nm (blue), 635 nm (red), and white light.

A lens set is usually required for real-world applications. The position and orientation of
multiple lenses in a lens set need to be accurately calibrated and positioned during assembly to
minimize optical errors.
[180, 181]
Here, we demonstrate the capability of zooming-focused MIP-VPP
in directly fabricating an in-situ assembled lens set. A Keplerian laser beam expander is used as a
representative example. Figure 44a shows the design of a 1:2 ratio laser beam expander. The focal
lengths of the fisheye lenses are 2.17 mm and 4.35 mm, respectively. The CAD model of the laser
beam expander is shown in Figure 44b. The 3D printed result, shown in Figure 44c, has two
fisheye lenses with self-aligned optical centers. The two printed lenses offer accurate dimension
control and smooth surface quality directly defined by the 3D-printing apparatus. The assembly
features were printed using the traditional MIP-VPP process (

= 0).
88

Figure 44: 3D printed beam expander testcases. a) Schematic illustration of the laser beam expander
with 1:2 expanding ratio. b) CAD model of the laser expander. c) Printing result of the 3D-printed laser
expander, scale bar: 1 mm.

The performance of the beam expander was studied by shooting a laser beam of 635 nm
wavelength through it. The laser beam through the beam expander project a larger round dot on a
grid light screen which is 100 mm away from the expander, as shown in Figure 45-left, compared
to the laser beam without expander shown in Figure 45-right. Therefore, 3D-printed optical
devices with complex structures, including mechanical fixtures, could be in-situ fabricated by
zooming-focused MIP-VPP, as it supports high surface smoothness and high dimensional accuracy
printing for both lenses and assembly features.
89

Figure 45: Beam expanding performance of the 3D-printed laser beam expander.

We demonstrate the customizability and adaptivity of the Zooming-Focused MIP-VPP by
printing two customized geometries. In Figure 46a, cone-shaped fibers are printed in a hexagonal
manner. Each fiber is 250um in diameter at its root and 50um in diameter at its tip. The length of
the fiber is 1.8mm. Smooth surface quality and sharp tips can be observed. And every fiber presents
a good accuracy of the length, thickness, and position. This test case shows the capability of
Zooming-Focused MIP-VPP in fabricating dense micro features. In Figure 46b, a cone gear with
four teeth is printed by Zooming-Focused MIP-VPP. The side surface of the Ridges and valleys
of the cone teeth are demonstrated in the top Figure 46b. The dimensional accuracy and the surface
quality can be verified from the top view and side view images. Super smooth of the slide surface
shows perfect elimination of the stair-stepping. The bottom Figure 46b shows a sharp lateral
dimensional control of the cross-sectional image in the printed cone gear.
90

Figure 46: Customized geometry printed by zooming-focused MIP-VPP. a) microscale cone fibers
structure front-view(top) and top-view (bottom). b) microscale cone gear structure with four teeth front-view(top)
and top-view (bottom). scalebar 200 μm.

4.1.7 Conclusion
A zooming-focused MIP-VPP-based 3D printing process has been presented for optical
lens fabrication. The zooming-focused MIP-VPP uses frustum layer stacking to reduce the
staircase stepping and improve the surface roughness so microscale dimensional accuracy and
nanoscale surface roughness can be achieved. Consequently, no post-processing, such as meniscus
coating and surface polishing, is needed. We demonstrated the capability of zooming-focused
MIP-VPP in 3D printing CPC elements and fisheye lenses with subwavelength surface roughness
(3.4 nm) and pixel scale dimensional accuracy (14.7 µm) within minutes, offering a low-cost
91

solution for the rapid fabrication of customized optical lenses. We further demonstrate the printing
of an in-situ assembled beam expander with a 1:2 expanding ratio, offering the advantage of
integrated optical device fabrication. The 3D-printed assembly enabled by zooming-focused MIP-
VPP features high surface quality and remarkable positioning accuracy over each optical lens.
Hence any assembly error or additional calibration effort during the lens assembly is avoided. With
the principle of zooming-focused MIP-VPP demonstrated, its fabrication scale can be adjusted by
enlarging or shrinking the sizes of the imaging system for lenses suitable for various engineering
systems. The mask projection system can also be improved, such as expanding the aperture size
of the objective lens to provide a larger mask image on the building platform. Furthermore, the use
of frustum layer stacking may open a new avenue for 3D printing optical components and other
devices that require an ultra-smooth surface finish and high dimensional accuracy.
As a result, zooming focused MIP-VPP eliminates the layer stepping aliasing and improves
the surface quality to the sub-micron level. This improvement of surface quality is achieved
without sacrificing the dimensional accuracy, printing speed, and fabrication scale.

92

4.2 Improving Lateral Resolution through Spatiotemporal MIP-VPP
The lateral resolution is defined as the smallest printable feature in a lateral plane, which
is limited by the discrete nature of the digital mask image projection process. The CAD model is
sliced by a set of parallel planes to form pixelized mask images for layer-by-layer stacking of the
thin sheets. The sliced mask images will be sent to DMD or LCD to project on the building area.
The resolution of the mask images, measured by the numbers of the pixels, are determined by
numbers of the micro-mirrors in DMD or the cells in LCD. For a specified mode of DMD or LCD,
the building area decreases when the size of pixels shrinks down, since the numbers of the pixels
is determined. A mask image tool with smaller pixel size has higher the resolution of mask images,
but smaller fabrication extent and slower speed for same objects with same volume. This is the
trade-off between process resolution and production efficiency. Traditional AM processes balance
the resolution and efficiency by tuning the processes either in the temporal domain (e.g., higher
speed in serial processes) or in the spatial domain (e.g., more tools in parallel processes). To
improve the resolution of the mask images without sacrificing efficiency, a data-driven mask
image planning method based on subpixel shifting in a split second by tuning the process in both
temporal and spatial domains is presented. A piezo-actuated shifting 3D printer incorporating our
subpixel shifting method was built. The corresponding slicing and optimizing algorithm for the
generation subpixel mask images was developed. Several test cases are built to demonstrate the
effectiveness in the lateral aliasing reduction.
In previous work [83], we know lateral aliasing is determined by both the mask images and
the photocuring behavior. An optimized pixel blending method was developed by canceling out
the aliasing from the pixelized mask images using the modelized aliasing from photocuring
behavior. an optimized pixel blending method to address the aliasing issue, which can significantly
93

improve the part accuracy and surface finish by intelligently setting mask images used in MIP-
VPP. Based on this optimized pixel blending method, we invent a novel spatiotemporal mask
image projection vat photopolymerization (sps-MIP-VPP) to further push the resolution limitation
to subpixel level by a subpixel shifting printer and the corresponding slicing algorithm of subpixel
mask image generation.
4.2.1 Modelling Non-Uniform Curing Behavior
In the optimized pixel blending method, the aliasing from the photocuring behavior is
modelized. The cutting behavior of a tool in CNC machining is uniform both across the cutter and
the part (i.e., the cutting depth under the cutting tool is the same while it is zero outside the cutting
tool). Unlike CNC machining, the material’s photocuring behavior in MIP-VPP is highly
nonuniform, including 1) nonuniform within each tool (a projection image pixel) and 2)
nonuniform across the whole layer.
1) Nonuniform in a pixel: the pixel of a projection image is Gaussian distributed rather than
uniformly distributed [27]. Hence, the curing depth varies at different locations according to the
Beer–Lambert law of absorption [28]. In particular, a single pixel generates a paraboloidal shape
instead of a cubic or cylindrical shape [1]. Figure 47a-left shows the Gaussian profile of a single
pixel, and Fig.47b-left shows the simulated curing profile in the paraboloidal shape (blue color)
for a single pixel. Also, each pixel spreads into neighboring pixels and causes additional energy
deposition, which is called pixel blending [20]. Figure 47a-right shows the energy-deposition
effect of multiple overlapped pixels, and the red curve shows the accumulated energy contributed
by all the associated pixels. Figure 47b-right shows the curing result of these pixels, and the blue
curve shows the curing profile with a top-hatted paraboloidal shape. Depending on the optic
configuration, the projection image pixels have different Gaussian profiles at the focal plane.
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Figure 47c-left shows the pixel profile of a highly focused image from a high-quality projector.
The crest and trough of the ripples correspond to the center of the micromirrors and the gap
between the adjacent micromirrors. The highly focused image is beneficial for high quality
fabrication in MIP-SL systems; however, the ripple profile will be replicated onto the printed part
with undesired marks [16,23]. In contrast, Figure 47c-right shows the pixel profile of a less
focused image, where the pixels are blurred and spread to neighboring pixels. According to the
curing profile (Figure 47b) the over-blurred image will affect the edge sharpness of the cured part
and the accuracy of the lateral dimension. Figure 47c-middle shows a pixel profile that achieves
a good balance between the fully focused and over-blurred profiles. Based on the projection image
calibration [29], this pixel profile will be used for the process planning in our study.

Figure 47: Simulation of the non-uniform polymerization behavior. a) the 2D (the projection of 3D) and
3D visualization of the Gaussian-shaped energy distribution of a pixel and the accumulated energy of multiple
95

pixels. b) The simulated cured shape based on the accumulated energy and curing threshold. c) The accumulated
energy of pixels with different shaped Gaussian distribution

2) Nonuniform across the whole layer: MIP-VPP is a multiple-tool process (using pixel
array), and the pixels have different geometries and intensities, due to the inherent optical defects
caused by spherical aberration, astigmatism, dispersion, spatial incoherence, and distortion.
Different gaussian functions need to be used to approximate the light intensity of all the pixels in
the work. For an actual projection system, these energy differences between pixels need to be
considered when curing behavior model is built. A grayscale image can be used to compensate the
nonuniformity between the pixels. Grayscale value from 0 (black) to 255(white) is applied to each
pixel according to its inherent gaussian function to make the light intensity uniform across the
layer.
4.2.2 Overview of the Spatiotemporal MIP-VPP Process
in the remainder of this section, we will discuss how the subpixel shifting technique can
address the energy dose control challenge in the MIP-VPP process. In the sps-MIP-VPP process,
the projection image is moved at a subpixel size distance (

, n ≥ 2) in a split second. Figure
48 shows the basic concept and hardware configuration of the sps-MIP-SL process for n = 2. For
each layer, we precompute a set of optimized images and then project the mask images in a
predefined sequence (from Mask1 to Mask4) with a controlled exposure time (

). Between
two consecutive exposures, the projection device is moved by a small distance

using a fast
and precise XY linear stage. The total exposure time of the layer is still equal to the regular curing
time for photopolymerizing a layer. Such a method extends the traditional static MIP-VPP process
from the pure spatial domain to the spatiotemporal domain by increasing the changing frequency
96

of the projection images by

times (in 1–10

Hz). Consequently, the process resolution of sps-
MIP-VPP can be significantly improved using better controlled light exposure in the curing
process.

Figure 48: Principle of subpixel shifting for n=2. Four mask images will be used to photocured a 2D
layer

Figure 49 shows the comparison of the mask images and the simulated curing profiles
between the static MIP-VPP process and the subpixel shifting method for a circular layer. Figure
49a shows the curing profile of the binary static mask. large lateral aliasing due to the pixelized
mask image can be observed from the simulation. Figure 49b shows curing profile of the static
mask with grayscale pattern. The gaussian distribution of the UV energy is controlled to generate
smoother boundary and less aliasing. Figure 49c shows the photomasks of sps-MIP-VPP process
with n = 2. Each pixel is divided into four subpixels with corresponding grayscale pattern. The
97

sps-MIP-VPP process with optimized mask images can achieve higher product surface quality
without sacrificing the process throughput.

Figure 49: Comparison of the mask image and simulated curing profile between static process and sps-
MIP-VPP process. a) mask image and simulated curing profile of binary MIP-VPP. b) mask image and simulated
curing profile of grayscale MIP-SL. c) mask image and simulated curing profile of sps-MIP-SL.

4.2.3 Photopolymerization Simulation of Spatiotemporal MIP-VPP
The quality of a manufacturing process is typically evaluated by the tightness control of
the dimensional tolerance and geometrical tolerance. The geometric tolerance includes positional
tolerance and shape tolerance (e.g., straightness, flatness, circularity, squareness, etc.). The sps-
MIP-VPP system can meet these geometric dimensioning and tolerancing (GD&T) criteria in the
following ways: a) Dimensional tolerance: although the pixels are Gaussian distributed and
overlap with each other, these properties can be taken advantage of by intelligently setting the
grayscale values of the projection images to tune the part dimension. As shown in Figure 50a, a
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boundary pixel merges with the neighboring pixels, so the cured boundary can be fine-tuned by
assigning the grayscale level of the boundary pixel. Accordingly, a tight dimension tolerance can
be achieved by the optimized pixel blending algorithm. The subpixel shifting technique provides
another degree of design freedom to control the dimension by moving the location of the boundary
pixel (green color in Figure 50d). As both the grayscale value and the location of the boundary
pixel can be adjusted in the near-continuous motion mode, the dimensional accuracy can be more
precisely controlled. The blue color curves in Figure 50a, 50d and, 50e show the predicted shapes
of the cured part by adjusting both the grayscale value and the location of the boundary pixel. b)
Positional tolerance: positional tolerance is crucial for part assemblies. Although the pixel blending
algorithm can achieve a reasonable control on the dimensional tolerance, the discretized nature of
the light projection device brings challenges to the positional tolerance control. It is almost
impossible to precisely align all the features with the pixels in a layer for a generic digital model.
This problem can be effectively addressed by subpixel shifting. An example is shown in Figure
50b, in which a slab part has six slot-hole features. Also, these holes are misaligned within the
pixels, and the size of the gap between the adjacent holes is not a multiple of the pixel size. Hence
the locational tolerance cannot be tightly controlled by adjusting the layout of the part. In
comparison, with subpixel shifting, the features can be fabricated by exposing the shifted
projection images that may be perfectly aligned with the features. c) Shape tolerance: the MIP-
VPP process fabricates parts by tessellating the target geometry with discretized unit cells. Such
approximation introduces shape errors. For example, as shown in Figure 50c, the inclined line
feature and curve feature lost the linearity and circularity during the tessellation process (top
figure). This problem can be addressed by moving the mask image in a subpixel distance to better
approximate the features (bottom figure). Essentially, shape tolerance is benefited by the increased
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resolution from the subpixel shifting motion. d) Surface finishing and sharpness: due to the
Gaussian distribution and pixel blending effect, it is challenging to achieve a high-contrast image,
i.e., the accumulated energy gradually increases from 0 to maximum value across the boundary
(the red curve in Figure 50a, 50d and, 50e)). It is desirable to plan the images such that the
accumulated energy curve has a steeper slope around the shape boundary. Hence: 1) a higher
contrast can facilitate the separation between the solid and liquid areas without resulting in the gel
state (partially cured), because the gelled features affect the surface finishing and part accuracy
during the post-processing stage. 2) According to the absorption law (

=

ln (/

)), the
curing depth is logarithmically proportional to the accumulated energy. Hence the higher contrast
results in a steeper side surface in the vertical direction, which is more desired in the layer-based
AM process. As shown in the simulated result in Figure 50e, the subpixel shifting-based image
projection can form a steeper energy curve (solid red curve) than the regular pixel blending images
(dashed red curve). Consequently, sharper side edges (blue curves) can be achieved for a high-
contrast layer boundary.
100

Figure 50: Control of the dimensional tolerance and geometrical tolerance of the sps-MIP-VPP. a) one
dimensional simulation shows the dimensional accuracy can be tuned by adjusting the grayscale value of the
boundary pixel. a) An example to show the pixel shifting technique can achieve high positional tolerance. c) Two
examples to show the pixel shifting technique (bottom) can achieve high shape tolerance. d) The dimensional
accuracy can also be tuned by the location of the boundary pixel in the pixel shifting system. e) The comparison
between the regular pixel blending and pixel shifting shows the pixel shifting approach can achieve higher-
contrast images and hence sharper and cleaner cured edges

The aforementioned discussion elucidated the basic principle of subpixel shifting to
improve MIP-VPP’s process resolution and fabrication accuracy without sacrificing production
throughput. However, the shapes of all the demonstrated cases are relatively simple, and the
forward problem (i.e., from mask images to the cured profile) is computationally inexpensive. The
real-world applications in the sps-MIP-VPP process are more complicated in terms of the given
geometric shape, the number of pixels, and the nonuniformity of the projection pixels. More
importantly, the mask image planning to compute a set of mask images from a given cured profile
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is an inverse problem. Such a backward problem is much more computationally intensive, typically
requiring an iteration-based optimization solver.
4.2.4 Optimizing Mask Image Planning for Spatiotemporal MIP-VPP
In this section, we will present a highly efficient optimization solver, namely error diffusion,
to address the complex mask image planning problem for sps-MIP-VPP and account for the non-
uniformity of the projection pixels.
As demonstrated to the Zhou et al.’s previous work [60], the pixel blending of light
intensity would present tremendous capability in selectively solidifying liquid resin into desired
shapes. However, a single Gaussian function was used to approximate the light intensity of all the
pixels in the work. The actual projection system, however, has inherent optical defects caused by
spherical aberration, astigmatism, dispersion, spatial incoherence, and distortion. Hence the energy
distribution is generally nonuniform. These geometric and energy differences in a projection image
would affect the pixel blending results and, therefore, part quality. Also, the original pixel blending
method can only solve a small-scale problem. It falls short when dealing with real-world CAD
models to be fabricated in the sps-MIP-VPP process. Zhou et al. presented a mask image
calibration method for MIP-VPP to address the nonuniformity of a DMD-based projector before
[83]. Our approach follows the same calibration method, i.e., a geometric calibration system that
can calibrate the position, shape, size, and orientation of a pixel, and an energy calibration system
that can calibrate the light intensity of a pixel.
By integrating the geometric and energy calibration results, we can compute an
approximation function for the light intensity of each calibrated pixel by using the following
equation:
102

(, ) = exp −
1
2

( −

) cos () − ( −

) sin ()

+
( −

) () − ( −

) cos ()

(27)

As the Figure 51 shows, (, ) is any point on the building platform, (

,

) is the center
of the calibrated pixel, is direction of the elliptic gaussian distribution of the light intensity,

,

are the derivations of the gaussian distribution of the light intensity. The parameters

,

,

,

, and θ are computed from the geometric calibration system, and the parameter A is
computed from the energy calibration system.

Figure 51: Light intensity model of each calibrated pixel

Accordingly, a database of such functions (, ) can be built by storing all the parameters
(

,

,

,

, , ). For a pixel not directly calibrated, an approximated function can be computed
by identifying its direct neighbors and, accordingly, using a simple linear interpolation to calculate
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the related parameters. Based on the calibrated approximation function (, ), we can derive a
new optimized pixel blending model shown as follows.

(28)

,
is the nominal photopolymerization design at point (p, q) on building platform.
,
is
the accumulated UV energy dosage at point (x, y).

is the grayscale applied to pixel (i, j). δ is
the curing threshold.
,

is the practical photopolymerization result at point (p, q) on building
platform.
Considering the eqn.(1), we have

,
=

exp−
1
2

− ̅

cos
̅

− −

sin
̅

+
− ̅

̅

− −

cos
̅

(29)

=

̅

,

=

,

=

(30)

=

255
, = 1,2 … 255
(31)

104

where ̅

,

,

,

,

are the calibrated parameters for native pixel (i, j) with full
light intensity (grayscale = 255 or

= 1).

,

,

are the Gaussian parameters for pixel (i,
j) with partial light intensity (i.e., grayscale <255 or

< 1). Compared with the original pixel
blending model eqn.(1), the objective function and the constraints are identical; however, the
Gaussian function (, ) has been replaced by the image calibration results. The new optimized
pixel blending model is more general than the original model; however, it is also more difficult to
solve since

,
may vary from pixel to pixel in the newly formulated mode.
4.2.5 Generation of Mask Images using Error Diffusion Algorithm
In the formulated model defined in Equation (1), the shape parameters are not linearly
decreasing with the light intensity, even though all the calibrated parameters are constant
coefficients. Thus, such an optimized pixel blending model cannot be solved by linear
programming solvers. An iteration-based optimization method called discrete direct search (DDS)
was developed in previous work [83]. The DDS method is independent of the properties of the
constraints, variables, and objective functions in the optimization model, and can only provide a
suboptimal solution for images with medium size. It cannot be used for large-scale mask image
planning required by the sps-MIP-VPP.
To plan the set of mask images for sps-MIP-VPP, we investigated two fast optimization
methods: boundary erosion and error diffusion, where error diffusion was built upon boundary
erosion. Experimental results demonstrated that both algorithms could solve the optimized pixel
blending problem in a reasonable time. However, the error diffusion method performed
substantially better than the boundary erosion method.
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The boundary erosion method is purely based on geometric information. The photocuring
process in MIP-VPP includes two phases: light irradiation and polymerization. In MIP-VPP
system each pixel will spread into its neighboring pixels. Also, when the light beam is projected
on the resin surface, the liquid particle will disperse photons in the lateral direction. The higher the
refractive index, the stronger the dispersion will be. Thus, the actual irradiation area of a projection
image is larger than that of an ideal projection image. Therefore, the built part usually comes with
a larger size than the designed one. During the polymerization process, the photo-initiator bonds
the monomer together once the input irradiation energy is larger than the critical exposure energy.
Both the curing depth and width are larger than the energy penetration depth and width. Thus, on
the other hand, it increases the size of the part in the XY direction. At the same time, it provides
the over-curing that is required to bond the newly cured layer with the previous layer. The curing
process analysis can guide us for better mask image planning results. An intuitive way to address
the dimensional problem is to shrink the contour or the mask image to compensate for the over-
curing size. Such a compensation-based method is called boundary erosion in this paper.
In the boundary erosion method, the boundary pixels are gradually eroded according to the
calibrated parameters. For the sake of simplicity, we demonstrated the process of applying the
boundary erosion method to a binary mask image with a circle shape. Figure 52a shows the
original binary image generated from the slicing algorithm, and Figure 52b shows the updated
mask image after one pixel was eroded along the boundary. Experimental results show that the
boundary erosion approach can solve the problem quickly and obtain good dimension accuracy
for parts with large sizes. However, two problems exist for this heuristic method: (1) It cannot
preserve sharp features, e.g., a sharp corner is rounded. (2) It cannot preserve thin features, e.g., a
thin wall is lost even other thicker walls have accurate dimensions. The main reason for this issue
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is the boundary erosion approach is based on the geometrical model and only considers the
geometrical information, including the size and dimension. However, it does not consider the
photocuring process, including the energy dosing, the polymerization, etc. As discussed before,
the light beam has the profile of Gaussian distribution instead of uniform top-hat distribution. Each
pixel is also spreading into the neighboring pixels. Thus, the sharp corner will be rounded if no
extra pixels in the neighborhood are used to help the increase of the light energy. Similarly, for the
thin features, the accumulated light energy is not saturated to reach the threshold. Thus, it is also
lost, or only partially cured gel is generated.

Figure 52: Illustration of boundary erosion method

Although boundary erosion works well for parts with large sizes, it has difficulty dealing
with sharp features and small features. To overcome this problem, we combined the boundary
erosion method and the pixel blending model, and developed a new method called error diffusion.
The error diffusion method is based on both geometrical and energy information. Unlike the
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optimization methods used before, the error diffusion method considers the error information and
compensates the energy distribution by diffusing the errors back to the planned mask images.
The basic idea of the pixel blending process is schematically shown in Figure 53, A 3D
solid model (a) is first sliced by a set of horizontal planes. Each slice is then converted into a 2D
image (b) by some sampling methods. In our case, we use super-sampling. That is, the sampling
resolution is n times higher than the projection image’s resolution. Hence, for each pixel of the
projecting image, we subdivide it into n x n subpixels and use the subpixel resolution as the input
image’s resolution. According to the target image (b), grayscale mask images is used to uniform
the light intensity in a mask image(c) by the sample points calibration. Since the energy distribution
each pixel follows a Gaussian function(d), the mask image will convolute with the Gaussian
function and get the accumulated intensity (e) as = ∗ . The accumulated intensity should be
same for every pixel of the mask images to get a uniform curing behavior. According to the
polymerization reaction process, the material will be cured only when the energy is higher than
the critical energy. Thus, the blending result (g) is computed as

= ( ), where T is the gate
function (f ) with the threshold of critical energy for a given resin, and

image(g) is a binary
image representing the blending result. Comparing the blending result

image(g) with the target
we get the error image (h) . The colorful pixels show the errors (red color denotes extra
portions and green color denotes missing portions). The objective of mask image planning is to
reduce the error as small as possible without violating the constraint.
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Figure 53: An open-loop pixel blending process

Although the theoretical pixel blending model is straightforward, solving it for the global
optimum is difficult, if not impossible in practice. It becomes even worse when the calibrated
database is integrated into the optimization model. Different optimization techniques such as the
gradient-based optimization approach and the iteration-based heuristics approach can be used.
Unfortunately, they cannot achieve an acceptable solution in a reasonable time. The main problem
is that they work in an open-loop manner without considering the essential property of the pixel
blending process. For example, the iteration-based DDS method arbitrarily assigns light intensity
to pixels without considering the direction and amount of the pixel change [83], while the gradient-
based method only relies on the mathematical model that is quite generic [60]. Here, a new error
diffusion method has been developed to solve the mask image planning problem. It updates the
light intensity according to the error changes, as shown in Figure 54, using similar notations in
Figure 53. In essence, the pixel blending process now evolves from an open-loop system to a
closed-loop system.
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Figure 54: Error diffusion algorithm based on closed-loop pixel blending process.

The error diffusion method is similar to other algorithms listed before, except that it
diffuses errors and updates the light intensity according to the diffused errors. An initial solution
from the geometric method or the aforementioned boundary erosion method is used as a starting
point of the error diffusion method. According to the current light intensity and the calibrated
Gaussian distribution of each pixel, the accumulated energy for each subpixel is calculated. Based
on the accumulated energy and the threshold energy, the error of each subpixel is judged based on
Eqn.(32).

=
⎩
⎪
⎨
⎪
⎧
0

=

−

( ,)∈

≠

(32)
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=

+

(,)∈

(33)

The (, ) is the position of small subpixels represented in XY coordinates. The ( , ) is
the position of the large pixels represented in XY coordinates.
( ,)
is the grayscale pattern applied
to the large pixels.

(
( ,)
) is the accumulated energy of the small subpixel at (, ) point
generated from the large pixels
( ,)
. The photopolymerization error

is determined by
difference between the accumulated energy and the threshold energy . Accordingly, the error is
diffused to its neighboring pixels to rectify their light intensity as in Eqn.(33).

is the
correction energy for large pixel at ( , ) generated by the small subpixel

. Note that, the error
is diffused with the weight proportional to the Gaussian distance as used for convolution. The
flowchart of the error diffusion method is discussed using examples as follows.
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Like other closed-loop systems in engineering problems, this algorithm is converged
asymptotically. A constant parameter λ can be used as a multiplier for the diffused error to control
the convergence speed. The larger the parameter, the faster the algorithm can converge, but it may
oscillate when approaching the steady state. In contrast, a smaller parameter can lead to slower but
smoother convergence. Figure 55 shows the error convergence with different coefficients. It
shows that the algorithm converges pretty fast in the initial steps. During these steps, the algorithm
erodes the boundary pixels to match the target dimensions. The algorithms converge much slower
in the following steps as it manipulates the pixel values to recover the sharp features during these
steps. In Figure 55a shows a portion of a typical target image, Figure 55b–55d shows the error
images corresponding to the target image at different optimization stages.
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Figure 55: Iteration process controlled by the optimization parameters

Figure 56 shows the detailed error diffusion procedure based on a simplified example.
Suppose the image size is 4 x 4, and the target geometric shape is a circle. In this example, the
super sampling level n= 3. According to the image coverage, a super sampled image with small
pixels is shown in Figure 56a. An initial mask image is generated by counting the ratio of small
white pixels, as shown in Figure 56b. After applying the Gaussian function to each big pixel and
distributing the light intensity to its neighborhood, the accumulated effect for each small pixel is
shown in Figure 56c. In this case, the Gaussian parameter σ =1.0. After comparing the
accumulated effect with the predefined threshold, the error of each small pixel can be calibrated
as

. The threshold is set as 1.8 in this example. The errors occur on the boundary of the
geometry. In this case, the circular feature is a small feature, and some portions of the boundary
are missing. Accordingly, the error of each super-sampled pixel will be diffused into its
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neighboring original pixels following the Gaussian function. Hence, all the diffused errors on each
original pixel will be accumulated together. The final result is shown in Figure 56d After adding
the diffused errors to each pixel of the previous mask image, the updated mask image is generated
in Figure 56e. Note that the grayscale value is truncated in the range of [0, 1]. In this example, as
the boundary has missing portions, the grayscale values of the updated mask image are higher than
the initial mask image. With the new mask image, the accumulated effect will be updated. This
process is repeated until the error is converged.

Figure 56: An example to show the error diffusion process

Once the parameters are obtained using the image calibration method, the error diffusion
method is ready to solve the optimization problem. However, the error diffusion method is still
iterative, and the efficiency could still be a problem if the image size is large. The convolution
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process shows that the pixels around the part boundaries are more important and more challenging
to solve in the mask image planning. Based on this observation, a more efficient error diffusion
method using adaptive sampling has been developed. Figure 57 shows two examples of adaptive
sampling. That is, only the regions that are close to the boundary (shown as blue and green dots)
are finely sampled. The adaptive error diffusion method can dramatically improve the mask image
planning efficiency without losing solution quality. Also, the algorithm complexity now only
depends on the length of the contour rather than the image area. Thus, the adaptive sampling-based
error diffusion is a more general and robust method for the sps-MIP-VPP process.

Figure 57: Adaptive error diffusion with improved efficiency

The subpixel shifting method significantly expands the searching space of mask images.
However, the dramatically increased amount of data brings significant challenges to the
computational algorithms. The error diffusion method will consider the interplay between the
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pixels and light–material interactions, the shifted mask images’ coordination, and the optical and
material properties. The revised optimization model incorporating the calibrated pixel blending
model can be augmented by replacing the single image (pixel location: ̅

,

) with a set of
images (pixel location: ̅

+

,

+

), where Δx and Δy are the shifted subpixel distances).
Similarly, the error diffusion algorithm can be revised by adding another loop to evaluate all the
subpixel images for each iteration. Note the super sampling level n can be the same or larger than
the subpixel shifting level.
4.2.6 Computational Results and Simulation Verification
The presented mask image planning algorithm based on adaptive error diffusion has been
implemented using Cpp programing language with Microsoft Visual Cpp compiler. The used test
platform was a commodity personal computer with a 3.2 GHz processor and 16 GB random-access
memory running Windows 10. A 3D simulation software was developed using Microsoft
Foundation Class (MFC) and OpenGL library to verify the effectiveness of the computation results.
The simulation software implemented the visualization of the profile of individual pixels (gold
color), accumulated energy (red color), and the cured part (blue color) for any planned mask image
or a set of shifted mask images (Figure 58a). Several test cases were designed to demonstrate the
effectiveness of the subpixel shifting technique based on adaptive error diffusion in controlling
dimensional and geometric tolerances
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Figure 58: Simulation of the Subpixel shifting process. a) The perspective view of the 3D simulation
software. b) The side view of the 3D simulation software shows the pixel shifting approach (bottom) can achieve
steeper accumulated energy and a sharper cured surface. c) The top view of the 3D simulation software shows
the pixel shifting approach (bottom) can achieve higher circularity of the printed feature than the static MIP-VPP
process (top) and the regular pixel blending approach (middle).

1) Shape Tolerance Control
A test case of a simple circular shape was used to verify the circularity control of the
developed method. Figure 58b shows the comparison of the accumulated energy (red color)
between the traditional image planning method (top) and the subpixel shifting method (bottom). It
can be seen that the mask images generated by the subpixel shifting method can achieve a much
smoother distribution of the accumulated energy. Figure 58c shows the top view of the
accumulated energy (red) and the cured profile (black) for the static MIP-VPP (top), the regular
pixel blending approach (middle), and the subpixel shifting approach (bottom), respectively.
Among them, the subpixel shifting approach has a smoother boundary and higher circularity. Due
to the Gaussian distribution and pixel blending in the layer curing process, the traditional MIP-
VPP process cannot preserve sharp features.
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Figure 59: Simulation of the Lateral aliasing elimination via subpixel shifting process a) Sharp feature
simulation: the mask image (left) and simulated cured profile in the lateral direction (right) for the grayscale-
based mask image planning method (top) and the error diffusion-based mask image planning method (bottom). b)
The benchmark test cases designed to simulate the mask image planning algorithms for sharp features and small
features. f ) Small feature simulation: the mask image (left) and simulated cured profile in the lateral direction
(right) for the grayscale-based mask image planning method (top) and the error diffusion-based mask image
planning method (bottom).

A set of squared features Figure 59b-left, magenta color) were designed to validate the
squareness control of the subpixel shifting technique. Figure 59a shows the planned images (left)
and the associated cured profile (right: magenta color) by the traditional grayscale image planning
approach (top) and the error diffusion-based image planning approach (bottom). Note the error
diffusion method generated optimized mask images with intriguing grayscale patterns around the
corners, which is nonintuitive to humans. The underlying reason is that the corner features are
located at the boundaries of the part; hence less exposure energy is allocated due to the pixel
blending effect. Therefore, a sharp corner will be lost where the light energy is less than the critical
exposure energy (Ec). To retrieve the missing portions, the optimized mask images using adaptive
error diffusion automatically assign positive grayscale values outside the part and around the
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corners to increase the accumulated energy at the sharp corners. However, the grayscale values of
the outside pixels should be delicately assigned to avoid curing the regions outside the CAD model.
In the optimized mask image result, higher grayscale values were assigned to the pixels close to
the corner. The simulated curing results Figure 59a-right clearly show the subpixel shifting
method (bottom) can preserve the sharp corners, where the static MIP-VPP process generates
undesired round corners (top).
2) Dimensional Tolerance Control
Previous section demonstrates adjusting the grayscale value of the boundary pixels and
using subpixel shifting can achieve tight dimensional control in principle. In this section, we will
mainly focus on the validation of the dimensional tolerance control for small features. Figure 59b-
right (red color) shows the test cases with a set of line strips with a varying width from one single
pixel to multiple pixels. We observed that the dimensional tolerance of large features (over 10
pixels) is easy to control for both the traditional MIP-VPP and sps-MIP-VPP processes. However,
the traditional MIP-VPP process will completely lose the small features with only a few pixels
(e.g., 1-pixel wide strip in Figure 59c-top). In comparison, the adaptive error diffusion method
can preserve such small features by automatically assigning positive grayscale values to the pixels
outside the small features (Figure 59c-bottom).
3) Positional Tolerance Control
In this section, we will validate the effectiveness of the error diffusion method for more
test cases. Figure 60 shows a test case with a micropillar array that consists of micropillars with
2-pixel width and the incremental gaps with a subpixel size (g.1). The traditional image planning
method generates rounded corners and incorrect gap sizes (g.2) due to the combination of the small
geometric features and the subpixel-size locations. Like the test cases of sharp-feature and small-
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feature, the error diffusion method intelligently tuned the grayscale values of the neighboring
pixels and computed non-intuitive mask images (two example mask images are shown in g.3 and
g.4). The light energy accumulation of the planned mask images can generate both sharp corners
and desired gap size (g.5).

Figure 60: Dimensional accuracy and positional accuracy simulation for multiple squares

4)Sharpness Control
We discussed the steeper accumulated energy with higher energy contrast could generate
sharper part boundary to enhance the surface smoothness in Section 2. In this section, we used the
same cylindrical feature used in the shape tolerance study to validate the sharpness control. Figure
49b shows the side view of the accumulated energy (red) and the cured part (blue). The subpixel
shifting approach (bottom) can achieve steeper accumulated energy and a sharper cured surface.
The improvement is mainly attributed to the larger searching design space introduced by the mask
image motion in the sps-MIP-VPP process. Hence the error diffusion method can intelligently
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assign the grayscale values to the mask image pixels to maximize the contrast of the accumulated
energy. Such optimized results are non-intuitive to users and cannot be achieved by simple mask
image planning methods.
4) Computational Cost Study
The adaptive error diffusion method considers the specific property of the MIP-VPP’s
mask image planning problem. Hence, it can significantly improve computational efficiency and
part quality. In this study, different mask image planning methods, including the linear
programming method, the DDS method, and the error diffusion method, were compared based on
various image sizes for one slicing layer of a Stanford dragon model [60]. In the tests, the iteration
number for the error diffusion method was set to 15, and the CPU time for optimizing each image
is less than 30 s for each layer. The results are shown in Table 1. In comparison, the linear
programming method can only solve very small-scale problems, which is not acceptable for
industrial applications. Both DDS and error diffusion methods can achieve satisfactory results in
a reasonable time. However, the error diffusion method as a closed-loop approach can achieve
smaller errors in much less computational time.
Table 2: Study of Computational Costs
Image size
Linear programming DDS Error diffusion
Error CPU [s] Error CPU [s] Error CPU [s]
80x60 0 35 14 5 10 1.2
800x600 N/A N/A 823 45 545 10
1024x768 N/A N/A 962 76 850 12
1280x1024 N/A N/A 1525 265 1345 15
1920x1200 N/A N/A 2152 425 1562 25
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4.2.7 Experimental Validation
The aforementioned discussion elucidated the basic principle of subpixel shifting to
improve MIP-VPP’s process resolution and fabrication accuracy without sacrificing production
throughput. However, the hardware change alone using the traditional image planning method
cannot achieve such performance improvement. There are two key challenges of the sps-MIP-SL
process: 1)how to shift mask image precisely to control the exposures at

locations. 2)how to
achieve the subpixel mask image shifting in a split second. To solve these two key challenges, a
piezo actuated subpixel shifting system is explored.
A prototype machine to achieve the sps-MIP-VPP process was built for physical tests
(Figure 61). In the prototype system, a DMD based projector from SprintRay Inc. (Los Angeles,
CA) was used as the mask image projection device. The pixel number of the projector was 1024 x
768, and the envelope size of the projection image was 160 x 120 mm
2
. Hence each pixel size was
0.156 mm. Commercial photocurable resin (Perfactory SI500 from EnvisionTec Inc., Dearborn,
MI) was used in the tests. The layer exposure time was 4 s base on the curing depth analysis.

Figure 61: The hardware of piezo-actuated subpixel shifting MIP-VPP system
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A high-performance eight-axis motion control board KFLOP + SnapAmp (Dynomotion
Inc., Calabasas, CA) was used to drive the linear stages. The piezo stage was controlled by a high-
precision and fast-speed piezo-driver (Viking Industrial Products, Marlboro, MA) with input
voltage from 0 to 200 V. For each layer with subpixel level n = 4, the piezo-driven XY linear stage
moves the projected mask images in a stepwise moving mode within a pixel (Figure 62a) The
increment size of each step is 1/4 of the pixel size (i.e., 0.039 mm) and the projection time of each
mask image is 4/16 s (i.e., 0.25 s or 4 Hz). The accuracy and resolution of the piezo-stage were
verified, as shown in Figure 62b for one voltage pulse to the Y-piezo.

Figure 62: Subpixel shifting implementation. b) The subpixel shifting sequence of Mask1–Mask16 for n
= 4. c) The mask image movement by a piezo-drive XY linear stage for one voltage pulse in the Y-axis.

A part built using MIP-VPP is usually larger than the desired size if no compensation is
applied. To verify the adaptive error diffusion method on building parts with accurate dimensions,
a test case with an incremental stair size was designed, as shown in Figure 63a. Furthermore, the
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part was duplicated six times and placed at six different positions within the building platform to
verify the algorithm’s effectiveness in considering the mask image calibration result. The size of
each built stair was measured using a micrometer in both X and Y directions. The measurement
results of the eight stairs are shown in Figure 63b. Almost all the dimensions are controlled within
50 μm (note the pixel size was 156 μm and the subpixel size was 39 μm), and the error percentage
is less than 1% (an average of 0.15% with the standard deviation of 0.2%).

Figure 63: Experimental results and analysis of the dimensional accuracy a) Dimensional accuracy
study. b) Thin feature verification.

1) Thin Feature Verification
The simple boundary erosion method may lose thin features, whereas the error diffusion
method is general for parts with different dimensions. This test case is to verify the error diffusion
method on building such thin features. The CAD model is composed of thin strips and cubes with
incremental pixel sizes shown Figure 63b. Note the same model was used in the simulation test
shown in Figure 63b, all the tiny features were successfully built in the tests, and the size variation
is within 50 μm. Figure 63b shows the built results of the error diffusion method, in which all the
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thin features are retrieved. In comparison, the same part built using the boundary erosion method
lost all the one-pixel size features.
2) Sharp Feature Verification
The boundary erosion method does not consider the energy information. Hence the
fabricated sharp features are rounded due to the pixel blending effect. In comparison, the error
diffusion algorithm considers both the geometric and energy information to recover the sharp
corners. The same squared features in the previous section were used in sharp feature tests. The
mask images generated by the error diffusion method were used in the building process. It can be
seen that all of the corners are rounded for the part built by the boundary erosion method (Figure
64a-left), while the corners are much sharper for the part built by the error diffusion method
(Figure 64a–right). Some extra materials exist at the built corners, which can be further optimized
by a better calibration and optimization procedures in the future.

Figure 64: Experimental results and analysis of the application. a) sharp feature verification. b) A real-
world case studies.

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3) Real-World Testcase Verification
Finally, the error diffusion-based subpixel shifting technique was tested using a real-world
test case. As shown in Figure 63b, a teeth model was fabricated using the traditional MIP-SL
process, the regular pixel blending technique,[20] and the subpixel shifting method. It is clear the
subpixel shifting method outperforms the other two approaches in terms of surface finishing and
process resolution, which verified the effectiveness of the data driven subpixel shifting method for
MIP-SL.
4.2.8 Printing Results Discussion
A critical barrier of current AM technologies is how to address the performance trade-off
between process resolution and production throughput. For example, for the material-deposition-
based AM techniques, the multi-nozzle jetting process has higher throughput than the single-
nozzle writing process. However, the resolution is affected by the limited gap size between
adjacent nozzles [50]. For the energy-deposition-based AM techniques, Envision Tec developed
the 3SP process to increase the single-tool SLA process by increasing the changing frequency of
the laser status up to 106 Hz and traveling speed over 200 m/s [86]. However, the resolution is
now affected by the synchronization of the laser status and the spinning mirror location at ultrahigh
scanning speed. In general, the laser-based SLA has lower throughput than the MIP-VPP
technology due to its multitool parallel photocuring nature. However, the resolution is now
affected by the fixed position of the micromirrors used in light projection [57]. One potential
solution to address the resolution and throughput trade-off is a marriage between the motion-driven
SLA processes (in the temporal domain) and the multitool-based MIP-VPP processes (in the
spatial domain) by moving the projected image with controlled motions. Once the spatially static
images are replaced by dynamic images assisted by linear movements, the process resolution will
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now be defined by the motion resolution in addition to the pixel resolution. In particular, the pixel
resolution is directly related to the ability to print fine features and to control dimensional tolerance,
whereas the motion resolution is directly related to the ability to print features at accurate locations
and to control geometric tolerance. Furthermore, an interesting property of the MIP-VPP process
is that the pixel energy takes a continuous value when varying the grayscale level and/or the
exposure time; hence the size of the cured material can be continuously tuned by adjusting the
pixel energy [60]. Consequently, the resolution contributed by the pixels can be infinitely high if
the light–matter interaction effect is modeled and optimized. Theoretically, the process resolution
of the sps-MIP-VPP system can be mainly determined by the motion resolution even though the
pixel resolution is limited.
In the past years, various motion-assisted processes have been studied by several research
groups. Ha et al. proposed a moving projector approach to expand the printing area for the micro-
MIP-VPP [85]. Lee et al. extended the moving-projection system for the micro-MIP-VPP process
and addressed the part size limitation by stitching the projected image [84]. Although high-
resolution was reserved in these systems, their discrete moving-projector systems have very low
changing-frequency in the temporal domain, even lower than the regular MIP-VPP. Emami et al.
introduced a scanning-projection-based MIP-VPP process by continuously moving the projection
device over the printing area to achieve large-scale fabrication with improved resolution [75].
Later they proposed an analytical model for the scanning-projection system to investigate the
scanning and curing properties [79]. Zheng et al. presented a novel scanning-projection based MIP-
VPP [70], whereas a customized F-theta lens was required that severely limits the printing area
size. Several commercial machines have also been developed based on the scanning-projection
technique, including the Moving Light 3D printers from Prodways Technology (Les Mureaux,
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France) and the Large Area Maskless Photopolymerization (LAMP) from DDM Systems (Atlanta,
Georgia). Compared with the discrete moving-projector approaches, these continuous scanning
projection systems have a much higher changing frequency. The number of the mask images is
increased from one to 1000 (defined by the projection width such as 1920 x 1280 for 1080P
projectors and the exposure time such as 2 s). Although the continuous scanning systems can
increase the throughput by projecting mask images in parallel with motion, the resolution is still
limited by the pixel resolution as the motion is only used to expand the printing area rather than to
improve the process resolution. Instead of moving the projection image with a distance defined by
the projection image width (for the discrete moving projector systems) or the pixel size (for the
continuous scanning projection systems), we considered another strategy of using subpixel shifting
in a split second in this article. This strategy was first studied in our previous work [83], in which
a novel video-projection MIP-VPP process was developed. In such a process, a set of mask images
are projected and synchronized with the one-axis linear motion to increase the process resolution.
However, the one-axis movement of the projection image in the diagonal direction limited the
number of mask images used in the process. The stepper motor-driven linear motion system also
limited the motion speed and the related image changing frequency. Recently, Yuan et al. reported
another interesting motion-assisted micro-MIP-VPP by oscillating the projection device within
one pixel to fabricate smooth micro lens arrays [57]. However, the same projected image was used
for a 2D layer during the vibration. Without increasing the changing frequency of mask images,
such a temporally static image planning approach may not improve the process resolution for more
general geometric shapes.
The comparison between our sps-MIP-VPP and other motion-assisted MIP-VPP in terms
of spatial and temporal frequency is given in the Figure 65. The developed sps-MIP-VPP system
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dynamically changes the used mask images during a layer exposure while maintains the total
exposure time the same as the regular MIP-VPP process. To achieve a high image changing
frequency, we used a piezo-driven XY linear stage to move the projection device. The piezo-stage
used in this work has sub-micrometer resolution and microsecond response time. In comparison,
the pixel size of a projection image is in hundreds of micrometers, and the photocuring time of a
mask image is in hundreds of milliseconds. Hence, both the time of moving the projection device
and the motion error of a piezo-stage can be omitted in the sps-MIP-VPP system. Compared with
the stepper motor-driven linear motion system used in the video-projection MIP-VPP process [83]
the piezo-stage provides several orders of magnitude higher motion speed and resolution.

Figure 65: The spatial and temporal domain control of various vat photopolymerization

Much more mask images are now enabled in the sps-MIP-VPP process due to the increased
hardware resolution and speed; however, the computational cost is also dramatically increased to
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compute the larger number of mask images. Accordingly, we developed a process planning
framework for the sps-MIP-VPP process in this work. Such an efficient mask image planning
method considers both the spatial and temporal domains in MIP-VPP and shows to be effective
for various test cases. Further development of such mask image planning methods for faster
projection image movement and higher mask image changing frequency is critical for the sps-
MIP-VPP process [65,68] and volumetric lithography [72]. Finally, although developed for the
sps-MIP-SL process, the presented data-driven process planning framework is applicable to other
energy deposition-based AM processes by considering their spatiotemporal properties.
4.2.9 Conclusion
AM processes suffer from limited tools available for material deposition and energy control.
Most AM processes are either planned in the temporal domain or the spatial domain to balance
process resolution and production throughput. In this work, we have presented a subpixel shifting-
based MIP-VPP process by fastmoving the projection device with a subpixel distance in a split
second, so the mask images used in the sps-MIP-VPP process can have significantly increased
design space to address limited tools available. Furthermore, an efficient error diffusion-based
optimization method based on calibrated light projection systems has been developed to fully
utilizes the spatiotemporal property of the MIP-VPP process. Both simulation and experimental
results show that the data-driven mask image planning method based on subpixel shifting can
significantly improve the process accuracy and resolution without losing production throughput.
Finally, the developed subpixel shifting technique is a generic method that can be integrated into
various MIP-VPP systems to advance their use in future industrial applications.

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4.3 Improving Surface Roughness through Vibration-Assisted MIP-VPP
In Chapter 4.1 and Chapter 4.2, we discussed how motion-assisted mask image projection
enhances the lateral and vertical feature resolution of mask image-based vat photopolymerization.
Apart from dimensional accuracy and feature resolution, surface quality is another crucial aspect
of the end product. Surface roughness, which evaluates surface imperfections like scratches, pits,
digs, and textures, can impact friction, wear resistance, corrosion resistance, fatigue, and optical
performance. This chapter will explore the motion-assisted mask image projection processes that
enhance the surface roughness of the fabricated parts.

Figure 66: Surface roughness requirements for various industrial products: (a) Optical glass featuring
surface roughness in the tens of nanometers range
[105, 182]
; (b) Turbine blades with surface roughness in the
micrometer range
[183]
; c) Cylinder liners exhibiting sub-micrometer surface roughness
[184]
; (d) Microneedle
arrays with surface roughness in the hundreds of nanometers range
[185]
.

Surface roughness requirements vary based on the specific application, ranging from
millimeter scale for regular structures and micrometer scale for surface appearance, to sub-
micrometer scale for products' mating requirements and nanometer scale for optical properties
[105,
182-186]
, as it is shown in Figure 66.Noticeably, some industrial products, such as optical lenses,
mirrors, sliding and fitting assemblies, demand sub-micron surface roughness on functional
surfaces but only require tens of microscale resolution in the fabrication process
[187, 188]
. This
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distinction means that the fabrication process may achieve sub-micron smoothness without
necessarily providing sub-micron resolution.
Although traditional VPP makes breakthrough in its achievable speed and resolution, the
poor surface quality has been regarded as a barrier for applying VPP to high precision application
[106-108]
. For example, the conflicts between building size, printing speed, and surface roughness
challenge researchers in developing pragmatic VPP processes
[109, 110]
. VPP’s surface roughness
highly depends on the adopted printing process. A 3D CAD model is first sliced into numbers of
thin layers, which are then used to sequentially fabricate each layer by projecting UV mask images
on the photocurable resin. Three types of common aliasing exist and deteriorate the surface
roughness in the traditional VPP processes. Firstly, the pixelated mask image projected on the
building platform generates the contour aliasing in every layers of the printed structure
[111]
.
Secondly, the nonuniform light intensity of the LCD mask image generates the “dark zone”
aliasing in the XY plane of every layer. Thirdly the staircase stepping aliasing generated by layer
wise stacking reduces the vertical surface roughness.
Chapter 4.2 examined staircase stepping aliasing and presented a solution through
zooming focus motion-assisted mask image projection (MIP-VPP), which achieved 50 nm vertical
resolution and 3.4 nm surface roughness on the vertical surface of printed parts. However, contour
aliasing and "dark zone" aliasing remain at the micron scale, necessitating further research to
improve VPP's surface roughness to sub-micron levels.
In previous chapters, we proposed a spatiotemporal MIP-VPP method to improve lateral
resolution to the sub-pixel level. However, surface roughness of this process is insufficient for
industrial products with sub-micron surface roughness, as evidenced by the jagged surfaces of
fabricated parts. The resolution of spatiotemporal MIP-VPP is determined by the number of split
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exposures in one layer, which is typically less than 10x10 due to projector refresh rate limitations.
The corresponding surface roughness is limited to tens of micron.
Though hardware constraints limit resolution, we found that increasing the number of split
exposures can improve surface roughness. To meet sub-micron surface roughness requirements,
we investigated a vibration-assisted photopolymerization method utilizing a high-frequency mask
image shifting technique. This method involves vibrating an unchanged LCD mask image using a
high-frequency piezo actuator during one layer fabrication. The vibration pattern is determined by
the geometric information of the printing feature, solving the "dark zone" aliasing while improving
surface roughness without sacrificing printing speed or scale.
In this section we will first introduce the contour aliasing and “dark zone” aliasing and
reason for these two types of aliasing. Secondly, we will discuss the principle of the vibration
assisted mask image projection and how it solves the contour and dark zone aliasing. Thirdly, we
will demonstrate the implementation of the vibration assisted mask image projection process. A
high frequency piezo linear stages are used to achieve accurate and fast vibration. Fourthly, we
will demonstrate the printing result using purposed process. The surface roughness, dimensional
accuracy, contour smoothness, and the optical transparency will be fully investigated. Finally, we
will discuss the significance of vibration assisted MIP-VPP on printing surface roughness. The
future work and potential application will be presented at the end.
In this section, we will systematically address the challenges and solutions associated with
surface roughness requirements in vat photopolymerization. Our discussion will be organized as
follows:
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1. Introduction to contour aliasing and "dark zone" aliasing: We will provide a detailed
explanation of these two types of aliasing, their causes, and their impact on surface
roughness and overall part quality.
2. Vibration-assisted mask image projection principle: We will discuss the underlying
principles of this novel method and explain how it resolves contour and "dark zone"
aliasing issues.
3. Implementation of vibration-assisted mask image projection: We will describe the setup
and execution of the vibration-assisted mask image projection process, including the use
of high-frequency piezo linear stages for accurate and rapid vibration.
4. Results from the proposed process: We will present comprehensive findings from the
vibration-assisted mask image projection process, focusing on surface roughness,
dimensional accuracy, contour smoothness, and optical transparency of the printed parts.
5. Significance of vibration-assisted MIP-VPP for surface roughness: We will discuss the
implications of our findings for the broader field of vat photopolymerization and the
potential applications of this technique in industries requiring high-precision parts.
6. Future work and potential applications: Finally, we will outline possible directions for
future research to further refine the vibration-assisted MIP-VPP method and expand its
range of applications. We will also discuss the potential impact of this research on the
development of advanced manufacturing technologies and industrial processes.
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4.3.1 Issue of Contour Aliasing and 'Dark Zone' Aliasing

Figure 67: The contour aliasing and ‘dark zone’ aliasing on LCD mask images: (a) the contour aliasing
of a circle mask images, scalebar: 250 μm. (b)Nonuniform light intensity distribution of the white mask image,
scalebar: 250 μm.

The LCD-based additive manufacturing process involves projecting LCD mask images
onto the building platform to cure liquid resin. The shape of the cured layer depends on the
geometry and light intensity of the mask images. Ideally, the mask image geometry should be
identical to the sliced layer, and the light intensity should be uniform. However, in practical MIP-
VPP manufacturing, this is not the case. The LCD mask images consist of an array of LCD
elements with characteristic sizes between 10-100 μm, with 50 μm elements typically used in most
VPP processes
[189]
. The sliced layers from CAD models are represented by these LCD elements,
determining the resolution of the layer images. Consequently, the contour of the layer images
exhibits pixelated aliasing within the size of one pixel (50 μm), as shown in Figure 67a. This
contour aliasing creates jagged surfaces on the printed parts. Although the spatiotemporal MIP-
VPP process divides pixels into sub-pixels and improves the resolution to less than 10 μm, contour
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aliasing at the 10 μm scale remains too large for fabricating sub-micron smooth surfaces. Industrial
applications demand a printing method capable of reducing surface roughness to less than 1 μm.
In addition to geometry aliasing of the mask image, light intensity uniformity significantly
influences the cured results. Experimental measurements were conducted to study the light
intensity of the LCD mask images. A CCD camera (Plugable USB Digital Microscope) was
mounted on top of the LCD to measure the light intensity of LCD elements during the printing
process. The convoluted light intensity of a full white mask image is shown in Figure 67b. From
the CCD camera images, it is apparent that the light intensity on the mask is not uniform. 'Dark
zone' areas between pixels exhibit weaker light intensity than the pixel centers. The light intensity
of the 'dark zone' area is insufficient to cure the liquid resin, resulting in pixelated and bumpy
surfaces in the XY plane. Additionally, anisotropic light intensity distribution in the X and Y
directions is observed due to the LCD elements' elliptical shape, with 45 μm in the X direction and
35 μm in the Y direction. This leads to anisotropic aliasing on the printed surface, with the Y
direction aliasing more significant than the X direction.
A verification test case of a cylindrical part was printed using a commercial VPP 3D printer
(Elegoo 2.0) to demonstrate the contour aliasing and the 'dark zone' aliasing. The parameters were
set as follows: 50 μm layer thickness, 5-second exposure time, and 50 μm pixel size. The printed
result is shown in Figure 68, where Figure 68adisplays an overview of the cylindrical test case,
and Figure 68b-68d presents zoomed-in views of the printed surface at three different locations.
The contour of the cylinder exhibits zigzag aliasing, as observed in Figure 68b and Figure 68c,
which is identical to the mask image results. The contour aliasing size is the same as the smallest
pixel size. Furthermore, the surface of the printed cylinder has grid textures with a 50 μm x 50 μm
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size, identical to the 'dark zone' in the mask images, confirming the relationship between surface
aliasing and light intensity distribution.

Figure 68: A cylindrical testcase printed by commercial VPP 3D printer: (a) overview of the printed
cylindrical testcase, scalebar: 1mm; (b-d) Zoom-in figure of the printed testcase, scalebar: 250 μm.

In our investigation, we determined that the 'dark zone' and contour aliasing present on the
mask images are responsible for the observed rough surface texture. To devise a strategy for
enhancing surface quality, we conducted an analysis of the physical light intensity distribution of
the LCD mask images and performed a numerical simulation of the convoluted light intensity
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distribution. This approach allowed us to establish the relationship between the single-pixel light
intensity distribution and the 'dark zone' in the convoluted mask image.
We measured the relative light intensity distribution of an individual pixel in the LCD mask
image, utilizing the grayscale values obtained from mask images captured by a CCD camera. The
resulting data is depicted in Figure 69. As shown in Figure 69a, the relative light intensity of a
single pixel reveals an anisotropic energy distribution in both the X and Y directions. According
to prior research, light intensity fluctuations exceeding 5% are known to yield substantial uncured
features. Figure 69b illustrates the light intensity distribution in the X direction, with a region of
stable light intensity spanning 40 μm. This stable region exhibits minor light intensity variations,
primarily attributed to the design of the LCD elements.
Figure 69c presents the light intensity distribution in the Y direction, where the region of
stable light intensity measures 30 μm in the X direction. This stable light intensity region impacts
the convoluted light intensity and uniformity of light intensity present in the mask images.
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Figure 69: The relative light intensity of a single pixel on the elegoo2.0 LCD photo mask: (a) The light
intensity distribution of a single pixel in XY plane. (b) The light intensity distribution of a single pixel in X
direction. (c) The light intensity distribution of a single pixel in Y direction.

The convoluted energy distribution across the mask image influences surface roughness in
the form of dark zone aliasing and contour aliasing. Our primary focus lies in understanding the
factors that contribute to the formation of this convoluted energy distribution. To this end, we
conducted a simulation of the convoluted light intensity based on the measured light intensity
distribution of a single pixel.
As illustrated in Figure 70, the convoluted energy (marked with a black line) is calculated
by accumulating the energy of individual pixels (marked with a red line). The gap between two
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pixels measures 50 μm. Figure 70a displays the convoluted light intensity in the X direction,
revealing energy variation between two pixels. The simulated energy in the X direction exhibits a
20% energy drop at the gap between two LCD elements in the X direction.
Figure 70b presents the convoluted light intensity in the Y direction, where the energy
drop between two LCD elements in the Y direction amounts to 60%. Both energy drops contribute
to the formation of uncured regions on the printed surface, consequently resulting in a rough
surface texture.

Figure 70: The convoluted light intensity distribution simulated from the single pixel: (a) the convoluted
light intensity distribution in X direction; (b) the convoluted light intensity distribution in Y direction.
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The simulated energy distribution is corroborated by the physical measurement of
convoluted light intensity on the white mask image. Figure 71 displays the practical measured
results, with convoluted light intensities in the X-X' and Y-Y' directions measured separately.
Figure 71b illustrates the convoluted light intensity on the LCD mask images measured in the X-
X' direction. The practical measured result follows the same pattern as the simulation results in
Figure 70a, with a 20% energy variation observed at the gap between two pixels in the X direction,
consistent with the simulation. Figure 71c presents the convoluted light intensity on the LCD mask
images measured in the Y-Y' direction. The measured result also exhibits a 50% light intensity
variation, confirming the accuracy of our simulation. Small spike features and distorted pixels can
be observed in the practical measured light intensity. These discrepancies may be attributed to
scratches and dust on the separation film, which deflect light and distort the mask image.
Based on this verification, we have validated the convoluted light intensity simulation
model derived from a single pixel. Additionally, we have established the relationship between the
shape of a single pixel and the nonuniformity of the light intensity in the mask images.
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Figure 71: The practical measured convoluted light intensity of the mask image: (a) The mask image of
a white picture on the building platform; b) The practical convoluted light intensity in X-X’ direction; (c) The
convoluted light intensity in Y-Y’ direction.

In this section, we have examined the contour aliasing and the 'dark zone' aliasing. The
LCD mask image is responsible for both types of aliasing: the pixel sizes of the LCD screen impose
limitations on the contour aliasing, while the nonuniform light intensity contributes to the 'dark
zone' aliasing. We have found that the nonuniform light intensity can be explained through
convoluted light intensity simulations derived from a single pixel. Building upon these findings,
we will propose a vibration-assisted MIP-VPP process in the next section, guided by the
convoluted light intensity simulation.
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4.3.2 Overview of the Vibration-Assisted MIP-VPP Process
To address the contour aliasing and the 'dark zone' aliasing issues and improve surface
smoothness, we propose a vibration-assisted mask image projection-based vat
photopolymerization guided by the convoluted light intensity simulation. Building on previous
work
[117]
, we note that the geometry and light intensity distribution of a mask image can be more
precisely controlled by dividing the one-shot UV exposure into multiple exposures, each with a
slight shift in the XY direction. Inspired by this, we determined that the nonuniformity of light
intensity between the pixels can be reduced by replacing a single exposure of the mask images
with a stack of multiple sub-mask images, each with a displacement shift. A precisely controlled
vibration actuator enables the implementation of this displacement. In this section, we present the
schema of the vibration-assisted MIP-VPP process, discuss essential parameters, such as vibration
frequency, amplitude, and pattern, and analyze the effectiveness of the vibration-assisted MIP-
VPP approach.
The schema of the vibration-assisted MIP-VPP process is illustrated in Figure 72 This
setup is based on a commercial LCD-based 3D printer. The UV light emitted from the LED light
source is masked by the LCD photomask, selectively curing the liquid resin on the build platform.
The primary hardware modification for the vibration-assisted MIP-VPP system involves mounting
the LCD photomask on two piezo actuators, enabling high-frequency and high-resolution vibration
in both the X and Y directions. The resin tank, build platform, LED UV light source, and Z stage
remain isolated from the vibration, ensuring that the only difference during the printing process is
the vibration of the mask images. A computer controls the motion of the Z stage, exposure of the
mask images, and vibration of the LCD photomasks.
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During the fabrication of a single layer, the LCD photomask vibrates in multiple cycles. In
each cycle, the LCD photomask shifts to several different positions, referred to as steps. The time
interval between two steps is determined by the frequency of the vibration. The shifting time of
the piezo actuator is negligible due to the fast response frequency (200ms) of the piezo actuator
compared to the applied vibration frequency (around 10Hz)
[190, 191]
. The mask image remains
unchanged during the fabrication of a single layer. Figure 72 presents an example of a potential
vibration pattern, where the LCD is shifted to six different positions, dividing one LCD pixel into
six sections. These six-step shifts constitute one cycle, and the fabrication of a single layer consists
of n cycles of mask image shifting. The vibration frequency, amplitude, and direction significantly
influence the surface roughness. We investigated these essential parameters using simulations of
convoluted light intensity energy distribution to optimize surface quality.

Figure 72: The schema of the vibration assisted MIP-VPP to improve the surface roughness.

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The vibration pattern of the LCD mask images is parameterized, as shown in Figure 73.
The LCD mask images vibrate in 'a' steps in the X direction and 'b' steps in the Y direction. The
convoluted light intensity without vibration is denoted as a 1-step exposure. An n-step exposure
divides the exposure into 'n' sub-exposures. Each sub-exposure takes an equal amount of time, and
the sum of the sub-exposure times remains the same as the original exposure time. Consequently,
the equivalent light intensity of each sub-exposure is 1/n of the full exposure.
The mask image of each step is shifted in the X and/or Y direction. The shifting distance
is determined by dividing the pixel size by 'n', where the pixel size is 50 μm for our test case. This
parameterization allows for precise control of the vibration pattern, enabling the optimization of
surface quality by adjusting the number of sub-exposures and the shifting distance in both the X
and Y directions. The mask image is shifted sequentially to positions in the order of 1, 2, 3, ... up
to ab. After reaching the ab position, the mask image returns to position 1 for the next step, creating
a cyclical vibration pattern.

Figure 73: Vibration pattern of the LCD mask images

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As discussed in Figure 70, the light intensity distribution of a single pixel in the LCD
photomask is anisotropic, with a 'dark zone' of 5 μm in the X direction and 15 μm in the Y direction.
Since vibrations in the X and Y directions are orthogonal, the convoluted light intensity in these
directions can be considered independent. To investigate the effect of mask image vibration on the
'dark zone' aliasing, we examine the X-direction and Y-direction vibrations separately.
We simulate the convoluted light intensity of the vibration assisted mask image from the
light intensity of a single pixel. Figure 74 illustrates the convoluted light intensity in the X
direction with and without vibration. The convoluted light intensity without vibration is denoted
as a one-step exposure. Convoluted light intensities of two, three, and four steps represent the
vibration patterns 2x1, 3x1, and 4x1, respectively. Figure 74 displays the light intensity of five
consecutive pixels to represent the light intensity across the entire mask image. The X-axis
represents the position and size of the pixels, while the Y-axis indicates the relative light intensity.
The relative light intensity of each single pixel is depicted in red, and the convoluted
relative light intensity is shown as a black line. The red solid line represents the original position
of the light intensity in the LCD mask image without vibration, and the red dashed line denotes
the shifted mask image positions. For a two-step exposure, there is only one shifted position, while
for three and four-step exposures, there are two and three shifted positions, respectively. The
period between two pixels is fixed at 50 μm, which is determined by the hardware.
From the simulation, we observed that the convoluted light intensity of a one-step exposure
(no vibration) exhibits energy variation at the gap between two neighboring pixels. However, the
intensity variation for a two-step exposure mask image is smaller than that of a one-step exposure.
This reduction in variation is due to the mask image shifting to the position between the gaps of
the original pixels, which mitigates the nonuniformity caused by the 'dark zone'. Additionally, we
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noticed that the convoluted light intensity variation continues to decrease as the number of
vibration steps increases. After four steps of mask image shifting, the light intensity variation is
reduced to less than 5%, which is uniform enough to achieve sub-micron smooth surface. The
residual light intensity variation observed after applying the vibration steps can be attributed to the
quality of the LCD elements, rather than the 'dark zone' aliasing. This indicates that the vibration-
assisted approach effectively mitigates the 'dark zone' aliasing,

Figure 74: The simulation of the convoluted light intensity in X direction: (a) one step exposure (without
vibration); (b) two steps exposure; (c) three steps exposure; (d) four steps exposure.

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The convoluted light intensity in the Y direction is similarly simulated, as shown in Figure
75. The black line represents the simulated convoluted light intensity of the mask image, while the
red line illustrates the light intensity of a single pixel. Due to the anisotropic energy distribution
within a single pixel, the convoluted light intensity of one-step exposure in the Y direction exhibits
a larger energy variation compared to that in the X direction. The simulation of one-step exposure
in Figure 75a reveals a 55% light intensity drop at the gap between pixels. By applying a two-step
vibration to the mask image, the valley in the simulated light intensity is reduced to 10%,
demonstrating the effectiveness of mask image vibration in uniforming the light intensity. Further
increasing the vibration steps to 3 and 4 steps reduces the energy variation to less than 5%.
Notably, the peak light intensity reduces to 80% of its original value after the mask image
is vibrated in the Y direction at 3 or more steps. This reduction occurs because the light intensity
of the original peak portion is shifted to the original valley portion to smooth the total energy
distribution. As the total energy does not change but the distribution becomes more uniform, the
peak light intensity decreases. The reduced light intensity can be easily compensated by increasing
the exposure to 125% of the original exposure time, ensuring proper curing while maintaining a
smoother surface.
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Figure 75: The simulation of the convoluted light intensity in Y direction: (a) one step exposure (without
vibration); (b) two steps exposure; (c) three steps exposure; (d) four steps exposure.

The simulations for both X and Y directions exhibit a trend in which light intensity
variation decreases as the number of vibration steps increases. This relationship between energy
variation and vibration steps is illustrated in Figure 76, where energy variation is calculated based
on the difference between maximum and minimum light intensity values.
In the X direction, light intensity variation starts at 17% for a 1-step exposure, drops to 5%
for a 3-step exposure, and further decreases to 2% when the vibration steps reach 6. Conversely,
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in the Y direction, light intensity variation begins at 55% and dramatically falls to 5% when the
number of vibration steps reaches 4. It continues to decline to 1% with a 6-step exposure.
The difference in the number of steps required to reach a variation of less than 5% in the
X and Y directions can be explained by the fact that the 'dark zone' in the Y direction is wider than
the 'dark zone' in the X direction. As a result, the Y direction needs more steps of shifting to fill
the gap. The smaller variation in the Y direction compared to the X direction is due to the fewer
spikes in light intensity for a single pixel in the Y direction, which is determined by the physical
design of the LCD element.
Since a 5% variation in light intensity is already sufficiently small for surface roughness
fabrication, a 5% variation threshold is used to select the vibration pattern. Following this model,
a 3-step vibration in the X direction and a 4-step vibration in the Y direction are selected to achieve
a smooth energy distribution with a variation of 5% or less.
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Figure 76: The relationship between the light intensity variation and the numbers of steps of vibration:
(a) The light intensity variation in X direction; (b) The light intensity variation in Y direction.

The mask image vibration progresses from step 1 to step 12, forming one cycle of the
vibration. Multiple cycles of vibration are employed in printing one layer to ensure a uniform
increase in UV energy across all shifting positions. The number of cycles in one layer fabrication
is another factor that affects the curing results. A higher number of cycles results in less sequential
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curing. However, the maximum number of cycles is limited by the vibration frequency of the
actuator. In this case, a 12Hz vibration frequency is selected to avoid hysteresis of the piezo
actuator. With an exposure time of one layer set at 5 seconds, the number of cycles is calculated
to be 5 cycles. The vibration pattern is illustrated in Figure 77.

Figure 77: The vibration pattern designed based on the light intensity simulation

In this section, we presented a vibration-assisted mask image projection-based vat
photopolymerization (MIP-VPP) process to address the contour aliasing and 'dark zone' aliasing
issues and improve surface smoothness. The proposed method replaces a single exposure of the
mask images with a stack of multiple sub-mask images, each with a displacement shift, enabled
by a precisely controlled vibration actuator. By analyzing the convoluted light intensity
simulations, we determined the optimal vibration pattern to achieve a smooth energy distribution
with a variation of 5% or less. A 3-step vibration in the X direction and a 4-step vibration in the Y
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direction were selected, with multiple cycles of vibration employed to ensure a uniform increase
in UV energy across all shifting positions. The effectiveness of the vibration-assisted MIP-VPP
approach will be demonstrated by the experiment results in later section.
4.3.3 Materials and Methods
In this section we describe the materials, equipment, and methods employed in our study
to implement, investigate, and evaluate the vibration-assisted MIP-VPP process.
Photopolymer: A commercial photopolymer (Anycubic clear) was used in the experiments.
The resin was chosen based on its compatibility with the LCD screen's UV wavelength and its
transparency to easily observe the surface texture. The curing time of the selected resin is 4~10s,
curing wavelength is 405nm.
LCD: The LCD screen used for the photomask was a Elegoo Mars 2pro 5.5 inches LCD
monochrome screen (Sharp04 SX04 2k LCD). The resolution of the LCD is 1440 x 2560 with
pixel size of 50 μm. The screen provided the necessary resolution and light transmission
capabilities for the vat photopolymerization process.
Piezo actuator: The piezo actuator used for the vibration was a Co-Dired Piezo Actuator
(PC4QQ, 18.0 μm Max Displacement, 6.5mm x 6.5mm x18.0mm). The maximum displacement
of the actuator is 18 μm with resonant frequency 65kHz. The load for maximum displacement is
400 N. The recommended drive voltage limit is 150 v. They are driven by the voltage driver (Piezo
Master, MODEL VP7206), which is controlled by a microcontroller (Arduino Due). The schema
and the practical photo of the piezo actuator and its controller are demonstrated in the Figure 78.
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Figure 78: The piezo actuator and its controller: (a) The schema of the piezo actuator and its
controller; (b) The practical connects of the piezo actuator and its controller.

Piezo amplifier: To enhance the displacement of the piezo actuator, a pair of two-stage
mechanical amplification structures were utilized as piezo amplifiers. These amplifiers can achieve
up to 5 times the maximum displacement of the piezo actuator, ensuring adequate range and
precision for mask image vibration. The CAD design and the practical assembly of the piezo
amplifier are illustrated in Figure 79. The CAD design demonstrates that the deformation of the
piezo actuators acts on a flexible-hinged diamond-shaped mechanical amplifier, which in turn
pushes the center plate in the target direction. This arrangement allows for high resolution-
controlled amplification of the piezo actuator's movement.
The physical assembly showcases two amplifiers stacked together and connected by the
center plate to facilitate shifting movement in both the X and Y directions. The bottom amplifier
is fixed to the optical table, while the top amplifier is attached to the LCD mount. This
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configuration ensures stability and precise control of the mask image vibration during the printing
process.

Figure 79: The design of piezo stage amplifier (a) The schema of the piezo stage amplifier; (b) The
practical connects of the piezo stage amplifier.

the resolution of the piezo actuator after amplification was investigated. The displacement
corresponding to voltage was measured using a series of 640x480px images of the piezo stage
taken under 386x magnification by a MicroVu SOL 161 optical microscope. Due to the limitations
of the Arduino Due controller utilized in the study, the analog output of the DAC pin ranged from
0.53V to 2.75V, translating to input voltages of the piezo actuator from 31.53V to 165V.
Initially, input voltages ranging from 31.53V to 138.30V were applied sequentially to
expand the piezo actuator. Subsequently, input voltages from 138.30V to 31.53V were applied to
shrink the piezo actuator. The position of the platform corresponding to each input voltage was
captured by the optical microscope under 386x magnification and measured through image
processing.
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The displacement of the piezo actuator in the XY direction is presented in Tables 3 and
Table 4. By controlling the input voltage, a 1 µm resolution of the displacement of the piezo linear
stage can be achieved, demonstrating the accuracy of the mask image vibration control.
Table 3: The X-Direction Resolution of the Piezo Actuator

Table 4: The Y-Direction Resolution of the Piezo Actuator

The UV projector was mounted on piezo actuators, which facilitated its shifting in both the
X and Y directions based on the input voltage signal. The controlled mask image shifting process
is illustrated in Figure 80. Figure 80a presents a schematic diagram of the piezo actuator control
mechanism using voltage signals. Two square waves are employed to manage mask image shifting
in the X and Y directions independently. Figure 80b and Figure 80c demonstrate the actual mask
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image shifting in the X and Y directions, respectively, as the voltage varies from 31.5V to 138.3V.
The camera remains stationary during the mask image shifting process. The measured distances of
relative mask image shifting correspond to the data presented in Tables 3 and Table 4

Figure 80: The mask image shifting controlled by input signal: (a) The voltage signal input to control
the mask image shifting; (b) The mask image shifting in X direction over the voltage change, scalebar: 300 μm;
(c) The mask image shifting in Y direction over the voltage change, scalebar: 300 μm.

Other hardware: The experimental hardware setup is depicted in Figure 81. It consists of
a vertically-oriented linear stage (MotorizeXSlide™ XN150 Series – Lead Screw) with a straight-
line accuracy of 0.003"/10" (0.076 mm/25 cm) and repeatability of 0.0001 inch (0.00254 mm). An
Elegoo Mars Pro 2 resin tank is mounted on a fixed optical table. An Arduino Due microcontroller
is employed to coordinate the motion of the linear stage, mask images, and the vibration of the
piezo stages. A 405nm UV LED module (UV_A MODULE 6565S) with collimator lenses are
used to generated uniform UV light source. Besides, an LCD control panel is incorporated into the
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setup for managing the experimental process. Additionally, a 350W 12V power supply is
integrated to provide sufficient power to the various components.

Figure 81: Schematic of the Vibration-Assisted MIP-VPP experimental hardware: (a) physical setup of
the vibration-assisted MIP-VPP printer; (b) CAD design of the essential component of the vibration-assisted
MIP-VPP printer.

Measurement and Characterization: Dimensional accuracy, contour roughness, and surface
roughness of the printed samples are evaluated using optical microscopy images captured by the
Micro-Vu Sol system. Transparency of the printed parts is characterized by calculating the
Modulation Transfer Function (MTF) from the images of a USAF 1951 resolution target.
4.3.4 Results of Vibration-Assisted MIP-VPP
In this section, we present the experimental results of the vibration-assisted MIP-VPP,
focusing on the key performance indicators of the printed parts. We will discuss the impact of
vibration on surface roughness, contour smoothness, dimensional accuracy, and optical
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transparency of the printed parts. By evaluating these characteristics, we aim to demonstrate the
effectiveness of the vibration-assisted MIP-VPP technique in improving the surface quality and
optical performance of the printed parts, as well as its potential applications in various industries.
To investigate the surface roughness of parts produced using vibration-assisted MIP-VPP,
a 9mm cylinder was printed, and the surface roughness of its top surface was measured. Figure 82
presents the CAD model of the 9mm cylinder, the sliced mask image corresponding to it, and an
example of the printed test case without vibration assistance. The surface roughness of the printed
example is evaluated by the microscope image at x100 magnification rate.

Figure 82: Example of the cylinder testcases to measure the surface roughness: (a) The CAD model of
the 9mm cylinder testcase; (b) The sliced mask image of the 9mm cylinder testcase, scale bar: 2 mm; (c) The
practical mask image measured from building platform, scale bar: 150 μm; (d) The example of the printed 9mm
cylinder testcase, scale bar: 2 mm; (e) The zoom in view of the characteristic surface of the printed cylinder
testcase, scale bar: 250 μm.

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In this study, we printed the 9mm cylinder test case using the vibration patterns discussed
in Section 4.3.2. As previously mentioned, the vibration pattern significantly impacts the surface
roughness of the printed parts, with an increase in vibration steps leading to a decrease in surface
roughness. The vibration pattern of the mask images is noted as a x b, where a is the numbers of
the steps in X direction and b is the numbers of the steps in Y direction. Vibration patterns of 1x1
(no vibration), 2x2, 2x3, 2x4, ..., 3x4, and 4x4 were tested under identical printing parameters. The
exposure time for each layer of the test case was set at 5 seconds, the resolution of the LCD
photomask was 50 µm, and the layer thickness of the test case was 50 µm. The printed samples
were measured using an optical microscope, and the results are demonstrated in the Figure 83.
Figure 83 displays the top surface of the cylinder test case as examined under an optical
microscope. The grid texture is clearly discernible in the images of the sample printed without
vibration. As the vibration steps increase, the grid texture becomes less prominent, suggesting that
surface roughness decreases with an increasing number of vibration steps. Improvements in
surface roughness are observed with increases in both the X and Y direction vibration steps, as
seen in the results for XY vibrations of 2x2, 2x3, 2x4, and 2x2, 3x2, 4x2. However, the increase
in vibration steps in the Y direction has a more significant impact on surface roughness reduction.
This observation can be explained by the light intensity simulation discussed in section
4.3.2. The 'dark zone' aliasing in the Y direction is larger than that in the X direction, so vibration
in the Y direction is more effective in reducing the size of the large 'dark zone' aliasing compared
to the X direction. Notably, the surface roughness of the 3x4 and 4x4 vibration patterns are similar.
This is because the X direction 'dark zone' aliasing has already been reduced to less than a 5%
variation at 3 steps vibration, consistent with our simulation results.
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Figure 83: Surface roughness of the printed cylinder testcase with different vibration pattern.

The contour smoothness is another parameter used to evaluate surface smoothness. Since
the VPP process is an anisotropic fabrication method, the surface roughness of the side surface is
determined by the contour aliasing. A triangle testcase was printed to measure the contour
smoothness of the bottom line and the slope line. Figure 84 presents a comparison of the contour
smoothness of parts fabricated using conventional VPP, grayscale VPP, vibration-assisted VPP,
and a combination of vibration-assisted and grayscale VPP. The results indicate that the
conventional VPP process produces parts with noticeable contour aliasing, leading to a rougher
side surface. In contrast, the grayscale VPP method shows an improvement in contour smoothness,
as it can partially address contour aliasing issues. The vibration assisted VPP process demonstrates
further improvement in contour smoothness, as the mask image vibration reduces the contour
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aliasing effect. Finally, the combination of vibration-assisted and grayscale VPP methods yields
the smoothest contours, as it takes advantage of both techniques to mitigate contour aliasing and
achieve optimal surface smoothness.

Figure 84: Contour smoothness of vibration-assisted VPP: (a) sample printed by binary mask image
VPP; (b) sample printed by grayscale mask image VPP; (c) sample printed by vibration-assisted VPP; (d) sample
printed by vibration+grayscale VPP, scale bar: 250 μm.

The contour smooth is quantified by the roughness of the bottom line and the slope line of
the triangle testcase. Figure 85 and Figure 86 quantify the contour smoothness of the bottom and
slope lines of the triangle test case, respectively. Figure 85 shows the bottom line smoothness of
samples printed without vibration (a), with grayscale mask image (b), with vibration-assisted VPP
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(c), and with both grayscale and vibration-assisted VPP (d). The standard deviation of the contour
to the neutral line is used to evaluate the contour aliasing. The standard deviation values for these
samples are 6.5 µm, 3.0 µm, 1.9 µm, and 1.1 µm, respectively. This indicates a significant
improvement in bottom line smoothness when incorporating both grayscale and vibration-assisted
VPP techniques.
Similarly, Figure 86 illustrates the slope line smoothness of samples printed without
vibration (a), with grayscale mask image (b), with vibration-assisted VPP (c), and with both
grayscale and vibration-assisted VPP (d). The standard deviation values for these samples are 8.0
µm, 6.2 µm, 2.0 µm, and 1.8 µm, respectively. The standard deviation of the slope line to the
neutral line is larger than that of the bottom line to the neutral line. These can be explained by the
pixelated error from the mask images. These results further confirm that the combination of
grayscale and vibration-assisted VPP techniques yields the smoothest contours, significantly
improving the slope line smoothness compared to other methods.
Compared the slope line ad bottom line contour smoothness, both grayscale and vibration
process have effect in smoothing the pixelated error and the slope surface. But the vibration
assisted VPP has more significant effect on smoothing the slope surface, given by the standard
deviation. In conclusion, the analysis of bottom and slope line smoothness demonstrates the
superior performance of combining grayscale and vibration-assisted VPP methods. This
combination yields the lowest standard deviation values and the smoothest contours, making it an
ideal solution for applications requiring high-quality surface finishes.
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Figure 85: The contour smoothness of the bottom line of the triangle testcase: (a) the bottom line
smoothness of sample printed without vibration; (b) the bottom line smoothness of sample printed by grayscale
mask image; (c) the bottom line smoothness of sample printed by vibration assisted VPP; (d) the bottom line
smoothness of sample printed by grayscale and vibration assisted VPP.

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Figure 86: The contour smoothness of the slope line of the triangle testcase: (a) the slope line
smoothness of sample printed without vibration; (b) the slope line smoothness of sample printed by grayscale
mask image; (c) the slope line smoothness of sample printed by vibration assisted VPP; (d) the slope line
smoothness of sample printed by grayscale and vibration assisted VPP.

Apart from surface roughness, dimensional accuracy is a crucial parameter in evaluating
the performance of printed parts. In this study, we assessed the dimensional accuracy of the
vibration assisted VPP process and compared it to the designed target dimensions. Circle and
triangle test cases were printed and measured to validate the dimensional accuracy of the process.
The test cases were designed with varying characteristic dimensions of 2000 µm, 1000 µm, 500
µm, and 200 µm.
Figure 87 illustrates the dimensional accuracy of the vibration assisted VPP process by
comparing printed samples with and without vibration. The figure shows printed triangle samples
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by the no vibration VPP (a), printed triangle samples by the vibration-assisted VPP (b), printed
circle samples by the no vibration VPP (c), and printed triangle samples by the vibration-assisted
VPP (d). upon comparison, there is no obvious difference in terms of dimensional accuracy
between the samples printed with and without vibration assistance.

Figure 87: The dimensional accuracy of the vibration-assisted VPP: (a) Printed triangle samples by no
vibration VPP; (b) Printed triangle samples by vibration-assisted VPP; (c) Printed circle samples by no vibration
VPP; (d) Printed triangle samples by vibration-assisted VPP.

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The dimensions of the printed parts were quantified by their characteristic lengths, which
are the length of the hypotenuse in the triangle test case and the diameter in the circle test cases.
Figure 88 demonstrates the quantification comparison of the dimensional accuracy of the
vibration-assisted VPP, showcasing the dimensions of the triangle test case (a) and the circle test
case (b). Firstly, the dimensions of parts printed with and without vibration assistance appear to be
generally similar. This indicates that the vibration-assisted VPP process does not have a significant
impact on the overall dimensional accuracy. For larger parts with characteristic dimensions greater
than 500 µm, it was observed that parts printed using the vibration-assisted VPP process were
approximately 50 µm larger than those printed with the conventional process. This discrepancy
can be attributed to the 50 µm amplitude of vibration in both the X and Y directions, which leads
to a dimensional change. In contrast, for smaller parts with characteristic dimensions of 200 µm,
the parts printed using vibration-assisted VPP were found to be smaller than those printed without
vibration assistance. This phenomenon can be explained by the vibration reducing the light
intensity and causing under-curing for small-sized samples. These findings suggest that the
vibration-assisted VPP process offers comparable dimensional accuracy to the conventional VPP
process, with some variation based on the size of the printed parts.
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Figure 88: The quantification comparison of the dimensional accuracy of the vibration assisted VPP:
(a) The dimension of the triangle testcase, (b) The dimension of the circle testcase.

The ability to create smooth surfaces with the vibration-assisted VPP process is particularly
advantageous for fabricating optical devices in situ. To evaluate the transparency of printed parts,
we use microscopy images of the USAF 1951 calibration target. The target is employed to measure
the modulation transfer function (MTF) of patterns passing through the printed parts, which is an
essential parameter in determining their suitability as optical components.
MTF is a widely used metric among optical designers to compare the performance of
optical systems. It measures the contrast between black and white gratings, as well as the variation
in contrast across different grating densities. A high MTF indicates a clear grating pattern image,
which is a testament to the excellent transparency of the printed part. Figure 88 illustrates the
setup for MTF calibration, where a printed flat disc is placed on the USAF 1951 calibration target.
By comparing the grating pattern photos taken through the printed discs, we can observe
that the grating pattern image appears distorted when printed without vibration assistance.
However, with the vibration-assisted VPP, the grating pattern image is significantly clearer. This
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improvement in transparency can be attributed to the differences in surface texture between the
two samples. The non-vibrated parts display a pixelated texture that reduces transparency, while
the surfaces of the vibrated parts are characterized by a smooth texture that allows for greater light
transmission and improved optical performance.

Figure 89: The transparency of the parts printed by the vibration assisted VPP: (a) The transparency
and the zooming figure of the surface texture of the parts printed without vibration; (b) The transparency and the
zooming figure of the surface texture of the parts printed with vibration;

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Figure 90 displays the Modulate Transfer Function (MTF) of the printed discs. Captured
using a MicroVu SOL161 at a magnification rate of 64x, Figure 90a and 90b focus on the interface
between the calibration target and the printed disc. The MTF calculated from these images is
presented in Figure 90c and Figure 90d

Figure 90: Modulate transfer function of the USAF 1951 calibration target: (a) The image of the USAF
1951 calibration target taken through the vibrated printed parts; (b) The image of the USAF 1951 calibration
target taken through the non-vibrated printed parts; (c) The modulate transfer function of the vibrated printed
parts over the grating density; (d) The modulate transfer function of the non-vibrated printed parts over the
grating density;

As the grating density increases, the MTF decreases for both vibrated and non-vibrated
printing. However, the vibrated printed disc exhibits a more gradual decrease in MTF compared
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to the non-vibrated printed disc, indicating superior optical performance. When the MTF falls
below 10%, the contrast of the grating pattern is deemed unrecognizable. Through the vibrated
printed disc, the smallest distinguishable feature of the calibration target is 32 lines/mm, while the
corresponding feature for the non-vibrated disc is only 7 lines/mm. This difference in MTF
performance across all grating patterns highlights the enhanced transparency of the vibration-
assisted printed parts.
In conclusion, the vibration-assisted VPP process improves the surface finish and
transparency of the printed parts. The higher MTF values achieved with vibration-assisted printing
demonstrate its potential for fabricating high-quality optical components, opening up new
possibilities for applications in the field of additive manufacturing.
In this section, we presented the experimental results of the vibration-assisted MIP-VPP
technique, focusing on key performance indicators such as surface roughness, contour smoothness,
dimensional accuracy, and optical transparency. The experiments revealed that the vibration-
assisted MIP-VPP process significantly improved the surface quality and optical performance of
the printed parts. Furthermore, the dimensional accuracy of the vibration-assisted VPP was
comparable to that of the conventional VPP process. The enhanced surface finish and transparency
of printed parts using the vibration assisted VPP process indicates its potential for fabricating high-
quality optical components.
4.3.5 Discussion
In this chapter, we discuss the experimental results of the vibration-assisted MIP-VPP
technique and its effects on key performance indicators. The aim is to demonstrate the
effectiveness of this technique in improving surface quality and optical performance, while also
exploring potential applications, limitations, and future work.
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The contour smoothness of parts printed using vibration-assisted VPP is significantly
improved compared to conventional and grayscale VPP methods. By increasing the number of
vibration steps, the grid texture on the surfaces becomes less prominent, resulting in a smoother
surface. This improvement is particularly evident in the Y direction, which has a more significant
impact on surface roughness reduction due to the larger 'dark zone' aliasing. The combination of
vibration-assisted and grayscale VPP offers the best results in terms of contour smoothness,
making it an ideal solution for applications requiring high-quality surface finishes.
Unlike other surface roughness improving processes, such as defocusing MIP-VPP
[111]
,
meniscus coating
[132, 192]
, the vibration-assisted MIP-VPP improves the surface roughness without
deteriorating the dimensional accuracy. This unique advantage sets it apart from other techniques
and further emphasizes its potential in various applications where both surface quality and
dimensional accuracy are of utmost importance.
Although the vibration-assisted MIP-VPP process demonstrates promising results, it also
presents certain limitations and challenges that need to be addressed. The printing process employs
XY vibrations to smooth the surface and reduce contour aliasing. However, these vibrations can
also introduce dimensional variations and shape distortion. For larger parts with characteristic
dimensions exceeding 500 µm, the process tends to produce parts that are slightly larger than those
fabricated using conventional methods, primarily due to the amplitude of the vibrations. To
optimize the performance of the vibration-assisted MIP-VPP process, further research and
development efforts are necessary to address these challenges and refine the technique for
producing high-quality parts with minimal dimensional variations and shape distortion.
The improved surface quality and transparency of printed parts created using vibration-
assisted MIP-VPP opens up new possibilities for applications in various industries. With the ability
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to create smooth surfaces and optically transparent components without compromising
dimensional accuracy, this technique can be particularly advantageous in the manufacturing of
optical devices, microfluidic components, and high-precision mechanical parts. The combination
of grayscale and vibration-assisted VPP methods offers even better surface finish and contour
smoothness, making it ideal for applications requiring high-quality surface finishes.
Future work in the field of vibration-assisted MIP-VPP should focus on addressing the
limitations and challenges identified in this study. This could include optimizing vibration patterns
to improve dimensional accuracy for different part sizes and further refining the combination of
grayscale and vibration-assisted methods for even better surface quality and contour smoothness.
Additionally, investigating the long-term performance and durability of printed parts produced
using this technique will be crucial for ensuring their suitability for various applications. Finally,
exploring potential new applications in areas such as bioprinting, microelectronics, and other
emerging fields will contribute to expanding the range of applications for this promising additive
manufacturing technique.
4.3.6 Conclusion
In this section, we have investigated the effectiveness of the vibration-assisted MIP-VPP
technique for improving surface quality and optical performance in additive manufacturing. We
have compared the results of this technique with conventional and grayscale VPP methods and
demonstrated its potential in various applications.
The key findings from the experiments are as follows:
1. Vibration-assisted MIP-VPP significantly improves the contour smoothness of
printed parts, resulting in a smoother surface and reduced surface roughness.
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2. The combination of vibration-assisted and grayscale VPP techniques offers the best
results in terms of contour smoothness, making it suitable for applications requiring
high-quality surface finishes.
3. The vibration-assisted MIP-VPP technique maintains dimensional accuracy,
making it an ideal solution for applications where both surface quality and
dimensional accuracy are crucial.
4. The improved surface quality and transparency of printed parts created using
vibration-assisted MIP-VPP open new possibilities for applications in various
industries, including optical devices, microfluidic components, and high-precision
mechanical parts.

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Chapter 5: Conclusion and Recommendation
5.1 Research Question and Hypothesis Testing Results
As stated in Section 3, we have the following three research questions to answer in this
dissertation to improve current resolution and surface quality in MIP-VPP:
Q1: How can motion-assisted MIP-VPP enhance vertical resolution without sacrificing
fabrication speed or building volume?
To answer this question, the following hypotheses are investigated:
Hypothesis 1.1: A frustum layer stacking compared to normal layer stacking can
effectively reduce the vertical staircase.
Hypothesis 1.2: A smooth change of mask images can be achieved by zooming the focal
distance of the MIP-VPP hardware.
Hypothesis 1.3: The combination of the continuous liquid interface production and the
continuous change of the mask images enable the frustum layer stacking

Q2: How can motion-assisted MIP-VPP improve lateral resolution without affecting
fabrication speed or building volume?
To answer this question, the following hypotheses are investigated:
Hypothesis 2.1: Higher resolution mask images can be achieved by replacing the
original mask image to a series of superimposed sub images with a shifted position.
Hypothesis 2.2: The sub-images can be generated computationally efficiently from the
original mask image by error diffusion.
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Hypothesis 2.3: The high-resolution mask image shifting can be achieved by A piezo
actuated pixel shifting mechanism.

Q3: How can motion-assisted MIP-VPP reduce surface roughness in MIP-VPP-
fabricated products without compromising process resolution, fabrication speed, or building
volume?

To answer this question, the following hypothesis are investigated:
Hypothesis 3.1: The nonuniform light intensity from the LCD mask image can be
uniformed by vibrating the mask image at certain frequency and amplitude.
Hypothesis 3.2: The contour aliasing from the LCD mask image can be reduced by
vibrating the mask image at certain frequency and amplitude.
Hypothesis 3.3: The vibration amplitude applied in the printing is so small that
dimensional accuracy of the products will not be affect.

In Chapter 4.1, we test Hypotheses 1.1, 1.2, and 1.3 by proposing a novel zooming-focused
MIP-VPP technique to 3D print products without staircase aliasing. By using frustum layer
stacking, we introduce slope angle control to the layer image, enabling the printing of frustum
layers within a single printing step, supporting Hypothesis 1.1. We adjust the focal distance of the
mask image projection system to achieve a continuous change in the magnification rate, ensuring
smooth transitions of the mask images and validating Hypothesis 1.2. Utilizing continuous liquid
projection on the PDMS building window, we accomplish ultra-thin (1µm) layer thickness. The
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combination of continuous liquid projection and mask image alteration is demonstrated by various
test cases, showing a reduction in nanoscale surface roughness in the vertical direction and
verifying Hypothesis 1.3. The simulation and physical results collectively confirm the
effectiveness of the zooming-focused MIP-VPP approach in minimizing vertical aliasing, thus
validating all three hypotheses.
Hypothesis 2.1, Hypothesis 2.2, and Hypothesis 2.3 are tested in chapter 4.2. We simulated
the shifted mask images and their superimposed intensity and verified a higher resolution of the
convoluted mask images via the pixel blending, which answered the hypothesis 2.1. We developed
an error erosion method to efficiently calculate the sub mask images accompanied by the mask
image shifting. The efficiency of the error erosion method is compared to the conventionally linear
programing, demonstrating a significant improvement, verifying the hypothesis 2.2. We built and
tested a piezo actuated mask image shifting system to achieve a high precision (5um) resolution
mask shifting of the subpixels. The physical experiment demonstrated the effectiveness of
reducing lateral aliasing and verified the hypothesis 2.3.
Hypothesis 3.1, Hypothesis 3.2, and Hypothesis 3.3 are tested in chapter 4.3 by
implementing a mask image vibration technique to reduce surface roughness in MIP-VPP-
fabricated products without compromising process resolution, fabrication speed, or building
volume. We studied and built a model of the nonuniform light intensity from the LCD mask image.
And we developed a vibration pattern which successfully mitigates intensity variations, verifying
Hypothesis 3.1. The contour aliasing of the printed parts is evaluated and reduced by the vibration
pattern, confirming Hypothesis 3.2. To support Hypothesis 3.3, the vibration amplitude applied in
the printing process is meticulously selected to ensure minimal impact on the dimensional accuracy
of the products. The effect of the vibration assisted MIP-VPP on the dimensional accuracy is
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evaluated by comparing with the parts printed via conventional MIP-VPP. The combined
simulation and experimental results demonstrate the efficacy of the mask image vibration
technique in reducing surface roughness, validating all three hypotheses.
5.2 Contribution
When seeking the solution of improving the process resolution and surface smoothness of
MIP-VPP process, new knowledge is discovered and listed as below:
1. In the study of zooming-focused MIP-VPP, we investigated and modeled the light
intensity of the projection-based MIP-VPP system. By understanding the variations
in light intensity, we proposed a dynamic compensation method that adjusts the
focal distance and magnification rate of the projection system, resulting in a more
uniform light intensity distribution and improved surface quality.
2. In the study of the spatiotemporal MIP-VPP, we developed an innovative mask
image shifting method that generates higher-resolution mask images by
superimposing a series of shifted sub-images. This technique increases lateral
resolution without affecting the fabrication speed or building volume.
3. In the study of the spatiotemporal MIP-VPP, we devised an error diffusion
algorithm that efficiently generates sub-mask images for the mask image shifting
method. This algorithm significantly improves the computational efficiency of
generating high-resolution mask images, making it a practical solution for MIP-
VPP systems.
4. In the study of the vibration-assisted MIP-VPP, we analyzed and modeled the light
intensity variation of the LCD mask image, providing a deeper understanding of
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the factors contributing to surface roughness. This analysis helps to identify the
critical parameters for further optimization of the MIP-VPP process.
5. In the study of the vibration-assisted MIP-VPP, we developed a mask image
vibration method that reduces both contour aliasing and light intensity variation.
By carefully selecting the vibration frequency and amplitude, our method enhances
surface smoothness without compromising dimensional accuracy, process
resolution, or fabrication speed.
These contributions advance our understanding of MIP-VPP processes to some extent and
provide novel techniques for enhancing resolution and surface quality in additive manufacturing.
5.3 Future Work
In the current research, we have made progress in improving the process resolution and
surface smoothness of the MIP-VPP process through the development of innovative techniques,
including zooming-focused MIP-VPP, spatiotemporal MIP-VPP, and vibration-assisted MIP-VPP.
These methods have shown promising results in addressing vertical aliasing, increasing lateral
resolution, and enhancing surface quality.
However, there is still some work left to further extend the proposed research goal. Firstly,
the zooming-focused MIP-VPP, enabled by moving the objective lens, has demonstrated the
effectiveness of improving vertical resolution. However, the fabrication scale of the current
process is limited to 5mm by the size of the mask image. To accommodate applications such as
camera lenses and other optical devices, enlarging the fabrication to a scale of hundreds of
millimeters is necessary. Developing an improved method to make the zooming-focused MIP-VPP
more applicable to large-scale applications is a crucial future direction.
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Secondly, the vibration-assisted MIP-VPP improves surface roughness by eliminating the
'dark zone' aliasing and contour aliasing. However, the dimensional accuracy of the printed parts
is slightly sacrificed (50µm). This sacrifice in dimensional accuracy is significant when the feature
size is smaller than 200µm. For ultra-thin features (50µm), the vibration-assisted MIP-VPP is not
applicable. It is essential to devise an innovative approach to improve surface roughness for thin
features without compromising dimensional accuracy.
Thirdly, more effort should be put into exploring potential applications for the proposed
high-quality printing processes. For example, the refined MIP-VPP techniques could be employed
in the fabrication of microfluidic devices, biomedical implants, and microscale mechanical
components, among others. By identifying and addressing the unique requirements and constraints
of these application areas, the proposed techniques can be optimized and adapted for broader use
in additive manufacturing.
The future work can be specified as following:
1. Extend the zooming focused MIP-VPP to large scale (100mm) fabrication of the
industrial products:
 Develop methods and hardware for scaling up the zooming-focused MIP-VPP
technology to accommodate larger fabrication sizes.
 Investigate the impact of increased fabrication scale on resolution, surface
roughness, and fabrication speed.
 Identify and address potential challenges in the implementation of large-scale MIP-
VPP systems, such as maintaining consistent light intensity and accurate mask
image alignment.
2. Extend the vibration assisted MIP-VPP to micro scale (50um) fabrication:
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 Develop a feature-dependent vibration method that adapts the vibration patterns
based on the specific characteristics of the microscale features being printed,
ensuring optimal surface roughness and dimensional accuracy.
 Investigate the relationship between vibration parameters, such as frequency and
amplitude, and the resulting quality of microscale features, in order to establish
guidelines for the most effective vibration patterns.
 Test and validate the feature-dependent vibration method on a range of microscale
structures with varying geometries and complexities, demonstrating its versatility
and applicability to different fabrication scenarios.
3. Explore the industrial application for the proposed process:
 The proposed zooming focused MIP-VPP can be used to fabricate complex
geometries with reduced staircase effect, making it suitable for applications in the
fields of optics, such as lenses and mirrors, as well as intricate mechanical
components that require smooth surfaces and precise dimensions.
 The proposed spatiotemporal MIP-VPP can be used to fabricate high-resolution
parts with fine features, such as microfluidic devices, miniature sensors, and
electronic components, which are essential in industries like healthcare,
telecommunications, and consumer electronics.
 The proposed vibration assisted MIP-VPP can be used to fabricate products with
improved surface quality and reduced contour aliasing, which is beneficial for
applications that require smooth surfaces and minimal post-processing, such as
aerospace components, medical implants, and precision-engineered parts.
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By addressing these future work directions, we can continue to advance the state-of-the-art
in MIP-VPP processes, making them even more effective and practical for a wide range of
applications in additive manufacturing.

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Abstract (if available)

Abstract Additive manufacturing (AM), which enables the direct fabrication of products with complex geometries from computer-aided design (CAD) models in a single step, has gained widespread adoption across various industries. However, The intrinsic discontinuities in pixels and layers constrain the resolution in both the lateral plane and the vertical direction, as well as the surface roughness of the printed parts. This research addresses lateral resolution, vertical resolution, and surface roughness limitations via motion-assisted vat photopolymeriztion. We developed a zooming-focused mask image projection vat photopolymerization (zooming-focused MIP-VPP) process to increase vertical resolution in additive manufacturing. We developed a spatiotemporal mask image projection vat photopolymerization (sps-MIP-VPP) process to increase lateral resolution. We developed a vibration-assisted mask image projection vat photopolymerization (vibration-assisted MIP-VPP) process to enhance surface quality in additive manufacturing. As a proof of concept, several industrial products were fabricated to verify the effectiveness and efficiency of the proposed motion-assisted MIP-VPP.

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Asset Metadata

Creator Xu, Han (author)

Core Title Motion-assisted vat photopolymerization: an approach to high-resolution additive manufacturing

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Industrial and Systems Engineering

Degree Conferral Date 2023-05

Publication Date 05/16/2024

Defense Date 04/25/2023

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag additive manufacturing,high resolution,OAI-PMH Harvest,vat photopolymerization

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Chen, Yong (committee chair), Gupta, Satyandra (committee member), Qiang, Huang (committee member), Zhao, Hangbo (committee member)

Creator Email hxu050@usc.edu,xuhan0226@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC113131598

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