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Optical forces near micro-fabricated devices
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Optical forces near micro-fabricated devices
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Content
OPTICAL FORCES NEAR MICRO-FABRICATED DEVICES
by
Camilo Andr´ es Mej´ ıa Prada
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
(PHYSICS)
December 2013
Copyright 2013 Camilo Andr´ es Mej´ ıa Prada
Acknowledgments
I would like to thank my advisor Dr. Michelle Povinelli for her support and guidance.
Her expertise and knowledge led us into great discussions concerning optical forces.
I thank Eric Jaquay and Dr. Luis Javier Martinez for developing the recipe for fabri-
cating photonic crystal devices. They know that the path was difficult, but worth it.
I would also like to acknowledge Ningfeng Huang, Chenxi Lin, and Jing Ma for
performing calculations and holding discussions relevant to this study.
I am grateful to Dr. Mehmet Solmaz for his mentoring in optical design and lab
protocols. He put the experimental side of the laboratory to work. I also thank Roshni
Biswas for performing relevant calculations and participating in discussions.
Finally, I thank Mia Ferrera Wiesenthal for rendering the schematic of LATS.
This work was funded by an Army Research Office Young Investigator Award under
award no. 56801-MS-YIP and an Army Research Office PECASE Award under award
no. 56801-MS-PCS. Computation for work described in this dissertation was supported
by the University of Southern California Center for High-Performance Computing and
Communications (www.usc.edu/hpcc).
ii
Contents
Acknowledgments ii
List of Figures v
Chapter 1 : Introduction 1
1.1 Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Mechanical forces . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Van der Waals force . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Optical forces . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.4 Thermal gradient force . . . . . . . . . . . . . . . . . . . . 12
Chapter 2 : Light Assisted templated self-assembly 14
2.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Optical force using guided modes . . . . . . . . . . . . . . . . . . . 16
2.3 Optical force maps and reconfiguration . . . . . . . . . . . . . . . . 18
2.4 Interparticle interaction . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Parametric dependency . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Experimental considerations . . . . . . . . . . . . . . . . . . . . . 23
Chapter 3 : Experimental demonstration of Light Assisted Templated Self-
assembly 25
3.1 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.1 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.2 Optical setup . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.3 Stiffness analysis . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.4 Force measurement . . . . . . . . . . . . . . . . . . . . . . 34
3.4.5 Force and potential calculation . . . . . . . . . . . . . . . . 35
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
iii
Chapter 4 : Optical trapping of metal-dielectric cluster near a microcavity 38
4.1 Trapping of gold nano-particles . . . . . . . . . . . . . . . . . . . . 39
4.2 Cluster formation . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Chapter 5 : Measurement of the bending modulus of a GUV using a dual-
beam optical trap 49
5.1 Stress profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.1.1 Surface stress using ray optics . . . . . . . . . . . . . . . . 51
5.1.2 Surface stress using the Maxwell Stress tensor . . . . . . . . 53
5.1.3 Comparison between RO and MST . . . . . . . . . . . . . . 54
5.2 Dual-beam optical trap setup . . . . . . . . . . . . . . . . . . . . . 57
5.3 GUV fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4 Trapping and stretching of GUVs . . . . . . . . . . . . . . . . . . . 60
5.4.1 Calibration of optical forces . . . . . . . . . . . . . . . . . 61
5.4.2 Temporal response to applied stress . . . . . . . . . . . . . 62
5.5 Measurement of lipid bilayer bending modulus . . . . . . . . . . . 63
5.6 POPC, POPC-chol and DPPC bending modulus . . . . . . . . . . . 68
Chapter 6 : Summary 74
Reference List 76
iv
List of Figures
2.1 Light-assisted templated self-assembly using a photonic crystal slab 15
2.2 (a) Normalized transmission of a silicon slab on a silica substrate for
a height of 0.6a and hole radius of 0.2a, (b) Force in the z direction
for a sphere of radius 0.1a placed 0.25a above the slab surface. The
sphere is positioned equidistant from the four closest holes. . . . . . 17
2.3 Normalized intensity for normally incidentx-polarized light for R1,
R2, and R3, respectively at a height of (a-c) 0.25a above the surface
of the slab and (d-f) 0.1a above the surface of the slab. White circles
indicate hole positions. . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 (a-c) Force in thex-z plane at y = 0, (d-f) Force in thex-y plane atz =
0.25a for R1, R2, and R3, respectively. The red dots represent stable
trapping points and the black circles indicate hole positions. . . . . . 19
2.5 (a) Pattern for x-polarization, (b) Pattern for y-polarization. . . . . . 20
2.6 (a) Schematic of the system (b) Transmission spectrum of photonic-
crystal slab with and without particles above (c)F
x
c= on particle
1 for 3 different positions of particle 2 (d)F
y
c= on particle 1 for 3
different positions of particle 2. The legend indicates the position of
particle 2 in (b), (c), and (d). . . . . . . . . . . . . . . . . . . . . . 21
2.7 (a) Vertical force as a function of for a sphere with radius 0.15a,
(b) Vertical force as a function of sphere radius for=3, (c) Vertical
force for R1 as a function of for a sphere radius of 0.35a. . . . . . 23
2.8 (a) Vertical force as a function of for a sphere with radius 0.15a,
(b) Vertical force as a function of sphere radius for=3, (c) Vertical
force for R1 as a function of for a sphere radius of 0.35a. . . . . . 24
v
3.1 Schematic picture of templated, light-assisted self assembly. Inci-
dent light from below excites a guided-resonance mode of a photonic-
crystal slab, giving rise to optical forces on nanoparticles in a solu-
tion. Under the influence of the forces, the nanoparticles self assem-
ble into regular patterns. . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 (a) SEM image of photonic-crystal slab. The scale bar in the inset
is 1 m. (b) 3D FDTD simulation of the magnetic field (H
z
) for
a normally-incident, x-polarized plane wave. Circles represent the
positions of holes; four unit cells are shown. (c) Measured transmis-
sion spectrum (log scale). . . . . . . . . . . . . . . . . . . . . . . . 27
3.3 Light-assisted, templated self assembly of 520 nm diameter parti-
cles above a photonic-crystal slab. The square lattice of the slab is
visible in the background, oriented at 45 with respect to the cam-
era. (a-c) Sequential snapshots taken with the light beam on. (d)
Snapshot taken after the beam is turned off. . . . . . . . . . . . . . 28
3.4 Trap stiffness for incident, x-polarized light. (a) Particle positions
(red dots) extracted from a 20-second video. Blue ellipses represent
two standard deviations in position. (b) Histogram of stiffness val-
ues in the direction parallel to the polarization of the incident light.
(c) Stiffness in the direction perpendicular to the incident polariza-
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 (a) Force map in one unit cell. The black circle represents the posi-
tion of a hole, and the color bar represents the vertical optical force
in dimensionless units ofFc=P , wherec is the speed of light, and
P is the incident optical power. A negative force indicates attraction
toward the slab. The arrows represent the magnitude of the lateral
optical force. The length of the white arrow at the bottom center of
the figure corresponds to the maximum in-plane value of 1.6. (b)
The potential through the center of the hole along the x-direction.
The green dashed lines indicate the position of the hole. (c) The
potential through the center of the hole along they-direction. . . . . 31
4.1 (a) Schematic of defect cavity in a hexagonal lattice of holes with
diameter d and thickness h, (b) Electric field distribution at the cen-
ter of the slab at resonance, (c) Electric field distribution at z=130
nm above the slab at resonance, (d) Electric field distribution at z =
130 nm above the slab slightly off resonance. . . . . . . . . . . . . 41
vi
4.2 Vertical force on gold nanoparticles of different sizes and a dielectric
particle withr = 80 nm and = 1.59. . . . . . . . . . . . . . . . . . 42
4.3 Gradient (a), and scattering and absorption forces (b) on a gold par-
ticle withr = 80nm using the dipole approximation. . . . . . . . . . 43
4.4 (a) Vertical force as a function of frequency for several different
vertical positions. Normalized force in the (b) horizontal and (c)
vertical directions as a function of position in the x-y plane for a
gold particle withr = 80nm. . . . . . . . . . . . . . . . . . . . . . 44
4.5 Electric field intensity in the midplane (a) and on a vertical cut (b)
of a trapped gold particle with r = 80 nm. Horizontal (c) and verti-
cal (d) optical forces as a function of position for a dielectric parti-
cle with r = 80 nm in the presence of a trapped gold particle (yel-
low circle). (e) Horizontal optical force as a function of position
for a dielectric particle near trapped gold (yellow) and dielectric
(blue) particles. The arrows next to the graph represent Fc=flux
= 0.002. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.6 Force on gold particle due to dielectric particle. . . . . . . . . . . . 48
5.1 Dual-beam optical trap . . . . . . . . . . . . . . . . . . . . . . . . 50
5.2 (a) Optical stress for multiple reflections on a spheroid with volume
equal to a 5 m sphere and e = 0.4 (n
1
= 1.335, n
2
= 1.38, w =
11m). RI of Red Blood Cells is close to the value chosen in this
case. (b) Total optical surface stress in the equatorial plane for single
beam, and (c) total optical surface stress for dual beam. . . . . . . . 52
5.3 Circularly polarized electric field intensity and corresponding stress
profile in the equatorial plane for (a) Single Beam (b) Dual beam. 54
5.4 Comparison of stress profile arising from single beam in the equato-
rial plane obtained from ray optics and MST for varying eccentric-
ities (b) single beam electric field intensity profile corresponding to
e = 0:0 ande = 0:9 (c) Dual beam. . . . . . . . . . . . . . . . . . . 55
5.5 Comparison of MST and RO for spheroids with e = 0.4 (a) refractive
index variation and (b) size variation. . . . . . . . . . . . . . . . . . 57
5.6 (a) Schematic of the stretching of a GUV using DBOT, (b) Optical
setup that incorporates a silicon chip for fiber-to-capillary alignment
and microfluidic adapters to couple the flow channel. . . . . . . . . 58
vii
5.7 (a) POPC-GUV optically trap at low power, (b) POPC-GUV opti-
cally trap at high power, (c) A plot of the contours fitted to both
stretching powers (blue = low power / low tension, green = high
power / high tension). The scale bar is 10m. . . . . . . . . . . . . 61
5.8 The drag forces in no-slip and perfect-slip cases are plotted against
the calculated optical forces (solid lines) from one beam for two
power levels, 50mW and 100mW. 50m corresponds to the center
of the channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.9 (a) The optical power (blue line; right axis) is suddenly increased
from 100 mW to 500 mW. The major axis strain is shown by the red
dots (left axis). (b) Micrograph of deforming GUV . The scale bar is
10m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.10 Measurement of the bending modulus of a pure POPC GUV . Time
axis is common to plots a-c, dotted guide lines show time points at
which laser power is increased. (a) Laser power as a function of
time. (b) 2D contour plot showing the radius as a function of angle
in the image plane as a function of time. (c) Percentage area strain
as a function of time. (d) Average stress on the GUV as a function
of eccentricity and base radius. (e) Average percentage area strain
for each laser power plotted versus the scaled lateral tension. is
the fitted value of the bending modulus. . . . . . . . . . . . . . . . 64
5.11 Apparent area strain versus ln(tension) plots for POPC and DPPC
GUVs with 20% cholesterol. The POPC lipid membrane is more
flexible than DPPC lipid membrane. The error bars represent one
standard deviation of area strain and lateral tension. The errors in
bending modulus values are calculated from the standard errors of
the slope using linear regression analysis. . . . . . . . . . . . . . . 69
5.12 Histograms of the bending modulus distributions of and DPPC-Chol,
POPC-Chol, and pure POPC lipid GUVs. . . . . . . . . . . . . . . 70
5.13 Laser power ramp-up and ramp-down experiments on two different
GUVs, (a) POPC and (b) DPPC-20%Chol. . . . . . . . . . . . . . . 73
viii
Chapter 1
Introduction
In recent years, new complex structures have been fabricated on the nano-scale which
exhibit novel optical properties. An example of such devices is a photonic crystal
(PhC). A PhC is a periodic refractive index perturbation in a high-index material. These
man-made structures can forbid light propagation within certain frequency ranges and
demonstrate novel dispersion characteristics (1).
In order to fabricate such complex structures, new techniques have been developed;
one particular method involves particle self-assembly. In the self-assembly method (2)
particles are deposited upon a substrate and organize themselves in a form determined
by their interparticle interactions. A fundamental constraint, however, in the self-
assembly method is that only energetically favorable structures are formed. Templated
self-assembly methods have been introduced (3; 4) to expand the range of structures
that can be fabricated using self-assembly. In this method, particle assembly occurs
on a template. The geometry of the template guides particle arrangement, allowing
the formation of diverse crystal structures. A fundamental problem with this method
is that the final structure is determined strongly by the geometry of the template.
Consequently, the formation of structures that possess different size particles and a
wide range of material, is not well controlled.
On the other hand, since the discovery of optical tweezers (5; 6), optical forces near
micro-photonic devices have been studied extensively. While standard optical tweezers
1
use a focused laser beam to trap and manipulate particles, structured light fields allow
even greater control. Large optical forces have been achieved by using the strong elec-
tromagnetic fields near nano-apertures (7), gratings (8), photonic-crystal micro-cavities
(9; 10), slot waveguides (11) and plasmonic structures (12; 13). Applications of such
particle manipulation have been found from the physical sciences to biology (14; 15).
However, much of this work has only considered single-particle traps.
In this dissertation, I study optical forces near micro-fabricated devices for multi-
particle manipulation. I consider particles of different sizes and compositions. In
particular, I focus my study on both dielectric and gold particles as well as Giant
Unilamellar Vesicles (GUVs).
First, I consider optical forces near a PhC and establish the feasibility of a technique
which we term Light-Assisted Templated Self-assembly (LATS). In contrast to previous
work on Fabry-Perot enhancement of trapping forces above a flat substrate (16), I
exploit the guided resonance modes of a PhC (17) to provide resonant enhancement of
optical forces. The guided mode forms spatially distinct trapping patterns at different
polarizations and wavelengths. I consider the possibility of assembling multi-particle
patterns. Specially, I show how optical forces near a photonic crystal slab can be used
to assist templated self-assembly and permit different particle pattern formation. Thus,
a PhC will serve as an optically reconfigurable template for multi-particle trapping.
Furthermore, in contrast to localized mode trapping techniques (which is necessarily
limited to a smaller area, and thus, fewer trap sites) the use of an extended mode has
obvious advantages. The fact that every trap operates simultaneously in a small area
eliminates the need for a spatial light modulator or beam scanning of the sample. This
2
is a unique feature of our extended-mode trapping system.
Combining the optical field created in our system with a microfluidic flow also
provides the potential for sorting particles via the creation of lock-in states (18).
Additionally, changing the shape of the hole within each unit cell, an asymmetric
potential could be created which could operate as a ratchet (19). We anticipate that
our system could be applied in self-assembly experiments and also to build clusters in
a reversible way; the later would enable the creation of reconfigurable, optical matter
(20). Our LATS system is able to assemble structures with arbitrary symmetries which
are ordinarily unavailable for assembly due to energy constraints. We demonstrate this
unique ability both theoretically and experientally. In addition, our structure could be
fabricated upon an active material, thereby merging the source and trap. With proper
electrical excitation, such structures could easily be adopted for various applications.
Finally, although previous work has examined light forces between different layers of
particles in colloidal PhC (21; 22), forces on colloidal layers above a micro-fabricated
template were not considered. I used full-vectorial electromagnetic simulations of
Maxwell’s equations in time domain to calculate optical forces on dielectric particles of
500 nm.
In Chapter 3, I present the experimental demonstration of LATS. I explain all
relevant setup and fabrication methods. Next, I use a particle tracking software to
record the position of a 520 nm polystyrene particle while trapped in a square lattice
PhC. In addition, I numerically calculate the optical force at the center of the trapping
pattern and the corresponding optical stiffness of the PhC trap.
3
In the following chapter, I calculate the optical force near a PhC micro-cavity for
gold and dielectric particles 50 nm in diameter. Next, I use the electromagnetic field
re-distribution around a trapped gold nano-particle to direct a metal-dielectric cluster
formation.
In the final chapter, I explore optical forces near a Dual Beam Optical Trap (DBOT)
and measure the bending modulus of a GUV 10 m. First, I present a method to
extract the bending modulus of the membrane from the area strain data. This method
incorporates three-dimensional ray-tracing to calculate the applied stress in the DBOT
within the ray optics approximation. I compare the optical force calculated using the ray
optics approximation and Maxwell Stress Tensor method to ensure the approximation’s
accuracy. Next, we apply this method to 3 populations of GUVs to extract the bending
modulus of membranes comprised of saturated and monounsaturated lipids in both gel
and liquid phases. We find that bending modulus values depend on lipid phase and are
normally distributed for the three populations we tested. Our results also indicate that
the addition of cholesterol to POPC membranes does not alter the respective bending
modulus. Moreover, the values we obtain for bending moduli agree well with those
found in literature. This technique is a promising route to obtaining statistically relevant
bending moduli data for lipid membranes with controlled compositions.
1.1 Forces
For a particle in water, we can consider the interplay of various forces: gravitational,
buoyant, brownian, electrostatic and those forces that develop from the interaction
between an external electromagnetic field and a particle. I’ll refer to the last one as an
4
optical force.
In general, the dynamics of the particle is governed by the following equation:
m
~ x =
_
~ x +
~
F
fluid
+
~
F
optical
+~ (t) (1.1)
where
is the friction coefficient, and
i
(t) are mutually uncorrelated white noises
that obey the fluctuation-dissipation relationh
i
(t)j
j
(t)i = 2k
B
T
ij
(tt
0
).
In this chapter, I describe the origin and order of magnitude for these forces acting
upon a 1m polystyrene particle.
1.1.1 Mechanical forces
In the particular case of a polystyrene particle in water (
water
= 1g=cm
3
;
water
=
1:3 10
4
cm
2
=s;
poly
= 2:5g=cm
3
), the gravitational force is on the order of 110
2
pN while the buoyancy force is 0:6 10
2
pN. Thus, the particle feels a net
gravitational force on the order of 1 10
3
pN.
Using the diffusion coefficient D = k
B
T=6r = 0:5m
2
=s as well as the drag
coefficient
= 6r, the average Brownian force can be shown to be 1 10
4
pN.
For this system the Reynolds number is 1 10
7
. At a low Reynolds number,
the inertia plays no role in the particle dynamics whatsoever. That means, what the
particle is doing at any given moment is entirely determined by the forces that are
5
exerted on particle at that moment, and by nothing in the past. Thus, Eq. (1.1) reduces to
_
~ x =
~
F
fluid
+
~
F
optical
+~ (t) (1.2)
1.1.2 Van der Waals force
The Van der waals force arises from electron cloud fluctuations surrounding the nucleus
of an electrically neutral atom. This force is a short range force and, in general, is
attractive.
For macroscopic bodies with known volumes and numbers of atoms (molecules
per unit volume), the total van der Waals force is often computed based on the micro-
scopic theory as the sum over all interacting pairs. It is necessary to integrate over the
total volume of the object, which makes the calculation dependent on the object’s shape.
The van der Waals force between two spheres of constant radiiR
1
andR
2
is then a
function of separationr
F
VW
=
AR
1
R
2
6r
2
(R
1
+R
2
)
(1.3)
where A is a constant that depends on both materials. For two 500 nm polystyrene
particles in water,A = 0:22 10
20
J andF = 1 10
3
pN at 300 nm of separation.
In the case of a sphere near a surface, the van der Waals force is
F
VW
=
AR
6r
2
(1.4)
6
For a 500 nm particle floating in water 100 nm from a silicon surfaceF = 1 10
3
pN.
1.1.3 Optical forces
Optical forces arise whenever an electromagnetic wave propagates in a medium. It
can be understood as a change in momentum of the propagating light due to a change
in refractive index; it is a force felt by the induces dipoles in the medium due to the
presence of an electromagnetic field.
Maxwell stress tensor method
For a small volumeV the induced charge will experience a mechanical force given by
the Lorentz relation (23):
~
F =
Z
V
(
~
E +
~
J
~
B)dV (1.5)
Expressing the charge, current and magnetic field in terms of the total electric field
~
E
we can obtain
~
F =
I
S
~
T
d~ a
d
~
S
dt
(1.6)
whereS is the surface enclosing the volumeV ,
~
S is the Poynting vector andT
is the
Maxwell stress tensor defined as
T
=D
E
1
2
(
~
D
~
E) +H
B
1
2
(
~
H
~
B) (1.7)
In general, there are two natural approximations often employed in such calculations:
either the particles is much smaller than the wavelength (dipole approximation), or much
larger (ray optics).
7
Optical force using dipole approximation
Let us write Eq. (1.6) in the volume form (24; 25)
h
~
Fi =
I
S
h
~
T
id~ a =
Z
d
3
~ rrhT (r)i (1.8)
For a small particle, the total field inside the particle can be expressed in terms of the
external field
~
E
0
(~ r) as
E(r) =E
0
(r) +G(rr
0
)E
0
(r) (1.9)
whereG is the Green function,
G
ij
(r) = (k
2
0
ij
+@
i
@
j
)
e
ik
0
r
4
0
r
(1.10)
and is the polarizability
=
0
1i
0
k
3
0
=(6
0
)
;
0
= 4
0
a
3
1
+ 1
: (1.11)
The time-averaged force, expressed in terms of the external electric field and the induced
dipole moment is (26)
h
~
Fi =
1
2
Re
"
X
i
p
i
rE
0i
#
=
1
2
Re
"
X
i
E
0i
rE
0i
#
(1.12)
8
using Faraday’s lawrE =i!
0
H and the relation
P
i
E
i
rE
i
= (Er)E
+E
(rE), the dipolar forceh
~
Fi can be expand as the sum of the gradient force, radiation
pressure and spin density of the light field as
h
~
Fi =
1
2
Re [E
0
rE
0
] +
scat
1
c
h
~
Si
+
scat
[crh
~
L
S
i] (1.13)
where
scat
k
0
Im()=
0
is the scattering cross section andh
~
L
S
i =
0
4i!
(
~
E
0
~
E
0
) is
the time averaged spin density.
The first term is the gradient force, the second term is the radiation pressure (propor-
tional to the Poynting vectorh
~
Si), and the last term is a curl force associated with the
nonuniform distribution of the spin density of the light field.
Optical binding
The dipole approximation made in Eq. (1.13) is valid for a single particle in an external
field. For multiple particles, the total electric field is the superposition of the incident
electric field and the field emitted by the induced dipoles.
Let us consider two dielectric particles with induced dipole moments ~ p(r
A
) and
~ p(r
B
) in an electric field
~
E such that,p
i
(r
AjB
) =
AjB
ij
E
j
(r
AjB
). In terms of the incident
field
~
E
0
, the total field can be rewritten as (27):
E
i
(r
A
) =E
0i
(r
A
) +G
ij
(r
A
;r
B
)
B
jk
E
k
(r
B
) (1.14)
E
i
(r
B
) =E
0i
(r
B
) +G
ij
(r
B
;r
A
)
A
jk
E
k
(r
A
):
The solution for particle B of this equation is E
i
(r
B
) = K
ij
E
0j
(r
B
) +
K
ij
G
jk
A
km
E
0m
(r
A
), whereK
ij
is the inverse tensor, [IG(r
B
;r
A
)
A
G(r
A
;r
B
)
B
]
1
.
9
For small particles we can approximateK
ij
=
ij
, thus, Eq. (1.12) for particleB can be
written as
F
(r
B
) =
1
2
Re[
B
ij
E
0j
(r
B
)@
B
E
0i
(r
B
) (1.15)
+
B
ij
G
jk
A
km
E
0m
(r
A
)@
B
E
0i
(r
B
)
+
B
ij
E
0j
(r
B
)@
B
G
iq
A
qn
E
0n
(r
A
)]:
The first term is the classical formula for the optical force which is determined solely
by the spatial distribution of the incident field. This force is usually further divided
into both a gradient force and scattering force. The second term is the interaction
between the incident field and the induced dipole from the scattered field. The third
term corresponds to the force acting upon the induced dipole from the scattered field by
the incident field.
In general, by using the Maxwell stress tensor method we incorporate the effect of
optical binding.
Optical force in the ray optics approximation
For particles much larger than the wavelength of the incident light, we can use the
Ray Optics approximation (RO). Under this assumption, rays undergo a change in
momentum after a reflection on a surface. This change in momentum induces an optical
force on the object.
Let us consider a ray propagating along the incident direction
^
i
0
. This ray carries a
momentum of~ p = (n
1
=c)P
^
i
0
, wheren
1
is the refractive index of the medium,P is the
optical power andc is the velocity of light in vacuum. Following a reflection, the force
10
acting upon a surface can be calculated as the difference in momentum carried by the
incident ray compared to that carried by both the reflected and transmitted rays (28; 29),
~
f =
n
1
P
c
^
i
0
(R
0
^ r
0
+
n
2
n
1
T
0
^
t
0
)
(1.16)
where ^ r
0
and
^
t
0
define the unit vectors along the reflected and transmitted directions,
respectively.R
0
andT
0
represent the Fresnel reflection and transmission coefficients.
For optical fibers, the intensity of light emanating is best approximated by a Gaussian
distribution. Thus, the power incident on a differential area s is related to the total
beam powerP
T
by the following equation,
P =
2P
T
w
2
exp
2r
2
w
2
scos (1.17)
wherew andr represent the beam width and the radial distance from the beam axis ^ z
at the interacting point, and is the angle between the surface normal and the beam
propagation direction. Using Eq. (1.16) and (1.17) the surface stress can be shown to be
~ =
n
1
cos
c
^
i
0
(R
0
^ r
0
+
n
2
n
1
T
0
^
t
0
)
2P
T
w
2
exp
2r
2
w
2
(1.18)
In the case of a close object, a transmitted beam experiences multiple internal reflections.
If we considerm internal reflections, the total optical surface stress is
~
m
=
2P
T
n
1
T
0
scos
w
2
cs
m
exp
2r
2
w
2
m1
Y
j=1
R
j
n
2
n
1
^
i
m
(
n
2
n
1
R
m
^ r
m
+T
m
^
t
m
)
(1.19)
The incident vector takes into account the divergence of the beam, meanwhile the
subsequent reflected and transmitted vectors can be computed using Snell’s law. Finally,
11
the total force is the integral of the stress over the surface.
1.1.4 Thermal gradient force
In a material that possesses a complex dielectric constant, the absorption of electromag-
netic energy induces a change in temperature which leads to a convective force due to
the fluid flow.
The dynamics of such a system can be studied by numerically solving the steady-
state, incompressible Navier-Stokes equation in the microfluidic environment together
with a heat transfer equation. The incompressible Navier-Stokes equations consist of a
momentum balance
@u
@t
+uru =rp +r
2
u +F (1.20)
This equation ignores variations in density with temperature. We add free convection
to the fluid flow with the Boussinesq approximation: F = g((T )(T
0
)). We
assume that the fluid density is independent of pressure and depends linearly on the
temperatureT :
0
=
0
C
p
(TT
0
), whereC
p
is the heat capacity.
The heat equation is an energy conservation equation. The change in energy is equal
to the heat source minus the divergence of the diffusive heat flux
C
p
@T
@t
+urT
+r (krT ) =Q (1.21)
12
where Q is a source term. The solution of the coupled system provides the velocity
field and temperature distribution within the fluid.
This effect is important for systems with absorption. A numerical simulation esti-
mates that, for a 1m particle, the convective force is 1 10
6
pN for the channel
and parameters used in this thesis (Chapter 3 and 4).
13
Chapter 2
Light Assisted templated self-assembly
Self-assembly methods (2) have been used to construct complex materials including
three dimensional photonic crystals (30). However, a fundamental constraint is that only
energetically favorable structures are formed. To overcome this problem, templated
self-assembly methods have been introduced (3; 4). The template guides particle
arrangement, allowing the formation of diverse albeit static crystal structures. Here
we consider how optical forces can be used to direct assembly and reconfiguration
of particles on a photonic crystal slab, which serves as an optically reconfigurable
template. Our calculations predict the formation of stably trapped crystal patterns
that depend on the wavelength and polarization of the light source. We envision that
the process of light-assisted templated self-assembly may be used for fabrication of
complex photonic materials and all-optically reconfigurable photonic devices.
Optical forces have been studied extensively since the discovery of optical tweezers
(5; 6). While standard optical tweezers use a focused laser beam to trap and manipulate
particles, structured light fields (14; 15) allow even greater control over particle
deflection, transport, and sorting. Recently, researchers have used light fields near
microphotonic devices to trap and manipulate particles. This work leverages the
strong electromagnetic gradients near devices such as nanoapertures (7), gratings (8),
photonic-crystal microcavities (9; 10) to generate large optical forces. However, much
of this work has considered single-particle traps.
14
x
y
z
Figure 2.1: Light-assisted templated self-assembly using a photonic crystal slab
Here we suggest the possibility of assembling multi-particle patterns. We propose
to use the guided resonance modes of photonic crystal slabs (17) to enhance the optical
forces acting upon particles in a solution. In contrast to previous work on Fabry-Perot
enhancement of trapping forces above a flat substrate (16), the photonic crystal is used
to form spatially distinct trapping patterns at different polarizations and wavelengths.
While previous work has examined light forces between different layers of particles
in colloidal photonic crystals (22), forces on colloidal layers above a microfabricated
template were not considered.
The system we consider is shown schematically in Fig 2.1. In an experiment,
light would illuminate the photonic crystal slab from below. The structured light
fields above the slab give rise to optical forces on particles in a solution, which can
potentially result in trapping. Due to the variation of the electromagnetic field over
spatial length scales comparable to the particle diameter, the dipole approximation
cannot be assumed. However, the optical force can be found by using full-vectorial
calculations of Maxwells equations to calculate the Maxwell Stress Tensor (22).
15
2.1 Design
For concreteness, we consider a silicon photonic crystal ( = 11.9) with a square lattice
of holes of radius 0.2a and thickness 0.6a resting on a silica substrate ( = 2.1), where
a is the lattice constant of the photonic crystal. The holes in the photonic crystal and
the region above the crystal are filled with water ( = 1.77). The incident light is an
x-polarized plane wave propagating in thez direction.
We calculate the transmission through the slab numerically using the finite-
difference time-domain (FDTD) method (31) with the freely available MIT MEEP
package (32). The computational resolution is 32 grid points per lattice constant.
Fig 2.2(a) shows the normalized transmission through the photonic-crystal slab. The
transmission exhibits three distinctive resonance peaks at frequencies(c=a) = 0.4721,
0.4958, and 0.5008. The characteristic shape of the resonance peaks is well known
from existing work in literature (17). We will refer to the three resonances shown as
R1, R2, and R3 respectively.
2.2 Optical force using guided modes
We calculate the force on polystyrene spheres ( = 2.28) of radius 0.1a. The center
of the sphere is placed at a height of 0.25a above the slab surface. We calculate the
Maxwell stress tensor on the surface of an integration box of size 0.3a centered on the
particle (23). The calculation is done in a computational cell that includes one period of
the photonic crystal slab. Periodic boundary conditions are used in the lateral directions
(x andy), and perfectly matched layer boundaries are used in thez direction. The forces
calculated in this manner represent the optical forces on a periodic array of polystyrene
16
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0
0.2
0.4
0.6
0.8
1
0.45 0.47 0.49 0.51 0.53
-0.03
-0.09
0.45 0.47 0.49 0.51 0.53
Frequency (c/a)
Fz c/φ Transmission
R3 R1 R2
0
(a)
(b)
-0.06
Figure 2.2: (a) Normalized transmission of a silicon slab on a silica substrate for a height of
0.6a and hole radius of 0.2a, (b) Force in thez direction for a sphere of radius 0.1a placed 0.25a
above the slab surface. The sphere is positioned equidistant from the four closest holes.
spheres above the photonic crystal slab.
We plot the dimensionless quantity F
z
c=, where F
z
is the z component of the
optical force, is the incident power per unit cell and c is the speed of light (see Fig
2.2(b)) for a sphere positioned equidistant from the four closest holes. For each of
the three resonance frequencies shown in Fig 2.2(a), we observe a peak in the optical
force. The sign of the force is negative, indicating that the particles are attracted toward
the slab. For comparison, we calculate the radiation pressure of a plane wave on a
spherical particle in the absence of the photonic crystal. We obtain a repulsive force of
F
z
c= 1 10
5
, three orders of magnitude smaller than the attractive optical force
above the photonic crystal. We expect that even larger optical forces can be achieved by
using a photonic-crystal slab with higher-Q resonances. Higher Q has been correlated
with decreasing hole size (17).
17
R1 R2 R3
(a) (b) (c)
x
y
x
y
x
y
0 0 0
6.6 3.3 40.3
|E|
2
/|E
0
|
2
(d) (e) (f)
0 0 0
35.6 22.6 158.6
x
y
x
y
x
y
Figure 2.3: Normalized intensity for normally incident x-polarized light for R1, R2, and R3,
respectively at a height of (a-c) 0.25a above the surface of the slab and (d-f) 0.1a above the
surface of the slab. White circles indicate hole positions.
The intensity for these resonances is shown in Fig. 2.3(a-c) at a height of 0.25a
above the surface of the slab. Fig 2.3(d-f) shows the intensity at a height of 0.1a (equal
to the particle radius). The intensityjEj
2
is normalized to the source intensityjE
0
j
2
.
We observe that each resonance has a different field profile. There is a slight change in
field profile with height, and the intensity increases approaching the slab.
2.3 Optical force maps and reconfiguration
We now look at the spatial dependence of the force on resonance. The force at each
point is calculated by placing the sphere at that point, calculating the full-vectorial
electromagnetic fields, and computing the Maxwell Stress Tensor. For numerical
reasons, we have calculated the forces at a height of 0.25a above the slab to insure that
the Maxwell Stress Tensor integration surface does not overlap with either the particle
or the slab boundaries. Figs 2.4(a), (b), and (c) show the force on a vertical (x-z) slice at
18
−0.5 0
−0.5
0
0.5
0.5 −0.5 0.5
−0.5
0
0.5
−0.5 0 0.5
0.3
0.4
0.5
0.6
−0.5 0 0.5
−0.5 0 0.5
−0.5
0
−0.5
0
0.5
0.5
(a) (b) (c)
(d) (e) (f)
x
x
x
x x
z z z
y y y
x
0
F c/φ
F c/φ
0.3
0.4
0.5
0.6
0.3
0.4
0.5
0.6
Figure 2.4: (a-c) Force in the x-z plane at y = 0, (d-f) Force in the x-y plane at z = 0.25a for
R1, R2, and R3, respectively. The red dots represent stable trapping points and the black circles
indicate hole positions.
y = 0. The length of the arrows next to the graphs represents a valueFc= = 0:2. The
vertical forces are attractive over the whole unit cell: this indicates that a particle near
the slab will be attracted to it.
Figs 2.4(d), (e), and (f) show the force on a horizontal (x-y) slice 0.25a above
the slab surface. The circles indicate hole positions. The arrows next to the graphs
represent a value ofFc= = 0:2, and the red circles represent the stable points to which
the particles are attracted. R1, R2 and R3 correspond to different stable patterns. Fig
2.4(d) has an additional stable point at the center of the unit cell, but it is not labeled
with a red dot because it has a weak lateral force compared to other trapping points.
Similarly, Fig 2.4(e) has two additional weakly stable points at the bottom (or top)
edge of the unit cell. We note that the spatial profile of the forces exerted on particles
touching the slab surface (height 0.1a) may differ slightly from those shown here due
to the weak variation in intensity with height (Fig 2.3).
19
R1 R2
(a)
(b)
R3
Figure 2.5: (a) Pattern for x-polarization, (b) Pattern for y-polarization.
Different light polarizations correspond to different self-assembled patterns, as
shown in Fig 2.5. Due to the symmetry of the photonic crystal, the patterns for
y-polarized light are the same as those for x-polarized light, only rotated =2 (Figs
2.5(a) and 2.5(b)).
In an experiment, particles may be trapped one by one, or a few at a time, in order
to build up ordered patterns. We believe that the red dots shown in Figs 2.4 and 2.5 are
good predictors of the patterns that will be formed in this manner. In our computations,
due to the boundary conditions on the unit cell, we are calculating the optical force on
periodic arrays of particles above the photonic crystal slab. However, since interparticle
interactions (27) are negligible in this system, the force on an individual particle above
the slab will be nearly identical.
2.4 Interparticle interaction
To show that interparticle interactions are weak compared to the force on a particle
from the photonic crystal, we considered the geometry shown in Fig 2.6(a). A first
20
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0
0.2
0.4
0.6
0.8
1
0.45 0.47 0.49 0.51 0.53
Transmission
R3 R1 R2
(a)
(b)
no particle
(-0.1a, 0.1a)
(0.02a, 0)
(-0.4a,-0.4a)
2
1
2
2
y
x
(c)
(d)
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-0.01
-0.03
0.45 0.47 0.49 0.51 0.53
Frequency (c/a)
Fx c/φ
0
-0.02
0.01
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-0.01
-0.03
0.45 0.47 0.49 0.51 0.53
Frequency (c/a)
Fy c/φ
0
-0.02
0.01
-0.04
0.02
Frequency (c/a)
Figure 2.6: (a) Schematic of the system (b) Transmission spectrum of photonic-crystal slab with
and without particles above (c) F
x
c= on particle 1 for 3 different positions of particle 2 (d)
F
y
c= on particle 1 for 3 different positions of particle 2. The legend indicates the position of
particle 2 in (b), (c), and (d).
particle (labeled 1, blue circle) was placed over the slab at a position of [0.3a, 0.3a,
0.25a]. A second particle was placed in one of three alternate positions (labeled 2,
green circles). Fig 2.6(b) shows that the presence of the particles does not visibly affect
the transmission spectrum. In Figs 2.6(c) and (d), we plot the lateral forces on particle
1 for different positions of particle 2. It can be seen that particle 2 has minimal effect
on the force.
As an additional check, we recalculated the force maps of Figs 2.4(d-f) using a
larger computational cell of size 2a 2a, with one particle per computational cell. No
changes were observed in the force maps.
Lastly, we checked that when particles were placed at each of the red dots in Fig
2.4(d-f), the configuration remained stable despite the perturbation. One particle per
unit cell was displaced, and we verified that the restoring force was in the necessary
direction to restore the initial pattern.
21
In the system studied here, the particles do not significantly affect the transmission
spectrum (Fig 2.6(b)). We have checked that even if the particles are placed inside the
holes in the slab or on the slab surface, the shift in the resonance is a small fraction of
the resonance width (15% or less). For larger particles, particles of higher index, and/or
higher Q slab resonances, particle trapping may begin to shift the resonance position.
This opens up the possibility for intriguing phenomena such as self-induced or bistable
trapping, effects which have previously been studied for particle trapping near photonic
cavities (9). This is an interesting direction for further research.
2.5 Parametric dependency
Fig 2.7(a) shows the vertical force as a function of the dielectric permittivity of the
sphere,, for a sphere of radius 0.15a. The magnitude of the force increases with, and
no shift is observed in the resonance peak. Similarly, Fig 2.7(b) shows the vertical force
as a function of sphere radius for a particle with = 3. The force increases with increas-
ing sphere radius and no shift is observed in the resonance peak. For larger sphere
radii, increasing will increase the force and shift the resonance peak, as shown in Fig
2.7(c). The figure shows the region of the spectrum near R1 for a sphere radius of 0.35a.
Changing the template will change the frequencies and quality factors of the
resonances, thus changing the optical force. Fig 2.8(a) shows the change in force due to
changing hole diameter for R1, = 2.28, slab thickness 0.6a and sphere radius of 0.1a.
The force peak shifts to higher frequencies as d is increased. The force peak widens,
and the peak amplitude decreases, corresponding to a decrease in the resonance quality
factor. Fig 2.8(b) shows the change in force with slab thickness for R1, = 2.28, hole
22
0.46 0.47 0.48 0.49 0.5 0.51
0
Frequency (c/a)
Fc/flux
-1.2
-1.6
-0.4
-0.8
0
-1.2
-1.6
-0.4
-0.8
Frequency (c/a)
Fc/flux
0.46 0.47 0.48 0.49 0.5 0.51
2
4
6
8
10
12
0.15a
0.25a
0.35a
Frequency (c/a)
0
5
-5
-15
-10
-20
0.46 0.464 0.468 0.472 0.476 0.48
Fc/flux
-25
ԑ=
2
4
6
8
10
12
ԑ=
r=
r=0.15 a
r=0.35 a
ε=3
Figure 2.7: (a) Vertical force as a function of for a sphere with radius 0.15a, (b) Vertical force
as a function of sphere radius for=3, (c) Vertical force for R1 as a function of for a sphere
radius of 0.35a.
diameter of 0.4a, and sphere radius of 0.1a.
2.6 Experimental considerations
The photonic-crystal slabs studied here are fabricated in a silicon-on-insulator wafer
using standard techniques (33). The lattice constant of the photonic crystal can be
scaled to place a given guided resonance at a particular wavelength of interest (1). For
a = 760nm, for example, R1 occurs at = 1609 nm, while R2 occurs at 1532 nm.
Fora= 375 nm, R1 is at 794 nm and R2 is at 748 nm. An upper bound on the power
required for trapping was obtained by setting the height of the potential depthjUj >
23
Frequency (c/a)
0
-0.02
-0.04
-0.06
-0.08
-0.1
Fc/flux
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
Fc/flux
Frequency (c/a)
0.45 0.46 0.47 0.48 0.49 0.5 0.44 0.46 0.48 0.5
0.55a
0.6a
0.65a
0.7a
0.28a
0.32a
0.36a
0.4a
0.44a
0.48a
0.52a
d=
h=
Figure 2.8: (a) Vertical force as a function of for a sphere with radius 0.15a, (b) Vertical force
as a function of sphere radius for=3, (c) Vertical force for R1 as a function of for a sphere
radius of 0.35a.
10K
B
T . The power density fora = 760 nm was 1 mW/unit cell for R1, which has a
modest Q of260. The required power decreases with increasing Q. Moreover, the
power calculated in this manner is likely to be overestimated. Our results indicate that
the particles will be attracted to the surface of the slab. As seen in Fig 2.3, the intensity
increases closer to the slab surface, producing a higher force than the values shown in
Fig 2.4, which were calculated for a height of 0.25a (for a particle touching the slab
surface, with a height of 0.1a).
In conclusion, we predict that optical forces above photonic crystal slabs will lead
to a variety of complex, stably trapped crystal patterns. Changing the wavelength or
polarization of the incident light may be used to reconfigure the patterns. We expect the
approach of light-assisted self-assembly, described here, to be of broad utility for fab-
rication of complex photonic materials, sensors, and filters. It is intriguing to consider
whether metamaterials based on metal nanoclusters (34), for example, could be assem-
bled in the manner we describe, using the optical response of the nanoclusters to tune or
adjust assembly.
24
Chapter 3
Experimental demonstration of Light
Assisted Templated Self-assembly
In this chapter, we demonstrate for the first time the trapping of multiple particles using
an extended mode in a two-dimensional photonic crystal slab. Our system improves
upon previous optical trapping work in three main ways. First, we have demonstrated
a compact, multiparticle trapping system in the present work; over 180 particles are
trapped in an area less than 150m
2
. The number of particles could be increased in a
straightforward way by proper design of the structure or by increasing the size of the
incident beam. Second, since the large array of particles is trapped by exciting a single
mode of the structure without the use of a spatial light modulator or any movement of
the beam or sample, it can be easily adapted for a variety of integrated, lab-on-a-chip
applications. Finally, by tuning the polarization of the incident light, we can tune the
preferential direction of the in-plane trapping stiffness of our structure.
The process of light-assisted templated self-assembly (LATS) is shown schemat-
ically in Fig 3.1. Light is incident from below on a photonic-crystal slab. The
photonic-crystal slab consists of a silicon device layer patterned with a periodic array
of air holes. The slab supports guided-resonance modes, electromagnetic modes for
which the light intensity near the slab is strongly enhanced. In chapter 2 (35; 36), we
have predicted theoretically that when the incident laser is tuned to the wavelength
of a guided-resonance mode, nanoparticles will be attracted toward the slab. The
25
Figure 3.1: Schematic picture of templated, light-assisted self assembly. Incident light from
below excites a guided-resonance mode of a photonic-crystal slab, giving rise to optical forces
on nanoparticles in a solution. Under the influence of the forces, the nanoparticles self assemble
into regular patterns.
attractive, optical force arises from a strong electric field gradient just above the slab
surface. In addition, the nanoparticles will experience lateral optical forces due to
the electromagnetic field structure of the guided mode. We expect the optical forces
to result in the assembly of particles into regular 2D arrays. Importantly, the array
patterns formed can be different from the triangular, close-packed structure formed by
traditional, colloidal, self assembly. Figure 1 depicts a square array of particles, as one
example. By exciting different resonant modes of the slab, or by designing different
slab templates, we expect that a variety of patterns can be produced. Moreover, these
patterns can reversibly be assembled or disassembled, suggesting the possibility of
”programmable optical matter”.
3.1 Design
We designed, fabricated, and characterized a photonic-crystal slab for use in the
LATS process. An electron micrograph is shown in Fig 3.2(a). The device was
fabricated in silicon using electron-beam lithography and reactive ion etching (see
26
Figure 3.2: (a) SEM image of photonic-crystal slab. The scale bar in the inset is 1m. (b) 3D
FDTD simulation of the magnetic field (H
z
) for a normally-incident, x-polarized plane wave.
Circles represent the positions of holes; four unit cells are shown. (c) Measured transmission
spectrum (log scale).
Methods in Sec.3.4). The dimension and spacing of the holes were designed to support
doubly-degenerate guided-resonance modes near 1.55 m. The magnetic field profile
resulting from an x-polarized, incident plane wave is shown in Fig 3.2(b). Fields were
calculated using the three-dimensional finite-difference time-domain method (FDTD).
The field-profile for a y-polarized incident wave is rotated by 90 degrees. Fig 3.2(c)
shows the measured transmission spectrum of the device. The guided-resonance mode
appears as a dip in the spectrum. The quality factor, Q, was determined to be 170 by
fitting to a Fano-resonance shape.
3.2 Results
We carried out the assembly process in a microfluidic chamber filled with 520-nm
diameter polystyrene particles, using a laser power of 64 mW. Fig 3.3 shows snapshots
of the LATS process. The photonic-crystal lattice is visible in the background of each
frame. When the laser beam is turned on, nanoparticles are attracted towards the slab,
and begin to occupy sites within the square lattice (Fig 3.3(a)). As time progresses,
27
Figure 3.3: Light-assisted, templated self assembly of 520 nm diameter particles above a
photonic-crystal slab. The square lattice of the slab is visible in the background, oriented at
45 with respect to the camera. (a-c) Sequential snapshots taken with the light beam on. (d)
Snapshot taken after the beam is turned off.
additional particles diffuse into the region where the beam intensity is high, and begin
to form a cluster (Fig 3.3(b)). Eventually, a regular array of particles is formed (Fig
3.3(c)). The square symmetry of the assembled particles is evident from the picture.
When the laser beam is turned off, the particles immediately begin to disperse and
diffuse away from the slab (Fig 3.3(d)). The frames in Fig 3.3 were recorded with
a dilute particle solution for clarity of imaging, and represent an elapsed time of
approximately one hour. Faster cluster formation occurs within solutions of higher
particle concentration.
Each site of the square lattice may be viewed as an optical trap. We used particle-
tracking software to analyze particle motion for fully-assembled clusters (see Sec. 3.4).
Fig 3.4 shows the recorded particle positions extracted from a 20-second video. The
incident light was polarized along the x-direction of the lattice. The figure shows that
the particles tend to stay above the holes in the photonic crystal, with some variation in
position over time. Each blue ellipse represents a fit to the data in a single unit cell. It
can be seen that the variation in particle position increases at the edge of the trapping
28
Figure 3.4: Trap stiffness for incident, x-polarized light. (a) Particle positions (red dots)
extracted from a 20-second video. Blue ellipses represent two standard deviations in position.
(b) Histogram of stiffness values in the direction parallel to the polarization of the incident light.
(c) Stiffness in the direction perpendicular to the incident polarization.
Angle
x
(pN nm
1
W
1
)
x
y
(pN nm
1
W
1
y
0
0
1.48 0.35 2.25 0.48
45
0
1.88 0.52 1.79 0.46
90
0
2.08 0.40 1.39 0.29
Table 3.1: Polarization dependence of trap stiffness.
region, due to the reduction in power away from the center of the beam.
The stiffness of each trap can be determined from the variance in particle position
(37). Figs 3.4(b) and 3.4(c) show histograms of the in-plane stiffness values extracted
from the videos. The stiffness of each trap was normalized to the local intensity in that
unit cell (see Sec. 3.4). We observe that the power-normalized stiffness over the array
of traps is normally distributed, both for the parallel and perpendicular stiffness. The
mean parallel stiffness is lower than the perpendicular stiffness, as shown in Table 3.1
(0
o
angle).
29
We observe that the trap stiffness can be tuned by rotating the direction of incident
light. When the incident light is polarized at 45
o
with respect to the lattice directions, the
stiffness values are approximately the same in the x- and y-directions (Table 3.1). This
is to be expected, since the incident light excites both of the doubly-degenerate modes
with equal strength. At 90
o
, the stiffness in the perpendicular direction (y) is again
lower than in the parallel (x) direction. The ability to tune the stiffness with incident
light indicates the strong optical nature of our traps. The mean values of stiffness
are comparable to those reported elsewhere in the literature for single particle traps (38).
Using the Stokes drag method (see Sec.3.4), we experimentally estimated the
maximum force exerted on the particles by the traps to be 0.3 pN.
3.3 Discussion
To understand how the optical forces result in the observed nanoparticle patterns, we
calculated the force numerically (see Sec. 3.4). Fig 3.5(a) shows the force on a 520 nm
diameter particle whose bottom edge is 25 nm above the surface of the photonic crystal
slab for x-polarized light. The background color represents the vertical force, where
a negative force indicates attraction toward the slab. There are two regions above and
below the hole where the force is slightly repulsive, but at any other position within the
unit cell the particle results in attraction towards the slab. The arrows represent in-plane
forces.
To determine the equilibrium position of the trapped particle, we calculate the
optical potential. Given the relative size of the particles (260 nm radius) and holes (150
30
Figure 3.5: (a) Force map in one unit cell. The black circle represents the position of a hole,
and the color bar represents the vertical optical force in dimensionless units ofFc=P , wherec
is the speed of light, andP is the incident optical power. A negative force indicates attraction
toward the slab. The arrows represent the magnitude of the lateral optical force. The length of
the white arrow at the bottom center of the figure corresponds to the maximum in-plane value
of 1.6. (b) The potential through the center of the hole along the x-direction. The green dashed
lines indicate the position of the hole. (c) The potential through the center of the hole along the
y-direction.
nm radius) in our experiment, the particle can be drawn into the hole by the attractive
vertical force. We calculate the optical potential as a function of x-y position. For
each x-y position, the vertical height of the particle is as small as possible, given the
geometrical constraints (see inset in Fig 3.5(b)). The result is shown by the blue line
(total potential) in Figs 3.5(b) and 3.5(c). It can be seen that the stable equilibrium
position is at the center of the hole (x = 0, y = 0), in agreement with experimental
observations.
The ability of the particle to sink into the hole is a key factor in determining the
equilibrium positions. For comparison, the red, dashed line (in-plane potential) in Fig
3.5(b) shows the optical potential calculated at a constant z height (bottom edge of
particle 25 nm above the slab surface). Two local minima are observed at the edges of
the hole, which are indicated by green, dashed lines. From inspection of Fig 3.5(a), it
31
can be seen that these points correspond to locations where the in-plane forces are zero.
However, these two minima are not stable equilibrium positions of the total potential,
(blue line, Fig 3.5(b-c)).
From our experiments, we determined that the threshold intensity for trapping was
134W per unit cell (see Sec. 3.4). For this intensity, the calculated potential depth is
4.5k
B
T .
3.4 Methods
3.4.1 Fabrication
The photonic-crystal device consists of a square lattice of holes etched into a SOI wafer.
The lattice constant a is 860 nm, and the hole radius is 0.174a (150 nm). The thickness
of the silicon device layer is 250 nm. The buried oxide layer is 3 m thick, and the
silicon substrate is 600 m thick. To fabricate the device, a layer of PMMA 4% in
anisole was spin coated onto an SOI wafer. Electron beam lithography at 30 kV was
used to write the photonic-crystal pattern into the resist. A modified Bosch process was
used to transfer the pattern into the Si device layer using a mixture of SF6 and C4F8
gases. Plasma-enhanced chemical vapor deposition was used to deposit a 193 nm layer
of SiNx onto the polished, back surface of the sample to reduce unwanted reflections.
The photonic-crystal sample is mounted on a circular glass slide and inserted in a
rotary stage. The sample is covered with a solution of 520 nm diameter polystyrene
particles (700 L of Thermo Scientific Fluoro-Max R520 particles in suspension,
diluted in 60 mL of deionized water). 600 L of 1% Triton-X was added to the
solution to minimize particle stiction to the sample surface. On a second glass slide, an
32
open-topped microfluidic chamber (4 mm x 4 mm) was fabricated in a 5 m layer of
PDMS using photolithography. The PDMS chamber was pressed on to the sample and
sealed inside the rotary stage.
3.4.2 Optical setup
A Santec tunable laser with a tuning range from 1500 nm to 1620 nm was connected
to an erbium-doped fiber amplifier and a tunable bandpass filter with a width of 1 nm.
An adjustable, neutral-density filter and polarization-control optics were used to control
the power and polarization of the beam. A fiber-to-free-space collimator directed the
beam to the entrance aperture of a 20X objective (NA = 0.26), which focused the beam
onto the top surface of the sample from the back side. A second 20X objective was
used in conjunction with a beam splitter to collect the light from the top surface and to
image the particle motion on a CMOS camera. Prior to each self-assembly experiment,
the transmission was measured in cross-polarization mode. Polarizers before and after
the sample were oriented at 90 degrees from one another, and the wavelength of the
transmission peak was identified. This laser was then tuned to the peak wavelength to
carry out light-assisted self assembly.
3.4.3 Stiffness analysis
After the assembly of a cluster, we recorded videos with a fixed exposure time of 33 ms,
the fastest value available in our experimental set-up. Typical videos were 600 or more
frames in length. Particle motion was analyzed using MATLAB algorithms written
by Blair and Dufresne (http://physics.georgetown.edu/matlab/), which are modified
33
versions of IDL routines written by Crocker and Grier (39).
The blue ellipses in Fig 3.4(a) are obtained by fitting the data in each unit cell.
The direction and relative lengths of the major and minor axis are determined from the
eigenvectors and eigenvalues of the scaled covariance matrix. The ellipse is drawn to
represent a 95% confidence interval: positions outside the ellipse will be observed only
5% of the time by chance if the underlying distribution is Gaussian.
The measured variances were corrected for motion blur due to the finite integration
time of the camera (40). We observed that the variance as a function of position within
the cluster had a 2D Gaussian distribution. This is to be expected due to the spatial
variation in intensity in the incident beam. We confirmed via 3D-FDTD simulations of a
finite-size structure with Gaussian beam excitation that the field intensity above the slab
has a Gaussian envelope. The stiffness values shown in Fig 3.4 are normalized to the
local power at each trapping site, as determined by fitting the experimentally-measured
variances to a 2D spatial Gaussian.
We directly measured the diffusion coefficient by first assembling a cluster, then
blocking the beam and performing a linear regression fit to the subsequent diffusion as
a function of time (39). A value of 0.56m
2
=s was obtained.
3.4.4 Force measurement
After assembling a cluster, we increased the flow speed until all particles were released.
We observed the release of the last, trapped particle at a fluid velocity of 30m=s for
a laser power of 64 mW. In the low Reynolds number regime, the particle velocityv is
34
related to the external force by the equationF = 6rv, where is the viscosity of the
medium, andr is the radius of the particle.
3.4.5 Force and potential calculation
For each particle position within the unit cell, we performed an FDTD simulation to
calculate the electromagnetic fields and obtained the optical force from an integral
of the Maxwell Stress Tensor over a box just large enough to include the particle.
The z dependence of the force inside the hole was obtained from an exponential fit
to the field intensity above the slab. The power threshold for stable trapping was
determined experimentally from the size of the stable cluster (16 unit cells, or 13.8m,
in diameter). The Gaussian distribution of energy in the mode, obtained from a fit of
the variance measurements, had a FWHM of 17m. Given the input power of 64 mW
we find a peak intensity of 240W per unit cell, and a trapping threshold intensity at
the edge of the cluster of 134W per unit cell.
3.5 Conclusions
In summary, we have demonstrated the technique that we term light-assisted, templated
self-assembly. A square array of 180 polystyrene particles was assembled above a
square-lattice photonic crystal. The assembly of the array resulted from the near field
of an extended mode in a photonic crystal slab. The optical force was enhanced at the
resonant frequency of the mode. We characterized the trapping stiffness of our system,
and we showed that the calculation of optical forces successfully predicts the observed
35
equilibrium trapping positions.
Our system can be re-designed to assemble larger clusters. One approach would
involve using a mode with a higher quality factor. The optical force scales with field
intensity, and higher-Q modes should have higher local field intensity in the near
field for a given input power. Furthermore, photons in higher-Q modes spread out
further within the slab due to their longer lifetimes, thereby increasing the trapping
area. Another approach is to re-design the photonic-crystal slab to increase the field
concentration, as shown previously (36). Expanding or re-shaping the incident beam
may also permit the formation of larger clusters.
One major advantage of our approach to self-assembly is the ability to assemble
complex structures with symmetries that are not usually available due to energy
concerns; here we have demonstrated just one. Different photonic-crystal templates
will lead to different particle patterns and more complex unit cells. For a given
template the particle patterns can be reconfigured by changing the wavelength or
polarization of the incident beam. The nature of the particle itself can also be changed;
interesting applications may result from the use of metal nanoparticles, quantum dots
or combinations of different species (35).
Our system also lends itself to compact integration on-chip. The only external
component in our experiment is the laser beam, and by fabricating the device in an
active material, the source itself could also be integrated. Using electrical excitation
of such a photonic-crystal laser would further miniaturize our structure in a way that
would enable a variety of interesting on-chip applications.
36
We also anticipate that our technique will find applications in the fabrication of
metamaterials and other photonic devices. After assembling a 2D array of particles,
polymerization could be used to transfer the array to another substrate. The technique
could thus be used to create 3D arrays layer-by-layer. With suitable design of the
structure it should also be possible to assemble 3D arrays in situ.
Finally, a variety of dynamic, real-time applications could be enabled by LATS. For
instance, LATS could be used to form reconfigurable photonic materials, such as a tun-
able photonic filter. The pattern of particles assembled can be altered in a reversible
way simply by changing the input polarization or wavelength. By combining the opti-
cal potential landscape with microfluidic flow channels, it should also be possible to
sort particles into different channels based upon size or refractive index (18; 19). The
manipulation of biological objects could be simplified in our system by allowing for
batch processing of tasks which typically use a single optical trap to examine a sin-
gle specimen. Furthermore, by trapping biological objects with controlled spacing, our
system allows for control over interactions.
37
Chapter 4
Optical trapping of metal-dielectric
cluster near a microcavity
Optical trapping and manipulation of particles has been demonstrated in a large number
of structures since the discovery of optical tweezers. Resonant structures have been
of great interest for trapping, sorting and cooling of particles with different optical
properties such as atoms (41), quantum dots (42), DNA (43), resonant particles (44; 45),
and dielectric particles (46). Different mechanisms are used to resonantly enhance the
electromagnetic field near a structure, in particular, waveguides (47), ring resonators
(48), nanoantennas (49; 50), plasmonic structures, photonic crystals and photonic
crystals cavities (51; 52) are the most popular choices.
However, in the dipole approximation, the gradient force neglects the field redistri-
bution inside the optical cavity due to the polarization of the trapped particle. In this
chapter, I use a FDTD method to solve Maxwell equations and compute the Maxwell
stress tensor for a gold particle on top of a micro-cavity. Next, I show a secondary trap
formation due to the field redistribution around the trapped gold particle and present an
optically induced metal-dielectric cluster formation.
38
4.1 Trapping of gold nano-particles
As mentioned before, in templated self-assembly methods the geometrical relation
between template and particles mainly defines the final structure. Thus, cluster forma-
tion of particles with the same geometry but different composition cannot be controlled
independently. The use of optical forces for particle trapping allows for differential
response based on composition, since dielectric and metal particles experience different
optical forces.
Strong optical forces can be obtained from the field enhancement near micropho-
tonic structures. In particular, the microcavity mode of a photonic crystal can be used
to create a nanoparticle trap (9; 10). In this section we present a controllable formation
of a metal/dielectric cluster. We propose to use optical forces on particles in a solution
above a photonic-crystal microcavity and the field redistribution due to the presence
of a trapped particle. First, The photonic-crystal microcavity forms an optical trap
for gold nano-particles. Field enhancement near the trapped gold nanoparticle then
forms secondary trapping sites for a pair of dielectric nanoparticles. The process allow
optically driven formation of a well-defined metal/dielectric cluster.
In contrast to previous work on optical binding where clustering comes from
interactions between dipoles (27), in this paper we are considering nondipolar colloidal
particles. Therefore, a trapped gold nano-particle on a photonic crystal microcavity acts
as an off-resonance plasmonic trap for dielectric particles instead of a dipole-like parti-
cle. While previous work has examined light forces on metallic particles (53; 54; 55),
between metallic nano-particles (56), and forces due to plasmonic traps (45; 57), optical
forces in a particle-like off-resonance plasmonic trap were not considered.
39
The structure we consider is shown in Fig 4.1(a). In an experiment, light would
illuminate the photonic crystal microcavity from below. The photonic crystal is formed
of a hexagonal lattice of holes in Si (n = 3.46). The lattice constant is 400 nm, the
hole radius is 140 nm, and the thickness of the silicon slab is 200 nm. The microcavity
is created by removing a hole and displacing its two nearest neighbors along the x
direction outward by 8 nm (9). The photonic crystal rests on a SiO
2
substrate (n =
1.45), and the holes in the lattice and the region above the crystal are filled with water
(n = 1.33).
The microcavity mode was simulated using the finite-difference time-domain
method (FDTD). A y-polarized plane wave illuminates the microcavity from below.
We take the computational cell to be 3.6m x 3.4m x 2m in size. PML boundary
conditions were used. A microcavity resonance value of 226.9 THz and quality factor
Q 60 were obtained. The microcavity strongly enhances the electric field near the
missing hole, and the field intensity is 80 times higher than without the microcavity
in the midplane of the slab. The field intensity in (V=m)
2
is plotted in Fig. 4.1(b)
for a source power of 1:84mW=m
2
. The intensity decays by a factor of 30 at a
height of 130 nm above the slab Fig 4.1(c). At this height, we have observed that
the strongest local intensity is achieved slightly off resonance (f = 229.1 THz) Fig 4.1(d).
A gold nano-particle, far from the plasmon resonance, can be seen as a particle
with a slowly varying, frequency dependent, imaginary dielectric constant, when the
size of the particle is bigger than the mean free path of the electrons in the metal
(58; 59). However, when the particle size is comparable to the electric field variation of
the optical trap, the force cannot be approximated as a sum of the gradient, scattering
and absorption forces, due to the strong field redistribution. Therefore, in order to
40
20
40
60
80
100
120
0.5
1
1.5
2
2.5
3
3.5
4
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
(b)
(c) (d)
y
x
z
(a)
h
r
d
Figure 4.1: (a) Schematic of defect cavity in a hexagonal lattice of holes with diameter d and
thickness h, (b) Electric field distribution at the center of the slab at resonance, (c) Electric field
distribution atz=130 nm above the slab at resonance, (d) Electric field distribution atz = 130 nm
above the slab slightly off resonance.
calculate the optical force on spherical gold particles we used full-vectorial calculations
of Maxwell’s equations and the Maxwell Stress Tensor method (22).
In Fig 4.2, we plot the normalized optical force in the vertical direction for gold
nanoparticles with several different radii. The bottom edge of each particle is 50 nm
above the surface of the slab, and each particle is centered on the microcavity. F
z
is the
z component of the optical force, c is the speed of light in vacuum, and the flux is the
total electromagnetic flux over the computational cell. A negative force indicates attrac-
tion toward the microcavity, whereas a positive force indicates repulsion. We observe
both strong size and frequency dependence of the optical force near the microcavity
resonance. For the largest gold particle shown (r = 160 nm), the force exhibits a Fano-
resonance-like shape and is repulsive (positive force). For the smallest gold particle
41
2.2 2.22 2.24 2.26 2.28 2.3 2.32 2.34 2.36
x 10
14
−0.02
0
0.02
0.04
0.06
0.08
Frequency (Hz)
Fz c/flux
80 nm
100 nm
120 nm
140 nm
160 nm
80 nm dielec
r =
Figure 4.2: Vertical force on gold nanoparticles of different sizes and a dielectric particle withr
= 80 nm and = 1.59.
shown (r = 80nm), the force exhibits a single dip and is attractive (negative force). For
particles of intermediate sizes, the force transforms from a Fano shape to a single dip.
Thus, only gold particles with radii smaller than 160 nm can be trapped. Furthermore,
the force on an 80nm gold particle is an order of magnitude larger than that felt by a
dielectric particle of the same size.
Qualitatively, we can understand the different shapes of the curves in Fig 4.2 using
insight from the dipole approximation. For trapping to occur, the attractive gradient
force must be larger than the sum of the repulsive scattering and absorption forces. For
small radii, the electric field polarizes the particle such that the gradient, scattering, and
absorption contributions to the optical force can be written as (54):
F
grad
=
jj
2
rjEj
2
(4.1)
F
sca
=
n
w
~ sC
sca
c
F
abs
=
n
w
~ sC
abs
c
42
2.2 2.25 2.3 2.35 2.4 2.45
x 10
14
−0.02
−0.015
−0.01
−0.005
Frequency
Fc/flux
2.15 2.2 2.25 2.3 2.35 2.4 2.45
x 10
14
0
1
2
3
4
5
6
x 10
−4
Frequency
Fc/flux
Fsca
Fabs
(a) (b)
Figure 4.3: Gradient (a), and scattering and absorption forces (b) on a gold particle with r =
80nm using the dipole approximation.
From the analytical expressions in Ec (4.1) we plot the forces in Fig 4.3. It can easily
be seen that the gradient force on a gold particle is attractive and has the form of a single
dip near resonance, while the scattering and absorption forces are repulsive and have a
Fano-like shape.
Looking at Fig 4.2, we observe that for small particles (e.g. r = 80 nm), the shape of
the optical force curve resembles that of a gradient force term. For larger particles (e.g.
r = 160 nm), the shape of the curve resembles that of scattering and absorption terms.
We infer that as the particle size decreases, the gradient force dominates the scattering
and absorption effects, leading to trapping.
In Fig 4.4, we plot the frequency-dependent vertical force on a gold particle with r
= 80 nm for several different heights, assuming the particle is centered over the micro-
cavity. The frequency at which the optical force peaks shifts as a function of particle
height, due to the interaction between the gold particle and the microcavity. This effect
is similar to the self-induced trapping effect previously observed for dielectric particles
43
−0.4 −0.2 0 0.2 0.4
−0.4
−0.2
0
0.2
0.4
x (μm)
y (μm)
x (μm)
Y (μm)
−0.4 −0.2 0 0.2 0.4
0.4
0.2
0
-0.2
-0.4
−15
−10
−5
0
5
10
(a) (b)
(c)
2.24 2.26 2.28 2.3 2.32 2.34
x 10
14
−0.025
−0.02
−0.015
−0.01
−0.005
0
F c/flux
50 nm
110 nm
170 nm
230 nm
290 nm
Frequency (Hz)
z=
0
4
8
12
16
x 10
-3
x 10
-3
z
y (μm)
Figure 4.4: (a) Vertical force as a function of frequency for several different vertical positions.
Normalized force in the (b) horizontal and (c) vertical directions as a function of position in the
x-y plane for a gold particle withr = 80nm.
(9). For a frequency of 229.1THz, the force is attractive for all heights shown.
We fix the excitation frequency at 229.1THz and consider the spatial dependence
of the optical force. In Figs 4.4(b) and 4.4(c), we show the normalized horizontal and
vertical forces (Fc=flux) on a gold particle whose bottom edge is located 50nm above
the slab. The color bars indicate the magnitude, and the arrows in Fig. 4.4(b) indicate
the direction. There is a stable trapping position at the center of the microcavity. The
trap is weakest along the y-direction and has a stiffness k
y
= 0.2 pNnm
1
W
1
. We
estimate a minimum trapping power of 1mW by requiring the stability number to be at
44
least one (60).
4.2 Cluster formation
A gold nanoparticle trapped at the center of the microcavity will redistribute the elec-
tromagnetic field, forming new optical trapping locations. Figs 4.5(a) and 4.5(b) show
the electric field intensity 130nm above the slab at the fixed frequency of 229.1THz for
the same source power as in Fig 4.1. The maximum value of intensity is 7 times higher
than without a gold particle. The high intensity regions can attract dielectric particles.
Figs 4.5(c) and 4.5(d) show the horizontal and vertical optical forces on a dielectric
particle with n =1.59 and r = 80 nm in the presence of the trapped gold nanoparticle,
indicated by the solid, yellow circle. The dielectric particle is stably trapped in the
position indicated by the dashed circles. In the Figures, the arrows next to the graphs
represent values ofFc=flux = 0.002. We estimate a minimum trapping power of 7 mW
with trapping stiffnesses ofk
x
= 0.08pNnm
1
W
1
,k
y
= 0.25pNnm
1
W
1
andk
z
=
0.09pNnm
1
W
1
.
By symmetry, a second possible trapping position exists on the opposite side of the
gold particle. Fig. 4.5(e) shows the optical force on a dielectric particle with r = 80 nm
after both a gold and dielectric particle are trapped in the locations shown by the solid
yellow and blue circles, respectively. The arrow next to the graph represents a value
ofFc=flux = 0.002. The dielectric particle is trapped in the position indicated by the
dashed line. The results of Fig 4.5 thus indicate that dielectric-metal-dielectric clusters
will be formed, consisting of three particles with composition and orientation as shown.
45
y (µm)
x (µm)
y (µm)
−0.2 0 0.2
0.2
0
-0.2
0 0.1 0.2 0.3
0
0.1
0.2
0.3
0.4
x (µm)
y (µm)
0 0.10.2 0.3
−0.4
−0.2
0
0.2
x (µm)
(a)
(c) (e)
|E|
2
5
10
15
20
25
30
35
0.1
0.2
0.3
0.4
0.5
0
0 0.1 0.2 0.3
y (µm)
z (µm)
y
|E|
2
(b)
(d)
10
20
30
40
50
60
70
z
−0.4 0.4
0.4
−0.4
(V/m)
2
(V/m)
2
Figure 4.5: Electric field intensity in the midplane (a) and on a vertical cut (b) of a trapped gold
particle with r = 80 nm. Horizontal (c) and vertical (d) optical forces as a function of position
for a dielectric particle withr = 80 nm in the presence of a trapped gold particle (yellow circle).
(e) Horizontal optical force as a function of position for a dielectric particle near trapped gold
(yellow) and dielectric (blue) particles. The arrows next to the graph representFc=flux = 0.002.
We note that the specific force values and the required power for trapping calculated
here assume normally-incident light distributed over an area of approximately 12m
2
.
In an experiment, the coupling efficiency between the incident light and the cavity can
likely be increased by adjusting the spot size of the normally-incident beam and/or
using a side-coupled waveguide to inject light into the microcavity mode, increasing
the optical forces and reducing the power required for trapping.
In an experiment, surface treatment of the gold nanoparticles, dielectric nanopar-
ticles, and photonic crystal may be required to prevent aggregation, adhesion, and/or
electrostatic effects. Previous experimental work in the literature has developed
46
techniques for preventing aggregation of gold (55) and dielectric (61) nanoparticles, as
well as adhesion between nanoparticles and microphotonic devices (51). Electrostatic
interactions in the system can be minimized using a phosphate buffer solution with a
regulated pH of 7.0 (47).
We estimate that the rms variation of the gold particle due to Brownian motion
is approximately 20nm for the trap stiffness and power given in the paper. We have
calculated the field profile for displacements of the gold particle on this order. We
observe that the variation in field intensity is less than 10%. While the symmetry of
the intensity distribution is broken slightly due to particle motion, both the qualitative
shape and magnitude is largely preserved. We therefore do not expect Brownian motion
to significantly affect the conditions for optical trapping.
We have also checked that the force exerted on the gold particle by the dielectric
particle is much smaller than the force exerted on the gold particle by the cavity.
The Figure below shows the optical forces on the gold (yellow) and dielectric (blue)
particles in the configuration at which the forces are strongest. The arrow next to the
graph indicates Fc=flux = 0.002. We calculate that the force on the gold particle
will displace the particle by less than 12nm from the center of the photonic-crystal
microcavity. As seen in our answer to point 4 above, displacements on this order will
not significantly change the field profile.
We finally, simulated the heating-induced fluid convection in COMSOL and used
Stokes law to estimate the drag force exerted on particles by flow in a 10m thick chan-
nel. The heating source in the simulation is calculated using the spatially-dependent
electric fields obtained from FDTD simulations. For an incident power of 10 mW, the
47
0
0.1
0.2
0 0.1 0.2 0.3
y (µm)
z (µm)
-0.2
-0.1
Figure 4.6: Force on gold particle due to dielectric particle.
maximum temperature rise is 7 K, and the maximum flow speed is 0.1 nm/s. The
normalized drag force (F
d
c / flux) on a 80 nm radius gold particle is estimated to be less
than 4 10
9
, which is negligible compared to the optical trapping force.
In summary, we have proposed a method for using optical forces to induce the for-
mation of metallo-dielectric clusters with controlled morphology. The method relies on
field-enhancement effects near gold particles trapped above a photonic-crystal micro-
cavity. If the self-assembly process is performed in a photopolymerizable solution, a
secondary laser spot could be used to locally solidify the region around the cluster.
Moreover, integration with a microfluidic channel could allow for controlled, sequen-
tial delivery of individual gold and metal particles, along with controlled collection of
metallo-dielectric clusters.
48
Chapter 5
Measurement of the bending modulus
of a GUV using a dual-beam optical
trap
The dual beam optical trap (DBOT) has seen growing interest in the field of optical
manipulation of biological cells, due to its ease of implementation and non-invasive
nature. Cells (70) and giant unilamillar vesicles (GUVs)(71), have been successfully
trapped and stretched in a DBOT platform, which provide an excellent tool to investi-
gate elastic properties. The DBOT provides a method for non-invasive application of
time-dependent forces in a device suitable to rapid, high-throughput measurements.
GUVs typically contain the same solution on the interior and exterior of the spheri-
cal, lipid bilayer, thus the refractive index difference required for trapping is not present.
Here, we used GUVs with different, osmotically-balanced solutions on the interior and
exterior of the bilayer. We construct a DBOT based on optical fibers integrated with a
capillary flow channel and demonstrate trapping and stretching of a GUV . We charac-
terize the response of the GUV to a step increase in stress. We further conduct exper-
iments in which the area strain of the GUV is measured as a function of applied stress
and develop a method to extract the bending modulus of the membrane from the data.
This method incorporates three-dimensional ray-tracing methods to calculate the applied
49
Figure 5.1: Dual-beam optical trap
stress in the DBOT. The value we obtain for the bending modulus agrees well with liter-
ature values. Unlike previous results in the literature (62; 63), we use a laser-wavelength
selected to minimize heating effects (64; 65) avoid the use of polyethylene glycol, which
is known to destabilize membranes (66), and model the mechanical properties of the
GUV as a lipid bilayer according to the accepted approach developed by Helfrich (67).
Our results demonstrate the potential of the DBOT for rapid, flow-through measure-
ments of membrane response to changing physiochemical environments, opening a path
for a wide range of biological experiments.
5.1 Stress profile
Counter-propagating beams emitted from two optical fibers form an optical trap due to
their Gaussian intensity profile, which results in a gradient force, and radiation pressure
(Fig 5.1)(72). Though the net force acting on the object in the stable trapping position
is zero, it experiences a non-zero surface stress (73). This surface stress deforms the
surface by an amount depending upon intrinsic elasticity. Thus, elastic objects like
cells and GUVs assume the shape of a prolate spheroid (67) when trapped, with an
eccentricity dependent upon the trapping power and elasticity. In order to quantify the
shape deformation of GUVs and cells, it is of utmost importance to calculate the surface
stress profile on spheroidal objects with a refractive index comparable to the cells and
50
GUVs. An exact calculation of optical stress profile on the surface of spheroidal objects
is an absolute requirement to quantify the resultant shape deformation of the object
based on its intrinsic elasticity.
5.1.1 Surface stress using ray optics
We consider a spheroidal dielectric object illuminated by a single Gaussian beam. The
spheroidal surface is resolved into grid points with corresponding azimuthal angle
varying from 0 to=2 and polar angle from 0 to 2. In the case of a closed object,
a transmitted beam experiences multiple internal reflections. If we considerm internal
reflections, the total optical surface stress is (29)
~
m
=
2P
T
n
1
T
0
scos
w
2
cs
m
exp
2r
2
w
2
m1
Y
j=1
R
j
n
2
n
1
^
i
m
(
n
2
n
1
R
m
^ r
m
+T
m
^
t
m
)
(5.1)
The incident vector takes in to account the divergence of the beam, meanwhile the
subsequent reflected and transmitted vectors can be computed using Snell’s law. The
stress due to the first reflection from the incident beam is computed and subsequent
surface stress arising from multiple reflection of the transmitted beam are also recorded.
Given the symmetries of the system and provided non-polarized light, only the stress
calculations in the equatorial plane are needed, any other cut of the spheroid would be a
rotation of the equatorial plane.
Fig 5.2, shows the optical stress from the incident beam (
1
) and the first four
internal reflections (
2
5
) for a sheroid of volume equal to 523.6m
3
ande = 0.4.
The angular range [0=2] corresponds to the illuminated part of the object while
[=2] is the shadowed region. For
1
there is a maximum stress in the shadowed
51
Figure 5.2: (a) Optical stress for multiple reflections on a spheroid with volume equal to a 5m
sphere and e = 0.4 (n
1
= 1.335,n
2
= 1.38,w = 11m). RI of Red Blood Cells is close to the
value chosen in this case. (b) Total optical surface stress in the equatorial plane for single beam,
and (c) total optical surface stress for dual beam.
region since a larger change in momentum is observed when a ray goes from a higher to
lower refractive index. Next, the surface stress from each bounce is added vectorialy to
obtain the total optical surface stress profile in the equatorial plane from a single beam
(Fig 5.2(b)). As we can see, each reflection introduces a spike in the total surface stress
near the region of illumination.
In an experiment, a low coherence source is useful for dual beam optical trapping
since it eliminates interference effects. The two trapping beams are incoherent with
respect to each other, since the path length difference between the two fibers are much
longer than the coherence length of the light. Thus, the stress contribution from the
dual beam trap is obtained via vectorial addition of the stress due to the two opposing
Gaussian beams (Fig 5.2(c))). As expected, the optical stress is symmetric and the
spikes represent interferences from each beam, but not between the beams. In order
to corroborate this result we performed the same calculation using the Maxwell Stress
tensor method.
52
5.1.2 Surface stress using the Maxwell Stress tensor
The Maxwell Stress tensor method (MST) express the optical force acting on an object
via the change in electromagnetic momentum across the surface of a close object. If we
want to consider the optical stress in the surface, we can consider two surfaces, one just
above the object interface and one just bellow (28). The total force is then
F
sur
=
I
Sout
T
out
^ nda
I
S
in
T
in
^ nda (5.2)
Expanding the stress tensor matrix in terms of the associated electric and magnetic field
components and applying continuity of fields across the surface, the time average surface
stress takes the following form
h
sur
i =
0
4
(n
2
2
n
2
1
)
jD
n
j
2
n
2
2
n
2
1
2
0
+jE
t
j
2
^ n (5.3)
D
n
andE
t
are the normal component of electric flux density and tangential component
of the electric field respectively and
0
is the permittivity of vacuum. In agreement with
the RO model, the surface stress at any point P is directed along the surface normal ^ n.
In our simulations, we performed a complete full wave solution of the Maxwell’s
equations using FDTD analysis to compute the electric fields. We simulated a circularly
polarized light to emulate time-averaged unpolarized light. We first simulated the single
beam case and computed the corresponding stress components. Using mirror symmetry
we flipped the electric field to simulate a counter propagating beam and calculated the
stress components. A scalar addition of these two stress components (since both are
directed along the surface normal) yields the total contribution from two incoherent
53
Figure 5.3: Circularly polarized electric field intensity and corresponding stress profile in the
equatorial plane for (a) Single Beam (b) Dual beam.
counter propagating beams.
Fig 5.3(a) shows the stress profile and the corresponding field intensity profile on
the equatorial plane of a spheroid object with refractive indexn
2
= 1:38 arising from
a single Gaussian beam propagating in a medium with refractive index n
1
= 1:335
(initial beam waistw
0
= 2:4m and propagation distance of 110m ). The dual beam
stress profile and the electric field intensity, obtained by incoherent sum of electric field
intensity arising from each beam as shown in Fig 5.3(b).
5.1.3 Comparison between RO and MST
We carried out a comparison in order to verify the stress profile obtained using RO
and MST. We varied the eccentricity, size of the spheroidal object and refractive index
contrast between the surrounding medium and the object. In each of the following
cases, wavelength is set at 808 nm and the initial beam waist is w
0
= 2.4 m and the
54
Figure 5.4: Comparison of stress profile arising from single beam in the equatorial plane
obtained from ray optics and MST for varying eccentricities (b) single beam electric field inten-
sity profile corresponding toe = 0:0 ande = 0:9 (c) Dual beam.
medium is assumed to be water which possesses a refractive index n = 1.335. The
choice of beam waist and wavelength agree with our experimental set up. In the cases of
refractive index and eccentricity variation, the trap center is at distance of 110m from
the beam waist, which makes the ratio of the beam waist, at the trap site, and radius
of an equal volume sphere close to 1. In this way, we establish similar illumination
conditions for all the cases studied. In the first two cases, the volume of the object is
kept equal to that of a sphere of radius 5m. In the case of the size variation, the beam
width at the trap center is adjusted by varying the location of the trap site to give a ratio
of beam waist to object radius equal to 1.
Fig 5.4 shows a comparison between RO and MST with spheroids of different
eccentricities and a refractive index contrast of 0.0105 (similar to what exits in case of
GUVs with sucrose and glucose solution inside and outside). Similar to observation
made by (28) for spheres, in the single beam case, we also notice that RO and MST
analysis matches exactly in the illuminated surface ( = =2 to ) for spheroids
with low eccentricity due to the lack of interference fringes. But in the shadowed
55
region ( = to 2) the presence of surface waves (74) and fold caustics (75) (region
on the shadowed surface where rays from two different angle of incidence strikes)
causes wave interference, resulting in the deviation of RO from MST. The first peak
in the RO stress profile arises from the first order fold caustics (i.e. on the first
reflection from the shadowed region) and as is evident from the figure it broadens
with eccentricity due to an enhanced focusing effect. For higher eccentric objects the
first order fold caustics exits over a larger region, therefore the interference is more
prominent (extreme eccentric case e = 0:9). Fig 5.4(b) illustrates the single beam
field profile for a sphere (e = 0) and highly eccentric prolate spheroid (e = 0:9)
illustrating the fact of enhanced focusing and interference effects. The dual beam
case (Fig 5.4(c)) is just an extension of the single beam stress profile and as expected
the interference effects now become predominant throughout the surface of the spheroid.
The correlation between RO and MST analysis with refractive index variation and
size variation is shown in Fig 5.5. The values of refractive index contrast chosen here
agree with those which exist for GUVs (n = 0:0105), RBCs (n = 0:045) and
polystyrene (n = 0:245). As refractive index contrast is increased, the angle of
refraction increases and rays undergo larger deviation from the incidence direction;
consequently the two ray inference zone (first order fold caustics) broadens. This is
evident in Fig 5.5(a) where the first peak in the RO analysis becomes wider as we
increase the refractive index contrast. This trend leads to enhanced interference effects
in the shadowed region, thus causing RO to further deviate from FDTD analysis.
However, in case of size variation, the interference effects increase marginally with
size, and thus the variation between RO from MST remains constant.
56
Figure 5.5: Comparison of MST and RO for spheroids with e = 0.4 (a) refractive index variation
and (b) size variation.
However, in all the cases studied, it is quite apparent that the area under the stress
curves for both the RO and MST remains approximately equivalent. Therefore it is
justified in using the RO approximation while calculating the total force on the object.
However, for the stress profile calculation, the interference effects on the spheroid sur-
face causes the RO and MST to deviate from one another substantially. Resultantly,
for cases where the point of interest is surface stress (as in the case for living cells and
GUVs trapped in DBOT), MST provides a more accurate result even if the spheroid size
is within the RO limit.
5.2 Dual-beam optical trap setup
A schematic of a dual-beam optical trap is shown in Fig 5.6. Light is emitted from two
optical fibers (OF) creating a counter-propagating Gaussian beam trap. The optical
fiber used is HI780 (Corning, USA) with 2:4m mode-field radius. The beam size
at the center of the flow channel (w
0
= 11:5m) was calculated from the size of the
optical fiber mode using the ABCD matrix method. Radiation pressure pulls the GUV
57
Figure 5.6: (a) Schematic of the stretching of a GUV using DBOT, (b) Optical setup that incor-
porates a silicon chip for fiber-to-capillary alignment and microfluidic adapters to couple the
flow channel.
into the center of the two beams, creating a stable trap (68; 69). In this position, the
surface stress (70; 72) can deform the GUV along the beam axis, as shown.
The micro-fluidics channel was created with a square capillary of 100 m inner
diameter (Vitrocom, USA) and a 100 m wall thickness. The optical fibers were
aligned perpendicularly to the micro-fluidics channel at a distance of 300m. The axis
of each optical fiber was aligned so that the trapping region was at the center of the
flow channel; thus trapped GUVs were not in physical contact with capillary walls.
The capillary-to-fiber union was coated with index matching liquid, and the capillary is
coupled to a peristaltic pump (Instech, USA) using microfluidic adapters (Upchurch,
USA).
A wavelength of 808 nm was used and a power output of 250 mW was measured
at each fiber. The wavelength was selected to minimize water absorption (76) in the
infrared and thereby avoid heating effects (64; 65). For a wavelength of 808 nm,
we have simulated the heating of our flow channel in COMSOL and found that the
58
temperature increases less than 1 K for 250 mW of laser power.
A microscope was used to observe the experiment and a video was recorded at 61 fps
by a GiGe camera with CCD image sensor (Basler AG, Germany). Both the laser and
camera were controlled through a customized Labview program (National Instruments).
5.3 GUV fabrication
To facilitate trapping, it is necessary to create a refractive index difference between the
interior and exterior of the GUV . To achieve this, we fabricated GUVs with an internal
sucrose solution and transferred them to an osmotically-balanced glucose solution. The
density difference between sucrose and glucose results in a refractive index difference.
GUVs with sucrose inside were fabricated by electroformation from pure
dioleoylphosphatidylcholine (DOPC) using a modified vertion of (77). The synthetic
lipid, 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC), 1,2-dipalmitoyl-sn-
glycero-3-phosphocholine (DPPC), and cholesterol were purchased from Avanti Polar
Lipids (Alabaster, AL). The lipids were dissolved in chloroform and deposited onto an
indium-tin oxide (ITO) coated glass slide from Delta Technologies (Loveland, CO).
The electroformation chamber was formed with two ITO slides, with the conducting
sides facing inwards and separated by a 2.5 mm thick silicone spacer. A 500 mM
sucrose solution in 4 mM HEPES titrated to pH 7.0 with sodium hydroxide was added
to the overnight vacuum-dried lipid film on the slides giving a final lipid concentration
of 0.25 mg/mL. A 2.65 V AC electric field, generated at 100 Hz by a function generator,
was applied to the chamber. The electroformation temperatures were chosen so that
the lipids remain in the liquid phase. Transfer to a glucose solution was achieved by
59
performing three two-fold serial dilutions on the GUV suspension using an identical
buffer solution, but with the sucrose swapped for an equimolar quantity of glucose.
Refractive indices were measured for the pure sucrose and glucose solutions using a
refractometer (PAL-RI Refractometer - Atago, Bellevue, WA). Values were obtained for
the pure glucose solution (RI = 1.3455 0.0003) and sucrose solution (GUV interior
RI = 1.3575 0.0003), and the dilution used in the experiment yielded a final refractive
index difference of 0.0105 0.0003, slightly less than the difference between the two
pure solutions. The prepared mixture was then left to sit for ten minutes in order to
allow GUVs to sediment to the bottom of the tube. Following this time, half of the total
liquid volume was carefully removed at the surface. This suspension was then directly
pumped into our microfluidic device.
5.4 Trapping and stretching of GUVs
GUVs were captured using minimal power (total power of 100 mW; 50 mW from each
fiber) while ensuring the flow was stopped and the GUV was resting at the bottom of
the channel. As a GUV is pulled up toward the optical axis, its initial circular shape
is drawn into a slightly prolate elliptical shape due to the stress profile created by the
optical stress. Axial deformation of a GUV made of POPC lipid is shown in Fig 5.7(a).
The total power was then increased to 500 mW, and the deformed shape at maximum
power is shown in Fig 5.7(b). The major axis is increased fromd = 11.53 0.05m to
d = 11.94 0.05m along the beam axis, while the minor axis decreased from 10.29
0.05m to 10.05 0.05m. Fig 5.7(c) shows the contours at both minimum and
maximum power. Stretching of the GUV along the beam axis can be clearly observed.
60
Figure 5.7: (a) POPC-GUV optically trap at low power, (b) POPC-GUV optically trap at high
power, (c) A plot of the contours fitted to both stretching powers (blue = low power / low tension,
green = high power / high tension). The scale bar is 10m.
5.4.1 Calibration of optical forces
The optical power calibration of our DBOT setup was done using shooting experiments.
We first trapped a polystyrene bead (n = 1:615) in water with equal power from both
beams and pushed it towards the one beam by increasing the power of the opposite
beam. We then blocked on the opposite beam by reducing the current supplied to
zero. The polystyrene bead accelerated towards the opposite beam and the distance
vs. time information were recorded. The velocity as a function of distance was then
calculated and used in drag force equation, F
drag
= 6v , where is the viscosity
of the water (0.0009 Pas), is the radius of polystyrene bead (4.97 m) and v is the
velocity of the particle. The following correction factor was multiplied to the drag force,
c =
1
1
9
16
b
+
1
8
b
3
45
256
b
4
1
16
b
5
2 +b
3 +b
(5.4)
whereb is the distance to the closest wall and
1
is the slip coefficient.
61
Figure 5.8: The drag forces in no-slip and perfect-slip cases are plotted against the calculated
optical forces (solid lines) from one beam for two power levels, 50mW and 100mW. 50 m
corresponds to the center of the channel.
The first term is associated with the proximity of the walls while the second term
takes into account the slip-flow boundary conditions on the surface of the particle [36].
The experiment was repeated with two different optical power levels, 50 mW and 100
mW. The corrected drag forces, assuming no-slip conditions (!1 ), are the upper
boundary of the force exerted on the bead. In the case of perfect slip ( ! 0), the
corrected drag force is the minimum force expected on the bead and therefore our lower
boundary. In Fig 5.8 these boundaries were plotted against the theoretical optical forces.
Our theoretical prediction of optical force is consistent within these boundaries while the
upper boundary matches the theoretical optical force within15% at the center of the
channel.
5.4.2 Temporal response to applied stress
We measured the response of a GUV to a step increase in applied stress. The total laser
power was increased from 100 mW to 500 mW, as shown in Fig 5.9(a) (blue line; right
axis).
62
Figure 5.9: (a) The optical power (blue line; right axis) is suddenly increased from 100 mW to
500 mW. The major axis strain is shown by the red dots (left axis). (b) Micrograph of deforming
GUV . The scale bar is 10m.
The power was held at its maximum value for 5 seconds and then decreased to its
initial value. The major axis strain is shown on the left axis (red dots). The major axis
strain was calculated by assuming that the shape of the GUV at maximum power is a
prolate spheroid. We take the diameter of a sphere with the same volume as the zero-
power value of the major axis. The major axis strain is the percent change in major axis
compared to the zero-power value. From Fig 5.9(a), it can be seen that the major axis
strain increases nearly instantaneously with the step increase in power. The initial strain
of 8.2 0.4% increases by 4.1 0.25%. Based on our frame rate of 61 fps, we are able
to capture 2-3 data points in the transition region between power levels.
5.5 Measurement of lipid bilayer bending modulus
In our experiment, we gradually ramped the laser power from 100 mW to 500 mW as
shown in Fig 5.10(a). The power was increased in 11 steps, with a holding time of 1 s
at each step. As the power increased, we observed axial deformation of the GUV and a
clear dampening of membrane fluctuations.
63
Figure 5.10: Measurement of the bending modulus of a pure POPC GUV . Time axis is common
to plots a-c, dotted guide lines show time points at which laser power is increased. (a) Laser
power as a function of time. (b) 2D contour plot showing the radius as a function of angle in the
image plane as a function of time. (c) Percentage area strain as a function of time. (d) Average
stress on the GUV as a function of eccentricity and base radius. (e) Average percentage area
strain for each laser power plotted versus the scaled lateral tension. is the fitted value of the
bending modulus.
We calculate the area strain of a GUV as follows. The grayscale image for each
frame was processed using our in-house Matlab code to trace the edge of the GUV . The
nominal edge is located at the grayscale center-of-mass in an 8-pixel region surrounding
the darkest pixel at each point along the GUV circumference. The edge coordinates
formed a closed contour for each frame. The nominal GUV center was located at the
center of this contour.
The resulting contour data set for each image frame consists of the location of the
contour center and a contour radius for each of 360 azimuthal angular points. Contour
64
radius can be expressed as a function of azimuthal angle (Fig 5.10(b)). We then pro-
ceeded to expand the contour data in Legendre polynomials. This strategy fits the shape
of the contour as an expansion of the equatorial plane of a base sphere with radius R.
This base sphere is the assumed relaxed shape of the untrapped GUV . To find R, the
GUV at minimum trapping power is assumed to be a prolate sphereoid. The volume of
this GUV is calculated asV
0
= 4=3a
2
b, where a is the semi-minor axis and b is the
semi-major axis. R is then the radius of a sphere with the same volume as this prolate
spheroid. Initial area is calculated as the surface area of this sphere,A
0
= 4R
2
. We
restrict our Legendre expansion to the second mode coefficient (u
2
), since this can be
used to represent the shape change of the contour from circle to increasingly eccentric
ellipse (78; 79). We reconstructed the contour using the equation,
r() =R +u
2
P
2
(cos) (5.5)
where
P
2
(cos) =
1
2
(3cos
2
1) (5.6)
andu
2
, using the ortho-normality of the Legendre polynomials, is given by the integral:
u
2
=
5
2
Z
P
2
(cos)[r(cos)R]d (5.7)
After reconstruction the contour, the top half the contour [0-] was numerically inte-
grated to generate a solid of rotation about its major axis corresponding to the surface
area of the GUV
A = 2
Z
0
rsin
s
r
2
+
(dr)
2
(d)
2
d (5.8)
65
Area strain is calculated using (AA
0
)=A
0
(Fig 5.10(c)). As expected, we observe an
increase in area strain with increasing laser power.
The bending modulus
B
of the GUV membrane can be obtained by calculating area
strain as a function of lateral tension. In the low-stress regime (80)
AA
0
A
0
=
kT
8
B
ln
h
0
(5.9)
where k is Boltzmanns constant, T is temperature,
h
is the lateral tension on the
membrane, and
0
is the intrinsic lateral tension.
In order to determine the lateral tension on the membrane at each power level
(81), it is necessary to calculate the surface stress on the GUV . Ray optics approaches
have previously been used to calculate the force on spherical (82) and spheroidal (83)
objects. We assume a spheroidal shape for the GUV , and calculate the total force on the
front and back surfaces. For each power level, we calculated the force on a spheroid
with major and minor axes equal to the average values over all image frames. We
included the effect of multiple reflections, up to 5 bounces. For each incident ray and
each bounce, we determine whether the bounce occurs on the front or back surface and
store the vector force. For each surface, we then add the force contributions vectorially
to determine the total force on the surface. The stress is calculated by dividing the total
force by the surface area.
The calculated average stress is shown in Fig 5.10(d). We present the results as
a function of eccentricity (e) and base radius (R). The optical power from each beam
was taken to be 250 mW and the refractive index difference (n) to be 0.0105. While
66
the average stress decreases slightly with eccentricity, it substantially decreases with
increasing base radius (R), in the 8-14m range.
The average stress is then translated to a lateral tension
h
(81; 78). After each
increase in optical power, the GUV reaches an equilibrium shape with a homogeneous
surface tension in the membrane. The pressure difference created inside (p) is equal
on the equator and the poles of the ellipsoid. If we assume that there is no applied stress
at the poles, given the ray optics approximation, we can use the Young-Laplace equation
to write
p = (c
1
+c
2
)
equ
h
T
equ
= (c
1
+c
2
)
pol
h
(5.10)
wherec
1
andc
2
are the principal curvature values at both the equator and the poles, and
T
equ
is the magnitude of stress exerted on the surface at the equator. We calculated the
curvatures from the extracted contours. The optical stressT
equ
was calculated using ray
optics. As mentioned before, the surface stress is zero at the poles and the symmetrical
discontinuities or spikes in the stress profile around the poles are a result of multiple
internal reflections. In our calculation, we included the effects of multiple bounces
(up to 5) of each incident ray inside the GUV . For the refractive indices of the inte-
rior and exterior of the GUV used in our experiment, the reflection is small (R=0.002%).
We calculate an initial tension (
0
) of 5.76 0.25 10
5
mN/m and plot area
strain as a function of the log of scaled lateral tension in Fig 5.10(e). The error bars on
both axes are equal to the standard deviation of the corresponding quantity, taken over
all images recorded at a fixed laser power. The slope is proportional to the bending
modulus, which is found to be 7.95 0.45 kT. The log-linear relationship indicates that
67
we are in the low stress regime and that area expansion of the membrane comes from
damping bending fluctuations, as opposed to direct stretching (i.e. area dilation) of the
membrane, as observed at higher stresses (84).
We note that the experimental data shown in Figs 5.10(a), 5.10(b), 5.10(c), and
5.10(e) is obtained from a single GUV . Moreover, we note that since the stress is not
uniform over the GUV surface, a more sophisticated model of vesicle deformation
would include the effects of stress non-uniformity on final shape. This is an interesting
area for further research.
5.6 POPC, POPC-chol and DPPC bending modulus
Under experimental conditions (at room temperature), both pure POPC and POPC with
20 mol% cholesterol are in the liquid phase whereas DPPC with 20 mol% cholesterol is
in the gel phase (85).
The bending modulus scales with membrane thickness squared multiplied by the
area compressibility modulus (h
2
K
a
)(86). Gel-phase bilayers are thicker (87), and
the bending modulus of a lipid bilayer in the gel phase is consequentially higher, than
that of the same membrane in the liquid phase (88; 89).
We first demonstrated that optical stretching is able to measure the difference in
bending modulus of lipid membranes with different phase states. Fig 5.11 presents the
dependence of relative area strain on applied tension for two GUV compositions: POPC
and DPPC with 20 mol% cholesterol. The initial tensions are at 7.53 0.73 10
5
68
Figure 5.11: Apparent area strain versus ln(tension) plots for POPC and DPPC GUVs with
20% cholesterol. The POPC lipid membrane is more flexible than DPPC lipid membrane. The
error bars represent one standard deviation of area strain and lateral tension. The errors in bend-
ing modulus values are calculated from the standard errors of the slope using linear regression
analysis.
mN/m and 1.61 0.19 10
5
mN/m, and the calculated bending modulus values are
9.16 0.72 kT and 24.89 2.47 kT for the POPC and DPPC membrane respectively.
To obtain population statistics, we measured the bending moduli of GUVs fabricated
from pure POPC (n = 56), POPC with 20 mol% cholesterol (n = 40), and DPPC with 20
mol% cholesterol (n = 39) respectively. We made fit normal distributions to the data to
obtain the following bending modulus means and standard deviations: 8.13 2.06 kT
for POPC vesicles, 8.50 1.83 kT for POPC with 20%Chol, and 27.24 7.69 kT for
DPPC with 20%Chol. Histograms of the population bending modulus data are given in
Fig 5.12.
We confirmed the normality of the populations collected using the Kolmogorov-
Smirnov test. We then performed the students t-test to compare the two POPC
69
Figure 5.12: Histograms of the bending modulus distributions of and DPPC-Chol, POPC-Chol,
and pure POPC lipid GUVs.
populations (POPC and POPC-20%Chol). The populations are not significantly differ-
ent (p > 0.05). This analysis confirmed the indistinguishability of POPC populations
with and without cholesterol, in contrast to other published results (90; 91). Cholesterol
is believed to increase by ordering the acyl chains. Marsh did an extensive comparison
of several methods and concluded that cholesterol increases mean bending moduli (92).
However recent reports have argued that the contribution of cholesterol to bending
modulus is not universal and unsaturated lipids (particularly 1,2-dioleoyl-sn-glycero-
3-phosphocoline, DOPC) are not affected by the addition of cholesterol (78; 93). Our
results agree with these recent reports. We were unable to produce high yields of pure
DPPC GUVs.
There are several factors that may influence the width of the distributions shown
in Fig 5.12. GUVs made from POPC may be photooxidized when exposed to bright
light during microscopy (94). However, in our optical stretching experiment, the
70
experimental duration for a single GUV was around 30s, and no fluorescent dye
was used, so we do not expect significant photooxidation to occur. The width of the
DPPC/cholesterol distribution may also be a result of compositional heterogeneity due
to non-uniform cholesterol incorporation during lipid film drying, as has been shown
previously (95).
We analyzed possible sources of systemic error in this experiment. One potential
error source comes from how we calculated vesicle surface area, by fitting the thermal
undulations. For the data presented in Fig 5.12, only the second Legendre coefficient is
used to calculate the area change. When we include the fourth mode, as suggested by
Gracia and coworkers (78), the bending modulus values tend to decrease by approxi-
mately 5%, within the reported error. Including the fourth mode does not decrease the
width of the population histograms, indicating that lack of precision in shape fitting is
not a major contributor to the width of the observed distributions.
Second, GUVs are trapped at a position slightly below the axis of the optical trap
due to gravity. For example, consider a spheroidal GUV with 10m equivalent sphere
volume and eccentricity of 0.4. The net gravitational force on the GUV is proportional
to the density difference between the solutions on the inside and outside and equals
1.3 pN. The trapping position can be calculated by balancing the gravitational force
with the optical gradient force. At the minimum trapping power of 100 mW, the GUV
is trapped2m below the optical axis. At a power of 500 mW, the offset decreases to
0.4m. The microscope is initially focused on the equatorial plane of the GUV . As
the GUV is pulled further upward into the trap, the segment of the vesicle in the focal
plane will shift.
71
To test whether this effect could have a significant impact on the results, we
calculated the volume of the GUV at every frame by assuming the captured contour
corresponded to the equatorial plane of a spheroidal vesicle and measuring the length of
the major and minor axes. If the upward shift of the GUV were significant, it would be
expected to result in consistently smaller measured GUV volumes as power increased.
This is because the true volume of the GUV (expected to remain constant through this
experiment) can be calculated from the true equatorial plane contour measured when the
microscope is initially focused at minimum trapping power. Contours corresponding
to other GUV cross-sections would result in a smaller calculated volume. A significant
shift of the vesicle along the optical axis would therefore result in an apparent (and
purely artifactual) decrease in volume. In fact, in no experiment did the average
calculated volume change exceed 1%, and there was no relationship between volume
and power.
Finally, we performed a set of experiments to test whether the GUV stretching
system exhibits hysteresis. The optical power was ramped up and down between
minimum and maximum trapping power. The bending modulus was obtained both as
the vesicle dilated and contracted. Fig 5.13 shows two examples of time-dependent
area strain and extracted area strain vs. surface tension, for pure POPC (Fig 5.13(a))
and DPPC-20%Chol (Fig 5.13(b)). For this POPC vesicle, the bending modulus values
obtained from the two slopes gave 6.52 0.32 kT for ramp-up and 6.58 0.35 kT
for ramp-down. For this DPPC-20%Chol vesicle, we obtained 27.47 3.29 kT for
ramp-up and 28.04 2.25 kT for ramp-up and ramp-down. In all cases tested, the
change in measured bending modulus for the two different deformation directions is
within error.
72
Figure 5.13: Laser power ramp-up and ramp-down experiments on two different GUVs, (a)
POPC and (b) DPPC-20%Chol.
In conclusion, we have used optical stretching in a microfluidic DBOT system to
investigate the elastic bending modulus of populations of POPC, POPC with 20% Chol,
and DPPC with 20% Chol GUVs. The difference in bending modulus between liquid
and gel-phase GUVs is unambiguous; gel-phase GUVs are significantly stiffer. The
bending elastic modulus of the membrane composed of the monounsaturated phospho-
lipid (POPC) is insensitive to Chol at concentrations up to 20%. This technique is a
simple and powerful approach to the measurement of membrane bending properties. It
has the potential to be widely deployed in efforts to understand relationships between
membrane composition and membrane mechanics.
73
Chapter 6
Summary
In this thesis, we predicted and demonstrated for the first time that optical forces above
photonic crystal slabs will lead to a variety of complex, stably trapped crystal patterns.
We demonstrated that wavelength or polarization of may be used to reconfigure the pat-
terns. Next, we trapped multiple particles using an extended mode in a two-dimensional
photonic crystal slab.
Our system improves upon previous optical trapping work in three main ways.
First, we have demonstrated a compact, multiparticle trapping system in the present
work; over 180 particles are trapped in an area less than 150 m
2
. The number of
particles could be increased in a straightforward way by proper design of the structure
or by increasing the size of the incident beam. Second, since the large array of particles
is trapped by exciting a single mode of the structure without the use of a spatial light
modulator or any movement of the beam or sample, it can be easily adapted for a
variety of integrated, lab-on-a-chip applications. Finally, by tuning the polarization
of the incident light, we can tune the preferential direction of the in-plane trapping
stiffness of our structure.
It is intriguing to consider whether metamaterials based on metal nanoclusters,
for example, could be assembled in the manner we describe. We have proposed a
method for using optical forces to induce the formation of metallo-dielectric clusters
with controlled morphology. The method relies on field-enhancement effects near gold
74
particles trapped above a photonic-crystal microcavity. We proved the formation and
showed that Brownian motion and heating-induced fluid convection do not significantly
affect the conditions for the optical formation of a metallo-dielectric clusters.
We expect the approach of light-assisted self-assembly, described here, to be of
broad utility for fabrication of complex photonic materials, sensors, and filters.
75
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86
Abstract (if available)
Abstract
In this dissertation, I study optical forces near micro-fabricated devices for multi-particle manipulation. I consider particles of different sizes and compositions. In particular, I focus my study on both dielectric and gold particles as well as Giant Unilamellar Vesicles (GUVs). ❧ First, I consider optical forces near a PhC and establish the feasibility of a technique which we term Light-Assisted Templated Self-assembly (LATS). I exploit the guided resonance modes of a PhC to provide resonant enhancement of optical forces. The guided mode forms spatially distinct trapping patterns at different polarizations and wavelengths. I consider the possibility of assembling multi-particle patterns. Specially, I show how optical forces near a photonic crystal slab can be used to assist templated self-assembly and permit different particle pattern formation. Thus, a PhC will serve as an optically reconfigurable template for multi-particle trapping. ❧ Next, I calculate the optical force near a PhC micro-cavity for gold and dielectric particles ∼ 50 nm in diameter. Next, I use the electromagnetic field re-distribution around a trapped gold nano-particle to direct a metal-dielectric cluster formation. ❧ Finally, I explore optical forces near a Dual Beam Optical Trap (DBOT) and measure the bending modulus of a GUV ∼ 10 μm. First, I present a method to extract the bending modulus of the membrane from the area strain data. This method incorporates three-dimensional ray-tracing to calculate the applied stress in the DBOT within the ray optics approximation. I compare the optical force calculated using the ray optics approximation and Maxwell Stress Tensor method to ensure the approximation’s accuracy. Next, we apply this method to 3 populations of GUVs to extract the bending modulus of membranes comprised of saturated and monounsaturated lipids in both gel and liquid phases.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Mejía Prada, Camilo Andrés
(author)
Core Title
Optical forces near micro-fabricated devices
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Publication Date
10/15/2013
Defense Date
09/23/2013
Publisher
University of Southern California
(original),
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(digital)
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OAI-PMH Harvest,optical forces,photonic crystal,photonics
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application/pdf
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Language
English
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Electronically uploaded by the author
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Advisor
Povinelli, Michelle L. (
committee chair
)
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cmejiaprada@gmail.com
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Mejía Prada, Camilo Andrés; Mejia Prada, Camilo Andres
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Tags
optical forces
photonic crystal
photonics