University of Southern California Dissertations and Theses

Laser manipulation of atomic and molecular flows

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Laser manipulation of atomic and molecular flows

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LASER MANIPULATION OF ATOMIC AND MOLECULAR FLOWS

by
Taylor C. Lilly

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
(AEROSPACE ENGINEERING)

August 2010

Copyright 2010 Taylor C. Lilly

ii

Dedication

To my Papa.
Thank you for your example.

iii

Acknowledgements
As with any long journey, the success of this doctorate began with small steps. I am very grateful
to my parents, for their support of my endeavors, confidence in my success, and graciousness in helping me
learn from my mistakes. All I have accomplished, can now claim, or shall pursue in the future, is a
reflection of their guidance, example, and love. I am fortunate to have a brother, a man of character and
compassion, and am grateful I may call him my kin. Though not a brother by blood, I am also indebted to a
man who is as good as family, Dr. Robert Mancini. I thank Robert for his stalwart dedication to our
friendship, his example on living a professional and responsible life, and the competition such an example
spurs. I look forward to growing and creating “our generation” with both of these men.
My professional life has been, in no small part, made possible by the philosophy and patience of
Dr. Andrew Ketsdever. Dr. Ketsdever’s dedication to student research, and practice of pushing his charges
to the extent of their potential, has created a rich series of opportunities for me, culminating in the
completion of this dissertation. I have been fortunate to work for, and with, Andrew for the better part of
nine years, and am grateful for the example he has given me; he has shown me a strong methodology for
research, professional contribution, and the formation of the next generation of engineers.
In support of my work, and professional growth, I am grateful to those individuals who have been
blessed with the patience to encourage and foster me. Dr. Ingrid Wysong and Prof. E. P. Muntz have
provided me with programmatic and academic support over a series of projects, starting early in my
undergraduate career. Through their confidence and guidance, I was shown the possibility of moving
beyond four years of school, and into a career. I would also like to make a special thank you to Dr.
Michael Shaffer, for both his extraordinary patience in guiding me to the finer points of optics and atomic
theory, and for his friendship outside of the lab.
I am also grateful to my compatriots and brothers in arms, so to speak. Mr. Riki Lee and Dr. Nate
Selden have proven to be invaluable and integral parts to my academic career. Whether a lab partner,
homework decipherer, or willing cohort for a 15 minute coffee break, I am grateful to both of these men for
their camaraderie. With Riki and Nate, I can include a long list of good people who have made my journey
through life, and academia so far, possible: Barry Cornella, Dr. Ryan Downey, Dr. Anthony Pancotti,

iv

Jordan Olliges, Miles Killingsworth, Capt. Ron Remsburg, Tom and Vivian Varga, Giuseppe Canzonieri,
Kevin Lax, Fr. Bill Messenger, Jenny and Zach Laney, to name a few of many. These people have made
working on my goals, throughout life, enjoyable and rewarding.

v

Table of Contents
Dedication ii
Acknowledgements iii
List of Tables vi
List of Figures vii
Nomenclature x
Abstract xii
Chapter 1 Motivation and Current Research 1
1.1 Introduction 1
1.2 Related Research and Experiments 7
Chapter 2 Theory 18
2.1 Laser Fields 19
2.2 Non-Resonant Interaction: Induced Dipole 21
2.3 Resonant Interaction: Quantum Mechanics and the semi-Classical Approach 22
2.4 Resonant Interaction: Optical Bloch Equations 32
2.5 Resonant Interaction: Forces 34
Chapter 3 Methodology 39
3.1 Numerical: TCL Code 40
3.2 Numerical: SMILE 49
3.3 Experimental 55
3.4 Summary 63
Chapter 4 Steering 64
4.1 Photon Scattering 64
4.2 Resonant Dipole 71
4.3 Non-Resonant Dipole 74
4.4 Summary 78
Chapter 5 Collimation 80
5.1 Photon Scattering 80
5.2 Resonant Dipole 84
5.3 Non-Resonant Dipole 87
5.4 Summary 92
Chapter 6 Optical Lattices 94
6.1 Atomic Acceleration 95
6.2 Neutral Gas Heating 98
6.3 Summary 107
Chapter 7 Conclusions 108
Bibliography 112
Appendix A Empirical Values and Their Sources 117

vi

List of Tables
Table 4-1: Conditions for photon scattering steering simulations ................................................................. 68
Table 4-2: Conditions for resonant dipole steering simulations .................................................................... 73
Table 4-3: Conditions for non-resonant dipole steering simulations ............................................................. 75
Table 5-1: Conditions for photon scattering collimation simulations ........................................................... 82
Table 5-2: Conditions for resonant dipole collimation simulations............................................................... 85
Table 5-3: Conditions for non-resonant dipole collimation simulations ....................................................... 88
Table 6-1: Conditions for resonant optical lattice acceleration simulations .................................................. 96
Table 6-2: Conditions for non-resonant optical lattice gas heating simulations .......................................... 101
Table A-1: Cs atomic weight ....................................................................................................................... 117
Table A-2: Cs vapor pressure (solid) ........................................................................................................... 117
Table A-3: Cs vapor pressure (liquid) ......................................................................................................... 117
Table A-4: Cs Ionization potential .............................................................................................................. 118
Table A-5: Cs-Cs total collision cross section ............................................................................................. 118
Table A-6: Cs 6p
2
P
3/2
↔ 6s
2
S
1/2
transition frequency ............................................................................... 118
Table A-7: Cs 6p
2
P
3/2
→ 6s
2
S
1/2
transition lifetime ................................................................................... 118
Table A-8: N static polarizability ................................................................................................................ 119
Table A-9: N
2
static polarizability ............................................................................................................... 119
Table A-10: Cs static polarizability ............................................................................................................. 120

vii

List of Figures
Figure 1-1: Notional diagram of A) laser-atom momentum transfer by photon scattering and B) species
trajectory deflection by the induced dipole force .............................................................................. 5
Figure 1-2: Notional diagram of species interaction within an optical lattice ................................................. 6
Figure 1-3: Notional diagrams of A) 3D optical molasses (Shaffer 2008) and B) spatial constriction in a
MOT (Shaffer 2008) ......................................................................................................................... 8
Figure 1-4: Notional diagram of Cs
2
trapping in a radial dipole trap utilizing a Gaussian beam (Takekoshi,
Patterson and Knize 1998) .............................................................................................................. 10
Figure 1-5: A) Rb potential in a LG beam (Kuga, et al. 1997) and B) a notional diagram of a dipole trap
utilizing a LG beam (Kuga, et al. 1997) .......................................................................................... 11
Figure 1-6: Diagram of dipole potential wells within an optical lattice, note the axial direction has been
stretched by a factor of 1500 (Alt 2004) ......................................................................................... 12
Figure 1-7: Deflection of I
2
and CS
2
by the radial dipole force from a pulsed laser (Sakai, et al. 1998) ...... 13
Figure 1-8: Deceleration of C
6
H
6
by the radial dipole force from a pulsed laser (Fulton, Bishop and Barker
2004) ............................................................................................................................................... 13
Figure 1-9: Diffraction of a Na atomic beam through an optical lattice (Moskowitz, et al. 1983) ................ 14
Figure 1-10: Optical lattice accelerated Cs A) transposition (Schrader, et al. 2001) and B) retention
(Schrader, et al. 2001) ..................................................................................................................... 15
Figure 1-11: Coherent Rayleigh Scattering signal versus lattice velocity measured by the laser frequency
offset (H. T. Bookey, et al. 2006) ................................................................................................... 16
Figure 1-12: Simulated energy deposition into CH
4
from a pulsed optical lattice for various gas pressures
(Ngalande, Gimelshein and Shneider 2007).................................................................................... 17
Figure 2-1: Non-resonant induced dipole force vs. radial position ............................................................... 22
Figure 2-2: Excited state probability vs. time for various δ, without spontaneous emission ......................... 28
Figure 2-3: Excited state probability vs. time, with spontaneous emission (Metcalf and van der Straten
1999) ............................................................................................................................................... 29
Figure 2-4: Natural transition probability line shape vs. frequency .............................................................. 30
Figure 2-5: Excited state probability vs. time using the OBE (Metcalf and van der Straten 1999) ............... 33
Figure 2-6: Resonant induced dipole force vs. axial position within an optical lattice ................................. 38
Figure 3-1: Notional diagram of TCL code domain ...................................................................................... 41
Figure 3-2: Axial velocity, v
z
, comparison at oven and skimmer planes ....................................................... 46
Figure 3-3: Radial velocity, v
x,y
, comparison at oven and skimmer planes ................................................... 46
Figure 3-4: Starting velocity distribution for TCL code simulations: A) v
axial
B) v
radial
................................. 47
Figure 3-5: Comparison of analytical and statistical simulation subroutines in the TCL code ..................... 49

viii

Figure 3-6: SMILE domain for A) starting surface and B) non-resonant dipole steering simulations .......... 52
Figure 3-7: SMILE baseline density [m
-3
] for A) starting surface and B) non-resonant dipole steering
simulations ...................................................................................................................................... 53
Figure 3-8: SMILE A) domain and B) baseline density [m
-3
] for non-resonant dipole collimation
simulations ...................................................................................................................................... 54
Figure 3-9: SMILE domain and laser intensity [W/m
2
] for neutral gas heating ............................................ 55
Figure 3-10: Diagram of experimental vacuum system (side view) .............................................................. 56
Figure 3-11: Diagram of experimental laser system (top view) continued in Figure 3-12 ............................ 57
Figure 3-12: Diagram of experimental laser system (top view) continued from Figure 3-11 ....................... 58
Figure 3-13: Saturated absorption signal vs. frequency sweep for experimental laser master ...................... 59
Figure 3-14: Comparison between saturated absorption setup and atomic beam fluorescence ..................... 60
Figure 3-15: Example of A) background and B) atomic beam fluorescence signals..................................... 61
Figure 3-16: Example of signal to noise for experimental atomic beam detection ....................................... 62
Figure 3-17: Example of A) background subtracted signal and B) vertical bin summing ............................ 63
Figure 4-1: Photon scattering force vs. incident laser power ........................................................................ 65
Figure 4-2: Photon scattering force vs. atom velocity ................................................................................... 66
Figure 4-3: Atomic beam spatial profile including both Cs ground states .................................................... 67
Figure 4-4: Diagram of notional experimental configuration represented by simulations in section 4.1 ...... 67
Figure 4-5: Atomic beam A) spatial and B) velocity profiles ....................................................................... 69
Figure 4-6: Atomic beam deflection vs. laser frequency for photon scattering steering ............................... 69
Figure 4-7: Numerical vs. experimental atomic beam deflection for photon scattering steering .................. 70
Figure 4-8: Atomic beam deflection vs. laser power using δ = 1.1 MHz for photon scattering steering ...... 71
Figure 4-9: Laser detuning vs. intensity for Ω/ δ = 0.2 .................................................................................. 72
Figure 4-10: Diagram of notional experimental configuration represented by simulations in section 4.2 .... 73
Figure 4-11: Atomic beam center vs. vertical laser position for resonant dipole steering ............................. 74
Figure 4-12: Diagram of notional experimental configuration represented by simulations in section 4.3 .... 74
Figure 4-13: Density [m
-3
]: A) baseline and after 100 pulses with laser center, y= B) 00 μm, C) 10 μm, D)
20 μm .............................................................................................................................................. 76
Figure 4-14: Time averaging for vertical density cross-sectional profile plots ............................................. 77
Figure 4-15: Cross sectional density [m
-3
] at for laser y= A) 00 μm, B) 10 μm, C) 20 μm ........................... 77
Figure 4-16: Flow center vs. laser position for non-resonant dipole steeri ng ................................................ 78

ix

Figure 5-1: Diagram of notional experimental configuration represented by simulations in section 5.1 ...... 81
Figure 5-2: Atomic beam width vs. laser frequency for photon scattering collimation ................................ 83
Figure 5-3: Atomic beam A) collimation and B) dispersion vs. laser power via photon scattering .............. 83
Figure 5-4: Diagram of notional experimental configuration represented by simulations in section 5.2 ...... 84
Figure 5-5: Atomic beam width vs. laser vertical position for resonant dipole collimation .......................... 86
Figure 5-6: Atomic beam width vs. laser power for resonant dipole collimation ......................................... 86
Figure 5-7: Diagram of notional experimental configuration represented by simulations in section 5.3 ...... 87
Figure 5-8: Density [m
-3
] initial condition used for non-resonant collimation .............................................. 88
Figure 5-9: Density [m
-3
] after 100 laser pulses with 0 ns intervening time ................................................. 89
Figure 5-10: Density [m
-3
] after 50 laser pulses with 5 ns intervening time ................................................. 90
Figure 5-11: Time averaging for vertical density cross-sectional profile plots ............................................. 90
Figure 5-12: Cross sectional density [m
-3
] vs. intervening time for pulsed dipole collimation ..................... 91
Figure 5-13: Cross sectional velocity [m/s] for two intervening times ......................................................... 91
Figure 5-14: Cross sectional density [m
-3
] vs. pulse energy for pulsed dipole collimation ........................... 92
Figure 6-1: Atomic beam A) spatial and B) velocity distribution after 10
6
m/s
2
acceleration for 14 μs ....... 96
Figure 6-2: Atomic beam A) spatial and B) velocity distribution for two laser diameters/powers ............... 97
Figure 6-3: Normalized free-molecular density perturbations due to a pulsed optical lattice ....................... 98
Figure 6-4: Free-molecular velocity distribution perturbations due to a pulsed optical lattice ................... 100
Figure 6-5: Translational and rotational temperatures after one pulse vs. radial position ........................... 102
Figure 6-6: Change in total domain translational energy after one pulse vs. initial gas temperature .......... 102
Figure 6-7: Overall temperature after 10 pulses vs. intervening time for multiple gases ............................ 103
Figure 6-8: N
2
overall temperature after 10 pulses vs. intervening time for multiple pressures ................. 103
Figure 6-9: Ar overall temperature vs. pulse for intervening time of 0 ns .................................................. 104
Figure 6-10: Ar overall temperature vs. pulse for intervening time A) 0.5 ns B) 10ns ............................... 105
Figure 6-11: Ar pressure vs. radial position, intervening time of 10 ns ...................................................... 105
Figure 6-12: N
2
overall temperature vs. pulse, intervening time of 1 ns ..................................................... 106
Figure A-1: Comparison of published Cs vapor pressures .......................................................................... 117
Figure A-2: Cs D
2
transition energies (Steck 2008) .................................................................................... 118
Figure A-3: Cs D
2
cycling transition ........................................................................................................... 119

x

Nomenclature
Symbol Units Description
AOM Acousto-Optic Modulator
c m/s speed of light in a medium
c
0
m/s speed of light in vacuum = 299792458
C
n
equation constant
c
n
proportionality constant
CCD Charge Coupled Device
CW continuous wave
e C charge, electron = 1.602176487 x 10
-19

DSMC Direct Simulation Monte Carlo
D
L
m diameter, laser (FWHM)
E V/m electric field, instantaneous amplitude
E
0
V/m electric field, oscillation amplitude
f Hz frequency
= J s Planck constant over 2 pi = 1.054571628 x 10
-34

ˆ
H
Hamiltonian operator
i square root of (-1)
k
botlz
J/K Boltzmann’s constant = 1.3806503 × 10
-23

k
L
rad/m wavenumber, laser
k
Δ
rad/m wavenumber, optical lattice (k
Δ
= k
L1
-k
L2
)
Kn n.d. Knudsen number
L m characteristic length scale
m kg mass
m
e
kg mass, electron = 9.10938215 x 10
-31

n m
-3
number density
OBE Optical Bloch Equations
p kg m s
-1
momentum
P W power
P
L
W power, laser
q
i
logarithmic derivative, imaginary
q
r
logarithmic derivative, real
r m radial variable
R J energy, recoil =( = k)
2
/(2m)
RWA Rotating Wave Approximation
s n.d. saturation parameter
s
0
n.d. saturation parameter, on-resonance
SMILE Statistical Modeling In Low-density Environments
t s temporal variable
TCL author’s initials
T K temperature
U J energy, potential
v m/s velocity
v
mp
m/s velocity, most probable
' v m/s velocity, average thermal
x,y,z m spatial variable
α C m
2
V
-1
polarizability, static
β s/m inverse of most probable speed in a gas
Γ rad/s radiative width of an electronic transition (FWHM)
Δl
c
m coherence length
Δ ν
L
Hz line width, laser
Δ ω
L
rad/s line width, laser

xi

Symbol Units Description
δ rad/s detuning from resonance, δ = ( ω
L
- ω
0
)
ε
0
A·s V
-1
·m
-1
electric constant = 8.854187817... x 10
-12

ˆ ε polarization vector
ϵ

J energy
ϵ
L

J energy, laser
ϵ
n

J energy, state
λ
0
m wavelength, transition
λ
mfp
m mean free path
ξ m/s velocity, optical lattice
ρ n.d. density matrix, quantum two-state populations
σ
T
m
2
collisional cross section
τ s mean lifetime of an electronic transition
τ
L
s pulse width, laser (FWHM)
φ wavefunction, instantaneous
Ψ wavefunction, time-dependant
Ω rad/s Rabi frequency
ω rad/s angular frequency
ω
0
rad/s angular frequency, transition ( ω
0
= ω
21
)
ω
L
rad/s angular frequency, laser
ω
Δ
rad/s angular frequency, optical lattice ( ω
Δ
= ω
L1
- ω
L2
)

xii

Abstract
The continuing advance of laser technology enables a range of broadly applicable, laser-based flow
manipulation techniques. The characteristics of these laser-based flow manipulations suggest that they may
augment, or be superior to, such traditional electro-mechanical methods as ionic flow control, shock tubes,
and small scale wind tunnels. In this study, methodology was developed for investigating laser flow
manipulation techniques, and testing their feasibility for a number of aerospace, basic physics, and micro
technology applications. Theories for laser-atom and laser-molecule interactions have been under
development since the advent of laser technology. The theories have yet to be adequately integrated into
kinetic flow solvers. Realizing this integration would greatly enhance the scaling of laser-species
interactions beyond the realm of ultra-cold atomic physics. This goal was realized in the present study. A
representative numerical investigation, of laser-based neutral atomic and molecular flow manipulations,
was conducted using near-resonant and non-resonant laser fields. To simulate the laser interactions over a
range of laser and flow conditions, the following tools were employed: a custom collisionless gas particle
trajectory code and a specifically modified version of the Direct Simulation Monte Carlo statistical kinetic
solver known as SMILE. In addition to the numerical investigations, a validating experiment was
conducted. The experimental results showed good agreement with the numerical simulations when
experimental parameters, such as finite laser line width, were taken into account. Several areas of interest
were addressed: laser induced neutral flow steering, collimation, direct flow acceleration, and neutral gas
heating. Near-resonant continuous wave laser, and non-resonant pulsed laser, interactions with cesium and
nitrogen were simulated. These simulations showed trends and some limitations associated with these
interactions, used for flow steering and collimation. The use of one of these interactions, the induced
dipole force, was extended beyond a single Gaussian laser field. The interference patterns associated with
counter-propagating laser fields, or “optical lattices,” were shown to be capable of both direct species
acceleration and gas heating. This study resulted in predictions for a continuous, resonant laser-cesium
flow with accelerations of 10
6
m/s
2
. For this circumstance, a future straightforward proof of principle
experiment has been identified. To demonstrate non-resonant gas heating, a series of pulsed optical lattices
were simulated interacting with neutral non-polar species. An optimum time between pulses was identified

xiii

as a function of the collisional relaxation time. Using the optimum time between pulses, molecular
nitrogen simulations showed an increase in gas temperature from 300 K to 2470 K at 1 atm, for 50
successive optical lattice pulses. A second proof of principle experiment was identified for future
investigation.

1

Chapter 1 Motivation and Current Research
1.1 Introduction
Laser interactions with neutral atomic and molecular flows offer a strong, tunable method for the
remote control of that species’ position, momentum, and energy. This control of flow trajectories enables
technologies and applications that are currently impossible without the precision, selectivity, and energy
density associated with modern laser fields. So far the use of laser flow manipulation has been primarily
limited to low-temperature atomic physics, where resonant laser interactions offer the ability to cool,
confine (Gordon and Ashkin 1980), and interrogate atoms and molecules with sub-micron precision
(Steuernagel 2005). The theory and techniques developed for these “cold” applications can be extended to
include intense laser fields for the steering, collimating, accelerating, and heating of flows with higher
densities and energies than current common practice. One impediment to this extension is the development
of a methodology for investigating the design space of potential laser modification scenarios beyond a
single laser-atom or laser-molecule interaction, i.e. integrating the interaction into a numeric kinetic flow
solver. In this study, a methodology for such an investigation has been developed. This includes the
incorporation of the laser-flow interaction into two numerical simulation tools, covering both free-
molecular and collisional gas flows. The methodology can be applied to a myriad of engineering
applications.
1.1.1 Hypersonics
One area where laser manipulation of a neutral flow has current critical significance is that of
hypersonics. Present limitations in creating high velocity (kinetic energy), high temperature (enthalpy)
flows require trading one of those parameters for the other. Due to this limitation, test facilities cannot
obtain the full measure of desired flow conditions; laser manipulation techniques can enable flow
conditions which are currently unrealizable. At the moment, the limitation with applying laser based flow
manipulation to hypersonics is the scale of the interaction. For applications discussed in this study, the
length scales of the flows are on the order of laser diameters (mm to cm) and column depths (cm to m).

2

The feasibility of scaling these techniques to the size of macroscopic test facilities, e.g. wind tunnels, will
require advances in laser affordability, power, and controllability.
The scale of the interaction does not diminish the advantages offered by laser-based flow
manipulation. Laser methods allow for remote interaction with the flow, separating acceleration and
heating interactions into independent processes. By decoupling the processes and not requiring the flow to
interact with mechanical surfaces, the manipulated flow volume is preserved until it is investigated. This
bypasses limitations associated material limits, such as melting. The methodology developed in this study
can aid in the design of hypersonic flows by offering tools for exploring laser and flow parameters and
relating them to the desired resultant flow velocity and temperature. The process of designing the
simulated hypersonic flow is then reduced to selecting the appropriate laser conditions for the flow
identified by the numerical investigation and procuring the hardware required to produce those fields.
One of the specific ways which laser manipulation enables high speed flow research and
applications is by overcoming the necessity of ionizing the flow in order to use Coulomb forces to alter
species’ trajectories (Livingston and Blewett 1962). A major drawback to ionization is the necessity of
neutralizing the flow in order to return to the desired neutral condition. Charged particle systems are
additionally complicated by space-charge limitations and hampered in charge exchange due to scattering
(Ketsdever 1995). Neutral laser manipulation eliminates these limitations since the flow is not ionized.
Another process currently used to accelerate neutral flows utilizes a laser heated nozzle expansion to trade
laser deposited enthalpy increase for directional kinetic energy (Caledonia, et al. 1990). In this process, a
high intensity laser breaks down the gas at the throat of a nozzle in order to ionize, and partially dissociate,
the species. The ions readily absorb energy from the laser field, which, in turn, heats the gas through a
process analogous to Joule heating in a resistor. The expanding gas quickly recombines charged species to
self-neutralize; however, the result is still a mixed flow of atomic, molecular and ionic components
(Minton, et al. 1991). This technique also suffers from the temperature limits of the apparatus, especially
the throat of the nozzle. By using lasers below the breakdown threshold of the gas, this study shows the
applicability of using laser manipulation for accelerating and steering a neutral flow without the byproducts
or limitations of ionization.

3

In addition to flow acceleration and steering, hypersonics involves significant energy deposition in
high density gases. Here, the interest is aimed at large, well characterized, temperature increases in the
target gas, without noticeable ionization. For orbital and ballistic reentry conditions, the understanding of
high-temperature, strongly non-equilibrium gas flows surrounding the vehicle, requires accurate data for
the dominant energy transfer processes and chemical reactions. This data is required in terms of both
energy dependent cross sections (Park 1990) and temperature dependent rates (Anderson 2006). Current
methods for investigating these flows involve the use of combustion, shocks (Jerig, Thielen and Roth
1990), laser pyrolysis, and arc jets (Matsui, et al. 2004). Combustion involves, and must account for,
unwanted chemical species and reactions. Shock facilities suffer from limitations associated with the
length of time available to make the desired measurement (da Silva, Guerra and Loureiro 2006). Laser
pyrolysis is a resonant phenomenon that is applicable to only a few specific species. And, arc jets rely on
ionization and plasma heating which may significantly affect the measurement of interest. This study
shows the feasibility of using specially shaped laser interference patterns to deposit energy into an arbitrary
gas, e.g. nitrogen, raising the gas to temperatures in excess of 2400 K. The deposition couples only with a
single translational energy mode, allowing for well characterized initial conditions invaluable to
experimental research.
1.1.2 Microengineering
In addition to the high enthalpy flows associated with hypersonics, micro-machining and micro-
engineering also deal with flows with very strict requirements on the control of energy and momentum.
These areas can benefit greatly from the controllability and precision of laser manipulation. The key
parameter in micro applications is the ability to affect a target flow over length scales which push the limits
of physical manufacture. The ability to create masks, beam blocks, controlling electric fields, and other
material-based flow interaction devices, is hindered on these scales; while, the ability to shape light at such
scales is routine. Electromagnetic processes are currently used to ionize and direct an ionic flow for
etching and deposition materials in micro applications. These processes suffer from the same issues
mentioned before, including particle-particle interactions and difficulty maintaining a neutral environment.
With laser manipulation methods, the precision and characteristics of the flow control is limited only by
one’s ability to control the laser field. This study provides several examples of using laser fields to steer

4

and collimate atomic and molecular flows, which are applicable to micro technologies. The methodology
developed in this study can be adapted for designing full-scale systems for: micro-machining, micro-
engineering, and lithography. The methodology can also be used for predicting the level of flow control
capable by integrating laser control into existing systems.
1.1.3 Atomic Physics
Laser manipulation has been a thoroughly exploited, well studied tool for an extremely selective
branch of atomic physics. Lasers are currently used to manipulate single atoms, usually resonantly coupled
alkalis. These atoms are cooled from a warm, low density atomic cloud into a high density, sub-Kelvin
ensemble. The ensemble is then used for spectroscopic and quantum mechanical experimentation. Even in
the field of its birth, laser manipulation can be extended. Directional control of atomic flows permits
investigations on controlled collision processes, such as state mixing rates and energy dependant cross
sections. Moreover, the emerging field of quantum computing depends heavily on the ability to shape
potential patterns to hold the quantum information carriers (Miroshnychenko, et al. 2006). The
methodology developed in this study takes the process of laser manipulation beyond the confines of low
temperature atomic physics and incorporates laser interactions into existing gas dynamic simulation
techniques.
1.1.4 Background Physics and Potential for Examination
The process by which a laser interacts with atomic and molecular species is directly related to the
distortion of that species’ electronic structure in response to the presence of an electric field. In modeling
the interaction, a laser’s influence is represented as a travelling wave with an oscillating electric field, and
the atom or molecule is represented as a quantum mechanical system. The interaction of a particular
species with the oscillating electric field manifests itself in a few different ways, depending on the
proximity of the laser frequency to resonances in the species’ electronic structure. The two primary
interaction processes are photon scattering and the induced dipole force. Photon scattering is only
prevalent when the laser is tuned near a resonance and is analogous to a coherent photon “wind” or photon
pressure. A conceptual diagram for this process can be found in Figure 1-1 (A). The second process is the
dipole gradient force which results from the potential energy of the species’ immersion in the electric field.

5

The potential is proportional to the square of the field strength which leads to a force imparted on the
species in a spatially inhomogeneous field, e.g. the Gaussian radial profile of a nominal laser field. In this
relation, the proportionality constant is the polarizability and is dependent on both proximity to resonance
and the orientation of the molecule relative to the electric field direction. For an ensemble of many atoms
or molecules, the polarizability of an individual atom or molecule can be reduced to a direction-
independent value by integration over all possible orientations. An atom or molecule which traverses a
Gaussian field would have its trajectory bent by the dipole force, like light rays through a lens. A notional
diagram of trajectory bending by the dipole force is represented in Figure 1-1 (B). This scenario leads to
the potential of focusing atomic particle beam using a light field, similar to the focusing a light beam using
a material lens.

A B
Figure 1-1: Notional diagram of A) laser-atom momentum transfer by photon scattering and B) species
trajectory deflection by the induced dipole force
The versatility of these interactions is enhanced by the versatility of shaping light fields. When
working with the induced dipole force, the magnitude and direction of the force is directly linked to the
shape of the laser field. One way to exploit this connection is by using the interference pattern formed by
two counter-propagating laser fields. The interfering fields take the shape of a standing wave or “optical
lattice.” This pattern has a periodicity that is half the constituent laser’s wavelength, on the order of 100’s
of nanometers, yielding a strong gradient in the electric field strength. Two applications for this potential
field configuration are direct flow acceleration, and neutral gas heating. In the case of a flow, where the
density is low enough that the atoms or molecules can be trapped within the lattice potentials, the bulk

6

velocity of the species becomes bound to the velocity of the lattice pattern, like balls on a roulette table
falling into the bins. By modulating the constituent laser frequencies, the lattice pattern can be put into
motion, accelerating the trapped species, much like spinning the roulette table back up, after the balls have
fallen. Experimentally, a single atom has been subjected to accelerations as high as 10
5
m/s
2
using this
method (Schrader, et al. 2001). If the density of the gas is sufficiently high, the process of intermolecular
collisions offers the ability to transfer energy from the laser field to the gas. Species within the optical
lattice are continually accelerated towards the points of least potential. These accelerated species
continually collide with one another. Collisions then redistribute the imparted kinetic energy to unexcited
energy modes, heating the gas. Previous numerical simulations suggest that this energy deposition can
increase the average translational temperature in the axial direction over 500 K for a single 1 ns laser pulse
(Ngalande, Gimelshein and Shneider 2007). A notional diagram of these two optical lattice processes can
be seen in Figure 1-2.

Figure 1-2: Notional diagram of species interaction within an optical lattice
The primary numerical tools used in this study were a custom collisionless particle trajectory code
written specifically for this application, and the collisional DSMC code SMILE (Ivanov, Kashkovsky, et al.
1997). The custom trajectory code, referred to as the “TCL” code, was written to simulate the resonant and
near-resonant effects of laser fields on a cesium atomic beam. The code can be easily updated to simulate
any alkali metal, meta-stable noble gas, or other atomic species which can be modeled as a “mutli” two-
level system as is done in the derivation of theory in Chapter 2. The cesium beam is representative of
resonant metal flows found in micro technology applications, such as lithography and neutral beam etching.
SMILE has been modified to add the capability of simulating non-resonant interactions between laser fields
and arbitrary molecules. The flows simulated using SMILE are representative of flows found in aerospace
applications, which involve gases such as oxygen and nitrogen or can be extended to model gases for
Potential Depth
Collision & Energy Transfer
Higher Density
Lower Density
Accelerating Potential
Stationary Potential
Oscillating Trapped Atom

7

entirely different planetary atmospheres. These tools were exercised by investigating various
configurations of laser fields interacting with atomic and molecular flows. The configurations were chosen
for their experimental relevance, and extrapolation to final applications. In addition to the numerical
investigations, and in order to validate the TCL code approach, an experiment was conducted to
demonstrate the resonant steering of a thermal [373 K] cesium atomic beam using photon scattering.
1.2 Related Research and Experiments
Fine control of atoms and molecules using laser fields has its roots in low temperature atomic
physics. Driving the research in this field are applications in quantum computing, fundamental research in
Bose-Einstein condensates (Stamper-Kurn, et al. 1998), and quantum collisions. Although the present
study addresses applications which produce opposite effects, fast and hot vs. slow and cold, any precedence
in the use of lasers for flow manipulation is helpful. To be familiar with these effects, understanding the
applicability, advantages, and shortcomings of various laser interaction processes, assists in grounding the
basis of the present study. A look at past and present laser-atom and laser-molecule interaction
experiments is presented below. There are two ways to categorize laser-atom interaction processes: by
purpose, and by laser interaction regime. Most laser-atom experiments are performed with one of two
purposes. The first is to remotely control the atom’s momentum and energy. These techniques act as
damping or dissipative tools to “cool” the atoms. The second purpose is that of spatial confinement or
transposition. These techniques are ideally conservative to avoid adding unintended energy to “cooled”
system. With regards to the interaction regime, the frequency of the laser in relation to resonances in the
atom has a strong impact on the form of the interaction. Therefore, processes may, be described as either
resonant (and near-resonant) or non-resonant.
1.2.1 Optical Molasses
The primary method for laser cooling of resonant atomic species is that of “optical molasses.”
This is a near-resonant process for momentum reduction. In this technique atoms are cooled by velocity-
selective absorption and spontaneous emission of photons, which are Doppler shifted in the atomic frame.
The name of the technique was first coined by experimenters who likened the process to a particle in a
viscous fluid (Chu, et al. 1985). The setup for 3D optical molasses entails six lasers arranged in three

8

orthogonal counter-propagating pairs. If all of the lasers are slightly detuned to the red side (lower
frequency / longer wavelength) of an electronic resonance in the species, the Doppler shift resulting from
an atom’s velocity towards a particular laser will cause the atom to absorb more photons from that laser
than its anti-parallel partner. The atom is slowed as the disparity in absorbed photons increases the
momentum transfer from the counter propagating laser to the atom. It is important to note that optical
molasses does not spatially confine, it affects only the velocity of the atom. This condenses the velocity
distribution while not directly affecting the spatial distribution. A notional diagram of this setup can be
found in Figure 1-3 (A).
Using only one laser, this technique, can be applied to slow an atomic beam instead of a three
dimensional cloud. An atomic beam of sodium was decelerated using a counter propagating laser by
frequency sweeping the laser to maintain an optimal Doppler shift throughout the process (Ertmer, et al.
1985). The beam was slowed from several hundred m/s to less than 90 m/s over a distance of several
meters. Conversely, there is no reason why an opposite frequency shift could not accelerate a beam. The
shortcoming of optical molasses is that the force, imparted on the atoms, is limited by the resonant photon
absorption/emission cycle, which saturates as the state populations (ground/excited) equalize. Thus, there
exists a theoretical maximum force, which is independent of the irradiating laser. This limits the minimum
time and distance required for acceleration and deceleration of an ensemble of atoms via the scattering
force (the force present in an optical molasses).

A B
Figure 1-3: Notional diagrams of A) 3D optical molasses (Shaffer 2008) and B) spatial constriction in a MOT
(Shaffer 2008)
1.2.2 Magneto-Optical Traps (MOT)
The addition of a strong magnetic field to optical molasses allows for the utilization of Zeeman
shifting of the atom’s magnetic sublevels. In this process a spatially gradient magnetic field is imposed on

9

an atom cloud. The magnetic field changes the energy levels of the atoms, depending on their location
within the field. This causes an atom to absorb photons from one of the two opposing laser fields based on
their position along the laser axis and the polarization of the laser light. Doppler shifting changes the
likelihood of photon absorption with respect to the atoms’ velocity. Zeeman shifting in an inhomogeneous
magnetic field changes the likelihood of absorption with respect to the atoms’ location. Therefore, this
configuration offers a method of adding spatial confinement to near-resonant momentum reduction. Such a
setup is referred to as a Magneto-Optical Trap (MOT). With a suitably shaped and sufficiently strong
magnetic field, an optical molasses / MOT laser configuration may act as both a cooling mechanism and a
confinement force. The MOT is a fairly standard apparatus in atomic physics experimentation; it cools and
constricts atoms from a background gas to central region, for use in other cold atom experiments. A
conceptual diagram of the MOT spatial confinement process along one axis can be seen in Figure 1-3 (B).
1.2.3 Dipole Traps
In the presence of a strong electric field a dipole potential is induced on a non-polar atom or
molecule. This dipole potential can be exploited in a spatially varying electric field, such as a focused
laser, to impart a useful force on the species. The dipole gradient force is non-dissipative, i.e. conservative,
and can be used for confinement and transposition of species without adding energy to the system. A
complete review of dipole traps can be found in (Grimm, Weidmüller and Ovchinnikov 1999).
1.2.3.1 Radial Dipole Traps
The radial intensity variation in a nominal laser field leads to a spatial gradient in the induced
dipole potential of an atom, along the field’s radial direction. This force, for red detuned lasers (lower
frequency than an electronic resonance), forces atoms towards areas of higher intensity (less potential). For
a Gaussian laser field, atoms are forced to the core of the radial profile. There is no cooling of the atom
due specifically to the dipole force, but other optical techniques, e.g. molasses or Sisyphus cooling
(Salomon, et al. 1990), can be used in conjunction with a dipole trap to increase the phase space density of
the trapped ensemble. This technique may be used with resonant lasers, but is routinely used with far-
from-resonance light to avoid complication due to photon scattering (Miller, Cline and Heinzen 1993).

10

Figure 1-4: Notional diagram of Cs
2
trapping in a radial dipole trap utilizing a Gaussian beam (Takekoshi,
Patterson and Knize 1998)
Cesium dimers (Cs
2
) have been trapped using a radial dipole trap (Takekoshi, Patterson and Knize
1998). A 17 W, 10.6 μm (non-resonant), CO
2
laser was focused to a waist diameter of 64 μm. The
pinching of the focus (axial direction) and the Gaussian shape of the field (radial direction) effectively
created a three dimensional force which pushed the molecules towards the focal point of the laser. Figure
1-4 shows a diagram of the experimental setup. The CO
2
beam is the radial dipole trap used to trap
molecules which have been cooled and isolated within a MOT. In this experiment other lasers were used to
accomplish molecule ejection and ionization for detection. The experiment was tuned far off-resonance to
reduce photon scattering, and interference with the MOT. By tuning far from a resonance, the magnitude
of the dipole force is reduced relative to the laser intensity; therefore, a powerful laser was required to
impart an appreciable force.
When the frequency of the laser used in a radial dipole trap was tuned to the blue side (higher
frequency) of an electronic transition, rather than the red, the sign of the force is reversed. An atom is then
pushed away from regions of high intensity towards those of low. Applying this to a traditional Gaussian
field profile, the atoms would be repelled from the center, and the laser would act as an anti-trap. However,
if the shape of the field were to look more like a doughnut, the minima in the center of the profile would act
as a good dipole trap. In addition to trapping, this shape has the advantage of placing the atoms in a region
of low laser intensity, which reduces photon scattering even near-resonance. An example of such a field
profile can be seen in Figure 1-5 (A). In this figure, the potential energy of a Rb atom within the beam is
given as an equivalent temperature, / TE k ∝ .

11

A B
Figure 1-5: A) Rb potential in a LG beam (Kuga, et al. 1997) and B) a notional diagram of a dipole trap utilizing
a LG beam (Kuga, et al. 1997)
Rb atoms have been trapped in such a “doughnut” trap using a Laguerre-Gaussian (LG) field
profile. In this experiment, a 600 mW laser was focused to a radius of 600 μm. The Rb atoms were first
cooled and loaded into a MOT, then transferred to the trap. Because there is no axial confinement, as with
a pinched (focused) red-detuned dipole trap, the main trap field was recovered, split, and brought through
the ends of the trap to act as plug fields. A diagram of the experimental trapping setup can be seen in
Figure 1-5 (B). This experiment reinforces the versatility of using tailored light as an atom manipulator.
Traditional cooling and trapping setups use red-detuned Gaussian lasers because the combination yields an
advantageous setup for cooling and spatial confinement. This experiment shows that convention does not
limit possibility. While designing a new experiment, it is important to build on the experience of other
experiments, including those which take paths that oppose common practice.
1.2.3.2 Axial Dipole Traps
For two, co-linearly polarized, counter-propagating lasers with the same frequency, the electric
field pattern formed by their interference is a standing wave when averaged over the fast oscillating
(optical) terms. The same dipole force which pushes the atom to the center of a red detuned radial dipole
trap now pushes the atoms towards the anti-nodes of the standing wave, where the light intensity is highest.
In comparison with radial gradients which vary over 100’s or 10’s of μm, the standing wave formed by
visible lasers varies over fractions of that distance. These strong gradients allow for trapping of atoms with
a spatial confinement on the order of 100’s of nm. A diagram of the axial periodic dipole potential of an
atom in such an interference pattern can be seen in Figure 1-6 with the axial direction stretched 1,500 times

12

to show individual wells. The three-dimensional shape of the wells comes from the interference pattern
(axial), the Gaussian shape of the two lasers (radial), and, if the lasers are also co-focal instead of
collimated, there is an additional shape due to axial variation in laser waist.

Figure 1-6: Diagram of dipole potential wells within an optical lattice, note the axial direction has been stretched
by a factor of 1500 (Alt 2004)
1.2.4 Atom Optics
Just as the dipole force is used to create long residence times within a dipole trap, the dipole force
has also been used for the purpose of controlling particle trajectories in a manner similar to controlling rays
of light. In a bit of role reversal, light is be used to diffract atomic beams much the same way that atoms
are used to diffract light. In an example setup, the atom is subjected to the laser for only a brief amount of
time (the transit time). In most cases there is little to no energy addition to the atom, it is simply redirected
(diffracted). Technically, in this manner, the dipole force is being applied as a momentum adjuster instead
of for spatial confinement. Much of the present study’s basics are found in this research on atom optics.
1.2.4.1 Radial Dipole Atom Optics
Iodine (I
2
) and carbon disulfide (CS
2
) molecules have been deflected by non-resonant pulsed
lasers using the radial dipole force(Stapelfeldt, et al. 1997). In these experiments two differing lasers were
used independently to deflect an off axis jet from a pulsed micro-nozzle (Sakai, et al. 1998). The molecules
where then ionized with a third laser through multi-photon ionization and their time of flight detected by a
set of multi-channel plates. One laser used was an Nd:YAG at 1064 nm with a pulse width of 14 ns and an
energy of 10 mJ. Focused to 7 μm, this laser yielded a peak intensity of ~9x10
15
W/m
2
. The second laser
was a CO
2
laser at 10.6 μm, with a pulse width of 70 ns and energy of 600 mJ. Focused to 35 μm, this laser
yielded a peak intensity of ~4.5x10
15
W/m
2
. The velocity profile of the deflected molecules can be seen in

13

Figure 1-7. The entrance trajectory of the molecules and the location of the ionization beam are also shown
in the figure.

Figure 1-7: Deflection of I
2
and CS
2
by the radial dipole force from a pulsed laser (Sakai, et al. 1998)
A similar work slowed benzene molecules with an off-resonance pulsed laser (Fulton, Bishop and
Barker 2004). The setup was similar to the above work with similar results; however, the goal was to cause
a linear deceleration, not a redirection. The atoms were directed through the center of the beam instead of
around the core, slowing molecules as they travelled to the beam radius from the center of the profile. An
example of their measured velocity profile can be seen in Figure 1-8. Although the pulse width was short,
the reported deceleration was upwards of 10
8
“g”. This indicated the potential use of dipole force for
accelerating atoms and molecules. This experiment also pointed out the boundary for using this technique,
where the laser intensity will breakdown the irradiated molecules, defeating the neutral goal of the process.

Figure 1-8: Deceleration of C
6
H
6
by the radial dipole force from a pulsed laser (Fulton, Bishop and Barker 2004)
1.2.4.2 Axial Dipole Atom Optics
The interference pattern from two counter-propagating lasers (optical lattice) can create a
diffraction grating through which to scatter the beam. A single 20 mW on-resonance laser was retro-

14

reflected and focused to 25 μm to diffract an atomic beam of sodium (Moskowitz, et al. 1983). The laser
standing wave added transverse momentum to the atoms dependent on the laser power (electric field
strength). An example of the diffraction pattern can be seen in Figure 1-9.

Figure 1-9: Diffraction of a Na atomic beam through an optical lattice (Moskowitz, et al. 1983)
1.2.5 Optical Lattices – Optical Conveyor
When the frequency of one of the two lasers which construct an optical lattice is slightly shifted,
the interference pattern is put in motion (Alt 2004). When the acceleration associated with that motion is
small enough, compared to the depth of the trap, atoms which have been trapped within the potentials
continue to be trapped and are accelerated with the pattern. (This process is the same as waiting for the ball
on a roulette table to slow enough to fall in the bin. After it is in the bin, if the wheel is spun up again, the
ball will be accelerated along with the wheel.) There is precedence for the use of optical lattices for
particle transposition (Kuhr, et al. 2001), when lasers are tuned far from resonance (Miroshnychenko, et al.
2006). One point to consider in these experiments is that either the acceleration or the transposition was
small. This mitigated the magnitude of frequency detuning or rapidity of frequency modulation required
for the constituent lasers. The two reasons for significantly detuning from resonance are firstly, the
scattering of photons, and thus random heating, is inversely related to laser detuning. Secondly, to make
the interference pattern move, the lasers must be detuned from each other. By tuning far from resonance,
the laser delta detuning from each other can be considered small in relation to the detuning from resonance.
This allows simplifying assumptions in the force derivation, by allowing one to treat the resonant coupling
in the system as if it were from only one laser frequency.

15

A B
Figure 1-10: Optical lattice accelerated Cs A) transposition (Schrader, et al. 2001) and B) retention (Schrader, et
al. 2001)
In (Schrader, et al. 2001) an atom is first loaded into a MOT before being transferred to a one
dimensional optical lattice. The lattice bin, with the atom, is then shifted from position A to position B.
The lattice bin is then shifted back from B to A and the atom is reloaded into the MOT. While in the MOT
(position A) and at again at position B the atom can be irradiated with resonant light to ascertain its
existence and position through fluorescence. A breakdown of the particle transposition can be seen in
Figure 1-10 (A). The acceleration imparted on the atom was varied over four orders of magnitude. This
gave reasonable indication to the usefulness of this technique for creating high velocity flows. While the
accelerations in this case were high, the magnitude of the transposition and the highest attained velocity are
small. The measured efficiency, given by the likelihood that the atom stayed in the lattice, can be seen in
Figure 1-10 (B).
1.2.6 Optical Lattices – Temperature Measurement
Recently, pulsed optical lattices have been used to measure the temperature of rarefied gases
(Bookey and Bishop 2006), flames (X. Pan, et al. 2001), and plasmas (X. Pan, P. F. Barker, et al. 2002), as
well as continuum gases (Pan, Shneider and Miles, Power spectrum of coherent Rayleigh-Brillouin
scattering in carbon dioxide 2005). In these experiments a pulsed optical lattice was used to create a
density gradient in the gas. The strength of the density gradient was related to the relative number density
of particles in a velocity group near the velocity of the lattice. The strength of scattered light from a probe
beam irradiating the grating was proportional to the density gradient. Therefore, the intensity of the
scattered light gave information on the velocity distribution of the gas. The density of the gas (rarefied

16

through continuum) changed the analytical tools required to analyze this information, namely the use of
Rayleigh or Rayleigh-Brillouin scattering. At higher densities, the concept has been shown over a range of
gas species and density combinations (Pan, Shneider and Miles, Coherent Rayleigh-Brillouin scattering in
molecular gases 2004).

Figure 1-11: Coherent Rayleigh Scattering signal versus lattice velocity measured by the laser frequency offset
(H. T. Bookey, et al. 2006)
1.2.7 Optical Lattices – Gas Heating
The investigation of energy and momentum deposition to a continuum gas using a pulsed optical
lattice has not been experimentally realized. It has been analytically investigated and numerically
simulated for several configurations. These simulations have attempted to deposit energy into the gas
(Ngalande, Gimelshein and Shneider 2007), as well as induce more effective transport and gas mixing
(Shneider, Barker and Gimelshein, Transport in room temperature gases induced by optical lattices 2006).
In these cases, the pulsed optical lattice was used as a series of deep potential wells which increase
intermolecular collisions. This increase in translational energy was then available for redistribution into
internal modes through collisional relaxation. The technique uniquely excited a single translational energy
mode (the laser propagation axis). It was then through collisional relaxation that energy was transferred,
making this technique well-suited for non-equilibrium experiments.

17

Figure 1-12: Simulated energy deposition into CH
4
from a pulsed optical lattice for various gas pressures
(Ngalande, Gimelshein and Shneider 2007)

18

Chapter 2 Theory
The two forces considered in this study are: forces derived from the absorption and re-emission of
photons (purely resonant), forces derived from the induced dipole potential (and near- and non-resonant).
The latter of these is associated with the species immersion in electric fields, i.e. those caused by a light
field. A laser referred to as “resonant” or “near-resonant” has a frequency in the proximity of an electronic
resonance within the species. For cases which refer to resonant forces, the example atom investigated in
this study was cesium and the transition of interest was the D
2
line (6s
2
S
1/2
↔ 6p
2
P
3/2
). Most of the theory
associated with the D
2
transition is applicable, with minor modifications, to the D
1
transition as well.
Cesium was called out in this investigation because its electronic structure, like other alkali metals, is
amenable to being treated as a two-level system. This offered a great simplification for the quantum
mechanical derivation of resonant forces. Cesium’s transitions are readily excited by available diode lasers
(852 nm for D
2
or 894 nm for D
1
) and an extensive experimental history was available (Steck 2008). For
the case of non-resonant forces, molecular nitrogen was investigated to show the feasibility of molecular,
non-polar, non-resonant, interactions which can be extended to arbitrary combinations of lasers and gas
species. The theories associated with near-resonant and non-resonant dipole forces differ greatly due to
proximity to an electronic resonance and the validity of simplifying assumptions. Therefore, the two
theories will be addressed separately to allow for a more in-depth description of the complex near-resonant
interaction.
Resonant interactions will be discussed based on the derivations of previous authors on atomic and
molecular manipulation by laser fields (Metcalf and van der Straten 1999) (Foot 2005) (Letokhov 2007).
Following these authors, the mechanics of atom-laser interaction will utilize the semi-classical approach.
The laser fields will be considered classical electromagnetic waves, while the atom will be approached as a
quantum mechanical system. Where possible, the equivalent understanding in the light-as-photons model
will be identified. The quantum mechanical derivations used are consistent with (Liboff 1991) (Cohen-
Tannoudji, Diu and Laloë 1977). The optical derivations are consistent with (Hecht 2002).

19

2.1 Laser Fields
In order to treat laser radiation as plane waves of a single frequency, it is assumed that the path
length of any experimental setup is much less than the coherence length of the laser used. As a real system,
a laser does not produce truly monochromatic light, but a bandwidth of frequencies, or line width, around a
center wavelength. Assuming that all photons of all frequencies start in phase, the length at which the
lowest frequency and the highest frequency will become 2 π out-of-phase is called the coherence length and
can be calculated by

2
c
LL
cc
l
π
νω
Δ= =
ΔΔ
(2.1)
For a narrow line, 852 nm, laser with a line width of approximately 1 MHz, the coherence length is
approximately 300 m. This means that for path lengths significantly less than 300 m, the light can be
considered monochromatic, i.e. plane waves with a frequency equal to that of the center frequency of the
laser.
Assuming a monochromatic plane wave, the electric field of a propagating laser field can be
denoted in several ways.

() ( )
00
() ( )
00
(,) sin( )
2
(,) cos( )
2
LL LL
LL LL
ik x t ik x t
LL
ikxt ikxt
LL
ee
Ext E k x t E
i
ee
ExtE kx tE
ωω
ωω
ω
ω
+− +
+−+
−
=+=
+
=+=
(2.2)
Nearly all of the work in this study considers light as a linearly polarized. This allows the theory
describing the electric field oscillation to be limited to one dimension. A major concern with the derivation
of the interactions between atoms and molecules will be the strength of the electric field as a function of
space and time; but, as mentioned, it will not be a function of orientation. The oscillation of the electric
field as a function of the axial direction is given by equation (2.2). The spatial profile as a function of the
radial direction is linked to the intensity profile of the emitting laser. The transverse spatial mode of most
lasers is the TEM
00
mode (Transverse ElectroMagnetic, fundamental mode), which has no electromagnetic
components in the direction of propagation, and forms a Gaussian radial distribution. The intensity profile
for a continuous wave laser can be mathematically represented as

20

2
2
2
11 2
2
4ln2
() ,
Cr L
L
PC
Ir Ce C C
D
π
= ⇐ =− =− (2.3)
For a pulsed laser with a Gaussian temporal envelope, the intensity profile can be can be represented as

22
23 0
() 23
11 2 3
32 2 2
4ln2 4ln2
(, ) , ,
Cr C t t
L
L
L
CC
Irt Ce e C C C
D τ
π
−
−
= ⇐ =− =− =−
ε
(2.4)
Both of these representations are normalized such that the integration over space and time results in the
laser power or pulse energy respectively. The electric field amplitude for a given intensity is given by

2
0
00
2
EI
c ε
= (2.5)
Another spatial distribution considered is that of two counter-propagating fields, i.e. collinear axes
with opposite wave vectors. In this case the oscillating fields will constructively and destructively interfere
to create a wave pattern. It should be noted that the polarization vectors, for these fields, is considered
linearly polarized and co-aligned. The simplest pattern created is that from two lasers of the same
frequency and intensity. This is accomplished by splitting one laser and bringing it back on itself or retro-
reflecting with a mirror. The pattern formed is that of a standing wave. The superposition principle
requires that their field amplitudes are added, not their intensity, thus the square of the summation of their
electric fields, i.e. intensity of the combined field, is given by

{}
2
2 22 2
00 0
( , ) sin( ) sin( ) 4 cos ( )sin ( )
LL LL L L
E xt E kx t E kx t E kx t ωω ω =++ −+ = (2.6)
It can be seen that the electric field is comprised of a spatial oscillation, cos
2
(k
L
x), and a temporal, sin
2
( ω
L
t),
oscillation. By averaging over the fast temporal oscillation which oscillates at the optical frequency, the
resulting electric field as a function of space is given by

()
222 2
00
( , ) 2 cos ( ) 1 cos(2 )
LL
Ext E kx E kx ≅=+ (2.7)
This represents a spatially oscillating electric field with a periodicity of half the constituent laser
wavelength.
A more complex pattern is that of a moving standing wave. Consider that the two crossing lasers
are not of the same frequency nor necessarily of the same intensity. In such a case, the frequency

21

difference between the two lasers causes the interference pattern to go into motion. The square of the sum
of their electric field amplitudes is given by
()( ) ()
()( ) ()
222 22
L1 L1 L1 L2 L2 L2
L1 L2 L1 L2 L1 L2
L1 L2 L1 L2 L1 L2
(,) cos( ) cos( )
cos
cos
Ext E k x t E k x t
EE k k x t
EE k k x t
ωω
ωω
ωω
=−+ −
+−−−
++−+
(2.8)
When k
L1
≈-k
L2
and ω
L1
≈ ω
L2
, the interference term of the field has two components: one with a relatively
long spatial and short temporal period and the other with a short spatial and long temporal period. Later
derivations will only be concerned with the spatial derivative of equation (2.8), thus the portion with the
long spatial period is neglected; additionally the fast oscillating terms, cos
2
is averaged over a temporal
oscillation as was done for equation (2.7). The resultant equation for the electric field amplitude as a
function of space and time is

()
2
L1 L2
(, ) 1 cos( ) Ext E E kx t ω
ΔΔ
≅+ − (2.9)
2.2 Non-Resonant Interaction: Induced Dipole
Due to its relative simplicity, the force on a non-polar species immersed in a light field tuned far
below any electronic resonances is addressed first. For such a field, the distortion of the species’ electronic
structure or polarizability, is reduced to a value equivalent to its value in a static field. This simplification
is valid for fields where ω
L
<< ω
0
, i.e. variations in the electric field are slow enough to be considered static
in the temporal frame of the electronic system. For an ensemble of atoms or molecules, the orientation
specific values for the polarizability (parallel and perpendicular components) can be integrated to a
directionally independent average value. Examples of these polarizabilities are given in the appendix. This
removes much of the complication originating from resonant interactions which must be addressed
quantum mechanically. The induced potential energy and force imparted on a molecule in a static electric
field given by (Boyd 1992)

2
1
2
UE α =− (2.10)

2
1
2
FU E α =−∇ = ∇ (2.11)

22

Equation (2.11) effectively translates a spatial gradient in the electric field to a force on the
species, driving it to the point of least potential. For a Gaussian intensity profile, and far red-detuned
(lower frequency than electronic resonance) fields, this leads to a force pushing the molecule towards the
point of highest intensity. Combining equations (2.5) and (2.11) the force on a molecule in a laser field
becomes

dip
0
() Ir
F
cr
α
ε
∂
=
∂
(2.12)
The analytical form of the force on the species is further specified by applying the intensity profile from
either a continuous wave laser, equation (2.3), or a pulsed laser, from equation (2.4). The force on a
nitrogen molecule, assuming equation (2.4), at the peak and on the axis of a 250 mJ, 250 μm laser pulse can
be seen in Figure 2-1. In this figure a negative force refers to a force on the molecule in the anti-radial
direction, i.e. towards the center of the Gaussian field.

Figure 2-1: Non-resonant induced dipole force vs. radial position
2.3 Resonant Interaction: Quantum Mechanics and the semi-Classical Approach
For interaction systems where the irradiating field is tuned near an electronic resonance within the
immersed species, the interaction mechanics are more complicated than for a non-resonant interaction;
therefore, a quantum mechanical approach must be used. Continuing along the basis of research which has
been done in the field of resonant laser interactions, an atomic system is considered in lieu of the more
complex molecular system. When relevant, the D
2
transition (6s
2
S
1/2
↔ 6p
2
P
3/2
) in cesium is used as an
example system.
2.3.1 Schrödinger and Hamiltonian
It is a postulate of quantum physics that for a well-defined observable, X, there exists an operator,
ˆ
X , such that a measurement of X produces values, x, which are eigenvalues of the relation
0 100 200 300
-2.5
-2
-1.5
-1
-0.5
0
x 10
-19
Radial Position [ μm]
Force [N]

23

ˆ
Xx ϕϕ = (2.13)
The quantum mechanical eigenvalue equation for the energy of a system is expressed by the time-
independent Schrödinger equation

ˆ
() () Hr r ϕϕ =
G G ε (2.14)
It is this equation that is solved for the electron-nucleus system to attain the electronic energy levels of an
atom in the n
th
electronic state,
n
ε . The wavefunction expresses the probability of an electron’s existence,
as a function of position and momentum in the atom-centric frame such that ,,, , ,
xy z
rxyzppp =< >
G . The
operator on the left hand side is the Hamiltonian operator and eigenvalues are the electronic energy levels.
Hamiltonian mechanics is an alternate formulation of classical (Newtonian) mechanics where the
Hamiltonian operator is an expression of the total energy, kinetic and potential, of the system. By virtue of
its formulation, the time rate of change of position (velocity) can be found by taking the derivative of the
Hamiltonian with respect to momentum. Likewise the time rate of change of momentum (force) can be
found by the negative derivative of the Hamiltonian with respect to position. This latter statement is of
particular importance as it represents the Hamiltonian equivalent of the force due to a spatially
inhomogeneous potential field, e.g. gravity, charge, or induced dipole potential.
The evolution of the wavefunciton, as a function of time, is given by the time-dependent
Schrödinger equation

ˆ
(,) (,) irt Hrt
t
ψψ
∂
=
∂
G G = (2.15)
This sets the relation between the time rate of change of the wavefunction and the Hamiltonian (energy) of
the system at that time. If it is assumed that the Hamiltonian is independent of time (as it relates to the total
energy of a closed system), the solutions to time-dependent Schrödinger equation (2.15) can be found by
separation of variables. The solution must include the spatial solution to the time-independent Schrödinger
equation (2.14), as well as the temporal solution to an ODE of the form f’(t)+f(t)=0. Thus eigenfunctions
(wavefunctions) of equation (2.15) will take the form
(, ) ( )
it
rt re ψϕ
−
=
= G G ε
(2.16)

24

By the principle of superposition, the total state of the atom is given by the sum of all the available states
(wavefunctions) scaled by a proportionality constant. This is expressed as
(,) () (,) () ( )
n
it
nn n n
nn
rt c t rt c t re ψψ ϕ
−
==
∑∑
= G GG ε
(2.17)
where |c
n
(t)|
2
is the probability of finding the system in n
th
state at time t. The n
th
state is in turn described
by the wavefunciton, ψ
n
.
At this point, consider a form of notation which operates on two states, giving the coupling
between them. In the “bra-ket” notation, the integration of the inner product of the n
th
state and modified
(by an operator) k
th
state over all of the atom-centric space is given by

*3
ˆˆ
() ()
nk n k
XrXrdr ϕϕ ϕ ϕ =
∫
G GG
(2.18)
Because the solutions to the Schrödinger equation are eigenfunctions of the problem and thus a basis for the
system, they are orthogonal. In this way < φ
n
| φ
k
> = 0 for n ≠ k. In other words, integrating over the inner
product of two distinct un-modified states results in 0. In order for the normalization of the wavefunctions
to make sense, the sum of the expectation value of all available states must equal 1. This is because the
square of the wavefunciton amplitude is a probability which must sum to unity for all possible states. In
order to do this one takes the bra and ket of the same state to get the square of the magnitude as it is in
essence taking the inner product of itself.
1
kk
k
ϕϕ =
∑
(2.19)
At this point, the stated equations are exact; no approximations have been made. However, the
case for an atom in a radiation field is unsolvable for anything but the most elementary cases, and must be
addressed using some form of approximation. Through the method of time-dependent perturbation theory,
an atom is initially assumed to be in an unperturbed state which satisfies the time-independent Schrödinger
equation, equation (2.14). A perturbation is turned on at time t = 0, and its effect on the energy of the
system is described by a perturbation Hamiltonian such that the total Hamiltonian for the system is given
by

0
ˆˆ ˆ
(,) ( ) (,) Hrt H r H rt ε ′ =+
G GG
(2.20)

25

where ε is a parameter of “smallness” and will be addressed momentarily. Start with equation (2.15) and
substitute in equations (2.17) and (2.20). Finally start from the left and integrate over all space and all
states to consider the atom as a whole. The interim equation is then given by

0
()
ˆˆ
() () (,) ()
n
n n
it
nk n
nk
it it
nkn n kn
nk nk
icte
t
H r cte H rt cte
ϕϕ
ϕϕ ε ϕ ϕ
−
−−
∂
=
∂
′ +
∑∑
∑∑ ∑∑
===
=GG
ε
εε
(2.21)
The n summation comes from equation (2.17) where n sums over the states which we are interested (all
possible) while the k summation comes from the integration over all possible states with which the n
th
state
may couple (again, all possible). On the left-hand side all terms < φ
n
| φ
k
> = 0 for n ≠ k. Additionally,
through the normalization condition (2.19), the sum of all n = k terms equals 1. On the right-hand side a
similar argument is made after exchanging the unperturbed Hamiltonian for a constant through the time-
independent Schrödinger equation (2.14). The perturbation term on the other hand does not satisfy the
orthogonality of eigenfunctions, because the wavefunctions are modified by the perturbation operator.
Using a final substitution to shorten the notation for the perturbation,

*
ˆˆ
() ( , ) ( ) ( , ) ( )
nk n k n k
H t Hrt rHrt rdr ϕϕ ϕ ϕ ′′ ′ ==
∫
G GG GG (2.22)
and the derivative product rule, the time rate of change of the coefficients c
n
(t) is given by
() () () ' () ()
nn n k
it i t it i t
nnn nn nkk
k
d
ie c t c te c te H tc te
dt
ε
−− − −
+= +
∑
== = = =εε ε ε
εε (2.23)
Equation (2.23)can be simplified by cancelling the identical term on either side of the equality and
multiplying both sides by the positive n
th
state exponential to simplify the exponential terms to one. After
replacing the energy with the Bohr angular frequency,
()
nk n k
ω =− = εε , equation (2.23) becomes
() ' () ()
nk
it
nnkk
k
d
ict H tcte
dt
ω
ε =
∑
= (2.24)
2.3.2 Two-Level Atom in an Oscillating Electric Field
At this point one of two further assumptions must be made about the perturbation. Either the
perturbation is assumed to be small, from either a short time evolution or weak field interaction, or the
system can be reduced to a two-level system, where the perturbation only couples between two energy

26

states. Under the former assumption a power expansion in ε is taken to converge the terms of c
n
(t) and
express the probability of being in an excited state as a first (or arbitrary) order approximation. This
method is only valid in cases where the probability of being in the excited state is small and is limited by
the complexity of taking the calculation to an arbitrary power, in the interest of accuracy. Under the second
assumption, ε is set equal to 1 and the indices are reduced to n,k = 1,2. Because the forces considered in
this study are acting over long periods of time, with respect to the perturbation of the electronic structure,
and an alkali atom readily behaves like a two-level system, the second assumption is made. This resonance
approximation is only appropriate when the perturbation frequency (laser frequency) is close to the
resonant frequency of the transition between states 1 and 2 (ground and excited: 1,2 → g,e) such that all
other states can be ignored. The pair of coupled equations thus derived from equation (2.24) is then given
by

0
0
() ' () ()
() ' () ()
it
ggee
it
eegg
d
ict H tcte
dt
d
ict H tcte
dt
ω
ω
−
=
=
= = (2.25)
2.3.3 Rabi Frequency
The next step is to apply the appropriate perturbation function. At this point the single laser
derivation and the derivation appropriate for the interference pattern of two counter-propagating lasers
diverge. However, regardless of the shape of the perturbing field, it can be assumed that it is made up of an
amplitude term and an oscillating term. Therefore the Hamiltonian for the perturbed atom can be separated
into similar parts, one for the energy of the perturbed state (amplitude) and the other for its oscillation in
time and space. The perturbation Hamiltonian is calculated as an induced electric dipole potential by using
the electric dipole approximation. It is assumed that the electric field does not vary over the space of the
atomic wavefunction , i.e. λ
L
>>r
atom
. By writing the (valence) electron-nucleus system as a dipole, the
Hamiltonian (potential) is

ˆ
(,) () Hrt eEt r ′ =− ⋅
G G G (2.26)
Next, reintegrate the Hamiltonian into equation (2.25) and the bra-ket notation. It is convenient to
define a scalar value for the interaction strength between the magnitude of the irradiating field, i.e. electric

27

field amplitude, and the dipole matrix element, i.e. the integration of the dipole moment over all space for a
given electric field orientation (integral represented by the bra ket). This scalar value is defined as the Rabi
frequency and is given by

0
12
ˆ
eE
r ϕε ϕ Ω= ⋅
G = (2.27)
It will be shown that this definition also yields a relevant characteristic of the atom in the radiant field: the
frequency of oscillation between the ground and excited states when taken independently of spontaneous
emission. If viewed from the perspective of the photon approach (wave-particle duality), this represents the
average rate of stimulated adsorption and subsequent stimulate emission which would cycle the atom
between the ground and excited states. Calculating the Rabi frequency analytically is non-trivial; however,
there are several relations that will be introduced which can lead to a value derived from experimentation.
Until then, it is a convenient way to book-keep the interaction Hamiltonian and the strength of the
interaction as it relates to the induced dipole potential.
2.3.4 State Probability – Coherent Evolution in a Laser Field
As an example, the probability coefficients given in equation (2.25) can be solved for a perturbing
field of one irradiating laser which takes the sine form of equation (2.2). Consider the more general form
where the laser wavelength is different than the transition wavelength, i.e. near-resonance but not on-
resonance. By differentiating equation (2.25) such that the equations can be decoupled, and substituting
equations (2.2), (2.26) and (2.27), the problem can be solved as an initial value problem. Given the initial
condition that the atom starts with a probability of being in the ground state equal to unity, the time
development of the probability amplitudes (where |c
g,e
(t)|
2
is the state probability/population) is given by

2
2
() cos sin
22
() sin
2
it
g
it
e
tt
ct i e
t
ct i e
δ
δ
δ +
−
⎛⎞ ′′ ΩΩ
⎟ ⎜
=− ⎟
⎜
⎟
⎜ ⎟
′ ⎝⎠ Ω
′ ΩΩ
=−
′ Ω
(2.28)

22
δ ′ Ω≡ Ω + (2.29)
To reach this solution, the Rotating Wave Approximation has been made. This approximation
neglects terms of the order 1/ ω
L
compared with terms of the order 1/ δ. This is only valid for near-resonant
situations where δ<< ω
L
. The derivation, so far, only addresses the stimulated forcing of atomic energy

28

states by the irradiating laser; but, does not address the quantum phenomenon of spontaneous emission,
which artificially resets the time development of the excited state probability. This evolution is shown for
Figure 2-2, without spontaneous emission. The dashed line represents the resonance condition of δ = 0, the
grey line represents δ = Γ, and the black line represents δ = 2.5 Γ. Time is given in units of 1/ Γ and all lines
assume Ω = Γ. Γ is the radiative width of the transition which will be defined in section 2.3.6 below.

Figure 2-2: Excited state probability vs. time for various δ, without spontaneous emission
2.3.5 Spontaneous Emission and State Lifetime
The time an atom will spend in an excited state before spontaneously releasing a photon and
relaxing to the ground state is an important characteristic of the transition. An approximation, or alternate
understanding, can be understood through classical mechanics. Consider the atom as a classical system
with an excited electron acting as a harmonic oscillator. The excited electron then exhibits an electric
dipole oscillating at an angular frequency. This oscillating dipole will radiate a power equal to

22 4
0
3
0
12
ex
P
c
ω
πε
= (2.30)
This radiated power in turn depletes the stored energy in the excited state. Consider the total energy of the
harmonic motion as

22
0
2
e
mx ω
= ε (2.31)
The rate of the energy lost from the system is equal to the power radiated such that

22
3
0
6
e
de
dt
mc
ω
τ
πε
=− = −
εε
ε (2.32)
0 2 4 6 8 10
0
0.2
0.4
0.6
0.8
1
1/ Γ
Excitation probability

δ=2.5 Γ
δ= Γ
δ=0

29

For strong transitions, the lifetime, τ, from the classical approach is very close to the quantum mechanical
result. While this is a useful way of thinking about the process of spontaneous emission, it is not an
accurate derivation to apply to our system.
Returning to the quantum mechanical view of the atom, the derivation of the spontaneous
emission of a photon from an atom becomes more complex. The same derivation for the stimulated
processes of an atom immersed in an oscillating electric field can be done for an atom in free space which
emits a single photon. This derivation requires the quantization of the emitted photon’s field and the
inclusion of vacuum fluctuations. The final result of that derivation gives the change in the probability of
being in the excited state as a function of time as

23
2
2122
3
0
1
() ()
2
3
de
ct r c t
dt
c
ω
ϕϕ
πε
=−
G = (2.33)
This yields a quantum mechanical lifetime equal to

23
2
12
3
0
1
3
e
r
c
ω
ϕϕ
τ
πε
=
G = (2.34)
Because the dipole transition matrix element (the bra-ket notation part) is difficult to analytically calculate,
its value is often derived through the relation in equation (2.34) and experimental measurement of the
lifetime. This element can be related to the element in the Rabi frequency to give a value to the bra-ket
notation in equation (2.27). After adding the probability of randomly relaxing to ground to the time
evolution given in Figure 2-2, the average of 1, 10, and 100 atoms is given in Figure 2-3 where Ω = Γ and δ
= - Γ.

Figure 2-3: Excited state probability vs. time, with spontaneous emission (Metcalf and van der Straten 1999)

30

2.3.6 Natural Width
Because of the finite lifetime of the excited state, the frequency of the emitted photon is not
completely determined as a discrete quantity. The shape of the probability, or spectrum, of frequencies
emitted by the atom during spontaneous emission is given by the Lorentzian shape

() ()
nw
2
2
0
1
()
2
2
f ω
π
ωω
⎡ ⎤
⎢ ⎥
Γ
⎢ ⎥
=
⎢ ⎥
⎢ ⎥ −+Γ
⎣ ⎦
(2.35)
Equation (2.35) is normalized such that the integral over all frequencies is equal to 1. The width, Γ, is
referred to as the natural line width and defines the radiative broadening of a spectral line. The root of this
width is derived from the Heisenberg uncertainty principle as it relates to time and energy.
t
tt
εε ω ΔΔ > ⇒ Δ ≈ ⇒ Δ ≈
ΔΔ
==
== (2.36)
Thus, the natural line width of the transition can be related to the lifetime of the transition by

1
τ
Γ≡ (2.37)
where τ is the mean lifetime of the transition. An example of the natural line shape of the cesium D
2
line
can be seen in Figure 2-4.

Figure 2-4: Natural transition probability line shape vs. frequency
An additional concept to note is that of fine and hyperfine splitting. These concepts relate to
electronic energy states and the addition of levels and sublevels within those states. The existence and
separation of the P
1/2
and P
3/2
excited energy states is an example of fine splitting. Additionally there are
sublevels to these two excited states. In the case of the P
3/2
energy state, there exists four hyperfine levels,
F = 3…5. Generally speaking, the line shape of any particular transition, e.g. the F = 4 ground state to the
P
3/2
F=5 excited state, has the same spectral width discussed before, e.g. Figure 2-4. For reference, the
-4 -2 0 2 4
0.1
0.2
0.3
0.4
0.5
0.6
( ω - ω
0
) / Γ
Prob. Density [s/rad]

31

hyperfine splitting associated with cesium D-lines can found in the appendix. Because of quantum
mechanical selection rules, transitions are only allowed between states of ±1 hyperfine level. This excludes
a transition from the F = 5 excited state and the F = 3 ground state. This requires that an atom, being
excited by a field tuned to the F = 4 ground to F = 5 excited state transition, will cycle between these two
hyperfine levels with no chance of decaying to the F = 3 ground state. There is, however, a finite
possibility of erroneous excitation to the F = 4 excited state which in turn can decay into the F = 3 ground
state, excluding the atom from further cycling. This concept plays a vital role in resonant phenomena
which rely on the cycling atoms between ground and excited states and the necessity of maintaining this
cycle.
2.3.7 Doppler and Recoil Effects
Under non-relativistic velocity transformations, the momentum of a photon is invariant. This
means that the magnitude of the change of momentum of the atom by spontaneous emission, stimulated
absorption, or stimulated emission is constant and is related to frequency by

0
recoil recoil
pmv k
c
ω
== =
= = (2.38)
Which leads to a change of momentum of an atom which absorbs or emits a photon is given by

0
mv k mv ±=
G G G = (2.39)
Since the photon is assumed to be absorbed or emitted, the frequency used to calculate the wave vector
must be equivalent to the transition frequency, ω
0
. Note that the transition frequency is a spectrum of
possible frequencies as stated above. Conservation of energy requires that the internal and kinetic energies
of the photon-atom system must be constant

22
00
11
22
mv mv ωω ±= ±
G G == (2.40)
If the atom is moving, the frequency of the photon, relative to the laboratory frame, will be shifted by the
Doppler Effect in the atom frame. For a photon which is absorbed, or conversely emitted, the frequency of
the photon in the laboratory frame will be given by

00
kv R ωω =+ ⋅ ±
G G == = (2.41)

32

Since it is assumed that the photon is being absorbed or emitted from the atom, it is necessary for the
photon to contain the energy equivalent to the atomic change in kinetic energy due to the change in
momentum from the recoil. The total detuning of an adsorbed photon from resonance includes both
Doppler and recoil effects and is given by

0 L
kv R δω ω =+ ⋅ − −
G G = (2.42)
The “Doppler effect” is a change in observed photon frequency due to the velocity of the emitting or
absorbing atom and the “recoil effect” is a shift in frequency due to kinetic energy transfer to the atom in
the process of absorption or emission. The frequency shift due to the recoil effect is, for most
applications, negligible. For the Cs D
2
line the ratio h Γ/2 πR equals 2.5x10
3
. Therefore, the recoil effect is
not considered in this study.
2.4 Resonant Interaction: Optical Bloch Equations
The density matrix is a way of describing the state of the atom at any particular time. The
diagonal terms of the density matrix give the probability (or population of an ensemble) of being in the two
states, ground and excited; the off-diagonal terms, or coherences, describe the coupling between the two
states within the field. The density matrix for a two-level atom is given by

()
2
*
1112 **
12
2
*
2
21 2
gg ge
eg ee
cccc
cc
c
cc c
ρρ
ρ
ρρ
⎛⎞
⎛⎞ ⎛ ⎞ ⎟ ⎜
⎟ ⎟⎟ ⎜⎜ ⎜
⎟ ⎟⎟
⎜⎜ ==⎜ =
⎟ ⎟⎟
⎜⎜ ⎜
⎟ ⎟⎟
⎟⎟ ⎜⎜ ⎜ ⎟ ⎝⎠ ⎝ ⎠
⎟ ⎜
⎝⎠
(2.43)
The time rate of change of the density matrix will be required to calculate the forces on the atom within the
field. The formulation of these rates (time derivative of the density matrix elements) is described by the
Optical Bloch Equations (OBE) which are optical equivalents to the Bloch equations for nuclear magnetic
resonance.
2.4.1 Optical Bloch Equations
A reformulation of the density matrix elements, as they relate proportionality constants derived in
equation (2.25), and inclusion of the spontaneous emission of photons as damping to the stimulated
excitation of the atom, ultimately results in the OBE:

33

()
()
()
()
*
*
*
2
2
22
22
gg
ee eg ge
ee
ee ge eg
ge
ge ee gg
eg
eg gg ee
d
i
dt
d i
dt
d
i
i
dt
d
i
i
dt
ρ
ρρ ρ
ρ
ρρ ρ
ρ
δρ ρ ρ
ρ
δρ ρ ρ
=+Γ + Ω − Ω
=−Γ + Ω − Ω
⎛⎞
Γ
⎟ ⎜
=− + ⎟ + Ω −
⎜
⎟
⎜ ⎟
⎝⎠
⎛⎞
Γ
⎟ ⎜
=− − ⎟ + Ω −
⎜
⎟
⎜ ⎟
⎝⎠

(2.44)
where
it
ij ij
e
δ
ρρ
−
≡ . For
*
eg ge
ρρ = the steady state solution to the OBE yields

()() 22 1
1
12(1)
eg
gg ee ee
i
is
s
ss
ρ
δ
ρρ ρ
Ω
=
Γ− +
−= ⇒ =
++
(2.45)
where

()
2
0
22 2
2
4
12
s
s
δ
δ
Ω
==
+Γ
+Γ
(2.46)
and

2
2
0sat
2sII =Ω Γ = (2.47)
It can be seen that as the strength of the external field increases ( Ω→∞, s>>1), the population difference
asymptotes, ρ
gg
- ρ
ee
→ 0. If these equations are used to calculate the probability of finding an atom in the
excited state, the resulting evolution can be seen in Figure 2-5. The OBE yield the same time evolution as
the evolution shown in Figure 2-3 using the average of an infinite number of atoms.

Figure 2-5: Excited state probability vs. time using the OBE (Metcalf and van der Straten 1999)

34

2.4.2 Saturation and Power Broadening
As defined in this derivation, the value for I
sat
is the intensity required for an on-resonance field to
cause the population of the excited state to be 1/4 of the total population. The transition at this point is said
to be “saturated” such that further increases in laser intensity does not result in comparable increases in
excited population. This intensity is directly calculable and given by

sat
3
0
3
hc
I
π
λτ
= (2.48)
It is now possible to explicitly calculate the emission rate of spontaneous photons as the product of the rate
of spontaneous emission and the likelihood of being in the excited state:

( )
0
2
0
2
12
pee
s
s
γρ
δ
Γ
=Γ =
++ Γ
(2.49)
When s
0
→ ∞, the maximum photon emission rate is given by Γ/2. This limitation is important for
processes which rely on the spontaneous emission of photons for momentum dispersion, such as optical
molasses. Note also that the rate of the photon scattering as a function of laser detuning has a shape with a
full width half maximum of Γ for s
0
<<1. Equation (2.49) can be re-written as

()
()
( )
00
2
2
0
0
2
0
22
4
12
11
1
ss
s
s
s
δ
δ
ΓΓ
=
⎛⎞
⎟ ⎜
++ Γ
⎟
⎜
++ ⎟
⎜
⎟
⎜
⎟ ⎜ Γ+
⎝⎠
(2.50)
Equation (2.50) shows that as the on-resonance saturation parameter increases, the effective width of the
profile increases as well. “Power broadening” affects the bandwidth of photon frequencies which excite
the atom for a given laser intensity.
2.5 Resonant Interaction: Forces
Through the Ehrenfest theorem, the quantum mechanical equivalent to the classical force due to a
spatial gradient in potential energy is given by the expectation value of the gradient of the Hamiltonian
such that

ˆ
H
F
x
∂
=−
∂
(2.51)
The expectation value of the Hamiltonian is calculated by

35

*
nk n k
nk
AccA ϕϕ =
∑∑
(2.52)
This force is related to the time dependant wavefunction describing the electronic structure of the atom. By
following equation (2.51) through with the OBE, the force on an atom can be described by

*
*
eg eg
F
xx
ρρ
⎛⎞
∂Ω ∂Ω
⎟ ⎜
⎟
=⎜ +
⎟
⎜
⎟ ⎟ ⎜∂∂
⎝⎠
= (2.53)
where x is the axis of interest, i.e. the axial laser direction or the radial laser direction. It is important to
note at this point, equation (2.53) describes all forces (scattering and dipole) on the atom within the laser
field as long as the matrix element, ρ, includes relaxation due to spontaneous emission. In order to
conveniently assess the variation of these forces with space, the gradient of the Rabi frequency, a measure
of intensity, is defined as
()
ri
qiq
x
∂Ω
=+ Ω
∂
(2.54)
where (q
r
+iq
i
) is the logarithmic derivative of Ω separated into its real and imaginary components. The
force is then separated into a real derivative corresponding to the amplitude gradient and the imaginary part
corresponding to the phase gradient. Substituting equation (2.54) into equation (2.53) yields

( ) ( )
** * *
r eg eg i eg eg
Fq iq ρρ ρ ρ =Ω +Ω + Ω −Ω == (2.55)
Equation (2.55) is general enough that it can be used for any situation where the OBE can be solved for the
density matrix elements. It is important to note that the notation of the complex Rabi frequency is for
book-keeping purposes only, and it should be made clear that the force is real. The following extensions of
the derivation give the forces a final form and application to the force.
2.5.1 Atom in a Single Laser Field
Following previous sections and substituting the value for ρ
ge
from equation (2.45) into equation
(2.55) the force can be given by

1
12
ri
s
Fq q
s
δ
⎛⎞
⎟ ⎜
=− +Γ⎟
⎜
⎟
⎜ ⎟
⎝⎠ +
= (2.56)
First, consider the force on an atom along the axial direction of the laser for a single traveling wave. In this
case, there is no amplitude gradient along the laser axis, but there is a phase gradient. Therefore, for the

36

case of the single travelling wave, given by equation (2.2), q
r
= 0 and q
i
= k. This results in a force on an
atom equal to

()
0
scat
2
0
2
12
ks
F
s δ
Γ
=
++ Γ
= (2.57)
This force is the scattering force due to photon absorption and spontaneous emission. Because the
derivative was taken along the laser propagation direction, there is no information for the force due to
gradients in the radial direction, which will be addressed next. This form of the equation properly
addresses the asymptotic behavior of the force with increasing laser power (s
0
→∞). The asymptote is equal
to one photon momentum ( ћk) times half the scattering rate. This results from the maximum steady state
excited population of ½, due to continual stimulated absorption and emission.
Starting with equation (2.51), take the derivative in the radial direction, instead of the axial. There
would be no component of phase change along the wave front as the laser light is assumed to be coherent.
Therefore all that remains is the gradient in the laser intensity due to the TEM
00
laser spatial mode.

()
2
dip
42
g
F
rr r
ε
δδ
⎛⎞ ∂Δ
∂Ω Ω∂Ω
⎟ ⎜
⎟
≅− =− ⎜ = −
⎟
⎜
⎟ ⎟ ⎜ ∂∂ ∂
⎝⎠
==
(2.58)
This force is the induced dipole force caused by the atoms emersion in the electric field. By combining
equations (2.3) and (2.47) the force on a near-resonant atom in a continuous wave Gaussian can be found
by substituting the derivative of the radial intensity profile into the following equation

2
dip
sat
() 1
28
Ir
F
rI r δδ
∂ Ω∂Ω Γ
≅− =−
∂∂
==
(2.59)
In this derivation a few points should be made. The radial derivative of intensity is negative in the positive
radial direction. This leads to a positive force (with the radial direction) for positive detuning (blue) and a
negative force for negative detuning (red). In other words, for red-detuned light, atoms would be drawn to
the center of the Gaussian profile and vice-versa for blue-detuned light.
2.5.2 Moving Atom in a Standing Wave
The interference pattern of two counter propagating laser fields is a slightly more complicated
endeavor. The motion of the atom within the interference pattern (optical lattice) must be addressed. In the

37

derivation so far, spatial gradients in the field have been ignored through the electric dipole approximation.
If the atom is stationary or the gradient of the field is not strong, this is a reasonable approximation. In
order to now account for the atomic motion within the optical lattice, and the spatial gradients traversed by
the moving atom, a first-order correction to the OBE is made such that the time derivatives of the
appropriate quantities are modified by the spatial gradient times the velocity. This transforms the spatial
gradient at a point into a time derivative. This first-order approximation is only appropriate if the velocity
of the atom relative to the standing wave is slow enough such that the state of the atom adiabatically
changes with the change in external parameters, e.g. gradient in field strength. Under such a condition, the
atom’s state at any time/position is considered steady state. These first order corrections are given by

x
v
tt x
∂Ω ∂Ω ∂Ω
=+
∂∂ ∂
(2.60)
and

eg eg eg
x
v
tt x
ρρ ρ ∂∂ ∂
=+
∂∂ ∂
(2.61)
In order to arrive at a general force for an atom in a standing wave, the following formulations are
combined: the electric dipole as the Hamiltonian [eqn. (2.26)], the electric dipole approximation [the field
does not vary over the atomic volume], standing wave [eqn. (2.7)], the definitions for the Rabi frequency
[eqn. (2.27)] and OBE [eqn. (2.44)], first order correction to the OBE [eqn. (2.60) and (2.61)], and the total
force on an atom in a laser field [eqn. (2.55)]. The force on a moving atom within a one dimensional
optical lattice is given by

222 2
22 2
(1 ) 2 ( 4)
1
1
(4)(1)
rxr
ss
s
Fq vq
s
s
δ
δ
δ
⎛⎞
−Γ − +Γ
⎟ ⎜
⎟
⎜
⎟ =− −
⎜
⎟
⎜
⎟
+
⎜ +Γ + Γ ⎟
⎝⎠
= (2.62)
From the cosine in electric field, the derivative terms are given by q
i
=0 and q
r
=-k tan(kx). In order for this
equation to satisfy the slow velocity assumption made during the first-order correction to the OBE, the
following inequality must be true

1
1
2
x
kv
π Γ
(2.63)

38

For a cesium atom, and a laser tuned near the D
2
transition, this inequality is unity for a velocity of
approximately 28 m/s. This means that the above derivation is only appropriate for relative velocities
between the lattice and the atom of around 3 m/s or less.
Using equation (2.62) and substituting values consistent with the center of two 100 mW,
collimated, 1 mm, lasers with a detuning δ = -50 Γ, the force and corresponding acceleration acting on the
atom at rest can be seen in Figure 2-6.

Figure 2-6: Resonant induced dipole force vs. axial position within an optical lattice
The shape is defined by the tangent in the derivative and the oscillation in the light intensity within the
lattice. The highest gradient is not at the point of highest intensity, where there is no gradient. Therefore
there is a bend at the anti-nodes where the intensity is high but the gradient is still weak.

-4 -2 0 2 4
x 10
-7
-1
-0.5
0
0.5
1
x 10
-18
Axial Position [m]
Force [N]
-4 -2 0 2 4
x 10
-7
-4
-2
0
2
4
x 10
6
Axial Position [m]
Accel. [m/s
2
]

39

Chapter 3 Methodology
In order to address selected examples of laser modification of atomic or molecular flows, two
numerical approaches and one experimental study were investigated. The two numerical approaches
addressed two disparate regimes; each approach representing atomic and/or molecular components of the
working gas. One approach allowed for intermolecular collisions (continuum and near-continuum flow);
the other approach considered the species as experiencing collisionless trajectories (rarefied flow). For a
detailed derivation of kinetic theory associated with collisional and collisionless flows, see (Bird 1994).
The simpler of these regimes, rarefied flow, is associated with atomic and molecular beam flows (Ramsey
1956). Neutral particle beams are characterized by collimation and flow direction which greatly reduces
the density and transverse temperature of the flow. Under these conditions, the influence of intermolecular
collisions can be neglected. Rarefied flows are prevalent in micro-machining and material processing
applications as well as space simulations and atomic physics research. The second regime, which is near-
continuum, must be addressed by a simulation package which can properly model intermolecular collisions
in the flow, as well as the internal modes and energy redistribution associated with the collisions. This is
particularly important for laser induced heating of a neutral gas. In this process, the flow of energy from
the laser field to translational and internal energy modes is the key process of interest and is entirely
dependent on the process of collisional relaxation (Ngalande, Gimelshein and Shneider 2007).
The first numerical approach used was a custom particle trajectory code, written specifically for
simulating the effect of a near-resonant laser field on an alkali atomic beam. The simplicity of rarefied gas
dynamic simulation, and the complexity of the resonant laser interaction, causes the required modification
of an existing collisional flow simulation package to be both excessive and difficult; therefore, a custom
code was written from the ground up. The code’s primary purpose was to offer a tool for managing the
complex forces described in Chapter 2 as they apply to spatially inhomogeneous systems such as Gaussian
laser fields, laser interference patterns, and near-Gaussian distributions associated with particle beams. The
simulated flows represent select examples relevant to the lithographic and atomic physics communities. As

40

a validation of writing the custom particle trajectory code, an experiment was conducted. A thermal [373
K] cesium beam was steered by means of the photon scattering force.
The second code employed in this study was a thoroughly validated DSMC code, SMILE (Ivanov
and Rogasinsky 1988), which was specifically modified to incorporate non-resonant forces induced on
molecules by pulsed laser fields. Modification and execution of this code represents the first simulations,
to the author’s knowledge, of molecular trajectory modification by a pulsed laser field for the purpose of
flow steering and collimation. The SMILE code was chosen for its long history of experimental validation,
established by historic fidelity in studies of rarefied flows (Ivanov, Kashkovsky, et al. 1997). The code was
of particular use in modeling flows where the temperature and density were sufficient to make the influence
of intermolecular collisions the dominant aspect of the flow, e.g. laser collimation of an effluent source and
neutral gas heating.
3.1 Numerical: TCL Code
The TCL code is a Monte Carlo particle trajectory code written specifically for this study. The
TCL code is a parallel Fortran 77 code which simulates a pre-determined number of particle trajectories
through a domain. On each processor, individual trajectories are simulated in series, independent of its
predecessor. At the end of the simulation, the final location of the recorded trajectories is gathered and
used for sampling the spatial profile of the particles on the sampling plane. The domain does not contain
any formal geometry with which the particle may interact. The domain is constructed such that a particle
passes from injection to sampling, having only interacted with a laser field. The domain is representative
of the following basic geometry: a stagnation plenum emptying into a differential pumping chamber with a
skimming orifice opposite the plenum orifice. The skimming orifice separates the differential pumping
chamber from, another, independently pumped region which houses the detection volume. If the inlet area
of the differential pump is much greater than the area of the orifices, the background pressure buildup in the
differential pumping chamber will be many orders of magnitude lower than the plenum pressure. The
number of particles which would pass directly from the stagnation chamber, through the differential
pumping chamber, and through the skimmer, would be on the same order or higher than the number of
background particles which reflect around the differential pumping chamber and on through the skimmer.
Additionally, the particles which pass directly from the plenum orifice, through the skimming orifice, are

41

highly directional, while those which reflect around the differential pumping chamber will be distributed
over nearly a full hemisphere. This means that, of the particles which reach the detection volume, the
majority will have originated from the plenum, having passed a straight line from the plenum through the
skimmer and onto the detection volume. This is the basis for geometrically skimming an effluent flow to
create a neutral particle beam. The injection surface of the TCL code represents the skimmer surface, the
origin of a directional flow.
The TCL code is a fully three dimensional code; the code allows for the simulation of any
combination of particle injection and laser field geometries. The code starts a trajectory by injecting the
particle on an injection surface. The particle’s injection characteristics, position and velocity, are obtained
by sampling from a set of distributions read into the program from a set of text files. This allows the
program to incorporate the input characteristics representative of any arbitrary distribution desired for a
given experimental setup, e.g. the geometric skimming example given above. The distribution used for this
study is discussed in section 3.1.1. The particle is assumed to be in free flight when not interacting with the
laser field; it is free of both intermolecular collisions and external forces. Therefore outside the interaction
region, the particle’s motion is simply assumed to continue along the velocity vector, or line-of-motion. A
diagram of the simulation domain can be seen in Figure 3-1. Note that the generic setup in the TCL
simulation is a circular profile cesium atomic beam interacting with a perpendicular circular profile laser
field; a setup which requires three dimensional simulation capabilities.

Figure 3-1: Notional diagram of TCL code domain
While within the interaction region the particle is simulated through one of a variety of simulation
methods discussed in section 3.1.2. Once the particle has left the interaction region, the intersection
between the particle’s final line-of-motion, and the sample plane, is found. In essence, this is a “drift”
portion of the simulation which translates the velocity distribution into a spatial distribution as would be

42

done in a notional experimental setup. At the sample plane, the particle is saved until a statistically
relevant number have been collected. Then histograms are taken of the sampled particles position,
velocity, and other attributes maintained by the code, e.g. electronic state, time of flight, number of
transition cycles, etc... The limits of the histograms are dynamically set such that the upper and lower
bounds are set ± 3 σ from the mean, with overflow bins counting those particles which are outside of the
histogram limits.
3.1.1 Atomic Beam Formation
The first requirement for the creation of a predictive simulation of the effect of a laser field on a
thermal atomic beam, is the representative creation of the initial un-perturbed atomic beam. For the
following example, the nominal conditions are as follows: cesium working fluid, an oven temperature of
350 K, an oven orifice diameter of 1 mm, a skimmer diameter of 1 mm, and a skimmer-oven separation of
100 mm. Experimentally, such an atomic beam can be created by heating pure cesium in an oven and
geometrically skimming the effluence to create a beam as described above. An orifice in the oven wall
would allow atoms from the evaporated metal to leave towards the test section. In order to numerically
simulate this condition, the phase space (position and velocity) distribution of the atoms as they cross the
orifice plane must be determined. The Knudsen number indicates which approach to flow dynamics will
govern the problem, from continuum (Kn → 0) where the flow would be treated as a sonic orifice
expanding into vacuum (Ashkenas and Sherman 1965) to free-molecular (Kn → ∞) where the flow would
be treated as a collisionless equilibrium flux across a plane (Bird 1994). The Knudsen number is defined
by the ratio of the mean distance between an atom’s or molecule’s successive intermolecular collisions and
a length scale of interest, in this case the diameter of the orifice and is given by

mfp
Kn
L
λ
≡ (3.1)
One simplifying assumption for the calculation of kinetic properties of a gas is that the
intermolecular potentials can be modeled as hard spheres (Bird 1994). For the low energies (<1 eV) and
pressures (<1 Torr) in the Cs oven, this approximation yields little difference from other, more complex,
intermolecular potentials. With this assumption, the expression for the mean free path is given by

43

mfp
1
2
T
n
λ
σ
= (3.2)
The number density can be found through the ideal gas law and the vapor pressure relation given in the
appendix. For the nominal condition stated, the Knudsen number at the oven orifice is approximately 9.
As the Knudsen number is greater than unity for the flow leaving the oven orifice, a free-molecular
approach to the flow is appropriate. Treated as such, the effect of intermolecular collisions is neglected.
The fluxal properties of the atoms leaving the orifice can be found by taking moments of the
velocity distribution within the gas. For a stationary gas in thermal equilibrium, i.e. a stagnation chamber
or oven, the velocity distribution is considered Maxwellian and is given in polar coordinates by

2
boltz
32
2 2
eq
boltz
1
() sin
2
m
v
kT
m
f vdv e v d ddv
kT
φφ θ
π
⎛⎞
⎟ ⎜
⎟ − ⎜
⎟
⎜
⎟ ⎟ ⎜
⎝⎠
⎛⎞
⎟ ⎜
⎟ = ⎜
⎟
⎜
⎟ ⎜
⎝⎠
GG
(3.3)
The angles are defined as θ measured from x → y in the xy plane and ϕ measured from z → v. In this
particular case it is convenient to define the z direction as the axial direction of the beam, normal to the
orifice plane. In order to simplify the presentation of this equation, a shorthand notation is given for the
inverse of the most probable speed,

boltz
1
2
mp
m
vkT
β≡= (3.4)
The same equilibrium distribution can be presented in Cartesian coordinates as the following

()
22 2 2
3
eq
32
()
xy z
vv v
xy z
f v dv e dv dv dv
β β
π
−++
=
GG
(3.5)
Since it is assumed that there are no intermolecular collisions over a length scale equivalent to the orifice
diameter, the equilibrium distribution defines the outflow conditions of the oven effluence. In effect, the
oven orifice acts as a portion of the stagnation volume without a wall on which to reflect.
Atoms which cross a surface are given preferential weighting based on their velocity in the
direction normal to the surface. Weighting higher velocity atoms accounts for the fact that, per unit time,
there is a larger volume from which faster atoms may originate, before crossing the surface. This
weighting is given by multiplying the equilibrium distribution by the velocity in the surface normal
direction, and is used for distributions which contribute to the fluxal properties crossing the orifice exit

44

plane. If the orifice is considered infinitely thin, thus all atoms entering the orifice exit the other side
unperturbed, fluxal values for the flow out of the oven (normal direction » ϕ = 0 ) are calculated per unit
time per unit area as

22
2
3
3
32
00
cos( )sin
v
QnA Qe v dddv
π
π
β
π
β
φφφθ
π
∞
−
−
=
∫∫ ∫
(3.6)
The number flux of atoms crossing that surface is given by setting Q = 1, which simplifies to

boltz
oven
8
'
44
kT nA nA
Nv
m π
==
(3.7)
The next step is to determine the velocity distribution of the atoms as they cross the surface. In order to
adequately represent the velocities of atoms which cross the surface, the velocity distribution in the normal
direction must be weighted and re -normalized. In Cartesian coordinates this results in

()
22 2 2
22
3
32
00
1
2
xy z
z
vvv
v
zzxy zz
v e dv dv dv v e dv
β
β
ββ
ππβ
π
∞∞ ∞ ∞
−++
−
−∞ −∞
==
∫∫ ∫ ∫
(3.8)

22
2
oven
() 2
z
v
zz z z
f vdv ve dv
β
β
−
= (3.9)
The velocity distribution in the radial direction (x or y axis) is unchanged from the equilibrium distribution
given by

22
x,y
oven x,y x,y eq x,y x,y ,
() ()
v
xy
f vdv f vdv e dv
β β
π
−
== (3.10)
This derivation is consistent with the atoms which cross a surface in an equilibrium gas. This does
not address the distribution of atoms which cross the skimmer surface and subsequently interact with the
laser field. The next step is to address the effect of geometric velocity skimming used to create a thermal
atomic beam. By a method of brute force computing, a Monte Carlo simulation of the atoms leaving the
oven orifice can be done to obtain the probability of directly passing through the skimmer. The ratio of
atoms which are populated at the oven orifice to those which pass through the skimmer is found to be
approximately 40,000 to 1 for the example being considered (1 mm orifices and 100 mm separation). This
means that in order to computationally simulate one atom which passes through the skimmer, and on to the
test section, it would be required to populate 40,000 atoms at the oven plane. It is much more efficient,

45

then, to calculate an analytical expression for the flux and velocity distributions at the skimmer surface and
begin the simulation at that point.
Calculating the flux of atoms which pass directly from the oven through the skimmer is performed
by changing the limits of integration on equation (3.6). The new limits of integration only include ϕ from 0
to the inverse tangent of the radius of the skimmer divided by the oven-skimmer separation. This
continuing derivation assumes that the oven can be reduced to a point source. These limits only consider
atoms whose velocity at the exit plane of the oven pass through a ring, “R” in radius, which is coaxial with
the oven orifice and downstream a distance “d”.

()
1
22
3
tan
3
skimmer
32
0
0
cos( )sin
Rd
v
NnA ev dddv
π
β
π
β
φφφθ
π
−
∞
−
−
=
∫∫∫
(3.11)

()
4
skimmer
22
2
nR
N
dR
π
β
=
+
(3.12)
For the continued nominal case, this analytically corresponds to a ratio of

( ) ( )
22 22
2
oven
2 4
skimmer
2
*40,000
2
dR dR
N nR
N
R nR
β
π
β
π
++
===
(3.13)
The Monte Carlo approach is thus consistent with the analytical model. In this case the velocity
distribution must be renormalized over the reduced limits of integration in order to satisfy a distribution
requirement that complete integration yields unity.

()
()
1
22
3 2
tan
3
32 22 0
0
cos( )sin
2
Rd
v
R
ev dddv
dR
π
β
π
β
φφφθ
βπ
π
−
∞
−
−
=
+
∫∫∫
(3.14)

()
22
22
4
skimmer
2
() 2 cos( )
v
dR
fv v e
R
β
β
φ
π
−
+
=
G (3.15)
Because of the more complicated limits of integration, it is easier to stay with spherical coordinates within
the code and convert to Cartesian components after sampling from the spherical distribution function. The
speed distribution of the atoms passing through the skimmer is given by

()
1
2
tan
2
skimmer skimmer
0
0
() ( ) sin
Rd
fvfvvdd
π
φφ θ
−
=
∫∫
G (3.16)

46

22
43
skimmer
() 2
v
fv ve
β
β
−
= (3.17)
And likewise, the angular distribution of φ is given by

()
22
skimmer
2
() 2cos sin
dR
f
R
φφφ
+
= (3.18)
The angular distribution of theta is even over 0 to 2 π.
The cumulative function for these distributions is not easy to solve for use by the inverse function
method of sampling from a distribution; therefore, it is easier to sample from these distributions using the
acceptance-rejection method. In this method, an evenly distributed variable is generated using the random
number generator and used to select a value for the desired variable, e.g. phi. Either a direct analytical
equation or a lookup table is used to assign a probability to that value, e.g. equation (3.18). If the ratio of
that value to the maximum for the distribution function is greater than another randomly generated number,
then the value is accepted. The advantage to this method is that it can be used for any distribution function
so long as an analytic expression or lookup table is provided. Following these derivations, the change in
the equilibrium distribution at the oven surface as it passes the skimmer can be seen in Figure 3-2 and
Figure 3-3.

Figure 3-2: Axial velocity, v
z
, comparison at oven and skimmer planes

Figure 3-3: Radial velocity, v
x,y
, comparison at oven and skimmer planes
0 200 400 600 800 1000
0
1
2
3
4
x 10
-3
Velocity [m/s]
Probability Density [s/m]

Oven
Skimmer
-500 0 500
0
0.5
1
1.5
2
2.5
x 10
-3
Velocity [m/s]
Probability Density [s/m]

Oven
-2 -1 0 1 2
0
0.5
1
Velocity [m/s]
Probability Density [s/m]

Skimmer

47

Note that the velocity profile in the radial direction has been trimmed significantly while the axial
direction has been shifted slightly faster. The radial trimming is important as it represents a mechanically
easy way of creating a continuous supply of relatively cool atoms with distributions which are
advantageous for interaction with resonant laser processes. Additionally, by analytically determining the
distributions at the skimmer surface, the computational effort required to provide for a realistic atomic
phase space distribution within the TCL code is greatly reduced. The velocity profiles for the injection
surface of the TCL code asre shown in Figure 3-4 as the sampled profiles from the simulation when no
laser is present (baseline simulation). Note that no difference is seen between the analytical equation and
the implemented injection distributions.

A B
Figure 3-4: Starting velocity distribution for TCL code simulations: A) v
axial
B) v
radial

3.1.2 Interaction Simulation Methods
After being injected into the domain, a particle moves along its particular velocity vector to the
interaction region between the flow and the laser fields. In the interaction region the particle’s trajectory is
simulated in one of several ways. The basic simulation routine is the direct numerical integration of the
equations of motion using the assumption of constant acceleration acting over a small time step. In this
routine the acceleration (force) on the particle is determined as a function of position, or as a function of
position and time, depending on the force being simulated. The position and velocity are then advanced
using the kinematic equations for constant acceleration.

(,)
i
i
Fxt
a
m
vv at
xx vt
=
=+ Δ
=+Δ
(3.19)
0 200 400 600 800 1000
0
1
2
3
4
x 10
-3
Velocity [m/s]
Probability Density [s/m]

Analytical
Sampled
-5 0 5
0
0.2
0.4
0.6
0.8
1
Velocity [m/s]
Probability Density [s/m]

Analytical
Sampled

48

In this routine, the time step must be small enough to adequately approximate the assumed constant
acceleration during a time step, but large enough to minimize rounding error and the time for each
simulation.
The second simulation scheme for the particle’s trajectory within the interaction region is to
numerically integrate the trajectory as a function of position using a Backwards Differentiation Formula, or
Gear method, of solving ordinary differential equations. For a CW laser field, the force acting on the
particle is only a function of position and not time. Therefore a system of first order differential equations
can be created with two variables, position and velocity, in each dimension. The first derivative of position
is given by velocity and the first derivative of velocity is given by acceleration (force). In the TCL Code,
this n variable, first order, system is numerically solved using “DC03 Ordinary differential equations:
Gear’s method, sparse Jacobian”, a numerical subroutine used with permission from the HSL Archive
*
.
This subroutine was validated against direct numerical integration for various academic systems of
equations and further validated by running a set of identical simulations, one using the ODE solver and the
other using direct numerical integration. The ODE solver returned the same results as the direct simulation
within 1% (the statistical accuracy based on simulated trajectories) at a savings of several orders of
magnitude of computational time.
In addition to simulating an analytical force on a particle, e.g. equation (2.57) or equation (2.59), a
statistical approach was used in order to address the issue of loosing particles to alternate ground states.
This loss originates from hyperfine splitting and erroneous excitation/relaxation discussed briefly in section
2.3.6. Because a cesium atom from an equilibrium source will be evenly split between the two hyperfine
ground states, and the forces derived require the resonant approximation between the laser field and a two-
level atom, the ground state which is not in resonance passes the interaction region without alteration.
Additionally, for a process like photon scattering, there exists a finite possibility that the atom will decay
into the unaffected ground state, instead of into the primary ground state from which it came, even for
systems being excited on the cycling transition.

*
HSL, A Collection of Fortran codes for large-scale scientific computation. See http://www.hsl.rl.ac.uk,
(2007)

49

This statistical subroutine simulates the trajectory of the particle by continually interrogating the
probability of absorbing or emitting a photon within a Monte Carlo scheme. If a photon was absorbed, the
particle gained ћk momentum in the direction of the laser propagation. If the excited atom emits a photon
through spontaneous emission, ћk was gained in a random direction evenly distributed over 4 π steradian.
In these simulations the transition strengths and likelihoods of absorbing into and out of individual
hyperfine levels was accounted for. This gives the possibility of tracking the individual atoms’ electronic
energy states as a function of time. A comparison between the spatial profile of a deflected atomic beam
given by this statistical approach and the analytical equations can be seen in Figure 3-5. This spatial profile
only includes atoms which reside in the particular ground state susceptible to laser modification at the
frequency used. Further comment on the inclusion or exclusion of a particular ground state is given in
section 4.1.

Figure 3-5: Comparison of analytical and statistical simulation subroutines in the TCL code
No significant difference is seen between the statistical quantum photon absorption method, the
direct integration of equations of motion, or the numerical solutions to the initial condition ODEs.
Therefore, as the fastest method is to solve the ODEs, the results from those simulations are shown when
referring to the TCL code results. The exception to this is the results as they pertain to the direct
acceleration of particles using an accelerating resonant optical lattice. In these simulations, the use of a
spatially and temporally dependant force complicates the ODE solver, thus the simulations are conducted
through direct integration of the equations of motion at the cost of computational time.
3.2 Numerical: SMILE
Due to the combination of simpler force derivations associated with the non-resonant interaction
and the importance of intermolecular collisions for many arbitrary molecular flows, the adaptation of an
-0.01 0 0.01 0.02 0.03
0
50
100
150
200
250
Position [m]
Probability Density [1/m]

Analytical
Statistical

50

existing gas dynamic code was deemed preferable to including collisions in the TCL code. Since the force
directly affects the velocity distributions of the flow, a kinetic approach must be used to model the laser-
species interaction. The kinetic code chosen was the DSMC code SMILE. The code has been modified to
include the non-resonant laser interactions described in section 2.2. These modifications were enacted by
approximating the force on the species as a temporally and spatially varying acceleration which is
considered constant over the duration of the time step. Time steps were therefore reduced such that the
species does not traverse appreciable fractions of the laser field or pulse width in one time step. The
SMILE code has been broadly applied and experimentally validated, see (Ivanov and Gimelshein 2003) and
the references therein. Some of the features which have contributed to its broad applicability and validation
(thus its choice for this study) are as follows: The majorant frequency scheme is employed for modeling
molecular collisions (Ivanov and Rogasinsky 1988). This feature in particular is important for maintaining
fidelity while reducing the time step to satisfy the laser interaction conditions. The VHS model is used for
modeling intermolecular interactions (Bird 1994). The Larsen-Borgnakke model (Borgnakke and Larsen
1975) with temperature-dependent rotational and vibrational relaxation numbers is utilized for rotation-
translation and vibration-translation energy transfer.
Since the SMILE code was used exclusively for non-resonant interactions, the laser fields modeled
were pulsed lasers in order to create a strong field to impart reasonable forces on the species. The flow
modification over the course of a single laser pulse (picoseconds to nanoseconds) is experimentally
measureable (Sakai, et al. 1998) but difficult to extend to application. Most applications require a more
consistent flow with which to interact. Therefore a train of pulses was simulated to create a quasi-steady
state flow condition for flow steering, collimation, and heating. As mentioned, time steps must be
sufficiently small in order to accurately model the laser interaction. This time step may be smaller than is
required for accurate gas dynamic evolution without the laser interaction; therefore, the time periods
without a laser pulse were simulated independently, but with the same molecules, cells, and domain
conditions as the time periods with a laser. The simulation time step was changed based on the laser
presence in order to find an optimized simulation condition which minimized runtime without reducing
laser interaction fidelity.

51

The process for simulating the interaction of a pulse train with the simulated flow was carried out
as follows: First, the domain was populated in the baseline configuration of the simulation with the
appropriate geometric flow constraints. This step provided the density and velocity profiles for the
geometries being tested in the absence of the laser field. The baseline flow fields were then available for
comparison with the laser manipulated flows. Additionally, this step populated the simulation domain to
reduce the time to a quasi-steady condition. After the domain was populated, the laser interaction was
simulated as a series of simulation pairs. Each pair consisted of either two simulations with active laser
interaction or a simulation with active laser manipulation followed by a simulation with re-configured
temporal parameters to simulate a period of “waiting” or “intervening” time. During the last 250 of 5000
time steps, cells were sampled to give the flow field values.
3.2.1 Simulations for Laser Flow Steering
The first simulation condition is that used for laser steering of an already directional flow. As
described above, a basic thermally accelerated directional flow can be created using an orifice (or slit) in a
stagnation chamber and one or more collimating orifices (or slits) downstream. In order to reduce the total
memory and complexity of the simulations, the domain for the simulations is a two-dimensional domain.
The diagrams are considered infinitely long into the page, resulting in a geometry which forms a sheet flow
instead of a round beam flow. This sheet then interacts with a round beam crossing into the infinite
direction. In these simulations, a stagnation condition is enforced at the left boundaries of the simulation
domain with vacuum outflow elsewhere. Aligned with, and 40 μm downstream from, a 2D slit in the
stagnation chamber, a similar 2D slit is placed to collimate and directionalize the flow. Both slits are
simulated to be 10 μm tall. Surfaces are all infinitely thin and simulated with fully diffuse reflections at a
wall temperature of 300 K. The stagnation condition was assumed to be molecular nitrogen at 1 Pa and
300 K which yields a stagnation number density of 2.41x10
20
m
-3
. A diagram of the two-dimensional
baseline simulation domain can be seen in Figure 3-6 (A). The baseline (no laser) density [m
-3
] flow field
for this simulation can be seen in Figure 3-7 (A).

52

A B
Figure 3-6: SMILE domain for A) starting surface and B) non-resonant dipole steering simulations
Since the area of interest is contained completely in the right third of the domain, it is inefficient to
simulate the stagnation chamber and plume expansion region for every test run. Therefore a starting
surface was taken at the collimating slit. A simulation was run using the domain in Figure 3-6 (A) and
averaged over 10’s of thousands of time steps in order to obtain the required statistical accuracy. The
resulting flow field was then sampled along the starting surface identified in Figure 3-6 (A) to ascertain
number density and directional values for the bulk velocities and gas temperatures as a function of position
along the surface. These conditions were subsequently enforced as the inflow conditions for the domain in
Figure 3-6 (B). The starting surface used an ellipsoidal distribution with independent temperatures in each
direction. The domain was subsequently populated as only the directional jet to the right of the second 2D
slit (collimating). In addition to reducing the simulation time, by reducing the simulated domain, the new
domain, with starting surface influx condition, had the effect of increasing the statistical results of the
simulation by increasing the portion of simulated particles in the area of interest for a fixed memory
allocation. The background cells used for sampling and basic collisions were divided such that there were
400 in the vertical direction and 200 along the horizontal. SMILE automatically subdivides collision cells
for accurate collisional simulation. This grid satisfies the requirement that a particle does not cross the cell
in one time step. Time steps were 2 picoseconds for laser simulations. The revised domain, which was
used for the non-resonant laser steering simulations, can be seen in Figure 3-6 (B). The resulting baseline
(no laser) density [m
-3
] flow field for this simulation can be seen in Figure 3-7 (B). The flow has become
significantly directional in comparison to the stagnation expansion, but still exhibits some divergence.
-40 -20 0 20 40 60 80
-40
-20
0
20
40
Location [ μm]
Location [ μm]
Inflow
Inflow
Inflow
Vacuum Outflow
Vacuum Outflow
Vacuum Outflow
Starting
Surface
40 60 80
-40
-20
0
20
40
Location [ μm]
Location [ μm]
Vacuum Outflow
Vacuum
Vacuum
Starting
Surface
Laser
Diam

53

A B
Figure 3-7: SMILE baseline density [m
-3
] for A) starting surface and B) non-resonant dipole steering simulations
3.2.2 Simulations for Laser Flow Collimation
Like the simulations for laser steering, the laser collimation simulations assume a stagnation
condition of molecular nitrogen at 1 Pa and 300 K within a two-dimensional simulation which creates a
sheet expansion, infinitely long in the dimension into the page. Unlike the collimation simulations, the
effect of intermolecular collisions on the region near the slit in the stagnation chamber make it sensible to
simulate both the expansion plume, which interacts with the laser, as well as the stagnation chamber
upstream. As with the steering simulations, the left boundaries are simulated as a stagnation condition with
vacuum outflow elsewhere. Again the surfaces were simulated with fully diffuse reflections at 300 K. The
background cells used for sampling and basic collisions were divided such that there were 400 in the
vertical direction and 600 along the horizontal. SMILE automatically subdivides collision cells for
accurate collisional simulation. This grid satisfies the requirement that a particle does not cross the cell in
one time step. Time steps ranged from 1 to 10 picoseconds depending on simulation. The domain and the
baseline (no laser) density [m
-3
] flow field for the laser collimation simulations can be seen in Figure 3-8.

54

A B
Figure 3-8: SMILE A) domain and B) baseline density [m
-3
] for non-resonant dipole collimation simulations
3.2.3 Simulations for Neutral Gas Heating
While the previous two cases considered two-dimensional simulations where the laser field
propagated orthogonal to the flow direction (into the infinite direction), the simulations for neutral gas
heating simulated the interaction of a stagnant gas with a pulsed optical lattice which has periodic features
in the axial laser direction. In this case the optical lattice geometry makes an axisymmetric simulation the
logical choice; the domain is modeled axisymmetrically around the optical axis of the two anti-parallel,
counter propagating laser pulses. The domain consists of two periodic boundaries, on the left and right,
which create a simulation consistent with the center of a series of optical lattice potentials. The third
boundary is the specularly reflecting symmetry axis for the simulation, and the fourth is an inflow condition
enforced as an equilibrium gas at ambient conditions, 300 K and 1 atm. The radial domain includes the
laser diameter and surrounding gas, enclosed sufficiently to ensure temperature and pressure perturbations
did not propagate from the symmetry axis to the equilibrium boundary over the simulated pulses. The
described domain assured no discontinuity at the boundary. The domain for the laser gas heating
simulations can be seen in Figure 3-9 with the symmetry axis at the top of the plot.
-40 -20 0 20 40 60 80
-40
-20
0
20
40
Position [ μm]
Position [ μm]
Inflow
Inflow
Inflow
Vacuum Outflow
Vacuum Outflow
Vacuum Outflow
Laser
Diam

55

Figure 3-9: SMILE domain and laser intensity [W/m
2
] for neutral gas heating
3.3 Experimental
The experimental apparatus was designed to provide demonstration and proof-of-concept for the
use of laser field-modification of an atomic beam. The experiment also offered a straightforward validation
for the TCL code simulations of resonant interactions with a thermal alkali atomic beam. The experiment
reproduced the numerical simulation of steering an atomic beam using the photon scattering force from a
near-resonant laser. The atomic beam was a geometrically skimmed thermal [373 K] cesium beam
interacting with a laser field tuned near the D
2
transition (6s
2
S
1/2
↔ 6p
2
P
3/2
) with a wavelength of 852.35
nm. The laser was further tuned to the cycling transition of F = 4 ground to F = 5 excited hyperfine levels.
The apparatus is discussed in three parts: the vacuum system which created the atomic beam, the laser
system which provided the interaction field, and the detection system which provided diagnostic
information about the laser and atomic beam.
3.3.1 Vacuum System
A diagram of the vacuum system can be seen in Figure 3-10. The atomic beam starts with a
cesium oven, similar in design to (Tompa, Lopes and Wohlram 1987). The coldest point of the oven,
which drives the vapor pressure of the cesium, was actively controlled to 100°C by a tape heater around the
oven and thermocouple located at the bottom of the reservoir. The shutoff/regulating valve and subsequent
piping was kept at least 10°C warmer by keeping the bottom of the reservoir exposed (uncovered by
insulating foil). The exit of the oven into the first chamber of the vacuum system terminated in a 1/4”
stainless steel tube centered on a vacuum feed-through. The flange, on which the oven was mounted, was

56

wrapped in another heater tape and monitored with a thermocouple in order to maintain the required 10°C
temperature increase over the rest of the oven, assuring that no condensation occurred once the cesium
evaporated from the bottom of the reservoir. The effluence from the oven was differentially pumped by
one of two Varian Turbo-V 81M pumps. On the opposite side of the differential pumping chamber was a
copper disk with a 1 mm orifice through the center. The disk acted as a collimating skimmer for the
effluence in order to create an atomic beam with a sufficiently small transverse velocity for the experiment.
The separation between the oven exit and the skimming orifice was approximately 260 mm.

Figure 3-10: Diagram of experimental vacuum system (side view)
After the skimming orifice, the atomic beam entered the interaction and drift regions. The
differential pumping chamber and the diagnostic chamber were connected by a gate valve, 2-3/4” ConFlat
6-way cube, and longer 2-3/4” ConFlat full nipple which acted as a drift tube. After passing through the
skimming orifice, the atomic beam passed through the 6-way cross, on which were mounted two laser
viewports. The viewports were anti-reflective-coated to reduce stray light within the chamber and are
manufactured to better than 10 arc seconds of parallelism, λ/4, and 20/10 scratch-dig. The scratch-dig
rating is a measure of the surface roughness and therefore a measure of the uniformity of the viewport.
20/10 refers to no scratches wider than 20 microns or digs/bubbles more than 10 microns in diameter. In
this cube, the atomic beam interacted with the first laser field, gaining transverse momentum. After the
interaction, the atomic beam drifted for approximately 450 mm before being irradiated by a second laser
field which induced fluorescence. This fluorescence was detected by a cooled silicon CCD camera which
indicated the beam’s position.
3.3.2 Laser System
The laser system originated with an 852 nm diode laser manufactured by Sacher Lasertechnik
GmbH. The laser has a maximum power of 1000 mW from a tapper amplified Littman/Metcalf diode laser
Drift Tube

57

setup. The line width of the laser is company rated to be approximately 1 MHz; however, the experimental
results suggest a width larger than that specified. The emitted laser light was spatially filtered through a
single-mode, polarization maintaining, optical fiber and subsequently passed through wave plates to create
a linearly polarized field horizontal to the table surface. The light was then passed through an acousto-
optic modulator in a double pass configuration (Donley, et al. 2005). The acousto-optic modulator, also
referred to as a Bragg cell, passes the light through a crystal on which is mounted a piezoelectric actuator.
Proportional to an RF signal sent to the pizeo device, acoustic waves pass through the crystal. The light
passing through then diffracts off of these waves through the processes of Bragg diffraction. The zero-th
order light passes through the device unaffected, however the 1
st
and -1
st
order diffractions are both
redirected by the Bragg angle and frequency shifted by the frequency of the acoustic waves. The frequency
shifted light was then passed to the experimental setup, rotated and split to act as both the interaction laser
field (first laser field), which caused the modification of the atomic beam, and the fluorescent field (second
laser field), which allowed for the detection of the atomic beam position. A diagram of the laser system
can be seen in Figure 3-11 and Figure 3-12.

Figure 3-11: Diagram of experimental laser system (top view) continued in Figure 3-12
1/2 WP
1/4 WP
To Figure 3-12

58

Figure 3-12: Diagram of experimental laser system (top view) continued from Figure 3-11
The originating laser system is a diode laser system which can emit over a range of wavelengths.
The active tunablity and narrow linewidth of the laser originates from the self-seeding of the laser diode in
the Littman/Metcalf setup (Stry, et al. 2006). By modifying the angle of a mirror which reflects part of the
emitted light from the diode onto a diffraction grating, the frequency of the self-seeding portion of the laser
can be changed. The retro-reflected light has a very narrow line width which in turn causes the diode
cavity (diode and facet) to emit that narrow light. Course adjustment is obtained using a servo-actuator
attached to the mirror, and fine and high frequency adjustment is obtained by using the piezoelectric
actuator at the tip of the servo. The first step in locking the laser to a particular frequency was to scan the
piezoelectric actuator across several hundred MHz of frequency space. The laser light from the master
diode (tunable diode) was split into two parts. Part of the light passed to the tapered amplifier where its
power was greatly increased at whatever wavelength was being emitted by the laser diode. The other
portion of the light went into a saturated absorption spectroscopic setup with an ambient temperature
cesium cell.
To Figure 3-11

59

Figure 3-13: Saturated absorption signal vs. frequency sweep for experimental laser master
The saturated absorption setup allows for atomic spectral interrogation of the cesium cell well
below the Doppler broadening limit caused by the atoms’ room temperature equilibrium velocities. The
experimentally recorded Doppler-free saturation absorption spectroscopic profile from the laser’s internal
absorption setup can be seen in Figure 3-13. Note the 5 distinct peaks and 1 shoulder of the profile. This
profile corresponds to transitions between the upper ground state (F = 4) and the 3 allowed hyperfine
excited states (F = 3…5) (see Figure A-2 for more information). The two end peaks and the shoulder in the
middle are the actual resonances associated with the electronic transitions given in the appendix. The other
three peaks are “crossover” peaks caused by the saturation absorption setup. These erroneous peaks are
exactly half way between each of the transitions and represent the combined signal from the respective
resonances. From this signal, the piezoelectric actuator is centered on the side of one of the peaks and the
output from the saturation absorption setup is fed into a loopback controller. As the intensity of the light
changes due to frequency shifts, the loopback controller changes the piezoelectric signal to compensate for
the variation in the laser frequency, and returns the laser to a frequency which produces the desired
intensity level. It should be noted that this setup is acted upon by a different light path than the one which
is amplified and emitted by the laser. Therefore this setup acts only to give stability to the frequency
seeding of the master diode. Through this method, the frequency stability of the laser is maintained
regardless of minor temperature, current, or stress/strain fluctuations in the laser diode. The drawback of
this method is the necessity of locking to the side of a transition instead of the peak for a simple control
circuit. For this experiment, this drawback is not significant since the light is being further modulated by
an AOM which can compensate for the small difference between center and side of an individual
resonance.
-300 -200 -100 0 100 200
-0.2
-0.15
-0.1
-0.05
0
0.05
Frequency [MHz]
Signal [arb. units]
~452 HMz
211 MHz
Shoulder

60

Figure 3-14: Comparison between saturated absorption setup and atomic beam fluorescence
The cycling transition for the upper ground state is the F = 4 → F = 5 transition, the far right and
smallest peak in Figure 3-13. The width of that transition for a perfectly monochromatic laser, in a truly
Doppler-free non-power broadened setup, would be given by the natural radiative width profile seen in
Figure 2-4, approximately 5.25 MHz. However, the zoomed in and normalized view of that peak from the
saturation absorption experiment, and the atomic beam fluorescent signal as a function of AOM detuning
from the experiment, are shown in Figure 3-14 to be wider than the natural width of the transition,
approximately 20 MHz. The fluorescent signal is obtained from the maximum intensity of the CCD image
of the cesium atomic beam as a function of the frequency of the laser given by the detuning of the AOM.
These two signals, from the saturated absorption setup and AOM detuning, are both wider than the natural
line width of the atom (5.25 MHz) which suggests that the laser may not have as narrow a line width as is
given in the manufacturer’s specification sheet.
3.3.3 Diagnostic System
The diagnostic system primarily consisted of a Hamamatsu, cooled, CCD camera mounted such
that the plane of its field of view is perpendicular to the atomic beam axis (lens and beam are coaxial). In
other words, the camera looked down stream of the atomic beam, through the atomic beam. Therefore, as a
swath of the beam is illuminated by a perpendicular laser field, the camera viewed the cross section of the
beam at the location of illumination. This concept was used for atomic beam detection and position
determination. Part of the laser light was split from the path leading to the interaction region of the
experiment and was directed to the diagnostic chamber. This light interacted with the atomic beam,
causing it to fluoresce. This fluorescence was imaged on the CCD and the cross section of the atomic beam
was measured. This technique only illuminated atoms which are in the F = 4 ground state, leaving F = 3
200 205 210 215 220 225 230
0.4
0.6
0.8
1
1.2
Frequency [MHz]
Signal [arb. units]

Fluorescence
Sat. Abs.

61

ground state atoms undetected. This is consistent with numerical simulations which only consider the
F = 4 ground state. The undetected atoms are also the atoms which are unaffected by the interaction laser
and are not germane to the laser interaction quantification.

A B
Figure 3-15: Example of A) background and B) atomic beam fluorescence signals
Figure 3-15 shows a false color image of the intensities detected by the cooled CCD. Image (A)
represents the background signal from the chamber with the fluorescence laser field irradiating the
detection region. The false color is linearly scaled from the highest intensity in the frame (red) to the
lowest intensity in the frame (blue). In this image, the background light included the scattered light off of
the windows, ambient light from the laboratory, and combinations of that light reflected off the stainless
steel of the chamber. For this image, the gate valve which separated the differential pumping chamber and
the diagnostic chamber is closed. This kept the cesium beam from entering the diagnostic chamber and
thus no fluorescence was present. Image (B) shows the same experimental setup but with the gate valve
open, allowing the cesium beam to enter the diagnostic region and subsequently fluoresce. The false color
is again linearly scaled from the highest intensity in the frame (red) to the lowest intensity in the frame
(blue); thus, image (A) and image (B) are scaled differently (different limits) in order to show features
within the field of view.

62

Figure 3-16: Example of signal to noise for experimental atomic beam detection
Figure 3-16 represents the intensity values found in Figure 3-15 (A) subtracted from Figure 3-15
(B). This gives the atomic beam profile cross section without the addition of scattered or background light.
The image is shown as a three-dimensional surface plot to expose the signal to noise ratio. In order to
determine the shift of the atomic beam due to the laser interaction, the center (or other geometric
parameter) of the atomic beam must be calculated. In order to be consistent with the numerical
simulations, the two-dimensional cross section was reduced to a one-dimensional plot of summed intensity
as a function of position along the horizontal axis. The total intensity of the vertical pixels along the
horizontal axis can be seen in Figure 3-17. From the vertically binned profile seen in Figure 3-17 (B), a
least-squares Gaussian fit was performed to determine the center of the profile. This method was
consistently used for the numerical and experimental profiles in order to ensure consistent determination of
the profile shifts.

63

A B
Figure 3-17: Example of A) background subtracted signal and B) vertical bin summing
3.4 Summary
Two numerical techniques were used in this study. The first technique was a custom particle
trajectory code which simulates a rarefied atomic beam interacting with a near-resonant laser field. This
code was written specifically for this study. The code uses analytical rarefied gas dynamic distributions to
create the initial conditions for the beam. The individual particle trajectories are then simulated by either
direct numerical integration of the equations of motion, solutions to a set of n-variable ordinary differential
equations by Gear’s Method, or a statistical approach to photon absorption and emission. In order to
provide validation for this new code an experiment was conducted on a demonstration case. The case
chosen was the horizontal steering of a thermal [373 K] cesium beam by photon scattering. The validating
experiment used a narrow line width, external cavity, tunable diode laser and acousto-optic modulator to
impart momentum to the cesium beam, as a function of the detuning of the laser from the cycling transition
between the 6S
1/2
F = 4 ground state and the 6P
3/2
F = 5 excited state. The position of the atomic beam was
detected by a cooled CCD. The second numerical approach used the DSMC code SMILE which had been
specifically modified to include the non-resonant interaction between a pulsed laser field and an arbitrary
molecular flow. This code was exercised in instances where the effects of intermolecular collisions were
deemed important to the qualitative application of laser modification of the flow.

200 400 600 800 1000 1200
0.5
1
1.5
2
x 10
5
Pixel [#]
Vert. Intensity Sum [Arb.]

64

Chapter 4 Steering
As discussed in Chapter 3, the two numerical methods employed in this study were a free
molecular particle trajectory code, TCL code, and a collisional statistical kinetic solver, SMILE. In this
chapter, both codes simulated flows which were already directional through geometric skimming. The
TCL code was employed to demonstrate some aspects of a thermal [350 K] cesium atomic beam interacting
with a near-resonant continuous wave laser. These simulations addressed both the force derived from the
scattering of photons as well as the resonantly enhanced induced dipole force. SMILE was employed to
demonstrate some aspects of a geometrically skimmed nitrogen effluence interacting with a non-resonant
pulsed laser. These simulations addressed the non-resonant version of the induced dipole force. For both
flows utilizing the induced dipole force, there was a significant limitation associated with the size of the
flow and the laser field. Since the induced dipole force, or dipole gradient force, was derived from a spatial
gradient in the laser field, increasing the size of the light field necessarily decreased the spatial gradient for
a fixed intensity. Thus, simulations involving the induced dipole force involved micro-scale flows.
The simulations using the TCL code demonstrate the applicability of laser manipulation for
steering an alkali atomic flow. Alkalis are readily described using the resonant theory derived in Chapter 2,
and offer electronic transitions which couple well to existing laser frequencies. Additionally, applications
for alkali flows have been identified in the lithography (Kreis, et al. 1996) and micro-machining
communities. These theories have also been extended to other metals, such as iron(te Sligte, et al. 2004).
In order to validate the collisional code, experimental results for the steering of a cesium flow using photon
scattering, were compared with the numerical results. The simulations using SMILE demonstrated the
applicability of laser manipulation for steering any flow, given that adequate laser fields can be generated
for the flow.
4.1 Photon Scattering
The force exerted on an atom as a result of photon scattering is dependent on both the intensity,
i.e. number of photons per unit time, and the frequency of the light in relation to electronic resonances

65

within the atom, i.e. likelihood of absorbing a particular photon. The latter of these two dependencies is
additionally complicated by Doppler shifting of the incident light in the atomic frame due to an atom’s
velocity component in the laser propagation direction. Referencing equation (2.57), the force exerted on a
cesium atom at the center of a 25.4 mm (1”) Gaussian laser locked to the D
2
cycling transition
(F = 4 → F = 5) as a function of the laser’s power can be seen in Figure 4-1.

Figure 4-1: Photon scattering force vs. incident laser power
The force saturates at a value of 1.28x10
-20
N. The asymptote originates from the limiting case of an atom
in a strong laser field, where the atom spends approximately half its time in the excited state and half its
time in the ground state, via continual absorption and stimulated emission. Only the time in the excited
state is eligible for spontaneous emission, thus the maximum force on the atom is one photon’s momentum
every two times the transition lifetime

scat
max
2
k
F
Γ
=
ħ
(4.1)
After the limiting necessity of atom-laser systems with isolated (cycling) transitions, the inability to
indefinitely increase the force on the atom is the greatest drawback to the use of the photon scatting force
for steering atomic trajectories.
In addition to incident power, another influence on the scattering force is the velocity of the atom
within the field, with relation to the frequency of the incident light. The change in frequency of a photon in
the atom frame due to the Doppler shift is discussed in section 2.3.7. . For a laser which is tuned to the
center of the transition, the velocity at which the atom is most likely to absorb a photon is 0 m/s. Doppler
shifting of the incident photon frequencies decreases the likelihood of absorption for all non-zero velocities,
by increasing the relative detuning. The dependence of the scattering force on the velocity of the atom, for
various laser powers, is shown in Figure 4-2. When the laser is detuned 20 MHz to the blue side (higher
10
-3
10
-2
10
-1
10
0
10
-20
Power [W]
Force [N]

66

frequency) of the transition, the most likely velocity to absorb a photon is +17.05 m/s, in the direction of
laser propagation. In the frame of the atom, the 20 MHz blue detuned light is red shifted by an atom
travelling away from the laser, moving it closer to resonance. This shift leads to a preferential region of
velocities which will absorb photons, and is the premise for all optical molasses based laser momentum
deposition or cooling.

Figure 4-2: Photon scattering force vs. atom velocity
A complication to the use of a resonant process with cesium is the splitting of the cesium energy
states, particularly the ground state. Until this point, the atom has been, theoretically, considered a two-
level system with brief mention of the fine and hyperfine splitting of the alkali energy levels. This
treatment is still applicable when the extension is made that the atom is a “multi-”two-level system with
independent systems for each ground-excited pair of hyperfine energy levels. In carefully planned resonant
and near-resonant applications, non-primary two-level systems have negligible impact due to the fairly
large distance between any two states. For example, the difference between the cesium ground hyperfine
states is 9,192,631,769,975 Hz (~9 GHz). This distance makes the coupling between the laser field and the
more distant systems associated with the other ground state very weak. Since only half the atoms will be in
the coupling ground state, only half of the cesium atoms from an equilibrium source are affected by a
technique such as photon scattering. It is possible to pass the cesium atoms through a microwave cavity
and pump all the atoms into the appropriate ground state, adding a level of complexity to an experiment.
As the ground state splitting of cesium is used as the time standard for the world, the availability of such
microwave sources is not prohibitive; however, such ground state pumping is beyond the scope of this
study.
-100 -50 0 50 100
0
5
10
15
x 10
-21
Velocity [m/s]
Force [N]

P=100 mW, δ=20 MHz
P=100 mW
P=10 mW
P=1 mW

67

Figure 4-3: Atomic beam spatial profile including both Cs ground states
As not to pass this point without consideration, simulations were conducted which accounted for a
50/50 equilibrium source (50% in F = 3 and 50% in F = 4 ground hyperfine states). The comparison of the
atomic beam’s spatial distribution, using the TCL code, which includes both deflected and un-deflected
atoms, can be seen in Figure 4-3. Detection methods which rely on resonant fluorescence (such as the
fluorescence and cooled CCD used in this study) would not show the profile seen in Figure 4-3. The
unaffected F = 3 atoms would not fluoresce, and therefore, not be detected. For simulations using the
statistical model of photon scattering (as opposed to the analytical model), the state of the cesium atom was
tracked, and its likelihood of being excited to the F = 4 upper state, and subsequently falling into the F = 3
ground state, was calculated. The existence of such a process is important for quantitative information on
the density of affected versus unaffected atoms. Since the purpose of this study was to look at the
deflection of the beam and not necessarily the likelihood of excitation, the unaffected atoms (F = 3) are not
considered in the following simulations.

Figure 4-4: Diagram of notional experimental configuration represented by simulations in section 4.1
The numerical simulations in this section were performed using the TCL custom particle trajectory
code, employing the ODE solver subroutine mentioned in section 3.1.2. The simulations represented an
-10 0 10 20
0
100
200
300
400
Position [mm]
Probability [n.d.]

P
L
=13 mW & δ=1.1 MHz
No Laser

68

atomic beam, with a round radial profile, interacting with a perpendicular laser field, with a round radial
profile. The beam deflection was along the laser propagation axis. A diagram of a notional experimental
configuration can be seen in Figure 4-4. To investigate the use of the resonant photon scattering force, for
steering a cesium atomic beam, the TCL Code simulated a range of laser parameters. The baseline
condition for the trajectory calculations can be seen in Table 4-1.

Table 4-1: Conditions for photon scattering steering simulations
With the unaffected atoms not considered in the deflected profile (only F = 4 atoms in the
simulation), a comparison between the spatial profiles of the atomic beam at the sample surface, with and
without the laser interaction, can be seen in Figure 4-5 (A). In this simulation the laser was assumed to be
13 mW, aligned perpendicular to the axial direction of the beam propagating from left to right in the figure,
and detuned to the blue side (higher frequency) of the D
2
transition by 1.1 MHz. The peak of the
distribution shifted approximately 5 mm, or five times the full width half maximum of the original
distribution. The magnitude of the spatial displacement was mainly a function of the amount of distance
the atoms are allowed to drift after the interaction region. The change in the velocity distribution function
is seen in Figure 4-5 (B). The center of the distribution shifted approximately 3.5 m/s, over 3 times the full
width half max of the original distribution. This corresponds to a Doppler shift of approximately 4 MHz.
The shift is in line with the edge of the bandwidth where an atom would have accelerated itself out of
resonance with the laser field, approximately half of the natural width (2.6 MHz) plus the laser detuning
(1.1 MHz). It should be noted that the laser was simulated as a monochromatic source, i.e. a laser source
with zero line width.
TCL Code
Sampled trajectories 1.0E6
Interaction Model ODE Solver
Laser Diameter 25.4 mm
Inflow Diameter 1.0 mm
Inflow Surface to Laser Center 265 mm
Laser Center to Sample Surface 445 mm

69

A B
Figure 4-5: Atomic beam A) spatial and B) velocity profiles
In order to provide a consistent procedure for comparing the influence of the laser, the resulting
spatial profile from the simulation was fit to a least-squares fit Gaussian profile. While the distributions
have been significantly perturbed from a Gaussian, a consistent means for comparing the deflection of the
beam was needed. The least-squares Gaussian fit was as convenient as the location of the peak, a
geometric mean, or other profile quantity and it attempted to average non-symmetric profiles for
comparison. In subsequent discussion of the displacement of the atomic beam spatial profile, the reference
is to the center position of the Gaussian least-squares fit, in reference to the same parameter for the baseline
simulation, without the laser interaction.

Figure 4-6: Atomic beam deflection vs. laser frequency for photon scattering steering
Figure 4-6 shows the center of the Gaussian least-squares fit, as a function of the incident laser
frequency, given by the relative detuning from the F = 4 → F = 5 cycling transition frequency. A factor
which limited the photon scattering force, for a constant frequency laser, was that the acceleration of the
atoms along the laser propagation axis eventually moved the atoms farther out of resonance, and reduced
the force as a function of velocity. This lead to an optimum frequency at which to set a constant-frequency
laser. By shifting the light slightly to the blue side (higher frequency) of the transition, a balance was
-10 0 10 20
0
100
200
300
400
Position [mm]
Probability [n.d.]

P
L
=13 mW & δ=1.1 MHz
No Laser
0 2 4 6
0
0.2
0.4
0.6
0.8
1
Velocity [m/s]
Probability [n.d.]

P
L
=13 mW & δ=1.1 MHz
No Laser
-40 -20 0 20 40
1
2
3
4
Detuning [MHz]
Deflection [mm]
0 2 4 6 8 10
0
1
2
3
4
5
Detuning [MHz]
Deflection [mm]

70

struck between the largest number of initial atoms to be affected by the laser field, and the frequency
bandwidth available for momentum addition before the atom was accelerated out of resonance.

A B
Figure 4-7: Numerical vs. experimental atomic beam deflection for photon scattering steering
In order to validate the TCL code approach, the effect of incident laser frequency on the deflection
of the atomic beam was experimentally demonstrated. The comparison between the numerical results and
the experiment can be seen in Figure 4-7 (A) with the magnitude of the deflection normalized by the
maximum value. In order to compare similar quantities, both numerical and experimental points represent
the center position of a least-squares Gaussian fit of the horizontal atomic beam spatial profile, with respect
to the unaffected atomic beam position. The most likely reason for the shift and broadening of the
experimental curve is the non-zero line width of the laser. The simulations assumed a monochromatic laser
field, a laser field whose line width is a delta function. If the laser was not a monochromatic system, there
would have been increased overlap between the frequencies of light emitted by the laser and the natural
width of the transition. This overlap broadened the bandwidth of center laser frequencies accepted by the
atom. This broadening caused an atom to stay within resonance through a wider range of velocities,
shifting and broadening the experimental profile to the blue side (higher frequency) of the transition. When
the frequency of the experimental profile in Figure 4-7 is normalized by the recorded Doppler-free
saturation absorption spectroscopic profile width, given in Figure 3-14, and the numerical frequency profile
is normalized by the assumed natural line width of 5.25 MHz, the experiment shows much closer
agreement with numerical predictions. The normalized profiles can be seen in Figure 4-7 (B). The
agreement between the peak in the experimental displacement and the numerically predicted peak validates
the general use of the TCL code as a predictive tool.
-50 0 50
0.2
0.4
0.6
0.8
1
Frequency [MHz]
Displacement [n.d.]

Numerical
Experimental
-5 0 5
0.2
0.4
0.6
0.8
1
Frequency [ Γ]
Displacement [n.d.]

Numerical
Experimental

71

Figure 4-8: Atomic beam deflection vs. laser power using δ = 1.1 MHz for photon scattering steering
In order to look at the effect of laser power on the steering of an atomic beam by the photon
scattering force, the magnitude of the displacement at the peak of the frequency dependant deflection curve
shown in Figure 4-6 is plotted as a function of laser power in Figure 4-8. The increase in laser power does
not exactly follow the more dramatic asymptotic shape of Figure 4-1 due to the Gaussian radial profile of
the laser field and the effect of power broadening. The region of the field where the atom is saturated, thus
diminishing returns on force, is not uniform due to the Gaussian intensity profile of the laser field.
Therefore an increase in laser power has a stronger effect on the outer extent of the laser diameter allowing
the total deflection to continue to increase even if the core of the laser has long reached saturation. In other
words, even for a constant FWHM of the beam, the length of time which the atom is accelerated grows
with increasing laser power since the portion of the laser field which can excite the atom grows.
Additionally, increasing laser power introduces the concept of “power broadening” discussed in section
2.4.2. The higher laser intensity, the broader the effective line width of the transition, and the broader the
bandwidth of accepted photons, causing increased deflection by allowing the atom to stay in resonance
longer.
4.2 Resonant Dipole
Unlike the photon scattering force, the use of the dipole force is not asymptotically limited as a
function of the laser intensity and less stringently tied to Doppler shifting. The dipole force offers the
potential for stronger forces than are capable through photon scattering. While this process is applicable to
all species subjected to an electric field, the magnitude of the effect is greatly enhanced when the
irradiating laser field is tuned near an atomic resonance. Tuning the laser near resonance comes with a
tradeoff; however, as the increased likelihood of photon absorption near resonance has a severe impact on
20 40 60 80 100
2
4
6
8
10
x 10
-3
Power [mW]
Deflection [mm]

72

the force. The derivation for the dipole force discussed in Chapter 2 is for the force on an atom in the
ground state; for an excited atom, the sign of the force is reversed. The sign reversal means that keeping
the atom in the ground state is important. For the resonant dipole force, equation (2.59) shows that the
dipole gradient force increases with increasing intensity gradient and decreasing detuning from resonance.
This suggests that there is an optimum combination of intensity (gradient) and detuning which provides for
the maximum force with an acceptable likelihood of excitation.

Figure 4-9: Laser detuning vs. intensity for Ω/ δ = 0.2
Figure 4-9 shows the required detuning versus laser power such that the ratio of the Rabi
frequency and the detuning ( Ω/ δ) is kept constant at 0.2, at the center of a 50 μm diameter laser. This ratio
represents a condition of a constant likelihood of excitation, approximately 1%. Since the dipole gradient
force increases linearly with increasing laser power but the Rabi frequency increases with the square root of
power; stronger forces may be obtained by increasing the laser power while adequately detuning the laser
to maintain the low probability of excitation. For the simulations performed in this section, a detuning of
approximately 511 GHz was used; this translates to 853.59 nm light. Another limitation on the induced
dipole force is its connection to the gradient in the laser field. For a Gaussian laser field, a strong gradient
requires the laser diameter to be small. This particular issue is addressed through laser interference in the
case of using optical lattices, as described in section 2.5.2.
10
-2
10
0
10
2
10
10
10
12
Power [W]
Detuning [Hz]

73

Figure 4-10: Diagram of notional experimental configuration represented by simulations in section 4.2
The numerical simulations in this section were performed using the TCL custom particle trajectory
code, employing the ODE solver subroutine mentioned in section 3.1.2. The simulations represented an
atomic beam, with a round radial profile, interacting with a perpendicular laser field, with a round radial
profile. The beam deflection was along the laser radial direction. A diagram of a notional experimental
configuration can be seen in Figure 4-10. The Gaussian intensity profile (TEM
00
spatial mode) is a suitable
spatially varying electric field, such that the force imparted on an atom passing through the field tends to
act like a lens. For a laser detuned to the red side (lower frequency) of the transition, the dipole gradient
force is in the opposite direction from the gradient of the electric field strength (intensity), thus atoms are
attracted to the center of the laser profile.

Table 4-2: Conditions for resonant dipole steering simulations
Simulations were conducted using the parameters in Table 4-2. The position of the laser relative
to the center of the atomic beam was varied from -50 μm to 50 μm, in the vertical direction, or
approximately a laser diameter in either direction, orthogonal to the laser axis and the atomic beam axis.
The deflection of the atomic trajectories versus laser position can be seen in Figure 4-11. As with
comparing atomic beam steering using the scattering force, the profile in Figure 4-11 represents the centers
TCL Code
Sampled trajectories 1.0E6
Interaction Model ODE Solver
Laser Diameter 50 μm
Inflow Diameter 10 μm
Inflow Surface to Laser Center 2.0 mm
Laser Center to Sample Surface 100 mm
Laser Power 100 W
Laser Detuning -511 GHz

74

of Gaussian least-squares fits of the spatial profiles of the deflected atomic beam. The shape of the plot
adequately traces the intensity gradient of the laser field in the vertical direction. It can also be seen that,
while the laser was near the centerline, there was a fairly linear relation between the position of the laser
and the deflection of the atomic beam. This is advantageous for micro-machining applications which
require fine control of the atomic flow for micron and sub-micron feature sizes.

Figure 4-11: Atomic beam center vs. vertical laser position for resonant dipole steering
4.3 Non-Resonant Dipole

Figure 4-12: Diagram of notional experimental configuration represented by simulations in section 4.3
The numerical simulations in this section were performed using the modified SMILE collisional
kinetic solver, employing the DSMC method described in section 3.2.1. The simulations represented a two
dimensional flow, a sheet of gas, interacting with a round laser field propagating perpendicular to the flow
direction. The flow deflection was along the laser radial direction. A diagram of a notional experimental
configuration can be seen in Figure 4-12. Moving from a resonant interaction between the laser and the
flow, to a non-resonant interaction, a tradeoff is made between the arbitrary applicability of the interaction
and the magnitude of the force. In this section, the flow has changed from cesium to nitrogen, a
representative arbitrary non-polar flow. As a consequence, the laser has changed, from a CW laser with
relatively low intensity, to a pulsed laser with much higher instantaneous intensity. This makes up for the
-50 0 50
-0.1
-0.05
0
0.05
0.1
Laser Position [ μm]
Position [mm]

75

significant reduction in the induced dipole force. An immediate consideration for this change is that the
system ceases to be easily simulated as a steady state phenomenon; it is instead simulated as a series of
laser pulses which form a quasi-steady system. The increased complexity of the simulation also required
that the domain be reduced from a fully three dimensional simulation to a two-dimensional simulation, as
described in section 3.2.1.

Table 4-3: Conditions for non-resonant dipole steering simulations
The simulations followed the procedure discussed in Chapter 3. The flow geometry was that of an
effluent flow which was geometrically skimmed. The laser interaction was simulated only on the portion
of the flow after the collimating slit, and therefore only on the portion of the flow which had already gained
some degree of directionality. The simulation consisted of a series of simulations which represented a train
of pulses interacting with the flow. An example of one of these pulse trains is a train of 5 ns pulses with
centers spaced 10 ns apart. The total train is 100 pulses long, with each pulse following the parameters
given in Table 4-3. With an average axial velocity in excess of 400 m/s, the amount of space traversed by a
molecule in the span of one pulse simulation (10 ns) was approximately 4 μm or 16% of the diameter of the
half-max of the laser field. On average, therefore, a molecule was influenced by approximately 6 pulses.
SMILE
Average Particles in Domain 50E6
Number of Time Steps 5000
Time Step 2 ps
Background Cells 80,000
Additional Sample Cells 720,000
Laser Wavelength 532 nm
Laser Diameter 25 μm
Laser Pulse Width 5 ns
Laser Pulse Energy 250 mJ

76

A B C D
Figure 4-13: Density [m
-3
]: A) baseline and after 100 pulses with laser center, y= B) 00 μm, C) 10 μm, D) 20 μm
The induced dipole force acted as a conservative potential around the center of the laser field.
Therefore, it bent the trajectory of a molecule like the lensing of light. When the center of the laser field
was positioned at the center of the molecular flow, the laser field acted to retard the expansion of the flow;
this is the basis for non-resonant dipole collimation in Chapter 5. When the center of the laser field was
positioned above or below the center of the flow, the molecular trajectories were bent by the field and the
flow was redirected, but only during the laser pulse portion of the simulation. The final density field for
several laser field center locations can be seen in Figure 4-13. In order to quantify the amount of deflection
imparted on the flow, the final set of density sample bins (x = 80 μm) from Figure 4-13 (B) are show in
Figure 4-14 as a function of the laser pulse set number; there were two pulses per pulse set and a total of 50
pulse sets, for 100 total pulses. The flow field cross section started in the initial condition and evolved as
the modified flow crossed the domain and reached the x=80 μm bins location. Because the flow field had
become a quasi-steady system after approximately 10 pulse sets (20 pulses total), the remaining flow fields
were be used to obtain better statistics. For the density profiles considered for comparison, the last 25 sets
of pulses (50 total pulses) were averaged, as seen in Figure 4-14.

77

Figure 4-14: Time averaging for vertical density cross-sectional profile plots
Figure 4-15 shows the result of time averaging the right-most vertical sampling bins for three
different laser locations. With the laser aligned with the collimating slit (displacement = 0 μm), the effect
of the laser was to act as a collimating lens. This concept is further investigated in Chapter 5. The peak
centerline number density of the flow increased by a factor of 5. With a laser displacement of 10 μm, the
flow shows a significant redirection towards the laser side of the centerline. In this case, the laser field
acted to both further collimate the flow and subsequently to steer it. Lastly, with a laser offset of 20 μm
(almost a full laser diameter), the flow shows influence from the laser on approximately half of the flow.
The flow has thus been subjected to a weak collimation and redirection proportional to laser intensity
which is overlapping the flow.
A B C
Figure 4-15: Cross sectional density [m
-3
] at for laser y= A) 00 μm, B) 10 μm, C) 20 μm
In order to compare the redirection of the flow by laser fields at various positions, the spatial
distribution of the flow number density at x = 80 μm was fit to a least-squares Gaussian curve. The centers
of those fits are shown in Figure 4-16, versus the position of the laser field along the y axis. The center of
the Gaussian curve fit was unaffected when the laser is aligned with the collimating slit. From there the
flow steering tracked the laser gradient as shown in the resonant corollary. As the laser was displaced from
-30 -20 -10 0 10 20 30
0
2
4
6
8
x 10
18
Vertical Position [ μm]
Number Density [m
-3
]

00 μm
Baseline
-30 -20 -10 0 10 20 30
0
1
2
3
4
x 10
18
Vertical Position [ μm]
Number Density [m
-3
]

10 μm
Baseline
-30 -20 -10 0 10 20 30
0
1
2
3
x 10
18
Vertical Position [ μm]
Number Density [m
-3
]

20 μm
Baseline

78

the centerline, the magnitude of the force at the centerline of the slit increased; the force was proportional
to the gradient of the intensity, not the intensity itself. Therefore the maximum steering was achieved for
the largest displacement coupled with the largest force. As shown in Figure 2-1, the largest force was
found at approximately 1 laser radii from the centerline, i.e. the half width half maximum. This, therefore,
was also the point in the laser field capable of providing the strongest steering force on the flow direction.
For laser displacements greater than this, the force was insufficient to adequately bend the flow and
therefore the steering of the flow returned to 0.

Figure 4-16: Flow center vs. laser position for non-resonant dipole steering
4.4 Summary
Three examples of using laser modification to steer a directional flow have been discussed. Two
of the examples show the steering of a thermal cesium beam by one of two resonant laser interactions,
photon scattering or the induced dipole force. Photon scattering is a straightforward and easily applied
force for the redirection of an atomic flow analogous to moving in a viscous fluid. However, photon
scattering suffers from drawbacks associated with its applicability for a particular atom-laser combination,
and a limitation in maximum force due to the transition lifetime of the atom. This interaction method has
the most potential for flows where some form of velocity selection is desired and can also be used to
separate atoms in particular ground states. The induced dipole force is less restricted by particular atom-
laser systems, but is greatly enhanced when in the proximity of a resonant electronic transition. By this
force, an atomic beam was shown to displace proportional to the intensity gradient of an irradiating laser
field with a Gaussian intensity profile. In a region around the centerline, approximately 1 laser radii in
either direction, the displacement of the atomic beam was fairly linear with the displacement of the laser,
leading to applications in micro-machining and deposition control where micron control of a neutral atomic
beam would be required. The third example was extending the induced dipole force simulations to an
-20 -10 0 10 20
-10
-5
0
5
10
Laser Position [ μm]
Flow Center [ μm]

79

arbitrary molecular species, nitrogen. Extending this force required the use of pulsed lasers to compensate
for the reduced effect of the induced dipole in non-resonant situations. Again, the displacement of the flow
followed the gradient of the irradiating laser. It showed a fairly linear region between the peaks of the
displacement, approximately 1 laser radii in either direction from centerline. Since the force on the
molecule is proportional to its polarizability, this process could be used for species separation based on
polarizability to mass ratio.

80

Chapter 5 Collimation
The next examples discussed are those of laser modification of a flow in order to collimate it. As
a potential advantage over geometric flow skimming techniques, the use of laser modification offers the
ability to truly collimate. Geometric skimming trades density inversely for velocity distribution width,
leading to the waste of most of an effluent source in order to get the desired velocity distribution profile.
By altering the species trajectories directly with a laser field, the flow density can be maintained while still
decreasing the width of the velocity distribution. Again, the two numerical methods discussed in Chapter 3
were employed to investigate these laser modification situations. Following the simulations for resonant
and near-resonant steering of an alkali atomic beam, the TCL code was used to investigate the use of
photon scattering and the induced dipole force to further collimate a thermal cesium beam. These
simulations continue to have applicability in the micro-machining and lithography communities as well as
atomic physics and spectroscopy. SMILE was again used to simulate the interaction between a flow of
molecular nitrogen and non-resonant laser pulses. For these simulations, the effect of intermolecular
collisions on the expansion flow was vital to assessing the use of laser modification for true flow
collimation. For both the resonant and non-resonant dipole forces, the concept of bending the species
trajectories is analogous to bending rays of light through a lens. By modifying the intensity of the laser
field (or the time between pulses), the focal point of the “molecular lens” is changed, although different for
each velocity group. This requires some optimization to be made for the axial velocity of the beam,
especially for thermal systems with wide axial velocity distributions.
5.1 Photon Scattering
The use of photon scattering for the collimation of alkali atomic beams has been documented
experimentally (Rehse, Bockel and Lee 2004). The process is known as a one-dimensional “optical
molasses” because the laser fields act like a viscous fluid, retarding the velocity of the particles along the
laser axis. In this configuration two, counter propagating, non-interfering (in reference to their polarization
vectors), laser fields are aligned perpendicular to an alkali atomic beam. When both lasers are slightly red-

81

detuned (lower frequency) from the atomic transition, atoms with velocities travelling against the
propagation direction of each laser will be preferentially selected for excitation, through Doppler shifting in
the atomic frame. Subsequently, this adds momentum counter to the atom’s velocity, slowing it down.
This setup is very similar to the situation described in section 4.1, with the addition of a counter
propagating field. This is accomplished, experimentally, by retro reflecting the initial beam off of a mirror.
The use of wave plates is required to assure the beam does not interfere with itself. It is important to note
that the position of the atoms is not affected by this technique. The “viscous” effect only slows the velocity
of the atoms but does not affect their location within the beam. Therefore this process cannot focus an
atomic beam; it simply collimates the beam, halting dispersion. Thus the minimum beam width obtained
by this technique is the initial beam diameter at the point the beam entered the laser field. However the
minimum velocity spread obtained by this technique is given by the recoil velocity of an atom after
emitting a photon, e.g. approximately 3 cm/s for cesium.

Figure 5-1: Diagram of notional experimental configuration represented by simulations in section 5.1
The numerical simulations in this section were performed using the TCL custom particle trajectory
code, employing the ODE solver subroutine mentioned in section 3.1.2. The simulations represented an
atomic beam, with a round radial profile, interacting with a counter-propagating, perpendicular laser fields,
with a round radial profile. The beam collimation was along the laser propagation axis. A diagram of a
notional experimental configuration can be seen in Figure 5-1. Table 5-1 gives the simulation conditions
for the collimation of a thermal atomic beam via photon scattering. For the sake of simplicity, the same
simulation conditions as the beam steering simulations in section 4.1 were used. Because the TCL code
does not incorporate the effects of intermolecular collisions, as is done in SMILE, the interaction with an

82

already partially collimated and free molecular scenario was considered instead of the higher density flow
associated with the beam source effluence (the oven).

Table 5-1: Conditions for photon scattering collimation simulations
Following the procedure used in section 4.1, the spatial distribution of a thermal cesium beam was
evaluated after interacting with the laser fields. These spatial distributions were fit with a least-squares
Gaussian fit in order to allow consistent comparison. Figure 5-2shows the width of these fits, for a cesium
beam which has crossed through a one-dimensional optical molasses, versus the frequency detuning of the
irradiating lasers, in relation to the cesium D
2
transition. The grey line represents the width of the atomic
beam simulated without passing through the laser interaction, the baseline width. When the lasers were
detuned too far to the red side (lower frequency) of the transition, none of the atoms, at any velocity, were
sufficiently in resonance to be strongly influenced by the lasers. In other words, too few photons were
scattered to cause significant momentum transfer from the lasers to the atomic beam. As the frequency
detuning was reduced (moving closer to the center of the graph), a larger fraction of the atoms were
brought into resonance and the atomic beam became more collimated. After an optimal detuning,
approximately Γ/2 or 2.62 MHz, the magnitude of the effect was reduced until it was reversed. This is due
to the lasers crossing to the blue side (higher frequency) of the transition. On the blue side, the relevant
velocity classes being Doppler shifted into resonance were reversed in sign. In this region, each laser
begins preferentially selecting atoms which were moving away from it, causing dispersion instead of
collimation.
TCL Code
Sampled trajectories 1.0E6
Interaction Model ODE Solver
Laser Diameter 25.4 mm
Inflow Diameter 1.0 mm
Inflow Surface to Laser Center 265 mm
Laser Center to Sample Surface 445 mm

83

Figure 5-2: Atomic beam width vs. laser frequency for photon scattering collimation
The root of the collimation process in a one dimensional optical molasses is a balance between
two opposing forces, one from each laser. In equilibrium, i.e. when the atom has zero velocity along the
laser propagation axis, the atom absorbs photons equally from each laser. As the power of the lasers is
increased, the rate at which velocity is damped from the system is increased; however, the magnitude of the
collimation is still limited by the initial spatial distribution upon entering the laser fields. Without spatial
selection along the atomic beam, the resulting width of the atomic beam cannot be reduced below the initial
width at the point where the beam enters the laser field. Additionally, increasing the power will
substantially degrade the molasses as the effects of stimulated emission become prevalent. If an atom were
to absorb a photon from laser “A” and have it subsequently stimulated out by laser “B”, the result would be
a net two photon momentum gain in the direction of laser “A’s” propagation. This process is random; it
would tend to cause dispersion (heating) of the flow instead of collimation.

A B
Figure 5-3: Atomic beam A) collimation and B) dispersion vs. laser power via photon scattering
Neglecting the effects of stimulated emission between the two lasers, the minimum and maximum
values for atomic beam width as a function of laser frequency, shown in Figure 5-2, can be seen in Figure
5-3 (A) and Figure 5-3 (B) respectively. The beam collimation and dispersion plots, depending on the
frequency detuning, show an asymptotic trend related to the limitations intrinsic to the photon scattering
-40 -20 0 20 40
1
1.5
2
2.5
Detuning [MHz]
Width [mm]
-10 -5 0 5 10
0
1
2
3
Detuning [MHz]
Width [mm]
20 40 60 80 100
0.5
0.55
0.6
Power [mW]
Width [mm]
20 40 60 80 100
1.6
1.8
2
2.2
Power [mW]
Width [mm]

84

process, namely the maximum rate of photon scattering. Features within the curves, such as the sudden
drop in Figure 5-3 (A) at 78 mW and dip in Figure 5-3 (B) at 54 mW, are artifacts from the finite number
of simulations performed. Each point in the graph represents the relative extrema (minimum or maximum)
from a set of simulations which swept a frequency space, as was done to create Figure 5-2. A perfectly
smooth line would be required to accurately find an extrema; this would in turn require an infinite number
of points (simulations). Therefore, without sufficient resolution in the frequency dependant profile (similar
to Figure 5-2) to adequately capture the true minima or maxima, artificial features relating to the position
and magnitude of those extrema appear in Figure 5-3 (A) and Figure 5-3 (B).
5.2 Resonant Dipole

Figure 5-4: Diagram of notional experimental configuration represented by simulations in section 5.2
The numerical simulations in this section were performed using the TCL custom particle trajectory
code employing the ODE solver subroutine mentioned in section 3.1.2. The simulations represented an
atomic beam, with a round radial profile, interacting with a perpendicular laser field, with a round radial
profile. The beam collimation was along the laser radial direction. A diagram of a notional experimental
configuration can be seen in Figure 5-4. The baseline conditions for these simulations can be seen in Table
5-2. In section 4.2 the resonant dipole force was shown to bend an atomic beam, as it passed through a
Gaussian light field, in a fashion that could analogously called “atom optics”. In this chapter the effect was
approached for the purpose of collimating an atomic beam. Like a lens used to bend light rays, this
configuration of the dipole force resulted in the focusing of the atomic beam. Unlike the spatially
independent scattering force, the position of the atomic trajectories was critical to obtaining the desired
focusing.

85

Table 5-2: Conditions for resonant dipole collimation simulations
The first example of flow collimation, by the radial dipole force, is the effect of the width of the
spatial profile versus the location of the crossing laser field, taken from the simulations performed for
section 4.2. It was shown in Figure 4-11 that the location of the laser field has a direct effect on the final
location of the center of the profile. Additionally, the simulations show a change in the width of the spatial
profile as a function of laser position. Figure 5-5 shows the width of the Gaussian least-squares fit of the
spatial profile, versus the position of the laser field in the vertical direction. The grey line represents the
un-perturbed width of the atomic beam, i.e. the width with no laser. As may be expected, the positions
which resulted in the most dispersion of the beam are the positions which imparted the most deflection on
the beam. At these positions, the force on an individual portion of the atomic flow was different according
to that portion’s entry into the field. This means that when atomic beam entered the laser such that the
greatest deflection was accomplished, there was also the greatest difference between the forces acting on
the top of the atomic beam and the forces on the bottom of the beam. When the position of the center of
the laser field was aligned with the axis of the beam, the divergence of the atomic beam was cancelled by
the bending of the atomic trajectories as they crossed the laser field.
TCL Code
Sampled trajectories 1.0E6
Interaction Model ODE Solver
Laser Diameter 50 μm
Inflow Diameter 10 μm
Inflow Surface to Laser Center 2.0 mm
Laser Center to Sample Surface 100 mm
Laser Power 100 W
Laser Detuning -511 GHz

86

Figure 5-5: Atomic beam width vs. laser vertical position for resonant dipole collimation
The collimation of the flow is dependent on the relation between the axial velocity of the atomic
beam, i.e. interaction time, and the power of the laser. Increasing the power of the laser increases the
magnitude of the dipole force attracting the atoms to the point of highest intensity within the profile. This
is analogous to selecting a faster lens for an optical system. However it is possible to pull so strongly that
the laser field imparts more than a cancelling amount of momentum to the flow, causing the flow to
become convergent. Figure 5-6 shows the width of the Gaussian least-squares fit of the spatial profile,
versus the power of the radiating laser field. For the set of simulations considered, the power of the laser
was not sufficiently increased to cause the focal point of the “atomic lens” to pass the simulation plane and
therefore begin to show dispersion of the flow.

Figure 5-6: Atomic beam width vs. laser power for resonant dipole collimation
-50 0 50
0.11
0.12
0.13
0.14
Laser Position [ μm]
Beam Width [mm]
20 40 60 80 100
0.105
0.11
0.115
0.12
Power [mW]
Width [mm]

87

5.3 Non-Resonant Dipole

Figure 5-7: Diagram of notional experimental configuration represented by simulations in section 5.3
The numerical simulations in this section were performed using the modified SMILE collisional
kinetic solver, employing the DSMC method described in section 3.2.1. The simulations represented a
two-dimensional flow, an expanding sheet of gas, interacting with a round laser field, propagating into the
infinite direction. The flow collimation was along the laser radial direction. A diagram of a notional
experimental configuration can be seen in Figure 5-7. Following the procedures in section 3.2.1, the
portion of the domain past the exit plane of the 2-D slit can be seen in Figure 5-8 for the unperturbed
baseline simulation. The stagnation condition was again N
2
at 300 K and 1 Pa. This flow field was the
steady state solution to the 2-D flow from the slit and was the initial condition for the simulation of the
laser perturbation. The black line represents the full width half max diameter of the laser field. The laser
was centered on the slit in the vertical direction and offset by one radius from the exit plane. The code only
applied the force from the laser on the portion of the domain shown in Figure 5-8, not further upstream in
the stagnation chamber (shown in Figure 3-8). In a physical experiment the laser field would be clipped by
the stagnation chamber and not propagate inside.

88

Figure 5-8: Density [m
-3
] initial condition used for non-resonant collimation
The characteristics of the laser interaction simulations are given in Table 5-3. The laser pulses
were assumed to have a Gaussian spatial profile in the plane of Figure 5-8 and a Gaussian temporal
envelope, with a full width half maximum of 5 ns. The simulations were conducted in pulse sets, with the
same molecules being used from set to set, as was done in the simulations for laser steering in section 4.3.
The first variable investigated was the effect of the time between pulses, on the collimation of the flow.
This was accomplished by simulating the domain with the laser pulse, in the first half of the step, and
without, for a second, and then repeating the process for the desired number of pulses in the train. The
simulations, with and without the laser, both simulated 5000 time steps with the total time of the second
half of the step ranging from 0 ns (back to back pulse train) to 50 ns in between pulses. Fifty pulses were
chained together for each simulation, except for the 0 ns case, where 100 back-to-back pulses were
simulated.

Table 5-3: Conditions for non-resonant dipole collimation simulations
SMILE
Total Particles in Domain 110E6
Number of Time Steps 5000
Time Step 1-10 ps
Background Cells 240,000
Additional Sample Cells 360,000
Laser Wavelength 532 nm
Laser Diameter 25 μm
Laser Pulse Width 5 ns
Laser Pulse Energy 250 mJ

89

Figure 5-9 shows the number density flow field after a sequence of 100 laser pulses (1,000 ns)
have been passed though the region. The false-color images use the same scale as Figure 5-8. The flow
field shows an increase in the centerline density, with local maxima appearing within the laser field at
approximately half a radii, and again after 1 radius. The exit from the slit also shows an increase in density
due to the lensing of the expansion profile towards the centerline. The two local maxima, on the right
originate from the deceleration of the flow by the laser field along the centerline. The axial velocity was
accelerated on the left side of the laser field and decelerated on the right. This caused a compression within
the laser field after the center of the laser. It should be noted that the sampling for these flow fields was
conducted during the last 5% of the 100
th
pulse, leading to increased statistical noise in the plot.

Figure 5-9: Density [m
-3
] after 100 laser pulses with 0 ns intervening time
Figure 5-10 also shows the number density after a set of simulations; in this case, 50 pulses were simulated
with 50 intervening times of 5 ns each. Therefore, the total interaction time was (50*10 ns) + (50*5 ns),
750 ns. Per unit time, however, the amount of laser interaction was less than the previous case by 30%.
Compared with Figure 5-9, the increase in axial density has been reduced. The false color scaling is the
same in Figure 5-10, as it was for Figure 5-8 and Figure 5-9.

90

Figure 5-10: Density [m
-3
] after 50 laser pulses with 5 ns intervening time
As was the case with non-resonant steering, the simulation resulted in a quasi-steady condition
after approximately 10 sets of pulses and wait times. Figure 5-11 shows the vertical profile from the last
(x = 40 μm) set of sampling cells in Figure 5-9 as a function of the pulse set simulated, one pulse set
included both the pulse, and the intervening time. The vertical density profile did not change significantly
as a function of further simulations. In order to increase the statistics of the results, the last 25 sets of
pulses were averaged to get the vertical density profile of the resulting laser-collimated flow.

Figure 5-11: Time averaging for vertical density cross-sectional profile plots
Figure 5-12 shows the comparison between several, averaged, vertical flow density profiles as a
function of the wait time between pulses, for constant laser pulse energy of 250 mJ. For the case of the 0
ns intervening, 100 back to back pulses were simulated. For all other intervening times, 50 pulses, and 50
periods without the laser, were simulated. As expected, the largest increase in the centerline number
density (thus collimation) was for the case of the least intervening time between pulses, 0 ns. However,

91

even the case of a pulse every 50 ns, which resulted in molecules interacting with only one pulse over the
course of their expansion, showed a factor of two increase in the centerline density when compared with
the case of no laser interaction.

Figure 5-12: Cross sectional density [m
-3
] vs. intervening time for pulsed dipole collimation
Figure 5-13 shows the first frame of Figure 5-12, for the vertical velocity instead of the density.
As an indication of collimation, the magnitude of the vertical velocity was reduced by approximately half
over the spatial width of the laser pulse. Outside the laser profile, the velocity profile trends along the same
linear relationship as the unperturbed case. As a note, this vertical velocity profile is given for a cross
section at (x = 40 μm) from the slit, approximately 1 laser diameter further downstream from the laser.

Figure 5-13: Cross sectional velocity [m/s] for two intervening times
The same sets of simulations, as were conducted for Figure 5-12, were run for the case of a
continuous train of pulses (0 ns intervening time), but varying laser pulse energies. Figure 5-14 shows
vertical flow density profiles for varying pulse energies. It should be noted that as the pulses reach an
-15 -10 -5 0 5 10 15
1
1.5
2
2.5
x 10
19
Position [ μm]
Density [1/m
3
]

Baseline
0 ns
5 ns
-15 -10 -5 0 5 10 15
1
1.5
2
2.5
x 10
19
Position [ μm]
Density [1/m
3
]

Baseline
0 ns
10 ns
-15 -10 -5 0 5 10 15
1
1.5
2
2.5
x 10
19
Position [ μm]
Density [1/m
3
]

Baseline
0 ns
25 ns
-15 -10 -5 0 5 10 15
1
1.5
2
2.5
x 10
19
Position [ μm]
Density [1/m
3
]

Baseline
0 ns
50 ns
-15 -10 -5 0 5 10 15
-100
0
100
Position [ μm]
Velocity [m/s]

Baseline
0 ns
5 ns
-10 -5 0 5 10
-100
-50
0
50
100
Position [ μm]
Velocity [m/s]

Baseline
0 ns
5 ns

92

energy of 500 mJ, the on axis intensity has reached 1.32x10
17
W/m
2
. At these intensities, the likelihood of
ionization has become appreciable. However, it should also be noted that the gain in collimation is not
significant between 200 mJ and 500 mJ. Therefore the necessity of such laser intensities is not paramount.

Figure 5-14: Cross sectional density [m
-3
] vs. pulse energy for pulsed dipole collimation
5.4 Summary
Three examples of using laser modification to collimate a flow have been discussed. Two of the
examples showed the collimation of a thermal cesium beam by one of two resonant laser interactions,
photon scattering (optical molasses) or the induced dipole force. Photon scattering is a straightforward and
easily applied force for the reduction of the perpendicular velocity of atomic beam; however, it suffers from
drawbacks associated with applicability and maximum force. As both an advantage and a disadvantage, the
photon scattering force is not spatially dependant in and of itself (but can be by way of adding a magnetic
field), meaning that it only retards the momentum of the atoms without altering their spatial position. The
minimum beam size is, therefore, dependant on the inflow beam size. The resonant dipole force is not so
-15 -10 -5 0 5 10 15
1
1.5
2
2.5
x 10
19
Position [ μm]
Density [1/m
3
]

Baseline
250 mJ
50 mJ
-15 -10 -5 0 5 10 15
1
1.5
2
2.5
x 10
19
Position [ μm]
Density [1/m
3
]

Baseline
250 mJ
100 mJ
-15 -10 -5 0 5 10 15
1
1.5
2
2.5
x 10
19
Position [ μm]
Density [1/m
3
]

Baseline
250 mJ
150 mJ
-15 -10 -5 0 5 10 15
1
1.5
2
2.5
x 10
19
Position [ μm]
Density [1/m
3
]

Baseline
250 mJ
200 mJ
-15 -10 -5 0 5 10 15
1
1.5
2
2.5
3
x 10
19
Position [ μm]
Density [1/m
3
]

Baseline
250 mJ
300 mJ
-15 -10 -5 0 5 10 15
1
1.5
2
2.5
3
x 10
19
Position [ μm]
Density [1/m
3
]

Baseline
250 mJ
500 mJ

93

limited. The atomic trajectories are bent like rays of light through a lens, which can lead to beam sizes
smaller than those which entered the laser field. The third example showed the collimation of a nitrogen
effluent flow using a train of high intensity laser pulses. In these simulations, the increase in centerline
density was shown versus the influences of laser pulse energy, for a rapid train of pulses and versus varying
intervening times between pulses, for constant pulse energy. In both cases, the more laser light which
interacted with the flow, the greater the increase in final centerline density. For the case of increasing laser
pulse energy, the limit is the intensity which causes breakdown and ionization of the expanding gas.

94

Chapter 6 Optical Lattices
The last two chapters have looked at the use of the induced dipole potential force, as it relates to
the Gaussian radial profile of a laser field. Because the induced dipole potential is derived from the spatial
gradient of the field, the strength of the force is directly related to the size and shape of the field.
Increasing the size of a Gaussian field reduces the spatial gradient, and reduces the intensity for a given
energy or power. Increasing the energy or power of the laser to compensate will increase the intensity, but
it does not address the decrease in the spatial gradient; therefore, more intensity is required to get the same
force for a larger Gaussian field. This problem can be overcome, in some cases, for applications which do
not require the force to be perpendicular to the laser direction. The interference pattern between two
counter-propagating laser fields, or optical lattice, creates an intensity pattern which will oscillate along the
laser propagation axis. The oscillation periodicity is half the constituent lasers’ wavelength. In this
fashion, an increase in the radial size of the laser field does not decrease the gradient along the laser
propagating axis; therefore, the force on the atom or molecule is a function of laser intensity and axial
position, not radial laser field diameter.
This chapter looks at the use of such interference patterns for two different applications. The first
application is the acceleration of atoms by trapping them within the optical potentials of an accelerating
optical lattice. Like a roulette ball falling into the bins on the table, an atom will become trapped within the
optical lattice if its velocity is close to the velocity of the lattice. Then, like the roulette ball trapped in the
table bin as the table is spun up, the atom will be accelerated as fast as the lattice is accelerated. The
second application is to use a pulsed optical lattice (an optical lattice formed by laser pulses instead of CW
fields) to heat a neutral continuum gas. In this application, the lattice is used to deposit kinetic energy into
the translational mode of the gas along the laser axis. Then, through a process of collisional relaxation, the
energy is transferred to other translational and internal modes.

95

6.1 Atomic Acceleration
A significant area of interest, for the use of laser modification of particle trajectories, is in the
direct acceleration of neutral species to hyperthermal velocities. Current external field methods of utilizing
the induced dipole force are limited by the magnitude and/or gradient of the electric field which can be
generated by electro-mechanical setups. Laser fields bypass this limitation and allow for electric field
strengths which approach the breakdown limit of the species and which vary over the wavelength of the
light, producing very strong gradients. These strong fields are necessary to impart high accelerations, such
that the velocity is increased in a laboratory scale distance. In order to reach 1 km/s in a distance of 1 m, at
least 5E5 m/s
2
, or 50,000 times the force of gravity, must be applied to the species. As demonstrated by
(Schrader, et al. 2001) and (Wilkinson, et al. 1996), acceleration, capable of obtaining high velocity neutral
atoms, is possible with frequency modification of an optical lattice. While the referenced experiments were
interested in the relocation of a single alkali atom, the approach can be applied to a flow (atomic beam), in
order to create a continuous stream of high velocity atoms.
The numerical simulations in this section were performed using the TCL custom particle trajectory
code employing the direct integration subroutine mentioned in section 3.1.2. These simulations follow the
resonant theory described in Chapter 2. The following simulations took the basic experimental setup
described in section 3.3 and extrapolated the concept for a potential proof of concept experiment. In these
simulations, a second laser field was counter propagating to the interaction field described section 3.3.
Additionally, a second acousto-optic modulator would be required to independently modulate the second
field frequency. The modulation of the two laser fields creates a moving standing wave, whose velocity is
dependent on the time-dependant frequency modulation. Therefore an accelerating optical lattice is formed
by modulating one, or both, of the two AOM frequencies from f
1
to f
2
repeatedly, over time. The design
space investigated by the simulations in this section consisted of modulating the lattice over a time period
between 4 and 14 μs, and the power of the laser between 0.1 and 1.1 W. In order to keep a constant
intensity and thus a constant maximum force, the laser diameter was increased from 1 mm to 11 mm
proportional to the laser power. In practice this would give the crossing thermal beam more interaction
time.

96

Table 6-1: Conditions for resonant optical lattice acceleration simulations
Table 6-1 gives some of the simulation conditions for the tangential acceleration of an atomic
beam, by optical lattice potentials. The acceleration was in the laser axial direction. The acceleration of
the lattice was fixed at 10
6
m/s
2
. Therefore the final velocity of the beam in the laser axial direction was
linearly related to the time of the interaction. Since the purpose of these simulations was to make some
predications about the feasibility of a proof of principle experiment, the acceleration is kept constant at a
value which is both experimentally plausible and within the range of moderate RF generators, required to
drive the acousto-optic modulators. For an 852.35 nm laser, the frequency detuning necessary for the
interference pattern to move at 4 m/s is 9.3 MHz. Therefore, the driving electronics must be able to
modulate from 0 MHz to 9.3 MHz over the course of a few microseconds while maintaining a clean, phase
locked signal. In the case of two, independently modulated, beams, each AOM may be driven by the same
signal but in opposite directions from its center frequency based on the m = +1 or m = -1 Bragg angle,
therefore reducing the electronic requirements by a factor of two.

A B
Figure 6-1: Atomic beam A) spatial and B) velocity distribution after 10
6
m/s
2
acceleration for 14 μs
Figure 6-1 shows the final spatial and velocity distributions for the atomic beam after it interacted
with an accelerating optical lattice, modulated such that it accelerated to 14 m/s every 14 μs. The simulated
laser field was assumed to be a Gaussian shape, therefore the intensity of the laser changed along the radial
-0.01 0 0.01 0.02 0.03 0.04
0
20
40
60
Position [m]
Probability [n.d.]
0 2 4 6 8 10 12
0
0.2
0.4
0.6
0.8
1
Velocity [m/s]
Probability [n.d.]

Accelerated
Skimmer
TCL Code
Sampled trajectories 1.0E6
Interaction Model Direct Integration
Simulated Acceleration 10
6
m/s
Inflow Diameter 1.0 mm
Inflow Surface to Laser Center 265 mm
Laser Center to Sample Surface 445 mm

97

direction. At the average axial velocity, it took approximately 4 μs to travel a millimeter. In this
simulation, the power and diameter of the laser have been increased proportionally such that the diameter
(FWHM) is approximately 11 mm. Therefore, an atom needed to be accelerated for the full duration of its
travel across the beam in order to be accelerated to 14 m/s. Since most of the atoms fail that requirement
due to bad timing, or limitations in the intensity profile, the average change in velocity was approximately
9.5 m/s. The grey line represents the velocity distribution of the atomic beam as it was at the injection
surface (the skimmer of an experiment), thus the unperturbed velocity distribution of the beam.

A B
Figure 6-2: Atomic beam A) spatial and B) velocity distribution for two laser diameters/powers
Now consider a simulation which accelerated the atoms over a time period of 4 μs, from 0 to 4
m/s. If the laser diameter was 1 mm, the shape of the disturbed velocity distribution would also show the
portion of atoms which were not fully accelerated to the maximum velocity. Figure 6-2 (A) shows the
velocity distribution of the atomic beam after crossing a 1 mm laser field, with a 4 μs modulation. In this
case the atoms had barely enough time to be effected by one modulation cycle before crossing the laser
field completely. Figure 6-2 (B) shows the velocity distribution for an 11 mm laser diameter. With more
time in the laser field, and therefore more modulation cycles to accelerate the flow, the velocity distribution
shows the entire atomic beam had been affected, leaving none of the atoms in their initial condition.
Furthermore, the atoms were distributed closely around an accelerated center velocity. While this velocity
is not the maximum velocity of the lattice (4 m/s), the entire atomic beam had been accelerated.
These simulations looked at extending the experimental setup used in this study for accelerating
the flow to a few meters per second, as a proof of principle. In order to show that such acceleration is
possible to higher velocities, the interaction time of the atoms within the laser must be increased. If the
laser is rotated such that it no longer intersected the atomic beam perpendicularly, the interaction time of
-2 -1 0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
Velocity [m/s]
Probability [n.d.]

Accelerated
Skimmer
-1 0 1 2 3
0.5
1
1.5
2
2.5
3
Velocity [m/s]
Probability [n.d.]

Accelerated
Skimmer

98

the atomic beam within the laser is greatly increased. An additional consideration is the need to modulate
over a much wider frequency range. The AOM used in the experiment could modulate 10-15 MHz to
either side of a 110 MHz center frequency. In order to reach a final velocity of 1 km/s, the laser frequency
must be modified by 2.3464 GHz. While there are no single acousto-optic modulators which can reach that
bandwidth, multiple passes through the same AOM can result in integer multiplications of the single AOM
bandwidth (Buchkremer, et al. 2000).
6.2 Neutral Gas Heating
In this section, the effects of a pulsed laser on a non-resonant continuum gas were investigated for
the potential and limitations of using an optical lattice for energy deposition. The laser-gas coupling was
limited to the translational energy mode, along the laser axis, and allowed for the production of a well
characterized non-equilibrium gas volume. Through the mechanism of intermolecular collisions, the
obtained kinetic energy, from the lattice, was subsequently transferred from the axial translational mode,
where it could potentially be returned to potential energy within the lattice, to other translational or internal
energy modes. These collisional relaxation phenomenon have been long studied and suggest a predicable
mechanism for energy redistribution. Without the effect of intermolecular collisions, the atoms would have
simply oscillated within the lattice potentials, trading potential for kinetic energy, and vice-versa.

Figure 6-3: Normalized free-molecular density perturbations due to a pulsed optical lattice
Such oscillations in a collisionless gas create a density gradient in the medium as the species
redistribute themselves within potential wells. Such a density gradient can be seen in Figure 6-3 for a free
molecular gas. This redistribution is the concept behind direct velocity distribution measurements of gas

99

species using coherent Rayleigh scattering (H. T. Bookey, et al. 2006). This technique has also been
extended into collisional regimes (Pan, Shneider and Miles, Coherent Rayleigh-Brillouin scattering in
molecular gases 2004) through the use of coherent Rayleigh-Brillion scattering analysis. Such temperature
measurement techniques could be used in a notional experimental setup for evaluating the energy
deposition into a gas. Another potential temperature measurement technique for diagnosing such an
experiment would be tunable diode laser spectroscopy (Zhou, Jeffries and Hanson 2007).
As this investigation is strongly dependant on the effect of an optical lattice on the velocity
distribution of a gas, a kinetic examination of the gas-lattice interaction must be conducted. As the non-
resonant dipole force is conservative, the difference in the maximum and minimum potential energy in the
field, or well depth ΔU, can be related to a maximum change in atomic or molecular velocity that
corresponds to a kinetic energy equal to ΔU. The maximum change in velocity is given by
2 vUm Δ= Δ (6.1)
This maximum change in velocity relates directly to the maximum amount of energy which can be
deposited into the gas relative to the gas temperature. If the average kinetic energy of the gas is on par with
the maximum energy addition, the likelihood of the potential well adding energy is about the same as the
potential well subtracting energy. Essentially, the lattice is in thermal equilibrium with the energy of the
gas. Reference (Shneider and Barker, Optical Landau damping 2005) describes the effect of an optical
lattice on the velocity distribution in the laser propagation direction as limited to ± Δv of the lattice velocity
given by

k
ω
ξ
Δ
Δ
= (6.2)
The magnitude of this window represents the maximum velocity which may start at the bottom of the
potential well and remain trapped within. This window causes perturbation in the velocity distribution
which tends towards a uniform distribution from ξ+ Δv to ξ- Δv. For lattice potentials corresponding to a
velocity range wider than the bulk of the velocity distribution (high intensity), the influence of the optical
lattice causes a general broadening of the distribution. To illustrate this effect, the evolving velocity
distribution in a free molecular gas due to a 200 m/s pulsed optical lattice can be seen in Figure 6-4.

100

Figure 6-4: Free-molecular velocity distribution perturbations due to a pulsed optical lattice
6.2.1 Pulse Width, Pulse Trains, and Gas Relaxation
For high intensity lasers, creating a stationary optical lattice, the general broadening of the velocity
distribution constitutes energy addition to the gas, which will be redistributed as the gas returns to
equilibrium, through collisions. For laser pulse durations on the order of nanoseconds or faster, there is
insufficient time for the gas outside the interaction region to travel into the interaction region, or vice-versa.
This limits the effect of thermal diffusion on the interaction process, over the length of one pulse. For the
interaction of one pulse with a gas, there is a limit to the amount of energy which can be deposited, namely
the difference between the well depth and the temperature of the gas. There is an additional restriction that
the temperature of relevance for this is the axial translational temperature. The average collision
frequencies of gases at standard temperature and pressure is on the order of 100 ps. Therefore the process
of redistributing the deposited energy will begin quickly and have a greater impact on pulses with pulse
widths greater than many collision times.
If the laser pulses are short, in order to allow the pulse intensity to increase without causing
breakdown, the ability for the gas to redistribute the energy from the excited translational mode to other
modes is degraded. This leads to the practical consideration of a train of pulses forming many sequential
optical lattices. If multiple pulses are used (either through the use of a high speed pulsed laser or an optical
cavity), consideration must be given to the time between pulses and the relaxation of the gas to equilibrium.
Because of the decrease in deposition efficiency with temperature, and the time required to relax to
equilibrium, there exists an optimum time between the pulses, in order to create the highest temperature at
the center of the interaction region. In order to create the optimum condition for successive pulses,
adequate time must be given for the excited axial translational mode to transfer energy to unexcited modes,
and relax to equilibrium, but not so much time as to allow thermal diffusion begins to carry the deposited
-1000 -800 -600 -400 -200 0 200 400 600 800 1000
0
0.5
1
1.5
x 10
-3
Velocity (m/s)
f

0 ns
10 ns
12.6 ns
20 ns

101

energy outward. Ultimately, by transferring energy out of the translational mode, a cooler gas is created in
the interaction region, which can more efficiently absorb energy from subsequent pulses.
The numerical simulations in this section were performed using the modified SMILE collisional
kinetic solver employing the DSMC method described in section 3.2.1. In order to cover a range of
species, argon, nitrogen, and methane were used as test gases. The polarizability to mass ratio of these
gases are, 2.715, 4.24 (Wolf, Briegleb and Stuart 1929), and 10.92 (Kang and Jhon 1982), [10
-15
C m
2
kg
-1

V
-1
] respectively. For monatomic species, the only consideration for thermal equilibrium is translational
relaxation. Assuming the variable hard sphere intermolecular potential for argon at 1 atm and 300 K, the
mean collision time is 137 ps (Bird 1994). Since a variable hard sphere gas needs 4 to 6 collisions to relax
from a large deviation from equilibrium, an optimum intervening time is expected to be about 700 ps. For
the same conditions, with nitrogen and methane, the time for translational relaxation is approximately 500
ps and 320 ps respectively; though, as molecules, internal modes must be also considered. For nitrogen at
300 K, 900 ps is estimated to bring the rotational energy within 1/e of the translational energy (Bird 1994).
Thus the optimal intervening time for nitrogen should be some time after the longest relaxation time (> 900
ps). The vibrational relaxation is estimated at a timescale much longer than translational or rotational, and
longer than thermal diffusion. Therefore at temperatures lower than the characteristic vibrational
temperature (for nitrogen ≈3000 K), vibrational relaxation does not influence the optimal intervening time.

Table 6-2: Conditions for non-resonant optical lattice gas heating simulations
6.2.2 Results and Discussion
Sampling was conducted over the last 5% of the simulation time. With between 5 and 15 million
simulated particles in the computational domain, the average statistical error for a given sampling cell was
approximately 2%. Mentioned temperatures are assumed to be at the centerline of the gas volume shown in
Figure 3-9. The simulation conditions are shown in Table 6-2. Since the pulses were modeled as Gaussian
in space, and the heating within the gas is related to the intensity of the laser field (depth of the potential
SMILE
Total Particles in Domain 5-15E6
Number of Time Steps 5000
Laser Wavelength 532 nm
Laser Diameter 100 μm
Laser Pulse Width 100 ps
Laser Pulse Energy 393 mJ

102

well), the final translational and rotational temperature profiles of the gas for a single pulse follow a trend,
proportional to the laser intensity. The temperature profiles of the test gases after a single optical lattice
interaction can be seen in Figure 6-5. It should be noted that the three test gases were tested independently;
this figure does not represent a mixture.

Figure 6-5: Translational and rotational temperatures after one pulse vs. radial position
The short duration of the laser pulse did not allow for thermal diffusion to influence the profile
shape, over a single pulse, nor allow for the internal energy modes to equilibrate with the excited
translational mode. The translational temperature mentioned in Figure 6-5 was the overall translational
temperature and was therefore an average of the three translational modes. As mentioned, due to the short
pulse width and relatively low temperatures (compared to vibrational characteristic temperatures), the
vibrational modes were unexcited over a single lattice pulse. The relative magnitude of the temperature
profile for each gas was consistent with the polarizability to mass ratio given above.

Figure 6-6: Change in total domain translational energy after one pulse vs. initial gas temperature
The efficiency of the energy transfer between the lattice and the gas was dependent on the
difference between the well depth and the translational velocity distribution. Figure 6-6 shows the change
in total translational energy in the domain, for one optical lattice pulse, for various initial gas temperatures,
relative to the change at 300 K. In each of these simulations, the laser characteristics were the same, only
the ambient gas temperature, and thus velocity distribution, changed. The root of this degradation in
0 20 40 60 80 100
300
400
500
600
Position [ μm]
Temperature [K]

Ar - trn
N
2
- trn
N
2
- rot
CH
4
- trn
CH
4
- rot
500 1000 1500 2000 2500
10
-1
10
0
Temperature [K]
Δ Energy [n.d.]

Ar
N
2
CH
4

103

deposition efficiency was that the hotter, broader initial velocity distribution was disturbed less
significantly than a cooler, narrower distribution for a given laser power and pulse duration.

Figure 6-7: Overall temperature after 10 pulses vs. intervening time for multiple gases
Figure 6-7 shows the final temperature (average of translational + internal) of multiple gases,
initially at 300 K, measured at the end of 10 sequential lattice pulses, with zero lattice velocity. The
highest temperature reached by the gas was dependent on the polarizability to mass ratio, which factors into
well depth, and the available energy modes in the gas. For a train of pulses with little intervening time,
shown by the left most point, nitrogen had a higher temperature than argon, due to its higher polarizability
to mass ratio. However, as deposited translational energy relaxed into internal modes, argon attained a
higher overall temperature since the energy was not distributed over as many modes. The simulated
optimum intervening time for argon, i.e. that which produces the highest overall temperature, was around
700 ps, consistent with the time for translational relaxation. The simulated optimum time for nitrogen was
around 1 ns, consistent with the time for rotational relaxation. The simulated optimum time for methane
was around 250 ps.

Figure 6-8: N
2
overall temperature after 10 pulses vs. intervening time for multiple pressures
Since the collision frequency was dependent on the number density of the gas, an inversely
proportional shift in the optimum intervening time was observed with changing gas pressure. This shift
was limited, for long intervening times, by the effect of thermal diffusion removing energy from the
10
-1
10
0
500
1000
1500
Intervening Time [ns]
Temperature [K]

Ar
N
2
CH
4
10
-1
10
0
10
1
600
800
1000
1200
Intervening Time [ns]
Temperature [K]

1.0 atm
0.5 atm
0.1 atm

104

centerline of the volume. Figure 6-8 shows the shift in optimal intervening time for the centerline
temperature of nitrogen originally at 300 K. The intervening time shows a peak shift between 1.0 and 0.5
atm, without a significant decrease in centerline temperature. At 0.1 atm, the peak should have been
approximately 10 ns according to the relaxation time; however, this was long enough for diffusion to
become a significant effect, reducing the centerline temperature. Therefore the reduction in temperature
due to thermal diffusion masked the effect of more efficient energy deposition.

Figure 6-9: Ar overall temperature vs. pulse for intervening time of 0 ns
In addition to inadequate time for thermal relaxation, rapidly applied lattice pulses actively
decelerate gas particles, for intervening times at or near zero. Figure 6-9 shows the argon centerline
temperature evolution as a function of pulse number with no intervening time. There was a large increase
in temperature for the first pulse, approximately 110 K. This was followed by a considerably smaller
energy addition for the second pulse, about 30 K, due to the lattice interaction with a velocity distribution
that had already been severely disturbed. However, the third pulse showed a marked decrease in the
temperature, about 30 K. A single 100 ps pulse was not sufficiently long enough to cause a large fraction
of particles to cross a distance equal to half of a lattice spatial period, 133 nm, in one pulse period, 100 ps.
Over one temporal pulse width, the particles were accelerated towards the anti-nodes. However, the axial
dipole force is periodic. By the third pulse, a large enough fraction of particles had travelled far enough,
that the sign of the dipole force had changed. They had crossed the anti-node and were now being
decelerated instead of accelerated. With a peak-to-peak time of 200 ps, the particles expected, on average,
one collision between pulses. After the fourth pulse, the spatial distribution of particles was sufficient that
there were a relatively equal number of accelerated and decelerated particles. Temperature increase was
then a function of the partial thermal relaxation which the gas undergoes pulse to pulse.
0 2 4 6 8 10
300
350
400
450
Pulse [n.d.]
Temperature [K]

105

A B
Figure 6-10: Ar overall temperature vs. pulse for intervening time A) 0.5 ns B) 10ns
Figure 6-10 (A) shows the temperature evolution, of the centerline of the gas volume, as a function
of the pulse number, for a 0.5 ns intervening time. Again, the test gas was argon. Temperature after the
pulse is shown with ( ◊), and the temperature after the intervening time (thus before the next pulse) is shown
as ( □). There was no significant temperature loss due to thermal diffusion during the intervening time. The
expected decrease in deposition efficiency with temperature was observed as a 20% smaller temperature
increase between pulse 9 and pulse 10, than between pulse 1 and pulse 2. If the intervening time was
increased sufficiently, thermal diffusion mitigated the temperature increase, as it allowed for the flow of
thermal energy away from the centerline. Figure 6-10 (B) shows the temperature evolution at the centerline
as a function of the pulse number, for a 10 ns intervening time. Intervening times, resulting in excess of
approximately 10 collisions, would impart more total energy to the gas volume, as the lattice is interacting
with a cooler gas, but yield a smaller maximum temperature. Figure 6-11 shows the evolution of the
pressure profile for the same test run. From pulse 4 to pulse 10, a linear propagation of the pressure profile
was observed, from a radial distance around 40 μm to 60 μm. The propagation of the pressure profile was
representative of diffusion, and the cause of the temperature loss seen in Figure 6-10 (B) past pulse four.

Figure 6-11: Ar pressure vs. radial position, intervening time of 10 ns
0 2 4 6 8 10
400
600
800
1000
1200
Pulse [n.d.]
Temperature [K]
0 2 4 6 8 10
400
600
800
1000
1200
Pulse [n.d.]
Temperature [K]
0 20 40 60 80 100
1
1.5
2
2.5
Position [ μm]
Pressure [atm]

Pulse 2
Pulse 4
Pulse 6
Pulse 8
Pulse 10

106

In order to assess the usefulness of the gas-lattice interaction as a method for creating high
temperature gases, a simulation was run over 50 pulses, with an optimal intervening time of 1 ns. Figure
6-12 shows the temperature evolution of nitrogen, initially at 300 K and 1 atm, as a function of pulse
number. The highest temperature reached was 2480 K. The reduction in deposited energy, as a function of
temperature is seen by the concave temperature profile. In addition to reduced deposition efficiency, there
was an increased effect of thermal diffusion, ultimately limiting the centerline temperature. The increase in
thermal diffusion at higher temperatures may change the optimal intervening time, as a function of the
number of pulses used as the benchmark. At higher temperatures shorter mean collision time and a higher
effect of thermal diffusion may require a shorter intervening time to reach higher temperatures.

Figure 6-12: N
2
overall temperature vs. pulse, intervening time of 1 ns
The following, notional, experimental setup has been identified. Splitting a seeded (narrow line width),
532 nm, pulsed Nd:YAG laser would provide the pump pulses which form the optical lattice. The
challenge is in creating a gas volume, which is both optically accessible and capable of being quantifiably
diagnosed. One solution to this challenge is a micro-channel, cut into two halves a quartz substrate. The
cut volume would be filled with the working gas and placed into a liquid nitrogen shielded, high vacuum
chamber. By sparsely mounting the micro-channel, the heat loss from the gas-substrate system, through
conduction and convection, could be made manageably small. The nitrogen shield would act as both a
constant temperature radiative surface, as well as a working gas chiller. The efficiency of the optical lattice
energy deposition increases as the temperature of the working gas decreases. Lastly, resistive strips could
be deposited on the substrate, allowing for resistive temperature measurements. The experimental
procedure would be to pass the pump pulses through the system orthogonally polarized; this will give the
baseline energy deposition from optical losses within the system, in the absence of the optical lattice.
When the pulses’ polarizations are aligned, and the optical lattice is formed, the additional increase in the
0 10 20 30 40 50
500
1000
1500
2000
2500
Pulse [n.d.]
Temperature [K]

107

substrate temperature can be used to calculate the energy deposition in to the gas via the radiative heat loss
from the substrate to the shield. Significant challenges are associated with the machining of the interaction
volume and temperature measurements; however, this setup attempts to maximize the energy deposition
efficiency while attempting to measure a straightforward experimental parameter, temperature.
6.3 Summary
When utilizing the induced dipole potential, it is the gradient of the electric field which is the
contributing characteristic of the field. Therefore the scalability of using the Gaussian radial profile of a
laser field is limited by the ramifications of reduced gradient for larger field diameters. The interference
pattern of two counter-propagating laser fields (an optical lattice) overcomes this limitation for applicable
situations, by creating a field oscillation in the axial direction, which oscillates over a span half the length
of the constituent laser wavelengths, significantly shorter than the diameter. For the case of accelerating
atoms to high velocities, a moving optical lattice can be used to trap and accelerate a flow of atoms. The
numerical setup used in these simulations was the same as for previous resonant interaction simulations,
and was based on the TCL code validating experiment mentioned in Chapter 4. This numerical setup was
identified as a possible experimental setup, for a proof of principle experiment, which would tangentially
accelerate a cesium atomic beam, to several meters per second, over a few microsecond timeframe. Such
acceleration is potentially capable of reaching km/s in a laboratory scale accelerator. The simulations show
the possibility of performing such a proof of principle experiment and predict the atomic beam profile after
the laser interaction. Optical lattices can also be used to heat a neutral gas. Simulations with the modified
SMILE code were performed, to show the effect of pulse spacing on the energy deposition of pulsed optical
lattices into argon, nitrogen, and methane gases. Optimum pulse spacing was found through a tradeoff
between relaxation time in the gas and thermal diffusion. With the optimum pulse spacing, a train of 50
pulses was able to raise the centerline temperature of a nitrogen volume to over 2400 K. This is a direct
improvement over processes such as laser pyrolysis, which requires resonant coupling to molecular internal
modes and is only capable of temperatures in the range of 1500-1800 K.

108

Chapter 7 Conclusions
A numerical study was conducted on resonant and non-resonant laser flow manipulations, for a
variety of laser and neutral flow conditions. A methodology for extending the use of laser fields, beyond
the control of ultra-cold experimentation in atomic physics, to relatively high energy flows, associated with
a myriad of engineering applications, was investigated. The representative test cases investigated in this
study, relate directly to flows associated with hypersonics, micro-machining, micro-engineering, and
continuing application in the area of atomic physics and spectroscopy. Two numerical approaches, with a
validating experimental effort, were exercised in this study, covering a range of flow conditions from free-
molecular to continuum. The simulations demonstrate the feasibility of laser flow manipulation in the
areas of flow steering, collimation, direct flow acceleration, and neutral gas heating.
Two numerical techniques were employed to investigate several, select, examples of laser flow
modification. The first technique was the TCL code, a custom particle trajectory code, which simulated a
rarefied thermal atomic beam interacting with a near-resonant laser field. The individual particle
trajectories were simulated by either direct numerical integration of the equations of motion, the solution to
a set of n-variable ordinary differential equations by Gear’s Method, or a statistical approach to photon
absorption and emission. In order to provide validation for this new code, a demonstration experiment was
conducted for the horizontal steering of a thermal [373 K] cesium beam by photon scattering. The
validating experiment used a narrow line width, external cavity, tunable diode laser to impart momentum to
the cesium beam. The laser was frequency shifted, by an acousto-optic modulator, to show atomic beam
steering as a function of detuning from the cycling transition, between the 6S
1/2
F = 4 ground state and the
6P
3/2
F = 5 excited state. The numerical simulations using the TCL code showed good agreement with the
validating experiment when experimental factors, such as the finite line width of the laser, were taken into
account. The second numerical approach used in the study was the DSMC code SMILE, which had been
specifically modified to include the non-resonant interaction between a pulsed laser field and an arbitrary
molecular flow. This code was used to investigate instances where the effects of intermolecular collisions
were deemed important to the qualitative application of laser modification to the flow.

109

With regards to steering a flow using laser manipulation, three cases were discussed. Two of the
cases demonstrated the steering of a thermal cesium beam by one of two resonant laser interactions: photon
scattering or the induced dipole force. Photon scattering is a straightforward and easily applied force for
the redirection of an atomic flow, analogous to being pushed by a viscous fluid; however, it suffers from
drawbacks associated with its applicability, for a particular atom-laser combination, and is limited in
maximum force by the transition lifetime. This interaction method has the most potential for flows where
some form of velocity selection is desired, as it relies on Doppler shifting, a velocity selective process. It
can also be used to separate atoms in particular ground states, as is done in high precision cesium beam
atomic clocks. The induced dipole force is less restricted by a particular atom-laser system, but is greatly
enhanced when the laser is in proximity of a resonant transition. An atomic beam was shown to displace
proportional to the intensity gradient of an irradiating laser field, using the dipole gradient force. In a
region around the centerline, approximately 1 laser radii in either direction of the flow center, the
displacement of the atomic beam was fairly linear with the displacement of the laser, leading to
applications in micro-machining and deposition control, where micron control of a neutral atomic flow
would be required. The third case of flow steering extended the induced dipole force simulations to an
arbitrary molecular species, nitrogen. Extending the dipole force to this flow scenario required the use of
pulsed lasers to compensate for the reduced magnitude of the induced dipole, in non-resonant situations.
Again, the displacement of the flow followed the gradient of the irradiating laser, with a fairly linear region
between the peaks of the displacement, approximately 1 laser radii in either direction from centerline of the
flow. Since the force on the molecule is proportional to its polarizability, it can be envisioned that this
process could be used for species separation, based on polarizability to mass ratio.
With regards to collimating a flow using laser manipulation, three cases, similar to those used for
flow steering, were discussed. Again, two of the cases involved the resonant interaction of a laser field and
an atomic cesium beam, while the other case extrapolated the use of the induced dipole force to a
representative arbitrary molecular flow, nitrogen. In the first case, photon scattering was shown to be a
straightforward and easily applied force for the reduction of the perpendicular velocity of an atomic beam;
however, it suffers from drawbacks associated with applicability and maximum force, as mentioned before.
As both an advantage and a disadvantage, the photon scattering force is not spatially dependant in and of

110

itself (but can be by way of adding a magnetic field), it only retards the momentum of the atoms without
altering their spatial position. This leads to a minimum beam size dependent on the inflow beam size. The
resonant dipole force is not so limited. Atomic trajectories are bent like rays of light through a lens,
focusing the atomic beam and leading to beam sizes smaller than those which entered the laser field.
However, this effect leads to dispersion on the other side of the focal point. The third case demonstrated
the collimation of a nitrogen vacuum effluent flow using a train of high intensity laser pulses. In these
simulations, an increase in centerline density was shown versus the influences of laser pulse energy, for a
rapid train of pulses, and versus varying intervening times between pulses, for constant pulse energy. In
both cases, the more laser light which interacted with the flow, the greater the increase in final centerline
density. For the trend of increasing laser pulse energy, the limit of the collimation becomes the intensity
which will cause breakdown and ionization of the expanding gas.
The last cases discussed in this study involve overcoming a counterproductive trade: increasing
beam diameter reduces the magnitude of the gradient, per unit intensity, associated with the induced dipole
force in the radial direction. The interference pattern of two counter-propagating laser fields (an optical
lattice), overcomes this limitation for applicable situations, by creating a field oscillation in the axial
direction which oscillates over a span half the length of the constituent laser wavelengths, significantly
shorter than the diameter of the field. For the case of accelerating atoms to high velocities, a moving
optical lattice was shown to trap and accelerate a flow of atoms. The numerical setup used in these
simulations was the same as for previous resonant interaction simulations, and was based on the TCL
code’s validating experiment, discussed in the chapter on flow steering. This numerical setup was
identified as a possible experimental setup for a proof of principle experiment, which would tangentially
accelerate a cesium atomic beam to several meters per second, over a few microsecond timeframe, an
acceleration of 10
6
m/s
2
. Such acceleration is potentially capable of reaching km/s in a laboratory scale
accelerator. The simulations showed the possibility of performing a proof of principle experiment, and
they predicted the atomic beam profile after the laser interaction. Optical lattices can also be used to heat a
neutral gas. Simulations with the modified SMILE code were performed, which showed the effect of pulse
spacing on the energy deposition, from pulsed optical lattices to representative non-polar monotonic,
diatomic, and polyatomic species. Optimum pulse spacing was found through a tradeoff between

111

relaxation time in the gas and thermal diffusion. With the optimum pulse spacing, a train of 50 pulses
raiseed the centerline temperature of a nitrogen volume to over 2400 K. This is a direct improvement over
processes such as laser pyrolysis, which requires resonant coupling to molecular internal modes and is only
capable of temperatures in the range of 1500-1800 K.

112

Bibliography
Alt, Wolfgang. "Optical control of single neutral atoms." Dissertation, Mathematisch-
Naturwissenschaftlichen Fakultät, Rheinischen Friedrich-Wilhelms-Universität Bonn, Bonn, 2004, 118.
Amini, Jason M., and Harvey Gould. "High Precision Measurement of the Static Dipole Polarizability of
Cesium." Physical Review Letters (The American Physical Society) 91, no. 15 (October 2003): 153001-1 -
153001-4.
Anderson, John David. Hypersonic and High Temperature Gas Dynamics. 2nd Edition. Edited by Joseph
A. Schetz. Reston, VA: American Institute of Aeronautics and Astronautics, Inc., 2006.
Ashkenas, Harry, and Frederick S. Sherman. "Structure and utilization of supersonic free jets in low density
wind tunnels." Prepared in cooperation with Calif. Univ., Berkeley, Jet Propulsion Laboratory, National
Aeronautics and Space Administration, 1965, 28.
Bird, G. A. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. 1st Edition. New York, New
York: Oxford University Press, 1994.
Bookey, H. T., A. I. Bishop, M. N. Shneider, and P. F. Barker. "Narrow-band coherent Rayleigh
scattering." Journal of Raman Spectroscopy (John Wiley & Sons) 37, no. 6 (May 2006): 655-662.
Bookey, Henry T., and Alexis I., Barker, P. F. Bishop. "Narrow-band coherent Rayleigh scattering in a
flame." Optics Express (Optical Society of America) 14, no. 8 (April 2006): 3461-3466.
Borgnakke, C., and P. S. Larsen. Journal of Computational Physics 18 (1975): 405-420.
Boyd, Robert W. Nonlinear Optics. 1st Edition. San Diego, CA: Academic Press, Inc., 1992.
Buchkremer, F. B. J., R. Dumke, Ch. Buggle, G. Birkl, and W. Ertmer. "Low-cost setup for generation of 3
GHz frequency difference phase-locked laser light." Review of Scientific Instruments (American Institute of
Physics) 71, no. 9 (September 2000): 3306-3308.
Caledonia, George E., Robert H. Krech, Byron D. Green, and Anthony N. Pirri. Source of high flux
energetic atoms. USA Patent 4,894,511. January 16, 1990.
Chu, S., L. Hollberg, J. E. Bjorkholm, A. Cable, and A. Ashkin. "3-dimensional viscous confinement and
cooling of atoms by resonance radiation pressure." Physical Review Letters (The American Physical
Society) 55, no. 48 (1985).
Cohen-Tannoudji, Claude, Bernard Diu, and Franck Laloë. Quantum Mechanics. 2nd Edition. Translated
by Susan Reid Hemley, Nicole Ostrowsky and Dan Ostrowsky. Vol. II. II vols. Paris: Hermann, 1977.
da Silva, M. Lino, V. Guerra, and J. Loureiro. "Nonequilibrium Dissociation Processes in Hyperbolic
Atmospheric Entries." Journal of Thermophysics and Heat Transfer (American Institute of Aeronautics and
Astronautics) 21, no. 2 (2006): 303-310.
Das, D., A. Banerjee, S. Barthwal, and V. Natarajan. "A rubidium-stabilized ring-cavity resonator for
optical frequency metrology: precise measurement of the D1 line in 133Cs." European Physical Journal D
(EDP Sciences) 38, no. 3 (2006): 545-552.

113

Donley, E. A., T. P. Heavner, F. Levi, M. O. Tataw, and S. R. Jefferts. "Double-pass acousto-optic
modulator system." Review of Scientific Instruments (American Institute of Physics) 76, no. 06 (June
2005): 063112-1 - 063112-6.
Ertmer, W., R. Blatt, J. L. Hall, and M. Zhu. "Laser Manipulation of Atomic Beam Velocities:
Demonstration of Stopped Atoms and Velocity Reversal." Physical Review Letters (The American Physical
Society) 54, no. 10 (March 1985): 996-999.
Estermann, I., S. N. Foner, and O. Stern. "The Mean Free Paths of Cesium Atoms in Helium, Nitrogen, and
Cesium Vapor." Physical Review (American Physical Society) 71, no. 4 (February 1947): 250-257.
Foot, Christopher J. Atomic Physics. New York, NY: Oxford University Press Inc., 2005.
Fulton, R., A. I. Bishop, and P. F. Barker. "Optical Stark Decelerator for Molecules." Physical Review
Letters (The American Physical Society) 93, no. 24 (December 2004): 243004-1 - 243004-4.
Gerginov, V., C. E. Tanner, S. Diddams, A. Bartels, and L. Hollberg. "Optical frequency measurements of
6s 2S1/2–6p 2P3/2 transition in a 133Cs atomic beam using a femtosecond laser frequency comb." Physical
Review A (The American Physical Society) 70, no. 4 (October 2004): 042505-1 - 042505-8.
Gordon, J. P., and A. Ashkin. "Motion of atoms in a radiation trap." Physical Review A (The American
Physical Society) 21, no. 5 (May 1980): 1606-1617.
Grimm, Rudolf, Matthias Weidmüller, and Yurii B. Ovchinnikov. Optical Dipole Traps for Neutral Atoms.
Vol. 42, in Advances in Atomic, Molecular, and Optical Physics, edited by Benjamin Bederson and Herbert
Walther, 95-170. Elsevier Science & Technology Books, 1999.
Hecht, Eugene. Optics. 4th Edition. Edited by Adam Black. San Francisco, CA: Pearson Education Inc.,
2002.
Heumann, K. G., et al. "Atomic Weights of the Elements 1993." Pure & Applied Chemistry (International
Union of Pure and Applied Chemistry) 66, no. 12 (1994): 2423-2444.
Ivanov, M. S., A. V. Kashkovsky, S. F. Gimelshein, and G. N. Markelov. Thermophysics and
Aeromechanics 4 (1997): 251-268.
Ivanov, M. S., and S. F. Gimelshein. Edited by A. D. Ketsdever and E. P. Muntz. 23rd International
Symposium on Rarefied Gas Dynamics. Whistler, Canada: American Institute of Physics, 2003. 339-348.
Ivanov, M. S., and S. V. Rogasinsky. Soviet Journal of Numerical and Analytical Mathamatical Modelling
3 (1988): 453-465.
Jerig, L., K. Thielen, and P. Roth. "High-Temperature Dissociation of Oxygen Diluted in Argon or
Nitrogen." AIAA Journal (American Institute of Aeronautics and Astronautics) 29, no. 7 (1990): 1136.
Kang, Young Kee, and Mu Shik Jhon. "Additivity of Atomic Static Polarizabilities and Dispersion
Coefficients." Theoretica Chimica Acta (Springer - Verlag) 61, no. 1 (January 1982): 41-48.
Ketsdever, Andrew David. "The Production of Energetic Atomic Beams via Charge Exchange for the
Simulation of the Low-Earth Orbit Environment." Dissertation, Aerospace Engineering, University of
Southern California, Los Angeles, CA, 1995, 158.
Kreis, M., et al. "Pattern generation with cesium atomic beams at nanometer scales." Applied Physics B
(Springer-Verlag) 63, no. 3 (December 1996): 649–652.

114

Kuga, Takahiro, Yoshio Torii, Noritsugu Shiokawa, and Takuya Hirano. "Novel Optical Trap of Atoms
with a Doughnut Beam." Physical Review Letters (The American Physical Society) 78, no. 25 (June 1997):
4713-4716.
Kuhr, Stefan, Wolfgang Alt, Dominik Schrader, Martin Müller, Victor Gomer, and Dieter Meschede.
"Deterministic Delivery of a Single Atom." Science, July 13, 2001: 278-280.
Letokhov, Vladilen S. Laser Control of Atoms and Molecules. New York, NY: Oxford University Press
Inc., 2007.
Liboff, Richard L. Introductory Quantum Mechanics. 2nd Edition. Reading, MA: Addison-Wesley
Publishing Co., Inc., 1991.
Lide, David R., and H. P. R. Frederikse, . CRC Handbook of Chemistry and Physics. 78. Boca Raton, FL:
CRC Press LLC, 1997.
Livingston, M. Stanley, and John P. Blewett. Particle Accelerators. New York, NY: McGraw-Hill, 1962.
Manista, Eugene J., and John W. Sheldon. "Measurement of the Cesium-Cesium Total Cross Section by
Atomic Beam Methods." Prepared for Journal of Chemical Physics, Lewis Research Center, National
Aeronautics and Space Administration, Cleveland, OH, 1964, 6.
Matsui, Makoto, Hiroki Takayanagi, Yasuhisa Oda, Kimiya Komurasaki, and Yoshihiro Arakawa.
"Performance of arcjet-type atomic-oxygen generator by laser absorption spectroscopy and CFD analysis."
Volume 73, no. 3 (April 2004): 341-346.
Metcalf, Harold J., and Peter van der Straten. Laser Cooling and Trapping. Edited by R. Stephen Berry,
Joseph L. Birman, Jeffrey W. Lynn, Mark P. Silverman, H. Eugene Stanley and Mikhail Voloshin. New
York, NY: Springer-Verlag New York, Inc., 1999.
Miller, J. D., R. A. Cline, and D. J. Heinzen. "Far-off-resonance optical trapping of atoms." Physical
Review A (The American Physical Society) 47, no. 6 (June 1993): R4567-R4570.
Minton, Timothy K., Christine M. Nelson, David E. Brinza, and Ranty H. Liang. Inelastic and Reactive
Scattering of Hyperthermal Atomic Oxygen From Amorphous Carbon. National Aeronautics and Space
Administration, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA: Jet
Propulsion Laboratory, 1991, 36.
Miroshnychenko, Y., et al. "Inserting Two Atoms into a Single Optical Micropotential." Physical Review
Letters (The American Physical Society) 97, no. 24 (December 2006): 243033-1 - 243033-4.
Moskowitz, Philip E., Phillip L. Gould, Susan R. Atlas, and David E. Pritchard. "Diffraction of an Atomic
Beam by Standing-Wave Radiation." Physical Review Letters (The American Physical Society) 51, no. 5
(August 1983): 370-373.
Ngalande, Cedrick G., Sergey F. Gimelshein, and Mikhail N. Shneider. "Gas mixing with pulsed optical
lattices." Physics of Fluids (American Institute of Physics) 19, no. 12 (December 2007): 128101-1 -
128101-4.
Ngalande, Cedrick, Sergey F. Gimelshein, and Mikhail N. Shneider. "Energy and momentum deposition
from pulsed optical lattices to nonionized gases." Applied Physics Letters (American Institute of Physics)
90, no. 12 (March 2007): 121130-1 - 121130-3.
Pan, X., P. F., Meschanov, A. V. Barker, R. B. Miles, and J. H. Grinstead. "Temperature Measurements in
Plasmas Using Coherent Rayleigh Scattering." Aerospace Sciences Meeting and Exhibit. Reno, NV:
American Institute of Aeronautics and Astronautics, 2001.

115

Pan, Xingguo, Mikhail N. Shneider, and Richard B. Miles. "Coherent Rayleigh-Brillouin scattering in
molecular gases." Physical Review A (The American Physical Society) 69, no. 03 (March 2004): 033814-1
- 033814-16.
Pan, Xingguo, Mikhail N. Shneider, and Richard B. Miles. "Power spectrum of coherent Rayleigh-
Brillouin scattering in carbon dioxide." Physical Review A (The American Physical Society) 71, no. 04
(April 2005): 045801-1 - 045801-2.
Pan, Xingguo, Peter F. Barker, A. Meschanov, Mikhail N. Shneider, and Richard B. Miles. "Temperature
measurements by coherent Rayleigh scattering." Optics Letters (Optical Society of America) 27, no. 3
(February 2002): 161-163.
Park, Chul. Nonequilibrium Hypersonic Aerothermodynamics. New York, NY: Wiley, 1990.
Rafac, R. J., C. E. Tanner, A. E. Livingston, K. W. Kukla, H. G. Berry, and C. A. Kurtz. "Precision lifetime
measurements of the 6p2P1/2,3/2 states in atomic cesium." Physical Review A (The American Physical
Society) 50, no. 3 (September 1994): R1976-R1979.
Rafac, Robert J., Carol E. Tanner, A. Eugene Livingston, and H. Gordon Berry. "Fast-beam laser lifetime
measurements of the cesium 6p 2P1/2,3/2 states." Physical Review A (The American Physical Society) 60,
no. 5 (November 1999): 3648-3662.
Ramsey, Norman F. Molecular Beams. Edited by R. J. Elliott, J. A. Krumhansl, W. Marshall and D. H.
Wilkinson. New York, NY: Oxford University Press, 1956.
Rehse, Steven J., Karen M Bockel, and Siu Au Lee. "Laser collimation of an atomic gallium beam."
Physical Review A (American Physical Society) 69, no. 06 (June 2004): 063404-1 - 063404-5.
Safronova, M. S., W. R. Johnson, and A. Derevianko. "Relativistic many-body calculations of energy
levels, hyperfine constants, electric-dipole matrix elements, and static polarizabilities for alkali-metal
atoms." Physical Review A (The American Physical Society) 60, no. 6 (December 1999): 4476-4487.
Sakai, Hirofumi, et al. "Optical deflection of molecules." Physical Review A (The American Physical
Society) 57, no. 4 (April 1998): 2794-2801.
Salomon, C., J. Dalibard, W. D. Phillips, A. Clairon, and S. Guellati. "Laser Cooling of Cesium Atoms
Below 3 μK ." Europhysics Letters (IOP Science) 12, no. 8 (August 1990): 683-688.
Schrader, D., S. Kuhr, W. Alt, M. Müller, V. Gomer, and D. Meschede. "An optical conveyor belt for
single neutral atoms." Applied Physics B (Springer) 73, no. 8 (November 2001): 819-824.
Shaffer, Michael K. "Photoassociative spectroscopy of ultracold metastable argon and study of dual species
trap loss in a rubidium - metastable argon MOT." Dissertation, Physics, Old Dominion University, Norfolk,
VA, 2008, 132.
Shneider, M. N., and P. F. Barker. "Optical Landau damping." Physical Review A (The American Physical
Society) 71, no. 05 (May 2005): 053403-1 - 053403-9.
Shneider, M. N., P. F. Barker, and S. F. Gimelshein. "Transport in room temperature gases induced by
optical lattices." Jounral of Applied Physics (American Institute of Physics) 100, no. 07 (October 2006):
074902-1 - 074902-6.
Soven, Paul. "Self-consistent linear response study of the polarizability of molecular nitrogen." Journal of
Chemical Physics (American Institute of Physics) 82, no. 7 (April 1985): 3289-3291.
Stamper-Kurn, D. M., et al. "Optical Confinement of a Bose-Einstein Condensate." Physical Review Letters
(The American Physical Society) 80, no. 10 (March 1998): 2027-2030.

116

Stapelfeldt, H., Hirofumi Sakai, E. Constant, and P. B. Corkum. "Deflection of Neutral Molecules using the
Nonresonant Dipole Force." Physical Review Letters (The American Physical Society) 79, no. 15 (October
1997): 2787-2790.
Steck, Daniel Adam. "Alkali D Line Data." Vers. 2.1. Daniel A. Steck. September 8, 2008.
http://www.steck.us/alkalidata/ (accessed October 6, 2008).
Steuernagel, Ole. "Coherent transport and concentration of particles in optical traps using varying
transverse beam profiles." Journal of Optics A (Institute of Physics Publishing) 7, no. 6 (May 2005): S392-
S398.
Stry, S., S. Thelen, J. Sacher, D Malmer, P. Hering, and M. Mürtz. "Widely tunable diffraction limited
1000 mW external cavity diode laser in Littman/Metcalf configuration for cavity ring-down spectroscopy."
Applied Physics B (Springer-Verlag) 85, no. 2-3 (2006): 365-374.
Takekoshi, T., B. M. Patterson, and R. J. Knize. "Observation of Optically Trapped Cold Cesium
Molecules." Physical Review Letters (The American Physical Society) 81, no. 23 (December 1998): 5105-
5108.
Tanner, C. E., et al. "Measurement of the 6p2P3/2 State Lifetime in Atomic Cesium." Physical Review
Letters (American Physical Society) 69, no. 19 (Novermber 1992): 2765-2767.
Taylor, John Bradshaw, and Irving Langmuir. "Vapor Pressure of Cesium by the Positive Ion Method."
Physical Review (American Physical Society) 51, no. 9 (May 1937): 753-760.
te Sligte, E., et al. "Atom lithography of Fe." Applied Physics Letters (American Institute of Physics) 85,
no. 19 (November 2004): 4493-4495.
Tompa, G. S., J. L. Lopes, and G. Wohlram. "Compact efficient modular cesium atomic beam oven."
Review of Scientific Instruments (American Institute of Physics) 58, no. 8 (August 1987): 1536-1537.
Wilkinson, S. R., C. F. Bharucha, K. W. Madison, Qian Niu, and M. G. Raizen. "Observation of Atomic
Wannier-Stark Ladders in an Accelerating Optical Potential." Physical Review Letters (The American
Physical Society) 76, no. 24 (June 1996): 4512-4514.
Wolf, K. L., G. Briegleb, and H. A. Stuart. Z. Phys. Chem. B 6 (1929): 163-209.
Zhou, Xin, Jay B. Jeffries, and Ronald K. Hanson. "Wavelength-Scanned Tunable Diode Laser
Temperature Measurements in a Model Gas Turbine Combustor." AIAA Journal (American Institute of
Aeronautics and Astronautics) 45, no. 2 (February 2007): 420-425.

117

Appendix A Empirical Values and Their Sources
* denotes use by this work.

Cesium Properties

Physical Properties
Atomic Weight Units Source
m=132.905 m =[amu] (Lide and Frederikse 1997) *
m=132.90543±(5.0x10
-5
) m =[amu] (Heumann, et al. 1994)
Table A-1: Cs atomic weight
The melting point of cesium is 28.44° C = 301.59 K (Lide and Frederikse 1997).
Equation Units Source
10 10
log 10.5460 1.00log 4150
S
PTT =− −
P
S
=[Torr], T=[K] (Taylor and Langmuir 1937) *
10
log 5.006 4.711 3999
S
PT =+ −
P
S
=[Pa], T=[K] (Lide and Frederikse 1997)
Table A-2: Cs vapor pressure (solid)
Equation Units Source
10 10
log 11.0531 1.35log 4041
L
PTT =− −
P
L
=[Torr], T=[K] (Taylor and Langmuir 1937) *
10
log 5.006 4.165 3830
L
PT =+ −
P=[Pa], T=[K] (Lide and Frederikse 1997)
Table A-3: Cs vapor pressure (liquid)

Figure A-1: Comparison of published Cs vapor pressures
280 300 320 340 360 380 400
Temperature [K]
Pressure [Pa]

Taylor and Langmuir
Lide and Frederikse

118

Ionization Potential
Cs 1
st
Ionization Potential Units Source
E
I
=3.893905373±(9.6x10
-7
) E
I
=[eV] (Steck 2008) *
Table A-4: Cs Ionization potential
Collision Cross Section
Cs-Cs Total Collision Cross Section Units Source
σ=2040±20%=2040±408 σ=[Å
2
] (Manista and Sheldon 1964) *
σ=2350±5%=2350±118 σ=[Å
2
] (Estermann, Foner and Stern 1947)
Table A-5: Cs-Cs total collision cross section
Transitions
Frequency Units Source
f
0
=351725718.509±0.055 f
0
=[MHz] (Das, et al. 2006)
f
0
=351725718.4744±0.0051 f
0
=[MHz] (Gerginov, et al. 2004) *
f
0
=351725718.50±0.11 f
0
=[MHz] (Steck 2008)
Table A-6: Cs 6p
2
P
3/2
↔ 6s
2
S
1/2
transition frequency
Lifetime Units Source
τ=30.55±0.27 τ=[ns] (Tanner, et al. 1992)
τ=30.499±0.070 τ=[ns] (R. J. Rafac, et al. 1994) *
τ=30.57±0.07 τ=[ns] (R. J. Rafac, et al. 1999)
τ=30.405±0.077 τ=[ns] (Steck 2008)
Table A-7: Cs 6p
2
P
3/2
→ 6s
2
S
1/2
transition lifetime

Figure A-2: Cs D
2
transition energies (Steck 2008)
351.725 718 50(11) THz
4.021 776 399 375 GHz (exact)
5.170 855 370 625 GHz (exact)
339.7128(39) MHz
188.4885(13) MHz
12.798 51(82) MHz
263.8906(24) MHz

119

Figure A-3: Cs D
2
cycling transition

Static Polarizabilities

Unit Conversion

2-1 6 3 16 3
0
(C·m ·V ) 4 10 (c 1.11265 m) 10 (cm) απ α α
−−
×= × = ε

324 3 243
0
(cm ) 10 ( ) 0.148184 10 ( Åa) αα α
−−
× ==

( )
23 αα α
⊥
=+
&

Nitrogen
Static Polarizability Units Source
α=1.224x10
-40
α =[Cm
2
V
-1
] (Lide and Frederikse 1997) *
Table A-8: N static polarizability
Static Polarizability Units Source
α=1.936x10
-40
α =[Cm
2
V
-1
] (Lide and Frederikse 1997) *
α=1.832x10
-40
α =[Cm
2
V
-1
] (Soven 1985)
Table A-9: N
2
static polarizability

120

Cesium
Static Polarizability Units Source
α=6.593x10
-39
α =[Cm
2
V
-1
] (Safronova, Johnson and Derevianko 1999)
α=6.611±0.009x10
-39
α =[Cm
2
V
-1
] (Amini and Gould 2003)
Table A-10: Cs static polarizability

Abstract (if available)

Abstract The continuing advance of laser technology enables a range of broadly applicable, laser-based flow manipulation techniques. The characteristics of these laser-based flow manipulations suggest that they may augment, or be superior to, such traditional electro-mechanical methods as ionic flow control, shock tubes, and small scale wind tunnels. In this study, methodology was developed for investigating laser flow manipulation techniques, and testing their feasibility for a number of aerospace, basic physics, and micro technology applications. Theories for laser-atom and laser-molecule interactions have been under development since the advent of laser technology. The theories have yet to be adequately integrated into kinetic flow solvers. Realizing this integration would greatly enhance the scaling of laser-species interactions beyond the realm of ultra-cold atomic physics. This goal was realized in the present study. A representative numerical investigation, of laser-based neutral atomic and molecular flow manipulations, was conducted using near-resonant and non-resonant laser fields. To simulate the laser interactions over a range of laser and flow conditions, the following tools were employed: a custom collisionless gas particle trajectory code and a specifically modified version of the Direct Simulation Monte Carlo statistical kinetic solver known as SMILE. In addition to the numerical investigations, a validating experiment was conducted. The experimental results showed good agreement with the numerical simulations when experimental parameters, such as finite laser line width, were taken into account. Several areas of interest were addressed: laser induced neutral flow steering, collimation, direct flow acceleration, and neutral gas heating. Near-resonant continuous wave laser, and non-resonant pulsed laser, interactions with cesium and nitrogen were simulated.

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Asset Metadata

Creator Lilly, Taylor C. (author)

Core Title Laser manipulation of atomic and molecular flows

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Aerospace Engineering

Publication Date 07/22/2010

Defense Date 05/26/2010

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

Tag atomic beams,cesium,Direct Simulation Monte Carlo,DSMC,experiment,laser flow manipulation,laser-atom interaction,laser-molecule interaction,molecular beams,numerical simulation,OAI-PMH Harvest

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Muntz, E. Phil (committee chair), Bickers, Nelson Eugene, Jr. (committee member), Gimelshein, Sergey (committee member), Ketsdever, Andrew (committee member), Redekopp, Larry G. (committee member)

Creator Email taylor.lilly@hotmail.com,tlilly@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3207

Unique identifier UC1317506

Identifier etd-Lilly-3829 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-352839 (legacy record id),usctheses-m3207 (legacy record id)

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Document Type Dissertation

Rights Lilly, Taylor C.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu