Index
A
axial translation, 151
elements, 151
paraxial propagation of Gaussian beams, 149
treatment of Gaussian beam propagation, 148–162
ABCD Propagation Law, 151, 152
Aberrated incident phase, 236
Aberration-free lens, 265
Alkali atom, 331
Ampere-Maxwell equation, 11
Ampere’s equation, 5
Angle of transmission, 257
Angular coordinate vs. radial coordinate, 220, 221, 222
Angular frequency, 25
Anisotropic crystals
light propagation, 87
optical beam profiler, 75
light propagation in anisotropic crystals, 57–59
Aperture(s). See also specific name e.g. Circular aperture
calculations for plane waves incident, 209–234
diffraction patterns, 326
on-axis irradiance vs. half angular, 246
various distances, 220, 221, 222
Aperture plane, 209
calculated Gaussian behavior, 279
incident Gaussian beam, 282
incident Gaussian field, 277
larger than aperture radius, 313
vs. normalized electric field, 210
vs. Poynting vector, 211, 213, 214, 215
Poynting vector vs. axial distance, 315
unperturbed beam half-width, 293
vs. wavelength ratio, 211
Aperture radius, 324
distance, 218
Aplanatic, 236
Aspheric lens, 301
Atomic dipole traps, 334
Atomic polarizability tensor, 331
Atomic resonance, 310
Atomic transition, 309
Atom traps
beyond circular aperture computational results, 334–341
bringing together and apart for two-qubit operations, 342–343
neutral, 306
Axial distance from aperture plane, 315
Axial intensity distribution, 185
Axial position
vs. electric field, 263
vs. on-axis irradiance, 247
Axial translation, 151
TEM beams, 170
Azimuthal polarization, 169–170
B
Barium sodium niobate, 67
Beam(s). See also specific name
azimuthal angle, 170
high-resolution intensity profile, 222
multi-mode, 173
parameter product, 146
polarization directions, 170
quality factor, 171
waist, 172
walk-off angle, 103
width of clipped
focused-Gaussian beams, 303
Beam distributions
aperture plane for circular aperture, 209–214
aperture plane for elliptical aperture, 221–225
aperture plane for square aperture, 227–230
beyond aperture plane for circular aperture, 215–216
beyond aperture plane for elliptical aperture, 226
beyond aperture plane for square aperture, 231–233
Beam half-widths
clipped focused-Gaussian beams, 285
intensity, 301
Beam intensity
clipping ratio, 290
effects of clipping, 287
focal plane, 286
Beam propagation model, 319–323
Beam propagation product (BPP), 146, 171–172, 174
characteristics of slow and fast waves, 67–72
light propagating, 81
Binomial expansion, 181
Birefringence, 73
Blackbody radiation
electromagnetic fields and origin of light, 15
intensity spectrum, 16
Blue-detuned atomic traps (BDT), 305, 311, 316, 318, 319, 325
diffraction, 344
properties, 330
Bohr’s classical model of atom, 14
Boltzmann’s constant, 334
Bose-Einstein condensate, 337
Brewster’s window, 166
Bright spot traps, 339
C
Carter and Williams’ integral form, 195
Cartesian axes, 9
unit vectors, 78
Cerenkov effect, 169
Cesium triborate, 67
Circular aperture
atom traps computational results, 334–341
beam distributions and aperture plane, 209–214
beam distributions beyond aperture plane, 215–216
calculated transmission function, 204
intensity distribution pattern, 320
Circularly polarized light right hand, 48
Clebsch-Gordan coefficients, 331
Clipped focused-Gaussian beams, 283, 297
beam half-width, 285
beam width, 303
maximum theoretical intensity, 298
Clipped Gaussian beams
calculations and measurements, 299–302
calculations using GHVDT, 282–288
maximum obtainable irradiance, 288
Clipping, 287
beam intensity profile, 290
calculated on-axis intensities, 300
effects for beam intensity distribution, 287
effects for off-focal plane axial location, 287
experimentally measured on-axis intensities, 301
maximum normalized intensity, 288
vs. peak intensities, 299
peak on-axis intensity, 289
Collimated beam, 56
input, 157
laser, 39
or focused TEM Gaussian beam, 158
Coma, 254
vector diffraction across curved interface, 266–269
Complex phase notation
commonly used, 352
engineer’s vs. physicist’s, 347–352
sinusoidal waves, 347
Composite units, 26
Compton’s experiments, 18
Condition, 132
Confocal parameter of Gaussian beam, 146
Coulomb Gauge, 6
Coulomb’s Law, 2
Coupling, 127
Critical angle, 125
Crystal(s)
anisotropic and light propagation, 87
anisotropic and optical beam profiler, 75
classes and light propagation, 59
Crystal coordinate systems, 108
Cutoff condition, 127
Cylindrical coordinates, 143
D
Dark spot traps, 339
Dashed potential energy minimum, 342
Detuning, 310
Diabatic trapping potential energy, 335, 338
Diamagnetic materials, 5
Dielectric axes, 100
Dielectric permittivity tensor, 58
Dielectric waveguide materials, 134
Diffracted beam shapes, 216
Diffracted-Gaussian regime, 278
Diffracted light just beyond circular aperture, 311–317
Diffracting aperture plane, 291–292
Diffraction
defined, 177
integrals, 291
light incident, 224
limited Rayleigh range, 174
theory model of mathematical approximations, 179
theory of volumes and surfaces, 180
Diffraction of Gaussian beams, 273–303
calculations and measurements, 299–302
calculations using GHVDT, 282–288
effect of diffraction on M2 parameter, 283–285
longitudinal field component, 289–290
maximum obtainable irradiance, 288
propagation using Luneberg’s vector diffraction theory, 291–293
radial and longitudinal intensities, 286–287
region between pure-diffraction and unperturbed behavior, 282
Diffraction patterns, 19
aperture, 326
MOT cloud, 321
radial and intensity profiles, 186
trap depths and trap dimensions, 327
Dimensionless parameters, 241
Dipole potential energy, 306
Dipole trap depths, 317
Discrete angles, 132
Discrete incident angles, 124
Displacement vector, 84
fast and slow waves, 82
Double-primes, 260
Double refraction, 73
E
Einstein’s theory of photons, 18
Electric dipole, 3
moment, 306
Electric displacement vector, 30, 57
Electric field
longitudinal component, 216–220
resultant vector, 44
vs axial position, 263
calculated on-axis values, 217
components and plane of incidence, 116
direction of components, 118
electromagnetic fields and origin of light, 2
global maxima, 253
integral expressions, 295
maximum calculated value, 249
normalized locations of global maxima, 255
oscillatory behavior, 46
plots of parts and modules squared, 250
Poynting vector vs. aperture plane, 211
rotational behavior, 48
vector, 38
waveguide model, 127
Electric potentials, 8
Electric susceptibility, 57
Electromagnetic boundary conditions, 119
Electromagnetic energy flow, 13
blackbody radiation, 15
boundary conditions, 117
Einstein’s theory of photons, 18
electric fields, 2
hertz vector potential, 8
magnetic fields, 3
particle-function of classical mechanics, 19
photoelectric effect, 17
Planck’s theory of light quanta, 16
Poynting vector, 13
quantum mechanical interlude, 15–28
radiation from classical atom, 14
radiation from orbiting charge, 9
Schrdinger equation, 22
units and dimensions, 26
vector and scalar potentials, 6–7
wavefunction of quantum mechanics, 20–21
wavefunctions of electrons in stable atom, 23
wave particle duality of matter, 19
Electromagnetic radiation, 13
linear polarization, 44
linear source-free media, 32–34
Maxwell’s equations in linear media, 29–31
polarization states of light, 42–49
Electron orbiting, 10
Electron oscillator motion, 58
Elliptical aperture, 221
beam distributions beyond aperture plane, 226
beam distributions in aperture plane, 221–225
Elliptically polarized light, 43
End coupling, 127
Engineering
complex notation, 352
vs. physicist’s complex phase notation, 347–352
Escape paths, 338
Evanescent waves
application, 139
light propagation in dielectric waveguide, 138
Experimental plane waves, 38–39
point source, 41
Extraordinary fields, 107
Extraordinary wave, 100
components, 105
light propagation in anisotropic crystals, 102–103
F
Faraday’s law, 4
Far-field divergence, 172
Far-field solid angular spread product, 173
Fast wave, 84
displacement vector and Poynting vector, 82
guided TE modes, 133
Field components, 66
double integral forms, 191
single integral forms, 192–196
Field integrals for spherical coordinate system, 257
Field units, 27
Finite difference time domain, 189
Flat-top intensity profile incident, 187
Focal plane, 266
calculated intensity distributions, 267
intensity distributions for lens surface, 266
values of on-axis intensity, 283
Focal spot, 265
Focused Gaussian beams, 283, 297
beam half-width, 285
beam width, 303
maximum theoretical intensity, 298
propagation and unperturbed beam, 296
Focused TEM Gaussian beam, 158
beam width, phase fronts, and far-field divergence angle, 147
beam width and radial intensity profile, 149
Focusing, 171
Gaussian beam by thin lens, 153–162
Fresnel approximation, 295
Fresnel diffraction region, 322
trapping cold atoms with laser light, 319–323
Fresnel-Kirchoff model, 182
diffraction integral, 181
Fresnel model, 184
and transmission coefficients, 116–120
wave propagation across interface, 116–120
Fresnel transmission
coefficients, 239
function, 257
polarized light, 260
G
Gaussian beam(s). See also specific name e.g. Clipped Gaussian beams
aperture plane, 282
collimated or focused TEM, 158
confocal parameter, 146
derivation of properties, 155
diverging, 154
diverging TEM, 154
focused, 158
focusing and thin lens, 153–162
through multiple optical elements, 150–152
Gaussian beam propagation, 141–176
ABCD matrix treatment, 148–162
paraxial propagation of Gaussian beams, 143–147
unperturbed beam, 296
using Luneberg’s vector diffraction theory, 291–293
Gaussian behavior
aperture plane, 279
normalized on-axis intensity, 284
Gaussian electric field, 274
Gaussian field aperture plane, 277
Gaussian hertz vector diffraction theory (GHVDT)
clipped Gaussian beams calculations, 282–288
diffraction of Gaussian beam, 273–275
diffraction of Gaussian beams, 276–281
plane-wave diffraction behavior, 281
Poynting vector, 280
Gaussian laser beam
incident, 154
incident field, 274
Gaussian light field, 293
Gaussian wave paraxial, 151
Geometrical optics, 252
Geometrical shadow region, 195
Global maxima, 252
Grating coupling, 127
Green’s theorem, 199
Guided waves
incident angles leading to transmitted angles, 126
H
Half angular aperture vs. normalized on-axis irradiance, 246
Half-width, 227
Handedness. See also specific type light field, 46
Helmholtz wave equation, 141
HVDT and KVDT, 216
Hermite Gaussian beams, 144, 172
paraxial propagation of Gaussian beams, 164
Hermite polynomials, 144
Hertz vector, 273
electromagnetic fields and origin of light, 8
Hertz vector diffraction theory (HVDT), 209
Helmholtz wave equation, 216
Maxwell’s equations, 216
scalar and vector diffraction theories, 190–196
Higher-order Gaussian beams, 163–168
High-resolution beam intensity profile, 222
Huygen’s vector principle, 240
Hyperfine splittings, 311
I
Incident angles leading to transmitted angles, 126
Incident beams
collimated, 159
converging, collimated or diverging, 153
waist, 160
Incident field
direction of components, 118
Gaussian laser beam, 274
Incident Gaussian beam, 282
Incident Gaussian field, 274
aperture plane, 277
converging or diverging, 293
Incident laser beam tilting, 337
Incident light beam, 113
Incident plane waves, 258
Incident wave, 131
Input beam collimated, 157
Integrand, 141
Intensity, 14
minimum beam half-widths, 301
polarized beams, 270
unperturbed beam, 302
values of normally incident plane wave, 265
Intensity distributions
focal plane, 267
pattern for metallic circular aperture, 320
Poynting vector, 223, 224, 225
purely Gaussian beam propagation, 279
Intensity patterns
diffraction patterns, 186
nanotrap, 328
rubidium trapped, 343
Intensity spectrum for blackbody, 16
Irradiance, 14
Isotropic crystals, 59
Isotropic medium, 3
K
Kinematics, 19
Kirchhoff vector diffraction theory (KVDT)
Helmholtz wave equation, 216
Maxwell’s equations, 216
scalar and vector diffraction theories, 197–200
Kirchoff approximations, 210
Kirchoff boundary conditions, 189
Poynting vector components, 212
Kirchoff vector, 313
Kronecker delta symbol, 58
L
Laboratory coordinate system, 107
light propagation in anisotropic crystals, 108
Laguerre Gaussian beams, 144
paraxial propagation of Gaussian beams, 165–168
Laguerre Gaussian laser modes, 166
Laguerre polynomials, 144
function, 168
Laser
beam diabatic trapping potential energy, 336, 340
beam tilting, 337
detuning, 309
dipole trap depths, 317
intensity incident, 317
Left hand circularly polarized light, 49
Left hand elliptically polarized, 47
Lens
and focal plane refractive index, 161
surface and focal plane intensity distributions, 266
Light
beam, 113
calculations of reactive intensity, 278
distributions high and low intensity, 314
fields and atom trap potentials, 315–317
handedness, 46
right hand elliptically polarized, 46
spherical surface for focusing, 245
trapping cold atoms with laser light, 315–317
wave incident illustration, 115
diffraction, 224
vector diffraction across curved interface, 260–261
across convex boundary into optically thicker medium, 237
across convex boundary into optically thinner medium, 237
anisotropic crystal, 87
biaxial crystals, 81
vector diffraction across curved interface, 237
Light propagation in anisotropic crystals, 53–111
allowed directions of and Ẽ in anisotropic medium, 61–62
characteristics of slow and fast waves in biaxial crystal, 67–72
crystal coordinate systems, 108
directions of D and E for slow and fast waves, 64–66
double refraction and optic axes, 73–84
expression relating ρ and θ, 104–105
expressions for components of angles θ, φ and Ω, 74–77
field directions of D and E vectors, 100–101
interim summary, 84
laboratory coordinate systems, 108
light propagation in anisotropic crystal, 60–66
non-plane waves, 56
ordinary wave, 106
plane waves, 55
principal coordinate axes, 58
principal refractive indices, 59
propagation along principal axes and planes, 85–99
propagation along principal axes X, Y, and Z, 85–86
propagation along principal plane XY, 94
propagation along principal plane YZ, 87
propagation equation in presence of walk-off, 107–112
ps and pf, 69
relating angle δ to Ω, θ and φ, 78–80
special cases, 106
summary of cases, 97
three crystal classes, 59
values of n for given propagation direction, 63
vectors associated with light propagation, 54–56
walk-off angles ps and pf, 83
Light propagation in dielectric waveguide, 123–140
conditions for guided waves, 123–126
evanescent waves, 138
field amplitudes for guided waves, 127–129
Light quanta Planck’s theory, 16
Linearly polarized, 44
example, 45
Linear media
Lithium triborate, 67
Longitudinal field component, 248
diffraction of Gaussian beams, 289–290
and radial extents, 248
unperturbed paraxial approximation, 289–290
Lorentz condition, 8
Lorentz Gauge, 7
Lorentzian form, 148
Lorentz model, 58
Low-numerical aperture lenses, 252
Luneberg’s method, 198
Luneberg’s variation, 294
Luneberg’s vector diffraction theory, 291–293
M
amplitude, 130
direction of components, 118
electromagnetic fields and origin of light, 3
global maxima, 254
normalized locations of global maxima, 256
oscillatory behavior, 46
rotational behavior, 48
waveguide model, 127
Magnetic polarization, 30
Magnetic substate, 339
Magneto-optical trap (MOT), 318
optics-free region, 323
projecting near-field diffraction patterns, 321
Maxwell’s equations, 1, 6, 31, 38, 54, 109, 198
HVDT and KVDT, 216
Metallic circular aperture, 320
Micro-electrical-mechanical systems, 178
calculating light fields, 238, 258, 261
electric field vs. axial position, 263
electric field vs. radial position, 263, 264
Minimum beam size, 142
Monochromatic plane waves, 61
solutions, 55
Movable atomic dipole traps, 334
Multi-mode beams, 173
N
Nanotrap intensity patterns, 328
Near-field diffraction patterns, 321
Near-field optics, 178
Negative uniaxial crystals, 60, 97, 100
orientation of fields, 102
Neumann boundary conditions, 274
Neutral atom traps, 306
Newton’s laws of motion, 14
Non-diffraction limited beams, 173
Nonmagnetic dielectric medium, 119
Non-plane waves, 56
Nonzero, 102
Normal incidence
calculations for vector
diffraction across curved interface, 262
frequencies, 337
Normalized electric field vs. aperture plane, 210
Normalized intensity clipping ratio, 288
Normalized longitudinal component, 251
Normalized on-axis intensity Gaussian behavior, 284
Normalized on-axis irradiance vs. half angular aperture, 246
Normally incident light, 254
vector diffraction across curved interface, 258–259
Normally incident plane wave, 265
O
Off-axis focusing, 254
vector diffraction across curved interface, 266–269
Off-focal plane axial location, 287
On-axis calculations, 215
On-axis distance vs. Poynting vector, 225–228
On-axis intensity
focal plane, 283
Gaussian behavior, 284
on-axis position, 287
On-axis irradiance vs. axial position, 247
On-axis light distributions, 254
On-axis position, 287
On-axis values for electric fields, 217
Optical beam profiler, 302
anisotropic crystals, 75
Optical dipole trapping potential energy
theory of polarization dependence, 331–333
Optically anisotropic, 53
Optically isotropic, 53
Optical properties, 113
Optical system and sequential matrix operations, 150
Optical traps
intensity calculations, 324
rubidium atoms, 262
trapping cold atoms with laser light, 306–310
Optic axes, 74
Snell’s laws, 185
Optics-free region, 323
Ordinary fields, 107
light propagation in anisotropic crystals, 106
Orthogonal, 38
Orthonormality, 333
Oscillating point charge, 49
Oscillator strength, 332
Oscillatory behavior for electric field, 46
P
Paramagnetic materials, 5
Paraxial Gaussian wave, 151
Paraxial propagation of Gaussian beams, 141–176
ABCD matrices, 149
ABCD matrix treatment of Gaussian beam propagation, 148–162
azimuthal and radial polarizations, 169–170
focusing Gaussian beam by thin lens, 153–162
Hermite-Gaussian beams, 164
higher-order Gaussian beams, 163–168
Laguerre-Gaussian (LG) beams, 165–168
M2 historical evolution, 172
paraxial wave equation, 142
propagation of Gaussian beam through multiple optical elements, 150–152
TEM00 Gaussian beam
propagation and
Paraxial-type of approximations, 236
Paraxial wave equation, 142
Particle-function
classical, 19
classical mechanics, 19
extracting desired information, 20
vs. wavefunction, 21
Particle-wave duality, 18
Peak intensities vs. clipping ratio, 299
Peak on-axis intensity, 289
Permeability, 273
Permittivity, 273
Photoelectric effect, 18
electromagnetic fields and origin of light, 17
sodium, 17
Photons, 18
Einstein’s theory, 18
Physicist’s complex notation, 352
Planck’s constant, 308
Planck’s theory of light quanta, 16
Plane of incidence, 114
electric field components, 116
Plane wave(s), 36, 38–39, 56, 61, 258
diffraction behavior, 281
electromagnetic waves in linear media, 36–41
incident apertures calculations, 209–234
light propagation in anisotropic crystals, 55
normally incident, 265
point source, 41
vs. Poynting vector, 314
radial beam profile, 268
radial intensity profiles, 187
solutions, 55
Plane waves incident calculations
beam distributions beyond aperture plane for circular aperture, 215–216
beam distributions beyond aperture plane for elliptical aperture, 226
beam distributions beyond aperture plane for square aperture, 231–233
beam distributions in aperture plane, circular aperture, 209–214
beam distributions in aperture plane, elliptical aperture, 221–225
beam distributions in aperture plane for square aperture, 227–230
diffracted beam shapes, 216
longitudinal component of electric field EZ, 216–220
on-axis calculations, 215
Point source and plane wave, 41
Polarization, 3
angle, 223
circular, 47
dependent atomic dipole traps, 320–343
paraxial propagation of Gaussian beams, 169–170
potential, 312
Polarization directions light, 42
TEM beams, 170
unit vector, 107
Polarized beams intensity, 270
Polarized light, 260
Polarizers, 169
Position vector, 36
Positive uniaxial crystals, 97
orientation of fields, 102
Potassium niobate, 67
Potential energy
diabatic trapping, 335, 336, 338, 340
operator, 331
Potential units, 27
Power transmission function, 203–204
vs. aperture plane, 211, 213, 214, 215
vs. axial distance from aperture plane, 315
calculated z-component, 213, 214, 215
calculated z-components, 219
component, 202
electromagnetic fields and origin of light, 13
fast and slow waves, 82
GHVDT, 280
intensity distribution, 223, 224, 225
Kirchoff boundary conditions, 212
oscillates, 203
vs. plane-wave, 314
slow and fast waves, 85
transverse components, 313
Primes, 260
light propagation in anisotropic crystals, 58
orientation, 68
propagation, 86
Principal dielectric permittivities, 58
Principal refractive indices, 53
define, 59
light propagation in anisotropic crystals, 59
Prism coupling, 127
Projected light distributions, 324
Projection of diffraction patterns MOT cloud, 321
near field, 321
trapping cold atoms with laser light, 318–330
Propagation, 15
across interface of two homogeneous media, 113–122
along principal axes and planes, 85–99
along principal axes X, Y, and Z, 85–86
along principal plane XY, 94
constant, 171
electromagnetic wave, 54
equation in presence of walk-off, 107–112
expressions for arbitrary directions, 97
factor, 171
multiple optical elements, 150–152
principal axes, 86
unit vector, 101
unperturbed beam focused Gaussian beam, 296
Pure diffraction behavior, 278
Purely Gaussian (PG)
beam propagation, 279
behavior, 278
intensity distributions, 279
Q
Quantum mechanics, 15
defined, 21
Quantum numbers, 24
R
Radial beam profiles, 184, 268
Radial coordinate vs. angular coordinate, 220, 221, 222
Radial extents, 248
Radial intensity distribution
diffraction patterns, 186
on-axis position, 287
plane waves, 187
Rayleigh-Sommerfeld model, 185
Radially polarized beam, 169
Radial position vs. electric field, 263, 264
Radiation
classical atom, 14
orbiting charge, 9
Radius of curvature for spherical boundary, 237
Raman scattering, 18
rate, 344
trap photon, 342
Rayleigh-Jeans theory, 16
diffraction-limited, 174
focused TEM Gaussian beam, 158, 159, 160
Rayleigh-Sommerfeld model, 182, 183, 184
calculation, 188
diffraction integral, 180
radial and axial intensity distribution, 185
Ray optics matrix methods, 149
Ray refraction, 152
Ray tracing, 252
Reactive intensity light field, 278
Red-detuned atomic traps (RDT), 305, 311, 316, 318, 319, 325
diffraction, 342
properties, 329
scattering rate, 335
Reflection
interface not normal to Cartesian axis, 121–122
Refraction
interface not normal to Cartesian axis, 121–122
focal plane, 162
lens and focal plane, 161
Resultant electric field vector, 44
Ridge waveguide, 123
Right hand
circularly polarized light, 48
elliptically polarized light, 48
elliptically polarized light field, 46
Rodrigue’s formula, 164
Rotational behavior, 49
electric field and magnetic field, 48
Rotational kinetic energy, 15
Rubidium
hyperfine ground states, 341
optically trap, 262
trapped intensity pattern, 343
trap properties, 341
S
Saturation field, 308
Scalar diffraction model calculations, 182–184
Scalar diffraction theory, 177–208
analytical on-axis expressions and calculations, 201–202
analytical on-axis expressions using HVDT, 201
analytical on-axis expressions using KVDT, 202
comparisons of scalar diffraction model calculations, 182–184
double integral forms for field components, 191
Fresnel-Kirchoff diffraction integral, 181
Hertz vector diffraction theory (HVDT), 190–196
Kirchhoff vector diffraction theory (KVDT), 197–200
power transmission function, 203–204
Rayleigh-Somerfeld diffraction integral, 180
scalar diffraction theories, 178–181
single integral forms for field components, 192–196
vector diffraction theories, 189
verification of Snell’s laws using diffraction, 185–188
Scalar intensity, 329
Scalar potentials, 1
electromagnetic fields and origin of light, 6–7
Schoch’s line integral form, 193
Schrdinger equation, 20
electromagnetic fields and origin of light, 22
Second order differential equation, 144
Separation of variables, 23
Sequential matrix operations, 150
Shanding scale, 287
Single central maxima, 228
Single integral forms for field components, 192–196
Sinusoidal waves, 347
Slab
dielectric waveguide, 124
Slow wave, 84
displacement vector and Poynting vector, 82
Snell’s Law, 187
angle of transmission, 257
refraction, 188
refraction and geometrical optics, 185
refraction and reflection, 115, 122
Spatial filter, 172
Spatial sinusoidal behavior, 46
Spatial variables, 33
Specular reflection, 169
Spherical aberration, 236, 254
vector diffraction across curved interface, 263–265
Spherical boundary and radius of curvature, 237
Spherical coordinate system, 50
field integrals, 257
Spherical harmonics, 23
Spherical surface
exit, 269
focusing for light, 245
source point, 239
Spherical waves
electromagnetic waves in linear media, 49–52
front, 52
Spot area angular spread product, 173
Square aperture
beam distributions beyond aperture plane, 231–233
beam distributions in aperture plane, 227–230
diffracting, 227
intensity distribution, 231–233
Stratton-Chu diffraction integral, 236
Stratton theory, 253
System matrix, 149
T
Tangential components, 119
Taylor series, 295
TEM00 beams, 149
azimuthal angle, 170
diverging, 154
field amplitudes, 133
focused, 158
light propagation in dielectric waveguide, 130–133
mode combinations, 171
polarization directions, 170
propagation and parameters, 143–147
Theory of Stratton, 253
Thin lens, 236
phase transformation, 322
Tightly focused beam, 159
Tilting of incident laser beam, 337
TM modes, 125
light propagation in dielectric waveguide, 134–137
Top hat laser beam, 39
Transmission function and circular aperture, 204
Transmitted wave, 127
Transverse component, 125
Transverse electric, 124
Transverse field, 248
Transverse magnetic, 124
Transverse propagation, 125
Trapping
atoms using light fields, 305
cold neutral atoms, 327
depths and dimensions, 327
photon Raman transitions, 342
photons, 335
properties, 341
rubidium hyperfine ground states, 341
Trapping cold atoms with laser light, 305–343
beam propagation model, Fresnel diffraction, 319–323
bringing traps together and apart, 342–343
computational results for beyond circular aperture, 334–341
diffracted light just beyond circular aperture, 311–317
light fields and atom trap potentials, 315–317
optical dipole trapping potential energy, 306–310
polarization-dependent atomic dipole traps, 320–343
projection calculations, 324–328
projection of diffraction patterns, 318–330
propagation model HVDT, 312–314
theory of polarization dependence, 331–333
trapping atoms using light fields, 305
Tunneling of trap photon Raman transitions, 342
U
Ultra-high vacuum, 318
Ultraviolet catastrophe, 16
light propagation in anisotropic crystals, 97–106
Unit vector
polarization directions, 107
possible directions, 107
propagation vector, 101
Unperturbed beam
focused Gaussian beam propagation, 296
half-width, 293
peak intensities, 302
V
Vector diffraction across curved interface, 235–272
calculation of electromagnetic fields and Poynting vectors, 245–251
light incident at angle, 260–261
light propagation across convex boundary into optically thicker medium, 237
light propagation across convex boundary into optically thinner medium, 237
normal incidence calculations, 262
normalization and simplification, 241–244
normally incident light, 258–259
off-axis focusing and coma, 266–269
theoretical setup, 237, 253–254
two beams, 269
vector diffraction theory at spherical surface, 238–240
Vector Diffraction theory, 177–208, 216
Vector electromagnetic wave, 189
Vector Huygen’s principle, 240
Vector polarization potential, 273
Vector potentials, 1
electromagnetic fields and origin of light, 6–7
Vectors associated with light propagation, 54–56
Vector units, 27
W
Walk-off angles, 56, 69, 70, 84, 98, 99, 101, 104
beam, 103
ps and pf, 83
Wavefunction
electrons in stable atom, 23
extracting desired information, 22
vs. particle-function, 21
Waveguide model, 127
Wavelength, 13
vs. aperture plane, 211
Wavenumber, 8
light, 324
Wave particle duality of matter, 19
Waveplates, 169
Wave propagation across interface of two homogeneous media, 113–122
Fresnel reflection and transmission coefficients, 116–120
reflection and refraction at interface, 121–122
reflection and refraction at planar interface, 113–115
Wave sum, 127
Wave theory, 17
Wave vector, 39
Well-collimated beam, 158