Introduction .................................................... 1
1 Optical plane waves in an unbounded medium ................. 4
1.1 Introduction to optical plane waves ........................ 4
1.1.1 Plane waves and Maxwell's equations ................. 4
(a) The y-polarized plane wave ..................... 5
(b) The x-polarized plane wave ..................... 6
1.1.2 Plane waves in an arbitrary direction ............... 7
1.1.3 Evanescent plane waves .............................. 9
1.1.4 Intensity and power ................................. 9
1.1.5 Superposition and plane wave modes ................. 10
(a) Plane waves with circular polarization ........ 10
(b) Interference of coherent plane waves .......... 10
(c) Representation by summation of plane waves .... 11
1.1.6 Representation of plane waves as optical rays ...... 13
1.2 Mirror reflection of plane waves .......................... 14
1.2.1 Plane waves polarized perpendicular to the plane
of incidence ....................................... 14
1.2.2 Plane waves polarized in the plane of incidence .... 15
1.2.3 Plane waves with arbitrary polarization ............ 15
1.2.4 The intensity ...................................... 15
1.2.5 Ray representation of reflection ................... 15
1.2.6 Reflection from a spherical mirror ................. 16
1.3 Refraction of plane waves ................................. 17
1.3.1 Plane waves polarized perpendicular to the plane
of incidence ....................................... 17
1.3.2 Plane waves polarized in the plane of incidence .... 19
1.3.3 Properties of refracted and transmitted waves ...... 20
(a) Transmission and reflection at different
incident angles ............................... 20
(b) Total internal reflection ..................... 21
(c) Refraction and reflection of arbitrary
polarized waves ............................... 21
(d) Ray representation of refraction .............. 21
1.3.4 Refraction and dispersion in prisms ................ 22
(a) Plane wave analysis of prisms ................. 22
(b) Ray analysis of prisms ........................ 24
(c) Thin prism represented as a transparent
layer with a varying index .................... 24
1.3.5 Refraction in a lens .............................. 25
(a) Ray analysis of a thin lens ................... 25
(b) Thin lens represented as a transparency with
varying index ................................. 27
1.4 Geometrical relations in image formation .................. 28
1.5 Reflection and transmission at a grating .................. 30
1.6 Pulse propagation of plane waves .......................... 31
Chapter summary ........................................... 32
2 Superposition of plane waves and applications ............. 34
2.1 Reflection and anti-reflection coatings ................... 34
2.2 Fabry-Perot resonance ..................................... 37
2.2.1 Multiple reflections and Fabry-Perot resonance ..... 37
2.2.2 Properties of Fabry-Perot resonance ................ 39
2.2.3 Applications of the Fabry-Perot resonance .......... 41
(a) The Fabry-Perot scanning interferometer ....... 41
(b) Measurement of refractive properties of
materials ..................................... 42
(c) Resonators for filtering and time delay of
signals ....................................... 43
2.3 Reconstruction of propagating waves ....................... 43
2.4 Planar waveguide modes viewed as internal reflected
plane waves ............................................... 46
2.4.1 Plane waves incident from the cladding ............. 46
2.4.2 Plane waves incident from the substrate ........... 48
(a) Incident plane waves with sin-1(nc/ns) < θS <
π/2 ........................................... 48
(b) Incident plane waves with 0 < θS <
sin-1(nc/ns) .................................. 48
2.4.3 Plane waves incident within the waveguide:
the planar waveguide modes ......................... 48
2.4.4 The hollow dielectric waveguide mode ............... 50
Chapter summary ........................................... 51
3 Scalar wave equation and diffraction of optical
radiation ................................................. 53
3.1 The scalar wave equation .................................. 54
3.2 The solution of the scalar wave equation: Kirchhoff's
diffraction integral ...................................... 55
3.2.1 Kirchhoff's integral and the unit impulse
response ........................................... 57
3.2.2 Fresnel and Fraunhofer diffractions ................ 57
3.2.3 Applications of diffraction integrals .............. 58
(a) Far field diffraction pattern of an aperture .. 58
(b) Far field radiation intensity pattern of
a lens ........................................ 60
(c) Fraunhofer diffraction in the focal plane
of a lens ..................................... 62
(d) The lens viewed as a transformation element ... 65
3.2.4 Convolution theory and other mathematical
techniques ......................................... 65
(a) The convolution relation ...................... 66
(b) Double slit diffraction ....................... 66
(c) Diffraction by an opaque disk ................. 67
(d) The Fresnel lens .............................. 67
(e) Spatial filtering ............................. 67
Chapter summary ........................................... 71
4 Optical resonators and Gaussian beams ..................... 73
4.1 Integral equations for laser cavities ..................... 74
4.2 Modes in confocal cavities ................................. 75
4.2.1 The simplified integral equation for confocal
cavities ........................................... 75
4.2.2 Analytical solutions of the modes in confocal
cavities ........................................... 77
4.2.3 Properties of resonant modes in confocal cavities .. 78
(a) The transverse field pattern .................. 78
(b) The resonance frequency ....................... 79
(c) A simplified analytical expression of the
field ......................................... 80
(d) The spot size ................................. 81
(e) The diffraction loss .......................... 81
(f) The line width of resonances .................. 82
4.2.4 Radiation fields inside and outside the cavity ..... 83
(a) The far field pattern of the ТЕМ modes ........ 84
(b) A general expression for the TEMlm Gaussian
modes ......................................... 84
(c) An example to illustrate confocal cavity
modes ......................................... 85
4.3 Modes of non-confocal cavities ............................ 86
4.3.1 Formation of a new cavity for known modes of
confocal resonator ................................. 86
4.3.2 Finding the virtual equivalent confocal resonator
for a given set of reflectors ...................... 88
4.3.3 A formal procedure to find the resonant modes in
non-confocal cavities .............................. 89
4.3.4 An example of resonant modes in a non-confocal
cavity ............................................. 91
4.4 The propagation and transformation of Gaussian beams
(the ABCD matrix) ......................................... 91
4.4.1 A Gaussian mode as a solution of Maxwell's
equation ........................................... 92
4.4.2 The physical meaning of the terms in the Gaussian
beam expression .................................... 94
4.4.3 The analysis of Gaussian beam propagation by
matrix transformation .............................. 95
4.4.4 Gaussian beam passing through a lens ............... 97
4.4.5 Gaussian beam passing through a spatial filter ..... 98
4.4.6 Gaussian beam passing through a prism ............. 100
4.4.7 Diffraction of a Gaussian beam by a grating ....... 102
4.4.8 Focusing a Gaussian beam .......................... 103
4.4.9 An example of Gaussian mode matching .............. 104
4.4.10 Modes in complex cavities ......................... 105
4.4.11 An example of the resonance mode in a ring
cavity ............................................ 106
Chapter summary .......................................... 107
5 Optical waveguides and fibers ............................ 109
5.1 Introduction to optical waveguides and fibers ............ 109
5.2 Electromagnetic analysis of modes in planar optical
waveguides ............................................... 112
5.2.1 The asymmetric planar waveguide ................... 112
5.2.2 Equations for ТЕ and TM modes ..................... 112
5.3 ТЕ modes of planar waveguides ............................ 113
5.3.1 ТЕ planar guided-wave modes ....................... 114
5.3.2 ТЕ planar guided-wave modes in a symmetrical
waveguide ......................................... 115
5.3.3 The cut-off condition of ТЕ planar guided-wave
modes ............................................. 117
5.3.4 An example of ТЕ planar guided-wave modes ......... 118
5.3.5 ТЕ planar substrate modes ......................... 119
5.3.6 ТЕ planar air modes ............................... 119
5.4 TM modes of planar waveguides ............................ 121
5.4.1 TM planar guided-wave modes ....................... 121
5.4.2 TM planar guided-wave modes in a symmetrical
waveguide ......................................... 122
5.4.3 The cut-off condition of TM planar guided-wave
modes ............................................. 123
5.4.4 An example of TM planar guided-wave modes ......... 123
5.4.5 TM planar substrate modes ......................... 124
5.4.6 TM planar air modes ............................... 125
5.4.7 Two practical considerations for TM modes ......... 126
5.5 Guided waves in planar waveguides ........................ 126
5.5.1 The orthogonality of modes ........................ 126
5.5.2 Guided waves propagating in the y-z plane ......... 127
5.5.3 Convergent and divergent guided waves ............. 127
5.5.4 Refraction of a planar guided wave ................ 128
5.5.5 Focusing and collimation of planar guided waves ... 129
(a) The Luneberg lens ............................ 129
(b) The geodesic lens ............................ 129
(c) The Fresnel diffraction lens ................. 130
5.5.6 Grating diffraction of planar guided waves ........ 131
5.5.7 Excitation of planar guided-wave modes ............ 134
5.5.8 Multi-layer planar waveguides ..................... 135
5.6 Channel waveguides ....................................... 135
5.6.1 The effective index analysis ...................... 136
5.6.2 An example of the effective index method .......... 140
5.6.3 Channel waveguide modes of complex structures ..... 141
5.7 Guided-wave modes in optical fibers ...................... 142
5.7.1 Guided-wave solutions of Maxwell's equations ...... 142
5.7.2 Properties of the modes in fibers ................. 144
5.7.3 Properties of optical fibers in applications ...... 145
5.7.4 The cladding modes ................................ 146
Chapter summary .......................................... 146
6 Guided-wave interactions ................................. 148
6.1 Review of properties of the modes in a waveguide ......... 149
6.2 Perturbation analysis .................................... 150
6.2.1 Derivation of perturbation analysis ............... 150
6.2.2 A simple application of perturbation analysis:
perturbation by a nearby dielectric ............... 152
6.3 Coupled mode analysis .................................... 153
6.3.1 Modes of two uncoupled parallel waveguides ........ 153
6.3.2 Modes of two coupled waveguides ................... 154
6.3.3 An example of coupled mode analysis: the grating
reflection filter ................................. 155
6.3.4 Another example of coupled mode analysis:
the directional coupler ........................... 160
6.4 Super mode analysis ...................................... 163
6.5 Super modes of two parallel waveguides ................... 163
6.5.1 Super modes of two well-separated waveguides ...... 164
6.5.2 Super modes of two coupled waveguides ............. 164
6.5.3 Super modes of two coupled identical waveguides ... 166
(a) Super modes obtained from the effective
index method ................................. 166
(b) Super modes obtained from coupled mode
analysis ..................................... 168
6.6 Directional coupling of two identical waveguides viewed
as super modes ........................................... 169
6.7 Super mode analysis of the adiabatic Y-branch and Mach-
Zehnder interferometer ................................... 170
6.7.1 The adiabatic horn ................................ 170
6.7.2 Super mode analysis of a symmetric Y-branch ....... 171
(a) A single-mode Y-branch ....................... 171
(b) A double-mode Y-branch ....................... 173
6.7.3 Super mode analysis of the Mach-Zehnder
interferometer .................................... 173
Chapter summary .......................................... 175
7 Passive waveguide devices ................................ 176
7.1 Waveguide and fiber tapers ............................... 176
7.2 Power dividers ........................................... 176
7.2.1 The Y-branch equal-power splitter ................. 177
7.2.2 The directional coupler ........................... 177
7.2.3 The multi-mode interference coupler ............... 178
7.2.4 The Star coupler .................................. 182
7.3 The phased array channel waveguide frequency
demultiplexer ............................................ 186
7.4 Wavelength filters and resonators ........................ 188
7.4.1 Grating filters ................................... 188
7.4.2 DBR resonators .................................... 189
7.4.3 The ring resonator wavelength filter .............. 189
(a) Variable-gap directional coupling ............ 190
(b) The resonance condition of the couple ring ... 191
(c) Power transfer ............................... 192
(d) The free spectral range and the Q-factor ..... 192
7.4.4 The ring resonator delay line ..................... 194
Chapter summary .......................................... 195
8 Active opto-electronic guided-wave components ............ 196
8.1 The effect of electro-optical χ .......................... 197
8.1.1 Electro-optic effects in plane waves .............. 197
8.1.2 Electro-optic effects in waveguides at low
frequencies ....................................... 198
(a) Effect of Δχ' ................................ 198
(b) Effect of Δχ" ................................ 199
8.2 The physical mechanisms to create Δχ ..................... 200
8.2.1 Δχ' ............................................... 200
(a) The LiNbO3 waveguide ......................... 202
(b) The polymer waveguide ........................ 203
(c) The III-V compound semiconductor waveguide ... 203
8.2.2 Δχ" in semiconductors ............................. 205
(a) Stimulated absorption and the bandgap ........ 205
(b) The quantum-confined Stark effect, QCSE ...... 206
8.3 Active opto-electronic devices ........................... 211
8.3.1 The phase modulator ............................... 211
8.3.2 The Mach-Zhender modulator ........................ 212
8.3.3 The directional coupler modulator/switch .......... 213
8.3.4 The electro-absorption modulator .................. 214
8.4 The traveling wave modulator ............................. 215
Chapter summary .......................................... 217
Appendix ...................................................... 219
Index ......................................................... 225
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