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APPENDIX. FREE AND FORCED OSCILLATIONS IN SLIGHTLY NONLINEAR SYSTEMS
1. LINEAR OSCILLATOR
Free vibrations of linear oscillators are solutions of the equation
y" + gy' + wo2y = 0, (AI)
which is satisfied by
y • y 1e-gt/ 2 cos {wt - 1> }, (A2)
a damped vibration which, if the damping is small, occurs very nearly at the natural frequency, w • Woo
Forced vibrations of a linear oscillator occur at the driving frequency; the differential equation describing a multiply-periodic forced oscillation,
y" + gy' + wo2y = I: Ei cos {Wit} , i
has the steady-state solution
where
y = I: Yi cos {Wit - ¢;}, i
is the amplitude of the ith Fourier component and
(A3)
(A4)
(AS)
(A6)
is the phase lag between driving force and the ith Fourier component of the displacement; it vanishes if there is negligible damping.
139
140 Appendix
2. PERTURBATION METHOD FOR TREATING SMALL NON LlNEARITIES
Nonlinearities encountered in optical problems are as a rule confined to second- and third-order terms with very small coefficients. The general equation for a free nonlinear oscillator with small second- and third-order terms is
(A7)
The condition
(AS)
allows one to employ a perturbation method of solution. The exact solution for y will not differ greatly from that found in Section 1. It will be called the zero-order solution. The first-order solution is obtained by solving the equation resulting from substitution of Eq. (A2) into Eq. (A7), which then has the form of an equation of forced oscillation.
3. FREE OSCILLATION WITH SMALL NONLINEARITIES OF SECOND AND THIRD ORDER
To illustrate the principle of the method simply, we first apply it to the undamped, free oscillator. The first-order equation for free vibrations
can be transformed, using the identities
2 cos2 11 = 1 + cos 211,
4 cos3 11 = cos 311 + 3 cos 11 ,
into
y" + wo2y = -la2Y12 - !3a3y13 cos {w1t}
-la2Y1 2 cos {2Wlt} - !a3Y13 cos {3w1t},
(AID)
(All)
(A12)
containing one constant and three harmonic driving terms. The first-order solution must therefore be
(A13)
By substitution and comparison of coefficients, it is found that
Appendix
Yo = -a2Y12/2w02,
W 12 = W 02 + (3a3JI2/4) ,
Y2 = -a2Y12/2(wo2 - 4W12) ,
Y3 = -a3Y13/4(w02 - 9W12) ,
141
(AI4)
(AI5)
(AI6)
(AI7)
so that the second-order term displaces the center of vibration and the thirdorder term increases the fundamental frequency by a small amount
(AI8)
dependent on the square of the amplitude of the fundamental. The displacement
Y • -(a2/2wo2)Y12 + Y1COS{W1t} + (a2/6wo2)JI2 cos{2w1t}
+ (a3/8wo2)JI3 cos {3w1t}, (A19)
contains second and third harmonics as well as the fundamental frequency component.
In the next order of approximation, this solution is substituted in the original differential equation (A9) to form a new differential equation, leading to a more refined solution, containing still higher harmonics.
The dc and second-harmonic components in Eq. (AI9) are both proportional to a2 ; this coefficient does not appear in the third-harmonic component. It is therefore permissible to treat the effect of each order of nonlinearity separately. This is further justified in practice by the experimental circumstance that when second-order nonlinearity can occur at all, the second harmonic can be made far more intense than the third harmonic. Moreover, index-matching techniques allow suppression of all but one of the several harmonics which may arise in optical media.
4. SECOND-ORDER NONLINEAR OSCILLATOR IN FORCED VIBRATION
We accordingly consider next the forced oscillator with only secondorder nonlinearity,
(A20)
The displacement, given in lowest order by
(A21)
142 Appendix
is used to approximate Y in the second-order term, a2y2. The resulting equation is satisfied by the solution
in which
Y = Yo + Y1 cos{wt + ¢} + Y2 cos {2(wt + ¢)},
Yo = -a2Y12/2wo2,
Yl = Eo/[(w2 - wo2)2- g2w2]1/2,
Y2 = a2Y12/2(4w2 - W 02) ,
¢ = tan-1 [-gw/(wo2 - w2 )].
When w ~ Wo and damping is negligible, these approximate to
Yl = Eo/w2 ,
Y2 = a2Y12/Sw2 = a2E02/Sw6.
(A22)
(A23)
(A24)
(A25)
(A26)
(A27)
(A2S)
(A29)
At resonance, the damping directly determines the fundamental amplitude and indirectly determines the amplitude of the second harmonic by determining the applicable value of Yl'
Higher orders of approximation lead to fourth, sixth, etc., harmonics, but no experimental need for this refinement has yet arisen in optics, as the intensities are extremely weak, and it is unlikely that special indexmatching techniques will be possible to enhance them, because of dispersion.
5. THIRD-ORDER NONLINEARITY IN FORCED VIBRATION
Third-order nonlinearity is next considered for an undamped oscillator. This is adequate except near resonance, as it corresponds in optics to the region of normal dispersion. We shall not require a phase difference ¢. The differential equation
(A30)
has the approximate solution
(A31)
Appendix 143
so that it is very nearly equivalent to the linear equation
and so is satisfied by
Y = Yl COS{wt} + Ys cos{3wt}, (A33)
in which, by comparing coefficients of cos {wt} and cos {3wt}, it is found that
(A34)
(A35)
When driving frequency w is far below the resonance frequency,
When w IS much higher than resonance,
(A36)
(A37)
(A38)
(A39)
Iteration of this solution, to obtain a more refined approximation, is not necessary in optics, for reasons already given.
6. SECOND-ORDER NONLINEAR OSCILLATOR WITH TWO IMPRESSED FREQUENCIES
This problem is more general. The differential equation is
with zero-order solution
Y = 2 £1 2 cos {WIt} + £2 cos {w2t} Wo - WI W02 - W22
= Yl cos{w1t} + Y2 COS{W2t} , (A41)
144
the square of which is
y2 = iY12[1 + cos {2WIt}] + iY22[1 + cos {2w2t}]
+YIh [COS{(WI + (2)t} + cos {(WI - ( 2 )t}].
Appendix
(A42)
In addition to the two fundamentals, two second harmonics and their combined dc effect, sum and difference frequencies appear in the solution. If the amplitudes and frequencies of the two forcing terms are such that they produce comparable fundamental amplitudes, the mixture terms are comparable in importance with the harmonics. One of them may produce the dominant effect if its frequency is near resonance. The dispersion of the medium assumes even greater importance when index-matching is considered.
Physically, one can regard the phenomenon as a consequence of nonlinear response to the beats between the unequal impressed frequencies. The generation of combination tones by the ear and the action of superheterodyne detection circuits are well-known examples of this phenomenon.
A single fundamental may be considered to beat with itself at sum (double) and difference (zero) frequencies. The second-harmonic and dc effects are therefore special cases of this problem.
This can be generalized to any number of impressed frequencies and to higher orders of nonlinearity, with an increasingly complex spectrum of oscillations in the response.
BIBLIOGRAPHY
Included here are sources which were found to be useful in the preparation of this book. It is not a comprehensive list of the thousands of research contributions which have been made.
General Literature on Lasers and Laser Phenomena
Tomiyasu, "The Laser Literature-An Annotated Guide," Plenum Publishing Corp., New York, 1968.
"Masers and Optical Pumping," a reprint volume published by The American Institute of Physics, New York, 1965, contains many of the original articles on lasers and nonlinear optics.
Optics and Electromagnetic Theory
Condon and Odishaw, "Handbook of Physics," McGraw-Hill Book Co., New York, 1958.
Stratton, "Electromagnetic Theory," McGraw-Hill Book Co., New York, 1941. Landau and Lifshitz, "Electrodynamics of Continuous Media," Addison-Wesley Publish
ing Co., Reading, Mass., 1960. Heitler, "The Quantum Theory of Radiation," Oxford University Press (Clarendon),
1954.
Popular Reviews of Nonlinear Optics
Terhune, "Nonlinear Optics," International Science and Technology (August 1964), p. 38. Giordmaine, "Nonlinear Optics," Sci. Am., 210 (4), 38 (April 1964).
Static Nonlinear Effects
Yariv, "Quantum Electronics," John Wiley & Sons, New York, 1967. Condon and Odishaw, "Handbook of Physics," McGraw-Hill Book Co., New York, 1958. Landau and Lifshitz, "Electrodynamics of Continuous Media," Addison-Wesley Publish-
ing Co., Reading, Mass., 1960. Jenkins and White, "Fundamentals of Physical Optics," McGraw-Hill Book Co., New
York, 1937. "American Institute of Physics Handbook," McGraw-Hill Book Co., New York, 1957.
General Nonlinear Optics Theory
Bloembergen, "Nonlinear Optics," Benjamin, Inc., New York, 1965. Franken and Ward, "Optical Harmonics and Nonlinear Phenomena," Rev. Mod. Phys.
35 (1), 23 (January 1963).
145
146 Bibliography
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Bloembergen and Pershan, "Light Waves at the Boundary of Nonlinear Media," Phys. Rev. 128 (2), 606 (October 15, 1962).
Pershan, "Nonlinear Optical Properties of Solids: Energy Considerations," Phys. Rev. 130 (3), 919 (May 1, 1963).
Ward, "Calculation of Nonlinear Optical Susceptibilities Using Diagrammatic Perturbation Theory," Rev. Mod. Phys. 37 (1), 1 (January 1965).
Yariv, "Quantum Electronics," John Wiley & Sons, New York, 1967. Kelley et al., "Physics of Quantum Electronics," McGraw-Hill Book Co., New York,
1965, 1966 (Proceedings of international conferences on quantum electronics). Giordmaine, "Nonlinear Optics," Physics Today 22 (1), 38 (January 1969).
Harmonic Generation
Franken and Ward, "Optical Harmonics and Nonlinear Phenomena," Rev. Mod. Phys. 35 (1), 23 (January 1963).
Armstrong, Bloembergen, Ducuing, and Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127 (6), 1918 (September 15, 1962).
Bloembergen and Pershan, "Light Waves at the Boundary of Nonlinear Media," Phys. Rev. 128 (2), 606 (October 15, 1962).
Terhune, Maker, and Savage, "Optical Harmonic Generation in Calcite," Phys. Rev. Letters 8 (10), 404 (May 15, 1962).
Optical Mixing
Armstrong, Bloembergen, Ducuing, and Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127 (6), 1918 (September 15, 1962).
Bass, Franken, Hill, Peters, and Weinreich, "Optical Mixing," Phys. Rev. Letters 8 (1), 18 (January 1, 1962).
Optical Rectification
Bass, Franken, and Ward, Phys. Rev. 138 (2A), 534 (1965). Bass, Franken, Ward, and Weinreich, "Optical Rectification," Phys. Rev. Letters 9 (11),
446 (December 11, 1962).
Raman Scattering
Eckhardt, Hellwarth, McClung, Schwarz, and Weiner, "Stimulated Raman Scattering from Organic Liquids," Phys. Rev. Letters 9 (11),455 (December 1, 1962).
Zeiger, Tannenwald, Kern, and Herendeen, "Two-Step Raman Scattering in Nitrobenzene," Phys. Rev. Letters 11 (9), 419 (November 1, 1963).
Jones and Stoicheff, "Inverse Raman Spectra: Induced Absorption at Optical Frequencies," Phys. Rev. Letters 13 (22), 657 (November 30, 1964).
Platonenko and Khokhlov, "On the Mechanism of Operation of a Raman Laser," Soviet Phys.-JETP 19 (2), 378 (August 1964).
Fain and Yashchin, "On the Theory of Stimulated Combination (Raman) Radiation," Soviet Phys. -JETP 19 (2), 474 (August 1964).
Bibliography 147
Bloembergen and Shen, "Coupling between Vibrations and Light Waves in Raman Laser Media," Phys. Rev. Letters 12 (18), 504 (May 4, 1964).
Bloembergen and Shen, "Multimode Effects in Stimulated Raman Emission," Phys. Rev. Letters 13 (24), 720 (December 14, 1964).
Brillouin Scattering
Chiao, Townes, and Stoicheff, "Stimulated Brillouin Scattering and Coherent Generation of Intense Hypersonic Waves," Phys. Rev. Letters 12 (21), 592 (May 25, 1964).
Brewer and Rieckhoff, "Stimulated Brillouin Scattering in Liquids," Phys. Rev. Letters 13 (II), 334 (September 14, 1964).
Compton and Allison, "X Rays in Theory and Experiment," D. Van Nostrand Co., Princeton, New Jersey, 1935, pp. 231-233.
Slater, "Interaction of Waves in Crystals," Rev. Mod. Phys. 30 (I), 197 (January 1958).
Plasma Generation
Meyerand and Haught, "Gas Breakdown at Optical Frequencies," Phys. Rev. Letters 11 (9), 401 (November I, 1963).
Minck, "Optical Frequency Electrical Discharges in Gases," J. Appl. Phys. 35, 252 (January 1964).
Meyerand and Haught, "Optical-Energy Absorption and High-Density Plasma Production," Phys. Rev. Letters, 13 (1), 7 (July 6, 1964).
Ramsden and Davies, "Radiation Scattered from the Plasma Produced by a Focused Ruby Laser Beam," Phys. Rev. Letters 13 (7), 227 (August 17, 1964).
Archbold, Harper, and Hughes, "Time-Resolved Spectroscopy of Laser-Generated Microplasmas," British J. Appl. Phys. 15, 1321 (1964).
Wright, "Theory of the Electrical Breakdown of Gases by Intense Pulses of Light," Proc. Phys. Soc. 84, 41 (1964).
Gold and Bebb, "Theory of Multiphoton Ionization," Phys. Rev. Letters 14 (3), 60 (January 18, 1964).
Kroll, Ron, and Rostoker, "Optical Mixing as a Plasma Density Probe," Phys. Rev. Letters 13 (3), 83 (July 20, 1964).
Linlor, "Some Properties of Plasma Produced by Laser Giant Pulse," Phys. Rev. Letters 12 (14), 383 (April 6, 1964).
Bebb, "Theory of Three-Photon Ionization of the Alkali Atoms," Phys. Rev. 123 (1), 23 (January 5, 1967).
Platzman and Buchsbaum, "Light-Off-Light Scattering in a Plasma," Phys. Rev. Letters 12, 573 (May 25, 1964).
Naiman, DeWolf, Goldblatt, and Schwartz, "Laser-Induced Prebreakdown and Breakdown Phenomena Observed in Cloud Chamber," Phys. Rev. 146 (1), 133 (June 3, 1966).
Two-Photon Spectroscopy
Kleinman, "Laser and Two-Photon Processes," Phys. Rev. 125 (1), 87 (January I, 1962). Braunstein, "Nonlinear Optical Effects," Phys. Rev. 125 (2), 475 (January 15, 1962). Kaiser and Garrett, "Two-Photon Excitation in CaF. ; Eu2+," Phys. Rev. Letters 7 (6),
229 (September 15, 1961).
148 Bibliography
Abella, "Optical Double-Photon Absorption in Cesium Vapor," Phys. Rev. Letters 9 (11), 453 (December 1, 1962).
Hopfield, Worlock, and Park, "Two-Quantum Absorption Spectrum of KJ," Phys. Rev. Letters 11 (9), 414 (November 1, 1963).
Photon-Electron Scattering
Milburn, "Electron Scattering by an Intense Polarized Photon Field," Phys. Rev. Letters 10 (3), 75 (February 1, 1963).
Fiocco and Thompson, "Thomson Scattering of Optical Radiation from an Electron Beam," Phys. Rev. Letters 10 (3), 89 (February 1, 1963).
Bartell, Roskos, and Thompson, "Reflection of Electrons by Standing Light Waves," Phys. Rev. 166 (5), 1505 (February 25, 1968).
Intensity-Dependent Refractive Index
Maker, Terhune, and Savage, "Intensity-Dependent Changes in the Refractive Index of Liquids," Rev. Letters 12 (18), 507 (May 4, 1964).
Self-Focusing of Laser Beams
Chiao, Garmire, and Townes, "Self-Trapping of Optical Beams," Phys. Rev. Letters 13 (15), 479 (October 12, 1964).
INDEX
Absorption, 7, 18, 19, 22, 30, 32-35, 37, 39, 40, 42-45, 54, 64, 65, 75, 77, 83, 99, 102, 103, 115, 119, 122, 132, 135, 136
multiphoton,137 stimulated, 121, 122 two-photon,7,137
Acoustical branch, 110-114 Acoustics, nonlinearity in 6, 100 Active medium, 109 Ampere, definition of, 12, 13
law, 13 Amplification, 5,28,87,88, 100-104, 109,
130 parametric, see Parametric
Amplitude. 16, 21, 22, 25-27, 30, 35, 36, 44, 62, 74, 76, 78, 79, 85-87, 95, 98, 101-103, 118, 129, 130, 139, 141, 142, 144
complex representation, 16, 101, 102 equations for growth, SHG, 78,87 equations for growth, Brillouin
scattering, 129, 130 equations for growth, optical mixing,
102,103 transition, 35, 36
Anti-Stokes, 67, 68,111,115,116,119-122 Argon, 135-137 Attenuation,22,23,25,33, 125, 127 Axis, 51, see Optic axis
of index ellipsoid, 54 of crystal, 58, 59, 100
Barium titanate, 82 Basov, 8 Benzene, 119, 123 Biaxial crystal, 51, 53, 58, 63 Birefringence, 48-54, 58, 64, 77, 81, 92,
104 contrasts with optical activity, 61, 63,71 in optical mixing, 100, 101 inSHG,88-92
Boundary conditions, 27, 28, 30, 34 for field variables, 23
in nonlinear medium, 92-95 Bragg scattering 47, 48
law of, 47,124,133 Breakdown, electrical, 7, 44
of gases by laser light, 7, 134-137 Bremsstrahlung, inverse, 137 Brewster angle, 26 Brillouin, 47
scattering, 7,47,110,111,124 stimulated, 72,110, 114, 124-132
Bromonaphthalene, 119
Calcite, 97 Charge, see also Electron
bound,18,33,48 density p, 13, 14 distribution, 17,37 free, 11, 18 motion of, 11-14, 17-21,31 polarization, see Polarization, charge
Clausius-Mosotti law, 41 Coherence, 8,9, 21, 92, 109
in harmonic generation, 72-79 in scattering, 44-46, 65-67, 111 length, 76-78, 90, 96, 98, 101
Combination frequency, 54, 55, 68, 81, 85, 99-101, 104, 117, 123, 144, see also Difference frequency, Sum frequency
Compton effect, 32, 48, 133 scattering, stimulated, 133
Conductivity 0', 18-23, 33 Conductor, 18-22, 26, 28 Constitutive coefficients, 18, 21, 31, 33,
48 relations, 18,31, 100
Continuity, equation of, 127, 128 of fields at boundaries, 23-25, 93
Continuous spectrum, 35, 122
149
150
Cotton-Mouton effect, 6, 64, see also Magneto-optical effect
Coulomb, definition, 13 unit, 11-13, 19
Coupled equations, 35, 85, 101, 102, 117, 129
Cross section q, 33, 136 Current density J, 11-14, 17, 27, 32
displacement D, 14, 19, 20 Cyanin, anomalous dispersion in, 40 Cyc1ohexane,119
Damping, in dispersion theory, 21, 42, 43 and line width, 30, 43 in reflection, 26 of vibrations 119, 128, 129, 131, 139,
142 DeBroglie, 4, 48 Debye-Sears effect, 45-48, 69, 114, 124 Density, 41, 44, 57, 58, 71,105,110,114,
124-130, see also Charge fluctuations, 46, 134 of modes, 28, 29, 37
Dielectric, 18, 19, 21, 23, 40, 44, 45, 56, 87; 98, 99, 112, 125
constant E, 20, 40, 49, 57, 84, 114, 125, 126, 129
susceptibility, see Susceptibility tensor, 49, 57, 59, 84
reciprocal 54, 59 Difference frequency, 7,68,99-103, 117,
125, 128, 144 Diffraction, 2, 5, 31, 106
of electrons by light, 133, see also Compton scattering
of light by sound, 46-48, 124, 125, see also Brillouin scattering, DebyeSears effect
of x-rays by crystals, 47, see also Bragg scattering
Dipole, electric, 12, 69, 126 magnetic, 11 moment, 19, 32, 34, 41, 56, 59, 83 oscillator, 41, 43, 44, 45
Dispersion, 6, 8, 77, 82, 84, 87, 88, 97, 100, 105, 109, 111, 112, 118, 120, 126, 142, 144
acoustical, 112, 126 anomalous, 40, 65, 77, 112 electrons, 6, 8, 21, 33, 41, 43, 44, 48,
59, 60, 64, 68, 79, 81, 83, 88, 117
Dispersion (continued)
formula, 41-43, 47, 50, 64 mechanical, 110, 112-114, 118 negative, 43 theory of, 3, 4,6,40-44,73 in rectification, 98
INDEX
Displacement, current, D, 14, 19, 20 electric, see Induction of charges, 19, 41, 48, 57, 60, 81
Dissipation, 21, 23 Doppler effect, 30, 46, 47, 124, 130, 133
frequency shift, 47 Double refraction, 6, 48, 58, 64, 72, see
also Birefringence Drude, 3
Einstein, 3 Electromagnetic theory of light, 3, 11-18
waves, 3, 15, 19-30 Electronics, nonlinearity in, 5, 23, 100 Electron, see also Charge
bound, 41, 54, 117, 132, 135 conduction, 33 diffraction of, by light waves, 133 dispersion, see Dispersion electron free, 11, 132, 133, 135-137 interaction with radiation, 34, 35, 132-
137 plasma, 132 spin systems, 110 unbound, 137
Electro-optical effect, 6, 8, 56, 58, 59, 64, 69, 89, 104, 116, see also Pockels, Kerr
quadratic, 58, 105 relation to rectification, 98
Electrostriction, 56-58, 105, 114, 124, 125, 134
coefficient y, 125, 126, 129, 130 pressure, 126-128
Emission, 7, 18, 32, 34, 35, 39, 42, 54, 64, 65, 83, 115
line, shape of, 30, 42 spontaneous, 37, 65, 109 stimulated, see Stimulated emission
Energy, 3, 23 conservation of, 35, 42, 47, 87, 88, 100-
103,111,124,133 in dielectric, 57, 126
INDEX
Energy (continued)
in electromagnetic field, 13, 16, 17,30, 126
operator, quantum-mechanical, 33 propagation or transport of, 5, 17, 30,
32,49,78,87,88 Exponential decay, 29, 30, 103
growth in stimulated Brillouin scattering, 130
growth in stimulated Raman scattering, 118
representation of waves, 16, 21, 61,74, 101
Extinction, 45 coefficient K, 22, 23, 26
Extraordinary wave, 53, 64, 88-95, 97, see also Polarization, wave
Fabry-Perot, see Resonator Faraday effect, 6, 63, 64, 72, see also
Magneto-optical effect inverse, 7, 99 law of induction, 13
Ferroelecric crystal, 77, 104 Ferromagnetic substance, 59 Field emission, 132 Field equations, 13,20,22,49 Field intensity, definitions, 11
electric E, 8, 11-29, 31, 32, 41, 44, 48-51,56-63,68,69,71-74,78-81, 83-86,88-92,95,98-100,105,117, 125, 126, 128, 129, 135
generation by light, see Rectification magnetic H, 11-26, 31, 41, 49, 54-56,
60, 63, 64, 68, 71, 72, 79 generation by light, 12, see also Fara
day effect, inverse Fluorescence, 64-69 Flux, energy, 17, 30, 32, 78
photon, 32, 37, 67, 68, 103, 115, 137 Fourier, 27, 30, 37, 68, 73, 125, 139 Fresnel, 2, 3
equations, 25, 26, 95 Fundamental wave or frequency, 73, 75,
76, 78, 79, 81, 85, 87-97, 99, 101, 117-119, 123, 125, 129, 141, 144
Gases, breakdown of, 7, 134-137 dispersion in, 41 scattering of light by, 44
Gases (continued)
susceptibility of, 134 Gauss, law, 13
error function, 106 Geometrical optics, 4, 30-31, 39 Gyration vector yk, 61, 63, 64
151
Hamiltonian function flC', 33, 83, 84 Harmonic generation in electronics, 6
optical, see Third harmonic, Second harmonic
oscillator, 34, 41, 48,68,73, 139 Heisenberg, 4 Helium, 135-137
helium-neon laser, see Laser Hertz, 3 Huyghens,2
Idler wave, 104, 118, 124, 125, 128-131, see also Parametric amplifier
Index, absorption, 40 ellipsoid, 54, 59, 92 matching, 76-79, 86, 88-92, 95-98, 100,
105, 116, 118-120, 131, 141, 142, 144
as momentum conservation, 96, 97 use of birefringence for, 77, 88-92,
95-97, 100, 101 use of optical activity for, 77 use of anomalous dispersion for, 77
of mode, 28, 29, 37 of refraction, see Refraction, index of indices, repeated, summation over, 49,
57 Induction, electric D, 12, 19, 20, 23, 31,
49-54, 57, 61-63, 88 electromagnetic, 16 Faraday, law of, 13 magnetic B, 11, 12, 23, 49
Intensity, attenuation of, 33 at focus of lens, 31 effects on optical properties, 7, 69, 71,
72,105 Interference, 2, 31, 44, 45, 77,134 Intermodulation, see Modulation Inverse bremsstrahlung, 137 Inversion, in electronics, 6
symmetry, 58, 80, 84,132,134 Ionization, by accelerated electrons, 136,
137
152
Ionization (continued)
by laser light, 5, 135 by multiphoton absorption, 137 potential, 135
KDP (potassium dihydrogen phosphate), 82, 88-92, 94-96, 104
Kerr, effect, 6, 58, 59, 64, 105, 107, see also Electro-optical effect
cell, 115, 116 Kronecker delta, 49
Larmor precession frequency, 6, 54-56 Laser, 1, 8, 9, 23, 51, 65, 66, 77, 88,
92,98,99, 106, 110, 115, 118-126, 131-137
gas, 72,134 He-Ne, 1,32 mechanism of excitation, ruby, 65 neodymium, 1,7, 104, 119, 123, 124 Q-switching of, 7. 115, 116 Raman, 115, 117, 118, 122-124 ruby, 1, 65, 90, 91, 95, 97, 116, 123,
124, 130, 13 4 Lithium niobate, 7, 104 Lorentz, 3
theory of Zeeman effect, 54, 55 Lorentzian distribution, 30
Magneto-optical effect, 6, 8, 56, 63, 64, 69 see also Cotton-Mouton effect, Fa'raday effect, Voigt effect, Zeeman effect
Manley-Rowe relations, 88, 125, see also Parametric amplifier, Energy, conservation of
Maser, 8 Matrix element, 36, 37, 42, 43,84,137 Maxwell, 3,13
field equations, 13,20,49,51 Methane, 134 Michelson-Morley experiment, 3 Mixing, optical, 5, 6, 99-104, see also
Combination frequency constitutive relations for, 100 momentum relations, 100 as photon coalescence, 100, 104
Mode, 26-29, 31, 34, 35, 50, 52, 54, 61, 63,78,88,99,106,118, 121, l34
density, 28, 29, 37
Modulation, inter-, 6, 99 frequency-, 68, 118 of light, 99
by acoustic vibrations, 125, 129
INDEX
by molecular vibrations, 109, 118 Modulus of elasticity p, 57,125,126,130 Momentum, 3
conservation of, 100,101,111,119,124, 136
of electromagnetic waves, 18 of photon lik, 32, 47, 96, 97, 100, 120,
121, 124 operator, 33
Naphthalene, 119 Newton, 2,128 Nitrobenzene, 64, 107, 116, 119 Nonlinearity, in acoustics, 6
and crystal structure, 79, 80 of ear, 6 in electronics, 5, 144 extrinsic, 7, 68 intrinsic, 7, 69 order of, 7 in optics, 6-9, 44, 69, 70 second order, 7, 74, 79, 98,140-142 third order, 97, 140-142
Optical actvity, 60-63, 77,104 contrast with birefringence, 61, 63, 71
Optical branches, 110, 112-114, 118, see also Acoustical branch, Dispersion, mechanical
Optic axis, 5-54, 58, 63, 72, 77, 88, 90, 92,94, 97
Ordinary wave, 53, 88-92, 94, 95, 97, see also Polarization, wave
Oscillator, electronic, 41-43 forced, 41,117,139,141,142 harmonic, 34, 41, 48, 68, 73,139 molecular, 117-119, 121 strength, 42, 43
Parametric amplifier, 88, 118, 124, 125 amplification of light, 99,115
Passive medium, 72 Perturbation, 9, 37, 72
method for nonlinearity, 72, 73, 129 theory, quantum-mechanical, 34, 42, 83,
84,137
INDEX
Phase, 8,16,21,24,44,48,49,60,65,67, 74-77, 86, 88, 91, 94, 95, 98, 101, 109, 112, 117, 118, 119, 120, 142
change on reflection, 25, 26 coherence and, 8, 65, 67, 112, 117 velocity, 15, 16, 21, 50, 62, 73, 74, 88,
104, 112, 113 Phonon, 47, 65, 110, 111, 119-121, 124,
132 Phosphorescence, 65 Photoelasticity, 48 Photoelectric effect, 3, 13 2
ionization, multiphoton, 137 Photon, 31-37,42,47,69,96,97, 100, 104,
109-111, 119-122, 124, 132, 133, 136, 137
flux, see Flux, photon Physical optics, 30, 31 Piezoelectric effect, 56-58, 69, 80, 83, 88,
89 relation to Pockels effect, 59 tensor '}'ijk, 57-59
Planck,3 constant fl, 32, 47, 48, 96,124
Plasma, 19 generation with laser pulses, 132, 134,
135 oscillations, 110, 132
Pockels effect, 6, 8, 58, 59, 84, 92, 104, see also Electro-optical effect
relation to optical rectification, 98, 99 use for modulation, 99
Polarizability a, 20, 59, 114, 117 Polarization, charge P, 19-21, 23, 32, 41,
44, 56, 57, 68, 69, 72-74, 79, 80, 83, 85, 92, 95, 110, 114, 117, 125, 126
dc, 98, 105 nonlinear, 73-75, 81-83, 85, 88, 89, 93,
100 Polarization, wave, 7, 15, 16, 25, 26, 37,
45, 48, 50-53, 60,71,79, 81, 86, 88,90-92,94,95,97-99, 101, 105
circular, 61-63, 99, 104 elliptical, 63 in rectification, 99 in SHG, 85, 86, 88,90,91,94,95 in Zeeman effect, 55
Potential energy, 48 scalar <p, 14
Potential energy (continued)
vector A, 14 18,32,34,35,37,43 well, 44, 48
153
Poynting vector S, 17, 26, 30, 49, 51, 78, 94
Probability, 135 interpretation of wave mechanics, 4, 34 transition, 4, 34-37, 42,137
Prokhorov, 8 Propagation, 2, 3, 8, 17, 30, 39, 48-51,
54,71,79,88,93 Propagation constant k, 21-26, 28, 29, 33,
47,49,50,52,61,62,74-76,86,87, 92, 111, 113, 118, 120, 129-131
of sound wave k8, 128, 129-131 Propagation equation, 49
coupled, 85, 101, 102 differential, 14, 15,20,27,62,72
Propagation vector k, 15, 19, 21-24, 26, 30, 32, 46-54, 61, 63, 64, 85, 88, 93, 9~ 9~ 9~ 101, 12~ 121
Pumping, 23,118,124,125,131 of laser, 66
Pump wave of parametric amplifier, 103, 104,128
Q-Switch, see Laser, Q-switching of Quantum, see also Photon, Phonon
mechanics, 4, 9, 33-37 number, 34,37,56 theory, 3, 6, 8, 31-37,115
Quartz, birefringence of, 63, 71 optical activity of, 63,71 SHG in, 90 susceptibility tensor, 82 refractive index, 91
Radioactivity, 3 Raman activity, 69,117,122
effect, 6, 64-69, 114, 115 inverse, 7, 121, 122
frequency shift, 69, 117 -119, 123 laser, 115,117,118,123,124 lines, 115, 116 scattering, 7, 66, 67,110,111,114-117
stimulated, 115-124 spectroscopy, 68, 115, 116, 122 susceptibility, 69, 115, 117
Ray, 30, 31 Rayleigh scattering, 44-46, 117, 134
154
Rectification, in electronics, 6, 144 of light, 7, 74,80,98,99,104
Reflection, 23-26, 30, 31, 92, 93, 96 coefficient, 26, 31 Fresnel's formulas for, 25, 26 induced,7 total internal, 26, 106
Refraction, 6,23-26,30, 31, 93 double, see Double refraction, see also
Birefringence index of, P, 21-23, 26, 40, 42, 44, 50,
52-54, 59, 62, 64, 71, 72, 76, 87, 90, 92-96, 100-103, 105, 106, 125, 129-131, see also Dispersion
Snell's law of, see Snell's law of refraction
Resonance denominator, 137 line, 64 in oscillation 117, 142-144 radiation, 65
Resonator, optical, 28, 29, 118, 120, 131 Roemer, 2
Scattering, Brillouin, see Brillouin scattering
coherent,42,44,45, 73,109,110 of photons, 32 Raman, see Raman scattering Rayleigh, see Rayleigh scattering Stimulated, 110, 124, see also Brillouin
scattering, Raman scattering Thomson, 133
Schroedinger, 4 equation, 33, 35, 56, 84
Second harmonic, 7, 72-78, 80-85, 93, 96, 99, 100, 104, 105, 123, 132, 134, 141, 142, 144
generation (SHG), 72-97 combined with Raman conversion,
123-124 as photon coalescence, 83,97
Selection rules, 56, 66, 115 Self-focusing, 104-107, 116, 118, 121, 127,
131,134 SHG, see Second harmonic generation Signal wave, 103, 104, 118, see also
Parametric amplifier Skin effect, 28 Snell's law of refraction, 25, 31, 92, 94,
106, see also Refraction, index of
INDEX
Stimulated Compton scattering, see Compton scattering, stimulated
Sound, dispersion of 113 scattering of light by, 46-48, 115, 124,
125, see also Debye-Sears effect, Brillouin scattering
waves, 46, 125-129 pumped by light, 5, 111, 114, 124-
129, see also Brillouin scattering, stimulated
velocity of, 112, 130 Spectroscope, 2, 39 Spectroscopy, Raman, 68, 134 Spectrum, 29, 30, 35, 43, 110, 122, 144
Raman, 122 Zeeman, 54-56
Spontaneous emission, 37, 65,109 Spontaneous scattering, 110, 124 Stark effect, 39, 56 State, equation of, 126
excited, 7, 43, 44,66,67,115,121,137 ground, 7,37,44,65,66,83,115,121 intermediate, 36, 37, 42, 43, 65, 66, 83 stationary, 34-36, 42-44, 56, 65, 66 superposition, 42
Stereoisomer, 60, 63 Stimulated Brillouin scattering,
see Brillouin scattering, stimulated Stimulated emission, 8,23,37,43,66, 109,
110, 116, 121, 124 scattering, 110, 124, see also Scattering
Stimulated Raman effect, see Raman scattering, stimulated
Stokes, anti-, see Anti-Stokes law of fluorescence, 65, 68, 111 -shifted line, 66-68, 111, 116-119, 121,
122, see also Raman effect theorem of vector analysis, 13
Stress, 56-58, 71 tensOrajk, 57
Sum frequency, 7, 68, 85, 99-103, 128, 144, see also Combination frequency, Mixing, optical
Superposition, of waves, 5, 7, 27, 31, 73 of states, 42
Susceptibility X, 20, 21, 23, 33, 41-44, 48, 58, 60, 68, 69, 72, 73, 84, 105, 109, 114,124,125
atomic, 44, 68 complex, 41, 42
INDEX
Susceptibility (continued)
tensor, linear, 48-53, 58, 63, 81, 86-nonlinear, 79, 81-84, 87, 89, 96-100,
102, 118, 134 Raman, 69,115, 117 units, 20, 83
Symmetry, of crystal structure, 51, 58-60, 79,80,104, lOS, 109
of dielectric tensor, 59, 60, 79 of light beam, 23, 27,30, 106 of piezoelectric tensor, 58 inversIon, 58, 80, 84, 132, 134
Tensor, 48-51, 53, 57, 58, 60, 63, 73, 77, 81,84,88,89,92,98, see also Susceptibility
piezoelectric, 57-59, 83 reciprocal dielectric, 53, 54 stress, 57
Thermal motion, effects of, 30, 45, 59, 64, 105, 117, 130, 133
Thermionic emission, 132 Third harmonic, 7, 72, 77, 80, 85,97,134,
140-143 Thomson scattering, 133 Threshold, for breakdown of gas, 134-137
for optical amplification, 103 for stimulated Brillouin effect, 125, 131 for stimulated Raman effect, 116
Toluene, 119 Townes, 8 Transition, allowed, 56
between states, 32, 35, 37, 42;-{i5-69 forbidden, 68, 115 indirect, 36, 37 probability, 4,35-37,42, 137
Transmission coefficient of an interface, 26,31
Uncertainty, 42 Ultrasonic,46, Ill, 131
research, application of Brillouin scattering, 126
Uniaxial crystal, 51, 52, 54, 58, 63, 88, 93 Units, 9, 10, 12, 13,56,63,83 Unit cell, 48, 60, 71,81,83,84,134
Vector, identities, 14, 17 gyration, see Gyration vector potential, see Potential, vector
Vector (continued)
Poynting, see Poynting vector propagation, see Propagation vector wave, see Propagation vector
155
Velocity, of electromagnetic waves, 3, 15 group, 112 of light c/l', 2, 17,21,32,47, 131 phase, see Phase velocity of sound, v •• 46, 47, 112, 125, 126, 128,
130,131 of test charge, 11
Verdet constant, V. 63 Voigt effect, 64, see also Magneto-optical
effect
Wave, electromagnetic, 3, 14-17, 19-30,69 equation, 14, 72, 100, 115, 117, 125,
128,129 front, 30, 31,106 function, 34-36, 83, 84 modes, see Mode nature of light, 2 number, ll9, 131 polarization, see Polarization, wave plane, 15, 16,21,26,28,50 propagation, 14, lll, 118 sound,46 standing, 28, 29, 133 traveling, amplifier, 87, 103, 104, 131 traveling, SHG, 84-88, 104 vector, see Propagation vector
Wavelength A, 22, 23, 28, 30, 31, 44, 45, 47,60,65,71,76,90,95, 106, 113, 119, 130, 134
conversion, 104, 123, 136, see also Second harmonic generation, Mixing, Raman laser
X-rays, 3 Bragg scattering of, 47, 48, 124 from scattering of light, 133
Young, 2
Zeeman effect, 6, 39, 54-56 and Faraday effect 64,
see also Magneto-optical effect Zero-point energy, 34, 37