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

Armstrong, Bloembergen, Ducuing, and Pershan, "Interactions between Light Waves in a Nonlinear Dielectric," Phys. Rev. 127 (6), 1918 (September IS, 1962).

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