Introduction to Applied Solid State Physics pp 257-307 | Cite as

# Physics and Applications of the Nonlinear Optical Properties of Solids

## Abstract

The aim of this chapter is a discussion of some of the physics and applications of the nonlinear optical properties of solids. It begins with a review of electromagnetic wave propagation in solids and a derivation of the familiar linear relation between dielectric polarization and electric field. A more realistic anharmonic oscillator model is next introduced and this is shown to yield a polarization that is a nonlinear (i.e., a quadratic) function of the electric field. A physical picture of the nonlinear polarization and some solid state physics factors affecting the magnitude of the nonlinear susceptibility are also given. The central topic of the chapter is the propagation and interaction of three electromagnetic waves in a nonlinear medium. This results in a set of equations describing the spatial variation of the electric fields of these waves as they move through the crystal. Finally, these equations are used to discuss several applications of nonlinear solids. These are optical second harmonic generation, frequency mixing and up-conversion, and parametric amplification of optical signals.

## Keywords

Harmonic Generation Nonlinear Optical Property Nonlinear Crystal Nonlinear Susceptibility Crystal Length## Preview

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## References and Comments

- 1.J. M. Stone,
*Radiation and Optics*, McGraw-Hill, New York (1963), Chapter 15Google Scholar - E. M. Purcell,
*Electricity and Magnetism*, McGraw-Hill, New York (1965), Chapters 9 and 10.Google Scholar - 2.C. Kittel,
*Introduction to Solid State Physics*, Fifth Edition, John Wiley, New York (1976), page 410.Google Scholar - 3.C. Kittel, Reference 2, pages 400-408.Google Scholar
- 4.J. F. Nye,
*Physical Properties of Crystals*, Oxford University Press (1957), Chapter 4.Google Scholar - 5.A. Sommerfeld,
*Electrodynamics*, Academic, New York (1952), page 33.MATHGoogle Scholar - 6.Y. R. Shen, “Recent Advances in Nonlinear Optics,”
*Reviews of Modern Physics*,**48**, 1–32 (1976).ADSCrossRefGoogle Scholar - 7.J. Ducuing, in
*Quantum Optics*, R. J. Glauber (editor), Proceedings of the International School of Physics “Enrico Fermi,” Course XLII, Academic Press, New York (1969), page 448.Google Scholar - 8.N. Bloembergen,
*Nonlinear Optics*, W. A. Benjamin, New York (1965).Google Scholar - 9.A. Yariv,
*Quantum Electronics*, Second Edition, John Wiley, New York (1975), pages 413–418.Google Scholar - 10.A. Yariv, in
*Topics in Solid State and Quantum Electronics*, W. D. Hershberger (editor), John Wiley, New York (1972), pages 280–290.Google Scholar - 11.C. G. B. Garrett, “Nonlinear Optics, Anharmonic Oscillators, and Pyroelectricity,”
*IEEE Journal of Quantum Electronics*,**QE-4**, 70–84 (1968).ADSCrossRefGoogle Scholar - 12.See, for example, C. Kittel, W. D. Knight, M. A. Ruderman, A. C. Helmholz, and B. J. Moyer,
*Mechanics*(Berkeley Physics Course, Volume 1), Second Edition, McGraw-Hill, New York (1973), pages 224–226.Google Scholar - 13.J. J. Stoker,
*Nonlinear Vibrations*, Interscience Publishers, New York (1950), Chapter 4.MATHGoogle Scholar - 14.L. A. Pipes and L. R. Harvill,
*Applied Mathematics for Engineers and Physicists*, Third Edition, McGraw-Hill, New York (1970), Chapter 15.Google Scholar - 15.G. C. Baldwin,
*An Introduction to Nonlinear Optics*, Plenum Press, New York (1969), page 141.CrossRefGoogle Scholar - 16.F. W. Constant,
*Theoretical Physics*, Addison-Wesley, Reading, Massachusetts (1954), Section 6-6, pages 93–97.MATHGoogle Scholar - 17.J. B. Marion,
*Classical Dynamics of Particles and Systems*, Second Edition, Academic Press, New York (1970), Section 5.5, pages 165–167.Google Scholar - 18.See, for example, C. Kittel, W. D. Knight, M. A. Ruderman, A. C. Helmholz and B. J. Moyer, Reference 12, page 226.Google Scholar
- 19.G. C. Baldwin, Reference 15, pages 98-99.Google Scholar
- 20.F. Seitz,
*Modern Theory of Solids*, McGraw-Hill, New York (1940), page 637.MATHGoogle Scholar - 21.C. G. B. Garrett and F. N. H. Robinson, “Miller’s Phenomenological Rule for Computing Nonlinear Susceptibilities,”
*IEEE Journal of Quantum Electronics*,**QE-2**, 328–329 (1966).ADSCrossRefGoogle Scholar - 22.A. Yariv, Reference 9, Appendix 4, and N. Bloembergern, Reference 8, Chapter
*2*, give quantum mechanical discussions.Google Scholar - 23.A. Yariv, Reference 10, pages 283-285, gives figures exhibiting the polarization components at the fundamental and second harmonic frequencies.Google Scholar
- 24.J. F. Nye, Reference 4, Chapter 7, pages 110-115.Google Scholar
- 25.A. Yariv, Reference 10, page 287.Google Scholar
- 26.A. Yariv, Reference 9, pages 410-411, gives a table of the form of the nonlinear susceptibility tensor for various crystal classes.Google Scholar
- 27.J. Ducuing, Reference 7, Sections 3.2.1-3.2.3, pages 435-439.Google Scholar
- 28.A. Yariv, Reference 9, page 416, gives a table of values of
*d*^{(2ω)}_{ijk}for several crystals using the contracted*d*_{il}notation mentioned earlier. Yariv explains this notation on page 409. J. F. Nye, Reference 4, page 113, also discusses it, as do T. S. Moss, G. J. Burrell, and B. Ellis,*Semiconductor Opto-Electronics*, John Wiley, New York (1973), page 249.Google Scholar - 29.B. F. Levine, “Bond-Charge Calculation of Nonlinear Optical Susceptibilities for Various Crystal Structures,”
*Physical Review B*,**7**, 2600–2626 (1973), gives values of nonlinear susceptibilities (with references) for a large number of crystals.ADSCrossRefGoogle Scholar - See also the paper by B. F. Levine and C. G. Bethea,
*Applied Physics Letters*,**20**, 272–274 (1972).ADSCrossRefGoogle Scholar - The book by Yariv, Reference 9, page 416, gives a table of values of
*d*^{(2ω)}in units of 1/9×10^{−22}MKS. (To convert*d*^{(2ω)}in these units to*d*^{(2ω)}in esu, divide*d*^{(2ω)}(MKS) by 3.68 × 10^{−15}, as described by F. Zernike and J. E. Midwinter,*Applied Nonlinear Optics*, John Wiley, New York (1973), pages 52–53.)Google Scholar - Finally, mention should be made of the article by S. K. Kurtz, “Measurement of Nonlinear Optical Susceptibilities,” in
*Quantum Electronics: A Treatise*, H. Rabin and C. L. Tang (editors), Academic Press, New York (1975), Volume 1 (Nonlinear Optics, Part A), Section 3, pages 209–281. This article describes methods of experimental measurement and gives values of*d*^{(2ω)}for a number of crystals important in nonlinear applications.Google Scholar - 30.R. C. Miller, “Optical Second Harmonic Generation in Piezoelectric Crystals,”
*Applied Physics Letters*,**5**, 17–19 (1964).ADSCrossRefGoogle Scholar - 31.
- 32.B. F. Levine, Reference 29, page 2608, Table III.Google Scholar
- 33.B. O. Séraphin and H. E. Bennett, in
*Semiconductors and Semimetals*, R. K. Willardson and A. C. Beer (editors), Volume 3, Academic Press, New York (1967), Chapter 12, pages 526, 520, and 511. The value of*H*quoted for GaAs is the average of the two values given in this reference.Google Scholar - 34.B. F. Levine, Reference 29, page 2621, Table XVI.Google Scholar
- 35.R. A. Smith,
*Semiconductors*, Cambridge University Press (1959), page 386.Google Scholar - 36.C. Kittel, Reference 2, pages 409-412.Google Scholar
- 37.Y. R. Shen, Reference 6, Section II, pages 2-4.Google Scholar
- 38.See, for example, C. Kittel, W. D. Knight, M. A. Ruderman, A. C. Helmholz, and B. J. Moyer, Reference 12, pages 67-70. A useful table of conversion factors is given by G. Joos,
*Theoretical Physics*, Second Edition, Hafner Publishing Co., New York (1950), page 836.Google Scholar - 39.N. Bloembergen, “The Stimulated Raman Effect,”
*American Journal of Physics*,**35**, 989–1023 (1967), page 996.ADSCrossRefGoogle Scholar - 40.The author thanks B. Black for pointing out this approach.Google Scholar
- 41.See, for example, J. Ducuing, Reference 7, Sections 2.3 and 3.3 pages 433-435 and 439-448. This paper also gives references to the original work, including that of J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,”
*Physical Review*,**127**, 1918–1939 (1962). This paper is reproduced in the book by N. Bloembergen, Reference 8.ADSCrossRefGoogle Scholar - 42.A. Yariv, Reference 10, pages 290-292.Google Scholar
- 43.See N. Bloembergen, Reference 39, page 995, for a treatment of the interaction of four waves (with frequencies ω
_{1}, ω_{2}, ω_{3}, ω_{4}) subject to the condition that ω_{1}+ ω_{4}= ω_{2}+ ω_{3}.Google Scholar - 44.J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,”
*Physical Review*,**127**, 1918–1939 (1962), page 1928.ADSCrossRefGoogle Scholar - 45.J. Ducuing, Reference 7, page 434, equation (31).Google Scholar
- 46.See, for example, A. Yariv, Reference 9, page 421, for a treatment of the case of a solid with a nonzero value of the electrical conductivity.Google Scholar
- 47.P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of Optical Harmonics,”
*Physical Review Letters*,**7**, 118–119 (1961).ADSCrossRefGoogle Scholar - 48.See, for example, A. Yariv, Reference 9, Section 16.6, for a discussion of second harmonic generation for the case of a depleted input wave.Google Scholar
- 49.See, for example, G. R. Fowles,
*Introduction to Modern Optics*, Second Edition, Holt, Rinehart and Winston, New York (1975), page 25.Google Scholar - 50.See, for example, G. R. Fowles, Reference 49, Section 5.4.Google Scholar
- 51.See, for example, G. C. Baldwin, Reference 15, Section 4.3; J. Ducuing, Reference 7, page 441.Google Scholar
- 52.F. Zernike and J. E. Midwinter,
*Applied Nonlinear Optics*, John Wiley, New York (1973), Chapter 3.Google Scholar - 53.See A. Yariv, Reference 10, pages 298-303, for a discussion.Google Scholar
- 54.S. A. Akhmanov, A. I. Kovrygin, and A. P. Sukhorukov, “Optical Harmonic Generation and Optical Frequency Multipliers,” in
*Quantum Electronics: A Treatise*, H. Rabin and C. L. Tang (editors), Academic Press, New York (1975), Volume 1 (Nonlinear Optics, Part B), Section 8, pages 475–586.Google Scholar - 55.See, for example, A. Yariv, Reference 9, pages 424-428; A. Yariv, Reference 10, pages 294-298, G. C. Baldwin, Reference 15, pages 88-97.Google Scholar
- 56.See, for example, M. Born and E. Wolf,
*Principles of Optics*, Fourth Edition, Pergamon Press, New York (1970), page 671.Google Scholar - 57.See, for example, G. R. Fowles, Reference 49, Section 6.7.Google Scholar
- 58.See, for example, A. Yariv, Reference 9, Section 5.4.Google Scholar
- 59.See, for example, F. Zernike and J. E. Midwinter, Reference 52, Chapter 6; A. Yariv, Reference 9, pages 454-456; A. Yariv, Reference 10, pages 321-325.Google Scholar
- 60.J. Warner, “Difference Frequency Generation and Up-Conversion,” in
*Quantum Electronics: A Treatise*, H. Rabin and C. L. Tang (editors), Academic Press, New York (1975), Volume 1 (Nonlinear Optics, Part B), Section 10, pages 703–737.Google Scholar - 61.F. Zernike and J. E. Midwinter, Reference 52, Section 6.4; J. Warner, Reference 60, pages 707-712. These references also discuss the situation when Δ
*k*given by (9.154) is not zero.Google Scholar - 62.A. Yariv, Reference 10, page 325.Google Scholar
- 63.F. Zernike and J. E. Midwinter, Reference 52, page 99.Google Scholar
- 64.A. Yariv, Reference 9, Table 16.2, page 416.Google Scholar
- 65.J. Warner, Reference 60, page 705.Google Scholar
- 66.See, for example, A. Yariv, Reference 9, Section 17.1; A. Yariv, Reference 10, pages 304-313; F. Zernike and J. E. Midwinter, Reference 52, Sections 7.1–7.3.Google Scholar
- 67.J. A. Giordmaine in
*Quantum Optics*, R. J. Glauber (editor), Proceedings of the International School of Physics “Enrico Fermi,” Course XLII, Academic Press, New York (1969), pages 499–502.Google Scholar - 68.See especially A. Yariv, Reference 10, pages 311-313, and F. Zernike and J. E. Midwinter, Reference 52, Section 7.3, for discussions of phase mismatching in parametric amplification.Google Scholar
- 69.J. A. Giordmaine, Reference 67, page 502, Table II.Google Scholar
- 70.A. Yariv, Reference 9, Table 16.2, page 416.Google Scholar
- 71.A. Yariv, Reference 10, page 311.Google Scholar
- 72.See, for example, A. Yariv, Reference 10, Sections 7.9–7.11.Google Scholar
- 73.R. L. Byer, “Optical Parametric Oscillators,” in
*Quantum Electronics: A Treatise*, H. Rabin and C. L. Tang (editors), Academic Press, New York (1975), Volume 1 (Nonlinear Optics, Part B), Section 9, pages 587–702.Google Scholar - 74.J. A. Giordmaine, Reference 67, pages 502-509.Google Scholar

## Suggested Reading

- G. R. Fowles,
*Introduction to Modern Optics*, Second Edition, Holt, Rinehart and Winston, New York (1975). Chapter 6 of this senior-level optics text provides an introduction to the optics of solids; our treatment of polarization and wave propagation in a linear dielectric is similar to Fowles’ sections 6.2–6.4.Google Scholar - A. Yariv, in
*Topics in Solid State and Quantum Electronics*, W. D. Hershberger (editor), John Wiley, New York (1972), Chapter 7. This collection of articles contains a chapter by Yariv on optical second harmonic generation which discusses, among other things, nonlinear polarization in solids. Our discussion parallels the treatment of Yariv.Google Scholar - A. Yariv,
*Quantum Electronics*, Second Edition, John Wiley, New York (1975). This advanced textbook covers many topics in quantum electronics including, in Chapters 16 and 17, a detailed discussion of nonlinear optics, second harmonic generation, and parametric amplification more extensive than that in Yariv’s article quoted above.Google Scholar - N. Bloembergen,
*Non-Linear Optics*, W. A. Benjamin, New York (1965). This lecture note and reprint volume discusses its subject, including the quantum theory of nonlinear susceptibilities, at the advanced level. It includes reprints of several important original papers.Google Scholar - G. C. Baldwin,
*Non-Linear Optics*, Plenum Press, New York (1969). This short introductory book has brief but useful discussions of many of the topics we have covered.Google Scholar - J. J. Stoker,
*Nonlinear Vibrations*, Interscience Publishers, New York (1950). This short book is a good introduction to nonlinear oscillations and is clear and readable.MATHGoogle Scholar - R. J. Glauber (editor),
*Quantum Optics*[Proceedings of the International School of Physics “Enrico Fermi,” Course LXII] Academic Press, New York (1969). This collection of tutorial lectures at the advanced level includes several on various aspects of nonlinear optics. Especially pertinent is that by J. Ducuing on nonlinear optical processes.Google Scholar - N. Bloembergen, “The Stimulated Raman Effect,”
*Am. J. Phys.*,**35**, 989–1023 (1967). A tutorial and review article discussing many topics in the physics of nonlinear optics.ADSCrossRefGoogle Scholar - H. Rabin and C. L. Tang (editors),
*Quantum Electronics: A Treatise*, Academic Press, New York (1975), Volume 1 on*Nonlinear Optics*, Parts A and B. For those wishing to delve deeply into nonlinear optics at the advanced level, these books offer articles on, among other topics, the measurement of nonlinear optical susceptibilities, optical harmonic generation, optical parametric oscillators, and frequency up-conversion.Google Scholar - F. Zernike and J. E. Midwinter,
*Applied Nonlinear Optics*, John Wiley, New York (1973). This short book is rather terse in style, but covers a wide variety of topics in the physics and applications of nonlinear optics.Google Scholar