Section: 12 | Nonlinear Optical Constants |
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One or more tables in this document differ to those in the book. This is due to space restrictions in the book.
Summary of table differences
The table 'TABLE 1. Second Harmonic Generation Coefficients of Nonlinear Optical Crystals*' has one or more different columns to those in the book version.
The table 'TABLE 2. Third Harmonic Generation Coefficients of Some Nonlinear Optical Materials*' has one or more different columns to those in the book version.
How to Cite this Reference
 The recommended form of citation is: John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), CRC Press/Taylor & Francis, Boca Raton, FL. If a specific table is cited, use the format: "Physical Constants of Organic Compounds," in CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.

# NONLINEAR OPTICAL CONSTANTS

H. P. R. Frederikse

The relation between the polarization density P of a dielectric medium and the electric field E is linear when E is small, but becomes nonlinear as E acquires values comparable with interatomic electric fields (105 to 108 V/cm). Under these conditions the relation between P and E can be expanded in a Taylor’s series

P = ε0χ(1) E + 2χ(2) E 2 + 4χ(3) E 3 + … (1)

where εo is the permittivity of free space, while χ(1) is the linear and χ(2), χ(3), etc., the nonlinear optical susceptibilities.

If we consider two optical fields, the first Ejω1 (along the j-direction at frequency ω1) and the second Ekω2 (along the k-direction at frequency ω2) one can write the second term of the Taylor’s series as follows

${P}_{i}\left({\omega }_{1}{\omega }_{2}\right)=2{\chi }_{ijk}^{{\omega }_{3}={\omega }_{1}±{\omega }_{2}}{E}_{j}^{{\omega }_{1}}{E}_{k}^{{\omega }_{2}}$

When ω1 ≠ ω2 the (parametric) mixing of the two fields gives rise to two new polarizations at the frequencies ω3 = ω1 + ω2 and ω3´ = ω1 – ω2. When the two frequencies are equal, ω1 = ω2 = ω, the result is Second Harmonic Generation (SHG): χijk (2ω, ω, ω), while equal and opposite frequencies, ω1 = ω and ω2 = –ω lead to Optical Rectification (OR): χijk (0, ω, –ω). In the SHG case the following convention is adopted: the second order nonlinear coefficient d is equal to one half of the second order nonlinear susceptibility

dijk = 1/2χ(2)

Because of the symmetry of the indices j and k one can replace these two by a single index (subscript) m. Consequently, the notation for the SHG nonlinear coefficient in reduced form is dim where m takes the values 1 to 6. Only noncentrosymmetric crystals can possess a nonvanishing dijk tensor (third rank). The unit of the SHG coefficients is m/V (in the MKSQ/SI system).

In centrosymmetric media the dominant nonlinearity is of the third order. This effect is represented by the third term in the Taylor’s series (Equation 1); it is the result of the interaction of a number of optical fields (one to three) producing a new frequency ω4 = ω1 + ω2 + ω3. The third order polarization is given by

${P}_{j}\left({\omega }_{1}{\omega }_{2}{\omega }_{3}\right)={g}_{4}{\chi }_{jklm}{E}_{k}^{{\omega }_{1}}{E}_{1}^{{\omega }_{2}}{E}_{m}^{{\omega }_{3}}$

Third Harmonic Generation (THG) is achieved when ω1 = ω2 = ω3 = ω. In this case the constant g4 = 1/4. The third order nonlinear coefficient C is related to the third order susceptibility as follows:

Cjklm = 1/4χjklm

This coefficient is a fourth rank tensor. In the THG case the matrices must be invariant under permutation of the indices k, l, and m; as a result the notation for the third order nonlinear coefficient can be simplified to Cjn . The unit of Cjn is m2·V­–2 (in the MKSQ/SI system).

Applications of second order nonlinear optical materials include the generation of higher (up to sixth) optical harmonics, the mixing of monochromatic waves to generate sum or difference frequencies (frequency conversion), the use of two monochromatic waves to amplify a third wave (parametric amplification) and the addition of feedback to such an amplifier to create an oscillation (parametric oscillation).

Third order nonlinear optical materials are used for THG, self-focusing, four wave mixing, optical amplification, and optical conjugation. Many of these effects — as well as the variation and modulation of optical propagation caused by mechanical, electric, and magnetic fields (see the preceding table on “Elasto-Optic, Electro-Optic, and Magneto-Optic Constants”) are used in the areas of optical communication, optical computing, and optical imaging.

# References

1. Handbook of Laser Science and Technology, Vol. 111, Part 1; Weber, M. J. Ed., CRC Press, Boca Raton, FL, 1986.
2. Dmitriev, V.G., Gurzadyan, G.G., and Nikogosyan, D., Handbook of Nonlinear Optical Crystals, Springer-Verlag, Berlin, 1991. [https://doi.org/10.1007/978-3-662-13830-4]
3. Shen, Y.R., The Principles of Nonlinear Optics, John Wiley, New York, 1984.
4. Yariv, A., Quantum Electronics, Third Edition, John Wiley, New York, 1988.
5. Bloembergen, N., Nonlinear Optics, W.A. Benjamin, New York, 1965.
6. Zernike F. and Midwinter, J.E., Applied Nonlinear Optics, John Wiley, New York, 1973.
7. Hopf, F.A. and Stegeman, G.I., Applied Classical Electrodynamics, Vol. 2: Nonlinear Optics, John Wiley, New York, 1986.
8. Nonlinear Optical Properties of Organic Molecules and Crystals, Chemla, D. S., and Zyss, J., Eds., Academic Press, Orlando, FL, 1987.
9. Optical Phase Conjugation, Fisher, R. A., Ed., Academic Press, New York, 1983.
10. Zyss, J., Molecular Nonlinear Optics: Materials, Devices and Physics, Academic Press, Boston, 1994.
11. Nonlinear Optics, 5 articles in Physics Today, (American Institute of Physics), Vol. 47, No. 5, May 1994.

## TABLE 1. Second Harmonic Generation Coefficients of Nonlinear Optical Crystals*

 Material Other names Formula CAS Reg. No. Symmetry class dim×1012﻿/m V-1 λ/µm Continued on next page... Aluminum phosphate Aluminum orthophosphate AlPO4 7784-30-7 32 d11 = 0.35 ± 0.03 1.058 Ammonium dihydrogen phosphate ADP NH4H2PO4 7722-76-1 4̅2 m d36 = 0.53 1.064 Ammonium dihydrogen phosphate ADP NH4H2PO4 7722-76-1 d36 = 0.85 0.694 Barium borate (β) BBO Ba(BO2)2 13701-59-2 3 m d22 = 2.22 ± 0.09 1.06 Barium borate (β) BBO Ba(BO2)2 13701-59-2 d31 = 0.16 ± 0.08 1.06 Barium sodium niobate BSN Ba2Na(NbO3)5 12323-03-4 mm 2 d33 = –17.6 ± 1.28 1.064 Barium sodium niobate BSN Ba2Na(NbO3)5 12323-03-4 d31 = –12.8 ± 1.28 1.064 Barium strontium niobate Barium niobium strontium oxide BaSr(NbO3)4 37185-09-4 4 mm d33 = 11.3 ± 3.3 1.064 Barium strontium niobate Barium niobium strontium oxide BaSr(NbO3)4 37185-09-4 d31 = 4.31 ± 1.32 1.064 Barium strontium niobate Barium niobium strontium oxide BaSr(NbO3)4 37185-09-4 d15 = 5.98 ± 2 1.064 Barium titanate Barium metatitanate BaTiO3 12047-27-7 4 mm d33 = 6.8 ± 1.0 1.064 Barium titanate Barium metatitanate BaTiO3 12047-27-7 d31 = 15.7 ± 1.8 1.064 Barium titanate Barium metatitanate BaTiO3 12047-27-7 d15 = 17.0 ± 1.8 1.064 Benzil Diphenylethanedione (C6H5CO)2 134-81-6 32 d11 = 3.6 ± 0.5 1.064 Bismuth germanate BGO Bi4(GeO4)3 12233-56-6 4̅3 m d14 = 1.28 1.064 Cadmium germanium arsenide CdGeAs2 4̅2 m d36 = 351 ± 105 10.6 Cadmium selenide Cadmoselite CdSe 1306-24-7 6 mm d33 = 54.5 ± 12.6 10.6 Cadmium selenide Cadmoselite CdSe 1306-24-7 d31 = –26.8 ± 2.7 10.6 Cadmium sulfide Greenockite CdS 1306-23-6 6 mm d33 = 25.8 ± 1.6 1.058
 *These data are taken from Refs. 1 and 2.

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