Section: 12 | Nonlinear Optical Constants |

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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

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 ε_o 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 E_j^ω1 (along the j-direction at frequency ω₁) and the second E_k^ω2 (along the k-direction at frequency ω₂) one can write the second term of the Taylor’s series as follows

$P_{i} (ω_{1} ω_{2}) = 2 χ_{i j k}^{ω_{3} = ω_{1} \pm ω_{2}} E_{j}^{ω_{1}} E_{k}^{ω_{2}}$

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): χ_ijk (2ω, ω, ω), while equal and opposite frequencies, ω₁ = ω and ω₂ = –ω lead to Optical Rectification (OR): χ_ijk (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

d_ijk = 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 d_im where m takes the values 1 to 6. Only noncentrosymmetric crystals can possess a nonvanishing d_ijk 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

$P_{j} (ω_{1} ω_{2} ω_{3}) = g_{4} χ_{j k l m} E_{k}^{ω_{1}} E_{1}^{ω_{2}} E_{m}^{ω_{3}}$

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:

C_jklm = 1/4χ_jklm

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 C_jn . The unit of C_jn 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

Handbook of Laser Science and Technology, Vol. 111, Part 1; Weber, M. J. Ed., CRC Press, Boca Raton, FL, 1986.
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]
Shen, Y.R., The Principles of Nonlinear Optics, John Wiley, New York, 1984.
Yariv, A., Quantum Electronics, Third Edition, John Wiley, New York, 1988.
Bloembergen, N., Nonlinear Optics, W.A. Benjamin, New York, 1965.
Zernike F. and Midwinter, J.E., Applied Nonlinear Optics, John Wiley, New York, 1973.
Hopf, F.A. and Stegeman, G.I., Applied Classical Electrodynamics, Vol. 2: Nonlinear Optics, John Wiley, New York, 1986.
Nonlinear Optical Properties of Organic Molecules and Crystals, Chemla, D. S., and Zyss, J., Eds., Academic Press, Orlando, FL, 1987.
Optical Phase Conjugation, Fisher, R. A., Ed., Academic Press, New York, 1983.
Zyss, J., Molecular Nonlinear Optics: Materials, Devices and Physics, Academic Press, Boston, 1994.
Nonlinear Optics, 5 articles in Physics Today, (American Institute of Physics), Vol. 47, No. 5, May 1994.

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

Material	Other names	Formula	CAS Reg. No.	Symmetry class	d_im×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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