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Origin and symmetry of optical nonlinearities
Boulanger, B. and Zyss, J., International Tables for Crystallography (2013). Vol. D, Section 1.7.2, p. [ doi:10.1107/97809553602060000906 ]
has to be taken in the upper half plane of the complex plane. The reality of R (1) implies that . 1.7.2.1.2.2. Second-order susceptibility | top | pdf | Substitution of (1.7.2.15) in (1.7.2.12) yields ...

Superposition of monochromatic waves
Boulanger, B. and Zyss, J., International Tables for Crystallography (2013). Vol. D, Section 1.7.2.1.3, p. [ doi:10.1107/97809553602060000906 ]
optical properties Optical fields are often superpositions of monochromatic waves which, due to spectral discretization, will introduce considerable simplifications in previous expressions such as (1.7.2.20) relating the induced ...

Symmetry properties
Boulanger, B. and Zyss, J., International Tables for Crystallography (2013). Vol. D, Section 1.7.2.2, p. [ doi:10.1107/97809553602060000906 ]
optical properties 1.7.2.2.1. Intrinsic permutation symmetry | top | pdf | 1.7.2.2.1.1. ABDP and Kleinman symmetries | top | pdf | Intrinsic permutation symmetry, as already discussed, imposes ...

nth-order susceptibility
Boulanger, B. and Zyss, J., International Tables for Crystallography (2013). Vol. D, Section 1.7.2.1.2.3, p. [ doi:10.1107/97809553602060000906 ]
optical properties Substitution of (1.7.2.15) in (1.7.2.14) provides where and . All frequencies must lie in the upper half complex plane and reality of χ (n) imposes Intrinsic permutation symmetry ...

Quadratic response
Boulanger, B. and Zyss, J., International Tables for Crystallography (2013). Vol. D, Section 1.7.2.1.1.2, p. [ doi:10.1107/97809553602060000906 ]
...

Manley–Rowe relations
Boulanger, B. and Zyss, J., International Tables for Crystallography (2013). Vol. D, Section 1.7.2.2.1.2, p. [ doi:10.1107/97809553602060000906 ]
the quadratic induced polarization P (2), Manley–Rowe relations for sum-frequency generation state Since, (1.7.2.40) leads to an energy conservation condition, namely, which expresses that the power generated at ω 3 is equal ...

Linear susceptibility
Boulanger, B. and Zyss, J., International Tables for Crystallography (2013). Vol. D, Section 1.7.2.1.2.1, p. [ doi:10.1107/97809553602060000906 ]
optical properties By substitution of (1.7.2.15) in (1.7.2.7), where In these equations, to satisfy the energy conservation condition that will be generalized in the following. In order to ensure convergence ...

Induced polarization and susceptibility
Boulanger, B. and Zyss, J., International Tables for Crystallography (2013). Vol. D, Section 1.7.2.1, p. [ doi:10.1107/97809553602060000906 ]
at frequency . 1.7.2.1.4. Conventions for nonlinear susceptibilities | top | pdf | 1.7.2.1.4.1. Classical convention | top | pdf | Insertion of (1.7.2.26) in (1.7.2.25) together with permutation ...

ABDP and Kleinman symmetries
Boulanger, B. and Zyss, J., International Tables for Crystallography (2013). Vol. D, Section 1.7.2.2.1.1, p. [ doi:10.1107/97809553602060000906 ]
significant variation of the susceptibility . It follows correspondingly that the susceptibility is invariant with respect to the permutation of Cartesian indices appearing only in the numerator of (1.7.2.36), regardless of frequency. This property, which ...

Classical convention
Boulanger, B. and Zyss, J., International Tables for Crystallography (2013). Vol. D, Section 1.7.2.1.4.1, p. [ doi:10.1107/97809553602060000906 ]
optical properties Insertion of (1.7.2.26) in (1.7.2.25) together with permutation symmetry provides where the summation over ω stands for all distinguishable permutation of, K being a numerical factor given ...

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