Expression of third-order effective nonlinear susceptibility for third-harmonic generation in crystals

Xue-lin Yang; Sheng-wu Xie

doi:10.1364/AO.34.006130

Applied Optics
Vol. 34,
Issue 27,
pp. 6130-6135
(1995)
•https://doi.org/10.1364/AO.34.006130

Expression of third-order effective nonlinear susceptibility for third-harmonic generation in crystals

Xue-lin Yang and Sheng-wu Xie

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Xue-lin Yang and Sheng-wu Xie, "Expression of third-order effective nonlinear susceptibility for third-harmonic generation in crystals," Appl. Opt. 34, 6130-6135 (1995)

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Abstract

Third-harmonic-generation processes in crystals are governed by the fourth-rank tensor $χ_{i j k l}^{(3)}$ , which reflects the crystal symmetry. In this case, the third-order nonlinear susceptibility tensor can be contracted to the compact matrix form $χ_{i m}^{(3)}$ . The matrices $χ_{i m}^{(3)}$ for isotropic media and all 32 crystallo-graphic point groups are presented. With these matrices, the analytic expressions of third-order effective nonlinear susceptibility can be easily derived.

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Crystal Class	Matrix of $χ_{i m}^{(3)}$
Triclinic ${1, \bar{1}}$	$[\begin{array}{l} χ_{11} & χ_{12} & χ_{13} & χ_{14} & χ_{15} & χ_{16} & χ_{17} & χ_{18} & χ_{19} & χ_{10} \\ χ_{21} & χ_{22} & χ_{23} & χ_{24} & χ_{25} & χ_{26} & χ_{27} & χ_{28} & χ_{29} & χ_{20} \\ χ_{31} & χ_{32} & χ_{33} & χ_{34} & χ_{35} & χ_{36} & χ_{37} & χ_{38} & χ_{39} & χ_{30} \end{array}]$ $[\begin{array}{l} χ_{11} & χ_{12} & χ_{13} & χ_{14} & χ_{15} & χ_{16} & χ_{17} & χ_{18} & χ_{19} & χ_{10} \\ χ_{19} & χ_{22} & χ_{23} & χ_{24} & χ_{25} & χ_{14} & χ_{10} & χ_{12} & χ_{18} & χ_{15} \\ χ_{17} & χ_{25} & χ_{33} & χ_{23} & χ_{24} & χ_{13} & χ_{16} & χ_{15} & χ_{10} & χ_{14} \end{array}] *$
Monoclinic {2, m, 2/m}	$[\begin{array}{l} χ_{11} & 0 & χ_{13} & 0 & χ_{15} & χ_{16} & χ_{17} & χ_{18} & 0 & 0 \\ 0 & χ_{22} & 0 & χ_{24} & 0 & 0 & 0 & 0 & χ_{29} & χ_{20} \\ χ_{31} & 0 & χ_{33} & 0 & χ_{35} & χ_{36} & χ_{37} & χ_{38} & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & χ_{13} & 0 & χ_{15} & χ_{16} & χ_{17} & χ_{18} & 0 & 0 \\ 0 & χ_{22} & 0 & χ_{24} & 0 & 0 & 0 & 0 & χ_{18} & χ_{15} \\ χ_{17} & 0 & χ_{33} & 0 & χ_{24} & χ_{13} & χ_{16} & χ_{15} & 0 & 0 \end{array}] *$
Orthorhombic {222, mm2, mmm}	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{18} & 0 & 0 \\ 0 & χ_{22} & 0 & χ_{24} & 0 & 0 & 0 & 0 & χ_{29} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{35} & 0 & χ_{37} & 0 & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{18} & 0 & 0 \\ 0 & χ_{22} & 0 & χ_{24} & 0 & 0 & 0 & 0 & χ_{18} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{24} & 0 & χ_{16} & 0 & 0 & 0 \end{array}] *$
Tetragonal ${4, \bar{4}, 4 / m}$	$[\begin{array}{l} χ_{11} & χ_{12} & 0 & χ_{14} & 0 & χ_{16} & 0 & χ_{18} & χ_{19} & 0 \\ - χ_{12} & χ_{11} & 0 & χ_{16} & 0 & - χ_{14} & 0 & - χ_{19} & χ_{18} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{35} & 0 & χ_{35} & 0 & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & χ_{12} & 0 & 0 & 0 & χ_{16} & 0 & χ_{18} & - χ_{12} & 0 \\ - χ_{12} & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & χ_{12} & χ_{18} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 & 0 \end{array}] *$
Tetragonal ${422, 4 m m, 4 / m m m, \bar{4} 2 m}$	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{18} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & χ_{18} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{35} & 0 & χ_{35} & 0 & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{18} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & χ_{18} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 & 0 \end{array}] *$
Hexagonal ${6, \bar{6}, 6 / m}$	$[\begin{array}{l} χ_{11} & χ_{12} & 0 & χ_{14} & 0 & χ_{16} & 0 & 1 / 3 χ_{11} & 1 / 3 χ_{12} & 0 \\ - χ_{12} & χ_{11} & 0 & χ_{16} & 0 & - χ_{14} & 0 & - 1 / 3 χ_{12} & 1 / 3 χ_{11} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{35} & 0 & χ_{35} & 0 & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & 1 / 3 χ_{11} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & 1 / 3 χ_{11} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 & 0 \end{array}] *$
Hexagonal ${622, 6 m m, 6 / m m, \bar{6} m 2}$	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & 1 / 3 χ_{11} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & 1 / 3 χ_{11} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{35} & 0 & χ_{35} & 0 & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & 1 / 3 χ_{11} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & 1 / 3 χ_{11} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 & 0 \end{array}] *$
Trigonal ${3, \bar{3}}$	$[\begin{array}{l} χ_{11} & χ_{12} & 0 & χ_{14} & χ_{15} & χ_{16} & - χ_{15} & 1 / 3 χ_{11} & 1 / 3 χ_{12} & χ_{10} \\ - χ_{12} & χ_{11} & 0 & χ_{16} & - χ_{10} & - χ_{14} & χ_{10} & - 1 / 3 χ_{12} & 1 / 3 χ_{11} & χ_{15} \\ χ_{31} & χ_{32} & χ_{33} & 0 & χ_{35} & 0 & χ_{35} & - χ_{31} & - χ_{32} & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & χ_{15} & χ_{16} & - χ_{15} & 1 / 3 χ_{11} & 0 & χ_{10} \\ 0 & χ_{11} & 0 & χ_{16} & - χ_{10} & 0 & χ_{10} & 0 & 1 / 3 χ_{11} & χ_{15} \\ - χ_{15} & - χ_{10} & χ_{33} & 0 & χ_{16} & 0 & χ_{16} & χ_{15} & χ_{10} & 0 \end{array}] *$
Trigonal ${3 m, \bar{3} m, 32}$	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & χ_{15} & χ_{16} & - χ_{15} & 1 / 3 χ_{11} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & - χ_{15} & 0 & 0 & 0 & 1 / 3 χ_{11} & χ_{15} \\ χ_{31} & 0 & χ_{33} & 0 & χ_{35} & 0 & χ_{35} & - χ_{31} & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & χ_{15} & χ_{16} & - χ_{15} & 1 / 3 χ_{11} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & - χ_{15} & 0 & 0 & 0 & 1 / 3 χ_{11} & χ_{15} \\ - χ_{15} & 0 & χ_{33} & 0 & χ_{16} & 0 & χ_{16} & - χ_{15} & 0 & 0 \end{array}] *$
Cubic {23, m3}	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{18} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{18} & 0 & 0 & 0 & 0 & χ_{16} & 0 \\ 0 & 0 & χ_{11} & 0 & χ_{16} & 0 & χ_{18} & 0 & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & χ_{16} & 0 \\ 0 & 0 & χ_{11} & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 & 0 \end{array}] *$
Cubic ${432 m, \bar{4} 3 m, m 3 m}$	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & χ_{16} & 0 \\ 0 & 0 & χ_{11} & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 & 0 \end{array}]$
Isotropic medium	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & 1 / 3 χ_{11} & 0 & 1 / 3 χ_{11} & 0 & 0 \\ 0 & χ_{11} & 0 & 1 / 3 χ_{11} & 0 & 0 & 0 & 0 & 1 / 3 χ_{11} & 0 \\ 0 & 0 & χ_{11} & 0 & 1 / 3 χ_{11} & 0 & 1 / 3 χ_{11} & 0 & 0 & 0 \end{array}]$

Crystal	PM Type	[e_m]^T
Negative	I	[a³, b³, 0, 0, 0, 0, 0, 3ab², −3a²b, 0]
	II	[a²bd, ab²d, 0, 0, −b²c, 0, −a²c, b³d
		−2a²bd, a³d − 2ab²d, 2abc]
	III	[ab²d², −a²bd², 0, −bc², 2abcd, ac²,
		−2abcd, a³d² − 2ab²d², 2a²bd²
		−b³d², 2b²cd − 2a²cd]
Positive	I	[b³d³, a³d³, −c³, 3ac²d, −3a²cd², 3bc²d,
		−3b²cd², 3a²bd³, 3ab²d³, −6abcd²]
	II	[ab²d², −a²bd², 0, −bc², 2abcd, ac²,
		−2abcd, a³d² −2ab²d², 2a²bd²
		−b³d², 2b²cd − 2a²cd]
	III	[a²bd, ab²d, 0, 0, −b²c, 0, −a²c, b³d
		−2a²bd, a³d − 2ab²d, 2abc]

Plane	PM Type	[e_m]^T
xoz(θ < Ω)	I	[0, 1, 0, 0, 0, 0, 0, 0, 0, 0]
	II	[0, 0, 0, 0, c, 0, 0, −d, 0, 0]
	III	[0, 0, 0, c², 0, 0, 0, 0, d², −2cd]
xoz(θ > Ω)	I	[d³, 0, −c³, 0, 0, 3c²d, −3cd², 0, 0, 0]
	II	[0, 0, 0, c², 0, 0, 0, 0, d², −2cd]
	III	[0, 0, 0, 0, −c, 0, 0, d, 0, 0]
yoz	I	[0, d³, −c³, 3c²d, −3cd², 0, 0, 0, 0, 0]
	II	[0, 0, 0, 0, 0, −c², 0, −d², 0, 2cd]
	III	[0, 0, 0, 0, 0, 0, −c, 0, d, 0]
xoy	I	[0, 0, −1, 0, 0, 0, 0, 0, 0]
	II	[0, 0, 0, b, 0, −a, 0, 0, 0, 0]
	III	[0, 0, 0, 0, −b², 0, −a², 0, 0, 2ab]

Crystal Class	Matrix of $χ_{i m}^{(3)}$
Triclinic ${1, \bar{1}}$	$[\begin{array}{l} χ_{11} & χ_{12} & χ_{13} & χ_{14} & χ_{15} & χ_{16} & χ_{17} & χ_{18} & χ_{19} & χ_{10} \\ χ_{21} & χ_{22} & χ_{23} & χ_{24} & χ_{25} & χ_{26} & χ_{27} & χ_{28} & χ_{29} & χ_{20} \\ χ_{31} & χ_{32} & χ_{33} & χ_{34} & χ_{35} & χ_{36} & χ_{37} & χ_{38} & χ_{39} & χ_{30} \end{array}]$ $[\begin{array}{l} χ_{11} & χ_{12} & χ_{13} & χ_{14} & χ_{15} & χ_{16} & χ_{17} & χ_{18} & χ_{19} & χ_{10} \\ χ_{19} & χ_{22} & χ_{23} & χ_{24} & χ_{25} & χ_{14} & χ_{10} & χ_{12} & χ_{18} & χ_{15} \\ χ_{17} & χ_{25} & χ_{33} & χ_{23} & χ_{24} & χ_{13} & χ_{16} & χ_{15} & χ_{10} & χ_{14} \end{array}] *$
Monoclinic {2, m, 2/m}	$[\begin{array}{l} χ_{11} & 0 & χ_{13} & 0 & χ_{15} & χ_{16} & χ_{17} & χ_{18} & 0 & 0 \\ 0 & χ_{22} & 0 & χ_{24} & 0 & 0 & 0 & 0 & χ_{29} & χ_{20} \\ χ_{31} & 0 & χ_{33} & 0 & χ_{35} & χ_{36} & χ_{37} & χ_{38} & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & χ_{13} & 0 & χ_{15} & χ_{16} & χ_{17} & χ_{18} & 0 & 0 \\ 0 & χ_{22} & 0 & χ_{24} & 0 & 0 & 0 & 0 & χ_{18} & χ_{15} \\ χ_{17} & 0 & χ_{33} & 0 & χ_{24} & χ_{13} & χ_{16} & χ_{15} & 0 & 0 \end{array}] *$
Orthorhombic {222, mm2, mmm}	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{18} & 0 & 0 \\ 0 & χ_{22} & 0 & χ_{24} & 0 & 0 & 0 & 0 & χ_{29} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{35} & 0 & χ_{37} & 0 & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{18} & 0 & 0 \\ 0 & χ_{22} & 0 & χ_{24} & 0 & 0 & 0 & 0 & χ_{18} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{24} & 0 & χ_{16} & 0 & 0 & 0 \end{array}] *$
Tetragonal ${4, \bar{4}, 4 / m}$	$[\begin{array}{l} χ_{11} & χ_{12} & 0 & χ_{14} & 0 & χ_{16} & 0 & χ_{18} & χ_{19} & 0 \\ - χ_{12} & χ_{11} & 0 & χ_{16} & 0 & - χ_{14} & 0 & - χ_{19} & χ_{18} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{35} & 0 & χ_{35} & 0 & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & χ_{12} & 0 & 0 & 0 & χ_{16} & 0 & χ_{18} & - χ_{12} & 0 \\ - χ_{12} & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & χ_{12} & χ_{18} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 & 0 \end{array}] *$
Tetragonal ${422, 4 m m, 4 / m m m, \bar{4} 2 m}$	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{18} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & χ_{18} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{35} & 0 & χ_{35} & 0 & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{18} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & χ_{18} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 & 0 \end{array}] *$
Hexagonal ${6, \bar{6}, 6 / m}$	$[\begin{array}{l} χ_{11} & χ_{12} & 0 & χ_{14} & 0 & χ_{16} & 0 & 1 / 3 χ_{11} & 1 / 3 χ_{12} & 0 \\ - χ_{12} & χ_{11} & 0 & χ_{16} & 0 & - χ_{14} & 0 & - 1 / 3 χ_{12} & 1 / 3 χ_{11} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{35} & 0 & χ_{35} & 0 & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & 1 / 3 χ_{11} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & 1 / 3 χ_{11} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 & 0 \end{array}] *$
Hexagonal ${622, 6 m m, 6 / m m, \bar{6} m 2}$	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & 1 / 3 χ_{11} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & 1 / 3 χ_{11} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{35} & 0 & χ_{35} & 0 & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & 1 / 3 χ_{11} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & 1 / 3 χ_{11} & 0 \\ 0 & 0 & χ_{33} & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 & 0 \end{array}] *$
Trigonal ${3, \bar{3}}$	$[\begin{array}{l} χ_{11} & χ_{12} & 0 & χ_{14} & χ_{15} & χ_{16} & - χ_{15} & 1 / 3 χ_{11} & 1 / 3 χ_{12} & χ_{10} \\ - χ_{12} & χ_{11} & 0 & χ_{16} & - χ_{10} & - χ_{14} & χ_{10} & - 1 / 3 χ_{12} & 1 / 3 χ_{11} & χ_{15} \\ χ_{31} & χ_{32} & χ_{33} & 0 & χ_{35} & 0 & χ_{35} & - χ_{31} & - χ_{32} & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & χ_{15} & χ_{16} & - χ_{15} & 1 / 3 χ_{11} & 0 & χ_{10} \\ 0 & χ_{11} & 0 & χ_{16} & - χ_{10} & 0 & χ_{10} & 0 & 1 / 3 χ_{11} & χ_{15} \\ - χ_{15} & - χ_{10} & χ_{33} & 0 & χ_{16} & 0 & χ_{16} & χ_{15} & χ_{10} & 0 \end{array}] *$
Trigonal ${3 m, \bar{3} m, 32}$	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & χ_{15} & χ_{16} & - χ_{15} & 1 / 3 χ_{11} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & - χ_{15} & 0 & 0 & 0 & 1 / 3 χ_{11} & χ_{15} \\ χ_{31} & 0 & χ_{33} & 0 & χ_{35} & 0 & χ_{35} & - χ_{31} & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & χ_{15} & χ_{16} & - χ_{15} & 1 / 3 χ_{11} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & - χ_{15} & 0 & 0 & 0 & 1 / 3 χ_{11} & χ_{15} \\ - χ_{15} & 0 & χ_{33} & 0 & χ_{16} & 0 & χ_{16} & - χ_{15} & 0 & 0 \end{array}] *$
Cubic {23, m3}	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{18} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{18} & 0 & 0 & 0 & 0 & χ_{16} & 0 \\ 0 & 0 & χ_{11} & 0 & χ_{16} & 0 & χ_{18} & 0 & 0 & 0 \end{array}]$ $[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & χ_{16} & 0 \\ 0 & 0 & χ_{11} & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 & 0 \end{array}] *$
Cubic ${432 m, \bar{4} 3 m, m 3 m}$	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 \\ 0 & χ_{11} & 0 & χ_{16} & 0 & 0 & 0 & 0 & χ_{16} & 0 \\ 0 & 0 & χ_{11} & 0 & χ_{16} & 0 & χ_{16} & 0 & 0 & 0 \end{array}]$
Isotropic medium	$[\begin{array}{l} χ_{11} & 0 & 0 & 0 & 0 & 1 / 3 χ_{11} & 0 & 1 / 3 χ_{11} & 0 & 0 \\ 0 & χ_{11} & 0 & 1 / 3 χ_{11} & 0 & 0 & 0 & 0 & 1 / 3 χ_{11} & 0 \\ 0 & 0 & χ_{11} & 0 & 1 / 3 χ_{11} & 0 & 1 / 3 χ_{11} & 0 & 0 & 0 \end{array}]$

Crystal	PM Type	[e_m]^T
Negative	I	[a³, b³, 0, 0, 0, 0, 0, 3ab², −3a²b, 0]
	II	[a²bd, ab²d, 0, 0, −b²c, 0, −a²c, b³d
		−2a²bd, a³d − 2ab²d, 2abc]
	III	[ab²d², −a²bd², 0, −bc², 2abcd, ac²,
		−2abcd, a³d² − 2ab²d², 2a²bd²
		−b³d², 2b²cd − 2a²cd]
Positive	I	[b³d³, a³d³, −c³, 3ac²d, −3a²cd², 3bc²d,
		−3b²cd², 3a²bd³, 3ab²d³, −6abcd²]
	II	[ab²d², −a²bd², 0, −bc², 2abcd, ac²,
		−2abcd, a³d² −2ab²d², 2a²bd²
		−b³d², 2b²cd − 2a²cd]
	III	[a²bd, ab²d, 0, 0, −b²c, 0, −a²c, b³d
		−2a²bd, a³d − 2ab²d, 2abc]

Abstract

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Tables (3)

Equations (23)

Applied Optics