Determining selected quadratic coefficients in noncentrosymmetric crystals

Mike Melnichuk; Lowell T. Wood

doi:10.1364/JOSAA.23.001236

Journal of the Optical Society of America A
Vol. 23,
Issue 5,
pp. 1236-1242
(2006)
•https://doi.org/10.1364/JOSAA.23.001236

Determining selected quadratic coefficients in noncentrosymmetric crystals

Mike Melnichuk and Lowell T. Wood

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Mike Melnichuk and Lowell T. Wood, "Determining selected quadratic coefficients in noncentrosymmetric crystals," J. Opt. Soc. Am. A 23, 1236-1242 (2006)

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Abstract

We present a systematic theoretical examination concerning the practical possibility of measuring components of the quadratic symmetry tensors for solid materials lacking inversion symmetry. We show that information concerning the quadratic effects in noncentrosymmetrical crystals can be obtained directly by methods that allow for the bypassing of the coexisting dominant linear effects. Here, we deal only with the particular case of the electro-optic effect; however, the results and conclusions are also valid for the elasto-optic effects.

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E	k	Electro-optic Coefficients
Monoclinic m
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{32}$ , $r_{42}$
	$x_{2}$	$R_{12}$ , $R_{32}$ , $R_{52}$
	$x_{3}$	$R_{12}$ , $R_{22}$ , $r_{62}$
Monoclinic 2
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{31}$ , $r_{41}$
	$x_{2}$	$R_{11}$ , $R_{31}$ , $R_{51}$
	$x_{3}$	$R_{21}$ , $R_{22}$ , $r_{61}$
$E_{3}$	$x_{1}$	$R_{23}$ , $R_{33}$ , $r_{43}$
	$x_{2}$	$R_{13}$ , $R_{33}$ , $R_{53}$
	$x_{3}$	$R_{13}$ , $R_{23}$ , $r_{63}$
$E_{1}$ , $E_{3}$	$x_{1}$	$R_{25}$ , $R_{35}$
	$x_{2}$	$R_{15}$ , $R_{35}$ , $R_{55}$
	$x_{3}$	$R_{15}$ , $R_{25}$
Orthorhombic 222
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{31}$ , $r_{41}$
	$x_{2}$	$R_{11}$ , $R_{31}$
	$x_{3}$	$R_{11}$ , $R_{21}$
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{32}$
	$x_{2}$	$R_{12}$ , $R_{32}$ , $r_{52}$
	$x_{3}$	$R_{12}$ , $R_{22}$
$E_{3}$	$x_{1}$	$R_{23}$ , $R_{33}$
	$x_{2}$	$R_{13}$ , $R_{33}$
	$x_{3}$	$R_{13}$ , $R_{33}$ , $r_{63}$
$E_{2}$ , $E_{3}$	$x_{1}$	$R_{44}$
$E_{1}$ , $E_{3}$	$x_{2}$	$R_{55}$
$E_{1}$ , $E_{2}$	$x_{3}$	$R_{66}$
Orthorhombic mm2
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{31}$
	$x_{2}$	$R_{11}$ , $R_{31}$ , $r_{51}$
	$x_{3}$	$R_{11}$ , $R_{21}$
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{32}$ , $r_{42}$
	$x_{2}$	$R_{12}$ , $R_{32}$
	$x_{3}$	$R_{12}$ , $R_{22}$
$E_{1}$ , $E_{2}$	$x_{3}$	$R_{66}$

E	k	Electro-optic Coefficients
Trigonal 32
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{11}$ , $R_{32}$ , $R_{31}$ , $R_{42}$ , $R_{41}$ , $R_{56}$
	$x_{2}$	$R_{12}$ , $R_{21}$ , $R_{32}$ , $R_{31}$ , $r_{52}$ , $r_{41}$
	$x_{3}$	$R_{12}$ , $R_{21}$ , $R_{22}$ , $R_{11}$ , $r_{62}$ , $r_{21}$ , $r_{11}$
$E_{3}$	$x_{1}$	$R_{23}$ , $R_{13}$ , $R_{33}$
	$x_{2}$	$R_{13}$ , $R_{23}$ , $R_{33}$
	$x_{3}$	HOC
$E_{2}$ , $E_{3}$	$x_{1}$	$R_{24}$ , $R_{14}$ , $R_{65}$ , $R_{44}$ , $R_{55}$
	$x_{2}$	$R_{14}$ , $R_{24}$ , $R_{65}$
	$x_{3}$	$R_{24}$ , $R_{14}$ , $R_{65}$
Trigonal 3m
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{12}$ , $R_{31}$ , $R_{32}$ , $R_{41}$ , $R_{42}$
	$x_{2}$	$R_{11}$ , $R_{22}$ , $R_{31}$ , $R_{32}$ , $r_{51}$ , $r_{42}$
	$x_{3}$	$R_{11}$ , $R_{22}$ , $R_{21}$ , $R_{12}$ , $r_{61}$ , $r_{22}$ , $r_{12}$
Hexagonal 622
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{12}$ , $R_{31}$ , $R_{32}$ , $r_{41}$ , $r_{52}$
	$x_{2}$	$R_{11}$ , $R_{22}$ , $R_{31}$ , $R_{32}$
	$x_{3}$	$R_{11}$ , $R_{22}$ , $R_{21}$ , $R_{12}$
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{11}$ , $R_{32}$ , $R_{31}$
	$x_{2}$	$R_{12}$ , $R_{21}$ , $R_{32}$ , $R_{31}$ , $r_{52}$ , $r_{41}$
	$x_{3}$	$R_{12}$ , $R_{21}$ , $R_{22}$ , $R_{11}$
$E_{3}$	$x_{1}$	$R_{23}$ , $R_{13}$ , $R_{33}$
	$x_{2}$	$R_{13}$ , $R_{23}$ , $R_{33}$
	$x_{3}$	HOC
$E_{2}$ , $E_{3}$	$x_{1}$	$R_{44}$ , $R_{55}$
$E_{1}$ , $E_{3}$	$x_{2}$	$R_{55}$ , $R_{44}$
$E_{1}$ , $E_{2}$	$x_{3}$	$R_{66}$
Hexagonal 6mm
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{12}$ , $R_{31}$ , $R_{32}$
	$x_{2}$	$R_{11}$ , $R_{22}$ , $R_{31}$ , $R_{32}$ , $r_{51}$ , $r_{42}$
	$x_{3}$	$R_{11}$ , $R_{22}$ , $R_{21}$ , $R_{12}$
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{11}$ , $R_{32}$ , $R_{31}$ , $r_{42}$ , $r_{51}$
	$x_{2}$	$R_{12}$ , $R_{21}$ , $R_{32}$ , $R_{31}$
	$x_{3}$	$R_{12}$ , $R_{21}$ , $R_{22}$ , $R_{11}$
$E_{1}$ , $E_{2}$	$x_{3}$	$R_{66}$

E	k	Electro-optic Coefficients
Hexagonal $- 6 m 2$
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{12}$ , $R_{31}$ , $R_{32}$
	$x_{2}$	$R_{11}$ , $R_{22}$ , $R_{31}$ , $R_{32}$
	$x_{3}$	$R_{11}$ , $R_{22}$ , $R_{21}$ , $R_{12}$ , $r_{61}$ , $r_{22}$ , $r_{12}$
$E_{3}$	$x_{1}$	$R_{23}$ , $R_{13}$ , $R_{33}$
	$x_{2}$	$R_{13}$ , $R_{23}$ , $R_{33}$
	$x_{3}$	HOC
$E_{1}$ , $E_{3}$	$x_{2}$	$R_{55}$ , $R_{44}$
	$x_{3}$	$R_{66}$
Hexagonal 6
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{12}$ , $R_{31}$ , $R_{32}$ , $r_{41}$ , $r_{52}$
	$x_{2}$	$R_{11}$ , $R_{22}$ , $R_{31}$ , $R_{32}$ , $r_{51}$ , $r_{42}$
	$x_{3}$	$R_{11}$ , $R_{22}$ , $R_{12}$ , $R_{21}$ , $R_{61}$ , $R_{62}$ , $R_{16}$ , $R_{26}$
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{11}$ , $R_{32}$ , $R_{31}$ , $r_{42}$ , $r_{51}$
	$x_{2}$	$R_{12}$ , $R_{21}$ , $R_{32}$ , $R_{31}$ , $r_{52}$ , $r_{41}$
	$x_{3}$	$R_{12}$ , $R_{21}$ , $R_{22}$ , $R_{11}$ , $R_{62}$ , $R_{61}$ , $R_{16}$ , $R_{26}$
$E_{1}$ , $E_{2}$	$x_{3}$	$R_{16}$ , $R_{26}$ , $R_{62}$ , $R_{61}$ , $R_{66}$
Hexagonal $- 6$
$E_{3}$	$x_{1}$	$R_{23}$ , $R_{13}$ , $R_{33}$
	$x_{2}$	$R_{13}$ , $R_{23}$ , $R_{33}$
	$x_{3}$	HOC
Tetragonal 422, $- 42 m$
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{12}$ , $R_{31}$ , $R_{32}$ , $r_{41}$ , $r_{52}$
	$x_{2}$	$R_{11}$ , $R_{22}$ , $R_{31}$ , $R_{32}$
	$x_{3}$	$R_{11}$ , $R_{22}$ , $R_{21}$ , $R_{12}$
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{11}$ , $R_{32}$ , $R_{31}$
	$x_{2}$	$R_{12}$ , $R_{21}$ , $R_{32}$ , $R_{31}$ , $r_{52}$ , $r_{41}$
	$x_{3}$	$R_{12}$ , $R_{21}$ , $R_{22}$ , $R_{11}$
$E_{3}$	$x_{1}$	$R_{23}$ , $R_{13}$ , $R_{33}$
	$x_{2}$	$R_{13}$ , $R_{23}$ , $R_{33}$
	$x_{3}$	$r_{63}$ (for $- 42 m$ )
$E_{2}$ , $E_{3}$	$x_{1}$	$R_{44}$ , $R_{55}$
$E_{1}$ , $E_{3}$	$x_{2}$	$R_{55}$ , $R_{44}$
$E_{1}$ , $E_{2}$	$x_{3}$	$R_{66}$

E	k	Electro-optic Coefficients
Tetragonal 4mm
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{12}$ , $R_{31}$ , $R_{32}$
	$x_{2}$	$R_{11}$ , $R_{22}$ , $R_{31}$ , $R_{32}$ , $r_{51}$ , $r_{42}$
	$x_{3}$	$R_{11}$ , $R_{22}$ , $R_{21}$ , $R_{12}$
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{11}$ , $R_{32}$ , $R_{31}$ , $r_{42}$ , $r_{51}$
	$x_{2}$	$R_{12}$ , $R_{21}$ , $R_{32}$ , $R_{31}$
	$x_{3}$	$R_{12}$ , $R_{21}$ , $R_{22}$ , $R_{11}$
$E_{1}$ , $E_{2}$	$x_{3}$	$R_{66}$
Tetragonal $4, - 4$
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{12}$ , $R_{31}$ , $R_{32}$ , $r_{41}$ , $r_{52}$
	$x_{2}$	$R_{11}$ , $R_{22}$ , $R_{31}$ , $R_{32}$ , $r_{51}$ , $r_{42}$
	$x_{3}$	$R_{11}$ , $R_{22}$ , $R_{21}$ , $R_{12}$ , $R_{61}$ , $R_{62}$
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{11}$ , $R_{32}$ , $R_{31}$ , $r_{51}$ , $r_{42}$
	$x_{2}$	$R_{12}$ , $R_{21}$ , $R_{32}$ , $R_{31}$ , $r_{52}$ , $r_{41}$
	$x_{3}$	$R_{12}$ , $R_{21}$ , $R_{22}$ , $R_{11}$ , $R_{61}$ , $R_{62}$
$E_{1}$ , $E_{2}$	$x_{3}$	$R_{16}$ , $R_{26}$ , $R_{66}$
Anaxial Noncentrosymmetric
E	k	Electro-optic Coefficients
Cubic 23, $- 43 m$
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{31}$ , $R_{32}$ , $R_{12}$ , $R_{13}$ , $R_{23}$ , $r_{41}$ , $r_{52}$ , $r_{63}$
	$x_{2}$	$R_{11}$ , $R_{31}$ , $R_{22}$ , $R_{33}$ , $R_{12}$ , $R_{23}$
	$x_{3}$	$R_{11}$ , $R_{12}$ , $R_{22}$ , $R_{33}$ , $R_{32}$ , $R_{13}$
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{32}$ , $R_{11}$ , $R_{33}$ , $R_{21}$ , $R_{13}$
	$x_{2}$	$R_{12}$ , $R_{32}$ , $R_{31}$ , $R_{23}$ , $R_{21}$ , $R_{13}$ , $r_{52}$ , $r_{41}$ , $r_{63}$
	$x_{3}$	$R_{12}$ , $R_{22}$ , $R_{31}$ , $R_{23}$ , $R_{11}$ , $R_{33}$
$E_{3}$	$x_{1}$	$R_{23}$ , $R_{33}$ , $R_{31}$ , $R_{12}$ , $R_{11}$ , $R_{22}$
	$x_{2}$	$R_{13}$ , $R_{33}$ , $R_{21}$ , $R_{32}$ , $R_{11}$ , $R_{22}$
	$x_{3}$	$R_{13}$ , $R_{23}$ , $R_{21}$ , $R_{32}$ , $R_{31}$ , $R_{12}$ , $r_{63}$ , $r_{52}$ , $r_{41}$
$E_{2}$ , $E_{3}$	$x_{1}$	$R_{44}$ , $R_{55}$ , $R_{66}$
$E_{1}$ , $E_{3}$	$x_{2}$	$R_{55}$ , $R_{44}$ , $R_{66}$
$E_{1}$ , $E_{2}$	$x_{3}$	$R_{66}$ , $R_{55}$ , $R_{44}$

E	k	Electro-optic Coefficients
Monoclinic m
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{32}$ , $r_{42}$
	$x_{2}$	$R_{12}$ , $R_{32}$ , $R_{52}$
	$x_{3}$	$R_{12}$ , $R_{22}$ , $r_{62}$
Monoclinic 2
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{31}$ , $r_{41}$
	$x_{2}$	$R_{11}$ , $R_{31}$ , $R_{51}$
	$x_{3}$	$R_{21}$ , $R_{22}$ , $r_{61}$
$E_{3}$	$x_{1}$	$R_{23}$ , $R_{33}$ , $r_{43}$
	$x_{2}$	$R_{13}$ , $R_{33}$ , $R_{53}$
	$x_{3}$	$R_{13}$ , $R_{23}$ , $r_{63}$
$E_{1}$ , $E_{3}$	$x_{1}$	$R_{25}$ , $R_{35}$
	$x_{2}$	$R_{15}$ , $R_{35}$ , $R_{55}$
	$x_{3}$	$R_{15}$ , $R_{25}$
Orthorhombic 222
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{31}$ , $r_{41}$
	$x_{2}$	$R_{11}$ , $R_{31}$
	$x_{3}$	$R_{11}$ , $R_{21}$
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{32}$
	$x_{2}$	$R_{12}$ , $R_{32}$ , $r_{52}$
	$x_{3}$	$R_{12}$ , $R_{22}$
$E_{3}$	$x_{1}$	$R_{23}$ , $R_{33}$
	$x_{2}$	$R_{13}$ , $R_{33}$
	$x_{3}$	$R_{13}$ , $R_{33}$ , $r_{63}$
$E_{2}$ , $E_{3}$	$x_{1}$	$R_{44}$
$E_{1}$ , $E_{3}$	$x_{2}$	$R_{55}$
$E_{1}$ , $E_{2}$	$x_{3}$	$R_{66}$
Orthorhombic mm2
$E_{1}$	$x_{1}$	$R_{21}$ , $R_{31}$
	$x_{2}$	$R_{11}$ , $R_{31}$ , $r_{51}$
	$x_{3}$	$R_{11}$ , $R_{21}$
$E_{2}$	$x_{1}$	$R_{22}$ , $R_{32}$ , $r_{42}$
	$x_{2}$	$R_{12}$ , $R_{32}$
	$x_{3}$	$R_{12}$ , $R_{22}$
$E_{1}$ , $E_{2}$	$x_{3}$	$R_{66}$

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Equations (24)

Journal of the Optical Society of America A