Reflection from noncentrosymmetric uniaxial crystals: a multipole approach

E. B. Graham; R. E. Raab

doi:10.1364/JOSAA.13.001239

Journal of the Optical Society of America A
Vol. 13,
Issue 6,
pp. 1239-1248
(1996)
•https://doi.org/10.1364/JOSAA.13.001239

Reflection from noncentrosymmetric uniaxial crystals: a multipole approach

E. B. Graham and R. E. Raab

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E. B. Graham and R. E. Raab, "Reflection from noncentrosymmetric uniaxial crystals: a multipole approach," J. Opt. Soc. Am. A 13, 1239-1248 (1996)

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Abstract

A multipole theory is used to describe reflection and refraction at the interface of an isotropic medium and a transparent noncentrosymmetric crystal. Both optically active and optically inactive crystals are considered. Fresnel reflection amplitudes, which satisfy the requirements of reciprocity, are obtained for the uniaxial symmetry classes when the optic axis is normal to the boundary and parallel to it. These amplitudes are related to the angle of incidence and also to macroscopic property tensors of the crystal within the electric quadrupole–magnetic dipole approximation. Experiments based on the differential reflection of orthogonal polarizations of the incident light beam are described for measuring certain crystal properties.

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Crystal Class	α_xx	α_yy	α_zz	A_xyx	A_xyz	A_yzx	A_yzz	A_xzx	A_xzz	T_xx	T_xy	T_xz	T_yx	T_yy	T_yz
(I) Optic axis perpendicular to the reflecting surface (i.e., 3∥z)^a
4, 3, 6	α₁₁	α₁₁	α₃₃	0	A₁	A₂	0	A₃	0	$\frac{1}{2} A_{1}$	−A₃	0	A₃	$\frac{1}{2} A_{1}$	0
422, 32, 622	α₁₁	α₁₁	α₃₃	0	A₁	A₂	0	0	0	$\frac{1}{2} A_{1}$	0	0	0	$\frac{1}{2} A_{1}$	0
$\bar{4}$	α₁₁	α₁₁	α₃₃	0	0	SA₃ − CA₄	0	CA₃ + SA₄	0	CA₄ − SA₃	CA₃ + SA₄	0	CA₃ + SA₄	SA₃ − CA₄	0
$\bar{4} 2 m$	α₁₁	α₁₁	α₃₃	0	0	−CA₄	0	SA₄	0	CA₄	SA₄	0	SA₄	−CA₄	0
4mm, 3m, 6mm	α₁₁	α₁₁	α₃₃	0	0	0	0	A₃	0	0	−A₃	0	A₃	0	0
$\bar{6}, \bar{6} m 2$	α₁₁	α₁₁	α₃₃	0	0	0	0	0	0	0	0	0	0	0	0
(II) Optic axis in reflecting surface and parallel to the plane of incidence (i.e., 3∥x)^b
4, 3, 6	α₃₃	α₁₁	α₁₁	0	A₂	A₁	0	0	∓A₃	$A_{2} - \frac{1}{2} A_{1}$	0	0	0	$\frac{1}{2} A_{1}$	∓A₃
422, 32, 622	α₃₃	α₁₁	α₁₁	0	A₂	A₁	0	0	0	$A_{2} - \frac{1}{2} A_{1}$	0	0	0	$\frac{1}{2} A_{1}$	0
$\bar{4}$	α₃₃	α₁₁	α₁₁	0	A₄	0	0	0	±A₃	0	0	0	0	A₄	±A₃
$\bar{4} 2 m$	α₃₃	α₁₁	α₁₁	0	A₄	0	0	0	0	0	0	0	0	A₄	0
4mm, 3m, 6mm	α₃₃	α₁₁	α₁₁	0	0	0	0	0	∓A₃	0	0	0	0	0	∓A₃
$\bar{6}, \bar{6} m 2$	α₃₃	α₁₁	α₁₁	0	0	0	0	0	0	0	0	0	0	0	0
(III) Optic axis perpendicular to the plane of incidence (i.e., 3∥y)^b
4, 3, 6	α₁₁	α₃₃	α₁₁	±A₃	A₂	A₂	∓A₃	0	0	$\frac{1}{2} A_{1}$	0	±A₃	0	$A_{2} - \frac{1}{2} A_{1}$	0
422, 32, 622	α₁₁	α₃₃	α₁₁	0	A₂	A₂	0	0	0	$\frac{1}{2} A_{1}$	0	0	0	$A_{2} - \frac{1}{2} A_{1}$	0
$\bar{4}$	α₁₁	α₃₃	α₁₁	±A₃	A₄	−A₄	±A₃	0	0	A₄	0	∓A₃	0	0	0
$\bar{4} 2 m$	α₁₁	α₃₃	α₁₁	0	A₄	−A₄	0	0	0	A₄	0	0	0	0	0
4mm, 3m, 6mm	α₁₁	α₃₃	α₁₁	±A₃	0	0	∓A₃	0	0	0	0	±A₃	0	0	0
$\bar{6}, \bar{6} m 2$	α₁₁	α₃₃	α₁₁	0	0	0	0	0	0	0	0	0	0	0	0

Polarization State	E_ip	E_is	Analyzed Intensity I
↔	1	0	$\frac{S_{0} + S_{1}}{4 μ_{0} c}$
↕	0	1	$\frac{S_{0} - S_{1}}{4 μ_{0} c}$
$⤢$	$1 / \sqrt{2}$	$1 / \sqrt{2}$	$\frac{S_{0} + S_{2}}{4 μ_{0} c}$
$⤡$	$1 / \sqrt{2}$	$- 1 / \sqrt{2}$	$\frac{S_{0} - S_{2}}{4 μ_{0} c}$
↻	$1 / \sqrt{2}$	$- i / \sqrt{2}$	$\frac{S_{0} + S_{3}}{4 μ_{0} c}$
↺	$1 / \sqrt{2}$	$i / \sqrt{2}$	$\frac{S_{0} - S_{3}}{4 μ_{0} c}$

Crystal Class	α_xx	α_yy	α_zz	A_xyx	A_xyz	A_yzx	A_yzz	A_xzx	A_xzz	T_xx	T_xy	T_xz	T_yx	T_yy	T_yz
(I) Optic axis perpendicular to the reflecting surface (i.e., 3∥z)^a
4, 3, 6	α₁₁	α₁₁	α₃₃	0	A₁	A₂	0	A₃	0	$\frac{1}{2} A_{1}$	−A₃	0	A₃	$\frac{1}{2} A_{1}$	0
422, 32, 622	α₁₁	α₁₁	α₃₃	0	A₁	A₂	0	0	0	$\frac{1}{2} A_{1}$	0	0	0	$\frac{1}{2} A_{1}$	0
$\bar{4}$	α₁₁	α₁₁	α₃₃	0	0	SA₃ − CA₄	0	CA₃ + SA₄	0	CA₄ − SA₃	CA₃ + SA₄	0	CA₃ + SA₄	SA₃ − CA₄	0
$\bar{4} 2 m$	α₁₁	α₁₁	α₃₃	0	0	−CA₄	0	SA₄	0	CA₄	SA₄	0	SA₄	−CA₄	0
4mm, 3m, 6mm	α₁₁	α₁₁	α₃₃	0	0	0	0	A₃	0	0	−A₃	0	A₃	0	0
$\bar{6}, \bar{6} m 2$	α₁₁	α₁₁	α₃₃	0	0	0	0	0	0	0	0	0	0	0	0
(II) Optic axis in reflecting surface and parallel to the plane of incidence (i.e., 3∥x)^b
4, 3, 6	α₃₃	α₁₁	α₁₁	0	A₂	A₁	0	0	∓A₃	$A_{2} - \frac{1}{2} A_{1}$	0	0	0	$\frac{1}{2} A_{1}$	∓A₃
422, 32, 622	α₃₃	α₁₁	α₁₁	0	A₂	A₁	0	0	0	$A_{2} - \frac{1}{2} A_{1}$	0	0	0	$\frac{1}{2} A_{1}$	0
$\bar{4}$	α₃₃	α₁₁	α₁₁	0	A₄	0	0	0	±A₃	0	0	0	0	A₄	±A₃
$\bar{4} 2 m$	α₃₃	α₁₁	α₁₁	0	A₄	0	0	0	0	0	0	0	0	A₄	0
4mm, 3m, 6mm	α₃₃	α₁₁	α₁₁	0	0	0	0	0	∓A₃	0	0	0	0	0	∓A₃
$\bar{6}, \bar{6} m 2$	α₃₃	α₁₁	α₁₁	0	0	0	0	0	0	0	0	0	0	0	0
(III) Optic axis perpendicular to the plane of incidence (i.e., 3∥y)^b
4, 3, 6	α₁₁	α₃₃	α₁₁	±A₃	A₂	A₂	∓A₃	0	0	$\frac{1}{2} A_{1}$	0	±A₃	0	$A_{2} - \frac{1}{2} A_{1}$	0
422, 32, 622	α₁₁	α₃₃	α₁₁	0	A₂	A₂	0	0	0	$\frac{1}{2} A_{1}$	0	0	0	$A_{2} - \frac{1}{2} A_{1}$	0
$\bar{4}$	α₁₁	α₃₃	α₁₁	±A₃	A₄	−A₄	±A₃	0	0	A₄	0	∓A₃	0	0	0
$\bar{4} 2 m$	α₁₁	α₃₃	α₁₁	0	A₄	−A₄	0	0	0	A₄	0	0	0	0	0
4mm, 3m, 6mm	α₁₁	α₃₃	α₁₁	±A₃	0	0	∓A₃	0	0	0	0	±A₃	0	0	0
$\bar{6}, \bar{6} m 2$	α₁₁	α₃₃	α₁₁	0	0	0	0	0	0	0	0	0	0	0	0

Polarization State	E_ip	E_is	Analyzed Intensity I
↔	1	0	$\frac{S_{0} + S_{1}}{4 μ_{0} c}$
↕	0	1	$\frac{S_{0} - S_{1}}{4 μ_{0} c}$
$⤢$	$1 / \sqrt{2}$	$1 / \sqrt{2}$	$\frac{S_{0} + S_{2}}{4 μ_{0} c}$
$⤡$	$1 / \sqrt{2}$	$- 1 / \sqrt{2}$	$\frac{S_{0} - S_{2}}{4 μ_{0} c}$
↻	$1 / \sqrt{2}$	$- i / \sqrt{2}$	$\frac{S_{0} + S_{3}}{4 μ_{0} c}$
↺	$1 / \sqrt{2}$	$i / \sqrt{2}$	$\frac{S_{0} - S_{3}}{4 μ_{0} c}$

Abstract

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Equations (84)

Journal of the Optical Society of America A