Noncritical acousto-optic Bragg phase matching: analysis of orthorhombic and monoclinic crystal systems

Konstantin B. Yushkov

doi:10.1364/AO.427199

Applied Optics
Vol. 60,
Issue 24,
pp. 7113-7121
(2021)
•https://doi.org/10.1364/AO.427199

Noncritical acousto-optic Bragg phase matching: analysis of orthorhombic and monoclinic crystal systems

Konstantin B. Yushkov

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Konstantin B. Yushkov, "Noncritical acousto-optic Bragg phase matching: analysis of orthorhombic and monoclinic crystal systems," Appl. Opt. 60, 7113-7121 (2021)

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Table of Contents Category
- Optical Devices, Sensors, and Detectors
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About this Article
History
- Original Manuscript: April 27, 2021
- Revised Manuscript: July 4, 2021
- Manuscript Accepted: July 4, 2021
- Published: August 12, 2021

Abstract

Anisotropic acousto-optic diffraction in crystals is the fundamental phenomenon that is used to design acousto-optic tunable filters. Noncritical and quasicollinear phase-matching geometries of Bragg acousto-optic diffraction are compared for acoustic symmetry planes in four crystal systems (tetragonal, trigonal, orthorhombic, and monoclinic). The results for uniaxial crystals are reviewed and generalized for biaxial crystals. It is shown that cubic frequency dependence on the Bragg angle exists in two symmetry planes of orthorhombic crystals and conditionally exists in the symmetry plane of monoclinic crystals. It is also shown that there are two points in the symmetry plane of monoclinic crystals where noncritical phase matching takes place in quasicollinear diffraction geometry that can be used to design high-resolution tunable filters. Phase-matching configurations in $\alpha$-iodic acid and potassium gadolinium tungstate crystals are analyzed.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.

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Birefringence Type	Uniaxial				Biaxial
Crystal system	Tetragonal			Trigonal		Orthorhombic		Monoclinic
Acoustic symmetry	4/m or $4 / m m m$ ^a			$\bar{3}$ or $\bar{3} m$ ^b		mmm		2/m
Crystals	${T e O}_{2}$ ,	KDP	${P b M o O}_{4}$ ,	${L i N b O}_{3}$ ,		${H I O}_{3}$ , ${T l}_{3} {A s S}_{4}$		$K G d ({W O}_{4})_{2}$ ,
	${H g}_{2} {C l}_{2}$		${C a M o O}_{4}$	${S i O}_{2}$				$K Y ({W O}_{4})_{2}$
Interaction plane	(110)	(100)	$X + φ_{0}$	(100)	(100)	$(001)$ ^c	(010)	(010)
Dielectric axes	$x_{1} x_{3}$	$x_{1} x_{3}$	$x_{1} x_{3}$	$x_{1} x_{3}$	$x_{2} x_{1}$	$x_{2} x_{3}$	$x_{3} x_{1}$	$x_{2} x_{3}$
Stiffness Coefficients, Eq. (4)
$c_{xx}$ [ $\times \cos^{2} α$ ]	$(c_{11} - c_{12}) / 2$	$c_{66}$	$c_{66}^{*}$	$c_{66}$	$c_{66}$	$c_{44}$	$c_{66}$	$c_{66}$
$c_{xy}$ [ $\times 2 \sin α \cos α$ ]	–	–	–	$c_{14}^{*}$	–	–	–	$c_{46}$
$c_{yy}$ [ $\times \sin^{2} α$ ]	$c_{44}$	$c_{44}$	$c_{44}$	$c_{44}$	$c_{55}$	$c_{55}$	$c_{44}$	$c_{44}$
Piezoelectric Coefficients, Eq. (5) ^d
$h_{xx}$ [ $\times \cos^{2} α$ ]	–	–	–	–	–	–	–	$e_{16}$
$h_{xy}$ [ $\times \sin α \cos α$ ]	–	–	–	–	$e_{25} + e_{36}$	$e_{14} + e_{25}$	$e_{14} + e_{36}$	$e_{14} + e_{36}$
$h_{xx}$ [ $\times \sin^{2} α$ ]	–	–	–	–	–	–	–	$e_{34}$
Photo-Elastic Constants, Eq. (8)
$p_{xx}$ [ $\times \sin ϑ \cos α$ ]	–	–	–	$p_{41}^{*}$	–	–	–	$p_{46}$
$p_{xy}$ [ $\times \sin ϑ \sin α$ ]	$p_{44}$	$p_{44}$	$p_{44}$	$p_{44}$	$p_{55}$	$p_{55}$	$p_{44}$	$p_{44}$
$p_{y x}$ [ $\times \cos ϑ \cos α$ ]	$(p_{11} - p_{12}) / 2$	$p_{66}$	$p_{66}^{*}$	$p_{66}^{*}$	$p_{66}$	$p_{44}$	$p_{66}$	$p_{66}$
$p_{yy}$ [ $\times \cos ϑ \sin α$ ]	–	–	–	$p_{14}^{*}$	–	–	–	$p_{64}$

		Curvature Radius, $r \times (λ / 2 π)$
Axis	Direction	Fast Mode	Sign	Slow Mode
$x_{1}$	$x_{2}$	$n_{1}^{2} / n_{2}$	$<$	$n_{3}$
	$x_{3}$	$n_{2}$	$>$	$n_{1}^{2} / n_{3}$
$x_{2}$	$x_{3}$	$n_{1}$	$<$ or $>$	$n_{2}^{2} / n_{3}$
	$x_{1}$	$n_{2}^{2} / n_{1}$	$>$ or $<$	$n_{3}$
$x_{3}$	$x_{1}$	$n_{3}^{2} / n_{1}$	$>$	$n_{2}$
	$x_{2}$	$n_{1}$	$<$	$n_{3}^{2} / n_{2}$

Birefringence Type	Uniaxial				Biaxial
Crystal system	Tetragonal			Trigonal		Orthorhombic		Monoclinic
Acoustic symmetry	4/m or $4 / m m m$ ^a			$\bar{3}$ or $\bar{3} m$ ^b		mmm		2/m
Crystals	${T e O}_{2}$ ,	KDP	${P b M o O}_{4}$ ,	${L i N b O}_{3}$ ,		${H I O}_{3}$ , ${T l}_{3} {A s S}_{4}$		$K G d ({W O}_{4})_{2}$ ,
	${H g}_{2} {C l}_{2}$		${C a M o O}_{4}$	${S i O}_{2}$				$K Y ({W O}_{4})_{2}$
Interaction plane	(110)	(100)	$X + φ_{0}$	(100)	(100)	$(001)$ ^c	(010)	(010)
Dielectric axes	$x_{1} x_{3}$	$x_{1} x_{3}$	$x_{1} x_{3}$	$x_{1} x_{3}$	$x_{2} x_{1}$	$x_{2} x_{3}$	$x_{3} x_{1}$	$x_{2} x_{3}$
Stiffness Coefficients, Eq. (4)
$c_{xx}$ [ $\times \cos^{2} α$ ]	$(c_{11} - c_{12}) / 2$	$c_{66}$	$c_{66}^{*}$	$c_{66}$	$c_{66}$	$c_{44}$	$c_{66}$	$c_{66}$
$c_{xy}$ [ $\times 2 \sin α \cos α$ ]	–	–	–	$c_{14}^{*}$	–	–	–	$c_{46}$
$c_{yy}$ [ $\times \sin^{2} α$ ]	$c_{44}$	$c_{44}$	$c_{44}$	$c_{44}$	$c_{55}$	$c_{55}$	$c_{44}$	$c_{44}$
Piezoelectric Coefficients, Eq. (5) ^d
$h_{xx}$ [ $\times \cos^{2} α$ ]	–	–	–	–	–	–	–	$e_{16}$
$h_{xy}$ [ $\times \sin α \cos α$ ]	–	–	–	–	$e_{25} + e_{36}$	$e_{14} + e_{25}$	$e_{14} + e_{36}$	$e_{14} + e_{36}$
$h_{xx}$ [ $\times \sin^{2} α$ ]	–	–	–	–	–	–	–	$e_{34}$
Photo-Elastic Constants, Eq. (8)
$p_{xx}$ [ $\times \sin ϑ \cos α$ ]	–	–	–	$p_{41}^{*}$	–	–	–	$p_{46}$
$p_{xy}$ [ $\times \sin ϑ \sin α$ ]	$p_{44}$	$p_{44}$	$p_{44}$	$p_{44}$	$p_{55}$	$p_{55}$	$p_{44}$	$p_{44}$
$p_{y x}$ [ $\times \cos ϑ \cos α$ ]	$(p_{11} - p_{12}) / 2$	$p_{66}$	$p_{66}^{*}$	$p_{66}^{*}$	$p_{66}$	$p_{44}$	$p_{66}$	$p_{66}$
$p_{yy}$ [ $\times \cos ϑ \sin α$ ]	–	–	–	$p_{14}^{*}$	–	–	–	$p_{64}$

		Curvature Radius, $r \times (λ / 2 π)$
Axis	Direction	Fast Mode	Sign	Slow Mode
$x_{1}$	$x_{2}$	$n_{1}^{2} / n_{2}$	$<$	$n_{3}$
	$x_{3}$	$n_{2}$	$>$	$n_{1}^{2} / n_{3}$
$x_{2}$	$x_{3}$	$n_{1}$	$<$ or $>$	$n_{2}^{2} / n_{3}$
	$x_{1}$	$n_{2}^{2} / n_{1}$	$>$ or $<$	$n_{3}$
$x_{3}$	$x_{1}$	$n_{3}^{2} / n_{1}$	$>$	$n_{2}$
	$x_{2}$	$n_{1}$	$<$	$n_{3}^{2} / n_{2}$

Abstract

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Tables (2)

Equations (37)

Applied Optics