Photoelastic properties of NaBi(MoO<sub>4</sub>)<sub>2</sub> crystals

Oleg Krupych; Marian Kushnirevych; Oksana Mys; Rostyslav Vlokh

doi:10.1364/AO.54.005016

Applied Optics
Vol. 54,
Issue 16,
pp. 5016-5023
(2015)
•https://doi.org/10.1364/AO.54.005016

Photoelastic properties of NaBi(MoO₄)₂ crystals

Oleg Krupych, Marian Kushnirevych, Oksana Mys, and Rostyslav Vlokh

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Oleg Krupych, Marian Kushnirevych, Oksana Mys, and Rostyslav Vlokh, "Photoelastic properties of NaBi(MoO₄)₂ crystals," Appl. Opt. 54, 5016-5023 (2015)

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Abstract

In this work we present the results of comprehensive studies of the piezo-optic and elasto-optic effects in $NaBi {({MoO}_{4})}_{2}$ crystals. The full matrices of the piezo-optic and elasto-optic coefficients for the $NaBi {({MoO}_{4})}_{2}$ crystals are obtained for the first time, to the best of our knowledge. The acousto-optic effect in the $NaBi {({MoO}_{4})}_{2}$ is analyzed for the case of interactions of optical waves with the slowest acoustic waves. It is shown that the crystals under analysis can be efficiently used as working elements for the collinear acousto-optic tunable filters.

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	Long Axis		Side Axis 1		Side Axis 2
Sample Number	Direction	Size (mm)	Direction	Size (mm)	Direction	Size (mm)
1	[100]	19.4	[010]	3.28	[001]	3.33
2	[001]	16.4	[100]	3.24	[010]	3.18
3	$(0; \frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	19.3	[100]	3.33	$(0; \frac{- 1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	3.17
4	[110]	19.8	[001]	3.28	$[1 \bar{1} 0]$	3.20
5	[210]	19.7	[001]	3.32	$[1 \bar{2} 0]$	3.29
6	$(\frac{1}{\sqrt{2}}; \frac{1}{2}; \frac{1}{2})$ ^a	19.3	$(0; \frac{- 1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	3.18	$(\frac{- 1}{\sqrt{2}}; \frac{1}{2}; \frac{1}{2})$ ^a	3.33

Sample #	$Π$	Stress Component	Light Propagation Direction	$S^{eff}$	Light Polarization Direction	Refractive Index
1	$Π_{1 A} = π_{11}$	$σ_{1}$	[010]	$S_{21} = S_{12}$	[100]	$n_{o}$
1	$Π_{1 B} = π_{31}$	$σ_{1}$	[010]	$S_{21} = S_{12}$	[001]	$n_{e}$
2	$Π_{2 A} = π_{33}$	$σ_{3}$	[010]	$S_{23} = S_{13}$	[001]	$n_{e}$
	$Π_{2 B} = π_{13}$	$σ_{3}$	[010]	$S_{23} = S_{13}$	[100]	$n_{o}$
	$Π_{2 C} = π_{33}$	$σ_{3}$	[100]	$S_{13}$	[001]	$n_{e}$
	$Π_{2 D} = π_{23} = π_{13}$	$σ_{3}$	[100]	$S_{13}$	[010]	$n_{o}$
3	$Π_{3 A} = (π_{11} + π_{13} + π_{31} + π_{33} + 2 π_{44}) / 4$	$σ_{2}$ , $σ_{3}$ , $σ_{4}$	$(0; \frac{- 1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	$(S_{11} + 2 S_{13} + S_{33} - S_{44}) / 4$	$(0; \frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	$n_{45}$ ^b
	$Π_{3 B} = (π_{12} + π_{13}) / 2$	$σ_{2}$ , $σ_{3}$ , $σ_{4}$	$(0; \frac{- 1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	$(S_{11} + 2 S_{13} + S_{33} - S_{44}) / 4$	[100]	$n_{o}$
	$Π_{3 C} = (π_{11} + π_{13}) / 2$	$σ_{2}$ , $σ_{3}$ , $σ_{4}$	[100]	$(S_{12} + S_{13}) / 2$	[010]	$n_{o}$
	$Π_{3 D} = (π_{31} + π_{33}) / 2$	$σ_{2}$ , $σ_{3}$ , $σ_{4}$	[100]	$(S_{12} + S_{13}) / 2$	[001]	$n_{e}$
4	$Π_{4 A} = (π_{11} + π_{12} + π_{66}) / 2$	$σ_{1}$ , $σ_{2}$ , $σ_{6}$	$[1 \bar{1} 0]$	$(2 S_{11} + 2 S_{12} - S_{66}) / 4$	[110]	$n_{o}$
4	$Π_{4 B} = π_{31}$	$σ_{1}$ , $σ_{2}$ , $σ_{6}$	$[1 \bar{1} 0]$	$(2 S_{11} + 2 S_{12} - S_{66}) / 4$	[001]	$n_{e}$
5	$Π_{5 A} = (17 π_{11} + 8 π_{12} + 8 π_{66} - 6 π_{16} - 12 π_{61}) / 25$	$σ_{1}$ , $σ_{2}$ , $σ_{6}$	$[\bar{1} 20]$	$(8 S_{11} + 17 S_{12} + 12 S_{16} - 4 S_{66}) / 25$	[210]	$n_{o}$
5	$Π_{5 B} = π_{31}$	$σ_{1}$ , $σ_{2}$ , $σ_{6}$	$[\bar{1} 20]$	$(S_{11} + 2 S_{13} + S_{33} - S_{44}) / 4$	[001]	$n_{e}$
6	$Π_{6 A} = (2 π_{11} + π_{12} + π_{13} + \sqrt{2} π_{16}) / 4$	$σ_{1}$ , $σ_{2}$ , $σ_{3}$ , $σ_{4}$ , $σ_{5}$ , $σ_{6}$	$(0; \frac{- 1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	$(S_{11} + 2 S_{12} + 4 S_{13} + S_{33} - S_{44} - \sqrt{2} S_{16}) / 8$	[100]	$n_{o}$
6	$Π_{6 B} = [π_{11} + π_{13} + π_{33} + 3 π_{31} + 2 (π_{12} + π_{44}) + \sqrt{2} (2 π_{45} - π_{16})] / 8$	$σ_{1}$ , $σ_{2}$ , $σ_{3}$ , $σ_{4}$ , $σ_{5}$ , $σ_{6}$	$(0; \frac{- 1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	$(S_{11} + 2 S_{12} + 4 S_{13} + S_{33} - S_{44} - \sqrt{2} S_{16}) / 8$	$(0; \frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	$n_{45}$

	Direction			Effective Coefficient, B
Sample #	Stress	Light Propagation	Light Polarization	$Q$	$D$	$K$	$Π$
1	[100]	[010]	[100]	$- 14.487 \pm 0.201$	$- 8.191 \pm 0.055$	$- 6.296 \pm 0.209$	$1.024 \pm 0.034$
1	[100]	[010]	[001]	$- 11.615 \pm 0.169$	$- 7.508 \pm 0.051$	$- 4.106 \pm 0.177$	$0.772 \pm 0.033$
2	[001]	[010]	[001]	$- 15.083 \pm 0.209$	$- 3.394 \pm 0.022$	$- 11.689 \pm 0.211$	$2.199 \pm 0.040$
	[001]	[010]	[100]	$- 13.445 \pm 0.189$	$- 3.703 \pm 0.024$	$- 9.742 \pm 0.191$	$1.585 \pm 0.031$
	[001]	[100]	[001]	$- 15.199 \pm 0.213$	$- 3.394 \pm 0.022$	$- 11.805 \pm 0.214$	$2.220 \pm 0.040$
	[001]	[100]	[010]	$- 13.012 \pm 0.198$	$- 3.703 \pm 0.024$	$- 9.310 \pm 0.199$	$1.514 \pm 0.032$
3	$(0; \frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	$(0; \frac{- 1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	$(0; \frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	$- 15.242 \pm 0.212$	$- 5.592 \pm 0.043$	$- 9.650 \pm 0.217$	$1.691 \pm 0.038$
	$(0; \frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	$(0; \frac{- 1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	[100]	$- 12.313 \pm 0.173$	$- 5.845 \pm 0.045$	$- 6.468 \pm 0.179$	$1.052 \pm 0.029$
	$(0; \frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	[100]	[010]	$- 13.374 \pm 0.192$	$- 5.947 \pm 0.030$	$- 7.427 \pm 0.194$	$1.208 \pm 0.032$
	$(0; \frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	[100]	[001]	$- 12.824 \pm 0.190$	$- 5.451 \pm 0.028$	$- 7.373 \pm 0.192$	$1.387 \pm 0.036$
4	[110]	$[1 \bar{1} 0]$	[110]	$- 7.478 \pm 0.112$	$- 5.738 \pm 0.047$	$- 1.740 \pm 0.121$	$0.283 \pm 0.020$
4	[110]	$[1 \bar{1} 0]$	[001]	$- 7.098 \pm 0.117$	$- 5.260 \pm 0.043$	$- 1.838 \pm 0.124$	$0.346 \pm 0.023$
5	[210]	$[\bar{1} 20]$	[210]	$- 5.800 \pm 0.094$	$- 2.934 \pm 0.021$	$- 2.866 \pm 0.097$	$0.466 \pm 0.016$
5	[210]	$[\bar{1} 20]$	[001]	$- 4.726 \pm 0.084$	$- 2.690 \pm 0.020$	$- 2.036 \pm 0.086$	$0.383 \pm 0.016$
6	$(\frac{1}{\sqrt{2}}; \frac{1}{2}; \frac{1}{2})$ ^a	$(0; \frac{- 1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	[100]	$- 12.480 \pm 0.175$	$- 7.254 \pm 0.029$	$- 5.227 \pm 0.177$	$0.850 \pm 0.029$
6	$(\frac{1}{\sqrt{2}}; \frac{1}{2}; \frac{1}{2})$ ^a	$(0; \frac{- 1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	$(0; \frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	$- 15.183 \pm 0.206$	$- 6.940 \pm 0.028$	$- 8.243 \pm 0.208$	$1.444 \pm 0.037$

EOC	$p_{11}$	$p_{12}$	$p_{13}$	$p_{31}$	$p_{33}$	$p_{44}$	$p_{66}$	$p_{45}$	$p_{16}$	$p_{61}$
Present work	$0.196 \pm 0.005$	$0.159 \pm 0.007$	$0.189 \pm 0.003$	$0.154 \pm 0.002$	$0.228 \pm 0.003$	$0.019 \pm 0.002$	$- 0.044 \pm 0.004$	$0.017 \pm 0.004$	$- 0.023 \pm 0.004$	$0.035 \pm 0.012$
Data of Ref. [13]	0.243	0.205	0.25	0.21	0.29	—	—	—	—	—

	Long Axis		Side Axis 1		Side Axis 2
Sample Number	Direction	Size (mm)	Direction	Size (mm)	Direction	Size (mm)
1	[100]	19.4	[010]	3.28	[001]	3.33
2	[001]	16.4	[100]	3.24	[010]	3.18
3	$(0; \frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	19.3	[100]	3.33	$(0; \frac{- 1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	3.17
4	[110]	19.8	[001]	3.28	$[1 \bar{1} 0]$	3.20
5	[210]	19.7	[001]	3.32	$[1 \bar{2} 0]$	3.29
6	$(\frac{1}{\sqrt{2}}; \frac{1}{2}; \frac{1}{2})$ ^a	19.3	$(0; \frac{- 1}{\sqrt{2}}; \frac{1}{\sqrt{2}})$ ^a	3.18	$(\frac{- 1}{\sqrt{2}}; \frac{1}{2}; \frac{1}{2})$ ^a	3.33

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Equations (15)

Applied Optics