Numerical calculations of rotational contributions to the scattering coefficient in a piezoelectric LiTaO<sub>3</sub> crystal

Tomasz Błachowicz

doi:10.1364/JOSAB.15.002599

Journal of the Optical Society of America B
Vol. 15,
Issue 10,
pp. 2599-2606
(1998)
•https://doi.org/10.1364/JOSAB.15.002599

Numerical calculations of rotational contributions to the scattering coefficient in a piezoelectric LiTaO₃ crystal

Tomasz Błachowicz

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Tomasz Błachowicz, "Numerical calculations of rotational contributions to the scattering coefficient in a piezoelectric LiTaO₃ crystal," J. Opt. Soc. Am. B 15, 2599-2606 (1998)

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Abstract

A systematic theoretical analysis is presented of rotational contributions to light scattering in a piezoelectric crystal with small birefringence. The influence of rotational contributions on the intensity of nonelastically scattered light in a ${LiTaO}_{3}$ crystal—a trigonal system with $3 m$ symmetry—is described. Rotation affects the optical signals that result from two quasi-transverse waves. Experimental confirmation of such rotational contributions employed a formalism based on looking for eigenvectors and eigenvalues of a so-called characteristic matrix, which is a function of the direction of acoustic-wave propagation and the elastic constants of the medium, in this case modified by the piezoelectric effect. All calculations were performed (for both the hypersonic and the ultrasonic acoustic ranges) for 514.5-nm light. Velocities of acoustic waves in the hypersonic region were calculated under elastic constants taken from 20.9 to 38.5 GHz. Hypersonic data appear more relevant for investigating the effect that consists in measuring the changes of photon frequency scattered on acoustical phonons lying at the beginning of the first Brillouin zone.

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Angle (deg)	Eigenvalues of Characteristic Matrix for Quasi-Transverse Waves		Polarization Vector of First Quasi-Transverse Wave			Polarization Vector of Second Quasi-Transverse Wave
Angle (deg)	$X_{1} (Pa \times 10^{10})$	$X_{2} (Pa \times 10^{10})$	$γ_{1}^{(1)}$	$γ_{2}^{(1)}$	$γ_{3}^{(1)}$	$γ_{1}^{(2)}$	$γ_{2}^{(2)}$	$γ_{3}^{(2)}$
118	9.33	8.79	0.87198	0.43831	-0.21800	0.42652	-0.89880	-0.10110
119	9.29	8.78	0.86863	0.45204	-0.20290	0.44131	-0.89200	-0.09800
126	9.06	8.77	0.82280	0.55880	-0.10355	0.55466	-0.82930	-0.06800
127	9.03	8.77	0.81300	0.57520	-0.09052	0.57185	-0.81800	-0.06220
128	9.00	8.78	0.80232	0.59180	-0.07780	0.58914	-0.80610	-0.05600
129	8.97	8.78	0.79087	0.60847	-0.06550	0.60647	-0.79360	-0.04940
130	8.95	8.79	0.77863	0.62519	-0.05360	0.62376	-0.78050	-0.04230
139	8.93	8.80	0.76560	0.64193	-0.04200	0.64100	-0.76680	-0.03470
140	8.95	8.79	0.77859	0.62530	-0.05351	0.62380	-0.78040	-0.04230
141	8.97	8.78	0.79081	0.60860	-0.06545	0.60660	-0.79350	-0.04930
142	9.00	8.78	0.80220	0.59191	-0.07770	0.58930	-0.80600	-0.05590
143	9.03	8.77	0.81288	0.57540	-0.09041	0.57200	-0.81790	-0.06210
144	9.06	8.77	0.82270	0.55900	-0.10340	0.55480	-0.82920	-0.06790

Angle (deg)	Frequencies of Quasi- Transverse Waves		Scattering Coefficient for First Quasi-Transverse Wave, $R_{1}$ (1/m)	Scattering Coefficient for Second Quasi-Transverse Wave, $R_{2}$ (1/m)	Quotient of Scattering Coefficient without Rotational Contributions, $(R_{1} / R_{2})^{ncor}$	Quotient of Scattering Coefficient with Rotational Contributions, $(R_{1} / R_{2})^{cor}$	Difference between Quotients, $\| (R_{1} / R_{2})^{ncor} - (R_{1} / R_{2})^{cor} \|$
Angle (deg)	$f_{1}$ (GHz)	$f_{2}$ (GHz)
118	21.50	20.87	$1.60 \times 10^{- 8}$	$6.92 \times 10^{- 8}$	0.24	0.23	0.01
119	21.46	20.87	$1.23 \times 10^{- 8}$	$6.81 \times 10^{- 8}$	0.19	0.18	0.01
126	21.20	20.86	$9.98 \times 10^{- 9}$	$3.69 \times 10^{- 8}$	0.23	0.27	0.04
127	21.16	20.86	$2.09 \times 10^{- 8}$	$3.06 \times 10^{- 8}$	0.61	0.68	0.07
128	21.13	20.87	$3.73 \times 10^{- 8}$	$2.44 \times 10^{- 8}$	1.40	1.53	0.13
129	21.10	20.88	$5.99 \times 10^{- 8}$	$1.86 \times 10^{- 8}$	2.98	3.22	0.24
130	21.07	20.89	$8.98 \times 10^{- 8}$	$1.33 \times 10^{- 8}$	6.31	6.77	0.46
139	21.05	20.90	$2.24 \times 10^{- 7}$	$3.68 \times 10^{- 8}$	5.81	6.09	0.28
140	21.08	20.89	$1.81 \times 10^{- 7}$	$6.22 \times 10^{- 8}$	2.77	2.92	0.15
141	21.11	20.88	$1.40 \times 10^{- 7}$	$9.66 \times 10^{- 8}$	1.36	1.45	0.09
142	21.14	20.88	$1.00 \times 10^{- 7}$	$1.41 \times 10^{- 7}$	0.66	0.71	0.05
143	21.17	20.87	$6.53 \times 10^{- 8}$	$1.97 \times 10^{- 7}$	0.30	0.33	0.03
144	21.21	20.87	$3.61 \times 10^{- 8}$	$2.66 \times 10^{- 7}$	0.12	0.14	0.02

Angle (deg)	Frequencies of Quasi-Transverse waves		Scattering Coefficient for First Quasi-Transverse Wave, $R_{1}$ (1/m)	Scattering Coefficient for Second Quasi-Transverse Wave, $R_{2}$ (1/m)	Quotient of Scattering Coefficient without Rotational Contributions, $(R_{1} / R_{2})^{ncor}$	Quotient of Scattering Coefficient with Rotational Contributions, $(R_{1} / R_{2})^{cor}$	Difference between Quotients, $\| (R_{1} / R_{2})^{ncor} - (R_{1} / R_{2})^{cor} \|$
Angle (deg)	$f_{1}$ (GHz)	$f_{2}$ (GHz)
9	22.14	21.71	$6.27 \times 10^{- 7}$	$6.86 \times 10^{- 8}$	9.18	9.14	0.04
10	22.18	21.69	$5.69 \times 10^{- 7}$	$7.96 \times 10^{- 8}$	7.18	7.15	0.03
11	22.21	21.67	$5.16 \times 10^{- 7}$	$8.79 \times 10^{- 8}$	5.90	5.87	0.03
12	22.25	21.66	$4.66 \times 10^{- 7}$	$9.37 \times 10^{- 8}$	5.00	4.97	0.03
13	22.30	21.64	$4.20 \times 10^{- 7}$	$9.77 \times 10^{- 8}$	4.33	4.30	0.03
14	22.34	21.63	$3.78 \times 10^{- 7}$	$1.00 \times 10^{- 7}$	3.81	3.78	0.03
15	22.38	21.62	$3.39 \times 10^{- 7}$	$1.01 \times 10^{- 7}$	3.38	3.35	0.03
16	22.43	21.61	$3.03 \times 10^{- 7}$	$1.01 \times 10^{- 7}$	3.02	2.99	0.03
17	22.48	21.61	$2.70 \times 10^{- 7}$	$1.00 \times 10^{- 7}$	2.72	2.69	0.03
73	22.48	21.61	$2.52 \times 10^{- 8}$	$9.38 \times 10^{- 9}$	2.72	2.69	0.03
74	22.43	21.61	$2.49 \times 10^{- 8}$	$8.33 \times 10^{- 9}$	3.02	2.99	0.03
75	22.38	21.62	$2.43 \times 10^{- 8}$	$7.27 \times 10^{- 9}$	3.38	3.35	0.03
76	22.34	21.63	$2.35 \times 10^{- 8}$	$6.22 \times 10^{- 9}$	3.81	3.78	0.03
77	22.30	21.64	$2.24 \times 10^{- 8}$	$5.21 \times 10^{- 9}$	4.33	4.30	0.03
78	22.25	21.66	$2.11 \times 10^{- 8}$	$4.24 \times 10^{- 9}$	5.00	4.97	0.03
79	22.21	21.67	$1.95 \times 10^{- 8}$	$3.32 \times 10^{- 9}$	5.90	5.87	0.03
80	22.18	21.69	$1.77 \times 10^{- 8}$	$2.48 \times 10^{- 9}$	7.18	7.15	0.03
81	22.14	21.71	$1.57 \times 10^{- 8}$	$1.72 \times 10^{- 9}$	9.18	9.14	0.04
110	22.56	21.75	$2.40 \times 10^{- 7}$	$3.75 \times 10^{- 8}$	6.35	6.39	0.04
111	22.59	21.74	$2.75 \times 10^{- 7}$	$3.66 \times 10^{- 8}$	7.46	7.51	0.05
112	22.61	21.73	$3.12 \times 10^{- 7}$	$3.52 \times 10^{- 8}$	8.82	8.88	0.06
158	22.61	21.73	$1.91 \times 10^{- 6}$	$2.15 \times 10^{- 7}$	8.82	8.88	0.06
159	22.59	21.74	$1.86 \times 10^{- 6}$	$2.48 \times 10^{- 7}$	7.46	7.51	0.05
160	22.56	21.75	$1.81 \times 10^{- 6}$	$2.83 \times 10^{- 7}$	6.35	6.39	0.04

Angle (deg)	Frequencies of Quasi-Transverse Waves		Scattering Coefficient for First Quasi-Transverse Wave, $R_{1}$ (1/m)	Scattering Coefficient for Second Quasi-Transverse Wave, $R_{2}$ (1/m)	Quotient of Scattering Coefficient without Rotational Contributions, $(R_{1} / R_{2})^{ncor}$	Quotient of Scattering Coefficient with Rotational Contributions, $(R_{1} / R_{2})^{cor}$	Difference between Quotients, $\| (R_{1} / R_{2})^{ncor} - (R_{1} / R_{2})^{cor} \|$
Angle (deg)	$f_{1}$ (GHz)	$f_{2}$ (GHz)
0	22.06	21.80	$9.37 \times 10^{- 6}$	$1.61 \times 10^{- 5}$	0.59	0.58	0.01
90	22.06	21.80	$2.01 \times 10^{- 14}$	$3.45 \times 10^{- 14}$	0.59	0.58	0.01
91	22.08	21.79	$3.27 \times 10^{- 9}$	$4.41 \times 10^{- 9}$	0.76	0.74	0.01
92	22.10	21.79	$1.44 \times 10^{- 8}$	$1.63 \times 10^{- 8}$	0.90	0.88	0.02
93	22.13	21.78	$3.43 \times 10^{- 8}$	$3.46 \times 10^{- 8}$	1.01	0.99	0.02
94	22.15	21.78	$6.30 \times 10^{- 8}$	$5.90 \times 10^{- 8}$	1.09	1.07	0.02
95	22.18	21.77	$1.00 \times 10^{- 7}$	$8.98 \times 10^{- 8}$	1.13	1.11	0.02
96	22.21	21.77	$1.45 \times 10^{- 7}$	$1.28 \times 10^{- 7}$	1.15	1.13	0.02
97	22.23	21.77	$1.95 \times 10^{- 7}$	$1.74 \times 10^{- 7}$	1.14	1.12	0.02
98	22.26	21.76	$2.51 \times 10^{- 7}$	$2.29 \times 10^{- 7}$	1.11	1.10	0.02
99	22.29	21.76	$3.10 \times 10^{- 7}$	$2.95 \times 10^{- 7}$	1.07	1.05	0.02
100	22.32	21.76	$3.71 \times 10^{- 7}$	$3.73 \times 10^{- 7}$	1.01	1.00	0.01
101	22.34	21.76	$4.33 \times 10^{- 7}$	$4.64 \times 10^{- 7}$	0.95	0.93	0.01
102	22.37	21.76	$4.93 \times 10^{- 7}$	$5.70 \times 10^{- 7}$	0.88	0.86	0.01
103	22.39	21.76	$5.49 \times 10^{- 7}$	$6.93 \times 10^{- 7}$	0.80	0.79	0.01
167	22.39	21.76	$1.03 \times 10^{- 5}$	$1.30 \times 10^{- 5}$	0.80	0.79	0.01
168	22.37	21.76	$1.09 \times 10^{- 5}$	$1.26 \times 10^{- 5}$	0.88	0.86	0.01
169	22.34	21.76	$1.15 \times 10^{- 5}$	$1.23 \times 10^{- 5}$	0.95	0.93	0.01
170	22.32	21.76	$1.20 \times 10^{- 5}$	$1.20 \times 10^{- 5}$	1.01	1.00	0.01
171	22.29	21.76	$1.24 \times 10^{- 5}$	$1.18 \times 10^{- 5}$	1.07	1.05	0.02
172	22.26	21.76	$1.27 \times 10^{- 5}$	$1.16 \times 10^{- 5}$	1.11	1.10	0.02
173	22.23	21.77	$1.30 \times 10^{- 5}$	$1.15 \times 10^{- 5}$	1.14	1.12	0.02
174	22.21	21.77	$1.31 \times 10^{- 5}$	$1.16 \times 10^{- 5}$	1.15	1.13	0.02
175	22.18	21.77	$1.31 \times 10^{- 5}$	$1.17 \times 10^{- 5}$	1.13	1.11	0.02
176	22.15	21.78	$1.29 \times 10^{- 5}$	$1.21 \times 10^{- 5}$	1.09	1.07	0.02
177	22.13	21.78	$1.25 \times 10^{- 5}$	$1.26 \times 10^{- 5}$	1.01	0.99	0.02
178	22.10	21.79	$1.18 \times 10^{- 5}$	$1.34 \times 10^{- 5}$	0.90	0.88	0.02
179	22.08	21.79	$1.08 \times 10^{- 5}$	$1.46 \times 10^{- 5}$	0.76	0.74	0.01
180	22.06	21.80	$9.37 \times 10^{- 6}$	$1.61 \times 10^{- 5}$	0.59	0.58	0.01

Constant	Experimental Frequency of Acoustic Wave or Range of Frequencies (GHz)	Measured Elastic Constants $(Pa \times 10^{10})$	Ultrasonic Data $(Pa \times 10^{10})$
$c_{11}$	$33.74 \pm 0.16$	$22.98 \pm 0.21$	23.3
$c_{33}$	$36.43 \pm 0.19$	$25.84 \pm 0.26$	27.5
$c_{44}$	$20.94 \pm 0.16$	$8.85 \pm 0.14$	9.4
$c_{66}$	$21.45 \pm 0.19$	$9.28 \pm 0.16$	9.3
$c_{12}$	(21.45–33.74)	$4.42 \pm 0.53$	4.7
$c_{14}$	(20.94–21.70)	$0.45 \pm 0.29$	-1.1
$c_{13}$	(20.94–36.43)	$5.12 \pm 0.45$	8.0

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Equations (27)

Journal of the Optical Society of America B