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 LiTaO3 crystal—a trigonal system with 3m 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.

© 1998 Optical Society of America

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References

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  1. T. Błachowicz and Z. Kleszczewski, “Influence of the rotational contribution to the scattering coefficient in piezoelectric crystals,” in Proceedings of the European Acoustics Association Symposium, 26th Winter School on Molecular and Quantum Acoustics, T. Blachowicz, ed. (Upper Silesian Division of the Polish Acoustical Society, Gliwice, Poland, 1997), pp. 13–16.
  2. T. Błachowicz, “Application of Brillouin light scattering to analysis of acoustic properties of piezoelectric crystals,” Ph.D. dissertation (Institute of Physics, Silesian Technical University, Gliwice, Poland, 1997).
  3. T. Błachowicz and Z. Kleszczewski, “Observation of hypersonic acoustic waves in a LiTaO3 crystal,” Arch. Acoust. 22, 351–357 (1997).
  4. D. F. Nelson and M. Lax, “New symmetry for acousto-optic scattering,” Phys. Rev. Lett. 24, 379–380 (1970).
    [CrossRef]
  5. D. F. Nelson and P. D. Lazay, “Measurement of the rotational contribution to Brillouin scattering,” Phys. Rev. Lett. 25, 1187–1191 (1970).
    [CrossRef]
  6. D. F. Nelson, P. D. Lazay, and M. Lax, “Rotational contribution to Brillouin scattering and higher order photoelastic interactions,” in Proceedings of the Second International Conference on Light Scattering in Solids, M. Balkanski, ed. (Flammarion, Paris, 1971), pp. 477–482.
  7. I. L. Fabielinskii, Molecular Scattering of Light (Plenum, New York, 1968).
  8. J. Xu and R. Stroud, Acousto-optic Devices: Principles, Design and Applications (Wiley, New York, 1992).
  9. T. Błachowicz and Z. Kleszczewski, “Elastic constants of the lithium tantalate crystal in the hypersonic range,” Acoust. Lett. 20, 221–223 (1997).

1997 (2)

T. Błachowicz and Z. Kleszczewski, “Observation of hypersonic acoustic waves in a LiTaO3 crystal,” Arch. Acoust. 22, 351–357 (1997).

T. Błachowicz and Z. Kleszczewski, “Elastic constants of the lithium tantalate crystal in the hypersonic range,” Acoust. Lett. 20, 221–223 (1997).

1970 (2)

D. F. Nelson and M. Lax, “New symmetry for acousto-optic scattering,” Phys. Rev. Lett. 24, 379–380 (1970).
[CrossRef]

D. F. Nelson and P. D. Lazay, “Measurement of the rotational contribution to Brillouin scattering,” Phys. Rev. Lett. 25, 1187–1191 (1970).
[CrossRef]

Blachowicz, T.

T. Błachowicz and Z. Kleszczewski, “Observation of hypersonic acoustic waves in a LiTaO3 crystal,” Arch. Acoust. 22, 351–357 (1997).

T. Błachowicz and Z. Kleszczewski, “Elastic constants of the lithium tantalate crystal in the hypersonic range,” Acoust. Lett. 20, 221–223 (1997).

Kleszczewski, Z.

T. Błachowicz and Z. Kleszczewski, “Elastic constants of the lithium tantalate crystal in the hypersonic range,” Acoust. Lett. 20, 221–223 (1997).

T. Błachowicz and Z. Kleszczewski, “Observation of hypersonic acoustic waves in a LiTaO3 crystal,” Arch. Acoust. 22, 351–357 (1997).

Lax, M.

D. F. Nelson and M. Lax, “New symmetry for acousto-optic scattering,” Phys. Rev. Lett. 24, 379–380 (1970).
[CrossRef]

Lazay, P. D.

D. F. Nelson and P. D. Lazay, “Measurement of the rotational contribution to Brillouin scattering,” Phys. Rev. Lett. 25, 1187–1191 (1970).
[CrossRef]

Nelson, D. F.

D. F. Nelson and P. D. Lazay, “Measurement of the rotational contribution to Brillouin scattering,” Phys. Rev. Lett. 25, 1187–1191 (1970).
[CrossRef]

D. F. Nelson and M. Lax, “New symmetry for acousto-optic scattering,” Phys. Rev. Lett. 24, 379–380 (1970).
[CrossRef]

Acoust. Lett. (1)

T. Błachowicz and Z. Kleszczewski, “Elastic constants of the lithium tantalate crystal in the hypersonic range,” Acoust. Lett. 20, 221–223 (1997).

Arch. Acoust. (1)

T. Błachowicz and Z. Kleszczewski, “Observation of hypersonic acoustic waves in a LiTaO3 crystal,” Arch. Acoust. 22, 351–357 (1997).

Phys. Rev. Lett. (2)

D. F. Nelson and M. Lax, “New symmetry for acousto-optic scattering,” Phys. Rev. Lett. 24, 379–380 (1970).
[CrossRef]

D. F. Nelson and P. D. Lazay, “Measurement of the rotational contribution to Brillouin scattering,” Phys. Rev. Lett. 25, 1187–1191 (1970).
[CrossRef]

Other (5)

D. F. Nelson, P. D. Lazay, and M. Lax, “Rotational contribution to Brillouin scattering and higher order photoelastic interactions,” in Proceedings of the Second International Conference on Light Scattering in Solids, M. Balkanski, ed. (Flammarion, Paris, 1971), pp. 477–482.

I. L. Fabielinskii, Molecular Scattering of Light (Plenum, New York, 1968).

J. Xu and R. Stroud, Acousto-optic Devices: Principles, Design and Applications (Wiley, New York, 1992).

T. Błachowicz and Z. Kleszczewski, “Influence of the rotational contribution to the scattering coefficient in piezoelectric crystals,” in Proceedings of the European Acoustics Association Symposium, 26th Winter School on Molecular and Quantum Acoustics, T. Blachowicz, ed. (Upper Silesian Division of the Polish Acoustical Society, Gliwice, Poland, 1997), pp. 13–16.

T. Błachowicz, “Application of Brillouin light scattering to analysis of acoustic properties of piezoelectric crystals,” Ph.D. dissertation (Institute of Physics, Silesian Technical University, Gliwice, Poland, 1997).

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Figures (8)

Fig. 1
Fig. 1

Graphic illustration of the four scattering configurations considered in this paper. Definitions of states of polarization for scattered αij and incident light βij are provided in matrix form. By the rotation of the frame of reference (crystal) in the range (0, 180) angle degrees, the acoustic wave can propagate in different crystallographic directions.

Fig. 2
Fig. 2

The A configuration. Scattering coefficients as a function of the angle between the [100] crystallographic direction and the direction of incident light for (a) the first quasi-transverse wave and (b) the second quasi-transverse wave. The plane of scattering is the (101) crystallographic one.

Fig. 3
Fig. 3

The A configuration. (a) Quotients of scattering coefficients as a function of the angle between the [100] crystallographic direction and the direction of incident light for quasi-transverse waves and (b) the difference between those quotients with and without rotational contributions. The plane of scattering is the (101) crystallographic one.

Fig. 4
Fig. 4

The B configuration. Scattering coefficients as a function of the angle between the [010] crystallographic direction and the direction of incident light for (a) the first quasi-transverse wave and (b) the second quasi-transverse wave. The plane of scattering is the (110) crystallographic one.

Fig. 5
Fig. 5

The B configuration. (a) Quotients of scattering coefficients as a function of the angle between the [010] crystallographic direction and the direction of incident light for quasi-transverse waves and (b) the difference between those quotients with and without rotational contributions. The plane of scattering is the (110) crystallographic one.

Fig. 6
Fig. 6

The C configuration. Scattering coefficients as a function of the angle between the [010] crystallographic direction and the direction of incident light for (a) the first quasi-transverse wave and (b) the second quasi-transverse wave. The configuration is not suitable for experiment: It is not possible to observe two quasi-transverse waves simultaneously. The plane of scattering is the (011) crystallographic one.

Fig. 7
Fig. 7

The D configuration. Scattering coefficients as a function of the angle between the [100] crystallographic direction and the direction of incident light for (a) the first quasi-transverse wave and (b) the second quasi-transverse wave. The plane of scattering is the (110) crystallographic one.

Fig. 8
Fig. 8

The D configuration. (a) Quotients of scattering coefficients as a function of the angle between the [100] crystallographic direction and the direction of incident light for quasi-transverse waves and (b) the difference between those quotients with and without rotational contributions. The plane of scattering is the (110) crystallographic one.

Tables (5)

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Table 1 Numerical Results for the A Configuration Based on Hypersonic Dataa

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Table 2 Numerical Results for the A Configuration Based on Hypersonic Dataa

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Table 3 Numerical Results for the B Configuration Based on Hypersonic Dataa

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Table 4 Numerical Results for the D Configuration Based on Hypersonic Dataa

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Table 5 Comparison of Hypersonic a and Ultrasonic b Values of the Elastic Constants for the LiTaO3 crystal

Equations (27)

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Tij,j=ρu¨i,
Tij,j=cijklSkl,j-iχjenijEn,j
Tij=cijklSkl-enijEn,
ui=u0i{exp[i(χr-ωt)]+exp(-i(χr-ωt)]}
En=E0n exp[i(χr-ωt)].
Dm=emklSkl+εmnSEn,
emklSkl,m+εmnSEn,m=0,
emklSkl,j+iχjεmnSEn,j=0
0=Dm,jδjm=(emklSkl,j+iχjεmnSEn,j)δjm,
En,j=iχjEn,
χjδjm=χm,
En,j=-emklSkl,miχmεmnS=-emklχkχmuliχmεmnS,
Skl,j=uk,lj=-χlχjuk,Skl,j=ul,kj=-χkχjul
Tij,j=-cijklχlχjuk-enijemklχjχnεmnSχmχn χlχmuk,
-cijklχjχluk-enijemklχnχmεmnSχmχn χjχluk=-ω2ρukδik,
cijkl+enijemklχnχmεmnSχmχnχjχl-ω2ρδik=0.
|Qikχjχl-Xδik|=0.
R=π2kTλ4X nink 1cos2 δi cos2 δk j,l(Φjlαijβkl)2,
R=1V dσdΩ,
p13[13]=0.5[(ε-1)33-(ε-1)11],
p13[31]=0.5[(ε-1)11-(ε-1)33],
p31[13]=0.5[(ε-1)33-(ε-1)11],
Φjlαijβkl=Φ11α11β11+Φ22α22β22+Φ33α33β33+Φ12α11β22++Φ21α22β11++Φ13α11β33̲+Φ31α33β11̲++Φ23α22β11̲+Φ32α33β22̲,
Φ13=no2ne2(p(13)(12)χ1γ2+p(13)13χ1γ3̲+p(13)(21)χ2γ1+p(13)31χ3γ1̲),
Φ31=no2ne2(p(31)(12)χ1γ2+p(31)13χ1γ3̲+p(31)(21)χ2γ1+p(31)31χ3γ1̲),
Φ23=no2ne2(p(23)(11)χ1γ1+p(23)(22)χ2γ2+p(23)23χ2γ3̲+p(23)32χ3γ2̲),
Φ32=no2ne2(p(32)(11)χ1γ1+p(32)(22)χ2γ2+p(32)23χ2γ3̲+p(32)32χ3γ2̲).

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