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Light diffraction in paratellurite single crystals on a fast transversal sound without polarization reversal
cm __@
17June
1996
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 2 16 ( 1996) 208-2 10
Light diffraction in paratellurite single crystals on a fast transversal sound without polarization reversal S.V. Akimov, V.M. Gorbenko, V.V. Savchenko Non-traditional
Technologies
Science and Engineering
Center
Received 7 December
1995; revised manuscript
received
” Elent A”, GogoIya Street IS. Dnepropetroosk 320044. Ukraine 14 March 1996; accepted for publication 28 March 1996
Communicated by V.M. Agranovich
Abstract For the first time light diffraction
in paratellurite
sound without polarization reversal theory with allowance for spatial dispersion.
single crystals on a fast transversal
was found. This effect can be explained in terms of the Nelson-Lax
We used the Sheffer-Bergman technique to study acousto-optical (AO) interaction in high qualitative paratellurite single crystals which were grown by us. We studied a crystal measuring 10 X 10 X 10 mm with (lOO), (OIO), (001) oriented planes. The light propagated along the [lOOI axis and the sound propagated in the (100) plane. The incident light was polarized along the [OOI] or [OIO] axes. Photographs of the light diffraction on fast transversal (FT) sound have been obtained. For no polarization of the incident light diffraction with polarization reversal on such sound was observed. That is, the components of the photoelasticity tensor P,,,,, P,,,,, P,,,,, P3232, or in two-index mean the component Pd4, do not work. So it is doubtful that the such nonzero components exist in a paratellurite [ 1I. For the first time light diffraction on lT sound without polarization reversal was photographed by the authors. A picture (Fig. 1) for such sound was obtained. The incident and diffracted light were polarized along the [OlO] axis. It is impossible to explain such a diffraction in paratellurite by the Pockels theory. According to the crystal symmetry the components of the photoelasticity tensor P,,,, and Pzzz3 (in two-index mean P,,) are equal to zero. The appearance of such nonzero components is not explained by the calculations of Nelson and Lax [2]. This problem was solved in terms of the Nelson-Lax theory, which was developed by the authors allowing for spatial dispersion [3]. It is possible to explain another unusual effect in paratellurite due to spatial dispersion [4]. We obtained such an equation for the nonlinear polarization for the A0 interaction [3], Pi(l)
1) = &o( Xijk/ + Xtfj,t, + ifijk,
- (2’ij,-ara.te.s,,
+2idij~a,a,gij,,k,A)/[&0apKpri(OA.
+ 2ibijrrnk~ara,e,k~ k”)a,])E,(l,
+ 2ivijrmkmAaraSe,kr
o)~,,,(o-
1).
(1)
,yijk, is the true fourth-rank photoelasticity tensor representing what Nelson and Lax [2] call the direct photoelastic effect. It consists of the symmetric and the antisymmetric parts. The symmetric part can be related 0375-9601/96/$12.00 Copyright PII SO375-9601(96)00298-S
0 1996 Published
by Elsevier Science B.V. All rights reserved.
S.V. Akimov ef al./ Physics Letters A 216 (19961208-210
209
to the Pockels photoelastic tensor [Z]. The antisymmetric part has no analog in the Pockels formulation. It arises due to the rotation of the volume elements in the acoustic wave. The xljk, and other terms also have no analogues in the Pockels formulation. We see from calculations 131that x:jk, arises due to the motion of the center of mass of the volume elements in the acoustic wave. We call this the dynamic photoelastic effect. The imaginary term izjjk, is the tensor function representing what we call the direct photoelastic effect all(~wing for spatial dispersion. This term describes the contribution to the A0 interaction, for example from the natural optical and acoustical activities, from the effect of piezogyration and from other effects which are connected with the inhomogeneity of the interacting waves in a medium of slight nonlocality. The terms x:Jnl and igijk, represent the direct photoelastic effect together with the term xijk.. They describe the immediate interaction between the optical and the acoustical waves. represents the indirect photoelastic effect. This effect consists of the successive The term 2dij,a,a,e,,,, action of the piezoelectric and electro-optical effects [2]. The term 2ibij,, kjj,a,as eskl represents the contribution to the A0 interaction from the successive action of the piezoelectric and electrogyration effects. The term 2iu.,,rm kAa m , a seJk, describes the contribution from the successive action of the piezoelectric and electro-optic effects in an inhomogeneous field. The optic wave interacts with the spatial inhomogeneous electric field which appears in a crystal due to the piezoelectric effect. The parameter of the i~omogeneity is the wave vector of the acoustic wave. The term 2idijra,a,gij,,k,A represents the contribution from the successive action of the flexoelectric effect and the electro-optical effect. It is evident from the analysis of the expression that in the present geometry there are nonzero tensor functions ilZS2 and j& both in the antisymmetric and in the symmetric parts of the increment J&,. tfowever, in the antisymmetric part these tensor functions (‘IF) are canceled. The symmetric part has the form 131
All symbols coincide with those used by Nelson and Lax 121.Only this expression from (1) gives the nonzero components in the presence of the selected interaction geometry. These are the seventh, eighth and ninth terms of (2). The appearance of these terms in the equation for nonlinear polarization is connected with the spatial dispersion of the light wave in the crystal. In particular, it is connected with the gyration of such a wave. On the photograph we can see gaps of the diffraction lines along the [OIO] and [OOl] axes. Some antisymmetricity (the gap near the [OOl] axis is broader than the one near the [OlO] axis) can be explained by an acoustic wave along .the [OlO] direction. The wave near the [OlOl axis is more powerful than other sound waves in the crystal.
XV. Akimou et d/Physics
210
Fig. 1. Light diffraction on FT sound without polarization The light is polarized along the [OIO] axis.
reversal.
Letters A 216 (1996) 208-210
The wave vector of the light is perpendicular
to the plane of the figure.
It is interesting that the light diffracted without polarization reversal is absent when the incident light has a polarization along the [OOl] axis. One has to note that, as follows from (2) a TF with the P3323 and P,,,, components also occurs. But these TFs are now connected with other tensor components. Obviously, that is how the presence of diffraction light when the incident light was polarized along the [OlO] axis and its absence when the incident light was polarized along the [OOl] axis can be explained. Due to the experimental results we can infer also that the additions of the components of the ninth term (2) are negligible for this geometry. One has to note that for this geometry the extra terms given are equal to zero for diffraction with polarization reversal. All other additions in (1) which are connected with spatial dispersion are also equal to zero. In conclusion we want to add that the successive action of the flexo-electric and electro-optical effects in an inhomogeneous field could add to the described diffraction. Such an effect, as follows from Ref. [3], can be discribed by a TF which has the form Vijrmk~a,a,gsk,n,k,A/&gapKP4(
*A?
kA)a,-
(3)
It is evident from the analysis that the nonzero components of such a TF arise due to the joint presence of the acoustic vectors, which are directed along the [OlO] and [OOl] axes. The gaps along the axes can be explained by the addition of such TFs.
References [l] [2] [3] [4]
Acoustic crystals D.F. Nelson and V.V. Savchenko, S.V. Akimov et
(Moscow, 1982). M. Lax, Phys. Rev. B 3 (1971) 2778. Appl. UNIT1 No. 6576-V87, Dnepropetrovsk al., JEPT Lett. 51 (1990) 25.
(1987).
17June
1996
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 2 16 ( 1996) 208-2 10
Light diffraction in paratellurite single crystals on a fast transversal sound without polarization reversal S.V. Akimov, V.M. Gorbenko, V.V. Savchenko Non-traditional
Technologies
Science and Engineering
Center
Received 7 December
1995; revised manuscript
received
” Elent A”, GogoIya Street IS. Dnepropetroosk 320044. Ukraine 14 March 1996; accepted for publication 28 March 1996
Communicated by V.M. Agranovich
Abstract For the first time light diffraction
in paratellurite
sound without polarization reversal theory with allowance for spatial dispersion.
single crystals on a fast transversal
was found. This effect can be explained in terms of the Nelson-Lax
We used the Sheffer-Bergman technique to study acousto-optical (AO) interaction in high qualitative paratellurite single crystals which were grown by us. We studied a crystal measuring 10 X 10 X 10 mm with (lOO), (OIO), (001) oriented planes. The light propagated along the [lOOI axis and the sound propagated in the (100) plane. The incident light was polarized along the [OOI] or [OIO] axes. Photographs of the light diffraction on fast transversal (FT) sound have been obtained. For no polarization of the incident light diffraction with polarization reversal on such sound was observed. That is, the components of the photoelasticity tensor P,,,,, P,,,,, P,,,,, P3232, or in two-index mean the component Pd4, do not work. So it is doubtful that the such nonzero components exist in a paratellurite [ 1I. For the first time light diffraction on lT sound without polarization reversal was photographed by the authors. A picture (Fig. 1) for such sound was obtained. The incident and diffracted light were polarized along the [OlO] axis. It is impossible to explain such a diffraction in paratellurite by the Pockels theory. According to the crystal symmetry the components of the photoelasticity tensor P,,,, and Pzzz3 (in two-index mean P,,) are equal to zero. The appearance of such nonzero components is not explained by the calculations of Nelson and Lax [2]. This problem was solved in terms of the Nelson-Lax theory, which was developed by the authors allowing for spatial dispersion [3]. It is possible to explain another unusual effect in paratellurite due to spatial dispersion [4]. We obtained such an equation for the nonlinear polarization for the A0 interaction [3], Pi(l)
1) = &o( Xijk/ + Xtfj,t, + ifijk,
- (2’ij,-ara.te.s,,
+2idij~a,a,gij,,k,A)/[&0apKpri(OA.
+ 2ibijrrnk~ara,e,k~ k”)a,])E,(l,
+ 2ivijrmkmAaraSe,kr
o)~,,,(o-
1).
(1)
,yijk, is the true fourth-rank photoelasticity tensor representing what Nelson and Lax [2] call the direct photoelastic effect. It consists of the symmetric and the antisymmetric parts. The symmetric part can be related 0375-9601/96/$12.00 Copyright PII SO375-9601(96)00298-S
0 1996 Published
by Elsevier Science B.V. All rights reserved.
S.V. Akimov ef al./ Physics Letters A 216 (19961208-210
209
to the Pockels photoelastic tensor [Z]. The antisymmetric part has no analog in the Pockels formulation. It arises due to the rotation of the volume elements in the acoustic wave. The xljk, and other terms also have no analogues in the Pockels formulation. We see from calculations 131that x:jk, arises due to the motion of the center of mass of the volume elements in the acoustic wave. We call this the dynamic photoelastic effect. The imaginary term izjjk, is the tensor function representing what we call the direct photoelastic effect all(~wing for spatial dispersion. This term describes the contribution to the A0 interaction, for example from the natural optical and acoustical activities, from the effect of piezogyration and from other effects which are connected with the inhomogeneity of the interacting waves in a medium of slight nonlocality. The terms x:Jnl and igijk, represent the direct photoelastic effect together with the term xijk.. They describe the immediate interaction between the optical and the acoustical waves. represents the indirect photoelastic effect. This effect consists of the successive The term 2dij,a,a,e,,,, action of the piezoelectric and electro-optical effects [2]. The term 2ibij,, kjj,a,as eskl represents the contribution to the A0 interaction from the successive action of the piezoelectric and electrogyration effects. The term 2iu.,,rm kAa m , a seJk, describes the contribution from the successive action of the piezoelectric and electro-optic effects in an inhomogeneous field. The optic wave interacts with the spatial inhomogeneous electric field which appears in a crystal due to the piezoelectric effect. The parameter of the i~omogeneity is the wave vector of the acoustic wave. The term 2idijra,a,gij,,k,A represents the contribution from the successive action of the flexoelectric effect and the electro-optical effect. It is evident from the analysis of the expression that in the present geometry there are nonzero tensor functions ilZS2 and j& both in the antisymmetric and in the symmetric parts of the increment J&,. tfowever, in the antisymmetric part these tensor functions (‘IF) are canceled. The symmetric part has the form 131
All symbols coincide with those used by Nelson and Lax 121.Only this expression from (1) gives the nonzero components in the presence of the selected interaction geometry. These are the seventh, eighth and ninth terms of (2). The appearance of these terms in the equation for nonlinear polarization is connected with the spatial dispersion of the light wave in the crystal. In particular, it is connected with the gyration of such a wave. On the photograph we can see gaps of the diffraction lines along the [OIO] and [OOl] axes. Some antisymmetricity (the gap near the [OOl] axis is broader than the one near the [OlO] axis) can be explained by an acoustic wave along .the [OlO] direction. The wave near the [OlOl axis is more powerful than other sound waves in the crystal.
XV. Akimou et d/Physics
210
Fig. 1. Light diffraction on FT sound without polarization The light is polarized along the [OIO] axis.
reversal.
Letters A 216 (1996) 208-210
The wave vector of the light is perpendicular
to the plane of the figure.
It is interesting that the light diffracted without polarization reversal is absent when the incident light has a polarization along the [OOl] axis. One has to note that, as follows from (2) a TF with the P3323 and P,,,, components also occurs. But these TFs are now connected with other tensor components. Obviously, that is how the presence of diffraction light when the incident light was polarized along the [OlO] axis and its absence when the incident light was polarized along the [OOl] axis can be explained. Due to the experimental results we can infer also that the additions of the components of the ninth term (2) are negligible for this geometry. One has to note that for this geometry the extra terms given are equal to zero for diffraction with polarization reversal. All other additions in (1) which are connected with spatial dispersion are also equal to zero. In conclusion we want to add that the successive action of the flexo-electric and electro-optical effects in an inhomogeneous field could add to the described diffraction. Such an effect, as follows from Ref. [3], can be discribed by a TF which has the form Vijrmk~a,a,gsk,n,k,A/&gapKP4(
*A?
kA)a,-
(3)
It is evident from the analysis that the nonzero components of such a TF arise due to the joint presence of the acoustic vectors, which are directed along the [OlO] and [OOl] axes. The gaps along the axes can be explained by the addition of such TFs.
References [l] [2] [3] [4]
Acoustic crystals D.F. Nelson and V.V. Savchenko, S.V. Akimov et
(Moscow, 1982). M. Lax, Phys. Rev. B 3 (1971) 2778. Appl. UNIT1 No. 6576-V87, Dnepropetrovsk al., JEPT Lett. 51 (1990) 25.
(1987).