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On the polarization of light diffracted by ultrasound
Ultrasonics 38 (2000) 575–580 www.elsevier.nl/locate/ultras
On the polarization of light diffracted by ultrasound E. Blomme a, *, G. Gondek b, T. Katkowski b, P. Kwiek b, A. Sliwinski b, O. Leroy c a Katholieke Hogeschool Zuid-West-Vlaanderen (KATHO), Dept. VHTI, Doorniksesteenweg 145, 8500 Kortrijk, Belgium b University of Gdansk, Institute of Experimental Physics, ul. Wita Stwosza 57, 80-952 Gdansk, Poland c Katholieke Universiteit Leuven, Campus Kortrijk, E. Sabbelaan, Kortrijk, Belgium
Abstract If light is diffracted by ultrasound in an isotropic medium with acoustically induced birefringence, the state of polarization is modified in each order of diffraction with respect to the initial state of polarization of the incident light wave. In the present paper, some polarization effects are discussed in the case of normal light incidence. In general a rotation of the main polarization plane occurs, together with a change of the ellipticity. However, while the former effect always takes place, the latter only occurs in the case of ultrasonic light diffraction of the intermediate type. Some experimental measurements are included in case of argon laser light being diffracted by an ultrasonic wave propagating in fused silica (SiO ). © 2000 Elsevier Science B.V. All rights reserved. 2 Keywords: Acousto-optics; Acoustically induced birefringence; Polarization; Ultrasonic light diffraction
1. Introduction It is well known in optics that the refractive index of anisotropic materials depends on the polarization of the light passing through the medium. Waves with different states of polarization travel at different velocities and hence undergo different phase shifts. As a consequence, the polarization ellipse is modified as the wave propagates through the medium. In this way, linearly polarized light can be transformed into circularly polarized light and vice versa (e.g. a quarter wave retarder). Under other conditions, the anisotropic material can act as a polarization rotator, turning the plane of polarization of linearly polarized light by a fixed angle, maintaining its linear polarized nature (e.g. a half wave retarder). Likewise, optically active materials have the natural ability to rotate the plane of polarization of linearly polarized light, and so do some anisotropic liquids under certain conditions (e.g. liquid crystals). None of the above mentioned effects can be established in isotropic materials. However, when an acoustic wave propagates in an isotropic medium (e.g. a cubic crystal ), birefringence may be induced and the medium may act as a uniaxial crystal. Many media frequently * Corresponding author. Tel.: +32-5626-4120; fax: +32-5621-9867. E-mail address: [email protected] (E. Blomme)
encountered in optical devices exhibit this property. If light is passing through a medium with acoustically induced anisotropy, all the above effects can be observed but they are much more complicated, due to diffraction by the acoustic grating. Light is redistributed among several diffraction orders, the polarization and intensity of which are a function of all light–sound interaction parameters. In the present paper, it is shown how the polarization of initially linearly polarized light is affected while passing normally through an isotropic medium with ultrasonically induced birefringence. In particular, it is demonstrated that the acousto-optic (AO) cell can act as a polarization rotator for each of the created diffraction orders and that part of the linearly polarized light can be transformed into circularly polarized light. Some of the phenomena are illustrated in the case of ultrasonic light diffraction ( ULD) in fused silica (SiO ). 2 2. Geometry and parameters Consider an acoustic plane wave of angular frequency V and wave number K traveling in the z-direction in an isotropic optically transparent medium of thickness L ( Fig. 1). The relative displacement at position z and time t is described by S(z, t)=S sin(Vt−Kz) 0
0041-624X/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S0 0 4 1 -6 2 4 X ( 9 9 ) 0 0 10 2 - X
(1)
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E. Blomme et al. / Ultrasonics 38 (2000) 575–580
Q is related to an additional phase shift in all diffraction orders [3,4] and is proportional to the square of the acoustic frequency. Diffracted light amplitudes w and corresponding n intensities I =|w |2 can be evaluated from general RN n n theory for each order of diffraction n, the nth order having frequency v+nV. The state of polarization of the nth order diffracted light wave is determined by the ellipticity e of the n polarization ellipse and the azimuth angle a . The former n represents the ratio of the minor to the major axis, the latter the direction of the major axis with respect to the x-axis. At normal light incidence, one has the symmetry property I =I , e =e and a =a . −n n −n n −n n Fig. 1. Geometry of the light–sound interaction.
3. Polarization effects where S is the amplitude. The strain S(z, t) creates a 0 perturbation of the refractive index with an amplitude which is different in the z-direction, with respect to the x- and y-direction: Dm =Dm =0.5m3 p S , Dm =0.5m3 p S x y 0 12 0 z 0 11 0
(2)
where m is the refractive index in the absence of sound 0 and p and p are Pockel’s photoelastic constants. 11 12 Hence the medium becomes birefringent and can be treated as a uniaxial crystal with its optical axis in the z-direction. Let us now consider a monochromatic and linearly polarized optical plane wave traveling along the xdirection with frequency v and wave number k=2p/l in vacuum. The light is diffracted by the acoustic wave and, due to the birefringence, each order of diffraction is decomposed into two orthogonal modes, an ordinary and an extraordinary ray. As a result of both birefringence and diffraction, the two modes are affected in phase and amplitude in a different way, and when they are combined, they in general form an elliptically polarized wave. From theory [1,2], it can be seen that the final diffraction pattern in the exit plane z=L is governed by the following parameters. First, one has the Raman– Nath (RN ) parameters v =kDm L, v =kDm L y y z z
(3)
each representing a phase shift due to the diffraction process and being associated with the ordinary and extraordinary diffracted light rays, respectively. Neither parameter varies independently, but according to the constant ratio v /v =p /p . Second, the diffraction y z 12 11 pattern also depends on the Klein–Cook parameter Q, defined by Q=LK2/m k. 0
(4)
Under RN conditions, i.e. ultrasonic frequencies in the lower megahertz region and short light–sound interaction lengths (Q#0), linearly polarized light remains so in all orders of diffraction [1,2,5,6 ], but the electric light vector is rotated. The effect is demonstrated in Fig. 2 with respect to zero order in the case of ULD in SiO ( p =0.121, p =0.270). It is seen that a rotation 2 11 12 of 90° is possible within a realistic range of v values z [Fig. 2(a)]. However, over the same range, the zero order light intensity I is decreasing [Fig. 2(b)]. 0 Fig. 3 shows a similar phenomenon with respect to the first order diffracted light wave. Again a rotation of 90° can be established, in combination with an increase of light intensity I up to 25%. 1 At higher sound frequencies, RN conditions are no longer fulfilled and, in addition to Bragg effects at first and higher order Bragg angles of incidence [7,8], additional polarization phenomena appear, even at normal light incidence [1,2,9]. In general, not only the rotation effect occurs, but the initial linear state of polarization is affected in all orders of diffraction. Fig. 4 shows what happens to the zero order diffracted light in the case of ULD in SiO at Q=4.19 (acoustic frequency 62 MHz). 2 By increasing the sound amplitude from zero to some value corresponding to v #1.4, the state of polarization z is changing from linear to circular in an approximately quadratic way [Fig. 4(a)]. Hence the AO cell behaves as a special retarder, where any ellipticity between 0 and 1 can be obtained at a suitable voltage applied to the transducer. It is worth mentioning that the chosen value Q=4.19 is the optimum one to achieve the maximum e =1. For Q≠4.19, the peak ellipticity is less than 1 0 [1,2]. The corresponding light intensity variation is plotted on the same figure. In this case, the orientation of the main axis of the polarization ellipse remains rather stable [Fig. 4(b)]. Note the jump of 90° at v #1.4, where the polarization goes through the circular z state and the major axis becomes the minor axis and
E. Blomme et al. / Ultrasonics 38 (2000) 575–580
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Fig. 2. RN diffraction (Q=0.01) in SiO at normal light incidence: a and I as a function of v (v /v =2.23). 2 0 0 z y z
Fig. 3. RN diffraction (Q=0.01) in SiO at normal light incidence: a and I as a function of v (v /v =2.23). 2 1 1 z y z
vice versa. Fig. 5 illustrates analogous polarization changes with respect to the first diffraction orders. As circular polarization is not possible here, discontinuities
in the azimuth angle do not occur. From Fig. 5(b) the angle is seen to remain rather constant up to v #1. z At frequencies corresponding to Q values lower than
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E. Blomme et al. / Ultrasonics 38 (2000) 575–580
Fig. 4. Diffraction in the intermediate regime (Q=4.19) in SiO at normal light incidence: I , e and a as a function of v (v /v =2.23). 2 0 0 0 z y z
Fig. 5. Diffraction in the intermediate regime (Q=4.19) in SiO at normal light incidence: I , e and a as a function of v (v /v =2.23). 2 1 1 1 z y z
4.19, circular polarization in the zero order diffracted light cannot be established, although ellipticities larger than 0.5 can be obtained. Simultaneously, rotations of 90° of the main axis of the polarization ellipse remain
possible with respect to the zero order. A good illustration is Fig. 6, showing the results of experimental measurements of e and e , a and a , and I and I , in 0 ±1 0 ±1 0 ±1 the case of argon laser light being diffracted by a
E. Blomme et al. / Ultrasonics 38 (2000) 575–580
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(a)
(b) Fig. 6. Diffraction in the intermediate regime (Q=2.6) in SiO at normal light incidence: theoretical (curves) and experimental (marks) 0 and ±1 2 order (a) intensities, (b) ellipticities and (c) azimuth angles.
48.5 MHz sound wave propagating in SiO (Q=2.6). 2 More results and a description of the experimental set-up and method can be found elsewhere [2]. For Q values higher than 4.2, diffraction and polariza-
tion effects in SiO at normal light incidence gradually 2 disappear while Bragg effects and accompanying polarization changes only appear at critical (Bragg) angles [1].
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E. Blomme et al. / Ultrasonics 38 (2000) 575–580
(c) Fig. 6. (continued )
4. Conclusions
References
The initial state of polarization of light diffracted by ultrasound in general is affected in all orders of diffraction, the final state of polarization being a function of acoustic frequency and intensity, and the light–sound interaction width. At normal light incidence, both the ellipticity and the orientation of the polarization ellipse change, except under RN conditions where only a rotation of the polarization plane occurs. The results obtained suggest the possibility of constructing AO cells as polarization rotators as well as compensators using the Raman–Nath and intermediate range of ULD.
[1] E. Blomme, O. Leroy, A. Sliwinski, Acoustica/Acta Acoustica 82 (1996) 464. [2] E. Blomme, G. Gondek, T. Katkowski, P. Kwiek, O. Leroy, A. Sliwinski, Acoustica/Acta Acoustica (1999) in press. [3] P. Kwiek, R. Reibold, Acoustica 80 (1994) 294. [4] P. Kwiek, W. Molkenstruck, R. Reibold, Ultrasonics 34 (1996) 801. [5] A. Alippi, Opt. Commun. 8 (1973) 397. [6 ] W.R. Klein, W.D. Fitts, IEEE SU-21 (1974) 204. [7] E. Blomme, O. Leroy, Acoustica 63 (1987) 83. [8] E. Blomme, P. Kwiek, O. Leroy, A. Sliwinski, in: Proc. Ultrasonics Int. ’93, Vienna, Butterworth Heinemann, 1993, p. 69. [9] E. Blomme, O. Leroy, A. Sliwinski, Acousto-optics and applications II, Proc. SPIE 2643 (1995) 2.
On the polarization of light diffracted by ultrasound E. Blomme a, *, G. Gondek b, T. Katkowski b, P. Kwiek b, A. Sliwinski b, O. Leroy c a Katholieke Hogeschool Zuid-West-Vlaanderen (KATHO), Dept. VHTI, Doorniksesteenweg 145, 8500 Kortrijk, Belgium b University of Gdansk, Institute of Experimental Physics, ul. Wita Stwosza 57, 80-952 Gdansk, Poland c Katholieke Universiteit Leuven, Campus Kortrijk, E. Sabbelaan, Kortrijk, Belgium
Abstract If light is diffracted by ultrasound in an isotropic medium with acoustically induced birefringence, the state of polarization is modified in each order of diffraction with respect to the initial state of polarization of the incident light wave. In the present paper, some polarization effects are discussed in the case of normal light incidence. In general a rotation of the main polarization plane occurs, together with a change of the ellipticity. However, while the former effect always takes place, the latter only occurs in the case of ultrasonic light diffraction of the intermediate type. Some experimental measurements are included in case of argon laser light being diffracted by an ultrasonic wave propagating in fused silica (SiO ). © 2000 Elsevier Science B.V. All rights reserved. 2 Keywords: Acousto-optics; Acoustically induced birefringence; Polarization; Ultrasonic light diffraction
1. Introduction It is well known in optics that the refractive index of anisotropic materials depends on the polarization of the light passing through the medium. Waves with different states of polarization travel at different velocities and hence undergo different phase shifts. As a consequence, the polarization ellipse is modified as the wave propagates through the medium. In this way, linearly polarized light can be transformed into circularly polarized light and vice versa (e.g. a quarter wave retarder). Under other conditions, the anisotropic material can act as a polarization rotator, turning the plane of polarization of linearly polarized light by a fixed angle, maintaining its linear polarized nature (e.g. a half wave retarder). Likewise, optically active materials have the natural ability to rotate the plane of polarization of linearly polarized light, and so do some anisotropic liquids under certain conditions (e.g. liquid crystals). None of the above mentioned effects can be established in isotropic materials. However, when an acoustic wave propagates in an isotropic medium (e.g. a cubic crystal ), birefringence may be induced and the medium may act as a uniaxial crystal. Many media frequently * Corresponding author. Tel.: +32-5626-4120; fax: +32-5621-9867. E-mail address: [email protected] (E. Blomme)
encountered in optical devices exhibit this property. If light is passing through a medium with acoustically induced anisotropy, all the above effects can be observed but they are much more complicated, due to diffraction by the acoustic grating. Light is redistributed among several diffraction orders, the polarization and intensity of which are a function of all light–sound interaction parameters. In the present paper, it is shown how the polarization of initially linearly polarized light is affected while passing normally through an isotropic medium with ultrasonically induced birefringence. In particular, it is demonstrated that the acousto-optic (AO) cell can act as a polarization rotator for each of the created diffraction orders and that part of the linearly polarized light can be transformed into circularly polarized light. Some of the phenomena are illustrated in the case of ultrasonic light diffraction ( ULD) in fused silica (SiO ). 2 2. Geometry and parameters Consider an acoustic plane wave of angular frequency V and wave number K traveling in the z-direction in an isotropic optically transparent medium of thickness L ( Fig. 1). The relative displacement at position z and time t is described by S(z, t)=S sin(Vt−Kz) 0
0041-624X/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S0 0 4 1 -6 2 4 X ( 9 9 ) 0 0 10 2 - X
(1)
576
E. Blomme et al. / Ultrasonics 38 (2000) 575–580
Q is related to an additional phase shift in all diffraction orders [3,4] and is proportional to the square of the acoustic frequency. Diffracted light amplitudes w and corresponding n intensities I =|w |2 can be evaluated from general RN n n theory for each order of diffraction n, the nth order having frequency v+nV. The state of polarization of the nth order diffracted light wave is determined by the ellipticity e of the n polarization ellipse and the azimuth angle a . The former n represents the ratio of the minor to the major axis, the latter the direction of the major axis with respect to the x-axis. At normal light incidence, one has the symmetry property I =I , e =e and a =a . −n n −n n −n n Fig. 1. Geometry of the light–sound interaction.
3. Polarization effects where S is the amplitude. The strain S(z, t) creates a 0 perturbation of the refractive index with an amplitude which is different in the z-direction, with respect to the x- and y-direction: Dm =Dm =0.5m3 p S , Dm =0.5m3 p S x y 0 12 0 z 0 11 0
(2)
where m is the refractive index in the absence of sound 0 and p and p are Pockel’s photoelastic constants. 11 12 Hence the medium becomes birefringent and can be treated as a uniaxial crystal with its optical axis in the z-direction. Let us now consider a monochromatic and linearly polarized optical plane wave traveling along the xdirection with frequency v and wave number k=2p/l in vacuum. The light is diffracted by the acoustic wave and, due to the birefringence, each order of diffraction is decomposed into two orthogonal modes, an ordinary and an extraordinary ray. As a result of both birefringence and diffraction, the two modes are affected in phase and amplitude in a different way, and when they are combined, they in general form an elliptically polarized wave. From theory [1,2], it can be seen that the final diffraction pattern in the exit plane z=L is governed by the following parameters. First, one has the Raman– Nath (RN ) parameters v =kDm L, v =kDm L y y z z
(3)
each representing a phase shift due to the diffraction process and being associated with the ordinary and extraordinary diffracted light rays, respectively. Neither parameter varies independently, but according to the constant ratio v /v =p /p . Second, the diffraction y z 12 11 pattern also depends on the Klein–Cook parameter Q, defined by Q=LK2/m k. 0
(4)
Under RN conditions, i.e. ultrasonic frequencies in the lower megahertz region and short light–sound interaction lengths (Q#0), linearly polarized light remains so in all orders of diffraction [1,2,5,6 ], but the electric light vector is rotated. The effect is demonstrated in Fig. 2 with respect to zero order in the case of ULD in SiO ( p =0.121, p =0.270). It is seen that a rotation 2 11 12 of 90° is possible within a realistic range of v values z [Fig. 2(a)]. However, over the same range, the zero order light intensity I is decreasing [Fig. 2(b)]. 0 Fig. 3 shows a similar phenomenon with respect to the first order diffracted light wave. Again a rotation of 90° can be established, in combination with an increase of light intensity I up to 25%. 1 At higher sound frequencies, RN conditions are no longer fulfilled and, in addition to Bragg effects at first and higher order Bragg angles of incidence [7,8], additional polarization phenomena appear, even at normal light incidence [1,2,9]. In general, not only the rotation effect occurs, but the initial linear state of polarization is affected in all orders of diffraction. Fig. 4 shows what happens to the zero order diffracted light in the case of ULD in SiO at Q=4.19 (acoustic frequency 62 MHz). 2 By increasing the sound amplitude from zero to some value corresponding to v #1.4, the state of polarization z is changing from linear to circular in an approximately quadratic way [Fig. 4(a)]. Hence the AO cell behaves as a special retarder, where any ellipticity between 0 and 1 can be obtained at a suitable voltage applied to the transducer. It is worth mentioning that the chosen value Q=4.19 is the optimum one to achieve the maximum e =1. For Q≠4.19, the peak ellipticity is less than 1 0 [1,2]. The corresponding light intensity variation is plotted on the same figure. In this case, the orientation of the main axis of the polarization ellipse remains rather stable [Fig. 4(b)]. Note the jump of 90° at v #1.4, where the polarization goes through the circular z state and the major axis becomes the minor axis and
E. Blomme et al. / Ultrasonics 38 (2000) 575–580
577
Fig. 2. RN diffraction (Q=0.01) in SiO at normal light incidence: a and I as a function of v (v /v =2.23). 2 0 0 z y z
Fig. 3. RN diffraction (Q=0.01) in SiO at normal light incidence: a and I as a function of v (v /v =2.23). 2 1 1 z y z
vice versa. Fig. 5 illustrates analogous polarization changes with respect to the first diffraction orders. As circular polarization is not possible here, discontinuities
in the azimuth angle do not occur. From Fig. 5(b) the angle is seen to remain rather constant up to v #1. z At frequencies corresponding to Q values lower than
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E. Blomme et al. / Ultrasonics 38 (2000) 575–580
Fig. 4. Diffraction in the intermediate regime (Q=4.19) in SiO at normal light incidence: I , e and a as a function of v (v /v =2.23). 2 0 0 0 z y z
Fig. 5. Diffraction in the intermediate regime (Q=4.19) in SiO at normal light incidence: I , e and a as a function of v (v /v =2.23). 2 1 1 1 z y z
4.19, circular polarization in the zero order diffracted light cannot be established, although ellipticities larger than 0.5 can be obtained. Simultaneously, rotations of 90° of the main axis of the polarization ellipse remain
possible with respect to the zero order. A good illustration is Fig. 6, showing the results of experimental measurements of e and e , a and a , and I and I , in 0 ±1 0 ±1 0 ±1 the case of argon laser light being diffracted by a
E. Blomme et al. / Ultrasonics 38 (2000) 575–580
579
(a)
(b) Fig. 6. Diffraction in the intermediate regime (Q=2.6) in SiO at normal light incidence: theoretical (curves) and experimental (marks) 0 and ±1 2 order (a) intensities, (b) ellipticities and (c) azimuth angles.
48.5 MHz sound wave propagating in SiO (Q=2.6). 2 More results and a description of the experimental set-up and method can be found elsewhere [2]. For Q values higher than 4.2, diffraction and polariza-
tion effects in SiO at normal light incidence gradually 2 disappear while Bragg effects and accompanying polarization changes only appear at critical (Bragg) angles [1].
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E. Blomme et al. / Ultrasonics 38 (2000) 575–580
(c) Fig. 6. (continued )
4. Conclusions
References
The initial state of polarization of light diffracted by ultrasound in general is affected in all orders of diffraction, the final state of polarization being a function of acoustic frequency and intensity, and the light–sound interaction width. At normal light incidence, both the ellipticity and the orientation of the polarization ellipse change, except under RN conditions where only a rotation of the polarization plane occurs. The results obtained suggest the possibility of constructing AO cells as polarization rotators as well as compensators using the Raman–Nath and intermediate range of ULD.
[1] E. Blomme, O. Leroy, A. Sliwinski, Acoustica/Acta Acoustica 82 (1996) 464. [2] E. Blomme, G. Gondek, T. Katkowski, P. Kwiek, O. Leroy, A. Sliwinski, Acoustica/Acta Acoustica (1999) in press. [3] P. Kwiek, R. Reibold, Acoustica 80 (1994) 294. [4] P. Kwiek, W. Molkenstruck, R. Reibold, Ultrasonics 34 (1996) 801. [5] A. Alippi, Opt. Commun. 8 (1973) 397. [6 ] W.R. Klein, W.D. Fitts, IEEE SU-21 (1974) 204. [7] E. Blomme, O. Leroy, Acoustica 63 (1987) 83. [8] E. Blomme, P. Kwiek, O. Leroy, A. Sliwinski, in: Proc. Ultrasonics Int. ’93, Vienna, Butterworth Heinemann, 1993, p. 69. [9] E. Blomme, O. Leroy, A. Sliwinski, Acousto-optics and applications II, Proc. SPIE 2643 (1995) 2.