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Transition to optical chaos in a photorefractive parametric oscillator with Bi12TiO20 crystal
1 August 1998
Optics Communications 153 Ž1998. 295–300
Transition to optical chaos in a photorefractive parametric oscillator with Bi 12TiO 20 crystal A. Apolinar-Iribe 1, N. Korneev 2 , J.J. Sanchez-Mondragon ´ ´ ´ (INAOE), Apdo. Postal 51 y 216, CP 72000, Puebla, Pue., Mexico Instituto Nacional de Astrofısica, Optica y Electronica ´ ´ Received 22 January 1998; revised 22 April 1998; accepted 11 May 1998
Abstract Frequency-shifted waves are generated in a Bi 12TiO 20 crystal from two nearly counterpropagating orthogonally polarized pump waves when the electric voltage applied to the crystal exceeds the threshold value. Near the voltage threshold the output beams are regular in shape, for higher voltages running fringes appear, whilst far from the threshold chaotically moving speckles are observed. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Many nonlinear optical systems Žlasers, nonlinear cavities, Kerr media with feedback, etc.. produce chaotic output in time or both in time and space w1x. For photorefractive devices chaotic behavior is known for phase-conjugate mirrors w2x, and for various oscillators with external feedback w3–6x. In the experiment with a ring cavity containing a photorefractive gain medium the transition from regular oscillation to chaotic behavior a dependence on the aperture size was observed w7x. Such a transition is highly interesting from the theoretical viewpoint. In mirrorless photorefractive oscillators unstable output patterns have been observed for LiNbO 3 and BaTiO 3 w8,9x, but to our knowledgement, the development of chaos from regular output has not been reported. Here we describe a simple mirrorless oscillator, which produces regular or irregular output depending on the applied voltage. The device we use is the modification of the scheme suggested for degenerate four-wave mixing in cubic photorefractive crystals w10x. In Ref. w10x it was shown that noise generation occurs in the Bi 12TiO 20 crys-
1
Permanent address: Departamento de Fısica, Universidad de ´ Sonora. 2 Corresponding author. E-mail: [email protected]
tal if two orthogonally polarized beams are counterpropagating within it and an external voltage is applied. We show that by changing the angle between the pump beams one can obtain generation with a transition from regular to chaotic profile. We believe that the study of the device can give some useful insights into the appearance of chaotic behavior, due to the one-dimensional character of the output pattern and the clearly seen transition to chaos.
2. Experimental setup The experimental setup is shown in Fig. 1. Two pump beams from independent He–Ne lasers Ž l s 633 nm, 10 mW power each. with 2 mm beam diameter are nearly counterpropagating and intersect in a Bi 12TiO 20 ŽBTO. crystal. In the experiment we use a sample of dimensions of 8 mm = 8 mm = 6 mm with polished faces of 8 mm = 8 mm and silver paste electrodes deposited onto the faces measuring 8 mm = 6 mm. Both beams are in a horizontal plane, but there is an angle Žtypically 0.1 rad. between them. An external electric field is applied vertically in the Ž110. crystallographic direction. When the external DC electric field is applied to the crystal it is quite difficult to obtain a uniform field distribution. To produce a high field in the He–Ne laser beam intersection region and avoid surface discharges, we have
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 2 7 3 - 9
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A. Apolinar-Iribe et al.r Optics Communications 153 (1998) 295–300
3. Measurements and results
Fig. 1. Experimental setup. K 1 and K 2 are wavevectors of nearly counterpropagating mutually incoherent pump waves; P1 and P2 are their polarizations. The inset Ža. depicts the front face of the crystal; the region illuminated by the green light is dashed and the intersection region of red beams is shown in black.
used additional illumination from a green frequency-doubled Nd:YAG laser Ž l s 532 nm, 150 mW power; Fig. 1.. The entire crystal was illuminated by the expanded green beam, and the obstacle was inserted in the beam at a distance of 10–50 cm from the crystal, so that a dark horizontal stripe 1 mm thick was created in the region of red beam intersection. As the photoconductivity for green light is much higher than for red, in this arrangement practically all the voltage applied to the crystal drops within the narrow intersection region of the He–Ne laser beams. For the standard cut used ŽFig. 1., there are two polarization modes for which photorefractive gratings are written with opposite signs. This occurs because of the special character of the electrooptic tensor in cubic BTO. Plane polarizations of two nearly counterpropagating pump beams are oriented to fit these modes; they are turned 458 with respect to the direction of the external field and form a 908 angle at the center of the crystal Žbecause of the optical activity of about 6 degreesrmm these relations cannot be satisfied for the entire crystal volume.. A narrow vertical slot Ž0.4 mm. can be additionally inserted into one of the pump beams at a distance of 5–40 cm from the crystal.
When the voltage is high enough, generation develops in the form of beams propagating above and below the horizontal plane ŽFig. 1.. If the first pump beam produces generation above the plane, the second beam produces it below, and both generated beams have a similar structure. The oscillation has a clearly pronounced threshold character ŽFig. 2., the threshold field value is about 15 kVrcm, and this can vary depending on the geometry. Generated beam projection on a screen first appears as a spot with a regular form elongated in the vertical direction ŽFig. 3.. With higher voltage, running fringes are seen. Raising the voltage even more produces fringes of poor regularity, and far from the threshold; the pattern looks like chaotically moving speckles ŽFigs. 3 and 4.. Introducing the angle between pump beams in the horizontal plane is the most important step for obtaining a regular output. If the beams are exactly counterpropagating, generation has an irregular structure even for voltages very close to the threshold value, and the profile of the generated beam is closer to circular. Regular output can be obtained even without inserting a vertical slot into one of the beams ŽFig. 4.. If the slot is inserted, fringes in the output which appear for higher voltages have a more regular shape; without a slot they are curved and the transition to chaos occurs for a lower external field. The two generated beams have a similar form, independent of the insertion of a slot in one of the pump beams. The angular intensity distribution of the generated beam averaged over several minutes is presented in Fig. 5. With higher voltages the intensity maximum is displaced into the region of smaller angles. For high voltage and small
Fig. 2. Intensity of the generated beam as a function of the voltage applied to the crystal. The angle between beams is u s 0.1 rad, the width of the dark stripe is 1 mm. Because of the additional green light illumination nearly all the voltage drops across the dark stripe.
A. Apolinar-Iribe et al.r Optics Communications 153 (1998) 295–300
297
The spacing of the running fringes, which first appear ŽFig. 3b. is the same as the spacing between the diffraction orders of the pump beam after passing the crystal. This, in turn, is determined by the size of the edge of the electric field distribution. The shadow stripe is produced by the obstacle placed into the beam, so the sharpness of the shadow edge depends on the distance between the obstacle and the crystal. By moving the obstacle back and forth we were able to change the spacing between the diffraction orders by a factor of 4; the spacing of the running fringes changed accordingly, so that equality was maintained. This relation was broken only for the largest spacings. In this case there was no formation of running fringes of definite spatial frequency, but the output was developed as moving spots of different sizes. The direction and speed of fringe movement depend on illumination conditions; by moving the dark stripe where red beams pass up and down we were able to invert the direction of fringe displacement. Generally, the running frequency was lower than 0.5 Hz, which is smaller than the characteristic frequency offset of the generated beam ŽFig. 6..
Fig. 3. Output patterns in the far field for different voltages applied to the crystal. The narrow vertical slot is inserted into the pump beam. Ža. 2470 V, Žb. 2530 V, Žc. 3100 V, Žd. 3300 V.
angle between pump beams, the total power in the generated beam can reach 35–40% of the pump power. There is a frequency shift between the pump and generated waves. We have measured this by mixing the generated beam with a split part of the pump at a photodetector and feeding the detector output to a spectrum analyzer. When the pattern is a regular spot, the beat frequency is very well defined ŽFig. 6, 1770 V.. For higher voltages, the signal is a combination of intensity changes in the generated field and the effects of its interference with the local oscillator. It is seen that for high voltages the photodetector output approaches noise in the 3–4 Hz band.
Fig. 4. The same as in Fig. 3, but without the vertical slot. The speckle pattern appears earlier and is more developed. The left picture corresponds to the voltage of 2100 V, the right one to the voltage of 3500 V.
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A. Apolinar-Iribe et al.r Optics Communications 153 (1998) 295–300
Fig. 5. Angular intensity distribution in the generated beam measured along the vertical line passing through the center of the output pattern.
Fig. 6. Spectra of a photodetector signal. The photodetector is placed at center of the pattern and the output beam is mixed with the local oscillator.
A. Apolinar-Iribe et al.r Optics Communications 153 (1998) 295–300
We have also realized the traditional geometry of a double phase-conjugate mirror w11x. For this both beams must have the same polarization, and intersect in a vertical plane. In this case, generation develops in part of a narrow cone passing through pump beam directions. The output is irregular even quite close to the threshold.
299
the external field becomes higher, more oscillation directions and frequencies become possible. If pump beams are exactly counterpropagating, oscillation can occur in the solid angle Žtwo-dimensional set of directions.. But if the pump beams are tilted, Bragg conditions of diffraction have to be satisfied for both of them, and this gives K p1 y K s1 s K s2 y K p2 .
Ž3.
4. Discussion and conclusions The oscillation mechanism for our setup is well known in the theory of four-wave mixing and double phase-conjugate mirror operation w11,12x. The noise wave is amplified due to the interaction with a pump beam. The second pump beam provides feedback. For this feedback to be positive in our geometry it is necessary to have crosspolarized pump beams. If the gain is high enough, generation develops. Degenerate oscillation Žfrequencies of all beams being equal. is impossible in our case. For the unshifted photorefractive grating which is formed in the external DC field under stationary conditions the noise wave cannot be amplified. However, the necessary phase shift between interference pattern and photorefractive grating can be maintained dynamically if the photorefractive grating is running. Note that in BTO, only the grating running with near resonance velocity in the external DC electric field Žor stationary grating for AC field. can provide the necessary gain; the diffusion mechanism is too weak there because of the small electrooptic coefficient. The amplitude of the running photorefractive grating in the external DC field is given by w13x: ESC s y
mE
Ž 1 y vt M KL E . y i vt M Ž 1 q K 2 L2D .
,
Ž1.
where E is the external electric field applied to the sample, m is the fringe contrast, K s 2prL Ž L is the fringe spacing., v is the frequency of the running grating, L E s mt E is the drift length Ž m is the mobility and t is the lifetime of a photoelectron., t M s ´´ 0re m n 0 is the Maxwell relaxation time Ž n 0 is the average steady sate density of photoelectrons, e is the electron charge, ´´ 0 is the dielectric constant., and L D s Ž Dt .1r2 is the photoelectron diffusion length Ž D is the diffusion coefficient.. In Eq. Ž1. the trap saturation is neglected. The condition for oscillation development can be written in the form: Im Ž ESC . rm ) ET ,
Eq. Ž3. means that possible directions of oscillation are now limited to the surface of the wide cone, which passes through directions given by the pump beams ŽFig. 1.. Thus, introducing the angle between pump beams we reduce the number of elemental gratings to a one-dimensional set. This constitutes a big advantage for direct computer simulations, which are practically limited to two spatial and one temporal coordinates. There are two types of chaotic behavior. The first one results from the large number of independent degrees of freedom. Another type, involving the formation of a strange attractor, can occur in relatively simple nonlinear systems where only a few interacting modes Žor degrees of freedom. exist. It is possible that in the proposed device the chaotic behavior is determined by independent outputs produced by different parts of the crystal, and thus belongs to the first type. The interaction region geometry clearly plays a role in the transition to chaos. Introducing a slot, which limits the interaction region and makes it more homogeneous, results in a more regular output. Nevertheless, we are more inclined to interpret the results as the competition of a relatively small number of modes. The experiments demonstrate that near the threshold the output can be considered as a single spatial mode. Running fringes can be possibly interpreted as the appearance of a second mode. The spacing of fringes that appear first is determined by the characteristic size of the strong nonuniformity Žedge of the electric field distribution., thus the instability develops from the strongest perturbation existing in the system. When the voltage is raised even more, higher modes appear, finally leading to a complicated moving pattern. If there is no slot, different vertical ‘slices’ of crystal are relatively independent; this produces the additional fringe curvature and breaking in the horizontal plane of the output. In conclusion, we have demonstrated that the transition from smooth regular distribution to chaotic behavior occurs in the photorefractive parametric oscillator when an external DC field is applied.
Ž2.
where E T is some critical value depending on the sample thickness and electrooptic coefficient. The imaginary part of Eq. Ž1. has a maximum as a function of K and v which is reached for nonzero values of these parameters. Immediately above the voltage threshold, the running space charge wave with corresponding K and v develops. When
Acknowledgements Financial support of CONACyT project 0354PE is gratefully acknowledged. We also thank D. Gale for his help with the manuscript.
300
A. Apolinar-Iribe et al.r Optics Communications 153 (1998) 295–300
References w1x F.T. Arecchi and R.G. Harrison ŽEds.., Selected Papers on Optical Chaos, SPIE Milestone Series, VMS75, 1994. w2x G.J. Gauthier, P. Narum, R.W. Boyd, Phys. Rev. Lett. 58 Ž1987. 1640. w3x F. Laeri, T. Tschudi, J. Albers, Optics Comm. 47 Ž1983. 587. w4x H. Rajbenbach, J.P. Huignard, Optics Lett. 10 Ž1985. 137. w5x J.L. de Bougrenet de la Tochnaye, P. Pellat-Finet, J.P. Huignard, J. Opt. Soc. Am. B 3 Ž1986. 315. w6x S.I. Stepanov, M.P. Petrov, S.L. Sochava, Ferroelectrics 92 Ž1989. 199. w7x F.T. Arecchi, G. Giacomello, P.L. Ramazza, S. Residori, Phys. Rev. Lett. 65 Ž1980. 2531.
w8x B.I. Sturman, S.G. Odulov, U. Van Olfan, et al., J. Opt. Soc. Am. B 11 Ž1994. 1700. w9x S. Odulov, R. Jungen, T. Tschudi, Appl. Phys. B 56 Ž1993. 57. w10x S.T. Stepanov, M.P. Petrov, M.V. Krasin’kova, Sov. Phys. Tech. Phys. 29 Ž1984. 703. w11x S. Weiss, S. Sternklar, B. Fischer, Optics Lett. 12 Ž1987. 114. w12x A. Yariv, D. Pepper, Optics Lett. 1 Ž1977.. w13x S.I. Stepanov, M.P. Petrov, in: P. Gunter, J.P. Huignard ¨ ŽEds.., Photorefractive Materials and their Applications I, Springer, 1988, Chap. 9, p. 263.
Optics Communications 153 Ž1998. 295–300
Transition to optical chaos in a photorefractive parametric oscillator with Bi 12TiO 20 crystal A. Apolinar-Iribe 1, N. Korneev 2 , J.J. Sanchez-Mondragon ´ ´ ´ (INAOE), Apdo. Postal 51 y 216, CP 72000, Puebla, Pue., Mexico Instituto Nacional de Astrofısica, Optica y Electronica ´ ´ Received 22 January 1998; revised 22 April 1998; accepted 11 May 1998
Abstract Frequency-shifted waves are generated in a Bi 12TiO 20 crystal from two nearly counterpropagating orthogonally polarized pump waves when the electric voltage applied to the crystal exceeds the threshold value. Near the voltage threshold the output beams are regular in shape, for higher voltages running fringes appear, whilst far from the threshold chaotically moving speckles are observed. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Many nonlinear optical systems Žlasers, nonlinear cavities, Kerr media with feedback, etc.. produce chaotic output in time or both in time and space w1x. For photorefractive devices chaotic behavior is known for phase-conjugate mirrors w2x, and for various oscillators with external feedback w3–6x. In the experiment with a ring cavity containing a photorefractive gain medium the transition from regular oscillation to chaotic behavior a dependence on the aperture size was observed w7x. Such a transition is highly interesting from the theoretical viewpoint. In mirrorless photorefractive oscillators unstable output patterns have been observed for LiNbO 3 and BaTiO 3 w8,9x, but to our knowledgement, the development of chaos from regular output has not been reported. Here we describe a simple mirrorless oscillator, which produces regular or irregular output depending on the applied voltage. The device we use is the modification of the scheme suggested for degenerate four-wave mixing in cubic photorefractive crystals w10x. In Ref. w10x it was shown that noise generation occurs in the Bi 12TiO 20 crys-
1
Permanent address: Departamento de Fısica, Universidad de ´ Sonora. 2 Corresponding author. E-mail: [email protected]
tal if two orthogonally polarized beams are counterpropagating within it and an external voltage is applied. We show that by changing the angle between the pump beams one can obtain generation with a transition from regular to chaotic profile. We believe that the study of the device can give some useful insights into the appearance of chaotic behavior, due to the one-dimensional character of the output pattern and the clearly seen transition to chaos.
2. Experimental setup The experimental setup is shown in Fig. 1. Two pump beams from independent He–Ne lasers Ž l s 633 nm, 10 mW power each. with 2 mm beam diameter are nearly counterpropagating and intersect in a Bi 12TiO 20 ŽBTO. crystal. In the experiment we use a sample of dimensions of 8 mm = 8 mm = 6 mm with polished faces of 8 mm = 8 mm and silver paste electrodes deposited onto the faces measuring 8 mm = 6 mm. Both beams are in a horizontal plane, but there is an angle Žtypically 0.1 rad. between them. An external electric field is applied vertically in the Ž110. crystallographic direction. When the external DC electric field is applied to the crystal it is quite difficult to obtain a uniform field distribution. To produce a high field in the He–Ne laser beam intersection region and avoid surface discharges, we have
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 2 7 3 - 9
296
A. Apolinar-Iribe et al.r Optics Communications 153 (1998) 295–300
3. Measurements and results
Fig. 1. Experimental setup. K 1 and K 2 are wavevectors of nearly counterpropagating mutually incoherent pump waves; P1 and P2 are their polarizations. The inset Ža. depicts the front face of the crystal; the region illuminated by the green light is dashed and the intersection region of red beams is shown in black.
used additional illumination from a green frequency-doubled Nd:YAG laser Ž l s 532 nm, 150 mW power; Fig. 1.. The entire crystal was illuminated by the expanded green beam, and the obstacle was inserted in the beam at a distance of 10–50 cm from the crystal, so that a dark horizontal stripe 1 mm thick was created in the region of red beam intersection. As the photoconductivity for green light is much higher than for red, in this arrangement practically all the voltage applied to the crystal drops within the narrow intersection region of the He–Ne laser beams. For the standard cut used ŽFig. 1., there are two polarization modes for which photorefractive gratings are written with opposite signs. This occurs because of the special character of the electrooptic tensor in cubic BTO. Plane polarizations of two nearly counterpropagating pump beams are oriented to fit these modes; they are turned 458 with respect to the direction of the external field and form a 908 angle at the center of the crystal Žbecause of the optical activity of about 6 degreesrmm these relations cannot be satisfied for the entire crystal volume.. A narrow vertical slot Ž0.4 mm. can be additionally inserted into one of the pump beams at a distance of 5–40 cm from the crystal.
When the voltage is high enough, generation develops in the form of beams propagating above and below the horizontal plane ŽFig. 1.. If the first pump beam produces generation above the plane, the second beam produces it below, and both generated beams have a similar structure. The oscillation has a clearly pronounced threshold character ŽFig. 2., the threshold field value is about 15 kVrcm, and this can vary depending on the geometry. Generated beam projection on a screen first appears as a spot with a regular form elongated in the vertical direction ŽFig. 3.. With higher voltage, running fringes are seen. Raising the voltage even more produces fringes of poor regularity, and far from the threshold; the pattern looks like chaotically moving speckles ŽFigs. 3 and 4.. Introducing the angle between pump beams in the horizontal plane is the most important step for obtaining a regular output. If the beams are exactly counterpropagating, generation has an irregular structure even for voltages very close to the threshold value, and the profile of the generated beam is closer to circular. Regular output can be obtained even without inserting a vertical slot into one of the beams ŽFig. 4.. If the slot is inserted, fringes in the output which appear for higher voltages have a more regular shape; without a slot they are curved and the transition to chaos occurs for a lower external field. The two generated beams have a similar form, independent of the insertion of a slot in one of the pump beams. The angular intensity distribution of the generated beam averaged over several minutes is presented in Fig. 5. With higher voltages the intensity maximum is displaced into the region of smaller angles. For high voltage and small
Fig. 2. Intensity of the generated beam as a function of the voltage applied to the crystal. The angle between beams is u s 0.1 rad, the width of the dark stripe is 1 mm. Because of the additional green light illumination nearly all the voltage drops across the dark stripe.
A. Apolinar-Iribe et al.r Optics Communications 153 (1998) 295–300
297
The spacing of the running fringes, which first appear ŽFig. 3b. is the same as the spacing between the diffraction orders of the pump beam after passing the crystal. This, in turn, is determined by the size of the edge of the electric field distribution. The shadow stripe is produced by the obstacle placed into the beam, so the sharpness of the shadow edge depends on the distance between the obstacle and the crystal. By moving the obstacle back and forth we were able to change the spacing between the diffraction orders by a factor of 4; the spacing of the running fringes changed accordingly, so that equality was maintained. This relation was broken only for the largest spacings. In this case there was no formation of running fringes of definite spatial frequency, but the output was developed as moving spots of different sizes. The direction and speed of fringe movement depend on illumination conditions; by moving the dark stripe where red beams pass up and down we were able to invert the direction of fringe displacement. Generally, the running frequency was lower than 0.5 Hz, which is smaller than the characteristic frequency offset of the generated beam ŽFig. 6..
Fig. 3. Output patterns in the far field for different voltages applied to the crystal. The narrow vertical slot is inserted into the pump beam. Ža. 2470 V, Žb. 2530 V, Žc. 3100 V, Žd. 3300 V.
angle between pump beams, the total power in the generated beam can reach 35–40% of the pump power. There is a frequency shift between the pump and generated waves. We have measured this by mixing the generated beam with a split part of the pump at a photodetector and feeding the detector output to a spectrum analyzer. When the pattern is a regular spot, the beat frequency is very well defined ŽFig. 6, 1770 V.. For higher voltages, the signal is a combination of intensity changes in the generated field and the effects of its interference with the local oscillator. It is seen that for high voltages the photodetector output approaches noise in the 3–4 Hz band.
Fig. 4. The same as in Fig. 3, but without the vertical slot. The speckle pattern appears earlier and is more developed. The left picture corresponds to the voltage of 2100 V, the right one to the voltage of 3500 V.
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A. Apolinar-Iribe et al.r Optics Communications 153 (1998) 295–300
Fig. 5. Angular intensity distribution in the generated beam measured along the vertical line passing through the center of the output pattern.
Fig. 6. Spectra of a photodetector signal. The photodetector is placed at center of the pattern and the output beam is mixed with the local oscillator.
A. Apolinar-Iribe et al.r Optics Communications 153 (1998) 295–300
We have also realized the traditional geometry of a double phase-conjugate mirror w11x. For this both beams must have the same polarization, and intersect in a vertical plane. In this case, generation develops in part of a narrow cone passing through pump beam directions. The output is irregular even quite close to the threshold.
299
the external field becomes higher, more oscillation directions and frequencies become possible. If pump beams are exactly counterpropagating, oscillation can occur in the solid angle Žtwo-dimensional set of directions.. But if the pump beams are tilted, Bragg conditions of diffraction have to be satisfied for both of them, and this gives K p1 y K s1 s K s2 y K p2 .
Ž3.
4. Discussion and conclusions The oscillation mechanism for our setup is well known in the theory of four-wave mixing and double phase-conjugate mirror operation w11,12x. The noise wave is amplified due to the interaction with a pump beam. The second pump beam provides feedback. For this feedback to be positive in our geometry it is necessary to have crosspolarized pump beams. If the gain is high enough, generation develops. Degenerate oscillation Žfrequencies of all beams being equal. is impossible in our case. For the unshifted photorefractive grating which is formed in the external DC field under stationary conditions the noise wave cannot be amplified. However, the necessary phase shift between interference pattern and photorefractive grating can be maintained dynamically if the photorefractive grating is running. Note that in BTO, only the grating running with near resonance velocity in the external DC electric field Žor stationary grating for AC field. can provide the necessary gain; the diffusion mechanism is too weak there because of the small electrooptic coefficient. The amplitude of the running photorefractive grating in the external DC field is given by w13x: ESC s y
mE
Ž 1 y vt M KL E . y i vt M Ž 1 q K 2 L2D .
,
Ž1.
where E is the external electric field applied to the sample, m is the fringe contrast, K s 2prL Ž L is the fringe spacing., v is the frequency of the running grating, L E s mt E is the drift length Ž m is the mobility and t is the lifetime of a photoelectron., t M s ´´ 0re m n 0 is the Maxwell relaxation time Ž n 0 is the average steady sate density of photoelectrons, e is the electron charge, ´´ 0 is the dielectric constant., and L D s Ž Dt .1r2 is the photoelectron diffusion length Ž D is the diffusion coefficient.. In Eq. Ž1. the trap saturation is neglected. The condition for oscillation development can be written in the form: Im Ž ESC . rm ) ET ,
Eq. Ž3. means that possible directions of oscillation are now limited to the surface of the wide cone, which passes through directions given by the pump beams ŽFig. 1.. Thus, introducing the angle between pump beams we reduce the number of elemental gratings to a one-dimensional set. This constitutes a big advantage for direct computer simulations, which are practically limited to two spatial and one temporal coordinates. There are two types of chaotic behavior. The first one results from the large number of independent degrees of freedom. Another type, involving the formation of a strange attractor, can occur in relatively simple nonlinear systems where only a few interacting modes Žor degrees of freedom. exist. It is possible that in the proposed device the chaotic behavior is determined by independent outputs produced by different parts of the crystal, and thus belongs to the first type. The interaction region geometry clearly plays a role in the transition to chaos. Introducing a slot, which limits the interaction region and makes it more homogeneous, results in a more regular output. Nevertheless, we are more inclined to interpret the results as the competition of a relatively small number of modes. The experiments demonstrate that near the threshold the output can be considered as a single spatial mode. Running fringes can be possibly interpreted as the appearance of a second mode. The spacing of fringes that appear first is determined by the characteristic size of the strong nonuniformity Žedge of the electric field distribution., thus the instability develops from the strongest perturbation existing in the system. When the voltage is raised even more, higher modes appear, finally leading to a complicated moving pattern. If there is no slot, different vertical ‘slices’ of crystal are relatively independent; this produces the additional fringe curvature and breaking in the horizontal plane of the output. In conclusion, we have demonstrated that the transition from smooth regular distribution to chaotic behavior occurs in the photorefractive parametric oscillator when an external DC field is applied.
Ž2.
where E T is some critical value depending on the sample thickness and electrooptic coefficient. The imaginary part of Eq. Ž1. has a maximum as a function of K and v which is reached for nonzero values of these parameters. Immediately above the voltage threshold, the running space charge wave with corresponding K and v develops. When
Acknowledgements Financial support of CONACyT project 0354PE is gratefully acknowledged. We also thank D. Gale for his help with the manuscript.
300
A. Apolinar-Iribe et al.r Optics Communications 153 (1998) 295–300
References w1x F.T. Arecchi and R.G. Harrison ŽEds.., Selected Papers on Optical Chaos, SPIE Milestone Series, VMS75, 1994. w2x G.J. Gauthier, P. Narum, R.W. Boyd, Phys. Rev. Lett. 58 Ž1987. 1640. w3x F. Laeri, T. Tschudi, J. Albers, Optics Comm. 47 Ž1983. 587. w4x H. Rajbenbach, J.P. Huignard, Optics Lett. 10 Ž1985. 137. w5x J.L. de Bougrenet de la Tochnaye, P. Pellat-Finet, J.P. Huignard, J. Opt. Soc. Am. B 3 Ž1986. 315. w6x S.I. Stepanov, M.P. Petrov, S.L. Sochava, Ferroelectrics 92 Ž1989. 199. w7x F.T. Arecchi, G. Giacomello, P.L. Ramazza, S. Residori, Phys. Rev. Lett. 65 Ž1980. 2531.
w8x B.I. Sturman, S.G. Odulov, U. Van Olfan, et al., J. Opt. Soc. Am. B 11 Ž1994. 1700. w9x S. Odulov, R. Jungen, T. Tschudi, Appl. Phys. B 56 Ž1993. 57. w10x S.T. Stepanov, M.P. Petrov, M.V. Krasin’kova, Sov. Phys. Tech. Phys. 29 Ž1984. 703. w11x S. Weiss, S. Sternklar, B. Fischer, Optics Lett. 12 Ž1987. 114. w12x A. Yariv, D. Pepper, Optics Lett. 1 Ž1977.. w13x S.I. Stepanov, M.P. Petrov, in: P. Gunter, J.P. Huignard ¨ ŽEds.., Photorefractive Materials and their Applications I, Springer, 1988, Chap. 9, p. 263.