- Email: [email protected]
Low reflectivity and instability of a self-pumped phase conjugator caused by reflection gratings in +c-face incidence
15 January 1998
Optics Communications 146 Ž1998. 371–378
Full length article
Low reflectivity and instability of a self-pumped phase conjugator caused by reflection gratings in qc-face incidence Hong Lin ) , Adnan Yousuf, David W. Weir, Laura S. Auger Department of Physics, Bates College, 44 Campus AÕenue, Lewiston, ME 04240, USA Received 13 August 1997; revised 15 September 1997; accepted 16 September 1997
Abstract We have explored nonlinear dynamics of a self-pumped phase conjugator when the input beam enters the crystal from the face to which the qc-axis of the crystal points. The pattern dynamics is similar to that in the geometry of a-face incidence, but the phase-conjugate power is much weaker and is always unstable. Two-wave coupling coefficients are calculated and functions of transmission and reflection gratings are discussed. Our analysis shows that reflection gratings are an obstacle to obtain high reflectivity because energy flows from the phase-conjugate wave back to the incident beam for this incident geometry. When qc-face incidence is used, a crystal having a lower number density of charge carriers can have a better phase-conjugate reflectivity. q 1998 Elsevier Science B.V.
1. Introduction The self-pumped phase conjugator ŽSPPC. has been a very active research topic in the past fifteen years. The interest in SPPC arises from both their many applications, such as image storage and processing, and their physical mechanisms. Among various geometries of the SPPC, the one using a single input beam and without external mirrors Žthe so called ‘‘cat’’ phase conjugator. is the simplest. In the cat phase conjugator, four-wave mixing interaction regions are formed via fanning and internal reflections at one corner of the crystal w1x. Another mechanism for obtaining the phase-conjugate wave is stimulated backscattering, in which the backward beam is amplified at the expense of the forward beam w2x. Sometimes these two mechanisms can coexist w3x. Barium titanate ŽBaTiO 3 . crystal is the first and the most often used material for self-pumped phase conjugation because of its high gain for wave coupling. To increase the reflectivity of the phase conjugator, both 08- and 458-cut crystal have been studied
)
Corresponding author. E-mail: [email protected].
and different dopings have been experimented with. When a 08-cut crystal is used as the self-pumped phase conjugator, the input light usually enters the crystal from the face parallel to the c-axis of the crystal Ž a-face incidence.. Some people tried another geometry – the entrance face is the one to which the c-axis is perpendicular Ž c-face incidence., as shown in Fig. 1. For example, Mullen et al. obtained phase conjugation with c-face incidence when they studied stimulated backscattering using a seeding beam, and the reflectivity tended to be lower and less stable in this geometry than when the beam entered the a-face of the crystal w4x. Chang and Selviah reported that when the light entered an undoped crystal from the qc-face Žthe face to which qc-axis points. phase conjugation was achieved over a wide range of lateral positions and angles of the incident beam, which provide a large effective numerical aperture giving high-resolution imaging w5x. Dou et al. studied two BaTiO 3:Ce crystals in a-face and qcface incidences w6x. In the geometry of qc-face incidence, phase-conjugate signal was generated in one sample with a lower doping concentration but not in the other sample that had a higher concentration, though both sample had high reflectivities for a-face incidence. Weaker fanning w4x and
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 7 . 0 0 5 3 5 - X
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H. Lin et al.r Optics Communications 146 (1998) 371–378
Fig. 1. Two different geometries of incidence. Ža. a-face incidence, and Žb. q c-face incidence.
higher backscattering gain factor w6x were suggested as the sources for the differences between the two geometries of incidence, but no detailed study has been done on the mechanism that causes the differences. It is already well known that BaTiO 3 , as well as other photorefractive crystals, can demonstrate a series of complex spatial and temporal instabilities w7–10x. The occurrence of instabilities depends on the incident angle, position, power, and beam diameter of the input beam w11–14x. The instability is believed to be seeded by scattered light w7x and to be generated by competitions among refractive index gratings w9,15x. Functions of different index gratings have been studied. Gruneisen et al. showed experimentally that the presence of reflection gratings leads to a decrease and fluctuation in the phase-conjugate reflectivity of a double phase conjugator w16x, while Cruz et al. found that backscattering and transmission gratings both contribute to
the phase-conjugate signal w17x. Lambelet et al. measured the contributions to phase conjugation made by transmission and reflection gratings respectively in a cobalt-doped BaTiO 3 using low-coherence reflectometry and found the reflectivity from reflection gratings is much weaker than that from the transmission gratings w18x. Nakagawa et al. demonstrated that the buildup of reflection gratings in a ring mirror using a Fe-doped KNbO 3 crystal has a damaging effect on performance w19x. In this paper we report our study on the self-pumped phase conjugator when qc-face incidence is used. We have observed that the power reflectivity of the phase conjugator is much weaker than that of a-face incidence and that the phase-conjugate wave is always temporally unstable. Sometimes unstable spatial patterns are also observed. The analysis of reflection gratings shows that energy is transferred from the backward wave to the forward beam, which is an obstacle to high reflectivity of the phase conjugator.
2. Experimental results Our experiments were carried out with two BaTiO 3 crystals of similar dimensions Ž5.8 = 5.8 = 5.9 mm3 and 5.7 = 5.4 = 5.3 mm3 . but very different number densities of charge carriers Ž2 = 10 16 Ž1rcm3 . and 2 = 10 17 Ž1rcm3 ... The crystal having the lower number density of charge carriers is termed Crystal A in this paper, and its c-axis is parallel to one of the sides of 5.8 mm. The other crystal is termed Crystal B, with its c-axis parallel to one of 5.7 mm sides. Both the crystals were purchased from Sanders Associates. The power reflectivities of the two
Fig. 2. Experimental setup, in which FI means Faraday isolator, BS is beam splitter, and L is the focusing lens.
H. Lin et al.r Optics Communications 146 (1998) 371–378
crystals are up to 40% and 32%, respectively, in a-face incidence. Their spatio-temporal dynamical behaviors when a-face incidence is used were reported in Ref. w13x. The experimental setup is illustrated in Fig. 2. The laser used in the experiment is a helium-neon laser with TEM 00 output and multi-longitudinal modes operating at 633 nm. A Faraday isolator is placed in front of the laser to shield any backward light. The polarization of the laser is changed by a half-wave plate to make the laser be an extraordinary beam with respect to the crystal. The power of the input beam is about 6 mW. The light is focused in the crystal by a lens of 40 cm focal length. The beam diameter at the entrance face is about 0.4 mm, so that intensity of the incident beam is about 4.8 Wrcm2. The backward signal generated by the crystal is detected by a power meter that
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is connected to a dual-channel digital oscilloscope. The beam transmitted through the crystal is received by another power meter. The beam distribution inside the crystal is recorded by a camera set above the crystal. The light can enter the crystal either from the left side of the normal or the right side of the normal of the qc-face. For convenience, we define the incident angle to be positive when the light enters the crystal from the left side of the normal, as the case shown in Fig. 1b. The incident position, d, is measured from the corner from which the light bends away. Crystal A generates phase-conjugate signals for y538 F u F y408 and 198 F u F 328 at d s 3 mm.The power of the phase-conjugate beam fluctuates with time. Sometimes the fluctuation seems quasi-periodic, but most of the time
Fig. 3. Average power of the phase-conjugate wave, Ppc , versus the incident angle, u , of Ža. Crystal A, and Žb. Crystal B. The error bar gives the amplitude of oscillation in each measurement. The power of the input beam is 6.2 mW and the incident position is 3 mm.
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it is irregular. We examined the average power within the above two regions of incident angles. The average power of the phase-conjugate wave increases with u for positive incidence Ž u ) 0., which is similar to that in a-face incidence w14x, but does not have obvious variation for negative incidence Ž u - 0.. However, the phase-conjugate power is much weaker than that obtained in a-face incidence. The maximum power reflectivity is less than 2%. For smaller incident positions, we did not see any signal. In Crystal B, the temporally fluctuating phase-conjugate signal is observed at d s 3 mm as u is varied from y508 to y348. The measured average power is 0.57 mW or less, meaning the phase-conjugate reflectivity is less than 0.05%. There is no signal for u ) 0. For a smaller d Ž d s 2 mm., the backward signal consists of several spots distributed in the direction perpendicular to the incident plane when u changes between y708 and y408. This type of unstable pattern has also been observed in the geometry of a-face incidence w13x and its origin was discussed in Ref. w14x. The measurement of the average power versus the incident angle is summarized in Fig. 3, in which error bars represent the amplitude of the fluctuation. For each data point, the power is averaged over 30 minutes after the
signal is built up. It can be seen that the signal of Crystal A is five to ten times stronger than that of Crystal B. The performance of the crystal is not symmetric about the normal of its qc-face. We also measured the temporal variation of the transmitted light and of the phase-conjugate wave simultaneously. In the geometry of a-face incidence, the transmitted light drops when the phase-conjugate wave is built up, as has been observed by many researchers ŽFig. 4a.. This is because the phase-conjugate signal is formed at the expense of the incident light. Contrary to that, the transmitted light increases as the phase conjugate wave builds up in the geometry of qc-face incidence ŽFigs. 4b and 4c.. The fanning configurations corresponding to positive and negative incidence are shown in Fig. 5. To compare the two geometries of incidence, the fanning beam formed for a-face incidence is given in Fig. 5a. For u ) 0, the configuration in Crystal A is similar to that in a-face incidence. That is, the fanning light is reflected at the lower corner of the crystal to form the standard ‘‘cat’’ loop. There is no loop or filament formed in Crystal B, thus it explains why phase-conjugate signal is not observed for u ) 0. In the case of u - 0, the two crystals demon-
Fig. 4. Temporal behavior of the transmitted light, Pt , when the phase-conjugate signal, Ppc , is built up. Ža. Geometry of a-face incidence is used and u s 508. Žb. qc-face incidence and u s 308. Žc. qc-face incidence and u s y458. The data was taken at d s 3 mm in Crystal A.
H. Lin et al.r Optics Communications 146 (1998) 371–378
strate the same fanning configuration. The fanning beams first bend toward the side face and then are reflected into different directions. Part of the fanning light forms a path including several beams approximately along the diagonal
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line of the crystal, but the beams do not converge to a bright filament. Another part is reflected at two adjacent faces of the crystal and then intersects with the diagonal path. The fanning light has a greater intensity in Crystal B.
Fig. 5. Top view of beam configurations in Ža. Crystal A in the geometry of a-face incidence at u s 358, Žb. Crystal A in qc-face incidence at Ž1. u s 308 and Ž2. u s y458, and Žc. Crystal B in qc-face incidence at Ž1. u s 458 and Ž2. u s y458. d s 3.0 mm for all the pictures. The direction of the c-axis is given by the arrows. The finger-like spot at the upper-right-hand corner in Crystal A is due to a damage at that corner.
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This stronger fanning is due to the higher number density of charge carriers in Crystal B. 3. Reflection grating and beam coupling in H c-face incidence The fanning loops suggest two different mechanisms for phase conjugation, as schematically illustrated in Fig. 6. One case is that there are two four-wave mixing regions and the necessary pump waves are generated via total internal reflection at a corner of the crystal, as proposed by Feinberg w1x. In the other situation, stimulated backward scattering is also involved in addition to four-wave mixing. In the diagonal path of Fig. 6b, the backward scattered light, E b , makes an obtuse angle with the qc-axis and therefore experiences a positive gain w2,20x. It provides the necessary pump wave for the four-wave mixing regions labelled 2, 3, and 4. The phase-conjugate wave of the reflected beam Er becomes one of the pump beams in region 1. The phase-conjugate wave is generated via fourwave mixing in regions 1 and 2. Because the light does not form a bright filament, the phase-conjugate wave is not a strong signal. In the four-wave mixing process, four types of index gratings are formed, as shown in Fig. 7. One is the transmission grating formed by the interference between the incident beam, E 1, and the forward fanning beam, E 2 . Another is the reflection grating between the backward beam, E 3 , and the incident beam, E 1. The third is another reflection grating formed by the phase-conjugate wave, E 4 , and the incident beam, which is also called 2 k grating. And the fourth is the 2 k grating formed by E 3 and E 2 . Among them, the first three can directly cause energy flowing out or into the incident beam by two-wave mixing. We studied the energy transfer by calculating two-wave
Fig. 7. Index gratings formed in a four-wave mixing process. Ža. Transmission grating formed by E 1 and E 2 . Žb. Reflection grating formed by E 1 and E 3 . Žc. Reflection grating Ž2 k grating. formed by E 4 and E 1 , where E 4 s E 1) . Žd. 2 k grating formed by E 2 and E 3 . E 1 is the input beam in self-pump phase conjugation. The angle between each beam and the q z-direction is given in terms of a 1 and a 2 .
coupling coefficients for the index gratings. Consider two beams, E 1 and E i Ž i s 2, 3, 4., with wave vector k 1 and k i , respectively, and assume that the beam with k i is much weaker than that with k 1. The grating vector is k g s k i y k 1. The saturated coupling gain for the weak beam E i is proportional to the real part of the coupling coefficient times the effective coupling length. When the coupling coefficient is positive, energy is transferred from E 1 to E i . When the coupling coefficient becomes negative, the direction of energy flow is reversed. In a barium titanate crystal without an applied electric field, the coupling coefficient, g , is real and is given by w1,21x
gs
v Esc reff 2 nc cos d
,
Ž1.
where v is the angular frequency of the beam, n is the index of refraction corresponding to the polarization of the beam, and c is the speed of light. For extraordinary light, the electro-optic coefficient, reff , is given by the following relation w21,22x, reff s 12 n 4o r13 Ž cos2 u y cos2 b . q 4 n2e n 2o r42 sin2b qn4e r 33 Ž cos2 u q cos2 b . cos b ,
Fig. 6. Schematic diagrams of two different mechanisms for self-pumped phase conjugation in the geometry of q c-face incidence. Ža. Two four-wave mixing regions. Žb. Both four-wave mixing and backscattering are involved.
Ž2.
where f is the half angle between the two coupling beams in the crystal, b is the angle between grating vector k g and the qc-axis, n o and n e are refractive indices of the ordinary and extraordinary beam. For l s 633 nm, n o s 2.41 and n e s 2.37. The unclamped crystal parameters are r 13 s 24 pmrV, r 33 s 80 pmrV, r42 s 1640 pmrV. When
H. Lin et al.r Optics Communications 146 (1998) 371–378
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where k bT is the thermal energy, eˆ1 and eˆ2 are unit vectors along beam polarizations and their dot product gives cos 2 f , k g is the magnitude of the grating vector and k 0 is a constant of the crystal that depends on the number density of charge carriers, N, and the relative dielectric constant ´ in the direction of grating vector. It is given by k 0 s Ž Nq 2r´ 0 ´ k bT .
1r2
,
where q s 1.6 = 10 C, ´ s ´aŽsin b . 2 q ´ c Žcos b . 2 with ´a s 3700 and ´c s 135. Using the number density of charge carriers determined for our crystals, we calculated g versus the fanning angle a 2 y a 1 for the transmission grating and reflection grating, respectively. g varies as a function of a 1 for the 2 k grating since the angle between the two beams is 1808. The value of g is positive as a 2 y a 1 changes from 08 to 908 for the transmission grating. In fact, this is why fanning is built up-energy is transferred from the incident beam to the fanning loops. However, g is negative for most incident angles when 2 k grating is considered. The g curves of the two crystals for 2 k grating are shown in Fig. 8. When the incident angle, a 1, in the crystal varies from 08 to 908, g becomes negative when a 1 is greater than 58. Experimentally, a 1 can be changed from 08 to 258 since a 1 s siny1 Žsin urn e .. The minimum a 1 for us to observe the phase-conjugate signal is 88. Thus g for 2 k grating is always negative in our experiments. This means that energy flows from the phase-conjugate signal back to the incident beam so that the reflectivity of the phase-conjugate mirror is decreased. In the a-face geometry, the value of g is positive, so the 2 k grating contributes to the phase-conjugate signal. This qualitatively explains the observation that the power reflectivity is much weaker in the case of qc-face incidence. This also explains why the transmitted light increases when the signal begins to grow. It is worth noting that g corresponding to the reflection grating is also negative in the qc-face incidence. Comparing Fig. 8a to Fig. 8b, one can see that the absolute value of g is approximately ten times greater in Crystal B than that in Crystal A. In other words, the energy transferred from the phase-conjugate signal to the incident beam in Crystal B is about ten times more than that in Crystal A. That is due to the difference in their number densities of charge carriers-NA is ten times less than NB . Experimentally, we did observe that the average power of the phase conjugate wave of Crystal B is about one order of magnitude lower than that of Crystal A for u - 0 ŽFig. 3.. Therefore, a high number density of charge carriers has a damaging effect in the geometry of qc-face incidence. y19
Fig. 8. Two-wave coupling coefficient, g , versus incident angle in the crystal, a 1 for Ža. Crystal A, N s 2=10 16 Ž1rcm3 ., and Žb. Crystal B, N s 2=10 17 Ž1rcm3 .. T s 294 K.
both the beams propagate in the forward direction and form a transmission grating Žshown in Fig. 7a., f s Ž a 2 y a 1 .r2 and b s 908 y Ž a 2 q a 1 .r2, where a 1 and a 2 are the angles between the qz-direction and the two beams, respectively, in the crystal. If one is a forward beam and the other is a backward beam as that in Fig. 7b Žreflection grating., f s 908 yŽ a 2 y a 1 .r2 and b s Ž a 2 q a 1 .r2. For two counterpropagating beams ŽFig. 7c., a 1 s a 2 , so that f s 908 and b s a 1. Correspondingly, d is equal to f in Fig. 7a, is 908 y f in Fig. 7b w23x, and becomes zero in Fig. 7c. Esc is the space-charge electric field induced by the interference of light and is given by Esc s
k bT
kg
q 1 q Ž k grk 0 . 2
eˆ1 P eˆ2) ,
Ž3.
4. Discussion and summary Though the calculation of coupling coefficients gives the same curve for positive incidence Ž u ) 0. and negative
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incidence Ž u - 0., asymmetric performance is observed in both crystals. The asymmetry is demonstrated by different signal powers and different beam configurations. The asymmetry may result from inhomogeneities andror defects in the crystal. Recently Xie et al. reported their numerical results of self-pump phase conjugation. They found that fanning configuration and reflectivity of the phase conjugator are sensitive to the seed level Žrepresenting inhomogeneities and defects. used in the numerical simulation w24x. Besides the low power reflectivity, all of the phase-conjugate waves fluctuate with time in qc-face incidence. We think that the fluctuation is caused by competition among index gratings and may also come from beating between phase-conjugate waves and other beams since the phaseconjugate wave can have a frequency shift from the input beam w11,13x. As discussed above, there is more than one four-wave mixing process for the self-pumped phase conjugation, sometimes along with backscattering. In each four-wave mixing region, four types of gratings are formed. In fact, it has been shown both theoretically w9,15x and experimentally w16x that the coexistence of gratings leads to instabilities. In summary, we have studied spatio-temporal dynamics of two BaTiO 3 self-pumped phase-conjugators when the input light enters the crystal from its qc-face. The phaseconjugate wave is much weaker than that obtained in a-face incidence. Our analysis of beam coupling coefficients shows that the reflection gratings cause a loss in the phase-conjugate signal. The higher the number density of charge carriers, the greater the loss. The analysis is in good agreement with the experimental observations.
Acknowledgements We thank Mordechai Segev for helpful discussions. Our work is supported by grant PHY-9423882 from the National Science Foundation and a Cottrell College Science Award from Research Corporation. Justin Freeman participated in part of the experiments.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x
J. Feinberg, Optics Lett. 7 Ž1982. 486. T.Y. Chang, R.W. Hellwarth, Optics Lett. 10 Ž1985. 408. Y. Lian, H. Cao, P. Ye, Appl. Phys. Lett. 63 Ž1993. R.A. Mullen, D.J. Vickers, L. West, D.M. Pepper, J. Opt. Soc. Am. B 9 Ž1992. 1726. C.C. Chang, D.R. Selviah, Optics Comm. 124 Ž1996. 481. S.X. Dou, H. Cao, J. Zhang, Y. Lian, Y. Zhu, X. Wu, C. Yang, P. Ye, J. Opt. Soc. Am. B 12 Ž1995. 1048. P. Gunter, E. Voit, M.Z. Zha, J. Albers, Optics Comm. 55 ¨ Ž1985. 210. A.M. Smout, R.W. Eason, M.C. Gower, Optics Comm. 59 Ž1986. 77. D.J. Gauthier, P. Narum, R.W. Boyd, Phys. Rev. Lett. 58 Ž1987. 1640. T. Honda, Optics Lett. 18 Ž1993. 598. M.C. Gower, P. Hribek, J. Opt. Soc. Am. B 5 Ž1988. 1750. A.V. Nowak, T.T. Moore, R.A. Fisher, J. Opt. Soc. Am. 5 Ž1988. 1864. J. Dai, H. Lin, Optics Comm. 113 Ž1994. 335. H. Lin, J.E. Templeton, J. Opt. Soc. Am. B 14 Ž1997. 99. B.Q. He, P. Yeh, C. Gu, R.R. Neurgaonkar, J. Opt. Soc. Am. B 9 Ž1992. 114. M.T. Gruneisen, E.D. Seeberger, J.F. Mileski, K. Koch, Optics Lett. 16 Ž1991. 596. S.-C.D.L. Cruz, S. MacCormack, J. Feinberg, Q.B. He, H.-K. Liu, P. Yeh, J. Opt. Soc. Am. B 12 Ž1995. 1363. P. Lambelet, R.P. Salathe, ´ M.H. Garrett, D. Rytz, Appl. Phys. Lett. 64 Ž1994. 1079. K. Nakagawa, M. Zgonik, P. Gunter, J. Opt. Soc. Am. B 14 ¨ Ž1997. 839. T. Honda, T. Yamashita, H. Matsumoto, Optics Comm. 103 Ž1993. 434. Y. Fainman, E. Klancnik, S.H. Lee, Opt. Eng. 25 Ž1986. 228. J. Feinberg, D. Heiman, A.R. Tanguary Jr., R.W. Hellwarth, J. Appl. Phys. 51 Ž1980. 1297. P. Yeh, Optics Comm. 45 Ž1983. 323. P. Xie, J.-H. Dai, P.-Y. Wang, H.-J. Zhang, Phys. Rev. A 55 Ž1997. 3092.
Optics Communications 146 Ž1998. 371–378
Full length article
Low reflectivity and instability of a self-pumped phase conjugator caused by reflection gratings in qc-face incidence Hong Lin ) , Adnan Yousuf, David W. Weir, Laura S. Auger Department of Physics, Bates College, 44 Campus AÕenue, Lewiston, ME 04240, USA Received 13 August 1997; revised 15 September 1997; accepted 16 September 1997
Abstract We have explored nonlinear dynamics of a self-pumped phase conjugator when the input beam enters the crystal from the face to which the qc-axis of the crystal points. The pattern dynamics is similar to that in the geometry of a-face incidence, but the phase-conjugate power is much weaker and is always unstable. Two-wave coupling coefficients are calculated and functions of transmission and reflection gratings are discussed. Our analysis shows that reflection gratings are an obstacle to obtain high reflectivity because energy flows from the phase-conjugate wave back to the incident beam for this incident geometry. When qc-face incidence is used, a crystal having a lower number density of charge carriers can have a better phase-conjugate reflectivity. q 1998 Elsevier Science B.V.
1. Introduction The self-pumped phase conjugator ŽSPPC. has been a very active research topic in the past fifteen years. The interest in SPPC arises from both their many applications, such as image storage and processing, and their physical mechanisms. Among various geometries of the SPPC, the one using a single input beam and without external mirrors Žthe so called ‘‘cat’’ phase conjugator. is the simplest. In the cat phase conjugator, four-wave mixing interaction regions are formed via fanning and internal reflections at one corner of the crystal w1x. Another mechanism for obtaining the phase-conjugate wave is stimulated backscattering, in which the backward beam is amplified at the expense of the forward beam w2x. Sometimes these two mechanisms can coexist w3x. Barium titanate ŽBaTiO 3 . crystal is the first and the most often used material for self-pumped phase conjugation because of its high gain for wave coupling. To increase the reflectivity of the phase conjugator, both 08- and 458-cut crystal have been studied
)
Corresponding author. E-mail: [email protected].
and different dopings have been experimented with. When a 08-cut crystal is used as the self-pumped phase conjugator, the input light usually enters the crystal from the face parallel to the c-axis of the crystal Ž a-face incidence.. Some people tried another geometry – the entrance face is the one to which the c-axis is perpendicular Ž c-face incidence., as shown in Fig. 1. For example, Mullen et al. obtained phase conjugation with c-face incidence when they studied stimulated backscattering using a seeding beam, and the reflectivity tended to be lower and less stable in this geometry than when the beam entered the a-face of the crystal w4x. Chang and Selviah reported that when the light entered an undoped crystal from the qc-face Žthe face to which qc-axis points. phase conjugation was achieved over a wide range of lateral positions and angles of the incident beam, which provide a large effective numerical aperture giving high-resolution imaging w5x. Dou et al. studied two BaTiO 3:Ce crystals in a-face and qcface incidences w6x. In the geometry of qc-face incidence, phase-conjugate signal was generated in one sample with a lower doping concentration but not in the other sample that had a higher concentration, though both sample had high reflectivities for a-face incidence. Weaker fanning w4x and
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 7 . 0 0 5 3 5 - X
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H. Lin et al.r Optics Communications 146 (1998) 371–378
Fig. 1. Two different geometries of incidence. Ža. a-face incidence, and Žb. q c-face incidence.
higher backscattering gain factor w6x were suggested as the sources for the differences between the two geometries of incidence, but no detailed study has been done on the mechanism that causes the differences. It is already well known that BaTiO 3 , as well as other photorefractive crystals, can demonstrate a series of complex spatial and temporal instabilities w7–10x. The occurrence of instabilities depends on the incident angle, position, power, and beam diameter of the input beam w11–14x. The instability is believed to be seeded by scattered light w7x and to be generated by competitions among refractive index gratings w9,15x. Functions of different index gratings have been studied. Gruneisen et al. showed experimentally that the presence of reflection gratings leads to a decrease and fluctuation in the phase-conjugate reflectivity of a double phase conjugator w16x, while Cruz et al. found that backscattering and transmission gratings both contribute to
the phase-conjugate signal w17x. Lambelet et al. measured the contributions to phase conjugation made by transmission and reflection gratings respectively in a cobalt-doped BaTiO 3 using low-coherence reflectometry and found the reflectivity from reflection gratings is much weaker than that from the transmission gratings w18x. Nakagawa et al. demonstrated that the buildup of reflection gratings in a ring mirror using a Fe-doped KNbO 3 crystal has a damaging effect on performance w19x. In this paper we report our study on the self-pumped phase conjugator when qc-face incidence is used. We have observed that the power reflectivity of the phase conjugator is much weaker than that of a-face incidence and that the phase-conjugate wave is always temporally unstable. Sometimes unstable spatial patterns are also observed. The analysis of reflection gratings shows that energy is transferred from the backward wave to the forward beam, which is an obstacle to high reflectivity of the phase conjugator.
2. Experimental results Our experiments were carried out with two BaTiO 3 crystals of similar dimensions Ž5.8 = 5.8 = 5.9 mm3 and 5.7 = 5.4 = 5.3 mm3 . but very different number densities of charge carriers Ž2 = 10 16 Ž1rcm3 . and 2 = 10 17 Ž1rcm3 ... The crystal having the lower number density of charge carriers is termed Crystal A in this paper, and its c-axis is parallel to one of the sides of 5.8 mm. The other crystal is termed Crystal B, with its c-axis parallel to one of 5.7 mm sides. Both the crystals were purchased from Sanders Associates. The power reflectivities of the two
Fig. 2. Experimental setup, in which FI means Faraday isolator, BS is beam splitter, and L is the focusing lens.
H. Lin et al.r Optics Communications 146 (1998) 371–378
crystals are up to 40% and 32%, respectively, in a-face incidence. Their spatio-temporal dynamical behaviors when a-face incidence is used were reported in Ref. w13x. The experimental setup is illustrated in Fig. 2. The laser used in the experiment is a helium-neon laser with TEM 00 output and multi-longitudinal modes operating at 633 nm. A Faraday isolator is placed in front of the laser to shield any backward light. The polarization of the laser is changed by a half-wave plate to make the laser be an extraordinary beam with respect to the crystal. The power of the input beam is about 6 mW. The light is focused in the crystal by a lens of 40 cm focal length. The beam diameter at the entrance face is about 0.4 mm, so that intensity of the incident beam is about 4.8 Wrcm2. The backward signal generated by the crystal is detected by a power meter that
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is connected to a dual-channel digital oscilloscope. The beam transmitted through the crystal is received by another power meter. The beam distribution inside the crystal is recorded by a camera set above the crystal. The light can enter the crystal either from the left side of the normal or the right side of the normal of the qc-face. For convenience, we define the incident angle to be positive when the light enters the crystal from the left side of the normal, as the case shown in Fig. 1b. The incident position, d, is measured from the corner from which the light bends away. Crystal A generates phase-conjugate signals for y538 F u F y408 and 198 F u F 328 at d s 3 mm.The power of the phase-conjugate beam fluctuates with time. Sometimes the fluctuation seems quasi-periodic, but most of the time
Fig. 3. Average power of the phase-conjugate wave, Ppc , versus the incident angle, u , of Ža. Crystal A, and Žb. Crystal B. The error bar gives the amplitude of oscillation in each measurement. The power of the input beam is 6.2 mW and the incident position is 3 mm.
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it is irregular. We examined the average power within the above two regions of incident angles. The average power of the phase-conjugate wave increases with u for positive incidence Ž u ) 0., which is similar to that in a-face incidence w14x, but does not have obvious variation for negative incidence Ž u - 0.. However, the phase-conjugate power is much weaker than that obtained in a-face incidence. The maximum power reflectivity is less than 2%. For smaller incident positions, we did not see any signal. In Crystal B, the temporally fluctuating phase-conjugate signal is observed at d s 3 mm as u is varied from y508 to y348. The measured average power is 0.57 mW or less, meaning the phase-conjugate reflectivity is less than 0.05%. There is no signal for u ) 0. For a smaller d Ž d s 2 mm., the backward signal consists of several spots distributed in the direction perpendicular to the incident plane when u changes between y708 and y408. This type of unstable pattern has also been observed in the geometry of a-face incidence w13x and its origin was discussed in Ref. w14x. The measurement of the average power versus the incident angle is summarized in Fig. 3, in which error bars represent the amplitude of the fluctuation. For each data point, the power is averaged over 30 minutes after the
signal is built up. It can be seen that the signal of Crystal A is five to ten times stronger than that of Crystal B. The performance of the crystal is not symmetric about the normal of its qc-face. We also measured the temporal variation of the transmitted light and of the phase-conjugate wave simultaneously. In the geometry of a-face incidence, the transmitted light drops when the phase-conjugate wave is built up, as has been observed by many researchers ŽFig. 4a.. This is because the phase-conjugate signal is formed at the expense of the incident light. Contrary to that, the transmitted light increases as the phase conjugate wave builds up in the geometry of qc-face incidence ŽFigs. 4b and 4c.. The fanning configurations corresponding to positive and negative incidence are shown in Fig. 5. To compare the two geometries of incidence, the fanning beam formed for a-face incidence is given in Fig. 5a. For u ) 0, the configuration in Crystal A is similar to that in a-face incidence. That is, the fanning light is reflected at the lower corner of the crystal to form the standard ‘‘cat’’ loop. There is no loop or filament formed in Crystal B, thus it explains why phase-conjugate signal is not observed for u ) 0. In the case of u - 0, the two crystals demon-
Fig. 4. Temporal behavior of the transmitted light, Pt , when the phase-conjugate signal, Ppc , is built up. Ža. Geometry of a-face incidence is used and u s 508. Žb. qc-face incidence and u s 308. Žc. qc-face incidence and u s y458. The data was taken at d s 3 mm in Crystal A.
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strate the same fanning configuration. The fanning beams first bend toward the side face and then are reflected into different directions. Part of the fanning light forms a path including several beams approximately along the diagonal
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line of the crystal, but the beams do not converge to a bright filament. Another part is reflected at two adjacent faces of the crystal and then intersects with the diagonal path. The fanning light has a greater intensity in Crystal B.
Fig. 5. Top view of beam configurations in Ža. Crystal A in the geometry of a-face incidence at u s 358, Žb. Crystal A in qc-face incidence at Ž1. u s 308 and Ž2. u s y458, and Žc. Crystal B in qc-face incidence at Ž1. u s 458 and Ž2. u s y458. d s 3.0 mm for all the pictures. The direction of the c-axis is given by the arrows. The finger-like spot at the upper-right-hand corner in Crystal A is due to a damage at that corner.
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This stronger fanning is due to the higher number density of charge carriers in Crystal B. 3. Reflection grating and beam coupling in H c-face incidence The fanning loops suggest two different mechanisms for phase conjugation, as schematically illustrated in Fig. 6. One case is that there are two four-wave mixing regions and the necessary pump waves are generated via total internal reflection at a corner of the crystal, as proposed by Feinberg w1x. In the other situation, stimulated backward scattering is also involved in addition to four-wave mixing. In the diagonal path of Fig. 6b, the backward scattered light, E b , makes an obtuse angle with the qc-axis and therefore experiences a positive gain w2,20x. It provides the necessary pump wave for the four-wave mixing regions labelled 2, 3, and 4. The phase-conjugate wave of the reflected beam Er becomes one of the pump beams in region 1. The phase-conjugate wave is generated via fourwave mixing in regions 1 and 2. Because the light does not form a bright filament, the phase-conjugate wave is not a strong signal. In the four-wave mixing process, four types of index gratings are formed, as shown in Fig. 7. One is the transmission grating formed by the interference between the incident beam, E 1, and the forward fanning beam, E 2 . Another is the reflection grating between the backward beam, E 3 , and the incident beam, E 1. The third is another reflection grating formed by the phase-conjugate wave, E 4 , and the incident beam, which is also called 2 k grating. And the fourth is the 2 k grating formed by E 3 and E 2 . Among them, the first three can directly cause energy flowing out or into the incident beam by two-wave mixing. We studied the energy transfer by calculating two-wave
Fig. 7. Index gratings formed in a four-wave mixing process. Ža. Transmission grating formed by E 1 and E 2 . Žb. Reflection grating formed by E 1 and E 3 . Žc. Reflection grating Ž2 k grating. formed by E 4 and E 1 , where E 4 s E 1) . Žd. 2 k grating formed by E 2 and E 3 . E 1 is the input beam in self-pump phase conjugation. The angle between each beam and the q z-direction is given in terms of a 1 and a 2 .
coupling coefficients for the index gratings. Consider two beams, E 1 and E i Ž i s 2, 3, 4., with wave vector k 1 and k i , respectively, and assume that the beam with k i is much weaker than that with k 1. The grating vector is k g s k i y k 1. The saturated coupling gain for the weak beam E i is proportional to the real part of the coupling coefficient times the effective coupling length. When the coupling coefficient is positive, energy is transferred from E 1 to E i . When the coupling coefficient becomes negative, the direction of energy flow is reversed. In a barium titanate crystal without an applied electric field, the coupling coefficient, g , is real and is given by w1,21x
gs
v Esc reff 2 nc cos d
,
Ž1.
where v is the angular frequency of the beam, n is the index of refraction corresponding to the polarization of the beam, and c is the speed of light. For extraordinary light, the electro-optic coefficient, reff , is given by the following relation w21,22x, reff s 12 n 4o r13 Ž cos2 u y cos2 b . q 4 n2e n 2o r42 sin2b qn4e r 33 Ž cos2 u q cos2 b . cos b ,
Fig. 6. Schematic diagrams of two different mechanisms for self-pumped phase conjugation in the geometry of q c-face incidence. Ža. Two four-wave mixing regions. Žb. Both four-wave mixing and backscattering are involved.
Ž2.
where f is the half angle between the two coupling beams in the crystal, b is the angle between grating vector k g and the qc-axis, n o and n e are refractive indices of the ordinary and extraordinary beam. For l s 633 nm, n o s 2.41 and n e s 2.37. The unclamped crystal parameters are r 13 s 24 pmrV, r 33 s 80 pmrV, r42 s 1640 pmrV. When
H. Lin et al.r Optics Communications 146 (1998) 371–378
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where k bT is the thermal energy, eˆ1 and eˆ2 are unit vectors along beam polarizations and their dot product gives cos 2 f , k g is the magnitude of the grating vector and k 0 is a constant of the crystal that depends on the number density of charge carriers, N, and the relative dielectric constant ´ in the direction of grating vector. It is given by k 0 s Ž Nq 2r´ 0 ´ k bT .
1r2
,
where q s 1.6 = 10 C, ´ s ´aŽsin b . 2 q ´ c Žcos b . 2 with ´a s 3700 and ´c s 135. Using the number density of charge carriers determined for our crystals, we calculated g versus the fanning angle a 2 y a 1 for the transmission grating and reflection grating, respectively. g varies as a function of a 1 for the 2 k grating since the angle between the two beams is 1808. The value of g is positive as a 2 y a 1 changes from 08 to 908 for the transmission grating. In fact, this is why fanning is built up-energy is transferred from the incident beam to the fanning loops. However, g is negative for most incident angles when 2 k grating is considered. The g curves of the two crystals for 2 k grating are shown in Fig. 8. When the incident angle, a 1, in the crystal varies from 08 to 908, g becomes negative when a 1 is greater than 58. Experimentally, a 1 can be changed from 08 to 258 since a 1 s siny1 Žsin urn e .. The minimum a 1 for us to observe the phase-conjugate signal is 88. Thus g for 2 k grating is always negative in our experiments. This means that energy flows from the phase-conjugate signal back to the incident beam so that the reflectivity of the phase-conjugate mirror is decreased. In the a-face geometry, the value of g is positive, so the 2 k grating contributes to the phase-conjugate signal. This qualitatively explains the observation that the power reflectivity is much weaker in the case of qc-face incidence. This also explains why the transmitted light increases when the signal begins to grow. It is worth noting that g corresponding to the reflection grating is also negative in the qc-face incidence. Comparing Fig. 8a to Fig. 8b, one can see that the absolute value of g is approximately ten times greater in Crystal B than that in Crystal A. In other words, the energy transferred from the phase-conjugate signal to the incident beam in Crystal B is about ten times more than that in Crystal A. That is due to the difference in their number densities of charge carriers-NA is ten times less than NB . Experimentally, we did observe that the average power of the phase conjugate wave of Crystal B is about one order of magnitude lower than that of Crystal A for u - 0 ŽFig. 3.. Therefore, a high number density of charge carriers has a damaging effect in the geometry of qc-face incidence. y19
Fig. 8. Two-wave coupling coefficient, g , versus incident angle in the crystal, a 1 for Ža. Crystal A, N s 2=10 16 Ž1rcm3 ., and Žb. Crystal B, N s 2=10 17 Ž1rcm3 .. T s 294 K.
both the beams propagate in the forward direction and form a transmission grating Žshown in Fig. 7a., f s Ž a 2 y a 1 .r2 and b s 908 y Ž a 2 q a 1 .r2, where a 1 and a 2 are the angles between the qz-direction and the two beams, respectively, in the crystal. If one is a forward beam and the other is a backward beam as that in Fig. 7b Žreflection grating., f s 908 yŽ a 2 y a 1 .r2 and b s Ž a 2 q a 1 .r2. For two counterpropagating beams ŽFig. 7c., a 1 s a 2 , so that f s 908 and b s a 1. Correspondingly, d is equal to f in Fig. 7a, is 908 y f in Fig. 7b w23x, and becomes zero in Fig. 7c. Esc is the space-charge electric field induced by the interference of light and is given by Esc s
k bT
kg
q 1 q Ž k grk 0 . 2
eˆ1 P eˆ2) ,
Ž3.
4. Discussion and summary Though the calculation of coupling coefficients gives the same curve for positive incidence Ž u ) 0. and negative
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incidence Ž u - 0., asymmetric performance is observed in both crystals. The asymmetry is demonstrated by different signal powers and different beam configurations. The asymmetry may result from inhomogeneities andror defects in the crystal. Recently Xie et al. reported their numerical results of self-pump phase conjugation. They found that fanning configuration and reflectivity of the phase conjugator are sensitive to the seed level Žrepresenting inhomogeneities and defects. used in the numerical simulation w24x. Besides the low power reflectivity, all of the phase-conjugate waves fluctuate with time in qc-face incidence. We think that the fluctuation is caused by competition among index gratings and may also come from beating between phase-conjugate waves and other beams since the phaseconjugate wave can have a frequency shift from the input beam w11,13x. As discussed above, there is more than one four-wave mixing process for the self-pumped phase conjugation, sometimes along with backscattering. In each four-wave mixing region, four types of gratings are formed. In fact, it has been shown both theoretically w9,15x and experimentally w16x that the coexistence of gratings leads to instabilities. In summary, we have studied spatio-temporal dynamics of two BaTiO 3 self-pumped phase-conjugators when the input light enters the crystal from its qc-face. The phaseconjugate wave is much weaker than that obtained in a-face incidence. Our analysis of beam coupling coefficients shows that the reflection gratings cause a loss in the phase-conjugate signal. The higher the number density of charge carriers, the greater the loss. The analysis is in good agreement with the experimental observations.
Acknowledgements We thank Mordechai Segev for helpful discussions. Our work is supported by grant PHY-9423882 from the National Science Foundation and a Cottrell College Science Award from Research Corporation. Justin Freeman participated in part of the experiments.
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