- Email: [email protected]
Ordinary polarized phase conjugator using the photovoltaic effect
1 October 2000
Optics Communications 184 (2000) 257±263
www.elsevier.com/locate/optcom
Ordinary polarized phase conjugator using the photovoltaic eect Hon Fai Yau a,*, Hong Chang Kung a, Hsiao Yi Lee b, Ching Cherng Sun a, Tzu Chiang Chen a, Chi Ching Chang c, Yuh Ping Tong c, Junewen Chen d a
Institute of Optical Sciences, National Central University, Chungli, Taoyuan 32054, Taiwan, ROC Department of Electronic Engineering, Minghsin Institute of Technology, Hsinchu 304, Taiwan, ROC c Department of Applied Physics, Chung Cheng Institute of Technology, Tahsi, Taoyuan 335, Taiwan, ROC Electro-Optics Section, Chung Shan Institute of Science and Technology, P.O. Box 90008-8-10, Lung-Tan 32500, Taiwan, ROC b
d
Received 22 May 2000; received in revised form 17 July 2000; accepted 24 July 2000
Abstract Self-pumped phase conjugators so far always work with extraordinary (e) polarized light waves. Theoretical analysis however has pointed out that the e-polarized waves and ordinary (o) polarized waves could be coupled together via the circular photovoltaic eect. We report here an experimental realization of this theoretical prediction and present an o-polarized conjugator. The core element of this setup is a 0°-cut undoped BaTiO3 crystal, which is aligned in a Catconjugator architecture with respect to an incident beam of e-polarized waves. Simultaneously we shine a beam of o-polarized waves on the crystal. Upon the appearance of the e-conjugate waves, we also observed the o-conjugate waves. The resolution of this ordinary polarized phase conjugator is comparable to that of the prevailing e-waves conjugators. Ó 2000 Published by Elsevier Science B.V. PACS: 72.40; 42.65.H; 42.70.M; 42.25.J Keywords: Photovoltaic eect; Photogalvanic eect; Photorefractive eect; Double phase conjugator; Ordinary polarized phase conjugator
Photorefractive crystals are materials capable of recording dynamic phase volume gratings [1]. By manipulating the writing and reading of these phase gratings, applications can be found in optical interconnection, optical computing [2], optical information processing [3,4], phase-conjugate interferometry [5,6], optical data storage [7±10] and biomedical applications [11].
* Corresponding author. Tel.: +886-3-426-3185; fax: +886-3425-2897. E-mail address: [email protected] (H.F. Yau).
Owing to its large electro-optic coecient r42 [12], BaTiO3 crystals have been extensively used to demonstrate the eects of two-wave mixing [13], four-wave mixing [14], and self-pumped phase conjugation [15]. It is well known that self-pumped phase conjugate waves can be generated through photorefractive four-wave mixing in a piece of high gain photorefractive BaTiO3 crystal [16,17]. In order to have a signi®cant wave mixing, the polarization state of the incident wave must be extraordinary with respect to the crystal because only under this condition can the mixing of the waves fully use the largest electro-optic coecient
0030-4018/00/$ - see front matter Ó 2000 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 ( 0 0 ) 0 0 9 1 4 - 7
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H.F. Yau et al. / Optics Communications 184 (2000) 257±263
r42 and thus leads to the production of a beam of conjugate waves [18,19]. Accordingly the reported self-pumped phase conjugators always work with e-polarized light beams. However, four-wave mixing can take place in a photorefractive crystal such as a piece of BaTiO3 crystal with a pair of counter-propagating e-polarized waves and a beam of ordinary polarized (o-polarized) light. The waves here are mixed via the circular photovoltaic eect [20±22]. The eect is usually small but it becomes observable when the propagating direction of the three light beams are normal to the c^-axis of a photorefractive BaTiO3 crystal. In this situation, the coupling parameter can be derived from the coupled-mode equations as [23] D * $ * E 1 c Cm ESC ee v eo ; *$ + $ e $ * e ; v r kg e0 e0
$
2
where Cm is a constant, ESC , the amplitude of the * * space charge ®eld in the BaTiO3 crystal; ee and eo , the unit vectors of polarization directions of the incident e-polarized and o-polarized beams, re$ spectively; v , the coupling tensor between the e$ polarized and o-polarized beams; e , the dielectric * $ tensor; r , the electro-optic tensor; kg , the unit vector of the grating, which is formed by the incident e-polarized and o-polarized beams *through the circular photovoltaic eect. When the kg vector is written as * 1 kg p a; 2 a b2
b;
0 ;
3
the coupling tensor between e-polarized and opolarized beams will be [23] 0 1 0 0 ar42 n2o n2e 1 $ v p @ 0 0 br42 n2o n2e A; a2 b2 ar n2 n2 br n2 n2 0 42 o e
42 o e
4 where a * and b are the components of the grating vector (kg ) along the x-axis and y-axis of the crystal, respectively. no and ne are the ordinary index of refraction and the extraordinary index of refraction, respectively. r42 is the largest element of
Fig. 1. Diagram showing the recording of gratings by the incident e-waves and o-waves via photovoltaic eect. The optical * axis (^ c) *is normal to the incident plane. k being the grating g * vector, ke and ko being the propagation vectors of the e-waves and the o-waves, respectively.
the contracted electro-optic tensor of the BaTiO3 crystal. The zero values in the diagonal elements of the coupling tensor imply that the coupling of those beams of same polarization state (e±e or o± o) is inhibited. Ecient coupling only occurs between waves of orthogonal polarizations. Since the polarization state of the incident e-polarized and incident o-polarized beams could be expressed as (see Fig. 1) *
ee 0; *
0;
eo cos h;
1 ; sin h;
0 :
The eective coupling parameter is [23] Cm c p ESC ar42 n2o n2e cos h br42 n2o n2e sin h; a2 b2 5 where h is the angle between the incident o-polarized beam and the y-axis of the BaTiO3 crystal. The ESC in Eq. (5) will be zero if the simple photorefractive eect was considered only in this process. It is not zero in reality, although small, because there is another eect, circular photovoltaic eect, participating in this process. Circular photovoltaic eect arises from a photovoltaic current, which can be described as follows [22]:
H.F. Yau et al. / Optics Communications 184 (2000) 257±263
jn bnkl Ek El0 ibnl E E0 l ;
6
where jn is the photovoltaic current, bnkl , the photovoltaic tensor, Ek and El0 , the electrical ®eld components of the two incident light waves and bnl , the contracted form of bnkl . In Eq. (6), the Einstein summation convention has been used and ÔiÕ is the unit imaginary number. This current is not zero in the present situation and it ®nally results in a non-zero ESC in Eq. (5). Accordingly, if one manages to achieve a setup of having a pair of counter-propagating e-polarized waves and a beam of o-polarized waves, and if the propagation directions of all of these three beams are normal to the c^-axis of the BaTiO3 crystal, four-wave mixing would take place and thus a conjugate replica of the incident o-polarized waves would be detected. In this paper, we report an experimental demonstration of this kind of phase conjugation with a 0°-cut high gain nominally undoped BaTiO3 crystal. The size of the crystal is 7 mm 7 mm 7 mm. The experimental setup is shown schematically in Fig. 2. The light source is an Ar ion laser (SPL 2080-15) equipped with an etalon; the operating wavelength is at 514 nm. Two beams of light derived from this source are directed to be incident upon the BaTiO3 crystal. The polarization of the ®rst one is parallel to the optical tabletop and that of the second one is perpendicular to the tabletop.
259
The c^-axis of the crystal has been set normal to the optical tabletop. Consequently, the ®rst incident light beam is an ordinary beam with respect to the crystal, and whose conjugate waves are what we are looking for. Since the polarization direction of the second incident beam is parallel to the crystalÕs c^-axis, this incident beam is thus an extraordinary beam to the crystal and assumes a con®guration of a Cat-conjugator, only not in its optimized condition because the beam direction is normal to the c^-axis. Nevertheless we have managed to make this light beam travel along the characteristic route of a Cat-conjugator. In other words, the light beam fans up along the direction of the c^-axis upon entering the crystal, and retrore¯ects via the upper right corner of the crystal (Fig. 3). The retrore¯ected light beam is proved to be the conjugate of the incident e-polarized wave by the conventional test of unscrambling a distorted object. We have used an US Air Force resolution chart as the test pattern in this test. A picture of it is shown in Fig. 4(a). Experimental result has shown that the distorted pattern has been unscrambled well. We have not shown the experimental result here, because it looks similar to those published elsewhere [24,25]. Since we have managed to make the propagation directions of the incident e-polarized waves and its conjugate replica normal to the c^-axis of the crystal, and since the propagation direction of
Fig. 2. Schematic of the experimental set-up. M: mirror, HWP: half-wave plate, PBS: polarizing beam splitter, CBS: cube beam splitter, NDF: neutral density ®lter, PD: photodetector.
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H.F. Yau et al. / Optics Communications 184 (2000) 257±263
Fig. 3. A side view photograph showing the light deployment in the crystal. Both of the e-waves and o-waves are incident horizontally upon the crystal from the lower half of the left side in the picture.
the incident o-polarized light beam is also normal to the c^-axis, the optical condition for these light beams meets the condition for the production of the conjugate waves of the incident o-waves through circular photovoltaic eect. In fact, we have observed a retrore¯ected beam of the incident o-polarized beam. This retrore¯ected beam is also proved to be a conjugate light beam of the incident o-polarized light beam by passing the test of undistorting a distorted object. The optical layout for this test is the same as that for the e-conjugate waves and the same test pattern shown in Fig. 4(a) is used again. Fig. 4(b) is the distorted image observed at the photodetector plane (PD3) after it passes through a phase distorter twice via an ordinary plane mirror located at the position of the BaTiO3 crystal. Fig. 4(c) is a similar recording except the ordinary plane mirror has been removed and the re¯ected image is carried by the oconjugate waves to the PD3 plane. It is seen in Fig. 4(c) that the distorted image has been unscrambled, and it looks similar to those obtained with the e-waves conjugator published elsewhere [25,26]. In other words, the resolution of the present o-waves conjugator is comparable to that of the prevailing e-wave conjugators. In detail, the present conjugator is able to resolve the lines in group 4 of the test pattern. On doing this experiment, the test pattern is imaged, with an imaging
Fig. 4. (a) The pattern for the testing of the o-conjugate waves. (b) The image recorded at the PD3 plane when the testing pattern passes the distorter twice via a re¯ection of a plane mirror. (c) A similar recording as in Fig. 4(b) when the BaTiO3 crystal is used instead of the plane mirror.
lens of diameter equal to 5.08 cm and focal length equal to 10 cm, into the BaTiO3 crystal unmagni®ed. According to the theory of circular photovoltaic eect, a beam of e-polarized waves and a beam of o-polarized waves can be coupled by this eect to write a phase grating. During the writing of the gratings, they are also read simultaneously by the
H.F. Yau et al. / Optics Communications 184 (2000) 257±263
conjugate of the e-polarized waves in the present situation. This process results in a beam of light traveling in the reverse direction to the incident o-polarized waves. This is in fact a kind of Bragg diraction. However, owing to the circular photovoltaic coupling, the polarization of the diracted light is ordinary polarized. As a result, this diffracted light turns out to be the conjugate waves of the incident o-waves. According to this theory, as the intensity of the incident e-waves is increased, the e-conjugate waves increase. Consequently, the intensity of the o-conjugate waves also increases because they result from the reading of the phase gratings by the e-conjugate waves. Experimental result complied with this expectation. In this measurement, we kept the power of the incident o-waves at 28.5 mW. The cross section of this beam is about 3 mm in diameter, and that of the incident e-polarized beam is a little larger. The power of the conjugate o-waves was found to increase from 0.82 to 2.79 lW as the power of the incident e-waves was increased from 51.5 to 171 mW. The increases of these two sets of power values are approximately equal, consistent with the above model. Speci®cally, the increase of the ®rst set of power values is approximately 3.4 and that of the second set is approximately 3.3. A summary of the experimental results is given in Table 1. Since the analysis of circular photovoltaic coupling predicts that the conjugate o-waves do not come from the incident o-waves but from the e-conjugate waves, and since the intensity of the econjugate waves is already much weaker than that of the incident beams, the re¯ectivity of the oconjugate waves is therefore much weaker than that of the e-conjugate waves. The re¯ectivities of
261
the two conjugate waves derived from the experimental results are in agreement with the general expectation of this analysis. Analysis of the circular photovoltaic coupling of the e-polarized waves and the o-polarized waves predicts that energy ¯ows to the e-waves from the o-waves in the wave mixing process [18,19]. Experiments were designed to see whether this phenomenon takes place in our conjugator. We examined the changes of the intensities of the transmitted e-waves and the transmitted o-waves, by the photodetector PD1 and PD2 respectively, as either the incident o-waves or the incident ewaves were blocked intermittently. The experimental results are shown in Figs. 5 and 6. It is seen in these ®gures that the intensity of the transmitted o-waves increased as the incident e-waves were blocked from entering the crystal (Fig. 5). On the other hand, the intensity of the transmitted ewaves decreased as the incident o-waves were blocked from incidence on the crystal (Fig. 6). The former result can be understood as the drainage of the incident o-waves to the e-waves has been cut o, while the latter can be envisaged as the in¯ow to the incident e-waves from the o-waves has been blocked. Both of these results therefore support the description that the o-conjugate waves result from the circular photovoltaic coupling of the incident o-polarized light beam, the incident e-polarized light beam and its conjugate replica. In summary, with the help of the incident ewaves and their conjugate waves of a Cat-conjugator, we have demonstrated experimentally that it is possible to yield a beam of o-polarized conjugate waves of an incident o-waves without using any external mirror. In other words, we have presented
Table 1 The relationship of the incident power ratio vs the re¯ectivities of the phase conjugate waves Ie (mW)
Io (mW)
Ie =Io
Poc (lW)
Pec (mW)
Ro
Re
51.5 88 115 140 171
28.5 28.5 28.5 28.5 28.5
1.807 3.088 4.035 4.912 6.0
0.82 1.09 1.87 2.1 2.79
1.51 2.9 4.0 6.0 8.0
9 10ÿ5 1:2 10ÿ4 2 10ÿ4 2:3 10ÿ4 3:08 10ÿ4
6:25 10ÿ2 7 10ÿ2 7:6 10ÿ2 9:4 10ÿ2 0.1
Ie denotes the incident power of the e-waves upon the crystal; Io denotes the incident power of the o-waves upon the crystal; Poc denotes the output power of the o-conjugate waves; Pec denotes the output power of the e-conjugate waves; Ro denotes the re¯ectivity of the o-conjugate waves; Re denotes the re¯ectivity of the e-conjugate waves.
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H.F. Yau et al. / Optics Communications 184 (2000) 257±263
Fig. 5. The correlation between the powers of the e-polarized transmitted light beam (Pe ) and o-polarized transmitted beam (Po ), as the e-polarized incident beam is blocked o intermittently. These powers are recorded with PD1 and PD2, respectively.
Fig. 6. The correlation between the powers of the e-polarized penetrating light beam (Pe ) and o-polarized penetrating beam (Po ), as the o-polarized incident beam is blocked o intermittently. These powers are recorded with PD1 and PD2, respectively.
H.F. Yau et al. / Optics Communications 184 (2000) 257±263
here a self-pumped ordinary polarized phase conjugator. The mechanism for producing this o-conjugate waves is the circular photovoltaic process. In order to achieve the result, the two incident beams must propagate perpendicularly to the c^-axis of the photorefractive crystal. The resolution of these o-conjugate waves is good; it is comparable to the current e-waves conjugators. More speci®cally, lines in group 4 of an US Air Force resolution chart are resolved by it. Acknowledgements We thank the NSC of ROC for the support of this study. Thanks are also due to Dr. Y. OuYang for his helpful discussions, and also to Mr. HsinChung Wang for his help in preparing the ®gures. References [1] D.M. Pepper, J. Feinberg, N.V. Kukhtarev, Sci. Amer. (1990) 34±40. [2] P. Yeh, A.E. Chiou, J. Hong, P. Beckwith, T. Chang, M. Khoshnevisan, Opt. Engng. 28 (1989) 328±343. [3] P. Yeh, A.E. Chiou, Real-Time Optical Information Processing, Academic Press, New York, 1994, p. 457. [4] H.F. Yau, H.Y. Lee, N.J. Cheng, Appl. Phys. B 68 (1999) 1055±1059. [5] J. Feinberg, Opt. Lett. 8 (1983) 569±571. [6] F.C. Jahoda, P.G. Weber, J. Feinberg, Opt. Lett. 9 (1984) 362±364.
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[7] E. Chuang, W. Liu, J.J.P. Drolet, D. Psaltis, Proc. IEEE 87 (1999) 1931±1940. [8] J.H. Hong, I. McMichael, T.Y. Chang, W. Christian, E.G. Paek, Opt. Engng. 34 (1995) 2193±2203. [9] S. Kawata, Proc. IEEE 87 (1999) 2009±2020. [10] G. Barbastathis, D.J. Brady, Proc. IEEE 87 (1999) 2098± 2120. [11] A.E. Chiou, Proc. IEEE 12 (1999) 2074. [12] K.R. Macdonald, J. Feinberg, J. Opt. Soc. Am. 73 (1983) 548±553. [13] P. Yeh, IEEE J. Quant. Elec. 25 (1989) 484±519. [14] M. Cronin-Golomb, B. Fischer, J.O. White, A. Yariv, Appl. Phys. Lett. 41 (1982) 689. [15] J. Feinberg, Opt. Lett. 7 (1982) 486±488. [16] Y. Zhu, C. Yang, M. Hui, X. Niu, J. Zhang, T. Zhou, X. Wu, Appl. Phys. Lett. 64 (1994) 2341±2343. [17] T.Y. Chang, R.W. Hellwarth, Opt. Lett. 10 (1985) 408± 410. [18] H.F. Yau, H.Y. Lee, P.J. Wang, Opt. Engng. 33 (1994) 4033±4036. [19] J.E. Ford, Y. Fainman, S.H. Lee, Appl. Opt. 28 (1989) 4808±4815. [20] A.M. Glass, D. Von der Linde, T.J. Negran, Appl. Phys. Lett. 25 (1976) 233. [21] N. Kukhtarev, G. Dovgalenko, G.C. Duree Jr., G.J. Salamo, E.J. Sharp, B.A. Wechsler, M.B. Klein, Phys. Rev. Lett. 71 (1993) 4330±4333. [22] N. Kukhtarev, G. Dovgalenko, J. Shultz, G.J. Salamo, E.J. Sharp, B.A. Wechsler, M.B. Klein, Appl. Phys. A 56 (1993) 303±305. [23] P. Yeh, Introduction to Photorefractive Nonlinear Optics, Wiley, New York, 1993, pp. 125±133. [24] J. Feinberg, Opt. Lett. 8 (1983) 480±482. [25] J.E. Ford, Y. Fainman, S.H. Lee, Appl. Opt. 28 (1989) 4808±4815. [26] C.C. Chang, H.F. Yau, N.J. Cheng, P.X. Ye, Appl. Opt. 38 (1999) 7206±7213.
Optics Communications 184 (2000) 257±263
www.elsevier.com/locate/optcom
Ordinary polarized phase conjugator using the photovoltaic eect Hon Fai Yau a,*, Hong Chang Kung a, Hsiao Yi Lee b, Ching Cherng Sun a, Tzu Chiang Chen a, Chi Ching Chang c, Yuh Ping Tong c, Junewen Chen d a
Institute of Optical Sciences, National Central University, Chungli, Taoyuan 32054, Taiwan, ROC Department of Electronic Engineering, Minghsin Institute of Technology, Hsinchu 304, Taiwan, ROC c Department of Applied Physics, Chung Cheng Institute of Technology, Tahsi, Taoyuan 335, Taiwan, ROC Electro-Optics Section, Chung Shan Institute of Science and Technology, P.O. Box 90008-8-10, Lung-Tan 32500, Taiwan, ROC b
d
Received 22 May 2000; received in revised form 17 July 2000; accepted 24 July 2000
Abstract Self-pumped phase conjugators so far always work with extraordinary (e) polarized light waves. Theoretical analysis however has pointed out that the e-polarized waves and ordinary (o) polarized waves could be coupled together via the circular photovoltaic eect. We report here an experimental realization of this theoretical prediction and present an o-polarized conjugator. The core element of this setup is a 0°-cut undoped BaTiO3 crystal, which is aligned in a Catconjugator architecture with respect to an incident beam of e-polarized waves. Simultaneously we shine a beam of o-polarized waves on the crystal. Upon the appearance of the e-conjugate waves, we also observed the o-conjugate waves. The resolution of this ordinary polarized phase conjugator is comparable to that of the prevailing e-waves conjugators. Ó 2000 Published by Elsevier Science B.V. PACS: 72.40; 42.65.H; 42.70.M; 42.25.J Keywords: Photovoltaic eect; Photogalvanic eect; Photorefractive eect; Double phase conjugator; Ordinary polarized phase conjugator
Photorefractive crystals are materials capable of recording dynamic phase volume gratings [1]. By manipulating the writing and reading of these phase gratings, applications can be found in optical interconnection, optical computing [2], optical information processing [3,4], phase-conjugate interferometry [5,6], optical data storage [7±10] and biomedical applications [11].
* Corresponding author. Tel.: +886-3-426-3185; fax: +886-3425-2897. E-mail address: [email protected] (H.F. Yau).
Owing to its large electro-optic coecient r42 [12], BaTiO3 crystals have been extensively used to demonstrate the eects of two-wave mixing [13], four-wave mixing [14], and self-pumped phase conjugation [15]. It is well known that self-pumped phase conjugate waves can be generated through photorefractive four-wave mixing in a piece of high gain photorefractive BaTiO3 crystal [16,17]. In order to have a signi®cant wave mixing, the polarization state of the incident wave must be extraordinary with respect to the crystal because only under this condition can the mixing of the waves fully use the largest electro-optic coecient
0030-4018/00/$ - see front matter Ó 2000 Published by Elsevier Science B.V. PII: S 0 0 3 0 - 4 0 1 8 ( 0 0 ) 0 0 9 1 4 - 7
258
H.F. Yau et al. / Optics Communications 184 (2000) 257±263
r42 and thus leads to the production of a beam of conjugate waves [18,19]. Accordingly the reported self-pumped phase conjugators always work with e-polarized light beams. However, four-wave mixing can take place in a photorefractive crystal such as a piece of BaTiO3 crystal with a pair of counter-propagating e-polarized waves and a beam of ordinary polarized (o-polarized) light. The waves here are mixed via the circular photovoltaic eect [20±22]. The eect is usually small but it becomes observable when the propagating direction of the three light beams are normal to the c^-axis of a photorefractive BaTiO3 crystal. In this situation, the coupling parameter can be derived from the coupled-mode equations as [23] D * $ * E 1 c Cm ESC ee v eo ; *$ + $ e $ * e ; v r kg e0 e0
$
2
where Cm is a constant, ESC , the amplitude of the * * space charge ®eld in the BaTiO3 crystal; ee and eo , the unit vectors of polarization directions of the incident e-polarized and o-polarized beams, re$ spectively; v , the coupling tensor between the e$ polarized and o-polarized beams; e , the dielectric * $ tensor; r , the electro-optic tensor; kg , the unit vector of the grating, which is formed by the incident e-polarized and o-polarized beams *through the circular photovoltaic eect. When the kg vector is written as * 1 kg p a; 2 a b2
b;
0 ;
3
the coupling tensor between e-polarized and opolarized beams will be [23] 0 1 0 0 ar42 n2o n2e 1 $ v p @ 0 0 br42 n2o n2e A; a2 b2 ar n2 n2 br n2 n2 0 42 o e
42 o e
4 where a * and b are the components of the grating vector (kg ) along the x-axis and y-axis of the crystal, respectively. no and ne are the ordinary index of refraction and the extraordinary index of refraction, respectively. r42 is the largest element of
Fig. 1. Diagram showing the recording of gratings by the incident e-waves and o-waves via photovoltaic eect. The optical * axis (^ c) *is normal to the incident plane. k being the grating g * vector, ke and ko being the propagation vectors of the e-waves and the o-waves, respectively.
the contracted electro-optic tensor of the BaTiO3 crystal. The zero values in the diagonal elements of the coupling tensor imply that the coupling of those beams of same polarization state (e±e or o± o) is inhibited. Ecient coupling only occurs between waves of orthogonal polarizations. Since the polarization state of the incident e-polarized and incident o-polarized beams could be expressed as (see Fig. 1) *
ee 0; *
0;
eo cos h;
1 ; sin h;
0 :
The eective coupling parameter is [23] Cm c p ESC ar42 n2o n2e cos h br42 n2o n2e sin h; a2 b2 5 where h is the angle between the incident o-polarized beam and the y-axis of the BaTiO3 crystal. The ESC in Eq. (5) will be zero if the simple photorefractive eect was considered only in this process. It is not zero in reality, although small, because there is another eect, circular photovoltaic eect, participating in this process. Circular photovoltaic eect arises from a photovoltaic current, which can be described as follows [22]:
H.F. Yau et al. / Optics Communications 184 (2000) 257±263
jn bnkl Ek El0 ibnl E E0 l ;
6
where jn is the photovoltaic current, bnkl , the photovoltaic tensor, Ek and El0 , the electrical ®eld components of the two incident light waves and bnl , the contracted form of bnkl . In Eq. (6), the Einstein summation convention has been used and ÔiÕ is the unit imaginary number. This current is not zero in the present situation and it ®nally results in a non-zero ESC in Eq. (5). Accordingly, if one manages to achieve a setup of having a pair of counter-propagating e-polarized waves and a beam of o-polarized waves, and if the propagation directions of all of these three beams are normal to the c^-axis of the BaTiO3 crystal, four-wave mixing would take place and thus a conjugate replica of the incident o-polarized waves would be detected. In this paper, we report an experimental demonstration of this kind of phase conjugation with a 0°-cut high gain nominally undoped BaTiO3 crystal. The size of the crystal is 7 mm 7 mm 7 mm. The experimental setup is shown schematically in Fig. 2. The light source is an Ar ion laser (SPL 2080-15) equipped with an etalon; the operating wavelength is at 514 nm. Two beams of light derived from this source are directed to be incident upon the BaTiO3 crystal. The polarization of the ®rst one is parallel to the optical tabletop and that of the second one is perpendicular to the tabletop.
259
The c^-axis of the crystal has been set normal to the optical tabletop. Consequently, the ®rst incident light beam is an ordinary beam with respect to the crystal, and whose conjugate waves are what we are looking for. Since the polarization direction of the second incident beam is parallel to the crystalÕs c^-axis, this incident beam is thus an extraordinary beam to the crystal and assumes a con®guration of a Cat-conjugator, only not in its optimized condition because the beam direction is normal to the c^-axis. Nevertheless we have managed to make this light beam travel along the characteristic route of a Cat-conjugator. In other words, the light beam fans up along the direction of the c^-axis upon entering the crystal, and retrore¯ects via the upper right corner of the crystal (Fig. 3). The retrore¯ected light beam is proved to be the conjugate of the incident e-polarized wave by the conventional test of unscrambling a distorted object. We have used an US Air Force resolution chart as the test pattern in this test. A picture of it is shown in Fig. 4(a). Experimental result has shown that the distorted pattern has been unscrambled well. We have not shown the experimental result here, because it looks similar to those published elsewhere [24,25]. Since we have managed to make the propagation directions of the incident e-polarized waves and its conjugate replica normal to the c^-axis of the crystal, and since the propagation direction of
Fig. 2. Schematic of the experimental set-up. M: mirror, HWP: half-wave plate, PBS: polarizing beam splitter, CBS: cube beam splitter, NDF: neutral density ®lter, PD: photodetector.
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H.F. Yau et al. / Optics Communications 184 (2000) 257±263
Fig. 3. A side view photograph showing the light deployment in the crystal. Both of the e-waves and o-waves are incident horizontally upon the crystal from the lower half of the left side in the picture.
the incident o-polarized light beam is also normal to the c^-axis, the optical condition for these light beams meets the condition for the production of the conjugate waves of the incident o-waves through circular photovoltaic eect. In fact, we have observed a retrore¯ected beam of the incident o-polarized beam. This retrore¯ected beam is also proved to be a conjugate light beam of the incident o-polarized light beam by passing the test of undistorting a distorted object. The optical layout for this test is the same as that for the e-conjugate waves and the same test pattern shown in Fig. 4(a) is used again. Fig. 4(b) is the distorted image observed at the photodetector plane (PD3) after it passes through a phase distorter twice via an ordinary plane mirror located at the position of the BaTiO3 crystal. Fig. 4(c) is a similar recording except the ordinary plane mirror has been removed and the re¯ected image is carried by the oconjugate waves to the PD3 plane. It is seen in Fig. 4(c) that the distorted image has been unscrambled, and it looks similar to those obtained with the e-waves conjugator published elsewhere [25,26]. In other words, the resolution of the present o-waves conjugator is comparable to that of the prevailing e-wave conjugators. In detail, the present conjugator is able to resolve the lines in group 4 of the test pattern. On doing this experiment, the test pattern is imaged, with an imaging
Fig. 4. (a) The pattern for the testing of the o-conjugate waves. (b) The image recorded at the PD3 plane when the testing pattern passes the distorter twice via a re¯ection of a plane mirror. (c) A similar recording as in Fig. 4(b) when the BaTiO3 crystal is used instead of the plane mirror.
lens of diameter equal to 5.08 cm and focal length equal to 10 cm, into the BaTiO3 crystal unmagni®ed. According to the theory of circular photovoltaic eect, a beam of e-polarized waves and a beam of o-polarized waves can be coupled by this eect to write a phase grating. During the writing of the gratings, they are also read simultaneously by the
H.F. Yau et al. / Optics Communications 184 (2000) 257±263
conjugate of the e-polarized waves in the present situation. This process results in a beam of light traveling in the reverse direction to the incident o-polarized waves. This is in fact a kind of Bragg diraction. However, owing to the circular photovoltaic coupling, the polarization of the diracted light is ordinary polarized. As a result, this diffracted light turns out to be the conjugate waves of the incident o-waves. According to this theory, as the intensity of the incident e-waves is increased, the e-conjugate waves increase. Consequently, the intensity of the o-conjugate waves also increases because they result from the reading of the phase gratings by the e-conjugate waves. Experimental result complied with this expectation. In this measurement, we kept the power of the incident o-waves at 28.5 mW. The cross section of this beam is about 3 mm in diameter, and that of the incident e-polarized beam is a little larger. The power of the conjugate o-waves was found to increase from 0.82 to 2.79 lW as the power of the incident e-waves was increased from 51.5 to 171 mW. The increases of these two sets of power values are approximately equal, consistent with the above model. Speci®cally, the increase of the ®rst set of power values is approximately 3.4 and that of the second set is approximately 3.3. A summary of the experimental results is given in Table 1. Since the analysis of circular photovoltaic coupling predicts that the conjugate o-waves do not come from the incident o-waves but from the e-conjugate waves, and since the intensity of the econjugate waves is already much weaker than that of the incident beams, the re¯ectivity of the oconjugate waves is therefore much weaker than that of the e-conjugate waves. The re¯ectivities of
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the two conjugate waves derived from the experimental results are in agreement with the general expectation of this analysis. Analysis of the circular photovoltaic coupling of the e-polarized waves and the o-polarized waves predicts that energy ¯ows to the e-waves from the o-waves in the wave mixing process [18,19]. Experiments were designed to see whether this phenomenon takes place in our conjugator. We examined the changes of the intensities of the transmitted e-waves and the transmitted o-waves, by the photodetector PD1 and PD2 respectively, as either the incident o-waves or the incident ewaves were blocked intermittently. The experimental results are shown in Figs. 5 and 6. It is seen in these ®gures that the intensity of the transmitted o-waves increased as the incident e-waves were blocked from entering the crystal (Fig. 5). On the other hand, the intensity of the transmitted ewaves decreased as the incident o-waves were blocked from incidence on the crystal (Fig. 6). The former result can be understood as the drainage of the incident o-waves to the e-waves has been cut o, while the latter can be envisaged as the in¯ow to the incident e-waves from the o-waves has been blocked. Both of these results therefore support the description that the o-conjugate waves result from the circular photovoltaic coupling of the incident o-polarized light beam, the incident e-polarized light beam and its conjugate replica. In summary, with the help of the incident ewaves and their conjugate waves of a Cat-conjugator, we have demonstrated experimentally that it is possible to yield a beam of o-polarized conjugate waves of an incident o-waves without using any external mirror. In other words, we have presented
Table 1 The relationship of the incident power ratio vs the re¯ectivities of the phase conjugate waves Ie (mW)
Io (mW)
Ie =Io
Poc (lW)
Pec (mW)
Ro
Re
51.5 88 115 140 171
28.5 28.5 28.5 28.5 28.5
1.807 3.088 4.035 4.912 6.0
0.82 1.09 1.87 2.1 2.79
1.51 2.9 4.0 6.0 8.0
9 10ÿ5 1:2 10ÿ4 2 10ÿ4 2:3 10ÿ4 3:08 10ÿ4
6:25 10ÿ2 7 10ÿ2 7:6 10ÿ2 9:4 10ÿ2 0.1
Ie denotes the incident power of the e-waves upon the crystal; Io denotes the incident power of the o-waves upon the crystal; Poc denotes the output power of the o-conjugate waves; Pec denotes the output power of the e-conjugate waves; Ro denotes the re¯ectivity of the o-conjugate waves; Re denotes the re¯ectivity of the e-conjugate waves.
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H.F. Yau et al. / Optics Communications 184 (2000) 257±263
Fig. 5. The correlation between the powers of the e-polarized transmitted light beam (Pe ) and o-polarized transmitted beam (Po ), as the e-polarized incident beam is blocked o intermittently. These powers are recorded with PD1 and PD2, respectively.
Fig. 6. The correlation between the powers of the e-polarized penetrating light beam (Pe ) and o-polarized penetrating beam (Po ), as the o-polarized incident beam is blocked o intermittently. These powers are recorded with PD1 and PD2, respectively.
H.F. Yau et al. / Optics Communications 184 (2000) 257±263
here a self-pumped ordinary polarized phase conjugator. The mechanism for producing this o-conjugate waves is the circular photovoltaic process. In order to achieve the result, the two incident beams must propagate perpendicularly to the c^-axis of the photorefractive crystal. The resolution of these o-conjugate waves is good; it is comparable to the current e-waves conjugators. More speci®cally, lines in group 4 of an US Air Force resolution chart are resolved by it. Acknowledgements We thank the NSC of ROC for the support of this study. Thanks are also due to Dr. Y. OuYang for his helpful discussions, and also to Mr. HsinChung Wang for his help in preparing the ®gures. References [1] D.M. Pepper, J. Feinberg, N.V. Kukhtarev, Sci. Amer. (1990) 34±40. [2] P. Yeh, A.E. Chiou, J. Hong, P. Beckwith, T. Chang, M. Khoshnevisan, Opt. Engng. 28 (1989) 328±343. [3] P. Yeh, A.E. Chiou, Real-Time Optical Information Processing, Academic Press, New York, 1994, p. 457. [4] H.F. Yau, H.Y. Lee, N.J. Cheng, Appl. Phys. B 68 (1999) 1055±1059. [5] J. Feinberg, Opt. Lett. 8 (1983) 569±571. [6] F.C. Jahoda, P.G. Weber, J. Feinberg, Opt. Lett. 9 (1984) 362±364.
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