- Email: [email protected]
Apodization effect of a reflection photorefractive hologram with linear absorption
1 September 2000
Optics Communications 183 Ž2000. 327–331 www.elsevier.comrlocateroptcom
Apodization effect of a reflection photorefractive hologram with linear absorption Yasuo Tomita) , Ryota Kuze Department of Electronics Engineering, UniÕersity of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan Received 4 May 2000; received in revised form 28 June 2000; accepted 4 July 2000
Abstract New apodization mechanism of a thick reflection hologram recorded in a lossy photorefractive medium is described and analyzed numerically. The result shows that linear absorption of mutually coherent writing beams in the medium gives rise to the hologram apodization which leads to substantive sidelobe suppression in the Bragg detuning curve of the diffraction efficiency. It is also shown that the effectiveness of the hologram apodization strongly depends on the incident direction of a readout beam with respect to the gain direction of beam coupling that takes place between two writing beams. q 2000 Elsevier Science B.V. All rights reserved.
Wavelength multiplexing of reflection-type volume holograms called orthogonal wavelength-multiplexed storage method is considered to be superior to the conventional angular-multiplexing method because the former gives less cross talk noise upon readout of information-bearing pages w1,2x. In addition, it was observed that such an orthogonal data storage method by using a photorefractive crystal gave further cross-talk noise reduction that was due to sidelobe suppression as a result of an effective hologram apodization whose origin was not fully understood w1x. Later, Zhou et al. w3x attributed the cause for the hologram apodization to beam coupling between two writing beams during recording. Another cause due to short coherence length of writing beams was suggested by Lande et al. w4x. The holo-
) Corresponding author. Fax: q81-424-43-5164; e-mail: [email protected]
gram apodization is also of increasing importance for constructing narrow-band photorefractive interference filters w5,6x that can be used in wavelength-division multiplexing systems. Hofmeister et al. w7x reported a theoretical investigation of the spectral response of such a photorefractive filter. They showed that the spectral structure of the filter is strongly influenced by the photorefractive phase shift and is further apodized in the case of nonzero linear absorption. In this paper we describe and numerically analyze the effect of linear absorption on the apodization of a thick reflection hologram recorded in a diffusion-dominant photorefractive medium Ži.e., the photorefractive phase shift is "pr2.. We show that linear absorption occurred by itself during recording results in the spatial modulation of the hologram amplitude, which leads to the substantive hologram apodization. We also show that such an effect is further pronounced in the presence of beam coupling during recording. In practice, non-negligi-
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 8 7 8 - 6
328
Y. Tomita, R. Kuze r Optics Communications 183 (2000) 327–331
ble linear absorption is inevitable in highly-doped photorefractive ferroelectrics that are often used to achieve refractive-index changes as large as possible w8x. Let us consider a lossy photorefractive crystal in which a volume hologram is recorded with mutually coherent counterpropagating beams Žsee Fig. 1.. Let A j Ž z .expŽyi k j z . Ž j s 1, 2. be a complex amplitude at z for the jth writing beam, where k j is the wavenumber. To appreciate the effect of linear absorption on the hologram apodization, we assume, for simplicity, that A1ŽyLr2. equals to A 2 Ž Lr2.. Then, it is straightforward to show that in the absence of beam coupling the modulation depth of the intensity interference pattern m defined as 2 A1) A 2rŽ< A1 < 2 q < A 2 < 2 . has its magnitude given by sechŽ a z ., where a is the absorption coefficient of the crystal. This means that < m < takes the maximum at the center of the crystal and decreases smoothly from the center toward two crystal edges. ŽNote that when incident intensities of the two beams differ from each other, the maximum position of < m < shifts from the center.. Such a spatial modulation of m does not take place in the transmission geometry. Considering that the steady-state amplitude of the photorefractive index grating is proportional to m under the linear approximation w9x, we may express a spatial distribution of the index grating as n b q w n1 sechŽ a z .expŽyi Kz q i f g . q c.c.xr2, where n b is the background index of refraction, n1 is the maximum index change, f g is the photorefractive phase shift and the grating vector K is given by l0r2 n b in which l0 is a wavelength of the two writing beams in vacuum. It is seen that the grating amplitude is modulated by the sechŽ a z . curve. Such a soft-aperture effect on the grating amplitude modifies the well-known Bragg detuning curve w10x since
the curve is essentially determined by the gratingvector spectrum which can be found by Fourier transform of the grating amplitude along the propagation direction Žthe z axis. w4x. As we will see later, the soft-aperture effect provides the hologram apodization so that sidelobe ripples in the Bragg detuning curve are suppressed effectively. We note that the apodization phenomenon described above is different from two-photon apodization w4x in which the gating light-intensity profile is used to achieve the hologram apodization. In our case it is a natural consequence of recording a reflection-type photorefractive hologram with linear absorption. We performed numerical calculations to investigate the effect of linear absorption on the Bragg detuning curve. We took into account of the beam coupling phenomenon which usually takes place during recording. The coupled wave equations describing the recording process are given by d A1 dz d A2 dz
s y 12 a A1 y i k 0
s q 12 a A 2 q i k 0
< A 2 < 2A1 I0 < A1 < 2A 2 I0
exp Ž yi f g . ,
Ž 1a .
exp Ž qi f g . ,
Ž 1b .
where k 0 is the coupling constant given by p n1rl0 and I0 is the total intensity at z in the crystal. In the presence of non-negligible linear absorption closed form solutions to Eq. Ž1. are not available except for f g s 0 w7x. In addition, approximate analytic solutions suggested by Yeh w11x are valid only for a < k 0 . Therefore, we employed the shooting method w12x to solve the above equations numerically. The result was then used to calculate m that is in general a function of z. The coupled wave equations describing the readout process with two boundary conditions of AX1ŽyLr2. s const. and AX2 ŽqLr2. s 0 Žwhere X denotes the readout process by using the beam 1 at a wavelength of l. are given by d AX1 dz
s y 12 a AX1 y i k m )AX2 exp Ž yiD kz y i f g . ,
Ž 2a . Fig. 1. Setup for recording a photorefractive volume hologram in the reflection geometry.
d AX2 dz
s q 12 a AX2 q i k mAX1exp Ž iD kz q i f g . ,
Ž 2b .
Y. Tomita, R. Kuze r Optics Communications 183 (2000) 327–331
where k is given by p n1rl and D k Ž' kX2 y kX1 y K . is a wavenumber mismatch that can also be expressed as y4p n b Ž l y l0 .rl20 for < l y l 0 < < l0 . In Eq. Ž2. we neglected an erasure of the written hologram and the self-enhancement phenomenon w13x during readout. These assumptions may be justified if the readout time is much faster than hologram erasurerrewriting time during readout. This can be realized in the usually encountered situation where the readout beam is much weaker than the writing beams. Fig. 2 illustrates spatial profiles of m as a function of the normalized crystal length zrL for a L s 0, 1, 2 and 3. In the calculation we took k 0 L s 1 corresponding to n1 ; 1 = 10y4 for l 0 s 641.7 nm and L s 2 mm that were used in Rakuljic et al.’s experiment w1x. Assuming the diffusion operation, we took f g s ypr2. The negative sign of f g corresponds to the amplification of the beam 1 during recording. We also set the beam intensity ratio r defined as < A1ŽyLr2.rA 2 Ž Lr2.< 2 to be unity. It is seen that the spatial profile of m is nonuniform inside the crystal even when linear absorption is absent. This nonuniformity is caused by photorefractive beam coupling between the two writing beams and partially suppresses the aperture effect along the propagation direction Ži.e., the discontinuity of the hologram at both crystal edges. w3,7x. The peak position of m tends to shift toward the center of the crystal as linear absorption increases. This shift leads to further suppression of the aperture effect since, as
Fig. 2. Modulation-depth profiles of the steady-state intensity interference pattern inside a photorefractive crystal for a Ls 0, 1, 2 and 3 with k 0 Ls1 and f g syp r2. The boundary condition is given by A1Žy Lr2. s A 2 Ž Lr2..
329
Fig. 3. Normalized diffraction efficiency h Ž l.rh Ž l 0 . versus wavelength detuning from l0 for k 0 Ls1, a Ls1 and r s1 with n b s 2.286, l0 s641.7 nm and Ls 2 mm. Ža. f g sqp r2 and Žb. f g syp r2.
also explained by Hofmeister et al. w7x, the hologram is truncated more gently at both crystal edges, making the hologram apodization more effective. Note that spatial profiles of m for f g s qpr2 can be obtained by inverting the curves shown in Fig. 2 with respect to z s 0. Fig. 3 shows spectral responses of the normalized diffraction efficiency h Ž l.rh Ž l0 . for f g s qpr2 wFig. 3Ža.x and ypr2 wFig. 3Žb.x with k 0 L s 1 and a L s 1. Our choice of a L s 1 results from a requirement that the optimum a L maximizing the Mra Ža measure for the multiplexing capability of a volume hologram. be 2 in the transmission geometry w8x. In our case, however, the optimum a L may be about unity because of the roundtrip propagation of the readout beam in the reflection geometry. Each dotted curve in Figs. 3Ža. and 3Žb. corresponds to the envelope of the maxima in the conventional Bragg detuning curve calculated from the Kogelnik’s formula w10,14x for nonphotorefractive holograms. It is seen that the hologram apodization caused by linear
330
Y. Tomita, R. Kuze r Optics Communications 183 (2000) 327–331
absorption and beam coupling during recording makes the Bragg detuning bandwidth narrower than the conventional one. We also observe that the amplitude of sidelobe ripples for f g s ypr2 wFig. 3Žb.x is larger than that for f g s qpr2 wFig. 3Ža.x. This interesting behavior can be explained as follows: When f g is qpr2, a major contribution to the diffraction efficiency comes from the Bragg diffraction from the holograms recorded in the region nearby z s yLr2. In this situation the destructive interference among diffracted beams at an offBragg wavelength is not as strong as the case for f g s ypr2 because diffracted beams from the holograms over crystal thickness substantively contribute to the diffraction efficiency with larger phase shifts for f g s ypr2. Fig. 4 shows a linear absorption dependence of the detuning bandwidth D l for f g s qpr2 wFig. 4Ža.x and ypr2 wFig. 4Žb.x. The diffraction efficiency decreased by y30dB from the peak diffraction efficiency h Ž l0 . was chosen as the criterion for Fig. 5. Peak diffraction efficiency h Ž l0 . Žsolid curves. and detuning bandwidth D l Ždotted curves. versus the incident beam ratio r for k 0 Ls1 and a Ls1 with n b s 2.286, l0 s641.7 nm and Ls 2 mm. Ža. f g sqp r2 and Žb. f g syp r2.
Fig. 4. Detuning bandwidth D l versus a L for several values of k 0 L with r s1, n b s 2.286, l0 s641.7 nm and Ls 2 mm. Ža. f g sqp r2 and Žb. f g syp r2. Note that D l’s for the nonphotorefractive hologram are 1.2, 1.9, 2.7, 3.6 and 4.5 nm for k 0 Ls1, 2, 3, 4 and 5, respectively.
D l. It is seen that while an increase of linear absorption leads to a gradual increase of D l for f g s qpr2 wFig. 4Ža.x, it leads to a monotonic decrease of D l for f g s ypr2 wFig. 4Žb.x. These trends can be explained by the spatial distribution of the grating amplitude as shown in Fig. 2. The increase of linear absorption reduces the effective interaction region that is substantively localized nearby z s yLr2 for f g s qpr2, leading to the spectral broadening of the diffraction efficiency. When f g is ypr2, however, the hologram apodization becomes effective in proportion to the increase of linear absorption because diffracted beams from the holograms over crystal thickness contribute to the diffraction efficiency. Fig. 5 illustrates h Ž l0 . and D l as a function of r for f g s qpr2 wFig. 5Ža.x and ypr2 wFig. 5Žb.x with k 0 L s 1 and a L s 1. It is seen that h Ž l0 . is peaked near r s 6 for f g s qpr2 wFig. 5Ža.x and near r s 0.1 for f g s ypr2 wFig. 5Žb.x, respectively. Such enhancement of h Ž l0 . occurs because
Y. Tomita, R. Kuze r Optics Communications 183 (2000) 327–331
the competing effect between beam coupling and linear absorption on the hologram recording results in more or less uniform grating amplitude over the crystal thickness near these values of r. For this reason a choice of r maximizing h Ž l0 . cannot make the hologram apodization effective: D l is slightly smaller than 1.2 nm Žthe value for the nonphotorefractive hologram. when r maximizing h Ž l0 . is chosen. It is also seen that the hologram apodization is more effective for f g s ypr2 than for f g s qpr2, independently of r. We note that the curves for h Ž l0 . and D l tend to entirely shift toward larger Žsmaller. values of r for f g s qpr2 Žypr2. as k 0 L andror a L increase further. In conclusion, we have investigated the role of linear absorption on the hologram apodization. It is found that recording of a reflection-type volume hologram in a lossy photorefractive medium automatically provides the hologram apodization. The effectiveness depends on the beam intensity ratio and the incident direction of the readout beam with respect to the gain direction because of the combined influence of linear absorption and beam coupling on the grating amplitude profile. Finally, we note that as similar to the work by Zhou et al. w3x our present analysis relys on the linear approximation in which the space-charge field is proportional to m. This approximation breaks down when m is close to unity. Although a nonlinear analysis of beam coupling w15x is required in this situation, we speculate that the large modulation effect enhances the hologram apodization since it gives a higher space-charge field in the diffusion regime as m approaches to unity w9x.
331
Acknowledgements This work was supported by the Ministry of Education, Science, Sports and Culture of Japan, Grant No. 11450026.
References w1x G.A. Rakuljic, V. Leyva, A. Yariv, Opt. Lett. 17 Ž1992. 1471. w2x A. Yariv, Opt. Lett. 18 Ž1993. 652. w3x H. Zhou, F. Zhao, F.T.S. Yu, Appl. Opt. 33 Ž1994. 4345. w4x D. Lande, S.S. Orlov, L. Hesselink, Opt. Lett. 23 Ž1998. 1399. w5x V. Leyva, G.A. Rakuljic, B. O’Conner, Appl. Phys. Lett. 65 Ž1994. 1079. w6x S. Breer, H. Vogt, I. Nee, K. Buse, Electron. Lett. 34 Ž1998. 2419. w7x R. Hofmeister, A. Yariv, S. Yagi, J. Opt. Soc. Am. A 11 Ž1994. 1342. w8x K. Peithmann, A. Wiebrock, K. Buse, Appl. Phys. B 68 Ž1999. 777. w9x L. Solymar, D. J Webb, A. Grunnet-Jepsen, The Physics and Applications of Photorefractive Materials, Clarendon Press, Oxford, 1996. w10x P. Yeh, Introduction to Photorefractive Nonlinear Optics, Wiley, New York, 1993. w11x P. Yeh, Opt. Commun. 45 Ž1983. 323. w12x Y.H. Ja, Opt. Quantum Electron. 15 Ž1983. 529. w13x Y. Tomita, S. Matsushima, J. Opt. Soc. Am. B 14 Ž1997. 2877. w14x H.W. Kogelnik, Bell Syst. Tech. J 48 Ž1969. 2909. w15x For example, K. Ratnam, P.P. Banerjee, Opt. Commun. 107 Ž1994. 522; M.R. Belic, ´ Opt. Lett. 21 Ž1996. 183.
Optics Communications 183 Ž2000. 327–331 www.elsevier.comrlocateroptcom
Apodization effect of a reflection photorefractive hologram with linear absorption Yasuo Tomita) , Ryota Kuze Department of Electronics Engineering, UniÕersity of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan Received 4 May 2000; received in revised form 28 June 2000; accepted 4 July 2000
Abstract New apodization mechanism of a thick reflection hologram recorded in a lossy photorefractive medium is described and analyzed numerically. The result shows that linear absorption of mutually coherent writing beams in the medium gives rise to the hologram apodization which leads to substantive sidelobe suppression in the Bragg detuning curve of the diffraction efficiency. It is also shown that the effectiveness of the hologram apodization strongly depends on the incident direction of a readout beam with respect to the gain direction of beam coupling that takes place between two writing beams. q 2000 Elsevier Science B.V. All rights reserved.
Wavelength multiplexing of reflection-type volume holograms called orthogonal wavelength-multiplexed storage method is considered to be superior to the conventional angular-multiplexing method because the former gives less cross talk noise upon readout of information-bearing pages w1,2x. In addition, it was observed that such an orthogonal data storage method by using a photorefractive crystal gave further cross-talk noise reduction that was due to sidelobe suppression as a result of an effective hologram apodization whose origin was not fully understood w1x. Later, Zhou et al. w3x attributed the cause for the hologram apodization to beam coupling between two writing beams during recording. Another cause due to short coherence length of writing beams was suggested by Lande et al. w4x. The holo-
) Corresponding author. Fax: q81-424-43-5164; e-mail: [email protected]
gram apodization is also of increasing importance for constructing narrow-band photorefractive interference filters w5,6x that can be used in wavelength-division multiplexing systems. Hofmeister et al. w7x reported a theoretical investigation of the spectral response of such a photorefractive filter. They showed that the spectral structure of the filter is strongly influenced by the photorefractive phase shift and is further apodized in the case of nonzero linear absorption. In this paper we describe and numerically analyze the effect of linear absorption on the apodization of a thick reflection hologram recorded in a diffusion-dominant photorefractive medium Ži.e., the photorefractive phase shift is "pr2.. We show that linear absorption occurred by itself during recording results in the spatial modulation of the hologram amplitude, which leads to the substantive hologram apodization. We also show that such an effect is further pronounced in the presence of beam coupling during recording. In practice, non-negligi-
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 8 7 8 - 6
328
Y. Tomita, R. Kuze r Optics Communications 183 (2000) 327–331
ble linear absorption is inevitable in highly-doped photorefractive ferroelectrics that are often used to achieve refractive-index changes as large as possible w8x. Let us consider a lossy photorefractive crystal in which a volume hologram is recorded with mutually coherent counterpropagating beams Žsee Fig. 1.. Let A j Ž z .expŽyi k j z . Ž j s 1, 2. be a complex amplitude at z for the jth writing beam, where k j is the wavenumber. To appreciate the effect of linear absorption on the hologram apodization, we assume, for simplicity, that A1ŽyLr2. equals to A 2 Ž Lr2.. Then, it is straightforward to show that in the absence of beam coupling the modulation depth of the intensity interference pattern m defined as 2 A1) A 2rŽ< A1 < 2 q < A 2 < 2 . has its magnitude given by sechŽ a z ., where a is the absorption coefficient of the crystal. This means that < m < takes the maximum at the center of the crystal and decreases smoothly from the center toward two crystal edges. ŽNote that when incident intensities of the two beams differ from each other, the maximum position of < m < shifts from the center.. Such a spatial modulation of m does not take place in the transmission geometry. Considering that the steady-state amplitude of the photorefractive index grating is proportional to m under the linear approximation w9x, we may express a spatial distribution of the index grating as n b q w n1 sechŽ a z .expŽyi Kz q i f g . q c.c.xr2, where n b is the background index of refraction, n1 is the maximum index change, f g is the photorefractive phase shift and the grating vector K is given by l0r2 n b in which l0 is a wavelength of the two writing beams in vacuum. It is seen that the grating amplitude is modulated by the sechŽ a z . curve. Such a soft-aperture effect on the grating amplitude modifies the well-known Bragg detuning curve w10x since
the curve is essentially determined by the gratingvector spectrum which can be found by Fourier transform of the grating amplitude along the propagation direction Žthe z axis. w4x. As we will see later, the soft-aperture effect provides the hologram apodization so that sidelobe ripples in the Bragg detuning curve are suppressed effectively. We note that the apodization phenomenon described above is different from two-photon apodization w4x in which the gating light-intensity profile is used to achieve the hologram apodization. In our case it is a natural consequence of recording a reflection-type photorefractive hologram with linear absorption. We performed numerical calculations to investigate the effect of linear absorption on the Bragg detuning curve. We took into account of the beam coupling phenomenon which usually takes place during recording. The coupled wave equations describing the recording process are given by d A1 dz d A2 dz
s y 12 a A1 y i k 0
s q 12 a A 2 q i k 0
< A 2 < 2A1 I0 < A1 < 2A 2 I0
exp Ž yi f g . ,
Ž 1a .
exp Ž qi f g . ,
Ž 1b .
where k 0 is the coupling constant given by p n1rl0 and I0 is the total intensity at z in the crystal. In the presence of non-negligible linear absorption closed form solutions to Eq. Ž1. are not available except for f g s 0 w7x. In addition, approximate analytic solutions suggested by Yeh w11x are valid only for a < k 0 . Therefore, we employed the shooting method w12x to solve the above equations numerically. The result was then used to calculate m that is in general a function of z. The coupled wave equations describing the readout process with two boundary conditions of AX1ŽyLr2. s const. and AX2 ŽqLr2. s 0 Žwhere X denotes the readout process by using the beam 1 at a wavelength of l. are given by d AX1 dz
s y 12 a AX1 y i k m )AX2 exp Ž yiD kz y i f g . ,
Ž 2a . Fig. 1. Setup for recording a photorefractive volume hologram in the reflection geometry.
d AX2 dz
s q 12 a AX2 q i k mAX1exp Ž iD kz q i f g . ,
Ž 2b .
Y. Tomita, R. Kuze r Optics Communications 183 (2000) 327–331
where k is given by p n1rl and D k Ž' kX2 y kX1 y K . is a wavenumber mismatch that can also be expressed as y4p n b Ž l y l0 .rl20 for < l y l 0 < < l0 . In Eq. Ž2. we neglected an erasure of the written hologram and the self-enhancement phenomenon w13x during readout. These assumptions may be justified if the readout time is much faster than hologram erasurerrewriting time during readout. This can be realized in the usually encountered situation where the readout beam is much weaker than the writing beams. Fig. 2 illustrates spatial profiles of m as a function of the normalized crystal length zrL for a L s 0, 1, 2 and 3. In the calculation we took k 0 L s 1 corresponding to n1 ; 1 = 10y4 for l 0 s 641.7 nm and L s 2 mm that were used in Rakuljic et al.’s experiment w1x. Assuming the diffusion operation, we took f g s ypr2. The negative sign of f g corresponds to the amplification of the beam 1 during recording. We also set the beam intensity ratio r defined as < A1ŽyLr2.rA 2 Ž Lr2.< 2 to be unity. It is seen that the spatial profile of m is nonuniform inside the crystal even when linear absorption is absent. This nonuniformity is caused by photorefractive beam coupling between the two writing beams and partially suppresses the aperture effect along the propagation direction Ži.e., the discontinuity of the hologram at both crystal edges. w3,7x. The peak position of m tends to shift toward the center of the crystal as linear absorption increases. This shift leads to further suppression of the aperture effect since, as
Fig. 2. Modulation-depth profiles of the steady-state intensity interference pattern inside a photorefractive crystal for a Ls 0, 1, 2 and 3 with k 0 Ls1 and f g syp r2. The boundary condition is given by A1Žy Lr2. s A 2 Ž Lr2..
329
Fig. 3. Normalized diffraction efficiency h Ž l.rh Ž l 0 . versus wavelength detuning from l0 for k 0 Ls1, a Ls1 and r s1 with n b s 2.286, l0 s641.7 nm and Ls 2 mm. Ža. f g sqp r2 and Žb. f g syp r2.
also explained by Hofmeister et al. w7x, the hologram is truncated more gently at both crystal edges, making the hologram apodization more effective. Note that spatial profiles of m for f g s qpr2 can be obtained by inverting the curves shown in Fig. 2 with respect to z s 0. Fig. 3 shows spectral responses of the normalized diffraction efficiency h Ž l.rh Ž l0 . for f g s qpr2 wFig. 3Ža.x and ypr2 wFig. 3Žb.x with k 0 L s 1 and a L s 1. Our choice of a L s 1 results from a requirement that the optimum a L maximizing the Mra Ža measure for the multiplexing capability of a volume hologram. be 2 in the transmission geometry w8x. In our case, however, the optimum a L may be about unity because of the roundtrip propagation of the readout beam in the reflection geometry. Each dotted curve in Figs. 3Ža. and 3Žb. corresponds to the envelope of the maxima in the conventional Bragg detuning curve calculated from the Kogelnik’s formula w10,14x for nonphotorefractive holograms. It is seen that the hologram apodization caused by linear
330
Y. Tomita, R. Kuze r Optics Communications 183 (2000) 327–331
absorption and beam coupling during recording makes the Bragg detuning bandwidth narrower than the conventional one. We also observe that the amplitude of sidelobe ripples for f g s ypr2 wFig. 3Žb.x is larger than that for f g s qpr2 wFig. 3Ža.x. This interesting behavior can be explained as follows: When f g is qpr2, a major contribution to the diffraction efficiency comes from the Bragg diffraction from the holograms recorded in the region nearby z s yLr2. In this situation the destructive interference among diffracted beams at an offBragg wavelength is not as strong as the case for f g s ypr2 because diffracted beams from the holograms over crystal thickness substantively contribute to the diffraction efficiency with larger phase shifts for f g s ypr2. Fig. 4 shows a linear absorption dependence of the detuning bandwidth D l for f g s qpr2 wFig. 4Ža.x and ypr2 wFig. 4Žb.x. The diffraction efficiency decreased by y30dB from the peak diffraction efficiency h Ž l0 . was chosen as the criterion for Fig. 5. Peak diffraction efficiency h Ž l0 . Žsolid curves. and detuning bandwidth D l Ždotted curves. versus the incident beam ratio r for k 0 Ls1 and a Ls1 with n b s 2.286, l0 s641.7 nm and Ls 2 mm. Ža. f g sqp r2 and Žb. f g syp r2.
Fig. 4. Detuning bandwidth D l versus a L for several values of k 0 L with r s1, n b s 2.286, l0 s641.7 nm and Ls 2 mm. Ža. f g sqp r2 and Žb. f g syp r2. Note that D l’s for the nonphotorefractive hologram are 1.2, 1.9, 2.7, 3.6 and 4.5 nm for k 0 Ls1, 2, 3, 4 and 5, respectively.
D l. It is seen that while an increase of linear absorption leads to a gradual increase of D l for f g s qpr2 wFig. 4Ža.x, it leads to a monotonic decrease of D l for f g s ypr2 wFig. 4Žb.x. These trends can be explained by the spatial distribution of the grating amplitude as shown in Fig. 2. The increase of linear absorption reduces the effective interaction region that is substantively localized nearby z s yLr2 for f g s qpr2, leading to the spectral broadening of the diffraction efficiency. When f g is ypr2, however, the hologram apodization becomes effective in proportion to the increase of linear absorption because diffracted beams from the holograms over crystal thickness contribute to the diffraction efficiency. Fig. 5 illustrates h Ž l0 . and D l as a function of r for f g s qpr2 wFig. 5Ža.x and ypr2 wFig. 5Žb.x with k 0 L s 1 and a L s 1. It is seen that h Ž l0 . is peaked near r s 6 for f g s qpr2 wFig. 5Ža.x and near r s 0.1 for f g s ypr2 wFig. 5Žb.x, respectively. Such enhancement of h Ž l0 . occurs because
Y. Tomita, R. Kuze r Optics Communications 183 (2000) 327–331
the competing effect between beam coupling and linear absorption on the hologram recording results in more or less uniform grating amplitude over the crystal thickness near these values of r. For this reason a choice of r maximizing h Ž l0 . cannot make the hologram apodization effective: D l is slightly smaller than 1.2 nm Žthe value for the nonphotorefractive hologram. when r maximizing h Ž l0 . is chosen. It is also seen that the hologram apodization is more effective for f g s ypr2 than for f g s qpr2, independently of r. We note that the curves for h Ž l0 . and D l tend to entirely shift toward larger Žsmaller. values of r for f g s qpr2 Žypr2. as k 0 L andror a L increase further. In conclusion, we have investigated the role of linear absorption on the hologram apodization. It is found that recording of a reflection-type volume hologram in a lossy photorefractive medium automatically provides the hologram apodization. The effectiveness depends on the beam intensity ratio and the incident direction of the readout beam with respect to the gain direction because of the combined influence of linear absorption and beam coupling on the grating amplitude profile. Finally, we note that as similar to the work by Zhou et al. w3x our present analysis relys on the linear approximation in which the space-charge field is proportional to m. This approximation breaks down when m is close to unity. Although a nonlinear analysis of beam coupling w15x is required in this situation, we speculate that the large modulation effect enhances the hologram apodization since it gives a higher space-charge field in the diffusion regime as m approaches to unity w9x.
331
Acknowledgements This work was supported by the Ministry of Education, Science, Sports and Culture of Japan, Grant No. 11450026.
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