- Email: [email protected]
Formation dynamics of nonvolatile crossed-beam photorefractive gratings in doubly doped LiNbO3 crystals: Theoretical investigation
ARTICLE IN PRESS
Optik
Optics
Optik 121 (2010) 457–461 www.elsevier.de/ijleo
Formation dynamics of nonvolatile crossed-beam photorefractive gratings in doubly doped LiNbO3 crystals: Theoretical investigation Xin Wanga,, Aimin Yanb, De’an Liub, Xiangyin Lia a
Physics Experimental Center, Nanjing University of Science & Technology, Nanjing, Jiang Su 210094, PR China Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, PR China
b
Received 27 March 2008; accepted 18 August 2008
Abstract The formation dynamics of crossed-beam photorefractive gratings formed by the method of two-center holographic recording in doubly doped LiNbO3 crystals is investigated in this paper based on the theoretical model combining the two-center band transport model with the two-dimensional coupled-wave theory. The numerical simulations are presented for two-center holographic recording crossed-beam photorefractive gratings in LiNbO3:Fe:Mn crystals. The temporal and spatial evolutions of the refractive index modulation and the diffraction efficiency are shown. The spatial variation of the wave intensity is also presented. r 2008 Elsevier GmbH. All rights reserved. Keywords: Two-center holographic recording; Crossed-beam photorefractive gratings; Two-center band transport model; Two-dimensional coupled-wave theory
1. Introduction The photorefractive gratings in photorefractive crystals have been the subject of intense study for some years and have recently been discussed owing to the importance to a number of applications in optical data storage, integrated optics, optical communications, etc. [1–3]. The crossed-beam photorefractive gratings are defined as a type of holographic gratings formed with the intersection of two beams of finite widths [4]. Recently, as the optical communication networks and integrated optical components are rapidly growing in number and size, more and more crossed-beam photorefractive gratings are encountered [5]. Therefore, Corresponding author.
E-mail address: [email protected] (X. Wang). 0030-4026/$ - see front matter r 2008 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2008.08.006
knowledge of holographic formation dynamics of these gratings would be valuable. Many interests have been attached to this subject and in most of the investigations the gratings are treated with no time–space variations [6–8]. Some researchers have introduced the analytic solution of the space– charge field (SCF) to the two-dimensional or threedimensional coupled-beam equations for analyzing the dynamic crossed-beam photorefractive gratings [9–12]. But these methods are not viable for the crossed-beam photorefractive gratings recording in doubly doped LiNbO3 crystals. To achieve nonvolatile photorefractive grating, a method of two-center holographic recording has been proposed by Buse et al. [13], and based on this method we have developed a theoretical model for analyzing the influence of recording conditions on the crossed-beam photorefractive gratings recorded in doubly doped
ARTICLE IN PRESS 458
X. Wang et al. / Optik 121 (2010) 457–461
LiNbO3 crystals, in which the two-center material equations and the two-dimensional coupled-wave equations are solved jointly [14]. In this paper, we investigate the photorefractive formation dynamics of nonvolatile crossed-beam photorefractive gratings formed by the method of two-center holographic recording in doubly doped LiNbO3 crystals, and the previous theoretical model is still used. In Section 2 the theoretical model and the fundamental equations are shown. The numerical calculations of the refractive index modulation, the intensity of the readout and diffracted beam and the diffraction efficiency for the nonvolatile crossed-beam photorefractive gratings in LiNbO3:Fe:Mn crystals are presented in Section 3. The temporal and spatial evolutions of them are analyzed, which enables us to investigate the process of formation dynamics of the crossed-beam photorefractive gratings in the doubly doped LiNbO3 crystal. Finally, the conclusions are given in Section 4.
2. Theory model The holographic recording geometry and the analytical model we use are shown in Fig. 1. Two ordinarypolarized plane waves (red light) E1 and E2 of finite widths w1 and w2 are used to construct crossed-beam photorefractive grating via the photorefractive effect in doubly doped LiNbO3 crystals. The c-axis of the crystal cˆ is oriented along the z direction. The crystal normal is parallel to the x direction. The incident angles of both the recording beams inside the crystal are equal to y. The complex field amplitudes of E1 and E2 are given by E j ðrÞ ¼ Aj ðx; zÞexpðiK j rÞ;
j ¼ 1; 2
(1)
where A1,2 and K1,2 are the complex amplitude and the wave vector of the two recording beams E1 and E2, respectively.
Firstly, let us consider the crossed-beam photorefractive gratings in the recording phase. The SCF produced by the migration of the charge carriers in this phase is determined by the two-center material equations that vary simultaneously with the x and z coordinates, following the model used in the previous work [14]. The complex amplitude of the SCF is E SC ¼
ie ðN þ N S1 þ N e1 Þ 0 kg D1
(2)
where the constants e and e are the electronic charge and the electron mobility, respectively. kg is the amplitude of the grating vector which takes value of 2k0n0 sin y, where k0 is the wave vector of the recording beams in free space and n0 is the average refractive index of the crystal. The variable quantities N D1 , N S1 and Ne1 are the fundamental components of the Fourier series of the electronic concentration in the deep trap, the shallow trap and the conduction band, respectively. The crystal is sensitized by the uniform UV light of infinite width illuminating along the x direction during the recording stage. As a result of the presence of the SCF, a refractive index modulation of the photorefractive grating is induced by means of the linear electro-optic effect, which can be written as n30 g13 E SC (3) 2 where jE is the complex angle of the SCF and g13 is the electro-optic coefficient. The diffraction properties of the crossed-beam photorefractive gratings during the grating recorded and fixed phase are analyzed by the two-dimensional coupledwave equations [14]. The diffraction efficiency, which is defined as the ratio of diffracted energy along the x direction to the total power including the diffracted energy along the x direction and the transmitted energy along the Z direction, has the form n1 expðijE Þ ¼
PSout (4) S PR out þ Pout Rw where PSout ¼ 0 2 jA2 ðx; w1 Þj2 dx csc 2y and PR out ¼ R w1 2 jA ðw ; ZÞj dZ csc 2y are the integral of the diffracted 1 2 0 beam intensities in the x direction and of the transmitted beam intensities in the Z direction, respectively. Coordinates x and Z are orthogonal to K1 and K2, respectively (shown in Fig. 1), which is defined to simplify the numerical calculation of the two-dimensional coupled-wave equations. DE ¼
3. Numerical calculations and discussions Fig. 1. Recording geometry for a crossed-beam photorefractive grating in the crystal.
In order to solve the two-center material equations and the two-dimensional coupled-wave equations
ARTICLE IN PRESS
2.0x10-4 1.5x10-4 1.0x10-4 5.0x10-5 0.0 0
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Fig. 3. The time variation of the average refractive index modulation.
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1500 1000 0.0 0.2 0.4 0.6 0.8 1.0 1.0
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0.4 0.6 m) m η(
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10000
) m m ξ(
)
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η(
Fig. 4. Spatial variations of the intensity at the end of fixing: (a) the readout beam and (b) the diffracted beam.
3.5x10-4 3.0x10-4 2.5x10-4 2.0x10-4 1.5x10-4 1.0x10-4
)
m
0.4 0.6 η (m m)
1.0 0.8 0.6 0.4 0.2 0.8
m
5.0x10-5 0.0 0.0 0.2
1.0 0.0
ξ(
Refractive index modulation n1
459
ξ(
numerically and to obtain the simultaneous solution, the numerical procedure is used, which is similar to that proposed by Ren et al. [15], except that instead of the fourth-order Runge–Kutta method used to solve the coupling differential equations numerically the Heun method is used (a special case of the Runge–Kutta method) [16]. This numerical procedure has been described in detail in the previous work [14]. In this work, the parameters are equal to the values provided in our previous work and in Ref. [15] except that the boundary widths of the two recording waves are 1.0 mm, and the incident angle of them is 251 outside the crystal. Fig. 2 shows the spatial distribution of the refractive index modulation at the fixing end. It can be seen that the spatial distribution of the refractive index modulation is nonuniform in the grating region. It fluctuates and decreases as the grating region widens, which is mostly because of the rapid absorption of the UV light in the crystal causing the weaker fixed grating in the area distant from the input boundary. To analyze the temporal dynamic property of the photorefractive grating, the average refractive index modulation is defined as the spatial arithmetical mean of the refractive change in each time interval. Fig. 3 shows the change of the average refractive index modulation with time, which indicates that the index modulation increases and then almost moves towards saturation during the recording phase, and during the fixing phase it decreases and the grating is fixed finally. The appreciable increase of the index modulation when time is close to the end of the recording phase may be due to the spatial nonuniformity of dynamic grating. Fig. 4(a) and (b) shows the intensity spatial variations of the readout beam E1 and the diffracted beam E2 in the grating region at the end of fixing, respectively. From Fig. 4(a), one can see that the amplitude of the readout beam decreases with the increase in the value of x for constant Z. We think it is mostly due to the energy transfer to the diffracted beam as the readout beam propagates through the grating region. Certainly, the
Average refractive index modulation n1
X. Wang et al. / Optik 121 (2010) 457–461
Fig. 2. Spatial variation of the refractive index modulation at the end of fixing.
absorption of the readout beam in the crystal partially works on the decrease of the intensity. Along the Z direction, the intensity fluctuates and decreases with the increase of Z but at the input boundary, which indicates the profile deformation of the uniform plane readout beam caused by the beams’ coupling [9]. Fig. 4(b) shows the increase in the intensity of the diffracted beam during its propagation through the grating region to higher Z values. The increase in energy comes from the energy transferring from the readout beam. The figure
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0 0
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750
20 0
Average Diffraction Efficiency %
X. Wang et al. / Optik 121 (2010) 457–461
Diffraction Efficiency %
460
900
Fig. 5. Change of diffraction efficiency and the average diffraction efficiency with time at the output boundary.
also shows that the energy decreases with the increase of x along the x direction, which demonstrates the fact that the stronger index modulation determined benefits the higher intensity at lower x values for constant Z value in the grating region [9]. Fig. 5 plots the time dependence of the diffraction efficiency at the output boundary of the grating. It can be seen that the diffraction efficiency increases to maximum and then moves towards saturation during the recording phase, and it decreases and is fixed finally during the fixing phase. The evolution is almost similar to that of one-dimensional photorefractive gratings [15]. The result indicates that the fixing efficiency with the definition of the ratio of the fixed diffraction efficiency to the saturation diffraction efficiency takes a value of 94.5%. In this figure, the calculation of the average diffraction efficiency (average DE) at the output boundary of the grating is also presented, which is calculated by using the average refractive index modulation and the diffraction efficiency equation proposed by Kenan [4]. The line of the average DE shows that the fixing efficiency is much higher than that of the diffraction efficiency. The difference between the diffraction efficiency and its mean is due to the latter ignoring the spatial nonuniformity of the dynamic grating.
4. Conclusions Based on the two-center band transport model and the two-dimensional coupled-wave theory, the holographic formation dynamics of nonvolatile crossedbeam photorefractive gratings formed by two-center holographic recording in doubly doped LiNbO3 crystals has been analyzed in this paper. The numerical calculations were presented, which show the spatial nonuniform properties of the refractive index modulation. The change of spatial mean of refractive index modulation with time indicates holographic formation
dynamics. The spatial variations of the intensity of the readout and diffracted beam indicate the energy transferring and coupling between the readout and the diffracted beam. The time dependence of the diffraction efficiency and its mean shows the higher fixing efficiency and also indicates the influence of the spatial heterogeneity for two-center holographic recording nonvolatile crossed-beam photorefractive gratings in doubly doped LiNbO3 crystals. The results provide quantitative predictions of the expected behavior of the formation dynamics of these gratings and can be helpful for the design of finite boundary photorefractive holographic optical elements.
Acknowledgment This work is supported by Nanjing University of Science & Technology (Grant no. AB41928).
References [1] R. Ince, H. Yu¨kselici, A.T. I˙nce, Crystal diffraction grating in LiNbO3:Fe derived from interferometric volume holography, Opt. Lasers Eng. 44 (2006) 509–519. [2] J.P. Burger, S. Dubovitsky, W.H. Steier, Kerr-like nonlinear mode converters for integrated optic device applications, Opt. Commun. 212 (2002) 251–266. [3] J.-W. An, N. Kim, K.-W. Lee, Volume holographic wavelength demultiplexer based on rotation multiplexing in the 901 geometry, Opt. Commun. 197 (2001) 247–254. [4] R.P. Kenan, Theory of crossed-beam diffraction gratings, IEEE. J. Quantum Electron. QE-14 (12) (1978) 924–930. [5] S.Q. Tao, B. Wang, G.W. Burr, J.B. Chen, Diffraction efficiency of volume gratings with finite size: corrected analytical solution, J. Mod. Opt. 51 (2004) 1115–1122. [6] A.Y. Hamad, J.P. Wicksted, Volume grating produced by intersecting Gaussian beams in an absorbing medium: a Bragg diffraction model, Opt. Commun. 138 (1997) 354–364. [7] P.P. Banerjee, M.R. Chatterjee, N. Kukhturer, T. Kukhtareva, Volume holographic recording and readout for 90-deg geometry, Opt. Eng. 43 (2004) 2053–2060. [8] A.M. Yan, L.Y. Wan, De’an Liu, L.R. Liu, Anisotropic Bragg diffraction of finite-sized volume holographic grating in photorefractive crystals, Optik 118 (2007) 418–424. [9] W. Kro´likowski, M. Cronin-Colomb, Photorefractive wave mixing with finite beams, Opt. Commun. 89 (1992) 88–98. + [10] D. Fluck, J.A. Weiss, S. Bru+ lisauar, P. Gunters, Twowave mixing of focused Gaussian beams in photorefractive wave guides, Opt. Lett. 19 (24) (1994) 2080–2082. [11] Zhifang Chai, De’an Liu, Aimin Yan, Ya’nan Zhi, Liren Liu, Effect of absorption on the diffraction efficiency in Fe:LiNbO3 crystal with 90 recording geometry, J. Opt. A: Pure Appl. Opt. 7 (2005) 711–716.
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[12] C.W. Tarn, Gaussian-beam profile deformation via anisotropic photorefractive gratings formed by diffusive, photovoltaic, and drift mechanisms: a system transfer function approach, Opt. Eng. 37 (1) (1998) 229–236. [13] K. Buse, A. Adibi, D. Pasltis, Non-volatile holographic storage in double doped lithium niobite crystals, Nature 393 (1998) 665–668. [14] X. Wang, A.M. Yan, L.R. Liu, D. Liu, Y.N. Zhi, Z.J. Hu, Influence of recording conditions on crossed-beam
461
photorefractive gratings in doubly doped LiNbO3 crystals, Appl. Opt. 45 (23) (2006) 5942–5949. [15] L.Y. Ren, L.L. Liu, J.F. Zu, Z. Luan, Recording and fixing dynamics of nonvolatile photorefractive holograms in LiNbO3:Fe:Mn crystals, J. Opt. Soc. Am. B 20 (2003) 2162–2173. [16] G. Notni, R. Kowarschik, Diffraction analysis of threedimensional volume gratings with arbitrary boundaries, J. Opt. Soc. A A6 (1989) 1682–1691.
Optik
Optics
Optik 121 (2010) 457–461 www.elsevier.de/ijleo
Formation dynamics of nonvolatile crossed-beam photorefractive gratings in doubly doped LiNbO3 crystals: Theoretical investigation Xin Wanga,, Aimin Yanb, De’an Liub, Xiangyin Lia a
Physics Experimental Center, Nanjing University of Science & Technology, Nanjing, Jiang Su 210094, PR China Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, PR China
b
Received 27 March 2008; accepted 18 August 2008
Abstract The formation dynamics of crossed-beam photorefractive gratings formed by the method of two-center holographic recording in doubly doped LiNbO3 crystals is investigated in this paper based on the theoretical model combining the two-center band transport model with the two-dimensional coupled-wave theory. The numerical simulations are presented for two-center holographic recording crossed-beam photorefractive gratings in LiNbO3:Fe:Mn crystals. The temporal and spatial evolutions of the refractive index modulation and the diffraction efficiency are shown. The spatial variation of the wave intensity is also presented. r 2008 Elsevier GmbH. All rights reserved. Keywords: Two-center holographic recording; Crossed-beam photorefractive gratings; Two-center band transport model; Two-dimensional coupled-wave theory
1. Introduction The photorefractive gratings in photorefractive crystals have been the subject of intense study for some years and have recently been discussed owing to the importance to a number of applications in optical data storage, integrated optics, optical communications, etc. [1–3]. The crossed-beam photorefractive gratings are defined as a type of holographic gratings formed with the intersection of two beams of finite widths [4]. Recently, as the optical communication networks and integrated optical components are rapidly growing in number and size, more and more crossed-beam photorefractive gratings are encountered [5]. Therefore, Corresponding author.
E-mail address: [email protected] (X. Wang). 0030-4026/$ - see front matter r 2008 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2008.08.006
knowledge of holographic formation dynamics of these gratings would be valuable. Many interests have been attached to this subject and in most of the investigations the gratings are treated with no time–space variations [6–8]. Some researchers have introduced the analytic solution of the space– charge field (SCF) to the two-dimensional or threedimensional coupled-beam equations for analyzing the dynamic crossed-beam photorefractive gratings [9–12]. But these methods are not viable for the crossed-beam photorefractive gratings recording in doubly doped LiNbO3 crystals. To achieve nonvolatile photorefractive grating, a method of two-center holographic recording has been proposed by Buse et al. [13], and based on this method we have developed a theoretical model for analyzing the influence of recording conditions on the crossed-beam photorefractive gratings recorded in doubly doped
ARTICLE IN PRESS 458
X. Wang et al. / Optik 121 (2010) 457–461
LiNbO3 crystals, in which the two-center material equations and the two-dimensional coupled-wave equations are solved jointly [14]. In this paper, we investigate the photorefractive formation dynamics of nonvolatile crossed-beam photorefractive gratings formed by the method of two-center holographic recording in doubly doped LiNbO3 crystals, and the previous theoretical model is still used. In Section 2 the theoretical model and the fundamental equations are shown. The numerical calculations of the refractive index modulation, the intensity of the readout and diffracted beam and the diffraction efficiency for the nonvolatile crossed-beam photorefractive gratings in LiNbO3:Fe:Mn crystals are presented in Section 3. The temporal and spatial evolutions of them are analyzed, which enables us to investigate the process of formation dynamics of the crossed-beam photorefractive gratings in the doubly doped LiNbO3 crystal. Finally, the conclusions are given in Section 4.
2. Theory model The holographic recording geometry and the analytical model we use are shown in Fig. 1. Two ordinarypolarized plane waves (red light) E1 and E2 of finite widths w1 and w2 are used to construct crossed-beam photorefractive grating via the photorefractive effect in doubly doped LiNbO3 crystals. The c-axis of the crystal cˆ is oriented along the z direction. The crystal normal is parallel to the x direction. The incident angles of both the recording beams inside the crystal are equal to y. The complex field amplitudes of E1 and E2 are given by E j ðrÞ ¼ Aj ðx; zÞexpðiK j rÞ;
j ¼ 1; 2
(1)
where A1,2 and K1,2 are the complex amplitude and the wave vector of the two recording beams E1 and E2, respectively.
Firstly, let us consider the crossed-beam photorefractive gratings in the recording phase. The SCF produced by the migration of the charge carriers in this phase is determined by the two-center material equations that vary simultaneously with the x and z coordinates, following the model used in the previous work [14]. The complex amplitude of the SCF is E SC ¼
ie ðN þ N S1 þ N e1 Þ 0 kg D1
(2)
where the constants e and e are the electronic charge and the electron mobility, respectively. kg is the amplitude of the grating vector which takes value of 2k0n0 sin y, where k0 is the wave vector of the recording beams in free space and n0 is the average refractive index of the crystal. The variable quantities N D1 , N S1 and Ne1 are the fundamental components of the Fourier series of the electronic concentration in the deep trap, the shallow trap and the conduction band, respectively. The crystal is sensitized by the uniform UV light of infinite width illuminating along the x direction during the recording stage. As a result of the presence of the SCF, a refractive index modulation of the photorefractive grating is induced by means of the linear electro-optic effect, which can be written as n30 g13 E SC (3) 2 where jE is the complex angle of the SCF and g13 is the electro-optic coefficient. The diffraction properties of the crossed-beam photorefractive gratings during the grating recorded and fixed phase are analyzed by the two-dimensional coupledwave equations [14]. The diffraction efficiency, which is defined as the ratio of diffracted energy along the x direction to the total power including the diffracted energy along the x direction and the transmitted energy along the Z direction, has the form n1 expðijE Þ ¼
PSout (4) S PR out þ Pout Rw where PSout ¼ 0 2 jA2 ðx; w1 Þj2 dx csc 2y and PR out ¼ R w1 2 jA ðw ; ZÞj dZ csc 2y are the integral of the diffracted 1 2 0 beam intensities in the x direction and of the transmitted beam intensities in the Z direction, respectively. Coordinates x and Z are orthogonal to K1 and K2, respectively (shown in Fig. 1), which is defined to simplify the numerical calculation of the two-dimensional coupled-wave equations. DE ¼
3. Numerical calculations and discussions Fig. 1. Recording geometry for a crossed-beam photorefractive grating in the crystal.
In order to solve the two-center material equations and the two-dimensional coupled-wave equations
ARTICLE IN PRESS
2.0x10-4 1.5x10-4 1.0x10-4 5.0x10-5 0.0 0
150
300 450 600 Time (min)
750
900
Fig. 3. The time variation of the average refractive index modulation.
2500
1500 1000 0.0 0.2 0.4 0.6 0.8 1.0 1.0
500
)
m
0.8
0.4 0.6 m) m η(
0.2
Intensity (Wm-2)
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m
0 0.0
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0.8
0.6
0.2
0.4
2000 0 0.0
Intensity (Wm-2)
10000
) m m ξ(
)
mm
η(
Fig. 4. Spatial variations of the intensity at the end of fixing: (a) the readout beam and (b) the diffracted beam.
3.5x10-4 3.0x10-4 2.5x10-4 2.0x10-4 1.5x10-4 1.0x10-4
)
m
0.4 0.6 η (m m)
1.0 0.8 0.6 0.4 0.2 0.8
m
5.0x10-5 0.0 0.0 0.2
1.0 0.0
ξ(
Refractive index modulation n1
459
ξ(
numerically and to obtain the simultaneous solution, the numerical procedure is used, which is similar to that proposed by Ren et al. [15], except that instead of the fourth-order Runge–Kutta method used to solve the coupling differential equations numerically the Heun method is used (a special case of the Runge–Kutta method) [16]. This numerical procedure has been described in detail in the previous work [14]. In this work, the parameters are equal to the values provided in our previous work and in Ref. [15] except that the boundary widths of the two recording waves are 1.0 mm, and the incident angle of them is 251 outside the crystal. Fig. 2 shows the spatial distribution of the refractive index modulation at the fixing end. It can be seen that the spatial distribution of the refractive index modulation is nonuniform in the grating region. It fluctuates and decreases as the grating region widens, which is mostly because of the rapid absorption of the UV light in the crystal causing the weaker fixed grating in the area distant from the input boundary. To analyze the temporal dynamic property of the photorefractive grating, the average refractive index modulation is defined as the spatial arithmetical mean of the refractive change in each time interval. Fig. 3 shows the change of the average refractive index modulation with time, which indicates that the index modulation increases and then almost moves towards saturation during the recording phase, and during the fixing phase it decreases and the grating is fixed finally. The appreciable increase of the index modulation when time is close to the end of the recording phase may be due to the spatial nonuniformity of dynamic grating. Fig. 4(a) and (b) shows the intensity spatial variations of the readout beam E1 and the diffracted beam E2 in the grating region at the end of fixing, respectively. From Fig. 4(a), one can see that the amplitude of the readout beam decreases with the increase in the value of x for constant Z. We think it is mostly due to the energy transfer to the diffracted beam as the readout beam propagates through the grating region. Certainly, the
Average refractive index modulation n1
X. Wang et al. / Optik 121 (2010) 457–461
Fig. 2. Spatial variation of the refractive index modulation at the end of fixing.
absorption of the readout beam in the crystal partially works on the decrease of the intensity. Along the Z direction, the intensity fluctuates and decreases with the increase of Z but at the input boundary, which indicates the profile deformation of the uniform plane readout beam caused by the beams’ coupling [9]. Fig. 4(b) shows the increase in the intensity of the diffracted beam during its propagation through the grating region to higher Z values. The increase in energy comes from the energy transferring from the readout beam. The figure
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X. Wang et al. / Optik 121 (2010) 457–461
Diffraction Efficiency %
460
900
Fig. 5. Change of diffraction efficiency and the average diffraction efficiency with time at the output boundary.
also shows that the energy decreases with the increase of x along the x direction, which demonstrates the fact that the stronger index modulation determined benefits the higher intensity at lower x values for constant Z value in the grating region [9]. Fig. 5 plots the time dependence of the diffraction efficiency at the output boundary of the grating. It can be seen that the diffraction efficiency increases to maximum and then moves towards saturation during the recording phase, and it decreases and is fixed finally during the fixing phase. The evolution is almost similar to that of one-dimensional photorefractive gratings [15]. The result indicates that the fixing efficiency with the definition of the ratio of the fixed diffraction efficiency to the saturation diffraction efficiency takes a value of 94.5%. In this figure, the calculation of the average diffraction efficiency (average DE) at the output boundary of the grating is also presented, which is calculated by using the average refractive index modulation and the diffraction efficiency equation proposed by Kenan [4]. The line of the average DE shows that the fixing efficiency is much higher than that of the diffraction efficiency. The difference between the diffraction efficiency and its mean is due to the latter ignoring the spatial nonuniformity of the dynamic grating.
4. Conclusions Based on the two-center band transport model and the two-dimensional coupled-wave theory, the holographic formation dynamics of nonvolatile crossedbeam photorefractive gratings formed by two-center holographic recording in doubly doped LiNbO3 crystals has been analyzed in this paper. The numerical calculations were presented, which show the spatial nonuniform properties of the refractive index modulation. The change of spatial mean of refractive index modulation with time indicates holographic formation
dynamics. The spatial variations of the intensity of the readout and diffracted beam indicate the energy transferring and coupling between the readout and the diffracted beam. The time dependence of the diffraction efficiency and its mean shows the higher fixing efficiency and also indicates the influence of the spatial heterogeneity for two-center holographic recording nonvolatile crossed-beam photorefractive gratings in doubly doped LiNbO3 crystals. The results provide quantitative predictions of the expected behavior of the formation dynamics of these gratings and can be helpful for the design of finite boundary photorefractive holographic optical elements.
Acknowledgment This work is supported by Nanjing University of Science & Technology (Grant no. AB41928).
References [1] R. Ince, H. Yu¨kselici, A.T. I˙nce, Crystal diffraction grating in LiNbO3:Fe derived from interferometric volume holography, Opt. Lasers Eng. 44 (2006) 509–519. [2] J.P. Burger, S. Dubovitsky, W.H. Steier, Kerr-like nonlinear mode converters for integrated optic device applications, Opt. Commun. 212 (2002) 251–266. [3] J.-W. An, N. Kim, K.-W. Lee, Volume holographic wavelength demultiplexer based on rotation multiplexing in the 901 geometry, Opt. Commun. 197 (2001) 247–254. [4] R.P. Kenan, Theory of crossed-beam diffraction gratings, IEEE. J. Quantum Electron. QE-14 (12) (1978) 924–930. [5] S.Q. Tao, B. Wang, G.W. Burr, J.B. Chen, Diffraction efficiency of volume gratings with finite size: corrected analytical solution, J. Mod. Opt. 51 (2004) 1115–1122. [6] A.Y. Hamad, J.P. Wicksted, Volume grating produced by intersecting Gaussian beams in an absorbing medium: a Bragg diffraction model, Opt. Commun. 138 (1997) 354–364. [7] P.P. Banerjee, M.R. Chatterjee, N. Kukhturer, T. Kukhtareva, Volume holographic recording and readout for 90-deg geometry, Opt. Eng. 43 (2004) 2053–2060. [8] A.M. Yan, L.Y. Wan, De’an Liu, L.R. Liu, Anisotropic Bragg diffraction of finite-sized volume holographic grating in photorefractive crystals, Optik 118 (2007) 418–424. [9] W. Kro´likowski, M. Cronin-Colomb, Photorefractive wave mixing with finite beams, Opt. Commun. 89 (1992) 88–98. + [10] D. Fluck, J.A. Weiss, S. Bru+ lisauar, P. Gunters, Twowave mixing of focused Gaussian beams in photorefractive wave guides, Opt. Lett. 19 (24) (1994) 2080–2082. [11] Zhifang Chai, De’an Liu, Aimin Yan, Ya’nan Zhi, Liren Liu, Effect of absorption on the diffraction efficiency in Fe:LiNbO3 crystal with 90 recording geometry, J. Opt. A: Pure Appl. Opt. 7 (2005) 711–716.
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[12] C.W. Tarn, Gaussian-beam profile deformation via anisotropic photorefractive gratings formed by diffusive, photovoltaic, and drift mechanisms: a system transfer function approach, Opt. Eng. 37 (1) (1998) 229–236. [13] K. Buse, A. Adibi, D. Pasltis, Non-volatile holographic storage in double doped lithium niobite crystals, Nature 393 (1998) 665–668. [14] X. Wang, A.M. Yan, L.R. Liu, D. Liu, Y.N. Zhi, Z.J. Hu, Influence of recording conditions on crossed-beam
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photorefractive gratings in doubly doped LiNbO3 crystals, Appl. Opt. 45 (23) (2006) 5942–5949. [15] L.Y. Ren, L.L. Liu, J.F. Zu, Z. Luan, Recording and fixing dynamics of nonvolatile photorefractive holograms in LiNbO3:Fe:Mn crystals, J. Opt. Soc. Am. B 20 (2003) 2162–2173. [16] G. Notni, R. Kowarschik, Diffraction analysis of threedimensional volume gratings with arbitrary boundaries, J. Opt. Soc. A A6 (1989) 1682–1691.