- Email: [email protected]
Effect of recording conditions on the anisotropic diffraction of volume holographic gratings
ARTICLE IN PRESS
Optik
Optics
Optik 118 (2007) 373–376 www.elsevier.de/ijleo
Effect of recording conditions on the anisotropic diffraction of volume holographic gratings Xin Wang, Aimin Yan, De’an Liu, Liren Liu, Zhijuan Hu Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, P.O. Box 800-211, Shanghai 201800, PR China Received 15 September 2005; accepted 28 February 2006
Abstract The anisotropic Bragg diffraction of the volume holographic gratings in photorefractive crystals are investigated based on the model of anisotropic coupled-wave theory. The effect of the initial intensity ratio and the recording angles of the two recording waves on the anisotropic Bragg diffraction properties is discussed. It is shown that both the ratio of the initial intensity and the incident angles of the recording waves are selective action for the anisotropic Bragg diffraction efficiency of the volume holographic gratings, while these two recording conditions are not selective action for the isotropic Bragg diffraction. Furthermore, the Bragg phase matching condition of anisotropic diffraction is analyzed when the recording angles change. r 2006 Elsevier GmbH. All rights reserved. Keywords: Volume holographic gratings; Anisotropic diffraction; Diffraction efficiency
1. Introduction For a long time, the volume holographic gratings in photorefractive materials have attracted the attention of people for their great potential for several applications in different areas such as information processing, phase conjugation, light deflection, etc. [1]. The photorefractive effect and two beam coupling in birefringent photorefractive material can be influenced strongly by the polarization of light because of the anisotropic property of material. Using the anisotropic diffraction of the volume gratings can improve signal-to-noise ratio due to the polarization orthogonality of the readout and diffracted readout beam. Therefore, the anisotropic diffraction properties of volume holographic gratings in anisotropic photorefractive materials have been Corresponding author. Fax: +86 21 6991 8096.
E-mail address: [email protected] (X. Wang). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.02.009
extensively studied in the past few years [2–5]. However, the effect of the recording conditions has been almost ignored in analyzing the anisotropic coupled-wave equations in previous work. In this paper, the intensity modulation of the two recording waves which is coherent to the grating length is introduced to the anisotropic coupled-wave equations and the effect of the initial intensity ratio and the recording angles of the recording waves on the anisotropic Bragg diffraction properties of the volume holographic gratings is discussed in detail. The diffraction efficiency with various intensity modulations and different polarization modes is calculated. It is found that the diffraction efficiency of the volume holographic gratings is strongly influenced by the initial intensity ratio and the incident angle of the recording waves. In addition, the Bragg phase matching condition of anisotropic diffraction is investigated when the recording angles change. The theoretical investigation can be useful
ARTICLE IN PRESS 374
X. Wang et al. / Optik 118 (2007) 373–376
for the design of photorefractive volume holographic optical elements.
2. Theoretical analyses It is important to note that the discussion is limited to the case of the transmission grating in this paper. The recording diagram of the volume holographic grating is shown in Fig. 1 in which two plane waves incident upon a photorefractive crystal and intersect in the crystal. Here K1 and K2 are the wave vectors of the recording wave E1 and E2, respectively. y is the recording angle of the recording waves measured in the crystal. The optical axis is normal to the incident plane. As a result of the photorefractive effect, the volume holographic grating is caused by the interference pattern with a grating vector of Kg ¼ K1–K2 and its amplitude is 4pno siny/l0, where l0 is the wave length of the recording waves in free space and no is the refractive indices for the ordinary wave in the crystal. For the readout of this volume holographic grating on the Bragg condition, the electric field amplitudes of the readout wave and the diffracted wave can be given with the forms as following: Er ðrÞ ¼ e^ r R expðiKr rÞ, Es ðrÞ ¼ e^ s S expðiKs rÞ,
ð1Þ
where R, S, Kr, Ks, eˆr and eˆs are the wave amplitudes, the wave vectors and the unit polarization vectors of the readout and diffracted readout wave, respectively. Considered the anisotropic property of the photorefractive crystal, the coupled-wave equations for the readout and diffracted readout wave can be given by [6]: dR ¼ kr mðxÞS, dx dS ¼ ks mðxÞR, cos ys dx cos yr
ð2Þ
where yr and ys are the readout and diffracted angle measured in the crystal which are defined to be the angles between the x-axis and the wave vectors of the readout and diffracted readout wave, respectively. m(x) is the intensity modulation of the recording waves in the y E1 x E2
Fig. 1. Geometry of recording volume holographic grating in a photorefractive crystal.
crystal, which is defined as m(x) ¼ 2(I1I2)1/2/(I1+I2), where I1 and I2 are the intensity of the recording wave E1 and E2, respectively. The coupling coefficients kr and ks have forms as following: pweff kr ¼ , 2nr l0 cos yr pweff , ð3Þ ks ¼ 2ns l0 cos ys where weff ¼ ð^e1 e^ 2 Þð^er dx^ e^ s Þ, dx^ ¼ ~ r~~ E sc ~. Here nr and ns are the index of the crystal for the readout and diffracted readout wave. eˆ1 and eˆ2 are the unit polarization vector of the recording wave E1 and E2, respectively. ~ is the second-rank optical dielectric tensor. r~~ is the third-rank electro-optic tensor and Esc is the space charge field in the crystal. Considering the boundary conditions as R(0) ¼ 1 and S(0) ¼ 0, we can solve Eq. (2) and obtain the diffraction efficiency of the volume holographic grating which is defined to be the ratio of the power flow in the diffracted readout wave normal to the surface at the grating output to the power flow in the incident beam normal to the surface at the grating input and has a form: h i Z ¼ sin2 jkr ks j1=2 uðdÞ , (4) where uðdÞ ¼ 4G1
pffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi m0 tan m0 eGd tan1 m0 .
Here G is the coupling coefficient of the recording waves and m0 is initial intensity ratio of the recording waves with a form of m0 ¼ I20/I10, where I10 and I20 are the initial intensity of the recording wave E1 and E2, respectively. In addition, it is important to analyze the Bragg condition. We consider the facts that the volume holographic gratings are recorded in the negative uniaxial photorefractive crystal and that the recording waves are ordinary wave. There are four types diffraction in this crystal: oo-type diffraction, oe-type diffraction, eo-type diffraction and ee-type diffraction. For ootype diffraction and ee-type diffraction, both the readout wave and the diffracted readout wave are ordinary wave and extraordinary wave. For oe-type diffraction, the readout wave is ordinary wave and the diffracted readout wave is extraordinary wave. For eo-type diffraction, the readout wave is extraordinary wave and the diffracted readout wave is ordinary wave. ootype diffraction and ee-type diffraction are isotropic diffractions and oe-type diffraction and eo-type diffraction are anisotropic diffractions [7]. Based on the recording geometry of the volume holographic grating shown in Fig. 1, the wave vector diagram in the crystal
ARTICLE IN PRESS X. Wang et al. / Optik 118 (2007) 373–376
ky
ky
k0 n0 k0 ne
1.0 m0 = 0.1
r
Kr
Kg
Ks
r
s
kx
Kg
Ks
Diffraction efficiency
k0 n0 k0 ne Kr
375
kx
m0 = 0.3
0.8
m0 = 0.5 m0 = 0.7
0.6 0.4 0.2
(a)
(b)
Fig. 2. Wave vector diagram for anisotropic diffraction of a volume holographic grating in a negative uniaxial crystal: (a) oe-type diffraction and (b) eo-type diffraction.
0.0
5
10
15
20
25
20
25
Recording angle (degree)
(a) 1.0
for anisotropic Bragg diffraction is given in Fig. 2, where the angle of readout and diffraction yr and ys are show by the following [6]: for oe-type diffraction n2o n2e 4n2o sin2 y , 4no ne sin y ne cos ys yr ¼ cos1 , no ys ¼ sin1
Diffraction efficiency
m0 = 0.1 m0 = 0.5 m0 = 0.7
0.6 0.4 0.2
ð5Þ 0.0
for eo-type diffraction n2e n2o 4n2o sin2 y , 4n2o sin y no cos ys yr ¼ cos1 , ne
m0 = 0.3
0.8
5
ys ¼ sin1
10
15
Recording angle (degree)
(b) 0.70
3. Calculated results and discussions Based on the analysis in Section 2, we present the calculated results for a volume holographic grating recorded in doped lithium niobate (LiNbO3) crystal. The ordinary and the extraordinary refractive indices no and ne are 2.2910 and 2.2005 for a wavelength l of 633 nm in doped LiNbO3 crystal, respectively. The electro-optic coefficients for doped LiNbO3 crystal have following values: r13 ¼ 8.6e12 m/V, r33 ¼ 30.8e12 m/V, r51 ¼ 28.0e12 m/V [8]. Fig. 3 shows the diffraction efficiency of the volume holographic grating on the Bragg condition. It can be seen from Figs. 3a and b that the diffraction efficiency can be zero at some recording angle while it can also be 100% when the initial intensity ratio of the recording waves changes, v.v. This suggests that both the initial intensity ratio and the recording angle of the recording waves are selective action for the anisotropic diffraction efficiency. Fig. 3c shows that the isotropic diffraction
Diffraction efficiency
ð6Þ
where ne is the refractive index of the crystal for the extraordinary wave.
m0 = 0.1
0.65
m0 = 0.3
0.60
m0 = 0.5
0.55
m0 = 0.7
0.50 0.45 0.40 0.35 0.30 0.25 5
(c)
10 15 20 Recording angle (degree)
25
Fig. 3. Diffraction efficiency of the volume holographic grating in doped LiNbO3 crystal (m0 ¼ I20/I10): (a) oe-type diffraction, (b) eo-type diffraction, and (c) oo-type diffraction.
efficiency decreases with the recording angle increase when the initial intensity ratio of the recording waves is various, which indicates that the two recording conditions are not selective action for the isotropic Bragg diffraction. In addition, the figures show that the maximum diffraction efficiency of oo-type diffraction is much smaller than that of anisotropic diffraction. This can be explained as following: when the optical axis is perpendicular to the incident plane and the grating
ARTICLE IN PRESS X. Wang et al. / Optik 118 (2007) 373–376
Bragg readoutangle (degree)
376
oe-type diffraction eo-type diffraction
40 35 30 25 20 15 5
(a)
10 15 20 Recording angle (degree)
25
Bragg diffracted (degree)
40 oe-type diffraction eo-type diffraction
30 20 10 0 -10 -20 -30 -40
(b)
5
10 15 20 Recording angle (degree)
anisotropic property of the material on the Bragg diffraction properties of the volume holographic gratings in birefringent photorefractive crystal. As a sample, the calculations for the volume holographic grating recorded in doped LiNbO3 crystal are presented and the diffraction efficiency with various modulations and different recording angle is calculated. Results show that both the initial intensity ratio of the recording waves and the recording angle are selective action for the anisotropic Bragg diffraction efficiency. But these two recording conditions are not selective action for the isotropic Bragg diffraction. It is also shown that the maximum diffraction efficiency of isotropic Bragg diffraction is much smaller than that of anisotropic diffraction due to the different electro-optic coefficients. In addition, the Bragg phase matching condition of anisotropic diffraction is investigated when the recording angles change. To the application, these analytical results should be useful in designing optical switch, time-delay networks, electro-optical modulations and deflectors.
25
Fig. 4. (a) Relation between the Bragg readout and recording angle in the crystal and (b) relation between the Bragg diffracted and the recording angle in the crystal.
vector is parallel to the y direction, the diffraction efficiency of the oo-type diffraction is dependent on r22 while for anisotropic diffraction, it is dependent on r51. The amplitude of r22 is much smaller than that of r51 in doped LiNbO3 crystal, which causes that the coupling coefficient kr and ks are much smaller for oo-type diffraction than that for anisotropic diffraction. The recording angle dependence of angles of readout and the diffraction yr and ys for anisotropic Bragg diffraction is shown in Figs. 4a and b, in which the recording angle changes from 21 to 251. From the figures, we can see that the readout angle is smaller for oe-type diffraction than that for eo-type diffraction and the evolutions of the readout angle yr for the two types diffraction are quite similar. The evolutions of the diffraction ys are different for oe-type diffraction and eo-type diffraction. For oe-type diffraction, there is a readout angle where the diffraction angle is exactly zero [9]. However, there is no such angle for eo-type type diffraction. For oe-type diffraction, the diffracted angle decreases with the increase of the recording angle, while for eo-type diffraction, it increases firstly and then decreases.
4. Conclusion Based on the anisotropic diffractive theory, we discussed the effect of the recording conditions and the
Acknowledgments The authors acknowledge the Science and Technology committee of China (Grant 2002CCA03500) and the National Nature Science Foundation of China (Grant 60177016).
References [1] H.J. Eichler, P. Gunter, D.W. Pohl, Laser-induced Dynamic Gratings, Spinger, NewYork, 1986, pp. 223–224. [2] M.P. Petrov, T.G. Pencheva, S.I. Stepanov, Light diffraction from volume phase holograms in electrooptic photorefractive crystals, J. Opt. 12 (1981) 287–292. [3] H. Zhou, F. Zhao, F.T.S. Yu, Angle-dependent diffraction efficiency in thick photorefractive hologram, Appl. Opt. 34 (1995) 1303–1309. [4] J.J. Butler, M.S. Malcuit, M.A. Rodriguez, Diffractive properties of highly birefringent volume gratings: investigation, J. Opt. Soc. Am. B 19 (2002) 183–189. [5] K. Zaitsev, K.Y. Hsu, S.H. Lin, Anisotropic diffraction of light by volume holographic grating in birefringent photorefractive crystals with extended wavelength range, Opt. Exp. 10 (2002) 204–209. [6] D.A. Temple, C. Warde, Anisotropic scattering in photorefractive crystals, J. Opt. Soc. Am. B 3 (1986) 337–341. [7] E.N. Glytsis, T.K. Gaylord, Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction, J. Opt. Soc. Am. A 7 (1990) 1399–1420. [8] R.S. Weis, T.K. Gaylord, Lithium niobate: summary of physical properties and crystal structure, Appl. Phys. A 37 (1985) 191–203. [9] P. Gunter, J.-P. Huignard, Photorefractive Materials and Their Applications I, Spinger, NewYork, 1988, pp. 32–35.
Optik
Optics
Optik 118 (2007) 373–376 www.elsevier.de/ijleo
Effect of recording conditions on the anisotropic diffraction of volume holographic gratings Xin Wang, Aimin Yan, De’an Liu, Liren Liu, Zhijuan Hu Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Science, P.O. Box 800-211, Shanghai 201800, PR China Received 15 September 2005; accepted 28 February 2006
Abstract The anisotropic Bragg diffraction of the volume holographic gratings in photorefractive crystals are investigated based on the model of anisotropic coupled-wave theory. The effect of the initial intensity ratio and the recording angles of the two recording waves on the anisotropic Bragg diffraction properties is discussed. It is shown that both the ratio of the initial intensity and the incident angles of the recording waves are selective action for the anisotropic Bragg diffraction efficiency of the volume holographic gratings, while these two recording conditions are not selective action for the isotropic Bragg diffraction. Furthermore, the Bragg phase matching condition of anisotropic diffraction is analyzed when the recording angles change. r 2006 Elsevier GmbH. All rights reserved. Keywords: Volume holographic gratings; Anisotropic diffraction; Diffraction efficiency
1. Introduction For a long time, the volume holographic gratings in photorefractive materials have attracted the attention of people for their great potential for several applications in different areas such as information processing, phase conjugation, light deflection, etc. [1]. The photorefractive effect and two beam coupling in birefringent photorefractive material can be influenced strongly by the polarization of light because of the anisotropic property of material. Using the anisotropic diffraction of the volume gratings can improve signal-to-noise ratio due to the polarization orthogonality of the readout and diffracted readout beam. Therefore, the anisotropic diffraction properties of volume holographic gratings in anisotropic photorefractive materials have been Corresponding author. Fax: +86 21 6991 8096.
E-mail address: [email protected] (X. Wang). 0030-4026/$ - see front matter r 2006 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2006.02.009
extensively studied in the past few years [2–5]. However, the effect of the recording conditions has been almost ignored in analyzing the anisotropic coupled-wave equations in previous work. In this paper, the intensity modulation of the two recording waves which is coherent to the grating length is introduced to the anisotropic coupled-wave equations and the effect of the initial intensity ratio and the recording angles of the recording waves on the anisotropic Bragg diffraction properties of the volume holographic gratings is discussed in detail. The diffraction efficiency with various intensity modulations and different polarization modes is calculated. It is found that the diffraction efficiency of the volume holographic gratings is strongly influenced by the initial intensity ratio and the incident angle of the recording waves. In addition, the Bragg phase matching condition of anisotropic diffraction is investigated when the recording angles change. The theoretical investigation can be useful
ARTICLE IN PRESS 374
X. Wang et al. / Optik 118 (2007) 373–376
for the design of photorefractive volume holographic optical elements.
2. Theoretical analyses It is important to note that the discussion is limited to the case of the transmission grating in this paper. The recording diagram of the volume holographic grating is shown in Fig. 1 in which two plane waves incident upon a photorefractive crystal and intersect in the crystal. Here K1 and K2 are the wave vectors of the recording wave E1 and E2, respectively. y is the recording angle of the recording waves measured in the crystal. The optical axis is normal to the incident plane. As a result of the photorefractive effect, the volume holographic grating is caused by the interference pattern with a grating vector of Kg ¼ K1–K2 and its amplitude is 4pno siny/l0, where l0 is the wave length of the recording waves in free space and no is the refractive indices for the ordinary wave in the crystal. For the readout of this volume holographic grating on the Bragg condition, the electric field amplitudes of the readout wave and the diffracted wave can be given with the forms as following: Er ðrÞ ¼ e^ r R expðiKr rÞ, Es ðrÞ ¼ e^ s S expðiKs rÞ,
ð1Þ
where R, S, Kr, Ks, eˆr and eˆs are the wave amplitudes, the wave vectors and the unit polarization vectors of the readout and diffracted readout wave, respectively. Considered the anisotropic property of the photorefractive crystal, the coupled-wave equations for the readout and diffracted readout wave can be given by [6]: dR ¼ kr mðxÞS, dx dS ¼ ks mðxÞR, cos ys dx cos yr
ð2Þ
where yr and ys are the readout and diffracted angle measured in the crystal which are defined to be the angles between the x-axis and the wave vectors of the readout and diffracted readout wave, respectively. m(x) is the intensity modulation of the recording waves in the y E1 x E2
Fig. 1. Geometry of recording volume holographic grating in a photorefractive crystal.
crystal, which is defined as m(x) ¼ 2(I1I2)1/2/(I1+I2), where I1 and I2 are the intensity of the recording wave E1 and E2, respectively. The coupling coefficients kr and ks have forms as following: pweff kr ¼ , 2nr l0 cos yr pweff , ð3Þ ks ¼ 2ns l0 cos ys where weff ¼ ð^e1 e^ 2 Þð^er dx^ e^ s Þ, dx^ ¼ ~ r~~ E sc ~. Here nr and ns are the index of the crystal for the readout and diffracted readout wave. eˆ1 and eˆ2 are the unit polarization vector of the recording wave E1 and E2, respectively. ~ is the second-rank optical dielectric tensor. r~~ is the third-rank electro-optic tensor and Esc is the space charge field in the crystal. Considering the boundary conditions as R(0) ¼ 1 and S(0) ¼ 0, we can solve Eq. (2) and obtain the diffraction efficiency of the volume holographic grating which is defined to be the ratio of the power flow in the diffracted readout wave normal to the surface at the grating output to the power flow in the incident beam normal to the surface at the grating input and has a form: h i Z ¼ sin2 jkr ks j1=2 uðdÞ , (4) where uðdÞ ¼ 4G1
pffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi m0 tan m0 eGd tan1 m0 .
Here G is the coupling coefficient of the recording waves and m0 is initial intensity ratio of the recording waves with a form of m0 ¼ I20/I10, where I10 and I20 are the initial intensity of the recording wave E1 and E2, respectively. In addition, it is important to analyze the Bragg condition. We consider the facts that the volume holographic gratings are recorded in the negative uniaxial photorefractive crystal and that the recording waves are ordinary wave. There are four types diffraction in this crystal: oo-type diffraction, oe-type diffraction, eo-type diffraction and ee-type diffraction. For ootype diffraction and ee-type diffraction, both the readout wave and the diffracted readout wave are ordinary wave and extraordinary wave. For oe-type diffraction, the readout wave is ordinary wave and the diffracted readout wave is extraordinary wave. For eo-type diffraction, the readout wave is extraordinary wave and the diffracted readout wave is ordinary wave. ootype diffraction and ee-type diffraction are isotropic diffractions and oe-type diffraction and eo-type diffraction are anisotropic diffractions [7]. Based on the recording geometry of the volume holographic grating shown in Fig. 1, the wave vector diagram in the crystal
ARTICLE IN PRESS X. Wang et al. / Optik 118 (2007) 373–376
ky
ky
k0 n0 k0 ne
1.0 m0 = 0.1
r
Kr
Kg
Ks
r
s
kx
Kg
Ks
Diffraction efficiency
k0 n0 k0 ne Kr
375
kx
m0 = 0.3
0.8
m0 = 0.5 m0 = 0.7
0.6 0.4 0.2
(a)
(b)
Fig. 2. Wave vector diagram for anisotropic diffraction of a volume holographic grating in a negative uniaxial crystal: (a) oe-type diffraction and (b) eo-type diffraction.
0.0
5
10
15
20
25
20
25
Recording angle (degree)
(a) 1.0
for anisotropic Bragg diffraction is given in Fig. 2, where the angle of readout and diffraction yr and ys are show by the following [6]: for oe-type diffraction n2o n2e 4n2o sin2 y , 4no ne sin y ne cos ys yr ¼ cos1 , no ys ¼ sin1
Diffraction efficiency
m0 = 0.1 m0 = 0.5 m0 = 0.7
0.6 0.4 0.2
ð5Þ 0.0
for eo-type diffraction n2e n2o 4n2o sin2 y , 4n2o sin y no cos ys yr ¼ cos1 , ne
m0 = 0.3
0.8
5
ys ¼ sin1
10
15
Recording angle (degree)
(b) 0.70
3. Calculated results and discussions Based on the analysis in Section 2, we present the calculated results for a volume holographic grating recorded in doped lithium niobate (LiNbO3) crystal. The ordinary and the extraordinary refractive indices no and ne are 2.2910 and 2.2005 for a wavelength l of 633 nm in doped LiNbO3 crystal, respectively. The electro-optic coefficients for doped LiNbO3 crystal have following values: r13 ¼ 8.6e12 m/V, r33 ¼ 30.8e12 m/V, r51 ¼ 28.0e12 m/V [8]. Fig. 3 shows the diffraction efficiency of the volume holographic grating on the Bragg condition. It can be seen from Figs. 3a and b that the diffraction efficiency can be zero at some recording angle while it can also be 100% when the initial intensity ratio of the recording waves changes, v.v. This suggests that both the initial intensity ratio and the recording angle of the recording waves are selective action for the anisotropic diffraction efficiency. Fig. 3c shows that the isotropic diffraction
Diffraction efficiency
ð6Þ
where ne is the refractive index of the crystal for the extraordinary wave.
m0 = 0.1
0.65
m0 = 0.3
0.60
m0 = 0.5
0.55
m0 = 0.7
0.50 0.45 0.40 0.35 0.30 0.25 5
(c)
10 15 20 Recording angle (degree)
25
Fig. 3. Diffraction efficiency of the volume holographic grating in doped LiNbO3 crystal (m0 ¼ I20/I10): (a) oe-type diffraction, (b) eo-type diffraction, and (c) oo-type diffraction.
efficiency decreases with the recording angle increase when the initial intensity ratio of the recording waves is various, which indicates that the two recording conditions are not selective action for the isotropic Bragg diffraction. In addition, the figures show that the maximum diffraction efficiency of oo-type diffraction is much smaller than that of anisotropic diffraction. This can be explained as following: when the optical axis is perpendicular to the incident plane and the grating
ARTICLE IN PRESS X. Wang et al. / Optik 118 (2007) 373–376
Bragg readoutangle (degree)
376
oe-type diffraction eo-type diffraction
40 35 30 25 20 15 5
(a)
10 15 20 Recording angle (degree)
25
Bragg diffracted (degree)
40 oe-type diffraction eo-type diffraction
30 20 10 0 -10 -20 -30 -40
(b)
5
10 15 20 Recording angle (degree)
anisotropic property of the material on the Bragg diffraction properties of the volume holographic gratings in birefringent photorefractive crystal. As a sample, the calculations for the volume holographic grating recorded in doped LiNbO3 crystal are presented and the diffraction efficiency with various modulations and different recording angle is calculated. Results show that both the initial intensity ratio of the recording waves and the recording angle are selective action for the anisotropic Bragg diffraction efficiency. But these two recording conditions are not selective action for the isotropic Bragg diffraction. It is also shown that the maximum diffraction efficiency of isotropic Bragg diffraction is much smaller than that of anisotropic diffraction due to the different electro-optic coefficients. In addition, the Bragg phase matching condition of anisotropic diffraction is investigated when the recording angles change. To the application, these analytical results should be useful in designing optical switch, time-delay networks, electro-optical modulations and deflectors.
25
Fig. 4. (a) Relation between the Bragg readout and recording angle in the crystal and (b) relation between the Bragg diffracted and the recording angle in the crystal.
vector is parallel to the y direction, the diffraction efficiency of the oo-type diffraction is dependent on r22 while for anisotropic diffraction, it is dependent on r51. The amplitude of r22 is much smaller than that of r51 in doped LiNbO3 crystal, which causes that the coupling coefficient kr and ks are much smaller for oo-type diffraction than that for anisotropic diffraction. The recording angle dependence of angles of readout and the diffraction yr and ys for anisotropic Bragg diffraction is shown in Figs. 4a and b, in which the recording angle changes from 21 to 251. From the figures, we can see that the readout angle is smaller for oe-type diffraction than that for eo-type diffraction and the evolutions of the readout angle yr for the two types diffraction are quite similar. The evolutions of the diffraction ys are different for oe-type diffraction and eo-type diffraction. For oe-type diffraction, there is a readout angle where the diffraction angle is exactly zero [9]. However, there is no such angle for eo-type type diffraction. For oe-type diffraction, the diffracted angle decreases with the increase of the recording angle, while for eo-type diffraction, it increases firstly and then decreases.
4. Conclusion Based on the anisotropic diffractive theory, we discussed the effect of the recording conditions and the
Acknowledgments The authors acknowledge the Science and Technology committee of China (Grant 2002CCA03500) and the National Nature Science Foundation of China (Grant 60177016).
References [1] H.J. Eichler, P. Gunter, D.W. Pohl, Laser-induced Dynamic Gratings, Spinger, NewYork, 1986, pp. 223–224. [2] M.P. Petrov, T.G. Pencheva, S.I. Stepanov, Light diffraction from volume phase holograms in electrooptic photorefractive crystals, J. Opt. 12 (1981) 287–292. [3] H. Zhou, F. Zhao, F.T.S. Yu, Angle-dependent diffraction efficiency in thick photorefractive hologram, Appl. Opt. 34 (1995) 1303–1309. [4] J.J. Butler, M.S. Malcuit, M.A. Rodriguez, Diffractive properties of highly birefringent volume gratings: investigation, J. Opt. Soc. Am. B 19 (2002) 183–189. [5] K. Zaitsev, K.Y. Hsu, S.H. Lin, Anisotropic diffraction of light by volume holographic grating in birefringent photorefractive crystals with extended wavelength range, Opt. Exp. 10 (2002) 204–209. [6] D.A. Temple, C. Warde, Anisotropic scattering in photorefractive crystals, J. Opt. Soc. Am. B 3 (1986) 337–341. [7] E.N. Glytsis, T.K. Gaylord, Three-dimensional (vector) rigorous coupled-wave analysis of anisotropic grating diffraction, J. Opt. Soc. Am. A 7 (1990) 1399–1420. [8] R.S. Weis, T.K. Gaylord, Lithium niobate: summary of physical properties and crystal structure, Appl. Phys. A 37 (1985) 191–203. [9] P. Gunter, J.-P. Huignard, Photorefractive Materials and Their Applications I, Spinger, NewYork, 1988, pp. 32–35.