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The coupling and superposition behavior of higher-order diffraction image based on photorefractive nematic liquid crystal materials
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Optik 120 (2009) 524–529 www.elsevier.de/ijleo
The coupling and superposition behavior of higher-order diffraction image based on photorefractive nematic liquid crystal materials Dewei Gong, Zhongxiang Zhou, Yanbo Pei, Jianlong Zhang, Fengfeng Yao Department of Physics, Harbin Institute of Technology, Harbin 150001, China Received 24 April 2007; accepted 28 November 2007
Abstract We report some photorefractive higher-order diffraction phenomena of two kinds of photorefractive nematic liquid crystal materials. In the experiment, we observed the superposition of higher-order diffraction images, the coupling of higher-order diffraction lights, and the coupling between higher-order diffraction lights and incident beams. The observed behavior suggests the complexity of the coupling process. The properties of higher-order diffraction images are discussed theoretically, which are in good agreement with experimental results. r 2008 Elsevier GmbH. All rights reserved. Keywords: Photorefractive effect; Higher-order diffraction
1. Introduction The photorefractive (PR) effect has received a great deal of attention with respect to optical information processing applications [1]. Usually, the included angle between the incident beams is large enough when people store information in PR materials, so that we can use the Bragg diffraction and thus avoid the influence of higherorder diffraction (Raman–Nath regime). However, we can use higher-order diffraction to realize image processing such as image rotation and amplification [2,3]. The higher-order diffraction was first predicted by Brillouin [4] as diffraction of light by acoustic waves, and was experimentally revealed by Debye and Sears [5] and Lucas and Biquard [6]. The difference of higherorder diffraction between acousto-optic effect and PR effect is that in the case of dynamic interaction between Corresponding author. Tel.: +86 451 86414132; fax: +86 451 86412549. E-mail address: [email protected] (D. Gong).
0030-4026/$ - see front matter r 2008 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.11.012
the beams and refractive index modulation. For acousto-optic effect, the refractive index gratings are founded by acoustic waves. However, the refractive index gratings are founded by the spatial variation of light intensity for PR effect. So the diffraction of PR effect is more complex than that of acousto-optic effect. However, the properties of the higher-order diffraction images in PR materials have not been well studied because the higher-order diffraction images are usually blurry for PR crystals [2]. The recent development of organic PR materials, which are more costefficient and easier to prepare and to be modified than their inorganic counterparts, has fueled a surge of study in optical signal processing applications [7–14]. So we have the opportunity to investigate the properties of higher-order diffraction images. In this paper, we report some new phenomena of coupling and superposition based on the higher-order diffraction at large modulation in two photorefractive nematic liquid crystal materials. The results indicate the complexity of coupling.
ARTICLE IN PRESS D. Gong et al. / Optik 120 (2009) 524–529
p
2. Experiments and analysis The experimental setup is shown in Fig. 1. An s-polarized He–Ne laser (l ¼ 632.8 nm) was used as the light source. A beam from the laser was split into the reference beam and the signal beam. The signal beam passed through a spatial light modulator (SLM) and then was focused by an f ¼ 550 mm focal-length Fourier lens. The sample was placed at the focal plane. The reference beam illuminated onto the same area at an angle of 301 with the normal of sample surface. The included angle between reference beam and the bisector of the signal beam was about 1.51. Holograms were built by the interference of signal and reference beams. The normal of the sample was in the plane of incident beams. PR nematic liquid crystal [15] and PR liquid crystal polymer composite [16] were used as recording materials. There are common materials for organic PR experiments. Firstly, the nematic liquid crystal was used as recording material, which is the mixture of pure pentycyanabiphenyl liquid crystal (5CB) doped with fullerene (C60). The weight ratio was 99.95% (5CB) and 0.05% (C60). 5CB is a wonderful molecule for PR effect. The PR nonlinearity of the sample is highly improved by heavily doping with C60. Homeotropic alignment of the liquid crystal was obtained by depositing a hexadecyl trimethyl ammonium bromide film onto indium tin oxide (ITO) glass. The sample thickness was controlled to be around 16.6 mm with the help of spacers. Fig. 2 shows the
He-Ne Laser M1 M2 Sample BS CCD FL2
SLM
FL1
SL
M3
Fig. 1. Experimental set-up of higher-order diffraction images. M – mirror; BS – beam splitter; SLF – spatial light filter; FL – Fourier lens; SLM – spatial light modulator.
Fig. 2. Incident beams transmitted through the sample directly.
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Fig. 3. Higher-order diffraction images of two-wave coupling when signal images were two horizontal lines.
incident beams transmitted directly through the sample when the applied electric field was 0. The two horizontal lines were signal beams whose size was 10.0 9.0 mm2 on SLM. In the experiment, the externally applied electric field was varied from 0 to 0.30 V/mm (DC). When the voltage exceeded 0.06 V/mm, the complex diffraction images could be observed obviously by a CCD camera behind the sample (Fig. 3). According to Ref. [3], the experimental result should only have the diffraction order marked as n, n0 , f, f0 , i, i0 , o, o0 , m, m0 . We believe that Fig. 3 gives more information about the coupling. According to diffraction theory, if the refraction index gratings are illuminated with a laser beam, there are two kinds of diffractions: Bragg diffraction and Raman– Nath diffraction. The diffraction regime of the phase gratings is determined by the modulation parameter g ¼ pDnd=l cos f and Q0 ¼ Q= cos f ¼ 2pld=n0 L2 cos f (l is the free-space wavelength, d is the grating thickness, n0 is the refractive index of the whole medium in its undisturbed state, Dn is the modulation amplitude of the refractive index, L is the grating period and f is the angle of incidence inside the grating). In the Bragg diffraction regime, QX10 and Q0 /gX10 [17]. In this regime, multiple scattering is not permitted and only fundamental diffraction light is produced. Conversely, Q0 gp1 and Q0 p2 [18] is the Raman–Nath diffraction regime. The angular spread of the grating wave vector is larger than that of the Bragg angle, and therefore many diffraction orders can be observed in this regime. After interacting with the grating, the incident light beam is split into several diffraction beams. In our experiment, Q0 gE0.13 and Q0 E0.15. It might be expected that we had Raman–Nath diffraction. In the Raman–Nath regime, maxima of the intensity due to variation of the refractive index occur in directions making angles, denoted by yk, with the vertical direction of the grating vector K [19,20]: sin yk sin f ¼ sinðjk þ fÞ sin f ¼
klR L
k ðan integerÞ,
(1)
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where jk is the angle between the diffraction beam and incident beam, f is the incident angle of reading beam and lR is the wavelength of reading beam. If refraction index gratings are built by two-wave coupling in PR materials, the grating period L is determined by the following expression: L¼
lW , 2 sin b
(2)
where lW is the wavelength of writing beam, 2b is the angle between two writing beams. Substituting Eq. (2) into (1), the following is obtained: sin yk sin f ¼ sinðjk þ fÞ sin f ¼
klR lR ¼ 2k sin b. lW =2 sin b lW
(3) In our experiments, the writing beams are reading beams, so lW ¼ lr , b ¼ f. Because the angles jk, f and 2b are very small, we can obtain sin yk ¼ sinðjk þ fÞ jk þ f, sin b b and sin f f, then Eq. (3) can be written as yk jk þ f 2kb þ f.
(4)
In experiment, the image was two horizontal lines. In order to simplify our analysis, Fourier optics is simplified to geometry optics. The signal beam can be decomposed into many individual thin light beams whose diffractions are consonant with Eq. (4). The appearance of n, n0 , f, f0 , i, i0 , o, o0 , m, m0 has been explained well by Ref. [3], so the emphasis of the paper is to explain the appearance of c, c0 , d, d0 , e, g, g0 , h, h0 , j, j0 , k, k0 , l, p, p0 . First of all, we explain the appearance of diffraction light c, c0 . The coupling of two parts of signal beam b, b0 generates a PR grating that is parallel with b, b0 . The higher-order diffractions are c and c0 , so they have the same size with b, b0 , where 2b is the included angle between b and b0 in PR materials. According to Eq. (4), the included angle between c0 , b0 and b, c are nearly equal to 2b too. Then the reference beam a illuminate the same gratings. Spots h, h0 are the higherorder diffraction of reference beam a, which diffracted by the same grating with different incident direction. The included angles between h0 , a and a, h are also equal with 2b. After emergence, higher-order diffraction h0 can also couple with signal beam b0 , and k0 , j0 , e, g are 2, 1, +2, +3 diffraction order of gratings of h0 and b0 . In our experiments, the sample was placed at the Fourier plane, so the width of the interference area DL ¼ 0.The magnifications of k0 , j0 , e, g are 2, 1, +2, +3 strictly. Although k0 , j0 , e, g, as shown in Fig. 3, are only amplified in size, in fact, the higher-order diffractions k0 , j0 are rotated 1801 compared with b, b0 . With the same method, k, j, e, g0 are 2, 1, +2, +3 diffraction order of beams h and b. The size magnification of 2, 1, +2, +3 orders is 2, 1, +2, +3. From the above discussion, we can see that e is
overlapped +2 diffraction order. We can explain the emergence of d0 , c0 , l, p as the 2, 1, +2, +3 orders of h0 , and i0 , d, c, l, p0 as the 2, 1, +2, +3 orders of h and i. To our knowledge, the coupling of higher-order diffraction lights is first reported. d0 is the +2 order of h and b0 , and d is the +2 order of h0 and b. The emergence of k, k0 can also be explained with the same method. j0 , i0 , I, j can be looked as higher-order diffraction and energy could be transported among them. k0 , o0 , l, o, k, m0 , p0 , p, m, c0 , b0 , b, c, d0 , n0 , e, n, d and f0 , g0 , g, f can be explained with the same method. From Fig. 3 we know that many higher-order diffraction lights overlapped as one mutually indistinguishable diffraction light. When the images were two vertical lines, the result is shown in Fig. 4. Our analysis can also explain the experiment. Fig. 5 shows the higher-order diffraction images when the signal image was a square. The signal image can be considered as the addition of two groups of parallel lines. The experimental result is in excellent agreement with the above discussion. However, when the square was placed as the addition of signal images in Figs. 3 and 4, the higher-order diffraction images were fewer than that in Fig. 5. The reason may be the orientational property of the PR nematic liquid crystal. In the experiment, we found that the gratings were unstable. Intensity of different order diffraction images and scattered light changed with external electric field and exposure time. With electric field increasing from 0.06 to 0.30 V/mm, the higher-order diffraction images peaked sharply and then submerged by strong scattered light. In order to achieve high quality of optical diffraction images, we investigated the effect of the electric field. We found that the best-quality optical diffraction images, Figs. 3–5, could be obtained when the voltage was evenly increased from 0 to 5 V in 10 s. However, the best-quality higher-order diffraction images could only be maintained for about 1 s at 5 V, and then they were scattered whether the external electric field was changed or not. We did the same experiments with a composite derived from the high-performance PR polymer consisting of poly-(N-vinylcarbazole) (PVK), 5CB and C60 (55.45 wt% PVK, 44.45 wt% 5CB, 0.10 wt% C60). PVKbased PR composite systems have been reported to be
Fig. 4. Higher-order diffraction images of two-wave coupling when signal images were two vertical lines.
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Fig. 5. Higher-order diffraction images of two-wave coupling when the signal image was a square.
Fig. 6. Higher-order diffraction images of PR liquid crystal polymer composite (100 mm).
one of the best organic PR materials. The function of PVK is the photoconductive agent and matrix, 5CB is the nonlinear chromophore and C60 is the sensitizer. The glass transition temperature of the sample is about 44 1C. The standard structures are sandwiches of the PR polymer between two ITO glasses. In order to optimize use of the material, we have to split the difference of the enhancement of orientational mobility versus translational mobility leading to phase separation. The temperature is fixed at about 22 1C. We prepared many liquid crystal polymer composites with different thickness, from 60 to 130 mm, but the influence of thickness is limited for our experiments. When the electric field was 60 V/mm (DC), a similar phenomenon to Fig. 5 was observed (Fig. 6). However, the higher-order diffraction images were fewer and darker than those appearing in nematic liquid crystals. In order to show the result clearly, we increased the brightness and contrast of higher-order diffraction. The difference between them may be the different mechanism of gratings formation, which induces the refractive-index modulation of the PR nematic liquid crystal larger than that of the PR liquid crystal polymer composite. For the composite, the grating was steady for any steady external electric field. The diffraction images changed little when the voltage was 40–100 V/mm. The sample was broken down as the voltage exceeded 100 V/mm. When the square image was replaced with a ring image (Fig. 7(a) and (b)), the results can also be explained with the above method. An interesting phenomenon was found that there were more higherorder diffraction images for the PR liquid crystal
Fig. 7. Higher-order diffraction images of two-wave coupling when the signal image was a ring: (a) PR nematic liquid crystal; (b) PR liquid crystal polymer composite.
polymer composite. The number and definition of diffraction order are reverse for square image and ring image in two samples. This result may be induced by the difference of Fourier spectrum between two images. The different Fourier spectra produce different refractive-index modulation. Further studies along these lines are proceeding. In Fig. 8, the higher-order diffraction of signal images themselves was observed in PR nematic liquid crystals (Fig. 8(a)) and the PR liquid crystal polymer composite (Fig. 8(b)). The experimental results confirmed the above discussion of higher-order diffraction. In Fig. 8(b), the perpendicular discontinuities are the result of electric field direction. In our experiments, the grating generation mechanism is ‘‘orientational photorefractive effect’’ [21–23]. The external electric field direction is horizontal, so they only have the grating wave vector in horizontal direction. The reason of the unequidistant diffraction images in our figures is that the sample was illuminated by two beams, tilted at a slant angle with respect to the normal of the layers and having small angular separation. Individual thin beams of the signal beam traverse the
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cation of a DC field. The complexity of higher-order diffraction images shows that there are many kinds of coupling in the nematic liquid crystal and liquid crystal polymer composite. We propose a model to explain the origin of all diffraction images in the coupling process. We also show the stability of the gratings that were used in our experiment.
Acknowledgments This research was supported by the National Natural Science Foundation of China (Grant no. 69977009) and the Foundation of Heilongjiang Province LC04C011.
References
Fig. 8. Higher-order diffraction of signal images themselves. (a) PR nematic liquid crystal; (b) PR liquid crystal polymer composite.
ITO glass with different incident angles. The refractive index of ITO glass is different from that of nematic liquid crystals and liquid crystal polymer composites, so the direction of diffraction light changed with the different incident angles. Although the phenomena were only observed in our samples, our theory of coupling is not restricted in our samples, so we believe that the phenomena of higherorder diffraction images are widely existent for organic PR materials. They may be observed in liquid crystal cells with photoconductive PVK polymer layers [13], liquids sandwiched with photoconductive polymer films across the insulated polymer films [14] and other similar organic PR materials such as dye-, fullerene- and carbon-nanotube-doped liquid crystals [24,25] for the grating generation mechanism is similar to processes occurring in our samples.
3. Conclusions In conclusion, we have shown the novel higher-order diffraction images of PR nematic liquid crystals and liquid crystal polymer composites, following the appli-
[1] G.W. Burr, C.M. Jefferson, H. Coufal, M. Jurich, J.A. Hoffnagle, R.M. Macfarlane, R.M. Shelby, Volume holographic data storage at an areal density of 250 gigapixels/in2, Opt. Lett. 26 (2001) 444–447. [2] Zhongxiang Zhou, Yongyuan Jiang, Chunfeng Hou, Baohong Yuan, Xiudong Sun, Image rotation and amplification based on the photorefractive higher-order grating, Opt. Laser Eng. 35 (2001) 233–238. [3] Dewei Gong, Zhongxiang Zhou, Qingxin Meng, Chunfeng Hou, Higher-order diffraction images in photorefractive materials, Opt. Laser Eng. 45 (2007) 170–174. [4] L. Brillouin, Diffusion de la lumiere et des rayons x par un corps transparent homogene influence de l’agitation thermique, Ann. Phys. 17 (1921) 88–122. [5] P. Debye, F.W. Sears, On the scattering of light by supersonic waves, Proc. Natl. Acad. Sci. 18 (1932) 409–414. [6] R. Lucas, P. Biquard, Proprietes optiques des milieux solids et liquides soumis aux vibrations elastiques ultra sonores, J. Phys. Radium 3 (1932) 464–477. [7] G.P. Wiederrecht, B.A. Yoon, M.R. Wasielewski, High photorefractive gain in nematic liquid crystals doped with electron donor and acceptor molecules, Science 270 (1995) 1794–1797. [8] E. Mecher, F. Gallego-Go´mez, H. Tillmann, H.-H. Ho¨rhold, J.C. Hummelen, K. Meerholz, Near infrared sensitivity enhancement of photorefractive polymer composites by pre illumination, Nature 418 (2002) 959–964. [9] I.C. Khoo, H. Li, Y. Liang, Observation of orientational photorefractive effects in nematic liquid crystals, Opt. Lett. 19 (1994) 1723–1725. [10] S. Bartkiewicz, K. Matczyszyn, J. Mys´ liwiec, A. Miniewicz, F. Kajzar, Mechanisms of photorefractivity in liquid crystals, Nonlinear Opt. Quant. Opt. 31 (2004) 319–332. [11] Junho Mun, Choon Sup Yoon, Hyun-Wuk Kim, Su-An Choi, Jong-Duk Kim, Orientational holographic grating observed in liquid crystals sandwiched with photoconductive polymer films, Appl. Phys. Lett. 79 (2001) 1336–1933. [12] P. Pagliusi, G. Cipparrone, Photorefractive effect due to a photoinduced surface-charge modulation in undoped liquid crystals, Phys. Rev. E 69 (2004) 061708.
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[13] M. Kaczmarek, A. Dyadyusha, S. Slussarenko, I.C. Khoo, The role of surface charge field in two-beam coupling in liquid crystal cells with photoconducting polymer layers, J. Appl. Phys. 96 (2004) 2616–2623. [14] Hiroshi Ono, Tomoyuki Sasaki, Akira Emoto, Emi Uchida, Nobuhiro Kawatsuki, Twisted phase gratings induced by photoregulated mesogenic molecules on the surface of photoreactive polymer liquid crystal film, Appl. Phys. Lett. 89 (2006) 261918. [15] Yanbo Pei, Fengfeng Yao, Chunfeng Hou, Xiiudong Sun, Zhongxiang Zhou, High diffraction efficiency and a quasi-permanent grating in photorefractive nematic liquid crystal at low temperature, Opt. Lett. 30 (2005) 631–633. [16] J. Zhang, K.D. Singer, Homogeneous photorefractive polymer/nematogen composite, Appl. Phys. Lett 72 (1998) 2948–2951. [17] M.G. Moharam, T.K. Gaylord, R. Magnusson, Criteria for Bragg diffraction by phase gratings, Opt. Commun. 32 (1980) 14–18. [18] M.G. Moharam, T.K. Gaylord, R. Magnusson, Criteria for Raman–Nath diffraction by phase gratings, Opt. Commun. 32 (1980) 19–23.
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[19] C.V. Raman, N.S. Nath, The diffraction of light by high frequency sound waves: Part I, Proc. Indian Acad. Sci. A 2 (1935) 406–412. [20] C.V. Raman, N.S. Nath, The diffraction of light by high frequency sound waves: Part II, Proc. Indian. Acad. Sci. A 2 (1935) 413–420. [21] Baohong Yuan, Xiudong Sun, Chunfeng Hou, Yan Li, Zhongxiang Zhou, Yongyuan Jiang Chunfei Li, Kinetics of the formation of space-charge field in photorefractive polymers, J. Appl. Phys. 88 (2000) 5562–5569. [22] W.E. Moerner, S.M. Silence, F. Hache, G. C, Orientationally enhanced photorefractive effect in polymers, J. Opt. Soc. Am. B 11 (1994) 320–330. [23] I.C. Khoo, Orientational photorefractive effects in nematic liquid crystal film, IEEE J. Quantum Electron. 32 (3) (1996) 525–534. [24] G.P. Wiederrecht, Photorefractive liquid crystals, Annu. Rev. Mater. Res. 31 (2001) 139–169. [25] I.C. Khoo, J. Ding, Y. Zhang, K. Chen, A. Diaz, Supranonlinear photorefractive response of single-walled carbon nanotube- and C60-doped nematic liquid crystal, Appl. Phys. Lett. 82 (2003) 3587–3589.
Optik
Optics
Optik 120 (2009) 524–529 www.elsevier.de/ijleo
The coupling and superposition behavior of higher-order diffraction image based on photorefractive nematic liquid crystal materials Dewei Gong, Zhongxiang Zhou, Yanbo Pei, Jianlong Zhang, Fengfeng Yao Department of Physics, Harbin Institute of Technology, Harbin 150001, China Received 24 April 2007; accepted 28 November 2007
Abstract We report some photorefractive higher-order diffraction phenomena of two kinds of photorefractive nematic liquid crystal materials. In the experiment, we observed the superposition of higher-order diffraction images, the coupling of higher-order diffraction lights, and the coupling between higher-order diffraction lights and incident beams. The observed behavior suggests the complexity of the coupling process. The properties of higher-order diffraction images are discussed theoretically, which are in good agreement with experimental results. r 2008 Elsevier GmbH. All rights reserved. Keywords: Photorefractive effect; Higher-order diffraction
1. Introduction The photorefractive (PR) effect has received a great deal of attention with respect to optical information processing applications [1]. Usually, the included angle between the incident beams is large enough when people store information in PR materials, so that we can use the Bragg diffraction and thus avoid the influence of higherorder diffraction (Raman–Nath regime). However, we can use higher-order diffraction to realize image processing such as image rotation and amplification [2,3]. The higher-order diffraction was first predicted by Brillouin [4] as diffraction of light by acoustic waves, and was experimentally revealed by Debye and Sears [5] and Lucas and Biquard [6]. The difference of higherorder diffraction between acousto-optic effect and PR effect is that in the case of dynamic interaction between Corresponding author. Tel.: +86 451 86414132; fax: +86 451 86412549. E-mail address: [email protected] (D. Gong).
0030-4026/$ - see front matter r 2008 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.11.012
the beams and refractive index modulation. For acousto-optic effect, the refractive index gratings are founded by acoustic waves. However, the refractive index gratings are founded by the spatial variation of light intensity for PR effect. So the diffraction of PR effect is more complex than that of acousto-optic effect. However, the properties of the higher-order diffraction images in PR materials have not been well studied because the higher-order diffraction images are usually blurry for PR crystals [2]. The recent development of organic PR materials, which are more costefficient and easier to prepare and to be modified than their inorganic counterparts, has fueled a surge of study in optical signal processing applications [7–14]. So we have the opportunity to investigate the properties of higher-order diffraction images. In this paper, we report some new phenomena of coupling and superposition based on the higher-order diffraction at large modulation in two photorefractive nematic liquid crystal materials. The results indicate the complexity of coupling.
ARTICLE IN PRESS D. Gong et al. / Optik 120 (2009) 524–529
p
2. Experiments and analysis The experimental setup is shown in Fig. 1. An s-polarized He–Ne laser (l ¼ 632.8 nm) was used as the light source. A beam from the laser was split into the reference beam and the signal beam. The signal beam passed through a spatial light modulator (SLM) and then was focused by an f ¼ 550 mm focal-length Fourier lens. The sample was placed at the focal plane. The reference beam illuminated onto the same area at an angle of 301 with the normal of sample surface. The included angle between reference beam and the bisector of the signal beam was about 1.51. Holograms were built by the interference of signal and reference beams. The normal of the sample was in the plane of incident beams. PR nematic liquid crystal [15] and PR liquid crystal polymer composite [16] were used as recording materials. There are common materials for organic PR experiments. Firstly, the nematic liquid crystal was used as recording material, which is the mixture of pure pentycyanabiphenyl liquid crystal (5CB) doped with fullerene (C60). The weight ratio was 99.95% (5CB) and 0.05% (C60). 5CB is a wonderful molecule for PR effect. The PR nonlinearity of the sample is highly improved by heavily doping with C60. Homeotropic alignment of the liquid crystal was obtained by depositing a hexadecyl trimethyl ammonium bromide film onto indium tin oxide (ITO) glass. The sample thickness was controlled to be around 16.6 mm with the help of spacers. Fig. 2 shows the
He-Ne Laser M1 M2 Sample BS CCD FL2
SLM
FL1
SL
M3
Fig. 1. Experimental set-up of higher-order diffraction images. M – mirror; BS – beam splitter; SLF – spatial light filter; FL – Fourier lens; SLM – spatial light modulator.
Fig. 2. Incident beams transmitted through the sample directly.
525
k
m
o
j
i
c
h
b
d
n
f
g
f
p
m
l
k o
j i
h
a
c
b
d
n
e f
g
Fig. 3. Higher-order diffraction images of two-wave coupling when signal images were two horizontal lines.
incident beams transmitted directly through the sample when the applied electric field was 0. The two horizontal lines were signal beams whose size was 10.0 9.0 mm2 on SLM. In the experiment, the externally applied electric field was varied from 0 to 0.30 V/mm (DC). When the voltage exceeded 0.06 V/mm, the complex diffraction images could be observed obviously by a CCD camera behind the sample (Fig. 3). According to Ref. [3], the experimental result should only have the diffraction order marked as n, n0 , f, f0 , i, i0 , o, o0 , m, m0 . We believe that Fig. 3 gives more information about the coupling. According to diffraction theory, if the refraction index gratings are illuminated with a laser beam, there are two kinds of diffractions: Bragg diffraction and Raman– Nath diffraction. The diffraction regime of the phase gratings is determined by the modulation parameter g ¼ pDnd=l cos f and Q0 ¼ Q= cos f ¼ 2pld=n0 L2 cos f (l is the free-space wavelength, d is the grating thickness, n0 is the refractive index of the whole medium in its undisturbed state, Dn is the modulation amplitude of the refractive index, L is the grating period and f is the angle of incidence inside the grating). In the Bragg diffraction regime, QX10 and Q0 /gX10 [17]. In this regime, multiple scattering is not permitted and only fundamental diffraction light is produced. Conversely, Q0 gp1 and Q0 p2 [18] is the Raman–Nath diffraction regime. The angular spread of the grating wave vector is larger than that of the Bragg angle, and therefore many diffraction orders can be observed in this regime. After interacting with the grating, the incident light beam is split into several diffraction beams. In our experiment, Q0 gE0.13 and Q0 E0.15. It might be expected that we had Raman–Nath diffraction. In the Raman–Nath regime, maxima of the intensity due to variation of the refractive index occur in directions making angles, denoted by yk, with the vertical direction of the grating vector K [19,20]: sin yk sin f ¼ sinðjk þ fÞ sin f ¼
klR L
k ðan integerÞ,
(1)
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where jk is the angle between the diffraction beam and incident beam, f is the incident angle of reading beam and lR is the wavelength of reading beam. If refraction index gratings are built by two-wave coupling in PR materials, the grating period L is determined by the following expression: L¼
lW , 2 sin b
(2)
where lW is the wavelength of writing beam, 2b is the angle between two writing beams. Substituting Eq. (2) into (1), the following is obtained: sin yk sin f ¼ sinðjk þ fÞ sin f ¼
klR lR ¼ 2k sin b. lW =2 sin b lW
(3) In our experiments, the writing beams are reading beams, so lW ¼ lr , b ¼ f. Because the angles jk, f and 2b are very small, we can obtain sin yk ¼ sinðjk þ fÞ jk þ f, sin b b and sin f f, then Eq. (3) can be written as yk jk þ f 2kb þ f.
(4)
In experiment, the image was two horizontal lines. In order to simplify our analysis, Fourier optics is simplified to geometry optics. The signal beam can be decomposed into many individual thin light beams whose diffractions are consonant with Eq. (4). The appearance of n, n0 , f, f0 , i, i0 , o, o0 , m, m0 has been explained well by Ref. [3], so the emphasis of the paper is to explain the appearance of c, c0 , d, d0 , e, g, g0 , h, h0 , j, j0 , k, k0 , l, p, p0 . First of all, we explain the appearance of diffraction light c, c0 . The coupling of two parts of signal beam b, b0 generates a PR grating that is parallel with b, b0 . The higher-order diffractions are c and c0 , so they have the same size with b, b0 , where 2b is the included angle between b and b0 in PR materials. According to Eq. (4), the included angle between c0 , b0 and b, c are nearly equal to 2b too. Then the reference beam a illuminate the same gratings. Spots h, h0 are the higherorder diffraction of reference beam a, which diffracted by the same grating with different incident direction. The included angles between h0 , a and a, h are also equal with 2b. After emergence, higher-order diffraction h0 can also couple with signal beam b0 , and k0 , j0 , e, g are 2, 1, +2, +3 diffraction order of gratings of h0 and b0 . In our experiments, the sample was placed at the Fourier plane, so the width of the interference area DL ¼ 0.The magnifications of k0 , j0 , e, g are 2, 1, +2, +3 strictly. Although k0 , j0 , e, g, as shown in Fig. 3, are only amplified in size, in fact, the higher-order diffractions k0 , j0 are rotated 1801 compared with b, b0 . With the same method, k, j, e, g0 are 2, 1, +2, +3 diffraction order of beams h and b. The size magnification of 2, 1, +2, +3 orders is 2, 1, +2, +3. From the above discussion, we can see that e is
overlapped +2 diffraction order. We can explain the emergence of d0 , c0 , l, p as the 2, 1, +2, +3 orders of h0 , and i0 , d, c, l, p0 as the 2, 1, +2, +3 orders of h and i. To our knowledge, the coupling of higher-order diffraction lights is first reported. d0 is the +2 order of h and b0 , and d is the +2 order of h0 and b. The emergence of k, k0 can also be explained with the same method. j0 , i0 , I, j can be looked as higher-order diffraction and energy could be transported among them. k0 , o0 , l, o, k, m0 , p0 , p, m, c0 , b0 , b, c, d0 , n0 , e, n, d and f0 , g0 , g, f can be explained with the same method. From Fig. 3 we know that many higher-order diffraction lights overlapped as one mutually indistinguishable diffraction light. When the images were two vertical lines, the result is shown in Fig. 4. Our analysis can also explain the experiment. Fig. 5 shows the higher-order diffraction images when the signal image was a square. The signal image can be considered as the addition of two groups of parallel lines. The experimental result is in excellent agreement with the above discussion. However, when the square was placed as the addition of signal images in Figs. 3 and 4, the higher-order diffraction images were fewer than that in Fig. 5. The reason may be the orientational property of the PR nematic liquid crystal. In the experiment, we found that the gratings were unstable. Intensity of different order diffraction images and scattered light changed with external electric field and exposure time. With electric field increasing from 0.06 to 0.30 V/mm, the higher-order diffraction images peaked sharply and then submerged by strong scattered light. In order to achieve high quality of optical diffraction images, we investigated the effect of the electric field. We found that the best-quality optical diffraction images, Figs. 3–5, could be obtained when the voltage was evenly increased from 0 to 5 V in 10 s. However, the best-quality higher-order diffraction images could only be maintained for about 1 s at 5 V, and then they were scattered whether the external electric field was changed or not. We did the same experiments with a composite derived from the high-performance PR polymer consisting of poly-(N-vinylcarbazole) (PVK), 5CB and C60 (55.45 wt% PVK, 44.45 wt% 5CB, 0.10 wt% C60). PVKbased PR composite systems have been reported to be
Fig. 4. Higher-order diffraction images of two-wave coupling when signal images were two vertical lines.
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Fig. 5. Higher-order diffraction images of two-wave coupling when the signal image was a square.
Fig. 6. Higher-order diffraction images of PR liquid crystal polymer composite (100 mm).
one of the best organic PR materials. The function of PVK is the photoconductive agent and matrix, 5CB is the nonlinear chromophore and C60 is the sensitizer. The glass transition temperature of the sample is about 44 1C. The standard structures are sandwiches of the PR polymer between two ITO glasses. In order to optimize use of the material, we have to split the difference of the enhancement of orientational mobility versus translational mobility leading to phase separation. The temperature is fixed at about 22 1C. We prepared many liquid crystal polymer composites with different thickness, from 60 to 130 mm, but the influence of thickness is limited for our experiments. When the electric field was 60 V/mm (DC), a similar phenomenon to Fig. 5 was observed (Fig. 6). However, the higher-order diffraction images were fewer and darker than those appearing in nematic liquid crystals. In order to show the result clearly, we increased the brightness and contrast of higher-order diffraction. The difference between them may be the different mechanism of gratings formation, which induces the refractive-index modulation of the PR nematic liquid crystal larger than that of the PR liquid crystal polymer composite. For the composite, the grating was steady for any steady external electric field. The diffraction images changed little when the voltage was 40–100 V/mm. The sample was broken down as the voltage exceeded 100 V/mm. When the square image was replaced with a ring image (Fig. 7(a) and (b)), the results can also be explained with the above method. An interesting phenomenon was found that there were more higherorder diffraction images for the PR liquid crystal
Fig. 7. Higher-order diffraction images of two-wave coupling when the signal image was a ring: (a) PR nematic liquid crystal; (b) PR liquid crystal polymer composite.
polymer composite. The number and definition of diffraction order are reverse for square image and ring image in two samples. This result may be induced by the difference of Fourier spectrum between two images. The different Fourier spectra produce different refractive-index modulation. Further studies along these lines are proceeding. In Fig. 8, the higher-order diffraction of signal images themselves was observed in PR nematic liquid crystals (Fig. 8(a)) and the PR liquid crystal polymer composite (Fig. 8(b)). The experimental results confirmed the above discussion of higher-order diffraction. In Fig. 8(b), the perpendicular discontinuities are the result of electric field direction. In our experiments, the grating generation mechanism is ‘‘orientational photorefractive effect’’ [21–23]. The external electric field direction is horizontal, so they only have the grating wave vector in horizontal direction. The reason of the unequidistant diffraction images in our figures is that the sample was illuminated by two beams, tilted at a slant angle with respect to the normal of the layers and having small angular separation. Individual thin beams of the signal beam traverse the
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cation of a DC field. The complexity of higher-order diffraction images shows that there are many kinds of coupling in the nematic liquid crystal and liquid crystal polymer composite. We propose a model to explain the origin of all diffraction images in the coupling process. We also show the stability of the gratings that were used in our experiment.
Acknowledgments This research was supported by the National Natural Science Foundation of China (Grant no. 69977009) and the Foundation of Heilongjiang Province LC04C011.
References
Fig. 8. Higher-order diffraction of signal images themselves. (a) PR nematic liquid crystal; (b) PR liquid crystal polymer composite.
ITO glass with different incident angles. The refractive index of ITO glass is different from that of nematic liquid crystals and liquid crystal polymer composites, so the direction of diffraction light changed with the different incident angles. Although the phenomena were only observed in our samples, our theory of coupling is not restricted in our samples, so we believe that the phenomena of higherorder diffraction images are widely existent for organic PR materials. They may be observed in liquid crystal cells with photoconductive PVK polymer layers [13], liquids sandwiched with photoconductive polymer films across the insulated polymer films [14] and other similar organic PR materials such as dye-, fullerene- and carbon-nanotube-doped liquid crystals [24,25] for the grating generation mechanism is similar to processes occurring in our samples.
3. Conclusions In conclusion, we have shown the novel higher-order diffraction images of PR nematic liquid crystals and liquid crystal polymer composites, following the appli-
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