Japanese Journal of Applied Physics Vol. 47, No. 10, 2008, pp. 7963–7967 #2008 The Japan Society of Applied Physics
Light Wave Propagation and Bragg Diffraction in Thick Polarization Gratings Hiroshi O NO, Takuya S EKIGUCHI, Akira E MOTO, Tatsutoshi S HIODA, and Nobuhiro K AWATSUKI1 Department of Electrical Engineering, Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka, Niigata 940-2188, Japan 1 Department of Material Science and Technology, University of Hyogo, 2167 Shosha, Himeji, Hyogo 671-2201, Japan (Received June 4, 2008; accepted June 26, 2008; published online October 17, 2008)
Light wave propagation in thick polarization gratings has been studied using the finite-difference time-domain (FDTD) method. The diffraction properties of the polarization Bragg gratings were strongly dependent on both the polarization state and incident angle of the probe beam. These characteristics were well explained by the FDTD calculation and the subsequent Fourier transformation. The FDTD calculation also revealed the light electric field near and/or in the polarization Bragg gratings. [DOI: 10.1143/JJAP.47.7963] KEYWORDS: liquid crystal, hologram, polarization, FDTD, grating, Bragg
1.
Introduction
The increasing need to manipulate optical signals, following the introduction of fiber optics into communications and computation, and the development of parallel optical information-processing systems, has stimulated interest in highly functionalized optical devices. Diffractive optical devices are of great importance for realizing numerical optical functions such as optical memory devices,1) integrated optical devices,2) and thin-film optical filters.3,4) It is possible to fabricate diffractive optical devices by exposing light-sensitive materials such as photopolymers to interference light. The resulting gratings can control the light propagation direction and can be used in optoelectronic equipments. In order for diffractive optical devices to play more important roles in optical information processing systems, more advanced functions should be added to optical gratings. It is believed that if anisotropic rather than isotropic materials for optical gratings are used, then polarization controllable diffractive devices can be developed. Thus, polarization gratings using photo-alignment materials have been widely investigated for optical data storage and highly-functionalized optical devices.5–19) Using Kogelnik’s coupled-waves model,20) we have to treat two cases of the grating conditions according to a following dimensionless parameter Q: Q ¼ 2d=n2 ;
ð1Þ
where let d be the thickness of the grating, which is the optical path length through the sample, is the wavelength of the pump and probe beams, n is the refractive index of the medium, and is the period of the grating. One defines the regime Q > 1 as the Bragg regime of optical diffraction. In this regime, multiple scattering is not permitted and only one order of diffraction of light is produced. On the other hand, another defines the regime Q < 1 as the Raman–Nath regime of optical diffraction. In this regime, the angular spread of the grating wave vector is much larger than the Bragg angle and therefore many orders of diffraction can be observed. Most of studies about polarization gratings have been performed in the Raman–Nath regime. Todorov et al. recorded and characterized the polarization gratings using the azo-dye-doped poly(vinyl alcohol) (PVA).5,6) They experimentally and theoretically clarified that the polarization
E-mail address:
[email protected]
holographic gratings diffracted laser beams and converted polarization states at the same time. Ebralidze et al. showed that Jones matrix method was very convenient to describe the optical properties of the thin polarization gratings.8) Nikolova et al. investigated thin polarization holographic gratings recorded with two waves with orthogonal linear polarizations inmaterials in which illumination with linearly/circularly polarized light gives rise to linear/circular birefringence.9) The side-chain liquid-crystalline azobenzene polyester was used as a photoanisotropic material and polarization properties of the recorded polarization gratings were explained considering the photoinduced linear and circular birefringence. Cipparrone et al. observed the permanent polarization gratings in photosensitive Langmuir– Blodgett films.11) The thin polarization gratings were recorded in azo-dye-containing Langnuir–Blodgett films and the resultant polarization gratings were stable. We have fabricated highly stable polarization gratings in photocrosslinkable polymer liquid crystals. The thin polarization gratings have been prepared in photocrosslinkable polymer liquid-crystal films by the use of two orthogonally polarized He–Cd laser beams and subsequent annealing. The resulting pure polarization gratings exhibiting thermal stability up to 150 C diffract the beam and convert the polarization state at the same time according to the theoretical expectation.12,13) In the past few years several groups have been actively engaged in demonstrating the use of azobenzene-containing polymer films for thick (Bragg-type) holographic recording.16,17) Ha¨ckel et al. synthesized a new type of polymer blend with azobenzene-containing block copolymers as stable rewritable volume holographic media.16) The 20 multiplex gratings were written and the temporal stability of the multiplex gratings was recorded over a period of three weeks. Saishoji et al. prepared optically transparent thick films of methacrylate copolymer containing donor–acceptortype azobenzene and cyanobiphenyl moieties.17) The Bragg gratings with angular and polarization multiplicity were formed and the data storage of 55 holograms with angular multiplicity were demonstrated. Ishiguro et al. also demonstrated the formation of holographic gratings in the thick azobenzene polymer films containing various tolane moiety contents in the side chain. The thick polarization gratings were generated by using the two orthogonal circularly polarized beams and showed the high diffraction efficiency and fast response.18) To apply the polarization Bragg gratings to highly functionalized optical devices or realize the
7963
Jpn. J. Appl. Phys., Vol. 47, No. 10 (2008)
H. ONO et al.
Phase shift δ δ= π
Writing beams δ= 0
δ= π/2
δ= 3π/2
δ= 2π y x
z
Fig. 1. Polarization modulation in holographic recording with two waves with orthogonal circular polarization.
40 µm
multiplex holographic recording system, theoretical consideration of the diffraction properties was very important. In the present paper we extensively studied the diffraction properties in the polarization Bragg gratings by means of the finite-difference time-domain (FDTD) algorithm. FDTD Method for Periodic Anisotropic Medium
FDTD solvers have been developed as one of the most effective tools for computational electromagnetics problems.21) FDTD methods present a robust and powerful approach to directly solve Maxwell’s curl equations both in time and space and are also applied for analysis of arbitrary, anisotropic periodic structures at oblique incidence. Another advantage of the FDTD techniques is its capability to visualize the real-time images of the electromagnetic wave. The liquid crystalline devices have also been analyzed by means of FDTD methods.22–32) Thick polarization gratings can be written using a holographic recording technique.16,17) As writing beams, we consider the polarization interference pattern formed by the interaction of two plane waves E1 and E2 , of equal amplitude E and orthogonal polarizations. In the case of orthogonal linear exposure (OLE), E1 is vertical-linearly polarized, while E2 is horizontal-linearly polarized, and the resulting two waves are linear-polarized waves with mutually perpendicular polarization directions. The resulting light field (E ¼ E1 þ E2 ) is described for the OLE case by E cos E¼ ; ð2Þ iE cos where the phase difference between the two incident waves is a function of the position x and the grating spacing , and can be expressed as ¼
2x :
ð3Þ
In the case of orthogonal circular exposure (OCE), E1 and E2 are left- and right-circularly polarized waves, respectively. The resulting light field is described for the OCE case by E cos E¼ : ð4Þ E sin In the present paper we consider the polarization gratings fabricated by OCE. According to eq. (4) Fig. 1 describes the polarization states of the two writing beams, and the resulting electric field and intensity distribution of the two beams coupling for OCE. As shown in Fig. 1, the interference shows a modulated polarization state and uniform intensity. If photoreactive materials, in which optical anisotropy can be induced by light exposure, are irradiated with polarization interference light, as shown in Fig. 1, periodic anisotropy is expected to be induced in the films.
Antireflection coating
17 µm
2.
Transmitted light Bragg reflection
d
Λ
Probe beam
+θ
Absorption Boundary Condition x
z y
Fig. 2. Schematic view of the two-dimensional FDTD computational space.
The dielectric tensor for a uniaxial medium is written in the xyz coordinate system as 0 1 "xx "xy "xz B C ð5Þ "ðx; zÞ ¼ @ "yx "yy "yz A; "zx "zy "zz with "xx ¼ n2o þ ðn2e n2o Þ cos2 cos2 ; "xy ¼ "xz ¼ "yy ¼ "yz ¼ "zz ¼
"yx ¼ ðn2e n2o Þ cos2 sin cos ; "zx ¼ ðn2e n2o Þ sin cos cos ; n2o þ ðn2e n2o Þ cos2 sin2 ; "zy ¼ ðn2e n2o Þ sin cos sin ; n2o þ ðn2e n2o Þ sin2 ;
ð5aÞ ð5bÞ ð5cÞ ð5dÞ ð5eÞ ð5fÞ
where no and ne are the ordinary and extraordinary refractive indices of the medium, respectively, is the angle between the optic axis (director in the case of liquid crystal gratings) and the z-axis, and is the angle between the projection of the optic axis on the xy plane and the x-axis. The geometry for FDTD simulation is schematically described in Fig. 2. An essential feature in FDTD methods is a proper absorbing boundary condition to truncate the simulation space without artificial reflections. In the present paper we used the perfectly matched layer (PML) introduced by Berenger.33) By collecting the output electric fields and applying a Fourier transform to them, the far field amplitude or intensity distribution can be obtained. All calculations were performed using a probe beam with a 633 nm of wavelength. 3.
Results and Discussion
It is well known that the diffraction efficiency of a Bragg grating is strongly dependent on the incident angle of the reading beam.34) The volume grating is defined as a piece of dielectric in which the permittivity varies continuously in
7964
H. ONO et al.
0.25
(a)
0.20 0.15 0.10 0.05 0.00 -40
-20
0
20
40
-20
0
20
40
-20
0
20
40
Incident angle θ [deg]
Diffraction efficiency [%]
1.2 1.0
(b)
0.8 0.6 0.4 0.2 0.0 -40
Diffraction efficiency [%]
space in a nearly periodic manner. The periodic variation of permittivity can be achieved in holographic materials by illuminating them for a certain time with interference light. Additionally, since anisotropic gratings are considered in the present work, the optical anisotropy has to be induced by the light irradiation. As a rubbing free process for multi-domain liquid crystal alignment, high-density holographic optical memory, and highly-functionalized polarization gratings, the photo-alignment technique has received much attention from the viewpoint of both scientific and practical interest, and many types of photo-alignment materials have been investigated as described in the introduction. Figure 3 shows the diffraction efficiency versus incident angle to the Bragg polarization gratings fabricated by OCE. As shown in Fig. 3, at certain values of the incident angle, the Bragg diffraction process is effective, and as a result, in the far field after the OCE Bragg gratings, only one diffraction spot appears in the case of a right-hand circularly polarized probe beam. As shown in Fig. 3, the peak value and the angular selectivity of the diffraction efficiency are affected by the thickness of the OCE Bragg grating. The mechanism of the diffraction in the OCE Bragg grating is linked with the selective reflection in the helical structure formed by OCE. Polarization states of the diffraction beam are of important for realizing advanced functions such as optical memory devices and multifunctionalized optical components. Figure 4 shows schematics of the spatial distribution of the optical anisotropy in the OCE Bragg grating. When the probe light incidents on the OCE Bragg grating with slanted direction as shown in Fig. 4(a), the probe light propagates through the helical structure as described in Fig. 4(b). In the helical structure the right-hand circularly polarized light is partially diffracted and the polarization state of the diffraction beam is simultaneously converted into left-hand circularly polarized light. Thus, in our calculation the Bragg diffracted light is left-hand circularly polarized when the incident probe light is right-hand circularly polarized, while left-hand circularly polarized light is converted into right-hand circularly polarized light. As shown in Fig. 3, the diffraction properties of the OCE Bragg grating were strongly affected by the thickness of the grating. Figure 5 shows dependence of the thickness on the peak efficiency of the diffraction in OCE Bragg grating. The peak efficiency is increased with increasing the thickness as shown in Fig. 5. Since the polarization gratings contain the optical anisotropy, the diffraction properties of the polarization Bragg gratings are strongly dependent not only on the beam incident angle but also on the polarization state of the probe beam. Figure 6 shows the diffraction efficiency versus incident angle of the probe beam on varying the polarization state. As shown in Fig. 6(a), both positive and negative sides of incident angles give the peak diffraction efficiency when the probe beam is s-polarized. When the probe beam is circularly polarized only one side of incident angle give the peak as shown in Figs. 6(b) and 6(c). The peak value of the diffraction efficiency for the circularly-polarized probe beam is twice as large as that for the s-polarized probe beam. When the probe beam is left-hand circularly polarized, the Bragg diffracted beam is right-hand circularly polarized as shown in Fig. 6(b). On the other hand, the Bragg diffracted
Diffraction efficiency [%]
Jpn. J. Appl. Phys., Vol. 47, No. 10 (2008)
2.5
Incident angle θ [deg]
(c)
2.0 1.5 1.0 0.5 0.0 -40
Incident angle θ [deg]
Fig. 3. Diffraction efficiency versus incident angles of the probe beam upon varying the grating thickness. The grating constant and birefringence are 1.0 mm and 0.002, respectively. The probe beam is right-hand circularly polarized. The grating thickness is set to (a) 5.0, (b) 10.0, and (c) 15.0 mm.
(a)
+θ Probe beam x
y z
(b)
Fig. 4. (a) Calculation model of the spatial distribution of the director in the polarization Bragg grating and (b) schematic view of the director modulation in the direction of the beam propagation.
7965
Jpn. J. Appl. Phys., Vol. 47, No. 10 (2008)
H. ONO et al.
Transmitted light
Bragg reflection
80
(a)
8
Air
60 6
Z [µm]
Diffraction efficiency [%]
100
40
4
Grating
+0.8
20
2
0
- 0.8
Air
0
0
2
4
6
8
10
12
20
14
15
10
5
0
X [µm]
Film thickness [µm] Transmitted light
Fig. 5. Dependence of the thickness on diffraction efficiency. The grating constant and birefringence are 1.0 mm and 0.021, respectively. The incident angle of the probe beam is set to +18.45 . The probe beam is right-hand circularly polarized.
8
(b)
Air
Z [µm]
6
Diffraction efficiency [%]
4
Grating
+0.8
100
2
(a) 80
0 20
- 0.8
Air 15
10
5
0
X [µm]
60
RCP
LCP
Fig. 7. Simulated FDTD images of near-electric-field distribution for (a) right- and (b) left-hand circularly polarized light. The grating constant and birefringence are 1.0 mm and 0.02, respectively. The grating thickness is 5.0 mm. The incident angle of the probe beam is set to +18.45 .
40 20 0 -40
-20
0
20
beam is left-hand circularly polarized when the probe beam is right-hand circularly polarized as shown in Fig. 6(c). The mechanism of these properties in OCE Bragg gratings is linked with helical structure. As schematically described in Fig. 4, the rotation direction of the helical structure in the OCE Bragg grating depends on the sign of the incident angle. We attributed the symmetric behavior of the diffraction for the right- and left-hand circularly polarized probe beams to the above-mentioned rotation direction of the helical structure which depends on the sign of the incident angle of the probe beam. Figures 7(a) and 7(b) represent the calculated amplitude of the light electric fields for the right- and left-hand circularly polarized probe beams, respectively. The component of the light electric field is gradually generated in the grating medium as shown in Fig. 7(a) and propagates in the Bragg reflected direction when the probe beam is right-hand circularly polarized. In contrast, the probe beam is not diffracted in the medium when the probe beam is left-hand circularly polarized as shown in Fig. 7(b).
40
Incident angle θ [deg]
Diffraction efficiency [%]
100
(b) RCP 80 60 40 20 0 -40
-20
0
20
40
Incident angle θ [deg]
Diffraction efficiency [%]
100
(c)
LCP
80 60 40
4.
20 0 -40
-20
0
20
40
Incident angle θ [deg] Fig. 6. Diffraction efficiency versus incident angles of the probe beam upon varying the polarization state of the probe beam. The grating constant and birefringence are 1.0 mm and 0.015, respectively. The grating thickness is 15.0 mm. RCP and LCP denote right- and left-hand circularly polarizations, respectively.
Conclusions
The light electric field propagating in and/or near polarization Bragg gratings has been extensively analyzed by an FDTD algorithm for anisotropic media. The polarization Bragg gratings used here can be fabricated by irradiating two coherent beams with orthogonal circular polarization to the photo-reactive materials such as photoalignment liquid crystals. The polarization Bragg grating angular-selectively diffracts the laser beam and simultaneously converts the polarization state. Additionally the angular-selectivity of the diffraction in the polarization
7966
Jpn. J. Appl. Phys., Vol. 47, No. 10 (2008)
H. ONO et al.
Bragg grating is strongly dependent on both the thickness of the grating and the polarization state of the probe beam. These characteristics of polarization Bragg gratings were qualitatively explained by the helical structure observed in the slanted direction, and were well explained by the FDTD calculation and the subsequent Fourier transform of the electric field near the grating. 1) J. Ashley, C. M. Jefferson, M.-P. Bernal, B. Marcus, G. W. Burr, R. M. Macfarlane, H. Coufal, R. M. Shelby, H. Guenther, G. T. Sincerbox, and J. A. Hoffnagle: IBM J. Res. Dev. 44 (2000) 341. 2) R. Hunsperger: Integrated Optics: Theory and Technology (Springer, Berlin, 1995) 4th ed. 3) N. Kawatsuki and M. Uetsuki: Jpn. J. Appl. Phys. 29 (1990) 2420. 4) K. Fujisawa and M. Uetsuki: Jpn. J. Appl. Phys. 34 (1995) 3583. 5) T. Todorov, L. Nikolova, and N. Tomova: Appl. Opt. 23 (1984) 4309. 6) T. Todorov, L. Nikolova, and N. Tomova: Appl. Opt. 23 (1984) 4588. 7) T. D. Ebralidze and N. A. Ebralidze: Appl. Opt. 31 (1992) 4720. 8) T. D. Ebralidze: Appl. Opt. 34 (1995) 1357. 9) L. Nikolova, T. Todorov, M. Ivanov, F. Andruzzi, S. Hvilsted, and P. S. Ramanujam: Appl. Opt. 35 (1996) 3835. 10) N. C. R. Holme, L. Nikolova, P. S. Ramanujam, and S. Hvilsted: Appl. Phys. Lett. 70 (1997) 1518. 11) G. Cipparrone, A. Mazzulla, S. P. Palto, S. G. Yudin, and L. M. Blinov: Appl. Phys. Lett. 77 (2000) 2106. 12) H. Ono, A. Emoto, F. Takahashi, N. Kawatsuki, and T. Hasegawa: J. Appl. Phys. 94 (2003) 1298. 13) H. Ono, A. Emoto, and N. Kawatsuki: J. Appl. Phys. 100 (2006) 013522.
14) C.-U. Bang, A. Shishido, and T. Ikeda: Maclomol. Rapid Commun. 28 (2007) 1040. 15) J.-C. Chao, W.-Y. Wu, and A. Y.-G. Fuh: Opt. Express 15 (2007) 16702. 16) M. Ha¨ckel, L. Kador, D. Kropp, and H.-W. Schmidt: Adv. Mater. 19 (2007) 227. 17) A. Saishoji, D. Sato, A. Shishido, and T. Ikeda: Langmuir 23 (2007) 320. 18) M. Ishiguro, D. Sato, A. Shishido, and T. Ikeda: Langmuir 23 (2007) 332. 19) X. Pan, C. Wang, C. Wang, and X. Zhang: Appl. Opt. 47 (2008) 93. 20) H. Kogelnik: Bell Syst. Tech. J. 48 (1969) 2909. 21) K. S. Yee: IEEE Trans. Antennas Propag. 14 (1966) 302. 22) E. E. Kriezis and S. J. Elston: Opt. Commun. 165 (1999) 99. 23) E. E. Kriezis and S. J. Elston: Opt. Commun. 177 (2000) 69. 24) E. E. Kriezis: Microwave Opt. Technol. Lett. 35 (2002) 437. 25) B. Wang, D. B. Chung, and P. J. Bos: Jpn. J. Appl. Phys. 43 (2004) 176. 26) D. K. Hwang and A. D. Rey: Appl. Opt. 44 (2005) 4513. 27) E. P. Kosmidou, E. E. Kriezis, and T. D. Tsiboukis: Opt. Quantum Electron. 37 (2005) 149. 28) V. Ilyina, S. J. Cox, and T. J. Sluckin: Opt. Commun. 260 (2006) 474. 29) E. P. Kosmidou, E. E. Kriezis, and T. D. Tsiboukis: J. Appl. Phys. 100 (2006) 043118. 30) C.-Y. Liu and L.-W. Chen: J. Nanomater. (2006) 52946. 31) C. Oh and M. J. Escuti: Opt. Express 14 (2006) 11870. 32) C.-L. Ting, T.-H. Lin, C.-C. Liao, and A. Y. Fuh: Opt. Express 14 (2006) 5594. 33) J. P. Berenger: J. Comput. Phys. 114 (1994) 185. 34) L. Solymar and D. J. Cooke: Volume Holography and Volume Gratings (Academic Press, New York, 1981).
7967
