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Highly controllable form birefringence in subwavelength-period grating structures fabricated by imprinting on polarization-sensitive liquid crystalline polymers Hiroshi Ono,1,* Masaya Nishi,1 Tomoyuki Sasaki,1 Kohei Noda,1 Makoto Okada,1,2 Shinji Matsui,1,2 and Nobuhiro Kawatsuki1,3 1
Department of Electrical Engineering, Nagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka, Niigata 940-2188, Japan 2 Graduate School of Science, LASTI, University of Hyogo, Ako, Hyogo 678-1205, Japan 3 Department of Materials Science and Chemistry, University of Hyogo, 2167 Shosha, Himeji, Hyogo 671-2280, Japan *Corresponding author:
[email protected] Received May 25, 2012; accepted June 16, 2012; posted July 19, 2012 (Doc. ID 169378); published August 10, 2012
A subwavelength-period grating made in an intrinsic anisotropic medium was experimentally fabricated by means of imprinting and subsequent photoinduced molecular alignment of photocrosslinkable polymer liquid crystals (PCLC). Optical properties including the total birefringence and optic axis were theoretically and experimentally investigated by varying the crossing angle between the grating vector and the polarization azimuth of linearly polarized ultraviolet light for the photoalignment of PCLC. The total birefringence and optic axis were wellcontrolled by both form birefringence due to the subwavelength-period grating structure and intrinsic birefringence induced by photoalignment of PCLC. The finite-difference time-domain (FDTD) method was an effective tool for characterizing the optical properties of a subwavelength-period grating made in an intrinsic anisotropic medium. © 2012 Optical Society of America OCIS codes: 160.4670, 160.3710, 260.1440, 260.5430.
1. INTRODUCTION Recent advances in the high quality nanoreplication techniques of plastic films have renewed interest in the use of grating structures in combination with optical systems [1–7]. Optical properties of imprinted gratings are expected to be strongly dependent on both light wavelength λ and grating period Λ and therefore to be dependent on Λ ∕ λ. Form birefringence is an interesting phenomenon in classical optics. A periodic layered medium consisting of alternative layers of isotropic materials with different indices of refraction is optically equivalent to a homogeneous uniaxial material when the period is much smaller than the wavelength of light (Λ ∕ λ ≪ 1) [8]. The electric fields parallel to the grating grooves (TE polarization) and perpendicular to the grating grooves (TM polarization) need to satisfy different boundary conditions, resulting in different effective refractive indices for TE- and TM-polarized waves. In the limit of Λ ∕ λ ≪ 1, effective medium theory (EMT), in which sufficiently large number of boundaries between the layers with different dielectric constants is assumed to be contained in unit wavelength, is proposed by Rytov and has been used to explain the form birefringence [9]. Form birefringence is of great use for realizing numerical optical functions such as polarization grating devices, polarization beam splitters, and thin-film optical filters. If, instead 0740-3224/12/092386-06$15.00/0
of isotropic media, intrinsic anisotropic material is used for form birefringent media, then the possibility arises of higher functionalized optical devices. From some theoretical considerations in past studies, although none of the experimental results to date have been reported yet, use of intrinsic anisotropic material for form birefringent media possesses some important advantages over use of isotropic material. Han and Kostuk theoretically investigated the form birefringence in a lamellar subwavelength-period grating composed with intrinsic uniaxial birefringent materials [10]. When the subwavelength-period grating is made in the uniaxial birefringent material with the grating vector parallel to the optical axis of the material, the total birefringence is increased. One advantage of this combination between the form birefringence and intrinsic birefringence is that the same phase shift can be achieved with smaller aspect ratio of the grating, which is preferable for the easy fabrication. Moreover, Emoto et al. calculated form birefringence combined with intrinsic anisotropic media as a function of the angle between the optic axis and the grating vector with respect to the intrinsic birefringence of medium by means of not only EMT but also the finitedifference time-domain (FDTD) method [11]. In this paper subwavelength-period gratings with intrinsic uniaxial birefringent materials were experimentally fabricated by means of © 2012 Optical Society of America
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imprinting and subsequent photoinduced molecular alignment of photocrosslinkable polymer liquid crystals (PCLC).
2. EXPERIMENT A. Sample Preparation In this article, we synthesized a photocrosslinkable poly (methylmethacrylate) liquid crystal with 4-(4-methoxycinnamoyloxy)biphenyl side groups (PCLC) in order to optically induce the intrinsic birefringence in the imprinted gratings. The chemical structure of the PCLC is shown in Fig. 1. The PCLC exhibits a nematic liquid crystalline phase, and the clearing temperature was more than 300 °C determined by polarization optical microscopy and differential scanning calorimetry; more details about synthesis can be found elsewhere [12]. According to our previous studies, the PCLC used here exhibits thermally stable reorientation of mesogenic side groups by the use of linearly polarized ultraviolet (LPUV) light and subsequent annealing at around 150 °C. The solvent-evaporate imprint lithography provides the PCLC gratings as schematically described in Fig. 2(a). The PCLC films with grating structures were prepared by spin casting a solution of the PCLC on the silicone rubber stump. After removing the solvent, the PCLC exhibited the amorphous structure and the film showed good optical quality. The intrinsic birefringence of the PCLC film was induced using an He–Cd laser emitting linearly polarized continuous wave with wavelength of 325 nm. The orientation direction of mesogenic molecules (ϕD : director) was controlled by rotating a halfwave plate. The intrinsic birefringence of the PCLC film was not observed after irradiating LPUV, while the intrinsic birefringence was induced during annealing at 150 °C for 15 min. Epoxy adhesive was spun on the resultant PCLC films after annealing and covered with a quartz plate. The epoxy adhesive was cured and the final PCLC grating with a quartz plate was released from the silicone rubber stump. The resultant grating was observed by scanning electron microscopy (SEM) from a vertical view as shown in Fig. 2(b). The grating period Λ, grating depth d, and thickness D of PCLC layer without grating structure were 520, 180, and 200 nm, respectively. B. Characterization of Optical Anisotropy To observe the optical anisotropy in the subwavelength grating structure formed with intrinsic birefringent PCLC, we set up two types of optical systems consisting of am Nd:YAG laser (λ 1064 nm, B&W TEK Inc., BWR-50E/55870), two polarizers, and a quarter-wave plate as schematically described in Fig. 3. To measure the optic axis ϕo of the index ellipsoid of the subwavelength-period grating structure formed with intrinsic birefringent PCLC, the sample was set up between the two crossed polarizers (crossed Nicols configuration) as shown in Fig. 3(a), and transmittance was measured by varying the azimuth angle θG of the grating vector. Moreover the circularly polarized probe beam, which was generated by the
Fig. 1. Chemical structure of a photocrosslinkable poly(methylmethacrylate) liquid crystal with 4-(4-methoxycinnamoyloxy)biphenyl side groups (PCLC).
Fig. 2. (Color online) (a) Schematics of imprinting and photoalignment processes for preparing the gratings with intrinsic birefringence, and (b) SEM observation of the cross section of the resulting gratings.
quarter-wave plate as shown in Fig. 3(b), was incident on the sample and transmitted elliptically polarized light was observed by varying the azimuth angle θA of the analyzer. The value of the birefringence was estimated from the ellipticity p ofpthe transmitted elliptically polarized light, and p is defined as I min ∕ I max , where I min and I max are the minimum and maximum intensity of the light transmitted through the Glan– Thompson polarizing prisms by rotating them.
3. RESULTS AND DISCUSSION Figure 4(a) shows the dependence of azimuth angle θG of the grating vector on transmitted intensity, as measured using the optical system described in Fig. 3(a). The sample was set under crossed Nicols configuration and the expression for light propagation of the total system is
Fig. 3. (Color online) Experimental setup for measuring; (a) the optic axis and (b) total birefringence.
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" Eout θG
0 0 0 1
#
"
R−θG − Δϕ
" # 1 × RθG Δϕ ; 0
exp−iΓ ∕ 2
0
0
expiΓ ∕ 2
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#
(1)
where Δϕ is a rotated angle due to the LPUV induced birefringence in the PCLC film and ϕo θG Δϕ. Γ and Rϕo are the phase difference of the medium possessing total birefringence Δn at a wavelength of λ and the rotation matrix, respectively, and are given by Γ
2π Δnd D; λ
Rϕo
cos ϕo − sin ϕo
sin ϕo : cos ϕo
(2)
(3)
TΓ is transmission matrix for the uniaxial medium with the optic axis aligned parallel to the horizontal axis and can be expressed as follows: exp−iΓ ∕ 2 0 : (4) TΓ 0 expiΓ ∕ 2 Under this configuration, the transmitted intensity is proportional to sin2 2ϕo according to Eq. (1), and the azimuth angle of the index ellipsoid can be estimated from θmax G , giving the peak intensity. Figure 4(a) shows the results for dependence of the azimuth angle θG of the grating vector on transmitted intensity by varying the polarization direction of LPUV, i.e., the azimuth angle ϕD of the director of the mesogenic molecules, as were measured using the optical system described in Fig. 3(a). As shown in Fig. 4, θmax giving the peak intensity is G strongly dependent on the azimuth angle ϕD of the director of the mesogenic molecules. This means that the optic axis of the total birefringence is rotated by enhancement or compensation of form birefringence originating in a subwavelengthperiod grating structure and intrinsic birefringence induced by LPUV irradiation and subsequent annealing on PCLC films. Figure 4(b) shows the typical results for dependence of the azimuth angle θA of the analyzer on transmitted intensity, as were measured using optical system described in Fig. 3(b). When the circularly polarized light, which was generated using a quarter-wave plate just before the sample, is incident on the sample, the expression for light propagation of the total system is 1 0 Eout θA R−θA RθA · R−ϕo · TΓ 0 0 1 1 · Rϕo p : (5) 2 i The repeated pattern of valleys and peaks, which are linked with the azimuth angle of the optic axis, confirms that dependence of the transmitted intensity on θA in Fig. 4(b) is trigonometrical functions, as expected for the uniaxial anisotropic medium as described by jEout j. The ellipticity p of the transmitted p elliptically polarized light, which is defined as I min ∕ I max , as indicated in Fig. 4, is linked with phase difference Γ as follows:
Fig. 4. (Color online) (a) Dependence of azimuth of grating vector on output intensity measured using optical system of Fig. 3(a) on varying director ϕD of the PCLC films. Open circles represent the experimental data and solid curves are obtained by theoretical fitting. (b) Dependence of azimuth of analyzer on output intensity measured using optical system of Fig. 3(b). Blue and red open circles represent experimental data for the sample before and after annealing, respectively, and solid curves are obtained by theoretical fitting.
p
r 1 − sin Γ : 1 sin Γ
(6)
Substituting Eq. (2) into Eq. (6), the total birefringence is given by Δn
λ 1 − p2 : sin−1 2πd D 1 p2
(7)
Before annealing the PCLC film, the intrinsic birefringence is not yet induced and form birefringence due to only subwavelength-period grating structure is effective. Therefore the optical axis is perpendicular to the grating vector (ϕo π ∕ 2) before the PCLC film is annealed, because the polymeric medium is isotropic. One can notice that the transmitted intensity is proportional to 1 − 2 cos θA sin θA by substituting ϕo π ∕ 2 into Eq. (4). This theoretical result indicates the maximum intensity should be obtained at ϕ 135° and is in good agreement with the experimental result indicated as a blue arrow in Fig. 4(b). Once the PCLC film is annealed and the intrinsic birefringence of the PCLC is induced, the azimuth angle θmax giving the peak intensity is shifted from a blue A
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arrow to a red one and the amplitude of I max − I min is increased as shown in Fig. 4(b). We attribute the behavior to the thermally organized molecular alignment, which induces the intrinsic birefringence in the PCLC film, rotates the optic axis, and enhances the phase difference due to the birefringence. In the limit of Λ ∕ λ ≪ 1(subwavelength region), consider a single period with high spatial frequency in a binary phase diffractive structure as schematically described in Fig. 5. In the subwavelength-period region, the spatial modulation of the light electric field is sufficiently slow because there are large numbers of planar boundaries between two different media per unit volume. The wavelength λ is much larger than the grating period Λ and the effective index can be defined as a power series of Λ ∕ λ. When the optic axis is parallel or perpendicular to the grating vector, the EMT is effective by simply using the two refractive indices of anisotropic polymeric medium, i.e., npTE and npTM for TE and TM polarizations, respectively. The second-order EMT can be used to calculate the effective indices, and the effective indices for TE and TM polarizations are given by n2TE
n2TM
n20TM
n20TE
π2 Λ 2 2 F 1 − F2 n2pTE − 12 ; 3 λ
(8)
2 π2 Λ 2 2 1 F 1 − F2 n60TM n20TE 2 − 1 ; (9) 3 λ npTM
where n20TE Fn2pTE 1 − F;
(10)
1 1 F 2 1 − F n20TM npTM
(11)
are effective indices calculated with the zero-order EMT for TE and TM polarizations, respectively, and F is the filling factor defined as the ratio between the polymeric medium within one period: F
w ; Λ
(12)
Fig. 5. (Color online) Geometry used to calculate the form birefringence. The TE- and TM-polarized light is incident normal to the plate gratings.
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where w is the width of the polymeric plate and the form birefringence is calculated as ΔnF nTE − nTM . Effects of angles between the optic axis of the anisotropic polymer and grating vector on the form birefringence are analyzed by means of FDTD methods. FDTD solvers have been developed as among the most effective tools for computational electromagnetics problems. FDTD methods present a robust and powerful approach to directly solve curl equations both in time and in space. Maxwell’s equations for an anisotropic medium can be written as follows: iωε0 ε~ E ∇ × H;
(13)
iωμ0 H −∇ × E;
(14)
where ω is the angular frequency, and ε0 and μ0 are the permittivity and permeability, respectively, of free space. ε~ is a permittivity tensor for an anisotropic media, and permeability tensor μ is assumed to be a unit matrix in the present study. A three-dimensionally slanted optic axis should be described by introducing two kinds of rotation angles φ and ϑ, corresponding to slanted angles for x- and z-axis, respectively. The rotation matrix for each transformation is given as 0! 0 ; 1
(15)
0 − sin ϑ ! 1 0 : 0 cos ϑ
(16)
Hφ
cos φ sin φ − sin φ cos φ 0 0
Sϑ
cos ϑ 0 sin ϑ
When the material has linear dielectric properties and uniaxial anisotropy, only two dielectric constants, i.e., εo and εe , are necessary to specify the tensor description ε~ : 0
εo ε~ H−φ · S−ϑ@ 0 0
0 εo 0
1 0 0 ASϑ · Hφ: εe
(17)
The FDTD calculations were performed using the twodimensional Yee grid defined as a projection of the original Yee grid in three dimensions on the x-z plane [13,14]. In the two-dimensional FDTD calculation, the Maxwell rot is written in discrete forms as functions of Δx and Δz, and the time differential as a function of Δt. The grid space (Δx × Δz) is set to be 20 nm × 20 nm, and is smaller than both grating spacing and λ ∕ 20. The time step (Δt) is set to be 0.047 fs, and cΔt p is shorter than 1 ∕ 1 ∕ Δx2 1 ∕ Δz2 , where c is the light velocity in free space. The essential feature in FDTD methods is a proper absorbing boundary condition to truncate the simulation space without artificial reflections. In the present study, we employed the perfectly matched layer (PML) introduced by Berenger [15]. Strictly speaking, spatial distributions of the director of mesogenic molecules are inhomogeneous, as schematically shown in Fig. 6(a), because mesogenic molecules tend to be parallel to the boundary between the air and the medium. In the surface layer with thickness of Δw, we assume that the azimuth angle of the director is linearly changed from π ∕ 2 to
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Fig. 6. (Color online) (a) Schematics of spatial distribution of the director of the mesogenic molecules within one plane in subwavelength-period gratings. (b) Geometry of the director distributions to calculate the birefringence.
ϕD as shown in Fig. 6(b). Figures 7(a) and 7(b) show dependence of the azimuth angles of the director on ellipticity p of the transmitted light observed in the optical system of Fig. 3(b) and azimuth angles ϕo of the index ellipsoid estimated from the θmax observed in the optical system of G Fig. 3(a), respectively. The ellipticity of the input circularly polarized light is unity and is reduced because of the phase difference in the uniaxial birefringent medium. The ellipticity p is decreased by increasing the angle of the director ϕD as shown in Fig. 7(a). The azimuth angles ϕo of index ellipsoid are increased by increasing the angle of the director ϕD as shown in Fig. 7(b) as a convex upward function, although ϕo ϕD in the case of d 0. We attribute the behavior described in Figs. 7(a) and 7(b) to the compensation or enhancement of the intrinsic PCLC birefringence and form birefringence originating in subwavelength-period grating structure, which is affected by grating vector and director ϕD . Theoretical calculations were performed using both EMT and FDTD, and both results were in good agreement with each other in the case of ϕD 0, π ∕ 2. FDTD calculations were performed by varying Δw of the surface layer, in which the director is rotated as shown in Fig. 6. As shown in Figs. 7(a) and 7(b), the theoretical calculations well-explained the experimental results by assuming Δw 60 nm. In the present study, subwavelength-period grating is formed on the uniaxial birefringent film with a thickness of D 200 nm as shown in Fig. 2(b) and as schematically described in Fig. 5. Therefore the total birefringence Δn contains both effects of subwavelength-period grating d and the uniaxial birefringent film D as described in Eq. (2). To consider the “pure” form birefringence ΔnF , the effects of the uniaxial birefringent film under the subwavelength-period gratings on experimental results have to be separated because experiments presented here were performed under transmission configurations as shown in Fig. 3. Using an exact Jones matrix analysis, we can represent this system by the matrix
Fig. 7. (Color online) (a) Dependence of azimuth angles ϕD of the director on ellipticity p of the transmitted light observed in the optical system of Fig. 3(b). (b) Dependence of azimuth angles ϕD of the director on azimuth angles ϕo of index ellipsoid estimated from the θmax G observed in the optical system of Fig. 3(a). Full circles represented to the experimental data. Dotted and solid curves are obtained from FDTD calculations in the case of Δw 0 nm and Δw 60 nm, respectively. Open squares shows the EMT calculation.
Eout R−ϕS · TΓS · RϕS · R−ϕF · TΓF · RϕF " # 1 1 ; (18) · p 2 i where indices “S” and “F” are related to the uniaxial birefringent layer under the grating and the grating layer, respectively. The light electric field just after transmitting the grating layer with a thickness d is given by EF RϕS · T−1 ΓS · R−ϕS · Eout :
(19)
The light electric field Eout can be experimentally estimated by observing the polarization state of the transmitted light in the optical system described in Fig. 3(b). The azimuth angle ϕS of the optic axis in the uniaxial birefringent layer under the grating should be matched with PCLC director ϕD , which is equal to the polarization direction of irradiated LPUV, and phase difference ΓS is equal to 2πΔnPCLC D ∕ λ, where ΔnPCLC is intrinsic birefringence of the photo-aligned PCLC and was estimated to be 0.15 from experimental observation and theoretical calculation described in Figs. 7(a) and 7(b). Thus the light electric field EF can be estimated from the experimental conditions and observation results. In result the pure form birefringence
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photoalignment PCLC films, are parallel, the total effect enhances the birefringence of the medium, while the total effect compensates the birefringence when the two axes are perpendicular. Both the total birefringence and the optic axis are well-controlled by selecting the crossing angle between the two axes, and the properties were well-explained by the FDTD calculations. The well-controllable fabrication of the anisotropic medium should stimulate the development of highly functionalized optical devices in the future.
REFERENCES
Fig. 8. Form birefringence ΔnF after removing the effects of birefringence in the uniaxial birefringent film under the subwavelengthperiod gratings by varying the azimuth angle ϕD of the director.
ΔnF of the subwavelength-period gratings formed by the intrinsic uniaxial medium, by separating the effects of the uniaxial birefringent layer under the grating, can be calculated using ellipticity pF of the light EF as follows: ΔnF
λ 1 − p2F : sin−1 2πd 1 p2F
(20)
Figure 8 summarizes form birefringence ΔnF by varying the azimuth angle ϕD of the director. The form birefringence ΔnF is increased by increasing ϕD as shown in Fig. 8. The optic axis of form birefringence originating in subwavelength-period structure is perpendicular to the grating vector. Therefore the total birefringence is maximized when the azimuth angle of the director is perpendicular to the grating vector (ϕD 90°) due to the enhancement, while that is minimized when ϕD 0° due to the compensation. Since intrinsic birefringence of the photo-aligned PCLC is estimated to be 0.15 from experimental observation and theoretical calculation described in Figs. 7(a) and 7(b), the total birefringence is compensated due to the subwavelength-period grating when the angle of the director is less than around 20° (ϕD < 20°), while that is enhanced in the region of ϕD > 20° as shown in Fig. 8.
4. CONCLUSIONS Subwavelength-period gratings with intrinsic uniaxial birefringent materials were experimentally fabricated by means of imprinting and photoinduced molecular alignment of PCLC. The total birefringence of the subwavelength-period gratings with intrinsic uniaxial materials was controlled by both grating parameters and intrinsic birefringence. When the two axes in two kinds of index ellipsoid, originating in form birefringence of subwavelength-period gratings and intrinsic birefringence
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