Time-domain analysis of optically controllable biphotonic gratings in azo-dye-doped cholesteric liquid crystals

Hui-Chen Yeh

doi:10.1364/OE.19.005500

1. Introduction

Photochromic materials have been attracting considerable attention for their potential application in all optical devices associated with the control of molecular alignment in liquid crystal (LC) phases [1,2]. Azobenzene is a representative photochromic molecule using light-induced molecular reorientation through photoactivated isomerization [3,4]. Photoexcited azobenzene undergoes a reversible conformational transformation between trans-longitudinal and unstable cis-excited isomers [5,6]. Repetition of the trans-cis-trans isomerization cycles may lead to changes in the chemical and physical properties of azo-dye-doped liquid crystals, such as the order of orientation, molecular orientation, and LC-isotropic phase transition temperature.

Previous studies have successfully utilized the effects of molecular reorientation to control the phototunable characteristics of azo-dye-doped liquid crystals [7–10]. In the process of photoexcitation, azo dye molecules in planar nematic LCs tend to orient the LC director in a direction orthogonal to the optical electric field and the propagation wave vector of the polarized pump beam in a direction that minimizes the free energy of the system [11]. However, a z modulation effect occurs in azo-dye-doped cholesteric liquid crystal (ADDCLC) systems [12]. Cholesteric liquid crystals (CLCs) are produced by adding chiral molecules to nematic liquid crystal media. Thus, the structure possesses a spontaneous twist around a helical axis normal to the director. Due to the helical arrangement of the directors, CLCs exhibit a number of beneficial optical properties, including strong optical rotatory powers and selective reflection of circularly polarized light [13]. In photoexcited ADDCLCs, the azo dyes in the helix structure realign toward the direction of the wave vector of the pump beam, thereby minimizing the probability of repeated photoexcitation, yielding a distorted cholesteric structure with a great number of LC molecules tilted toward the wave vector. Photoisomerization and a concomitant thermal effect may also alter the helical pitch of CLCs [14,15]. Combining the unique optical properties of CLCs with the reversible photochemical reaction of azo derivatives has increased the importance of optically controllable devices based on ADDCLCs in recent years [12,15–19].

In a previous paper, the authors developed optically switchable biphotonic gratings (BGs) based on azo-dye-doped CLCs. The diffraction performance of BGs was related to the relative intensity of green and red beams [16]. With further investigation, it was noticed that the diffraction performance of BGs was closely related to the helical pitch of the cholesteric liquid crystal and the polarization state of the probe beam. The present study used the finite-difference time-domain (FDTD) method to analyze the relationship among the diffraction performance of BGs, the relative intensity of green and red beams, and the polarization state of the probe beam. The finite-difference time-domain method is a rigorous analysis that considers lateral light scattering and diffraction effects [20]. The transmission spectra of ADDCLC cells irradiated with green pump beams were first measured to infer LC director configurations corresponding to configurations in dark regions of red interference fields inducing BGs. Then, LC director configurations in bright regions of the red interference fields were calculated using the two-dimensional FDTD implementation to fit experimental data of the first order diffraction efficiencies. The calculated configurations of the LC director were examined using the polarization states of the first order diffracted beams.

2. Experiments

2.1. Sample preparation and experimental setup

The host LC in the present study was nematic BL009 (n_o = 1.5266, Δn = 0.2915 at 25°C, Merck). The chiral agent CB15 (Merck), which induces right-handed helical structures, was dissolved in the host LC to produce a cholesteric phase. Two cholesteric samples with different reflection bands were prepared from the homogeneous mixtures. The mixture ratios of BL009 and CB15 in sample 1 and sample 2 were 9.4 and 30.7 wt%, respectively. The photoresponsive material was bisazobenzene dichroic dye D2 of Merck. The concentration of dye dopant in both samples was 1wt%. Each mixture was injected into glass cells with a gap of 38 μm. Two indium-tin-oxide (ITO)-glass substrates were coated with polyvinyl alcohol (PVA) alignment film and rubbed in the same direction to produce planar-oriented cholesteric samples. Using a Fourier-transform infrared ray spectrometer, the center wavelength of the selective reflection band of sample 1 was confirmed to be 2400 nm, which is the same as the selective reflection band used in a previous study [16]. The center wavelength of the reflection band of sample 2 was selected in the visible region to measure changes in spectral characteristics on photoirradiation using a spectrometer for visible regions. The optically controllable biphotonic gratings were conducted under the illumination of one green beam with the simultaneous irradiation of a spatially intensity-modulated interference field generated by two coherent red beams [16]. Experiments proceeded at room temperature without thermal isolation of the cell.

The experimental setup is schematically illustrated in Fig. 1 . Two s-polarized coherent red beams from a He-Ne laser with an operating wavelength of 633 nm were used as a pump laser to create a spatially intensity-modulated interference field. The two pump beams with equal intensity intersected on the cell at an angle of 2.84° in air, resulting in a grating period Λ of approximately 12.8 μm. The normalized intensity distribution of the interference field along the x axis is:

I (x) = \cos^{2} (\frac{π x}{Λ}) .

One s-polarized green beam derived from an Ar⁺ laser with an operating wavelength of 514 nm impinged simultaneously on the interference region. The bright region of the red interference field was called “region R”, and the dark region “region G”. The diffraction properties of the induced BGs were probed using a He-Ne laser with a wavelength of 633 nm and an intensity of 1mW/cm² at normal incidence. To characterize the BGs, the polarization state of the probe beam was varied using a quarter- or half-wave plate.

Fig. 1 Schematic illustration of the biphotonic gratings in azo-dye-doped cholesteric liquid crystal films. (a) The photoinduced variations in the CLC structure created by one s-polarized green beam and two coherent s-polarized red beams. (b) The intensity distribution of the interference field along the x axis.

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2.2. Results

The present study revealed that the diffraction performance of the BGs depended heavily on the polarization state of the probe beam and relative intensity of the green and red beams. Figure 2 illustrates the variations in the first order diffraction efficiency of BGs with the intensity of the green beam for sample 1 using four polarization states of the probe beam.

Fig. 2 Dependence of the first order diffraction efficiency on the intensity of the green beam for sample 1 probed using left-handed circularly (LHC)-, s-, p- and right-handed circularly (RHC)-polarized beams. The intensity of each red pump beam is 909 mW/cm².

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In the Bragg condition, the left-handed circularly (LHC) polarized beam transmitted through the right-handed CLC. The reflection band of sample 1 was in the infrared region; therefore, the transmittance under different handedness of the probe beam was nearly equal before illumination with the green beam. However, diffraction performance clearly differed following the formation of the BGs. To investigate the polarization dependent properties, BGs were created based on sample 2 and variations in the texture of the CLC on photoirradiation were inferred according to the spectral characteristics measured using a spectrometer for the visible region (Ocean Optics USB4000). The wavelength of the probe beam was out of the reflection band of sample 2. As with sample 1, the transmittance for different handedness of the probe beam was nearly equal before illumination with the green beam. Figure 3 illustrates the diffraction properties of sample 2 versus the intensity of the green beam for four polarization states of the probe beam under varying intensities of red pump beams.

Fig. 3 Dependences of the first order diffraction efficiency on the intensity of the green beam for sample 2, probed using LHC-, s-, p- and RHC-polarized beams. The intensity of each red pump beam is (a) 909 mW/cm² and (b) 340 mW/cm².

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BGs were formed by green-beam-induced dye reorientation through trans-cis isomerization in the dark regions (region G) of the red interference field and by red-beam-induced suppression of dye reorientation via cis-trans back isomerization in the bright regions (region R) [16,21,22]. An increase in the intensity of the green beam resulted in an increase in distortion and thus a greater difference between textures in the two regions, thereby increasing the first order diffraction efficiency [Fig. 3 (a)]. However, the first order diffraction efficiency declined when the red beams were too weak to suppress the strong green-beam-induced reorientation of the dyes and LCs in region R [Fig. 3(b)]. The Stokes parameters S ₀, S ₁, S ₂, and S ₃ were measured using the circular method [23] to determine the polarization state of the first order diffracted beam. Figure 4 illustrates the measured normalized Stokes parameters S _1,2,3’ = S _1,2,3/S ₀ of the first order diffracted beam of the BGs versus the intensity of the green beam.

Fig. 4 Dependence of the normalized Stokes parameters of the first order diffracted beam versus the intensity of the green beam for (i) LHC-, (ii) s-, (iii) p-, and (iv) RHC-polarized probe beams. The filled and open symbols represent the experimental and simulation results, respectively. The intensity of each red pump beam is (a) 909 mW/cm² and (b) 340 mW/cm².

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3. FDTD analysis

3.1. FDTD algorithm for periodic anisotropic structures

The FDTD method is a powerful approach to solve Maxwell’s time-dependent curl equations, used to govern the propagation of light waves. For sourceless nonmagnetic anisotropic media, discrete expressions of electric and magnetic fields in Maxwell’s equations can be approximated using the central difference scheme based on the Yee grid [24] as follows:

E^{n} = E^{n - 1} + \frac{Δ t}{ε_{0}} {\tilde{ε}}^{- 1} \cdot \nabla \times H^{n - 1 / 2}

and

H^{n + 1 / 2} = H^{n - 1 / 2} - \frac{Δ t}{μ_{0}} \cdot \nabla \times E^{n}

where n represents the time points, Δt is the time step, and

{\tilde{ε}}^{- 1}

is the inverse of the spatially varying relative permittivity tensor. For a uniaxial LC medium, the relative permittivity tensor

\tilde{ε}

is written in the xyz coordinate system as:

\tilde{ε} = [\begin{matrix} ε_{x x} & ε_{x y} & ε_{x z} \\ ε_{y x} & ε_{y y} & ε_{y z} \\ ε_{z x} & ε_{z y} & ε_{z z} \end{matrix}]

with [25]

\begin{array}{l} ε_{x x} = n_{o}^{2} + (n_{e}^{2} - n_{o}^{2}) \cos^{2} θ \cos^{2} φ \\ ε_{x y} = ε_{y x} = (n_{e}^{2} - n_{o}^{2}) \cos^{2} θ \sin φ \cos φ \\ ε_{x z} = ε_{z x} = (n_{e}^{2} - n_{o}^{2}) \cos θ \sin θ \cos φ \\ ε_{y y} = n_{o}^{2} + (n_{e}^{2} - n_{o}^{2}) \cos^{2} θ \sin^{2} φ \\ ε_{y z} = ε_{z y} = (n_{e}^{2} - n_{o}^{2}) \cos θ \sin θ \sin φ \\ ε_{z z} = n_{o}^{2} + (n_{e}^{2} - n_{o}^{2}) \sin^{2} θ \end{array}

where n_o and n_e are the ordinary and extraordinary refractive indices, θ is the tilt angle between the LC director and the surface of the substrate (xy-plane), and ϕ is the twist angle between the projection of the LC director on the xy-plane and the x axis. In the CLC structure, the twist angle ϕ = 2πz/P, where P is the helical pitch. Once

\tilde{ε}

is defined at every grid point within the computational space, the sequence of light propagation in the structure can proceed, ruled by the time updating algorithm from Maxwell’s equations.

The present study implemented the FDTD method in a two-dimensional (2D) computational space. Even though the problem was reduced to a 2D structure, all components of electric and magnetic fields should be considered due to the coupling between all field components induced by the anisotropy of the media. To suppress artificial reflections, the computational space was terminated using two Berenger’s perfectly matched layers (PMLs) normal to the z-axis. Transverse direction boundaries were truncated by imposing Bloch periodic boundary conditions, in consideration of the periodic structure. The incident plane wave was launched into the computational space based on the total-field/scattered-field technique. Dual-time excitation was utilized to define the polarization of the input continuous wave. Transforming the calculated near-field results at the exit face of the device to far-field provided far-field information. The far-field diffraction of gratings yielded discrete diffracted orders, corresponding to the Fourier series of the field distribution across an infinitely periodic structure [26]. With normal incidence, the diffraction angle θ_m of the m ^th-order maximum was determined using the diffraction equation

Λ \sin θ_{m} = m λ

where λ is the wavelength of incident light and Λ is the grating period. Far-field optical disturbance was obtained using the vectorial Fourier transform of the near-fields E_x,y,z along the near-field collection line, spanning the extent of the grating period immediately following the structure, as follows [27]:

E_{/ /}^{m} (t) = \frac{1}{Λ} \int_{0}^{Λ} [E_{x} (t, x) \cos θ_{m} - E_{z} (t, x) \sin θ_{m}] \exp (j \frac{2 π m}{Λ} x) d x

E_{⊥}^{m} (t) = \frac{1}{Λ} \int_{0}^{Λ} E_{y} (t, x) \exp (j \frac{2 π m}{Λ} x) d x

where

E_{/ /}^{m}

and

E_{⊥}^{m}

denote the far-field components of the m ^th-order diffraction beam parallel and perpendicular to the 2D computational space, respectively. The intensity of the m ^th-order diffracted beam was proportional to the time-averaged value of the square of the instantaneous electric field over an interval T equal to one period [26]:

I^{m} = \frac{1}{T} \int_{t - T / 2}^{t + T / 2} c^{2} ε_{0} ({| E_{/ /}^{m} (t) |}^{2} + {| E_{⊥}^{m} (t) |}^{2}) d t .

Diffraction efficiency was calculated as the ratio of the intensity of the m ^th-order diffracted beam to the total intensity of the transmitted light.

3.2. Simulation results and discussion

The FDTD method was adopted for optical wave calculations due to the transverse variation in the structure of BGs, for which the stratified medium approach was inadequate. To clarify the LC director configuration under photoirradiation and define the permittivity tensor for the FDTD implementation, the spectral characterization was measured. Figure 5 illustrates the transmission spectra of sample 2 under irradiation using various intensities of the green beam. Following the initiation of excitation, variations in the reflection band at the long-wavelength edge were clearer than those of the reflection band at the short-wavelength edge [Fig. 5(a)], due to the photoinduced tilt of the director in the CLC medium toward the z axis, confirmed by implementing FDTD calculations. Similar to an electric field [28], the tilt angle of the LC director in the photoinduced distorted structure was zero at the inner surfaces of the two glass plates due to the boundary conditions, reaching its maximum θ_m, the midlayer tilt angle, at the center of the cell. The photoinduced θ_m depended on the intensity of the green beam and the electric-field-induced θ_m depended on the strength of the applied field. With an increase in irradiation time, the reflection band redshifted and the bandwidth shortened [Fig. 5(b)], indicating an elongation in the pitch of the CLC and a decrease in the birefringence of LC caused by photoisomerization and the concomitant thermal effect [15]. Figure 6 demonstrates the effects of photoirradiation on the midlayer tilt angle θ_m, the helical pitch P, and the birefringence Δn for sample 2.

Fig. 5 Transmission spectra of sample 2 with varying intensities of the green beam (a) at the initiation of excitation and (b) in photostationary equilibrium (approximately 3 min of irradiation).

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Fig. 6 The effects of photoirradiation on (a) the midlayer tilt angle θ_m, (b) the helical pitch P, and the birefringence Δn in the inset for sample 2. The green diamonds correspond to the results fitting the transmission spectra in Fig. 5 under irradiation with the green beam. The blue and red diamonds correspond to the results fitting the first order diffraction efficiencies in Figs. 3(a) and 3(b), respectively.

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Midlayer tilt angles were first calculated by fitting the transmission spectra in Fig. 5(a), assuming no variation in the helical pitch and the refractive indices at the beginning of excitation. These results were used to determine the helical pitches and refractive indices by fitting the transmission spectra in Fig. 5(b), assuming that the midlayer tilt angles in the photostationary equilibrium were the same as the midlayer tilt angles at the beginning of excitation. The calculated midlayer tilt angle, the helical pitch, and the birefringence under irradiation with the green beam were used to calculate the relative permittivity tensor $\tilde{ε}$ in the center of region G of the BGs. To obtain the configuration of the LC director in region R of the BGs, the midlayer tilt angle and the helical pitch in the center of region R were calculated by fitting the experimental diffraction efficiencies in Fig. 3. The refractive indices n_e and n_o in region R were assumed to be the same as the indices in region G due to heat diffusion. The geometry of the FDTD simulation is schematically illustrated in Fig. 7 . For simplicity, the ITO contacts were assumed to possess the optical behavior of the glass substrate. Table 1 summarizes the numerical parameters used for the simulation.

Fig. 7 (a) Schematic layout of the two-dimensional FDTD simulation space. (b) The tilt angle θ of the LC director along the z axis. (c) The midlayer tilt angle θ_m along the x axis in one spatial period. θ_mR and θ_mG correspond to the midlayer tilt angle in the centers of regions R and G, respectively.

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Table 1. Overview of FDTD Simulation Parameters

View Table

The tilt angle distribution along the z axis was set as a sinusoidal function under irradiation subject to boundary conditions and electrostatic energy density. The midlayer tilt angle θ_m and the helical pitch P, modulated by the intensity distribution of the red interference field along the x axis, were approximated as:

θ_{m} (x) = θ_{m R} + (θ_{m G} - θ_{m R}) \sin (\frac{π x}{Λ})

P (x) = P_{R} + (P_{G} - P_{R}) \sin^{2} (\frac{π x}{Λ})

where θ_mR and θ_mG correspond to the midlayer tilt angles in the centers of regions R and G, respectively. P_R and P_G correspond to the pitches. Differences in the structure between regions R and G increased with an increase in the intensity of the green beam. Differences in the structure between regions R and G diminished under a red pump beam of low intensity when the red beams were unable to suppress the effects of the strong green beam on the structure of the CLC (Fig. 6). The results support the proposed model for the formation of the BGs [16]. To examine the configuration of the simulated LC director, the Stokes parameters of the first order diffracted beam were calculated and plotted in Fig. 4. The calculated results are in good agreement with the experimental data.

Figure 8 illustrates the calculated development of the electric field at the near-field collection line in one temporal period for four types of polarization of the probe beam. The instantaneous electric fields E_x and E_y along the x axis were modulated most deeply for the LHC polarized probe beam, resulting in the highest first order diffraction efficiency for the LHC polarized probe beam among the four types of polarization under the same excitation conditions. To better understand the physical mechanisms involved, Fig. 9 depicts the polarization of the transmitted light at the centers of regions R and G for four types of polarization of the incident beam. The electric fields of the incident beams are represented as:

s : E_{y} = \cos ω t

p : E_{x} = \sin ω t

LHC : E_{x} = \sin ω t, E_{y} = - \cos ω t

RHC : E_{x} = \sin ω t, E_{y} = \cos ω t

where ω is the angular temporal frequency. Due to optical activity [29], the polarization of a linearly polarized beam passing through the CLC rotates through an angle [Figs. 9(b) and 9(c)]. The incident circularly polarized beams are regarded as the superposition of p- and s-polarized beams. According to the principle of superposition, the corresponding outgoing light [Figs. 9(a) and 9(d)] of LHC- and RHC-polarized beams are the subtraction and addition of the corresponding outgoing light [Figs. 9(c) and 9(b)] of p- and s-polarized beams, respectively. These results indicate that the LHC-polarized beam exhibited a greater difference in amplitude between the centers of regions R and G than the RHC-polarized beam did. Because the optical rotatory power is pitch dependent [29], CLC samples with different helical pitches resulted in different rotation angles of linearly polarized beams through the CLCs, which in turn lead to differences in the diffraction performance for different polarization states of probe beam (Figs. 2 and 3).

Fig. 8 Calculated development of (i) the x component, (ii) the y component, and (iii) the z component of the electric field at the near-field collection line in one temporal period for (a) LHC-, (b) s-, (c) p-, and (d) RHC-polarized probe beams with the same intensity. The intensities of the red and green beams forming BGs are 909 mW/cm² and 306 mW/cm², respectively. Axes x and t are scaled in the grid spacing Δx and the time step Δt, respectively.

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Fig. 9 Polarization of the transmitted light at the centers of regions R (red lines) and G (green lines) for (a) LHC-, (b) s-, (c) p-, and (d) RHC-polarized probe beams. The asterisks indicate the electric fields at the beginning of the temporal period.

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4. Conclusion

The present study demonstrated optically controllable BGs based on ADDCLC cells, and analyzed the diffraction characteristics using the FDTD method. The BGs were formed under the illumination of one green beam with the simultaneous irradiation of an interference field generated by two coherent red beams. The green beam induced elongation of the helical pitch and dye reorientation in the direction of the wave vector via trans-cis isomerization and the red beams suppressed the elongation of the helical pitch and dye reorientation via cis-trans back isomerization in the dark and bright regions of the red interference field. The first order diffraction efficiency increased with an increase in the intensity of the green beam, but decreased when the red beams were unable to suppress dye reorientation induced by the strong green beam. Application of the FDTD algorithm demonstrated that the model of photoinduced distortion of the CLC structure explained the diffraction performance of the BGs well. The electric fields of LHC polarized probe beams in one spatial period were modulated most deeply with four polarized probe beams passing through the gratings. Therefore, the first order diffraction efficiency under the LHC polarized probe beam was the largest among the four polarized probe beams under the same excitation conditions.

Acknowledgments

The author would like to thank two anonymous reviewers and the editor for their comments.

References and links

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Simulation Parameter	Value
Speed of light in free space, c	3 × 10⁸ m/s
Wavelength of the plane wave source in free space, λ	633 nm
Grid spacing, Δx = Δz = λ/20	31.65 nm
Time step, Δt = Δx/2c	5.275 × 10⁻¹⁷ s
PML thickness	0.633 μm
Cell gap, d	38 μm
Glass thickness	3.165 μm
Refractive index of glass	1.5
Grating period, Λ	12.8 μm

Time-domain analysis of optically controllable biphotonic gratings in azo-dye-doped cholesteric liquid crystals

Abstract

1. Introduction

2. Experiments

2.1. Sample preparation and experimental setup

2.2. Results

3. FDTD analysis

3.1. FDTD algorithm for periodic anisotropic structures

3.2. Simulation results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Tables (1)

Equations (15)

Optics Express