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We propose a liquid crystal (LC) polarization grating that conserves the polarization state of incident light, wherein the variation range of the twist angle is 2π. The design scheme for theoretically 100% diffraction efficiency of the first-diffraction order is derived, and a prototype LC grating is evaluated. Under zero voltage, the fabricated LC grating exhibits high efficiency of the first-order diffraction, validating the proposed design scheme. The high efficiency of the second-order diffraction can also be achieved under a high voltage so that the LC director in the midplane is vertical to the substrate plane. The circular polarization sense of the second-order diffraction is identical to that of the incident light as in the case of the first-order diffraction. This grating functions as a beam deflector, steering the input beam in three different directions (zeroth-, first-, and second-order diffractions) by adjusting the applied voltage.
©2012 Optical Society of America
1. Introduction
Liquid crystal (LC) polarization gratings [1–16] exhibit high first-order diffraction efficiency (theoretically 100%) and excellent polarization splitting properties. Because of these advantages, their potential applications are projection-type displays [17], polarization converters [18], and Stokes polarimeters [19–21]. LC polarization gratings can be grouped into two categories: the homogeneous or hybrid orientation-based grating and the twisted nematic (TN) orientation-based one. The former grating has spatially-distributed homogeneous or hybrid orientation domains, which are equivalent to a collection of phase plates whose optic axes are spatially distributed from −π /2 to π/2, where the retardation has to equal R = (N + 1/2)λ (N and λ are an integer and wavelength, respectively). On the other hand, the latter grating is composed of spatially distributed TN domains whose orientation state is spatially varied. In this case, the phase of the output light depends on the azimuthal angles of surface LC directors on entrance and exit substrates. So far, an LC polarization grating with spatially distributed twist angle demonstrated a good beam deflecting function similar to a blazed grating, where the retardation equals R = (N/2 + 1/4)λ and the incident light is linearly polarized [1]. This TN-LC polarization grating exhibits theoretically 100% diffraction efficiency. Additionally, the LC polarization grating consisting of two oppositely twisted orientation layers demonstrates an excellent beam deflection performance with nearly 100% diffraction efficiency and almost achromatic characteristics [12].
Another feature of LC polarization gratings is unique polarization conversion properties; that is, the polarization state of the incident light is converted into the corresponding polarization state. For example, the sense of the incident circularly polarized light flips by passing through an LC polarization grating. This polarization conversion is inevitable in conventional LC polarization gratings. If the polarization conservation is achieved, it enables flexible design of a planned optical system. In this paper, we propose an LC polarization grating with spatially distributed TN orientation domains in which the polarization state of the incident light is conserved. First, we derive a condition for theoretically 100% diffraction efficiency in the TN-LC polarization grating whose twist angle is spatially varied. Next, the measurement result of diffraction efficiency of the proposed TN-LC grating is described and voltage-dependent diffraction properties are discussed together with an LC molecular orientation model.
2. Theoretical condition for 100% diffraction efficiency
In this section, we consider the situation wherein the polarization state of the incident light is invariant when the light passes through an LC cell. When the Jones vector of the incident light is given as Ein = (Ex, Ey), the transmitted light is expressed as Eout = WTNEin, where WTN is the Jones matrix of a TN-LC cell [22, 23]. Furthermore, because the polarization state is conserved, the Jones vector of the transmitted light should be expressed as Eout = aEin, where a is an imaginary number whose absolute value is unity (aa* = 1), indicating that there is no energy loss of light. Consequently, the following eigen equation has to be satisfied.
where Ein and a are considered an eigenvector and an eigenvalue, respectively. Similar optical analysis of general TN-LC cells based on the concept of the eigen equation was previously introduced [24]. We apply here the eigen equation-based analysis to the design of TN-LC polarization gratings. Two eigenvalues aj (j = 1 or 2) are obtained by solving the eigen equation |WTN − aI| = 0, where I is a unit matrix.where In Eq. (4), Δn is birefringence, d is the cell thickness, and Φ is the twist angle defined to be Φ = Ψo − Ψi, as shown in Fig. 1 . Because the matrix WTN is a unitary matrix, the absolute value of the eigenvalue has to be unity (ajaj* = 1), indicating conservation of the light energy. Therefore, the eigenvalue should be rewritten as follows:
Fig. 1 Definitions of twist angle (Φ) and azimuthal angle of surface LC directors on input and output substrates (Ψi and Ψo). Twist angle is defined to be Φ = Ψo − Ψi.
Because the transmitted light is expressed as Eout = ajEin, (−1)jϕ indicates the phase shift between the transmitted and incident light beams. By comparing the real parts of Eqs. (2) and (6), we can obtain the following equation.
To satisfy this equation for an arbitrary twist angle Φ,
has to be satisfied because u > 0, where N is an integer. Therefore, we obtainThis indicates that the phase of the transmitted light can be controlled by the twist angle Φ and Eq. (8) is a sufficient condition for 100% diffraction efficiency. If the LC layer is sufficiently thick (u >> 1) in order to satisfy Mauguin’s condition, the condition Θ = Nπ is reduced to
The eigenvector Ein(j), which is identical to the Jones vector of the incident light, can be derived from the equation (WTN − ajI)Ein(j) = 0 as follows:
where Ein(j) is normalized so that |Ein(j)| = 1. The above equation indicates that the incident light has to be circularly polarized light. If right-handed (or left-handed) circularly polarized light is incident on the TN-LC cell, the index j has to be determined so that j = 1 (or j = 2). Finally, the Jones vector of the transmitted light is easily derived as follows:From Eqs. (11) and (12), it is clear that unlike conventional LC polarization gratings, the rotation senses of the circularly polarized incident and transmitted light beams are identical.
The phase of the transmitted light can be controlled by the twist angle Φ at will. When the twist angle varies spatially and periodically, the phase of the transmitted light also vibrates. If the phase variation Φ is proportional to the positional coordinate from −π to π, the TN-LC grating with the spatially distributed TN domains functions as a complete blazed grating; that is, only first-order diffraction is produced (100% efficiency). Because the phases of the transmitted light for j = 1 and 2 are −Φ and Φ, respectively, the two output beams Eout(1) and Eout(2) deflect oppositely. Additionally, the rotation sense of Eout(1) is opposite to that of Eout(2), as in the case of conventional polarization gratings.
3. Experimental verification of LC polarization grating with high efficiency
According to the discussion in the previous section, we propose a design scheme of TN-LC polarization gratings with high diffraction efficiency. The conditions of this design scheme are i) the LC layer is sufficiently thick in order to satisfy Mauguin’s condition, ii) retardation R = Nλ, iii) the twist angle linearly varies in a range of 2π, and iv) the incident light is circularly polarized. To verify the proposed design scheme, the LC polarization grating with the spatially distributed twist orientation was fabricated by the microrubbing process [25, 26]. Besides microrubbing, we may employ some methods to form a spatially distributed TN orientation, e.g., photoalignment, microscale imprinting [27], and nanorubbing by atomic force microscopy [1]. Until now, good beam deflection properties of the LC polarization grating with the TN orientation have been demonstrated [1, 11, 12, 16]. In the previous grating, a microscale periodic alignment process was carried out only on one of the two substrate surfaces and another surface was uniformly processed in order to vary the twist angle [1, 11, 16]. On the other hand, we introduce a different concept, in which the microrubbing patterns were formed on both substrate surfaces, as shown in Figs. 2(a) and 2(b). In the previous grating, the highly twisted orientation area ( ± π twist) is somewhat unstable owing to the high concentration of elastic free energy and it tends to transition to other stable orientation states. However, this combination of the microrubbing pattern is better for achieving the highly twisted orientation state ( ± π twist); that is, we were able to repeatedly produce the periodic distribution of the ± π twisted orientations.
Fig. 2 (a) Microrubbing pattern and (b) surface LC molecular orientation of the LC polarization grating, where the grating period is Λ = 80 µm.
In the fabrication of the LC grating, we adopted the microrubbing process owing to its thermal stability, minimal aging performance, and photostability. First, polyimide films (SE2170, Nissan Chemical Industries) were rubbed using a tiny diamond stylus with a radius of curvature of 25 µm. The pushing load was 1 mN. The pitch of the stylus scan was 3 µm, where the typical thickness of the rubbed line was 4 µm. The microrubbing pattern was approximated by eight steps to save the processing time. Next, the two rubbed substrates were oppositely combined using glass rod spacers (20 µm diam.). Finally, nematic LC (5CB: 4-cyano-4'-pentylbiphenyl) was injected into the fabricated empty cell, followed by edge sealing using epoxy resin.
A complicated and random disclination pattern was observed just after injecting the nematic LC. However, the periodic disclination pattern appeared after applying a sufficiently large voltage of 10 V (1 kHz, square wave), as shown in Fig. 3(a) . The period of the thick disclination lines is equal to that of the microrubbing (Λ = 80 µm). The possible LC molecular orientation state and the distribution profile of the twist angle Φ are also shown in Fig. 3(b) and 3(c), respectively. The variation range of the twist angle (2π) is equivalent to the phase variation range of the transmitted light. Because the twist angle at x = Λ/2 is considerably small (almost homogeneous orientation), the transmittance under the crossed polarizers at this position varies depending on the retardation. The twist angles at x = Λ/4 and x = 3Λ/4 are almost π/2, and the surface LC director on the entrance substrate is parallel or perpendicular to the transmission axis of the polarizer; consequently, this area constantly appears as a bright area. The absolute value of the twist angle at x = 0 and Λ is very large, i.e., |Φ| = π and is discontinuous, causing a high concentration of the elastic free energy. The disclination should thicken to avoid the concentration of the elastic free energy.
Fig. 3 (a) Microscope image of the LC polarization grating under crossed polarizers (Λ = 80 µm). (b) LC molecular orientation model. (c) Spatial distribution profile of twist angle Φ(x), where profile is approximated by eight steps.
When the LC cell is heated above the nematic-isotropic transition temperature and then gradually cooled to room temperature, we can obtain a more stable LC orientation state with low energy in which the twist angle varies in space from −π/2 to π/2 [16]. Consequently, two thin disclination lines occur within a period. This low energy state (stable state) tends to transition to the higher energy state (metastable state) including the ± π twist with the application of 10 V; namely, the elastic free energies of the two states are the same at 10 V because the LC directors are almost homeotropically oriented. The transition between the two states must be accompanied by the movement of the disclination lines. Such disclination movement is slightly observed at the edge of the microrubbing area. It should be noted that a thick disclination line splits into two thin disclination lines as shown in Fig. 3(a). We believe that some sort of pinning effect of the disclination might be induced at the edge of the microrubbing area.
As described in the previous section, the thick LC layer (u >> 1) and the LC grating being equivalent to a λ plate (R = Nλ) is a sufficient condition for high diffraction efficiency. To discuss the validity of this design scheme, we investigated the temperature dependence of the diffraction efficiency, as shown in Fig. 4 . First, voltage dependence of the normalized retardation R/λ was evaluated as shown in Fig. 4(a), where the inset shows the temperature dependence of the normalized retardation. In the measurement process of the transmitted light intensity, the light beam (He-Ne laser) was incident on a homogeneous orientation area that is outside the microrubbing area. The retardation at zero voltage is R = 4.92λ, resulting in u = 4.92 for Φ = π; accordingly, the former condition u >> 1 is roughly satisfied, whereas the latter condition R = Nλ is almost satisfied.
Fig. 4 (a) Relationship between normalized retardation R/λ and applied voltage, where λ = 633 nm and T = 24°C. Inset shows temperature dependence of normalized retardation. Temperature dependence of diffraction efficiency when (b) right- and (c) left-handed circularly polarized light beams are incident.
Figure 4(b) and 4(c) shows the temperature dependences of the diffraction efficiency when right- and left-handed circularly polarized light beams (RCP and LCP) are incident on the LC grating, respectively. In the case of RCP incidence, the strong diffraction of the positive first order is generated particularly around 24°C, at which the retardation is almost R = 5λ (exactly R = 4.92λ). As the temperature increases, the positive first-order diffraction is extinguished, the negative first-order diffraction almost remains constant, and the zeroth order diffraction increases. Particularly, the LC cell hardly operates as an LC grating around 29°C because the retardation is roughly regarded as R = 4.5λ (specifically, R = 4.36λ). The eigenvalue under the conditions of u >> 1 and R = (N + 1/2)λ is aj = i(−1)j [derived from Eq. (2)]. Accordingly, Eout = i(−1)jEin, indicating that the phase of the transmitted light is independent of the twist angle Φ and is spatially uniform. Therefore, the LC cell never functions as a grating. When the LCP is incident on the LC grating, strong, negative first-order diffraction is obtained at 24°C [Fig. 4(c)], and the diffraction decreases as the temperature increases. These characteristics for the LCP incidence are complementary to those for the RCP incidence.
Although the insertion loss of the LC grating was compensated in the evaluation process of the diffraction efficiency, the maximum diffraction efficiency of the fabricated grating (~60%) was rather lower than the theoretical prediction of 100% [Fig. 4(b) and 4(c)]. The most significant factor reducing the efficiency is possibly the light scattering due to the thick disclination lines with a thickness of ~15 µm, which is not negligible compared with the grating period Λ = 80 µm. Given that the light passed through the disclination never contributes to the first-order diffraction efficiency, the maximum efficiency can be evaluated on the basis of the following Fourier transform:
where m is an integer corresponding to the diffraction order, Λd is the thickness of the disclination lines, and Eox and Eoy are the respective x and y polarization components of the output light that can be obtained from Eq. (12). In the case of j = 1, the maximum efficiency of the first order is then obtained as follows:which is identical to the expression incorporating the so called “fly back length” [28]. Therefore, by substituting Λ = 80 μm and Λd = 15 μm, the maximum diffraction efficiency is evaluated to be 66%.One of the other factors is that the distribution profile of the twist angle is approximated by eight level steps, as shown in Fig. 3(c). The theoretical diffraction efficiency of a blazed grating approximated by a stepwise phase profile is expressed as follows [29]:
where L is the level of the approximation (the number of steps). Because η = 0.95 for L = 8, the diffraction efficiency should decrease from 66% to 63%, which is similar to the measured results (~60%) shown in Figs. 4(b) and 4(c).The diffraction efficiency of the fabricated LC grating is controlled in response to the applied electric field, as shown in Fig. 5 . Figure 5(a) and 5(b) shows the voltage dependence of the diffraction efficiency for the RCP and LCP incidences, respectively. The operation region is divided into three regimes: regime I (V < 0.5), II (0.5 ≤ V < 1.5), and III (V ≥ 1.5). In regime I, retardation is almost constant, as shown in Fig. 4(a), resulting in a roughly constant diffraction efficiency because the applied voltage is lower than the threshold voltage of Freedericksz transition. When the applied voltage exceeds 0.5 V (regime II), the LC director tends to be perpendicular to the substrate and the twisted orientation vanishes, causing the deterioration of the diffraction properties of the TN-LC grating.
Fig. 5 Relationship between diffraction efficiency and applied voltage when incident light is (a) right- and (b) left-handed circularly polarized, where λ = 633 nm and T = 24°C.
From Fig. 5, it is revealed that considerably high second-order diffraction efficiencies are achieved in regime III; that is, η+2 = 0.69 for RCP and η−2 = 0.74 for LCP. To explain this curious diffraction phenomenon, we present LC molecular orientation models in regimes I and II, as shown in Fig. 6(a) and 6(b), respectively. In regime I, the LC molecules in the LC bulk layer are parallel to the substrate plane, whereas the azimuthal angle (twist angle) is spatially varied in the x-direction. The large jump of the twist angle is produced at x = Λ/2, resulting in the ± π reverse twist disclination. The retardation in regime I is almost R = 5λ (correctly, R = 4.92λ).
Fig. 6 LC molecular orientation models in regimes (a) I and (b) II. Rubbing directions on the upper and lower polyimide surfaces are the same (parallel). Light propagation behavior for RCP and LCP incidences is also illustrated.
On the other hand, in regime III the LC molecules in the bulk layer begin to incline as the applied voltage rises, and then, the molecules in the midplane become almost vertical to the substrate plane, as shown in Fig. 6(b). The direction of inclination of the upper- and lower-half layers are governed by the pretilt direction on the upper and lower polyimide surfaces because the LC directors in the midplane are reoriented almost vertically. Furthermore, the thick disclinations that deteriorate the diffraction properties desirably disappear because the twist deformation is no longer maintained in the bulk layer. In this regime, the LC cell is considered a stack of two hybrid orientation cells with an equal retardation of R/2. When the retardation equals R/2 = λ/2, the upper- and lower-half layers function as a hybrid orientation-type LC polarization grating [3]. The diffraction efficiencies take maximum values at 2.16 V [Fig. 5(a) and 5(b)], and the retardation becomes a full wavelength at the same voltage, as shown in Fig. 4(a), ensuring R/2 = λ/2. The maximum values of the diffraction efficiency are η−2 = 0.69 and 0.74 for RCP and LCP incidences, respectively. These values are ~20% greater than the maximum diffraction efficiency in regime I. It is likely that this increase in the diffraction efficiency is achieved by the elimination of the thick disclinations; the thickness (~15 µm) is ~19% of the grating period (Λ = 80 µm).
The two sublayers operate as independent LC polarization gratings, as mentioned above. Now, we consider the case when RCP light is incident on the LC grating. After passing through the first layer, the light beam is deflected at a certain angle α and the polarization state is converted into LCP [Fig. 6(b)]. Because the azimuthal angles of the LC directors in the second layer vary oppositely, the LCP light is further deflected at an angle of 2α; consequently, strong second-order diffraction is produced. Furthermore, because the polarization state turns into RCP again, the polarization state is also maintained in regime III.
In addition to the operations in regimes I and III, no diffraction mode is achieved by applying a sufficiently high voltage; hence, the LC orientation is almost homeotropic. In this situation, only the zeroth order diffraction occurs strongly. Therefore, this LC grating can deflect an input circularly polarized light toward three different angles (0, α, and 2α). This feature should be advantageous for some applications such as optical interconnects. Furthermore, it should be noted that unlike the conventional LC polarization grating, this LC grating never exhibits polarization conversion whenever the incident light is deflected toward the three directions.
As for the switching speeds, typical values for a TN-LC grating (d = 8 μm) have been measured to be τon = 1.7 ms and τoff = 14.5 ms in the grating area, and τon = 3.1 ms and τoff = 25.7 ms in the homogeneous orientation area (outside the grating area), where the applied voltage was switched between 0 and 10 V (square wave, 1 kHz). This result suggests that the micropatterned LC orientation reduces the switching time in the TN-LC grating. In fact, in regular LC polarization gratings with no twist deformation, the rise time is reduced owing to the decrease in the threshold voltage of the Freedericksz transition induced by the micropatterned LC orientation [30].
5. Conclusion
The LC polarization grating that conserves the polarization state of the incident light is achieved by periodically distributing the twist angle of the LC molecular orientation in the in-plane direction. We derived a design scheme for theoretically 100% diffraction efficiency; its conditions are i) the LC layer is sufficiently thick in order to satisfy Mauguin’s condition, ii) retardation R = Nλ, iii) the twist angle linearly varies over 2π, and iv) the incident light is circularly polarized.
The diffraction efficiency of the LC grating exhibits a complicated voltage dependence; that is, the LC cell functions as a single grating producing a certain diffraction angle α (regime I), and it operates as a stack of two hybrid-orientation-type LC polarization gratings with equal diffraction angles. Consequently, the total diffraction angle is 2α (regime III). The diffraction angle decreases to zero as a sufficiently high voltage is applied because the LC molecular orientation becomes homeotropic. The polarization state is conserved in these three situations. Therefore, we can switch the input beam direction among three different angles (0, α, and 2α). The nature of polarization conservation and the wide angular change of the diffracted light beam from 0 to 2α in this unique LC grating would be advantageous for more flexible designs of practical optical systems.
Acknowledgments
The authors thank Dr. Ryota Ito for his assistance in measuring the response time of the TN-LC polarization grating.
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