Switchable Polarization-Independent Liquid- Crystal Fabry-Perot Filter

Transcription

1 Kent State University From the SelectedWorks of Philip J. Bos December 18, 2008 Switchable Polarization-Independent Liquid- Crystal Fabry-Perot Filter Enkh-Amgalan Dorjgotov Achintya K. Bhowmik Philip J. Bos, Kent State University - Kent Campus Available at:

2 Switchable polarization-independent liquid-crystal Fabry Perot filter Enkh-Amgalan Dorjgotov, 1, * Achintya K. Bhowmik, 2 and Philip J. Bos 1 1 Chemical Physics Interdisciplinary Program, Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA 2 Intel Corporation, Santa Clara, California 95052, USA *Corresponding author: edorjgot@lci.kent.edu Received 23 June 2008; revised 12 November 2008; accepted 14 November 2008; posted 19 November 2008 (Doc. ID 97768); published 18 December 2008 A new approach to a polarization-independent twisted liquid-crystal (LC) structure, where phase difference between orthogonal eigenmodes is tuned to be an integer multiple of 2π, is demonstrated with a numerical model. For select wavelengths, polarization-independent operation can be achieved by tuning the twist rate and thickness of the LC cavity. Applications can be found in polarization- independent switches and field sequential wavelength selection devices Optical Society of America OCIS codes: , /09/ $15.00/ Optical Society of America 1. Introduction Nematic liquid crystal (LC) is characterized by an orientational ordering that is defined by unit vector (the director) along the average direction of elongated molecules long axis [1]. Orientational ordering of LC results in macroscopic optical and electrical anisotropy, which is exploited in a variety of applications of electro-optic devices. Although LC devices have a compact, nonmechanical structure and low power consumption, polarization sensitivity is inherent in an optically anisotropic LC medium in which light polarized along and perpendicular to the director experiences different indices of refraction. Previously in the field of wavelength selection devices, a tunable LC Fabry Perot interferometer (FPI) was proposed as an alternative to mechanical devices [2]. A typical LC FPI consists of two parallel reflecting mirrors sandwiching the optical cavity filled with tunable LC material. If the Fabry Perot (FP) mirrors are identical, then the transmission can be calculated as T ¼ τ 2 =½ð1 ρþ 2 þ 4ρsin 2 ðδþš, where τ and ρ are transmittance and reflectance of the mirrors and δ ¼ 2πnd=λ þ φ, with n, d, and ϕ being the cavity index, cavity thickness, and phase change on reflection, respectively [3]. In this case, the resonance condition for transmission maximum is met when δ ¼ mπ, where m ¼1; 2; 3, with m being the order of the transmission maximum. When the FP cavity is filled with homogeneous nontwist LC, the eigenmodes of light propagation are planar-polarized waves parallel (extraordinary mode) and perpendicular (ordinary mode) to the director, and the FP transmittance is the sum of transmittances of these modes. In general, transmission maximums of ordinary and extraordinary modes do not overlap, and only the latter can be tuned with application of external field across the FP cavity. Such polarization-dependent tunability is not desirable in wavelength selection applications because of losses and polarization fluctuations in real devices. Although previous methods to reduce polarization sensitivity of LC FPI significantly differed from one another, the underlying principle was to reduce the phase difference between orthogonal eigenmodes traversing through the LC medium. For example, Patel and Lee [4] proposed to use a 90 twist LC FPI in the high field regime, where halfway through the LC medium, distinct phase retardations experienced by two eigenmodes are switched, thus canceling the effective phase difference between these modes. Similarly, Morita and Johnson [5] suggested having two crossed quarter-wave retarders sandwiching the 74 APPLIED OPTICS / Vol. 48, No. 1 / 1 January 2009

3 homogeneous LC layer (whose optic axis is 45 to theirs) within the FP cavity. In this configuration, linear polarization of the eigenmodes gets rotated 90 passing through the quarter-wave retarder upon reflection from the mirrors, resulting in equivalent path length for two eigenmodes as they pass through the cavity twice. In these methods where the phase difference between the eigenmodes is minimized, resonance peaks of the eigenmodes overlap and are simultaneously tuned as the LC is switched with external field. It is also possible to have polarizationinsensitive operation, where resonance peaks of the eigenmodes do not overlap and only one of them is continuously tuned with an external field. This is what Lee et al. [6] achieved with hybrid anchored LC FPI, where the LC molecules are homeotropic on one surface and axially homogeneous on the other. All these methods have advantages such as continuous tunability and high resolution that make them good candidates for wavelength division multiplexing applications. Another approach to polarization-insensitive operation is to have a very high twist rate, where mode mixing reduces the difference between effective indices of the eigenmodes. In effect, when the twist rate increases such that the wavelength of light is much greater than the pitch of the helical structure, LC becomes optically isotropic. Although it is possible to switch the high twist rate LC FPI by unwinding the helix with an external field, the switching threshold field is large, and the resulting director deformation is not uniform. 2. New Device Concept We report a new approach in achieving polarization insensitivity for twisted LC FPI. The polarization-insensitivity requirement in our approach is not to reduce the phase difference between the eigenmodes but to simultaneously satisfy the resonance conditions of each eigenmode independently. At intermediate twist angles of LC (larger than waveguide regime [7] but smaller than highly twisted state that is optically isotropic), the eigenmodes are generally elliptically polarized waves with ellipticity defined by parameters such as thickness, pitch, and indices of the LC. In the rotating frame, polarization states of the eigenmodes do not change, because elliptically polarized eigenmodes follow the helical structure of the director such that effective indices of individual eigenmodes stay constant. It was shown earlier that the phase condition for a resonance peak in FPI is an integer multiple of the half-wave. As in a priori publications on the subject, this condition is satisfied when phases of the eigenmodes are equal and are an integer multiple of halfwave mπ, nπ, where m ¼ n, and m and n are integers. But this condition is also satisfied even when phases of the eigenmodes are not equal as long as the individual phases of the eigenmodes are different integer multiples of half-wave mπ, nπ, where m n. Since phase retardation of the eigenmodes increase with Fig. 1. Diagram of acquired phases of eigenmodes upon reflection from the mirrors. increasing twist angle, it is possible to find a twist angle where individual phases of the eigenmodes fulfill an integer multiple of the half-wave requirement. If such a twist angle, where each eigenmode reaches an integer multiple of the half-wave (mπ and nπ) simultaneously, is found, then polarizationindependent nonsplit resonance peak is expected. However, we must also consider the effect of polarization change upon reflection from the FP mirrors. When an elliptically polarized eigenmode, E 1, travels through the FP cavity, it picks up mπ phase, and upon reflection from one of the mirrors, it will be decomposed into two eigenmodes, E 1 and E 2. This is due to the fact that, unlike linearly polarized light, when an elliptically polarized eigenmode reflects off the mirror, its polarization state changes such that it is no longer a pure eigenmode but becomes a combination of pure eigenmodes. These two eigenmodes will pick up additional mπ and nπ phases as they travel back through the cavity, making the total phases 2 mπ and ðm þ nþπ for E 1 and E 2, respectively. These waves will be further decomposed into separate eigenmodes as they reflect from the other mirror and will pick up additional phases. The same is true for the second eigenmode as it too will be decomposed into separate eigenmodes upon reflection from the FP mirrors. Figure 1 illustrates how the eigenmodes are decomposed upon reflection from the mirrors and the total phase shift acquired as they travel through the cavity. From Fig. 1 it can be seen that, unless n ¼ m þ 2lðl ¼ 1; 2; 3 Þ is true, reflected eigenmodes will be out of phase with each other, which destroys the resonance condition. This means that the relative phase difference between the eigenmodes in the local frame must be an integer multiple of 2π in order to satisfy the resonance condition. In summary, the requirements for polarizationinsensitive operation of intermediate twist rate LC FPI are based on the resonance conditions of transmittance maximum and the interference conditions of reflected eigenmodes. The first condition is met when the phases of the eigenmodes are different integer multiples of the half-wave, while the second condition is met when the relative phase difference between the eigenmodes reaches the integer multiple of the full wave. For a given cavity thickness and LC material parameters, the twist angle that satisfies these requirements must be found. 3. Numerical Simulation and Results As mentioned in Section 2, phases of the eigenmodes as they travel through the LC medium must be tuned 1 January 2009 / Vol. 48, No. 1 / APPLIED OPTICS 75

4 to specific conditions. In this section we introduce the numerical procedures to find the twist angle that satisfies polarization-independent operation requirements. First, polarizations of the eigenmodes for a given twist LC medium are found using the Poincaré sphere method [8], and then Jones calculus [9] for light propagation through birefringent media is introduced for phase calculations. Once methods are introduced, we numerically simulate hypothetical LC FPI that can be used as a color switch in the visible spectrum. Poincaré sphere representation of polarization states of light as it traverses through birefringent optically active media is visually illustrative in probing the evolution of the eigenmodes in twist LC structure. It can be found in Bigelow and Kashnow s paper that the ellipticity of the eigenmodes is calculated from the Poincaré sphere representation as ω ¼ arctan½2d=ðpuþš=2, where d and p are thickness and pitch of the LC, and u ¼ Δnd=λ [10]. Here the ellipticity, ω, is related to the long and short axis of the polarization ellipse as tanðωþ ¼b=a. When calculating state of polarization and the intensity of the light as it travels through optically active birefringent media, it is convenient to represent electric vectors in terms of Jones vectors and optically active mediums as 2 2 matrices called Jones matrices. If the long and short axes of the polarization ellipse are known, then the Jones vector representations of orthogonal electric fields are E ~ 1 ¼ða; ibþ and E ~ 2 ¼ ð ib; aþ [11]. p We can let j E ~ j 2 ¼ a 2 þ b 2 ¼ 1, in which case a ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=ðtan 2 ω þ 1Þ and b ¼ a tanðωþ. Twisted LC structure can be thought of as an infinitesimally small N number of birefringent slabs, each placed in a ϕ=n angle with respect to its neighboring slab. In this case, Jones matrix representation of an individual birefringent slab is given by M ¼ expðiγe Þ 0 : ð1þ 0 expðiγ o Þ Here, Γ e;o ¼ 2πn e;o d=ðλnþ. The rotated Jones matrix in the laboratory frame becomes M 0 ¼ MRðφ=NÞ, where Rðϕ=NÞ is a rotation matrix given by cosðϕ=nþ RðϕÞ ¼ sinðϕ=nþ sinðϕ=nþ : ð2þ cosðϕ=nþ Output electric vectors in the rotating frame are calculated using ð ~ E 0 Þ 1;2 out ¼ðM0 Þ N~ E 1;2 in : ð3þ Electric vectors can be rewritten in the laboratory frame by multiplying with a rotation matrix: ð ~ EÞ 1;2 out ¼ Rð φþð~ E 0 Þ 1;2 out : ð4þ Note that, in general, electric vectors are of the form ~E ¼ðE x expðiθþ; E y Þ, and the phase difference between input and output vectors of an eigenmode is calculated using Δθ 1;2 ¼ θ 1;2 out θ1;2 in : ð5þ Similarly, relative phase difference between the eigenmodes is calculated using Δθ ¼ θ 1 out θ2 out : ð6þ Using the numerical methods introduced, we model hypothetical LC FPI, where the LC layer thickness is 0:45 μm, and ordinary and extraordinary indices are 1.5 and 1.9, respectively. Dielectric mirrors were modeled as a stack of six alternating layers of TiO 2 ðn ¼ 2:27Þ and MgFlðn ¼ 1:38Þ. The optical thickness of each layer is tuned to be near λ=4 thick at 550 nm to get average reflectance of around 80%. The transmittance spectrum of the LC FPI for coherent normally incident light source was calculated using the Berreman 4 4 matrix formulation [12,13], which is a direct solve of Maxwell s equations in one dimension. In this method, individual optical elements are represented by a 4 4 matrix that depends on the dielectric and other optical parameters of the material. In our model, the LC FPI consists of 12 layers of isotropic dielectric material and 1000 layers of anisotropic birefringent slabs, each with thickness 0:45 nm, representing the LC medium. The transmittance was normalized by the total unpolarized incident light, and we ignored the effects of glass substrates, conducting and alignment layers, because they can be minimized and/or compensated in real devices. First, homogeneous LC FPI, where the eigenmodes are linearly polarized lights parallel and perpendicular to the director, is simulated. We modeled two separate LC FPI, one with a polarizer along the director axis and the other perpendicular to the director axis, and transmitted intensities are shown in Fig. 2. A highly twisted LC FPI light sees an approximately isotropic medium with index equal to the average of ordinary and extraordinary indices. In this case, there is a single polarization-independent transmission peak at a wavelength where the resonance condition is met (Fig. 3). However, the LC twist rate must be very high in order for the effective index to equal to the average of ordinary and extraordinary indices. Instead, the resonance condition for transmission maximum at that wavelength can be found for each eigenmode at an intermediate twist angle. Once eigenmodes were determined using the Poincaré sphere, the twist LC structure is discretized (N ¼ 1000) and output electric vectors in the rotating frame are calculated using Eq. (3). The relative phase difference between the eigenmodes in the rotating frame is calculated using Eq. (6) and plotted as a function of the twist angle in Fig. 4. The acquired phase of each eigenmode as light travels through the LC cavity is calculated at a twist angle where phase difference between the eigenmodes reaches 76 APPLIED OPTICS / Vol. 48, No. 1 / 1 January 2009

5 Fig. 2. Transmittance of a nontwist LC FPI (optical structure: ðn L n H Þ 3 n LC ðn H n L Þ 3, where n L ¼ 1:38, n H ¼ 2:27, n LC ¼ 1:5=1:9, d L 95 nm, d H 57 nm, d LC ¼ 0:45 μm, and twist angle ¼ 0). 2π. In the rotating frame, output electric vectors from discrete birefringent layers can be calculated as ð E ~ 0 iþ 1;2 out ¼ðM0 Þ i ð EÞ ~ 1;2 in, where i ¼ 1; 2; 3 N, and the phase at thickness x ¼ðd=NÞi is calculated using Eq. (5). Phases of the eigenmodes in the local frame and transmitted intensity for the 0:45 μm cavity 169 twist LC FPI are shown in Fig. 5. In order to confirm our analysis, we modeled a LC FPI where phases of the eigenmodes did not satisfy the resonance conditions for polarization insensitive transmission peak. For example, the left plot in Fig. 5 shows that, when the thickness is at around 0:23 μm (twist angle 0:23 169=0:45), phases of the eigenmodes reach 1π and 2π in the rotating frame. Although this satisfies the resonance condition for transmittance maximum for each eigenmode, the interference condition is not met, and as a result, peak splitting occurs (see Fig. 6). Fig. 3. Transmittance of a highly twisted LC FPI (d LC ¼ 0:45 μm, n LC ¼ 1:5=1:9, and twist angle ¼ 3600 ). Fig. 4. Plot of relative phase difference between the eigenmodes as a function of twist angle for a 525 nm wave (d LC ¼ 0:45 μm and n LC ¼ 1:5=1:9). Note that the phase difference reaches 2π at a 169 twist angle. Alternatively from Fig. 4, the twist angle where relative phase difference between the eigenmodes reaches 4π in the rotating frame can be found. At this twist angle, the phases of the eigenmodes in the local frame and the transmitted intensity are calculated and plotted in Fig. 7. We numerically calculated the LC director configuration as a function of applied voltage by minimizing the free energy, which consists of Frank Oseen elastic energy and electric energy. Material parameters of the modeled LC are n e ¼ 1:9, n o ¼ 1:5, ε == ¼ 21, ε ¼ 6, γ 0:083PaS, K p N, K 22 7 p N, and K p N. Here K 11 to K 33 are elastic constants, and γ is the rotational viscosity of the LC material. We assumed infinite anchoring energy such that the LC molecules on the substrate surfaces are fixed permanently with a 2 pretilt angle. A detailed numerical method for calculating the director configuration of the twisted LC structure can be found in Berreman s paper on LC twist cell dynamics [14]. Figure 8 shows what happens when the external field is applied across the FP cavity and the twist LC structure is unwound. As the twist is unwound, the transmittance peak splits and moves toward shorter the wavelength region before finally reaching the single peak at around 480 nm. 4. Discussion If the FP cavity, the thickness, and the LC parameters are known, then the polarization-insensitive twist angle can be determined from the plot of the relative phase difference between the eigenmodes in the rotating frame (Fig. 4). At this twist angle, the phases of the eigenmodes in the rotating frame reach 4π and 2π (left plot in Fig. 5), which satisfies the resonance conditions for the polarizationindependent transmittance peak. The corresponding effective indices in the rotating frame are calculated using ðn eff Þ 1;2 ¼ ϕ 1;2 λ=2πd, where φ 1;2 are 4π and 2π 1 January 2009 / Vol. 48, No. 1 / APPLIED OPTICS 77

6 Fig. 5. Phases of the eigenmodes in the local frame and the transmitted intensity of the twisted LC FPI (d LC ¼ 0:45 μm, n LC ¼ 1:5=1:9, λ phase ¼ 525 nm, and twist angle ¼ 169 ). and are found to be 2.28 and 1.11, respectively. Note that the effective indices can also be calculated analytically as ðn 1;2 Þ 2 ¼ ε ==; þ α 2 ð1 2ε=δÞ [15], where ε ==; are dielectric permittivities along and perpendicular to the LC director, α ¼ λ=p, ε ¼ðε þ ε Þ=2, and δ ¼ðε == ε Þ=2. The results of the analytical calculation agree well with that of the numerical solution up to the second decimal point. The analytical equation for effective indices can also be used to calculate the phase difference between the eigenmodes for the twist LC structure and get the same result as shown in Fig. 4. Note that even before effective indices of the eigenmodes reach the average index of the LC, the resonance condition is met, and the transmittance spectrum looks the same as that of isotropic FPI with average index of the LC (see the right plot in Figs. 5 and 3). Figure 6 shows transmittance plots of two cases where peak splitting in the resonance peaks indicates polarization sensitivity. In this FPI, the phases of the eigenmodes reach 1π and 2π in the rotating frame (from the left plot in Fig. 5). However, this condition does not satisfy the polarization-insensitive Fig. 6. Transmitted intensity of the twisted LC FPI (d LC ¼ 0:23 μm, n LC ¼ 1:5=1:9, and twist angle ¼ 86 ). resonance peak requirements, and as a result, large peak splitting occurs. Alternatively, a twist angle at which phases of the eigenmodes reach 1π and 5π (left plot in Fig. 7), can be found from Fig. 4, where the polarization-insensitive transmittance peak reappears (right plot in Fig. 7). Figure 8 shows tunability of the LC FPI as an external field is applied across the cavity. As the LC twist is unwound, resonance conditions break down, and the transmission peak splits. We note here that although peak splitting occurs, the transmission peak does not completely separate into two peaks, and instead, when the field strength is high enough so that the director aligns perpendicular to the cavity surface, peak splitting disappears. This would not be the case in a high-finesse etalon, where the transmission peaks are sharper and the separation of adjacent peaks is higher. In such a case, there will be complete peak splitting of the eigenmodes in the intermediate voltage range. The same explanation applies to the order of the peak where, in the case of Fig. 8, low-order peak, which has wider transmission bandwidth, has less splitting as compared to the higher-order peak. Possible applications of such a device are tunable optical filters and polarization-independent light modulators in display systems. For example, the switch shown in Fig. 8 can be used as modulator for a field sequential color system where three light sources, one for each of the R, G, and B wavelength regions, illuminate the modulator in a time sequential manner. Depending on the wavelength of the illuminated light, the modulator either transmits or blocks the light. In such wide transmission bandwidth applications, polarization peak splitting due to slight variations in the twist angle would not hurt the performance, because the distance between the split peaks is much shorter than the transmission bandwidth. Generally, LC etalons in optical filters are considered high-order narrow bandwidth transmittance filters whose transmittance is not very high, and as a result, they are not suited for application mentioned 78 APPLIED OPTICS / Vol. 48, No. 1 / 1 January 2009

7 Fig. 7. Phases of the eigenmodes in the local frame and the transmitted intensity of the twisted LC FPI (d LC ¼ 0:45 μm, n LC ¼ 1:5=1:9, λ phase ¼ 525 nm, and twist angle ¼ 340 ). earlier. For example, in Patel and Lee s 90 twist cell, significant tunability requires high-order etalon as a result of field-on operation. Similarly, additional retarders within the LC cavity as suggested by Morita and Johnson also increases the etalon order. In the cases where the resonance peaks of the eigenmodes do not overlap, as in the Lee et al. method, the tunable light throughput is the same as that of the polarization-dependent modulator, which is not acceptable for devices with a high light transmission requirement. 5. Conclusion In conclusion, we determined the requirements for polarization-independent LC FPI for intermediate twist LC structure. These are as follows: 1. Phases of each eigenmode in the local frame must be an integer multiple of π. 2. Relative phase difference between the eigenmodes in the local frame must be an integer multiple of 2π. Fig. 8. Transmitted intensity of the twisted LC FPI as a function of applied voltage (d LC ¼ 0:45 μm, n LC ¼ 1:5=1:9, and twist angle ¼ 169 ). The first requirement comes from the resonance condition of the transmission peak in the FPI, whereas the second requirement comes from the interference condition between the eigenmodes as they reflect from the FP mirrors. References 1. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, 1993). 2. J. S. Patel, M. A. Saifi, D. W. Berreman, C. Lin, N. Andreadakis, and S. D. Lee, Electrically tunable optical filter for infrared wavelength using liquid crystals in a Fabry Perot etalon, Appl. Phys. Lett. 57, (1990). 3. M. Born and E. Wolf, Principles of Optics (Pergamon, 1965). 4. J. S. Patel and S. D. Lee, Electrically tunable and polarization insensitive Fabry Perot etalon with a liquid-crystal film, Appl. Phys. Lett. 58, (1991). 5. Y. Morita and K. M. Johnson, Polarization-insensitive tunable liquid crystal Fabry Perot filter incorporating polymer liquid crystal waveplates, Proc. SPIE 3475, (1998). 6. J. H. Lee, H. R. Kim, and S. D. Lee, Polarization-insensitive wavelength selection in an axially symmetric liquid-crystal Fabry Perot filter, Appl. Phys. Lett. 75, (1999). 7. C. Mauguin, Sur la représentation gémoétrique de Poincaré relative aux propriétés optiques des piles de lames, Bull. Soc. Fr. Mineral. 34, 6 15 (1911). 8. J. H. Poincaré, Theorie Mathematique de la Lumiere (Saint-Andre-des-Arts, 1892), Vol. 2, p R. C. Jones, New calculus for the treatment of optical systems. I. Description and discussion of the calculus, J. Opt. Soc. Am. 31, (1941). 10. J. E. Bigelow and R. A. Kashnow, Poincaré sphere analysis of liquid crystal optics, Appl. Opt. 16, (1977). 11. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (Wiley, 1999). 12. D. W. Berreman, Optics in stratified and anisotropic media: 4 4-formulation, J. Opt. Soc. Am. 62, (1972). 13. D. W. Berreman, Optics in smoothly varying anisotropic planar structures: application to liquid-crystal twist cells, J. Opt. Soc. Am. 63, (1973). 14. D. W. Berreman, Liquid crystal twist cell dynamics and backflow, J. Appl. Phys. 46, (1975). 15. H. de Vries, Rotatory power and other optical properties of certain liquid crystals, Acta Crystallogr. 4, (1951). 1 January 2009 / Vol. 48, No. 1 / APPLIED OPTICS 79