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Total Internal Reflections in Liquid Crystals Optics and Photonics Presented in Partial Fulfillment of the Second Midterm Clinton Braganza Liquid Crystal Institute, K.S.U. 4/4/2004 Reflection Coefficients SIGMA Polarization H E H q rq i 1 2 r Er qt n1 cosq i n2 cosq t r12 n1 cosq i n2 cosq t Reflection Coefficients Pi Polarization Ei Hi 1 2 Eqri qr Hq r t Ht n2 cosq i n1 cos qt r12 n2 cosq i n1 cos qt Et Total Internal Reflection SIGMA polarization Writing qt explicitly using Snell’s law r12 If n12 2 n1 cosq i n2 1 2 sin q i n2 n12 2 n1 cosq i n2 1 2 sin q i n2 n1 > n2, all the incident power is reflected if the incident angle is greater than arcsin (n2 / n1). TIR in Liquid Crystals: Glass to LC. Note that liquid crystals are birefringent, therefore reflections will depend of the orientation of the liquid crystal with respect to the direction of light propagation. Let us consider this liquid crystal: ne = 1.7 no = 1.5 nglass = 1.7 De > 0 TIR in LC’s: Orientation We will consider the following configurations: A homeotropic With sigma and pi polarized light incident on the cell surface A planar z y z y cell cell – With director parallel to y-axis and sigma and pi polarized light incident on the cell surface – With director parralel to x-axis And sigma and pi polarized light incident on the cell surface. TIR in LC’s Homeotropic Cells E This Ei, S - polarization encounters no, therefore when the incident angle is greater than arcsin (no/ng) = 61.9º , all the light is reflected. TIR in LC’s Homeotropic Cells Ei, P - polarization This encounters neff This ne no ne2 cos2 no2 sin 2 n increase from no to ne, which is the same as glass, therefore TIR does not take place. k TIR in LC’s Planar Cell : director parallel to y-axis z Ei, S - polarization y Here the electric field always encounters no, therefore if the incident angle is greater than the critical angle we have 100% reflectance. TIR in LC’s Planar Cell : director parallel to y-axis Ei, Pi - polarization As the incidence angle is increase, the refractive index decreases from ne to no as the electric field becomes parallel to the director, therefore TIR happens here. TIR in LC’s Planar Cell : Director parallel to x-axis Ei, S - polarization z y Here the electric field is always parallel to ne, therefore we do not have TIR TIR in LC’s Planar Cell : Director parallel to x-axis Ei, Pi - polarization Here the electric field always encounters no, therefore TIR occurs at incident angles greater than the critical angle. TIR in LC’s: LC to Glass Let us consider a different liquid crystal no =1.5 ne =1.8 Therefore the critical angle is arcsin(nglass/ne) = 70.81º TIR in LC’s : LC to Glass Homeotropic cell E Ei, S - polarization In the liquid crystal the light encounters no, which is less than nglass, therefore no TIR occurs here. TIR in LC’s : LC to Glass Homeotropic cell Ei, p - polarization In the liquid crystal the light encounters neff, which increases from no to ne. Therefore TIR occurs. TIR in LC’s: LC to Glass Planar Cell: director parallel to y-axis Ei, S - polarization z y In the liquid crystal the light encounters no, therefore no TIR occurs TIR in LC’s: LC to Glass Planar Cell: director parallel to y-axis z Ei, p - polarization y In the liquid crystal the light encounters neff, which decreases to no as the incident angle increase, therefore no TIR occurs TIR in LC’s: LC to Glass Planar Cell: director parallel to x-axis Ei, S - polarization z y In the liquid crystal the light encounters ne, therefore TIR occurs Ei, p - polarization In the liquid crystal the light encounters no, therefore no TIR occurs TIR in ChLC’s: Glass to LC Knowing the orientation of the liquid crystal at the boundary we treat the planar texture as the previous nematic cases. Planar Texture Focal Conic Texture I expect a periodic behavior here, for example, for s- polarization: If director is parallel to cell normal we have TIR Focal Conic Texture if director is parallel to polarization of light we will have no TIR. Some applications Switchable fiber optic cables –too expensive A more economical use would be for optical switches. – Shown below is a telecom optical switch designed by Baker, that can switch light to two different positions without changing the polarization. ITO ITO Conclusions Total internal reflection was solved by carefully analyzing the orientation of the liquid crystal director with respect to light propagation. It would be nice to get a general solution for TIR in LC’s, without first knowing the director orientation. For the case of cholesterics this problem would involve studying the effect of the evanescent wave from one chiral layer to another. References Yang, D-K, J. Opt. A: Pure and Appl. Opt, 5(2003) 402-408 Baker, A. P., 1998 Liquid Crystal Optical Switch Having Reduced Crosstalk, USA Patent # 4,720,171 Xianyu, H., et al, Optics Letters, 28 10 (2003) Boiko, Y., et al, Optics Letters, 27 19 (2002)