二極體在電子的領域中，扮演著使電流從一方向通過，而從另一方向則被阻擋的一種重要元件；而在光學中，我們可以將能使光從一方向通過而另一方向被阻擋的元件稱為光學二極體。本文中，使用兩個不同螺距的膽固醇液晶夾入向列型液晶材料做為一種非等向性複合結構(optical hetero-junction anisotropic structure, OHAS)，使其達到光學二極體的效果。
膽固醇型液晶獨特的圓偏振光二色性，會使得在光子帶隙範圍中，和膽固醇行液晶旋性相同的圓偏振光被反射，而旋性不同的圓偏振光通過。在本文中，使用有限元素法(finite element method, FEM)來分析膽固醇液晶的頻譜圖形，有限元素法的好處在於，所計算出來的特徵矩陣為稀疏矩陣，所以在計算的時間上，能夠大幅度的縮短。
當光通過非等向性複合結構時，從結構前方和後方入射的光，在頻譜圖上帶隙的形狀和位置是不同的；當綠光波長範圍的光波從前方入射結構時會被反射、而從後方入射結構時則會通過，紅光波長範圍的光波則反之；對於中央的非等向性材料缺陷層，折射率的改變，也會影響頻譜的圖形形狀。而通過計算的結果，我們可以利用這樣的特性，來設計作為一個彩色濾光片的設計。

The diode, an essential element in most electronic circuits, permits a current in one direction while blocking it in the opposite one. An optical diode transmits light in one direction while blocking it in reverse. We present a new optical hetero-junction anisotropic structure (OHAS) consisting of an anisotropic layer sandwiched between two cholesteric (CLC) layers with different periodicity of helix.

Cholesteric liquid crystals show the unusual optical property of selective reflection of circularly polarized light, i.e., light with the same handedness as the CLC helix cannot propagate with the frequencies in the photonic band gap (PBG). In this study, we investigate the Bragg reflection properties of CLCs using the finite element method (FEM), a well-recognized numerical method. It has the advantages of easy convergence in the space domain. Moreover, the parameter matrices of the differential equation are sparse in the FEM. By take advantage of sparse matrix, the calculation time can be reduced significantly.

The transmission spectra of forward and backward light propagation in OHAS are different. The green light will be reflected for forward propagation, whereas it will be transmitted for backward propagation, and the red light is opposite. The refractive indexes of anisotropic layer also change the transmission spectra. Therefore, we are capable of separating visible lights and accordingly design new color filters.

[1] E. Yablonovitch, “Inhibited Spontaneous Emission in Solid-State Physics and Electronics,” Phys. Rev. Lett. 58(20), 2059(1987)
[2] S. John, “Strong Localization of Photonic in Certain Disordered Dielectric Super lattices,” Phys. Rev. Lett. 58(23), 2486(1987)
[3] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University, Princeton (1995)
[4] D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “All-Dielectric One-Dimensional Periodic Structures for Total Omnidirectional Reflection and Partial Spontaneous Emission Control,” J. Lightwave Technol. 17(11), 2018 (1999)
[5] V. F. Rodríguez-Esquerre, M. Koshiba, and H. E. Hernández-Figueroa, “Finite-Element Time-Domain Analysis of 2-D Photonic Crystal Resonant Cavities,” IEEE Photon. Tech. Lett. 16(3), 816 (2004)
[6] Ch. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, “Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals,” Appl. Phys. Lett. 82(17), 2767 (2003)
[7] M. Bayindir, B. Temelkuran, and E. Ozbay, “Photonic-crystal-based beam splitters,” Appl. Phys. Lett. 77(24), 3902 (2000)
[8] S. Noda M. Imada, M. Okano, S. Ogawa, M. Mochizuki, and A. Chutinan, “Semiconductor Three-Dimensional and Two-Dimensional Photonic Crystals and Devices,” IEEE J. Quantum Electron. 38(7), 726 (2002)
[9] I.-C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena, John Wiley & Sons, New York (1995)
[10] P. Tran, “All-optical switching with a nonlinear chiral photonic bandgap structure,” J. Opt. Soc. Am. B 16(16), 70 (1999)
[11] L. Gilles and P. Tran, “Optical switching in nonlinear chiral distributed Bragg reflectors with defect layers,” J. Opt. Soc. Am. B 19(4), 630 (2002)
[12] M. F. Moreira, I. C. S. Carvalho, W. Cao, C. Bailey, B. Taheri, and P. Palffy-Muhoray, “Cholesteric liquid-crystal laser as an optic fiber-based temperature sensor,” Appl. Phys. Lett. 85(14), 2691 (2004)
[13] T. Woliński, “Calorific Effect of the Magnetically Induced Phase Transition in Induced Chiral Nematic Systems,” Mol. Cryst. Liq. Cryst. 162B, 171 (1988)
[14] J. Adams, W. Hass, and J. Dailey, “Cholesteric Films as Optical Filters,” J. Appl. Phys. 42(10), 4096 (1971)
[15] P. Yeh and C. Gu, Optics of Liquid Crystal Displays, Wiley, New York (1999)
[16] S. Kurihara, T. Kanda, T. Nagase, and T. Nonaka, “Photochemical color switching behavior of induced cholesteric liquid crystals for polarizer free liquid crystalline devices,” Appl. Phys. Lett. 73(15), 2081 (1998)
[17] Y. Semenova, Y. Panarin, G. Farrell, and S. Dovgalets, “Liquid Crystal Based Optical Switches,” Mol. Cryst. Liq. Cryst. 413, 385 (2004)
[18] J. Hwang, M. H. Song, B. Park, S. Nishimura, T. Toyooka, J. W. Wu, Y. Takanishi, K. Ishikawa, and H. Takezoe, “Electro-tunable optical diode based on photonic bandgap liquid-crystal heterojunctions,” Nature Mater. 4, 383 (2005)
[19] Y.-C. Yang, C.-S. Kee, J.-E. Kim, H.Y. Park, J.-C. Lee, and Y.-J. Jeon, “Photonic defect modes of cholesteric liquid crystals,” Phys. Rev. E 60(6), 6852 (1999).
[20] I. R. Matías, I. D. Villar, F. J. Arregui, and R. O. Claus, “Development of an optical refractometer by analysis of one-dimensional photonic bandgap structures with defects,” Opt. Lett. 28(13), 1099 (2003)
[21] V. I. Kopp and A. Z. Genack, “Twist Defect in Chiral Photonic Structures,” Phys. Rev. Lett. 89(3), 033901 (2002)
[22] J. Schmidtke and W. Stille, “Photonic defect modes in cholesteric liquid crystal films,” Eur. Phys. J. E 12, 553 (2003)
[23] I. J. Hodgkinson, Q. H. Wu, K. E. Thorn, A. Lakhtakia, and M. W. McCall,” Spacerless circular-polarization spectral-hole filters using chiral sculptured thin films: theory and experiment” opt. Commun. 184 (2002) 57
[24] M. H. Song, K.-C. Shin, B. Park, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. M. Swager, and H. Takezoe, “Polarization characteristics of phase retardation defect mode lasing in polymeric cholesteric liquid crystals,” Sci. Tech. Adv. Mater. 5, 437 (2004)
[25] M. H. Song, B. Park, K.-C. Shin, T. Ohta, Y. Tsunoda, H. Hoshi, Y. Takanishi, K. Ishikawa, J. Watanabe, S. Nishimura, T. Toyooka, Z. Zhu, T. M. Swager, and H. Takezoe, “Effect of Phase Retardation on Defect-Mode Lasing in Polymeric Cholesteric Liquid Crystals,” Adv. Mater. 16(9-10), 779 (2004)
[26] D. W. Berreman and T. J. Scheffer, “Bragg Reflection of Light from Single-Domain Cholesteric Liquid-Crystal Films,” Phys. Rev. Lett. 25(9), 577 (1970)
[27] D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4 × 4 Matrix Formulation,” J. Opt. Soc. Am. 62(4), 502 (1972)
[28] D. W. Berreman, “Optics in smoothly varying anisotropic planar structures : Application to liquid-crystal twist cell,” J. Opt. Soc. Am. 63(11), 1374 (1973)
[29] H. Wöhler, G. Haas, M. Fritsch, and D. A. Mlynski, “Faster 4×4 matrix method for uniaxial inhomogeneous media,” J. Opt. Soc. Am. A 5(9), 1554 (1988)
[30] W. D. St. John, W. J. Fritz, Z. J. Lu, and D.-K. Yang, “Bragg reflection from cholesteric liquid crystals,” Phys. Rev. E 51(2), 1191 (1995)
[31] Q.Hong, T.X. Wu, and S.-T. Wu, “Optical wave propagation in a cholesteric liquid crystal using the finite element method,” Liq. Cryst. 30(3), 367 (2003)
[32] A. Suryanto, E. Van Groesen, and M. Hammer, “Finite Element Analysis of Optical Bistability in One-Dimensional Nonlinear Photonic Band Gap Structures with a Defect,” J. Nonlinear Opt. Phys. & Materials 12(2), 187 (2003)
[33] A. Suryanto, E. Van Groesen, M. Hammer, and H. J. W. M. Hoekstra, “A finite element scheme to study the nonlinear optical response of a finite grating without and with defect,” Opt. Quant. Electron. 35, 313 (2003)
[34] V. Ilyina, S. J. Cox, and T. J. Sluckin, “FDTD Method for Light Interaction with Liquid Crystals,” Mol. Cryst. Liq. Cryst. 422, 271 (2004)
[35] J. Jin, The Finite Element Method in Electromagnetics, Wiley, New York (2002)
[36] J. P. Berenger, “A Perfectly Matched Layer for the Absorption of Electromagnetic Waves,” J. Comput. Phys. 114, 185 (1994)
[37] C. Oldano, “Electromagnetic-wave propagation in anisotropic stratified media,” Phys. Rev. A 40(10), 6014 (1989)
[38] S. Mujumdar and H. Ramachandran, “Use of a graded gain random amplifier as an optical diode,” Opt. Letters 26(12), 929 (2001)
[39] S.O. Konorov, D.A. Sidorov, I. Bugar, M.J. Bloemer V.I. Beloglazov, N.B. Skibina, D.Chorvat, M.Scalora and A.M. Zheltikov, “Experimental demonstration of a photonic-crystal-fiber optical diode” Appl. Phys. B 78, 547 (2004)