一维光子晶体全向反光镜的设计
基于一维全反射原理的光子晶体球面反射镜
Photonic crystal spherical mirrors based onthe one-dimension omnidirectional reflection of photonic crystal Liang Wang, Shuping Li*, Chengxin Cai, Xugang Cui, Yongbin Niu and Peixue Wu School of Information Engineering, Tibet Institute for NationailitiesXianyang, Shaanxi, China*e-mail: gxglsp@Abstract—A photonic crystal spherical mirror based on the one-dimension omnidirectional reflection of photonic crystalis presented. The results show that when the diffractive index of two materials is 1.6 (polystyrene) and 4.6 (tellurium) respectively, and the corresponding optimized thickness is 0.75a and 0.25a. And a very high reflectance over wide frequency ranges is observed. In the omnidirectional reflection band, the more than 70 and 80 percent of the light are contained within the central core and circle bounded by the second dark ring of he diffraction pattern respectively.Keywords- Photonic crystal; Spherical mirror; Omnidirectional reflection bandI. I NTRODUCTIONThe spherical mirror is one of the most prevalent of optical devices which are used for imaging and solar energy collection and in laser cavities. The common spherical mirror is metallic mirror which has a broad rangeof frequencies incident from arbitrary angles, however, there are inacceptable absorption at infrared and optical frequencies. In other hand, energy is concentrated on the paper-thin of metallic surface, it makes the surface temperature increasing so that the surface becomes deformed, and the quality of the spherical mirror is affected strongly. Multilayer dielectric spherical mirrors can be extremely low absorption in narrow range of frequencies incident from a particular angle. It is too sensitive to the incident frequency and angle. For arbitrary angle incident light, one-dimension photonic crystal has the omnidirectional reflection band [1,2], because the dielectric materials has low absorption, the photonic crystal reflection mirror, which is made of the dielectric materials, has extremely small loss [3-5]. In this paper, a photonic crystal spherical mirror based on the one-dimension omnidirectional reflection of photonic crystal is presented, it has high reflection in the broad frequency band. And the focusing properties of the photonic crystal spherical mirror with different parameters are studied.II. S IMULATION AND RESULTSAccording to principle of the one-dimension omnidirectional reflection of photonic crystal, the multilayer dielectric spherical mirror is designed as following:The reported results showed that high ratio of refractive indices of two layers can lead to high photonic band gaps [6]. Fig.1 shows a simulation model of a photonic crystal spherical mirror, which is consisted of thedielectric layers of two materials, the multilayer dielectricFigure 1. Schematic of the multilayer dielectric spherical mirror. spherical mirror is consist of alternating first layer and second layers, where the first layer refractive index is 4.6 (tellurium), and the second layer refractive index is 1.6 (polystyrene).When materials indices are defined, the structure parameters are optimized, and the crystal spherical mirror with the optimum design has the widest relative bandwidth. In Fig.2, when the film thickness h1 (the refractive index 1.6) is 0.75a, the relative bandwidth is maximum value, where a is the lattice constant of the 1D photonic crystal. So the corresponding thickness of tellurium film (refractive index 4.6) is 0.25a. In the following discussions, the optimization parameters are used.When a light is incident on the periodical multilayer stack with alternative indices, the normalized omnidirectional reflection band for the case of normal incidence is 0.145-0.28[c/a] [6], where c is the speed of light in vacuum..Figure 2. Relative bandwidth for different thin film thickness2010 International Conference on Optics, Photonics and Energy Engineering978-1-4244-5236-1/10/$26.00 ©2010 IEEE OPEE 2010Figure 3. The reflection spectrum for the photonic crystal sphericalmirror with different curvature radii.However, for the photonic crystal spherical mirror with the optimization parameters, the normalized frequency band of the omnidirectional reflection is 0.16-0.285[c/a] which is shown in Fig.3. In the following, the transmission performance of spherical mirror in this frequency band is studied by numerical simulations using finite-difference time-domain method.It is well known that when a light beam is focused by lens, the position of axial intensity maximum of the diffracted wave deviates from the geometrical focus, but it is closed to the lens. It is also fit to the spherical mirror. A light is incident on the photonic crystal spherical mirror, the waist of the beam emerging from the spherical mirror dose not at the geometric focus, the location of the waist is called diffraction focus, the plane local at this point is called diffraction focal plane. The fraction of the total energy contained within central core of the diffraction pattern in the geometric focal plane and diffraction focal plane for the photonic crystal spherical mirror with different curvature radii are shown in Fig.4 and Fig.5 respectively.Figure 4. The fraction of the total energy contained within central core of the diffraction pattern in the geometric focal plane for the photoniccrystal spherical mirror with different curvature radii.Figure 5. The fraction of the total energy contained within central core of the diffraction pattern in the diffraction focal plane for the photoniccrystal spherical mirror with different curvature radii.For the different curvature radii, it is very clearly that the more than 70 percent of the light are contained within the central core in the diffraction focal plane, and the less than 70 percent of the light are contained within the central core in the geometric focal plane except for curvature radius of 5 micron. And the curves in the diffraction focal plane are almost flat in the omnidirectional reflection band. However, in this band, the curves looks like step curves in geometric focal plane when the curvature radius is larger than 25, and with the radius of curvature increasing, the step point move to the high frequency.Fig.6 and Fig.7 show that the more than 80 percent of the light are contained within circle bounded by the second dark ring of he diffraction pattern in the diffraction focal plane, and energy ratio is less than 80 percent of the light in the geometric focal plane except for curvature radius of 5 micron. And in the omnidirectional reflection band, the curves in the diffraction focal plane are almost flat, but the curves looks like step curves in geometric focal plane when the curvature radius is larger than 25, and the step point moves to the high normalization frequency with thecurvature radius increasing.Figure 6. The fraction of the total energy contained within circle bounded by the second dark ring of he diffraction pattern in the geometric focal plane for the photonic crystal spherical mirror withdifferent curvature radii.Figure 7. The fraction of the total energy contained within circlebounded by the second dark ring of he diffraction pattern in thediffraction focal plane for the photonic crystal spherical mirror withdifferent curvature radiiIII. C ONCLUSIONA photonic crystal spherical mirror is consist of alternating first layer and second layers, where the first layer refractive index is 4.6 (tellurium), and the second layer refractive index is 1.6 (polystyrene), the corresponding optimized film thickness is 0.25a and 0.75a respectively. And for these parameters, a broad normalization frequency band is observed, and the transmission performance is studied. Compared with the focusing performance curves for photonic crystal spherical mirrors with different parameters, in the omnidirectional reflection band, in the diffraction focal plane, the more than 70 and 80 percent of the light are contained within the central core and circle bounded by the second dark ring of he diffraction pattern respectively. It has high focusing capacity and a wide range of frequencies and at the same time is of low loss.R EFERENCES[1] J. N. Winn, Y.Fin, S.Fan and J. D. Joannopoulos, “Omnidirectionalreflection from a one-dimensional photonic crystal,” Opt. Letters, vol. 23, pp. 1573-1575, October 1998[2] Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulosand E. L. Thomas, “A Dielectric Ominidirectional Reflector,”Science, vol. 282, pp. 1679-1682, November 1998.[3] E. Yablonovitch., “Inhibited spontaneous emission in solid-statephysics and electronics,” Phys. Rev. Lett., vol. 58, pp. 2059-2062, May 1987.[4] S. John, “Strong localization of photons in certain disordereddieletric superlatices,” Phys. Rev. Lett., ,vol. 58, pp. 2486-2489, June 1987[5] W. H. Southwell, “Omnidirectional mirror design with quarter-wave dieletric stacks,” Appl. Opt. , vol. 38, pp. 5464-5467, September 1999.[6] B. Temelkuran, E. L. Thomas, J. D. J oannopoulos,and Y. Fink., “Low-loss infrared dielectric material system for broadband dual-range onmidirectional reflectivity,” Opt. Lett., vol. 26, pp. 1370-1372, September 2001。
基于一维超导体光子晶体的可调光波反射器的设计
基于一维超导体光子晶体的可调光波反射器的设计黄蓓;王晓华【摘要】文章用半导体Si和超导体YBa2 Cu3 O7 (YBCO)组成一维超导体光子晶体(AB)N,利用传输矩阵法对其反射特性进行了理论推导与数值模拟仿真研究.由于该一维超导体光子晶体组成材料中,仅有超导体YBCO的相对介电常数会受外部温度的调节,通过调整超导体YBCO的厚度、温度、电磁波的入射角和极化关系等参数,利用传输矩阵法计算其反射率,经过数值仿真发现其反射率对超导体YBCO的厚度、电磁波的入射角和极化关系等参数变化明显,而对外部温度的变化不明显.我们希望这些数值模拟仿真的结果能为设计可调的光波反射器提供理论研究基础.【期刊名称】《太原师范学院学报（自然科学版）》【年(卷),期】2018(017)003【总页数】5页(P63-67)【关键词】超导体;光子晶体;反射器;可调的【作者】黄蓓;王晓华【作者单位】盐城师范学院新能源与电子工程学院,江苏盐城224007;盐城盐兴机动车检测有限公司,江苏盐城224015;盐城师范学院新能源与电子工程学院,江苏盐城224007【正文语种】中文【中图分类】O431.10 前言1987年,Yablonovitich和John分别从“抑制自发辐射”和产生光波的“局域化”的角度出发[1,2],几乎同时独立地提出了“光子晶体”．光子晶体就是不同折射率的材料呈周期性排列的人工微结构,根据其空间尺寸,可以分为一维、二维和三维等光子晶体．由于电磁波在其内会产生布拉格散射干涉,从而对入射的电磁波就会形成光子禁带的特性,利用这个特性,对光子晶体的研究日益增多[3-5]．不过,很多光子晶体一旦制成,其光子禁带就不可调．并且,从制备角度出发,一维光子晶体要比二维或三维的光子晶体要简单．因此,对一维可调控的光子禁带的研究变得十分有意义．2004年,Hojo和Mase把等离子体和介质材料组成一维等离子体光子晶体,利用等离子体的折射率可以受其电子密度影响,通过对一维等离子体光子晶体的色散关系求解,从而得到其可调控的光子禁带[6]．2013年,章海锋把超导体引入不同结构的一维光子晶体中,形成一维超导体光子晶体,利用超导体的介电常数可以受温度调控,从而得到温度调控的光子禁带或高反射率[7,8]．利用超导体的低损耗和介电常数可调性,很多研究人员把超导体和不同折射率材料,如超材料[9]、半导体[10,11]或普通介质材料[12,13],组成各种一维超导体光子晶体,用传输矩阵法计算其反射率或透射率,实现了在不同频率范围内可调控的反射器或滤波器,为制备新型的光子晶体应用器件提供了理论研究基础．图1 一维超导体光子晶体(AB)N反射器的结构示意图我们课题组也先后把单负材料[14]、石墨烯[15]或半导体GaAs[16]分别组成不同的一维光子晶体,通过传输矩阵法计算其透射率,利用这些材料的介电常数或磁导率受外部参数的可调控性,从而得到可调控的多通道或单通道的滤波器．另外,我们用超导体YBa2Cu3O7(YBCO)和半导体Ge组成的隐身斗篷,利用超导体YBCO的介电常数受外部温度的调节,通过散射取消的隐身理论计算和电磁软件的数值仿真,实现了温度调节的宽频隐身[17]．本文,我们用超导体YBCO和半导体Si组成一维超导体光子晶体,通过调整超导体YBCO的厚度、电磁波的入射角和极化关系等外部参数,还利用超导体YBCO的介电常数受外部温度的调节,利用传输矩阵法计算其反射率,经过数值仿真从而实现可调控的光波反射器．1 理论模型图1是本文的一维超导体光子晶体(AB)N反射器的结构示意图,其中A是半导体Si,B是超导体YBCO,N是周期数．在可见光频率范围内,半导体Si的折射率是3.3．根据文献[17],基于双流体模型,超导体YBCO是高温超导体,并且其损耗可以忽略不计,则可以推导出超导体YBCO的相对介电常数εr随外部温度T和波长λ的变化关系公式如下:(1)根据文献[17]可知公式(1)的参数:居里温度Tc=93 K,伦敦穿透浓度在T=0 K时λL(0)=145 nm和高温超导体p=2．当电磁波以θ角斜入射到一维超导体光子晶体中,其任意层l的传输矩阵的表达式如下[18]:(2)这里,是波长,c是光速度．另外,对于不同模式的电磁平面波有:(3)那么,对于多层材料的传输矩阵的表达式:(4)式中多层的特征矩阵M=M1M2…MN-1MN．可以得到该一维超导体光子晶体的反射系数r与反射率R:(5)该一维超导体光子晶体放置在空气中,因此,2 结果与讨论在可见光波长范围(350-800 nm)内,一维超导体光子晶体(AB)N反射器结构中,半导体Si的介电常数并不随外部温度变华而变化．根据公式(1),可以绘出图2是无损超导体YBCO的相对介电常数被外部温度进行调节的关系图．由图2可以发现,在超导体YBCO的居里温度下,其相对介电常数的在同一波长下,随着外部温度增加而略微增加;且其相对介电常数在外部温度不变的情况下,随着波长增加而略微减小．说明在超导体YBCO的居里温度下,其相对介电常数是可以受外部温度进行略微调节的．图2 超导体YBCO的相对介电常数随外部温度与波长变化的关系图图3 在部温度T/Tc=0.5时,超导体YBCO厚度的变化对(AB)N结构的正入射电磁波反射率影响关系图本文是用传输矩阵法研究一维超导体光子晶体air/(AB)N/air在可见光波长范围内的反射特性．因此,先设定在外部温度T/Tc=0.5与电磁波正入射时,半导体Si厚度为60 nm,周期数N=15,超导体YBCO厚度的变化对该结构反射率的影响,如图3所示．由图3可见,该结构对正入射的电磁波能产生较宽的光子禁带,也就是近于1的全反射波宽:在dYBCO=50 nm时,λ=260 nm;在dYBCO=60 nm时,λ=318 nm;在dYBCO=70 nm时,λ=338 nm．同样,可以发现随超导体YBCO厚度增大,光子禁带是向长波的方向偏移,也就是发生红移．因此,该一维超导体光子晶体air/(AB)N/air可以用来制成宽频的光波反射器．根据图3,取超导体YBCO厚度为60 nm,其他条件与图3一样,下面讨论电磁波入射角的变化对一维超导体光子晶体air/(AB)N/air结构反射率的影响,计算结果见图4．由图4(a)可以发现,用该结构制成的反射器对TE模式的任何入射角电磁波能产生很好的较宽的光子禁带,也就是近于1的全反射波宽λ≥305 nm,并且随入射角增大,光子禁带是向低波的方向偏移,也就是发生蓝移;同时图4(b)可以发现,用该结构制成的反射器对TM模式电磁波能在低入射角时也能产生较宽的光子禁带,随入射角增大,与TE模式电磁波的变化趋势一样:光子禁带也是向低波的方向略微偏移,也就是发生略微蓝移．不过随入射角增大,光子禁带的波宽迅速减小．这就表明,用该结构制成的反射器对入射电磁波的极化较敏感．图4 在部温度T/Tc=0.5时,电磁波入射角对(AB)N结构的反射率影响关系图:(a)TE 波;(b)TM波根据上面的讨论,取电磁波入射角θ=45°,其它条件与图4一样,下面讨论外部温度的变化对一维超导体光子晶体air/(AB)N/air结构反射率的影响,计算结果见图5．在组成该结构的材料中,只有超导体YBCO的介电常数能被外部温度进行调节．由图5可以发现,该结构对45°入射角的两种模式电磁波都能产生较好的较宽的光子禁带,也就是近于1的全反射波宽:在TE波,λ=309 n m;在TM波,λ=157 nm．虽然,TE 波比TM波产生的光子禁带要宽,不过,两种模式产生的光子禁带对外部温度的变化很不明显．在图2中,分析认为超导体YBCO的相对介电常数虽能被外部温度调节,不过变化很小,也就是超导体YBCO的折射率随外部温度变化很小,因此造成该结构的反射率对外部温度的变化不明显,这个结论与文献[12]一致．图5 在电磁波入射角θ=45°时,外部温度对(AB)N结构的反射率影响关系图:(a)TE 波;(b)TM波3 结论本文用半导体Si和超导体YBCO组成一维超导体光子晶体,利用传输矩阵法对air/(AB)N/air(其中A是Si和B是YBCO)的反射特性进行了理论推导与数值模拟仿真研究．通过数值模拟仿真,发现该结构对正入射的电磁波能产生较宽的光子禁带,即近于1的全反射波宽随着超导体YBCO的厚度增加而增加,且发生红移．同样,数值模拟仿真结果表明电磁波的任何入射角对TE波都能产生大于305 nm的光波光子禁带,而TM波的光子禁带波宽随入射角增大会迅速减小．这就表明,用该结构制成的光波反射器对入射电磁波的极化较敏感．另外,该一维超导体光子晶体组成材料中,仅有超导体YBCO的相对介电常数会受外部温度的调节,不过由于其变化很小,也就是超导体YBCO的折射率随外部温度变化很小,因此仿真结果表明其反射率对外部温度的变化不明显．我们希望这些数值模拟仿真的结果能为设计可调的光波反射器提供理论研究基础．参考文献：【相关文献】[1] YABLONOVITCH E.Inhibited spontaneous emission in solid-state physics and electronics[J].Physical Review Letters,1987,58(20):2059-2062[2] JOHN S.Strong localization of photons in certain disordered dielectricsuperlattices[J].Physical Review Letters,1987,58(23):2486-2489[3] MAIGYTE L,STALIUNAS K.Spatial filtering with photonic crystals[J].Applied Physics Reviews,2015,2(1):1-17[4] DOLAN J A,WILTS B D,Vignolini S,et al.Optical properties of gyroid structured materials: from photonic crystals to metamaterials[J].Advanced Optical Materials,2015,3(1):12-32 [5] KURAMOCHI E.Manipulating and trapping light with photonic crystals from fundamental studies to practical applications[J].Journal of Materials ChemistryC,2016,4(47):11032-11049[6] HOJO H,MASE A.Dispersion relation of electromagnetic waves in one-dimensional plasma photonic crystals[J].Journal of Plasma and Fusion Research,2004,80(2):89-90 [7] ZHANG H F,LIU S B,YangH.Omnidirectional photonic band gap in one-dimensionalternary superconductor-dielectric photonic crystals based on a new thue-morse aperiodic structure[J].Journal of Superconductivity and Novel Magnetism,2013,27(3):41-52[8] ZHANG H F,LIU S B,Yang H,et al.Investigating the omnidirectional photonic band gap in one-dimensional superconductor-dielectric photonic crystals with a modified ternary fibonacci quasiperiodic structure[J].Journal of Superconductivity and Novel Magnetism,2013,26(12):3391-3400[9] SRIVASTAVA S K,AGHAJAMALI A.Analysis of reflectance properties in 1D photonic crystal containing metamaterial and high-temperature superconductor[J].Journal of Superconductivity and Novel Magnetism,2017,30(2):343-351[10] SRIVASTAVE S K,AGHAJAMALI A.Investigation of reflectance properties in 1D ternary annular photonic crystal containing semiconductor and high-Tc superconductor[J].Journal of Superconductivity and Novel Magnetism,2016,29(6):1423-1431[11] HERRERA A Y,CALERO J M,Porras-Montenegro N.Pressure, temperature, and thickness dependence of transmittance in a 1D superconductor-semiconductor photonic crystal[J].Journal of Applied Physics,2018,123(3):033101-1-033101-5[12] GONZLEZ L E,ORDOEZ J E, ZAMBRANO G,et al.YBa2Cu3O7-x/BaTiO3 1D superconducting photonic crystal with tunable broadband response in the visiblerange[J].Journal of Superconductivity and Novel Magnetism,2018,31(7):2003-2009 [13] ELSAYED H A,ALY A H.Terahertz frequency superconductor-nanocomposite photonic band gap[J].International Journal of Modern Physics B,2018,32(12):1850056-1-1850056-10 [14] LI P,LIU Y.Multichannel filtering properties of photonic crystals consisting of single-negative materials[J].Physics Letters A,2009,373(21):1870-1873[15] FENG Y,LIU Y,WANG X,et al.Tunable multichannel plasmonic filter based on a single graphene Sheet on a fibonacci quasiperiodic structure[J].Plasmonics,2018,13(3):653-659 [16] XUE Q,WANG X,LIU C,et al.Pressure-controlled terahertz filter based on 1D photonic crystal with a defective semiconductor[J].Plasma Science andTechnology,2018,20(035504):1-7[17] WANG X,Liu Y,FENG Y.Temperature controlled infrared broadband cloaking with the bilayer coatings of semiconductor and superconductor[J].Physica C:Superconductivity and Its Applications,2015,513(3):13-17[18] 王辉,李永平.用特征矩阵法计算光子晶体的带隙结构[J].物理学报,2001,50(11):2172-2178。
一维液晶光子晶体全向反射带
Tunable omni-directional mirror based on one-dimensional photonic structure using twisted nematic liquid crystal:the anchoring effectsCarlos G.Avendaño*and Daniel Martínez Universidad Autónoma de la Ciudad de México, Corona No.320, Palma,C.P.07160,G.A.Madero,MéxicoD.F.,Mexico*Corresponding author:caravelo2000@Received14March2014;revised25May2014;accepted15June2014;posted18June2014(Doc.ID208247);published16July2014We present the tunability of the omnidirectional band gap(OBG)in a one-dimensional photonic structure consisting of N nematic liquid-crystal slabs in a twisted configuration.It is alternated by N isotropic dielectric layers by a dc electric field aligned along the periodicity axis.We consider the general case for which arbitrary anchoring of the nematic at the boundaries is taken into account.The threshold elec-tric field for which the OBG is created decreases as the anchoring strength diminishes.The width of this band gets larger as the electric field increases and its center frequency is modified slightly.©2014 Optical Society of AmericaOCIS codes:(230.3720)Liquid-crystal devices;(230.5298)Photonic crystals./10.1364/AO.53.0046831.IntroductionOver recent years photonic crystals(PCs)have received increasing interest from the scientific com-munity for their applications in photonic devices such as tunable optical filters and optical switches. PCs are periodic structures of materials with differ-ent dielectric constants that can prohibit the propa-gation of electromagnetic waves within a certain frequency range.This is called photonic band gap (PBG)if the PC forbids the propagation for both polarizations of incident light,and it is named pho-tonic stop band(PSB)if the structure prohibits the transmission for only one polarization[1–3].This particular reflection property in one-dimensional (1D)PCs is usually called Bragg reflector or Bragg mirror.Tuning and switching of PCs can be achieved by taking advantage of optical nonlinearities caused by intensive laser illumination[4],incorporating floating magnetic particles into ordered structures [5],using a semiconductor[6,7],and using a liquid crystal(LC)[8–12].To the best of our knowledge,the first study re-ported on the tunability of the PBG in PCs with LC is attributed to Busch and John[8].In1999they demonstrated that when a nematic LC is infiltrated into the void regions of an inverse opal PC,the resulting composite exhibits a completely tunable three-dimensional PBG under the action of an external electric field,which rotates the axis of the nematic molecules relative to the inverse opal ter,Leonard et al.[11]infiltrated a two-dimensional macroporous Si PC with a nematic LC and showed that temperature tuning can be accom-plished by provoking a phase transition of the LC from the nematic to the isotropic phase.Recently, Molina et al.[13]considered a1D structure,consist-ing of nematic LC slabs in a twisted configuration alternated by transparent isotropic dielectric films.1559-128X/14/214683-08$15.00/0©2014Optical Society of America20July2014/Vol.53,No.21/APPLIED OPTICS4683They found a strong dependence of electric field on the PBG for incident waves of left-and right-circular polarization at arbitrary incidence angles.A complete PBG requires that there be no states in the given frequency range for propagation in any di-rection in the structure and for both polarizations, which implies the total reflectivity for all incident an-gles.This phenomenon is known as omnidirectional band gap(OBG).If,however,the light propagation is prohibited for only one polarization,then the term omnidirectional stop band(OSB)is used.In1D PCs,the range of OBG is determined by the contrast of refractive indices between the composites[14–16]. Abdulhalim[17]showed theoretically that a struc-ture consisting of alternating anisotropic layers where the optic axis direction of the layers rocks back and forth around the normal to the layers can act as an omnidirectional reflector.Subsequently,it was ex-hibited that this structure can be used to build a re-flective Fabry–Perot resonator showing reflection peaks with complete polarization conversion[18]. Ha et al.[19]theoretically demonstrated the tunabil-ity of the omnidirectional reflection band of a1D PC consisting of alternating isotropic dielectric and nem-atic LC layers by an electric field.They showed that the width of this band becomes wider as the external electric field increases,but the center frequency is changed little.Surface anchoring plays an essential role in the operation of LC devices.In the last decades many surface treatment techniques have been imple-mented to build appropriate anchoring conditions such as rubbing,deposition of surfactants,or oblique evaporation of SiO[20].The structure of LCs near the interface is different than that in the bulk,and this structure in the surface changes the boundary conditions and influences the behavior of the LC in the bulk.There are two cases of surface anchoring. In the hard anchoring case the director near the surface adopts a rigidly fixed orientation and the an-choring energy is very large.In the soft anchoring case the surface strengths are not strong enough to impose a well-defined orientation at the interface and the expression for the anchoring energy is some finite function that depends on the LC properties at the surface and the surface properties.The latter case represents the majority of systems.In addition, if there are external agents(pressure,electric field, magnetic field,and temperature)the LC’s director at the surface deviates from the initial orientation[21]. In their seminal work,Rapini and Papoular[22] introduced a simple phenomenological expression for the interfacial energy per unit area for a1D de-formation.Zhao et al.[23]studied the surface anchor-ing of nematic LCs in twisted configuration(TNLC) and,through spherical harmonic expansion of the surface energy,they derived a two-term expression for the anchoring energy,which is highly symmetric. In this paper we theoretically show that the sys-tem studied in[13]is able to produce an OBG that can be electrically controlled for circularly polarized incident waves.We take into account both the strongand the weak anchoring conditions of the nematic atthe boundaries.The1D structure considered here consists of N nematic LC slabs in a twisted configu-ration alternated by N transparent isotropic dielec-tric films.For each of the TNLC cells,the nematic issandwiched between two dielectric layers in such a way that its director is aligned parallel in both fron-tiers.A twist is then imposed on the LC by rotating acertain given angle2φt,one of the dielectric layers,about its own normal direction.2.Electromagnetic Wave PropagationWe are focused in the reflection and transmission ofplane waves due to a1D structure consisting of Nlayers of thickness d h[see Fig.1(a)]subjectedto a dc electric field E dc 0;0;E dc applied along the periodicity axis,which is parallel to z axis.Thehalf-spaces z<0and z>N d h are taken to befree space.Now we consider that an electromagnetic wave is obliquely incident parallel to the x–z plane on the system from the half-space z<0.To study this system we use a matrix formalism in which a boun-dary-value problem has to be established in order to determine the reflection and transmission coeffi-cients[24].If we define the four-vectorΨ x;z;t ψ z exp ik x x−iωt e x;e y;h x;h y exp ik x x−iωt ,with ωthe angular frequency of the propagating wave and k x the transversal component of wavevector,Fig.1.Schematic of(a)1D structure consisting of an N nematic LC in a twisted configuration alternated by N isotropic dielectric films with thicknesses d and h,respectively.For the first twisted nematic cell,the twist angle is equal to−φt at z 0,andφt at z d.This configuration is exactly the same for all twisted nem-atic cells.(b)Wave vector k is an oblique incident parallel to the x–z plane and makes an angleθwith respect to the z axis.Here a L;R represent the left-and right-circularly polarized components of the incident plane waves,which are assumed to be known,whereas the reflection amplitudes r R;L and transmission amplitudes t L;R are not known.4684APPLIED OPTICS/Vol.53,No.21/20July2014we can express Maxwell ’s equations,inside a nonmagnetic medium,in the following form:∂ψ∂zik 0A z ψ;(1)where the 4×4matrix A z is given byA z 0B B BBB @−k x εzx k 0εzz−k x εzyk 0εzz0−k 2x 20εzz0000−εyx k 2x 20−εyy 0k x εyz k 0εzzεxx εxy−k x εxz k0εzz1C C C C C A0B B B @0010−10εyz εzx εzzεyz εzy εzz 00−εxzεzxεzz −εxzεzy εzz1C C C A ;(2)with εij (i ,j x ,y ,z )the elements of dielectric tensor in the sample.For the dielectric films the matrix A z is independent of the position,whereas,for each TNLC layer A z depends on the local orientation of the principal axis of the LC molecule.For these layers,εij ε⊥δij ε∥−ε⊥ n i n j ,where δij is the Kronecker delta,ε⊥and ε∥denote the dielectric per-mittivity perpendicular and parallel to the nematic axis,respectively ,and n is the director representing the average local orientation of LC;k 0 2π∕λis the wave number in free space,λdenotes the wave length and θis the incidence angle made between the wave vector and the z axis.In writing Eq.(1)we have de-fined the dimensionless electric e and magnetic hfields related to E and H fields as e Z −1∕2E and h Z 1∕20H ,with Z 0 μ0∕ε0p the free space imped-ance.Finally ε0and μ0are the permittivity and permeability of the vacuum,respectively .The general solution of Eq.(1)for a wave propagat-ing from z 0to z N d h can be written asψ N d h M ψ 0≡exp ik 0Z N d h 0A z 0 d z 0ψ 0 ;(3)where the transfer matrix M relates the wave vector ψat the left side of the structure to that of the right side.Note that the problem of obtaining M is reduced to find a method to integrate Eq.(3)on the whole structure.Because of the nonhomogeneity of the medium proposed here,we consider it as broken up into many thin parallel layers and treat each as if they had homogeneous anisotropic optical parameters.In this way M is obtained by multiplying iteratively the matrix for each sublayer from z 0to z N d h .Knowing the incident wave vector ψ 0 and the transfer matrix M ,we can solve the four simultaneous linear equations for the transmittedand reflected wave vectors.Numerical and analytical propagation matrix methods for solving Eq.(3)have been detailed in [25]and [26],respectively .As previously stated,let us consider that an electromagnetic wave is obliquely incident parallel to x –z plane on the system from the half-space z <0,for which the wave vector is defined by k k x ;k y ;k z k 0 sin θ;0;cos θ [see Fig.1(b)].In this region of space,for the arbitrary polarization state,the solutions of Maxwell ’s equations for e z e x ;e y ;e z and h z h x ;h y ;h z can be expressed ase hleftB B B @a L i u −v exp ik z z−r Li u −v − exp −ik z z ;−ia L i u −v exp ik z z ir L i u −v − exp −ik z z1CC C A 0B B B @−a Ri u vexp ik zz r Ri u v − exp −ik z z ;−ia R i u v exp ik z z ir R i u v − exp −ik z z1CC C A;(4)where the unit vectors u and v are defined asu u y2p ;v∓cos θu x sin θu z2p ;(5)and u x ;u y ;u z is the triad of Cartesian unit vectors.With the assumption that the half-space z ≥N d h is also empty,the transmitted fields in that half-space may be set down as e htrt L i u −v exp ik z z −N d h ;−it L i u −v exp ik z z −N d h− t R i u v exp ik z z −N d h ;it R i u v exp ik z z −N d h:(6)In these equations,the complex-valued amplitudes a L;R of the left-and right-circularly polarized (LCP and RCP)components of incident plane wave are assumed to be known,whereas the reflection ampli-tudes r R;L and transmission amplitudes t L;R are not known.Notice that from Eq.(4),the incident elec-tric field which is assumed as e z e x ;e y ;e z a L i u −v −a R i u v exp ik z z ,is a vector on the perpendicular plane to the wave vector k [see Fig.1(b)].Indeed,if the mutually perpendicular vec-tors u and v define a plane in a point of space,a perpendicular vector to this plane can be obtained by doing the cross product;the operation gives u ×v sin θ;0;cos θ ,which is a vector parallel to k .Notice also that the vectors i u −v and i u v form a complete base for expanding the circularly polarized electric field.Continuity of the tangential components of the electric and magnetic fields across the planes z 020July 2014/Vol.53,No.21/APPLIED OPTICS4685and z N d h leads to the prescriptions of the boundary valuesψ 0Q2p·B B@a Ra Lr Rr L1C CA;ψ N d hQ2p·B B@t Rt L1C CA;(7)with the4×4matrixQ 0B B@cosθcosθcosθcosθ−i i i−ii cosθ−i cosθi cosθ−i cosθ11−1−11C CA:(8)By virtue of Eqs.(3)and(7)we obtain the matrix algebraic equation0 B B@t Rt L1C CA U 0;N d h ·B B@a Ra Lr Rr L1C CA;(9)whereU 0;N d h Q−1·M·Q;(10) is the transfer matrix relating amplitudes a L,a R,r L, and r R(from z≤0)to transmitted amplitudes t L,t R [for z≥N d h ].The result of solving Eq.(9)is best arranged as0 B B@t Rt Lr Rr L1C CAB B@t RR t RLr LR r LLt RR t RLr LR r LL1C CAa Ra L;(11)where t LR,etc.,are the transmission coefficients and r LR,etc.,are the reflection coefficients.The co-polarized transmittances are denoted by T LL j t LL j2and T RR j t RR j2,and the cross-polarized onesby T LR j t LR j2and T RL j t RL j2,and similarly for the reflectances R RR j r RR j2,etc.3.Twisted Nematic ConfigurationThe equations governing the equilibrium configura-tion of each TNLC layer under the influence of a dc electric field by taking into account the interaction between the nematic LC and the confining surface are obtained by minimization of the total free energy of the system[27,28],which is given by F F c−12ZRe f E dc·D g d V12ZK1 ∇·n 2 K2 n·∇×n 2 K3 n×∇×n 2−K4∇· n×∇×n n∇·n d V12ZS0g L d S1ZS dg R d S−1ZRe f E dc·D g d V;(12)where the electric energy density− 1∕2 Re f E dc·D g takes into account the interaction of the electric field E dc with the LC and it satisfies the dielectric constit-utive relation D i ε0εij E;j.The elastic moduli K1, K2,and K3describe splay,twist,and bend bulk deformations,respectively,while K4is the surface splay bending elastic constant representing the coefficient of a divergence term,which can be trans-formed to a surface integral by using Gauss’s theorem.This term and the surface ones,given by 1ZS0g L d S and1ZS dg R d S;provide the interaction between the TNLC layer and the confining surface[23].Here we assume arbitrary anchoring condition so that g L and g R denote the strength of the elastic surface interactions in units of energy per area,corresponding to the left and right substrates of each nematic cell.Finally,n is the director.Here we take n cosα z cosφ z ;cosα z sinφ z ; sinα z ,and we propose analogous symmetric expressions for the anchoring energy of each slab presented in[23]given byg L Wαsin2αL m Wφcos2αL m sin2 φL m φt ;g R Wαsin2αR m Wφcos2αR m sin2 φR m−φt ;(13)whereα z andφ z are the polar and azimuthal angles made by n with the x–y plane and the x axis, respectively.αL m andφL m are the polar and azimuthal angles at the left boundary of each TNLC,at the right boundary,the corresponding angles areαR m,φR m.Finally,Wαand Wφare the polar and azimuthal anchoring strengths at the interfaces.It is worth mentioning that in the case of hard anchoring conditions,the angles at the boundaries of each layer adopt the valuesαL m αR m 0,φL m −φt and φR m φt,beingφt the twist angle(see Figs.1and2). Also,for very large strengths Wα→∞and Wφ→∞. The dependence of the director inclinationsα z andφ z on the position is obtained by minimization of the free-energy F given by Eq.(12).We perform this minimization by a free-end-point variation for which n is subjected to a finite force at the cell bor-ders[29].This direct procedure gives the following set of two coupled bulk equations[28]:4686APPLIED OPTICS/Vol.53,No.21/20July20140 f α d 2αdz2 12df α d α d αdz 2−12dg α d α d φdz2d −2σ2sin αcos α;(14)0 g α d 2φdz2dg α d αd αdz d φdz ;(15)subjected to the set of four arbitrary boundary con-ditions at each layer:d αdz z z L m1d Γαsin 2α 1−Γsin 2 φ φt f α z z L m ;(16)d αdz z z R m −1d Γαsin 2α 1−Γsin 2 φ−φt f α z z R m ;(17)d φdz z z L m Γd Γαcos 2αsin 2 φ φt g α z z L m ;(18)d φdz z z R m −Γd Γαcos 2αsin 2 φ−φt g α z z R m ;(19)where f α and g α are given byf α cos 2αK 3K 1sin 2α;(20)g αK 2K 1cos 2α K3K 1sin 2αcos 2α;(21)and the dimensionless parameter σ2 ε0 ε∥−ε⊥ E 2dc ∕K 1d−2represents the ratio of the electric and elastic energies.Here z L m m −1 d h and z Rm m −1 d h d ,with m 1;2;3;…;N ,represent the positions of left and right boundaries of the N nematic layers,respectively.We have defined the following four dimensionless parameters Γα 1∕γα,Γ γφ∕γα,γα W αd ∕K 1,and γφ W φd ∕K 1,which are related to the polar and azimuthal anchoring strengths at the substrates.The parameter Γrepresents the ratio between the azimuthal and polar surface forces.For very large surface forces γα→∞and γϕ→∞,so that Γα→0and,consistently ,the set of arbitrary boundary conditions Eqs.(16)–(19)is reduced to the set of hard anchoring boundary conditions,for whichα z L m α z R m 0,φ z L m −φt ,and φ z Rm φt .In this limit notice that the coupled Eqs.(14)and (15)are the same of those obtained in [28]as expected.The equilibrium configuration of each TNLC layer as a function of parameter σis obtained by solving Eqs.(14)and (15)for α z and φ z subjected to the conditions expressed in Eqs.(16)–(19).Then this con-figuration is substituted into Eq.(3)to obtain the transfer matrix M as a function of σand therefore,the reflection and transmission coefficients can be computed.4.Numerical Results and DiscussionNumerical calculations were performed by consider-ing an LC phase 5CB for which K 1 0.62×10−11N ,K 2 0.39×10−11N ,and K 3 0.82×10−11N [30]refractive indices at optical frequencies n e ε∥p 1.717and n oε⊥p 1.53,and twist angle 2φt 90°.The isotropic dielectric medium is ZnS,so that its refractive index is equal to n d 2.35.Equations (14)and (15)were solved by using the shooting method [31].Due to the competition between orientation produced by surface anchoring and by electric-field effects,only above a certain critical value σc of parameter σ,the deformation of nematic LC occurs [28].In this way ,the maximum value of critical elec-tric field is expected to be reached for the case of hard anchoring conditions;whereas,for the weak anchor-ing case,we expect that σc will diminish as the sur-face forces get smaller.In what follows we present numerical computations for strong and weak anchor-ing conditions taking σc as parameter of reference.A.Hard Anchoring ConditionsFor hard surface anchoring we plot the co-polarizedtransmission band-edge frequencies of the first PSB for incident RCP waves (T RR )as a function of inci-dent angle for σ σc 3.16(upper curve of Fig.3)and for σ 3σc (lower curve of Fig.3).Here,we con-sider h ∕d n o ∕n d 0.65,N 20and the frequency normalized in the units of d ∕λ.The band-edge frequencies are taken so that the transmission coef-ficient becomes smaller than 0.5%.Notice that the OSB is absent for σ σc (corresponding to a critical voltage V c 3.4V),whereas for V 3V c thisFig.2.Schematic of the polar αand azimuthal φangles made by the director n with the x –y plane and the x axis,respectively ,at the boundaries of the first LC slab.The twist angle is given by φt .At the middle of the TNLC,α φ 0.The configuration is the same for all nematic layers.20July 2014/Vol.53,No.21/APPLIED OPTICS4687phenomenon occurs.Figure 3suggests that there ex-ists a threshold electric field σth ,for which above this value the OSB is present.This band takes place only when the difference between the upper-and lower-edge frequencies at θ 0°and θ 85°,respectively,are positive.Figure 4shows the dependence of these edge frequencies as a function of the external dc elec-tric field σparametrized by σc .One can see that the OSB appears only above σth 1.097σc (correspond-ing to V th 3.73V).For larger values than this threshold,the OSB ’s width becomes wider as the electric field increases due to a simultaneous aug-ment and decrement of upper-and lower-edge frequencies.It can be seen from an inspection of Fig.4that the width of this band augments rapidly for val-ues near the threshold electric field,and slowly for larger ones.For σ 4σc (or V 13.6V),the OSB ’swidth equals 0.014.Note that the center frequency of the OSB is maintained with slight change.The strong dependence of the dielectric tensor of LC on the external dc electric field applied gives rise to an increment of the impedance ratio between LC and isotropic slabs so that the width of the PSB gets wider as the electric field augments (Fig.3)and it produces the OSB as well (Fig.4).We also computed the co-polarized and cross-polarized transmittance band-edge frequencies of the first PSB for incident LCP and RCP waves,denoted by T LL ,T RL ,and T LR ,respectively .The results showed that these band-edge frequencies are equal to those obtained for T RR to numerical accuracy (plots are not shown here).As the transmittance is forbidden for both po-larizations at any incident angle,we can affirm that the structure considered here is able to produce an OBG.B.Soft Boundary ConditionsIf the strength of the E dc is high enough,soft boun-dary conditions have to be taken into account.It is experimentally found that for an LC phase 5CB the polar anchoring γαis of the order of 101and this value is,in turn,one or two orders stronger than the azimuthal anchoring γφ[32].With this in mind,we plot the dependence of the edge frequencies as a func-tion of σ∕σc for specific dimensionless anchoring parameters Γ 0.1,Γα 0.1(Fig.5)and Γα 0.5(Fig.6).Numerical results show that under this conditions,the critical values for which the deforma-tion of the nematic LC occurs are σc 2.86(or V c 3.07V)for Γα 0.1,and σc 2.135(or V c 2.28V)for Γα 0.5.As can be seen in Fig.5,the OSB takes place only above the threshold field σth 1.132σc (corresponding to V th 3.47V).In Fig.6the corresponding minimum value is σth 1.112σc (or V th 2.53V).It is worth noticing that these threshold values are smaller than that for hard anchoring conditions,which is consistent with the fact that for weaker-surface anchoringstrengths,Fig.3.Co-polarized transmittance band-edge frequencies (units of d ∕λand solid lines)of the first PSB for incident RCP waves (T RR )on the photonic structure as function of incident angle θwhen σ σc 3.16(upper curve)and σ 3σc (lower curve)by assuming hard anchoring conditions.We have considered an LC phase 5CB and h ∕d n o ∕n d 0.65,N 20.The gray area in the lower curve represents the OSB.The magnitude of this band is determined by the positive difference between the upper-and lower-edge frequen-cies at θ 0°and θ 85°,respectively .As the co-polarized and cross-polarized transmittance band-edge frequencies of the first PSB for incident LCP and RCP waves equal those obtained for T RR to numerical accuracy ,the plots are not shown here.Since the transmittance is forbidden for both polarizations at any incident angle,the photonic structure considered here is able to produce anOBG.Fig.4.Dependence of edge frequencies as function of ¯Edc of the first PSB for incident RCP waves and hard anchoring conditions.The dot-dashed line represents the upper edge at θ 0°and the dashed line depicts the lower edge at θ 85°.The solid line shows the center of the OSB.The threshold for which the OSB is created occurs at σth 1.097σc .4688APPLIED OPTICS /Vol.53,No.21/20July 2014the external field is also able to change the configu-ration at the boundaries of TNLC cells for smaller values of it.We can observe in Fig.5that for σ 2.7σc (or V 8.29V)the width of the OSB is 0.014,and in Fig.6the same breadth is reached at σ 1.57σc (or V 3.58V).The solid curves presented in plots 5and 6show that due to the concurrent augment and decrement of upper-and lower-edge frequencies,the center frequency of the OSB is kept with small change.As occurred in the hard anchoring conditions,the computations show that to numerical accuracy ,the band-edge frequen-cies of the first PSB for co-polarized and cross-polarized transmittances for incident LCP and RCP waves are equal to those obtained for T RR ,in such a way that the PC produces an OBG under soft boundary conditions.The results presented here demonstrate the pos-sibility of controlling the width of the OBG by an ex-ternal electric field.Numerical computations show that large values of these breadths can be achieved at smaller values of the surface strengths.Indeed,we observe in Figs.4–6that the width 0.014is attained at voltages of 13.6,8.29,and 3.58V ,respectively ,in such a way that optical devices based on our model will operate at lower voltages due to the boundary anchoring effects.5.Concluding RemarksWe have demonstrated that the phenomenon of OBG in 1D photonic structure using twisted nematic LCs can be created and tuned by applying an external dc electric field aligned parallel to the periodicity axis.We take into account both the hard and the soft an-choring conditions of the nematic at the boundaries.Here we performed computations for an LC phase 5CB.For hard anchoring boundaries,we showed that the OBG can be created and tuned only when the voltage V overcomes the threshold voltage V th 3.73V.For soft anchoring conditions,the corresponding threshold voltage is V th 3.47V when surface strengths are Γ Γα 0.1,and V th 2.53V,if we take Γ 0.1and Γα 0.5.Hence,we found that the critical voltage for which the OBG is created is smaller for soft anchoring conditions than that for hard anchoring ones.We observed that the width of the OBG gets larger as the voltage increases and its center frequency is modified little.Numerical computations showed that large values of OBG ’s breadths can be achieved at smaller values of the surface strengths.We hope that these properties can motivate the fabrication of new types of optical filters and optical switches.References1.E.Yablonovitch,“Inhibited spontaneous emission in solid-state physics and electronics,”Phys.Rev .Lett.58,2059–2062(1987).2.J.D.Joannopoulos,P .R.Villeneuve,and S.Fan,“Photonic crystals:putting a new twist on light,”Nature 386,143–149(1997).3.S.G.Johnson and J.D.Joannopoulos,Photonic Crystals:The Road from Theory to Practice (Springer,2002).4.M.Scalora,J.P .Dowling,C.M.Bowden,and M.J.Bloemer,“Optical limiting and switching of ultrashort pulses in nonlin-ear photonic band gap materials,”Phys.Rev .Lett.73,1368–1371(1994).5.M.Golosovsky ,Y.Saado,and D.Davidov ,“Self-assembly of floating magnetic particles into ordered structures:a promis-ing route for the fabrication of tunable photonic band gap materials,”Appl.Phys.Lett.75,4168–4170(1999).6.P .Halevi and F .Ramos-Mendieta,“Tunable photonic crystals with semiconducting constituents,”Phys.Rev .Lett.85,1875–1878(2000).7.Y.Yi,P .Bermel,K.Wada,X.Duan,J.D.Joannopoulos,and L.C.Kimerling,“Tunable multichannel optical filter based on silicon photonic band gap materials actuation,”Appl.Phys.Lett.81,4112–4114(2002).8.K.Busch and J.John,“Liquid-crystal photonic-band-gap materials:the tunable electromagnetic vacuum,”Phys.Rev .Lett.83,967–970(1999).9.K.Yoshino,Y.Shimoda,Y.Kawagishi,K.Nakayama,and M.Osaki,“Temperature tuning of the stop band in transmission spectra of liquid-crystal infiltrated synthetic opal as tunable photonic crystal,”Appl.Phys.Lett.75,932–934(1999).10.Q.B.Meng,C.H.Fu,S.Hayami,Z.Z.Gu,O.Sato,and A.Fujishima,“Effects of external electric field upon the photonic band structure in synthetic opal infiltrated with liquid crys-tal,”J.Appl.Phys.89,5794–5796(2001).11.S.W .Leonard,J.Mondia,H.van Driel,O.Toader,S.John,K.Busch,A.Birner,and U.Gosele,“Tunable two-dimensional photonic crystals using liquid crystal infiltration,”Phys.Rev .B 61,R2389–R2392(2000).12.J.A.Reyes,J.A.Reyes-Avendaño,and P .Halevi,“Electricaltuning of photonic crystals infilled with liquid crystals,”mun.281,2535–2547(2008).Fig.6.Similar to Fig.4but for soft anchoring boundaries.The threshold for which the OSB is observed occurs at σth 1.112σc .Anchoring parameters were taken Γ 0.1and Γα 0.5.Fig.5.Same as Fig.4but for soft anchoring boundaries.The threshold for which the OSB is observed occurs at σth 1.132σc .Anchoring parameters were taken Γ Γα 0.1.20July 2014/Vol.53,No.21/APPLIED OPTICS4689。
级联一维光子晶体全方位反射器的带宽最大化 图2
级联一维光子晶体全方位反射器的带宽最大化图2（b）剩下a，c均可通过改变光学厚度实现。
带隙偏移不太理解d，e，f，未画。
同上文献图3（b）斜入射（a）为TM波，仅计算参数不同同上文献图545度入射角（中间图）的TM波未画85度入射角绿线为TM波，蓝线为TE波0度，两偏振态重合关于全方位光子带隙与其交叠问题不太懂，不知是否指图中最宽的部分。
故PCove PC1/PC2ove PC1/PC2opt三张图也未画另一元三维光子晶体的偏振滤波特性研究（刘文莉）中三元光子晶体的图还有些问题。
尚未找到处理办法na=3.3;nb=2.34;nc=1.5;n1=1;n2=1;for d=400:800;c4=0;c1=asin(n1*sin(c4)/na);c2=asin(na*sin(c1)/nb);c3=asin(nb*sin(c2) /nc);d1=350;a=d1/(4*na);b=d1/(4*nb);c=d1/(4*nc);Ba=2*pi*na*a*cos(c1)/d;Bb=2*pi*nb*b*cos(c2)/d;Bc=2*pi*nc*c*cos(c3)/df=4*pi*1e-7;e=1e-9/(36*pi);m=sqrt(e/f);za=m*cos(c1)*na;zb=m*cos(c2)*nb;zc=m*cos(c3)*nc;s1=cos(Bc);s2=-i*sin(Bc)/zc;s3=-i*zc*sin(Bc);s4=cos(Bc);W=[s1 s2;s3 s4];p1=cos(Bb);p2=-i*sin(Bb)/zb;p3=-i*zb*sin(Bb);p4=cos(Bb);P=[p1 p2;p3 p4];q1=cos(Ba);q2=-i*sin(Ba)/za;q3=-i*za*sin(Ba);q4=cos(Ba);Q=[q1 q2;q3 q4];O=[Q*P*W]*[W*P*Q];O1=O^6;O11=O1(1,1);O12=O1(1,2);O13=O1(2,1);O14=O1(2,2);z1=sqrt(e/f)*n1*cos(c4);z2=sqrt(e/f)*n2*cos(c4);t=2*z1/(z1*(O11+z2*O12)+O13+z2*O14);t1=abs(t);s=d-399;k(1,s)=t1endd=400:800;plot(d,k)。
一种基于一维光子晶体背反射镜的倒置型半透明聚合物太阳能电池及其制备方法[发明专利]
![一种基于一维光子晶体背反射镜的倒置型半透明聚合物太阳能电池及其制备方法[发明专利]](https://img.taocdn.com/s3/m/35ce30a8be1e650e53ea99b1.png)
专利名称：一种基于一维光子晶体背反射镜的倒置型半透明聚合物太阳能电池及其制备方法
专利类型：发明专利
发明人：沈亮,于文娟,阮圣平,刘彩霞,郭文滨,董玮,张歆东,陈维友,陆斌武
申请号：CN201210544338.4
申请日：20121214
公开号：CN103000811A
公开日：
20130327
专利内容由知识产权出版社提供
摘要：本发明属于聚合物太阳能电池技术领域，具体涉及一种基于高低折射率材料WO及LiF构成的一维光子晶体作为背反射镜的倒置型半透明聚合物太阳能电池及其制备方法。
首先，在导电玻璃衬底上生长一层均匀致密的N型TiO薄膜，然后旋涂上一层二氯苯溶解的P3HT:PCBM溶液，退火，再依次生长WO和Ag；最后，在半透明银电极上，再生长[WO/LiF]一维光子晶体。
本发明制备的半透明聚合物太阳能电池，解决了传统半透明太阳能电池透过率高、效率低的问题。
[WO/LiF]一维光子晶体的高反射膜结构有利于提高特定波长光的反射和吸收，解决了半透明电池效率与透过率之间的矛盾，有效地提高了能量转换效率。
申请人：吉林大学,无锡海达安全玻璃有限公司
地址：130012 吉林省长春市前进大街2699号
国籍：CN
代理机构：长春吉大专利代理有限责任公司
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一维光子晶体全向带隙限光特性的研究
井冈山大学学报(自然科学版)26文章编号：1674-8085(2012)03-0026-05一维光子晶体全向带隙限光特性的研究*李棚1，张明存1，叶飞1,2（1．六安职业技术学院，安徽，六安237100 2．合肥工业大学，安徽，合肥230009）摘要：采用传输矩阵的计算方法研究了一维光子晶体结构对光传输特性的影响，利用MATLAB绘制不同结构参数的一维光子晶体透射率图谱。
通过绘图发现，改变一维光子晶体的结构参数，能够实现带隙宽度的最大化，同时，可以实现入射角在0到90度之间的全方向带隙限光。
选择适当的结构参数能够实现在1550nm光波附近的宽屏全向带隙限光。
关键词：光子晶体;传输矩阵法;数值分析;透射谱;带隙限光中图分类号：O436 文献标识码：A DOI:10.3969/j.issn.1674-8085.2012.03.006 RESEARCH ON THE CHARACTERISTICS OF OPTICAL TRANSMISSION WITH OMNIDIRECTIONAL BANDGAP CONTROL IN 1D PHOTONICCRYSTALLI Peng1，ZHANG Ming-cun1，YE Fei1.2( 1. Lüan V ocation Technology College, Lüan, Anhui 237100, China; 2.Hefei University of Technology, Hefei , Anhui 230009, China) Abstract: The effect of one-dimensional photonic crystal structure for optical transmission property was studied by transfer matrix computing method. MATLAB is used to draw One-dimensional photonic crystal transmittance map of different structure parameter. Through drawing, we find the larges bandgap width when the one-dimensional photonic crystal structure parameter is changed. Omni directional band gap limit of light will be implemented in angle from 0o to 90o. Broadband Omni directional bandgap limit light in a 1550nm optical wavelength near be achieved by select the appropriate structural parametersKey words: photonic crystals; transfer matrix method; numerical analysis; transmission spectrum; bandgap limit lightS. John [1]及E.Yablonovitch [2]几乎同时指出，光子系统中的光子会受到晶格周期性结构的散射，部分波段会因为干涉而形成能隙，进而在传输的光信号中出现能带。
一维光子晶体全向反射镜的研究进展
一维光子晶体全向反射镜的研究进展
张芬;肖峻;谢康
【期刊名称】《光通信技术》
【年(卷),期】2009(33)7
【摘要】光子晶体最显著的特点就是其具有光子带隙结构,可以通过抑制某些频率电磁波的传播来实现在该频段上的全反射,利用光子晶体的这一特性可做成全向反射镜,文章介绍了目前一维光子晶体全向反射镜的基本理论以及在几个主要波段上的实现.
【总页数】3页(P60-62)
【作者】张芬;肖峻;谢康
【作者单位】电子科技大学,光电信息学院,成都,610054;电子科技大学,光电信息学院,成都,610054;电子科技大学,光电信息学院,成都,610054
【正文语种】中文
【中图分类】TN929.11
【相关文献】
1.含负折射率介质一维光子晶体的全向反射镜 [J], 倪重文;沈小明;金铱;唐丽;是度芳
2.一维光子晶体全向带隙限光特性的研究 [J], 李棚;张明存;叶飞
3.一维光子晶体全向带隙限光特性的研究 [J], 李棚;张明存;叶飞
4.含各向异性单负材料的一维光子晶体的全向带隙 [J], 董丽娟;石云龙
5.一维光子晶体的全向反射特性 [J], 黄正逸;金铱;马骥;徐雷;陈宪锋
因版权原因，仅展示原文概要，查看原文内容请购买。
一维光子晶体滤波器的设计及性能研究的开题报告
一维光子晶体滤波器的设计及性能研究的开题报告一、研究背景光子晶体是一种具有周期性介电常数分布的光学材料，其具有在光子能带禁止带内的完全反射特性，因此被广泛地应用于光学滤波器、光学调制器和光学传感器等领域。
一维光子晶体是最简单的光子晶体结构，具有制备简单、制作工艺成熟、调控精度高的优点，因此得到了广泛关注。
本课题在此基础上，着重研究一维光子晶体滤波器的设计及其性能研究，期望在滤波器的制备和性能优化方面探索新思路、新方法，为实际应用提供基础研究支持。
二、研究内容1. 设计一维光子晶体滤波器的基础理论和方法，包括设计滤波器结构、计算材料的介电常数、对光源特性的处理等。
2. 制备一维光子晶体滤波器板，并进行材料性质测试，在此基础上优化制备工艺，实现滤波器的高效性能。
3. 对所制备的一维光子晶体滤波器进行测试，并分析其频率响应和滤波效果，并探究影响滤波器性能的主要因素。
4. 根据上述研究结果，对一维光子晶体滤波器的使用条件和应用范围进行总结和分析，并提出优化设计建议。
三、研究意义一维光子晶体滤波器作为新型光学器件，在信息通信和光学传感等领域具有广泛应用前景。
本课题旨在通过对其制备和性能的研究，提高其在实际应用中的效率和稳定性，为相关领域探索新的技术路径。
四、研究方法本研究主要采用理论分析、实验测试和数据分析等方法，其中具体方法包括：1. 采用光学传输矩阵（TMM）方法，进行滤波器结构的计算和参数优化。
2. 利用电感耦合等离子体（ICP）刻蚀技术，制备一维光子晶体滤波器。
3. 利用紫外可见分光光度计、椭偏仪、扫描电子显微镜等实验设备，对制备材料进行分析测定。
4. 运用软件对实验获得的数据进行分析处理，得出滤波器性能的相关参数。
五、研究计划本研究计划分为三个阶段：1. 初期阶段，完成相关文献综述和理论分析，确定一维光子晶体滤波器的设计方案。
2. 中期阶段，进行一维光子晶体制备和材料性能测试，并结合实验数据，对制备工艺和滤波器性能进行初步优化。
一维和二维光子晶体全向禁带的实现与展宽的开题报告
一维和二维光子晶体全向禁带的实现与展宽的开题报告题目：一维和二维光子晶体全向禁带的实现与展宽引言：光子晶体是一种具有周期性介电常数的材料，可以用来控制光的传播性质，例如引导、支持和禁止光在特定频率范围内传播。
其中，全向禁带（omnidirectional band gap，OBG）是指光子晶体在所有方向上都能禁止光的传播，在光子晶体的设计和应用中具有重要意义。
本文将研究如何实现和展宽一维和二维光子晶体的全向禁带，并探讨其在光学器件中的应用。
一、一维光子晶体全向禁带的实现与展宽1.1 实现方法一维光子晶体是由周期性介电常数分布的一系列层组成的。
其全向禁带的实现方法通常是通过调整层厚度和介电常数来控制光子晶体的带隙。
实现一维光子晶体全向禁带的基本步骤：（1）选择一个适当的周期（周期越大，带隙越宽）。
（2）确定各层厚度和介电常数的分布规律，通常采用布拉格反射定律和逆反射率法计算。
（3）制备光子晶体样品，通常采用电子束或光刻技术进行制备。
1.2 展宽方法实际制备过程中，不可避免地存在一些微观结构上的偏差和缺陷，这些结构缺陷会对带隙的宽度和形状产生显著影响，降低全向禁带的品质。
因此，如何展宽带隙，提高全向禁带的质量，一直是研究重点。
展宽方法一般有以下几种：（1）改变层厚度或介电常数分布形状，通过优化光子晶体结构，提高带隙品质。
（2）利用多重光子晶体结构，构造超光子晶体，并采用光子带隙的分离和共振耦合等技术增强带隙。
（3）采用非晶态光子晶体、共轭光子晶体等一些新兴结构，实现更广泛的带隙宽度，并克服传统光子晶体中晶格缺陷这一不足之处。
二、二维光子晶体全向禁带的实现与展宽2.1 实现方法二维光子晶体是由周期性介电常数分布的二维图案组成。
实现二维光子晶体全向禁带与一维光子晶体类似，不同之处在于二维光子晶体需要控制的是TE和TM两个偏振方向上的光子晶体带隙。
实现二维光子晶体全向禁带的基本步骤：（1）确定光子晶体的结构类型，通常采用正方形、三角形等基本元胞实现二维光子晶体的周期性结构。
一维光子晶体的结构设计及光学特性研究的开题报告
一维光子晶体的结构设计及光学特性研究的开题报
告
一、选题背景
光子晶体是一种由周期性折射率介质构成的材料，在光学中具有重
要的应用前景。
它的独特之处在于，它的能带结构可以被设计用来控制
材料的光学性质。
一维光子晶体是其中特殊的一种类型，它的周期性只
有一个方向，并且具有周期性折射率分布结构。
一维光子晶体的研究和应用已经被广泛关注，特别是在光学传感、
光电子学和激光技术等领域。
因此，对一维光子晶体的结构设计和光学
特性的研究具有重要的意义。
二、研究目的
本研究旨在设计一种具有优异光学特性的一维光子晶体，并且探究
其光学性能。
通过合理设计材料的折射率分布结构和周期，达到对光的
控制和操纵，使其具有特定的波长选择性和传输性能。
三、研究内容
1. 综述一维光子晶体的基本原理、结构设计方法和制备工艺。
2. 确定一维光子晶体的结构参数，即周期和材料的折射率分布结构，并通过理论计算得到其光学能带结构和传输性质。
3. 根据设计的一维光子晶体结构参数，采用制备方法制备出具有设
计要求的样品，例如溶胶-凝胶法、干膜法等。
4. 实验测试样品的光学特性，例如反射谱、透射谱、色散曲线等，
与理论分析结果进行对比分析。
五、预期成果
合理设计的一维光子晶体结构参数，通过制备工艺制备出具有特定光学性能的样品。
通过实验测试样品的光学特性，并与理论计算结果进行对比，验证结构设计和制备工艺的可行性。
最终，将在一维光子晶体的结构设计和光学特性的研究领域做出独到的贡献。
一维光子晶体全方向反射带的研究
一维光子晶体全方向反射带的研究
刘文莉;唐婷婷;何修军
【期刊名称】《人工晶体学报》
【年(卷),期】2014(43)3
【摘要】设计了由SiO2和Te构成的一维光子晶体结构,对该结构的全向反射带特性进行分析。
找到了TE和TM模的全向反射带范围,TE模的反射带在785 nm到1258 nm内变化,TM模的反射带在785 nm到1099 nm范围内变化。
讨论了晶格常数,晶格周期对该全向反射带的影响。
结果表明,TE模的全向反射带始终比TM 模的宽。
晶格常数增大,TE和TM模的全向反射带均向归一化频率较高(对应长波)的方向移动。
晶体周期增大,TM模的全向反射带会变窄,而TE模的全向反射带变化不大。
此外还发现,参数变化对TM模的反射带的影响明显大于对TE模的影响。
【总页数】5页(P704-707)
【关键词】光子晶体;全方向反射带;晶格常数;归一化频率
【作者】刘文莉;唐婷婷;何修军
【作者单位】成都信息工程学院光电技术学院
【正文语种】中文
【中图分类】O734
【相关文献】
1.高双折射全内反射光子晶体光纤的特性研究 [J], 李未;徐旭明;何玉平;邱深玉;李健文
2.厚度的阶梯性变化对远红外高反射光子晶体带隙的调制研究 [J], 王超;张凤祥;邵东旭;程冰心;石敏;黄茂松;许遥
3.高功率光纤激光器全内反射型大模场光子晶体光纤设计 [J], 刘骁;陈建国;韩敬华;张彬;崔旭东
4.全内反射型光子晶体光纤横向负荷及扭曲特性研究 [J], 杨晓辰;饶云江;朱涛;唐庆涛
5.利用一维电介质-磁光子晶体设计宽带全方向全反镜 [J], 强海霞;蒋立勇;李相银因版权原因，仅展示原文概要，查看原文内容请购买。
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0.285
0.3
角频率
0.25
0.2
0.15
0.1
TM
TE TE
0.05
0 -1
-0.8
-0.6
-0.4
-0.2
0 0.2 波矢量切分量
0.4
0.6
0.8
1
角频率与波矢量切向分量关系
全角度反光镜的设计 • 选用的高折射率材料Ge(折射率为nH=4.375)，L表 示选用的低折射率材料ZnS(折射率为nL=2.2)，其 中，中心波长 3.5μm。
0.9 0.7 0.5 0 30 角度 60
TE TM
90
1.0
禁带边沿
0.8 0.6 0.4 0 30 角度 60 90
数值结果与分析
0.5
90°
90°
0.45
0.4
0.35
相 对 宽 度 ：
波 长 ：
归 一 化 频 率
： 0.1475(2πc/a)~0.205(2πc/a)
3000~4000nm
0.35
归 一 化 波 长 ：
0.306(2πc/a) ~0.322(2πc/a)
0.3
角频率
0.25
0.2
TM
0.15 0.1
TE
0.05
0 -1
-0.8
-0.6
-0.4
-0.2
0 波矢量切分量
0.2
0.4
0.6
0.8
1
角频率与波矢量切向分量关系
1 0.8
反射率R
0 80
0.6 0.4 0.2 0 200 250 300 350 400 波长（nm) 450 500
光子晶体1
光子晶体2
x a b
x a b
a b
a
b
A：高折射率材料Hf02 (折射率nH=209)； B：低折射率材料SiO2 (折射率nL=1.472)； 中心波长 ：365nm.
•
•
•
未组合前的光子禁带
0.5
90°
90°
波 长 ： 300-375nm
0.45
0.4
相 对 带 宽 ： 5.09%
| ( A D) / 2| 1
| ( A D) / 2 | 1
角度对禁带分布影响
da db da db da db da db da db da db
A
B
A
B
A
B
A
B
A
B
A
B
一维光子晶体平面示意图
A代表高折射率材料聚苯乙烯 (折射na=4.6)； B代表低折射率材料氟化镁(折射率nb=1.35)， 其中，中心波长 2μm。
反射率
反射率
2 3 归一化波长g
4
5
(c)入射角为60°
(d)入射角为85°
不同入射角TM反射率与归一化波长关系
入射光为TE波
1 0.8 0.6 0.4 0.2 0 1 2 3 归一化波长g 4 5 1 0.8 0.6 0.4 0.2 0 1 2 3 归一化波长g 4 5
反射率
反射率
(a)入射角为0°
4500
5000
TM波反射率R随波长的变化图
• 计算结果与禁带分析的结果一致，进一步证明了 所设计的结构具有一个全反带。 • 对3000~4000nm的入射光能实现全反功能。
禁带的展宽 • 紫外光区可选择的材料比较少，材料的折射率 最大不超过2.4，最小不小于1.46，很难实现 全反镜的制作。 • 要采用禁带展宽的方法去实现。 • 目前通常禁带展宽的方法有3种。 • 本论文采用角域叠加。
2 p0 t (m11 m21 p1 ) p0 m21 m22 p1
光子晶体的理论计算方法-3
一维光子晶体具有周期结构，可以用Bloch波理 论来解释
K ( , ) (1/ )cos1[( A D) / 2]
| ( A D) / 2 | 1
传播带 禁带 边界
毕业 论文
一维光子晶体全向反光镜的设计
姓名： 班级： 指导老师：
主要内容
研究的背景及意义 光子晶体的理论计算方法 角度对禁带分布影响 全角度反射镜的设计
结论
研究的背景及意义
金属 反射镜
传统介质 反射镜
光子晶体 反射镜
全方向反射
效率高 对角度明显
无趋肤效应 全方向反射
吸收大
趋肤效应
光子晶体的理论计算方法-1
• 入射光为TM光
1 0.8
1 0.8 0.6 0.4 0.2 0
反射率
0.4 0.2 0 1 2 3 归一化波长g 4 5
反射率
0.6
1
2 3 归一化波长g
4
5
(a)入射角为0°
1 0.8 0.6 0.4 0.2 0 1 2 3 归一化波长g 4 5 1 0.8 0.6 0.4 0.2 0 1
(b)入射角为30°
ab
d
光子晶体的平面结构示意图
1
0.8
0° 30° 60° 85°
0.6
反射率R
0.4 0.2 0
2000
2500
3000
3500 4000 波 长 （ nm)
4500
5000
TE波反射率R随波长的变化图
1
0.8 0° 30° 60° 0.6
反射率R
85°
0.4
0.2
0
2000
2500
30方法不能求解
传输矩阵法 平面波展开法
时域有限差分法
有限元法 ……………
光子晶体的理论计算方法-2
传输矩阵法的实质：
把界面两边的场利用一个二维矩阵联系起来：
EI H M j I
a
b
EIII H III
A
B
d
光子晶体的理论计算方法-2
光通过所有N-1层之后的传输方程：
EN m11 m12 EN E1 M1M 2 M N 1 H H m m 1 N 21 22 H N
光子晶体的反射系数（r）和透射系数（t）
r (m11 m12 p1 ) p0 m21 m22 p1 (m11 m12 p1 ) p0 m21 m22 p1
1 0.8 0.6 0.4 0.2 0 1 2 3 归一化波长g 4 5
(b)入射角为30°
1 0.8 0.6 0.4 0.2 0 1 2 3 归一化波长g 4 5
反射率
(c)入射角为60°
反射率
(d)入射角为85°
不同入射角TE反射率与归一化波长关系
角度对禁带分布影响
1.3 1.1 TM TE
禁带宽度
组合后TE波反射率R随波长的变化图
1 0.8
反射率R
0 80
0.6 0.4 0.2 0 200 250 300 350 400 波长(nm) 450 500
组合后TM波反射率R随波长的变化图
结论 一．采用数值模拟与理论分析相结合的方法 二．分析了光子晶体禁带特性 三. 研究了两种不同模式偏振光入射情况下光子晶 体带隙随入射角演化的特点。 四. 设计了单一光子晶体结构和组合光子晶体结构 ，实现了3-4μm和330-350nm波段的任意偏振光的 全角度反射，反射率都达到了99%，禁带相对宽度 分别为28.5%和5.09%。
谢 谢
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