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## Two-dimensional function photonic crystal

Xiao Li, Lei Tian-Yu, Liang Yu, Zhao Min, Liu Hui, Zhang Si-Qi, Li Hong, Ma Ji, Wu Xiang-Yao
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• #### 摘要

光子晶体是由两种或两种以上不同介电常数材料所构成的周期性光学纳米结构. 光子晶体结构可分为一维、二维和三维, 其中二维光子晶体已成为研究的热点. 可调带隙的二维光子晶体可以设计出新型的光学器件, 因此, 对它的研究具有重要的理论意义和应用价值. 本文提出的二维新型函数光子晶体可以实现光子晶体带隙的可调性. 所谓二维函数光子晶体, 即组成它的介质柱的介电常数是空间坐标的函数, 它不同于介电常数为常数的二维常规光子晶体. 二维函数光子晶体是通过光折变非线性光学效应或电光效应使介质柱的介电常数成为空间坐标的函数. 运用平面波展开法给出了TE和TM波的本征方程, 由傅里叶变换得到二维函数光子晶体介电常数 (r) 的傅里叶变换 (G), 其傅里叶变换比常规二维光子晶体的复杂. 计算发现当介质柱介电常数为常数时, 其傅里叶变换与常规二维光子晶体的相同, 因此二维常规光子晶体是二维函数光子晶体的特例. 在此基础上具体研究了二维函数光子晶体TE波和TM波的带隙结构, 其介质柱介电常数函数形式取为 (r) = kr + b, 其中k, b为可调的参数. 并与二维常规光子晶体TE波和TM波的带隙结构进行了比较, 发现二维函数光子晶体与二维常规光子晶体TE波和TM波的带隙结构有明显的区别, 二维函数光子晶体的带隙数目、位置以及宽度随参数k的变化而发生改变. 从而实现了二维函数光子晶体带隙结构的可调性, 为基于二维光子晶体的光学器件的设计提供了新的设计方法和重要的理论依据.

#### Abstract

Photonic crystal is a kind of periodic optical nanostructure consisting of two or more materials with different dielectric constants, which has attracted great deal of attention because of its wide range of potential applications in the field of optics. Photonic crystal can be fabricated into one-, or two-, or three- dimensional one. Among them, the two-dimensional photonic crystal turns into a hot focus due to its fantastic optical and electrical properties and relatively simple fabrication technique. Since the tunable band gaps of two-dimensional photonic crystals are beneficial to designing the novel optical devices, to study their optical and electrical properties for controlling the electromagnetic wave is quite valuable in both theoretical and practical aspects. In this work, we propose a new type of two-dimensional function photonic crystal, which can tune the band gaps of photonic crystals. The two-dimensional function photonic crystal is different from the traditional photonic crystal composed of medium columns with spatially invariant dielectric constants, since the dielectric constants of medium column are the functions of space coordinates. Specifically, the photorefractive nonlinear optical effect or electro-optic effect is utilized to turn the dielectric constant of medium column into the function of space coordinates, which results in the formation of two-dimensional function photonic crystal. We use the plane-wave expansion method to derive the eigen-equations for the TE and TM mode. By the Fourier transform, we obtain the Fourier transform form (G) for the dielectric constant function (r) of two-dimensional function photonic crystal, which is more complicated than the Fourier transform in traditional two-dimensional photonic crystal. The calculation results indicate that when the dielectric constant of medium column is a constant, the Fourier transforms for both of them are the same, which implies that the traditional two-dimensional photonic crystal is a special case for the two-dimensional function photonic crystal. Based on the above theory, we calculate the band gap structure of two-dimensional function photonic crystal, especially investigate in detail the corresponding band gap structures of TE and TM modes. The function of dielectric constant can be described as (r) = kr + b, in which k and b are adjustable parameters. Through comparing the calculation results for both kinds of photonic crystals, we can find that the band structures of TE and TM modes in two-dimensional function photonic crystals are quite different from those in traditional two-dimensional photonic crystal. Adjusting parameter k, we can successfully change the number, locations and widths of band gaps, indicating that the band gap structure of two-dimensional function photonic crystal is tunable. These results provide an important design method and theoretical foundation for designing optical devices based on two-dimensional photonic crystal.

#### 作者及机构信息

###### 通信作者: 吴向尧, wuxy2066@163.com
• 基金项目: 吉林省科技发展计划项目(批准号: 20130101031JC)资助的课题.

#### Authors and contacts

###### Corresponding author: Wu Xiang-Yao, wuxy2066@163.com
• Funds: Project supported by Scientific and Technological Development Foundation of Jilin Province (Grant No. 20130101031JC).

#### 参考文献

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#### 施引文献

•  [1] Yablonovitch E 1987 Phys. Rev. Lett. 58 2059 [2] John S 1987 Phys. Rev. Lett. 58 2486 [3] Benistyh H, Weisbuch C, Olivier S 2004 SPIE 5360 119 [4] Lou S Q, Wang Z, Ren G B 2004 Acta Opt. Sin. 24 313(in Chinese) [娄淑琴, 王智, 任国斌 2004 光学学报 24 313] [5] Shang P G, Sacharia A 2003 Opt. Express 11 167 [6] Yin J L, Huang X G, Liu S H 2007 Chin. J. Lasers 34 671 (in Chinese)[殷建玲, 黄旭光, 刘颂豪 2007 中国激光 34 671] [7] Wang H, Li Y P 2001 Acta Phys. Sin. 50 2172 (in Chinese) [王辉, 李永平 2001 物理学报 50 2172] [8] Zhao Y H, Qian C J, Qiu K S, Gao Y N, Xu X L 2015 Opt. Express 23 9211 [9] Li Z J, Zhang Y, Li B J 2006 Opt. Express 14 3887 [10] Geng T, Wu N, Dong X M, Gao X M 2016 Acta Phys. Sin. 65 014213 (in Chinese) [耿滔, 吴娜, 董祥美, 高秀敏 2016 物理学报 65 014213] [11] Susa N 2002 J. Appl. Phys. 91 3501 [12] Yu J L, Shen H J, Ye S, Hong Q S 2012 Acta Opt. Sin. 32 1106003 (in Chinese) [余建立, 沈宏君, 叶松, 洪求三 2012 光学学报 32 1106003] [13] Wang X, Chen L C, Liu Y H, Shi Y L, Sun Y 2015 Acta Phys. Sin. 64 174206 (in Chinese) [王晓, 陈立潮, 刘艳红, 石云龙, 孙勇 2015 物理学报 64 174206] [14] Klitzing V, Klaus 1986 Rev. Mod. Phys. 58 519 [15] Zhang X, Zhang H J, Wang J, Felser C, Zhang S C 2012 Science 335 1464 [16] Seng F L, Sebastian K, Wen X, Hui C 2015 Phys. Rev. A 91 023811 [17] Lu C, Li W, Jiang X Y, Cao J C 2014 Chin. Phys. B 23 097802 [18] Francesco M, Andrea A 2014 Chin. Phys. B 23 047809 [19] Zhang H Y, Gao Y, Zhang Y P, Wang S F 2011 Chin. Phys. B 20 094101 [20] Dai Y, Liu S B, Wang S Y, Kong X K, Chen C 2014 Chin. Phys. B 23 065202
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##### 出版历程
• 收稿日期:  2016-01-30
• 修回日期:  2016-04-17
• 刊出日期:  2016-07-05

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