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构成1维光子晶体的原始材料的折射率只在1维方向上周期性地变化,一个周期可以由几层不同折射率的原始材料组成。二次元1维光子晶体的周期是最简单的,仅由A和B两种原始材料组成,它们的折射率分别为nA和nB,整个光子晶体是由很多个同样的周期重复排列而成,如ABAB…ABAB,记为(AB)L,L为周期数。由于原料的折射率变化是严格周期性的,因此这种光子晶体可称之为1维本征光子晶体或1维无掺杂光子晶体。目前,这种光子晶体一般附着于玻璃等透明衬底上而不能独立存在,其结构如图 1所示。
分析光子晶体的光学特性的方法有许多[3, 11-18],而对于1维光子晶体来说,采用传输特征矩阵法是非常合适的[12, 15, 19],假定光波从折射率为n0的入射介质入射到由均匀、各向同性、非磁性的介质组成的1维光子晶体(多层膜系)的表面上,光子晶体制作在折射率为ng的透明衬底上(如图 1中所示的玻璃衬底),则由传输特征矩阵法得到光子晶体表面的能量反射率为:
$ R=\left|\frac{\eta_{0}-Y}{\eta_{0}+Y}\right|^{2} $
(1) 式中, η0为入射介质的导纳, Y是光子晶体与衬底的组合体的等效导纳。
$ Y=\frac{C}{B} $
(2) 式中, B和C构成光子晶体与衬底的组合传输矩阵的矩阵元素,表示为:
$ \left[\begin{array}{l} B \\ C \end{array}\right]=\left\{\prod \limits_{j=1}^{2 L}\left[\begin{array}{cc} \cos \delta_{j} & \mathrm{i} \sin \delta_{j} / \eta_{j} \\ \mathrm{i} \eta_{j} \sin \delta_{j} & \cos \delta_{j} \end{array}\right]\right\}\left[\begin{array}{l} 1 \\ \eta_{\mathrm{g}} \end{array}\right] $
(3) 式中, ηj和ηg分别是第j层膜和衬底介质的导纳。在垂直入射的情况下,导纳与折射率相等:ηi=c/vi(i=0, j, g), vi为入射介质中、第j层膜中和衬底介质中的光速,c为真空中光速。当ηg=1时,Y是纯光子晶体的导纳; 在可见光、近红外波段,Y为光子晶体的折射率。
$ \boldsymbol{T}_{j}=\left[\begin{array}{cc} \cos \delta_{j} & \operatorname{isin} \delta_{j} / \eta_{j} \\ \mathrm{i} \eta_{j} \sin \delta_{j} & \cos \delta_{j} \end{array}\right] $
(4) 式中,Tj为第j层薄膜的特征传输矩阵, δj为第j层膜的位相厚度,定义为:
$ {\delta _j} = \frac{{2{\rm{ \mathit{ π} }}{n_j}{d_j}}}{\lambda }\cos {\theta _j} $
(5) 式中,λ为入射光波的波长, dj为第j层膜的厚度, θj为夹角, nj为第j层膜的折射率, nj=c/vj。
对于图 1所示的结构,每个周期内有两层膜,则周期的传输矩阵变为:
$ \boldsymbol{M}=\boldsymbol{T}_{\mathrm{A}} \boldsymbol{T}_{\mathrm{B}}=\left[\begin{array}{cc} m_{11} & \mathrm{i} m_{12} \\ \mathrm{i} m_{21} & m_{22} \end{array}\right] $
(6) 矩阵元分别为:
$ \left\{\begin{array}{l} m_{11}=\cos \delta_{A} \cos \delta_{\mathrm{B}}-\frac{\eta_{\mathrm{B}}}{\eta_{\mathrm{A}}} \sin \delta_{\mathrm{A}} \sin \delta_{\mathrm{B}} \\ m_{12}=\frac{\cos \delta_{\mathrm{A}} \sin \delta_{\mathrm{B}}}{\eta_{\mathrm{B}}}+\frac{\sin \delta_{\mathrm{A}} \cos \delta_{\mathrm{B}}}{\eta_{\mathrm{A}}} \\ m_{21}=\eta_{\mathrm{A}} \sin \delta_{\mathrm{A}} \cos \delta_{\mathrm{B}}+\eta_{\mathrm{B}} \cos \delta_{\mathrm{A}} \sin \delta_{\mathrm{B}} \\ m_{22}=\cos \delta_{\mathrm{A}} \cos \delta_{\mathrm{B}}-\frac{\eta_{\mathrm{A}}}{\eta_{\mathrm{B}}} \sin \delta_{\mathrm{A}} \sin \delta_{\mathrm{B}} \end{array}\right. $
(7) 很明显,关系式m11m22+m12m21=1成立。则对于结构如图 1所示的1维光子晶体,(3)式变为:
$ \left[\begin{array}{l} B \\ C \end{array}\right]=\boldsymbol{M}^{L}\left[\begin{array}{c} 1 \\ \eta_{\mathrm{g}} \end{array}\right] $
(8) 式中, L是指数,表示周期,即存在2L层膜。令E表示单位矩阵,即:
$ \boldsymbol{E}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] $
(9) 由(8)式递推得:ML=UL-1(χ)M-UL-2(χ)E,从而有:
$ \begin{array}{c} \boldsymbol{M}^{L}= \\ {\left[\begin{array}{cc} m_{11} U_{L-1}(\chi)-U_{L-2}(\chi) & \mathrm{i} m_{12} U_{L-1}(\chi) \\ \mathrm{i} m_{21} U_{L-1}(\chi) & m_{22} U_{L-1}(\chi)-U_{L-2}(\chi) \end{array}\right]} \end{array} $
(10) 式中, UL-1(χ)、UL-2(χ)是第L-1阶次、第L-2阶次第2类切比雪夫多项式。于是(3)式变为:
$ \begin{array}{c} \left[\begin{array}{c} B \\ C \end{array}\right]=\\ \left[\begin{array}{c} {\left[m_{11} U_{L-1}(\chi)-U_{L-2}(\chi)\right]+\operatorname{i} \eta_{\mathrm{g}} m_{12} U_{L-1}(\chi)} \\ \operatorname{i}m_{21} U_{L-1}(\chi)+\eta_{\mathrm{g}}\left[m_{22} U_{L-1}(\chi)-U_{L-2}(\chi)\right] \end{array}\right] \end{array} $
(11) 代入(2)式得:
$ \begin{array}{c} Y = \frac{C}{B} = \\ \frac{{{\eta _{\rm{g}}}\left[ {{m_{22}}{U_{L - 1}}(\chi ) - {U_{L - 2}}(\chi )} \right] + {\rm{i}}{m_{21}}{U_{L - 1}}(\chi )}}{{\left[ {{m_{11}}{U_{L - 1}}(\chi ) - {U_{L - 2}}(\chi )} \right] + {\rm{i}}{\eta _{\rm{g}}}{m_{12}}{U_{L - 1}}(\chi )}} \end{array} $
(12) 由上式可见,等效导纳Y是复数。而Y的实部为:
$ \begin{array}{*{20}{c}} {{Y_{{\rm{real }}}} = }\\ {\frac{{{\eta _{\rm{g}}}\left[ {{U_{L - 1}}^2(\chi ) - 2\chi {U_{L - 1}}(\chi ){U_{L - 2}}(\chi ) + {U_{L - 1}}^2(\chi )} \right]}}{{{{\left[ {{m_{11}}{U_{L - 1}}(\chi ) - {U_{L - 2}}(\chi )} \right]}^2} + {{\left[ {{\eta _{\rm{g}}}{m_{12}}{U_{L - 1}}(\chi )} \right]}^2}}}} \end{array} $
(13) 利用第2类切比雪夫多项式通项表示式和各阶次第2类切比雪夫多项式之间的递推关系[20],(13)式变为:
$ \begin{array}{c} Y_{\text {real }}= \\ \frac{\eta_{\mathrm{g}}\left[U_{L-1}^{2}(\chi)-U_{L}(\chi) U_{L-2}(\chi)\right]}{\left[m_{11} U_{L-1}(\chi)-U_{L-2}(\chi)\right]^{2}+\left[\eta_{\mathrm{g}} m_{12} U_{L-1}(\chi)\right]^{2}} \end{array} $
(14) 由上面的讨论可知,等效导纳Y及其实部Yreal都是波长λ的函数,即Y=Y(λ),Yreal=Yreal(λ)。结合第2类切比雪夫多项式的特性,通过(14)式或(13)式可见,1维光子晶体的周期数达到一定数值时,等效导纳的实部Yreal(λ)在某波长范围内的取值将趋近于或等于零。
二次元多周期1维光子晶体带隙的实质
The essence of the band gap of 1-D photonic crystal with period consisting of two elements
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摘要: 为了讨论1维光子晶体带隙的本质, 利用传输特征矩阵法分析了二次元多周期1维光子晶体的反射率, 并推导出等效折射率实部的表达式。以氟化镁(MgF2)和硫化锌(ZnS)构成的二次元多周期的1维光子晶体为例, 进行了数值计算, 绘制了反射率及等效折射率实部的曲线, 并进行了分析。结果表明, 反射率为1.0的波长区间与等效折射率实部为零的波长区间相同; 对于带隙范围内的光波而言, 1维光子晶体的等效折射率的实部等于或趋近于零时, 1维光子晶体是虚等效折射率材料。该研究对二次元多周期1维光子晶体的研究是有帮助的。Abstract: In order to discuss the essence of 1-D photonic crystal band gap, the reflectivity of 1-D photonic crystal consisting of two elements was analyzed, and the expression of the real part of the equivalent refractive index was derived by using the transmission characteristic matrix method. Taking the 1-D photonic crystal composed of magnesium fluoride (MgF2) and zinc sulfide (ZnS) as an example, the analytical numerical calculations of the reflectivity and the real part of the equivalent refractive index were done and the corresponding curves were drawn. It was the conclusion that, wavelength range with reflectivity of 1.0 is the same as that with real part of equivalent refractive index of zero. For the light waves in the band gap, the real part of the equivalent refractive index of 1-D photonic crystal is equal to or close to zero, namely, 1-D photonic crystal is virtual equivalent index material in the extent of band gap. In other words, the optical essence of photonic crystal band gap is that photonic crystal becomes virtual equivalent index material. This study is helpful for the study of 1-D photonic crystal with multi periods consisting of two elements.
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Key words:
- physical optics /
- photonic crystal /
- band gap /
- imaginary refractive index
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[1] YABLONOVICH E. Inhibited spontaneous emission in solid state physics and electronics[J]. Physical Review Letters, 1987, 58(20): 2059-2061. doi: 10.1103/PhysRevLett.58.2059 [2] JOHN S. Strong localization of photons in certain disordered dielectric superlattices[J]. Physical Review Letters, 1987, 58(23): 2486-2489. doi: 10.1103/PhysRevLett.58.2486 [3] JOANNOPOULOS J D, MEADE R D, WINN J N. Photonic crystals: Molding the flow of light[M]. Princeton, USA: Princeton University Press, 1995: 7. [4] KAVOKIN A V, SHELYH I A, MALPUEC H A. Lossless interface modes at the boundery between two periodic dielectric structures[J]. Physics Review, 2005, B72(23): 233102. [5] JIANG Y, ZHANG W L, ZHU Y Y. Optical Tamm state theory study on asymmetric[J]. Acta Physica Sinica, 2013, 62(16): 167303(in Chinese). doi: 10.7498/aps.62.167303 [6] CHEN Y, DONG J, LIU T, et al. Study on the sensing mechanism of photonic crystal containing metal layer based on the coupling analysis of optical Tamm state[J]. Chinese Journal of Lasers, 2015, 42(11): 111400(in Chinese). [7] SHUKLA M K, DAS R. Tamm-plasmon polaritons in one-dimensional photonic quasi-crystals[J]. Optics Letters, 2018, 43(3): 362-365. doi: 10.1364/OL.43.000362 [8] CHEN X F, LI Sh J, ZHANG Y, et al. The wide-angle perfect absorption based on the optical Tamm states[J]. Optoelectronics Lett-ers, 2018, 10(4): 317-320. [9] ALBERT J P, JOUANIN C, CASSAGE D. Photonic crystal modeling using a tight-binding Wannier function method[J]. Optical and Quantum Electronics, 2002, 34(1/3): 251-263. doi: 10.1023/A:1013393918768 [10] TANG J F, GU P F, LIU X, et al. Modern optical thin film techno-logy[M]. Hangzhou: Zhejiang University Press, 2006: 20-33(in Chinese). [11] ALIVIZATOS E G, ALIVIZATOS E G, CHREMMOS I D, et al. Green's-function method for the analysis of propagation in holey fibers[J]. Journal of the Optical Society of America, 2004, A21(5): 847-857. [12] YABLONOVITCH E, GMITTER T J, LEUNG K M. Photonic band structure: The face-centered-cubic case employing nonspherical a-toms[J]. Physical Review Letters, 1991, 67(17/21): 2295-2297. [13] PENDRY J B. Calculation of photon dispersion relations[J]. Physical Review Letters, 1992, 69(19): 2772-2776. doi: 10.1103/PhysRevLett.69.2772 [14] WANG H, LI Y P. An eigen matrix method for obtaining the band structure of photonic crystals[J]. Acta Physica Sinica, 2011, 50(11): 2172-2178(in Chinese). [15] WU X Y, MA J, LIU X J, et al. Quantum theory of photonic crystals[J]. Physica, 2014, E59(5): 174-180. [16] FAN X Zh, YI Y Y, CHEN Q M, et al. On the equivalence of the transfer matrix method for investigating one dimensional photonic crystal with the fresnel coefficient matrix method for analyzing optical thin film[J]. Laser Journal, 2014, 35(7): 26-29(in Chinese). [17] ZHU H X, YE T, ZHANG K F. Study on high sensitivity pressure sensing characteristics of photonic crystal fiber[J]. Laser Technology, 2019, 43(4): 511-516(in Chinese). [18] PENG R R, LIU B, CHEN J. Study on side-hole surface plasmon resonance refractive index sensing based on single-core photonic crystal optical fiber[J]. Laser Technology, 2018, 42(5): 713-717(in Chinese). [19] BORN M, WOLF E. Principles of optics[M]. 7th ed. London, UK: Cambridge University Press, 1999: 38-74. [20] SHI B J. Determinants of RFPrLrR circulant matrices of the Chebyshev polynomials[J]. Pure and Applied Mathematics, 2016, 32(3): 304-317(in Chinese). [21] JOHNSON P B, HRISTY R W C. Optical constants of the noble metals[J]. Physics Review, 1972, B6(12): 4370-4379. [22] ORDAL M A, LONG L L, BELL R J, et al. Optical properties of the metals Al, Co, Cu, Au, Fe, Pb, Ni, Pd, Pt, Ag, Ti, and W in the infrared and far infrared[J]. Applied Optics, 1983, 22(7): 1099-1120. doi: 10.1364/AO.22.001099