亚波长介质光栅设计简化模式法的修正及相关研究
|
摘要
亚波长介质光栅是伴随纳米制造技术发展起来的一种新型光栅。它具有高衍射效率的特点，因此在光通讯、光计算、全息摄影、光学度量、激光脉冲压缩等光学元器件中有重要应用,相关研究受到广泛关注。为提高亚波长介质光栅设计的计算效率，提高其结构性能的可预见性，本文重点开展了其简化算法的研究。
在对影响严格耦合波法（RCWA）计算精度的因素进行分析的基础上，提出采用该方法所获光栅衍射效率作为光栅设计理论值的条件及误差参考。
对已有简化模式法（SMM）进行了深入分析并与RCWA计算结果进行了对比研究，找到了已有的SMM算法计算误差较大的原因；以三角形介质光栅为例，提出了一种从多层膜反射和薄膜导纳刻画各个模式反射率的修正算法（MSMM），利用该新型物理模型修正后的计算结果与RCWA的计算结果对比分析表明，修正后的MSMM方法与RCWA的计算偏差大大缩小。
利用该方法，通过计算光栅导纳随光栅周期的变化规律，分别对矩形和三角形光栅的衍射效率进行了研究。矩形光栅导纳随着占宽比变化仅在空气导纳附近较小范围内，说明亚波长矩形介质光栅的反射率和光栅的占宽比相关性较小；通过分析得出矩形介质光栅无法得到接近100%的透射效率。给出了入射角一定的情况下，不同周期的三角形光栅反射率的变化规律。以此为基础设计出TE偏振和TM偏振同时能达到衍射效率超过99.9%的三角形介质光栅。
应用MSMM分析了光栅衍射效率和光栅结构参数之间的关系，并优化设计出双端口光栅分束器。研究了光栅的矩形结构、三角形结构在偏振无关、偏振相关情形下的分束性能。设计得到的偏振无关光栅分束器在1020nm-1100nm波段，TE波和TM波的0级和-１级衍射效率都在50%左右；偏振相关分束器在该波段的消光比也都在10以上。
对于亚波长三角形、正弦形和矩形介质光栅，该方法将衍射效率表达成相位差偏移和光栅与空气导纳之差引起的反射，比RCWA更直观的展示了光栅衍射的物理过程。相比只提供初值的SMM，MSMM可以直接用来进行优化计算。因此该算法对于加深对该类介质光栅的认识以及设计优化具有积极作用。
Dielectric subwavelength grating is a new type grating developing with evolutionof nano technology. Because of high efficiency, it is used frequently in opticalcommunication, computation, metrology, holography, laser pulse compression andso on. Many researchers focused on this issue. For the purpose of increasing thespeed of calculation in designing dielectric subwavelength gratings and giving agood prediction of grating structure, the simplified method will be researched in thispaper.
In this paper, the factors of computational accuracy are analyzed when therigorous coupled wave analysis (RCWA) method is used to design a grating. Thisanalysis shows the result that RCWA could be used to be a reference as a theoreticalvalue for other algorithm in grating design.
The simplified modal method (SMM) is analyzed and the deviation betweenSMM and RCWA is calculated to find the reason of its big error. Take a triangulargrating as an example, which is seen as multilayer films and the each layer owns anew effective index. The reflection and transmission of grating could becharacterized by optical admittance of the films. Then formulas of gratingtransmission in SMM are modified by reflection. The calculation by this modifiedsimplified modal method (MSMM) is a better approximation to RCWA than othermethods.
Based on MSMM, for a rectangular grating, although the admittance changeswith duty cycle, its variation range is very near the admittance of air. This means thereflection of rectangular grating will not change a lot when duty cycle is changed.And a rectangular grating cann’t get transmission efficiency near100%. Then themode indices and admittance versus the period of triangular gratings and the dutycycle of rectangular grating are analyzed. A grating with transmission efficiencymore than99.9%for both TE and TM-polarized light is designed.
The relations between grating structure and transmission efficiency areanalyzed by MSMM. The splitting result of rectangular and triangular grating areanalyzed for both TE and TM polarization.Based on these calculations,1×2beamsplitters are designed and optimized by MSMM. A polarization-independent splittergrating will get a good splitting result at the band1020nm-1100nm. Thetransmission efficiency of0th and-1st order for both TE and TM polarization is nearto50%at this band. The extinction ratio of a polarization-dependent splitter is biggerthan10at this band.
Contrast to RCWA, MSMM gives a clear physical explanation of gratingdiffraction by considering the transmission efficiency as the result of phase shift ofmodes and difference of grating admittance and aerial admittance. This method canbe applicable to triangular, sinusoidal and rectangular gratings. Because it can beused to optimize the grating parameters directly, MSMM is better than SMM whichonly gives starting value of grating parameters. Thus MSMM gives a goodunderstanding on dielectric subwavelength gratings and is useful in designing andoptimizing them.
在对影响严格耦合波法（RCWA）计算精度的因素进行分析的基础上，提出采用该方法所获光栅衍射效率作为光栅设计理论值的条件及误差参考。
对已有简化模式法（SMM）进行了深入分析并与RCWA计算结果进行了对比研究，找到了已有的SMM算法计算误差较大的原因；以三角形介质光栅为例，提出了一种从多层膜反射和薄膜导纳刻画各个模式反射率的修正算法（MSMM），利用该新型物理模型修正后的计算结果与RCWA的计算结果对比分析表明，修正后的MSMM方法与RCWA的计算偏差大大缩小。
利用该方法，通过计算光栅导纳随光栅周期的变化规律，分别对矩形和三角形光栅的衍射效率进行了研究。矩形光栅导纳随着占宽比变化仅在空气导纳附近较小范围内，说明亚波长矩形介质光栅的反射率和光栅的占宽比相关性较小；通过分析得出矩形介质光栅无法得到接近100%的透射效率。给出了入射角一定的情况下，不同周期的三角形光栅反射率的变化规律。以此为基础设计出TE偏振和TM偏振同时能达到衍射效率超过99.9%的三角形介质光栅。
应用MSMM分析了光栅衍射效率和光栅结构参数之间的关系，并优化设计出双端口光栅分束器。研究了光栅的矩形结构、三角形结构在偏振无关、偏振相关情形下的分束性能。设计得到的偏振无关光栅分束器在1020nm-1100nm波段，TE波和TM波的0级和-１级衍射效率都在50%左右；偏振相关分束器在该波段的消光比也都在10以上。
对于亚波长三角形、正弦形和矩形介质光栅，该方法将衍射效率表达成相位差偏移和光栅与空气导纳之差引起的反射，比RCWA更直观的展示了光栅衍射的物理过程。相比只提供初值的SMM，MSMM可以直接用来进行优化计算。因此该算法对于加深对该类介质光栅的认识以及设计优化具有积极作用。
Dielectric subwavelength grating is a new type grating developing with evolutionof nano technology. Because of high efficiency, it is used frequently in opticalcommunication, computation, metrology, holography, laser pulse compression andso on. Many researchers focused on this issue. For the purpose of increasing thespeed of calculation in designing dielectric subwavelength gratings and giving agood prediction of grating structure, the simplified method will be researched in thispaper.
In this paper, the factors of computational accuracy are analyzed when therigorous coupled wave analysis (RCWA) method is used to design a grating. Thisanalysis shows the result that RCWA could be used to be a reference as a theoreticalvalue for other algorithm in grating design.
The simplified modal method (SMM) is analyzed and the deviation betweenSMM and RCWA is calculated to find the reason of its big error. Take a triangulargrating as an example, which is seen as multilayer films and the each layer owns anew effective index. The reflection and transmission of grating could becharacterized by optical admittance of the films. Then formulas of gratingtransmission in SMM are modified by reflection. The calculation by this modifiedsimplified modal method (MSMM) is a better approximation to RCWA than othermethods.
Based on MSMM, for a rectangular grating, although the admittance changeswith duty cycle, its variation range is very near the admittance of air. This means thereflection of rectangular grating will not change a lot when duty cycle is changed.And a rectangular grating cann’t get transmission efficiency near100%. Then themode indices and admittance versus the period of triangular gratings and the dutycycle of rectangular grating are analyzed. A grating with transmission efficiencymore than99.9%for both TE and TM-polarized light is designed.
The relations between grating structure and transmission efficiency areanalyzed by MSMM. The splitting result of rectangular and triangular grating areanalyzed for both TE and TM polarization.Based on these calculations,1×2beamsplitters are designed and optimized by MSMM. A polarization-independent splittergrating will get a good splitting result at the band1020nm-1100nm. Thetransmission efficiency of0th and-1st order for both TE and TM polarization is nearto50%at this band. The extinction ratio of a polarization-dependent splitter is biggerthan10at this band.
Contrast to RCWA, MSMM gives a clear physical explanation of gratingdiffraction by considering the transmission efficiency as the result of phase shift ofmodes and difference of grating admittance and aerial admittance. This method canbe applicable to triangular, sinusoidal and rectangular gratings. Because it can beused to optimize the grating parameters directly, MSMM is better than SMM whichonly gives starting value of grating parameters. Thus MSMM gives a goodunderstanding on dielectric subwavelength gratings and is useful in designing andoptimizing them.
引文
[1] M. C. Hutley, Diffraction Gratings[M]. New York: Academic Press,1982,p3-5
[2] George R. Harrison and George W. Stroke，Interferometric control of gratingruling with continuous carriage advance[J]. J. Opt. Soc. Am.1995，45(2):112-121
[3] A. Labeyrie and J. Flamand. Spectrographic performance of holographically madediffraction gratings[J]. Opt. Comm.，1969,1(l):5-8
[4]A.Labeyrie and J. Flamand. Diffraction gratings through hologrpahy[J]. Opt.Spectra.1969:1-50
[5]T. Kita and T. Harada. Ruling engine using a piezoelectric device for large andhigh-groove density gratings[J]. Appl. Opt,1992,31:1399.
[6]时轮，郝德阜，齐向东.高精度衍射光栅刻划机的最新技术进展[J].仪器仪表学报[J].2001,22(4):438一439.
[7]巴音贺希格.衍射光栅色散理论与光栅设计、制作和检验方法研究[D]：［博士学位论文］．长春：中国科学院长春光学精密机械与物理研究所，2004
[8]王鹏.蚀刻光栅的矢量耦合波分析与菲涅耳衍射理论[D]：［博士学位论文］.上海：中国科学院上海光学精密机械研究所，1999
[9]伊德尔,严瑛白,谭峭峰等.亚波长光栅用于实现宽光谱消色散1/4波片的研究[J].中国激光,2003,30(5):405-408
[10]D. C. Flanders. Submicrometer periodicity grating as artificial anisotropicdielectrics [J]. Appl. Phys. Lett.,1983,42(6):492-494
[11] R. C. Enger, S. K. Case. Optical elements with ultrahigh spatial-frequencysurface corrugations [J]. Appl. Opt.,1983,22(20):3220-3228
[12] L. H. Cescato, E. Gluch, N. Streibl. Holographic quarterwave plates[J]. Appl.Opt.,1990,29(22):3286-3290
[13] E. Hasman, Z. Bomzon, A. Niv et al.. Polarization beam-splitters and opticalswitches based on space-variant computer-generated subwavelength quasi-periodic structures[J]. Opt. Commun.,2002,209(1-3):45~54
[14]卢向东,傅克祥,王植恒等.用亚波长正弦介质光栅设计1/4波片[J].激光技术,2003,27(4):368-370
[15] Z. Bomzon, G. Biener, V. Kleiner et al.. Radially and azimuthally polarizedbeams generated by space-variant dielectric subwavelength gratings[J]. Opt. Lett.,2002,27(5):285-287
[16]A. Niv, Biener, G. Kleiner V. Kleiner et al.. Spiral phase elements obtained by useof discrete space-variant subwavelength gratings[J]. Opt. Commun.,2005,251(4-6):306-314
[17] G. Biener, A. Niv, V. Kleiner et al.. Near-field Fourier transform polarimetry byuse of a discrete space-variant subwavelength grating[J]. J. Opt. Soc. Am. A,2003,20(10):1940-1948
[18] L. F. Li. Multilayer modal method for diffraction gratings of arbitrary profile,depth and permittivity [J]. J. Opt. Soc.Am. A,1993,10(12):2581-2591
[19] G. Cincotti. Polarization gratings: design and applications[J]. IEEE J.Quantum.Electron.,2003,39(12):1645-1652
[20] J. A. Davis, J. Adachi, R. Carlos et al.. Polarization beam splitters usingpolarization diffraction gratings[J]. Opt. Lett.,2001,26(9):587-589
[21] L. F. Li. Formulation and comparison of two recursive matrix algorithms formodeling layered diffraction gratings[J].J. Opt. Soc. Am. A,1996,13(5):1024-1035
[22] Z. Bomzon, V. Kleiner, E. Hasman et al.. Computer-generated space-variantpolarization elements with subwavelength metal stripes[J]. Opt. Lett.,2001,26(1):33-35
[23] F. Gori. Measuring Stokes parameters by means of a polarization grating[J]. Opt.Lett.,1999,24(9):584-586
[24] H. Lajunen, J. Tervo, J. Turunen. High-efficiency broadband diffractive elementsbased on polarization gratings [J]. Opt. Lett.,2004,29(8):803-805
[25]张亮,李承芳,刘文等.一种亚波长偏振分波/合波器的研制[J].光学学报,2006,26(7):1048-1052
[26] Z. Bomzon, G. Biener, V. Kleiner et al.. Real-time analysis of partially polarizedlight with a space-variant subwavelength dielectric grating[J]. Opt. Lett.,2002,27(3):188-190
[27] A. G. Lopez, H. G. Graighead. Wave-plate polarizing beam splitter based on aform-birefringent multilayer grating[J].Opt. Lett.,1998,23(20):1627-1629
[28] T.Clausnitzer, J. Limpert, K. Z llner, H. Zellmer, H.Fuchs, E.Kley,A.Tünnermann, M. Jupé, and D. Ristau, Highly efficient transmission gratings infused silica for chirped-pulse amplification systems[J], Appl. Opt.2003,42:6934-6938
[29] T. Wu, C. Zhou, J. Zheng, J. Feng, H. Cao, L. Zhu, and W. Jia, Generation ofdouble femtosecond pulses by using two transmissive gratings[J], Appl. Opt.2010,49:4506-4513
[30] R. C. Enger and S. K. Case, Optical elements with high ultrahighspatial-frequency surface corrugations[J],Appl. Opt.1983,22:3220–3228.
[31].J.Neauport,E.Lavastre,G.Razé,G.Dupuy,N.Bonod,M.Balas,G.deVillele,J.Flamad,S.Kaladgew,andF.Desserouer, Effect of electric field on laser induced damagethreshold of multilayer dielectric gratings[J], Opt. Express,2007,15:2508–12522
[32] F. J. Wen and P. S. Chung, Design of a high-efficiency seven-port beam splitterusing a dual duty cycle grating structure[J], Appl. Opt.2011,50(19):4506-4513
[33]M. W. Farn, Binary gratings with increased efficiency[J],Appl. Opt.1992,31:4453–4458
[34]D. L. Brundrett, E. N. Glytsis, and T. K. Gaylord, Subwavelength transmissiongrating retarders for use at10:6m[J],Appl. Opt.1996,35:6195–6202
[35]季家镕.高等光学教程[M]科学出版社.2009.p153
[36].P.C.Waterman，Scattering by Periodic surfaces[J]，J.Acoust.Soe.Am.,1975.57:791-802.
[37]S.L.Chuang and J.A.Kong, Wave scattering from a period dielectric surface for ageneral angle of incidence[J]，Radio Science，1982，17(2):545-557·
[38]S.Savaidis，P.Frangos，D.L.Jaggard，K.Hizanidis，Scattering from fractallycorrugated surfaces with use of the extended boundary condition method[J]，J.Opt.Soc.Am.A，1997，14:475-485.
[39]Y.Ohkawa，Y.Tsuji，and M.Koshiba，Analysis of anisotropic dieleetric gratingdiffraction using the finite-element method[J]， J.Opt.Soc.Am.A，1996，13:1006-1012.
[40]B.Rieherzhagen，Finite element ray tracing: A new method for ray tracing ingradient-index media[J]，Appl.Opt，1996，35(31):6186-6189.
[41]X.Peng，A.Asundi，Y.Chen，Zh.Xiong，Study of the mechanical properties of Nd:YVO4Crystal by use of laser interferometry and finite-element analysis[J]，Appl.Opt，2001，40(9):1396-1403.
[42]J.Chandezon，M.T.Dupuis，G.Cornet，and D.Maystre. Multicoated gratings: adifferential formalism applicable in the entire optical region[J]. J.Opt.Soc.Am，1982，72(7):839-846
[43]L.Li，J.Chandezon，G.Granet，and J.Plumey, Rigorous and efficientgrating-analysis method made easy for optical engineers[J]. Appl.Opt.1999，38(2):304-313
[44]M. G. Moharam and T. K. Gaylord, Diffraction analysis of dielectricsurface-relief gratings[J], J. Opt. Soc. Am.1982,72:1385-1392
[45]M. G. Moharam and T. K. Gaylord, Rigorous coupled-wave analysis of metallicsurface-relief gratings[J], J. Opt. Soc. Am. A,1986,3:1780-1787
[46] N. M. Lyndin, O. Parriaux,and A.V. Tishchenko, Modal analysis and suppressionof the Fourier modal method instabilities in highly conductive gratings [J], J. Opt.Soc. Am. A.2007,24(12):3781-3788
[47] S. Campbell, L. C. Botten,R. C. McPhedran, and C. M. Sterke. Modal method forconical diffraction by slanted lamellar gratings [J], J. Opt. Soc. Am. A.2009,26(4):938-948.
[48]M. G. Moharam, D. A. Pommet, E. B. Grann and T. K. Gaylord, Stableimplementation of the rigorous coupled-wave analysis for surface-relief gratings:enhanced transmittance matrix approach[J], J. Opt. Soc. Am. A.1995,12:1077-1086
[49]L. Li and C. W. Haggans, Convergence of the coupled-wave method for metalliclamellar diffraction gratings[J], J. Opt. Soc.Am. A,1993,10:1184-1189.
[50]P. Lalanne and G. M. Morris,Highly improved convergence of the coupled-wavemethod for TM polarization[J], J. Opt. Soc.Am. A,1996,13:779-784.
[51] L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, and J. R. Andrewartha,The dielectric lamellar diffraction grating[J], Opt. Acta1981,28:413-428.
[52] L. C. Botten, M. S. Craig, R. C. Mcphedran, et al., The finitely conductinglamellar diffraction grating, Opt. Acta,[J]28:1087-1102
[53] D.Taillaert, P. Bienstman, and R. Baets, Compact efficient broadband gratingcoupler for silicon-on-insulator waveguides[J]. OPT. Lett.2004,29:2749-2751
[54] M. R. Zunoubi and H. A. Kalhor, Diffraction of electromagnetic waves byperiodic arrays of rectangular cylinders[J],2006, J. Opt. Soc. Am. A,23(2):306-313
[55]樊叔维，周庆华，李红槽型衍射光栅结构参数优化设计研究[J].光学学报,2010,30(11):3133-3139
[56] L. Zhang, H. Lin,C. Jin, H. Zhou and T. Huo, Broadband multilayer-coatednormal incidence blazed grating with~10%diffraction efficiency through the13–16nm wavelength region [J]. OPT. Lett.2009,34(6):818-820
[57] J. Wang, Y. Jin, J. Ma, T. Sun, and X.Jing, Design and analysis of broadbandhigh-efficiency pulse compression gratings [J].APPL.OPT.2010,49(16),2969-2978
[58] H. Haidner, J.T.Sheridan and N. Streibl, Dielectric binary blazed gratings[J],APPL.OPT.1993,33(22):4276-4278
[59] A. V. Tishchenko, Phenomenological representation of deep and high contrastlamellar gratings by means of the modal method[J], Opt. QuantumElectron.,2005,37:309-330.
[60] T.Clausnitzer, T.K mpfe, E.-B.Kley, et al., An intelligible explanation ofhighly-efficient diffraction in deep dielectric rectangular transmission gratings[J],Opt. Express,2005,13:10448-10456
[61] T. Clausnitzer, T. K mpfe, E.-B. Kley, A. Tünnermann, A. Tishchenko, and O.Parriaux, Investigation of the polarization-dependent diffraction of deep dielectricrectangular transmission gratings illuminated in Littrow mounting[J], Appl. Opt.2007,46:819-826
[62] B. Wang, C. Zhou, J. Feng, H. Ru, and J. Zheng, Wideband two-port beamsplitter of a binary fused-silica phase grating[J], APPL.OPT.2008,47(22):4004-4008
[63] J. Zheng, C. Zhou, B. Wang, and J. Feng, Beam splitting of low-contrast binarygratings under the second Bragg angle incidence[J], J.Opt. Soc. Am. A2008,25(5),1075-1083.
[64] J. Zheng, C. Zhou, J. Feng, and B. Wang, Polarizing beam splitter of deep-etchedtriangular-groove fused silica gratings [J], Opt. Lett.2008,33(14):1554-1556
[65] C. Zhou, Simplified modal method for subwavelength gratings [J] Proc.SPIE,2012,8564:856412-1.
[66] J.Zheng, C. Zhou, J. Feng, H. Cao and P.Lu, A metal-mirror-based reflectingpolarizing beam splitter[J], J. Opt. A: Pure Appl. Opt.2009,11:015710
[67] J. Feng, C. Zhou, H. Cao, and P. Lv, Deep-etched sinusoidal polarizing beamsplitter grating [J],Appl. Opt.,2010,49(10):1739-1743
[68] J. Feng, C. Zhou, J. Zheng, H. Cao, and P. Lv, Design and Fabrication of apolarization-independent two-port beam splitter [J].Appl. Opt.2009,48(29):5636-5641.
[69] H. Cao, C. Zhou, J. Feng, P. Lv, and J. Ma, Design and fabrication of apolarization-independent wideband transmission fused-silica grating [J] Appl. Opt.2010,49(21):4108-4112.
[70] J. Feng, C. Zhou, J. Zheng, H. Cao, and P. Lv, Dual-function beam splitter of asubwavelength fused silica grating [J] Appl. Opt.2009,48(14):2697-2701.
[71] J. Feng, C. Zhou, J. Zheng and B. Wang, Modal analysis of deep-etchedlow-contrast two-port beam splitter grating [J], Opt. Commun.2008,281:5298-5301.
[72] A. Hu, C. Zhou, H. Cao, J. Wu, J. Yu and W. Jia, Modal analysis ofhigh-efficiency wideband reflective gratings[J], J. Opt.2012,14:055705
[73] W. Sun, P. Lv, C. Zhou, J. Wu and S. Wang, Modal analysis of the0th ordernulled phase masks [J],Chin. Opt. Lett.2013,11(7):070502
[74] A. Hu, C. Zhou, H. Cao, J. Wu, J. Yu, W. Jia, Polarization-independent widebandmixed metal dielectric reflective gratings[J],Appl. Opt.2012,51(20):4902-4906.
[75] J. Wu, C. Zhou, H. Cao, A. Hu,W. Sun, and W. Jia, Simplified mode analysis ofguided mode resonance gratings with asymmetric coatings [J],Chin. Opt. Lett.2013,11(6):060501
[76] J. Feng, C. Zhou, B. Wang, J. Zheng, W. Jia, H. Cao, and P. Lv,Three-port beamsplitter of a binary fused silica grating[J]Appl. Opt.2008,47(35):6638-6643.
[77] J. Feng, C. Zhou, H. Cao, and P. Lu, Unified design of sinusoidal groove fusedsilica grating[J], Appl. Opt.2010,49(30):5697-5704.
[78] P. Lv, C. Zhou, J. Feng, and H. Cao, Unified Design of Wavelength-IndependentDeep-Etched Fused-Silica Gratings [J] Opt. Commun.2010,283:4135-4140.
[79] H. Cao, C. Zhou, J. Feng, and J. Ma,“Polarization-independent triangular-groovefused-silica gratings with high efficiency at a wavelength of1550nm [J],Opt.Commun.2010,283:4271-4273.
[80] X. Jing, J. Zhang, S. Jin, P. Liang, and Y. Tian, Design of highly efficienttransmission gratings with deep etched triangular grooves[J], Appl. Opt.2012,51(33):7920-7933.
[81] J. Liu, H. Gao, J. Zhou, D. Liu, The design of a polarizing beam splitter madefrom a dielectric rectangular-groove grating[J], Opt. Laser Technol.2009,41:622–626
[82] B. Wang, L. Lei, L. Chen, J. Zhou, Connecting-layer-based polarizing beamsplitter grating with high efficiency for both TE and TM polarizations[J], Opt.Laser Technol.2009,44:2145-2148
[83] B. Wang, L. Chen, L. Lei, J. Zhou, Two-layer dielectric grating forpolarization-selective beam splitter with improved efficiency and bandwidth[J],Opt. Commun.2013,294:329-332.
[84] B. Wang, L. Chen, L. Lei, J. Zhou, Two-Layer Dielectric Grating as Two-PortBeam Splitter[J],IEEE Potonic. Tech. L.2013,25(9):863-866
[85] H. Zhao and D. Yuan, Design of fused-silica rectangular transmissiongratings for polarizing beam splitter based on modal method[J], Appl. Opt.2010,49(5):759-763.
[86] X, Jing, S. Jin, J. Zhang, et al., Enhancement of the accuracy of the simplifiedmodal method for designing a subwavelength triangular grooves grating[J], Opt.Lett.2013,38(1):10-12
[87] X. Jing, J. Zhang, Y. Tian, and S. Jin, Improvement of the validity of thesimplified modal method for designing a subwavelength dielectric transmissiongrating [J], APPL.OPT.2014,53(2):259-268
[88] W. Sun, P. Lv, C. Zhou, H. Cao, and J. Wu, Multireflection modal method forwideband fused-silica transmission gratings, APPL.OPT.52,2800-2807,2013
[89]E.H.Cho, H.S.Kim, B.H.Cheong, et al.,Two-dimensional photonic crystal colorfilter development[J], Opt. Express.2009,17(10):8621-8629
[90]Y.Yeh, S.H.Park, Fiber-optic tunable filter with a concave mirror [J], Opt. Lett.2012,37(4):626-628
[91]R.Magnusson, Y.Ding, MEMS tunable resonant leaky mode filers [J],IEEEPhoton Technol. Lett.2006,18(14):1479-1481
[92] P. Sheng, R. S. Stepleman, and P. N. Sanda, Exact eigenfunctions forsquare-wave gratings: application to diffraction and surface-plasmoncalculations[J] Phys. Rev. B26(6),2907–2917(1982).
[93]唐晋发，顾培夫，刘旭等，现代光学薄膜技术[M]浙江大学出版社，2006，p22.
[94] J. Kim, J. R. Birge, V. Sharma, et al., Ultrabroadband beam splitter with matchedgroup-delay dispersion[J], Opt. Lett.2005,30:1569–1571
[2] George R. Harrison and George W. Stroke，Interferometric control of gratingruling with continuous carriage advance[J]. J. Opt. Soc. Am.1995，45(2):112-121
[3] A. Labeyrie and J. Flamand. Spectrographic performance of holographically madediffraction gratings[J]. Opt. Comm.，1969,1(l):5-8
[4]A.Labeyrie and J. Flamand. Diffraction gratings through hologrpahy[J]. Opt.Spectra.1969:1-50
[5]T. Kita and T. Harada. Ruling engine using a piezoelectric device for large andhigh-groove density gratings[J]. Appl. Opt,1992,31:1399.
[6]时轮，郝德阜，齐向东.高精度衍射光栅刻划机的最新技术进展[J].仪器仪表学报[J].2001,22(4):438一439.
[7]巴音贺希格.衍射光栅色散理论与光栅设计、制作和检验方法研究[D]：［博士学位论文］．长春：中国科学院长春光学精密机械与物理研究所，2004
[8]王鹏.蚀刻光栅的矢量耦合波分析与菲涅耳衍射理论[D]：［博士学位论文］.上海：中国科学院上海光学精密机械研究所，1999
[9]伊德尔,严瑛白,谭峭峰等.亚波长光栅用于实现宽光谱消色散1/4波片的研究[J].中国激光,2003,30(5):405-408
[10]D. C. Flanders. Submicrometer periodicity grating as artificial anisotropicdielectrics [J]. Appl. Phys. Lett.,1983,42(6):492-494
[11] R. C. Enger, S. K. Case. Optical elements with ultrahigh spatial-frequencysurface corrugations [J]. Appl. Opt.,1983,22(20):3220-3228
[12] L. H. Cescato, E. Gluch, N. Streibl. Holographic quarterwave plates[J]. Appl.Opt.,1990,29(22):3286-3290
[13] E. Hasman, Z. Bomzon, A. Niv et al.. Polarization beam-splitters and opticalswitches based on space-variant computer-generated subwavelength quasi-periodic structures[J]. Opt. Commun.,2002,209(1-3):45~54
[14]卢向东,傅克祥,王植恒等.用亚波长正弦介质光栅设计1/4波片[J].激光技术,2003,27(4):368-370
[15] Z. Bomzon, G. Biener, V. Kleiner et al.. Radially and azimuthally polarizedbeams generated by space-variant dielectric subwavelength gratings[J]. Opt. Lett.,2002,27(5):285-287
[16]A. Niv, Biener, G. Kleiner V. Kleiner et al.. Spiral phase elements obtained by useof discrete space-variant subwavelength gratings[J]. Opt. Commun.,2005,251(4-6):306-314
[17] G. Biener, A. Niv, V. Kleiner et al.. Near-field Fourier transform polarimetry byuse of a discrete space-variant subwavelength grating[J]. J. Opt. Soc. Am. A,2003,20(10):1940-1948
[18] L. F. Li. Multilayer modal method for diffraction gratings of arbitrary profile,depth and permittivity [J]. J. Opt. Soc.Am. A,1993,10(12):2581-2591
[19] G. Cincotti. Polarization gratings: design and applications[J]. IEEE J.Quantum.Electron.,2003,39(12):1645-1652
[20] J. A. Davis, J. Adachi, R. Carlos et al.. Polarization beam splitters usingpolarization diffraction gratings[J]. Opt. Lett.,2001,26(9):587-589
[21] L. F. Li. Formulation and comparison of two recursive matrix algorithms formodeling layered diffraction gratings[J].J. Opt. Soc. Am. A,1996,13(5):1024-1035
[22] Z. Bomzon, V. Kleiner, E. Hasman et al.. Computer-generated space-variantpolarization elements with subwavelength metal stripes[J]. Opt. Lett.,2001,26(1):33-35
[23] F. Gori. Measuring Stokes parameters by means of a polarization grating[J]. Opt.Lett.,1999,24(9):584-586
[24] H. Lajunen, J. Tervo, J. Turunen. High-efficiency broadband diffractive elementsbased on polarization gratings [J]. Opt. Lett.,2004,29(8):803-805
[25]张亮,李承芳,刘文等.一种亚波长偏振分波/合波器的研制[J].光学学报,2006,26(7):1048-1052
[26] Z. Bomzon, G. Biener, V. Kleiner et al.. Real-time analysis of partially polarizedlight with a space-variant subwavelength dielectric grating[J]. Opt. Lett.,2002,27(3):188-190
[27] A. G. Lopez, H. G. Graighead. Wave-plate polarizing beam splitter based on aform-birefringent multilayer grating[J].Opt. Lett.,1998,23(20):1627-1629
[28] T.Clausnitzer, J. Limpert, K. Z llner, H. Zellmer, H.Fuchs, E.Kley,A.Tünnermann, M. Jupé, and D. Ristau, Highly efficient transmission gratings infused silica for chirped-pulse amplification systems[J], Appl. Opt.2003,42:6934-6938
[29] T. Wu, C. Zhou, J. Zheng, J. Feng, H. Cao, L. Zhu, and W. Jia, Generation ofdouble femtosecond pulses by using two transmissive gratings[J], Appl. Opt.2010,49:4506-4513
[30] R. C. Enger and S. K. Case, Optical elements with high ultrahighspatial-frequency surface corrugations[J],Appl. Opt.1983,22:3220–3228.
[31].J.Neauport,E.Lavastre,G.Razé,G.Dupuy,N.Bonod,M.Balas,G.deVillele,J.Flamad,S.Kaladgew,andF.Desserouer, Effect of electric field on laser induced damagethreshold of multilayer dielectric gratings[J], Opt. Express,2007,15:2508–12522
[32] F. J. Wen and P. S. Chung, Design of a high-efficiency seven-port beam splitterusing a dual duty cycle grating structure[J], Appl. Opt.2011,50(19):4506-4513
[33]M. W. Farn, Binary gratings with increased efficiency[J],Appl. Opt.1992,31:4453–4458
[34]D. L. Brundrett, E. N. Glytsis, and T. K. Gaylord, Subwavelength transmissiongrating retarders for use at10:6m[J],Appl. Opt.1996,35:6195–6202
[35]季家镕.高等光学教程[M]科学出版社.2009.p153
[36].P.C.Waterman，Scattering by Periodic surfaces[J]，J.Acoust.Soe.Am.,1975.57:791-802.
[37]S.L.Chuang and J.A.Kong, Wave scattering from a period dielectric surface for ageneral angle of incidence[J]，Radio Science，1982，17(2):545-557·
[38]S.Savaidis，P.Frangos，D.L.Jaggard，K.Hizanidis，Scattering from fractallycorrugated surfaces with use of the extended boundary condition method[J]，J.Opt.Soc.Am.A，1997，14:475-485.
[39]Y.Ohkawa，Y.Tsuji，and M.Koshiba，Analysis of anisotropic dieleetric gratingdiffraction using the finite-element method[J]， J.Opt.Soc.Am.A，1996，13:1006-1012.
[40]B.Rieherzhagen，Finite element ray tracing: A new method for ray tracing ingradient-index media[J]，Appl.Opt，1996，35(31):6186-6189.
[41]X.Peng，A.Asundi，Y.Chen，Zh.Xiong，Study of the mechanical properties of Nd:YVO4Crystal by use of laser interferometry and finite-element analysis[J]，Appl.Opt，2001，40(9):1396-1403.
[42]J.Chandezon，M.T.Dupuis，G.Cornet，and D.Maystre. Multicoated gratings: adifferential formalism applicable in the entire optical region[J]. J.Opt.Soc.Am，1982，72(7):839-846
[43]L.Li，J.Chandezon，G.Granet，and J.Plumey, Rigorous and efficientgrating-analysis method made easy for optical engineers[J]. Appl.Opt.1999，38(2):304-313
[44]M. G. Moharam and T. K. Gaylord, Diffraction analysis of dielectricsurface-relief gratings[J], J. Opt. Soc. Am.1982,72:1385-1392
[45]M. G. Moharam and T. K. Gaylord, Rigorous coupled-wave analysis of metallicsurface-relief gratings[J], J. Opt. Soc. Am. A,1986,3:1780-1787
[46] N. M. Lyndin, O. Parriaux,and A.V. Tishchenko, Modal analysis and suppressionof the Fourier modal method instabilities in highly conductive gratings [J], J. Opt.Soc. Am. A.2007,24(12):3781-3788
[47] S. Campbell, L. C. Botten,R. C. McPhedran, and C. M. Sterke. Modal method forconical diffraction by slanted lamellar gratings [J], J. Opt. Soc. Am. A.2009,26(4):938-948.
[48]M. G. Moharam, D. A. Pommet, E. B. Grann and T. K. Gaylord, Stableimplementation of the rigorous coupled-wave analysis for surface-relief gratings:enhanced transmittance matrix approach[J], J. Opt. Soc. Am. A.1995,12:1077-1086
[49]L. Li and C. W. Haggans, Convergence of the coupled-wave method for metalliclamellar diffraction gratings[J], J. Opt. Soc.Am. A,1993,10:1184-1189.
[50]P. Lalanne and G. M. Morris,Highly improved convergence of the coupled-wavemethod for TM polarization[J], J. Opt. Soc.Am. A,1996,13:779-784.
[51] L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, and J. R. Andrewartha,The dielectric lamellar diffraction grating[J], Opt. Acta1981,28:413-428.
[52] L. C. Botten, M. S. Craig, R. C. Mcphedran, et al., The finitely conductinglamellar diffraction grating, Opt. Acta,[J]28:1087-1102
[53] D.Taillaert, P. Bienstman, and R. Baets, Compact efficient broadband gratingcoupler for silicon-on-insulator waveguides[J]. OPT. Lett.2004,29:2749-2751
[54] M. R. Zunoubi and H. A. Kalhor, Diffraction of electromagnetic waves byperiodic arrays of rectangular cylinders[J],2006, J. Opt. Soc. Am. A,23(2):306-313
[55]樊叔维，周庆华，李红槽型衍射光栅结构参数优化设计研究[J].光学学报,2010,30(11):3133-3139
[56] L. Zhang, H. Lin,C. Jin, H. Zhou and T. Huo, Broadband multilayer-coatednormal incidence blazed grating with~10%diffraction efficiency through the13–16nm wavelength region [J]. OPT. Lett.2009,34(6):818-820
[57] J. Wang, Y. Jin, J. Ma, T. Sun, and X.Jing, Design and analysis of broadbandhigh-efficiency pulse compression gratings [J].APPL.OPT.2010,49(16),2969-2978
[58] H. Haidner, J.T.Sheridan and N. Streibl, Dielectric binary blazed gratings[J],APPL.OPT.1993,33(22):4276-4278
[59] A. V. Tishchenko, Phenomenological representation of deep and high contrastlamellar gratings by means of the modal method[J], Opt. QuantumElectron.,2005,37:309-330.
[60] T.Clausnitzer, T.K mpfe, E.-B.Kley, et al., An intelligible explanation ofhighly-efficient diffraction in deep dielectric rectangular transmission gratings[J],Opt. Express,2005,13:10448-10456
[61] T. Clausnitzer, T. K mpfe, E.-B. Kley, A. Tünnermann, A. Tishchenko, and O.Parriaux, Investigation of the polarization-dependent diffraction of deep dielectricrectangular transmission gratings illuminated in Littrow mounting[J], Appl. Opt.2007,46:819-826
[62] B. Wang, C. Zhou, J. Feng, H. Ru, and J. Zheng, Wideband two-port beamsplitter of a binary fused-silica phase grating[J], APPL.OPT.2008,47(22):4004-4008
[63] J. Zheng, C. Zhou, B. Wang, and J. Feng, Beam splitting of low-contrast binarygratings under the second Bragg angle incidence[J], J.Opt. Soc. Am. A2008,25(5),1075-1083.
[64] J. Zheng, C. Zhou, J. Feng, and B. Wang, Polarizing beam splitter of deep-etchedtriangular-groove fused silica gratings [J], Opt. Lett.2008,33(14):1554-1556
[65] C. Zhou, Simplified modal method for subwavelength gratings [J] Proc.SPIE,2012,8564:856412-1.
[66] J.Zheng, C. Zhou, J. Feng, H. Cao and P.Lu, A metal-mirror-based reflectingpolarizing beam splitter[J], J. Opt. A: Pure Appl. Opt.2009,11:015710
[67] J. Feng, C. Zhou, H. Cao, and P. Lv, Deep-etched sinusoidal polarizing beamsplitter grating [J],Appl. Opt.,2010,49(10):1739-1743
[68] J. Feng, C. Zhou, J. Zheng, H. Cao, and P. Lv, Design and Fabrication of apolarization-independent two-port beam splitter [J].Appl. Opt.2009,48(29):5636-5641.
[69] H. Cao, C. Zhou, J. Feng, P. Lv, and J. Ma, Design and fabrication of apolarization-independent wideband transmission fused-silica grating [J] Appl. Opt.2010,49(21):4108-4112.
[70] J. Feng, C. Zhou, J. Zheng, H. Cao, and P. Lv, Dual-function beam splitter of asubwavelength fused silica grating [J] Appl. Opt.2009,48(14):2697-2701.
[71] J. Feng, C. Zhou, J. Zheng and B. Wang, Modal analysis of deep-etchedlow-contrast two-port beam splitter grating [J], Opt. Commun.2008,281:5298-5301.
[72] A. Hu, C. Zhou, H. Cao, J. Wu, J. Yu and W. Jia, Modal analysis ofhigh-efficiency wideband reflective gratings[J], J. Opt.2012,14:055705
[73] W. Sun, P. Lv, C. Zhou, J. Wu and S. Wang, Modal analysis of the0th ordernulled phase masks [J],Chin. Opt. Lett.2013,11(7):070502
[74] A. Hu, C. Zhou, H. Cao, J. Wu, J. Yu, W. Jia, Polarization-independent widebandmixed metal dielectric reflective gratings[J],Appl. Opt.2012,51(20):4902-4906.
[75] J. Wu, C. Zhou, H. Cao, A. Hu,W. Sun, and W. Jia, Simplified mode analysis ofguided mode resonance gratings with asymmetric coatings [J],Chin. Opt. Lett.2013,11(6):060501
[76] J. Feng, C. Zhou, B. Wang, J. Zheng, W. Jia, H. Cao, and P. Lv,Three-port beamsplitter of a binary fused silica grating[J]Appl. Opt.2008,47(35):6638-6643.
[77] J. Feng, C. Zhou, H. Cao, and P. Lu, Unified design of sinusoidal groove fusedsilica grating[J], Appl. Opt.2010,49(30):5697-5704.
[78] P. Lv, C. Zhou, J. Feng, and H. Cao, Unified Design of Wavelength-IndependentDeep-Etched Fused-Silica Gratings [J] Opt. Commun.2010,283:4135-4140.
[79] H. Cao, C. Zhou, J. Feng, and J. Ma,“Polarization-independent triangular-groovefused-silica gratings with high efficiency at a wavelength of1550nm [J],Opt.Commun.2010,283:4271-4273.
[80] X. Jing, J. Zhang, S. Jin, P. Liang, and Y. Tian, Design of highly efficienttransmission gratings with deep etched triangular grooves[J], Appl. Opt.2012,51(33):7920-7933.
[81] J. Liu, H. Gao, J. Zhou, D. Liu, The design of a polarizing beam splitter madefrom a dielectric rectangular-groove grating[J], Opt. Laser Technol.2009,41:622–626
[82] B. Wang, L. Lei, L. Chen, J. Zhou, Connecting-layer-based polarizing beamsplitter grating with high efficiency for both TE and TM polarizations[J], Opt.Laser Technol.2009,44:2145-2148
[83] B. Wang, L. Chen, L. Lei, J. Zhou, Two-layer dielectric grating forpolarization-selective beam splitter with improved efficiency and bandwidth[J],Opt. Commun.2013,294:329-332.
[84] B. Wang, L. Chen, L. Lei, J. Zhou, Two-Layer Dielectric Grating as Two-PortBeam Splitter[J],IEEE Potonic. Tech. L.2013,25(9):863-866
[85] H. Zhao and D. Yuan, Design of fused-silica rectangular transmissiongratings for polarizing beam splitter based on modal method[J], Appl. Opt.2010,49(5):759-763.
[86] X, Jing, S. Jin, J. Zhang, et al., Enhancement of the accuracy of the simplifiedmodal method for designing a subwavelength triangular grooves grating[J], Opt.Lett.2013,38(1):10-12
[87] X. Jing, J. Zhang, Y. Tian, and S. Jin, Improvement of the validity of thesimplified modal method for designing a subwavelength dielectric transmissiongrating [J], APPL.OPT.2014,53(2):259-268
[88] W. Sun, P. Lv, C. Zhou, H. Cao, and J. Wu, Multireflection modal method forwideband fused-silica transmission gratings, APPL.OPT.52,2800-2807,2013
[89]E.H.Cho, H.S.Kim, B.H.Cheong, et al.,Two-dimensional photonic crystal colorfilter development[J], Opt. Express.2009,17(10):8621-8629
[90]Y.Yeh, S.H.Park, Fiber-optic tunable filter with a concave mirror [J], Opt. Lett.2012,37(4):626-628
[91]R.Magnusson, Y.Ding, MEMS tunable resonant leaky mode filers [J],IEEEPhoton Technol. Lett.2006,18(14):1479-1481
[92] P. Sheng, R. S. Stepleman, and P. N. Sanda, Exact eigenfunctions forsquare-wave gratings: application to diffraction and surface-plasmoncalculations[J] Phys. Rev. B26(6),2907–2917(1982).
[93]唐晋发，顾培夫，刘旭等，现代光学薄膜技术[M]浙江大学出版社，2006，p22.
[94] J. Kim, J. R. Birge, V. Sharma, et al., Ultrabroadband beam splitter with matchedgroup-delay dispersion[J], Opt. Lett.2005,30:1569–1571