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有效介质理论在利用人工微结构材料拓展光学参数方面具有重要意义. 本文对电介质光子晶体等具有非局域性质的人工微结构材料发展了一种新的赝局域有效介质理论, 通过局域的有效介电常数
${\overleftrightarrow \varepsilon ^{\rm{p}}}\left( \omega \right)$ 、局域的有效磁导率${\overleftrightarrow \mu ^{\rm{p}}}(\omega)$ 、以及额外的波矢${{{k}}_a}$ 来描述其光学性质. 研究发现, 该赝局域有效介质兼具局域和非局域介质的性质, 在与${{{k}}_a}$ 垂直的晶面上表现出局域介质的光学性质, 而在与${{{k}}_a}$ 平行的晶面则表现出非局域介质的光学性质, 如负折射、全反射等. 进一步研究表明, 对于所有入射角的光波在穿过拥有奇数层结构单元的赝局域有效介质时, 都会出现额外的$\text{π}$ 相位差, 基于此设计了一种全角度相位光栅. 相对于传统的光学材料, 赝局域介质具有更加丰富有趣的光学性质, 有望在未来应用到更多的新型光学器件设计之中.Effective medium theory is of great importance for using the artificial microstructure materials to extend the optical parameters. In this article, we develop a new kind of effective medium theory for artificial microstructures with nonlocal effects, like photonic crystals, which we name the pseudo-local effective medium theory. The optical properties of the pseudo-local effective medium are described by effective local permittivity${\overleftrightarrow \varepsilon ^{\rm{p}}}\left( \omega \right)$ and permeability${\overleftrightarrow \mu ^{\rm{p}}}\left( \omega \right)$ , together with an additional wave vector${{{k}}_a}$ . We find that the pseudo-local medium exhibits a unique blend of local and nonlocal characteristics. On the surface normal to${{{k}}_a}$ , the pseudo-local medium is optically equivalent to its local medium counterpart. While on the surface parallel to${{{k}}_a}$ , the abnormal wave phenomena induced by inherent nonlocality, such as negative refraction and total reflection, may occur. Furthermore, it is found that a$\text{π}$ phase shift is added to transmission wave through the pseudo-local medium composed of odd number of unit cells under all incident angles. Based on this unique feature, an all-angle phase grating is proposed. Our work opens a route towards the advanced optical devices based on the pseudo-local effective media.[1] Pendry J B 2000 Phys. Rev. Lett. 85 3966
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图 1 (a) 用于实现PLM的电介质光子晶体结构单元; (b) 横电偏振下光子晶体的能带图; (c)第二支能带对应的等频率曲线, 其中红色曲线为
$fa/c = 0.3{{4}}27$ 时的等频率曲线; (d) 光子晶体的赝局域有效参数Fig. 1. (a) Illustration of the unit cell of the dielectric photonic crystal for the realization of PLM; (b) band structures of the photonic crystal for transverse-electric polarization; (c) equal frequency contours in the second band; the red lines denote the equal frequency contour at
$fa/c = 0.3{{4}}27$ ; (d) pseudo-local effective parameters of the photonic crystal.
图 2 横电偏振的高斯波以30°入射角从空气照射到(a) PLM板和(b) LMC板上的电场分布图(工作频率为
$fa/c = 0.3427$ )Fig. 2. Electric field-distribution for a transverse electric-polarized Gaussian beam incident from air onto the (a) PLM plate and (b) LMC plate under 30°-incidence at
$fa/c = 0.3427$ .
图 3 (a) 不同厚度的PLM板在LMC背景下的透射率随入射角的变化; (b) LMC背景下将一电单极光源置于PLM板左侧时的电场分布图
Fig. 3. (a) Transmittance through the PLM plate as the function of the incident angle in the LMC background; (b) electric field-distribution when an electric monopole source is placed on the left side of the PLM plate in the LMC background
图 4 在频率(a)
$fa/c = 0.3427$ 和(b)$fa/c = 0.3{556}$ 下, 光子晶体构造的PLM (红色曲线)和背景介质(灰色曲线)的等频率曲线(左图), 以及横电偏振的高斯光以25°入射角照射时的电场分布图(右图)Fig. 4. Left: equal frequency contours of the photonic crystal-based PLM (red) and the background medium (gray) at (a)
$fa/c = 0.3427$ and (b)$fa/c = 0.3{556}$ . Right: electric fields-distribution for a transverse electric-polarized Gaussian beam incident from the background medium onto the PLM under 25°-incidence at (a)$fa/c = 0.3427$ and (b)$fa/c = 0.3{556}$ .
图 5 (a) 基于PLM的全角度相位光栅示意图; (b) 横电偏振的平面波在10° (左)、45° (中)和60° (右)入射角下的电场分布图
Fig. 5. (a) Illustration of an all-angle phase grating based on the PLM; (b) electric field-distributions for transverse electric-polarized plane waves incident onto the phase grating under 10°- (left), 45°- (middle) and 60°- (right) incidences.
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[1] Pendry J B 2000 Phys. Rev. Lett. 85 3966
Google Scholar
[2] Smith D R, Pendry J B, Wiltshire M C 2004 Science 305 788
Google Scholar
[3] Liu Y, Zhang X 2011 Chem. Soc. Rev. 40 2494
Google Scholar
[4] Zheludev N I 2010 Science 328 582
Google Scholar
[5] Lai Y, Ng J, Chen H, Han D, Xiao J, Zhang Z, Chan C T 2009 Phys. Rev. Lett. 102 253902
Google Scholar
[6] Liberal I, Engheta N 2017 Nat. Photonics 11 149
Google Scholar
[7] Niu X, Hu X, Chu S, Gong Q 2018 Adv. Opt. Mater. 2018 1701292
Google Scholar
[8] Luo J, Lu W, Hang Z, Chen H, Hou B, Lai Y, Chan C T 2014 Phys. Rev. Lett. 112 73903
Google Scholar
[9] Luo J, Hang Z H, Chan C T, Lai Y 2015 Laser Photonics Rev. 9 523
Google Scholar
[10] Luo J, Liu B, Hang Z H, Lai Y 2018 Laser Photonics Rev. 2018 1800001
Google Scholar
[11] Luo J, Li J, Lai Y 2018 Phys. Rev. X 8 31035
Google Scholar
[12] Chu H, Li Q, Liu B, Luo J, Sun S, Hang Z H, Zhou L, Lai Y 2018 Light-Sci. Appl. 7 50
Google Scholar
[13] Yu N, Genevet P, Kats M A, Aieta F, Tetienne J, Capasso F, Gaburro Z 2011 Science 334 333
Google Scholar
[14] Ni X, Emani N K, Kildishev A V, Boltasseva A, Shalaev V M 2012 Science 335 427
Google Scholar
[15] Sun S, He Q, Xiao S, Xu Q, Li X, Zhou L 2012 Nat. Mater. 11 426
Google Scholar
[16] Sun S, Yang K, Wang C, Juan T, Chen W T, Liao C Y, He Q, Xiao S, Kung W, Guo G, Zhou L, Tsai D P 2012 Nano Lett. 12 6223
Google Scholar
[17] Sun W, He Q, Sun S, Zhou L 2016 Light-Sci. Appl. 5 e16003
Google Scholar
[18] Wang S, Wu P C, Su V, Lai Y, Chu C H, Chen J, Lu S, Chen J, Xu B, Kuan C, Li T, Zhu S, Tsai D P 2017 Nat. Commun. 8 187
Google Scholar
[19] Xu Y, Fu Y, Chen H 2016 Nat. Rev. Mater. 1 16067
Google Scholar
[20] He Q, Sun S, Xiao S, Zhou L 2018 Adv. Opt. Mater. 2018 1800415
Google Scholar
[21] Joannopoulos J D, Villeneuve P R, Fan S 1997 Nature 386 143
Google Scholar
[22] Yablonovitch E 1987 Phys. Rev. Lett. 58 2059
Google Scholar
[23] John S 1987 Phys. Rev. Lett. 58 2486
Google Scholar
[24] Yao Z, Luo J, Lai Y 2016 Opt. Lett. 41 5106
Google Scholar
[25] Luo J, Yang Y, Yao Z, Lu W, Hou B, Hang Z H, Chan C T, Lai Y 2016 Phys. Rev. Lett. 117 223901
Google Scholar
[26] Yao Z, Luo J, Lai Y 2017 Opt. Express 25 30931
Google Scholar
[27] Luo J, Lai Y 2019 Opt. Express 27 15800
Google Scholar
[28] Li S, Wang Y, Zhang W, Lu W, Hou B, Luo J, Lai Y 2020 New J. Phys. 22 023033
Google Scholar
[29] Huang X, Lai Y, Hang Z H, Zheng H, Chan C T 2011 Nat. Mater. 10 582
Google Scholar
[30] Moitra P, Yang Y, Anderson Z, Kravchenko I I, Briggs D P, Valentine J 2013 Nat. Photonics 7 791
Google Scholar
[31] Li Y, Kita S, Muñoz P, Reshef O, Vulis D I, Yin M, Lončar M, Mazur E 2015 Nat. Photonics 9 738
Google Scholar
[32] Maxwell G J C 1904 Philos. Trans. R. Soc. London, Ser. A 203 385
Google Scholar
[33] Bruggeman D A G 1935 Ann. Phys.-Berlin 416 636
Google Scholar
[34] Wu Y, Li J, Zhang Z Q, Chan C T 2006 Phys. Rev. B 74 85111
Google Scholar
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