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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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[2] Smith D R, Pendry J B, Wiltshire M C 2004 Science 305 788
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[3] Liu Y, Zhang X 2011 Chem. Soc. Rev. 40 2494
Google Scholar
[4] Zheludev N I 2010 Science 328 582
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[5] Lai Y, Ng J, Chen H, Han D, Xiao J, Zhang Z, Chan C T 2009 Phys. Rev. Lett. 102 253902
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[6] Liberal I, Engheta N 2017 Nat. Photonics 11 149
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[7] Niu X, Hu X, Chu S, Gong Q 2018 Adv. Opt. Mater. 2018 1701292
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[8] Luo J, Lu W, Hang Z, Chen H, Hou B, Lai Y, Chan C T 2014 Phys. Rev. Lett. 112 73903
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[9] Luo J, Hang Z H, Chan C T, Lai Y 2015 Laser Photonics Rev. 9 523
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[10] Luo J, Liu B, Hang Z H, Lai Y 2018 Laser Photonics Rev. 2018 1800001
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[11] Luo J, Li J, Lai Y 2018 Phys. Rev. X 8 31035
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[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
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[14] Ni X, Emani N K, Kildishev A V, Boltasseva A, Shalaev V M 2012 Science 335 427
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[15] Sun S, He Q, Xiao S, Xu Q, Li X, Zhou L 2012 Nat. Mater. 11 426
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[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
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[17] Sun W, He Q, Sun S, Zhou L 2016 Light-Sci. Appl. 5 e16003
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[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
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[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
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[27] Luo J, Lai Y 2019 Opt. Express 27 15800
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[28] Li S, Wang Y, Zhang W, Lu W, Hou B, Luo J, Lai Y 2020 New J. Phys. 22 023033
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[29] Huang X, Lai Y, Hang Z H, Zheng H, Chan C T 2011 Nat. Mater. 10 582
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[30] Moitra P, Yang Y, Anderson Z, Kravchenko I I, Briggs D P, Valentine J 2013 Nat. Photonics 7 791
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[31] Li Y, Kita S, Muñoz P, Reshef O, Vulis D I, Yin M, Lončar M, Mazur E 2015 Nat. Photonics 9 738
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[32] Maxwell G J C 1904 Philos. Trans. R. Soc. London, Ser. A 203 385
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[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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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Figure 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
$fa/c = 0.3427$ )Figure 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
Figure 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)Figure 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
Figure 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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