Microscopic analysis of surface Bloch modes on periodically perforated metallic surfaces and their relation to extraordinary optical transmission

Xin Zhang, Haitao Liu, and Ying Zhong

Phys. Rev. B 89, 195431 – Published 21 May 2014

Abstract

We build up a microscopic description of the surface Bloch mode (SBM) on metallic surfaces patterned with a periodic array of subwavelength holes. The SBM also is called a spoof surface plasmon when metals approach perfect electric conductors at terahertz or microwave frequencies. In the description, the SBM is found to be composed of surface waves on flat metallic surfaces, the surface plasmon polariton (SPP), and the quasicylindrical wave (QCW) launched by every individual hole in the array, which shines new light on the design of perforated metallic surfaces as plasmonic metamaterials. We also establish explicit relations between the macroscopic SBM picture and the microscopic surface-wave picture for explaining the extraordinary optical transmission (EOT) through subwavelength metallic hole arrays. Our analysis shows that the SBM is related tightly to the EOT through a complex pole of the in-plane wave vector at which the SBM and the EOT possess similar field distributions expressed as superpositions of the SPPs and QCWs.

Received 27 February 2014
Revised 3 May 2014

DOI:https://doi.org/10.1103/PhysRevB.89.195431

Authors & Affiliations

Xin Zhang¹, Haitao Liu^1,*, and Ying Zhong²

¹Key Laboratory of Optical Information Science and Technology, Ministry of Education, Institute of Modern Optics, Nankai University, Tianjin 300071, China
²State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Tianjin 300072, China

^*Corresponding author: liuht@nankai.edu.cn

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Vol. 89, Iss. 19 — 15 May 2014

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Figure 1
(a) Microscopic description of the SBM supported by a metallic surface pierced with a two-dimensional periodic array of infinite-depth subwavelength holes. $P_{0}$ and $Q_{0}$ are the coefficients of the right-going and the left-going HWs that originate from every $y$ -periodic chain of holes in the array. (b) HW (black solid curves), its SPP component (blue dashed curves), and the residual QCW component (green dashed-dotted curves), launched on an air/gold interface by a subwavelength line scatterer at $x$ = $z$ = 0. The magnetic-field component Re( $H_{y}$ ) is shown for λ = 0.73, 1.04, 3.13, and 10.4 μm from top to bottom (vertically shifted by 4 for clarity). The complex value of $H_{y}$ at $x$ = λ is normalized to be 1. The refractive indices of gold at λ = 0.73, 1.04, 3.13, and 10.4 μm are 0.16 + 4.37 $i$ , 0.27 + 7.16 $i$ , 1.76 + 19.4 $i$ , and 13.2 + 56.9 $i$ , respectively [33]. (c) Scattering coefficients τ and ρ that characterize the transmission and reflection of HWs at a chain of holes. (d) Scattering coefficients β(± $k_{x}$ ) and $t$ that characterize the excitation of HWs and of the fundamental chain mode by the incident plane wave. (e) Scattering coefficients α and $r$ that characterize the excitation of HWs by an incident fundamental chain mode and its reflection.
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Figure 2
(a1) and (a2) Modulus of the transmission coefficient $t_{A}$ from an incident plane wave in air (polarized in the $x$ - $z$ plane) to the fundamental mode of a periodic array of subwavelength holes in gold substrate (see the inset). (b1) and (b2) Modulus of the reflection coefficient $r_{A}$ of the fundamental mode of the hole array (see the inset). The data in (a) and (b) are obtained for different wavelengths λ and incident angles θ. The array period is $a$ = 0.94 μm, and the side length of the square holes is 0.2 μm. The superimposed are the SBM dispersion curves of Re( $k_{SBM}$ ) for different wavelengths obtained with the fully vectorial FBMM (red circles), the HW model (yellow pluses), and the SPP model (green triangles). The dashed white lines represent the air light lines. The dispersion curves for Re( $k_{SBM}$ ) < 0, which are not shown in the figure, are symmetrical with those for Re( $k_{SBM}$ ) > 0 about the vertical axis Re( $k_{SBM}$ ) = 0. (c) Modulus of the imaginary part of the SBM propagation constant $k_{SBM}$ for different wavelengths λ. There are Im( $k_{SBM}$ ) > 0 and Im( $k_{SBM}$ ) < 0 for the right-going and the left-going SBMs, respectively. The results are obtained with the FBMM (red circles), the HW model (black solid curve), and the SPP model (blue dashed curve). The horizontal black dashed line in the inset represents the cross position between the right-going and the left-going SBMs [at which Re( $k_{SBM}$ ) = 0]. (d) Field ratio γ between the right-going and the left-going waves contained in the SBMs, which is plotted for different wavelengths. The data are obtained with the HW model (black solid curves) and the SPP model (blue dashed curves). The left and the right branches correspond to the left-going and the right-going SBMs, respectively.
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Figure 3
(a) and (b) Distribution of the magnetic field | $H_{y}$ ( $x$ , $a$ /2,0)| on a perforated gold surface ( $z$ = 0) and distribution of | $H_{y}$ ( $a$ /2, $a$ /2, $z$ )| along the vertical $z$ direction (at $x$ = $y$ = $a$ /2), which are obtained for right-going SBMs at different wavelengths (λ = 0.73, 1.04, 3.13, and 10.4 μm from top to bottom). The array period is $a$ = 0.9λ, and the side length of the square holes is 0.19λ. The total field of the SBM (red) and the SPP field (blue) contained in the SBM are obtained with the fully vectorial FBMM (circles) and the HW model (solid curves). The vertical green dashed lines in (a) represent the hole boundaries, and the vertical black dashed lines in (b) represent the air/gold interface. (c1) and (c2) $x$ - $y$ distribution of the SBM field on the gold surface ( $z$ = 0) obtained with the FBMM and the HW model, respectively. (c3) and (c4) Scattered fields at the resonance peaks of $t_{A}$ and $r_{A}$ , respectively, at which $k_{x} = Re (k_{SBM})$ and EOT happens. The results in (c1)–(c4) are obtained for λ = 1.04 μm and $a$ = 0.9λ.
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