Properties of spoof plasmon in thin structures

Abstract

Spoof surface plasmon polariton (SSPP) is an exotic electromagnetic state that confines light at a subwavelength scale at a design-specific frequency. It has been known for a while that spoof plasmon mode can exist in planar, thin structures with dispersion properties similar to that of its wide three-dimensional structure counterpart. We, however, have shown that spoof plasmons in thin structures possess some unique properties that remain unexplored. Our analysis reveals that the field interior to SSPP waveguide can achieve an exceptional hyperbolic spatial dependence, which can explain why spoof plasma resonance incurs red-shift with the reduction of the waveguide thickness, whereas common wisdom suggests frequency blue-shift of a resonant structure with its size reduction. In addition, we show that strong confinement can be achieved over a wide band in thin spoof plasmon structure, ranging from the spoof plasma frequency up to a lower frequency considerably away from the resonant point. The nature of lateral confinement in thin SSPP structures may enable interesting applications involving fast modulation rate due to enhanced sensitivity of optical modes without compromising modal confinement.

Keywords: band modulation; dispersion; frequency red-shift; planar structure; spoof plasmon.

Figure 1.

(a) Schematics of a 3D SSPP structure standing alone and a two-dimensional (2D) SSPP structure on substrate. (b) Planar SSPP seen from a lateral viewpoint. The groove is very long along z direction. (c)(i) Illustration of effective thickness of SSPP groove. The optical mode formed between the pair of conductors extends beyond the physical thickness of metal. (ii) 3D simulation result of the mode formation inside the groove, justifying the concept of effective thickness.

Figure 2.

(a) Comparison between dispersion diagram of three different SSPP structure: 3D SSPP (with large thickness), planar 2D SSPP with finite thickness (t = 2a), and thin film SSPP (with $t\to 0$). The other geometric parameters, i.e. groove length (h), groove width a, and periodicity d are taken as h = d ≈ 10a, and the external environment in all three cases consists of air. (b) Comparison between dispersion diagram of thin film SSPP structure ($t\to 0$) for two different cases: waveguide placed in air, and placed on a silicon substrate. The other geometric parameters are taken as h = d ≈ 10a. The strong impact of a substrate on shaping the dispersion characteristics of a thin film SSPP structure is vivid. The solid lines are from the developed theoretical model in this paper, while the discrete dots (filled circle) and (open circle) are obtained by FDTD numerical simulation in COMSOL Multiphysics, v. 5.2; Comsol, Inc.

Figure 3.

(a) Illustration of confined E_x field profile in 2D transverse Y Z plane for a unit cell of SSPP waveguide, while the mode propagates along X direction. Field profile along Y axis is labelled as laterally confined field (red in colour), and that along Z direction as vertically confined field (blue in colour). (b) Simulation result of the profile of electric field component (E_x) taken along a line crossing through an SSPP unit cell of 8 μm thickness at a frequency ω_p/2, where ω_p = πc/2h. Note the peculiar hyperbolic field distribution of cosh(|P|y) spatial dependence inside the groove. (c) Modulation of bandwidth of a thin film ($t\to 0$) SSPP structure and an infinitely thick ($t\to \mathrm{\infty }$) SSPP structure with the change of the refractive index of the substrate/external environment. For thin planar structure, we vary the index of the substrate underneath the waveguide, whereas for infinitely thick structure, refractive index of the dielectric half-space outside of the SSPP waveguide is varied. The diagrams are drawn for groove length h = 10d ,and groove width $a=\frac{d}{10}$. Solid lines are obtained by the theory established in this paper, while the discrete circles (o) and triangles (Δ) are obtained via numerical simulation in COMSOL Multiphysics [36]