Analysis of Resonance Absorption in Multilayered Thin-Film Bi-Grating

The resonance absorption of a multilayered bi-grating which consists of thin-film corrugated periodically in two directions is investigated. The absorption in a multilayered thin-film bi-grating has been of considerable interest since we can expect more complex behaviors in the absorption pheno-men by virtue of the presence of double periodicity and multilayer structure. In solving the problem, we employed a computational technique based on modal expansion. Taking a sandwiched structure /Ag/SiO 2 /Ag/ for an example, we observed: 1) excitation of a single-interface surface plasmon mode at the lit surface of the 1 st Ag layer with strong field enhancement for thick enough Ag layer case; 2) excitation of coupled short-range or long-range surface plasmon modes at each surface between vacuum and Ag layers with strong field enhancements for thin enough Ag layer cases no matter with the thickness of SiO 2 layers; 3) enhancements of field at surfaces between Ag and SiO 2 layers in some cases related with the thickness of SiO 2 layers. The coupled plasmon modes were resulted by the resonance waves on four surfaces in these cases.


Introduction
Periodically corrugated thin metal films have an interesting property such as the partial or total absorption of incident light energy. The absorption is associated with the excitation of the surface plasmons and is then termed the resonance absorption [1] [2] [3]. Most of studies on the resonance absorption have mainly dealt with a thin metal film grating whose surfaces are periodic in one direction [4] [5] [6].
In our previous study, we examined the excitation of coupled plasmon modes in a thin-film grating made of a metal [7]. When the metal is thick, e.g., more than ten times the skin depth, the plasmon can be excited on the lit surface alone. This is termed a single-interface surface plasmon (SISP). When the thickness is decreased, the plasmon can be seen also on the other surface of the film.
The two plasmon waves interact with each other to form two coupled plasmon modes called short-range and long-range surface plasmon (SRSP and LRSP) [8].
In the present research, we consider a sandwiched structure: metal/dielectric/metal, which is interesting for the application in development of optical equipment, for example improving sensitivity of clinical sensing [9], surface enhanced phenomena such as Ramman scattering, and solar cells.

Formulation and Method of Solution
In this section we first formulate the problem of diffraction by multilayered thin-film bi-grating shown in Figure 1(a). After formulating the problem, we state a method of solution based on a modal-expansion approach.

Incident Wave
The electric and magnetic field of an incident light is given by where i e and i h are the electric-and magnetic-field amplitude; is the incident wave vector with = π and 0 n is the relative refractive index of region V 0 ; P = (X, Y, Z) is an observation point; λ is the wavelength of the incident wave; θ is the incident angle between the Z-axis and the incident wave-vector; ϕ is the azimuth angle between the X-axis and the plane of incidence.
The amplitude of the incident electric field can be decomposed into TE-(TM-) component, which means the electric (or magnetic) field is perpendicular to the plane of incidence. To do this, we define two unit vectors TE e and TM e that span a plane orthogonal to i k . Hence, the amplitude i e in (1) (2) where the symbol δ is the polarization angle between i e and TE e shown in Figure 1(b).

Diffracted Wave
We seek for the diffracted fields E l (P) and H l (P) in each region. These should satisfy the following requirements.
where, α and β are the phase constants in X and Y.
(C4) Boundary conditions: The tangential components of electric and magnetic fields are continuous across the boundaries S l .

Geometry of the Multilayered Thin-Film Bi-Grating
The multilayered thin-film bi-grating which is laminating L − 1 grating layers has period d in both Xand Y-directions shown in Figure 1(a). The semi-infinite region over the multilayered bi-grating and the substrate are denoted by V 0 and V L , respectively. Moreover, each region inside the multilayered bi-grating, which is numbered starting from the incident side, is denoted by and homogeneous media with refractive indices n  , and a permeability of each region is equal with that of the vacuum 0 µ . The interface between V The profile of S  is sinusoidal and given by ( ) where ω  denotes an average distance between S  and

Method of Solution
We solve the problem above using Yasuura's method of modal expansion [10]. To do this, we first define the set of modal functions; next we construct approximate solutions in terms of finite modal expansions with unknown coefficients; and, finally we determine the coefficients applying the boundary conditions. Modal functions: Because the diffracted waves have both TE-and TM-components, we need TE and TM vector modal functions in constructing the solutions. Here we employ the functions derived from the Floquet modes (separated solutions of the Helmholtz equations satisfying the periodicity (C3) and the radiation conditions (C2) if necessary). The modal functions for electric fields for each region are given by where, , 0, 1, 2, m n = ± ±  and 0,1, 2, , l L =  , and wave vectors in (6) and [ ] ( )  TE  TM  TE  TM  d  TE  TM  ,  ,   TE  TM  TE  TM  TE  TM  , where N denotes the truncation number.
Boundary matching: Because the approximate solutions satisfy the require- are determined such that the solutions satisfy the boundary conditions (C4) in an approximate sense. In the Yasuura's method, the least-squares method is employed to fit the solution to the boundary conditions [10] [11]. That is, we find the coefficients that minimize the weighted mean-square error by where 1 S′ denotes one-period cells of the interface S  , Γ  is the intrinsic impedance of the medium in V  and v is a unit normal vector of each boundary. To solve the least-squares problem on a computer, we need a discretized form of the problem. We first discretize the weighted mean-square error I N by applying a two-dimensional trapezoidal rule where the number of sampling points is chosen as 2(2N + 1) [9]. We then employ orthogonal decomposition methods [singular-value decomposition (SVD) and QR decomposition (QRD)] in solving the discretized problem [12] [13].
It is known that the solutions obtained by Yasuura's method have proof of convergence [13] [14]. We, therefore, can employ the coefficients ( ) The coefficient defined above is the power carried away by propagating diffraction orders normalized by the incident power.

Numerical Results
The multilayered bi-grating is made by 3 layers: /Ag/SiO 2 /Ag/. The incident light is a TM-polarized plane wave with a 650 nm wavelength. The relative refractive index of Vacuum = 1, n Ag = 0.07+4.2i and SiO e = 27.8 nm. We will then calculate the diffraction efficiencies and field distributions of these gratings.  1 st Ag layer is enough thick, the oscillation near the upper surface does not reach the lower surface and, hence, the field below the 1 st Ag layer is zero. This indicates the excitation of SISP.

Conclusion
We solved the problems for 3 thin-films bi-grating. By calculating the diffraction efficiency and field distributions, we showed that the SPR phenomenon excited and we observed: 1) excitation of a SISP mode at the lit surface of the 1 st Ag layer with strong field enhancement for thick enough layer case; 2) excitation of coupled SPR modes (SRSP or LRSP) at each surface between vacuum and Ag layers with strong field enhancements for thin enough Ag layer cases no matter with the thickness of SiO 2 layers; 3) enhancements of field at surfaces between Ag and SiO 2 layers in some cases related with the thickness of SiO 2 layers. The coupled plasmon modes were resulted by the resonance waves excited on four surfaces in these cases. In future, we plan to study applications for multilayered bi-grating such as improving the sensitivity of a bio-sensor by determining changes of SPRs excited at different layers' surfaces.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.