Our current understanding of the optical properties of metallic nanostructures has benefited from the development and commercialization of powerful electromagnetic simulation tools. Simulations allow the relationships between the elements of a system to be analyzed in detail and multiple design options to be compared before implementation. In this section, we first give a brief introduction of the rigorous coupled-wave analysis (RCWA) technique that is used in the study of the far-field response of the planar/grating multilayer system, we then discuss the variables of the multilayer structures in the spectral response simulations.
4.3.1
Rigorous Coupled-Wave Analysis (RCWA)
The rigorous coupled-wave analysis (RCWA), also known as Fourier modal method, was formulated by Moharam and Gaylord to model the diffraction of EM waves by periodic 1-D and 2-D gratings [147]. It is a frequency-domain method based on expanding the fields and material permittivities inside the corrugated region in terms of (finite) Fourier series that are solutions to Maxwell’s equations. Matching boundary conditions at the grating interfaces through the calculation of the eigenvalues and eigenvectors of the system equations gives field distributions inside and outside the grating region, which yields the desired complex diffraction efficiencies. The method gives rigorous results in sense that the full set of vector Maxwell’s equations are solved with only two simplifying assumptions: 1) a piecewise-linear approximation to the grating construction and 2) a truncation parameter for the Fourier series representation of the permittivity within each grating layer.
The RCWA algorithm calculates the diffraction efficiencies of a stack of lamellar grating layers between two semi-infinite half spaces: the substrate and the superstrate. More complicated grating profiles can be constructed layer by layer so that the permittivity does not vary in the y-direction, as illustrated in Fig. 4.12. The accuracy of the calculation improves as the number of layers used in the step-like approximation increases, although this could result in increasing the computation time. When the model exactly matches the real grating profile, the accuracy is only limited by the number of the diffracted orders for which the diffraction efficiencies are calculated.
The RCWA method is considered to be the most efficient approach for lamellar gratings [149]. As the calculation essentially includes the zeroth order of diffraction, the technique can be used to analyze the optical responses
§ 4. EOT Through Planar Metal Films Coupled to Metallic Gratings 72
Figure 4.12: Schematic representation of a lamellar approximation to a triangular grating profile using the RCWA method. The linear polarizations TE and TM are also defined.
of the subwavelength gratings, which is particularly useful for the study of EOT phenomenon. It is also interesting to note that there has been some discussion in the literature regarding the convergence of the technique. In the TE-polarized case, the unknown functions are Ez and Hx, both of which
are continuous across the grating surface for non-magnetic media. Thus the convergence of their Fourier representation is relatively fast. However, for TM polarization, Ex is discontinuous when crossing the corrugated surface.
Therefore, to obtain convergence in the calculation, it requires a high number of Fourier components in expansion of the field for highly conducting grating materials [150], which generally involves long computation times and large memory. Subsequently, a reformulation of the eigenproblem of RCWA has been proposed by Lalanne and Morris [151] to improve convergence rates for TM polarization.
In this study, a RCWA-based commercial grating analysis software, called G-Solver [148], has been used for analyzing the spectral response of the multilayer structures. In addition to the classical two-dimensional gratings (e.g., lamellar, blaze, sinusoidal and triangular), GSolver also handles three-
dimensional grating (cross-gratings which have periodic diffraction structures along both x- and z-dimensions) cases, which is useful for the analyzing EM responses of the structures studied in this chapter and the following chapter (a planar metal film coupled to a hole-array layer).
4.3.2
Variables of the Structure
A major advantage of the multilayer system here is it allows the direct observation of the controlled excitation/coupling processes within the structure. Figure 4.13 shows the variables used in the simulations, which can be categorized into the following three groups: 1) illuminating conditions, including: polarization (TE or TM), wavelength λ and angle of incidence θi;
2) geometric parameters, including: the thickness of the Ag grating hg, its
period P and duty cycle DC (DC = w/P ); the thickness of the dielectric spacer hs; and the thickness of the planar Ag film hAg; 3) material parameters,
including: the refractive index of the dielectric spacer nd and that of the
substrate ns. By varying a single parameter at a time in each series of
simulations and analyze the response of the structure, we can easily identify the origins of the different resonances within the structure.
Although several metals, such as Au, Ag, Cu and Al, can support SP excitation at optical wavelengths, Au and Ag are the most commonly used metals. Furthermore, SPP waves propagating along the surfaces of Ag are less attenuated and exhibits higher localization of electromagnetic field in the dielectric than SPPs supported by Au [11]. We therefore use Ag in our study of EOT. In the simulations, the complex dielectric constants of the Ag are taken from Ref. [41].
§ 4. EOT Through Planar Metal Films Coupled to Metallic Gratings 74
Figure 4.13: Variables of the planar-grating multilayer structure used in the spectral response simulations.