Elsevier

Optik

Volume 126, Issue 24, December 2015, Pages 5504-5508
Optik

Comprehensive analysis of diffraction efficiency of conical beam from sub-wavelength binary metallic grating using rigorous coupled-wave theory

https://doi.org/10.1016/j.ijleo.2015.09.111Get rights and content

Abstract

Rigorous coupled-wave theory has been implemented to study comprehensively diffraction efficiency of binary metallic gratings. We have investigated diffraction efficiency for TE and TM polarizations and conical incident beam in more detail than the other reported studies. As a general case, we study diffraction efficiency with respect to incident angles and grating period. Calculated first-order diffraction efficiencies for one-dimensional binary metallic gratings have been shown to converge to the correct value with an effect of increasing number of space harmonics over a wide range of grating depth. More field number of space harmonics is shown to be required for the convergence of gratings with larger grating periods, deeper gratings, TE and TM polarization, and conical diffraction.

Introduction

Rigorous coupled-wave analysis has been already applied to planar phase gratings [1], [2], surface-relief phase gratings [3], planar absorption gratings [4]. As well as, rigorous coupled-wave analysis (RCWA) is used to solve problems of diffraction on surface-relief absorption gratings [5]. Concerning this technique, the structures are divided into several regions and permeability changes, so electric field and magnetic field in each layer can be expanded. This analysis has been studied for binary grating structure [6] and approximation is given for other structures like as sinusoidal or saw-tooth. The rigorous coupled-wave analysis has also been developed for dielectric surface-relief gratings [7] and also extended to gratings having an arbitrary complex permittivity. This analysis showed that dielectric structure requires less orders than metallic structures that need more orders to get accurate results.

Gratings with rectangular grooves metallic in diffraction optical lens elements are used for infrared telescopes [8], [9]. Attention about metallic gratings lead to wide ability at construction techniques like electron-beam lithography or holographic and gratings with deep grooves and small period [10].

Recently, we tried to consider interaction of TE and TM polarization in more detailed [11], [12] Whereas, conical-diffraction configuration has not been considered. Thus, we aimed to implement the numerical of the RCWA for the three-dimensional rectangular groove binary metallic gratings and present shown the Formulations for the conical-diffraction configuration. Then results have been compared with results of planer diffraction configuration. The effect of the numbers of the field space-harmonic expansion on the convergence of the diffraction efficiency at deeper gratings has been investigated.

Section snippets

Mathematical approach

Three-dimensional binary grating diffraction is schematically shown in Fig. 1. A linearly polarized electromagnetic wave is obliquely incident at an arbitrary angle of incidence θ and azimuthal angle ϕ upon a binary metallic grating. The grating is subdivided into two different regions with relative permittivity ϵI and ϵII. In the grating regions (0 < z < d) the periodic relative permittivity is expandable in a Fourier series as

ϵ(x)=hϵh(z)exp(jhk0x),where Λ is the grating period, ϵh is the hth

Representation of fields at conical mount

Total electric field in region I is the sum of incident and reflected waves. The incident normalized electric field vector in region I may be written as

Einc=uexp[jk0ϵI(sinθcosϕx+sinθsinϕy+cosθz)],andu=(cosψcosθcosϕsinψsinϕ)xˆ+(cosψcosθcosϕsinψcosϕ)yˆcosψsinθzˆ,where ψ is the angle between the electric-field vector and the plane of the incidence. Normalized solutions in region I (0 < z) and in region II (d < z) are given by

EI=Einc+mRmexp[j(kxmx+kyykI,zmz)],Rm is the normalized vector

Simulation works

Diffraction characteristics for rectangular-groove metallic gratings with equal groove and ridge widths are presented for free space wavelengths of 0.5 μm (in the visible region) and 1 μm, 10 μm (in the infrared region) as a function of period, groove depth and angle of incidence. To illustrate the stability of the present method, the diffraction efficiency is plotted versus the normalized grating depth and shown in Fig. 3, Fig. 4, Fig. 5. In addition, losses of TE, TM Polarization and conical

Conclusions and outlook

Herein, rigorous coupled-wave analysis has been applied to surface-relief absorption gratings. For metallic gratings, using of the complex permittivity avoids using the infinite-conductivity approximation. Conical diffraction and TE, TM polarizations and arbitrary angles of incidence have been treated. Diffraction characteristics of rectangular-groove gold gratings were presented for free-space wavelengths of 0.5, 1.0 and 10.0 μm for all diffracted orders as a function of period, groove depth,

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