Numerical Methods for Scientists and Engineers

A rigorous coupled-wave analysis for metallic surface-relief gratings is presented. This approach allows an arbitrary complex permittivity to be used for the material and thus avoids the infinite conductivity (perfect-conductor) approximation. Both TE and TM polarizations and arbitrary angles of incidence are treated. Diffraction characteristics for rectangular-groove gold gratings with equal groove and ridge widths are 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, polarization, and angle of incidence. Results include the following: (1) TM-polarization diffraction characteristics vary more rapidly than do those for TE polarization, (2) 95% first-order diffraction efficiency occurs for TM polarization at 10.0 μm, (3) 50% absorption of incident power occurs at 0.5 μm, and (5) the perfect-conductor approximation is not valid for TM polarization at any of the wavelengths and is not valid for TE polarization at 0.5 μm.

Rigorous coupled-wave analysis of metallic surface-relief gratings

We show that silicon-based metagratings capable of large-angle, multifunctional performance can be realized using inverse freeform design. These devices consist of nonintuitive nanoscale patterns and support a large number of spatially overlapping optical modes per unit area. The quantity of modes, in combination with their optimized responses, provides the degrees of freedom required to produce high-efficiency devices. To demonstrate the power and versatility of our approach, we fabricate metagratings that can efficiently deflect light to 75° angles and multifunctional devices that can steer beams to different diffraction orders based on wavelength. A theoretical analysis of the Bloch modes supported by these devices elucidates the spatial mode profiles and coupling dynamics that make high-performance beam deflection possible. This approach represents a new paradigm in nano-optical mode engineering and utilizes different physics from the current state-of-the-art, which is based on the stitching of nonint...

Large-Angle, Multifunctional Metagratings Based on Freeform Multimode Geometries

Diffractive optical structures for increasing the efficiency of crystalline silicon solar cells are discussed. As a consequence of the indirect band gap, light absorption becomes very ineffective near the band edge. This can be remedied by use of optimized diffraction gratings that lead to light trapping. We present blazed gratings that increase the optically effective cell thickness by approximately a factor of 5. In addition we present a wideband antireflection structure for glass that consists of a diffraction grating with a dielectric overcoat, which leads to an average reflection of less than 0.6% in the wavelength range between 300 and 2100 nm.

Submicrometer gratings for solar energy applications.

The existing theories which aim to explain the extraordinary optical transmission of a metallic film pierced by a two-dimensional subwavelength hole array [T.W. Ebbesen, H.J. Lezec, H.F. Ghaemi, T. Thio, and P.A. Wolff, Natural (London) 391, 667 (1998)] all have in common the following feature: instead of studying the two-dimensional crossed grating resulting from the hole array, they consider a one-dimensional grating with infinite slits. We show that such a simplification introduces an efficient channel for light transmission in lamellar gratings, which does not exist for hole arrays. Therefore in order to explain the relatively high transmission observed by Ebbesen et al., it is necessary to take into account the existence of the holes in the array. In this paper we develop a two-dimensional analysis of the experiment performed by Ebbesen et al. No simplification is introduced. This allows us to obtain theoretically the long-wavelength peak reported by Ebbesen et al. with the same grating thickness as the one used by these authors. We also review and study in detail the various contributions devoted to this very surprising effect.

/pdf/theory-of-light-transmission-through-subwavelength-periodic-2kaxlsw2or.pdf

Theory of light transmission through subwavelength periodic hole arrays

A rigorous modal theory describing the diffraction properties of a dielectric lamellar grating is presented. The numerical implementation is shown to be suited to modelling the behaviour of high refractive index gratings. This suggests that an approach of this type may be successfully applied to the problem of lossy metallic lamellar gratings.

The Dielectric Lamellar Diffraction Grating

A rigorous modal theory describing the diffraction properties of a finitely conducting lamellar grating is presented. The method used is the generalization to lossy structures of an earlier formalism for the dielectric lamellar grating. Sample results of the method are given, demonstrating its accuracy and its ability to deal with problems intractable by the widely used integral-equation formalisms of diffraction grating theory.

The Finitely Conducting Lamellar Diffraction Grating

The problem of diffraction by a perfectly conducting lamellar grating is examined numerically and analytically by use of a modal expansion technique. At various complex wavelengths, with a positive imaginary part, individual mode amplitudes have poles which give rise to resonance effects in the orders at nearby real wavelengths. These are the well-known resonance anomalies. We determine the trajectories of these poles as functions of groove depth, finding that as this parameter increases they move in the complex wavelength plane to a mode threshold, or cut-off wavelength—from a Rayleigh wavelength in the case of S -polarization and from zero wavelength in the case of P -polarization. It is emphasized that the modal expansion technique is particularly valuable in gaining an understanding of the dynamics of diffraction gratings, and that the resonance poles are perhaps the basic dynamical objects determining their behaviour.

Resonance Anomalies in the Lamellar Grating

The study of the infinitely conducting lamellar grating resonance anomalies is extended to arbitrary wavelengths and angles of incidence, and the behaviour of the trajectories is found theoretically and numerically. The discussion of resonances for varying wavelength is unified with that for varying angle of incidence. The phenomena of ‘repulsion of anomalies’ and ‘exchange of identity’ are explained. Infinite conductivity results are compared with those for various finite conductivities, and it is shown that there is a close similarity between them, suggesting that the ideas developed for the former case have a wider validity. This is seen to be so when a brief survey is made of results for other grating profiles and arbitrary values of conductivity.

Further Properties of Lamellar Grating Resonance Anomalies

A modal expansion formalism is presented to solve the scalar-wave diffraction problem for a perfectly conducting reflection grating whose grooves are semi-circular in cross-section. The theory is given for both of the fundamental polarizations, together with efficiency calculations for the first-order Littrow mounting. Similarity between the results and those of other profile shapes is observed and the groove radius which produces optimum blazing performance is determined. Resonance anomalies are observed in the efficiency spectra and are attributed to complex poles in the mode amplitudes. The positions of these poles are traced as a function of groove radius.

J.R. Andrewartha

Papers

The Dielectric Lamellar Diffraction Grating

The Finitely Conducting Lamellar Diffraction Grating

Resonance Anomalies in the Lamellar Grating

Further Properties of Lamellar Grating Resonance Anomalies

A Modal Theory Solution to Diffraction from a Grating with Semi-circular Grooves