The rigorous coupled-wave analysis technique for describing the diffraction of electromagnetic waves by periodic grating structures is reviewed. Formulations for a stable and efficient numerical implementation of the analysis technique are presented for one-dimensional binary gratings for both TE and TM polarization and for the general case of conical diffraction. It is shown that by exploitation of the symmetry of the diffraction problem a very efficient formulation, with up to an order-of-magnitude improvement in the numerical efficiency, is produced. The rigorous coupled-wave analysis is shown to be inherently stable. The sources of potential numerical problems associated with underflow and overflow, inherent in digital calculations, are presented. A formulation that anticipates and preempts these instability problems is presented. The calculated diffraction efficiencies for dielectric gratings are shown to converge to the correct value with an increasing number of space harmonics over a wide range of parameters, including very deep gratings. The effect of the number of harmonics on the convergence of the diffraction efficiencies is investigated. More field harmonics are shown to be required for the convergence of gratings with larger grating periods, deeper gratings, TM polarization, and conical diffraction.

http://circuit.ucsd.edu/~zhaowei/ECE212C/notes/Lecture6_reading_materials.pdf

Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings

An enhanced, numerically stable transmittance matrix approach is developed and is applied to the implementation of the rigorous coupled-wave analysis for surface-relief and multilevel gratings. The enhanced approach is shown to produce numerically stable results for excessively deep multilevel surface-relief dielectric gratings. The nature of the numerical instability for the classic transmission matrix approach in the presence of evanescent fields is determined. The finite precision of the numerical representation on digital computers results in insufficient accuracy in numerically representing the elements produced by inverting an ill-conditioned transmission matrix. These inaccuracies will result in numerical instability in the calculations for successive field matching between the layers. The new technique that we present anticipates and preempts these potential numerical problems. In addition to the full-solution approach whereby all the reflected and the transmitted amplitudes are calculated, a simpler, more efficient formulation is proposed for cases in which only the reflected amplitudes (or the transmitted amplitudes) are required. Incorporating this enhanced approach into the implementation of the rigorous coupled-wave analysis, we obtain numerically stable and convergent results for excessively deep (50 wavelengths), 16-level, asymmetric binary gratings. Calculated results are presented for both TE and TM polarization and for conical diffraction.

Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach

The recent reformulation of the coupled-wave method by Lalanne and Morris [ J. Opt. Soc. Am. A13, 779 ( 1996)] and by Granet and Guizal [ J. Opt. Soc. Am. A13, 1019 ( 1996)], which dramatically improves the convergence of the method for metallic gratings in TM polarization, is given a firm mathematical foundation in this paper. The new formulation converges faster because it uniformly satisfies the boundary conditions in the grating region, whereas the old formulations do so only nonuniformly. Mathematical theorems that govern the factorization of the Fourier coefficients of products of functions having jump discontinuities are given. The results of this paper are applicable to any numerical work that requires the Fourier analysis of products of discontinuous periodic functions.

https://mcgrating.com/references/ref11.pdf

Use of Fourier series in the analysis of discontinuous periodic structures

Holographic techniques allow for the construction of 3D images by controlling the wave front of light beams. Huang et al. develop ultrathin plasmonic metasurfaces to provide 3D optical holographic image reconstruction in the visible and near-infrared regions for circularly polarized light.

/pdf/three-dimensional-optical-holography-using-a-plasmonic-3psfczrfn0.pdf

Three-dimensional optical holography using a plasmonic metasurface

Nature routinely produces nanostructured surfaces with useful properties, such as the self-cleaning lotus leaf, the colour of the butterfly wing, the photoreceptor in brittlestar and the anti-reflection observed in the moth eye. Scientists and engineers have been able to mimic some of these natural structures in the laboratory and in real-world applications. Here, we report a simple aperiodic array of silicon nanotips on a 6-inch wafer with a sub-wavelength structure that can suppress the reflection of light at a range of wavelengths from the ultraviolet, through the visible part of the spectrum, to the terahertz region. Reflection is suppressed for a wide range of angles of incidence and for both s- and p-polarized light. The antireflection properties of the silicon result from changes in the refractive index caused by variations in the height of the silicon nanotips, and can be simulated with models that have been used to explain the low reflection from moth eyes. The improved anti-reflection properties of the surfaces could have applications in renewable energy and electro-optical devices for the military.

/pdf/improved-broadband-and-quasi-omnidirectional-anti-reflection-28262vbtzi.pdf

Improved broadband and quasi-omnidirectional anti-reflection properties with biomimetic silicon nanostructures

Techniques for the design of continuously tapered two-dimensional (2D) subwavelength surface-relief grating structures for broadband antireflection surfaces are investigated. It has been determined that the Klopfenstein taper [ Proc. IRE44, 31 ( 1956)] produces the optimum graded-index profile with the smallest depth for any specified minimum reflectance. A technique is developed to design the equivalent tapered subwavelength surface-relief grating structure by use of 2D effective-medium theory. An optimal Klopfenstein tapered 2D subwavelength grating is designed to reduce the Fresnel reflections by 20 dB over a broad band from an air–substrate (ns = 3.0) interface. The performance is verified by use of both a 2D effective-medium-theory simulation algorithm and rigorous coupled-wave analysis. These structures are also shown to achieve this low reflectance over a wide field of view (θFOV > 110°). The pyramidal spatial profile, which has generally been assumed to produce the optimal broadband antireflection grating structure, is shown to require a significantly larger depth to achieve the same performance as a Klopfenstein-designed tapered antireflection grating structure.

Optimal design for antireflective tapered two-dimensional subwavelength grating structures

Two-dimensional symmetric and asymmetric subwavelength binary gratings are investigated. A method for determining the three effective indices of a two-dimensional (2-D) subwavelength grating is presented, as well as a theoretical formalization for the effective index parallel with the normal to the surface. It is shown that a 2-D asymmetric binary grating on the surface of a dielectric substrate is analogous to a biaxial thin film. If the grating is symmetric, then the two effective indices perpendicular to the normal are equal, and the grating is analogous to a uniaxial thin film. Using these effective indices and the quarter-wave Tschebyscheff synthesis technique, we designed two- and three-level binary gratings to suppress reflections over a broad band. It is shown that for a substrate index of ns = 3.0 a three-level 2-D binary grating reduced reflections below 0.1% from 8 μm to 12 μm.

Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings

The range of validity and the accuracy of scalar diffraction theory for periodic diffractive phase elements (DPE’s) is evaluated by a comparison of diffraction efficiencies predicted from scalar theory to exact results calculated with a rigorous electromagnetic theory. The effects of DPE parameters (depth, feature size, period, index of refraction, angle of incidence, fill factor, and number of binary levels) on the accuracy of scalar diffraction theory is determined. It is found that, in general, the error of scalar theory is significant (∊ > ±5%) when the feature size is less than 14 wavelengths (s < 14λ). The error is minimized when the fill factor approaches 50%, even for small feature sizes (s = 2λ); for elements with an overall fill factor of 50% the larger period of the DPE replaces the smaller feature size as the condition of validity for scalar diffraction theory. For an 8-level DPE of refractive index 1.5 analyzed at normal incidence the error of the scalar analysis is greater than ±5% when the period is less than 20 wavelengths (Λ < 20λ). The accuracy of the scalar treatment degrades as either the index of refraction, the depth, the number of binary levels, or the angle of incidence is increased. The conclusions are, in general, applicable to nonperiodic as well as other periodic (trapezoidal, two-dimensional) structures.

Eric B. Grann

Papers

Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings

Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach

Optimal design for antireflective tapered two-dimensional subwavelength grating structures

Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary gratings

Limits of scalar diffraction theory for diffractive phase elements