Simulation of light propagation by local spherical interface approximation

A new local elementary interface approximation is introduced for the modeling of wave propagation through interfaces between homogeneous media. The incident wave and the surface profile are approximated locally by a spherical wave and a spherical surface, respectively. The wave field travels through the modulated structure according to the laws of geometrical optics, being refracted by the surface and propagating to the output plane locally as a geometric spherical wave. Diffraction theory is applied to propagate the field from the output plane onwards. We provide comparisons of the method with the thin-element approximation, the local plane-wave and interface approach, and rigorous diffraction theory using a sinusoidal surface-relief grating as an example. We illustrate the power of the new method by applying it to the analysis of a diffractive beam splitter.     

On the zero-thickness model of diffractive optical elements

In a recent paper [J. Opt. Soc. Am. A 16, 113 (1999)] a thin-element approximation of diffractive optical elements was used to describe diffraction of oblique incident wave fronts. This expression motivated by a ray optical analysis is shown to be incorrect. I discuss how the thin-element approximation can be generalized to arbitrary diffraction geometries. This includes an intuitive interpretation of the results.     

Efficiency analysis of diffractive lenses

Multilevel diffractive optical elements are necessary for achieving high-efficiency performance. Here the diffraction efficiency of a multilevel phase-only diffractive lens is analyzed. Approximate, as well as more accurate, approaches are presented. Both plane-wave and Gaussian illumination are discussed. It is shown that for many practical cases the diffraction efficiency can be determined by only a single parameter that takes into account the spatial bandwidth product as well as the focal length of the lens and the illumination wavelength. The analysis is based on the scalar theory and the thin-element approximation. Justification for doing this is presented. The results are valid for lenses with at least F/5.     

Vectorial thin-element approximation: a semirigorous determination of Kirchhoff's boundary conditions

A semirigorous model is presented that bridges the gap between classical scalar diffraction theory on the one hand and fully rigorous simulation models on the other. It falls back on the basic ideas of scalar diffraction theory, especially the Kirchhoff approximation. In contrast to classical techniques, however, the boundary values are determined by rigorous methods of the stratified medium theory in the scope of a fully vectorial formulation. By this means the proposed approach takes vertical rigorous coupling effects inside the grating into account while neglecting the lateral ones. We therefore call this method semirigorous and use the name vectorial thin-element approximation. A direct comparison with rigorous coupled-wave analysis as a fully rigorous simulation model allows a detailed discussion of its range of validity and demonstrates a reduction of computation time of the order of 3 magnitudes. In addition, it also reveals deeper insight into the details of the electrodynamics inside diffracting structures. Some examples will demonstrate this benefit.     

Design of non-paraxial array illuminators by step-transition perturbation approach

Abstract

We introduce an efficient Fourier-domain formulation of an approximate method to model non-paraxial diffractive elements. The method is based on evaluation of local field perturbations caused by abrupt surface-profile transitions. It facilitates fast parametric optimization of binary and four-level diffractive array illuminators in the non-paraxial domain of diffractive optics. Comparison with rigorous electromagnetic theory of gratings shows that optimization with the perturbation method gives accurate results if the smallest feature size in the surface profile is larger than one wavelength. Some binary designs are demonstrated using electron beam lithography.     

Efficient optimization of diffractive optical elements based on rigorous diffraction models.

An efficient optimization strategy for the design of diffractive optical elements that is based on rigorous diffraction theory is described. The optimization algorithm combines diffraction models of different degrees of accuracy and computational complexity. A fast design algorithm for diffractive optical elements is used to yield estimates of the optimum surface profile based on paraxial diffraction theory. These estimates are subsequently evaluated with a rigorous diffraction model. This scheme allows one to minimize the need to compute diffraction effects rigorously, while providing accurate design. We discuss potential applications of this scheme as well as details of an implementation based on a modified Gerchberg-Saxton algorithm and the finite-difference time-domain method. Illustrative examples are provided in which we use the algorithm to design Fourier array illuminators.     

Collimating cylindrical diffractive lenses: rigorous electromagnetic analysis and scalar approximation

Practical collimating diffractive cylindrical lenses of 2, 4, 8, and 16 discrete levels are analyzed with a sequential application of the two-region formulation of the rigorous electromagnetic boundary-element method (BEM). A Gaussian beam of TE or TM polarization is incident upon the finite-thickness lens. F/4, F/2, and F/1.4 lenses are analyzed and near-field electric-field patterns are presented. The near-field wave-front quality is quantified by its mean-square deviation from a planar wave front. This deviation is found to be less than 0.05 free-space wavelengths. The far-field intensity patterns are determined and compared with the ones predicted by the approximate Fraunhofer scalar diffraction analysis. The diffraction efficiencies determined with the rigorous BEM are found to be generally lower than those obtained with the scalar approximation. For comparison, the performance characteristics of the corresponding continuous Fresnel (continuous profile within a zone but discontinuous at zone boundaries) and continuous refractive lenses are determined by the use of both the BEM and the scalar approximation. The diffraction efficiency of the continuous Fresnel lens is found to be similar to that of the 16-level diffractive lens but less than that of the continuous refractive lens. It is shown that the validity of the scalar approximation deteriorates as the lens f-number decreases.     

Rigorous and approximate analysis of metallic gratings in soft X-ray and EUV regions

Abstract

In deep ultraviolet and soft X-ray regions the complex refractive indices of metals differ substantially from their values in the visible region of the spectrum. The implications of this fact are analysed for metallic gratings illuminated by electromagnetic fields with wavelengths ranging from soft X‐rays to near infrared, concentrating on short wavelengths. In particular, we study metal-stripe gratings (linear polarizers for visible light) by rigorous diffraction theory to determine the short-wavelength region in which a combination of the geometrical thin-element approximation and the theory of single-layer films can be applied. Then we study inductive grid filters for protection of X-ray detectors from infrared radiation in space applications.     

Rigorous electromagnetic design of finite-aperture diffractive optical elements by use of an iterative optimization algorithm

We propose a rigorous electromagnetic design of two-dimensional and finite-aperture diffractive optical elements (DOEs) that employs an effective iterative optimization algorithm in conjunction with a rigorous electromagnetic computational model: the finite-difference time-domain method. The iterative optimization process, the finite-difference time-domain method, and the angular spectrum propagation method are discussed in detail. Without any approximation based on the scalar theory, the algorithm can produce rigorous design results, both numerical and graphical, with fast convergence, reasonable computational cost, and good design quality. Using our iterative algorithm, we designed a diffractive cylindrical lens and a 1-to-2-beam fanner for normal-incidence TE-mode illumination, thus showing that the optimization algorithm is valid and competent for rigorously designing diffractive optical elements. Concerning the problem of fabrication, we also evaluated the performance of the DOE when the DOE profile is discrete.     

Design of a Binary Grating with Subwavelength Features that Acts as a Polarizing Beam Splitter

A binary diffractive optical element, acting as a polarizing beam splitter, is proposed and analyzed. It behaves like a transmissive blazed grating, working on the first or the second diffraction order, depending on the polarization state of the incident radiation. The grating-phase profile required for both polarization states is obtained by means of suitably sized subwavelength groups etched in an isotropic dielectric medium. A rigorous electromagnetic analysis of the grating is presented, and numerical results concerning its performances in terms of diffraction efficiency as well as frequency and angular bandwidths are provided.     

Integral equation method with parametrization of grating profile theory and experiments

Abstract

Theories and their numerical implementations to rigorously solve Maxwell's equations are fundamental in diffractive optics. Various rigorous methods have been developed in the past which allow the analysis and synthesis of diffraction gratings. The basis of the great majority of rigorous methods is the Fourier series expansion of the permittivity function. Unfortunately the Fourier expansion can easily give poor results for the TM mode. By using an integral method with parametrization of the grating profile this problem can be overcome for surface modulated gratings. The theory and a comparison of theoretical and experimental results are presented.     

Analysis of the influence of the passive facet of blazed transmission gratings in the intermediate diffraction regime

Blazed diffraction gratings are of enormous practical importance for imaging and analyzing hybrid optical systems. The intermediate diffraction regime is characterized by the transition from the scalar to the rigorous electromagnetic theory. An effect known as shadowing occurs and reduces the diffraction efficiency. Based on rigorous calculations for optimized sawtooth-shaped and binary-multilevel blaze profiles, we deduce a semianalytical model describing the shadowing phenomenon for the general case of oblique incidence. We discuss illumination both from air and from the substrate. Though a multilevel blaze possesses a discrete substructure, our shadowing model remains valid, if substructural effects are neglected. We find that electromagnetic effects due to the passive blaze facet lead to the efficiency reduction, and the blazing efficiency shows a linear dependence on the ratio of blaze wavelength to grating period. Our shadowing model is applied to predict the performance of a Littrow-like blazing condition in transmission geometry as, e.g., for a diffractive solid immersion lens.     

Wave-optical structure design with the local plane-interface approximation

Abstract

The design of an optical element profile with specified transmission function for a given incident wave is of fundamental concern in optical design. A well-known example is the design of an aspherical surface in order to realize a spherical phase-only transmission. Various wave-optical system design methods lead to transmission functions as a first step. Then, often the thin-element approximation is applied in a second step to obtain an element structure with the desired transmission. However, if the refraction at the optical interface cannot be neglected, the thin-element approximation is not valid. In this case, a higher version of the local plane-interface approximation can be used for the element structure design. An algorithm for this model is introduced and its characteristics are discussed for the example of a non-paraxial Gaussian-to-tophat beam shaping element.     

Beam shaping in the nonparaxial domain of diffractive optics

We address the problem of shaping the radiant intensity distribution of a highly nonparaxial coherent field by means of a diffractive element located in the plane of the beam waist. To be capable of wide-angle energy redistribution the element must necessarily contain wavelength-scale transverse features, and consequently it must be designed on the basis of rigorous diffraction theory. We consider, in particular, wide-angle Gaussian to flat-top beam shaping in one dimension. Scalar designs are provided and their validity is evaluated by rigorous diffraction theory, which is also used for optimization deep inside the nonparaxial domain, where the scalar designs fail. Experimental verification is provided by means of electron-beam lithography.     

Binary blazed reflection gratings

A reflection grating with a binary surface profile is presented that has high diffraction efficiency. The measured intensity for the + 1st diffracted order was 77%. The binary grating is composed of a minilattice with feature sizes comparable with the wavelength of the incident light. The overall structure is designed in such a way that it imitates a conventional blazed grating. The grating also has interesting polarization properties. The main part of the TE-polarized light is diffracted into the 1st diffracted order, and most of the TM-polarized light remains in the 0th diffracted order. The measurements of the grating are compared with rigorous diffraction theory and found to be in reasonable agreement.     

Born approximation for the nonparaxial scalar treatment of thick phase gratings

The conventional design of phase gratings or kinoforms with a paraxial transmission function is restricted to the paraxial domain and thin elements. Therefore, the design and analysis of thick phase-relief structures require a nonparaxial theory, as given by the Born approximation. The Born approximation is derived as an extension of the scalar thin-element theory, which is applicable for thick elements with large propagation angles. As an example, general prism gratings on curved surfaces are treated.     

二元光学元件横向加工误差对衍射效率的影响

横向加工误差包括对准误差和线宽误差,加工误差对衍射效率的影响是二元光学元件研究的重要问题.本文基于标量衍射理论,提出了一种与MATLAB相结合的分层计算方法,重点分析了4阶和8阶二元光学元件横向加工误差和衍射效率的关系.结果表明,控制8阶二元光学元件对准误差,可抑制衍射效率的迅速下降,适当调整线宽能够减小对准误差造成的...     

Zone-boundary optimization for direct laser writing of continuous-relief diffractive optical elements

Enhancing the diffraction efficiency of continuous-relief diffractive optical elements fabricated by direct laser writing is discussed. A new method of zone-boundary optimization is proposed to correct exposure data only in narrow areas along the boundaries of diffractive zones. The optimization decreases the loss of diffraction efficiency related to convolution of a desired phase profile with a writing-beam intensity distribution. A simplified stepped transition function that describes optimized exposure data near zone boundaries can be made universal for a wide range of zone periods. The approach permits a similar increase in the diffraction efficiency as an individual-pixel optimization but with fewer computation efforts. Computer simulations demonstrated that the zone-boundary optimization for a 6 microm period grating increases the efficiency by 7% and 14.5% for 0.6 microm and 1.65 microm writing-spot diameters, respectively. The diffraction efficiency of as much as 65%-90% for 4-10 microm zone periods was obtained experimentally with this method.     

Coupling of light from an LED into a thin light guide by diffractive gratings

Ring-shaped and radial diffractive gratings are designed with rigorous diffraction theory to couple light of a nearly monochromatic LED into a thin planar light guide on the bottom side. The theoretical coupling efficiencies for ring-shaped and radial gratings are 41% and 66%, respectively. Optimized diffractive elements are manufactured with direct electron-beam lithography and reactive-ion-etching into SiO2 substrates. Good agreement between experimental and theoretical results for selected radial gratings is reached. Furthermore, the mass production tests using injection molding are carried out with good replicability.     

Diffractive exit-pupil expander for display applications

Two-dimensional binary diffraction gratings can be used in wearable display applications as exit-pupil expanders (EPEs) (or numerical-aperture expanders) to increase the size of the display exit pupil. In retinal scanning displays the EPE is placed at an intermediate image plane between the scanners and the display exit pupil. A focused spot scans across the diffractive EPE and produces multiple diffraction orders at the exit pupil. The overall luminance uniformity across the exit pupil as perceived by the viewer is a function of the uniformity of the diffraction-order intensities, focused-spot size, grating period, scanning-beam profile, and the viewer's eye-pupil size. The design, the diffraction-order uniformity, and the effects of the grating phase angle on the uniformity for binary diffraction gratings are discussed. Also discussed are the display exit-pupil uniformity and the impact of the diffractive EPE on the point-spread function and the modulation transfer function of the display. Both theoretical and experimental results are presented.