Thin Element Approximation Research Articles

Fast simulation of the influence of a refractive free-form microstructure on a wave field based on scalar diffraction theory

We present a novel fast simulation approach to simulate the influence of refractive freeform microstructures on a wave field. The FRISP (Finite Refractive Index Selective Propagation) method combines the Rayleigh-Sommerfeld diffraction integral with a thin element approximation and provides a comprehensive framework for understanding the optical properties of these microstructures. The main advantage of this method is its reduced complexity, which leads to a remarkable reduction in computation time by more than two orders of magnitude compared to finite-difference time-domain (FDTD) methods. This efficiency facilitates the iterative optimization of refractive microstructures and thus represents a practical tool to improve this type of microstructures. The verification of the FRISP method is realized by comparing the focal position and spot size of refractive microstructures. For this purpose, we compare FDTD, Mie theory and experimental data on microspheres with the predictions of FRISP. This comparison demonstrates the robustness and reliability of the approach, emphasizes its validity and demonstrates it as a valuable tool for the design and analysis of microstructures.

Validity of Thin Element Approximation in diffractometry of opaque thick objects

Validity of Thin Element Approximation in diffractometry of opaque thick objects

Thickness optimization algorithm to improve multilayer diffractive optical elements performance.

The diffractive zone thicknesses of conventional diffractive optical elements (DOEs) are generally obtained using the thin element approximation (TEA). However, the TEA yields inaccurate results in the case of thick multilayer DOEs (MLDOEs). The extended scalar theory (EST) is an alternative thickness optimization method that depends on the diffractive order and the optimization wavelength. We developed an algorithm to research suitable EST input parameters. It combines ray-tracing and Fourier optics to provide a performance estimate for each EST parameter pair. The resulting "best" MLDOE designs for three different material combinations are analyzed using rigorous finite-difference time-domain. Compared to the TEA, the proposed algorithm can provide performing zone thicknesses.

Multilayer diffractive optical element material selection method based on transmission, total internal reflection, and thickness

The polychromatic integral diffraction efficiency (PIDE) metric is generally used to select the most suitable materials for multilayer diffractive optical elements (MLDOEs). However, this method is based on the thin element approximation, which yields inaccurate results in the case of thick diffractive elements such as MLDOEs. We propose a new material selection approach, to the best of our knowledge, based on three metrics: transmission, total internal reflection, and the optical component's total thickness. This approach, called "geometric optics material selection method" (GO-MSM), is tested in mid-wave and long-wave infrared bands. Finite-difference time-domain is used to study the optical performance (Strehl ratio) of the "optimal" MLDOE combinations obtained with the PIDE metric and the GO-MSM. Only the proposed method can provide MLDOE designs that perform. This study also shows that an MLDOE gap filled with a low index material (air) strongly degrades the image quality.

Hybrid ray-tracing/Fourier optics method to analyze multilayer diffractive optical elements.

The performance (paraxial phase delay) of conventional diffractive optical elements is generally analyzed using the analytical scalar theory of diffraction, based on thin-element approximation (TEA). However, the high thickness of multilayer diffractive optical elements (MLDOEs) means that TEA yields inaccurate results. To address this, we tested a method based on ray-tracing simulations in mid-wave and long-wave infrared wave bands and for multiple f-numbers, together with the effect of MLDOE phase delay on a collimated on-axis beam with an angular spectrum method. Thus, we accurately generate optical figures of merit (point spread function along the optical axis, Strehl ratio at the "best" focal plane, and chromatic focal shift) and, by using a finite-difference time-domain method as a reference solution, demonstrate it as a valuable tool to describe and quantify the longitudinal chromatic aberration of MLDOEs.

Modeling infrared behavior of multilayer diffractive optical elements using Fourier optics.

In this paper, we propose to explore the infrared (IR) behavior of multilayer diffractive optical elements (MLDOEs). IR MLDOEs are designed for the development of space instruments dedicated to Earth observation. The phase effect of the MLDOE on a paraxial plane wave is studied using exact kinoform shapes for each layer. The modeling of the optical path difference uses thin element approximation. Until now, MLDOEs have been designed and simulated on ray-tracing software with binary diffractive layers. In this study, after passing through the MLDOE, the field is propagated using a method that utilizes the angular spectrum of plane waves. The Strehl ratio is used to determine the "best focus" plane, where it is shown that the focalization efficiency is above 95% for the working order in the mid- and long-wave IR bands. This result, along with the very low energy content of the other orders, proves the strong imaging potential of MLDOEs for dual-band applications. It is also demonstrated that the MLDOE has the same chromatic behavior as standard DOEs, making it a very useful component for IR achromatization.

Light pattern generation with hybrid refractive microoptics under Gaussian beam illumination

The generation of wide-angle diffraction patterns can be done in different ways using either thin diffractive optical elements with small features sizes or arrays of microoptics with large optical paths that are thick diffractive optical elements. Our aim is to create as many high contrast diffraction-limited dots in the far-field as possible with a uniform intensity distribution. As a model system, we use a sinusoidal phase grating and as a peculiarity, we introduce non-uniform illumination using a Gaussian beam illumination. By making use of the self-imaging phenomenon, a large number of peaks with uniform distribution are generated for a defined range of the phase grating thicknesses due to the sinusoidal curvature. For very high structures, the pattern distribution is not uniform and it demonstrates that very thick sinusoidal phase gratings are not suitable pattern generators. For simulation, we compare thin element approximation, fast Fourier transform beam propagation method, and the rigorous finite difference time domain method. The large-angle diffraction is considered using a high numerical aperture propagator for far-field simulation. We demonstrate that the beam propagation and the Fraunhofer approximation are not accurate enough. Also, our rigorous near-field calculation versus phase grating thickness confirms the significant influence of reflection of thick structures on the far-field distribution, especially on pattern uniformity. Finally, experiments were carried out to confirm our findings and a good agreement between the simulation and experimental far-field distributions confirms our approach.

Diffuse scattering due to stochastic disturbances of 1D-gratings on the example of line edge roughness.

Diffuse scattering of optical one-dimensional gratings becomes increasingly critical as it constrains the performance, e.g., of grating spectrometers. In particular, stochastic disturbances of the ideal grating structure provoke straylight. In this paper, the straylight spectrum of stochastically disturbed gratings is examined. First, a 1D-method is presented that allows to calculate 2D-diffuse scattering of arbitrarily polarized light originating from stochastic disturbances of the grating geometry on the basis of standard optical simulation tools. Within the scope of this method an enormous reduction of computational effort is achieved compared to the full 2D-simulation approach, i.e., the computation time can be reduced by several orders of magnitude. Hence, the method also allows to address even large period gratings that are not possible to calculate within a full 2D-approach. In analogy to scattering theories for surface roughness the method relies on typical characteristics of straylight originating from small disturbances, that the angle resolved scattering (ARS) can be separated into a product of the power spectral density describing the 2D stochastic process and additional factors depending on the undisturbed 1D grating structure. In a second part, an analytical model within Fourier optics utilizing thin element approximation (TEA) describing the wide angle scattering of lamellar gratings disturbed by line edge roughness (LER) for TE-polarized light is derived and verified by applying the 1D-simulation method. For shallow gratings, we find an excellent agreement between simulation and TEA over the whole transmission half space. In addition, this model allows a descriptive understanding of the underlying physical effects and, accordingly, the influence of relevant parameters (grating geometry, refractive indices, illumination) onto the scattering spectra is discussed. Further, it is shown that LER-scattering can be described within a modified Rayleigh-Rice-ARS usually found within the frame of surface roughness.

Diffraction symmetry of binary Fourier elements with feature sizes on the order of the illumination wavelength and effects of fabrication errors

When building spot array binary Fourier diffractive optical elements (DOEs) having feature sizes on the order of the wavelength, we noticed remarkable variations in the experimental diffraction efficiency compared to the simulation results. Even with the use of high-cost electron beam lithography and rigorous Fourier modal method simulations, there appear to be no publications, to the best of our knowledge, showing close agreement in diffraction efficiency between the simulation and experimental results. In this Letter, we show that the diffraction symmetry of binary Fourier DOEs can be an efficient and consistent metric for evaluating the limit of the thin-element approximation and the effects of fabrication errors.

Rotationally symmetric formulation of the wave propagation method-application to the straylight analysis of diffractive lenses.

We introduce a modified formulation of the wave propagation method for the efficient simulation of rotationally symmetric micro-optical components. The reformulated algorithm provides an increased computational performance of approximately two orders of magnitude and strongly reduced memory requirements, in comparison to the original formulation. This enables the efficient wave optical simulation of extended micro-optical structures beyond the common thin-element approximation. As a prototypical example, we assess the modified algorithm for the evaluation of straylight induced by diffractive lenses. We find an excellent accuracy, while comparing to rigorous simulations, which justifies the ability to overcome the limitations of the thin-element approximation.

Parabasal thin-element approximation approach for the analysis of microstructured interfaces and freeform surfaces.

The thin-element approximation (TEA) approach is an efficient algorithm to analyze microstructured interfaces, e.g., diffractive optical elements or scattering surfaces. However, the classical approach is valid only under the condition of paraxial illumination. We hereby develop an extended algorithm to include parabasal illumination, which is characterized by low divergence with arbitrary propagation direction. The extended approach is named as the parabasal TEA approach. In this paper, we present the algorithm of the parabasal TEA approach and compare the results with that of a rigorous calculation in order to demonstrate its validity. We also discuss the role of the parabasal TEA approach in a more general concept for modeling light propagating through freeform surfaces.

Calculation of diffraction of laser radiation by a two-dimensional (cylindrical) axicon with the high numerical aperture in various models

We consider the diffraction of Gaussian beams by a cylindrical axicon whose numerical aperture (NA) is close to or higher than a limiting value (when the incident wave is assumed not to pass through an element). Three models of diffraction were considered: the ray approach, the vector wave theory with a thin optical element approximation and the solution of Maxwell’s equations by the finite elements method. Although in the ray approach the limiting NA corresponds to the total internal reflection (TIR), the analysis of the ray path has shown that with increasing NA (narrowing axicon’s angle) a proportion of energy leaks through the lateral sides, forming a diverging beam. In the wave theory, energy dissipation in lateral directions also occurs, but evanescent waves play a special role in the element’s near-field zone. In this case, the analytical estimations for the electric field components have been obtained in a thin element approximation. Application of the finite element method to Maxwell’s equations has shown that for optimal concentration of energy at the refractive element’s tip its NA should be increased (by narrowing the axicon’s angle or by increasing the material refractive index) only until the TIR occurs. The further increase of the NA results in both the reflection of rays from the flat surface and their output from the lateral sides.

Analytical design of freeform optical elements generating an arbitrary-shape curve

A method is proposed for designing refractive optical elements focusing a collimated incident beam into a curve with specified shape. A general relationship for the freeform surface of the optical element is derived as an envelope of a parametric family of hyperboloids of revolution that focuses the incident beam into the points on the curve. Using the thin optical element approximation, the calculation of the hyperboloid parameters providing required irradiance distribution along the curve is reduced to the solution of an explicit first order differential equation. Optical elements generating line segment focus and circular arc focus are designed. The simulation results demonstrate generation of high-quality curves.

Holographic method for topography measurement of highly tilted and high numerical aperture micro structures

This paper presents an analysis of the topography characterization of a tilted sample by interferometric measurement systems in transmission and reflection configurations. The developed analytical expressions permit the direct reconstruction of the three-dimensional shape of a tilted sample. The first method presented in this paper, the so-called Thin Tilted Element Approximation (TTEA) method, is an extension of the well-known Thin Element Approximation for tilted geometry, which can be applied to the case of large sample tilts, but it requires the numerical aperture (local shape gradients) of the sample to be low. The second method presented here, the so-called Tilted Local Ray Approximation (TLRA) algorithm, is based on the analysis of local ray transition through a measured object. The method can be applied for the accurate shape characterization of tilted samples with high shape gradient. The developed algorithms require focused images, while in the inclined sample configuration the image plane is tilted relatively to the detection plane. To overcome this problem we propose the application of an efficient algorithm for the numerical propagation between tilted planes. The accuracy of TTEA and TLRA algorithms is successfully proven with simulations and experimental measurements for the case of low and high NA microlenses. ► Shape measurement of inclined samples in digital holographic microscopy is investigated. ► Methods for an accurate topography reconstruction of tilted samples are proposed. ► Two reconstruction algorithms are developed for the shapes of low and high NA. ► An efficient algorithm for numerical refocusing to tilted plane is presented.

Analytical design of refractive optical elements generating a curve

A method for designing refractive optical elements focusing collimated incident beam into a curve with specified shape is proposed. A general relationship for the freeform surface of the optical element is derived as an envelope of a parametric family of hyperboloids of revolution which focus the incident beam into the points on the curve. Using the thin optical element approximation, the calculation of the hyperboloid parameters providing required irradiance distribution along the curve is reduced to the solution of an explicit first order differential equation. The application for the proposed method to focus into the set of points on the curve is presented. The optical element generating line segment focus and the composite optical element to focus into the curve that consists of two circular arcs are designed. The optical elements to focus into the set of points on the line-segment and on the curve that consists of two circle arcs.

Analogy between generalized Coddington equations and thin optical element approximation

Local wavefront curvature transformations at an arbitrarily shaped optical surface are commonly determined by generalized Coddington equations that are developed here via a local thin optical element approximation. Eikonal distributions of the incident and refracted beams are calculated and related by an eikonal transfer function of a local thin optical element located in close proximity to a given point at a tangent plane of an optical surface. Main coefficients and terms involved in the generalized Coddington equations are derived and explained as a local nonparaxial generalization for the customary paraxial wavefront transformations.

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.

Electromagnetic approach to the thin element approximation

Electromagnetic approach to the thin element approximation

Electromagnetic approach to the thin element approximation

We show that the rigorous Fourier modal method yields the thin element approximation when the feature sizes and the period tend to infinity. The transversal eigenmodes of the structure can be divided into sets whose effective refractive index corresponds to the refractive indices of the grating. Each set can be treated as the thin element approximation with constant amplitude and piecewise continuous phase transmission. We also demonstrate that the primary reason for the rapid failure of the thin element approximation with thick binary structures is attributable to destructive interference of the waveguide modes.

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

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.