Lamellar Gratings Research Articles

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.

Rigorous calculations and synchrotron radiation measurements of diffraction efficiencies for tender X-ray lamellar gratings: conical versus classical diffraction.

When reflection gratings are operated at grazing incidence in the extreme off-plane configuration and the incident beam trajectory is parallel to the grooves, the diffraction into the first order can be more efficient than in the classical orientation. This situation is referred to as the conical diffraction case. In the classical configuration the grooves are perpendicular to the incident beam and thus an efficiency-reducing shadowing effect will be observed at very grazing angles. It was recently shown that a laminar grating could provide symmetric and relatively high efficiencies in conical diffraction for diffraction even of photons with large energies of the order of 4 and 6 keV. For photon energies in the tender X-ray range, accurate computing tools for the calculation of diffraction efficiencies from gratings with simple coatings have not been available. Promising results for this spectral range now require the development of tools for modelling the diffraction efficiency expected in optical instrumentation, in which the provision of high efficiency in the indicated spectral range is mandatory. This is the case when weak sources are to be investigated, like in space science. In this study it will be shown that scalar calculations are not appropriate for this purpose, while newly introduced rigorous calculations based on the boundary integral equation method, implemented in the PCGrate® code, can provide predictions that are in agreement with observed diffraction efficiencies. The agreement is achieved by modelling the exact surface profile. This applies for both the conical diffraction configuration and for the classical in-plane configuration, in which a significantly lower efficiency was obtained. Even though the profile of the presented grating was not perfect, but significantly distorted, the calculations show that efficiency-wise the structure provided already more than 75% of the ideally expected efficiency for conical diffraction. This is a very promising result for further optimization of diffraction gratings for use in the tender X-ray range.

Study of lamellar multilayer grating near B K-edge and Si L-edge

We present an efficient design of lamellar multilayer grating (LMG) for the extreme ultraviolet/soft X-ray region near both the Si L-edge and B K-edge. The designs of LMGs have been performed by exploring different materials combinations, which are capable of providing high resolution and peak reflectivity. The performance analysis of the designed LMGs is made analytically using the coupled wave theory for single-order regime. That provides an estimation of diffraction efficiency and resolution. The effect of structural imperfections such as tapering of the lamellar profile and interfacial width on the optical properties are also analysed to describe non-ideal LMG structures.

Matched coordinates in the framework of polynomial modal methods for complex metasurface modeling.

The polynomial modal method (PMM) is one of the most powerful methods for modeling diffraction from lamellar gratings. In the present work, we show that applying it to the so-called matched coordinates leads to important improvement of convergence for crossed lamellar gratings with patterns that are not parallel to the coordinates' axes. After giving the new formulation of the PMM under matched coordinates in the general framework of biperiodic structures, we provide numerical examples to demonstrate the effectiveness of the proposed approach.

Element sensitive reconstruction of nanostructured surfaces with finite elements and grazing incidence soft X-ray fluorescence.

The geometry of a Si3N4 lamellar grating was investigated experimentally with reference-free grazing-incidence X-ray fluorescence analysis. While simple layered systems are usually treated with the matrix formalism to determine the X-ray standing-wave field, this approach fails for laterally structured surfaces. Maxwell solvers based on finite elements are often used to model electrical field strengths for any 2D or 3D structures in the optical spectral range. We show that this approach can also be applied in the field of X-rays. The electrical field distribution obtained with the Maxwell solver can subsequently be used to calculate the fluorescence intensities in full analogy to the X-ray standing-wave field obtained by the matrix formalism. Only the effective 1D integration for the layer system has to be replaced by a 2D integration of the finite elements, taking into account the local excitation conditions. We will show that this approach is capable of reconstructing the geometric line shape of a structured surface with high elemental sensitivity. This combination of GIXRF and finite-element simulations paves the way for a versatile characterization of nanoscale-structured surfaces.

Characteristic diffuse scattering from distinct line roughnesses

Lamellar gratings are widely used diffractive optical elements; gratings etched into Si can be used as structural elements or prototypes of structural elements in integrated electronic circuits. For the control of the lithographic manufacturing process, a rapid in-line characterization of nanostructures is indispensable. Numerous studies on the determination of regular geometry parameters of lamellar gratings from optical and extreme ultraviolet (EUV) scattering highlight the impact of roughness on the optical performance as well as on the reconstruction of these structures. Thus, a set of nine lamellar Si gratings with a well defined line edge roughness or line width roughness were designed. The investigation of these structures using EUV small-angle scattering reveals a strong correlation between the type of line roughness and the angular scattering distribution. These distinct scattering patterns open new paths for the unequivocal characterization of such structures by EUV scatterometry.

Reconstructing detailed line profiles of lamellar gratings from GISAXS patterns with a Maxwell solver

Laterally periodic nanostructures have been investigated with grazing-incidence small-angle X-ray scattering (GISAXS) by using the diffraction patterns to reconstruct the surface shape. To model visible light scattering, rigorous calculations of the near and far field by numerical solution of Maxwell's equations with a finite-element method are well established. The application of this technique to X-rays is still challenging, owing to the discrepancy between the incident wavelength and the finite-element size. This drawback vanishes for GISAXS because of the small angles of incidence, the conical scattering geometry and the periodicity of the surface structures, which allows a rigorous computation of the diffraction efficiencies with sufficient numerical precision. To develop metrology tools based on GISAXS, lamellar gratings with line widths down to 55 nm were produced by state-of-the-art electron-beam lithography and then etched into silicon. The high surface sensitivity of GISAXS in conjunction with a Maxwell solver allows the detailed reconstruction of the grating line shape for thick non-homogeneous substrates as well. The reconstructed geometric line-shape models are statistically validated by applying a Markov chain Monte Carlo sampling technique which reveals that GISAXS is able to reconstruct critical parameters like the widths of the lines with sub-nanometre uncertainty.

Conformal spray-deposited fluorine-doped tin oxide for mid- and long-wave infrared plasmonics

Nanocrystalline spray-deposited fluorine-doped tin oxide (FTO) was investigated for mid- and long-wave infrared plasmonics. Silicon lamellar gratings were conformally coated with FTO, and the excitation of surface plasmon polaritons (SPP) was investigated via their angle and wavelength-dependent reflectivity. Photon-to-SPP coupling efficiency as a function of grating parameters, and in comparsion to gallium-doped zinc oxide (GZO) gratings, was quantitatively analyzed based on a figure of merit related to the sharpness and depth of the coupling resonance. Conformal spray-deposited FTO would be useful in mid- and long-wave infrared plasmonic channel wave guides.

Split-And-Delay Unit for FEL Interferometry in the XUV Spectral Range

In this work we present a reflective split-and-delay unit (SDU) developed for interferometric time-resolved experiments utilizing an (extreme ultraviolet) XUV pump–XUV probe scheme with focused free-electron laser beams. The developed SDU overcomes limitations for phase-resolved measurements inherent to conventional two-element split mirrors by a special design using two reflective lamellar gratings. The gratings produce a high-contrast interference signal controlled by the grating displacement in every diffraction order. The orders are separated in the focal plane of the focusing optics, which enables one to avoid phase averaging by spatially selective detection of a single interference state of the two light fields. Interferometry requires a precise relative phase control of the light fields, which presents a challenge at short wavelengths. In our setup the phase delay is determined by an in-vacuum white light interferometer (WLI) that monitors the surface profile of the SDU in real time and thus measures the delay for each laser shot. The precision of the WLI is 1 nm as determined by optical laser interferometry. In the presented experimental geometry it corresponds to a time delay accuracy of 3 as, which enables phase-resolved XUV pump–XUV probe experiments at free-electron laser (FEL) repetition rates up to 60 Hz.

Polynomial modal analysis of slanted lamellar gratings.

The problem of diffraction by slanted lamellar dielectric and metallic gratings in classical mounting is formulated as an eigenvalue eigenvector problem. The numerical solution is obtained by using the moment method with Legendre polynomials as expansion and test functions, which allows us to enforce in an exact manner the boundary conditions which determine the eigensolutions. Our method is successfully validated by comparison with other methods including in the case of highly slanted gratings.

PiC code KARAT simulations of Coherent THz Smith-Purcell Radiation from diffraction gratings of various profiles

Generation of coherent THz Smith-Purcell radiation by single electron bunch or multi-bunched electron beam was simulated for lamellar, sinusoidal and echelette gratings. The dependences of the CSPR intensity of the corrugation gratings depth were investigated. The angular and spectral characteristics of the CSPR for different profiles of diffraction gratings were obtained. It is shown that in the case of femtosecond multi-bunched electron beam with 10 MeV energy sinusoidal grating with period 292 μm and groove depth 60 μm has the uniform angular distribution with high radiation intensity.

Radiative heat transfer between metallic gratings using Fourier modal method with adaptive spatial resolution

We calculate the radiative heat transfer between two identical metallic one-dimensional lamellar gratings. To this aim we present and exploit a modification to the widely-used Fourier modal method, known as adaptive spatial resolution, based on a stretch of the coordinate associated to the periodicity of the grating. We first show that this technique dramatically improves the rate of convergence when calculating the heat flux, allowing to explore smaller separations. We then present a study of heat flux as a function of the grating height, highlighting a remarkable amplification of the exchanged energy, ascribed to the appearance of spoof-plasmon modes, whose behavior is also spectrally investigated. Differently from previous works, our method allows us to explore a range of grating heights extending over several orders of magnitude. By comparing our results to recent studies we find a consistent quantitative disagreement with some previously obtained results going up to 50\%. In some cases, this disagreement is explained in terms of an incorrect connection between the reflection operators of the two gratings.

Spectral tailoring of nanoscale EUV and soft x-ray multilayer optics

Extreme ultraviolet and soft X-ray (XUV) multilayer optics have experienced significant development over the past few years, particularly on controlling the spectral characteristics of light for advanced applications like EUV photolithography, space observation, and accelerator- or lab-based XUV experiments. Both planar and three dimensional multilayer structures have been developed to tailor the spectral response in a wide wavelength range. For the planar multilayer optics, different layered schemes are explored. Stacks of periodic multilayers and capping layers are demonstrated to achieve multi-channel reflection or suppression of the reflective properties. Aperiodic multilayer structures enable broadband reflection both in angles and wavelengths, with the possibility of polarization control. The broad wavelength band multilayer is also used to shape attosecond pulses for the study of ultrafast phenomena. Narrowband multilayer monochromators are delivered to bridge the resolution gap between crystals and regular multilayers. High spectral purity multilayers with innovated anti-reflection structures are shown to select spectrally clean XUV radiation from broadband X-ray sources, especially the plasma sources for EUV lithography. Significant progress is also made in the three dimensional multilayer optics, i.e., combining micro- and nanostructures with multilayers, in order to provide new freedom to tune the spectral response. Several kinds of multilayer gratings, including multilayer coated gratings, sliced multilayer gratings, and lamellar multilayer gratings are being pursued for high resolution and high efficiency XUV spectrometers/monochromators, with their advantages and disadvantages, respectively. Multilayer diffraction optics are also developed for spectral purity enhancement. New structures like gratings, zone plates, and pyramids that obtain full suppression of the unwanted radiation and high XUV reflectance are reviewed. Based on the present achievement of the spectral tailoring multilayer optics, the remaining challenges and opportunities for future researches are discussed.

Simulation for Small Lamellar Grating FTIR Spectrometer for Passive Remote Sensing

A miniaturized FTIR spectrometer based on lamellar grating interferometry is being developed for passive remote-sensing. Consisting of a pair of micro-mirror arrays, the lamellar grating can be fabricated using MEMS technology. This paper describes a method to compute the optical field in the interferometer to optimize the design parameters of the lamellar grating FTIR spectrometer. The lower limit of the micro-mirror width in the grating is related to the formation of a Talbot image in the near field and is estimated to be about 100 μm for the spectrometer to be used for the wavelength range of 7-14 μm. In calculating the far field at the detection window, the conventional Fraunhofer equation is inadequate for detection distance of our application, misleading the upper limit of the micro-mirror width to avoid interference from higher order diffractions. Instead, the far field is described by the unperturbed plane-wave combined with the boundary diffraction wave. As a result, the interference from the higher order diffractions turns out to be negligible as the micro-mirror width increases. Therefore, the upper limit of the micro-mirror width does not need to be set. Under this scheme, the interferometer patterns and their FT spectra are successfully generated.

Polynomial modal analysis of lamellar diffraction gratings in conical mounting.

An efficient numerical modal method for modeling a lamellar grating in conical mounting is presented. Within each region of the grating, the electromagnetic field is expanded onto Legendre polynomials, which allows us to enforce in an exact manner the boundary conditions that determine the eigensolutions. Our code is successfully validated by comparison with results obtained with the analytical modal method.

Fluorine-doped tin oxides for mid-infrared plasmonics

Fluorine-doped tin oxides (FTO) were investigated for infrared plasmonic applications. Nano-crystalline FTO thin films were grown by the SPEED chemical-spray deposition method. Complex permittivity spectra were measured from 1.6 to 12 μm wavelength. These spectra were used to calculate materials parameters, which compared well with values from transport measurements, and to predict characteristics of surface plasmon polaritons (SPP). Reflectivity spectra for lamellar FTO gratings revealed SPP coupling resonances in good agreement with predictions. The FTO film studied here is well suited for plasmonic applications in the important 3-5 μm wavelength range.

Fourier method for modeling slanted lamellar gratings of arbitrary end-surface shapes in conical mounting.

An efficient modal method for numerically modeling slanted lamellar gratings of isotropic dielectric or metallic media in conical mounting is presented. No restrictions are imposed on the slant angle and the length of the lamellae. The end surface of the lamellae can be arbitrary, subject to certain restrictions. An oblique coordinate system that is adapted to the slanted lamella sidewalls allows the most efficient way of representing and manipulating the electromagnetic fields. A translational coordinate system that is based on the oblique Cartesian coordinate system adapts to the end-surface profile of the lamellae, so that the latter can be handled simply and easily. Moreover, two matrix eigenvalue problems of size 2N × 2N, one for each fundamental polarization of the electromagnetic fields in the periodic lamellar structure, where N is the matrix truncation number, are derived to replace the 4N × 4N eigenvalue problem that has been used in the literature. The core idea leading to this success is the polarization decomposition of the electromagnetic fields inside the periodic lamellar region when the fields are expressed in the oblique translational coordinate system.

Infrared surface polaritons on bismuth

Optical constants for evaporated bismuth (Bi) films were measured by ellipsometry and compared with those published for single crystal and melt-cast polycrystalline Bi in the wavelength range of 1 to 40 μm. The bulk plasma frequency ωp and high-frequency limit to the permittivity ε∞ were determined from the long-wave portion of the permittivity spectrum, taking previously published values for the relaxation time τ and effective mass m . This part of the complex permittivity spectrum was confirmed by comparing calculated and measured reflectivity spectra in the far-infrared. Properties of surface polaritons (SPs) in the long-wave infrared were calculated to evaluate the potential of Bi for applications in infrared plasmonics. Measured excitation resonances for SPs on Bi lamellar gratings agree well with calculated resonance spectra based on grating geometry and complex permittivity.

Platinum germanides for mid- and long-wave infrared plasmonics.

Platinum germanides (PtGe) were investigated for infrared plasmonic applications. Layers of Pt and Ge were deposited and annealed. X-ray diffraction identified PtGe(2) and Pt(2)Ge(3) phases, and x-ray photo-electron spectroscopy determined vertical atomic composition profiles for the films. Complex permittivity spectra were measured by ellipsometry over the 2 to 15 μm wavelength range. Surface plasmon polariton (SPP) characteristics such as propagation length and field penetration depth were calculated. Photon-to-SPP couplers in the form of 1D lamellar gratings were fabricated and characterized in the range 9 - 10.5 μm via wavelength-dependent specular reflection spectra for multiple angles of incidence. The observed resonances compare well with calculated spectra for SPP excitation on PtGe(2). Platinum germanides are CMOS compatible and may serve as SPP hosts for on-chip mid-IR plasmonic components with tighter field confinement than noble-metal hosts.

Casimir-Lifshitz force out of thermal equilibrium between dielectric gratings

We calculate the Casimir-Lifshitz pressure in a system consisting of two different 1D dielectric lamellar gratings having two different temperatures and immersed in an environment having a third temperature. The calculation of the pressure is based on the knowledge of the scattering operators, deduced using the Fourier Modal Method. The behavior of the pressure is characterized in detail as a function of the three temperatures of the system as well as the geometrical parameters of the two gratings. We show that the interplay between non-equilibrium effects and geometrical periodicity offers a rich scenario for the manipulation of the force. In particular, we find regimes where the force can be strongly reduced for large ranges of temperatures. Moreover, a repulsive pressure can be obtained, whose features can be tuned by controlling the degrees of freedom of the system. Remarkably, the transition distance between attraction and repulsion can be decreased with respect to the case of two slabs, implying an experimental interest for the observation of repulsion.