Lamellar Gratings Research Articles

Design and Fabrication of Diffractive Light-Collecting Microoptical Device with 1D and 2D Lamellar Grating Structures

This paper presents the optimal design method of diffractive light-collecting microoptical device and its fabrication method by E-beam lithography, fast atom beam etching, and hot-embossing processes. The light-collecting device proposed in the paper is comprised of 9 (3 × 3) blocks of optical elements: 4 blocks of 1D lamellar grating structures, 4 blocks of 2D lamellar grating structures, and a single block of nonpatterned element at the center, which acts for lens to be able to collect the diffracted and transmitted lights from the lamellar grating structures into the focus area. The overall size of the light-collecting device is 300 × 300 μm2, and the size of each block was practically designed as 100 × 100 μm2. The performance of 1D and 2D lamellar grating structures was characterized in terms of diffraction efficiency and diffraction angle using a rigorous coupled-wave analysis (RCWA) method, and those geometric parameters, depth, pitch, and orientation, were optimized to achieve a high light-collecting efficiency. The master molds for the optimized structures were fabricated on Si substrate by E-beam lithography and fast atom beam etching processes. The 100 μm thick patterned polymethyl methacrylate (PMMA) film was then replicated by a hot-embossing process. As a result, the patterned PMMA film collected 63.0% more incident light than a nonpatterned one.

A stable approach to conical diffraction of nearly lossless metallic gratings

We develop a multidomain pseudospectral method to analyze lamellar gratings for general conical diffraction from TE/TM diffraction. The validity and usefulness of this approach are demonstrated through normal metallic lossy and highly conductive lossless gratings. The developed method provides excellent numerical stability and convergence over commonly used rigorous couple wave analysis (RCWA) when the nearly lossless metal is involved. In the case, RCWA suffers from instability problem, while our method still performs well. Several examples are demonstrated.

Optimized grating as an ultra-narrow band absorber or plasmonic sensor

Lamellar gratings are investigated via temporal coupled-mode theory and numerical simulations. Total absorption can be achieved by an optimized grating with shallow grooves under normal incidence and the full width at half-maximum (FWHM) is only 0.4 nm. For certain wavelengths, the structure shows high absorption only within an ultra-narrow angle, which suggests that it can be used as a highly directional thermal emitter according to Kirchhoff's law. Besides, the resonant wavelength is sensitive to the refractive index of the environmental dielectric. The large sensitivity (1400 nm/RIU) and simultaneous small FWHM result in a huge figure-of-merit of 2300/RIU, which enables the structure to have great potential in plasmonic sensing.

Efficient implementation of B-spline modal method for lamellar gratings

The B-spline modal method (BMM) as applied to lamellar gratings analysis is revisited, and a new implementation is presented. The main difference with our previous work is that we now take into account discontinuities by putting a spline with a degenerate knot on them. Our new approach is compared with other implementations of the BMM and is shown to be superior in terms of numerical convergence.

Pseudospectral modal method for conical diffraction of gratings

For lamellar gratings and other layered periodic structures, the modal methods (including both analytic and numerical ones) are often the most efficient, since they avoid the discretization of one spatial variable. The pseudospectral modal method (PSMM) previously developed for in-plane diffraction problems of one-dimensional gratings achieves high accuracy for a small number of discretization points, and it outperforms most other modal methods. In this paper, an extension of the PSMM to conical diffraction problems is presented and implemented. Numerical examples are used to demonstrate the high accuracy and excellent convergence property of this method for both dielectric and metallic gratings.

Perfect Broadband Terahertz Antireflection by Deep‐Subwavelength, Thin, Lamellar Metallic Gratings

Perfect broadband terahertz antireflection coatings are demonstrated by deep-subwavelength, thin, lamellar metallic gratings both in theory and experiment. Abrupt phase changes introduced by grating to terahertz wave traversing the coated surface can be tuned from 0 to π as a function of grating parameters. At π/2, the impedance difference between two dielectrics is matched and perfect antireflection occurs.

Optical biosensing of bacteria and cells using porous silicon based, photonic lamellar gratings

We report on a method to extend the optical sensing capabilities of conventional RIFTS (reflective interferometric Fourier transform spectroscopy) biosensors for real-time detection of large microorganisms, such as bacteria and cells. Using macro porous silicon based 2D arrays of phase (lamellar) grating, we demonstrate that the zero-order optical reflectivity exhibits a similar interference pattern to that obtained for ordinary RIFTS biosensors, which can be Fourier transformed into optical thickness and exploited for biosensing. The sensing capabilities are demonstrated for Escherichia coli bacteria that were captured inside the macro-pores. The entrapment process is monitored and verified by confocal laser scanning microscopy.

Efficient implementation of Fourier modal slice absorption method for lamellar crossed gratings

The slice absorption method (SAM) is a numerical parallel solution reported by Rumpf et al. for analyzing large-scale devices with a complex structure. We use this method to simulate diffraction of the lamellar crossed gratings, and we use the wave matrix for the Fourier modal method (FMM) for SAM formulation. For this formulation, gratings with arbitrary profiles can be viewed as a stack of lamellar gratings via a staircase approximation, which is different from the difference approximation in the original SAM. In addition, an improved cascading-doubling algorithm is implemented, and not only for long uniform regions or repeated units. This algorithm provides a concurrent result with the FMM and promises an efficient application in scatterometry for multilevel gratings.

Extended theory of soft x-ray reflection for realistic lamellar multilayer gratings

An extended set of coupled wave equations were derived to describe non-idealized lamellar multilayer grating structures with properties as obtained with state-of-the-art fabrication techniques. These generalized equations can include all relevant effects describing the influence of passivation and contamination layers, non-rectangular lamel profiles and sidewall scalloping. The calculations showed that passivation and contamination plays an important role in that it may significantly reduce peak reflectivity. However, we also derived a condition for layer thicknesses having negligible effects. Slightly positive tapered lamel profiles are shown to further reduce the bandwidth as compared to a rectangular lamel profile, whereas negative tapers significantly increased the bandwidth. The influence of intriguing effects, such as the sidewall scalloping caused by Bosch Deep Reactive Ion Etching, are also modeled. We identified the signature of such scalloping as additional side peaks in the reflectivity spectrum and present parameters with which these can be effectively suppressed.

Wavelet element method for lamellar gratings

A wavelet element method is developed for analyzing lamellar diffraction gratings or grating stacks. The eigenmodes of the grating layers are accurately calculated by this method, and then the diffraction efficiencies of the gratings are calculated by the S-matrix algorithm. The method proposed in this paper consists in mapping each homogeneous layer to a wavelet element, and then matching them according to the boundary conditions between the layers. By this method the boundary conditions are satisfied rigorously and the Gibbs phenomenon in the Fourier modal method (FMM) can be avoided. The method performs better than the standard FMM for gratings involving metals. It can also be applied to analyze other discontinuous structures.

On the contribution of gratings with laterally graded groove depths to the design and performances of SOLEIL soft X-ray monochromators

The so called varied groove depth (VGD) etching technology, developed by Horiba-Jobin-Yvon, which consists in laterally grading the groove depth of a lamellar grating, is exploited in the monochromators of the seven soft x-ray beamlines of SOLEIL. With a VGD profile a single grating can perform in more than a decade wide energy band and some control is given over the harmonic content. It also requires a specific, but easily achieved design of the beamline optics to match the lateral footprint of the beam to the depth variation. Performances predicted by computation with electromagnetic propagation code are supported by flux measurements of the two first harmonics in the experimental chamber of a photoemission beamline, and by measurements of grating efficiencies on SOLEIL etrology beamline.

High efficiency multilayer gratings for monochromators in the energy range from 500 eV to 2500 eV

Multilayer (ML) gratings have been prepared by coating shallow ion etched lamellar gratings with a Mo2C/B4C multilayer having a layer thickness close to the groove depth. It was shown that such a structure behaves as a 2D synthetic crystal and can reach very high efficiencies when the Bragg condition is satisfied. A ML coated grating has been characterized at the SOLEIL Metrology beamline between 700 and 1700 eV and a beamline of PTB at BESSY II between 1750 and 3500 eV. A peak diffraction efficiency of nearly 27 % was measured at 2200 eV. The measured efficiencies are well reproduced by numerical simulations made with the electromagnetic propagation code CARPEM, This grating will be used in the Deimos beamline at SOLEIL together with a matched multilayer mirror.

MODAL METHOD BASED ON SUBSECTIONAL GEGENBAUER POLYNOMIAL EXPANSION FOR LAMELLAR GRATINGS: WEIGHTING FUNCTION, CONVERGENCE AND STABILITY

The Modal Method by Gegenbauer polynomials Expan- sion (MMGE) has been recently introduced for lamellar gratings by Edee (8). This method shows a promising potential of outstanding convergence but still sufiers from instabilities when the number of polynomials is increased. In this work, we identify the origin of these instabilities and propose a way to remove them. Among the numerical methods developed for the analysis of lamellar difiraction gratings, modal methods play an important role because of their great versatility and relative efiectiveness. In the classical modal method (1,2), the eigenvalues are obtained by solving a transcendental equation. In other modal methods, the eigenmodes and propagation constants are generally obtained by searching the eigenvalues and eigenvectors (3{8) of a matrix which is derived from the Maxwell's equations by using the method of moments (9). Mathematically speaking, for one-dimensional gratings with piecewise homogeneous media, and plane wave excitation, the eigenmodes are solutions of the Helmholtz equation subject to boundary conditions at the interfaces between two media and to the pseudo-periodicity condition. Numerically, the rate of convergence of the method depends on how the matrix from which eigenvalues are sought takes into account the continuity relations. Indeed, one of the main difierences between

Single-order operation of lamellar multilayer gratings in the soft x-ray spectral range

We demonstrate single-order operation of Lamellar Multilayer Gratings in the soft x-ray spectral range. The spectral resolution was found to be 3.8 times higher than from an unpatterned multilayer mirror, while there were no significant spectral sideband structures adjacent to the main Bragg peak. The measured spectral bandwidths and peak reflectivities were in good agreement with our theoretical calculations.

Arrays of doped and un-doped semiconductors for sensor applications

This numerical investigation proposes to use a lamellar grating of doped semiconductors as the active region of a nanoplasmonic biosensing device. Working with highly doped semiconductors instead of a metal allows controlling the value of the plasma frequency. It is possible to reach a plasma frequency close to the range of detection of the sensor to improve its sensitivity. A red shift of the plasmonic resonance of 10.2 nm for a 10−2 refractive index unit increase can be achieved.

Quasianalytical modal approach for computing Casimir interactions in periodic nanostructures

We present an almost fully analytical technique for computing Casimir interactions between periodic lamellar gratings based on a modal approach. Our method improves on previous work on Casimir modal approaches for nanostructures by using the exact form of the eigenvectors of such structures, and computing eigenvalues by solving numerically a simple transcendental equation. In some cases eigenvalues can be solved for exactly, such as the zero frequency limit of gratings modeled by a Drude permittivity. Our technique also allows us to predict analytically the behavior of the Casimir interaction in limiting cases, such as the large separation asymptotics. The method can be generalized to more complex grating structures, and may provide a deeper understanding of the geometry-composition-temperature interplay in Casimir forces between nanostructures.

Fourier-matching pseudospectral modal method for diffraction gratings: comment

Recently two variants of a pseudospectral modal method were developed for analyzing lamellar diffraction gratings: [J. Lightwave Technol. 27, 5151 (2009)] and [J. Opt. Soc. Am. A 28, 613 (2011)]. Both of them divide the computational domain into nonoverlapping subdomains and replace the spatial derivative in the Helmoltz equation by a differentiation matrix at the Chebyshev collocation points. The authors of the second reference claim that their method is more robust and accurate because they match the Fourier coefficient at the interfaces between the layers and drop some computed eigenmodes. We challenge these two ideas. Instead, we numerically demonstrate that by keeping all computed eigenmodes and by also numerically computing eigenmodes in homogeneous regions, the pseudospectral method performs better.

Analytical study of resonance conditions in planar resonators

We propose a study of a generalized Airy-like formula of the transmittance through planar resonators. A complete and analytical analysis of total transmission conditions is given according to physical parameters. The homogeneous problem related to free oscillations of resonators, which leads to complex resonance frequencies and quality factors, is also resolved. After a quick validation on well-known parallel-plate dielectric layers behaving as Fabry-Perot resonators, our discussion is applied on subwavelength metallic lamellar gratings from an analytical modal theory assuming perfectly electric conductors.

Luminescence efficiency of phosphors with periodically patterned surface

AbstractWe propose a principal scheme of external efficiency enhancement of phosphor luminescence, in particular chalcogenide semiconductor CdS. Green, orange, red, and near infrared frequencies can be enhanced and filtered from a continuous emission spectra of CdS phosphor on the basis of two different effects: (i) radiation stimulated formation of optically active defects in the phosphor structure; (ii) resonance diffraction of emitted photons on a patterned (regular system of etching grooves or/and metal lamellar grating) structure surface. We calculate the reflection spectra of an one dimensional resonant diffraction grating with single (CdS) and binary (CdS/Au) strip‐like rulings by the Riccati equation method. The gratings will control the emission angle cone and couple exponentially decaying (evanescent) part of phosphor radiance to air, which is a new mechanism of external efficiency enhancement (© 2012 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Growth rate and start current in Smith–Purcell free-electron lasers

This letter reports a theory to calculate the growth rate and start current of a Smith–Purcell free-electron laser, which is a promising radiation source in the terahertz domain. A two-dimensional model was used to investigate the interaction between a sheet electron beam and the surface wave above a lamellar grating. After deriving the growth rate from the dispersion equation, the start current was carefully estimated by considering the power flow above the grating. The agreement between the predictions of our theory and the results from the particle-in-cell simulations is acceptable.