The paper reports on development of an integral and nondestructive technique of characterization of low-dimensional
periodically arranged nanocrystals (LDPAN) by spectroscopic scatterometry in the UV-IR ranges. Some approaches to
the solution of direct and inverse problems in scatterometry are addressed. For the solution of the direct problem, the
author has chosen the universal method of boundary integral equations, which has demonstrated a broad range of
applicability and a high accuracy. Cases are analyzed in which a complicated three-dimensional diffraction problem
involving 2D gratings can be reduced to a two-dimensional one with 1D gratings, or multilayer mirrors with plane
boundaries. An algorithm is proposed for the solution of a system of nonlinear operator equations with an arbitrary, but
limited set of unknown LDPAN structural parameters, and a given set of measured values of diffraction efficiency. The
functional to be minimized in the course of solution of the inverse problem is identified, and methods of its
regularization and for monitoring the accuracy of the solution are proposed. A Fortran code written with the use of the
Löwenberg-Markwardt gradient method has turned out an efficient way to the solution of model problems for a Si
grating with a trapezoidal profile.
Multilayer coated blazed gratings with high groove density are the best candidates for use in high resolution EUV and
soft x-ray spectroscopy. Theoretical analysis shows that such a grating can be potentially optimized for high dispersion
and spectral resolution in a desired high diffraction order without significant loss of diffraction efficiency. In order to
realize this potential, the grating fabrication process should provide a perfect triangular groove profile and an extremely
smooth surface of the blazed facets. Here we report on recent progress achieved at the Advanced Light Source (ALS) in
fabrication of high quality multilayer coated blazed gratings. The blazed gratings were fabricated using scanning beam
interference lithography followed by wet anisotropic etching of silicon. A 200 nm period grating coated with a Mo/Si
multilayer composed with 30 bi-layers demonstrated an absolute efficiency of 37.6% in the 3rd diffraction order at 13.6
nm wavelength. The groove profile of the grating was thoroughly characterized with atomic force microscopy before and
after the multilayer deposition. The obtained metrology data were used for simulation of the grating efficiency with the
vector electromagnetic PCGrate-6.1 code. The simulations showed that smoothing of the grating profile during the
multilayer deposition is the main reason for efficiency losses compared to the theoretical maximum. Investigation of the
grating with cross-sectional transmission electron microscopy revealed a complex evolution of the groove profile in the
course of the multilayer deposition. Impact of the shadowing and smoothing processes on growth of the multilayer on the
surface of the sawtooth substrate is discussed.
The present work deals with a comprehensive numerical analysis of x-ray grazing-incidence scattering from single- and
double-boundary, finite-conducting rough surfaces with asperities of different statistics, performed with the use of a
mid-end workstation in a reasonable computation time. Multiple and multi-wave diffraction, refraction, absorption, and
resonances influence significantly x-ray and neutron scattering. These are pure dynamic effects, which require
application of a rigorous theory to correctly describe the power change in the specular order and to describe nonspecular
distribution. Despite the impressive progress attained in developing a rigorous theory with account for random
roughness, the author is aware only of approximate and asymptotic approaches in the case of neutron and x-ray
scattering even by 1D surfaces, such as the Born approximation, the distorted-wave Born approximation, parabolic
equation methods, etc. The PCGrate®-SX v.6.3 software developed on the basis of a modified boundary integral
equation method and the Separating solver allows one to operate with exact models, e.g., those involving Maxwell's
equations and rigorous boundary conditions, and appropriate radiation conditions. In order to compute the scattering
properties of a rough surface using the forward electromagnetic solver, Monte Carlo simulation is employed to average
the deterministic scattered power due to individual surfaces over an ensemble of realizations. The difference between
approximate and rigorous approaches can be clearly seen in cases where grazing incidence occurs at close to or larger
than the critical angle. This difference may give rise to wrong estimates of rms roughness and correlation length if they
are determined by comparing experimental data with calculations. Besides, the rigorous approach permits taking into
account any known roughness statistics, including quasi-periodicity of quantum dot ensembles.
The rigorous integral equation method (viz MIM) in which the border structure is represented by a multilayer 1-D grating working at small wavelength-to-period ratios is used for taking into account electromagnetic scattering from different-types of nano-irregularities, such as periodical, random, self-organized, and their combinations. An example of the third type is multiple quantum dot (QD) ensembles and quantum molecules. The borders may contain a few or a large number of asperities of any kind. The program deals with a structure that is a grating from a mathematical point of view but that can model a rough surface if the groove spacing becomes large compared with the width (or correlation length) of asperities. This classical model for scattered light computation of bulk or few-border rough mirrors and gratings at visible and IR wavelengths is applied in PCGrate(R)-SX v.6.2 software based on the MIM to calculate multilayer structures in the X-ray-VUV range, which is a very difficult problem for any rigorous method, even for 1-D surfaces. Border profiles of most realistic types including real ones (e.g. AFM-measured) have both periodical and random components, and some ensemble averaging is required to obtain exact specular and nonspecular (diffuse) scattering intensities. The PCGrate results are compared with the data obtained by approximated approaches and measurements. Calculated intensities of scattering from gratings and rough mirrors as well as multiple QD structures can differ from those obtained for ideal and average border profiles or from approximations by a few percent up to a few orders of magnitude. The method can be applied both to forward computations of efficiencies over a wide range of angles and to fitting of the border metrological performance by comparing to measured data.
Reflecting large-blaze-angle diffraction gratings operating in the off-plane grazing configuration can be used to advantage as high-efficiency tools permitting separation and focusing of a 2%-wavelength about 13.5-nm. A cooled multiple grating with fan-groove geometry is used as a model to select the desired spectral range and obtain a record-high efficiency and stigmatic image focusing for a high dispersion in adjacent orders. A maximal relative efficiency of 0.96-0.99 and reflectances of 0.71-0.96 can be readily obtained with an off-plane fine-pitch grating intercepting at incidence angles of 70-85° a converging unpolarized light beam from an EUV collector, a figure higher than that of a grating in an in-plane mount. An absolute efficiency of ~0.72, calculated with a PCGrate-SX code using the AFM-measured groove profile, is reached in the -1st order of a Mo-coated 200-nm-pitch Si test grating in unpolarized light.
The diffraction efficiencies of a Fresnel zone plate (ZP), fabricated by Xradia Inc. using the electron-beam writing technique, were measured using polarized, monochromatic synchrotron radiation in the extreme ultraviolet wavelength range 3.4-22 nm. The ZP had 2 mm diameter, 3330 zones, 150 nm outer zone width, and a 1 mm central occulter. The ZP was supported by a 100 nm thick Si3N4 membrane. The diffraction patterns were recorded by CMOS imagers with phosphor coatings and with 5.2 μm or 48 μm pixels. The focused +n orders (n=1-4), the diverging -1 order, and the undiffracted 0 order were observed as functions of wavelength and off-axis tilt angle. Sub-pixel focusing of the +n orders was achieved. The measured efficiency in the +1 order was in the 5% to 30% range with the phase-shift enhanced efficiency occurring at 8.3 nm where the gold bars are partially transmitting. The +2 and higher order efficiencies were much lower than the +1 order efficiency. The efficiencies were constant when the zone plate was tilted by angles up to ±1° from the incident radiation beam. This work indicates the feasibility and benefits of using zone plates to measure the absolute EUV spectral emissions from solar and laboratory sources: relatively high EUV efficiency in the focused +1 order, good out-of-band rejection resulting from the low higher-order efficiencies and the ZP focusing properties, insensitivity to (unfocused) visible light scattered by the ZP, flat response with off-axis angle, and insensitivity to the polarization of the radiation based on the ZP circular symmetry. EUV sensors with Fresnel zone plates potentially have many advantages over existing sensors intended to accurately measure absolute EUV emission levels, such as those implemented on the GOES N-P satellites that use transmission gratings which have off-axis sensitivity variations and poor out-of-band EUV and visible light rejection, and other solar and laboratory sensors using reflection gratings which are subject to response variations caused by surface contamination and oxidation.
The validity of approximating the efficiency of a multilayer grating operating at close to normal incidence in the soft-X-ray-EUV range with a product of the relative grating efficiency by the reflectance of its multilayer coating has been studied by the rigorous integral method. The widely used approximated approach has until recently been considered accurate enough for analysis of short-wavelength normal-incidence multilayer-coated gratings. Real gratings employed in the soft-X-ray-EUV range are used to demonstrate the inapplicability of this approximation to an analysis of precise positions of efficiency maxima for the external (n > 0) and internal (n < 0) diffraction orders, despite the small ratios of wavelength and groove depth to period. The present authors have performed an analysis of the accuracy inherent in a derived simple expression for spectral separation of the same plus and minus orders with respect to the wavelength, order's number, incident angle, period, and groove depth. The reason for the observed substantial (a few Angstrom or even nm) wavelength separation between the maxima of positive and negative orders is related to oblique, close-tonormal incidence of radiation on a grating operating in the short-wavelength spectral region and different angles of deviation of respective orders. The modeling carried out with the commercial code PCGrate-S(X) v.6.1 permitted not only prediction of the separation between positive and negative orders for a multilayer Mo/Si 4200-gr/mm grating with FM-measured trapezoidal groove profile, which is designed for operation in the EIS spectrometer on the Solar-B spacecraft, but obtaining a good agreement with synchrotron radiation measurements, including high orders as well. A conclusion is drawn that high-precision calculations of the efficiency of multilayer normal-incidence soft-X-ray-EUV range gratings have in some cases to be performed, although this may require increasing the computation time by several times compared to the commonly used approximate approach.
Efficiency measurements of a grazing-incidence diffraction grating, planned for the Constellation-X Reflection Grating Spectrometer (RGS), were performed using polarized synchrotron radiation at the NRL Brookhaven beamline X24C. The off-plane TM and TE efficiencies of the 5000 groove/mm MIT test grating, patterned on a silicon wafer, were measured and compared to the efficiencies calculated using the PCGRATE-SX code. The calculated and measured efficiencies are in agreement when using groove profiles derived from AFM measurements. The TM and TE efficiencies differ, offering the possibility of performing unique astrophysical science studies by exploiting the polarization sensitivity of the off-plane gratings. The grating calibrations demonstrate the importance of using polarized synchrotron radiation and code calculations for the understanding of the Constellation-X grating performance, in particular the effects of the groove profile and microroughness on the efficiency. The optimization of grazing incidence gratings, for both the off-plane and in-plane mounts, planned for the RGS and x-ray spectrometers on other missions will require detailed synchrotron measurements and code calculations.
The computational design of multilayer-coated diffraction gratings for the extreme ultraviolet (EUV) wavelength region and the experimental performance of the coated gratings depend on the optical constants of the layer materials. While accurate optical constants are available for many commonly used materials, the EUV optical constants can in practice differ significantly from the tabulated values. This is generally true near absorption edges, for reactive materials that may be subject to oxidation or contamination, and for the longer EUV wavelengths (>30 nm) where molecular effects can be important. Normal-incidence gratings with Mo/Si coatings operating in the 17-21 nm and 25-29 nm wavelength ranges were successfully designed and fabricated for the Extreme Ultraviolet Imaging Spectrometer (EIS) on the Solar-B mission, the first satellite instrument to carry a multilayer grating. Examples of multilayer gratings designed and fabricated for wavelengths <12 nm and >40 nm, using materials other than Mo/Si, will be given that have in many cases required the experimental determination of the optical constants owing to inaccuracies in the tabulated values.
Both classical (in-plane) and conical (off-plane) grating configurations can be used in the spectrometer being developed for the Spectroscopy X-ray Telescope (SXT), which is assigned for the Constellation-X mission. Rigorous absolute efficiency calculations of gold-coated diffraction gratings with ideal triangular, trapezoidal, and polygonal profiles have been carried out for both possible spectrometer mountings by the PCGrate-SX program based on a modified integral method, with due account of random roughness. Optimum grating parameters and spectrometer configuration providing maximum theoretical efficiency were determined. Rigorous calculations performed with optimization showed that blazed grating absolute efficiency for the in-plane configuration similar to that employed in the XMM-Newton X-ray telescope cannot exceed 0.2-0.3 at the maxima in the minus first diffraction order within the relevant range of grazing angles, frequencies, and blaze angles. By contrast, using a grazing off-plane mounting permits one to compute gratings with a few times higher theoretical absolute efficiency in first diffraction orders, both at the maxima and on the average, for much higher grating frequencies and blazing angles. Unlike the classical mount, conical diffraction gives rise to noticeable polarization effects and Rayleigh anomalies in TM polarization. In view of the possibility of fabricating almost ideal triangular grooves by anisotropic etching of smooth graze-cut (111) silicon wafers by interference lithography and of compensating aberrations by properly modifying the frequency and/or grating groove curvature, the off-plane grating configuration may turn out preferable, particularly if a high spectral resolving power can be reached. A comparison with efficiency calculations and measurements is presented.
A rigorous modified integral method applicable for diffraction grating analysis working from x-ray -up to millimeter range is presented. The changes have been made both in theory, and in numerical realization. In theory special attention has been given to power balance criterion generalization for the case of absorbing gratings and to forms of Green's function representations. in comparison with the well known integral method formulated by Maystre, a lot of fundamental improvements have been made in the following numerical sections: the forms of representation of a groove profile, choice of points for calculation of unknown functions, integration method, choice of numbers of collocation points and Green's function expansion terms and their derivatives. For the first time stable convergence for all types of gratings and wavelengths, including those with very deep profiles, high conductivity, small wavelength-to-period ratios, and, especially, for TM polarization has been achieved and demonstrated. Examples of results are given for a wide range of transmission and reflection gratings and parameters of light. Diffraction efficiencies calculated with the help of the developed method of analysis are compared with published data and calculations performed by other researchers. All results have been obtained using ordinary PC and commercially available program PCGrateTM 2000X.
The deep research of all types of echelle gratings, working from low (8) up to very high (143 1) diffraction orders with use of the rigorous modified integral method of the analysis is presented. The modified integral approach allows one, with the help of the standard program (PCGrateTM 2000X) and a rather small PC, to simulate one ofthe most hard-to-converge diffraction efficiency problems, what the behaviour of echelle is. In comparison with detailed paper of B. Loewen et al. "Echelles: scalar, electromagnetic, and real-groove properties" the significant difference was found in calculation values for some examples in TM polarization. The difference between the compared theoretical data with the same refractive index for 3 1 6 gr/mm r-2 echelle at 632.8 nm in the 9 order and for TM polarization is up to 25% of absolute efficiency. The difference between calculated curve and measured data for the same grating and polarization at 441.6 nm in 12 and 13 orders is small (one-two percents) in opposite to the data of E. Loewen et al. , where the difference is many times more because of weak convergence of their method. The appreciable difference also exists for the medium and high orders. The presented results for the given refractive indices (basically, taken from the book E. Palik) have the best coincidence to experiment in all cases. Numerical research of two largest monolithic echelles, made on the project SOFIA also is included. The new record of rigorous calculations for r-1O EXES echelle, working in 143 1 order was achieved at 1 0.6 tm. Because of the very small a wavelength-to-period ratios (O.OO1) it is necessary to increase truncation parameter for such a case up to such value, that in result the matrices with the order about three thousands turn out.
The normal incidence efficiencies of a 2400 groove/mm master grating and a replica gratin were measured using synchrotron radiation in the extreme UV region. As a result of the replication process, the replica grating had an aluminum surface with an oxide layer. An additional thin SiO2 coating was applied to the oxidized aluminum surface for the purpose of reducing the microroughness. The efficiency of the replica grating as a function of wavelength had an oscillatory behavior that resulted from a thin-film interference effect associated with the SiO2 coating. The measured efficiencies were compared to the efficiencies calculated by a computer program that is based on the modified integral approach. The calculation accounts for the groove profile as determined from atomic force microscopy, the optical properties of the grating surface and the three layers, and the polarization of the incident radiation. The measured and calculated efficiencies are in good overall agreement.
Application of rigorous integral method for computing the efficiency of arbitrary profile relief gratings used in all the optical spectral range is presented in this paper. The main progress of the method and the programs lays in numerical solution algorithm. In particular, an approximation of Green-function and its normal derivative is used providing a sufficient accuracy for common practice simultaneously with satisfactory computation time. There is a very important peculiarity of the algorithm, namely both distribution of points of collocation and choice of the number of terms in Green-function expansion are used. These characteristics are different for each special case: perfect conductivity, finite conductivity, transmission gratings and gratings for X-ray and XUV. Such programs can be used as a mathematical model to design and calculate complex multielement optical systems with diffraction gratings.
A comparison of various rigorous methods of analysis for soft x ray and XUV surface-relief gratings is presented. Example results for a wide range of groove width-to-groove-spacing ratios and angles of incidence are presented for lamellar gratings. Diffraction efficiencies for gold lamellar gratings obtained from the integral and the characteristic wave methods of analysis are compared with previously published numerical results. The accuracy, the convergence, and the stability of the numerical methods are discussed.
It was found that grating efficiency in short spectral range has unpredictable changes both under the alterations in groove geometry, groove frequency, angle of incidence and wavelength; and under replacement of reflectance, i.e. under the use of another coating material with the stand point of scalar theory. As is shown, this non-scalar effect becomes greatly apparent only for high-frequency gratings (up to 3600 g/mm) and includes two aspects. They are: (1) it is impossible to predict grating efficiency under the changing of coating material (without changes of all the other parameters) by multiplication on the ratio of coating material Fresnel reflectance; (2) optimum parameters (angle of incidence, groove depth, groove width for lamellar grating) found for one kind of coating material are not optimum for another.
Differential formalism gives a rapid convergence of accurate results even under conditions of propagation of several hundred spectral orders and grazing angles of incidence for the gratings with low groove-depth-to-groove-spacing ratio (h/d << 1) which are typical for this wavelength region. In the same way it was found that the rigorous results on the region of low (delta) /d ratios under the short time of computation may also be done through the modified integral method. By way of the programs created on the base of integral equations, the results done through the differential method were repeated and a good coincidence with them was found. These data allow us to speak about the equivalence of both methods -- integral and differential -- and about the convergence of algorithms.
There were examined mechanisms of arsenic sulphide films dissolution In water and spirit solutions of alkalis and in protonic and aprotonic solutions of aliphatic amines as well. Optimum conditions for the arsenic sulphide films processing without formation of a sediment and ratio of dissolution velosity of exposed and as-deposited parts of film less then of thousand units were found. 1 . INTIODUCTION It is widely known that holografic gratings (HG) have many advantages over engineruled gratings: they are free from " ghosts" have a lower level of scattered light it is possible to obtain gratings with frequency beyond 3000 1/mm The usial technique of HG manufacturing uses organic photoresists. The photoresists based on the low-molecular phenol formaldehyde resists with positive type of development are a most common choice. It is known that these resists do not dissolve in water but does dissolve in alkaline solutions The dissolution proceeds by molecular fragments each consisting of 7-10 phenolic cycles2through the stage of swelling which leads to a marked ruoghness of the etched surface and the distortions of the relief profile when the value of resolutions is beyond 3000 1/mm. It is indicated in Fig. 1 obtained with scanfling tunnel microscope. The resolution of the relief shown in the picture equals 3600 1/mm. It can be seen that at such values of resolution the obtained relief is distorted thus implying that the use of
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