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Refractive Indices
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Refractive Indices
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The optical constants for the metal listed have been measured by a number of investigators, and there is a large scatter in the reported values for any one metal. Instrumental sources beyond the investigator’s control, e.g. second order radiation or scattered light from the monochromator, are responsible for some of the scatter. Other contributing factors arise from errors in quantities measured and from genuine differences in the samples caused by the methods of preparation. – Dr. David W. Lynch and Dr. William R. Hunter in11.  
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In conjunction with such fundamental characteristics of a medium as chemical composition, density, conductance or viscosity, two optical constants, namely refractive index n and absorption coefficient k, are also of paramount importance. These parameters, which describe interaction between an electromagnetic field and the medium, are closely responsive to any changes of composition and structure of the medium. Computational design and manufacture of single- and multi-layer coatings require exact knowledge of the optical constants of the layer materials in a working wavelength region. Note that the complex refractive indices of coating materials for a real grating may disagree significantly with tabulated values due to both chemical purity of the substances and peculiarities of the technological process of substrate and layers fabrication. Properties of the thin layers may also differ from properties of the bulk material, since different kinds of nonhomogeneities like impurities, interdiffusion and roughness usually occur in the process of the layer growth.  
Moreover, the real and imaginary parts of the refractive indices (RIs) for some materials may vary appreciably depending on the RI library used. For example, the real part of RI for MgF2 has a spread of about 15 percent in the wavelength range 70-200 nm, according to different tabulated sources (Fig.1). As to the imaginary part of RI for MgF2, the discrepancy in Fig.1 is much higher around the absorption edge in the vicinity of 110 nm, where the corresponding values are close to zero and very sensitive to the edge's position.  
Note also that the purely mathematical procedure of RIs interpolation has an appreciable effect on the efficiency curves. In some cases the interpolation-caused difference for the efficiencies can reach several dozen percent9. This is due to both fast oscillation of the real part of RIs (Palik's data) for MgF2 in the range of 100 to 130 nm and the low density of table data to the right of λ= 115 nm (Fig.2).  
Measurement and prediction of the optical constants for different materials can be made by a variety of methods, including combinations of experimental and computational techniques. Thus, for example, for some materials and wavelength regions use of the well-known Drude model to predict the optical constants brings us to a good agreement with data of measurements, especially for the real part of the refractive index. However, for many other materials and wavelengths the values calculated by the Drude model depart widely from the measured ones. This is especially true in regard to the resonance domains in the soft X-ray and in the EUV-VUV regions. The optical constants in the neighborhood of the absorption edges are very sensitive to the configuration of an atom (or a molecule) in its environment.  
Although accurate optical constants are available for many commonly used materials, in practice they can differ significantly from the tabulated values, especially in the EUV-VUV ranges. This is generally true in proximity to absorption edges, for reactive materials subjected to oxidation or contamination, and for longer EUV wavelengths (>30 nm) where molecular effects can be of considerable importance. A systematic study of candidate materials that may be suitably transmissive for multilayers operating at wavelengths 40-90 nm was presented recently in1. This study implements a novel multilayer-coated photodiode technique for measuring the optical constants of the reactive materials protected by stable capping layers. The elements studied thus far are Sc, Ti, La, and Tb. Several other rare-earth elements are predicted to have transmission windows for wavelengths > 40 nm associated with strong transitions to partially filled outer electron shells. The goal of the study is to develop high reflectance multilayer coatings and gratings for the wavelength > 40 nm region.  
A comparison of the grating efficiencies calculated accurately using, for instance, the PCGrate®-S(X) program, with their measured values allows us not only to locate discrepancies between the tabulated and true values of the optical constants. In this regard we can also solve the inverse problem of finding the refractive index of a layer from the grating efficiency. The idea which underlies the method proposed by us is actually based on the non-scalar properties of the grating efficiency which are inherent in certain modes of its operation. If the relation between absolute grating efficiencies for different coating materials is distinct from the relation between reflection coefficients for the corresponding materials at the same values of the remaining parameters, the efficiency behavior for such grating cannot be described by the scalar theory of diffraction or by the perfect conductivity approximation.74  
For the exact grating groove profile specified, the PCGrate-S(X) program can be used to determine, for example, the efficiency of a bulk grating which is in operation at some incidence and wavelengths as a function of the imaginary and real parts of the refractive index of its material. Then, from the known experimental value of the efficiency one can readily derive the refractive index of interest by the least square procedure. Such a mode of finding the refractive indices is made feasible under two conditions, namely, (1) when an exact method based on accurate vector theories is used, and (2) when one knows the precise layer boundaries, which can be measured in one way or another6,13. When appropriate RI libraries are not available, the required modeling data can be obtained by an indirect route, for example, by comparing experimental evidence with theoretical results found for different values of the refractive indices. In some instances, this approach may be a viable alternative to an intricate experimental and theoretical work on determining the refractive indices of the layer materials obtained in some specific conditions.  
The above approach to deriving the optical constants from the grating efficiency is similar to the methods which are commonly used for determining the optical constants from the reflectance (transmittance) of bulk and multilayer mirrors. However, the approach is absolutely independent and appears to be more accurate due to the resonant nature of grating reflection (or transmission) into a principal order close to the blaze condition and/or in the vicinity of anomalies. It is essential that the optical constants derived from calculation and measurements of a bulk grating grazing-incidence efficiency are suited for thin layers as well, because the grating operates here close to the total external (or internal in the soft X-ray-EUV range) reflection regime. Furthermore, the condition of total internal reflection for gratings in the EUV range, which differs from that for mirrors, can be used for extracting the imaginary part of the refractive index - a method which may be considered as complementing the above approach. Although this approach can be applied advantageously to multilayer gratings coated by two or more different materials, the relevant calculation process becomes more complex.  
The effect of small changes in the refractive indices on the calculated absolute efficiency of bulk and multilayer gratings together with examples of how RI data are derived from different sources or from the theoretical efficiency values are presented below.  
The value nAl = 1.09 + i5.31 was borrowed from the reference14, while the values nAl = 0.93+i6.33 and nAl = 1.21+i6.92, for wavelengths of 578 and 632.8 nm, respectively, were taken from the handbook12, and nAl = 1.37+i7.62 for the wavelength 632.8 nm, from the handbook11. Values of nAl = 0.77+i5.8, nAl = 0.93+i5.8 and nAl = 1.09+i5.8 were obtained through the use of computer modeling of the absolute efficiency by changing the imaginary part of the refractive index. The modeling followed the foregoing procedure and was performed to suit the best agreement with the data of efficiency measurements in the (-9)th order for variable values of the real part of the refractive index.  
As is obvious from the efficiency curves shown in Fig.3, tabulated values of refractive indices of aluminum taken from various sources11,12,14 do not agree well with the experiment. A better agreement is observed15 with the use of the refractive index taken from12 for a wavelength of 578 nm. The best agreement with the data of experiments is achieved using the least-square technique for nAl = 0.93+i5.8. For this specific value of the refractive index, the best agreement of the predicted values of efficiency with experiments was also observed in the (-8)th order, which is evidence in favor of the approach proposed. For both orders TM efficiencies predicted with nAl = 0.93+i5.8 differ by no more than 10% from the measured ones throughout the entire range of incidence angles. The result may be thought of as good, particularly, if it is remembered that the ideal groove profile was used in the model.  
A comparison of the accurate values of the efficiency for this grating which were obtained using different refractive indices data demonstrates a strong dependence of the calculated results on the imaginary part of the refractive index. As the imaginary part of the optical constant varies several dozen percent, the absolute efficiency in the basic order of the grating also varies several dozen percent. However, when the real part of the index varies several this much, the efficiency changes a few percent.  
The grating G185M with the frequency 5870 gr/mm, designed for work in the VUV-NUV wavelength range by the project COS/HST16, was covered with a layer of Al + MgF2 to increase reflection and to protect from oxidation. The grating profile was measured3 by AFM before and after applying the coating Al + MgF2. As may be seen from Fig.4, the groove depth decreased after applying the coating approximately 2.05 times (46.4 nm and 22.6 nm, respectively), and its form changed appreciably, that is to say, a non-conformal layer is achieved in this grating.  
The computational model3 with a non-conformal layer between the boundaries having a real profile and with the refraction indices taken from Palik's Handbook correlates well in efficiency with experimental data in the wavelength range exceeding 180 nm (Fig.5). At the same time, RI's for MgF2 taken from well-known sources11,12, have zero absorption after 110 nm and incorrectly describe the efficiency of the grating in the short-wave part of the wavelength range, namely, for wavelengths lower than 160 nm (Fig.5). The refractive indices of MgF2 (Fig.1), which have non-zero imaginary parts for wavelengths larger than 110 nm17, describe adequately qualitative trends in the efficiency behavior throughout the entire range of interest from 120 to 255 nm (Fig.5). Therefore, any further improvement of the model should involve obtaining correct refractive indices for MgF2.  
When deriving the grating model with new refractive indices for MgF2, we apply the results obtained by18. For Al, we make use of refractive indices taken from11. Taking into account that the best agreement with the measured efficiencies over the entire wavelength range was achieved with the refractive indices for MgF2, which have non-zero imaginary parts for wavelengths larger than 110 nm (Fig.1), for the new grating model we leave the real part of the refractive index unchanged. Let us seek the unknown imaginary part of the refractive index in the form of a piecewise continuous function with interpolation nodes spaced at intervals of 10 nm and starting with 120 nm. As absorption in the MgF2 layer obtained from comparison between the calculated values and experiment after 170 nm is small, we can put the imaginary part of the refractive index for MgF2 equal to zero, starting from 170 nm and further, in the long-wave region. Thereafter, we can determine the first derivative of the sought-for piecewise linear function, which represents the imaginary parts of the refractive indices in the range between 120 nm and 170 nm. Then, for four different experimental values of the efficiency, the calculated curve was approximated by least square technique at intervals of 0.01 for the imaginary part of the refractive index. Final values of the imaginary part of MgF2 refractive index thus found are presented in the Table. For smoothing out the required function, we replaced the zero imaginary part of MgF2 refractive index at 160 nm, which was found from modeling by a value of 0.001, resulting essentially in no effect on the efficiency. The efficiency for the grating model with MgF2 refractive indices presented in the Table is shown in Fig.5.  
Table. Derived MgF2 Refracrive Indices (RIs) from Efficiency Modeling  
λ, nm  
Analysis of the results shows that predicted and experimental values of the efficiency are in the closest agreement over the entire wavelength range just at the parameters chosen for the presented model. In this case, the corresponding difference between theoretical predictions and experiment for all wavelengths at which the grating measurements were carried out was no more than 10%.  
In conclusion, we can demonstrate the effect of the imaginary part of MgF2 refractive index on efficiency modeling of G185M gratings. Let us create a library, similar to one in the Table, in which the imaginary parts of the refractive indices decrease linearly between 0.1 and zero in the range from 120 nm to 170 nm. This function is an average of the function presented in the Table and, as is easy to see, differs from it in all points by no more than 0.02 throughout. Fig.5 shows the efficiency curve plotted on the basis of both the approximate MgF2 refractive indices and all the remaining parameters of the model developed. Comparison of this curve with the curve constructed using accurately predicted refractive indices shows that the results of efficiency modeling differ significantly in those portions of the wavelength range in which there exist rather small differences in the imaginary parts of the refractive indices. The comparison again indicates the need for comprehensive numerical modeling of efficiency to obtain accurate values of the unknown refractive indices. It is also indicative of possible large errors in the predicted efficiency when extrapolating or interpolating the existing data on the optical constants.  
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