Combined interpolation, scale change, and noise reduction in spectral analysis

Significance 

In physics, a physical quantity is said to have a discrete spectrum if it assumes only distinct values, with gaps between one value and the next. Ideally, discrete spectral data equally spaced in wavelength are incompatible with standard methods of analysis, which are based on data equally spaced in energy. The most probable resolution of this shortfall is through interpolation- defined as any method where the interpolating function passes through the data themselves, leaving them unchanged. The most common technique is linear interpolation, where intermediate values are proximity-weighted averages of the nearest two values. Other sophisticated approaches are available; among which the ‘brick wall’ filter/cardinal spline – which is based on the sinc function and represents the continuous spectrum exactly – is the most explored. Nonetheless, this technique suffers from slow convergence, which ultimately makes it impractical. Unfortunately, since data typically include noise, noise is interpolated as well. Noise can be reduced by using interpolating filters, where the functions are not required to pass through the data. Approaches that offer significant improvements over interpolation to circumvent this drawback are available; however, they are either not optimal, insufficiently general, unnecessarily complex or optimization is done by inspection.

Therefore, there is a need for a conversion method that is simple, convenient, accurate and delivers noise reduction effects that are quantified. With this in view, researchers Long V. Le (PhD candidate), Prof. Tae J. Kim, and Prof. Young D. Kim of the Department of Physics at Kyung Hee University in the Republic of Korea, and Prof. David E. Aspnes of North Carolina State University in the US, developed a new but simple, convenient and accurate noise-reduction approach for interpolating spectra. Their goal was to deliver a practical approach to convert spectra available as discrete points equally spaced in wavelength, acquired, for example, by a photodiode-array detector, to equivalent spectra equally spaced in energy, as needed for analysis. Their work is currently published in Journal of Vacuum Science & Technology B.

The approach is based on numerical integration of the data weighted locally by a continuum Gaussian kernel. The researchers opted for Gaussian kernels because they are analytic, and minimize the reciprocal space (RS)-direct space (DS) uncertainty product, thereby providing relatively sharp cutoffs in both RS and DS. This is advantageous for filtering out noise while preserving information. In addition, when applied to Gaussian functions, trapezoidal-rule integration is accurate to fourth order in the ratio of point separation to Gaussian width. Therefore, the simplest and most convenient numerical-integration method is also the most accurate, significantly better than any other.

The authors pointed out that the continuum theory that they use confers significant additional advantages because it allows extensions to other continuum operations such as differentiation. Interestingly, approximations are limited to the numerical evaluation of a single integral.

In summary, the study by Kyung Hee and North Carolina State University scientists present a method based on Gaussian kernels that combines interpolation, scale change, noise reduction, convenience, and the possibility of additional processing such as differentiation (if desired) in a single step. In an interview with Advances in Engineering, Prof. Aspnes emphasized that their approach can be applied not only to the ellipsometric spectra discussed here, but also to those obtained by infrared spectroscopy, x-ray photoemission spectroscopy, and Raman scattering.

Combined interpolation, scale change, and noise reduction in spectral analysis - Advances in Engineering
Figure 1. Conversion of raw data (black circles) obtained linear in wavelength to equivalent information (red points) equally spaced in energy. Note the closer spacing of the data on the left. Conversion was done with a fourth-order extended-Gauss filter generating an output point spacing of 10 meV.

About the author

Long V. Le received his B.Sc. degree in materials science from the Vietnam National University in 2013. Following graduation, he worked as a researcher in the Institute of Materials Science (IMS), Vietnam Academy of Science and Technology (VAST). Since March 2015 he has been in the combined masters-PhD program on physics in the Nano Optical Property Laboratory, Department of Physics, Kyung Hee University, Korea. In addition to exploring methods of optimizing the extraction of information from spectra, current research interests include determining the optical properties of anisotropic materials and applications of spectroscopic ellipsometry to biophysics. He has 15 publications and has made 44 presentations at international conferences.

About the author

Young Dong Kim is Dean of the College of Sciences at Kyung Hee University in Seoul, Korea. He obtained his PhD at the University of Illinois Urbana/Champaign. After a postdoctoral position at the same institution, he joined the Department of Physics at Kyung Hee University in 1994. He has made numerous contributions in the use of spectroscopic ellipsometry for materials analysis, particularly in II-VI and III-V materials and their alloys.

Recent work has focused on using spectroscopic ellipsometry to determine optical properties of a wide range of materials at both room and cryogenic temperature, and adapting the approach to analyze biological processes at interfaces.

About the author

Tae Jung Kim is an Assistant Professor at Kyung Hee University in Seoul, Korea. He obtained his Ph.D. at Kyung Hee University in 2007. After a term as a postdoctoral research scientist, he joined the Department of Physics as an Assistant Professor in 2010. He has specialized in materials analysis, using spectroscopic ellipsometry to determine the optical properties of a wide variety of materials, in particular oxides and two-dimensional materials such as MoS2, MoSe2, and WS2 at temperatures ranging from 25 to 650 K.

His current research is focused on the determination of the optical properties of heterostructural transition-metal dichalcogenides.

About the author

D. E. Aspnes is Distinguished University Professor of Physics at North Carolina State University. He obtained Bachelors and Masters Degrees in Electrical Engineering at the University of Wisconsin, Madison, and a Ph.D. in Physics from the University of Illinois Urbana-Champaign. After 25 years at Bell Laboratories and Bellcore, he joined the Department of Physics at NCSU. He has had a long-term interest in theory and experiment of linear and nonlinear optical properties of materials and the use of optical methods to characterize surfaces, interfaces, thin films, and bulk materials.

He has published over 500 papers in technical journals and has served in numerous capacities in the professional community, including President of the American Vacuum Society in 2005. Recognition includes election to the National Academy of Sciences in 1998.

Reference

Van L. Le, Tae J. Kim, Young D. Kim, David E. Aspnes. Combined interpolation, scale change, and noise reduction in spectral analysis. Journal of Vacuum Science & Technology B, volume 37, 052903 (2019).

Go To Journal of Vacuum Science & Technology B

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