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.
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).