Development of pseudorandom binary arrays for calibration of surface profile metrology tools

Optical metrology tools, especially for short wavelengths (extreme ultraviolet and x-ray), must cover a wide range of spatial frequencies from the very low, which affects figure, to the important mid-spatial frequencies and the high spatial frequency range, which produces undesirable scattering. A major difficulty in using surface profilometers arises due to the unknown point-spread function (PSF) of the instruments [G. D. Boreman, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE, Bellingham, WA, 2001)] that is responsible for distortion of the measured surface profile. Generally, the distortion due to the PSF is difficult to account for because the PSF is a complex function that comes to the measurement via the convolution operation, while the measured profile is described with a real function. Accounting for instrumental PSF becomes significantly simpler if the result of measurement of a profile is presented in the spatial frequency domain as a power spectral density (PSD) distribution [J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, Englewood, CO, 2005)]. For example, measured PSD distributions provide a closed set of data necessary for three-dimensional calculations of scattering of light by the optical surfaces [E. L. Church et al., Opt. Eng. (Bellingham) 18, 125 (1979); J. C. Stover, Optical Scattering, 2nd ed. (SPIE Optical Engineering Press, Bellingham, WA, 1995)]. The distortion of the surface PSD distribution due to the PSF can be modeled with the modulation transfer function (MTF), which is defined over the spatial frequency bandwidth of the instrument. The measured PSD distribution can be presented as a product of the squared MTF and the ideal PSD distribution inherent for the system under test. Therefore, the instrumental MTF can be evaluated by comparing a measured PSD distribution of a known test surface with the corresponding ideal numerically simulated PSD. The square root of the ratio of the measured and simulated PSD distributions gives the MTF of the instrument. The applicability of the MTF concept to phase map measurements with optical interferometric microscopes needs to be experimentally verified as the optical tool and algorithms may introduce nonlinear artifacts into the process. In previous work [V. V. Yashchuk et al., Proc. SPIE 6704, 670408 (2007); Valeriy V. Yashchuk et al., Opt. Eng. (Bellingham) 47, 073602 (2008)] the instrumental MTF of a surface profiler was precisely measured using reference test surfaces based on binary pseudorandom (BPR) gratings. Here, the authors present results of fabricating and using two-dimensional (2D) BPR arrays that allow for a direct 2D calibration of the instrumental MTF. BPR sequences are widely used in engineering and communication applications such as global position systems and wireless communication protocols. The ideal BPR pattern has a flat “white noise” response over the entire range of spatial frequencies of interest. The BPR array used here is based on the uniformly redundant array (URA) prescription[E. E. Fenimore and T. M. Cannon, Appl. Opt. 17, 337 (1978)] initially used for x-ray and gamma ray astronomy applications. The URA’s superior imaging capability originates from the fact that its cyclical autocorrelation function very closely approximates a delta function, which produces a flat PSD.

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