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• W1601420565 abstract "Two dimensional power spectral density measurements of X-ray optics with the Micromap interferometric microscope Valeriy V. Yashchuk*, Andrew D. Franck, Steve C. Irick, Malcolm R. Howells, Alastair A. MacDowell, Wayne R. McKinney Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA, USA 94720 ABSTRACT A procedure and software have been developed to transform the area distribution of the residual surface heights available from the measurement with the Micromap interferometric microscope into a two-dimensional (2D) power spectral density (PSD) distribution of the surface height. The procedure incorporates correction of one of the spectral distortions of the PSD measurement. The distortion appears as a shape difference between the tangential and sagittal PSD spectra deduced from the 2D PSD distribution for an isotropic surface. A detailed investigation of the origin of the anisotropy was performed, and a mathematical model was developed and used to correct the distortion. The correction employs a modulation transfer function (MTF) of the detector deduced analytically based on an experimentally confirmed assumption about the origin of the anisotropy due to the asymmetry of the read-out process of the instrument’s CCD camera. The correction function has only one free parameter, the effective width of the gate-shaped apparatus function which is the same for both directions. The value of the parameter, equal to 1.35 pixels, was found while measuring the 2D PSD distribution of the instrument self-noise, independent of spatial frequency. The effectiveness of the developed procedure is demonstrated with a number of PSD measurements with different X-ray optics including mirrors and a grating. Keywords: interferometric microscope, power spectral density, X-ray optics, optical metrology 1. INTRODUCTION The interferometric microscope, such as the Micromap-570 1 , has become a basic metrology tool for highly accurate testing of the surface finish of X-ray optics with sub-Angstrom rms roughness. The standard list of output parameters of an interferometric microscope measurement includes values of roughness averaged over an area and along a sample line. However the task of designing high performance low scatter X-ray optical systems 2 requires the development of sophisticated X-ray scattering calculations based on rigorous information about the optics. One of the most insightful approaches to these calculations is based on the two-dimensional (2D) power spectral density (PSD) distribution of the surface height, 3 allowing for the evaluation of three-dimensional distributions of X-rays scattered by the optics. 4 The 2D PSD function S 2 may be viewed as a Fourier decomposition of the 2D surface height distribution h ( x , y ) into harmonic basis functions: 5 S 2 ( s x , s y ) = lim A → ∞ A L y / 2 − L y / 2 ∫ dy ∫ h ( x , y ) e − L x / 2 L x / 2 − 2 π i ( u x + v y ) dx where L x and L y are the tangential and sagittal dimensions of the measured surface region, A = L x L y ; u and v are the spatial frequency variables corresponding to the tangential, x , and sagittal, y , coordinates. In the case of discreet measurements with pixel dimensions ∆ x and ∆ y , M and N pixels in the tangential and sagittal directions, respectively, the 2D PSD distribution can be evaluated from the height distribution h m , n via equation S 2 ( l , k ) = M N ∆ x ∆ y F l , k ," @default.
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• W1601420565 date "2005-05-12" @default.
• W1601420565 modified "2023-09-23" @default.
• W1601420565 title "Two dimensional power spectral density measurements of X-ray optics with the Micromap interferometric microscope" @default.
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