Statistical self-calibrating algorithm for three-sample phase-shift interferometry

Abstract

A new self-calibrating algorithm is described that succeeds in reconstructing an almost error-free wavefront from only three interferograms. The algorithm is based on the assumption that the optical phase, taken modulo 2π , is quasi-uniformly distributed in the range [0, 2π) over the field of the interferograms. When the actual reference phases differ from those considered in the phase computation program a non-uniform histogram of the computed phase results. An analysis of this histogram allows a fitting procedure to find the actual phase shifts. Eventually an accurate shape of the wavefront can be calculated or a corrected signal can be sent to the phase-shifting device.

Quadrature component analysis of interferograms with random phase shifts

“…200 simulations are performed by choosing different β. In each simulation, the NOD is obtained according to equation (11) and the peak-to-valley (PV) and root-mean-square (RMS) values of the residual phase error are calculated. Then the relation between them is obtained and shown in figure 1.…”

“…However, random phase shifts may happen in PSI because of mechanical vibration and air turbulence. Therefore, many iterative methods [4][5][6][7][8][9][10] and non-iterative methods [11][12][13][14][15][16] have been proposed to deal with this problem. These algorithms [4][5][6][7][8][9][10][11][12][13][14][15][16] firstly, estimate the phase shifts and then use them as the inputs to a standard least squares algorithm to obtain the final phase distribution.…”

“…Recently, Vargas et al [17,18] proposed a new phaseshifting demodulation method based on principal component analysis (PCA). This PCA algorithm obtains the phase distribution directly from a sequence of randomly phase-shifted interferograms without estimation of the phase shifts, which is different from the mentioned algorithms [4][5][6][7][8][9][10][11][12][13][14][15][16]. This PCA algorithm works fast and effectively thus attracting much attention [19][20][21][22][23] since its publication.…”

Fourier-transform method of phase-shift determination

“…Authors Kinnstaetter et al, 19 and later Han and Kim, 20 and Kim et al 21 have developed iterative techniques to reduce a global error function that incorporates the global phase steps as global-free parameters. Statistical approaches that exploit the nonlinear response of the phase-retrieval algorithms have been described by Dobroiu et al 22,23 : These procedures also rely on iteration. The compound problem of translational and tilt-shift errors during phase shifting has been addressed with an iterative solution by Chen et al 24 The method described here falls into this latter category but operates deterministically by examining the Fourier domain without the necessity of iterative error minimization.…”

<title>Interferogram analysis for flatness metrology</title>

“…cos(2r • (vx + vy) + ço) = I + C1 • ei2X (vx+vyY)+ C • C1 = -I • e",C = -I • e''(4) For expressing digitally recorded fringe patterns we use the discrete form of this intensity distribution,…”