Phase-shifting algorithms for a finite number of harmonics: first-order analysis by solving linear systems

Abstract

From generalized phase-shifting equations, we propose a simple linear system analysis for algorithms with equally and nonequally spaced phase shifts. The presence of a finite number of harmonic components in the fringes of the intensity patterns is taken into account to obtain algorithms insensitive to these harmonics. The insensitivity to detuning for the fundamental frequency is also considered as part of the description of this study. Linear systems are employed to recover the desired insensitivity properties that can compensate linear phase shift errors. The analysis of the wrapped phase equation is carried out in the Fourier frequency domain.

Compensation of the two-stage phase-shifting algorithms in the presence of detuning and harmonics

“…(21) will be valid if the orthogonality conditions imposed by Eqs. (21)-(27) in [17] to the phase shifts α j;l and the coefficients B j;l , A j;l , j 1, 2, and l p; k are satisfied.…”

“…(9), we have Δ sin τ j α j ∕τ j;r l1 l sinτ j α j;l1 ∕τ j;r − sinτ j α j;l ∕τ j;r and Δ cos τ j α j ∕τ j;r l1 l cosτ j α j;l1 ∕τ j;r − cosτ j α j;l ∕τ j;r for j 1, 2 and l k or l p. Applying the orthogonality conditions given in Eqs. (21)-(27) in [17] to the shifts α j;l and the coefficients B j;l , A j;l , the last equations reduce:ĝ…”

“…Differential phase-shifting algorithms are susceptible to systematic errors, such as phase miscalibration or the presence of higher-order harmonics in the fringe profile, which have been recently demonstrated [4,5]. In previous publications [15][16][17], we have studied the advantages of considering consecutive interferogram differences in the generation of generalized phaseshifting algorithms [18][19][20], and following these approaches, we can implement similar equations in the analysis of the TSPS algorithms. When two-stage algorithms are employed to recover phase sums or phase differences, two sets of intensity patterns are used; these intensity patterns can be assumed to be of the form …”

“…(8)- (12) and (16) or Eqs. (16) and (30)-(33), as described with detail in [17]. The left index in B 1;k and A 1;k indicates that these coefficients are used for the set fμ 1;k g to recover ϕ.…”

“…(3) are chosen by solving the linear system of Eqs. (30)-(33) proposed in [17], the resultant algorithm has properties of insensitivity with respect to detuning and harmonics. Of course, this insensitivity will be with respect to the two principal reference frequencies employed for an interferometric TSPS algorithm.…”