Systematic spectral analysis of Phase Shifting Interferometry (PSI) algorithms was first proposed in 1990 by Freischlad and Koliopoulos (F&K). This analysis was proposed with the intention that “in a glance” the main properties of the PSI algorithms would be highlighted. However a major drawback of the F&K spectral analysis is that it changes when the PSI algorithm is rotated or its reference signal is time-shifted. In other words, the F&K spectral plot is different when the PSI algorithm is rotated or its reference is time-shifted. However, it is well known that these simple operations do not alter the basic phase demodulation properties of PSI algorithms, except for an unimportant piston. Here we propose a new way to analyze the spectra of PSI algorithms which is invariant to rotation and/or reference time-shift among other advantages over the nowadays standard PSI spectral analysis by F&K.
©2009 Optical Society of America
Spectral analysis of PSI algorithms was not systematically addressed before the paper of Freischlad and Koliopoulos (F&K) . F&K have correctly interpreted that PSI algorithms are actually complex quadrature filters in the temporal domain  (see also [2,3]) and used the Fourier transform to find the spectra of these filters. In addition to this, they have proposed a spectral analysis in order to graphically interpret the main features of these quadrature filters (PSI algorithms). However the main drawback of this spectral analysis is that it changes when the PSI filter is rotated in the complex plane and/or its reference is time-shifted. This analysis changes so much with rotation that one may have the false impression of dealing with a different PSI algorithm. However it is well known that the estimated phase of a rotated or time-shifted PSI algorithm remains unchanged, except for an unimportant piston phase [4,5].
Here we propose a new way of to analyze the spectra of PSI algorithms based on the frequency transfer function  of the complex filter associated with the PSI algorithm. This PSI filter’s spectrum is invariant to rotation and/or reference time-shift. Also this new spectral representation has zero response at frequencies rejected by the PSI quadrature filter.
2. The aim of a quadrature filter in phase shifting interferometry
Let us begin by showing the usual mathematical model of a temporal interferometric signal as,Eq. (3) from the interferogram are:1], one normally uses the following equivalent set of requirements,
3. Rotation and reference time-shift of PSI algorithms
In this section we discuss the rotation and reference time-shift of PSI algorithms. Malacara et. al. , page 239, and Schmit et. al.  have discussed this before. As it is well known , a PSI algorithm (or quadrature filter) may be expressed in terms of the real hr(t) and imaginary hi(t) parts of a complex impulse response filter h(t) = hr(t) + i hi(t) as,4], page 239. Rotating h(t) by Δ0 one obtains,Eq. (10) shows, rotation and/or reference time-shift linearly mix up the real and imaginary parts of the original PSI filter h(t). This give apparently “new” PSI algorithms that looks quite different albeit being the same one [4,5]. The family of PSI algorithms obtained from the rotated (or time shifted Δ0 = -ω 0 t 0) quadrature filter according to Eq. (10) looks now as,
4. The Freischlad and Koliopoulos spectral analysis
The spectral analysis of PSI algorithms published by Freishlad et. al.  consist on finding the Fourier transform of the real and imaginary parts of h(t), separately, that is1], and in this way obtain two real functions to plot.
But as we said, if we rotate the PSI algorithm one may easily run into trouble, because now the rotated spectra have the following form,Eq. (12), and may give the false impression that they correspond to two different PSI algorithms. In other words, the F&K spectral representation would give different spectral plots for each rotation of the same PSI algorithm. In our herein presented spectral representation, this drawback does not exist.
5. Our proposed spectral analysis
As revised in the previous section, the spectral analysis in  take the Fourier transform of the real hr(t) and the imaginary hi(t) components of h(t) separately; Eq. (12). This is followed by a rule to obtain two real functions to plot and analyze. On the other hand, we propose to analyze the spectra of the complex sum h(t) = hr(t) + i hi(t). So we only take the Fourier transform of h(t), i.e. H(ω) = F[h(t)]. This new way of analyzing the spectra of PSI algorithms have (as shown here) interesting and useful consequences.
It is well known that the spectra H(ω)  is the ratio of the output’s signal spectra Ic(ω) to the input’s signal spectra I(ω),4]. The output of any PSI filter is the complex analytical signal Ic(t) in Eq. (3) associated with the interferogram’s real signal I(t). Therefore, as it is shown in Fig. 1 we may have an infinite number of transfer functions H(ω) that comply with Eq. (5). In Fig. 1 we show two block diagrams of linear quadrature PSI filters that may be used to obtain our searched analytical signal Ic(t) at –ω 0 from the real interferometric signal I(t).
As we said, most PSI algorithms published to date have a constant phase shift Δ(ω) = Δ0 , therefore in these cases the plot of |H(ω)| may be used as its spectrum, which is given by6]; the rotated h(t)exp[iΔ0]; the time-shifted h(t-t 0), and the original PSI filter h(t) have all the same magnitude |H(ω)|,
In this section, using two examples, we illustrate the rotational/time-shifting ambiguity of the F&K analysis and how our new PSI spectral analysis based on |H(ω)| avoids them.
6.1 Two four-step PSI algorithms
Let us begin with two popular 4-step PSI algorithms ,1], one would need to analyze the following spectra,Eq. (15), one can easily find the magnitude of the Fourier transform of h1(t) and h2(t) and realize that both are proportional,Fig. 2 . We can see that in this representation both filters have proportional spectra and therefore (except for a piston phase equals to π/4 introduced by H2(ω) ) have identical phase demodulating properties.
6.2 Five-step Schwider-Hariharan PSI algorithm
Let us take another popular PSI algorithm; the Schwider-Hariharan  5-step one, which is,1] one would need to plot and analyze the following spectra,5]),Eq. (23) and Eq. (27), and according to the analysis proposed in  one may easily think that we are dealing with two different PSI algorithms!. However it is easy to see that the magnitude of H(ω) = F[h(t)] of both algorithms are (of course) identicalFig. 3 , where we can see that this 5-step PSI filter will eliminate the complex exponential at ω = 1.0 and the background of the interferogram I(t) at ω = 0.
We now list some advantages of our new way of analyzing the spectra of PSI algorithms.
- 1. The magnitude of the frequency response of the filter is invariant to the PSI filter rotations and/or constant time-shift of the reference signal (local oscillator).
- 2. The signals at frequencies ω 1,…,ωn that the PSI filter rejects are clearly seen as zeroes over the frequency axis i.e. |H(ω 1)| = … = |H(ωn)| = 0.
- 3. The properties of the PSI algorithms in the neighborhood of the rejected frequencies are also clearly shown. For example the detuning robustness of the Schwider-Hariharan 5-step algorithm is shown as a zero for the first derivative of H(ω) at ω = 1.0.
- 4. In  we show that the phase noise in a PSI algorithm is proportional to the integral of |H(ω)|2. So at a glance one may estimate the noise rejection of two “competing” PSI algorithms by their area under |H(ω)|2 for the same output signal’s energy.
In this paper we have proposed a new way to analyze the spectra of PSI quadrature filters which is invariant to PSI algorithm rotation or reference time-shift. As it is well known [4,5] a simple rotation and/or time-shift of a PSI algorithm do not alter the PSI phase but for an irrelevant piston. However, as simple as these operations are, according to the analysis proposed in  it gives different spectral plots, giving the false impression that the PSI algorithm have somehow changed. Additionally our new PSI spectral analysis clearly shows as zeroes in H(ω) the rejected frequencies, and no need for further interpretation is required. On top of this, the spectral analysis based on H(ω) is the most usual (and intuitive) manner to graphically represent the spectra of any optical and/or electrical filter in engineering.
We acknowledge the valuable support of the Mexican Science Council, CONACYT.
References and links
1. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]
2. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997). [CrossRef]
4. D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical Testing, 2th ed., (Marcel Deker, 2003).
6. J. G. Proakis, and D. G. Manolakis, Digital Signal Processing, 4th-ed., (Prentice Hall, 2007). [PubMed]