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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 19 — Sep. 14, 2009
  • pp: 16423–16428
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Spectral analysis of phase shifting algorithms

M. Servin, J. C. Estrada, and J. A. Quiroga  »View Author Affiliations


Optics Express, Vol. 17, Issue 19, pp. 16423-16428 (2009)
http://dx.doi.org/10.1364/OE.17.016423


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Abstract

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 OSA

1. Introduction

Spectral analysis of PSI algorithms was not systematically addressed before the paper of Freischlad and Koliopoulos (F&K) [1

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]

]. F&K have correctly interpreted that PSI algorithms are actually complex quadrature filters in the temporal domain [1

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]

] (see also [2

2. D. W. Phillion, “General methods for generating phase-shifting interferometry algorithms,” Appl. Opt. 36(31), 8098–8115 (1997). [CrossRef]

,3

3. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef] [PubMed]

]) 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

4. D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical Testing, 2th ed., (Marcel Deker, 2003).

,5

5. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef] [PubMed]

].

Here we propose a new way of to analyze the spectra of PSI algorithms based on the frequency transfer function [6

6. J. G. Proakis, and D. G. Manolakis, Digital Signal Processing, 4th-ed., (Prentice Hall, 2007). [PubMed]

] 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,
I(x,y,t)=a(x,y)+b(x,y)cos[φ(x,y)+ω0t].
(1)
Where a(x,y), b(x,y), and φ(x,y) are the background, contrast and the searched phase. Finally ω 0 is the temporal carrier (in radians) of the phase shifted interferograms. Rewriting it as,
I(x,y,t)=a+b2exp[i(φ+ω0t)]+b2exp[i(φ+ω0t)].
(2)
The spatial dependence of the functions a(x,y), b(x,y) and φ(x,y) were omitted for clarity. The aim of a PSI algorithm is to filter out a(x,y), and one of the two complex exponentials. Therefore the output signal Ic(t) is,
Ic(t)=[b(x,y)/2]exp{i[φ(x,y)+ω0t]}=h(t)*I(x,y,t).
(3)
Where the symbol * denotes a one-dimensional convolution, and h(t) is the PSI filter’s complex (quadrature) impulse response,
h(t)=hr(t)+ihi(t).
(4)
Denoting by H(ω) = F[h(t)] its Fourier transform. The minimal conditions on the frequency transfer function H(ω) to obtain the complex signal Ic(t) in Eq. (3) from the interferogram are:
H(ω0)0,    and      H(ω0)=H(0)=0.
(5)
These conditions on H(ω) are not normally used in the field of few-steps PSI interferometry. Instead due to [1

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]

], one normally uses the following equivalent set of requirements,

{F[hr(t)]iF[hi(t)]   }|ω=ω0=0,      and      F[hr(t)]|ω=0=F[hi(t)]|ω=0=0.
(6)

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. [4

4. D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical Testing, 2th ed., (Marcel Deker, 2003).

], page 239, and Schmit et. al. [5

5. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef] [PubMed]

] have discussed this before. As it is well known [1

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]

], 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,
tan[φ(x,y)]=hi(t)*I(x,y,t)hr(t)*I(x,y,t)|t=0.
(7)
The rotation (by Δ0) and the reference time-shift (t-t 0) of this complex linear filter are formally represented by,
h(t)exp[iΔ0],    and    h(tt0).
(8)
The most general impulse response of a linear-quadrature digital-filter (PSI algorithm) is, exp[iω0t]{   nanδ(tnT)} where exp[iw 0 t] is the reference signal (local oscillator), T is the sampling rate, and the coefficients an are weighting (possible complex) constants. Time shifting the reference one obtains,
h(tt0)=exp[iω0(tt0)]{   nanδ(tnT)}     =h(t)exp[iω0t0].
(9)
Therefore, time-shifting the reference signal also implies filter’s rotation, with Δ0 = -ω 0 t 0. This was first observed in Malacara et. al. [4

4. D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical Testing, 2th ed., (Marcel Deker, 2003).

], page 239. Rotating h(t) by Δ0 one obtains,
h(t)exp[iΔ0]=hr(t)cos(Δ0)hi(t)sin(Δ0)+i[hr(t)sin(Δ0)+hi(t)cos(Δ0).
(10)
As 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

4. D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical Testing, 2th ed., (Marcel Deker, 2003).

,5

5. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef] [PubMed]

]. 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,

tan[φ(x,y)+Δ0]=[hr(t)sin(Δ0)+hi(t)cos(Δ0)]*I(t)[hr(t)cos(Δ0)hi(t)sin(Δ0)]*I(t)|t=0.
(11)

4. The Freischlad and Koliopoulos spectral analysis

The spectral analysis of PSI algorithms published by Freishlad et. al. [1

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]

] consist on finding the Fourier transform of the real and imaginary parts of h(t), separately, that is
Hi(ω)=F[hi(t)]Hr(ω)=F[hr(t)].
(12)
Having these Fourier transforms, F&K recommend to analyze these spectra without constant or common phase factors [1

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]

], 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,
Hi(ω)=F[hr(t)sin(Δ0)+hi(t)cos(Δ0)]Hr(ω)=F[hr(t)cos(Δ0)hi(t)sin(Δ0)].
(13)
The spectral analysis that results from this rotated PSI algorithm is clearly different to the one in 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

|H(ω)|=|F[h(t)]|=|F{h(t)exp[iΔ0]}|=|F[h(tt0)]|. (16).

6. Examples

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

6.2 Five-step Schwider-Hariharan PSI algorithm

6.3 Discusion

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 [7

    7. M. Servin, J. C. Estrada, J. A. Quiroga, J. F. Mosiño, and M. Cywiak, “Noise in phase shifting interferometry,” Opt. Express 17(11), 8789–8794 (2009). [CrossRef] [PubMed]

    ] 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.

7. Conclusions

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

4. D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical Testing, 2th ed., (Marcel Deker, 2003).

,5

5. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef] [PubMed]

] 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 [1

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]

] 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.

Acknowledgements

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]

3.

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef] [PubMed]

4.

D. Malacara, M. Servin, and Z. Malacara, Interferogram analysis for Optical Testing, 2th ed., (Marcel Deker, 2003).

5.

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef] [PubMed]

6.

J. G. Proakis, and D. G. Manolakis, Digital Signal Processing, 4th-ed., (Prentice Hall, 2007). [PubMed]

7.

M. Servin, J. C. Estrada, J. A. Quiroga, J. F. Mosiño, and M. Cywiak, “Noise in phase shifting interferometry,” Opt. Express 17(11), 8789–8794 (2009). [CrossRef] [PubMed]

OCIS Codes
(120.2650) Instrumentation, measurement, and metrology : Fringe analysis
(120.3180) Instrumentation, measurement, and metrology : Interferometry

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: May 26, 2009
Revised Manuscript: August 14, 2009
Manuscript Accepted: August 15, 2009
Published: August 31, 2009

Citation
M. Servin, J. C. Estrada, and J. A. Quiroga, "Spectral analysis of phase
shifting algorithms," Opt. Express 17, 16423-16428 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16423


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