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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 11 — May. 23, 2011
  • pp: 10692–10697
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Design of phase-shifting algorithms by fine-tuning spectral shaping

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


Optics Express, Vol. 19, Issue 11, pp. 10692-10697 (2011)
http://dx.doi.org/10.1364/OE.19.010692


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Abstract

To estimate the modulating wavefront of an interferogram in Phase Shifting Interferometry (PSI) one frequently uses a Phase Shifting Algorithm (PSA). All PSAs take as input N phase-shifted interferometric measures, and give an estimation of their modulating phase. The first and best known PSA designed explicitly to reduce a systematic error source (detuning) was the 5-steps, Schwider-Hariharan (SH-PSA) PSA. Since then, dozens of PSAs have been published, designed to reduce specific data error sources on the demodulated phase. In Electrical Engineering the Frequency Transfer Function (FTF) of their linear filters is their standard design tool. Recently the FTF is also being used to design PSAs. In this paper we propose a technique for designing PSAs by fine-tuning the few spectral zeroes of a PSA to approximate a template FTF spectrum. The PSA’s spectral zeroes are moved (tuned) while gauging the plot changes on the resulting FTF’s magnitude.

© 2011 OSA

1. Introduction

Schwider et al. [1

1. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983). [CrossRef] [PubMed]

] derived the first (and best known) 5-steps detuning-robust PSI algorithm, and afterwards Hariharan et al. [2

2. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef] [PubMed]

] further analyzed its properties. In the book of Malacara et al. [3

3. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, CRC, 2005).

] an encyclopedic number of PSI algorithms are presented, including the motivation behind many of them; their spectral plots are also shown according to Freischlad and Koliopoulos (F&K) [4

4. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]

]. Spectral analysis of PSAs was popularized only after the F&K paper [4

4. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]

] was published. Nowadays, spectral analysis is the gold-standard reference to compare among several PSAs. Most people uses this spectra to visualize and gauge among several promising PSAs, and choose the best one for their needs [4

4. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]

8

8. J. Schmit and K. Creath, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. 35(28), 5642–5649 (1996). [CrossRef] [PubMed]

,11

11. J. Burke, “Extended averaging phase-shifting schemes for Fizeau interferometry on high-numerical aperture spherical surfaces,” Proc. of SPIE 7790, (2010).

,12

12. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

].

More recently Surrel proposed an x-polynomial P(x) associated with a PSA named the Characteristic Polynomial (CP) [9

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

]. The x-polynomial closely follows the z-transform, or in Surrel’s words [9

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

]: “This introduction of a polynomial is similar to what is underlying in the z-transform theory”. That is probably why Surrel preferred the use of x instead of z for his P(x) polynomial. Then, Surrel proposed the CP-diagram, to visualize some properties of a PSA based on the (discrete) angular location over the unit circle of these zeroes and their multiplicities [9

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

]. Surrel proposed no continuous spectral plot to gauge his x-polynomials [9

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

], as the F&K spectral plot does provide [4

4. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]

]. He did not follow the standard continuous spectral analysis of z-transformed digital filters [10

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

].

In 2010, Burke [11

11. J. Burke, “Extended averaging phase-shifting schemes for Fizeau interferometry on high-numerical aperture spherical surfaces,” Proc. of SPIE 7790, (2010).

] efficiently combined both perspectives: the discrete CP-diagram [9

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

], and the continuous F&K spectral analysis [4

4. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]

]. Burke did this to point-out the importance of fine-tuning the zeroes of a x-polynomial of symmetric PSAs. That is, Burke generated a new visualizing-gauging technique, by combining the discrete CP-diagram [9

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

], with the F&K continuous spectrum [4

4. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]

,11

11. J. Burke, “Extended averaging phase-shifting schemes for Fizeau interferometry on high-numerical aperture spherical surfaces,” Proc. of SPIE 7790, (2010).

], while finely tuning the PSA’ symmetric spectral zeroes.

However, as demonstrated in [12

12. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

], one drawback of the F&K’s spectral analysis [4

4. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]

] is that it changes when the PSI algorithm’s reference carrier (the local oscillator) is rotated. Another important drawback of the F&K spectrum is that only symmetrical PSAs may be F&K spectrally analyzed. That is because, F&K analysis needs to: “analyze the spectrum without constant or common phase factors [1

1. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983). [CrossRef] [PubMed]

]”, and in this way obtain two real functions to plot. We repeat, the F&K spectral analysis can only be made if the PSA is symmetric, such as the ones analyzed by Burke [11

11. J. Burke, “Extended averaging phase-shifting schemes for Fizeau interferometry on high-numerical aperture spherical surfaces,” Proc. of SPIE 7790, (2010).

], and Larkin et al. [5

5. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992). [CrossRef]

], and others [3

3. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, CRC, 2005).

]. As a consequence, general (non-symmetric) PSAs simply cannot be spectrally analyzed using F&K. Briefly, the F&K spectral plot limitations are: a) The spectral plot changes when the local oscillator is phase shifted [12

12. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

], and b) F&K cannot plot the PSA’s spectra of general non-symmetric PSAs. These two drawbacks are serious limitation of the F&K spectral analysis technique.

A different perspective of the PSA’s theory just discussed was proposed by Servin et al. [12

12. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

]. In this paper a general theory of PSAs based on the Frequency Transfer Function (FTF) H(ω) is given. The FTF is just the Fourier transform of the impulse response of a digital filter h(t), that is H(ω)=F[h(t)] [10

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

]; where F[·] is the Fourier transform. The use of the FTF however new in PSI, has been the standard way of spectral analysis in signal processing engineering for decades [10

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

]. The spectral analysis of PSAs based on the FTF does not have the limitations of the F&K spectral analysis.

In this paper, just to build a conceptual bridge, we have adopted a combined CP-FTF visualization. But given that the CP-diagram is a visual subset of our |H(ω)| plot, we would not need the CP-diagram, all the PSA’s information is within our |H(ω)| continuous plot.

2. Spectral analysis of PSI algorithms based on the FTF

For the reader’s convenience, we briefly review the FTF approach in PSI [12

12. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

]. Let us show the standard mathematical model of a set of N phase-shifted interferometric data as,
I(x,y,t)=k=0N1{a(x,y)+b(x,y)cos[φ(x,y)+ω0k]}δ(tk).
(1)
where the background illumination is a(x,y), the fringe contrast is b(x,y), and the carrier frequency is ω 0 (radians/interferogram). The Fourier transform F[•] of this signal over t is,
I(x,y,ω)=aδ(ω)+b2exp[iφ]δ(ω+ω0)+b2exp[iφ]δ(ωω0).
(2)
To know the interesting phase φ(x,y) one needs a filter h(t), to wipe-out the (ω) term and one b/2 term. To this end, the measured signal I(t) is introduced into a general N-steps PSA as,
tan[φ(x,y)]=k=0N1aksin(ω0t)I(k)k=0N1akcos(ω0t)I(k).
(3)
This PSA may be seen as the following N-steps, quadrature-filter h(t) tuned at ω 0 rad/sample,
h(t)={   k=0N1akδ(tk)}eiω0t,     H(ω)=F[h(t)]=k=0N1akeik(ωω0).
(4)
The data I(t) in Eq. (1) and the quadrature-filter h(t) may be convolved as,
S=[I(t)h(t)]t=N1=k=0N1akeiω0kI(k).
(5)
The complex number S is the value of I(t)*h(t) evaluated at the middle-point t=N-1, in which the filter h(t) and the data I(t) fully overlap. Finally, the searched phase is φ(x,y)=angle[S].

We visualize and gauge the PSA spectrum’s amplitude shape by plotting |H(ω)| as [12

12. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

],
|H(ω)|=|F[h(t)]|=Re(ω)2+Im(ω)2.
(6)
where H(ω)=Re(ω)+i Im(ω), being Re(ω) and Im(ω) real-valued functions. The magnitude of H(ω) has the good properties of being invariant to local oscillator’s (exp[ 0 t]) phase-shifts, and have a well-defined spectral plot for either a symmetric or non-symmetric PSAs.

3. Characteristic polynomial (CP) associated to a PSI algorithms

The CP proposed by Surrel [9

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

] is defined from the right hand side of Eq. (5) as,
P(x)=k=0N1akeiω0kxk=k=0N2(xdk).
(7)
P(x) is the CP x-polynomial, the data I(k) in Eq. (5) is formally substituted by xk [9

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

]. As Surrel shows, it is convenient to express P(x) as a product of N-1 monomials (x-dk). M roots at dk means that the PSA is robust to detuning at dk up to order M. Finally, the zeroes dk, and their multiplicities are plotted in a CP-diagram [9

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

]. First-order zeroes are plotted as small solid disks, and their multiplicities with greater circles around them [9

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

].

4. Fine-tuning spectral-zeroes for approximating a desired PSA spectrum

A first-order linear system that generates a (first-order) spectral zero at ω 0 is [10

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

],
h(t)=[δ(t)δ(t1)]eiω0t.
(9)
This may readily be seen by Fourier transforming h(t) as,
H(ω)=F[h(t)]=1ei(ω+ω0).
(10)
This is the basic building-block of our zero-based, spectral-shaping design of PSAs. Note that, the equivalent x-monomial is P(x)=[x-exp(-0)]. We will combine several first-order blocks (Eq. (10)), freely tuning their zeroes, to approximate a desired PSA spectral template.

As a first example, suppose we want a 5-step PSA’s having a flatter rejecting-band around -π/2 than a 5-step Schwider-Hariharan (SH) PSA [1

1. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983). [CrossRef] [PubMed]

,2

2. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef] [PubMed]

]. The SH-PSA has the following FTF,
HSH(ω)=(1eiω)[1ei(ω+π/2)]2[1ei(ω+π)].
(11)
The x-polynomial of the SH-PSA is (x-1)(x+i)2(x+1). The CP-diagram and the plot of |HSH(ω)| are shown in Fig. 1
Fig. 1 Magnitude of HSH(ω) Eq. (11), and its CP-diagram. The PSA’ spectral plot |HSH(ω)| has two first-order zeroes at 0, and π, and a second-order one at -π/2.
. This PSA has a second-order zero at –π/2, and two first-order zeroes at 0 and π. The second-order zero gives (second-order) detuning robustness at -π/2.

We now freely move and fine-tune (by trial and error) one zero at -π/2, and the one at π, to flatten the response around -π/2. This is done while gauging the |H 5(ω)| plot. After some iterations we settle on the following FTF, which corresponds to a non-symmetric PSA,
H5(ω)=(1eiω)[1ei(ω+0.45ω0)][1ei(ω+ω0)][1ei(ω+1.4ω0)].
(12)
The equivalent (CP) x-polynomial isP5(x)=(x1)(xei0.225π)(xei0.5π)(xei0.7π). The CP-diagram and the plot of |H 5(ω)| are shown in Fig. 2
Fig. 2 Magnitude of H 5(ω), and its CP-diagram. The PSA spectrum, has been considerably flattened around π/2 with respect to HSH(ω) for the same measured interferograms. From the CP-diagram alone, the spectral shape outside the 4 zeroes shown is absent. In contrast the plot of |H 5(ω)| shows it clearly. One must be aware of the small ripples within the stop-band.
. By inverse-Fourier transforming H 5(ω), or expanding P5(x), (both options being equally easy) we find the desired non-symmetric PSA as,

tan[ϕ]=2.4I(π/2)+2.9I(π)+0.57I(3π/2)I(2π)I(0)1.2I(π/2)2.3I(π)+2.7I(3π/2)0.23I(2π).
(13)

In Fig. 3
Fig. 3 Simulated speckle-like interferograms applied to our modified PSA (Eq. (13)). The 5 interferograms used in Eq. (13) may also be used for the SH-PSA. However the PSA detuning robustness is higher in our modified PSA (Eq. (13)) than in the SH-PSA (Eq. (11)).
we show an application for our modified PSA (Eq. (13)) to simulated speckle-like interferograms. For the sake of visual clarity we only show 3 interferograms (out of 5) along with the wrapped phase obtained from Eq. (13).

Let us continue with another example, a 9-steps PSA. We want high detuning robustness, (6-zeroes) at ω0=-π/2, and also robustness to bias illumination’s variation; 2-zeroes at ω=0. In Fig. 4
Fig. 4 Magnitude of P 9(e) and its CP-diagram. The resulting spectral rejection band is wide (6th order) around ω 0, and 2nd order around the origin.
the CP-diagram of P 9(x)=(x-1)2(x+i)6, and the plot of |P 9(e)| are shown,

Let us widen even further P 9(e) around -ω 0. A flatter than P 9(e) spectral response is obtained by spreading-out its 8 available zeroes. This is done while gauging the |H 9(ω)| plot. After some trial-and-error iterations we settle on the following non-symmetric PSA,

H9(ω)=[1ei(ω0.3ω0)][1eiω][1ei(ω+0.3ω0)][1ei(ω+0.6ω0)][1ei(ω+ω0)]                             [1ei(ω+1.4ω0)][1ei(ω+1.7ω0)][1ei(ω+2ω0)].
(14)

As Fig. 5
Fig. 5 CP diagram, and magnitude of H 9(ω). The resulting PSA’ spectral shape has been further flattened, with respect to P 9(e) around ω 0, and around of the origin. . From the CP-diagram alone, one may only wonder the spectral amplitude outside the zeroes shown, making almost impossible the fine-tuning task performed herein.
shows, we have further flattened the rejected-band with respect to that of P 9(e). The 8 zeroes have been spread-out around the unit circle. The designer may widen even more the rejection-band at a cost of tolerating bigger ripples; the signal amplitude at ω 0 is 82 times bigger than the highest ripple. This plot clearly shows the detailed spectral amplitude including the small ripples which cannot be seen in the CP-diagram. Fourier-inverse transforming H 9(ω) we find h 9(t) and from it, the searched 9-steps non-symmetric PSA.

5. Signal-to-Noise power ratio (S/N) in PSA designs

Given that the CP-diagram is blind to the continuous spectral’ amplitude, we cannot see if our PSA would perform better against noise with a slight change of the data carrier ω 0. By looking at Fig. 5 (for example) we see that the desired signal at ω=π/2, do not coincide with the spectra’s peak. Let us compute the S/N power-ratio at ω 0, and at its peak 1.16ω 0 [12

12. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

],
S/N(ω0)=|H9(ω0)|212πππ|H9(ω)|2dω=5.3,   S/N(1.16ω0)=|H9(1.16ω0)|212πππ|H9(ω)|2dω=6.6.
(15)
That is, increasing our carrier ω 0 from 0.5π (radians/sample) to 0.58π one obtains a 24.5% gain in the S/N power-ratio, i.e. (6.6/5.3)=1.245. We have suggested this carrier increase because we can seen the continuous plot of |H 9(ω)| in Fig. 5. In contrast, observing the CP-diagram alone, it is impossible to see where the location of the PSA’s spectral maximum is. Also, the F&K’ spectral plot [4

4. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]

] cannot be used, because it does not exist for non-symmetrical PSAs.

6. Conclusions

We have presented a zero-based, fine-tuning, spectrum-shaping technique for designing N-steps PSAs, based on the visualization and gauging of its continuous FTF spectral magnitude |H(ω)|. This technique gives more possibilities to approximate a target spectrum while keeping the size N of the PSA unchanged. This PSA’s design method starts by specifying a desired PSA spectrum’s template. Afterwards, the available (N-1) first-order zeroes are freely moved (fine-tuned), to approximate it. Finally, the inverse transform of the FTF (h(t)) is found, and from it the desired (in general non-symmetric) PSA. We remark, given that we freely move the PSA’s (N-1) zeroes, we end-up (with probability 1) synthesizing non-symmetric PSAs. These non-symmetric PSAs cannot be analyzed using the F&K spectral-plotting technique.

Finally we remark that, the target PSA spectrum may be determined by taking the Fourier transform of actual interferometric measurements of the kind of fringes being analyzed. This real-data spectral estimation and the desired phase noise-rejection are the key to estimate how many samples N a given PSA needs, and also where its (N-1) zeroes may be located.

References and links

1.

J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983). [CrossRef] [PubMed]

2.

P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef] [PubMed]

3.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, CRC, 2005).

4.

K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]

5.

K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992). [CrossRef]

6.

J. Schwider, O. Falkenstörfer, H. Schreiber, H. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993). [CrossRef]

7.

P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995). [CrossRef] [PubMed]

8.

J. Schmit and K. Creath, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. 35(28), 5642–5649 (1996). [CrossRef] [PubMed]

9.

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

10.

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

11.

J. Burke, “Extended averaging phase-shifting schemes for Fizeau interferometry on high-numerical aperture spherical surfaces,” Proc. of SPIE 7790, (2010).

12.

M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (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: March 7, 2011
Revised Manuscript: April 25, 2011
Manuscript Accepted: April 25, 2011
Published: May 17, 2011

Citation
A. Gonzalez, M. Servin, J. C. Estrada, and J. A. Quiroga, "Design of phase-shifting algorithms by fine-tuning spectral shaping," Opt. Express 19, 10692-10697 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10692


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References

  1. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983). [CrossRef] [PubMed]
  2. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26(13), 2504–2506 (1987). [CrossRef] [PubMed]
  3. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, CRC, 2005).
  4. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]
  5. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992). [CrossRef]
  6. J. Schwider, O. Falkenstörfer, H. Schreiber, H. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993). [CrossRef]
  7. P. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995). [CrossRef] [PubMed]
  8. J. Schmit and K. Creath, “Window function influence on phase error in phase-shifting algorithms,” Appl. Opt. 35(28), 5642–5649 (1996). [CrossRef] [PubMed]
  9. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef] [PubMed]
  10. J. G. Proakis, and D. G. Manolakis, Digital Signal Processing, 4th ed. (Prentice Hall, 2007). [PubMed]
  11. J. Burke, “Extended averaging phase-shifting schemes for Fizeau interferometry on high-numerical aperture spherical surfaces,” Proc. of SPIE 7790, (2010).
  12. M. Servin, J. C. Estrada, and J. A. Quiroga, “The general theory of phase shifting algorithms,” Opt. Express 17(24), 21867–21881 (2009). [CrossRef] [PubMed]

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