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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 2 — Jan. 17, 2011
  • pp: 638–648
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Two-step self-tuning phase-shifting interferometry

J. Vargas, J. Antonio Quiroga, T. Belenguer, M. Servín, and J. C. Estrada  »View Author Affiliations


Optics Express, Vol. 19, Issue 2, pp. 638-648 (2011)
http://dx.doi.org/10.1364/OE.19.000638


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Abstract

A two-step self-tuning phase-shifting method is presented. The phase-step between the two interferograms is not known when the experiment is performed. Our demodulating method finds, in a robust way, this unknown phase-step. Once the phase-step is estimated we proceed to phase demodulate the interferograms. Moreover our method only requires the fringe patterns to have a constant unknown phase-shift between them. As a consequence, this technique can be used to demodulate open and closed-fringed patterns without phase-sign ambiguity. The method may be regarded as a self-tuning quadrature filter, which determines the phase-shift between the two fringe patterns and finally estimates the demodulated phase map. The proposed technique has been tested with simulated and real interferograms obtaining satisfactory results.

© 2011 OSA

1. Introduction

Phase-Shifting Interferometry (PSI) is a useful technique in optical metrology for measuring the modulated phase of interferograms [1

1. D. Malacara, M. Servín, and Z. Malacara, “Interferogram analisis for optical testing”, Cambridge University Press, (2004), Marcel Dekker, Inc, (1998)

]. In standard PSI, a sequence of N interferograms is obtained having a known temporal carrier. Usually a minimum of three phase-shifted interferograms are need to retrieve the phase. Other phase-shifting methods have been reported as the four-step, five-step or the least-squares algorithm, for example. A good general overview of the phase-shifting algorithms can be found in [2

2. 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]

]. The use of fewer interferograms in PSI simplifies the computation process and the processing speed. Moreover, it increases the robustness against uncontrolled mechanical vibrations, air turbulence or temperature changes in the measuring system. In the case of open-fringe interferograms, there have been proposed many single-frame demodulating methods but in the case of closed-fringes, it is needed at least-two frames to solve the local phase-sign ambiguity. Note that a global sign ambiguity in the retrieved phase is irrelevant.

In the past there have been reported works about phase reconstruction with only two-frames. Ref [3

3. F. Mendoza-Santoyo, D. Kerr, and J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).

]. proposes a method for demodulating two interferograms with a known phase-shifting of π2 between them. In Refs [4

4. S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42(12), 3524–3531 (2003). [CrossRef]

,5

5. Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) 119(9), 424–428 (2008). [CrossRef]

] are proposed two-step phase-shifting algorithms that don’t need knowing the phase step between interferograms. The main drawbacks of [4

4. S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42(12), 3524–3531 (2003). [CrossRef]

,5

5. Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) 119(9), 424–428 (2008). [CrossRef]

] are that the methods don’t work if the phase-shift is close to π radians and in the case of interferograms with low spatial frequency fringes. Ref [6

6. X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007). [CrossRef]

]. shows an alternative approach that uses the least-squares method to extract the unknown phase-shift between two interferograms and then obtain the phase distribution. This method requires many iterations to converge (typically 17 iterations) and is sensitive to noise as only two frames are used. More recently, Ref [7

7. X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008). [CrossRef] [PubMed]

]. shows a method to extract the arbitrary unknown phase-shift between two interferograms and then reconstruct the complex object wave without iteration process. This technique only can be used in amplitude division interferometry as it needs to measure, additionally with the two phase-shifted frames, the intensity of the reference beam. On the other hand, this method requires the reference intensity to be constant along the full aperture that can be a non realistic case when we deal with real interferograms. In addition, the method assumes that the intensity of the reference beam is greater than the maximum of the object beam that can be problematic in some cases.

In this work, we propose a self-tuning two-step phase-shifting algorithm that doesn’t need to known the phase-step between interferograms. This phase-step can be any value between [-π, π]. The proposed method doesn’t need to iterate, it can be used to any kind of fringe pattern as it doesn’t need to measure the intensity of the reference beam and no suppositions are made to the fringe spatial frequency, background and modulation maps.

2. Proposed method

In PSI, a discrete temporal interferogram can be described using the following expression,
g(x,y,t)=a(x,y)+b(x,y)cos[Φ(x,y)+ω0t]
(1)
where a(x,y) is the background illumination, b(x,y) and Φ(x,y) are the modulation and phase maps, and ω0 is the temporal carrier frequency. The integer index t denotes the t th interferogram obtained at time t. In our case t = [0, 1]. For simplicity in the following we will drop the spatial dependence. Expression (1) can be rewritten as follow,
g(t)=a+b2[ei[Φ(x,y)+ω0t]+ei[Φ(x,y)+ω0t]]
(2)
The Fourier transform of expression (2) is,
G(ω)=aδ(ω)+b2[δ(ω+ω0)eiΦ+δ(ωω0)eiΦ]
(3)
where δ is the Dirac delta function. In order to obtain the desired phase map, we can define a complex quadrature filter with frequency response H(ω) and with the following requirementsH(0)=0, H(ω0)=0 and H(ω0)0 [2

2. 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]

]. The filtered complex function, G˜(ω), is obtained by multiplication of G(ω) and H(ω) as,
G˜(ω)=b2H(ω0)δ(ω+ω0)eiΦ
(4)
Performing the inverse Fourier transform of expression (4) we obtain the complex intensity function as,
g˜(t)=b2H(ω0)eiΦeiωot
(5)
Finally, the phase and modulation maps are acquired evaluating g˜(t) at t = 0 obtaining the angle and the norm of g˜(0) respectively as,
Φ=tan1(Im{g˜(0)}Re{g˜(0)})
(6)
and,

|g˜(0)|=b2|H(ω0)|
(7)

The mathematical analysis shown above assumes that the temporal frequency ω0 is known. On the other hand, as we said, in order to built a practical quadrature filter that achieves the requirement H(0)=0 and H(ω0)=0, it is necessary at least three temporal interferograms [1

1. D. Malacara, M. Servín, and Z. Malacara, “Interferogram analisis for optical testing”, Cambridge University Press, (2004), Marcel Dekker, Inc, (1998)

].

g(x,y,t)=cos[Φ(x,y)+ω0t]
(8)

The normalizing method used in this work has been reported in [9

9. J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003). [CrossRef]

]. The method shown in [9

9. J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003). [CrossRef]

] consists in employing an isotropic operator. This method is robust to wide-band fringe spatial frequencies, obtaining good solutions.

The proposed self-tuning phase-shifting algorithm is based on analyzing |g˜(0)|2 parameter when a detuning error exists. This detuning error appears when the quadrature filter H(ω) is tuned in a temporal frequency ωf different to the assumed one ω0 [2

2. 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]

]. Figure 1
Fig. 1 Scheme of the detuning error that appears when ωfω0. (a) In this case ωf coincides with ω0 and there is non detuning error. (b) In this case ωfω0 and there is non-detuning
shows a scheme of a H(ω) filter when there is non detuning error (Fig. 1 (a)) and when there is (Fig. 1 (b)). In Fig. 1(a) it is shown the case where ωf coincides with ω0 and there is non detuning error. In Fig. 1(b) it is shown the case where ωfω0 and therefore, H(ω0)0 and H(ω0)0. In this case, a detuning error does exist and expression (4) is not longer valid.

This expression is rewritten in the case of ωfω0 as,
G˜(ω)=12[H(ω0)δ(ω+ω0)eiΦ+H(ω0)δ(ωω0)eiΦ]
(9)
and the intensity complex function is given as,
g˜(t)=12[H(ω0)eiΦeiωot+H(ω0)eiΦeiωot]
(10)
In this case |g˜(0)| is affected by a detuning error and it is given by,
|g˜(0)|2=14[A(ω0)+B(ω0)+(C(ω0)cos[2Φ]D(ω0)sin[2Φ])]
(11)
with, A(ω0)=|H(ω0)|2, B(ω0)=|H(ω0)|2, C(ω0)=Re[H(ω0)H(ω0)] and D(ω0)=Im[H(ω0)H(ω0)]. The standard deviation σ of (11) is given by,
σ[|g˜(0)|2]=14σ[C(ω0)cos[2Φ]D(ω0)sin[2Φ]]
(12)
From a practical point of view, the standard deviation is obtained from |g˜(0)|2 map as,
σ[|g˜(x,y,0)|2]=1Mxy(|g˜(x,y,0)|2g˜m)2
(13)
where, M is the number of pixels in the map and g˜m is the mean of |g˜(x,y,0)|2. Note that in (13) we have introduced the spatial dependence of g˜ for the sake of clarify. Additionally, g˜(0) can be obtained from the convolution product between g(t) and h(t) as g˜(0)=g(t)h(t)dt, where h(t)=12πH(ω)eiωtdω. In expression (12) there is no contribution of A(ω0) and B(ω0) terms because they are constants and the standard deviation is invariant under translation changes. If ωf=ω0, B(ω0)=0, C(ω0)=0, D(ω0)=0 and σ[|g˜(0)|2] equals to zero in expression (12). In any other case, σ[|g˜(0)|2] obtains a positive value different of zero. Using this property, it is possible to construct a filter H(ω) that is tuned at different temporal frequencies ωf and look for the temporal frequency which returns a minimum value in σ[|g˜(0)|2]. Using this filter tuned at ω0, the phase map Φ is determined using expressions (4) to (6).

This method uses exhaustive search, thus, it is as fast as the size of the discrete searching tuning frequency set. This is not a serious concern given that the computational cost of exploring a sampled one-dimensional space [π,π] exhaustively with, say 600 samples, is not high.

3. Simulations

In order to show the performance of the proposed method, we have tested it with a simulation using closed-fringe interferograms. The utility of this method has special interest in the case of closed-fringe interferograms as two-frames are the minimum number of interferograms to reconstruct the phase without local phase-sign ambiguity. In Fig. 2
Fig. 2 Two fringe patterns used in the first simulation
are shown the two fringe patterns of the simulation. As can be seen from Fig. 2, the interferograms have a size of 411x411 pixels2. The background and modulation maps of the fringe patterns are equal to zero and 125 a.u (arbitrary units) in every pixel. The noise is Gaussian with zero mean and a variance of 13 a.u. The temporal frequency ω0 is equals to 1.2 rad.

In Fig. 3
Fig. 3 Plot between the standard deviation of the modulation map σ[|g˜(0)|2] and the different temporal frequenciesωf obtained in the first simulation
, it is shown the plot between the standard deviation of the modulation map σ[|g˜(0)|2] and the different temporal frequenciesωf. The discrete searching tuning-frequency set is composed by 600 frequencies equally spaced between [0, 2π] and the processing time is of 5 s using a 1.6 GHz laptop and MATLAB. The detected minimum corresponds to ω0 = 1.203 rad and it is marked in the figure with a red square.

Observe in Fig. 3 that are obtained two minimums, one at ω0 and other at 2πω0. Note that it is irrelevant to select one or the other as the only consequence will be a global phase-sign change in the reconstructed phase Φ. In Fig. 4
Fig. 4 Obtained wrapped phase in the first simulation
, it is shown the obtained wrapped phase. Finally, in Fig. 5
Fig. 5 Reconstructed phase (a) and computed error between actual and computed phases (b)
are shown the reconstructed phase and the computed error between actual and obtained phases. The rms (root-mean-square) error of the difference between the actual and measured phases is 0.08 rad.

In order to show the performance of the two-step self-tuning algorithm to noisy interferograms, we have obtained for different signal to noise ratios the computed temporal frequencyω^0, the absolute difference between the actual and computed temporal frequencies |ω0ω^0|, the rms error between the actual and computed phase maps and the processing times. The computed phase map is retrieved by interferograms without noise in order to show the detuning error caused by the mismatch between the actual and obtained temporal frequencies. In Table 1

Table 1. Obtained temporal frequenciesω^0, absolute difference between the actual and computed temporal frequencies |ω0ω^0|, rms errors obtained between the actual and computed phase maps (with the phase maps retrieved by interferograms without noise and from the computed temporal frequencies) and processing times for different signal to noise ratios

table-icon
View This Table
, the first row corresponds to the different signal to noise ratios that quantify how much the phase has been corrupted by noise. As can be seen from Table 1, the proposed two-step self-tuning algorithm gives appropriate results even in the case of interferograms with high noise.

4. Experimental results

We have applied the proposed algorithm to real interferograms. In order to check the obtained results, we have compared the retrieved phase from our method with the phase obtained from 10 phase-shifted interferograms, and using the least-squares phase demodulation method [10

10. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004). [CrossRef] [PubMed]

]. In Fig. 6
Fig. 6 (a), (b) Real Interferograms (c), (d) Resultant interferograms after the normalization process.
are shown the two interferograms used in our approach and the normalized version of them.

In Fig. 7(a)
Fig. 7 Computed wrapped phase obtained with the two-step proposed method (b) Wrapped phase obtained by the least-squares demodulation method using 10 interferograms
and 7(b) are shown the acquired wrapped phase obtained from our method and with the least-squares method using 10 interferograms. The processing times using the proposed method and the least-squares method are 12 s and 30 s respectively.

As can be seen from Figs. 7(a) and 7(b) both wrapped phases are similar. In Figs. 8(a)
Fig. 8 Reconstructed phases with the proposed method (a) and with the least-squares method (b).
and 8(b) are shown the reconstructed phases with both methods and finally in Fig. 9
Fig. 9 Difference between the reconstructed phases shown in Fig. 12 using the proposed method and the least-squares method
it is shown the difference between the reconstructed phases.

As can be seen from Fig. 9 there is a good agreement between both measures. The rms (root-mean-square) error of the difference between both reconstructed phases is 0.3 rad. On the other hand, the temporal frequency between interferograms obtained by the proposed and the least-squares methods is ω0 = 0.7 rad and ω0 = 0.85 rad respectively.

5. Conclusions

References and links

1.

D. Malacara, M. Servín, and Z. Malacara, “Interferogram analisis for optical testing”, Cambridge University Press, (2004), Marcel Dekker, Inc, (1998)

2.

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]

3.

F. Mendoza-Santoyo, D. Kerr, and J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).

4.

S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42(12), 3524–3531 (2003). [CrossRef]

5.

Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) 119(9), 424–428 (2008). [CrossRef]

6.

X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007). [CrossRef]

7.

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008). [CrossRef] [PubMed]

8.

M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express 17(19), 16423–16428 (2009). [CrossRef] [PubMed]

9.

J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003). [CrossRef]

10.

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004). [CrossRef] [PubMed]

OCIS Codes
(120.2650) Instrumentation, measurement, and metrology : Fringe analysis
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: September 8, 2010
Revised Manuscript: October 14, 2010
Manuscript Accepted: October 15, 2010
Published: January 5, 2011

Citation
J. Vargas, J. Antonio Quiroga, T. Belenguer, M. Servín, and J. C. Estrada, "Two-step self-tuning phase-shifting interferometry," Opt. Express 19, 638-648 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-638


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References

  1. D. Malacara, M. Servín, and Z. Malacara, “Interferogram analisis for optical testing”, Cambridge University Press, (2004), Marcel Dekker, Inc, (1998)
  2. 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]
  3. F. Mendoza-Santoyo, D. Kerr, and J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).
  4. S. Almazán-Cuéllar and D. Malacara-Hernandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42(12), 3524–3531 (2003). [CrossRef]
  5. Y. Zhu, L. Liu, Z. Luana, and J. Sun, “Discussions about FFT-based two-step phase-shifting algorithm,” Optik (Stuttg.) 119(9), 424–428 (2008). [CrossRef]
  6. X. F. Xu, L. Z. Cai, Y. R. Wanga, X. F. Meng, H. Zhang, G. Y. Dong, and X. X. Shen, “Blind phase shift extraction and wavefront retrieval by two-frame phase-shifting interferometry with an unknown phase shift,” Opt. Commun. 273(1), 54–59 (2007). [CrossRef]
  7. X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008). [CrossRef] [PubMed]
  8. M. Servin, J. C. Estrada, and J. A. Quiroga, “Spectral analysis of phase shifting algorithms,” Opt. Express 17(19), 16423–16428 (2009). [CrossRef] [PubMed]
  9. J. A. Quiroga and M. Servín, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003). [CrossRef]
  10. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004). [CrossRef] [PubMed]

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