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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 2 — Jan. 17, 2011
  • pp: 584–596
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Phase-shifting interferometry based on induced vibrations

J. Vargas, J. Antonio. Quiroga, A. Álvarez-Herrero, and T. Belenguer  »View Author Affiliations


Optics Express, Vol. 19, Issue 2, pp. 584-596 (2011)
http://dx.doi.org/10.1364/OE.19.000584


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Abstract

The presence of uncontrolled mechanical vibrations is typically the main precision-limiting factor of a phase-shifting interferometer. We present a method that instead of trying to insolate vibrations; it takes advantage of their presence to produce the different phase-steps. The method is based on spatial and time domain processing techniques to compute first the different unknown phase-steps and then reconstruct the phase from these tilt-shifted interferograms. In order to compensate the camera movement, it is needed to perform an affine registration process between the different interferograms. Simulated and experimental results demonstrate the effectiveness of the proposed technique without the use of any phase-shifter device.

© 2011 OSA

1. Introduction

Several methods have been proposed to eliminate the tilt-shift error. In Ref [5

5. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000). [CrossRef]

]. an algorithm immune to both translational and tilt-shift errors is presented. In this case, the principle of the algorithm is to use a first order Taylor series expansion of the phase-shifting errors instead of the original nonlinear phase-error equations; then by iteration the phase-shift plane is determined. In Ref [6

6. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002). [CrossRef] [PubMed]

], it is presented a phase-shifting demodulating method based on dividing the interferograms into small regions, where the phase-shifting can be considered approximately constant; a self-calibrating approach is applied in each block. In [7

7. J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008). [CrossRef]

] a combined spatial and temporal demodulating method is presented. First, a spatial demodulation is performed by a FFT based method. Second, the previously calculated phases are subtracted with respect to the first one, in order to obtain the different tilt-shift errors with respect to the first phase. Finally, by a least-square iterative minimization process the true phase map and the different tilt errors are determined. In Ref [8

8. Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE 4416, 152–157 (2001). [CrossRef]

]. a self-reference method is proposed for phase-shifting interferometry when translation and tilt errors exist. Another strategy to deal with this kind of problems can be using a one-shot phase-shifting interferometry method [9

9. G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. 33(23), 2788–2790 (2008). [CrossRef] [PubMed]

]. This approach allows recovering at the same time different interferograms at the camera detector. The main drawbacks of this method are that, on one hand it is necessary to share the camera detector between N interferograms and therefore the spatial resolution of the measured phase will be decreased. On the other hand, this method requires a matching procedure between the different interferograms that can be problematic in some cases

The mentioned methods [5

5. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000). [CrossRef]

8

8. Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE 4416, 152–157 (2001). [CrossRef]

] are not directly applicable in the case of a measuring system with severe mechanical vibrations. The method presented in Ref [5

5. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000). [CrossRef]

]. is based on a first order Taylor approximation of the tilt-shift errors so it is assumed that these tilt-shift errors are small compared with the problem phase. In [6

6. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002). [CrossRef] [PubMed]

] it is needed to establish beforehand the size of the processing block. If the phase-shifting error is large compared with the problem phase, the size of processing blocks must be very small and therefore the result of the iterative self-calibration algorithm can be not trustful. In presence of severe vibrations, a new problematic that appears and that cannot be solved with any of the methods exposed above [3

3. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001). [CrossRef]

8

8. Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE 4416, 152–157 (2001). [CrossRef]

] is the phase-error introduced by the movement of the camera. Note that if these vibrations introduce a camera movement the corresponding interferogram pixels will not match. In this case, the camera oscillation introduces a set of unknown affine transformations between interferograms in temporal phase-shifting. In typical mechanical stabilized temporal phase-shifting interferometry these affine transformations can be neglected, but this is not the case if severe vibrations are presented. An example where it is interesting to obtain interferometric data but it is impossible using the typical temporal phase-shifting methods is inside a thermal-vacuum chamber. The use of interferometry inside these chambers is important because it permits to obtain optical quality measures of optical components at different ranges of temperature and pressure that for example, it is necessary in space applications.

In this paper, we present a method that instead of trying to insolate vibrations; it takes advantage of their presence to produce the different phase-shifts. The method is based on spatial and temporal demodulation and can be divided in three steps. First, a set of interferograms are acquired in different times. A spatial fringe demodulation process by a Fourier transform method is achieved over each interferogram. Second, using the initial estimation of the phase, the different tilt-shifts and affine transformations are obtained. Finally, using the previously computed affine transformations, the different interferograms are referred to a unique camera frame and then a general demodulation technique based on a least-squares iterative minimization process is performed using the previously computed tilt-shifts. Note that using a temporal sequence of N interferograms to obtain the modulating phase by a TPI technique is more accurate that obtaining the phase from a single interferogram by the Fourier transform method. On one hand, the TPI method reduces errors in phase calculations when noisy interferograms are involved [2

2. D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical testing, (Marcel Dekker, Inc, 1998)

]. Additionally, the Fourier method suffers from border effects that introduce additional errors in the recovered phase. Our method makes possible to obtain interferometric measurements in very hostile environments as thermal-vacuum chambers, where severe shock vibrations with a typical frequency of 0.5 Hz and induced displacements in the range of millimeters appear caused by the cryogenic pump. Additionally, the proposed method also permits to acquire interferometric measures without the need of any phase-shifter device.

2. Proposed method

The proposed method is divided in three steps. First, it is determined the tilt-shift in each interferogram. Second, it is obtained the affine transformations between interferograms and finally, the phase is computed by a general temporal demodulation method based on the least-squares method.

2.1 Tilt-shift determination by the Fourier transform method

For the sake of clarify; we first assume that we have a mechanically stabilized interferometer in which the phase-shifts are introduced by a PZT device. In this case, the intensity at camera pixel (x,y) of the n th interferogram can be represented as,
In(x,y)=A(x,y)+B(x,y)cos[Φ0(x,y)+dn]
(1)
where, In is the theoretical intensity of the nth interferogram, A(x,y) is the background intensity, B(x,y) is the modulation of fringe pattern, dn is the phase-shift introduced by the PZT device and Φ0(x,y) is the phase distribution under test. If the PZT device interferometer has orientations errors during the shift, the expression (1) has to be extended to,
Int(x,y)=A(x,y)+B(x,y)cos[Φ0(x,y)+anx+bny+cn+dn]
(2)
where, an and bn are the spatial carrier frequency at x and y directions and cn denotes the additional piston value introduced by the tilt-shift. In the case of an interferometer submitted to severe vibrations, it is possible to perform phase-shifting measurements without any phase-shifter device. In this case, expression (2) is rewritten taking into account the camera oscillation to,
In(xn,yn,tn)=A(xn,yn)+B(xn,yn)cos[Φ0(xn,yn)+a(tn)xn+b(tn)yn+c(tn)]
(3)
with
[xn, yn, 1]T=M(tn)[x, y, 1]T
(4)
where, In(xn,yn,tn) denotes the theoretical intensity at camera pixel (xn,yn) and time tn, andM(tn)is a 3 × 3 affine transformation matrix which coefficients are time-dependent and that models the camera movement [10

10. R. Hartley, and A. Zisserman, Multiple View Geometry in Computer Vision, (Cambridge University Press, 2004).

].

In order to obtain the objective phaseΦ0, first a Fourier based spatial demodulation method is performed over the different interferograms obtained along time. Using this method it is obtained,
Φn(xn,yn)=Φ0(xn,yn)+a(tn)xn+b(tn)yn+c(tn)
(5)
As can be seen from Eq [5

5. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000). [CrossRef]

], the different phases Φn(xn,yn) have a tilt aberration that is time dependent. In order to subtract these tilt aberrations, a plane fit is performed for each phase Φn(xn,yn). A new phase is obtained by the difference between the different fitted planes and phases Φn(xn,yn) as,
Φ0(xn,yn)=Φn(xn,yn)Pn(xn,yn)
(6)
where, Pn(xn,yn) corresponds to the different tilt-shifts obtained from the plane fits,
Pn(xn,yn)=[a(tn)+a0]xn+[b(tn)+b0]yn+[c(tn)+c0]
(7)
Note that Φ0(xn,yn) in expression (6) is different than Φ0(xn,yn) in expression (5); this is because Φ0(xn,yn) has no tilt-aberrations because suppression of coefficientsa0, b0 c0 that are constants for all phases Φn(xn,yn). Observe that this is not an inconvenient because typically the tilt aberrations don’t introduce any relevant information in a measured phase distribution.

2.2 Affine registration between interferograms

Once that the tilt-shifts,Pn(xn,yn) have been obtained, it is necessary to perform an affine registration process to refer the different interferograms to a unique camera reference frame. The proposed process consists of determining a set of common control points in the computed phases Φ0(xn,yn). The control points are determined binarizing the local orientation of Φ0(xn,yn). The local orientation is obtained from the normalized gradient operator, n[Φ0]=Φ0|Φ0|, and it is defined as Θ[Φ0]=n[Φ0]yn[Φ0]x, where n[Φ0]k with k equal to x or y, denotes the x or y component of n respectively. The binarization process is obtained assigning the pixels with positive local orientation a value equal to true and to negative pixels a value equal to false. In the binarized map, the critical points —points where the gradient of Φ0 vanishes— appear as corner points, that can be detected with sub-pixel accuracy by a Harris corner detector [11

11. C. Harris, and M. J. Stephens, “A combined corner and edge detector,” in Alvey Vision Conference, pp. 147–152 (1988).

]. Figure 2(a)
Fig. 2 (a) phase example, (b) phase example transformed by an affine transformation corresponding to a translation of −5 and 12 pixels, a rotation of 0.5 rad and a scale change of 1.2 and 0.98 pixels in the x and y axis respectively, (c) corresponding binarized map of (a), (d) corresponding binarized map of (b), (e) phase example with the detected control points, (f) phase example transformed with the detected control points
and 2(b) show images of a phase example and of the same phase transformed by an affine transformation respectively. The applied affine transformation corresponds to a translation of −5 and 12 pixels in the x and y axis respectively, a rotation of 0.5 rad and a scale change of 1.2 and 0.98 pixels in the x and y axis respectively. In Figs. 2(c) and 2(d) it is shown the corresponding binarized maps of Figs. 2(a) and 2(b) respectively and the detected critical points by the corner detector. These points are shown as red rectangles. Finally, in Figs. 2(e) and 2(f) are shown the phases with the detected control points.

Once that the corresponding control points have been detected, the affine transformation matrix Mn between the nth interferogram and the first one (used as reference) can be obtained by at least three control-point correspondences [8

8. Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE 4416, 152–157 (2001). [CrossRef]

]. These matrixes Mn are refined by a non-linear iterative process minimizing the absolute difference between the fist interferogram and the affine rectificated nth interferogram using the Levenberg and Marquardt algorithm. Note that if is not possible to detect three control points, always is possible to obtain the affine transformation Mn by a non-linear iterative minimization process. In this case, the initial guess for the affine transformation consist in a translation of 0 pixel in both axis, a rotation of 0 rad and a scale factor of 1 pixel in both axis. This starring guess is then refined by a Levenberg and Marquardt iterative minimizing algorithm. Finally, by these obtained affine transformations, the corrected interferograms and tilt-shifts are computed by,
In(x,y,tn)=A(x,y)+B(x,y)cos[Φ^0(x,y)+Pn(x,y)]
(8)
with,

Pn(x,y)=[a(tn)+a0]x+[b(tn)+b0]y+[c(tn)+c0]
(9)

2.3 General temporal demodulation

After In(x,y,tn) and Pn(x,y) have been determined, the phase distribution Φ^0(x,y) can be obtained by a least squares method. Expression (8) can be rewritten as,
In(x,y,tn)=α0+α1cos[Pn(x,y)]+α2sin[Pn(x,y)]
(10)
where, α0=A(x,y), α1=B(x,y)cos[Φ^0(x,y)] and α2=B(x,y)sin[Φ^0(x,y)]. If we denote the measured intensity distribution asIn(x,y,tn), the sum of squared differences between the theoretical and measured intensity distributions is given by,
S=n(InIn)2
(11)
The unknown parametersα0, α1 and α2 can be determining minimizing S using the least-squares criterion that imposes Sα0=0, Sα1=0 and Sα2=0; this give us a linear system as,
Aα=b
(12)
where, α=[α0,α1,α2]T and
A=[Nncos[Pn(x,y))]nsin[Pn(x,y))]ncos[Pn(x,y))]ncos2[Pn(x,y))]ncos[Pn(x,y))]sin[Pn(x,y))]nsin[Pn(x,y))]ncos[Pn(x,y))]sin[Pn(x,y))]nsin2[Pn(x,y))]]
(13)
and,
b=[nInnIncos[Pn(x,y)]nInsin[Pn(x,y)]]
(14)
with N the total number of interferograms. From expressions (13) and (14) we finally have,
α=A1·b
(15)
and,

Φ^0=tan1(α2α1)
(16)

3. Simulations

In order to compare the results obtained by the proposed method, a computer simulation have been carried out to shown the effectiveness of the method. In our simulation, the actual phase map Φ^0(x,y) is characterized by the following Zernike coefficients in the Wyant notation [12

12. J. C. Wyant, and K. Creath, “Basic Wavefront Aberation Theory of Optical Metrology,” in Applied Optics and Optical Engineering, Vol. XI, Chapter 1, Academic Press (1992).

], Z4=1, Z5=0.8, Z6=1.2, Z10=0.4, Z12=0.5, Z13=1.1, with pv (peak to valley) of 5.3 rad and rms (root mean square) of 0.7 rad. The background is A(x,y)=140exp[9(x2+y2)], and the modulation amplitude is B(x,y)=110exp[7(x2+y2)], where 1x1 and 1y1. Additionally, it has been added additive white Gaussian noise to the interferograms shown in Eq. (8) with a mean of zero and a variance of 0.001 in arbitrary units. The introduced affine transformations and tilt-shifts between interferograms are given in Tables 1

Table 1. Affine transformations introduced to the actual phase map

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and 2

Table 2. , Introduced tilt-shifts to generate the different interferograms in the simulation

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respectively.

Figure 3
Fig. 3 Phase map described in the simulation section affected by the affine transformations shown in Table 1
shows images of the different phase maps affected by the affine transformations shown in Table 1. Figure 4
Fig. 4 Resultant interferogram of the phase shown in Fig. 2(b)
shows the resultant interferogram of the phase shown in Fig. 3(b).

In Fig. 5
Fig. 5 Different phasesΦ0(xn,yn) shown in Fig. 2, recovered using the Fourier transform demodulation method with the detected control points
, it is shown the different phases Φ0(xn,yn) recovered using the Fourier transform demodulation method and the obtained control points. With these control points the affine transformations between the first interferogram and the rest of them are computed.

In Fig. 6
Fig. 6 Difference between the phases after the affine registration process and the phase corresponding to the first interferogram shown in Fig. 2(a) that is used as reference
it is shown the difference between the phases after affine registration process Φ0(x,y) and the phase corresponding to the first interferogram that is used as reference. As can be seen from Figs. 6(a), (b), (c), (d) and (e) the affine registration process works correctly and the differences come from the typical ripple artifacts. The rms (root mean square error) of the difference between the phases after affine rectification Φ0(x,y) and the phase corresponding to the first interferogram is shown in Table 3

Table 3. , rms (root mean square error) of the difference between the phases after affine rectification and the phase corresponding to the first interferogram that is used as reference

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. In Table 3 it is also shown the relative error (err) of the different affine rectificated phases obtained from the division between the computed rms, shown in Table 3, and the pv of the actual phase map that is 5.3 rad. As can be seen from Table 3, the mean of the rms and relative errors are 0.096 rad and 0.28% respectively. After determining the different affine transformations, the different interferograms and tilt-shifts are affine registered with respect to the first one.

In Table 4

Table 4. , Recovered tilt-shifts in the simulation

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are given the recovered tilt-shifts. As can be seen from Tables 2 and 4, the recovered tilt-shifts are totally consistent with respect to the theoretical ones. The differences in the DC or independent terms in Tables 2 and 3 are because the actual absolute fringe orders are unknown when a homodyne interferometry method is used [1

1. K. Creath, “Phase-shifting interferometry techniques,” Prog. Opt. 26, 350–393 (1988).

,2

2. D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical testing, (Marcel Dekker, Inc, 1998)

]. The actual fringe orders and therefore the independent terms in Tables 2 and 3 can be obtained using heterodyne interferometry [1

1. K. Creath, “Phase-shifting interferometry techniques,” Prog. Opt. 26, 350–393 (1988).

,2

2. D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical testing, (Marcel Dekker, Inc, 1998)

] but note that these DC terms don’t introduce any relevant information.

Finally, in Fig. 7
Fig. 7 Recovered phase Φ0(x,y) after the general temporal demodulation method
it is shown the recovered phase Φ0(x,y) after the general temporal demodulation method and in Fig. 8
Fig. 8 Difference between the recovered phase and the theoretical phase
it is shown the difference between the recovered phase and the theoretical phase. The rms (root mean square error) of the difference between both phases is of about 0.054 rad. As can be seen from Table 3, the rms values obtained from the Fourier transform demodulating method are in all cases larger than the rms error recovered from the temporal demodulation method and using all interferograms.

4. Experimental results

For further verification of the performance of the proposed algorithm, we have applied it to experimental interferograms. We have used a Mach-Zehnder interferometer where the phase-shifts are introduced by induced vibrations without the need of any phase-shifter device. With this interferometer and using the proposed method we have characterized a high quality λ/100 glass plate. On the other hand the test plate has been also measured by using a commercial Zygo GPI Fizeau phase-shifting interferometer with a vibration-isolating platform. Images of the results obtained by the Zygo and Mach-Zenhder interferometers are shown in Figs. 9
Fig. 9 Results obtained by the Zygo GPI interferometer for the λ/100 high quality glass plate
and 10
Fig. 10 Result obtained by the Mach-Zenhder interferometer using the proposed method for the λ/100 high quality glass plate
respectively. As can be seen from Figs. 9 and 10, the general shape of the phase is similar in both cases.

The obtained rms (root mean square) and pv (peak to valley) of the wave-front error measured by the Zygo interferometer using a four step phase-shifting algorithm and with the Mach-Zenhder interferometer using the proposed method with ten interferograms is shown in Table 5

Table 5. , rms (root mean square) and pv (peak to valley) of the wave-front error measured by the Zygo interferometer and with the Mach-Zenhder interferometer for the λ/100 high quality glass plate

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.

As can be seen from Table 5, the results obtained with both instruments are very similar. The difference in the rms measurement is of about 0.0009 waves. On the other hand, the difference in the pv measurement is 0.026 waves.

5. Conclusions

We have proposed an interferometric method that is capable to obtain accurate measures when severe vibrations are present, that it is the case when it is desired to perform an object interferometric characterization inside a thermal-vacuum chamber, for example. The method doesn’t need any phase-shifter device and it takes advantage of these vibrations to produce the different phase-steps that are used to demodulate the phase. The proposed method is based on spatial and time domain processing techniques to compute first the different unknown phase-shifts and then reconstruct the phase from these tilt-shifted interferograms. In order to compensate the camera movement, it is needed to perform an affine registration process between the different interferograms. Simulated results and experiments demonstrate the effectiveness of the proposed method. This method can be used in a very hostile environment as thermo-vacuum chambers to obtain accurate interferometric tests. On the other hand costly phase-shifting devices are not longer required for steady-state measures.

References and links

1.

K. Creath, “Phase-shifting interferometry techniques,” Prog. Opt. 26, 350–393 (1988).

2.

D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical testing, (Marcel Dekker, Inc, 1998)

3.

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001). [CrossRef]

4.

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]

5.

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000). [CrossRef]

6.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002). [CrossRef] [PubMed]

7.

J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008). [CrossRef]

8.

Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE 4416, 152–157 (2001). [CrossRef]

9.

G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. 33(23), 2788–2790 (2008). [CrossRef] [PubMed]

10.

R. Hartley, and A. Zisserman, Multiple View Geometry in Computer Vision, (Cambridge University Press, 2004).

11.

C. Harris, and M. J. Stephens, “A combined corner and edge detector,” in Alvey Vision Conference, pp. 147–152 (1988).

12.

J. C. Wyant, and K. Creath, “Basic Wavefront Aberation Theory of Optical Metrology,” in Applied Optics and Optical Engineering, Vol. XI, Chapter 1, Academic Press (1992).

OCIS Codes
(050.5080) Diffraction and gratings : Phase shift
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: May 18, 2010
Revised Manuscript: August 2, 2010
Manuscript Accepted: August 4, 2010
Published: January 5, 2011

Citation
J. Vargas, J. Antonio. Quiroga, A. Álvarez-Herrero, and T. Belenguer, "Phase-shifting interferometry based on induced vibrations," Opt. Express 19, 584-596 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-584


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References

  1. K. Creath, “Phase-shifting interferometry techniques,” Prog. Opt. 26, 350–393 (1988).
  2. D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical testing, (Marcel Dekker, Inc, 1998)
  3. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001). [CrossRef]
  4. 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]
  5. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39(22), 3894–3898 (2000). [CrossRef]
  6. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002). [CrossRef] [PubMed]
  7. J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. 10(7), 075011 (2008). [CrossRef]
  8. Z. Ge and M. Takeda, “Self-reference method for phase-shift interferometry,” Proc. SPIE 4416, 152–157 (2001). [CrossRef]
  9. G. Rodriguez-Zurita, N. I. Toto-Arellano, C. Meneses-Fabian, and J. F. Vázquez-Castillo, “One-shot phase-shifting interferometry: five, seven, and nine interferograms,” Opt. Lett. 33(23), 2788–2790 (2008). [CrossRef] [PubMed]
  10. R. Hartley, and A. Zisserman, Multiple View Geometry in Computer Vision, (Cambridge University Press, 2004).
  11. C. Harris, and M. J. Stephens, “A combined corner and edge detector,” in Alvey Vision Conference, pp. 147–152 (1988).
  12. J. C. Wyant, and K. Creath, “Basic Wavefront Aberation Theory of Optical Metrology,” in Applied Optics and Optical Engineering, Vol. XI, Chapter 1, Academic Press (1992).

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