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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 11 — May. 26, 2008
  • pp: 7806–7817
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One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms

Gustavo Rodriguez-Zurita, Cruz Meneses-Fabian, Noel-Ivan Toto-Arellano, José F. Vázquez-Castillo, and Carlos Robledo-Sánchez  »View Author Affiliations


Optics Express, Vol. 16, Issue 11, pp. 7806-7817 (2008)
http://dx.doi.org/10.1364/OE.16.007806


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Abstract

An experimental setup for optical phase extraction from 2-D interferograms using a one-shot phase-shifting technique able to achieve four interferograms with 90° phase shifts in between is presented. The system uses a common-path interferometer consisting of two windows in the input plane and a phase grating in Fourier plane as its pupil. Each window has a birefringent wave plate attached in order to achieve nearly circular polarization of opposite rotations one respect to the other after being illuminated with a 45° linear polarized beam. In the output, interference of the fields associated with replicated windows (diffraction orders) is achieved by a proper choice of the windows spacing with respect to the grating period. The phase shifts to achieve four interferograms simultaneously to perform phase-shifting interferometry can be obtained by placing linear polarizers on each diffraction orders before detection at an appropriate angle. Some experimental results are shown.

© 2008 Optical Society of America

1. Introduction.

Phase-shifting interferometry is a reliable technique to extract phase information from interferograms [1

1. J. E. Greivenkamp and J. H. Bruning, “C.14 Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (John Wiley & Sons, 1992), pp. 501–598.

]. The technique is based on a system of linear equations which uses at least three different interferograms obtained from the same phase distribution [2

2. J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1990), pp. 271–359.

]. In order to achieve these interferograms, certain phase shift values are introduced between the signal wave and the reference wave. Among other variants, using four interferograms has been proved to be very useful for the case of interferograms having an adequate signal-to-noise ratio and good contrast [3

3. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1998), 26 pp. 349–393.

]. A constant phase shift value of α=90° is widely used for a case of N=4 interferograms [4

4. M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 44, 4439–4442 (1985). [CrossRef]

]. In this communication, we will focus ourselves on this variant.

For N interferograms, N-1 shifts must be carried out. For static phase distributions, this means that, for the case of N=4, four shots in sequence must be done. However, when a time-varying phase distribution is to be extracted, a suitable technique able to get four shifted interferograms at the same time is therefore needed. Some approaches to perform this task have been already demonstrated. One of them is a spatial phase stepping method with a holographic element, used for transient deformation measurements with electronic speckle pattern interferometry (ESPI) [5-6

5. B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, “Transient Deformation Measurement with Electronic Speckle Pattern interferometry by Use of a Holographic Optical Element for Spatial Phase Stepping,” Appl. Opt. 38, 5944–5947 (1999). [CrossRef]

]. A screen filter with an array of polarizing modulators under circular polarization has been also shown for optical testing applications [7

7. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44, 6861–6868 (2005). [CrossRef] [PubMed]

], but these polarizers have to be so small that its size is as tiny as a camera pixel size; the authors have named it as micropolarizer. Other similar case that uses micropolarizers is described in the patents from the references 8

8. James E Miller et al. “Methods and Apparatus for Splitting, Imaging, and Measuring Wavefronts in Interferometry,” US Pat. 6552808B2 (2002).

and 9

9. James E Miller et al. “Methods and Apparatus for Splitting, Imaging, and Measuring Wavefronts in Interferometry,” US Pat. 20030053071A1 (2003).

. In both cases, it is necessary the high precision technology and a reference arm interference. However, although these methods are versatile, they need rather special components.

In this communication, a simplified one-shot phase-shifting experimental method using grating interferometry is proposed and shown. It is based on a phase grating placed as the pupil of a 4f Fourier optical system.

With two windows in the object plane with a proper spacing in between, so that superposition of diffraction orders with no shear are achieved in the image plane [10

10. C. Meneses-Fabian, G. Rodríguez-Zurita, and V. Arrizón, “Common-path phase-shifting interferometer with binary grating,” Opt. Commun. 264, 13–17 (2006). [CrossRef]

]. Such a system performs as a common-path interferometer when the optical phase associated to one window can be taken as reference and the other one as the signal, thus having great mechanical stability [11

11. C. Meneses-Fabian, G. Rodríguez-Zurita, and V. Arrizón, “Optical Tomography of Transparent Objects with Phase-Shifting Interferometry and Stepping Wise Shifted Ronchi Ruling,” J. Opt. Soc. Am. A 23, 298–305 (2006). [CrossRef]

]. With the addition of wave plates for each window and linear polarizing filters for each diffraction order of interest, one-shot four interferograms 90° phase apart can be proven feasible. Other possible variants are also briefly discussed.

Two windows placed in the object plane of an imaging optical system were early employed in all-optical character recognition applications, where a space modulated Young interference fringe pattern is to be expected in the frequency plane when the fields within each of the windows in the object plane have the same relative distribution. An inspection after appropriate average of the resulting fringes of two similar or dissimilar fields has been reported to give useful information [12

12. H. Weinberger and U. Almi, “Interference Method for Pattern Comparison,” Appl. Opt. 10, 2482–2487 (1971). [CrossRef] [PubMed]

]. Placing a grating with its fundamental spatial frequency matching that of the Young interference pattern would form also a moiré pattern, as is the case in an interferometer reported for testing planarity [13

13. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, and R. Spolaczyk, “Semiconductor Wafer and Technical Flat Planeness Testing Interferometer,” Appl. Opt. 25, 1117–1121 (1986). [CrossRef] [PubMed]

]. In this interferometer, shifting of such moiré fringes can be achieved by displacements of the grating and can be used for tuning the reference phase [13

13. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, and R. Spolaczyk, “Semiconductor Wafer and Technical Flat Planeness Testing Interferometer,” Appl. Opt. 25, 1117–1121 (1986). [CrossRef] [PubMed]

]. Two windows are also used in the optical joint transform correlator [14

14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), pp. 243–246.

]. In this case, however, a hologram is recorded with the two characters to be compared and placed in the Fourier plane of some lens for reconstruction. Another predecessor of grating interferometry with two windows is the system for image subtraction to perform subtraction of both photographs from a stereoscopic pair in order to generate contours of equal elevation to obtain topography maps, as described in ref.15

15. P. W. Remijan, Processing Stereo Photographs by Optical Subtraction, PhD Thesis, University of Rochester (1978).

and references therein. The conveniences of using phase gratings were also remarked there. Polarizing filters in the Fourier plane to extract 180° phase images contained in grating diffraction orders for image subtraction are also described in [16

16. S. R. Dashiell and A. W. Lohmann, “Image Subtraction by Polarization-Shifted Periodic Carrier,” Opt. Commun. 8, 100–102 (1973). [CrossRef]

]. Probably due to the applications they were aimed to, none of these systems have incorporated the benefits of phase shifting. To our knowledge, the first attempts to explicitly suggest such a possibility can be drawn to ref. 10

10. C. Meneses-Fabian, G. Rodríguez-Zurita, and V. Arrizón, “Common-path phase-shifting interferometer with binary grating,” Opt. Commun. 264, 13–17 (2006). [CrossRef]

, 11

11. C. Meneses-Fabian, G. Rodríguez-Zurita, and V. Arrizón, “Optical Tomography of Transparent Objects with Phase-Shifting Interferometry and Stepping Wise Shifted Ronchi Ruling,” J. Opt. Soc. Am. A 23, 298–305 (2006). [CrossRef]

, 17

17. V. Arrizón and D. Sánchez-De-La-Llave, “Common-Path Interferometry with One-Dimensional Periodic Filters,” Opt. Lett. 29, 141–143 (2004). [CrossRef] [PubMed]

.

In the following sections, a double-window grating-interferometer featuring retardation-wave plates under linearly polarized light is first analyzed. Departures from the exact retardation values are considered in order to introduce an experimental set-up. Finally, some experimental results using it are shown.

2. Basic considerations

The Fig. 1 shows the arrangement of an ideal one-shot phase-shifting grating interferometer incorporating modulation of polarization. A combination of a quarter-wave plate Q and a linear polarizing filter P generates linearly polarized light at an appropriate azimuth angle (45°) entering the interferometer. Two quarter-wave plates (QL and QR) with their fast axes setted each other orthogonally are placed in front of the two windows of the common-path interferometer so as to generate left and right circularly polarized light as the corresponding beam leaves each window. A phase grating is placed at the system’s Fourier plane as the pupil. In the image plane, superimposition of diffraction orders, causing replicated images to interfere.

Fig. 1. One-shot phase-shifting grating interferometer with modulation of polarization. A, B: windows. Side view: ψi: polarizing angles (with i=1,2,3,4 to obtain phase-shifts ξi=0°, 90°, 180°, 270° respectively). Involved interference order superpositions are indicated from -2 to +2.

2.1. Interference patterns with polarizing filters and retarding plate.

When using birefringent plates which do not perform exactly as quarter-wave plates for the wavelength employed, the polarization angles of the linear polarizing filters to obtain 90° phase-shifts must change. To calculate the phase shifts induced in a more general polarization states by linear polarizers, consider two fields whose Jones vectors are described respectively by

JL(x,y)=12(1eiα)JR(x,y)=12(1eiα)eiϕ(x,y) .

These vectors represent the polarization states of two beams emerging from a retarding plate with phase retardation ±α′. Each beam enters the plate with linear polarization at ±45° with respect to the plate fast axis. Due to their orientations, the beams rotate in opposite directions, thereby the indices L and R are used. The beam with subscript R is supposed to carry a phase distribution ϕ(x, y). When each field is observed through a linear polarizing filter whose transmission axis is at an angle ψ, the new polarization states are

J=JψlJLJ=JψlJR
(1)

where the spatial dependence has been dropped for simplicity and with the linear polarizing transmission matrix given by

Jψl=(cos2ψsinψcosψsinψcosψsin2ψ).
(2)

If the new fields are left to interfere, the resulting irradiance can be written as

JT2=J+J2
=1+cosϕ·cos2ψ+sin2ψ·cosα+sin2ψ·cos[αϕ(x,y)]
+sin2ψ·cos[2αϕ(x,y)]
=1+sin2ψ·cosα+A(ψ,α)cos[ξ(ψ,α)ϕ(x,y)]
(3)

where

ξ(ψ,α)=tan1[sin2ψ·sinα+sin2ψ·sin2αcos2ψ+sin2ψ·cos2α+sin2ψ·cosα]
(4)

and

A(ψ,α)=(cos4ψ+sin4ψ+(1+12cos2α)·sin22ψ+2sin2ψ·cosα)12.
(5)

Plots of ξ(ψ, α′) and A(ψ, α′) A are shown in Fig. 2 for several values of α′.

Fig. 2. (a) Phase shift ξ(ψ, α′) as a function of ψ for several values of α′. Insert: α′ for ideal retardation and experimental retardation. (b) Amplitude A(ψ,α′) as a function of ψ for several values of α′.

For the special ideal case of α=π2 (quarter-wave plate), it is readily found that

ξ(ψ,π2)=2ψ,A2(ψ,π2)=1,
(6)

which can be verified in Fig. 2. Then, the irradiance of Eq. (3) reduces to

JT2=1+cos[2ψϕ(x,y)],
(7)

which is a well known expression used already for phase shifting [9

9. James E Miller et al. “Methods and Apparatus for Splitting, Imaging, and Measuring Wavefronts in Interferometry,” US Pat. 20030053071A1 (2003).

]. In fact, by denoting four irradiances at four different angles as

Ji2=1+cos[2ψiϕ(x,y)],
(8)

with i=1…4, the relative phase can be calculated as [5

5. B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, “Transient Deformation Measurement with Electronic Speckle Pattern interferometry by Use of a Holographic Optical Element for Spatial Phase Stepping,” Appl. Opt. 38, 5944–5947 (1999). [CrossRef]

]

tanϕ=J12J32J22J42
(9)

where ‖J⃗ 12, ‖J⃗ 22, ‖J⃗ 32 and ‖J⃗ 42 are the intensity measurements with the values of ψ given by ψ 1=0, ψ 2=π/4, ψ 3=π/2, ψ 4=3π/4. When the phase retardation is different from π/2, Eq.3 to Eq.5 must be used. In those cases, the value of ψ can be determined from Eq.4 looking for ξ=0,π/2,π,3π/2. For ξ=0 it is easy to see from the Fig. 2(a) that ψ 1=0. Cases ξ=π/2,3π/2 lead to the condition

cos2ψ+sin2ψ·cos2α+sin2ψ·cosα=0,
(10)

which can be transformed in the following second degree equation for cos2 ψ

{(1cos2α)2+4cos2α}cos4ψ+{2(1cos2α)·cos2α4cos2α}cos2ψ+cos22α=0
(11)

with two solutions ψa,b given by

cos2(ψ)=
{2cos2α(1cos2α)cos2α}±{(1cos2α)cos2α2cos2α}2{(1cos2α)2+4cos2α}cos22α(1cos2α)2+4cos2α
(12.a)

which enables the choosing of

ψ2=ψa,ψ4=ψb+π,
(12.b)

where ψa and ψb are two meaningful different solutions arising from Eq. (12.a) and |ψb|<ψa. For the case ξ=π, it is found that the following condition must be fulfilled

sin2ψ·sinα′+sin2 ψ·sin2α′=0

which leads to

ψ3=nπ+arctan(sec(α)).
(12.c)

The value with n=1 can be chosen.

2.2. Interferometer with phase-grating.

The phase object under test is placed in one of the two windows in the object plane. Thus, the Jones vector in the object plane can be written as

O(x,y)=JL(x+12x0,y)·w(x+12x0,y)+JR(x12x0,y)·w(x12x0,y)
(13)

A grating of spatial period d=λf/X 0 is placed in the Fourier plane. Then, the corresponding transmittance is given by

G(μ,v)=GP(μ)*n=δ(μnX0)
(14)

with δ(μ) denoting the Dirac delta function and * the convolution operation. μ=u/λf and ν=v/λf are the frequency coordinates scaled to the relevant wavelength λ and the focal length f. The actual frequency coordinates are thus u and v. The grating’s profile for a period is given by GP(μ). The point spread function of a system with such a pupil can be obtained with the inverse Fourier transform of G(μ,ν), which results in

G~(x,y)=X0·n=G~P(n·X0)·δ(xn·X0,y).
(15)
Fig. 3. Relative positions of diffraction orders from windows A and B. The case of order +1 out of phase by π with respect to the others is shown. Two interferograms with inverse contrast result. X 0=x 0.

The system’s image plane is the convolution of O⃗(x, y) and (x, y), which is basically the replication of each window at distances X 0. The replications of each window are displaced by ±12x0 with respect to the origin, so they are superimposed if Nx 0=X 0. Figure 3 shows the case of N=1. The case of similar amplitudes for each spectral diffraction orders (from order -2 to order +2) is presented, a similar situation can be found in phase gratings [15

15. P. W. Remijan, Processing Stereo Photographs by Optical Subtraction, PhD Thesis, University of Rochester (1978).

]. Also, in the Fig. 3 one of the four orders shown, the +1, is depicted π out of phase with respect to the others. This effect can be obtained with phase gratings because odd order amplitudes are proportional to Bessel functions of odd orders, which have in turn odd parity, whereas even order amplitudes follow Bessel functions of even parity. Thus, diffraction orders as described are expected to be obtained with phase gratings due to their particular distribution (x, y).

3. Experimental set-up

A green laser light with λ=532 nm was employed to illuminate the system of Fig. 1. Figure 4 shows four interferograms from the system as a preliminary observation. They were obtained before placing retardation plates and polarizing filters. The patterns show the relative phases of the diffraction orders as discussed in Sec.2.2. Two rightmost interferograms have the same fringe contrast. Such contrast appears to be the complementary one of the remaining leftmost pair of interferograms. The Fourier spectrum of the grating behaves as the one of Fig. 3 (with a π phase difference between even and odd single orders). In fact, the contrast of the remaining patterns (not shown) follows changes that can be explained in agreement with the parity properties of the Bessel functions.

Fig. 4. Image plane from a system as depicted in Fig. 1. Neither plate retardation plates nor polarizing filters were used. Two opposite fringe contrast can be seen. Compare with Fig. 3.

To complete the set-up, off-the-shelf retarding plates designed as quarter-wave plates for λa=514.5 nm were used in the windows. Thus, a nominal retardation of

α=π2λaλ=1.519rad
(16)

is calculated. Introducing this value in the solutions described in sec.2.1, the results ψ 1=0 ψ 2=46.577° ψ 3=92.989° ψ 4=136.42° were obtained. In the experimental setup, these values must be changed to ψ1, ψ2, ψ3, and ψ4 due to the additional 180° phase difference, as described later on.

The phase grating employed (110 ln/mm) generates five diffraction orders of similar but not equal average irradiance (Fig. 4), as expected. Because the respective irradiances do vary due both to the diffraction order amplitude and the variations of pattern amplitude (Eq. (5)), each interferogram was subject to a normalization process to each maximum of its irradiance before using Eq. (9). The separation between window centers was of x 0≈10mm. Other parameters used were focal lengths of f≈160mm, aw=6mm and bw=10mm.

4. Experimental results

4.1. Static distributions

Two test objects were prepared evaporating magnesium fluoride (MgF2) on a glass substrate: a disk or phase dot and a phase step. When each object was placed separately in one of the windows using the interferometer of Fig. 1 with polarizers P1, P2, P3 and P4, using the previously calculated angles ψ1, ψ2, ψ3, and ψ4, the interferograms of Fig. 5 were obtained. For each object, the four interferograms are shown together with the unwrapped phase calculated with Eq.9 at the right (in 256 grey levels). Examples, some typical raster lines for each unwrapped phase are shown in Fig. 6 (in arbitrary phase units).

Fig. 5. Upper row: phase dot. Four 90° phase-shifted interferograms and unwrapped phase. Lower row: phase step. Four 90° phase-shifted interferograms and unwrapped phase.
Fig. 6. Unwrapped calculated phases along typical raster lines of each object of Fig. 5. Scale factor: 0.405 rad.

These observations suggest a further simplification for the polarizing filters array for the case of the phase shift of π from a phase grating. It consists of using only one enough big filter to cover the two central interferograms at the same angle of ψ 2′ (ψ 2′=ψ 2=ψ 3) instead of two separate filters at ψ 2, ψ 3 respectively. Thus, only three linear polarizing filters have to be used. The transmission axes of the filters P1 and P4 can be both horizontally oriented (ψ 1′=ψ 1=ψ 4), see Fig. 7.

Fig. 7. Simplify polarizer filters array

This only alters the order of the shifted interferograms. Because the departure from an ideal case for the experimental conditions is relatively small, this configuration could be experimentally tested with the described set-up resulting in qualitative good results.

Fig. 8. Typical four 90° phase-shifted interferograms from oil flowing (Media 1)

4.2. Moving distributions

Immersion oil was applied to a glass microscope slide and allowed to flow under the effect of gravity by tilting the slide slightly. The slide was put in front of one of the object windows of the system of Fig. 1. Figure 8 shows a typical sequence of four shifted interferograms from the oil flow in arbitrary units. Figure 9 shows the resulting unwrapped phase evolution of another oil flow.

Fig. 9. typical unwrapped phase from interferograms of oil flowing (Media 2).

5. Final remarks

Acknowledgments

One of the authors (NITA) expresses sincere appreciation to CONACyT for grant 165912. Partial support from VIEP-BUAP (grant 33/EXC/06-I) is acknowledged. We thank V. Arrizon for lending a phase grating.

References and links

1.

J. E. Greivenkamp and J. H. Bruning, “C.14 Phase shifting interferometry,” in Optical Shop Testing, D. Malacara, ed. (John Wiley & Sons, 1992), pp. 501–598.

2.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1990), pp. 271–359.

3.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1998), 26 pp. 349–393.

4.

M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 44, 4439–4442 (1985). [CrossRef]

5.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, “Transient Deformation Measurement with Electronic Speckle Pattern interferometry by Use of a Holographic Optical Element for Spatial Phase Stepping,” Appl. Opt. 38, 5944–5947 (1999). [CrossRef]

6.

B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, “Spatial Phase-Stepped Interferometry Using a Holographic Optical Element,” Opt. Eng. 38, 2069–2074 (1999). [CrossRef]

7.

M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, “Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer,” Appl. Opt. 44, 6861–6868 (2005). [CrossRef] [PubMed]

8.

James E Miller et al. “Methods and Apparatus for Splitting, Imaging, and Measuring Wavefronts in Interferometry,” US Pat. 6552808B2 (2002).

9.

James E Miller et al. “Methods and Apparatus for Splitting, Imaging, and Measuring Wavefronts in Interferometry,” US Pat. 20030053071A1 (2003).

10.

C. Meneses-Fabian, G. Rodríguez-Zurita, and V. Arrizón, “Common-path phase-shifting interferometer with binary grating,” Opt. Commun. 264, 13–17 (2006). [CrossRef]

11.

C. Meneses-Fabian, G. Rodríguez-Zurita, and V. Arrizón, “Optical Tomography of Transparent Objects with Phase-Shifting Interferometry and Stepping Wise Shifted Ronchi Ruling,” J. Opt. Soc. Am. A 23, 298–305 (2006). [CrossRef]

12.

H. Weinberger and U. Almi, “Interference Method for Pattern Comparison,” Appl. Opt. 10, 2482–2487 (1971). [CrossRef] [PubMed]

13.

J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, and R. Spolaczyk, “Semiconductor Wafer and Technical Flat Planeness Testing Interferometer,” Appl. Opt. 25, 1117–1121 (1986). [CrossRef] [PubMed]

14.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), pp. 243–246.

15.

P. W. Remijan, Processing Stereo Photographs by Optical Subtraction, PhD Thesis, University of Rochester (1978).

16.

S. R. Dashiell and A. W. Lohmann, “Image Subtraction by Polarization-Shifted Periodic Carrier,” Opt. Commun. 8, 100–102 (1973). [CrossRef]

17.

V. Arrizón and D. Sánchez-De-La-Llave, “Common-Path Interferometry with One-Dimensional Periodic Filters,” Opt. Lett. 29, 141–143 (2004). [CrossRef] [PubMed]

OCIS Codes
(050.5080) Diffraction and gratings : Phase shift
(070.6110) Fourier optics and signal processing : Spatial filtering
(120.2650) Instrumentation, measurement, and metrology : Fringe analysis
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(260.5430) Physical optics : Polarization

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: February 25, 2008
Revised Manuscript: April 29, 2008
Manuscript Accepted: April 29, 2008
Published: May 15, 2008

Citation
Gustavo Rodriguez-Zurita, Cruz Meneses-Fabian, Noel-Ivan Toto-Arellano, José F. Vázquez-Castillo, and Carlos Robledo-Sánchez, "One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms," Opt. Express 16, 7806-7817 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-11-7806


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References

  1. J. E. Greivenkamp and J. H. Bruning, "C.14 Phase shifting interferometry," in Optical Shop Testing, D. Malacara, ed., (John Wiley and Sons, 1992), pp. 501-598.
  2. J. Schwider, "Advanced evaluation techniques in interferometry," in Progress in Optics, E. Wolf, ed., (North-Holland, 1990), pp. 271-359.
  3. K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, E. Wolf, ed., (North-Holland, 1998), 26 pp. 349-393.
  4. M. P. Kothiyal and C. Delisle, "Shearing interferometer for phase shifting interferometry with polarization phase shifter," Appl. Opt. 44, 4439-4442 (1985). [CrossRef]
  5. B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Transient deformation measurement with electronic speckle pattern interferometry by use of a holographic optical element for spatial phase stepping," Appl. Opt. 38, 5944-5947 (1999). [CrossRef]
  6. B. Barrientos-García, A. J. Moore, C. Pérez-López, L. Wang, and T. Tschudi, "Spatial Phase-stepped Interferometry using a holographic optical element," Opt. Eng. 38, 2069-2074 (1999). [CrossRef]
  7. M. Novak, J. Millerd, N. Brock, M. North-Morris, J. Hayes, and J. Wyant, "Analysis of a micropolarizer array-based simultaneous phase-shifting interferometer," Appl. Opt. 44, 6861-6868 (2005). [CrossRef] [PubMed]
  8. J. E Miller,  et al., "Methods and apparatus for splitting, imaging, and measuring wavefronts in Interferometry," U. S. Patent 6552808B2 (2002).
  9. J. E. Miller,  et al., "Methods and apparatus for splitting, imaging, and measuring wavefronts in Interferometry," U. S. Patent 20030053071A1 (2003).
  10. C. Meneses-Fabian, G. Rodríguez-Zurita, and V. Arrizón, "Common-path phase-shifting interferometer with binary grating," Opt. Commun. 264, 13-17 (2006). [CrossRef]
  11. C. Meneses-Fabian, G. Rodríguez-Zurita, and V. Arrizón, "Optical Tomography of transparent objects with Phase-Shifting Interferometry and Stepping Wise Shifted Ronchi Ruling," J. Opt. Soc. Am. A 23, 298-305 (2006). [CrossRef]
  12. H. Weinberger and U. Almi, "Interference method for pattern comparison," Appl. Opt. 10, 2482-2487 (1971). [CrossRef] [PubMed]
  13. J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, and R. Spolaczyk, "Semiconductor Wafer and Technical Flat Planeness Testing Interferometer," Appl. Opt. 25, 1117-1121 (1986). [CrossRef] [PubMed]
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), pp. 243-246.
  15. P. W. Remijan, Processing stereo photographs by Optical Subtraction, PhD Thesis, University of Rochester (1978).
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