## One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms

Optics Express, Vol. 16, Issue 11, pp. 7806-7817 (2008)

http://dx.doi.org/10.1364/OE.16.007806

Acrobat PDF (1564 KB)

### 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.

*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]

*f*Fourier optical system.

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]

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]

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

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]

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]

## 2. Basic considerations

_{L}and Q

_{R}) 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.

_{i},

*i*=1…4, results after placing a linear polarizer to each one of the interference patterns generated on each diffracting orders in the exit plane (P

_{1}, P

_{2}, P

_{3}, P

_{4}). Each polarizing filter transmission axis is adjusted at different angle ψ

*, so we obtain the desired phase shift*

_{i}*ξ*for each pair of orders. For a 90° phase-shift ξ

_{i}_{i}between interfering fields, the polarization angles

*ψ*in each diffraction order must be 0°, 45°, 90° and 135° for the case of ideal quarter-wave retardation (

_{i}*α*′=90°). In the next sections, some particularities arising from the optical components available for our set-up are discussed. Among these, the calculation of

*ψ*for the case of a non exact quarter-wave retardation is considered.

_{i}### 2.1. Interference patterns with polarizing filters and retarding plate.

*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

*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]

*J⃗*

_{1}‖

^{2}, ‖

*J⃗*

_{2}‖

^{2}, ‖

*J⃗*

_{3}‖

^{2}and ‖

*J⃗*

_{4}‖

^{2}are the intensity measurements with the values of

*ψ*given by

*ψ*

_{1}=0,

*ψ*

_{2}=

*π*/4,

*ψ*

_{3}=

*π*/2,

*ψ*

_{4}=3

*π*/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*,3

*π*/2. For

*ξ*=0 it is easy to see from the Fig. 2(a) that

*ψ*

_{1}=0. Cases

*ξ*=

*π*/2,3

*π*/2 lead to the condition

^{2}

*ψ*

*ψ*given by

_{a,b}*ψ*and

_{a}*ψ*are two meaningful different solutions arising from Eq. (12.a) and |

_{b}*ψ*|<

_{b}*ψ*. For the case

_{a}*ξ*=

*π*, it is found that the following condition must be fulfilled

*ψ*·sin

*α*′+sin

^{2}

*ψ*·sin2

*α*′=0

*n*=1 can be chosen.

### 2.2. Interferometer with phase-grating.

*d*=

*λf*/

*X*

_{0}is placed in the Fourier plane. Then, the corresponding transmittance is given by

*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

*G*(

_{P}*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

*O⃗*(

*x*,

*y*) and

*G̃*(

*x*,

*y*), which is basically the replication of each window at distances

*X*

_{0}. The replications of each window are displaced by

*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]. 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

*G̃*(

*x*,

*y*).

*π*/2, the previous sections justify the use of an interferometer consisting of two birefringent windows separated by

*x*

_{0}in the object plane and a phase grating in the Fourier plane. The interference of the fields of each window is obtained in the image plane when superposing itself the appropriate orders of diffraction. Linear polarizers in front of each order at the proper angle

*ψ*would give the values of phase shifts according with Eq. (3). In order to superpose orders +1 +2, 0+1, -1 0 and -2 -1 it is necessary to fulfill the condition

*d*=

*λf*/

*x*

_{0}.

## 3. Experimental set-up

*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.

*λ*=514.5

_{a}*nm*were used in the windows. Thus, a nominal retardation of

*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

*ψ*′

_{1},

*ψ*′

_{2},

*ψ*′

_{3}, and

*ψ*′

_{4}due to the additional 180° phase difference, as described later on.

*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}≈10

*mm*. Other parameters used were focal lengths of

*f*≈160

*mm*,

*a*=6

_{w}*mm*and

*b*=10

_{w}*mm*.

## 4. Experimental results

*ψ*′

_{1}=

*ψ*

_{1}and

*ψ*′

_{2}=

*ψ*

_{2}) to achieve mutual phase differences of Δ

*ξ*=

*ξ*(

*ψ*′

_{2},

*α*′)-

*ξ*(

*ψ*′

_{1},

*α*′)=

*π*/2. The remaining two axes were adjusted taking into account the additional phase shift of 180°. Using the same angles

*ψ*′

_{3}=

*ψ*

_{1}and

*ψ*′

_{4}=

*ψ*

_{2}, the two required phase differences Δ

*ξ*=

*ξ*(

*ψ*′

_{3},

*α*′)+

*π*-

*ξ*(

*ψ*′

_{1},

*α*′)=

*π*and Δ

*ξ*=

*ξ*(

*ψ*′

_{4},

*α*′)+

*π*-

*ξ*(

*ψ*′

_{1},

*α*′)=3

*π*/2 can be obtained.

### 4.1. Static distributions

_{2}) 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 P

_{1}, P

_{2}, P

_{3}and P

_{4}, using the previously calculated angles

*ψ*′

_{1},

*ψ*′

_{2},

*ψ*′

_{3}, and

*ψ*′

_{4}, 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).

*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 P

_{1}and P

_{4}can be both horizontally oriented (

*ψ*

_{1}′=

*ψ*

_{1}=

*ψ*

_{4}), see Fig. 7.

### 4.2. Moving distributions

## 5. Final remarks

## Acknowledgments

## References and links

1. | J. E. Greivenkamp and J. H. Bruning, “C.14 Phase shifting interferometry,” in |

2. | J. Schwider, “Advanced evaluation techniques in interferometry,” in |

3. | K. Creath, “Phase-measurement interferometry techniques,” in |

4. | M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. |

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. |

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. |

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. |

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. |

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 |

12. | H. Weinberger and U. Almi, “Interference Method for Pattern Comparison,” Appl. Opt. |

13. | J. Schwider, R. Burow, K.-E. Elssner, J. Grzanna, and R. Spolaczyk, “Semiconductor Wafer and Technical Flat Planeness Testing Interferometer,” Appl. Opt. |

14. | J. W. Goodman, |

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. |

17. | V. Arrizón and D. Sánchez-De-La-Llave, “Common-Path Interferometry with One-Dimensional Periodic Filters,” Opt. Lett. |

**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

- 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.
- J. Schwider, "Advanced evaluation techniques in interferometry," in Progress in Optics, E. Wolf, ed., (North-Holland, 1990), pp. 271-359.
- K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, E. Wolf, ed., (North-Holland, 1998), 26 pp. 349-393.
- M. P. Kothiyal and C. Delisle, "Shearing interferometer for phase shifting interferometry with polarization phase shifter," Appl. Opt. 44, 4439-4442 (1985). [CrossRef]
- 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]
- 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]
- 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]
- J. E Miller, et al., "Methods and apparatus for splitting, imaging, and measuring wavefronts in Interferometry," U. S. Patent 6552808B2 (2002).
- J. E. Miller, et al., "Methods and apparatus for splitting, imaging, and measuring wavefronts in Interferometry," U. S. Patent 20030053071A1 (2003).
- C. Meneses-Fabian, G. Rodríguez-Zurita, and V. Arrizón, "Common-path phase-shifting interferometer with binary grating," Opt. Commun. 264, 13-17 (2006). [CrossRef]
- 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]
- H. Weinberger and U. Almi, "Interference method for pattern comparison," Appl. Opt. 10, 2482-2487 (1971). [CrossRef] [PubMed]
- 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]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), pp. 243-246.
- P. W. Remijan, Processing stereo photographs by Optical Subtraction, PhD Thesis, University of Rochester (1978).
- S. R. Dashiell and A. W. Lohmann, "Image subtraction by Polarization-Shifted Periodic Carrier," Opt. Commun. 8, 100-102 (1973). [CrossRef]
- V. Arrizón and D. Sánchez-De-La-Llave, "Common-path Interferometry with one-dimensional periodic filters," Opt. Lett. 29, 141-143 (2004). [CrossRef] [PubMed]

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