## Phase-shifting interferometry based on the lateral displacement of the light source |

Optics Express, Vol. 21, Issue 14, pp. 17228-17233 (2013)

http://dx.doi.org/10.1364/OE.21.017228

Acrobat PDF (2753 KB)

### Abstract

A simple and inexpensive optical setup to phase-shifting interferometry is proposed. This optical setup is based on the Twyman-Green Interferometer where the phase shift is induced by the lateral displacement of the point laser source. A theoretical explanation of the induced phase by this alternative method is given. The experimental results are consistent with the theoretical expectations. Both, the phase shift and the wrapped phase are recovered by a generalized phase-shifting algorithm from two or more interferograms with arbitrary and unknown phase shift. The experimental and theoretical results show the feasibility of this unused phase-shifting technique.

© 2013 OSA

## 1. Introduction

1. D. Malacara, ed., *Optical Shop Testing*, 3rd ed. (John Wiley & Sons, 2007) [CrossRef] .

2. D. Malacara, M. Servin, and Z. Malacara, *Interferogram Analysis for Optical Testing*, 2nd ed. (Taylor & Francis Group, 2005) [CrossRef] .

3. G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A **8**, 822–827 (1991) [CrossRef] .

4. A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Optics and Lasers in Engineering **43**, 475–490 (2005) [CrossRef] .

5. R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Optics and Lasers in Engineering **51**, 626–632 (2013) [CrossRef] .

6. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. **13**, 2693–2703 (1974) [CrossRef] [PubMed] .

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

8. Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. **24**, 3049–3052 (1985) [CrossRef] [PubMed] .

10. G. Rodriguez-Zurita, C. Meneses-Fabian, N.-I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express **16**, 7806–7817 (2008) [CrossRef] [PubMed] .

11. M. Atlan, M. Gross, and E. Absil, “Accurate phase-shifting digital interferometry,” Opt. Lett. **32**, 1456–1458 (2007) [CrossRef] [PubMed] .

12. J. Min, B. Yao, P. Gao, R. Guo, J. Zheng, and T. Ye, “Parallel phase-shifting interferometry based on michelson-like architecture,” Appl. Opt. **49**, 6612–6616 (2010) [CrossRef] [PubMed] .

13. T. Pfeifer, R. Tutsch, J. Evertz, and G. Weres, “Generalized aspects of multiple-wavelength techniques in optical metrology,” {CIRP} Annals - Manufacturing Technology **44**, 493–496 (1995) [CrossRef] .

14. L. Bruno and A. Poggialini, “Phase-shifting interferometry by an open-loop voltage controlled laser diode,” Opt. Commun. **290**, 118–125 (2013) [CrossRef] .

15. M. Vannoni, A. Sordini, and G. Molesini, “He-ne laser wavelength-shifting interferometry,” Opt. Commun. **283**, 5169–5172 (2010) [CrossRef] .

16. C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett. **36**, 2417–2419 (2011) [CrossRef] [PubMed] .

17. J. Vargas, J. A. Quiroga, A. Álvarez Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express **19**, 584–596 (2011) [CrossRef] [PubMed] .

## 2. Principles of the phase-shifting by lateral source displacement

*d⃗*=

*dd̂*, where

*d̂*is a unit vector perpendicular to the optical axis, as it is shown in Fig. 1(b). The spherical wavefront which reaches the input plane of the lens

*L*is given by where

*p⃗*= (

*x*,

*y*) is a spatial variable with Cartesian components

*x*and

*y*,

*r*= ||

*r⃗*|| = ||

*f⃗*+

*p⃗*−

*d⃗*|| (where ||·|| denoting the Euclidean norm) is the distance between the point source at the point

*p⃗*in the input plane of the lens

*L*,

*A*

_{0}is a real value amplitude,

*k*= 2

*π*/

*λ*is the wavenumber,

*λ*is the wavelength, and

*i*is the imaginary unit (

*i*

^{2}= −1). We consider the paraxial approximation of

*A*(

*p⃗*,

*d⃗*) as: where

*B*

_{0}= (

*A*

_{0}/

*f*)exp[

*ikf*] is a complex amplitude and

*p*= ||

*p⃗*||. Due to the propagation through the lens

*L*, the amplitude

*A*(

*p⃗*,

*d⃗*) left its quadratic phase term exp[

*ikp*

^{2}/(2

*f*)] out. Thus, at the output plane of

*L*we have the tilted plane wavefront: which has a direction of propagation given by with respect to the optical axis as it is shown in Fig. 1(b).

*B*(

*p⃗*,

*d⃗*) as shown in Fig. 1(c) or, equivalently, Fig. 1(d). It is not loss of generality to suppose that the collimated beam

*B*(

*p⃗*,

*d⃗*) in the output plane of lens

*L*is at the plane of the mirror

*M*because the additional constant phase due to the propagation from

_{r}*L*to

*M*can be dropped. Thus, for the point

_{r}*p⃗*in the observation plane

*OP*, we have the intensity

*I*(

*p⃗*,

*d⃗*) due to the interference of the beams

*B*(

*p⃗*

_{2},

*d⃗*) and

*B̃*(

*p⃗*

_{2},

*d⃗*) as with | · | denoting the module, as shown in Fig. 1(d), and where the beam

*B̃*(

*p⃗*

_{2},

*d⃗*) is the beam

*B*(

*p⃗*

_{1},

*d⃗*) reflected from the test mirror, namely where the additional phase terms are associated with the length path

*ρ*= 2

*D*[1+(

*d / f*)

^{2}]

^{1/2}and the test mirror’s aberrations

*ϕ*(

*p⃗*

_{0}). Since

*p⃗*

_{2}=

*p⃗*

_{1}+

*σ⃗*, where

*σ⃗*=

*σd̂*with

*σ*= −2

*Dd / f*is a translation, we can rewrite the beam

*B*(

*p⃗*

_{2},

*d⃗*) as Considering the Eqs. (5), (6) and

*p⃗*

_{0}=

*p⃗*−

*τ⃗*, and going through some algebraic operations, we can rewrite the Eq. (4) as where

*δ*(

*d*) = −

*kρ*−

*kσd / f*is the phase shift, and

*τ⃗*is an linear image translation given by Substituting the variables

*ρ*and

*σ*in

*δ*(

*d*), and since

*d*

^{2}/

*f*

^{2}≪ 1, the phase shift

*δ*(

*d*) can be approximated to where the constant offset phase −2

*Dk*was omitted.

*d⃗*changes, two effects appear simultaneously in the interference intensity

*I*(

*p⃗*,

*d⃗*). The first one is the quadratic phase shift

*δ*(

*d*), given by Eq. (9). The second one is the linear translation

*τ⃗*, given by Eq. (8), of the fringe-pattern. Both effects are depicted in Fig. 2.

*D*. For example, for a lateral source displacement of 2 mm, a lens with

*f*= 0.5 m, and a laser source with

*λ*= 633 nm; we can obtain a phase shift of 2

*π*rad if the distance

*D*is set to

*δf*

^{2}/(

*kd*

^{2}) = 3.96 cm. If more phase shift gain is required, a greater distance

*D*is need and vice versa.

## 3. Optical experiment

*λ*= 633 nm. The laser beam was expanded and filtered by a microscope objective and a pinhole, respectively. The pinhole was located at the focal point of a collimating lens with focal length of

*f*= 0.5 m.

*M*and test

_{r}*M*mirrors. The test mirror surface was deformed in order to obtain a distorted wavefront. The fringe pattern due to the interference of these two reflected beams was observed on a screen on the observation plane

_{t}*OP*. The fringe patterns was acquired by a gray-scale 8-bit CCD camera with a resolution of 768 × 1008 pixels.

*μ*m. We considered a lateral source displacement of

*d*= ±2 mm with steps of 100

*μ*m. We choose the relative distance

*D*= 3.96 and 7.91 cm between the mirrors in order to obtain a phase shift

*δ*(2 mm) = 2

*π*and 4

*π*rad, respectively. The observation plane is placed to the distance

*g*= 11 cm from the reference mirror

*M*. For each progressive displacement step, a phase shifted interferogram was recorded. Thus, for each value of

_{r}*D*, 41 interferograms were acquired. Of these interferograms, we show two adjacent interferograms in Fig. 3 as an example.

5. R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Optics and Lasers in Engineering **51**, 626–632 (2013) [CrossRef] .

*μ*m (with standard deviation of 11.42

*μ*m) for the image translation. These results are good considering that the displacement is induced manually.

5. R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Optics and Lasers in Engineering **51**, 626–632 (2013) [CrossRef] .

*τ*= 37.82 and 29.92

*μ*m for

*D*= 7.91 and 3.96 cm, respectively). In addition, a scaling of this translation is performed by the camera’s imaging system. In our particular case, the size of the interferograms was of 3 × 3.9 cm and the target’s size of 2.7 × 3.5 mm. Thus, the image translation is reduced to 3.4 and 2.7

*μ*m for

*D*= 7.91 and 3.96 cm, respectively. But, since the pixel size is 3.5

*μ*m, such translations are not observable. Moreover, for large translations, because the translation is a linear function of the displacement, the numerical correction is very simple and consists of a translation of all the pixels of the interferogram by a certain number of pixels.

## 4. Conclusion

## Acknowledgments

## References and links

1. | D. Malacara, ed., |

2. | D. Malacara, M. Servin, and Z. Malacara, |

3. | G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A |

4. | A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Optics and Lasers in Engineering |

5. | R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Optics and Lasers in Engineering |

6. | J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. |

7. | P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. |

8. | Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. |

9. | C. Tay, C. Quan, and H. Shang, “Shape identification using phase shifting interferometry and liquid-crystal phase modulator,” Optics & Laser Technology |

10. | G. Rodriguez-Zurita, C. Meneses-Fabian, N.-I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express |

11. | M. Atlan, M. Gross, and E. Absil, “Accurate phase-shifting digital interferometry,” Opt. Lett. |

12. | J. Min, B. Yao, P. Gao, R. Guo, J. Zheng, and T. Ye, “Parallel phase-shifting interferometry based on michelson-like architecture,” Appl. Opt. |

13. | T. Pfeifer, R. Tutsch, J. Evertz, and G. Weres, “Generalized aspects of multiple-wavelength techniques in optical metrology,” {CIRP} Annals - Manufacturing Technology |

14. | L. Bruno and A. Poggialini, “Phase-shifting interferometry by an open-loop voltage controlled laser diode,” Opt. Commun. |

15. | M. Vannoni, A. Sordini, and G. Molesini, “He-ne laser wavelength-shifting interferometry,” Opt. Commun. |

16. | C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett. |

17. | J. Vargas, J. A. Quiroga, A. Álvarez Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express |

18. | E. Hecht, |

**OCIS Codes**

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 7, 2013

Manuscript Accepted: June 11, 2013

Published: July 11, 2013

**Citation**

Carlos Robledo-Sanchez, Rigoberto Juarez-Salazar, Cruz Meneses-Fabian, Fermin Guerrero-Sánchez, L. M. Arévalo Aguilar, Gustavo Rodriguez-Zurita, and Viridiana Ixba-Santos, "Phase-shifting interferometry based on the lateral displacement of the light source," Opt. Express **21**, 17228-17233 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-17228

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### References

- D. Malacara, ed., Optical Shop Testing, 3rd ed. (John Wiley & Sons, 2007). [CrossRef]
- D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing, 2nd ed. (Taylor & Francis Group, 2005). [CrossRef]
- G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A8, 822–827 (1991). [CrossRef]
- A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Optics and Lasers in Engineering43, 475–490 (2005). [CrossRef]
- R. Juarez-Salazar, C. Robledo-Sanchez, C. Meneses-Fabian, F. Guerrero-Sanchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Optics and Lasers in Engineering51, 626–632 (2013). [CrossRef]
- J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt.13, 2693–2703 (1974). [CrossRef] [PubMed]
- P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt.26, 2504–2506 (1987). [CrossRef] [PubMed]
- Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt.24, 3049–3052 (1985). [CrossRef] [PubMed]
- C. Tay, C. Quan, and H. Shang, “Shape identification using phase shifting interferometry and liquid-crystal phase modulator,” Optics & Laser Technology30, 545–550 (1998).
- G. Rodriguez-Zurita, C. Meneses-Fabian, N.-I. Toto-Arellano, J. F. Vázquez-Castillo, and C. Robledo-Sánchez, “One-shot phase-shifting phase-grating interferometry with modulation of polarization: case of four interferograms,” Opt. Express16, 7806–7817 (2008). [CrossRef] [PubMed]
- M. Atlan, M. Gross, and E. Absil, “Accurate phase-shifting digital interferometry,” Opt. Lett.32, 1456–1458 (2007). [CrossRef] [PubMed]
- J. Min, B. Yao, P. Gao, R. Guo, J. Zheng, and T. Ye, “Parallel phase-shifting interferometry based on michelson-like architecture,” Appl. Opt.49, 6612–6616 (2010). [CrossRef] [PubMed]
- T. Pfeifer, R. Tutsch, J. Evertz, and G. Weres, “Generalized aspects of multiple-wavelength techniques in optical metrology,” {CIRP} Annals - Manufacturing Technology44, 493–496 (1995). [CrossRef]
- L. Bruno and A. Poggialini, “Phase-shifting interferometry by an open-loop voltage controlled laser diode,” Opt. Commun.290, 118–125 (2013). [CrossRef]
- M. Vannoni, A. Sordini, and G. Molesini, “He-ne laser wavelength-shifting interferometry,” Opt. Commun.283, 5169–5172 (2010). [CrossRef]
- C. Meneses-Fabian and U. Rivera-Ortega, “Phase-shifting interferometry by wave amplitude modulation,” Opt. Lett.36, 2417–2419 (2011). [CrossRef] [PubMed]
- J. Vargas, J. A. Quiroga, A. Álvarez Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express19, 584–596 (2011). [CrossRef] [PubMed]
- E. Hecht, Optics, 4th ed. (Addison Wesley, 2002).

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