## A single-shot common-path phase-stepping radial shearing interferometer for wavefront measurements |

Optics Express, Vol. 19, Issue 5, pp. 4703-4713 (2011)

http://dx.doi.org/10.1364/OE.19.004703

Acrobat PDF (1036 KB)

### Abstract

A single-shot common-path phase-stepping radial shearing interferometer (RSI) is proposed for wavefront measurements. In the proposed RSI, three quarter-wave plates are used as phase shifters to produce four spatially separated phase-stepping fringe patterns that are recorded simultaneously by a single CCD camera. The proposed RSI can measure the wavefront under test in real-time, and it is also insensitive to environmental vibration due to its common-path structure. Experimentally the proposed RSI is applied to detect the distorted wavefronts generated by a liquid crystal spatial light modulator. The measured aberrations are in good agreement with that obtained with (by) a Hartmann-Shack wavefront sensor, indicating that the proposed RSI is a useful tool for wavefront measurements.

© 2011 OSA

## 1. Introduction

30. P. L. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. **29**(22), 3271–3279 (1990). [CrossRef] [PubMed]

32. J. P. Liu and T. C. Poon, “Two-step-only quadrature phase-shifting digital holography,” Opt. Lett. **34**(3), 250–252 (2009). [CrossRef] [PubMed]

## 2. Experimental setup and theoretical description

*f*

_{5}= 250

*mm*and

*f*

_{6}= 300

*mm*, respectively. One light beam with unknown polarization state is polarized when it pass through the polarizer P2 and is induced into the cyclic RSI system. After then, the polarized light beam is split into two beams with orthogonal polarization directions (vertical direction of the reflected light beam (defined as Beam1) and the horizontal direction of the transmitted light beam (defined as Beam2)). The Beam1 traverses the path PBS1-L5-M1-M2-L6-PBS1, and the size of it is magnified because

*f*

_{5}is less than

*f*

_{6}. Similarly, the size of the Beam2 which traverses along the opposite path is demagnified. Finally, the magnified beam and the demagnified beam are reflected and transmitted into FCPPS. However, no interference fringe would be observed because these two concentric beams have orthogonal polarization states. A complex amplitude

*E*

_{0}(

*x*,

*y*) can be used to describe the tested light beam, and

*A*

_{0}(

*x*,

*y*) and

*s*=

*f*

_{6}/

*f*

_{5}> 1, and the complex amplitudes of the two beams exiting from the PBS1,

*E*

_{1}(

*x*,

*y*) and

*E*

_{2}(

*x*,

*y*), can be written by

*E*

_{1}(

*x*,

*y*) and

*E*

_{2}(

*x*,

*y*) respectively, and their definitions are shown as

*D*to denote the size of the tested laser beam, the sizes of the magnified and demagnified beams which ejected from the cyclic RSI are

*D*×

*s*and

*D/s*, respectively. The aperture sizes of these two beams are limited by the diaphragm A2,and then the magnified and demagnified beams with

*D/s*aperture size are projected into FCPPS (shown in Fig. 1).

33. C. Dunsby, Y. Gu, and P. French, “Single-shot phase-stepped wide-field coherencegated imaging,” Opt. Express **11**(2), 105–115 (2003). [CrossRef] [PubMed]

33. C. Dunsby, Y. Gu, and P. French, “Single-shot phase-stepped wide-field coherencegated imaging,” Opt. Express **11**(2), 105–115 (2003). [CrossRef] [PubMed]

*P*

_{0}, a vertical linear polarizer,

*P*

_{90}, and a quarter-wave plate with horizontal fast axis,

*Q*

_{0}, and 45°fast axis,

*Q*

_{45}, are respectively [33

33. C. Dunsby, Y. Gu, and P. French, “Single-shot phase-stepped wide-field coherencegated imaging,” Opt. Express **11**(2), 105–115 (2003). [CrossRef] [PubMed]

**11**(2), 105–115 (2003). [CrossRef] [PubMed]

*i*= 1,2.

23. C. Paterson and J. Notaras, “Demonstration of closed-loop adaptive optics with a point-diffraction interferometer in strong scintillation with optical vortices,” Opt. Express **15**(21), 13745–13756 (2007). [CrossRef] [PubMed]

35. J. Strand and T. Taxt, “Performance evaluation of two-dimensional phase unwrapping algorithms,” Appl. Opt. **38**(20), 4333–4344 (1999). [CrossRef]

36. T. M. Jeong, D. K. Ko, and J. Lee, “Method of reconstructing wavefront aberrations by use of Zernike polynomials in radial shearing interferometers,” Opt. Lett. **32**(3), 232–234 (2007). [CrossRef] [PubMed]

## 3. The relation of the fringe visibility with the angle of P2 and the radial shear

*θ*of polarizer P2 with respect to horizontal direction. For simplicity, we assume that the light intensity has a uniform distribution in all path of our configuration, and the constant number

*I*

_{0}is used to denote the light intensity at the output of P2. As our description in section 2, we can take the PBS1 as a special polarizer which comprises of two independent polarizers with orthogonal polarization directions, i.e. the horizontal direction and the vertical direction. We use

*I*

_{t}and

*I*

_{r}to define the light intensity distributions which transmitted (

*i.e.*Beam2 defined as before) and reflected (

*i.e.*Beam1) by PBS1 respectively. According the Malus’ Law [34], we can get

*η*. According the principle of conservation of energy, we can getThe intensity of one of four interferograms, which is defined as

*I*, can be expressed aswhere

*K*can be written asFrom Eqs. (11) and (13), the relation between the fringe visibility

*K*and the angle

*θ*can be shown asFrom Eq. (14), we can get the peak value of the fringe visibility

*K*when the relation between the shear ratio

*s*and the angle

*θ*satisfy

*θ*= 0°or 90°respectively, the fringe visibility will be equal to zero. In the other word, no fringe patterns can be observed when

*θ*equals to 0°or 90°. It can be explained easily in this two extremely situations. If

*θ*is equal to 0°(or 90°), there is no Beam1 (or Beam2) but only Beam2 (or Beam1), so four identical intensity distribution but not four interferograms will be obtained simultaneously.

*s*= 1.2, and the change of fringe visibility

*K*with the angle

*θ*is plotted in Fig. 2 . From Fig. 2, the best angle of P2 with respect to horizontal direction for the best fringe visibility should be 55.22°in our practical system. From Eq. (15), the best angle

*θ*of P2 is different for different radial shear

*s*, and the change of the best angle

*θ*with different radial shear

*s*is shown in Fig. 3(

*a*) . From this curve, we can get that a larger angle

*θ*of P2 is required when the system has a larger radial shear

*s*. The fringe visibility

*K*is not very sensitive to the angle

*θ*when the system has a small radial shear

*s*, and for a large radial shear

*s*, the fringe visibility

*K*is very sensitive to a little change of the angle

*θ*. For clarity, we plot the relations between the fringe visibility

*K*and the angle

*θ*in Fig. 3(

*b*) for some different radial shears (here are 1.1, 1.4, 1.7, 2.0, 3.0, 5.0, respectively). From Fig. 3(

*b*), we can get the change of sensitivity of the fringe visibility with the changing of radial shear

*s*.

## 4. Experiment results and analysis

*λ*= 632.8

*nm*. Four fringe patterns are recorded with one 8 bit, 576 × 768 pixels CCD camera. The HS WFS has a 32 × 32 micro-lens array, and the aberrated wavefronts are generated by a 512 × 512 pixel LC SLM.

*b*) .

36. T. M. Jeong, D. K. Ko, and J. Lee, “Method of reconstructing wavefront aberrations by use of Zernike polynomials in radial shearing interferometers,” Opt. Lett. **32**(3), 232–234 (2007). [CrossRef] [PubMed]

*b*)). The maximum order of Zernike polynomials is set to 45. The reconstructed result is shown in Fig. 6(

*a*) . Let

*RMS*and

*PV*represent the root-mean-square value and peak-to-valley value of the wavefront under test. The

*RMS*value and the

*PV*value of the reconstructed wavefronts are 0.4753

*λ*and 3.3222

*λ*, respectively. When we normalize the

*RMS*value of each order Zernike polynomial to 1.0

*λ*, the modal decomposition coefficients of the wavefront under test can be obtained by the least square method, and it is shown in Fig. 6(

*b*).

*a*) , and the

*RMS*value and the

*PV*value of it are 0.4798

*λ*and 3.5953

*λ*, respectively. The corresponding coefficients for each order Zernike polynomial of the wavefront under test measured by HS WFS is plotted in Fig. 7(

*b*).

*a*) , and the

*RMS*value and the

*PV*value are 0.0348

*λ*and 0.3149

*λ*, respectively. The modal decomposition coefficients for each order Zernike polynomial of the residual error is shown in Fig. 8(

*b*).

*i.e.*Beam1) must be aligned with the demagnified beam (i.e. Beam2). In the other word, if a lateral offset is existing between Beam1 and Beam2, the under test wavefront would be hard to reconstructed accurately.

## 5. Conclusion

## Acknowledgments

## References and links

1. | P. Hariharan and D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. |

2. | D. Liu, Y. Yang, and Y. Shen, “System optimization of radial shearing interferometer for aspheric testing,” Proc. SPIE |

3. | M. Wang, B. Zhang, and N. Shouping, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE |

4. | T. Kohno, D. Matsumoto, and T. Yazawa, “Radial Shearing Interferometer For In-process Measurement of Diamond Turning,” Proc. SPIE |

5. | B. Zhang, L. Ma, M. Wang, and A. He, “Aspheric lens testing by means of compact radial shearing interferometer with two zone plates,” Laser Technol. |

6. | W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part I – theoretical consideration,” Optik (Stuttg.) |

7. | W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part II – experiment results,” Optik (Stuttg.) |

8. | W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) |

9. | D. Cheung, T. Barnes, and T. Haskell, “Feedback interferometry with membrane mirror for adaptive optics,” Opt. Commun. |

10. | T. Shirai, “Liquid-crystal adaptive optics based on feedback interferometry for high-resolution retinal imaging,” Appl. Opt. |

11. | T. Shirai, T. H. Barnes, and T. G. Haskell, “Adaptive wave-front correction by means of all-optical feedback interferometry,” Opt. Lett. |

12. | D. Li, P. Wang, X. Li, H. Yang, and H. Chen, “Algorithm for near-field reconstruction based on radial-shearing interferometry,” Opt. Lett. |

13. | Y. Yang, Y. Lu, and Y. Chen, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE |

14. | D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. |

15. | C. Hernandez-Gomez, J. L. Collier, S. J. Hawkes, C. N. Danson, C. B. Edwards, D. A. Pepler, I. N. Ross, and T. B. Winstone, “Wave-front control of a large-aperture laser system by use of a static phase corrector,” Appl. Opt. |

16. | M. Murty, “A Compact Radial Shearing Interferometer Based on the Law of Refraction,” Appl. Opt. |

17. | M. V. Murty and R. P. Shukla, “Radial shearing interferometers using a laser source,” Appl. Opt. |

18. | Q.-S. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, “Constant radial shearing interferometry with circular gratings,” Appl. Opt. |

19. | D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. |

20. | R. N. Smartt, “Zone plate interferometer,” Appl. Opt. |

21. | R. K. Mohanty, C. J. Joenathan, and R. S. Sirohi, “High sensitivity tilt measurement by speckle shear interferometry,” Appl. Opt. |

22. | C. Joenathan and R. Torroba, “Simple electronic speckle-shearing-pattern interferometer,” Opt. Lett. |

23. | C. Paterson and J. Notaras, “Demonstration of closed-loop adaptive optics with a point-diffraction interferometer in strong scintillation with optical vortices,” Opt. Express |

24. | F. Bai and C. Rao, “Experimental validation of closed-loop adaptive optics based on a self-referencing interferometer wavefront sensor and a liquid-crystal spatial light modulator,” Opt. Commun. |

25. | O. Bryngdahl, “Reversed-Radial-Shearing Interferometry,” J. Opt. Soc. Am. |

26. | E. Mihaylova, M. Whelan, and V. Toal, “Simple phase-shifting lateral shearing interferometer,” Opt. Lett. |

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

28. | C. Y. Chung, K. C. Cho, C. C. Chang, C. H. Lin, W. C. Yen, and S. J. Chen, “Adaptive-optics system with liquid-crystal phase-shift interferometer,” Appl. Opt. |

29. | D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express |

30. | P. L. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. |

31. | X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. |

32. | J. P. Liu and T. C. Poon, “Two-step-only quadrature phase-shifting digital holography,” Opt. Lett. |

33. | C. Dunsby, Y. Gu, and P. French, “Single-shot phase-stepped wide-field coherencegated imaging,” Opt. Express |

34. | D. Yu, and H. Tan, |

35. | J. Strand and T. Taxt, “Performance evaluation of two-dimensional phase unwrapping algorithms,” Appl. Opt. |

36. | T. M. Jeong, D. K. Ko, and J. Lee, “Method of reconstructing wavefront aberrations by use of Zernike polynomials in radial shearing interferometers,” Opt. Lett. |

37. | X. Li and C. Wang, “H. XIAN and W. JIANG, “Real-time modal reconstruction algorithm for adaptive optics systems,” Laser and Part. Beams |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(040.1880) Detectors : Detection

(110.1650) Imaging systems : Coherence imaging

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: January 5, 2011

Revised Manuscript: January 28, 2011

Manuscript Accepted: January 29, 2011

Published: February 24, 2011

**Citation**

Naiting Gu, Linhai Huang, Zeping Yang, and Changhui Rao, "A single-shot common-path phase-stepping radial shearing interferometer for wavefront measurements," Opt. Express **19**, 4703-4713 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-5-4703

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### References

- P. Hariharan and D. Sen, “Radial shearing interferometer,” J. Sci. Instrum. 38(11), 428–432 (1961). [CrossRef]
- D. Liu, Y. Yang, and Y. Shen, “System optimization of radial shearing interferometer for aspheric testing,” Proc. SPIE 6834, 68340U_1–8 (2007).
- M. Wang, B. Zhang, and N. Shouping, “Radial shearing interferometer for aspheric surface testing,” Proc. SPIE 4927, 673–676 (2002). [CrossRef]
- T. Kohno, D. Matsumoto, and T. Yazawa, “Radial Shearing Interferometer For In-process Measurement of Diamond Turning,” Proc. SPIE 3173, 280–285 (1997). [CrossRef]
- B. Zhang, L. Ma, M. Wang, and A. He, “Aspheric lens testing by means of compact radial shearing interferometer with two zone plates,” Laser Technol. 31, 37–46 (2007) (in Chinese).
- W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part I – theoretical consideration,” Optik (Stuttg.) 113(1), 39–45 (2002). [CrossRef]
- W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part II – experiment results,” Optik (Stuttg.) 113(1), 46–50 (2002). [CrossRef]
- W. Kowalik, B. Garncarz, and H. Kasprzak, “Corneal topography measurement by means of radial shearing interference: Part III – measurement errors,” Optik (Stuttg.) 114(5), 199–206 (2003). [CrossRef]
- D. Cheung, T. Barnes, and T. Haskell, “Feedback interferometry with membrane mirror for adaptive optics,” Opt. Commun. 218(1-3), 33–41 (2003). [CrossRef]
- T. Shirai, “Liquid-crystal adaptive optics based on feedback interferometry for high-resolution retinal imaging,” Appl. Opt. 41(19), 4013–4023 (2002). [CrossRef] [PubMed]
- T. Shirai, T. H. Barnes, and T. G. Haskell, “Adaptive wave-front correction by means of all-optical feedback interferometry,” Opt. Lett. 25(11), 773–775 (2000). [CrossRef]
- D. Li, P. Wang, X. Li, H. Yang, and H. Chen, “Algorithm for near-field reconstruction based on radial-shearing interferometry,” Opt. Lett. 30(5), 492–494 (2005). [CrossRef] [PubMed]
- Y. Yang, Y. Lu, and Y. Chen, “A Radial Shearing Interference system of Testing Laser Pulse Wavefront Distortion and the Original Wavefront Reconstructing,” Proc. SPIE 5638, 200–204 (2005). [CrossRef]
- D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. 46(34), 8305–8314 (2007). [CrossRef] [PubMed]
- C. Hernandez-Gomez, J. L. Collier, S. J. Hawkes, C. N. Danson, C. B. Edwards, D. A. Pepler, I. N. Ross, and T. B. Winstone, “Wave-front control of a large-aperture laser system by use of a static phase corrector,” Appl. Opt. 39(12), 1954–1961 (2000). [CrossRef]
- M. Murty, “A Compact Radial Shearing Interferometer Based on the Law of Refraction,” Appl. Opt. 3(7), 853–858 (1964). [CrossRef]
- M. V. Murty and R. P. Shukla, “Radial shearing interferometers using a laser source,” Appl. Opt. 12(11), 2765–2767 (1973). [CrossRef] [PubMed]
- Q.-S. Ru, N. Ohyama, T. Honda, and J. Tsujiuchi, “Constant radial shearing interferometry with circular gratings,” Appl. Opt. 28(16), 3350–3353 (1989). [CrossRef] [PubMed]
- D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt. 11(11), 2613–2624 (1972). [CrossRef] [PubMed]
- R. N. Smartt, “Zone plate interferometer,” Appl. Opt. 13(5), 1093–1099 (1974). [CrossRef] [PubMed]
- R. K. Mohanty, C. J. Joenathan, and R. S. Sirohi, “High sensitivity tilt measurement by speckle shear interferometry,” Appl. Opt. 25(10), 1661–1664 (1986). [CrossRef] [PubMed]
- C. Joenathan and R. Torroba, “Simple electronic speckle-shearing-pattern interferometer,” Opt. Lett. 15(20), 1159–1161 (1990). [CrossRef] [PubMed]
- C. Paterson and J. Notaras, “Demonstration of closed-loop adaptive optics with a point-diffraction interferometer in strong scintillation with optical vortices,” Opt. Express 15(21), 13745–13756 (2007). [CrossRef] [PubMed]
- F. Bai and C. Rao, “Experimental validation of closed-loop adaptive optics based on a self-referencing interferometer wavefront sensor and a liquid-crystal spatial light modulator,” Opt. Commun. 283(14), 2782–2786 (2010). [CrossRef]
- O. Bryngdahl, “Reversed-Radial-Shearing Interferometry,” J. Opt. Soc. Am. 60(7), 915–917 (1970). [CrossRef]
- E. Mihaylova, M. Whelan, and V. Toal, “Simple phase-shifting lateral shearing interferometer,” Opt. Lett. 29(11), 1264–1266 (2004). [CrossRef] [PubMed]
- M. P. Kothiyal and C. Delisle, “Shearing interferometer for phase shifting interferometry with polarization phase shifter,” Appl. Opt. 24(24), 4439–4447 (1985). [CrossRef] [PubMed]
- C. Y. Chung, K. C. Cho, C. C. Chang, C. H. Lin, W. C. Yen, and S. J. Chen, “Adaptive-optics system with liquid-crystal phase-shift interferometer,” Appl. Opt. 45(15), 3409–3414 (2006). [CrossRef] [PubMed]
- D. N. Naik, T. Ezawa, Y. Miyamoto, and M. Takeda, “3-D coherence holography using a modified Sagnac radial shearing interferometer with geometric phase shift,” Opt. Express 17(13), 10633–10641 (2009). [CrossRef] [PubMed]
- P. L. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. 29(22), 3271–3279 (1990). [CrossRef] [PubMed]
- X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31(10), 1414–1416 (2006). [CrossRef] [PubMed]
- J. P. Liu and T. C. Poon, “Two-step-only quadrature phase-shifting digital holography,” Opt. Lett. 34(3), 250–252 (2009). [CrossRef] [PubMed]
- C. Dunsby, Y. Gu, and P. French, “Single-shot phase-stepped wide-field coherencegated imaging,” Opt. Express 11(2), 105–115 (2003). [CrossRef] [PubMed]
- D. Yu, and H. Tan, Engineering Optics, China machine press, Beijing, chapter 15th, pp: 310, 422, 440–444(2006)(in Chinese).
- J. Strand and T. Taxt, “Performance evaluation of two-dimensional phase unwrapping algorithms,” Appl. Opt. 38(20), 4333–4344 (1999). [CrossRef]
- T. M. Jeong, D. K. Ko, and J. Lee, “Method of reconstructing wavefront aberrations by use of Zernike polynomials in radial shearing interferometers,” Opt. Lett. 32(3), 232–234 (2007). [CrossRef] [PubMed]
- X. Li and C. Wang, “H. XIAN and W. JIANG, “Real-time modal reconstruction algorithm for adaptive optics systems,” Laser and Part. Beams 11, 53–56 (2002) (in Chinese).

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