## Orientation dependent wavefront correction system under grazing incidence |

Optics Express, Vol. 21, Issue 18, pp. 20497-20505 (2013)

http://dx.doi.org/10.1364/OE.21.020497

Acrobat PDF (1386 KB)

### Abstract

Making use of the stretching effect of grazing incident laser beam, a novel method of wavefront correction was promoted. Without adding any extra beam expanding components, aberrations of wavefront could achieve satisfying correction by two grazing reflections along orthogonal directions on the deformable mirrors. The stretching effect expanded the beam size along grazing direction and the orientation dependent varying aberrations were well compensated as more actuators took effect in the correction process. Analysis showed that the fitting coefficient of all the first 30 order Zernike polynomials could be controlled within 5% by this method.

© 2013 Optical Society of America

## 1. Introduction

20. T. E. Carlsson, N. H. Abramson, and K. H. Fischer, “Automatic measurement of surface height with the interferoscope,” Opt. Eng. **35**(10), 2938–2942 (1996). [CrossRef]

24. H. Zimer, K. Albers, and U. Wittrock, “Grazing-incidence YVO_{4}-Nd:YVO_{4} composite thin slab laser with low thermo-optic aberrations,” Opt. Lett. **29**(23), 2761–2763 (2004). [CrossRef] [PubMed]

25. F. He, M. Gong, L. Huang, Q. Liu, Q. Wang, and X. Yan, “Compact TEM_{00} grazing-incidence Nd:GdVO_{4} laser using a folded cavity,” Appl. Phys. B **86**(3), 447–450 (2007). [CrossRef]

## 2. Grazing incidence wavefront correction system

*θ*, the amplifying factor

*γ*can be expressed as

*θ*changing, the incident beam could be continuously stretched to infinite size as the incidence angle approaching 90°. In practical application, the amplifying factor within 10 could be reached as the incidence angle changed from 0° to 84.3°

*Δh*.

*Δφ*between the two beams was

*|EG|-|CD|*

*cosθ*. Realization of the enlargement of incident beam sacrificed the modulation depth of DM. However, the modified range were among several micrometers which was small compared with the deformation range of DM. Thus, scarification within certain range in the modulation depth was acceptable in practical application.

_{1}and DM

_{2}consecutively. A

*4f*optical system was employed to arrange the two reflection planes in conjugate positions. The grazing angles were

*θ*and

_{1}*θ*and the grazing directions were along X-axis and Y-axis respectively. The Hartmann Shack senser was employed to detected the output wavefront. After necessary data analysis, the control signals of DMs were generated and the whole close-loop AO system was established.

_{2}26. W. Jiang and H. Li, “Hartmann-Shack wavefront sensing and wavefront control algorithm,” SPIE **1271**, 82–93 (1990). [CrossRef]

27. X. Li, C. Wang, H. Xian, X. Wu, and W. Jiang, “Zernike modal compensation analysis for an adaptive optics system using direct-gradient wavefront reconstruction algorithm,” SPIE **3762**, 116–124 (1999). [CrossRef]

*(x*on each pixel

_{a},y_{a})*(x*could be expressed as

_{i},y_{i})*θ*was the grazing incidence angle,

*r*was the influence radius of actuators,

*α*and

*β*were parameters measured from experiment. The influence on pixels induced by each actuator was set in one column, by conjugating all columns together, the influence matrix

*H*was obtained. The control signal

*A*was a column vector with elements equal to the number of actuators. The relationship between control signal and gradient vector

*W*measured by wavefront senser can be expressed as

*m*was the pixels’ number of incident beam and

*n*was the actuators’ number. As the pixels’ number

*m*was greatly larger than the actuators’ number

*n*, in most cases, the linear equations were inconsistent and had no exact solution. However, solution at the least-square sense still could be found as

*H*was the left inverse of

^{+}*H*,

*A**was the least square solution to Eq. (5). If one laser beam with wavefront of

*W*passed through the grazing reflected AO system with incidence angles of

*θ*and

_{x}*θ*, the control signal

_{y}*A*and residual wavefront error

_{x}**e*after grazing reflection along X-axis can be written as

_{x}*A*was the function of both grazing anglesThe residual error after two-sections correction can be written as the function of

_{y}**θ*and

_{x}*θ*

_{y}## 3. Characteristics of grazing incidence wavefront correction system

### 3.1 Orientation dependent characteristics of grazing incidence

*e*and

_{x}(θ_{x})*e*were the residual wavefront errors defined in Eq. (8). RMS was the root mean square value of the residual errors. The relationship between improvement factor and grazing angle was demonstrated in Fig. 6.

_{x}(0°)### 3.2 Effectiveness after correction in orthogonal directions

*ξ*

*Z*was the

_{j}(x,y)*j*order Zernike polynomial and

^{th}*e(x,y)*was the residual wavefront error after correction. The fitting coefficient

*ξ*measured the ratio of uncompensated aberration to the total aberration of original Zernike polynomial.

## 4. Conclusion

## Acknowledgments

## References and links

1. | M. L. Gong, Y. Qiu, L. Huang, Q. Liu, P. Yan, and H. T. Zhang, “Beam quality improvement by joint compensation of amplitude and phase,” Opt. Lett. |

2. | W. Lubeigt, G. Valentine, and D. Burns, “Enhancement of laser performance using an intracavity deformable membrane mirror,” Opt. Express |

3. | H. Baumhacker, G. Pretzler, K. J. Witte, M. Hegelich, M. Kaluza, S. Karsch, A. Kudryashov, V. Samarkin, and A. Roukossouev, “Correction of strong phase and amplitude modulations by two deformable mirrors in a multistaged Ti:sapphire laser,” Opt. Lett. |

4. | S. Piehler, B. Weichelt, A. Voss, M. A. Ahmed, and T. Graf, “Power scaling of fundamental-mode thin-disk lasers using intracavity deformable mirrors,” Opt. Lett. |

5. | F. Druon, G. Chériaux, J. Faure, J. Nees, M. Nantel, A. Maksimchuk, G. Mourou, J. C. Chanteloup, and G. Vdovin, “Wave-front correction of femtosecond terawatt lasers by deformable mirrors,” Opt. Lett. |

6. | C. Valentin, J. Gautier, J.-P. Goddet, C. Hauri, T. Marchenko, E. Papalazarou, G. Rey, S. Sebban, O. Scrick, P. Zeitoun, G. Dovillaire, X. Levecq, S. Bucourt, and M. Fajardo, “High-order harmonic wave fronts generated with controlled astigmatic infrared laser,” J. Opt. Soc. Am. B |

7. | X. Lei, B. Xu, P. Yang, L. Dong, W. Liu, and H. Yan, “Beam cleanup of a 532-nm pulsed solid-state laser using a bimorph mirror,” Chin. Opt. Lett. |

8. | X. Lei, S. Wang, H. Yan, W. Liu, L. Dong, P. Yang, and B. Xu, “Double-deformable-mirror adaptive optics system for laser beam cleanup using blind optimization,” Opt. Express |

9. | S. Hu, B. Xu, X. Zhang, J. Hou, J. Wu, and W. Jiang, “Double-deformable-mirror adaptive optics system for phase compensation,” Appl. Opt. |

10. | R. Zacharias, E. Bliss, S. Winters, R. Sacks, M. Feldman, A. Grey, J. Koch, C. Stolz, J. Toeppen, L. Van Atta, and B. Woods, “Wavefront control of high-power laser beams in the National Ignition Facility (NIF),” Proc. SPIE |

11. | R. Zacharias, E. Bliss, M. Feldman, A. Grey, M. Henesian, J. Koch, J. Lawson, R. Sacks, T. Salmon, J. Toeppen, L. Van Atta, S. Winters, B. Woods, C. Lafiandra, and D. G. Bruns, “The National Ignition Facility(NIF) wavefront control system,” Proc. SPIE |

12. | O. Solgaard, F. S. A. Sandejas, and D. M. Bloom, “Deformable grating optical modulator,” Opt. Lett. |

13. | T. Sato, H. Ishida, and O. Ikeda, “Adaptive PVDF piezoelectric deformable mirror system,” Appl. Opt. |

14. | P. Yang, Y. Liu, W. Yang, M.-W. Ao, S.-J. Hu, B. Xu, and W.-H. Jiang, “Adaptive mode optimization of a continuous-wave solid-state laser using an intracavity piezoelectric deformable mirror,” Opt. Commun. |

15. | Q. Xue, L. Huang, P. Yan, M. Gong, Z. Feng, Y. Qiu, T. Li, and G. Jin, “Research on the particular temperature-induced surface shape of a National Ignition Facility deformable mirror,” Appl. Opt. |

16. | M. Kasprzack, B. Canuel, F. Cavalier, R. Day, E. Genin, J. Marque, D. Sentenac, and G. Vajente, “Performance of a thermally deformable mirror for correction of low-order aberrations in laser beams,” Appl. Opt. |

17. | M. A. Arain, W. Z. Korth, L. F. Williams, R. M. Martin, G. Mueller, D. B. Tanner, and D. H. Reitze, “Adaptive control of modal properties of optical beams using photothermal effects,” Opt. Express |

18. | D. Brousseau, E. F. Borra, and S. Thibault, “Wavefront correction with a 37-actuator ferrofluid deformable mirror,” Opt. Express |

19. | N. Abramson, “The interferoscope: a new type of interferometer with variable fringe separation,” Optik (Stuttg.) |

20. | T. E. Carlsson, N. H. Abramson, and K. H. Fischer, “Automatic measurement of surface height with the interferoscope,” Opt. Eng. |

21. | X. Colonna de Lega, J. F. Biegen, D. Stephenson, and P. J. de Groot, “Characterization of a geometrically desensitized interferometer for flatness testing,” Proc. SPIE |

22. | P. de Groot, “Diffractive grazing-incidence interferometer,” Appl. Opt. |

23. | H. Nüger and J. Schwider, “Measurement of curvature and thickness variations of plane surfaces by grazing incidence interferometry,” Optik (Stuttg.) |

24. | H. Zimer, K. Albers, and U. Wittrock, “Grazing-incidence YVO |

25. | F. He, M. Gong, L. Huang, Q. Liu, Q. Wang, and X. Yan, “Compact TEM |

26. | W. Jiang and H. Li, “Hartmann-Shack wavefront sensing and wavefront control algorithm,” SPIE |

27. | X. Li, C. Wang, H. Xian, X. Wu, and W. Jiang, “Zernike modal compensation analysis for an adaptive optics system using direct-gradient wavefront reconstruction algorithm,” SPIE |

28. | H. Yang, G. Liu, C. Rao, Y. Zhang, and W. Jiang, “Combinational-deformable-mirror adaptive optics system for compensation of high-order modes of wavefront,” Chin. Opt. Lett. |

**OCIS Codes**

(010.3310) Atmospheric and oceanic optics : Laser beam transmission

(140.3300) Lasers and laser optics : Laser beam shaping

**ToC Category:**

Optical Devices

**History**

Original Manuscript: May 28, 2013

Revised Manuscript: August 16, 2013

Manuscript Accepted: August 16, 2013

Published: August 26, 2013

**Citation**

Xingkun Ma, Lei Huang, Mali Gong, Qiao Xue, Zexin Feng, Ping Yan, and Qiang Liu, "Orientation dependent wavefront correction system under grazing incidence," Opt. Express **21**, 20497-20505 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-18-20497

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### References

- M. L. Gong, Y. Qiu, L. Huang, Q. Liu, P. Yan, and H. T. Zhang, “Beam quality improvement by joint compensation of amplitude and phase,” Opt. Lett.38(7), 1101–1103 (2013). [CrossRef] [PubMed]
- W. Lubeigt, G. Valentine, and D. Burns, “Enhancement of laser performance using an intracavity deformable membrane mirror,” Opt. Express16(15), 10943–10955 (2008). [CrossRef] [PubMed]
- H. Baumhacker, G. Pretzler, K. J. Witte, M. Hegelich, M. Kaluza, S. Karsch, A. Kudryashov, V. Samarkin, and A. Roukossouev, “Correction of strong phase and amplitude modulations by two deformable mirrors in a multistaged Ti:sapphire laser,” Opt. Lett.27(17), 1570–1572 (2002). [CrossRef] [PubMed]
- S. Piehler, B. Weichelt, A. Voss, M. A. Ahmed, and T. Graf, “Power scaling of fundamental-mode thin-disk lasers using intracavity deformable mirrors,” Opt. Lett.37(24), 5033–5035 (2012). [CrossRef] [PubMed]
- F. Druon, G. Chériaux, J. Faure, J. Nees, M. Nantel, A. Maksimchuk, G. Mourou, J. C. Chanteloup, and G. Vdovin, “Wave-front correction of femtosecond terawatt lasers by deformable mirrors,” Opt. Lett.23(13), 1043–1045 (1998). [CrossRef] [PubMed]
- C. Valentin, J. Gautier, J.-P. Goddet, C. Hauri, T. Marchenko, E. Papalazarou, G. Rey, S. Sebban, O. Scrick, P. Zeitoun, G. Dovillaire, X. Levecq, S. Bucourt, and M. Fajardo, “High-order harmonic wave fronts generated with controlled astigmatic infrared laser,” J. Opt. Soc. Am. B25(7), 161–166 (2008). [CrossRef]
- X. Lei, B. Xu, P. Yang, L. Dong, W. Liu, and H. Yan, “Beam cleanup of a 532-nm pulsed solid-state laser using a bimorph mirror,” Chin. Opt. Lett.10(2), 021401 (2012). [CrossRef]
- X. Lei, S. Wang, H. Yan, W. Liu, L. Dong, P. Yang, and B. Xu, “Double-deformable-mirror adaptive optics system for laser beam cleanup using blind optimization,” Opt. Express20(20), 22143–22157 (2012). [CrossRef] [PubMed]
- S. Hu, B. Xu, X. Zhang, J. Hou, J. Wu, and W. Jiang, “Double-deformable-mirror adaptive optics system for phase compensation,” Appl. Opt.45(12), 2638–2642 (2006). [CrossRef] [PubMed]
- R. Zacharias, E. Bliss, S. Winters, R. Sacks, M. Feldman, A. Grey, J. Koch, C. Stolz, J. Toeppen, L. Van Atta, and B. Woods, “Wavefront control of high-power laser beams in the National Ignition Facility (NIF),” Proc. SPIE3889, 332–343 (2000). [CrossRef]
- R. Zacharias, E. Bliss, M. Feldman, A. Grey, M. Henesian, J. Koch, J. Lawson, R. Sacks, T. Salmon, J. Toeppen, L. Van Atta, S. Winters, B. Woods, C. Lafiandra, and D. G. Bruns, “The National Ignition Facility(NIF) wavefront control system,” Proc. SPIE3492, 678–692 (1999). [CrossRef]
- O. Solgaard, F. S. A. Sandejas, and D. M. Bloom, “Deformable grating optical modulator,” Opt. Lett.17(9), 688–690 (1992). [CrossRef] [PubMed]
- T. Sato, H. Ishida, and O. Ikeda, “Adaptive PVDF piezoelectric deformable mirror system,” Appl. Opt.19(9), 1430–1434 (1980). [CrossRef] [PubMed]
- P. Yang, Y. Liu, W. Yang, M.-W. Ao, S.-J. Hu, B. Xu, and W.-H. Jiang, “Adaptive mode optimization of a continuous-wave solid-state laser using an intracavity piezoelectric deformable mirror,” Opt. Commun.278(2), 377–381 (2007). [CrossRef]
- Q. Xue, L. Huang, P. Yan, M. Gong, Z. Feng, Y. Qiu, T. Li, and G. Jin, “Research on the particular temperature-induced surface shape of a National Ignition Facility deformable mirror,” Appl. Opt.52(2), 280–287 (2013). [CrossRef] [PubMed]
- M. Kasprzack, B. Canuel, F. Cavalier, R. Day, E. Genin, J. Marque, D. Sentenac, and G. Vajente, “Performance of a thermally deformable mirror for correction of low-order aberrations in laser beams,” Appl. Opt.52(12), 2909–2916 (2013). [CrossRef] [PubMed]
- M. A. Arain, W. Z. Korth, L. F. Williams, R. M. Martin, G. Mueller, D. B. Tanner, and D. H. Reitze, “Adaptive control of modal properties of optical beams using photothermal effects,” Opt. Express18(3), 2767–2781 (2010). [CrossRef] [PubMed]
- D. Brousseau, E. F. Borra, and S. Thibault, “Wavefront correction with a 37-actuator ferrofluid deformable mirror,” Opt. Express15(26), 18190–18199 (2007). [CrossRef] [PubMed]
- N. Abramson, “The interferoscope: a new type of interferometer with variable fringe separation,” Optik (Stuttg.)30, 56–71 (1969).
- T. E. Carlsson, N. H. Abramson, and K. H. Fischer, “Automatic measurement of surface height with the interferoscope,” Opt. Eng.35(10), 2938–2942 (1996). [CrossRef]
- X. Colonna de Lega, J. F. Biegen, D. Stephenson, and P. J. de Groot, “Characterization of a geometrically desensitized interferometer for flatness testing,” Proc. SPIE3520, 284–292 (1998). [CrossRef]
- P. de Groot, “Diffractive grazing-incidence interferometer,” Appl. Opt.39(10), 1527–1530 (2000). [CrossRef] [PubMed]
- H. Nüger and J. Schwider, “Measurement of curvature and thickness variations of plane surfaces by grazing incidence interferometry,” Optik (Stuttg.)111, 319–327 (2000).
- H. Zimer, K. Albers, and U. Wittrock, “Grazing-incidence YVO4-Nd:YVO4 composite thin slab laser with low thermo-optic aberrations,” Opt. Lett.29(23), 2761–2763 (2004). [CrossRef] [PubMed]
- F. He, M. Gong, L. Huang, Q. Liu, Q. Wang, and X. Yan, “Compact TEM00 grazing-incidence Nd:GdVO4 laser using a folded cavity,” Appl. Phys. B86(3), 447–450 (2007). [CrossRef]
- W. Jiang and H. Li, “Hartmann-Shack wavefront sensing and wavefront control algorithm,” SPIE1271, 82–93 (1990). [CrossRef]
- X. Li, C. Wang, H. Xian, X. Wu, and W. Jiang, “Zernike modal compensation analysis for an adaptive optics system using direct-gradient wavefront reconstruction algorithm,” SPIE3762, 116–124 (1999). [CrossRef]
- H. Yang, G. Liu, C. Rao, Y. Zhang, and W. Jiang, “Combinational-deformable-mirror adaptive optics system for compensation of high-order modes of wavefront,” Chin. Opt. Lett.5, 435–437 (2007).

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