## Interferometric adaptive optics testbed for laser pointing, wave-front control and phasing

Optics Express, Vol. 17, Issue 19, pp. 16696-16709 (2009)

http://dx.doi.org/10.1364/OE.17.016696

Acrobat PDF (503 KB)

### Abstract

Implementing the capability to perform fast ignition experiments, as well as, radiography experiments on the National Ignition Facility (NIF) places stringent requirements on the control of each of the beam’s pointing, intra-beam phasing and overall wave-front quality. In this article experimental results are presented which were taken on an interferometric adaptive optics testbed that was designed and built to test the capabilities of such a system to control phasing, pointing and higher order beam aberrations. These measurements included quantification of the reduction in Strehl ratio incurred when using the MEMS device to correct for pointing errors in the system. The interferometric adaptive optics system achieved a Strehl ratio of 0.83 when correcting for a piston, tip/tilt error between two adjacent rectangular apertures, the geometry expected for the National ignition Facility. The interferometric adaptive optics system also achieved a Strehl ratio of 0.66 when used to correct for a phase plate aberration of similar magnitude as expected from simulations of the ARC beam line. All of these corrections included measuring both the upstream and downstream aberrations in the testbed and applying the sum of these two measurements in open-loop to the MEMS deformable mirror.

© 2009 OSA

## 1. Introduction

4. J. P. Zou, A. M. Sautivet, J. Fils, L. Martin, K. Abdeli, C. Sauteret, and B. Wattellier, “Optimization of the dynamic wavefront control of a pulsed kilojoule/nanosecond-petawatt laser facility,” Appl. Opt. **47**(5), 704–710 (2008). [CrossRef] [PubMed]

5. C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, B. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan Petawatt—an ultra-high-intensity interaction facility,” Nucl. Fusion **44**(12), 239 (2004). [CrossRef]

_{r}, of 1, then they would deliver 5.6 kJ in a 40 μm diameter circle, exceeding the requirements by 1.6 kJ. When each of the two beams formed from a single NIF beam are co-phased, but with random piston errors between the four beam pairs, the encircled energy requirement of 4 kJ is exceeded 90% of the time provided the Strehl ratio is greater than 0.72 [6]. This latter statement was determined by analyzing the results from a thousand simulations to evaluate the system performance with random piston errors, random tip/tilt errors and a random realization of a residual turbulence profile applied to the eight beams. The random realization of the residual turbulence profile consisted of a Von Karman turbulence profile with the Fried parameter

*r*=

_{o}*D*/8 and the outer scale length set to

*D*/2 with

*D*representing the longest aperture dimension. In the case of tip/tilt errors, normally-distributed pseudo-random numbers with a mean of zero and a standard deviation of 1 μrad were assigned to both the tip and tilt components of the phase. The Von Karman turbulence parameters were chosen based upon residual wave-front measurements taken on one of the NIF beam lines after a low order deformable mirror was utilized to pre-correct for wave-front aberrations in the rod and disk amplifiers caused primarily by heat deposition from the flashlamps [6]. A Von Karman turbulence profile is also expected from the gas turbulence inside the beam tubes due the natural truncation of the outer scalelength of the turbulence spectrum by the beam tubes enclosing the optics.

8. K. L. Baker, E. A. Stappaerts, D. Gavel, S. C. Wilks, J. Tucker, D. A. Silva, J. Olsen, S. S. Olivier, P. E. Young, M. W. Kartz, L. M. Flath, P. Kruelevitch, J. Crawford, and O. Azucena, “High-speed horizontal-path atmospheric turbulence correction with a large-actuator-number microelectromechanical system spatial light modulator in an interferometric phase-conjugation engine,” Opt. Lett. **29**(15), 1781 (2004. [CrossRef] [PubMed]

9. J. Notaras and C. Paterson, “Point-diffrction interferometer for atmospheric daptive optics in strong scintillation,” Opt. Commun. **281**(3), 360–367 (2008). [CrossRef]

10. G. Love, T. Oag, and A. Kirby, “Common path interferometric wavefront sensor for extreme adaptive optics,” Opt. Express **13**(9), 3491–3499 (2005). [CrossRef] [PubMed]

11. K. L. Baker, “Interferometric Wavefront Sensors for High Contrast Imaging,” Optics Express **14**, 10970 (2006). [CrossRef] [PubMed]

## 2. Testbed layout

## 3. Measurements on the testbed

^{−4}change in the focal length. The change in tip/tilt between the individual beams focusing on the targeting sphere and the end of the fast ignition cone is slight, α(Δ

*x*/

*f.l.*), where α is the angle of the guide star beam from the optical axis of the ARC pupil, f.l. is the focal length of the parabola and Δx is the distance between the fiber backlighter and the end of the fast ignition cone. For each of the beam pairs forming the ARC pupil this corresponds to an applied tilt of 18.8 μrad or a peak-to-valley phase tip/tilt of 6.9 μm. The difference in the focal length imposed on the MEMS devices to make the required changes to the focus of the parabola is a peak-to-valley defocus of 0.66 μm. The application of the tip/tilt phase on the MEMS device causes a slight reduction in the Strehl ratio and this reduction was studied on the testbed, as well as, analytically.

^{2}, using the Marechal approximation,

*S*(-

_{r}~exp*σ*

^{2}) [12

12. K. L. Baker, E. A. Stappaerts, S. C. Wilks, D. Gavel, P. E. Young, J. Tucker, S. S. Olivier, D. A. Silva, and J. Olsen, “Performance of a phase-conjugate engine implementing a finite-bit phase correction,” Opt. Lett. **29**(9), 980–982 (2004). [CrossRef] [PubMed]

^{n}. Each level of correction covers a phase region of φ

_{o}=2π/2

^{n}radians. The phase variance within each of the correction levels is given bywhere m represents the number of phase levels and φ

_{avg}is the central phase within a particular correction level. Assuming a uniform distribution of phases, Eq. (1) can be approximated byUsing the “extended Marechal approximation”, the Strehl ratio,

*S*, can be expressed as

_{r}*S*~

_{r}*exp*{-(π

^{2}/3)/2

^{2n}}. A more formal derivation, yields the expression

*S*= sinc

_{r}^{2}(π/2

^{n}) [13

13. D. Gordon, “Love, Nigel Andrews, Philip Burch, David Buscher, Peter Doel, Colin Dunlop, John Major, Richard Myers, Alan Purvis, Ray Sharples, Andrew Vick, Andrew Zadrozny, Sergio R. Restaino, and Andreas Glindemann, Binary adaptive optics: atmospheric wave-front correction with a half-wave phase shifter,” Appl. Opt. **34**(27), 6058 (1995).

^{n}, is equal to the number of waves of tilt across the MEMS device,

*Amp*, divided by the number of actuator rows, 32. The Strehl ratio can then be expressed as

*S*~

_{r}*exp*{-(π

^{2}/3)(

*Amp*/32)

^{2}}.

*S*~0.95, to be achieved with no tilt applied to the MEMS deformable mirror. In Fig. 5a the number of waves of tilt applied to the MEMS deformable mirror from left to right was 0, 4, 8 and 12 waves, respectively. As the tilt applied to MEMS deformable mirror increases, it becomes a phase grating putting energy into multiple orders as seen in Fig. 5a. As the number of waves of tilt applied to the MEMS deformable mirror reaches 16, the MEMS device becomes a phase grating with a 180 degree phase shift between adjacent rows of pixels and an efficiency in the +1 order is ~43%. Figure 5b illustrates the Strehl ratio as a function of the waves of tilt applied across the MEMS device. The squares represent the derived Strehl ratios from the measured far-field patterns shown in Fig. 5a. The solid black line represents the theoretical curve derived above and the solid grey curve is simply a vertical displacement of the analytical curve to account for the small level of aberrations present on the MEMS device and on the optical train of the testbed. The Strehl ratio was determined by comparing the peak of the measured far-field intensity distribution to simulated far-field patterns chosen to match the lobe pattern of the measured far-field distribution. For the expected amplitude of 6.9 microns of applied tilt required to move the beam from the fiber backlighter to the back of the fast ignition cone, the Strehl ratio would be degraded by ~0.13 which is a significant fraction of the error budget and so will be delegated to a separate tip/tilt mirror. The results of this study will be presented in an auxiliary paper [14].

_{r}15. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. **23**(4), 713–720 (1988). [CrossRef]

17. K. L. Baker and D. A. Silva, “Evaluation of Two-Dimensional Phase Unwrapping Algorithms for Interferometric Characterization of Liquid-Crystal Spatial Light Modulators,” The Open Optics Journal **2**, 48 (2008). [CrossRef]

*S*=0.83.

_{r}15. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. **23**(4), 713–720 (1988). [CrossRef]

17. K. L. Baker and D. A. Silva, “Evaluation of Two-Dimensional Phase Unwrapping Algorithms for Interferometric Characterization of Liquid-Crystal Spatial Light Modulators,” The Open Optics Journal **2**, 48 (2008). [CrossRef]

*S*=0.087, while the Strehl ratio of the corrected far-field pattern was found to be

_{r}*S*=0.66.

_{r}*D*

_{φ}(r), of the phase aberration. A Von Karman spectrum was fit to the structure function as shown in Fig. 8 . Simulations of the ARC beam line have indicated that the Strehl ratio will be close to a value of,

*S*~0.09 [18]. This result was obtained via simulations of the ARC beamline using the PROP92 beam propagation code [19]. The Von Karman turbulence profile fit to the structure function calculated from the measured phase profile in Fig. 7b consisted of a Fried parameter equal to

_{r}*r*=

_{o}*D*/18 and the outer scale length equal to

*L*=1.8

_{o}*D*. This is consistent with the values expected on the ARC beam line and produce Strehl ratios close to the values expected from the ARC simulations. The Strehl ratio from the applied phase plate, as determined by analyzing Fig. 7c, was

*S*=0.087.

_{r}*S*= 0.88 at 1.053 μm [6]. This is due almost entirely to sharp gradients near the edges of the device arising from the polishing procedure used on the current MEMS device. The MEMS manufacturer has several methods to significantly reduce the phase gradients nears the edges of the MEMS mirrors and this will significantly reduce the reduction in Strehl ratio in future devices. On the current device, the reduction in Strehl ratio can be significantly reduced by using the inner 26x26 pixels.

_{r}## 4. Piston-removed phase variance

*d*is the length of one side of the square aperture for the piston-removed phase variance and the sub-aperture in the case of the wave-front fitting error. By converting to cylindrical coordinates, x = rcos(

*θ*) and y = rsin(

*θ*) and integrating over one octant of the aperture, the piston-removed phase variance for a square aperture can be expressed aswhere

*d*is the length of one side of the square aperture. This integral may be evaluated by expanding the modified Bessel function of the second kind, contained in the structure function, into a power series representation and performing the integration. The resulting power series solution for Eq. (6) can be written as

_{o}=0, this expression reduces to the result previously obtained for the piston-removed phase variance with a Kolmogorov turbulence spectrum, which also corresponds to the fitting error obtained with a piston-only MEMS device with a square actuators [7].

*r*=

_{o}*D*/18 and an outer scale length equal to

*L*=1.8

_{o}*D*, where κ

_{o}= 2π/

*L*. The phase variance should be approximately σ

_{o}^{2}=1.31(

*d/r*)

_{o}^{5/3}(1-0.68(

*d*κ

_{o})

^{1/3}) or σ

^{2}=0.34 rad

^{2}. The maximum Strehl ratio achievable with this phase plate, using a piston only MEMS device, is then

*S*= 0.71 which agrees well with the measured Strehl ratio of

_{r}*S*= 0.66.

_{r}## 5. Strehl ratio improvements

## 6. Summary

## Acknowledgements

## References and links

1. | R. A. Zacharias, N. R. Beer, E. S. Bliss, S. C. Burkhart, S. J. Cohen, S. B. Sutton, R. L. Van Atta, S. E. Winters, J. T. Salmon, M. R. Latta, C. J. Stolz, D. C. Pigg, and T. J. Arnold, “Alignment and wavefront control systems of the National Ignition Facility Opt,” Eng. |

2. | D. M. Pennington, C. G. Brown, T. E. Cowan, S. P. Hatchett, E. Henry, S. Herman, M. Kartz, M. Key, J. Koch, A. J. Mackinnon, M. D. Perry, T. W. Phillips, M. Roth, T. C. Sangster, M. Singh, R. A. Snavely, M. Stoyer, B. C. Stuart, and S. C. Wilks, Report No. UCRL-JC-140019, 2000. |

3. | R. Hartley, M. Kartz, W. Behrenclt, A. Hines, G. Pollock, E. Bliss, T. Salmon, S. Winters, B. V. Wonterghem, and R. Zacharias, “Wavefront correction for static and dynamic aberrations to within 1 second of the system shot in the NIF Beamlet demonstration facility,” SPIE |

4. | J. P. Zou, A. M. Sautivet, J. Fils, L. Martin, K. Abdeli, C. Sauteret, and B. Wattellier, “Optimization of the dynamic wavefront control of a pulsed kilojoule/nanosecond-petawatt laser facility,” Appl. Opt. |

5. | C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, B. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan Petawatt—an ultra-high-intensity interaction facility,” Nucl. Fusion |

6. | K. L. Baker, E. A. Stappaerts, D. C. Homoelle, M. A. Henesian, E. S. Bliss, C. W. Siders, and C. P. J. Barty, Interferometric adaptive optics for high power laser pointing, wave-front control and phasing Journal of Micro/Nanolithography MEMS, and MOEMS (2009). |

7. | W. J. Hardy, |

8. | K. L. Baker, E. A. Stappaerts, D. Gavel, S. C. Wilks, J. Tucker, D. A. Silva, J. Olsen, S. S. Olivier, P. E. Young, M. W. Kartz, L. M. Flath, P. Kruelevitch, J. Crawford, and O. Azucena, “High-speed horizontal-path atmospheric turbulence correction with a large-actuator-number microelectromechanical system spatial light modulator in an interferometric phase-conjugation engine,” Opt. Lett. |

9. | J. Notaras and C. Paterson, “Point-diffrction interferometer for atmospheric daptive optics in strong scintillation,” Opt. Commun. |

10. | G. Love, T. Oag, and A. Kirby, “Common path interferometric wavefront sensor for extreme adaptive optics,” Opt. Express |

11. | K. L. Baker, “Interferometric Wavefront Sensors for High Contrast Imaging,” Optics Express |

12. | K. L. Baker, E. A. Stappaerts, S. C. Wilks, D. Gavel, P. E. Young, J. Tucker, S. S. Olivier, D. A. Silva, and J. Olsen, “Performance of a phase-conjugate engine implementing a finite-bit phase correction,” Opt. Lett. |

13. | D. Gordon, “Love, Nigel Andrews, Philip Burch, David Buscher, Peter Doel, Colin Dunlop, John Major, Richard Myers, Alan Purvis, Ray Sharples, Andrew Vick, Andrew Zadrozny, Sergio R. Restaino, and Andreas Glindemann, Binary adaptive optics: atmospheric wave-front correction with a half-wave phase shifter,” Appl. Opt. |

14. | D. Homoelle, K. L. Baker, E. Utterback, C. W. Siders, and C. P. J. Barty, presented at the Ultrafast Optics, France, 2009 (unpublished). |

15. | R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. |

16. | C. Dennis, Ghiglia and Mark D. Pritt, |

17. | K. L. Baker and D. A. Silva, “Evaluation of Two-Dimensional Phase Unwrapping Algorithms for Interferometric Characterization of Liquid-Crystal Spatial Light Modulators,” The Open Optics Journal |

18. | Private Communication from M.A. Henessian, (2008). |

19. | R. A. Sacks, M. A. Henesian, S. W. Haney, and J. B. Trenholme, Report No. UCRL-LR-105821–96, 1996. |

20. | G. C. Valley, “Long- and short-term Strehl ratios for turbulence with finite inner and outer scales,” Appl. Opt. |

21. | P. M. Harrington and B, M. Welsh, “Frequency-domain analysis of an adaptive optical system's temporal response,” Opt. Eng. |

22. | R. F. Lutomirski and H. T. Yura, “Wave Structure Function and Mutual Coherence Function of an Optical Wave in a Turbulent Atmosphere,” Journal of the Optical Society of America |

**OCIS Codes**

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

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: July 2, 2009

Revised Manuscript: August 4, 2009

Manuscript Accepted: August 10, 2009

Published: September 3, 2009

**Citation**

K. L. Baker, D. Homoelle, E. Utternback, E. A. Stappaerts, C. W. Siders, and C. P. J. Barty, "Interferometric adaptive optics testbed for laser pointing, wave-front control and phasing," Opt. Express **17**, 16696-16709 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16696

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### References

- R. A. Zacharias, N. R. Beer, E. S. Bliss, S. C. Burkhart, S. J. Cohen, S. B. Sutton, R. L. Van Atta, S. E. Winters, J. T. Salmon, M. R. Latta, C. J. Stolz, D. C. Pigg, and T. J. Arnold, “Alignment and wavefront control systems of the National Ignition Facility Opt,” Eng. 43(12), 2873 (2004).
- D. M. Pennington, C. G. Brown, T. E. Cowan, S. P. Hatchett, E. Henry, S. Herman, M. Kartz, M. Key, J. Koch, A. J. Mackinnon, M. D. Perry, T. W. Phillips, M. Roth, T. C. Sangster, M. Singh, R. A. Snavely, M. Stoyer, B. C. Stuart, and S. C. Wilks, Report No. UCRL-JC-140019, 2000.
- R. Hartley, M. Kartz, W. Behrenclt, A. Hines, G. Pollock, E. Bliss, T. Salmon, S. Winters, B. V. Wonterghem, and R. Zacharias, “Wavefront correction for static and dynamic aberrations to within 1 second of the system shot in the NIF Beamlet demonstration facility,” Proc. SPIE 3047, 294 (1997).
- J. P. Zou, A. M. Sautivet, J. Fils, L. Martin, K. Abdeli, C. Sauteret, and B. Wattellier, “Optimization of the dynamic wavefront control of a pulsed kilojoule/nanosecond-petawatt laser facility,” Appl. Opt. 47(5), 704–710 (2008). [CrossRef] [PubMed]
- C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, B. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, “Vulcan Petawatt—an ultra-high-intensity interaction facility,” Nucl. Fusion 44(12), 239 (2004). [CrossRef]
- K. L. Baker, E. A. Stappaerts, D. C. Homoelle, M. A. Henesian, E. S. Bliss, C. W. Siders, and C. P. J. Barty, Interferometric adaptive optics for high power laser pointing, wave-front control and phasing Journal of Micro/Nanolithography MEMS, and MOEMS (2009).
- W. J. Hardy, Adaptive Optics for Astronomical Telescopes. (Oxford University Press, Oxford, 1998).
- K. L. Baker, E. A. Stappaerts, D. Gavel, S. C. Wilks, J. Tucker, D. A. Silva, J. Olsen, S. S. Olivier, P. E. Young, M. W. Kartz, L. M. Flath, P. Kruelevitch, J. Crawford, and O. Azucena, “High-speed horizontal-path atmospheric turbulence correction with a large-actuator-number microelectromechanical system spatial light modulator in an interferometric phase-conjugation engine,” Opt. Lett. 29(15), 1781 (2004. [CrossRef] [PubMed]
- J. Notaras and C. Paterson, “Point-diffrction interferometer for atmospheric daptive optics in strong scintillation,” Opt. Commun. 281(3), 360–367 (2008). [CrossRef]
- G. Love, T. Oag, and A. Kirby, “Common path interferometric wavefront sensor for extreme adaptive optics,” Opt. Express 13(9), 3491–3499 (2005). [CrossRef] [PubMed]
- K. L. Baker, “Interferometric Wavefront Sensors for High Contrast Imaging,” Opt. Express 14, 10970 (2006). [CrossRef] [PubMed]
- K. L. Baker, E. A. Stappaerts, S. C. Wilks, D. Gavel, P. E. Young, J. Tucker, S. S. Olivier, D. A. Silva, and J. Olsen, "Performance of a phase-conjugate engine implementing a finite-bit phase correction," Opt. Lett. 29(9), 980-982 (2004). [CrossRef] [PubMed]
- G. D. Love, N. Andrews, P. Burch, D. Buscher, P. Doel, C. Dunlop, J. Major, R. Myers, A. Purvis, R. Sharples, A. Vick, A. Zadrozny, S. R. Restaino, and A. Glindemann, "Binary adaptive optics: atmospheric wave-front correction with a half-wave phase shifter," Appl. Opt. 34(27), 6058 (1995).
- D. Homoelle, K. L. Baker, E. Utterback, C. W. Siders, and C. P. J. Barty, presented at the Ultrafast Optics, France, 2009 (unpublished).
- R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988). [CrossRef]
- G. C. Dennis and M. D. Pritt, Two-Dimensional Phase Unwrapping. (John Wiley & Sons, Inc., New York, 1998).
- K. L. Baker and D. A. Silva, “Evaluation of Two-Dimensional Phase Unwrapping Algorithms for Interferometric Characterization of Liquid-Crystal Spatial Light Modulators,” The Open Optics Journal 2, 48 (2008). [CrossRef]
- Private Communication from M.A. Henessian, (2008).
- R. A. Sacks, M. A. Henesian, S. W. Haney, and J. B. Trenholme, Report No. UCRL-LR-105821–96, 1996.
- G. C. Valley, “Long- and short-term Strehl ratios for turbulence with finite inner and outer scales,” Appl. Opt. 18(7), 984 (1979). [CrossRef] [PubMed]
- P. M. Harrington and B. M. Welsh, “Frequency-domain analysis of an adaptive optical system's temporal response,” Opt. Eng. 33(7), 2336 (1994). [CrossRef]
- R. F. Lutomirski and H. T. Yura, “Wave Structure Function and Mutual Coherence Function of an Optical Wave in a Turbulent Atmosphere,” J. Opt. Soc. Am. 61(4), 482 (1971). [CrossRef]

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