## Calibration and testing with real turbulence of a pyramid sensor employing static modulation

Optics Express, Vol. 17, Issue 9, pp. 7186-7195 (2009)

http://dx.doi.org/10.1364/OE.17.007186

Acrobat PDF (213 KB)

### Abstract

The pyramid sensor (PS) is an interesting alternative to the Shack-Hartmann wavefront sensor (SH WFS) for astronomical Adaptive Optics (AO) because of its potential advantages in sensitivity and applicability to novel wavefront sensing schemes. The PS uses a pyramidal prism to perform a knife-edge test in two dimensions simultaneously and relies on modulating the position of the prism to increase the linear dynamic range. It has been suggested that this could also be accomplished by a static diffusing element. We test this idea and show that the diffuser produces a modulation effect. We compare the results of our PS to a SH WFS measuring spatial and temporal properties of real turbulence produced in the lab with a hot-air turbulence generator.

© 2009 Optical Society of America

## 1. Introduction

22. O. Keskin, L. Jolissaint, and C. Bradley, “Hot-air optical turbulence generator for the testing of adaptive optics systems: principles and characterization,” Appl. Opt. **45**(20), 4888–4897 (2006). [PubMed]

## 2. Modulation and the pyramid sensor

*U*

^{±}

_{p}(

*x*,

*y*)∣

^{2}are the intensity in the upper and lower pupils respectively and the signal is normalized by the total intensity, ∣

*A*∣

^{2}. The diffraction theory of the Foucault knife-edge test[23, 24, 25, 26] gives the expression for the illumination in the re-imaged pupils

*α*is the modulation angle and

*f*is the spatial frequency. The result shows two distinct behaviors. [6] For

*f*<

*α*/

*λ*the sensor signal in the Fourier domain is proportional to the spatial frequency indicating that the PS acts as a slope sensor. In the regime

*f*> α/

*λ*, the PS behaves as a phase sensor. [6] Thus, the size of the modulation angle determines the dynamic range over which the PS gives a linear response to tilt.

## 3. Experiment

## 4. Results

^{-1}and 0.89 ±0.01 mrad

^{-1}respectively. The sensitivity is roughly a factor of two lower for the 1.0° diffuser as it produces a blur spot on the PS apex which is larger by the same factor. The use of the diffuser clearly results in a linear response of the intensity in the re-imaged pupil to an incoming tilt. This directly demonstrates that modulation can be achieved with a static diffusing element. The 0.5° diffuser was used to make the measurements of the turbulent wavefronts produced by the turbulator.

*r*

_{0}, the outer scale,

*L*

_{0}, and the inner scale,

*l*

_{0}. The mini-wavescope software does a reconstruction of the wavefront and a projection onto Zernike modes. The variance of the Zernike coefficients of the wavefront are calculated from the time series of measured coefficients and fit to the expected theoretical values from the Hills-Andrew model giving estimates for

*r*

_{0},

*L*

_{0}, and

*l*

_{0}. This is shown in Fig. 5(a). The values for

*r*

_{0},

*L*

_{0}, and

*l*

_{0}are 3.40 ±0.03 mm, 170 ± 10 mm, and 11.57 ± 0.07 respectively where the error is that given by

*χ*

^{2}minimization and estimated from the error on the variance of the coefficients. For the PS, we have 1d sensor and forgo the wavefront reconstruction. Instead the wavefront

*slope*is projected onto Zernike modes and the variance of these coefficients is calculated from the measured time series. At this point a similar fit is performed to the expected values of the coefficients of the wavefront

*slope*including the damping effects of the inner and outer scale again using the Hill’s-Andrew model. The PS case is shown in Fig. 5(b) and the values of

*r*

_{0},

*L*

_{0}, and

*l*

_{0}are 2.55 ± 0.08 mm, 70. ± 9 mm, and 4.6 ± 0.1 mm. Both sensors, although estimating slightly different parameters, are consistent with other spatial characterizations hot-air turbulence generators. [21, 34

34. E. Masciadri and J. Vernin, “Optical technique for inner-scale measurement: possible astronomical applications,” Appl. Opt. **36**, 1320–1327 (1997). [PubMed]

*T*

^{-6/5}. The relationship between

*r*

_{0}and the refractive index structure constant,

*C*

^{2}

_{N}, is given by

*C*

^{2}

_{N}profile has been replaced by the constant

*C*

^{2}

_{N}Δ

*h*. [37]

*C*

^{2}

*N*Δ

*h*is proportional to Δ

*T*

^{2}by Gladstone’s law [21] and thus the expected dependence of the Fried parameter is

*r*

_{0}∝= Δ

*T*

^{-6/5}. However, it has been observed that

*C*

^{2}

_{N}Δ

*h*depends linearly on Δ

*T*in the turbulator [21] and in this case

*r*

_{0}∝ Δ

*T*

^{-3/5}. The Fried parameter was measured for several temperature differences (by the procedure described above) and Fig. 6 shows a comparison of the results for both sensors. Power law fits to the data give an exponent of

*b*= 0.89 ± 0.08 and

*b*= 0.91 ± 0.09 for the mini-Wavescope and PS respectively. The two sensors agree and the exponent is between the -6/5 and -3/5 mentioned above. The PS does, however, systematically underestimate

*r*

_{0}compared to the mini-Wavescope.

*r*

_{0}due, in part, to the way in which the diffuser works. The diffuser is designed to induce a Gaussian blur corresponding to its diffusion angle on a beam which fills its clear aperture (~25 mm). For a given pencil of rays corresponding to each subaperture the effect of the diffuser may be a slightly more or less than the overall diffusion angle. This implies that the sensitivity of the PS varies with each subaperture. This is in fact the case and Fig. 7 shows a histogram of the measured sensitivities for the PS subapertures. The distribution has a mean value of 1.76 mrad

^{-1}close to the global calibration constant of 1.62 mrad

^{-1}. The standard deviation of the sensitivity is 0.9 mrad

^{-1}. The variation of sensitivity would appear in the wavefront slope measurements as a high order aberration and thus systematically serve to reduce

*r*

_{0}. The sensitivity of a given subaperture depends on the alignment of the system and due to the practical necessity to make adjustments to our prototype sensor, the subaperture sensitivities changed from run to run unlike the global calibration constant which is virtually independent of alignment. This effect could be calibrated out in future work with a re-designed, second generation sensor.

## 5. Conclusion and outlook

*r*

_{0}observed with the PS compares well to that measured by the SH WFS displaying the expected power law dependence on Δ

*T*. The

*r*

_{0}values observed with the PS are consistently lower than those of the SH WFS, which we believe to be due to variations in the sensitivity over the pupil of the PS due to the diffuser. This effect will be calibrated out in future work. Overall the spatial characterization of the turbulence with both sensors compare well to the expected range from other characterizations of hot air turbulence generators available in the literature.

## References and links

1. | R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. |

2. | J. M. Beckers, “Adaptive optics for astronomy - Principles, performance, and applications,” Annu. Rev. Astron. Astrophys. |

3. | R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic Wave Front sensor in closed loop Adaptive Optics,” Astron. Astrophys. |

4. | S. Esposito, A. Riccardi, and O. Feeney, “Closed-loop performance of pyramid wavefront sensor,” in Proc. SPIE |

5. | S. Esposito and A. Riccardi, “Pyramid Wavefront Sensor behavior in partial correction Adaptive Optic systems,” Astron. Astrophys. |

6. | C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. |

7. | A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, “Laboratory characterization of a Foucault-like wavefront sensor for adaptive optics,” in |

08. | S. Esposito, O. Feeney, and A. Riccardi, “Laboratory test of a pyramid wavefront sensor,” in Proc. SPIE |

09. | R. Ragazzoni, A. Ghedina, A. Baruffolo, E. Marchetti, J. Farinato, T. Niero, G. Crimi, and M. Ghigo, “Testing the pyramid wavefront sensor on the sky,” in Proc. SPIE |

10. | A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, “On Sky Test of the Pyramid Wavefront Sensor,” in Proc. SPIE |

11. | J. B. Costa, S. Hippler, M. Feldt, S. Esposito, R. Ragazzoni, P. Bizenberger, E. Puga, and T. F. E. Henning, “PYRAMIR: a near-infrared pyramid wavefront sensor for the Calar Alto adaptive optics system,” in Proc. SPIE |

12. | J. B. Costa, M. Feldt, K. Wagner, P. Bizenberger, S. Hippler, H. Baumeister, M. Stumpf, R. Ragazzoni, S. Esposito, and T. Henning, “Status report of PYRAMIR: a near-infrared pyramid wavefront sensor for ALFA,” in Proc. SPIE |

13. | R. Ragazzoni, “Adaptive optics for giant telescopes: NGS vs. LGS,” in |

14. | R. Ragazzoni, J. Farinato, and E. Marchetti, “Adaptive optics for 100-m-class telescopes: new challenges require new solutions,” in Proc. SPIE |

15. | E. Diolaiti, R. Ragazzoni, and M. Tordi, “Closed loop performance of a layer-oriented multi-conjugate adaptive optics system,” Astron. Astrophys. |

16. | J. M. Beckers, “Increasing the Size of the Isoplanatic Patch with Multiconjugate Adaptive Optics,” in |

17. | J. M. Beckers, “Detailed compensation of atmospheric seeing using multiconjugate adaptive optics,” in Proc. SPIE |

18. | R. Ragazzoni, E. Diolaiti, J. Farinato, E. Fedrigo, E. Marchetti, M. Tordi, and D. Kirkman, “Multiple field of view layer-oriented adaptive optics. Nearly whole sky coverage on 8 m class telescopes and beyond,” Astron. Astrophys. |

19. | D. Peter, M. Feldt, B. Dorner, T. Henning, S. Hippler, and J. Aceituno, “PYRAMIR: Calibration and Operation of a Pyramid Near-Infrared Wavefront Sensor,” Publ. Astron. Soc. Pac. |

20. | R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. |

21. | O. Keskin, L. Jolissaint, C. Bradley, S. Dost, and I. Sharf, “Hot-air turbulence generator for multiconjugate adaptive optics,” in Proc. SPIE |

22. | O. Keskin, L. Jolissaint, and C. Bradley, “Hot-air optical turbulence generator for the testing of adaptive optics systems: principles and characterization,” Appl. Opt. |

23. | Rayleigh, “On methods for detecting small optical retardations, and on the theory of Foucault’s test,” Philos. Mag. |

24. | E. H. Linfoot, “A Contribution to the Theory of the Foucault Test,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences |

25. | E. H. Linfoot, “On the theory of the Zonal Foucault Test,” Monthly Notices of the Royal Astronomical Society |

26. | E. H. Linfoot, “On the Interpretation of the Foucault Test,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences |

27. | J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Vérinaud, M. Feldt, S. Esposito, E. Puga, and J. Farinato, “Is there need of any modulation in the pyramid wavefront sensor?” in Proc. SPIE |

28. | J. B. Costa, M. Stumpf, and M. Feldt, “Testing a nonmodulated pyramid wavefront sensor,” in Proc. SPIE |

29. | P. Pugh, D. Lobb, D. Walker, and T. Williams, “Pupil-imaging wavefront gradient sensor,” Proc. SPIE |

30. | R. Clare and R. Lane, “Comparison of wavefront sensing with the Shack-Hartmann and pyramid sensors,” Proc. SPIE |

31. | R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. |

32. | D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A |

33. | L. C. Andrews, “An analystical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. |

34. | E. Masciadri and J. Vernin, “Optical technique for inner-scale measurement: possible astronomical applications,” Appl. Opt. |

35. | A. Fuchs, M. Tallon, and J. Vernin, “Focusing on a Turbulent Layer: Principle of the “Generalized SCIDAR”,” Publ. Astron. Soc. Pac. |

36. | C. Innocenti and A. Consortini, “Estimate of characteristics scales of atmospheric turbulence by thin beams: comparison between the von Karman and Hill-Andrews models,” J. Mod. Opt. |

37. | R. Tyson, |

**OCIS Codes**

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

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(010.7060) Atmospheric and oceanic optics : Turbulence

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

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: February 26, 2009

Revised Manuscript: April 11, 2009

Manuscript Accepted: April 13, 2009

Published: April 15, 2009

**Citation**

Jeffrey LeDue, Laurent Jolissaint, Jean-Pierre Véran, and Colin Bradley, "Calibration and testing with real turbulence of a pyramid sensor employing static modulation," Opt. Express **17**, 7186-7195 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7186

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### References

- R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
- J. M. Beckers, "Adaptive optics for astronomy - Principles, performance, and applications," Annu. Rev. Astron. Astrophys. 31, 13-62 (1993).
- R. Ragazzoni and J. Farinato, "Sensitivity of a pyramidic Wave Front sensor in closed loop Adaptive Optics," Astron. Astrophys. 350, L23-L26 (1999).
- S. Esposito, A. Riccardi, and O. Feeney, "Closed-loop performance of pyramid wavefront sensor," Proc. SPIE 4034, 184-189 (2000).
- S. Esposito and A. Riccardi, "Pyramid Wavefront Sensor behavior in partial correction Adaptive Optic systems," Astron. Astrophys. 369, L9-L12 (2001).
- C. Vérinaud, "On the nature of the measurements provided by a pyramid wave-front sensor," Opt. Commun. 233, 27-38 (2004).
- A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, "Laboratory characterization of a Foucault-like wavefront sensor for adaptive optics," Proc. SPIE 3353, 941-951 (1998).
- S. Esposito, O. Feeney, and A. Riccardi, "Laboratory test of a pyramid wavefront sensor," Proc. SPIE 4007, 416-422 (2000).
- R. Ragazzoni, A. Ghedina, A. Baruffolo, E. Marchetti, J. Farinato, T. Niero, G. Crimi, and M. Ghigo, "Testing the pyramid wavefront sensor on the sky," Proc. SPIE 4007, 423-430 (2000).
- A. Ghedina, M. Cecconi, R. Ragazzoni, J. Farinato, A. Baruffolo, G. Crimi, E. Diolaiti, S. Esposito, L. Fini, M. Ghigo, E. Marchetti, T. Niero, and A. Puglisi, "On Sky Test of the PyramidWavefront Sensor," Proc. SPIE 4839, 869-877 (2003).
- J. B. Costa, S. Hippler, M. Feldt, S. Esposito, R. Ragazzoni, P. Bizenberger, E. Puga, and T. F. E. Henning, "PYRAMIR: a near-infrared pyramid wavefront sensor for the Calar Alto adaptive optics system," Proc. SPIE 4839, 280-287 (2003).
- J. B. Costa, M. Feldt, K. Wagner, P. Bizenberger, S. Hippler, H. Baumeister, M. Stumpf, R. Ragazzoni, S. Esposito, and T. Henning, "Status report of PYRAMIR: a near-infrared pyramid wavefront sensor for ALFA," Proc. SPIE 5490, 1189-1199 (2004).
- R. Ragazzoni, "Adaptive optics for giant telescopes: NGS vs. LGS," in Proceedings of the Backaskog workshop on extremely large telescopes, pp. 175-180 (2000).
- R. Ragazzoni, J. Farinato, and E. Marchetti, "Adaptive optics for 100-m-class telescopes: new challenges require new solutions," Proc. SPIE 4007, 1076-1087 (2000).
- E. Diolaiti, R. Ragazzoni, and M. Tordi, "Closed loop performance of a layer-oriented multi-conjugate adaptive optics system," Astron. Astrophys. 372, 710-718 (2001).
- J. M. Beckers, "Increasing the Size of the Isoplanatic Patch with Multiconjugate Adaptive Optics," in Proceedings of a ESO Conference on Very Large Telescopes and their Instrumentation, held in Garching, March 21-24, 1988,Garching: European Southern Observatory, pp. 693-703 (1988).
- J. M. Beckers, "Detailed compensation of atmospheric seeing using multiconjugate adaptive optics," Proc.SPIE 1114, 215-217 (1989).
- R. Ragazzoni, E. Diolaiti, J. Farinato, E. Fedrigo, E. Marchetti, M. Tordi, and D. Kirkman, "Multiple field of view layer-oriented adaptive optics. Nearly whole sky coverage on 8 m class telescopes and beyond," Astron. Astrophys. 396, 731-744 (2002).
- D. Peter, M. Feldt, B. Dorner, T. Henning, S. Hippler, and J. Aceituno, "PYRAMIR: Calibration and Operation of a Pyramid Near-Infrared Wavefront Sensor," Publ. Astron. Soc. Pac. 120, 872-886 (2008). 0808.0137.
- R. Ragazzoni, E. Diolaiti, and E. Vernet, "A pyramid wavefront sensor with no dynamic modulation," Opt. Commun. 208, 51-60 (2002).
- O. Keskin, L. Jolissaint, C. Bradley, S. Dost, and I. Sharf, "Hot-air turbulence generator for multiconjugate adaptive optics," Proc. SPIE 5162, pp. 49-57 (2003).
- O. Keskin, L. Jolissaint, and C. Bradley, "Hot-air optical turbulence generator for the testing of adaptive optics systems: principles and characterization," Appl. Opt. 45, 4888-4897 (2006). [PubMed]
- Rayleigh, "On methods for detecting small optical retardations, and on the theory of Foucault’s test," Philos. Mag. 33, 161-178 (1917).
- E. H. Linfoot, "A Contribution to the Theory of the Foucault Test," Proc. R. Soc. London Ser. A, Math. Phys. Scie. 186, 72-99 (1946).
- E. H. Linfoot, "On the theory of the Zonal Foucault Test," Mon. Not. R. Astron. Soc. 108, 428-445 (1948).
- E. H. Linfoot, "On the Interpretation of the Foucault Test," Proc. R. Soc. London. Series A, Math. Phys. Scie. 193, 248-259 (1948).
- J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. V’erinaud, M. Feldt, S. Esposito, E. Puga, and J. Farinato, "Is there need of any modulation in the pyramid wavefront sensor?" Proc. SPIE 4839, 288-298 (2003).
- J. B. Costa, M. Stumpf, and M. Feldt, "Testing a nonmodulated pyramid wavefront sensor," Proc. SPIE 5490, 1304-1314 (2004).
- P. Pugh, D. Lobb, D. Walker and T. Williams, "Pupil-imaging wavefront gradient sensor," Proc. SPIE 2534, 312-317 (1995).
- R. Clare and R. Lane, "Comparison of wavefront sensing with the Shack-Hartmann and pyramid sensors," Proc. SPIE 5490, 1211-1222 (2004).
- R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976).
- D. M. Winker, "Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence," J. Opt. Soc. Am. A 8, 1568-1573 (1991).
- L. C. Andrews, "An analystical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere," J. Mod. Opt. 39, 1849-1853 (1992).
- E. Masciadri and J. Vernin, "Optical technique for inner-scale measurement: possible astronomical applications," Appl. Opt. 36, 1320-1327 (1997). [PubMed]
- A. Fuchs, M. Tallon, and J. Vernin, "Focusing on a Turbulent Layer: Principle of the "Generalized SCIDAR,"Publ. Astron. Soc. Pac. 110, 86-91 (1998).
- C. Innocenti and A. Consortini, "Estimate of characteristics scales of atmospheric turbulence by thin beams: comparison between the von Karman and Hill-Andrews models," J. Mod. Opt. 51, 333-342 (2004).
- R. Tyson, Principles of Adaptive Optics, 2nd ed. (Academic press, 1998).

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