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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 9 — Apr. 25, 2011
  • pp: 8135–8150
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Laboratory demonstrations on a pyramid wavefront sensor without modulation for closed-loop adaptive optics system

Shengqian Wang, Changhui Rao, Hao Xian, Jianlin Zhang, Jianxin Wang, and Zheng Liu  »View Author Affiliations


Optics Express, Vol. 19, Issue 9, pp. 8135-8150 (2011)
http://dx.doi.org/10.1364/OE.19.008135


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Abstract

The feasibility and performance of the pyramid wavefront sensor without modulation used in closed-loop adaptive optics system is investigated in this paper. The theory concepts and some simulation results are given to describe the detection trend and the linearity range of such a sensor with the aim to better understand its properties, and then a laboratory setup of the adaptive optics system based on this sensor and the liquid-crystal spatial light modulator is built. The correction results for the individual Zernike aberrations and the Kolmogorov phase screens are presented to demonstrate that the pyramid wavefront sensor without modulation can work as expected for closed-loop adaptive optics system.

© 2011 OSA

1. Introduction

At present most of the PWFS are used with modulation based on oscillating optical component in order to give a linear measurement of the local tilt, and it has been studied in the laboratory and successfully operated on the sky based on the modulation mode. Modulation plays a central role in the geometrical optical approximation to improve the linear range of the PWFS, which is very small in the nonmodulated case. Burvall discussed the approximate linear relationship between the wavefront derivative and the sensor response with modulation [9

9. A. Burvall, E. Daly, S. R. Chamot, and C. Dainty, “Linearity of the pyramid wavefront sensor,” Opt. Express 14(25), 11925–11934 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-11925. [CrossRef] [PubMed]

], and Vérinaud’s analysis explained the effects of modulation according to the diffraction optics theory [10

10. C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233(1-3), 27–38 (2004). [CrossRef]

]. However, the sensitivity of the PWFS will decrease with the modulation amplitude and the moving parts increase the complexity of the system. There is an attractive idea that rise with the use of the PWFS is how to work without any dynamic modulation, which would greatly simplify the optical and mechanical design of the adaptive optics system and also give highest sensitivity as expected to be achieved. Costa presented a study showing that the higher order uncompensated residuals of the atmospheric turbulence are equivalent to a modulation for the lower compensated modes [11

11. J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, and J. Farinato, “Is there need of modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003). [CrossRef]

,12

12. J. B. Costa, “Modulation effect of the atmosphere in a pyramid wave-front sensor,” Appl. Opt. 44(1), 60–66 (2005). [PubMed]

]. Ragazzoni introduced a static modulation method based on a light diffusing element, which causes a blur of the spot on the pyramid pin, in a similar way to the dynamic modulation [13

13. R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208(1-3), 51–60 (2002). [CrossRef]

], and then LeDue tested this idea using a holographic diffuser [14

14. J. LeDue, L. Jolissaint, J.-P. Véran, and C. Bradley, “Calibration and testing with real turbulence of a pyramid sensor employing static modulation,” Opt. Express 17(9), 7186–7195 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7186. [CrossRef] [PubMed]

]. Korkiakoski has found an iterative reconstruction method that can be used to evaluate the significance of the nonlinearities in the nonmodulated PWFS [15

15. V. Korkiakoski, C. Vérinaud, M. Le Louarn, and R. Conan, “Comparison between a model-based and a conventional pyramid sensor reconstructor,” Appl. Opt. 46(24), 6176–6184 (2007). [CrossRef] [PubMed]

].

The aim of this paper is to clearly present the choices that can be made to design the PWFS without modulation using in the closed-loop adaptive optics system. Firstly, the special behavior of the PWFS without modulation is described with theoretical analysis and numerical simulations. The detection trend of the PWFS is studied to illustrate that the correction orientation of the nonmodulated PWFS is right to have the successive iteration process for the closed-loop adaptive optics system. (According to this paper, the detection trend is the normalized multiplication of the wavefront gradient with the PWFS signal. If the signal of the detection trend is a positive number, the polarity of the PWFS signal is accord with the wavefront slope and the closed-loop correction along this orientation will be effective.) At the same time, the behavior of the small linear range for the PWFS without modulation is considered in this paper. Finally the laboratory experimental setup based on this sensor and the liquid-crystal spatial light modulator (LC SLM) has been developed and the experimental results demonstrate the feasibility of the PWFS without modulation used in the closed-loop adaptive optics system.

2. Theory and simulation of the PWFS without modulation

The principle of the PWFS has been well described in many other papers so that we just refer the conclusions important to this work. When the PWFS is used as a wavefront sensor, four images of the pupil are created on the detector and the detection signals of the PWFS are calculated with the following formula [1

1. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996). [CrossRef]

]:.
Sx(x,y)=I1(x,y)+I4(x,y)I2(x,y)I3(x,y)I1(x,y)+I2(x,y)+I3(x,y)+I4(x,y)
(1)
Sy(x,y)=I1(x,y)+I2(x,y)I3(x,y)I4(x,y)I1(x,y)+I2(x,y)+I3(x,y)+I4(x,y)
(2)
whereIi(x,y)is the intensity distribution corresponding to the quadrant i of the detector. For circular modulation with angular amplitude θ (assuming the angular amplitude is bigger than the gradient of the wavefront aberration), it follows that the detection signals are proportional to the phase derivatives from the simple geometrical optics considerations.

If no modulation is applied, the PWFS only yields the sign of the wavefront slope based on the knife-edge test, that is to say the linear range of the signal is very small according to the geometrical optics theory. Using the diffraction theory of the Foucault knife-edge test, the relationship between the wavefront phase ϕ(x,y) and the PWFS signal Sx(x,y) is given by the following expression [9

9. A. Burvall, E. Daly, S. R. Chamot, and C. Dainty, “Linearity of the pyramid wavefront sensor,” Opt. Express 14(25), 11925–11934 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-11925. [CrossRef] [PubMed]

] (here the PWFS is simply replaced by a roof prism and interferences between the two pupils are neglected, and the same formula to the PWFS signal Sy(x,y))
Sx(x,y)=1πB(y)B(y)p.v.sin(ϕ(x,y)ϕ(x,y))xxdx
(3)
where B(y) is the edge point of the chord passing the point (x,y) in the pupil. In the integral the contribution of the sine-term is weighted with the distance between the integration point (x,y) and the measured point(x,y). Obviously, there is not a linear relation between the PWFS signal and the phase differences according to this formula.

It is well known that the system can achieve successful closed-loop operation if the system satisfies two important conditions. The first one is that the detection trend of signal must be right during the closed-loop process, which means that the estimated correction is in the right direction to cause the residual variance of the closed-loop error to decrease continuously by successive iterations. The other one is that a small linear range is exist when the loop is closed successfully, which makes the closed-loop system to retain such a status in the closed-loop process. Due to the complexity of the theoretical formula (especially considering other impact factors, for example the interferences between the four pupils of the PWFS, the theoretical formula could become more complicated and hard to analyze), hence we adopt simulation method in the following discussion and present the preliminary simulation results to explain that it is possible to use the PWFS without modulation in the closed-loop adaptive system.

The simulated figures (shown from Fig. 2
Fig. 2 Results of the Z3 Zernike polynomial.
9
Fig. 9 Results of the Z10 Zernike polynomial.
) of PWFS without modulation for a series of Zernike modes (from Z3 to Z10, with RMS = 0.3λ) are obtained according to the above simulation modal. The Zernike order used is the Noll ordering without the piston mode [16

16. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]

]. As shown in these figures, Ii(x,y) (i = 1,2,3,4) corresponds to the normalized intensity map created through the i-th quadrant of the PWFS, and Wx denotes the normalized local derivatives on the wavefront (where Wx=W(x,y)/x), and Sx indicates the signal of the PWFS given by formula (1), and Trend X represents the normalized multiplication of the wavefront local tilt Wxwith the PWFS signal Sx (the same to Wy, Sy and Trend Y). With the qualitative point of view, some characteristics are found from these figures: firstly, the figure Sx has saturation effect due to the small linear range, but the outline of Sx image is identical with the wavefront difference Wx; secondly, for the signal Trend and Trend Y0, this is in agreement with the comment previously stated that the detection trend of the phase difference is right when using Sx and Sy as the local wavefront derivative. To study the general behavior of these characteristics, the simulated images for the Kolmogorov phase screen with RMS = 0.47λ (according to D/r0 = 12, where D is the input aperture diameter, and r0 is the Fried parameter) composed of 3 to 40 Zernike modes are shown in Fig. 10
Fig. 10 Results of the Kolmogorov phase screen (D/r0 = 12).
, and the same results can be obtained.

Fig. 3 Results of the Z4 Zernike polynomial
Fig. 4 Results of the Z5 Zernike polynomial.
Fig. 5 Results of the Z6 Zernike polynomial.
Fig. 6 Results of the Z7 Zernike polynomial.
Fig. 7 Results of the Z8 Zernike polynomial.
Fig. 8 Results of the Z9 Zernike polynomial.

3. Experiment results

The closed-loop optical layout of the adaptive optics system based on the PWFS without modulation and the LC SLM in the laboratory setup is sketched in Fig. 13
Fig. 13 Optical layout of the system based on the PWFS without modulation and the LC SLM.
. The pyramid optical part is the double refractive pyramid that behaves as a single pyramid having a base angle equal to the difference between the two base angles of these two pyramids. The first pyramid component is made of BAK6 glass, and the other one is made of ZK9 glass. A polarized He-Ne laser with 632.8 nm wavelength is used as the light source for the experiment. After passing through the lens and the pinhole, the collimated plane laser output illuminates the LC SLM, and then the beam (with an aperture diameter of 4.61mm) is reflected from the SLM and incidences into the PWFS. By using the signal from the images of the PWFS, the reconstructed wavefront is loaded onto the SLM and the wave aberration is corrected. In this configuration the LC SLM has two functions. First it is used to introduce the static aberration into the system and afterwards to correct the distorted wavefront as the deformable mirror. The lens 1 forms an F/217 beam that is focused on the tip of the glass pyramid, and the choice of the F-ratio leads to a spot size of approximately 330μm on the pyramid tip, which reduces the influence of the light loss due to the pyramid edges and the roof-shaped. Then the four beams formed by the four facets of the pyramid are re-imaged on the CCD 1 through the lens 2. Each of the four pupils has an optically defined diameter of 0.277mm. The lens 3 and the CCD 2 form the far-field imaging optical layout, so that the actual shape of the far-field spot can be seen at any time.

The plots in Fig. 14
Fig. 14 Plots of the RMS and SR for defocus, astigmatism and coma aberration.
represent the iteration process in closed-loop compensation for the defocus, astigmatism, and coma aberration, with four different initial values of RMS (0.1λ, 0.2λ, 0.3λ and 0.4λ) respectively. The results shown are for 20 correction loops, and it can be seen that after 15 loops the RMS (root-mean-square) and SR (strehl ratio) of the surface measurements have a good convergence property.

The results of the laboratory measurements using the PWFS without modulation for the Kolmogorov turbulence phase screens (D/r0 = 40 and D/r0 = 80, without global slope) are obtained in Fig. 15
Fig. 15 Plot of the RMS and SR for the turbulence phase-screen (D/r0 = 40, without global slope); the initial and corrected wavefront and far-field images.
and Fig. 16
Fig. 16 Plot of the RMS and SR for the turbulence phase-screen (D/r0 = 80, without global slope); the initial and corrected wavefront and far-field images.
respectively. According to these figures, the corrected wavefront has a RMS of about 0.05λ in each case. Referring to initial RMS values (1.300λ and 2.750λ) of these two turbulence phase screens, it is clear that the wavefront correction using the PWFS without modulation is feasible.

Two points must be emphasized about the experimental results. Firstly, the LC SLM is used as the deformable mirror with the main advantage to generate the modes with high precision at small amplitudes, so the interaction matrix that connects the wave-space to the PWFS is determined inside the linear range. Additionally, the aberration applied to the SLC is static without change in real-time like the practice atmospheric turbulence, namely the experiment is just a “static closed-loop” so the temporal characteristics of atmospheric turbulence are not considered in the calculation of the data. It is planned in future to complement the laboratory system with simulated turbulence phase screen generated by the Hot-Air Turbulence Generator.

4. More considerations for the PWFS used in the practical astronomical AO system

The result in Fig. 17
Fig. 17 Plot of the RMS with the number of iteration (calibration amplitude is 0.1λ, the random adding phase perturbation to each of the loop iteration has the RMS about 0.1λ).
is according to the calibration matrix with amplitude 0.1λ, and the gain factor is changed from 0.5 to 3. With the aim to represent the real astronomical condition, one way to achieve this is by adding some phase perturbation of the proper amplitude to each of the loop iteration, so we add the random phase perturbation to each of the loop iteration with the RMS about 0.1λ. It is clear with the proper gain it is possible to close the loop without modulation. The result with the random phase perturbation about 0.15λ is illustrated in Fig. 18
Fig. 18 Plot of the RMS with the number of iteration (calibration amplitude is 0.1λ, the random adding phase perturbation to each of the loop iteration has the RMS about 0.15λ).
and the same conclusion is obtained.

We know that the calibration amplitude plays an important role for the closed loop performance. Especially for a non-modulated PWFS this amplitude is critical because of the small linear regime, so we do the same simulation but with the calibration amplitude 0.3 (the results is shown in Fig. 19
Fig. 19 Plot of the RMS with the number of iteration (calibration amplitude is 0.3λ, the random adding phase perturbation to each of the loop iteration has the RMS about 0.1λ).
and Fig. 20
Fig. 20 Plot of the RMS with the number of iteration (calibration amplitude is 0.3λ, the random adding phase perturbation to each of the loop iteration has the RMS about 0.15λ).
). It can be seen from the figures that although the calibration amplitude is out of the linearity range of the sensor, with the proper gain factor the system can also be closed.

The modulation linearizes the PWFS behavior and reduces the number of iterations required to converge to the proper estimate of the wavefront, and the PWFS without modulation will take more iterations to close the loop. This is because the PWFS is saturated in the beginning and therefore the correction applied to the deformable mirror is too small, so it takes more iterations to “pull” the PWFS out of saturation and to adapt for changes in the atmosphere. The simulation is analyzed according to different modulation amplitude and is shown in Fig. 21
Fig. 21 Plot of the RMS with the number of iteration (calibration amplitude is 0.1λ, the random adding phase perturbation to each of the loop iteration has the RMS about 0.15λ).
(the modulation amplitudes are 1 times diffraction limit width, 2 times, 3 times and 4 times, respectively). As can be seen from this figure, along with the increase of the modulation amplitude the number of iterations required to achieve the certain closed-loop effect will gradually decrease and apparently the condition without modulation needs maximum number of iterations.

5. Conclusions and future work

The detection trend and the linearity range of the PWFS without modulation are explained in this paper and an optical system has been built in the laboratory to analyze the characteristics of the PWFS without modulation. For the defocus, astigmatism, coma and the atmospheric turbulence phase screens, the corresponding results are obtained during the closed-loop process. The experimental results demonstrate the feasibility of PWFS without modulation used in the adaptive optics system. The final stage of the experiment will be developed to work with the dynamic turbulence phase screen and the other deformable mirror. The actual astronomical observation AO system must consider a series of other issues, for example the atmospheric turbulence seeing, the turbulence time feature, the detector photon noise and read noise. So the PWFS without modulation applied in practical astronomical observation still needs a lot of work to do.

Acknowledgments

The project is supported by the National Natural Science Foundation of China (Grant No 61008038), and is also supported by West Light Foundation of The Chinese Academy of Sciences.

References and links

1.

R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996). [CrossRef]

2.

R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. 350, 23–26 (1999).

3.

T. Y. Chew, R. M. Clare, and R. G. Lane, “A comparison of the Shack-Hartmann and pyramid wavefront sensors,” Opt. Commun. 268(2), 189–195 (2006). [CrossRef]

4.

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). [CrossRef]

5.

S. Esposito, A. Tozzi, A. Puglisi, E. Pinna, A. Richardi, S. Busoni, L. Busoni, P. Stefanini, M. Xompero, D. Zanotti, and F. Pieralli, “First light AO system for LBT: toward on-sky operation,” Proc. SPIE 6272, 62720A, 62720A-9 (2006). [CrossRef]

6.

M. Feldt, D. Peter, S. Hippler, Th. Henning, J. Aceituno, and M. Goto, “PYRAMIR: first on-sky results from an infrared pyramid wavefront sensor,” Proc. SPIE 6272, 627218, 627218-6 (2006). [CrossRef]

7.

S. E. Egner, W. Gaessler, R. Ragazzoni, B. LeRoux, T. M. Herbst, J. Farinato, E. Diolaiti, and C. Arcidiacono, “MANU-CHAO: a laboratory ground-layer adaptive optics experiment,” Proc. SPIE 6272, 62724X, 62724X-12 (2006). [CrossRef]

8.

S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, and P. Stefanini, “Pyramid sensor for segmented mirror alignment,” Opt. Lett. 30(19), 2572–2574 (2005). [CrossRef] [PubMed]

9.

A. Burvall, E. Daly, S. R. Chamot, and C. Dainty, “Linearity of the pyramid wavefront sensor,” Opt. Express 14(25), 11925–11934 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-11925. [CrossRef] [PubMed]

10.

C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233(1-3), 27–38 (2004). [CrossRef]

11.

J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, and J. Farinato, “Is there need of modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003). [CrossRef]

12.

J. B. Costa, “Modulation effect of the atmosphere in a pyramid wave-front sensor,” Appl. Opt. 44(1), 60–66 (2005). [PubMed]

13.

R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208(1-3), 51–60 (2002). [CrossRef]

14.

J. LeDue, L. Jolissaint, J.-P. Véran, and C. Bradley, “Calibration and testing with real turbulence of a pyramid sensor employing static modulation,” Opt. Express 17(9), 7186–7195 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7186. [CrossRef] [PubMed]

15.

V. Korkiakoski, C. Vérinaud, M. Le Louarn, and R. Conan, “Comparison between a model-based and a conventional pyramid sensor reconstructor,” Appl. Opt. 46(24), 6176–6184 (2007). [CrossRef] [PubMed]

16.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]

17.

J. B. Costa, “Development of a new infrared pyramid wavefront sensor,” PhD thesis, Ruperto-Carola University of Heidelberg, Germany (2005).

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(010.7350) Atmospheric and oceanic optics : Wave-front sensing
(050.1960) Diffraction and gratings : Diffraction theory
(230.3720) Optical devices : Liquid-crystal devices
(230.6120) Optical devices : Spatial light modulators

ToC Category:
Adaptive Optics

History
Original Manuscript: November 9, 2010
Revised Manuscript: January 15, 2011
Manuscript Accepted: March 30, 2011
Published: April 14, 2011

Virtual Issues
Vol. 6, Iss. 5 Virtual Journal for Biomedical Optics

Citation
Shengqian Wang, Changhui Rao, Hao Xian, Jianlin Zhang, Jianxin Wang, and Zheng Liu, "Laboratory demonstrations on a pyramid wavefront sensor without modulation for closed-loop adaptive optics system," Opt. Express 19, 8135-8150 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8135


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References

  1. R. Ragazzoni, “Pupil plane wavefront sensing with an oscillating prism,” J. Mod. Opt. 43(2), 289–293 (1996). [CrossRef]
  2. R. Ragazzoni and J. Farinato, “Sensitivity of a pyramidic wave front sensor in closed loop adaptive optics,” Astron. Astrophys. 350, 23–26 (1999).
  3. T. Y. Chew, R. M. Clare, and R. G. Lane, “A comparison of the Shack-Hartmann and pyramid wavefront sensors,” Opt. Commun. 268(2), 189–195 (2006). [CrossRef]
  4. 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). [CrossRef]
  5. S. Esposito, A. Tozzi, A. Puglisi, E. Pinna, A. Richardi, S. Busoni, L. Busoni, P. Stefanini, M. Xompero, D. Zanotti, and F. Pieralli, “First light AO system for LBT: toward on-sky operation,” Proc. SPIE 6272, 62720A, 62720A-9 (2006). [CrossRef]
  6. M. Feldt, D. Peter, S. Hippler, Th. Henning, J. Aceituno, and M. Goto, “PYRAMIR: first on-sky results from an infrared pyramid wavefront sensor,” Proc. SPIE 6272, 627218, 627218-6 (2006). [CrossRef]
  7. S. E. Egner, W. Gaessler, R. Ragazzoni, B. LeRoux, T. M. Herbst, J. Farinato, E. Diolaiti, and C. Arcidiacono, “MANU-CHAO: a laboratory ground-layer adaptive optics experiment,” Proc. SPIE 6272, 62724X, 62724X-12 (2006). [CrossRef]
  8. S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, and P. Stefanini, “Pyramid sensor for segmented mirror alignment,” Opt. Lett. 30(19), 2572–2574 (2005). [CrossRef] [PubMed]
  9. A. Burvall, E. Daly, S. R. Chamot, and C. Dainty, “Linearity of the pyramid wavefront sensor,” Opt. Express 14(25), 11925–11934 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-25-11925 . [CrossRef] [PubMed]
  10. C. Vérinaud, “On the nature of the measurements provided by a pyramid wave-front sensor,” Opt. Commun. 233(1-3), 27–38 (2004). [CrossRef]
  11. J. B. Costa, R. Ragazzoni, A. Ghedina, M. Carbillet, C. Verinaud, M. Feldt, S. Esposito, E. Puga, and J. Farinato, “Is there need of modulation in the pyramid wavefront sensor,” Proc. SPIE 4839, 288–298 (2003). [CrossRef]
  12. J. B. Costa, “Modulation effect of the atmosphere in a pyramid wave-front sensor,” Appl. Opt. 44(1), 60–66 (2005). [PubMed]
  13. R. Ragazzoni, E. Diolaiti, and E. Vernet, “A pyramid wavefront sensor with no dynamic modulation,” Opt. Commun. 208(1-3), 51–60 (2002). [CrossRef]
  14. J. LeDue, L. Jolissaint, J.-P. Véran, and C. Bradley, “Calibration and testing with real turbulence of a pyramid sensor employing static modulation,” Opt. Express 17(9), 7186–7195 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7186 . [CrossRef] [PubMed]
  15. V. Korkiakoski, C. Vérinaud, M. Le Louarn, and R. Conan, “Comparison between a model-based and a conventional pyramid sensor reconstructor,” Appl. Opt. 46(24), 6176–6184 (2007). [CrossRef] [PubMed]
  16. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]
  17. J. B. Costa, “Development of a new infrared pyramid wavefront sensor,” PhD thesis, Ruperto-Carola University of Heidelberg, Germany (2005).

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