## Beam shaping for laser-based adaptive optics in astronomy |

Optics Express, Vol. 22, Issue 11, pp. 12994-13013 (2014)

http://dx.doi.org/10.1364/OE.22.012994

Acrobat PDF (1185 KB)

### Abstract

The availability and performance of laser-based adaptive optics (AO) systems are strongly dependent on the power and quality of the laser beam before being projected to the sky. Frequent and time-consuming alignment procedures are usually required in the laser systems with free-space optics to optimize the beam. Despite these procedures, significant distortions of the laser beam have been observed during the first two years of operation of the Gemini South multi-conjugate adaptive optics system (GeMS). A beam shaping concept with two deformable mirrors is investigated in order to provide automated optimization of the laser quality for astronomical AO. This study aims at demonstrating the correction of quasi-static aberrations of the laser, in both amplitude and phase, testing a prototype of this two-deformable mirror concept on GeMS. The paper presents the results of the preparatory study before the experimental phase. An algorithm to control amplitude and phase correction, based on phase retrieval techniques, is presented with a novel unwrapping method. Its performance is assessed via numerical simulations, using aberrations measured at GeMS as reference. The results predict effective amplitude and phase correction of the laser distortions with about 120 actuators per mirror and a separation of 1.4 m between the mirrors. The spot size is estimated to be reduced by up to 15% thanks to the correction. In terms of AO noise level, this has the same benefit as increasing the photon flux by 40%.

© 2014 Optical Society of America

## 1. The need for laser beam optimization

1. G. Rousset, “Wave-front
Sensors,” in *Adaptive Optics in
Astronomy*, F. Roddier, ed. (Cambridge University,
1999), pp. 91–130. [CrossRef]

*σ*

_{meas}) is a function of the spot size and the number of photons received per subaperture and per frame according to [1

1. G. Rousset, “Wave-front
Sensors,” in *Adaptive Optics in
Astronomy*, F. Roddier, ed. (Cambridge University,
1999), pp. 91–130. [CrossRef]

*θ*

_{image}is the full width at half maximum (FWHM) of the spot image in the focal plane of a SH subaperture, and

*N*

_{ph}is the average number of photons received in a SH subaperture each frame. Therefore, the smaller the spot image or the greater the number of received photons, the better the measurement accuracy. This article analyzes the case of the Gemini South multi-conjugate AO system (GeMS) [2

2. F. Rigaut, B. Neichel, M. Boccas, C. d’Orgeville, F. Vidal, M. A. van Dam, G. Arriagada, V. Fesquet, R. L. Galvez, G. Gausachs, C. Cavedoni, A. W. Ebbers, S. Karewicz, E. James, J. Lührs, V. Montes, G. Perez, W. N. Rambold, R. Rojas, S. Walker, M. Bec, G. Trancho, M. Sheehan, B. Irarrazaval, C. Boyer, B. L. Ellerbroek, R. Flicker, D. Gratadour, A. Garcia-Rissmann, and F. Daruich, “Gemini multiconjugate adaptive optics system review - I. Design, trade-offs and integration,” Mon. Not. R. Astron. Soc. **437**, 2361–2375 (2014). [CrossRef]

*θ*

_{image}depends on various parameters, among which the laser spot size in the sodium layer, the elongation of the seen spot due to the sodium layer thickness, the diffraction produced by the SH subaperture size and the atmospheric seeing.

3. B. Neichel, F. Rigaut, F. Vidal, M. A. van Dam, V. Garrel, E. Rodrigo Carrasco, P. Pessev, C. Winge, M. Boccas, C. d’Orgeville, G. Arriagada, A. Serio, V. Fesquet, W. N. Rambold, J. Lührs, C. Moreno, G. Gausachs, R. L. Galvez, V. Montes, T. B. Vucina, E. Marin, C. Urrutia, A. Lopez, S. J. Diggs, C. Marchant, A. W. Ebbers, C. Trujillo, M. Bec, G. Trancho, P. McGregor, P. J. Young, F. Colazo, and M. L. Edwards, “Gemini multi-conjugate adaptive optics system review II: Commissioning, operation and overall performance,” Mon. Not. R. Astron. Soc., in press (2014). [CrossRef]

4. C. d’Orgeville, S. Diggs, V. Fesquet, B. Neichel, W. Rambold, F. Rigaut, A. Serio, C. Araya, G. Arriagada, R. Balladares, M. Bec, M. Boccas, C. Duran, A. Ebbers, A. Lopez, C. Marchant, E. Marin, V. Montes, C. Moreno, E. Petit Vega, C. Segura, G. Trancho, C. Trujillo, C. Urrutia, P. Veliz, and T. Vucina, “Gemini South multi-conjugate adaptive optics (GeMS) laser guide star facility on-sky performance results,” Proc. SPIE **8447**, 84471Q (2012). [CrossRef]

*θ*

_{image}is reduced by 15%, it is equivalent for a given AO noise level to increase the number of photons by 40%. Laser-based AO systems usually being designed with little margin in terms of expected photons return, it appears all the more important to minimize the spot size.

*M*

^{2}= 1 [5

5. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE **1224**, 2–14 (1990). [CrossRef]

*M*

^{2}values greater than 1. An estimation of the

*M*

^{2}of a laser can be computed using a SH to measure the beam irradiance and wavefront at waist [6

6. D. R. Neal, W. J. Alford, J. K. Gruetzner, and M. E. Warren, “Amplitude and phase beam
characterization using a two-dimensional wavefront
sensor,” Proc. SPIE **2870**, 72–82
(1996). [CrossRef]

*x*and

*y*axis margin distribution of the beam intensity, can be obtained. Using the notations

*D*and

_{x}*θ*for this quantity along

_{x}*x*–direction at the waist and in the far-field respectively, the

*M*

^{2}is estimated by

*M*

^{2}factors on GeMS were below 1.5 and 1.3, in

*x*and

*y*respectively [4

4. C. d’Orgeville, S. Diggs, V. Fesquet, B. Neichel, W. Rambold, F. Rigaut, A. Serio, C. Araya, G. Arriagada, R. Balladares, M. Bec, M. Boccas, C. Duran, A. Ebbers, A. Lopez, C. Marchant, E. Marin, V. Montes, C. Moreno, E. Petit Vega, C. Segura, G. Trancho, C. Trujillo, C. Urrutia, P. Veliz, and T. Vucina, “Gemini South multi-conjugate adaptive optics (GeMS) laser guide star facility on-sky performance results,” Proc. SPIE **8447**, 84471Q (2012). [CrossRef]

*M*

^{2}close to 1 has not been achieved so far with GeMS laser. The

*M*

^{2}values even worsened during the first half of 2013, revealing significant degradation of the laser beam quality with time. The laser beam has been measured right at the output of the laser bench using a dedicated Shack-Hartmann sensor. The obtained irradiance maps are shown in Fig. 1 for sets of measurements taken in April and October 2013. Note that these measurements were both made after several days of alignment optimization,

*i.e.*on the best laser shape at each time. From the data of April 2013,

*M*

^{2}factors of 2.21 and 1.21 along

*x*and

*y*directions respectively were found. In October 2013, the corresponding

*M*

^{2}factors were 2.23 and 1.15.

*x*-axis, also visible in all plots of Fig. 1. The dotted line represents the half of the maximum to quantify the FWHM and it shows that the beam diameter is between 1.5 and 2 times larger along

*x*than along

*y*. The beam quality is always worse in the

*x*-axis with GeMS because it corresponds to the unguided dimension of the amplification stages in the waveguide amplifier module [4

4. C. d’Orgeville, S. Diggs, V. Fesquet, B. Neichel, W. Rambold, F. Rigaut, A. Serio, C. Araya, G. Arriagada, R. Balladares, M. Bec, M. Boccas, C. Duran, A. Ebbers, A. Lopez, C. Marchant, E. Marin, V. Montes, C. Moreno, E. Petit Vega, C. Segura, G. Trancho, C. Trujillo, C. Urrutia, P. Veliz, and T. Vucina, “Gemini South multi-conjugate adaptive optics (GeMS) laser guide star facility on-sky performance results,” Proc. SPIE **8447**, 84471Q (2012). [CrossRef]

*M*

^{2}values over the length of the runs has also been noticed, suggesting that the system suffers from constant misalignment [4

**8447**, 84471Q (2012). [CrossRef]

7. C. Béchet, A. Guesalaga, B. Neichel, V. Fesquet, and D. Guzman, “A two deformable-mirror concept to improve the laser efficiency of Gemini South MCAO,”, in Proceedings of the Third AO4ELT Conference, S. Esposito and L. Fini, eds. (INAF - Osservatorio Astrofisico di Arcetri, Firenze, Italy, 2013), 13344.

8. C. d’Orgeville, F. Daruich, G. Arriagada, M. Bec, M. Boccas, S. Bombino, C. Carter, C. Cavedoni, F. Collao, P. Collins, E. James, S. Karewicz, M. Lazo, D. Maltes, R. Mouser, G. Perez, F. Rigaut, R. Rojas, M. Sheehan, G. Trancho, V. Vergara, and T. Vucina, “The Gemini South MCAO laser guide star
facility: getting ready for first light,”
Proc. SPIE **7015**, 70152P (2008). [CrossRef]

9. R. Holzlöhner, D. Bonaccini Calia, and W. Hackenberg, “Physical optics modeling and optimization of laser guide star propagation,” Proc. SPIE **7015**, 701521 (2008). [CrossRef]

9. R. Holzlöhner, D. Bonaccini Calia, and W. Hackenberg, “Physical optics modeling and optimization of laser guide star propagation,” Proc. SPIE **7015**, 701521 (2008). [CrossRef]

3. B. Neichel, F. Rigaut, F. Vidal, M. A. van Dam, V. Garrel, E. Rodrigo Carrasco, P. Pessev, C. Winge, M. Boccas, C. d’Orgeville, G. Arriagada, A. Serio, V. Fesquet, W. N. Rambold, J. Lührs, C. Moreno, G. Gausachs, R. L. Galvez, V. Montes, T. B. Vucina, E. Marin, C. Urrutia, A. Lopez, S. J. Diggs, C. Marchant, A. W. Ebbers, C. Trujillo, M. Bec, G. Trancho, P. McGregor, P. J. Young, F. Colazo, and M. L. Edwards, “Gemini multi-conjugate adaptive optics system review II: Commissioning, operation and overall performance,” Mon. Not. R. Astron. Soc., in press (2014). [CrossRef]

- 0.61” for a perfect Gaussian beam with FWHM of 25 cm.
- 0.69” and 0.71” considering only amplitude aberrations measured in April 2013 and in October 2013 respectively.
- 0.74” considering both amplitude and phase aberrations of the laser beam. The same value 0.74” is obtained for April 2013 and for October 2013.

**8447**, 84471Q (2012). [CrossRef]

3. B. Neichel, F. Rigaut, F. Vidal, M. A. van Dam, V. Garrel, E. Rodrigo Carrasco, P. Pessev, C. Winge, M. Boccas, C. d’Orgeville, G. Arriagada, A. Serio, V. Fesquet, W. N. Rambold, J. Lührs, C. Moreno, G. Gausachs, R. L. Galvez, V. Montes, T. B. Vucina, E. Marin, C. Urrutia, A. Lopez, S. J. Diggs, C. Marchant, A. W. Ebbers, C. Trujillo, M. Bec, G. Trancho, P. McGregor, P. J. Young, F. Colazo, and M. L. Edwards, “Gemini multi-conjugate adaptive optics system review II: Commissioning, operation and overall performance,” Mon. Not. R. Astron. Soc., in press (2014). [CrossRef]

11. Y. Feng, L. R. Taylor, and D. B. Calia, “25 W Raman-fiber-amplifier-based 589 nm laser for laser guide star,” Opt. Express **17**(21), 19021–19026 (2009). [CrossRef]

*M*

^{2}quality and high power. There is still few on-sky experience to guarantee that this new technology can solve all the problems for an AO system like GeMS, requiring a 50-Watt laser. Therefore the study of an automatic technique to improve the beam shape of the current laser in GeMS is considered highly desirable.

12. A. Guesalaga, B. Neichel, M. Boccas, C. d’Orgeville, F. Rigaut, D. Guzman, and J. Anguita, “Improving stability, robustness, and performance of laser systems,” Proc. SPIE **8447**, 84474M (2012). [CrossRef]

7. C. Béchet, A. Guesalaga, B. Neichel, V. Fesquet, and D. Guzman, “A two deformable-mirror concept to improve the laser efficiency of Gemini South MCAO,”, in Proceedings of the Third AO4ELT Conference, S. Esposito and L. Fini, eds. (INAF - Osservatorio Astrofisico di Arcetri, Firenze, Italy, 2013), 13344.

## 2. 2-DM concept for laser beam shaping

*e.g.*[14

14. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. **4**, 1400–1403 (1965). [CrossRef]

18. M. T. Eismann, A. M. Tai, and J. N. Cederquist, “Iterative design of a holographic beamformer,” Appl. Opt. **28**, 2641–2650 (1989). [CrossRef]

15. J. W. Ogland, “Mirror system for uniform beam transformation in high-power annular lasers,” Appl. Opt. **17**, 2917–2923 (1978). [CrossRef] [PubMed]

17. P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. **19**, 3545–3553 (1980). [CrossRef] [PubMed]

*et al.*[18

18. M. T. Eismann, A. M. Tai, and J. N. Cederquist, “Iterative design of a holographic beamformer,” Appl. Opt. **28**, 2641–2650 (1989). [CrossRef]

19. 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), 1781–1783 (2004). [CrossRef] [PubMed]

20. M. C. Roggemann and D. J. Lee, “Two-Deformable-Mirror Concept for Correcting Scintillation Effects in Laser Beam Projection Through the Turbulent Atmosphere,” Appl. Opt. **37**(21), 4577–4585 (1998). [CrossRef]

24. R. Kizito, M. C. Roggemann, T. J. Schulz, and Y. Zhang, “Image sharpness metric-based deformable
mirror control for beam projection systems operating in strong
scintillation,” Proc.
SPIE **5160**, 406–416
(2004). [CrossRef]

*et al.*[13]. The first DM corrects for the amplitude and the second one corrects for the phase. The concept has later been revised by Roggemann

*et al.*[20

20. M. C. Roggemann and D. J. Lee, “Two-Deformable-Mirror Concept for Correcting Scintillation Effects in Laser Beam Projection Through the Turbulent Atmosphere,” Appl. Opt. **37**(21), 4577–4585 (1998). [CrossRef]

25. M. C. Roggemann and S. Deng, “Scintillation compensation for laser beam projection using segmented deformable mirrors,” Proc. SPIE **3763**, 29–40 (1999). [CrossRef]

26. M. C. Roggemann and A. C. Koivunen, “Wavefront sensing and deformable mirror control in strong scintillation,” J. Opt. Soc. Am. A **17**, 911–919 (2000). [CrossRef]

*mono-static*because the receiving and projecting apertures are the same. The phase and amplitude corruption induced by the atmosphere in the optical path from the target above the aperture is supposed to be measured with a suitable device. It could be a SH sensor, from which outputs need to be processed to compute the received amplitude and the phase of the field in the aperture plane of the telescope. The measurements are analyzed to deduce the commands to be sent to the DMs in order to shape the laser beam to the conjugate of the received field, and project it. First, the laser collimated beam is supposed to fall upon DM1, where a phase shape

*ϕ*

_{1}is added to the wave. This applied phase aims at modifying the amplitude shape of the wave in the Fraunhofer region, after the Fourier transforming mirror, when the wave is incident on DM2. The second DM, DM2, is optically conjugated to the pupil plane of the beam projecting aperture, so it only modifies the phase distribution of the output beam. A novel modification of the latter approach to near-field correction has been proposed by Barchers

*et al.*[21

21. J. D. Barchers and D. L. Fried, “Optimal control of laser beams for propagation through a turbulent medium,” J. Opt. Soc. Am. A **19**, 1779–1793 (2002). [CrossRef]

27. J. D. Barchers and B. L. Ellerbroek, “Improved compensation of turbulence-induced amplitude and phase distortions by means of multiple near-field phase adjustments,” J. Opt. Soc. Am. A **18**(2), 399–411 (2001). [CrossRef]

29. J. D. Barchers, “Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations,” J. Opt. Soc. Am. A **19**, 926–945 (2002). [CrossRef]

*z*(see Fig. 2(b)). DM2 is in the near-field of the DM1 (

*i.e.*in the Fresnel region), at a separation distance

*z*of the order of 1 meter. DM2 remains conjugated to the pupil plane of the beam projecting aperture as in Fig. 2(a)). The near-field configuration should allow matching the space constraints of the laser systems on the telescope, although maintaining the ability to correct for amplitude and phase of the field.

*U*. The phase patterns to be applied to the DMs are determined by a phase retrieval method [39

_{t}39. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

*e.g.*image plane, conjugated plane), as far as they are linked by a propagation relation. The first phase retrieval methods were developed by Gerchberg and Saxton [40], Gonsalves [41

41. R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. **66**, 961–964 (1976). [CrossRef]

42. J. R. Fienup, “Reconstruction of an object from the modulus of its fourier transform,” Opt. Lett. **3**, 27–29 (1978). [CrossRef] [PubMed]

39. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

21. J. D. Barchers and D. L. Fried, “Optimal control of laser beams for propagation through a turbulent medium,” J. Opt. Soc. Am. A **19**, 1779–1793 (2002). [CrossRef]

*Gerchberg-Saxton algorithms*, are applications of what is known as the method of sequential projections onto constraints sets [43]. The algorithm presented in Sect. 3 is a version of Gerchberg-Saxton method adapted to account for the near-field separation of the DMs and to unwrap the phases to be applied on DMs.

## 3. Algorithm for amplitude and phase correction

**x**is the coordinate vector in the plane of DM1, and

*u*(

_{l}**x**) and

*ϕ*(

_{l}**x**) account for the aberrated amplitude and phase in this plane respectively. The field leaving DM1 is noted

*U*

_{1}and is expressed as where

*ϕ*

_{1}is the phase pattern applied to DM1 and

*m*

_{1}represents the reflection mask for the mirror, in order to account for the finite extension of the device.

*U*

_{1}, at distance

*z*, is described by the linear unitary transformation

*T*such that [27

_{z}27. J. D. Barchers and B. L. Ellerbroek, “Improved compensation of turbulence-induced amplitude and phase distortions by means of multiple near-field phase adjustments,” J. Opt. Soc. Am. A **18**(2), 399–411 (2001). [CrossRef]

**f**

*is the spatial frequency vector,*

_{x}*λ*is the laser wavelength and

*U*is the incident field on DM2. After DM2 phase correction

_{z}*ϕ*

_{2}, the output field is where

*m*

_{2}represents the mask of this mirror pupil. DM2 is conjugated to the projecting aperture such that

*U*

_{2}is assumed to be the transmitted field.

*U*, noted where

_{t}*u*is its amplitude and

_{t}*ϕ*its phase. This desired output field is a constraint in the algorithm. Its choice is discussed in [7

_{t}7. C. Béchet, A. Guesalaga, B. Neichel, V. Fesquet, and D. Guzman, “A two deformable-mirror concept to improve the laser efficiency of Gemini South MCAO,”, in Proceedings of the Third AO4ELT Conference, S. Esposito and L. Fini, eds. (INAF - Osservatorio Astrofisico di Arcetri, Firenze, Italy, 2013), 13344.

*U*needs to be the conjugate of the field which would be received in a mono-static configuration like in Fig. 2(a) [27

_{t}27. J. D. Barchers and B. L. Ellerbroek, “Improved compensation of turbulence-induced amplitude and phase distortions by means of multiple near-field phase adjustments,” J. Opt. Soc. Am. A **18**(2), 399–411 (2001). [CrossRef]

*ϕ*only.

_{b}*U*and

_{t}*U*known, the goal of the 2DM concept is to determine

_{l}*ϕ*

_{1}and

*ϕ*

_{2}to be applied to DM1 and DM2 respectively in order to obtain the field

*U*on the projecting aperture [20

_{t}20. M. C. Roggemann and D. J. Lee, “Two-Deformable-Mirror Concept for Correcting Scintillation Effects in Laser Beam Projection Through the Turbulent Atmosphere,” Appl. Opt. **37**(21), 4577–4585 (1998). [CrossRef]

*ϕ*

_{2}and

*ϕ*

_{1}to satisfy the intensity constraints fixed by

*U*and

_{l}*U*respectively. The main steps of the algorithm, similar to the one described by Roggemann and Lee [20

_{t}**37**(21), 4577–4585 (1998). [CrossRef]

### 3.1. Forward iterative step for DM2

*ϕ*

_{1}, the incoming field on DM2 is computed Since we want the field after DM2 to be equal to the desired output

*U*, the required phase to apply on DM2

_{t}*ϕ*

_{2}is constrained to satisfy

*ϕ*

_{2}is obtained from Eq. (9), we compute an additional parameter

*α*, which is the ratio of the incident energy

*E*

_{2}on DM2 over the energy

*E*of the desired output field

_{t}*U*. This factor

_{t}*α*is used later in Eq. (11) to rescale the field to be propagated backward in the iterative step for determination of

*ϕ*

_{1}. The value of

*α*is mathematically defined by

### 3.2. Backward iterative step for DM1

*ϕ*

_{1}to be applied to DM1 [20

**37**(21), 4577–4585 (1998). [CrossRef]

*. Indeed, for any given complex field*

_{z}*U*,

*ϕ*

_{1}is obtained applying the constraint that the phase and amplitude after backpropagation beyond DM1 matches the input laser field phase

*ϕ*and amplitude

_{l}*u*. Thus,

_{l}45. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A **15**(10), 2759–2768 (1998). [CrossRef]

*ϕ*

_{1}and

*ϕ*

_{2}. In order to obtain phase functions

*ϕ*

_{1}and

*ϕ*

_{2}likely to be efficiently reproduced by the continuous sheet of a DM, the phases must be unwrapped and branch points must be avoided.

### 3.3. Phase unwrapping

12. A. Guesalaga, B. Neichel, M. Boccas, C. d’Orgeville, F. Rigaut, D. Guzman, and J. Anguita, “Improving stability, robustness, and performance of laser systems,” Proc. SPIE **8447**, 84474M (2012). [CrossRef]

**G**is the discrete representation of the gradient operator as described by Fried [45

45. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A **15**(10), 2759–2768 (1998). [CrossRef]

*𝒫𝒱*symbol stands for the principal-value operator which output is an angle that falls in the range ±

*π*(modulus 2

*π*),

**W**

*is a diagonal array of weights,*

_{g}**C**is a discrete representation of the curvature operator and

*μ*is a scalar weighting the curvature regularization term

**C**

^{T}

**C**.

*π*. This is why the unwrapper in Eq. (14) involves the gradient operator on both sides of the equation and the principal value operator on the right hand side where the wrapped phase is. In addition,

*ϕ*

_{2}and

*ϕ*

_{1}are computed from Eqs. (9) and (13) only where the field amplitude

*u*in the argument is numerically non zero. This leads in both cases to wrapped phase maps

*ϕ*, and the wrapped phase samples are arbitrarily set to zero where the amplitude is also zero,

*i.e.*where degeneracy occurs.

**C**

^{T}

**C**(in Eq. (14)), which favors solutions of

*ϕ̂*with low curvature. The relative importance of this penalty term can be tuned thanks to the scalar

*μ*in Eq. (14). In practice, we have found that

*μ*of the order of 10

^{−3}is apropriate to influence the solution only in areas of negligible amplitude.

### 3.4. Convergence criteria

**37**(21), 4577–4585 (1998). [CrossRef]

*i*iterations of the method, where the notation

*i*-th estimations of

*ϕ*

_{1}and

*ϕ*

_{2}are applied to DM1 and DM2 respectively. The scalar

*S*and it is sensitive to the phase matching between the reached output field and the desired output one. It is written where index (

*i*) again refers to the

*i*–th iteration of the procedure. This criterion is inspired from the far-field intensity criterion used in the literature [27

**18**(2), 399–411 (2001). [CrossRef]

*S*is expected to increase with iterations toward unity.

## 4. Simulations

*z*between the mirrors; the number of iterations; and the grid sampling step of the propagated fields and phase estimates.

### 4.1. Amplitude and phase correction

12. A. Guesalaga, B. Neichel, M. Boccas, C. d’Orgeville, F. Rigaut, D. Guzman, and J. Anguita, “Improving stability, robustness, and performance of laser systems,” Proc. SPIE **8447**, 84474M (2012). [CrossRef]

*z*= 1.8 m.

*i.e. ϕ*

_{1}=

*ϕ*

_{2}= 0, the input distortions lead to output amplitude very similar to the input one, slightly modified by the propagation in the near-field (

*z*= 1.8 m). This output amplitude is represented in Fig. 3(a), over the 1 cm diameter of the aperture of DM2. Cuts along

*x*and

*y*central axes are represented with solid curves in Fig. 3(b). It can be observed that the amplitude of the beam is far from being Gaussian.

*x*and

*y*central axes in Fig. 3(b). The desired output phase has been arbitrarily chosen to be a defocus (see dotted curve in Fig. 4(b)).

*x*and

*y*are plotted with solid curves, in Fig. 3(d) for the amplitude and in Fig. 4(b) for the phase. The convergence of the algorithm along the 400 iterations is illustrated in Fig. 5. Both corrections of amplitude and phase work perfectly.

*ϕ*

_{1}and

*ϕ*

_{2}computations, the estimation of the resulting phase is initially wrapped and only at non zero amplitude locations. The field amplitude (shown in Fig. 3(c)) for radii greater than 4 mm is so small that the phase cannot be estimated with accuracy at these locations. The solid curve in Fig. 4 is thus the result of the iterative unwrapper applied to the output phase estimate. In the outer ring area, the regularization of the iterative phase unwrapper detailed in Sect. 3.3 mainly drives the estimation of the phase. However, whatever the phase estimate in this area, it has no effect on the beam shaping performance because the field amplitude is almost zero. Note also that the criterion

*S*in Fig. 5 is not notably affected by the phase discrepancy in this area because of the amplitude weighting in Eq. (17).

*S*already jumping to 85% after a single iteration. Amplitude correction drives the global convergence. The influence of the DMs separation distance on the convergence speed is further studied in Sect. 4.2.

*ϕ*

_{1}and

*ϕ*

_{2}is identical to the resolution

*δ*of the introduced aberrations

*U*and of the desired output field

_{l}*U*, with

_{t}*δ*≃ 161

*μ*m. The desired output amplitude is a perfect Gaussian with FWHM of 1.7 mm, and the DMs pupils are considered to be 1 cm in diameter. The phase distortions

*ϕ*

_{1}and

*ϕ*

_{2}which should be applied on DM1 and DM2 in order to obtain such correction are presented in Fig. 6.

*y*for

*ϕ*

_{1}is shown in dotted line for the wrapped solution (between −

*π*and

*π*) and in solid line once unwrapped. The limit of the mirrors pupil diameter are represented by the vertical dashed lines. Again it can clearly be noticed that in the center of the slice (where the amplitude of the field is larger), the unwrapper accurately unwraps the phase estimates. On the pupil edges however, where the amplitude vanishes, the regularizing term of the unwrapper becomes significant and the unwrapped estimate is extrapolated, penalizing local curvature of the phase.

### 4.2. Influence of DM separation

*M*

^{2}factor of the output beam. This criterion is applicable in this case because the desired output amplitude is a Gaussian shape. The results of the correction performance after 400 iterations are presented in Fig. 8.

*z*between the DMs. The dotted horizontal lines stand for the

*U*. When the beam shaping system is considered, the

_{l}*M*

^{2}factor is computed after the DM2, which means that even if no DM correction is applied, the output beam is slightly affected in the near-field. Note that for

*z*≤ 1 m,

*M*

^{2}factor values as low as 1.02. Such values mean very good quality Gaussian beams.

*z*, its influence on the convergence speed is also studied. In Fig. 9 (left), the relative amplitude error of the output beam

*z*= 1 m, where convergence is reached after more than 350 iterations. The convergence is accelerated progressively when the separation

*z*is increased. For

*z*= 5 m, less than 20 iterations suffice.

*z*increases. This is illustrated on the right of Fig. 9, where the highest curve is the phase deformation

*ϕ*

_{1}required for separation

*z*= 1 m. The required deformation is reduced by approximately a factor equal to

*p*when the separation is changed from

*z*= 1 m to

*z*=

*p*, with

*p*in the range of 1 to 5 meters. It is thus an important parameter to take into account in the design of the beam shaping system, depending on the mechanical properties of the DMs to be used.

### 4.3. Influence of number of degrees of freedom

*ϕ*

_{1}and

*ϕ*

_{2}on the performance of the beam shaping is required. We specify this constraint in terms of the number of estimated phase samples across the DMs diameter,

*n*, such that a DM with ∼ 3/4

*n*

^{2}degrees of freedom over the pupil area is expected to be able to reproduce it satisfactorily.

*u*as the ones measured from GeMS in April and October 2013 (Fig. 1). Again, we do not consider any phase aberrations

_{l}*ϕ*for the input. The simulation grid sampling has been chosen equal to the resolution of the laser wavefront sensor measurement at GeMS (Fig. 1),

_{l}*i.e.*

*δ*= 198

*μ*m. The DMs aperture diameter is asumed to be 4.9 mm. The desired output amplitude is set to a perfect Gaussian with FWHM of 1.7 mm and a flat output phase.

*z*is represented in the plots of Fig. 10 for various values of

*n*. Input amplitude distortions measured in April 2013 are used for the plot on the left and the ones measured in October 2013 are used on the right plot. With a large number of degrees of freedom for the correction, it is possible to reduce the error to a few percentage points of its original value placing the DMs at short distances (

*z*below 1 m). At such short separations, the error reduction significantly degrades when the number of phase samples

*n*across the pupil is too small.

*z*,

*i.e.*beyond 1.2 m. Note that the distortions measured in October 2013 appear more difficult to correct than the ones observed in April 2013. For separations

*z*beyond 2 meters, there is no remarkable benefit in using 31 × 31 phase samples compared to using 11 × 11.

*n*= 31, the relative error

*z*= 2 m, is degraded by about 10 percentage points compared to the achieved error with

*z*= 80 cm. However, this is mainly due to the normalization value in the denominator of

*z*. In order to clarify this, the same simulations results are presented in Fig. 11 but showing the amplitude error

*z*between the DMs. This effect simply results from the propagation of the beam in the near-field. This absolute representation reveals that good correction quality is obtained if

*z*is greater than 1.4 m for all three cases (

*n*= 11, 17 and 31). If

*n*is greater than 11, shorter separations

*z*can be chosen for approximately the same good correction quality.

### 4.4. Benefit of correcting GeMS amplitude and phase distortions

## 5. Conclusion

38. V. Fesquet, C. Araujo, V. Garrel, A. Serio, F. Vidal, G. Arriagada, M. Boccas, F. Collao, S. Diggs, J. Donahue, C. D’Orgeville, G. Gausachs, C. Marchant, V. Montes, C. Moreno, B. Neichel, R. Oram, P. Pessev, W. Rambold, C. Urrutia, and T. Vucina, “Review of Gemini South Laser Guide Star Facility performance and upgrades,” in Proceedings of the Third AO4ELT Conference, S. Esposito and L. Fini, eds. (INAF - Osservatorio Astrofisico di Arcetri, Firenze, Italy, 2013), 16120.

## Acknowledgments

## References and links

1. | G. Rousset, “Wave-front
Sensors,” in |

2. | F. Rigaut, B. Neichel, M. Boccas, C. d’Orgeville, F. Vidal, M. A. van Dam, G. Arriagada, V. Fesquet, R. L. Galvez, G. Gausachs, C. Cavedoni, A. W. Ebbers, S. Karewicz, E. James, J. Lührs, V. Montes, G. Perez, W. N. Rambold, R. Rojas, S. Walker, M. Bec, G. Trancho, M. Sheehan, B. Irarrazaval, C. Boyer, B. L. Ellerbroek, R. Flicker, D. Gratadour, A. Garcia-Rissmann, and F. Daruich, “Gemini multiconjugate adaptive optics system review - I. Design, trade-offs and integration,” Mon. Not. R. Astron. Soc. |

3. | B. Neichel, F. Rigaut, F. Vidal, M. A. van Dam, V. Garrel, E. Rodrigo Carrasco, P. Pessev, C. Winge, M. Boccas, C. d’Orgeville, G. Arriagada, A. Serio, V. Fesquet, W. N. Rambold, J. Lührs, C. Moreno, G. Gausachs, R. L. Galvez, V. Montes, T. B. Vucina, E. Marin, C. Urrutia, A. Lopez, S. J. Diggs, C. Marchant, A. W. Ebbers, C. Trujillo, M. Bec, G. Trancho, P. McGregor, P. J. Young, F. Colazo, and M. L. Edwards, “Gemini multi-conjugate adaptive optics system review II: Commissioning, operation and overall performance,” Mon. Not. R. Astron. Soc., in press (2014). [CrossRef] |

4. | C. d’Orgeville, S. Diggs, V. Fesquet, B. Neichel, W. Rambold, F. Rigaut, A. Serio, C. Araya, G. Arriagada, R. Balladares, M. Bec, M. Boccas, C. Duran, A. Ebbers, A. Lopez, C. Marchant, E. Marin, V. Montes, C. Moreno, E. Petit Vega, C. Segura, G. Trancho, C. Trujillo, C. Urrutia, P. Veliz, and T. Vucina, “Gemini South multi-conjugate adaptive optics (GeMS) laser guide star facility on-sky performance results,” Proc. SPIE |

5. | A. E. Siegman, “New developments in laser resonators,” Proc. SPIE |

6. | D. R. Neal, W. J. Alford, J. K. Gruetzner, and M. E. Warren, “Amplitude and phase beam
characterization using a two-dimensional wavefront
sensor,” Proc. SPIE |

7. | C. Béchet, A. Guesalaga, B. Neichel, V. Fesquet, and D. Guzman, “A two deformable-mirror concept to improve the laser efficiency of Gemini South MCAO,”, in Proceedings of the Third AO4ELT Conference, S. Esposito and L. Fini, eds. (INAF - Osservatorio Astrofisico di Arcetri, Firenze, Italy, 2013), 13344. |

8. | C. d’Orgeville, F. Daruich, G. Arriagada, M. Bec, M. Boccas, S. Bombino, C. Carter, C. Cavedoni, F. Collao, P. Collins, E. James, S. Karewicz, M. Lazo, D. Maltes, R. Mouser, G. Perez, F. Rigaut, R. Rojas, M. Sheehan, G. Trancho, V. Vergara, and T. Vucina, “The Gemini South MCAO laser guide star
facility: getting ready for first light,”
Proc. SPIE |

9. | R. Holzlöhner, D. Bonaccini Calia, and W. Hackenberg, “Physical optics modeling and optimization of laser guide star propagation,” Proc. SPIE |

10. | B. L. Ellerbroek, “Adaptive Optics for the Thirty Meter Telescope,” in Proceedings of the Third AO4ELT Conference, S. Esposito and L. Fini, eds. (INAF - Osservatorio Astrofisico di Arcetri, Firenze, Italy, 2013), 13199. |

11. | Y. Feng, L. R. Taylor, and D. B. Calia, “25 W Raman-fiber-amplifier-based 589 nm laser for laser guide star,” Opt. Express |

12. | A. Guesalaga, B. Neichel, M. Boccas, C. d’Orgeville, F. Rigaut, D. Guzman, and J. Anguita, “Improving stability, robustness, and performance of laser systems,” Proc. SPIE |

13. | F. Y. Kanev and V. P. Lukin, “Amplitude phase beam control with the help of a two-mirror adaptive system,” Atmos. Opt. |

14. | B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. |

15. | J. W. Ogland, “Mirror system for uniform beam transformation in high-power annular lasers,” Appl. Opt. |

16. | D. Shafer, “Gaussian to flat-top intensity distributing lens,” Optics & Laser Technology |

17. | P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. |

18. | M. T. Eismann, A. M. Tai, and J. N. Cederquist, “Iterative design of a holographic beamformer,” Appl. Opt. |

19. | 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. |

20. | M. C. Roggemann and D. J. Lee, “Two-Deformable-Mirror Concept for Correcting Scintillation Effects in Laser Beam Projection Through the Turbulent Atmosphere,” Appl. Opt. |

21. | J. D. Barchers and D. L. Fried, “Optimal control of laser beams for propagation through a turbulent medium,” J. Opt. Soc. Am. A |

22. | M. A. Vorontsov and V. Kolosov, “Target-in-the-loop beam control: basic considerations for analysis and wavefront sensing,” J. Opt. Soc. Am. A |

23. | Z. Zhao, S. D. Lyke, and M. C. Roggemann, “Adaptive Optical Communication through Turbulent Atmospheric Channels,” in Proceedings of IEEE Conference on Communications, 2008. ICC ’08 (Institute of Electrical and Electronics Engineers, New York, 2008), 5432–5436. |

24. | R. Kizito, M. C. Roggemann, T. J. Schulz, and Y. Zhang, “Image sharpness metric-based deformable
mirror control for beam projection systems operating in strong
scintillation,” Proc.
SPIE |

25. | M. C. Roggemann and S. Deng, “Scintillation compensation for laser beam projection using segmented deformable mirrors,” Proc. SPIE |

26. | M. C. Roggemann and A. C. Koivunen, “Wavefront sensing and deformable mirror control in strong scintillation,” J. Opt. Soc. Am. A |

27. | J. D. Barchers and B. L. Ellerbroek, “Improved compensation of turbulence-induced amplitude and phase distortions by means of multiple near-field phase adjustments,” J. Opt. Soc. Am. A |

28. | J. D. Barchers, “Evaluation of the impact of finite-resolution effects on scintillation compensation using two deformable mirrors,” J. Opt. Soc. Am. A |

29. | J. D. Barchers, “Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations,” J. Opt. Soc. Am. A |

30. | F. Roddier, |

31. | R. A. Gonsalves, “Compensation of scintillation with a phase-only adaptive optic,” Opt. Lett. |

32. | O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. |

33. | L. Pueyo and N. Kasdin, “Polychromatic compensation of propagated aberrations for high-contrast imaging,” Astrophys. J. |

34. | L. Pueyo, J. Kay, N. J. Kasdin, T. Groff, M. McElwain, A. Give’on, and R. Belikov, “Optimal dark hole generation via two deformable mirrors with stroke minimization,” Appl. Opt. |

35. | D. Gavel, M. Ammons, B. Bauman, D. Dillon, E. Gates, B. Grigsby, J. Johnson, C. Lockwood, K. Morzinski, D. Palmer, M. Reinig, and S. Severson, “Visible light laser guidestar
experimental system (Villages): on-sky tests of new technologies for visible
wavelength all-sky coverage adaptive optics
systems,” Proc. SPIE |

36. | A. P. Norton, D. Gavel, M. Helmbrecht, C. J. Kempf, E. L. Gates, K. Chloros, D. Redel, and D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” accepted for Proc. SPIE (2014). |

37. | 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 |

38. | V. Fesquet, C. Araujo, V. Garrel, A. Serio, F. Vidal, G. Arriagada, M. Boccas, F. Collao, S. Diggs, J. Donahue, C. D’Orgeville, G. Gausachs, C. Marchant, V. Montes, C. Moreno, B. Neichel, R. Oram, P. Pessev, W. Rambold, C. Urrutia, and T. Vucina, “Review of Gemini South Laser Guide Star Facility performance and upgrades,” in Proceedings of the Third AO4ELT Conference, S. Esposito and L. Fini, eds. (INAF - Osservatorio Astrofisico di Arcetri, Firenze, Italy, 2013), 16120. |

39. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

40. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik |

41. | R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. |

42. | J. R. Fienup, “Reconstruction of an object from the modulus of its fourier transform,” Opt. Lett. |

43. | H. Stark and Y. Yang, |

44. | T. M. Venema and J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express |

45. | D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A |

46. | J. D. Schmidt, |

47. | C. C. Beckner Jr. and D. W. Oesch, “Implementation of a projection-on-constraints algorithm for beam intensity redistribution,” Proc. SPIE |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(100.5070) Image processing : Phase retrieval

(140.3300) Lasers and laser optics : Laser beam shaping

(100.5088) Image processing : Phase unwrapping

(110.1080) Imaging systems : Active or adaptive optics

**ToC Category:**

Adaptive Optics

**History**

Original Manuscript: March 27, 2014

Revised Manuscript: May 12, 2014

Manuscript Accepted: May 13, 2014

Published: May 21, 2014

**Citation**

Clémentine Béchet, Andrés Guesalaga, Benoit Neichel, Vincent Fesquet, Héctor González-Núñez, Sebastián Zúñiga, Pedro Escarate, and Dani Guzman, "Beam shaping for laser-based adaptive optics in astronomy," Opt. Express **22**, 12994-13013 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-12994

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### References

- G. Rousset, “Wave-front Sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University, 1999), pp. 91–130. [CrossRef]
- F. Rigaut, B. Neichel, M. Boccas, C. d’Orgeville, F. Vidal, M. A. van Dam, G. Arriagada, V. Fesquet, R. L. Galvez, G. Gausachs, C. Cavedoni, A. W. Ebbers, S. Karewicz, E. James, J. Lührs, V. Montes, G. Perez, W. N. Rambold, R. Rojas, S. Walker, M. Bec, G. Trancho, M. Sheehan, B. Irarrazaval, C. Boyer, B. L. Ellerbroek, R. Flicker, D. Gratadour, A. Garcia-Rissmann, F. Daruich, “Gemini multiconjugate adaptive optics system review - I. Design, trade-offs and integration,” Mon. Not. R. Astron. Soc. 437, 2361–2375 (2014). [CrossRef]
- B. Neichel, F. Rigaut, F. Vidal, M. A. van Dam, V. Garrel, E. Rodrigo Carrasco, P. Pessev, C. Winge, M. Boccas, C. d’Orgeville, G. Arriagada, A. Serio, V. Fesquet, W. N. Rambold, J. Lührs, C. Moreno, G. Gausachs, R. L. Galvez, V. Montes, T. B. Vucina, E. Marin, C. Urrutia, A. Lopez, S. J. Diggs, C. Marchant, A. W. Ebbers, C. Trujillo, M. Bec, G. Trancho, P. McGregor, P. J. Young, F. Colazo, M. L. Edwards, “Gemini multi-conjugate adaptive optics system review II: Commissioning, operation and overall performance,” Mon. Not. R. Astron. Soc., in press (2014). [CrossRef]
- C. d’Orgeville, S. Diggs, V. Fesquet, B. Neichel, W. Rambold, F. Rigaut, A. Serio, C. Araya, G. Arriagada, R. Balladares, M. Bec, M. Boccas, C. Duran, A. Ebbers, A. Lopez, C. Marchant, E. Marin, V. Montes, C. Moreno, E. Petit Vega, C. Segura, G. Trancho, C. Trujillo, C. Urrutia, P. Veliz, T. Vucina, “Gemini South multi-conjugate adaptive optics (GeMS) laser guide star facility on-sky performance results,” Proc. SPIE 8447, 84471Q (2012). [CrossRef]
- A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990). [CrossRef]
- D. R. Neal, W. J. Alford, J. K. Gruetzner, M. E. Warren, “Amplitude and phase beam characterization using a two-dimensional wavefront sensor,” Proc. SPIE 2870, 72–82 (1996). [CrossRef]
- C. Béchet, A. Guesalaga, B. Neichel, V. Fesquet, D. Guzman, “A two deformable-mirror concept to improve the laser efficiency of Gemini South MCAO,”, in Proceedings of the Third AO4ELT Conference, S. Esposito, L. Fini, eds. (INAF - Osservatorio Astrofisico di Arcetri, Firenze, Italy, 2013), 13344.
- C. d’Orgeville, F. Daruich, G. Arriagada, M. Bec, M. Boccas, S. Bombino, C. Carter, C. Cavedoni, F. Collao, P. Collins, E. James, S. Karewicz, M. Lazo, D. Maltes, R. Mouser, G. Perez, F. Rigaut, R. Rojas, M. Sheehan, G. Trancho, V. Vergara, T. Vucina, “The Gemini South MCAO laser guide star facility: getting ready for first light,” Proc. SPIE 7015, 70152P (2008). [CrossRef]
- R. Holzlöhner, D. Bonaccini Calia, W. Hackenberg, “Physical optics modeling and optimization of laser guide star propagation,” Proc. SPIE 7015, 701521 (2008). [CrossRef]
- B. L. Ellerbroek, “Adaptive Optics for the Thirty Meter Telescope,” in Proceedings of the Third AO4ELT Conference, S. Esposito, L. Fini, eds. (INAF - Osservatorio Astrofisico di Arcetri, Firenze, Italy, 2013), 13199.
- Y. Feng, L. R. Taylor, D. B. Calia, “25 W Raman-fiber-amplifier-based 589 nm laser for laser guide star,” Opt. Express 17(21), 19021–19026 (2009). [CrossRef]
- A. Guesalaga, B. Neichel, M. Boccas, C. d’Orgeville, F. Rigaut, D. Guzman, J. Anguita, “Improving stability, robustness, and performance of laser systems,” Proc. SPIE 8447, 84474M (2012). [CrossRef]
- F. Y. Kanev, V. P. Lukin, “Amplitude phase beam control with the help of a two-mirror adaptive system,” Atmos. Opt. 4, 878–881 (1991).
- B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4, 1400–1403 (1965). [CrossRef]
- J. W. Ogland, “Mirror system for uniform beam transformation in high-power annular lasers,” Appl. Opt. 17, 2917–2923 (1978). [CrossRef] [PubMed]
- D. Shafer, “Gaussian to flat-top intensity distributing lens,” Optics & Laser Technology 14, 159–160 (1982). [CrossRef]
- P. W. Rhodes, D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545–3553 (1980). [CrossRef] [PubMed]
- M. T. Eismann, A. M. Tai, J. N. Cederquist, “Iterative design of a holographic beamformer,” Appl. Opt. 28, 2641–2650 (1989). [CrossRef]
- 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, 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), 1781–1783 (2004). [CrossRef] [PubMed]
- M. C. Roggemann, D. J. Lee, “Two-Deformable-Mirror Concept for Correcting Scintillation Effects in Laser Beam Projection Through the Turbulent Atmosphere,” Appl. Opt. 37(21), 4577–4585 (1998). [CrossRef]
- J. D. Barchers, D. L. Fried, “Optimal control of laser beams for propagation through a turbulent medium,” J. Opt. Soc. Am. A 19, 1779–1793 (2002). [CrossRef]
- M. A. Vorontsov, V. Kolosov, “Target-in-the-loop beam control: basic considerations for analysis and wavefront sensing,” J. Opt. Soc. Am. A 22(1), 126–141 (2005). [CrossRef]
- Z. Zhao, S. D. Lyke, M. C. Roggemann, “Adaptive Optical Communication through Turbulent Atmospheric Channels,” in Proceedings of IEEE Conference on Communications, 2008. ICC ’08 (Institute of Electrical and Electronics Engineers, New York, 2008), 5432–5436.
- R. Kizito, M. C. Roggemann, T. J. Schulz, Y. Zhang, “Image sharpness metric-based deformable mirror control for beam projection systems operating in strong scintillation,” Proc. SPIE 5160, 406–416 (2004). [CrossRef]
- M. C. Roggemann, S. Deng, “Scintillation compensation for laser beam projection using segmented deformable mirrors,” Proc. SPIE 3763, 29–40 (1999). [CrossRef]
- M. C. Roggemann, A. C. Koivunen, “Wavefront sensing and deformable mirror control in strong scintillation,” J. Opt. Soc. Am. A 17, 911–919 (2000). [CrossRef]
- J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence-induced amplitude and phase distortions by means of multiple near-field phase adjustments,” J. Opt. Soc. Am. A 18(2), 399–411 (2001). [CrossRef]
- J. D. Barchers, “Evaluation of the impact of finite-resolution effects on scintillation compensation using two deformable mirrors,” J. Opt. Soc. Am. A 18, 3098–3109 (2001). [CrossRef]
- J. D. Barchers, “Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations,” J. Opt. Soc. Am. A 19, 926–945 (2002). [CrossRef]
- F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999). [CrossRef]
- R. A. Gonsalves, “Compensation of scintillation with a phase-only adaptive optic,” Opt. Lett. 22, 588–590 (1997). [CrossRef] [PubMed]
- O. Guyon, “Phase-induced amplitude apodization of telescope pupils for extrasolar terrestrial planet imaging,” Astron. Astrophys. 404, 379–387 (2003). [CrossRef]
- L. Pueyo, N. Kasdin, “Polychromatic compensation of propagated aberrations for high-contrast imaging,” Astrophys. J. 666, 609–625 (2007). [CrossRef]
- L. Pueyo, J. Kay, N. J. Kasdin, T. Groff, M. McElwain, A. Give’on, R. Belikov, “Optimal dark hole generation via two deformable mirrors with stroke minimization,” Appl. Opt. 48, 6296–6312 (2009). [CrossRef] [PubMed]
- D. Gavel, M. Ammons, B. Bauman, D. Dillon, E. Gates, B. Grigsby, J. Johnson, C. Lockwood, K. Morzinski, D. Palmer, M. Reinig, S. Severson, “Visible light laser guidestar experimental system (Villages): on-sky tests of new technologies for visible wavelength all-sky coverage adaptive optics systems,” Proc. SPIE 7015, 70150G (2008). [CrossRef]
- A. P. Norton, D. Gavel, M. Helmbrecht, C. J. Kempf, E. L. Gates, K. Chloros, D. Redel, D. Dillon, “Laser guidestar uplink correction using a MEMS deformable mirror: on-sky test results and implications for future AO systems,” accepted for Proc. SPIE (2014).
- X. Lei, S. Wang, H. Yan, W. Liu, L. Dong, P. Yang, B. Xu, “Double-deformable-mirror adaptive optics system for laser beam cleanup using blind optimization,” Opt. Express 20, 22143–22157 (2012). [CrossRef] [PubMed]
- V. Fesquet, C. Araujo, V. Garrel, A. Serio, F. Vidal, G. Arriagada, M. Boccas, F. Collao, S. Diggs, J. Donahue, C. D’Orgeville, G. Gausachs, C. Marchant, V. Montes, C. Moreno, B. Neichel, R. Oram, P. Pessev, W. Rambold, C. Urrutia, T. Vucina, “Review of Gemini South Laser Guide Star Facility performance and upgrades,” in Proceedings of the Third AO4ELT Conference, S. Esposito, L. Fini, eds. (INAF - Osservatorio Astrofisico di Arcetri, Firenze, Italy, 2013), 16120.
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
- R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
- R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. 66, 961–964 (1976). [CrossRef]
- J. R. Fienup, “Reconstruction of an object from the modulus of its fourier transform,” Opt. Lett. 3, 27–29 (1978). [CrossRef] [PubMed]
- H. Stark, Y. Yang, Vector Space Projections, a Numerical Approach to Signal and Image Processing, Neural Nets and Optics., Wiley Series in Telecommunications and Signal Processing (John Wiley., 1998).
- T. M. Venema, J. D. Schmidt, “Optical phase unwrapping in the presence of branch points,” Opt. Express 16, 6985–6998 (2008). [CrossRef] [PubMed]
- D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15(10), 2759–2768 (1998). [CrossRef]
- J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with examples in MATLAB (SPIE Press, 2010).
- C. C. Beckner, D. W. Oesch, “Implementation of a projection-on-constraints algorithm for beam intensity redistribution,” Proc. SPIE 6711, 67110L (2007)

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