## Laser beam complex amplitude measurement by phase diversity |

Optics Express, Vol. 22, Issue 4, pp. 4575-4589 (2014)

http://dx.doi.org/10.1364/OE.22.004575

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### Abstract

The control of the optical quality of a laser beam requires a complex amplitude measurement able to deal with strong modulus variations and potentially highly perturbed wavefronts. The method proposed here consists in an extension of phase diversity to complex amplitude measurements that is effective for highly perturbed beams. Named camelot for Complex Amplitude MEasurement by a Likelihood Optimization Tool, it relies on the acquisition and processing of few images of the beam section taken along the optical path. The complex amplitude of the beam is retrieved from the images by the minimization of a Maximum a Posteriori error metric between the images and a model of the beam propagation. The analytical formalism of the method and its experimental validation are presented. The modulus of the beam is compared to a measurement of the beam profile, the phase of the beam is compared to a conventional phase diversity estimate. The precision of the experimental measurements is investigated by numerical simulations.

© 2014 Optical Society of America

## 1. Introduction

2. J. Primot, “Three-wave lateral shearing interferometer,” Appl. Opt. **32**, 6242–6249 (1993). [CrossRef] [PubMed]

7. S. Fourmaux, S. Payeur, A. Alexandrov, C. Serbanescu, F. Martin, T. Ozaki, A. Kudryashov, and J. C. Kieffer, “Laser beam wavefront correction for ultra high intensities with the 200 tw laser system at the advanced laser light source,” Opt. Express **16**, 11987–11994 (2008). [CrossRef] [PubMed]

8. C. Roddier and F. Roddier, “Combined approach to the Hubble space telescope wave-front distortion analysis,” Appl. Opt. **32**, 2992–3008 (1993). [CrossRef] [PubMed]

9. R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta **21**, 709–720 (1974). [CrossRef]

10. J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl. Opt. **32**, 1747–1767 (1993). [CrossRef] [PubMed]

11. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. **32**, 1737–1746 (1993). [CrossRef] [PubMed]

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

17. S. T. Thurman and J. R. Fienup, “Complex pupil retrieval with undersampled data,” J. Opt. Soc. Am. A **26**, 2640–2647 (2009). [CrossRef]

*Complex Amplitude Measurement by a Likelihood Optimization Tool*, is described in Section 2. In Section 3 we validate it experimentally and assess its performance on a laboratory set-up designed to shape the complex amplitude of a laser beam and record images of the focal spot at several longitudinal positions. In particular, a cross-validation with conventional phase diversity is presented. Finally, experimental results are confronted to carefully designed simulations taking into account many error sources in Section 4. In particular, the impact of photon, detector and quantization noises on the estimation precision is studied.

## 2. camelot

### 2.1. Problem statement

*denote the complex amplitude in plane*

_{k}*P*. Ψ

_{k}*is decomposed onto a finite orthonormal spatial basis with basis vectors {*

_{k}*b*(

_{j,k}*x*,

*y*)}

_{j=[1,Nk]}: The coefficients of this decomposition are stacked into a single column vector, denoted by

*ψ*= [

_{k}*ψ*]

_{j,k}_{j=[1, Nk]}∈ ℂ

^{Nk}. In the following, a pixel basis is used without loss of generality.

*P*

_{0}, is supposed to be the unknown.

*P*

_{0}is called hereafter the estimation plane. We assume that the phase diversity is performed by measuring intensity distributions in

*N*planes, perpendicular to the propagation axis.

_{P}*P*(1 ≤

_{k}*k*≤

*N*) refer to these planes. The transverse intensity distributions of the field are measured by translating the image sensor along the optical axis around the focal plane. The measured signal in plane

_{P}*P*is a two dimensional discrete distribution concatenated formally into a single vector of size

_{k}*N*denoted by

_{k}*i*. As the detection of the images is affected by several noise sources, denoting

_{k}*n*the noise vector, the direct model reads: where |

_{k}*X*|

^{2}=

*X*⊙

*X*

^{*}and ⊙ represents a component-wise product. The component-wise product of two complex column vectors of size

*N*denoted

*ψ*can be expressed as a linear transformation (a transfer) of

_{k}*ψ*

_{0}and therefore be described by the product of the propagation matrix

*M*∈ ℂ

_{k}^{N0×Nk}by

*ψ*

_{0}.

*ψ*

_{0}by a differential shift phasor

*s*: The

_{k}*k*-th differential shift phasor is decomposed on the Zernike tip and tilt polynomials

*Z*

_{2}and

*Z*

_{3}expressed in the pixel basis [18

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

*a*

_{2,}

*and*

_{k}*a*

_{3,}

*being their respective coefficients:*

_{k}*a*is defined as:

*a*= {

*a*}. Without loss of generality, the first measurement plane (k=1) is chosen as the reference plane, so that

_{i,k}*a*

_{2,1}=

*a*

_{3,1}= 0.

_{1}being the far field of Ψ

_{0}⊙

*s*

_{1}:

*k*= 1) and plane

*P*

_{k}_{>1}is simulated by a Fresnel propagation performed in Fourier space: where

*d*

_{1}

*is the distance between plane 1 and plane*

_{k}*k*,

*ν*is the norm of the spatial frequency vector in discrete Fourier space, and IDFT the inverse of the DFT operator.

### 2.2. Inverse problem approach

*ψ*

_{0}from the set of measurements, {

*i*}, the basic idea is to invert the image formation model, i.e., the direct model. For doing so, we adopt the following maximum a Posteriori (map) framework [19

_{k}19. J. Idier, ed., *Bayesian Approach to Inverse Problems*, Digital Signal and Image Processing Series (ISTE, 2008). [CrossRef]

*ψ̂*

_{0}and misalignment coefficients

*â*are the ones that maximize the conditional probability of the field and misalignment coefficients given the measurements, that is the posterior likelihood

*P*(

*ψ*

_{0},

*a*|{

*i*}). According to Bayes’ rule: where

_{k}*P*(

*ψ*

_{0}), respectively

*P*(

*a*), embody our prior knowledge on

*ψ*

_{0}, respectively

*a*.

*C*= 〈

_{k}*n*·

_{k}*n*〉 is the covariance matrix of the noise on the pixels recorded in plane

_{k}^{T}*k*(diagonal if the noise is white) and −ln

*P*(

*ψ*) and −ln

*P*(

*a*) are regularization terms that embody our prior knowledge on their arguments.

*P*(

*ψ*) =

*P*(

*a*) =constant, and the MAP metric of Eq. (11) reduces to a Maximum-Likelihood (ML) metric.

11. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. **32**, 1737–1746 (1993). [CrossRef] [PubMed]

### 2.3. Minimization

*J*is a non-linear real-valued function of Ψ

_{0}and

*a*. In order to perform its minimization a method based on a quasi-Newton algorithm, called variable metric with limited memory and bounds method (VMLM-B) [20] is used. The VMLM-B method requires the analytical expression of the gradient of the criterion. The complex gradient of

*J*with respect to

*ψ*[21, 22], denoted ∇

*J*(

*ψ*), is defined in Eq. (12) as the complex vector having the partial derivative of

*J*with respect to ℜ(

*ψ*) (respectively ℑ(

*ψ*)) as its real (respectively imaginary) part.

- the computation of the difference between measurements and the direct model;
- weighing this difference by the result of the direct model;
- whitening the noise to take into account noise statistics;
- a reverse propagation that enables the projection of the gradient into the space of the unknowns and takes into account differential tip/tilts.

## 3. Experiment

### 3.1. Principle of the experiment

*ψ*=

_{c}*A*

_{C}e^{iφC}in the following. The beam reflected on the beam-splitter is used to record the intensity distribution

*I*with an image sensor conjugated with the aperture plane.

_{M}*A*is compared with

_{C}*A*, called measured modulus of the field hereafter, which is computed as the square-root of image

_{M}5. L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” in *Advances in Imaging and Electron Physics*, P. Hawkes, ed. (Elsevier, 2006), Vol. 141, Chap. 1, pp. 1–76. [CrossRef]

5. L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” in *Advances in Imaging and Electron Physics*, P. Hawkes, ed. (Elsevier, 2006), Vol. 141, Chap. 1, pp. 1–76. [CrossRef]

6. J.-F. Sauvage, T. Fusco, G. Rousset, and C. Petit, “Calibration and pre-compensation of non-common path aberrations for extreme adaptive optics,” J. Opt. Soc. Am. A **24**, 2334–2346 (2007). [CrossRef]

23. B. Paul, J.-F. Sauvage, and L. M. Mugnier, “Coronagraphic phase diversity: performance study and laboratory demonstration,” Astron. Astrophys. **552**, 1–11 (2013). [CrossRef]

24. A. Blanc, L. M. Mugnier, and J. Idier, “Marginal estimation of aberrations and image restoration by use of phase diversity,” J. Opt. Soc. Am. A **20**, 1035–1045 (2003). [CrossRef]

*A*and the data used are two of the three near focal plane intensity distributions: the focal plane image and the first defocused image.

_{M}### 3.2. Control of the field

25. V. Bagnoud and J. D. Zuegel, “Independent phase and amplitude control of a laser beam by use of a single-phase-only spatial light modulator,” Opt. Lett. **29**, 295–297 (2004). [CrossRef] [PubMed]

*λ*= 650nm injected into a 4.6μm core single-mode fiber. At the exit of the fiber, the beam is collimated, linearly polarized and then reflected on the SLM. The clear aperture diameter is

*D*= 3mm on the surface of the SLM. It is conjugated with a unit magnification onto the clear aperture plane. The spatial modulation and filtering are designed to control 15×15 resolution elements in the clear aperture i.e., 15 cycles per aperture. The effective result of the control in the clear aperture plane is called the true field. Its modulus is denoted

*A*.

_{T}*ψ*. Its modulus, denoted

_{S}*A*, is presented on the left of Fig. 3. It shows smooth variations as can be typically observed on an intense pulsed laser [7

_{S}7. S. Fourmaux, S. Payeur, A. Alexandrov, C. Serbanescu, F. Martin, T. Ozaki, A. Kudryashov, and J. C. Kieffer, “Laser beam wavefront correction for ultra high intensities with the 200 tw laser system at the advanced laser light source,” Opt. Express **16**, 11987–11994 (2008). [CrossRef] [PubMed]

*Z*

_{7}) of 11 rad Peak-to-Valley (PV), typical of a strong misalignment of a parabola for instance.

*A*, is presented at the center of Fig. 3.

_{M}*A*with the measured one

_{S}*A*, the following distance metric is defined:

_{M}*A*and

_{S}*A*have been normalized in flux

_{M}*A*and

_{S}*A*is presented on the right of Fig. 3. It has been multiplied by a factor 2.5 to present a dynamic comparable to

_{M}*A*.

_{M}*A*and

_{S}*A*enables to identify a clear cut separation between spatial frequencies below the cut-off frequency of the spatial modulation (15 cycles per aperture) and higher spatial frequencies. Due to the spatial filtering by the control module, the low spatial frequencies part of the difference between

_{M}*A*and

_{S}*A*can be attributed mostly to model errors between the simple simulation performed to compute

_{M}*A*and the effective experimental setup of the control module. For instance in the simulation, the spatial filter is assumed to be centered on the optical axis, the SLM illumination is assumed to be perfectly homogeneous, and lenses and optical conjugations are supposed to be perfect. This part of

_{S}*A*measurement. It is evaluated to 4 10

_{M}^{−3}. This gives insight on the measurement precision of

*A*, which should thus be of the order of 4 10

_{M}^{−3}.

### 3.3. camelot*estimation*

#### 3.3.1. Practical implementation of phase diversity measurement

*λ*PV is often chosen as it maximizes the difference between the focal plane intensity distribution and the defocused image. Therefore, the amplitude of defocus between the position of each plane is fixed to

*λ*PV. Considering such a defocus, it appears that no less than three different measurement planes are required with camelot. The first image is located in the focal plane, the second one is at a distance corresponding to

*λ*of defocus PV, the third one at a distance of 2

*λ*PV from the focal plane. The relation between the PV optical path difference

*δ*in the aperture plane and the corresponding translation distance between two successive planes

_{OPD}*d*

_{kk}_{+1}is: For the experiment, the focusing optics focal length is

*f*= 100mm and the aperture diameter is

*D*= 3mm. Consequently the translation distances are

*d*

_{12}=

*d*

_{23}= 5.78mm between the successive recording planes.

*R*

^{2}). Its characteristics are the following: a pixel size and spacing

*s*= 6.452μm, a readout noise standard deviation

_{pixel}*σ*= 6

_{ron}*e*

^{−}rms, a full well capacity of 18000

*e*

^{−}, and a 12 bit digitizer.

*λf*/(2

*s*) = 1.68 for

_{pixel}D*λ*= 650nm.

*N*= 5 × 10

_{phe}^{7}total photo-electrons are sufficient. Due to the limited full well capacity of the sensor,

*p*= 10 short exposures images are added to reach this number. For each image, background influence has been removed by a subtraction of an offset computed from the average of pixels located on the side of the images (hence not illuminated).

*C*is approximated by a diagonal matrix whose diagonal terms, [

_{k}*C*]

_{k}*, correspond to the sum of the variances of the photon noise, read out noise and quantization noise: where*

_{jj}*q*is the quantization step. In practice, the readout noise variance map

*j*[26

26. L. M. Mugnier, T. Fusco, and J.-M. Conan, “MISTRAL: a myopic edge-preserving image restoration method, with application to astronomical adaptive-optics-corrected long-exposure images.” J. Opt. Soc. Am. A **21**, 1841–1854 (2004). [CrossRef]

*N*= 214 × 214 pixels. For the figure, a region of interest of 140×140 pixels, centered on the optical axis, is selected. From left to right the focal plane image, the first defocused image and the second one are displayed. The colorscale is logarithmic.

_{pix}#### 3.3.2. Results

*N*

_{0}= 64 × 64. The current implementation of the algorithm is written in the IDL language.

*i*and the model that can be computed from the estimated field through Eq (6). This latter map is presented on the middle row of Fig. 5. The moduli of the differences between the measurements and the direct model, that is to say estimation residuals, are displayed on the bottom row of the same figure. These residuals are below 1% of the maximum of the measurements on the three planes.

_{k}*P*

_{2}and

*P*

_{3}respectively.

*A*, is presented on the left of Fig. 6. It has been normalized in flux to enable the comparison with the measured modulus

_{C}*A*, shown in the center of the figure. For the comparison,

_{M}*A*has been sub-sampled and resized to 64 pixels. The modulus of the difference between

_{M}*A*and

_{C}*A*is represented on the right. The main spatial structures of

_{M}*A*are well estimated by the method. The estimation residuals are below 20% of the maximum of the measured modulus, even in the zones where the flux is low (top right corner). The distance between the two moduli is

_{M}*A*and

_{M}*A*reported in Section 3.2 which was found to be 0.044. camelot thus delivers a modulus estimation that is several times closer to

_{S}*A*than

_{M}*A*is.

_{S}*ψ*,

_{C}*φ*, is presented on the left of the figure, the phase of the conventional phase diversity,

_{C}*φ*, is in the center of the figure, the modulus of their difference is presented on the right. For comparison of these phase maps, their differential piston has been set to 0. In the zones where the modulus is greater than 10% of the maximum modulus of

_{PD}*ψ*, the maximum of the phase residuals is below 2

_{M}*π*/10 rad.

*A*as the pupil transmittance instead of

_{C}*A*enables a better fit to the data with a 5% smaller criterion at convergence of the minimization, which is yet another indicator of the quality of the modulus

_{M}*A*estimated by camelot.

_{C}## 4. Performance analysis by simulations

*ψ*while taking into account the main disturbances that affect image formation: misalignments, photon and detector noises, limited full well capacity, quantization and miscalibration.

_{C}*N*, then for each pixel a Poisson occurrence is computed, and a Gaussian white noise occurrence with variance

_{phe,k}*N*, the corresponding image is computed as the addition of as many “short-exposures” as needed in order to take into account the finite well capacity, and each of these short exposures is corrupted with photon noise, readout noise and a 12 bit quantization noise. The same number of photo-electrons is attributed to each image :

_{phe,k}*N*=

_{phe,k}*N*where

_{phe}/N_{P}*N*designates the total number of photo-electrons.

_{phe}*N*

_{phe}_{,1}= 1.6 10

^{7}photo-electrons by adding 10 short exposures. It can be visually compared with the experimental focal plane image recorded in comparable conditions (right): the similarity between the two images illustrates the relevance of the image simulation.

*N*small compared to 5 10

_{phe}^{8},

*N*≤ 5 10

_{phe}^{8}, the average flux on illuminated pixels, that can be approximated by

*N*/(

_{phe}*N*

_{P}*n*) where

_{i}*n*is the average number of illuminated pixels per image plane (

_{i}*n*≈ 100), is smaller than the total readout noise contribution for one image plane, that is

_{i}*N*≥ 5 10

_{phe}^{8}, the photon noise contribution becomes predominant and the noise propagation of

*N*= 5 10

_{phe}^{6}photo-electrons, then starts to follow a

*N*= 5 10

_{phe}^{6}corresponds to the flux necessary to saturate the well capacity of the sensor. Above this limit, images are added to emulate the summation of “short-exposures” images. Hence, the noise level in the measurements starts to depend on the number of summed images, that is to say on

*N*, with a

_{phe}*A*too is imperfect i.e., is only an estimate of the true modulus

_{M}*A*, due to experimental artefacts, notably differential optical defects that affect image formation on the aperture plane imaging setup and noise influence. This claim is also supported by the fact that the phase estimated by conventional phase diversity fits the measurements better when the pupil transmittance is set to the modulus estimated by camelot,

_{T}*A*, instead of the measured modulus

_{C}*A*. The Fourier-based analysis mentioned at the end of Section 3.2 delivers an estimate of the measurement precision of

_{M}*A*that is evaluated to 4 10

_{M}^{−3}.

## 5. Conclusion

## Acknowledgments

## References and links

1. | R. B. Shack and B. C. Plack, “Production and use of a lenticular Hartmann screen (abstract),” J. Opt. Soc. Am. |

2. | J. Primot, “Three-wave lateral shearing interferometer,” Appl. Opt. |

3. | D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D Appl. Phys. |

4. | R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. |

5. | L. M. Mugnier, A. Blanc, and J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” in |

6. | J.-F. Sauvage, T. Fusco, G. Rousset, and C. Petit, “Calibration and pre-compensation of non-common path aberrations for extreme adaptive optics,” J. Opt. Soc. Am. A |

7. | S. Fourmaux, S. Payeur, A. Alexandrov, C. Serbanescu, F. Martin, T. Ozaki, A. Kudryashov, and J. C. Kieffer, “Laser beam wavefront correction for ultra high intensities with the 200 tw laser system at the advanced laser light source,” Opt. Express |

8. | C. Roddier and F. Roddier, “Combined approach to the Hubble space telescope wave-front distortion analysis,” Appl. Opt. |

9. | R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta |

10. | J. R. Fienup, J. C. Marron, T. J. Schulz, and J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl. Opt. |

11. | J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. |

12. | H. Stark, ed., |

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

14. | S. M. Jefferies, M. Lloyd-Hart, E. K. Hege, and J. Georges, “Sensing wave-front amplitude and phase with phase diversity,” Appl. Opt. |

15. | P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. |

16. | M. Agour, P. Almoro, and C. Falldorf, “Investigation of smooth wave fronts using slm-based phase retrieval and a phase diffuser,” J. Eur. Opt. Soc. Rapid Publ. |

17. | S. T. Thurman and J. R. Fienup, “Complex pupil retrieval with undersampled data,” J. Opt. Soc. Am. A |

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

19. | J. Idier, ed., |

20. | E. Thiébaut, “Optimization issues in blind deconvolution algorithms,” Proc. SPIE, 4847, 174–183. |

21. | K. B. Petersen and M. S. Pedersen, “The matrix cookbook” (2008), Version 20081110. |

22. | K. Kreutz-Delgado, “The Complex Gradient Operator and the CR-Calculus,” ArXiv e-prints (2009). |

23. | B. Paul, J.-F. Sauvage, and L. M. Mugnier, “Coronagraphic phase diversity: performance study and laboratory demonstration,” Astron. Astrophys. |

24. | A. Blanc, L. M. Mugnier, and J. Idier, “Marginal estimation of aberrations and image restoration by use of phase diversity,” J. Opt. Soc. Am. A |

25. | V. Bagnoud and J. D. Zuegel, “Independent phase and amplitude control of a laser beam by use of a single-phase-only spatial light modulator,” Opt. Lett. |

26. | L. M. Mugnier, T. Fusco, and J.-M. Conan, “MISTRAL: a myopic edge-preserving image restoration method, with application to astronomical adaptive-optics-corrected long-exposure images.” J. Opt. Soc. Am. A |

27. | J. J. Dolne, P. Menicucci, D. Miccolis, K. Widen, H. Seiden, F. Vachss, and H. Schall, “Advanced image processing and wavefront sensing with real-time phase diversity,” Appl. Opt. |

28. | T. Nishitsuji, T. Shimobaba, T. Sakurai, N. Takada, N. Masuda, and T. Ito, “Fast calculation of fresnel diffraction calculation using amd gpu and opencl,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2011), p. DWC20. |

**OCIS Codes**

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

(100.3190) Image processing : Inverse problems

(100.5070) Image processing : Phase retrieval

(140.3295) Lasers and laser optics : Laser beam characterization

(280.4788) Remote sensing and sensors : Optical sensing and sensors

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: November 22, 2013

Revised Manuscript: January 17, 2014

Manuscript Accepted: January 17, 2014

Published: February 20, 2014

**Citation**

Nicolas Védrenne, Laurent M. Mugnier, Vincent Michau, Marie-Thérèse Velluet, and Rudolph Bierent, "Laser beam complex amplitude measurement by phase diversity," Opt. Express **22**, 4575-4589 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4575

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### References

- R. B. Shack, B. C. Plack, “Production and use of a lenticular Hartmann screen (abstract),” J. Opt. Soc. Am. 61, 656 (1971).
- J. Primot, “Three-wave lateral shearing interferometer,” Appl. Opt. 32, 6242–6249 (1993). [CrossRef] [PubMed]
- D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics: I. Test calculations,” J. Phys. D Appl. Phys. 6, 2200–2216 (1973). [CrossRef]
- R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982). [CrossRef]
- L. M. Mugnier, A. Blanc, J. Idier, “Phase diversity: a technique for wave-front sensing and for diffraction-limited imaging,” in Advances in Imaging and Electron Physics, P. Hawkes, ed. (Elsevier, 2006), Vol. 141, Chap. 1, pp. 1–76. [CrossRef]
- J.-F. Sauvage, T. Fusco, G. Rousset, C. Petit, “Calibration and pre-compensation of non-common path aberrations for extreme adaptive optics,” J. Opt. Soc. Am. A 24, 2334–2346 (2007). [CrossRef]
- S. Fourmaux, S. Payeur, A. Alexandrov, C. Serbanescu, F. Martin, T. Ozaki, A. Kudryashov, J. C. Kieffer, “Laser beam wavefront correction for ultra high intensities with the 200 tw laser system at the advanced laser light source,” Opt. Express 16, 11987–11994 (2008). [CrossRef] [PubMed]
- C. Roddier, F. Roddier, “Combined approach to the Hubble space telescope wave-front distortion analysis,” Appl. Opt. 32, 2992–3008 (1993). [CrossRef] [PubMed]
- R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974). [CrossRef]
- J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble space telescope characterized by using phase-retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993). [CrossRef] [PubMed]
- J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993). [CrossRef] [PubMed]
- H. Stark, ed., Image Recovery: Theory and Application (Academic, 1987).
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
- S. M. Jefferies, M. Lloyd-Hart, E. K. Hege, J. Georges, “Sensing wave-front amplitude and phase with phase diversity,” Appl. Opt. 41, 2095–2102 (2002). [CrossRef] [PubMed]
- P. Almoro, G. Pedrini, W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605 (2006). [CrossRef] [PubMed]
- M. Agour, P. Almoro, C. Falldorf, “Investigation of smooth wave fronts using slm-based phase retrieval and a phase diffuser,” J. Eur. Opt. Soc. Rapid Publ. 7, 12046 (2012). [CrossRef]
- S. T. Thurman, J. R. Fienup, “Complex pupil retrieval with undersampled data,” J. Opt. Soc. Am. A 26, 2640–2647 (2009). [CrossRef]
- R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976). [CrossRef]
- J. Idier, ed., Bayesian Approach to Inverse Problems, Digital Signal and Image Processing Series (ISTE, 2008). [CrossRef]
- E. Thiébaut, “Optimization issues in blind deconvolution algorithms,” Proc. SPIE, 4847, 174–183.
- K. B. Petersen, M. S. Pedersen, “The matrix cookbook” (2008), Version 20081110.
- K. Kreutz-Delgado, “The Complex Gradient Operator and the CR-Calculus,” ArXiv e-prints (2009).
- B. Paul, J.-F. Sauvage, L. M. Mugnier, “Coronagraphic phase diversity: performance study and laboratory demonstration,” Astron. Astrophys. 552, 1–11 (2013). [CrossRef]
- A. Blanc, L. M. Mugnier, J. Idier, “Marginal estimation of aberrations and image restoration by use of phase diversity,” J. Opt. Soc. Am. A 20, 1035–1045 (2003). [CrossRef]
- V. Bagnoud, J. D. Zuegel, “Independent phase and amplitude control of a laser beam by use of a single-phase-only spatial light modulator,” Opt. Lett. 29, 295–297 (2004). [CrossRef] [PubMed]
- L. M. Mugnier, T. Fusco, J.-M. Conan, “MISTRAL: a myopic edge-preserving image restoration method, with application to astronomical adaptive-optics-corrected long-exposure images.” J. Opt. Soc. Am. A 21, 1841–1854 (2004). [CrossRef]
- J. J. Dolne, P. Menicucci, D. Miccolis, K. Widen, H. Seiden, F. Vachss, H. Schall, “Advanced image processing and wavefront sensing with real-time phase diversity,” Appl. Opt. 48, A30–A34 (2009). [CrossRef]
- T. Nishitsuji, T. Shimobaba, T. Sakurai, N. Takada, N. Masuda, T. Ito, “Fast calculation of fresnel diffraction calculation using amd gpu and opencl,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2011), p. DWC20.

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