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

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References

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    [Crossref]
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    [Crossref]
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    [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]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  15. P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605 (2006).
    [Crossref] [PubMed]
  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. 7, 12046 (2012).
    [Crossref]
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    [Crossref]
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    [Crossref]
  20. E. Thiébaut, “Optimization issues in blind deconvolution algorithms,” Proc. SPIE, 4847, 174–183.
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    [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]
  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).
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  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]
  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. 48, A30–A34 (2009).
    [Crossref]
  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.

2013 (1)

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

2012 (1)

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. 7, 12046 (2012).
[Crossref]

2009 (2)

2008 (1)

2007 (1)

2006 (1)

2004 (2)

2003 (1)

2002 (1)

1993 (4)

1982 (2)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[Crossref]

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

1976 (1)

1974 (1)

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

1973 (1)

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]

1971 (1)

R. B. Shack and B. C. Plack, “Production and use of a lenticular Hartmann screen (abstract),” J. Opt. Soc. Am. 61, 656 (1971).

Agour, M.

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. 7, 12046 (2012).
[Crossref]

Alexandrov, A.

Almoro, P.

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. 7, 12046 (2012).
[Crossref]

P. Almoro, G. Pedrini, and W. Osten, “Complete wavefront reconstruction using sequential intensity measurements of a volume speckle field,” Appl. Opt. 45, 8596–8605 (2006).
[Crossref] [PubMed]

Bagnoud, V.

Blanc, A.

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]

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]

Conan, J.-M.

Dolne, J. J.

Falldorf, C.

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. 7, 12046 (2012).
[Crossref]

Fienup, J. R.

Fourmaux, S.

Fusco, T.

Georges, J.

Gerchberg, R. W.

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

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[Crossref]

Hege, E. K.

Idier, J.

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]

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]

Ito, T.

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.

Jefferies, S. M.

Kieffer, J. C.

Kreutz-Delgado, K.

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

Kudryashov, A.

Lloyd-Hart, M.

Marron, J. C.

Martin, F.

Masuda, N.

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.

Menicucci, P.

Miccolis, D.

Misell, D. L.

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]

Mugnier, L. M.

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

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]

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]

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]

Nishitsuji, T.

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.

Noll, R. J.

Osten, W.

Ozaki, T.

Paul, B.

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

Payeur, S.

Pedersen, M. S.

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

Pedrini, G.

Petersen, K. B.

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

Petit, C.

Plack, B. C.

R. B. Shack and B. C. Plack, “Production and use of a lenticular Hartmann screen (abstract),” J. Opt. Soc. Am. 61, 656 (1971).

Primot, J.

Roddier, C.

Roddier, F.

Rousset, G.

Sakurai, T.

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.

Sauvage, J.-F.

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

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]

Schall, H.

Schulz, T. J.

Seiden, H.

Seldin, J. H.

Serbanescu, C.

Shack, R. B.

R. B. Shack and B. C. Plack, “Production and use of a lenticular Hartmann screen (abstract),” J. Opt. Soc. Am. 61, 656 (1971).

Shimobaba, T.

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.

Stark, H.

H. Stark, ed., Image Recovery: Theory and Application (Academic, 1987).

Takada, N.

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.

Thiébaut, E.

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

Thurman, S. T.

Vachss, F.

Widen, K.

Zuegel, J. D.

Appl. Opt. (8)

Astron. Astrophys. (1)

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

J. Eur. Opt. Soc. Rapid Publ. (1)

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. 7, 12046 (2012).
[Crossref]

J. Opt. Soc. Am. (2)

R. B. Shack and B. C. Plack, “Production and use of a lenticular Hartmann screen (abstract),” J. Opt. Soc. Am. 61, 656 (1971).

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

J. Opt. Soc. Am. A (4)

J. Phys. D Appl. Phys. (1)

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]

Opt. Acta (1)

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

Opt. Eng. (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Other (7)

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]

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 and M. S. Pedersen, “The matrix cookbook” (2008), Version 20081110.

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

H. Stark, ed., Image Recovery: Theory and Application (Academic, 1987).

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.

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Figures (9)

Fig. 1
Fig. 1 Schematic diagram for phase diversity measurement.

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Fig. 2
Fig. 2 Experimental setup for camelot validation.

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Fig. 3
Fig. 3 Left: modulus of the simulated field (AS), center: measured modulus (AM), right: 2.5 × |AM − AS|.

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Fig. 4
Fig. 4 Phase of the simulated field.

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Fig. 5
Fig. 5 Left: focal plane images (k = 1); center: first defocused images (k = 2); right: second defocused image (k = 3). Top: experimental images; middle: direct model result for |Mkψc|2; bottom: residuals, i.e., |ik − |Mkψc|2|. Logarithmic scale, 140 × 140 pixels ROI centered on the optical axis.

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Fig. 6
Fig. 6 From left to right: Ac, AM and |AC − AM|.

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Fig. 7
Fig. 7 From left to right: φc, φPD and |φC − φPD|.

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Fig. 8
Fig. 8 Focal plane images, left: simulation (Nphe,1 = 1.6 107, 10 short exposures), right: experiment.

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Fig. 9
Fig. 9 ε A 2 as a function of average photo-electrons per pixel Nphe. Black diamonds (⋄): perfect detector, green triangles (▵): finite well capacity and quantization, red asterisk (*): experiment result, dotted line: N phe − 2, dashed line: N phe − 1.

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Equations (16)

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Ψ k ( x , y ) = ∑ j = 1 N k ψ j , k b j , k ( x , y ) .
i k = | ψ k | 2 + n k ,
X ⊙ Y = [ ( X j Y j ) ] j = [ 1 , N ] T ,
ψ k = M k [ ψ 0 ⊙ s k ] .
s k = e i ( a 2 , k Z 2 + a 3 , k Z 3 ) .
i k = | M k [ ψ ⊙ s k ] | 2 + n k .
ψ 1 = DFT [ ψ 0 ⊙ s 1 ]
ψ k > 1 = exp ( i 2 π λ d 1 k ) i λ d 1 k IDFT [ DFT [ ψ 1 ] ⊙ exp ( i π λ d 1 k ν 2 ) ] .
P ( ψ 0 , a | { i k } ) ∝ P ( { i k } | ψ 0 , a ) P ( ψ 0 ) P ( a )
( ψ ^ 0 , a ^ ) = arg min ψ , a J ( ψ 0 , a )
J ( ψ 0 , a ) = ∑ k = 1 N 1 2 ( i k − | M k [ ψ 0 ⊙ s k ] | 2 ) T C k − 1 ( i k − | M k [ ψ 0 ⊙ s k ] | 2 ) − ln P ( ψ 0 ) − ln P ( a ) ,
∇ J ( ψ ) = ∂ J ∂ ℜ ( ψ ) + j ∂ J ∂ ℑ ( ψ ) = − ∑ k = 1 N 2 s k * ⊙ M k * T ︸ 4 ( C k − 1 ︸ 3 [ ( M k [ ψ ⊙ s k ] ) ︸ 2 ⊙ ( i k − | M k [ ψ ⊙ s k ] | 2 ) ︸ 1 ] ) .
∂ J ∂ a i , k = − ℑ [ ( ψ ⊙ s k ⊙ Z i ) T ( 2 M k * T C k − 1 [ ( M k [ ψ ⊙ s k ] ) ⊙ ( i k − | M k [ ψ ⊙ s k ] | 2 ) ] ) * ] .
ε M S 2 = ∑ j = 1 N 0 | A M , j − A S , j | 2 ∑ j = 1 N 0 | A S , j | 2 .
d k k + 1 = 8 ( f / D ) 2 δ O P D .
[ C k ] j j = [ σ p h , k 2 ] j + p ( σ ron 2 + q 2 / 12 )

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