Abstract

To retrieve the high-spatial-frequency information of atmospherically distorted wave fronts, it is desirable to use an adaptive optics system with a high degree of compensation. However, in the visible only partial compensation is attainable. We analyze the photoevent statistics corresponding to wave fronts with partial compensation. It is shown that the photoevent statistics evolve from a Bose–Einstein distribution to a Laguerre distribution as the number of corrected Zernike polynomials increases. Furthermore, the photoevent statistics are also analyzed as a function of the position at the image plane.

© 1999 Optical Society of America

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References

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  1. J. Y. Wang, J. K. Markey, “Modal compensation of atmospheric turbulence phase distortion,” J. Opt. Soc. Am. 68, 78–87 (1978).
    [CrossRef]
  2. F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).
  3. M. C. Roggemann, “Limited degree-of-freedom adaptive optics and image reconstruction,” Appl. Opt. 30, 4227–4233 (1991).
    [CrossRef] [PubMed]
  4. M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  5. M. P. Cagigal, V. F. Canales, “Speckle statistics in partially corrected wave fronts,” Opt. Lett. 23, 1072–1074 (1998).
    [CrossRef]
  6. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  7. V. F. Canales, M. P. Cagigal, “Rician distribution to describe speckle statistics in adaptive optics partial correction,” Appl. Opt. 38, 766–771 (1999).
    [CrossRef]
  8. J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford UK, 1993).
  10. B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).
  11. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  12. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989).

1999 (1)

1998 (1)

1991 (2)

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

M. C. Roggemann, “Limited degree-of-freedom adaptive optics and image reconstruction,” Appl. Opt. 30, 4227–4233 (1991).
[CrossRef] [PubMed]

1990 (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

1978 (1)

1976 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford UK, 1993).

Cagigal, M. P.

Canales, V. F.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989).

Fontanella, J. C.

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Gaffard, J. P.

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).

Lena, P.

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Markey, J. K.

Merkle, F.

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Noll, R. J.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989).

Rigaut, F.

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Roddier, N.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Roggemann, M. C.

Rousset, G.

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

Saleh, B. E. A.

B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989).

Wang, J. Y.

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford UK, 1993).

Appl. Opt. (2)

Astron. Astrophys. (1)

F. Rigaut, G. Rousset, J. C. Fontanella, J. P. Gaffard, F. Merkle, P. Lena, “Adaptive optics on a 3.6-m telescope: results and performance,” Astron. Astrophys. 250, 280–290 (1991).

J. Opt. Soc. Am. (2)

Opt. Eng. (1)

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Opt. Lett. (1)

Other (5)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1989).

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford UK, 1993).

B. E. A. Saleh, Photoelectron Statistics (Springer-Verlag, Berlin, 1978).

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Figures (3)

Fig. 1
Fig. 1

Values of the residual phase variance of the compensated wave front as a function of the position of the observation point at the image plane. Solid curve, theoretical values; circles, simulated values obtained by our fitting the light intensity PDF obtained from simulation to the theoretical one described by Eq. (12).

Fig. 2
Fig. 2

Photoelectron probability distribution at the PSF core for (a) 11, (b) 21, (c) 41, and (d) 81 corrected polynomials. Solid curves, theoretical exact values; dots, simulated values; triangles, theoretical approximated values.

Fig. 3
Fig. 3

Photoelectron probability distribution as a function of the position at the image plane for 80 corrected polynomials. Solid curves, theoretical exact values; dots, simulated values; (a) corresponds to the PSF core; (b)–(f ) correspond to points with a separation of λf/4D.

Equations (22)

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Φ(r, θ)=i=1aiZi(r, θ),
Δj=i=j+1|ai|2=coef(j)Dr05/3,
P(ϕ)=12πΔj1/2 exp[-ϕ2/(2Δj)].
Mϕ(ω)=- exp(jωϕ)P(ϕ)dϕ=exp-Δjω22.
P(Ar, Ai)=1(2πσr2)1/2 exp[-(Ar-Ar)2/2σr2]×12πσi2 exp(-Ai2/2σi2),
P(I, θ)=12 1(2πσr2)1/2 exp-(I cos θ-Ar)22σr2×1(2πσi2)1/2 exp-(I sin θ)22σi2,
P(I)=-ππP(I, θ)dθ.
Ar=αM¯ϕ(1),
σr2=(α2¯/2)[1+Mϕ(2)]-(α¯)2Mϕ2(1)/N,
σi2=(α2¯/2)[1-Mϕ(2)]/N,
P(I)=12σ2 exp-I+a22σ2I0aIσ2.
a2=[Ar4+2Ar2(σi2-σr2)-(σi2-σr2)2]1/2,
2σ2=σr2+σi2+Ar2-[Ar4
+2Ar2(σi2-σr2)-(σi2-σr2)2]1/2.
a2Mϕ(I)2=exp(-Δj),
2σ2=I-a2I-Mϕ(1)2,
P(I)=1I¯-a2 exp-I+a2I¯-a2I02 aII¯-a2.
P(I)1I¯-a2 exp-I+a2I¯-a21+aII¯-a22.
Δj(X, Y)=Δj(0, 0)+XD2 f2+YD2 f2,
P(n)=(I¯-a2)n(1+I¯-a2)n+1×exp-a21+I¯-a2Ln-a2/(I¯-a2)1+I¯-a2.
P(n)=0 1I¯-a2 exp-I+a2I¯-a21+aII¯-a22×In exp(-I)n! dI.
P(n)=(I¯-a2)n(1+I¯-a2)n+1 1+(n+1)a2(I¯-a2)(1+I¯-a2)×exp-a2I¯-a2.

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