Abstract

A new method to simulate turbulent phase is investigated in this paper. The goal of this method is to be able to simulate very long exposure times as well as time evolving turbulence conditions. But contrary to existing methods, our method allows to simulate such effects without using the whole memory space required by the simulated exposure time, making it particularly suited to the simulation of adaptive optics systems for very large apertures telescopes.

© 2006 Optical Society of America

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References

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  1. D. L. Fried, “Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and Very Short Exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  2. F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Optics 19, 281–376 (1981).
    [CrossRef]
  3. F. Roddier, Adaptive optics in astronomy (Cambridge University Press, 1999).
    [CrossRef]
  4. M. Carbillet, C. Vérinaud, B. Femenía, A. Riccardi, A. and L. Fini, “Modelling astronomical adaptive optics - I. The software package CAOS,” Mon. Not. R. Astron. Soc. 356, 1263–1275 (2005).
    [CrossRef]
  5. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  6. B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image processing, Proc. SPIE, 225–233 (1976).
  7. G. Sedmak, “Implementation of Fast-Fourier-Transform-Based Simulations of Extra-Large Atmospheric Phase and Scintillation Screens,” Appl. Opt. 43, 4527–4538 (2004)
    [CrossRef] [PubMed]
  8. C. M. Harding, R. A. Johnston and R. G. Lane, “Fast Simulation of a Kolmogorov Phase Screen,” Appl. Opt. 38, 2161–2170 (1999).
    [CrossRef]
  9. R. G. Lane, A. Glindemann and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  10. A. Ziad, J. Borgnino, F. Martin, J. Maire and D. Mourard, “Towards the monitoring of atmospheric turbulence model,” Astron. Astrophys. 414, 33–36 (2004)
    [CrossRef]
  11. A. Tokovinin, “From Differential Image Motion to Seeing,” Publ. Astron. Soc. Pac. 114, 1156–1166 (2002)
    [CrossRef]
  12. D. L. Fried, “Statistics of a Geometric Representation of Wavefront Distortion,” J. Opt. Soc. Am. 55, 1427–1435 (1965).
    [CrossRef]
  13. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical recipes in C. The art of scientific computing. 2nd ed., (Cambridge University Press, 1992).
  14. N. Takato N. and I. Yamaguchi, “Spatial correlation of Zernike phase-expansion coefficients for atmospheric turbulence with finite outer scale,” J. Opt. Soc. Am. A 12, 958–963 (1995).
    [CrossRef]

Appl. Opt.

Astron. Astrophys.

A. Ziad, J. Borgnino, F. Martin, J. Maire and D. Mourard, “Towards the monitoring of atmospheric turbulence model,” Astron. Astrophys. 414, 33–36 (2004)
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Mon. Not. R. Astron. Soc.

M. Carbillet, C. Vérinaud, B. Femenía, A. Riccardi, A. and L. Fini, “Modelling astronomical adaptive optics - I. The software package CAOS,” Mon. Not. R. Astron. Soc. 356, 1263–1275 (2005).
[CrossRef]

Opt. Eng.

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

Proc. SPIE

B. L. McGlamery, “Computer simulation studies of compensation of turbulence degraded images,” in Image processing, Proc. SPIE, 225–233 (1976).

Prog. Optics

F. Roddier, “The Effects of Atmospheric Turbulence in Optical Astronomy,” Prog. Optics 19, 281–376 (1981).
[CrossRef]

Publ. Astron. Soc. Pac.

A. Tokovinin, “From Differential Image Motion to Seeing,” Publ. Astron. Soc. Pac. 114, 1156–1166 (2002)
[CrossRef]

Waves Random Media

R. G. Lane, A. Glindemann and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Other

F. Roddier, Adaptive optics in astronomy (Cambridge University Press, 1999).
[CrossRef]

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical recipes in C. The art of scientific computing. 2nd ed., (Cambridge University Press, 1992).

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Figures (5)

Fig. 1.
Fig. 1.

Phase structure function (left) and variance of the Zernike polynomials coefficients (right) computed from 15000 generated phase screens using Eq. 1, for several values of Ncol : 1 (top), 2 (middle), and 4 (bottom). A D = 8 meter pupil was simulated on a square grid of 64×64 pixels, and the outer scale was equal to L 0 = 16 meters. The broken line on the figures on the left corresponds to the theoretical phase structure function, whereas the full line corresponds to the computed one. The full line on the figures on the right shows the theoretical variance of the coefficients of the Zernike polynomials 2 to 66, and the squares the computed values.

Fig. 2.
Fig. 2.

Phase structure function (left) and variance of the Zernike polynomials coefficients (right) computed from 15000 generated phase screens using Eq. 1, for two values of Ncol : 2 (top) and 4 (bottom). A D = 8 meter pupil was simulated on a square grid of 64×64 pixels, and the outer scale was equal to L 0 = 64 meters (L 0/D = 8).

Fig. 3.
Fig. 3.

Phase structure function (left) and variance of the Zernike polynomials coefficients (right) computed from 15000 generated phase screens using Eq. 1, for two values of Ncol : 2 (top) and 4 (bottom). A D = 4 meter pupil was simulated on a square grid of 64×64 pixels, and the outer scale was equal to L 0 = 16 meters (L 0/D = 4)

Fig. 4.
Fig. 4.

Phase structure function (left) and variance of the Zernike polynomials coefficients (right) computed from 15000 generated phase screens using Eq. 1, for two values of Ncol : 2 (top) and 4 (bottom). A D = 4 meter pupil was simulated on a square grid of 64×64 pixels, and the outer scale was this time equal to L 0 = 64 meters (L 0/D = 4)

Fig. 5.
Fig. 5.

Comparison (left), error (middle), and histogram of the error (right) between the the theoretical and observed Fried parameter r 0. The top panels correspond to a phase screen where r 0 is equal to 15 cm in the first half of the screen and oscillating between 10 and 20 cm in the second half. The bottom panels correspond to a phase screen with constant r 0 = 15 cm. The black line on the left figure shows the expected r 0, whereas the red line shows the r 0 value deduced from Zernike polynomials variance, both as a function of the iteration number. The dashed blue line on the r 0 error histogram shows the result of a fit of the error by a Gaussian distribution

Equations (12)

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X = AZ + B β
XZ T = A ZZ T
A = XZ T ZZ T 1
D φ ( r ) = ( L 0 r 0 ) 5 / 3 × 2 1 / 6 Γ ( 11 6 ) π 8 / 3 [ 24 5 Γ ( 6 5 ) ] 5 / 6 × [ Γ ( 5 6 ) 2 1 / 6 ( 2 π r L 0 ) 5 / 6 K 5 / 6 ( 2 π r L 0 ) ]
C φ ( r ) = ( L 0 r 0 ) 5 / 3 × Γ ( 11 6 ) 2 5 / 6 π 8 / 3 [ 24 5 Γ ( 6 5 ) ] 5 / 6 × ( 2 π r L 0 ) 5 / 6 K 5 / 6 ( 2 π r L 0 )
XX T = AZZ T A T + AZ β T B T + B β Z T A T + B ββ T B T
XX T = A ZZ T A T + BB T BB T = XX T A ZZ T A T
A T = ZZ T 1 ZX T
BB T = XX T A ZZ T ZZ T 1 ZX T = XX T A ZX T
BB T = UWU T = ULL T U T = ( UL ) ( UL T )
C φ ( r ) = ( L 0 r 0 ) 5 / 3 × Γ ( 11 6 ) 2 5 / 6 π 8 / 3 [ 24 5 Γ ( 6 5 ) ] 5 / 6 × ( 2 π r L 0 ) 5 / 6 K 5 / 6 ( 2 π r L 0 )
BB T = XX T A ZZ T ZZ T 1 ZX T = XX T A ZX T

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