Abstract

We describe new solutions permitting us to overcome the well-known problems encountered when employing the two main classical methods for numerical modeling of atmospherically perturbed phase screens. The first method, the fast-Fourier-transform-based numerical method, suffers from a lack of low frequencies. Subharmonics adding is an already-known solution to this problem, but no criterion has been defined up to now in order to precisely determine how many subharmonics are necessary for each given case of physical and numerical characteristics. We define two criteria and show their practical efficiency. The second, Zernike-based, method suffers, a contrario, from bad behavior of the phase screens at high spatial frequencies. To overcome this problem, due to numerical instability, we developed an algorithm based on an alternative definition of the Zernike polynomials, involving the recurrence definition of the Jacobi polynomials, as well as the relationship between the Zernike and the Jacobi polynomials. The methods described and used in this paper have been implemented within the freely distributed software package CAOS.

© 2010 Optical Society of America

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  1. R. G. Lane, A. Glindeman, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
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    [CrossRef]
  3. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
    [CrossRef]
  4. P. N. Regagnón, “Bispectral imaging in astronomy,” Ph.D.dissertation (Imperial College, 1995).
  5. 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]
  6. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  7. N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  8. R. W. Press, B. P. Flannery, S. A. Teutolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1986).
  9. M. Born and E. Wolf, Principles of Optics (Pergamon, 1985).
  10. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966).
  11. M. Carbillet, C. Vérinaud, B. Femenía, A. Riccardi, and L. Fini, “Modelling astronomical adaptive optics: I. the software package CAOS,” Mon. Not. R. Astron. Soc. 356, 1263–1275(2005).
    [CrossRef]
  12. M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
    [CrossRef]

2005

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

2004

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

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]

1998

1992

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

1990

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

1981

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

1976

Anconelli, B.

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Bertero, M.

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Boccacci, P.

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1985).

Carbillet, M.

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

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Dainty, J. C.

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

Femenía, B.

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

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Fini, L.

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

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Flannery, B. P.

R. W. Press, B. P. Flannery, S. A. Teutolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1986).

Glindeman, A.

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

Guarracino, M.

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Lane, R. G.

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

Lardière, O.

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Le Roux, B.

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Magnus, W.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966).

Noll, R.

Oberhettinger, F.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966).

Press, R. W.

R. W. Press, B. P. Flannery, S. A. Teutolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1986).

Puglisi, A. T.

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Regagnón, P. N.

P. N. Regagnón, “Bispectral imaging in astronomy,” Ph.D.dissertation (Imperial College, 1995).

Riccardi, A.

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

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

Roddier, N.

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

Sedmak, G.

Soni, R. P.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966).

Teutolsky, S. A.

R. W. Press, B. P. Flannery, S. A. Teutolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1986).

Vérinaud, C.

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

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Vetterling, W. T.

R. W. Press, B. P. Flannery, S. A. Teutolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1986).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1985).

Appl. Opt.

J. Opt. Soc. Am.

Mon. Not. R. Astron. Soc.

M. Carbillet, C. Vérinaud, B. Femenía, A. Riccardi, 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

M. Carbillet, C. Vérinaud, M. Guarracino, L. Fini, O. Lardière, B. Le Roux, A. T. Puglisi, B. Femenía, A. Riccardi, B. Anconelli, M. Bertero, and P. Boccacci, “CAOS—a numerical simulation tool for astronomical adaptive optics (and beyond),” Proc. SPIE 5490, 637–648 (2004).
[CrossRef]

Prog. Opt.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

Waves Random Media

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

Other

P. N. Regagnón, “Bispectral imaging in astronomy,” Ph.D.dissertation (Imperial College, 1995).

R. W. Press, B. P. Flannery, S. A. Teutolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1986).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1985).

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, 1966).

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Figures (4)

Fig. 1
Fig. 1

Integrated power ratio α Φ versus the number of subharmonics to be added n (left) and number of subharmonics to be added n versus the ratio L / L 0 (right). The integrated power ratio α Φ is computed for a screen length L equal to the wavefront outer scale L 0 . Note the values 0.561 for n = 0 and 0.99 for n = 2 , reported in the text. The number of subharmonics to be added is computed for a requested accuracy of 1% ( α Φ = 0.99 ). Note that n = 0 for L 0 / L 0.1 , and it becomes nearly 6 for L 0 / L 100 .

Fig. 2
Fig. 2

Structure function ratio α D versus the number of subharmonics to be added n and the screen physical length L, for the Kolmogorov model (left) and the von Karman model (right, with L 0 = 20 m ).

Fig. 3
Fig. 3

Zernike radial function computed for n = 44 (corresponding to j = ( n + 1 ) ( n + 2 ) 2 = 1035 ). Left: polynomial computation (classical method) result. Right: recursive algorithm (method presented here) result.

Fig. 4
Fig. 4

Left: example of a Kolmogorov/Zernike phase screen. Right: the associated theoretical (straight line) and simulated (crosses) structure functions, in logarithmic scale.

Equations (19)

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Φ φ ( ν ) = 0.0229 r 0 5 3 ( ν 2 + 1 L 0 2 ) 11 6 ,
Φ φ ( ν ) = lim L ( | φ ˜ L ( ν ) | 2 L 2 ) | φ ˜ L ( ν ) | L r 0 5 6 0.0228 ( ν 2 + 1 L 0 2 ) 11 12 ,
φ L ( i , j ) = 2 0.0228 ( L r 0 ) 5 6 { FFT 1 [ ( k 2 + l 2 + ( L L 0 ) 2 ) 11 12 exp { ı θ ( k , l ) } ] } ,
φ 3 n L ( i , j ) = 2 0.0228 ( L r 0 ) 5 6 1 3 n { DFT 1 [ ( k S 2 + l S 2 + ( L L 0 ) 2 ) 11 12 exp { ı θ ( k S , l S ) } ] } ,
Φ φ ( ν x , ν y ) d ν x d ν y = 2 π 0 ν Φ φ ( ν ) d ν = 2 π 0.0228 r 0 5 3 0 ν ( ν 2 + 1 L 0 2 ) 11 6 d ν = 6 π 5 0.0228 ( L 0 r 0 ) 5 3 .
ν x , lim ν y , lim Φ φ ( ν x , ν y ) d ν x d ν y = 2 π 0.0228 r 0 5 3 ν lim ν ( ν 2 + 1 L 0 2 ) 11 6 d ν = 6 π 5 0.0228 ( L 0 r 0 ) 5 3 ( 1 + ν lim 2 L 0 2 ) 5 6 ,
α Φ = ( 1 + ν lim 2 L 0 2 ) 5 6 .
α Φ 6 5 = 1 + ( L 0 3 n L ) 2 n 1 ln 3 ln [ L 0 L ( α Φ 6 5 1 ) 1 2 ] ,
D φ ( ρ ) = 2 Φ φ ( ν ) [ 1 cos ( 2 π ν ρ ) ] d ν ,
D φ ( ρ ) = 6.88 ( ρ r 0 ) 5 3 ,
α D ( n , L ) = 0.0131 L 5 3 1 3 n L ν 8 3 [ 1 cos ( π L ν ) ] d ν .
φ ( r , θ ) = j = 2 j max c j Z j ( r R , θ ) ,
R n m ( ρ ) = ρ m P ( n m ) / 2 ( 0 , m ) ( 2 ρ 2 1 ) ,
P k α , β ( x ) = [ C 1 P k 1 α , β ( x ) C 2 P k 2 α , β ( x ) ] C 0 ,
P 0 α , β ( x ) = 1 ,
P 1 α , β ( x ) = ( α + β + 2 ) / 2 x + ( α β ) / 2 ,
C 0 = 2 k ( α + β + k ) ( α + β + 2 k 2 ) ,
C 1 = ( 2 k + α + β 2 ) ( 2 k + α + β 1 ) ( 2 k + α + β ) x + ( α 2 β 2 ) ( 2 k + α + β 1 ) ,
C 2 = 2 ( k + α 1 ) ( 2 k + α + β ) ( k + β 1 ) .

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