Abstract

The phase screen method is a well-established approach to take into account the effects of atmospheric turbulence in astronomical seeing. This is of key importance in designing adaptive optics for new-generation telescopes, in particular in view of applications such as exoplanet detection or long-exposure spectroscopy. We present an innovative approach to simulate turbulent phase that is based on stochastic realization theory. The method shows appealing properties in terms of both accuracy in reconstructing the structure function and compactness of the representation.

© 2008 Optical Society of America

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References

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  1. ESO, "The Very Large Telescope Project," http://www.eso.org/projects/vlt/.
  2. M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, "New challenges for adaptive optics: extremely large telescopes," Mon. Not. R. Astron. Soc. 317, 535-544 (2000).
    [CrossRef]
  3. P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, and N. Yaitskova, "The eye of the beholder: designing the OWL," Proc. SPIE 4840, 151-170 (2003).
    [CrossRef]
  4. A. N. Kolmogorov, "Dissipation of energy in the locally isotropic turbulence," C. R. (Dokl.) Acad. Sci. URSS 32, 16-18 (1941).
  5. A. N. Kolmogorov, "The local structure of turbulence in incompressible viscous fluid for very large Reynold's numbers," C. R. (Dokl.) Acad. Sci. URSS 30, 301-305 (1941).
  6. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).
  7. H. G. Booker, T. A. Ratcliffe, and D. H. Schinn, "Diffraction from an irregular screen with applications to ionospheric problems," Philos. Trans. R. Soc. London, Ser. A 242, 579-609 (1950).
    [CrossRef]
  8. F. Assémat, R. W. Wilson, and E. Gendron, "Method for simulating infinitely long and nonstationary phase screens with optimized memory storage," Opt. Express 14, 988-999 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  15. B. L. Ho and R. E. Kalman, "Effective construction of linear state-variable models from input/output functions," Regelungstechnik 14, 545-548 (1966).
  16. S. Y. Kung, "A new identification and model reduction algorithm via singular value decomposition," in Proceedings of the 12th Asilomar Conference on Circuits, Systems and Computers (IEEE, 1978), pp. 705-714.
  17. H. Akaike, "Stochastic theory of minimal realization," IEEE Trans. Autom. Control 19, 667-674 (1974).
    [CrossRef]
  18. H. Akaike, "Markovian representation of stochastic processes by canonical variables," SIAM J. Control 13, 162-173 (1975).
    [CrossRef]
  19. U. B. Desai and D. Pal, "A realization approach to stochastic model reduction and balanced stochastic realizations," in Proceedings of IEEE Conference on Decision and Control (IEEE, 1982), pp. 1105-1112.
  20. F. Roddier, "The effects of atmospheric turbulence in optical astronomy," Prog. Opt. 19, 281-376 (1981).
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  21. S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability (Springer-Verlag, 1993).
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  23. R. Conan, "Modélisation des effets de l'échelle externe de cohérence spatiale du front d'onde pour l'observation à Haute Résolution Angulaire en Astronomie," Ph.D. thesis (Université de Nice-Sophia Antipolis, 2000).
  24. N. Takato and I. Yamagughi, "Spatial correlation of Zernike phase-expansion coefficients for atmospheric turbulence with finite outer scale," J. Opt. Soc. Am. A 12, 958-963 (1995).
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2006 (1)

2003 (1)

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, and N. Yaitskova, "The eye of the beholder: designing the OWL," Proc. SPIE 4840, 151-170 (2003).
[CrossRef]

2002 (1)

A. Tokovinin, "From differential image motion to seeing," Publ. Astron. Soc. Pac. 114, 1156-1166 (2002).
[CrossRef]

2000 (1)

M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, "New challenges for adaptive optics: extremely large telescopes," Mon. Not. R. Astron. Soc. 317, 535-544 (2000).
[CrossRef]

1996 (1)

A. Lindquist and G. Picci, "Canonical correlation analysis, approximate covariance extension, and identification of stationary time series," Automatica 32, 709-733 (1996).
[CrossRef]

1995 (1)

1991 (1)

1981 (1)

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

1976 (1)

1975 (1)

H. Akaike, "Markovian representation of stochastic processes by canonical variables," SIAM J. Control 13, 162-173 (1975).
[CrossRef]

1974 (2)

H. P. Zeiger and A. J. McEwen, "Approximate linear realization of given dimension via Ho's algorithm," IEEE Trans. Autom. Control 19, 153 (1974).
[CrossRef]

H. Akaike, "Stochastic theory of minimal realization," IEEE Trans. Autom. Control 19, 667-674 (1974).
[CrossRef]

1966 (1)

B. L. Ho and R. E. Kalman, "Effective construction of linear state-variable models from input/output functions," Regelungstechnik 14, 545-548 (1966).

1965 (1)

1950 (1)

H. G. Booker, T. A. Ratcliffe, and D. H. Schinn, "Diffraction from an irregular screen with applications to ionospheric problems," Philos. Trans. R. Soc. London, Ser. A 242, 579-609 (1950).
[CrossRef]

1941 (2)

A. N. Kolmogorov, "Dissipation of energy in the locally isotropic turbulence," C. R. (Dokl.) Acad. Sci. URSS 32, 16-18 (1941).

A. N. Kolmogorov, "The local structure of turbulence in incompressible viscous fluid for very large Reynold's numbers," C. R. (Dokl.) Acad. Sci. URSS 30, 301-305 (1941).

Automatica (1)

A. Lindquist and G. Picci, "Canonical correlation analysis, approximate covariance extension, and identification of stationary time series," Automatica 32, 709-733 (1996).
[CrossRef]

C. R. (Dokl.) Acad. Sci. URSS (2)

A. N. Kolmogorov, "Dissipation of energy in the locally isotropic turbulence," C. R. (Dokl.) Acad. Sci. URSS 32, 16-18 (1941).

A. N. Kolmogorov, "The local structure of turbulence in incompressible viscous fluid for very large Reynold's numbers," C. R. (Dokl.) Acad. Sci. URSS 30, 301-305 (1941).

IEEE Trans. Autom. Control (2)

H. P. Zeiger and A. J. McEwen, "Approximate linear realization of given dimension via Ho's algorithm," IEEE Trans. Autom. Control 19, 153 (1974).
[CrossRef]

H. Akaike, "Stochastic theory of minimal realization," IEEE Trans. Autom. Control 19, 667-674 (1974).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Mon. Not. R. Astron. Soc. (1)

M. Le Louarn, N. Hubin, M. Sarazin, and A. Tokovinin, "New challenges for adaptive optics: extremely large telescopes," Mon. Not. R. Astron. Soc. 317, 535-544 (2000).
[CrossRef]

Opt. Express (1)

Philos. Trans. R. Soc. London, Ser. A (1)

H. G. Booker, T. A. Ratcliffe, and D. H. Schinn, "Diffraction from an irregular screen with applications to ionospheric problems," Philos. Trans. R. Soc. London, Ser. A 242, 579-609 (1950).
[CrossRef]

Proc. SPIE (1)

P. Dierickx, J. L. Beckers, E. Brunetto, R. Conan, E. Fedrigo, R. Gilmozzi, N. Hubin, F. Koch, M. Le Louarn, E. Marchetti, G. Monnet, L. Noethe, M. Quattri, M. Sarazin, J. Spyromillo, and N. Yaitskova, "The eye of the beholder: designing the OWL," Proc. SPIE 4840, 151-170 (2003).
[CrossRef]

Prog. Opt. (1)

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

Publ. Astron. Soc. Pac. (1)

A. Tokovinin, "From differential image motion to seeing," Publ. Astron. Soc. Pac. 114, 1156-1166 (2002).
[CrossRef]

Regelungstechnik (1)

B. L. Ho and R. E. Kalman, "Effective construction of linear state-variable models from input/output functions," Regelungstechnik 14, 545-548 (1966).

SIAM J. Control (1)

H. Akaike, "Markovian representation of stochastic processes by canonical variables," SIAM J. Control 13, 162-173 (1975).
[CrossRef]

Other (8)

U. B. Desai and D. Pal, "A realization approach to stochastic model reduction and balanced stochastic realizations," in Proceedings of IEEE Conference on Decision and Control (IEEE, 1982), pp. 1105-1112.

S. Y. Kung, "A new identification and model reduction algorithm via singular value decomposition," in Proceedings of the 12th Asilomar Conference on Circuits, Systems and Computers (IEEE, 1978), pp. 705-714.

R. Conan, "Modélisation des effets de l'échelle externe de cohérence spatiale du front d'onde pour l'observation à Haute Résolution Angulaire en Astronomie," Ph.D. thesis (Université de Nice-Sophia Antipolis, 2000).

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability (Springer-Verlag, 1993).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
[CrossRef]

M. Aoki, State Space Modeling of Time Series, 2nd ed. (Springer-Verlag, 1991).

ESO, "The Very Large Telescope Project," http://www.eso.org/projects/vlt/.

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Figures (12)

Fig. 1
Fig. 1

Two points r 1 and r 2 at distance r on the aperture plane.

Fig. 2
Fig. 2

Plot of the singular values of the stochastic realization model. In this case we set the parameter values to ν = 10 , m = 64 ; hence the size of the A matrix before the reduction step (and the number of the singular values) is ν m = 640 .

Fig. 3
Fig. 3

Phase structure function along the wind direction. A comparison of the theoretical values (dashed curve) and those obtained with (i) the dynamic model identified with the procedure in Subsection 3A (dashed–dotted curve) and (ii) the dynamic model identified with the procedure in Subsection 3B (solid curve). The values of the parameters are set to L 0 = 2 m , r 0 = 0.2 m , D = 8 m , p s = 0.125 m .

Fig. 4
Fig. 4

Phase structure function along the direction orthogonal to the wind. A comparison of the theoretical values (dashed curve) and those obtained with (i) the dynamic model of Section 3 (solid curve) and (ii) the method of Assémat et al. (dashed–dotted curve). The values of the parameters are set to L 0 = 16 m , r 0 = 8 m , D = 8 m , p s = 0.125 m .

Fig. 5
Fig. 5

Phase structure function along the wind direction. A comparison of the theoretical values (dashed curve) and those obtained with (i) the dynamic model of Section 3 (solid curve) and (ii) the method of Assémat et al. (dashed–dotted curve). The values of the parameters are set to L 0 = 16 m , r 0 = 8 m , D = 8 m , p s = 0.125 m .

Fig. 6
Fig. 6

Variances of the Zernike coefficients. A comparison of the theoretical values (dashed curve) and those obtained with (i) the dynamic model of Section 3 (solid curve) and (ii) the method of Assémat et al. (dashed–dotted curve). The values of the parameters are set to L 0 = 16 m , r 0 = 8 m , D = 8 m , p s = 0.125 m .

Fig. 7
Fig. 7

Phase structure function along the wind direction. A comparison of the theoretical values (dashed curve) and those obtained with (i) the dynamic model of Section 3 (solid curve) and (ii) the method of Assémat et al. (dashed–dotted curve). The values of the parameters are set to L 0 = 64 m , r 0 = 8 m , D = 8 m , p s = 0.125 m .

Fig. 8
Fig. 8

Variances of the Zernike coefficients. A comparison of the theoretical values (dashed curve) and those obtained with (i) the dynamic model of Section 3 (solid curve) and (ii) the method of Assémat et al. (dashed–dotted curve). The values of the parameters are set to L 0 = 64 m , r 0 = 8 m , D = 8 m , p s = 0.125 m .

Fig. 9
Fig. 9

Phase structure function along the wind direction. A comparison of the theoretical values (dashed curve) and those obtained with (i) the dynamic model of Section 3 (solid curve) and (ii) the method of Assémat et al. (dashed–dotted curve). The values of the parameters are set to L 0 = 64 m , r 0 = 4 m , D = 4 m , p s = 0.0625 m .

Fig. 10
Fig. 10

Variances of the Zernike coefficients. A comparison of the theoretical values (dashed curve) and those obtained with (i) the dynamic model of Section 3 (solid curve) and (ii) the method of Assémat et al. (dashed–dotted curve). The values of the parameters are set to L 0 = 64 m , r 0 = 4 m , D = 4 m , p s = 0.0625 m .

Fig. 11
Fig. 11

Phase structure function along the wind direction. A comparison of the theoretical values (dashed curve) and those obtained with (i) the dynamic model of Section 3 (solid curve) and (ii) the method of Assémat et al. (dashed–dotted curve). The values of the parameters are set to L 0 = 3.5 m , r 0 = 0.3 m , D = 8 m , p s = 0.125 m .

Fig. 12
Fig. 12

Phase structure function along the wind direction. A comparison of the theoretical values (dashed curve) and those obtained with (i) the dynamic model of Section 3 (solid curve) and (ii) the method of Assémat et al. (dashed–dotted curve). The values of the parameters are set to L 0 = 1.6 m , r 0 = 0.15 m , D = 8 m , p s = 0.125 m .

Equations (50)

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D ϕ ( r ) = ϕ ( r 1 ) ϕ ( r 2 ) 2 .
D ϕ ( r ) = 2 ( σ ϕ 2 C ϕ ( r ) ) ,
D ϕ ( r ) = ( L 0 r 0 ) 5 3 c [ Γ ( 5 6 ) 2 1 6 ( 2 π r L 0 ) 5 6 K 5 6 ( 2 π r L 0 ) ] ,
c = 2 1 6 Γ ( 11 6 ) π 8 3 [ 24 5 Γ ( 6 5 ) ] 5 6 .
C ϕ ( r ) = ( L 0 r 0 ) 5 3 c 2 ( 2 π r L 0 ) 5 6 K 5 6 ( 2 π r L 0 ) .
ϕ ( u , v , t ) = i = 1 l γ i ψ i ( u , v , t ) ,
E { ψ i ( u , v , t ) ψ j ( u , v , t ) } = 0 , 1 i l , 1 j l , j i , 1 u , v m , 1 u , v m .
ψ i ( u , v , t + k T ) = ψ i ( u v i , u k T , v v i , v k T , t ) , i = 1 , , l ,
{ x t + 1 = A x t + K e t , y t = C x t + e t } ,
{ Λ 1 = C G Λ 2 = C A G Λ 2 ν 1 = C A 2 ν 2 G } ,
Λ i = E { y t + i y t T } = E { ( ϕ t + i m ϕ ) ( ϕ t m ϕ ) T } = C ϕ ( i η ) .
H [ Λ 1 Λ 2 Λ ν Λ 2 Λ 3 Λ ν + 1 Λ ν Λ ν + 1 Λ 2 ν 1 ]
= [ C G C A G C A ν 1 G C A G C A 2 G C A ν G C A ν 1 G C A ν G C A 2 ν 2 G ]
= [ C C A C A ν 1 ] [ G A G A ν 1 G ] .
T = [ Λ 0 Λ 1 Λ 2 Λ ν 1 Λ 1 T Λ 0 Λ 1 Λ ν 2 Λ 2 T Λ 1 T Λ 0 Λ ν 3 Λ ν 1 T Λ ν 2 T Λ ν 3 T Λ 0 ] ,
H ̂ L 1 H L T ;
H = L H ̂ L T .
H ̂ = U S V T = U S 1 2 S 1 2 V T ,
H ̂ U n ¯ S n ¯ V n ¯ T = U n ¯ S n ¯ 1 2 S n ¯ 1 2 V n ¯ T ,
{ U n ¯ = U ( : , 1 : n ¯ ) S n ¯ = S ( 1 : n ¯ , 1 : n ¯ ) V n ¯ = V ( : , 1 : n ¯ ) } .
H ̂ U n ¯ S n ¯ V n ¯ T .
{ C ρ 1 ( H ) L T V n ¯ S n ¯ 1 2 G ( ρ 1 ( H T ) L T U n ¯ S n ¯ 1 2 ) T } ,
σ ( H ) = [ Λ 2 Λ 3 Λ ν + 1 Λ 3 Λ 4 Λ ν + 2 Λ ν + 1 Λ ν + 2 Λ 2 ν ] .
[ C C A C A ν 1 ] A [ G A G A ν 1 G ] = σ ( H ) ,
A S n ¯ 1 2 U n ¯ T L 1 σ ( H ) L T V n ¯ S n ¯ 1 2 .
Σ t + 1 = A Σ t A T + ( G A Σ t C T ) R 1 ( G A Σ t C T ) T ,
Σ = A Σ A T + ( G A Σ C T ) ( Λ 0 C Σ C T ) 1 ( G T C Σ A T ) ,
{ Λ ¯ 0 , Λ ¯ 1 , Λ ¯ 2 , , Λ ¯ 2 ν 1 } ,
{ Λ ¯ 0 Λ 0 Λ ¯ 1 C G Λ 1 Λ ¯ 2 C A G Λ 2 Λ ¯ 2 ν 1 C A 2 ν 2 G Λ 2 ν 1 } .
{ Λ ¯ 0 , Λ ¯ 1 , Λ ¯ 2 , , Λ ¯ 2 ν 1 , Λ ¯ 2 ν , }
Λ ¯ i C A i 1 G , i 2 ν .
x t + 1 = A x t + K e t .
H ̂ = H = U S V T = U S 1 2 S 1 2 V T .
y = A ̃ z + B ̃ β ,
Σ y z E { y z T } = A ̃ E { z z T } ,
Σ y E { y y T } = A ̃ E { z z T } A ̃ T + B ̃ B ̃ T .
A ̃ = Σ y z Σ z 1 ,
B ̃ B ̃ T = Σ y A ̃ Σ z A ̃ T ,
{ x t + 1 = A x t + B w t y t = C x t } ,
A = [ A ̃ 1 A ̃ 2 I ( ν 1 ) m 0 ] = [ A ̃ I ( ν 1 ) m 0 ] ,
B = [ B ̃ 0 0 ] ,
C = [ I m 0 0 ] ,
Σ = [ Λ 0 Λ 1 Λ ν 1 Λ 1 Λ 0 Λ ν 2 Λ ν 1 Λ ν 2 Λ 0 ] .
A ̃ = [ Λ 1 Λ 2 Λ ν ] Σ 1 .
Σ = A Σ A T + B B T = [ A ̃ I ( ν 1 ) m 0 ] Σ [ A ̃ T I ( ν 1 ) m 0 ] + [ Q 0 0 0 ] .
Z i ( r ) = { n + 1 R n m ( r ) 2 cos ( m γ ) if m 0 , i even n + 1 R n m ( r ) 2 sin ( m γ ) if m 0 , i odd n + 1 R n m ( r ) if m = 0 } ,
R n m ( r ) = k = 0 ( n m ) 2 ( 1 ) k ( n k ) ! k ! ( n + m 2 k ) ! ( n m 2 k ) ! r n 2 k
ϕ ( r ) = i = 0 + a i Z i ( r D 2 ) , r D 2 ,
a i = R 2 Π ( r D 2 ) Z i ( r D 2 ) ϕ ( r ) d r .
E { a i a i } = { = 2 Γ ( 11 6 ) π 3 2 [ 24 5 Γ ( 6 5 ) ] 5 6 ( D r 0 ) 5 3 ( n + 1 ) ( n + 1 ) ( 1 ) ( n + n 2 m ) 2 × δ m m h = 0 ( 1 ) h h ! { ( π D f 0 ) 2 h + n + n 5 3 } × Γ [ h + 1 + n + n 2 , h + 2 + n + n 2 , h + 1 + n + n 2 , 5 6 h n + n 2 3 + h + n + n , 2 + h + n , 2 + h + n ] + ( π D f 0 ) 2 h Γ { [ n + n 2 h 5 6 , h + 7 3 , h + 17 6 , k + 11 6 n + n 2 + h + 23 6 , n n 2 + h + 17 6 , n n 2 + h + 17 6 ] } if m = m , m 0 , m 0 , i + i even ; or m = m = 0 = 0 otherwise } .

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