Abstract

A model of a nonmodulated pyramid wavefront sensor (P-WFS) based on Fourier optics has been presented. Linearizations of the model represented as Jacobian matrices are used to improve the P-WFS phase estimates. It has been shown in simulations that a linear approximation of the P-WFS is sufficient in closed-loop adaptive optics. Also a method to compute model-based synthetic P-WFS command matrices is shown, and its performance is compared to the conventional calibration. It was observed that in poor visibility the new calibration is better than the conventional.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. F. Roddier, Adaptive Optics in Astronomy (Cambridge U. Press, 1999).
    [CrossRef]
  2. R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
    [CrossRef]
  3. C. Vérinaud, "On the nature of the measurements provided by a pyramid wave-front sensor," Opt. Commun. 233, 27-38 (2004).
    [CrossRef]
  4. O. Wulff and D. Looze, "Nonlinear control for pyramid sensors in adaptive optics," Proc. SPIE 6272, 62721S (2006).
    [CrossRef]
  5. D. W. Phillion and K. Baker, "Two-sided pyramid wavefront sensor in the direct phase mode," Proc. SPIE 6272, 627228 (2006).
    [CrossRef]
  6. M. Le Louarn, C. Verinaud, V. Korkiakoski, and E. Fedrigo, "Parallel simulation tools for AO on ELTs," Proc. SPIE 5490, 705-712 (2004).
    [CrossRef]
  7. C. Vérinaud, M. Le Louarn, V. Korkiakoski, and M. Carbillet, "Adaptive optics for high-contrast imaging: pyramid sensor versus spatially filtered Shack-Hartmann sensor," Mon. Not. R. Astron. Soc. 357, L26-L30 (2005).
    [CrossRef]
  8. C. Arcidiacono, "Beam divergence and vertex angle measurements for refractive pyramids," Opt. Commun. 252, 239-246 (2005).
    [CrossRef]
  9. A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, "Laboratory characterization of a Foucault-like wavefront sensor for adaptive optics," Proc. SPIE 3353, 941-951 (1998).
    [CrossRef]

2006 (2)

O. Wulff and D. Looze, "Nonlinear control for pyramid sensors in adaptive optics," Proc. SPIE 6272, 62721S (2006).
[CrossRef]

D. W. Phillion and K. Baker, "Two-sided pyramid wavefront sensor in the direct phase mode," Proc. SPIE 6272, 627228 (2006).
[CrossRef]

2005 (2)

C. Vérinaud, M. Le Louarn, V. Korkiakoski, and M. Carbillet, "Adaptive optics for high-contrast imaging: pyramid sensor versus spatially filtered Shack-Hartmann sensor," Mon. Not. R. Astron. Soc. 357, L26-L30 (2005).
[CrossRef]

C. Arcidiacono, "Beam divergence and vertex angle measurements for refractive pyramids," Opt. Commun. 252, 239-246 (2005).
[CrossRef]

2004 (2)

M. Le Louarn, C. Verinaud, V. Korkiakoski, and E. Fedrigo, "Parallel simulation tools for AO on ELTs," Proc. SPIE 5490, 705-712 (2004).
[CrossRef]

C. Vérinaud, "On the nature of the measurements provided by a pyramid wave-front sensor," Opt. Commun. 233, 27-38 (2004).
[CrossRef]

1998 (1)

A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, "Laboratory characterization of a Foucault-like wavefront sensor for adaptive optics," Proc. SPIE 3353, 941-951 (1998).
[CrossRef]

1996 (1)

R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

J. Mod. Opt. (1)

R. Ragazzoni, "Pupil plane wavefront sensing with an oscillating prism," J. Mod. Opt. 43, 289-293 (1996).
[CrossRef]

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

C. Vérinaud, M. Le Louarn, V. Korkiakoski, and M. Carbillet, "Adaptive optics for high-contrast imaging: pyramid sensor versus spatially filtered Shack-Hartmann sensor," Mon. Not. R. Astron. Soc. 357, L26-L30 (2005).
[CrossRef]

Opt. Commun. (2)

C. Arcidiacono, "Beam divergence and vertex angle measurements for refractive pyramids," Opt. Commun. 252, 239-246 (2005).
[CrossRef]

C. Vérinaud, "On the nature of the measurements provided by a pyramid wave-front sensor," Opt. Commun. 233, 27-38 (2004).
[CrossRef]

Proc. SPIE (4)

O. Wulff and D. Looze, "Nonlinear control for pyramid sensors in adaptive optics," Proc. SPIE 6272, 62721S (2006).
[CrossRef]

D. W. Phillion and K. Baker, "Two-sided pyramid wavefront sensor in the direct phase mode," Proc. SPIE 6272, 627228 (2006).
[CrossRef]

M. Le Louarn, C. Verinaud, V. Korkiakoski, and E. Fedrigo, "Parallel simulation tools for AO on ELTs," Proc. SPIE 5490, 705-712 (2004).
[CrossRef]

A. Riccardi, N. Bindi, R. Ragazzoni, S. Esposito, and P. Stefanini, "Laboratory characterization of a Foucault-like wavefront sensor for adaptive optics," Proc. SPIE 3353, 941-951 (1998).
[CrossRef]

Other (1)

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Illustration of the nonmodulated pyramid sensor and its signal composition. α is tangent of the pyramid divergence angle [8, 9], f is the focal length, and f α / ( 2 π ) is half of the distance between the subbeam centers.

Fig. 2
Fig. 2

Descriptive illustration of an iterative, derivative-based, numerical inversion of a monotonic nonlinear function.

Fig. 3
Fig. 3

Illustration of the phase discretization for the model. An illumination threshold is required to choose which grid elements belong to the aperture and which are left outside.

Fig. 4
Fig. 4

Illustration of the simulated system.

Fig. 5
Fig. 5

Long exposure Strehl as a function of loop gain. Solid curves represent two-sided P-WFS, dotted curves the four-sided P-WFS. Dot markers show the conventional MVM, curves without markers the JR. The error bars are computed using the standard error from five sets of simulations. The simulations are made with λ W F S = 0.5 μ m , N = 160 , N i t = 1 .

Fig. 6
Fig. 6

Radially averaged power spectral densities of the residual phases in two-sided P-WFS simulations with optimal loop gains at λ W F S = 0.5 μ m . JR was used with N = 160 and N i t = 1 .

Tables (2)

Tables Icon

Table 1 Simulation Parameters

Tables Icon

Table 2 Optimal Strehl Ratios

Equations (43)

Equations on this page are rendered with MathJax. Learn more.

[ S x ( r ) S y ( r ) ] = [ I 0 , 0 ( r ) I 1 , 0 ( r ) + I 0 , 1 ( r ) I 1 , 1 ( r ) I t I 0 , 0 ( r ) I 0,1 ( r ) + I 1,0 ( r ) I 1 , 1 ( r ) I t ] ,
S x ( x , y ) [ S x 1 ( ϕ 1 ,  …  ,  ϕ M ) S x M ( ϕ 1 ,  …  ,  ϕ M ) ] ,
S y ( x , y ) [ S y 1 ( ϕ 1 ,  …  ,  ϕ M ) S y M ( ϕ 1 ,  …  ,  ϕ M ) ] .
J n , m ( ϕ ) = [ J x n , m ( ϕ ) J y n , m ( ϕ ) ] = [ S x x i , y i ( ϕ 1 ,  …  ,  ϕ M ) ϕ x j , y j S x x i , y i ( ϕ 1 ,  …  ,  ϕ M ) ϕ x j , y j ] ,
2 [ π ( N 2 ) 2 ] [ π ( N 2 ) 2 ] = 1 8 π 2 N 4 ,
elements ( J full ( 0 ) J m ( 0 ) ) 2 / elements ( J full ( 0 ) ) 2 ,
S = J ( ϕ ) ϕ w
ϕ = [ J ( 0 ) ] X m S ,
v n = C m 1 a n ϕ n ,
C m ( i , j ) = f i ( r ) f j ( r ) d r ,
a n ( i ) = g n ( r ) f i ( r ) d r .
c 2 = C m 1 [ a n a M ] [ J ( 0 ) ] X m S ,
Ψ p ( r ) = Ψ ( r ) 1 { T pyr ( f ) } ,
T pyr ( f ) = n = 0 1 m = 0 1 T n , m ( f ) = n = 0 1 m = 0 1 H ( ( 1 ) n f x , ( 1 ) m f y ) × exp { i α [ ( 1 ) n f x c + ( 1 ) m f y c ] } ,
Ψ p ( r ) = n = 0 1 m = 0 1 Ψ n , m ( r ) ,
Ψ n , m ( r ) = Ψ ( x ( 1 ) n α 2 π , y ( 1 ) m α 2 π ) 1 { H ( f ) } ( ( 1 ) n x , ( 1 ) m y ) ,
1 { H ( f ) } ( ( 1 ) n x , ( 1 ) m y ) = 1 4 δ ( x , y ) ( 1 ) n + m 4 π 2 p . v .   1 x y + i 4 π [ ( 1 ) n ( p . v .   1 x δ ( y ) ) + ( 1 ) m ( δ ( x ) p . v .   1 y ) ] ,
I p ( r ) = Ψ ( r ) Ψ * ( r ) = n = 0 1 m = 0 1 Ψ n , m ( r ) Ψ n , m * ( r ) + 2 n = 0 1 m = 0 1 n = 0 n n 1 m = 0 m m 1 Re [ Ψ n , m ( r ) Ψ n , m * ( r ) ] .
I p ( r ) n = 0 1 m = 0 1 I n , m ( r ) ,
I n , m ( r ) = Ψ n , m ( r ) Ψ n , m * ( r )
I t S x ( x , y ) = 1 π { R Ψ ( x , y ) [ I Ψ ( x , y ) ( p . v .   1 x δ ( y ) ) ] I Ψ ( x , y ) [ R Ψ ( x , y ) ( p . v .   1 x δ ( y ) ) ] } 1 π 3 { [ R Ψ ( x , y ) p . v .   1 x y ] × [ I Ψ ( x , y ) ( δ ( x ) p . v .   1 y ) ] [ I Ψ ( x , y ) p . v .   1 x y ] × [ R Ψ ( x , y ) ( δ ( x ) p . v .   1 y ) ] } ,
I t S y ( x , y ) = 1 π { R Ψ ( x , y ) [ I Ψ ( x , y ) ( δ ( x ) p . v .   1 y ) ] I Ψ ( x , y ) [ R Ψ ( x , y ) ( δ ( x ) p . v .   1 y ) ] } 1 π 3 { [ R Ψ ( x , y ) p . v .   1 x y ] × [ I Ψ ( x , y ) ( p . v .   1 x δ ( y ) ) ] [ I Ψ ( x , y ) p . v .   1 x y ] × [ R Ψ ( x , y ) ( p . v .   1 x δ ( y ) ) ] } ,
S x x i , y i ( ϕ ) = m x x i , y i ( ϕ ) + c x x i , y i ( ϕ ) ,
S y x i , y i ( ϕ ) = m y x i , y i ( ϕ ) + c y x i , y i ( ϕ ) ,
m x x i , y i ( ϕ ) = 1 π { P ( x i , y i ) cos ϕ x i , y i C s x ( x i , y i ) P ( x i , y i ) sin ϕ x i , y i C c x ( x i , y i ) } ,
m y x i , y i ( ϕ ) = 1 π { P ( x i , y i ) cos ϕ x i , y i C s y ( x i , y i ) P ( x i , y i ) sin ϕ x i , y i C c x ( x i , y i ) } ,
c x x i , y i ( ϕ ) = 1 π 3 { C c x y ( x i , y j ) C s y ( x i , y j ) C s x y ( x i , y j ) C c y ( x i , y j ) } ,
c y x i , y i ( ϕ ) = 1 π 3 { C c x y ( x i , y j ) C s x ( x i , y j ) C s x y ( x i , y j ) C c x ( x i , y j ) } ,
C s x ( x i , y i ) = x , y P ( x , y ) sin ϕ x , y 1 x i x × δ ( y i y ) d x d y ,
C c x ( x i , y i ) = x , y P ( x , y ) cos ϕ x , y 1 x i x × δ ( y i y ) d x d y ,
C s y ( x i , y i ) = x , y P ( x , y ) sin ϕ x , y 1 y i y × δ ( x i x ) d x d y ,
C c y ( x i , y i ) = x , y P ( x , y ) cos ϕ x , y 1 y i y × δ ( x i x ) d x d y ,
C s x y ( x i , y j ) = x , y P ( x , y ) sin ϕ x , y × 1 ( x i x ) ( y i y )  d x d y ,
C c x y ( x i , y j ) = x , y P ( x , y ) cos ϕ x , y × 1 ( x i x ) ( y i y )  d x d y .
m x x i , y i ( ϕ ) ϕ x j , y j = { 1 π { P cos ϕ x i , y i cos ϕ x j , y j D m x ( i , j ) + P sin ϕ x i , y i sin ϕ x j , y j D m x ( i , j ) } , if   x i x j , y i = y j , 1 π { P  sin  ϕ x i , y i C s x ( x i , y i ) P cos ϕ x i , y i C c x ( x i , y i ) } , if  x i = x j , y i = y j , 0 , o t h e r w i s e
m y x i , y i ( ϕ ) ϕ x j , y j = { 1 π { P cos ϕ x i , y i cos ϕ x j , y j D m y ( i , j ) + P sin ϕ x i , y i sin ϕ x j , y j D m y ( i , j ) } , if   y i y j , x i = x j , 1 π { P  sin  ϕ x i , y i C s y ( x i , y i ) P cos ϕ x i , y i C c y ( x i , y i ) } , if  y i = y j , x i = x j , 0 , o t h e r w i s e ,
D m x ( i , j ) = x r ( x j , y j ) P ( x , y ) 1 x i x d x ,
D m y ( i , j ) = y r ( x j , y j ) P ( x , y ) 1 y i y d y ,
D m x ( i , j ) = 1 d { [ ( x i x j + d ϵ ) log | x i x j + d ϵ | ( x i x j ϵ ) log | x i x j ϵ | ] [ ( x i x j + ϵ ) × log | x i x j + ϵ | ( x i x j d + ϵ ) × log | x i x j d + ϵ | ] } ,
c x x i , y i ( ϕ ) ϕ x j , y j = { 1 π 3 { C s x y ( x i , y i ) sin ϕ x j , y j D m y ( i , j ) + C c x y ( x j , y j ) cos ϕ x j , y j D m y ( i , j ) } , if  x i = x j , 1 π 3 { cos ϕ x j , y j D m x ( i , j ) D m y ( i , j ) C c y ( x i , y i ) sin ϕ x j , y j D m x ( i , j ) D m y ( i , j ) C s y ( x i , y i ) } , if   x i x j , }
c y x i , y i ( ϕ ) ϕ x j , y j = { 1 π 3 { C s x y ( x i , y i ) sin ϕ x j , y j D m x ( i , j ) + C c x y ( x j , y j ) cos ϕ x j , y j D m x ( i , j ) } , if   y i = y j , 1 π 3 { cos ϕ x j , y j D m x ( i , j ) D m y ( i , j ) C c x ( x i , y i ) sin ϕ x j , y j D m x ( i , j ) D m y ( i , j ) C s x ( x i , y i ) } , if   y i y j . }
J x n , m ( ϕ ) = m x x i , y i ( ϕ ) ϕ x j , y j + c x x i , y i ( ϕ ) ϕ x j , y j ,
J y n , m ( ϕ ) = m y x i , y i ( ϕ ) ϕ x j , y j + c y x i , y i ( ϕ ) ϕ x j , y j .

Metrics