Abstract

Adaptive optics supported solar speckle imaging requires the calibration of the source's Fourier amplitudes with the transfer function of atmosphere and optics. We present analytical models for the relevant transfer functions of an adaptive optics systems. The models include the effect of an arbitrary correction as well as anisoplanatism. The proposed models have been compared with observational data using measurements of α-Orionis and of the solar surface delivering both a direct and indirect method (using the spectral ratio technique) for validation. We find that measurements and model agree to a satisfactory degree.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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  17. K. G. Puschmann and M. Sailer, "Speckle reconstruction of photometric data observed with adaptive optics," Astron. Astrophys. 454, 1011-1019 (2006).
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    [CrossRef]
  27. J. Weiner, W. C. Danchi, D. D. S. Hale, J. McMahon, C. H. Townes, J. D. Monnier, and P. G. Tuthill, "Precision measurements of the diameters of α Orionis and o Ceti at 11 microns," Astrophys. J. 544, 1097-1100 (2000).
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  28. S. Harder and A. Chelli, "Estimating the point spread function of the adaptive optics system ADONIS using the wavefront sensor measurements," Astron. Astrophys. 142, 119-135 (2000).
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2007

C. Denker, A. Tritschler, T. R. Rimmele, K. Richards, S. L. Hegwer, and F. Wöger, "Adaptive optics at the Big Bear Solar Observatory: instrument description and first observations," Publ. Astron. Soc. Pac. 119, 170-182 (2007).
[CrossRef]

2006

T. Berkefeld, D. Soltau, and O. von der Lühe, "Multi-conjugate solar adaptive optics with the VTT and GREGOR," Proc. SPIE 6272, 627205 (2006).
[CrossRef]

T. Rimmele, K. Richards, J. Roche, S. Hegwer, and A. Tritschler, "Progress with solar multi-conjugate adaptive optics at NSO," Proc. SPIE 6272, 627206 (2006).
[CrossRef] [PubMed]

K. Mikurda and O. von der Lühe, "High resolution solar speckle imaging with the extended Knox Thompson algorithm," Sol. Phys. 235, 31-53 (2006).
[CrossRef]

K. G. Puschmann and M. Sailer, "Speckle reconstruction of photometric data observed with adaptive optics," Astron. Astrophys. 454, 1011-1019 (2006).
[CrossRef]

"GNU Scientific Library, v1.8," http://www.gnu.org/software/gsl/ (2006).
[PubMed]

2004

J. J. Fuensalida, S. Chueca, J. M. Delgado, B. Garcia-Lorenzo, J. M. Rodriguez-Gonzalez, C. K. Hoegemann, E. G. Mendizabal, M. Reyes, M. Verde, and J. Vernin, "Vertical structure of the turbulence above the observatories of the Canary Islands: parameters and statistics for adaptive optics," Proc. SPIE 5490, 749-757 (2004).
[CrossRef]

2003

O. von der Lühe, D. Soltau, T. Berkefeld, and T. Schelenz, "KAOS: adaptive optics system for the Vacuum Tower Telescope at Teide Observatory," Proc. SPIE 4853, 187-193 (2003).
[CrossRef]

D. Ren, S. L. Hegwer, T. Rimmele, L. V. Didkovsky, and P. R. Goode, "Optical design of high-order adaptive optics for the NSO Dunn Solar Telescope and the Big Bear Solar Observatory," Proc. SPIE 4853, 593-599 (2003).
[CrossRef]

G. B. Scharmer, P. M. Dettori, M. G. Lofdahl, and M. Shand, "Adaptive optics system for the new Swedish solar telescope," Proc. SPIE 4853, 370-380 (2003).
[CrossRef] [PubMed]

2000

J. Weiner, W. C. Danchi, D. D. S. Hale, J. McMahon, C. H. Townes, J. D. Monnier, and P. G. Tuthill, "Precision measurements of the diameters of α Orionis and o Ceti at 11 microns," Astrophys. J. 544, 1097-1100 (2000).
[CrossRef]

S. Harder and A. Chelli, "Estimating the point spread function of the adaptive optics system ADONIS using the wavefront sensor measurements," Astron. Astrophys. 142, 119-135 (2000).

1997

D. Bonaccini, E. Prieto, P. Corporon, D. Le Mignant, P. Prado, R. Gredel, N. Hubin, and J. Christou, "Performance of the ESO AO system, ADONIS, at La Silla 3.6-m telescope," Proc. SPIE 3126, 589-594 (1997).
[CrossRef]

G. Molodij and G. Rousset, "Angular correlation of Zernike polynomials for a laser guide star in adaptive optics," J. Opt. Soc. Am. A 14, 1949-1966 (1997).
[CrossRef]

1996

R. G. Paxman, J. H. Seldin, M. G. Loefdahl, G. B. Scharmer, and C. U. Keller, "Evaluation of phase-diversity techniques for solar-image restoration," Astrophys. J. 466, 1087-1099 (1996).
[CrossRef] [PubMed]

1993

O. von der Lühe, "Speckle imaging of solar small scale structure. I--methods," Astron. Astrophys. 268, 374-390 (1993).
[PubMed]

1992

M. F. Bilmont, M. C. Roggemann, D. W. Tyler, M. A. von Bokern, D. G. Voelz, and J. P. Albetski, "Effects of predetection atmospheric compensation and post-detection image processing on imagery collected at a ground-based telescope," Proc. SPIE 1688, 489-500 (1992).
[CrossRef]

1984

1983

1978

J. Y. Wang and J. K. Markey, "Modal compensation of atmospheric turbulence phase distortion," J. Opt. Soc. Am. 68, 78-87 (1978).
[CrossRef] [PubMed]

G. P. Lepage, "A new algorithm for adaptive multidimensional integration," J. Comput. Phys. 27, 192-203 (1978).
[CrossRef]

1976

1974

K. T. Knox and B. J. Thompson, "Recovery of images from atmospherically degraded short-exposure photographs," Astrophys. J. 193, L45-L48 (1974).
[CrossRef]

1973

1970

A. Labeyrie, "Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images," Astron. Astrophys. 6, 85-87 (1970).

1966

Appl. Opt.

Astron. Astrophys.

A. Labeyrie, "Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images," Astron. Astrophys. 6, 85-87 (1970).

K. G. Puschmann and M. Sailer, "Speckle reconstruction of photometric data observed with adaptive optics," Astron. Astrophys. 454, 1011-1019 (2006).
[CrossRef]

O. von der Lühe, "Speckle imaging of solar small scale structure. I--methods," Astron. Astrophys. 268, 374-390 (1993).
[PubMed]

S. Harder and A. Chelli, "Estimating the point spread function of the adaptive optics system ADONIS using the wavefront sensor measurements," Astron. Astrophys. 142, 119-135 (2000).

Astrophys. J.

J. Weiner, W. C. Danchi, D. D. S. Hale, J. McMahon, C. H. Townes, J. D. Monnier, and P. G. Tuthill, "Precision measurements of the diameters of α Orionis and o Ceti at 11 microns," Astrophys. J. 544, 1097-1100 (2000).
[CrossRef]

R. G. Paxman, J. H. Seldin, M. G. Loefdahl, G. B. Scharmer, and C. U. Keller, "Evaluation of phase-diversity techniques for solar-image restoration," Astrophys. J. 466, 1087-1099 (1996).
[CrossRef] [PubMed]

K. T. Knox and B. J. Thompson, "Recovery of images from atmospherically degraded short-exposure photographs," Astrophys. J. 193, L45-L48 (1974).
[CrossRef]

J. Comput. Phys.

G. P. Lepage, "A new algorithm for adaptive multidimensional integration," J. Comput. Phys. 27, 192-203 (1978).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. SPIE

M. F. Bilmont, M. C. Roggemann, D. W. Tyler, M. A. von Bokern, D. G. Voelz, and J. P. Albetski, "Effects of predetection atmospheric compensation and post-detection image processing on imagery collected at a ground-based telescope," Proc. SPIE 1688, 489-500 (1992).
[CrossRef]

T. Berkefeld, D. Soltau, and O. von der Lühe, "Multi-conjugate solar adaptive optics with the VTT and GREGOR," Proc. SPIE 6272, 627205 (2006).
[CrossRef]

T. Rimmele, K. Richards, J. Roche, S. Hegwer, and A. Tritschler, "Progress with solar multi-conjugate adaptive optics at NSO," Proc. SPIE 6272, 627206 (2006).
[CrossRef] [PubMed]

O. von der Lühe, D. Soltau, T. Berkefeld, and T. Schelenz, "KAOS: adaptive optics system for the Vacuum Tower Telescope at Teide Observatory," Proc. SPIE 4853, 187-193 (2003).
[CrossRef]

D. Ren, S. L. Hegwer, T. Rimmele, L. V. Didkovsky, and P. R. Goode, "Optical design of high-order adaptive optics for the NSO Dunn Solar Telescope and the Big Bear Solar Observatory," Proc. SPIE 4853, 593-599 (2003).
[CrossRef]

G. B. Scharmer, P. M. Dettori, M. G. Lofdahl, and M. Shand, "Adaptive optics system for the new Swedish solar telescope," Proc. SPIE 4853, 370-380 (2003).
[CrossRef] [PubMed]

J. J. Fuensalida, S. Chueca, J. M. Delgado, B. Garcia-Lorenzo, J. M. Rodriguez-Gonzalez, C. K. Hoegemann, E. G. Mendizabal, M. Reyes, M. Verde, and J. Vernin, "Vertical structure of the turbulence above the observatories of the Canary Islands: parameters and statistics for adaptive optics," Proc. SPIE 5490, 749-757 (2004).
[CrossRef]

D. Bonaccini, E. Prieto, P. Corporon, D. Le Mignant, P. Prado, R. Gredel, N. Hubin, and J. Christou, "Performance of the ESO AO system, ADONIS, at La Silla 3.6-m telescope," Proc. SPIE 3126, 589-594 (1997).
[CrossRef]

Publ. Astron. Soc. Pac.

C. Denker, A. Tritschler, T. R. Rimmele, K. Richards, S. L. Hegwer, and F. Wöger, "Adaptive optics at the Big Bear Solar Observatory: instrument description and first observations," Publ. Astron. Soc. Pac. 119, 170-182 (2007).
[CrossRef]

Sol. Phys.

K. Mikurda and O. von der Lühe, "High resolution solar speckle imaging with the extended Knox Thompson algorithm," Sol. Phys. 235, 31-53 (2006).
[CrossRef]

Other

"GNU Scientific Library, v1.8," http://www.gnu.org/software/gsl/ (2006).
[PubMed]

W. Press, S. Teukolsky, W. Vetterling, and B. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, 1992).

C. Denker, N. Deng, T. R. Rimmele, A. Tritschler, and A. Verdoni, "Field-dependent adaptive optics correction derived with the spectral ratio technique," Sol. Phys. (to be published).
[PubMed]

F. Roddier, "The effects of atmospheric turbulence in optical astronomy," in Progress in Optics, Vol. XIX, E. Wolf, ed. (Elsevier, 1981), pp. 283-376.

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Figures (8)

Fig. 1
Fig. 1

Model transfer functions and spectral ratios for various values of r 0 and the field angles 0 and 2 .

Fig. 2
Fig. 2

Model transfer functions and spectral ratios for various values of r 0 and the field angles 4 and 8 .

Fig. 3
Fig. 3

Model transfer functions and spectral ratios for various values of r 0 and the field angles 16 and 32 .

Fig. 4
Fig. 4

β i values for KAOS. The solid curve represents the result of the evaluation of the KAOS WFS data. The other curves were calculated by the multiplication of the solid curve with correlation coefficients calculated according to Subsection 2C.

Fig. 5
Fig. 5

Sample speckle reconstruction of the solar granulation in the Fraunhofer G band. The relevant short exposed speckle images were acquired using KAOS. The lockpoint has been indicated by the rectangle.

Fig. 6
Fig. 6

Spectral ratio at different viewing angles from the lockpoint. In all curves, the solid curve is the measured data, the dashed curve its best fit. The viewing angles are (a) 0 , (b) 3 , (c) 12 , and (d) 32 .

Fig. 7
Fig. 7

(a) Field distribution of ζ and (b) rms contrast within subfields of Fig. 6.

Fig. 8
Fig. 8

Result of a direct comparison between measured and modeled data. In (a), ADONIS was turned off. The atmospheric conditions have been estimated to a value of r 0 / D = 0.062 ( λ = 550   nm ) . In (b), ADONIS had been turned on.

Equations (42)

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W ( r ) = { 1 π , if | r | D / 2 0 , if | r | > D / 2 .
S ( q D ) = d r W ( r + 1 2 q D ) W ( r 1 2 q D ) × exp [ i ( φ ( r + 1 2 q D ) φ ( r 1 2 q D ) ) ] .
S ( s ) = d r W ( r + 1 2 s ) W ( r 1 2 s ) × exp [ i { ( φ ( r + 1 2 s ) ϕ ( r + 1 2 s ) ) ( φ ( r 1 2 s ) ϕ ( r 1 2 s ) ) } ] .
φ ( r ) = i = 2 a i F i ( r )
ϕ ( r ) = i = 2 N β i a i F i ( r ) .
+ := ( r + 1 2 s ) , := ( r 1 2 s ) .
S ( s ) = d r W + W exp [ 1 2 D ( s ) + K ( r , s ) + L ( r , s ) ] ,
K ( r , s ) = i = 2 N j = 2 N β j ( 1 1 2 β i ) a i a j ( F i + F i ) ( F j + F j ) ,
L ( r , s ) = i = N + 1 j = 2 N β j a i a j ( F i + F i ) ( F j + F j ) .
+ := ( r + 1 2 s ) , := ( r 1 2 s ) .
| S ( s ) | 2 = d r d r W + W W + W
× exp [ i { ( φ + ϕ + ) ( φ ϕ ) ( φ + ϕ + ) + ( φ ϕ ) } ] .
| S ( s ) | 2 = d r d r W + W W + W × exp [ D ( s ) D ( Δ r ) + 1 2 { D ( Δ r + s ) + D ( Δ r s ) } + K ( r , s ) + L ( r , s ) + K ( r , s ) + L ( r , s ) K ˜ ( r , r , s ) L ˜ ( r , r , s ) K ˜ ( r , r , s ) L ˜ ( r , r , s ) ] ,
K ˜ ( r , r , s ) = i = 2 N j = 2 N β j ( 1 1 2 β i ) a i a j ( F i + F i ) × ( F j + F j ) ,
L ˜ ( r , r , s ) = i = N + 1 j = 2 N β j a i a j ( F i + F i ) ( F j + F j ) .
a j ( γ ) a j ( 0 ) = 3.895 ( n + 1 ) ( r 0 / D ) 5 / 3 × 0 d h C n 2 ( h ) I n , m ( 2 γ h / D ) 0 d h C n 2 ( h ) ,
I n , m ( x ) = s n , m 0 d κ ( κ ) 14 / 3 J n + 1 2 ( κ ) ( J 0 ( κ x ) + k j J 2 m ( κ x ) ) ,
k j = { 0 , if   m = 0 ( 1 ) j , if   m 0 ,     s n , m = { 1 , if   m = 0 ( 1 ) n m , if   m 0 .
a j ( γ ) a j ( 0 ) a j ( 0 ) a j ( 0 ) = 0 d h C n 2 ( h ) I n , m ( 2 γ h / D ) 0 d h C n 2 ( h ) I n , m ( 0 ) ,
I = d u f ( u ) = d n x f ( x 1 ,     ,   x n ) ,
E = 1 N i = 1 N f ( u i ) .
lim N 1 N i = 1 N f ( u i ) = I .
β i = 1 σ i ,res 2 σ i ,orig 2 .
ζ = γ fit γ sub
V ( x ) = 2 J 1 ( 2 π r x ) ( 2 π r x )
S ( s ) = d r W + W   exp [ 1 2 D ( s ) 1 2 ( ϕ + ϕ ) 2 + ( φ + φ ) ( ϕ + ϕ ) ] ,
ϕ + = i = 2 N β i a i F i + ,     ϕ = i = 2 N β i a i F i ,
φ + = i = 2 a i F i + ,     φ = i = 2 a i F i .
  T ( r , s ) = [ φ + φ ] [ ϕ + ϕ ] 1 2 [ ϕ + ϕ ] 2
= i = 2 N j = 2 N β j a i a j ( F i + F i ) ( F j + F j )
    + i = N + 1 j = 2 N β j a i a j ( F i + F i ) ( F j + F j )
    1 2 i = 2 N j = 2 N β i β j a i a j ( F i + F i ) ( F j + F j )
= i = 2 N j = 2 N β j ( 1 1 2 β i ) a i a j ( F i + F i ) ( F j + F j )
    + i = N + 1 j = 2 N β j a i a j ( F i + F i ) ( F j + F j )
= K ( r , s ) + L ( r , s ) ,
S ( s ) = d r W + W exp [ 1 2 D ( s ) + K ( r , s ) + L ( r , s ) ] .
| S ( s ) | 2 = d r d r W + W W + W × E .
E = exp [ ( φ + φ ) ( φ + φ ) ( ϕ + ϕ ) + ( ϕ + ϕ ) ] , = exp [ 1 2 { ( φ + φ ) ( φ + φ ) ( ϕ + ϕ ) + ( ϕ + ϕ ) } 2 ] , = exp [ 1 2 ( φ + φ ) 2 + ( φ + φ ) 2 + ( ϕ + ϕ ) 2 + ( ϕ + ϕ ) 2 + 2 { ( ( φ + φ ) ( φ + φ ) ) ( φ + φ ) ( ϕ + ϕ ) + ( φ + φ ) ( ϕ + ϕ ) + ( φ + φ ) ( ϕ + ϕ ) ( φ + φ ) ( ϕ + ϕ ) } ( ϕ + ϕ ) ( ϕ + ϕ ) ( ϕ + ϕ ) × ( ϕ + ϕ ) ] .
( φ + φ ) ( φ + φ ) = 1 2 ( φ + φ + ) 2 + ( φ + φ ) 2 + ( φ φ + ) 2 ( φ φ ) 2 = D ( Δ r ) + 1 2 { D ( Δ r + s ) + D ( Δ r s ) } ,
E = exp [ { D ( s ) + 1 2 ( ϕ + ϕ ) 2 + 1 2 ( ϕ + ϕ ) 2 + D ( Δ r ) 1 2 ( D ( Δ r + s ) + D ( Δ r s ) ) ( φ + φ ) ( ϕ + ϕ ) + ( φ + φ ) ( ϕ + ϕ ) + ( φ + φ ) × ( ϕ + ϕ ) ( φ + φ ) ( ϕ + ϕ ) 1 2 ( ϕ + ϕ ) ( ϕ + ϕ ) 1 2 ( ϕ + ϕ ) × ( ϕ + ϕ ) } ] = exp [ D ( s ) D ( Δ r ) + 1 2 { D ( Δ r + s ) + D ( Δ r s ) } 1 2 ( ϕ + ϕ ) 2 + ( φ + φ ) × ( ϕ + ϕ ) 1 2 ( ϕ + ϕ ) 2 + ( φ + φ ) × ( ϕ + ϕ ) + 1 2 ( ϕ + ϕ ) ( ϕ + ϕ ) ( φ + φ ) ( ϕ + ϕ ) + 1 2 ( ϕ + ϕ ) × ( ϕ + ϕ ) ( φ + φ ) ( ϕ + ϕ ) ] = exp [ D ( s ) D ( Δ r ) + 1 2 { D ( Δ r + s ) + D ( Δ r s ) } + K ( r , s ) + L ( r , s ) + K ( r , s ) + L ( r , s ) K ˜ ( r , r , s ) L ˜ ( r , r , s ) K ˜ ( r , r , s ) L ˜ ( r , r , s ) ] ,
K ˜ ( r , r , s ) = i = 2 N j = 2 N β j ( 1 1 2 β i ) a i a j ( F i + F i ) ( F j + F j ) ,
L ˜ ( r , r , s ) = i = N + 1 j = 2 N β j a i a j ( F i + F i ) ( F j + F j ) ,

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