Abstract

Adaptive optics can be used to improve the quality of optical imaging instruments otherwise limited in resolution by atmospheric turbulence. To achieve a given level of resolution improvement at the lowest cost, design of adaptive mirrors must be carefully optimized. We have calculated the mean value of the optical transfer function (OTF) of an instrument corrected by adaptive optics using the OTF as a criterion for resolution optimization and describing perturbations through their phase structure function. A four-parameter numerical program has been written to compute the OTF. Assuming that the optics diameter and the perturbation Fried's parameter are known, it then becomes possible to optimize the system by a proper choice of the number of actuators and of the influence diameter, a parameter which characterizes the size of the required corrections.

© 1987 Optical Society of America

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References

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  1. J. C. Fontanella, “Analyse de surface d'onde, deconvolution et optique active,” J. Opt. Paris 16, 257 (1985).
    [CrossRef]
  2. R. H. Freeman, J. E. Pearson, “Deformable Mirrors for All Seasons and Reasons,” Appl. Opt. 21, 580 (1982).
    [CrossRef] [PubMed]
  3. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 484–490.
  4. P. Nisenson, R. Barakat, “Partial Atmospheric Correction with Active Optics,” to be published.
  5. System titus Framatome, Tour Fiat, 92400 Courbevoie, France.

1985 (1)

J. C. Fontanella, “Analyse de surface d'onde, deconvolution et optique active,” J. Opt. Paris 16, 257 (1985).
[CrossRef]

1982 (1)

Barakat, R.

P. Nisenson, R. Barakat, “Partial Atmospheric Correction with Active Optics,” to be published.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 484–490.

Fontanella, J. C.

J. C. Fontanella, “Analyse de surface d'onde, deconvolution et optique active,” J. Opt. Paris 16, 257 (1985).
[CrossRef]

Freeman, R. H.

Nisenson, P.

P. Nisenson, R. Barakat, “Partial Atmospheric Correction with Active Optics,” to be published.

Pearson, J. E.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 484–490.

Appl. Opt. (1)

J. Opt. Paris (1)

J. C. Fontanella, “Analyse de surface d'onde, deconvolution et optique active,” J. Opt. Paris 16, 257 (1985).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1970), pp. 484–490.

P. Nisenson, R. Barakat, “Partial Atmospheric Correction with Active Optics,” to be published.

System titus Framatome, Tour Fiat, 92400 Courbevoie, France.

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Figures (6)

Fig. 1
Fig. 1

Simplified scheme of continuous deformable mirror.

Fig. 2
Fig. 2

Scheme of an adaptive optics control loop.

Fig. 3
Fig. 3

Pattern of the actuators axis for a nineteen-actuator deformable mirror.

Fig. 4
Fig. 4

Shape of the deformation of the mirror plate produced by an actuator.

Fig. 5
Fig. 5

(a)–(c) Influence of the wavelength band on the optical transfer function for a fixed number of actuators.

Fig. 6
Fig. 6

(a)–(d) Influence of the deformation plate shape on the optical transfer function for a fixed number of actuators.

Tables (1)

Tables Icon

Table I Values of the Fried's Parameter on a Good Astronomical Site for Various Wavelengths

Equations (19)

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y ( ρ ) = σ d r E ( r ) E * ( r + ρ ) σ d r | E ( r ) | 2 ,
( r ) = ψ ( r ) j h j Φ j ( r ) ,
y ( ρ ) = 1 σ d r | A ( r ) | 2 × σ d r A ( r ) A ( r + ρ ) exp i [ ( r ) ( r + ρ ) ] .
y ( ρ ) ¯ = 1 σ d r | A ( r ) | 2 × σ d r A ( r ) A ( r + ρ ) exp i [ ( r ) ( r + ρ ) ] ¯ .
y ( ρ ) ¯ = 1 σ d r | A ( r ) | 2 × σ d r A ( r ) A ( r + ρ ) exp ½ [ ( r ) ( r + ρ ) ] 2 ¯ .
y ( ρ ) ¯ = exp ½ D ( ρ ) σ d r | A ( r ) | 2 × σ d r A ( r ) A ( r + ρ ) exp δ ( r , ρ ) , δ ( r , ρ ) = [ 1 2 j k [ Φ j ( r ) Φ j ( r + ρ ) ] [ Φ k ( r ) Φ k ( r , ρ ) ] h j h k ¯ + j [ Φ j ( r ) Φ j ( r + ρ ) ] h j [ ψ ( r ) ( r + ρ ) ] ¯ ] ,
D ( ρ ) = [ ψ ( r ) ψ ( r , ρ ) ] 2 ¯ = 6.88 ( ρ / r 0 ) 5 / 3 ,
y ( ρ ) ¯ = exp ½ D ( ρ ) σ d r | A ( r ) | 2 × σ d r A ( r ) A ( r + ρ ) exp [ ½ d | H ̅ | d + d h ¯ ] .
η 2 = σ d r [ ψ ( r ) ψ j h j Φ j ( r ) ] 2 ,
σ d r [ ψ ( r ) ψ ] Φ j ( r ) = k h k σ d r Φ j ( r ) Φ k ( r ) .
| p = Δ | h .
Δ j k = σ d r Φ j ( r ) Φ k ( r ) ,
H ¯ = Δ 1 P ¯ Δ 1 .
P ¯ j k = σ d r d r [ Φ j ( r ) Φ k ( r ) ] × [ ψ ( r ) ψ ] [ ψ ( r ) ψ ] ¯ .
P ¯ j k = 1 2 σ d r d r Φ j ( r ) Φ k ( r ) ] D ( r r ) 1 2 σ 2 σ d r Φ j ( r ) σ d r Φ j ( r ) × σ d r d r D ( r r ) + 1 2 σ σ d r Φ j ( r ) σ d r d r Φ k ( r ) D ( r r ) + 1 2 σ σ d r Φ k ( r ) σ d r d r Φ j ( r ) D ( r r ) .
| h ¯ = Δ 1 | p ¯ .
P ¯ j = σ d r Φ j ( r ) [ ψ ( r ) ψ ] [ ψ ( r ) ψ ( r + ρ ) ] ¯ .
P ¯ j = 1 2 σ d r Φ j ( r ) [ D ( r r ) D ( r r + ρ ) ] + 1 2 σ σ d r Φ j ( r ) σ d r [ D ( r r ) D ( r r + ρ ) ] .
Φ ( r ) = 1 [ r R d ] 2 + 2 [ r R d ] 2 log [ r R d ] .

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