Abstract

The applicability of wave-front correction by means of a bimorph mirror in conjunction with a curvature sensor is described. We use Zernike polynomials to describe the quality of the atmospheric-turbulence correction analytically. The match is limited by boundary conditions of the mirror and by the discreteness of the electrodes. The correction is limited by coupling of lower- and higher-order Zernike polynomials and necessitates an interfacing computer between the wave-front sensor and the bimorph mirror.

© 1994 Optical Society of America

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References

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    [CrossRef]
  2. S. P. Timoshenko, S. Woinowsky-Kriger, The Theory of Plates and Shells, 2nd ed. (McGraw-Hill, New York, 1959), Chap. 4, Sec. 24, pp. 94–97.
  3. S. A. Kokorowsky, “Analysis of adaptive optical elements made from piezoelectric bimorphs,”J. Opt. Soc. Am. 69, 181–187 (1979).
    [CrossRef]
  4. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part 2, Chap. 10, pp. 1175–1215.
  5. W. C. Young, Roarks Formulas for Stress and Strain, 6th ed. (McGraw-Hill, New York, 1989), Chap. 10, pp. 443–448.
  6. P. Jagourel, P. Y. Madec, M. Sechaud, “Adaptive optics: a bimorph mirror for wave front correction,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 394–405 (1990).
    [CrossRef]
  7. F. Roddier, “A new concept in adaptive optics: curvature sensing and compensation,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef] [PubMed]
  8. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), App. VII, pp. 767–772.
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    [CrossRef] [PubMed]
  12. N. Roddier, F. Roddier, “Curvature sensing and compensation: a computer simulation,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 92–96 (1989).
    [CrossRef]
  13. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,”J. Opt. Soc. Am. 56, 1372–1379 (1966).
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  14. F. Roddier, “Wavefront curvature sensing and compensation methods in adaptive optics,” in Propagation Engineering Fourth in a Series, L. R. Bissonnette, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1487, 123–128 (1991).
    [CrossRef]
  15. F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
    [CrossRef]
  16. D. P. Greenwood, “Mutual coherence function of a wave front corrected by zonal adaptive optics,”J. Opt. Soc. Am. 69, 549–554 (1979).
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  17. R. Cubalchini, “Modal wave-front estimation from phase derivative measurements,”J. Opt. Soc. Am. 69, 972–977 (1979).
    [CrossRef]

1991 (1)

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

1988 (1)

1980 (1)

1979 (4)

1978 (1)

1976 (1)

1966 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), App. VII, pp. 767–772.

Cubalchini, R.

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part 2, Chap. 10, pp. 1175–1215.

Fried, D. L.

Graves, J. E.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Greenwood, D. P.

Jagourel, P.

P. Jagourel, P. Y. Madec, M. Sechaud, “Adaptive optics: a bimorph mirror for wave front correction,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 394–405 (1990).
[CrossRef]

Kokorowsky, S. A.

Lipson, S. G.

Madec, P. Y.

P. Jagourel, P. Y. Madec, M. Sechaud, “Adaptive optics: a bimorph mirror for wave front correction,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 394–405 (1990).
[CrossRef]

Markey, J. K.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part 2, Chap. 10, pp. 1175–1215.

Noll, R. J.

Northcott, M.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Roddier, F.

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

F. Roddier, “A new concept in adaptive optics: curvature sensing and compensation,” Appl. Opt. 27, 1223–1225 (1988).
[CrossRef] [PubMed]

N. Roddier, F. Roddier, “Curvature sensing and compensation: a computer simulation,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 92–96 (1989).
[CrossRef]

F. Roddier, “Wavefront curvature sensing and compensation methods in adaptive optics,” in Propagation Engineering Fourth in a Series, L. R. Bissonnette, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1487, 123–128 (1991).
[CrossRef]

Roddier, N.

N. Roddier, F. Roddier, “Curvature sensing and compensation: a computer simulation,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 92–96 (1989).
[CrossRef]

Sechaud, M.

P. Jagourel, P. Y. Madec, M. Sechaud, “Adaptive optics: a bimorph mirror for wave front correction,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 394–405 (1990).
[CrossRef]

Silva, D. E.

Steinhaus, E.

Timoshenko, S. P.

S. P. Timoshenko, S. Woinowsky-Kriger, The Theory of Plates and Shells, 2nd ed. (McGraw-Hill, New York, 1959), Chap. 4, Sec. 24, pp. 94–97.

Wang, J. Y.

Woinowsky-Kriger, S.

S. P. Timoshenko, S. Woinowsky-Kriger, The Theory of Plates and Shells, 2nd ed. (McGraw-Hill, New York, 1959), Chap. 4, Sec. 24, pp. 94–97.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), App. VII, pp. 767–772.

Young, W. C.

W. C. Young, Roarks Formulas for Stress and Strain, 6th ed. (McGraw-Hill, New York, 1989), Chap. 10, pp. 443–448.

Appl. Opt. (2)

J. Opt. Soc. Am. (7)

Publ. Astron. Soc. Pac. (1)

F. Roddier, M. Northcott, J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Other (7)

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part 2, Chap. 10, pp. 1175–1215.

W. C. Young, Roarks Formulas for Stress and Strain, 6th ed. (McGraw-Hill, New York, 1989), Chap. 10, pp. 443–448.

P. Jagourel, P. Y. Madec, M. Sechaud, “Adaptive optics: a bimorph mirror for wave front correction,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 394–405 (1990).
[CrossRef]

S. P. Timoshenko, S. Woinowsky-Kriger, The Theory of Plates and Shells, 2nd ed. (McGraw-Hill, New York, 1959), Chap. 4, Sec. 24, pp. 94–97.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), App. VII, pp. 767–772.

F. Roddier, “Wavefront curvature sensing and compensation methods in adaptive optics,” in Propagation Engineering Fourth in a Series, L. R. Bissonnette, W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1487, 123–128 (1991).
[CrossRef]

N. Roddier, F. Roddier, “Curvature sensing and compensation: a computer simulation,” in Active Telescope Systems, F. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 92–96 (1989).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic drawing of a bimorph mirror. A different voltage can be applied to each electrode to control the local curvature.

Fig. 2
Fig. 2

(a) Open-loop adaptive-optics system (measurement of wave fronts). (b) Closed-loop operation (measurement of wavefront residuals).

Fig. 3
Fig. 3

Centrally supported bimorph mirror.

Fig. 4
Fig. 4

Correction of the first isotropic orders (approximate treatment). x is the ratio of the optical aperture to the electrode size. The Zernike radial and azimuthal orders are denoted by (n, m). The calculation is more accurate for higher x values. The correction is drawn relative to the full correction (no aberrations) for each order.

Equations (60)

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4 Z = - A 2 V ,
4 Z = q H ,
2 M = - q ,
2 Z = - M H ,
M = M x + M y 1 + ν ,
M = A H V ,
2 Z = - A V .
2 Z = - f ( r , θ )
Z ( r , θ ) = 1 2 π 0 2 π d ϕ 0 r ρ d ρ { ln ( R r ) - k = 1 1 k ( ρ k r k R 2 k - ρ k r k ) cos [ k ( ϕ - θ ) ] } f ( ρ , ϕ ) + 1 2 π 0 2 π d ϕ r R ρ d ρ { ln ( R ρ ) - k = 1 1 k ( ρ k r k R 2 k - r k ρ k ) cos [ k ( ϕ - θ ) ] } f ( ρ , ϕ ) .
Z ( r ) = A V 4 ( R 2 - r 2 ) .
A = 12 d 13 ( t 1 + t 2 ) t 1 3 k ,
k = 4 + 6 ( t 2 t 1 ) + 4 ( t 2 t 1 ) 2 + E 2 t 2 3 ( 1 - ν 1 ) E 1 t 1 3 ( 1 - ν 2 ) + E 1 t 1 ( 1 - ν 2 ) E 2 t 2 ( 1 - ν 1 ) ,
A 1.5 d 13 t 2 .
A V = { 2 W + δ ( ρ - a ) W ρ ρ a 0 ρ > a ,
Z ( r , θ ) = 1 2 π 0 2 π d ϕ 0 r ρ d ρ { ln ( R r ) + k = 1 1 k ( ρ r ) k cos [ k ( ϕ - θ ) ] } 2 W + 1 2 π 0 2 π d ϕ r a ρ d ρ { ln ( R ρ ) + k = 1 1 k ( r ρ ) k cos [ k ( ϕ - θ ) ] } 2 W + 1 2 π 0 2 π d ϕ a { ln ( R a ) + k = 1 1 k ( r a ) k cos [ k ( ϕ - θ ) ] } W ρ | ρ = a .
Z ( r ¯ ) = 1 4 π f ( ρ ¯ ) G ( r ¯ / ρ ¯ ) d 2 ρ + 1 4 π c [ G ( r ¯ / ρ ¯ ) W ρ | ρ = a - W ( a ) G ρ | ρ = a ] d l ,
G ρ | ρ = a = const .
G ( r ¯ / ρ ¯ ) = { - 2 ln ( r ) + 2 k = 1 1 k ( ρ r ) k cos [ k ( ϕ - θ ) ] r > ρ - 2 ln ( ρ ) + 2 k = 1 1 k ( r ρ ) k cos [ k ( ϕ - θ ) ] r < ρ .
G ρ | ρ = a = - 2 a { 1 + k ( r a ) k cos [ k ( ϕ - θ ) ] } ,
1 2 π k = 1 ( r a ) k 0 2 π d ϕ cos [ k ( ϕ - θ ) ] Z ( a , ϕ ) ,
Ψ ( r , θ ) = j = 1 a j Z j ( r , θ ) ,
1 2 π k 0 2 π d ϕ r k cos [ k ( ϕ - θ ) ] cos ( m ϕ ) 2 ( n + 1 )
( n + 1 ) 2 r m cos ( m θ ) .
Φ c ( m ) = j = i a j [ Z j ( m ) + k j Z i ( m ) ] ,
k j = { 1 2 ( n + 1 m + 1 ) 1 / 2 m > 0 [ ( n + 1 ) 2 ] m = 0 .
2 d 2 ρ ( Ψ - Φ c ) 2 w ( ρ ) ,
2 = Ψ 2 - j a j 2 + j , j a j a j * k j k j ,
2 0.25 ( D r 0 ) 5 / 3 .
2 0.049 ( D r 0 ) 5 / 3 ,
1 2 π 0 2 π d ϕ 0 a ρ d ρ { k = 1 1 k ( ρ r a 2 ) k cos [ k ( ϕ - θ ) ] } 2 W ( ρ , ϕ ) ,
1 2 π 0 2 π d ϕ a { k = 1 1 k ( ρ r R 2 ) k cos [ k ( ϕ - θ ) ] } W ρ | ρ = a .
a 2 1 2 ( a R ) 2 r cos ( θ ) ,
Z ( r , θ ) = 1 2 π 0 2 π d ϕ r R ρ d ρ { ln ( 1 ρ ) + k = 1 1 k ( r ρ ) k cos [ k ( ϕ - θ ) ] } f ( ρ , ϕ ) + 1 2 π 0 2 π d ϕ R r ρ d ρ { ln ( 1 r ) + k = 1 1 k ( ρ r ) k cos [ k ( ϕ - θ ) ] } f ( ρ , ϕ ) ,
f = A V = { 2 Z + δ ( ρ - a ) W ρ | ρ = a + δ ( ρ - b ) W ρ | ρ = a b ρ a 0 ρ < b , ρ > a ,
k 2 Z ^ ( k ¯ ) = k 2 W ^ ( k ¯ ) I ^ 2 ( k ¯ ) ,
I ^ ( k ) = 2 J 1 ( π k d ) π k d ,
0.0046 ( D r 0 ) 5 / 3 ( n + 1 ) 0 d k k - 8 / 3 [ J n + 1 ( 2 π k ) k ] 2 × [ 2 J 1 ( 2 π k / x ) ( 2 π k / x ) ] 4 ,
2 0.111 ( D r 0 ) 5 / 3 .
2 0.111 ( d r 0 ) 5 / 3 .
v = MDs ,
Z j even = 2 ( n + 1 ) R n m ( r ) cos ( m θ ) , Z j odd = 2 ( n + 1 ) R n m ( r ) sin ( m θ ) , Z j = ( n + 1 ) R n m ( r )             for m = 0.
R n m ( r ) = s = 0 ( n - m ) / 2 ( - 1 ) s ( n - s ) ! s ! [ ( n + m ) / 2 - s ] ! [ ( n - m ) / 2 - s ] ! r n - 2 s ,
0 d 2 r w ( r ) Z j Z j = δ j j ,
w ( r ) = { 1 π r 1 0 r > 1 .
Ψ ( R ρ , θ ) = j a j Z j ( ρ , θ ) ,
a j = d 2 ρ w ( ρ ) Ψ ( R ρ , θ ) Z j ( ρ , θ ) .
w ( ρ ) Z j ( ρ , θ ) = d 2 k Q j ( k , ϕ ) exp ( - 2 π i k ¯ · ρ ¯ ) ,
Q j even ( k , ϕ ) = 2 ( n + 1 ) J n + 1 ( 2 π k ) π k ( - 1 ) ( n - m ) / 2 i m cos ( m ϕ ) , Q j odd ( k , ϕ ) = 2 ( n + 1 ) J n + 1 ( 2 π k ) π k ( - 1 ) ( n - m ) / 2 i m sin ( m ϕ ) , Q j ( k , ϕ ) = ( n + 1 ) J n + 1 ( 2 π k ) π k ( - 1 ) n / 2             for m = 0.
Ψ ^ ( k ) = 0.023 r 0 - 5 / 3 k - 11 / 3 ,
r 0 = 1.68 [ ( 2 π λ ) 2 d z C n 2 ( z ) ] - 5 / 3 .
a j a j * = { 0.1524 ( D r 0 ) 5 / 3 ( - 1 ) ( n + n - 2 m ) / 2 [ ( n + 1 ) ( n + 1 ) ] 1 / 2 δ m m × Γ ( 14 3 ) Γ ( n + n - 5 / 3 2 ) Γ ( n - n + 17 / 3 2 ) Γ ( n - n + 17 / 3 2 ) Γ ( n + n + 23 / 3 2 ) j - j even 0 j - j odd .
2 = d 2 ρ ( Ψ - Φ c ) 2 w ( ρ ) .
2 = w ( ρ ) d 2 ρ [ Ψ - j = 2 a j ( Z j + k j Z i ) ] × [ Ψ * - j = 2 a j ( Z j + k j Z i ) ] = σ Ψ 2 - w ( ρ ) d 2 ρ × { Ψ j = 2 a j * ( Z j + k j Z i ) - Ψ * j = 2 a j ( Z j + k j Z i ) + j , j = 2 a j a j * ( Z j Z j + k j Z i Z j + k j Z i Z j + k j k j Z i 2 ) } ,
2 = σ Ψ 2 - j = i a j 2 - j = i k j ( a j * a i + a i * a j ) + j = i k j ( a j a i * ) + j = i k j ( a i a j * ) + j , j = i k j k j ( a j * a j ) = σ Ψ 2 - j = i a j 2 + j , j = i k j k j ( a j * a j ) .
a j * a j = d ρ ¯ d ρ ¯ w ( ρ ¯ ) w ( ρ ¯ ) Z j ( ρ ¯ , θ ) C ( R ρ ¯ , R ρ ¯ ) Z j ( ρ ¯ , θ ) ,
C ( R ρ ¯ , R ρ ¯ ) = Ψ ( R ρ ¯ ) Ψ * ( R ρ ¯ ) .
a j * a j = d k ¯ d k ¯ Q j * ( k ¯ ) Ψ ^ ( k R k R ) Q j ( k ¯ ) ,
Ψ ^ ( k R k R ) = 0.023 ( R r 0 ) 5 / 3 k - 11 / 3 δ ( k - k ) ,
Q j ( k ) = n + 1 ( - 1 ) n / 2 J n + 1 ( 2 π k ) π k [ 2 J 1 ( 2 π k / x ) 2 π k / x ] 2 ,
a j 2 = 0.0046 ( D r 0 ) 5 / 3 ( n + 1 ) d k k - 8 / 3 [ J n + 1 ( 2 π k ) k ] 2 × [ 2 J 1 ( 2 π k / x ) 2 π k / x ] 4 .

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