Abstract

Adaptive optics correction of a wave front by a deformable mirror that acts as a lossless spatial filter is studied. The decomposition of the wave front into Zernike polynomials provides a means for deriving the rms error of a corrected wave front in analytic form. The spatial filter is given in a functional form related to deformable mirror characteristics. A step filter approximation is derived and the conditions where the approximation holds are examined. An example is provided to demonstrate the utility of the spatial filtering concept for adaptive optics systems analysis.

© 1982 Optical Society of America

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References

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  1. J. E. Pearson, S. Hansen, J. Opt. Soc. Am. 67, 325 (1977).
    [Crossref]
  2. J. Feinleib, S. J. Lipson, P. F. Cone, Appl. Phys. Lett. 25, 311 (1974).
    [Crossref]
  3. J. Y. Wang, J. K. Markey, J. Opt. Soc. Am. 68, 78 (1978).
    [Crossref]
  4. J. E. Pearson, Opt. Lett. 2, 7 (1978).
    [Crossref] [PubMed]
  5. R. K. Tyson, Proc. Soc. Photo-Opt. Instrum. Eng. 270, 142 (1981).
  6. J. E. Pearson, R. H. Freeman, H. C. Reynolds, “Adaptive Optical Techniques for Wave-Front Correction,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Vol. 7, Chap. 8, pp. 245–340.
    [Crossref]
  7. D. L. Fried, J. Opt. Soc. Am. 67, 370 (1977).
    [Crossref]
  8. R. E. Wagner, “Imagery Utilizing Multiple Focal Planes,” Ph.D. Dissertation, U. Arizona, 1976.
  9. J. E. Harvey, G. M. Callahan, Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50 (1978).
  10. R. K. Tyson, D. M. Byrne, Proc. Soc. Photo-Opt. Instrum. Eng. 228, 21 (1980).
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 9 and Appendix 7.
  12. A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), Chap. 12.
  13. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 318.
  14. R. J. Noll, J. Opt. Soc. Am. 66, 207 (1976).
    [Crossref]
  15. H. R. Garcia, L. D. Brooks, Proc. Soc. Photo-Opt. Instrum. Eng. 141, 74 (1978).
  16. R. J. Roark, Formulas for Stress and Strain (McGraw-Hill, New York, 1965), p. 216.
  17. J. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).
  18. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972).

1981 (1)

R. K. Tyson, Proc. Soc. Photo-Opt. Instrum. Eng. 270, 142 (1981).

1980 (1)

R. K. Tyson, D. M. Byrne, Proc. Soc. Photo-Opt. Instrum. Eng. 228, 21 (1980).

1978 (4)

J. Y. Wang, J. K. Markey, J. Opt. Soc. Am. 68, 78 (1978).
[Crossref]

J. E. Pearson, Opt. Lett. 2, 7 (1978).
[Crossref] [PubMed]

J. E. Harvey, G. M. Callahan, Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50 (1978).

H. R. Garcia, L. D. Brooks, Proc. Soc. Photo-Opt. Instrum. Eng. 141, 74 (1978).

1977 (2)

1976 (1)

1974 (1)

J. Feinleib, S. J. Lipson, P. F. Cone, Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 9 and Appendix 7.

Brooks, L. D.

H. R. Garcia, L. D. Brooks, Proc. Soc. Photo-Opt. Instrum. Eng. 141, 74 (1978).

Byrne, D. M.

R. K. Tyson, D. M. Byrne, Proc. Soc. Photo-Opt. Instrum. Eng. 228, 21 (1980).

Callahan, G. M.

J. E. Harvey, G. M. Callahan, Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50 (1978).

Cone, P. F.

J. Feinleib, S. J. Lipson, P. F. Cone, Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

Feinleib, J.

J. Feinleib, S. J. Lipson, P. F. Cone, Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

Freeman, R. H.

J. E. Pearson, R. H. Freeman, H. C. Reynolds, “Adaptive Optical Techniques for Wave-Front Correction,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Vol. 7, Chap. 8, pp. 245–340.
[Crossref]

Fried, D. L.

Garcia, H. R.

H. R. Garcia, L. D. Brooks, Proc. Soc. Photo-Opt. Instrum. Eng. 141, 74 (1978).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 318.

Gradshteyn, J. S.

J. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Hansen, S.

Harvey, J. E.

J. E. Harvey, G. M. Callahan, Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50 (1978).

Lipson, S. J.

J. Feinleib, S. J. Lipson, P. F. Cone, Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

Markey, J. K.

Noll, R. J.

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), Chap. 12.

Pearson, J. E.

J. E. Pearson, Opt. Lett. 2, 7 (1978).
[Crossref] [PubMed]

J. E. Pearson, S. Hansen, J. Opt. Soc. Am. 67, 325 (1977).
[Crossref]

J. E. Pearson, R. H. Freeman, H. C. Reynolds, “Adaptive Optical Techniques for Wave-Front Correction,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Vol. 7, Chap. 8, pp. 245–340.
[Crossref]

Reynolds, H. C.

J. E. Pearson, R. H. Freeman, H. C. Reynolds, “Adaptive Optical Techniques for Wave-Front Correction,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Vol. 7, Chap. 8, pp. 245–340.
[Crossref]

Roark, R. J.

R. J. Roark, Formulas for Stress and Strain (McGraw-Hill, New York, 1965), p. 216.

Ryzhik, I. M.

J. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

Tyson, R. K.

R. K. Tyson, Proc. Soc. Photo-Opt. Instrum. Eng. 270, 142 (1981).

R. K. Tyson, D. M. Byrne, Proc. Soc. Photo-Opt. Instrum. Eng. 228, 21 (1980).

Wagner, R. E.

R. E. Wagner, “Imagery Utilizing Multiple Focal Planes,” Ph.D. Dissertation, U. Arizona, 1976.

Wang, J. Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 9 and Appendix 7.

Appl. Phys. Lett. (1)

J. Feinleib, S. J. Lipson, P. F. Cone, Appl. Phys. Lett. 25, 311 (1974).
[Crossref]

J. Opt. Soc. Am. (4)

Opt. Lett. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (4)

R. K. Tyson, Proc. Soc. Photo-Opt. Instrum. Eng. 270, 142 (1981).

J. E. Harvey, G. M. Callahan, Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50 (1978).

R. K. Tyson, D. M. Byrne, Proc. Soc. Photo-Opt. Instrum. Eng. 228, 21 (1980).

H. R. Garcia, L. D. Brooks, Proc. Soc. Photo-Opt. Instrum. Eng. 141, 74 (1978).

Other (8)

R. J. Roark, Formulas for Stress and Strain (McGraw-Hill, New York, 1965), p. 216.

J. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965).

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972).

R. E. Wagner, “Imagery Utilizing Multiple Focal Planes,” Ph.D. Dissertation, U. Arizona, 1976.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), Chap. 9 and Appendix 7.

A. Papoulis, The Fourier Integral and Its Applications (McGraw-Hill, New York, 1962), Chap. 12.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), p. 318.

J. E. Pearson, R. H. Freeman, H. C. Reynolds, “Adaptive Optical Techniques for Wave-Front Correction,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), Vol. 7, Chap. 8, pp. 245–340.
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

Adaptive optics correction function defined in Eq. (17) vs actuator spacing for values of the Zernike orders |nn′| = 0.

Fig. 2
Fig. 2

Adaptive optics correction function defined in Eq. (17) vs actuator spacing for values of the Zernike orders |nn′| = 2 and |nn′| = 8.

Fig. 3
Fig. 3

Adaptive optics correction function defined in Eq. (17) vs actuator spacing for values of the Zernike orders |nn′| = 4.

Fig. 4
Fig. 4

Adaptive optics correction function defined in Eq. (17) vs actuator spacing for values of the Zernike orders |nn′| = 6.

Tables (3)

Tables Icon

Table I Correction of Tilt and Defocus Example Using Step Filter Method

Tables Icon

Table II Correction of Tilt and Defocus Example with One Actuator

Tables Icon

Table III Correction of Tilt and Defocus Example with Five Actuators

Equations (42)

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Φ c ( x , y ) = Φ u ( x , y ) - 2 cos θ i M ( x , y ) ,
σ 2 = aperture [ Φ c ( x , y ) ] 2 d x d y [ Φ c ( x , y ) ] 2 d x d y = minimum .
σ 2 = A [ Φ ( x , y ) ] 2 d x d y A d x d y - [ A Φ ( x , y ) d x d y A d x d y ] 2 ,
I [ Φ ( x , y ) ] I ( 0 ) ~ 1 - ( 2 π λ ) 2 σ 2 ,
σ 2 = - PSD ( ξ , η ) d ξ d η ,
PSD ( η , ξ ) = FT [ Φ ( x , y ) ] * FT [ Φ ( x , y ) ] A ¯ ,
σ c 2 = 0 2 π 1 2 a PSD u ( ρ , ϕ ) ρ d ρ d ϕ ,
FT ( Φ c ) = FT ( Φ u ) [ 1 - γ FT ( M ) FT ( Φ u ) ] ,
G ( ρ , ϕ ) 2 = 1 - γ FT ( M ) FT ( Φ u ) - γ FT ( M ) * FT ( Φ u ) * + γ 2 [ FT ( M ) ] 2 FT ( Φ u ) 2 ,
G ( ρ , ϕ ) = 1 - γ FT ( M ) FT ( Φ u ) .
Φ ( r , θ ) = A o o + 1 2 n = 2 A n o R n o ( r R ) + n = 1 m = 1 n × ( A n m cos m θ + B n m sin m θ ) R n m ( r R ) ,
FT ( Φ ) = n = 2 A n o π R 2 r 2 J n + 1 ( 2 π ρ R ) π ρ R i n + n = 1 m = 1 n J n + 1 ( 2 π ρ R ) π ρ R π R 2 i n ( A n m cos m ϕ + B n m sin m ϕ ) ,
PSD ( Φ ) = 1 π ρ 2 n = 1 m = 0 n n = 1 m = 0 n ( A n m cos m ϕ + B n m sin m ϕ ) ( A n m cos m ϕ + B n m sin m ϕ ) 1 + δ m o 1 + δ m o × J n + 1 ( 2 π ρ R ) J n + 1 ( 2 π ρ R ) i n ( - i ) n .
σ 2 = n = 1 m = 0 n A n m 2 + B n m 2 2 ( n + 1 ) , n - m = even , B n o = 0.
σ c 2 = 0 2 π 1 2 a PSD ρ d ρ d ϕ = 0 2 π 0 PSD ρ d ρ d ϕ - 0 2 π 0 1 2 a PSD ρ d ρ d ϕ , or σ c 2 = σ u 2 - σ d 2 .
σ d 2 = n = 1 m = 0 n n = 1 m = 0 n ( A n m A n m + B n m B n m ) m = m only × 0 1 2 a i n ( - i ) n J n + 1 ( 2 π ρ R ) J n + 1 ( 2 π ρ R ) ρ d ρ .
Z ( R a , n , n ) = 0 1 2 a J n + 1 ( 2 π ρ R ) J n + 1 ( 2 π ρ R ) ρ = k = 0 ( - 1 ) k ( n + n + 2 k + 1 ) ! ( π 2 R a ) n + n + 2 k + 2 ( n + 1 + k ) ! ( n + 1 + k ) ! ( n + n + k + 2 ) ! k ! .
σ c 2 = n = 1 m = 0 n A n m 2 + B n m 2 2 ( n + 1 ) - n = 1 m = 0 n n = 1 m = 0 n × ( - 1 ) ( n - n ) / 2 ( A n m A n m + B n m B n m ) Z ( R a , n , n ) ,
I i ( x , y ) = exp { ln κ a 2 [ ( x - x i ) 2 + ( y - y i ) 2 ] } ,
M ( x , y ) = i = 1 N L i I i ( x , y ) ,
Φ c = A o o + 1 2 n = 2 R n o ( r R ) + n = 1 m = 1 n × ( A n m cos m θ + B n m sin m θ ) R n m ( r R ) - γ L o - γ i = 1 N L i exp { ln κ a 2 [ ( x - x i ) 2 + ( y - y i ) 2 ] } .
[ I ] [ L ] = [ P ]
I j k = γ aperture I j ( x j , y j ) I k ( x k , y k ) d x d y ,
P k = aperture Φ u ( x , y ) I k ( x k , y k ) d x d y .
FT [ I i ( x , y ) ] = π a 2 - ln κ exp ( π 2 ln κ ρ 2 ) × exp [ - i 2 π ( x i ξ + y i η ) ] ,
Φ c = ( A o o - C o o ) + 1 2 n = 2 ( A n o - C n o ) × R n o ( r R ) + n = 1 m = 1 n [ ( A n m - C n m ) cos m θ + ( B n m - D n m ) sin m θ ] R n m ( r R ) ,
Φ M ( r , θ ) = γ L o + γ i = 1 N L i × exp { ln κ a 2 [ ( x - x i ) 2 + ( y - y i ) 2 ] } = C o o + 1 2 n = 2 C n o R r o ( r R ) + n = 1 m = 1 n × ( C n m cos m θ + D n m sin m θ ) R n m ( r R ) .
C o o = 0 2 π 0 R Φ M r d r d θ 0 2 π 0 R r d r d θ ,
C n o = 0 2 π 0 R Φ M R n o ( r R ) 2 r d r d θ 0 2 π 0 R R n o ( r R ) 2 2 r d r d θ ,
C n m = 0 2 π 0 R Φ M cos m θ R n m ( r R ) r d r d θ 0 2 π 0 R [ R n m ( r R ) ] 2 cos 2 m θ r d r d θ ,
D n m = 0 2 π 0 R Φ M sin m θ R n m ( r R ) r d r d θ 0 2 π 0 R [ R n m ( r R ) ] 2 sin 2 m θ r d r d θ .
Φ u ( r , θ ) = 1 2 [ 2 ( r R ) 2 - 1 ] + 2 ( r R ) cos θ .
FT ( Φ u ) = R ρ [ - J 3 ( 2 π ρ R ) 2 + 2 i J 2 ( 2 π ρ R ) cos ϕ ] ,
PSD ( Φ ) = 1 π ρ 2 [ A 11 2 cos 2 ϕ J 2 2 ( 2 π ρ R ) + A 20 2 J 3 2 ( 2 π ρ R ) 2 ] .
G ( ρ , ϕ ) = 1 - γ FT ( M ) FT ( Φ u ) = 1 - γ L 0 δ ( ρ ) π ρ + γ L 1 π R 2 - ln κ exp ( π 2 ln κ ρ 2 ) R ρ [ - J 3 ( 2 π ρ R ) 2 + 2 i J 2 ( 2 π ρ R ) cos θ ] ,
σ 2 = n = 0 m = 0 n n = 0 m = 0 n 0 2 π 0 ( A n m A n m cos m ϕ cos m ϕ + B n m B n m sin m ϕ sin m ϕ + A n m B n m cos m ϕ sin m ϕ + A n m B n m cos m ϕ sin m ϕ ) × i n ( - i ) n 1 + δ m o 1 + δ m o × J n + 1 ( 2 π ρ R ) J n + 1 ( 2 π ρ R ) π ρ 2 ρ d ρ d ϕ .
0 J n + 1 ( 2 π ρ R ) J n + 1 ( 2 π ρ R ) ρ d ρ = 1 2 ( n + n 2 + 1 ) ( n - n 2 ) ! ( n - n 2 ) ! = 1 2 ( n + 1 ) and n = n .
0 2 π A n m B n m cos m ϕ sin m ϕ d ϕ = 0 for all m and m , 0 2 π A n m B n m cos m ϕ cos m ϕ d ϕ = 0 if m m = π A n m A n m if m = m 0 = 2 π A n m A n m if m = m = 0 , 0 2 π B n m B n m sin m ϕ sin m ϕ d ϕ = 0 if m m = π B n m B n m if m = m 0 = 2 π B n m B n m if m = m = 0.
1 + δ m o 1 + δ m o = 2 if m = 0 = 1 if m 0 , σ 2 = n = 1 m = 0 n 1 π 1 2 ( n + 1 ) ( A n m 2 + B n m 2 ) π , σ 2 = n = 1 m = 0 n A n m 2 + B n m 2 2 ( n + 1 ) .
Z ( R a , n , n ) = 0 1 2 a J n + 1 ( 2 π ρ R ) J n + 1 ( 2 π ρ R ) ρ d ρ .
Z = 0 1 2 a ( π ρ R ) n + n + 2 ρ k = 0 ( - 1 ) k ( n + n + 2 + 2 k ) ! ( π ρ R ) 2 k ( n + 1 + k ) ! ( n + 1 + k ) ( n + n + 2 + k ) ! k ! d ρ ,
Z ( R a , n , n ) = k = 0 × ( - 1 ) k ( n + n + 2 k + 1 ) ! ( π R 2 a ) n + n + 2 k + 2 ( n + 1 + k ) ! ( n + 1 + k ) ! ( n + n + k + 2 ) ! k ! .

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