Abstract

An approximate model of an adaptive optic element is presented which works well for components having many actuators. This zonal model is used to gain insight into the general behavior of systems that correct for high spatial frequency errors. The model is derived from the method of least squares, and considers the nonshift invariant properties of adaptive mirrors. It may be implemented with Fourier transform techniques and is, therefore, easy to program. The relationship between the zonal model and the simpler bandpass filter model of Harvey and Callahan is discussed.

© 1990 Optical Society of America

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References

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  1. J. E. Pearson, R. H. Freeman, H. C. Reynolds, “Adaptive Optical Techniques for Wave-Front Correction,” in Applied Optics and Optical Engineering, Volume 7, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), pp. 245–340.
  2. R. E. Wagner, “Imagery Utilizing Multiple Focal Planes,” Ph.D. dissertation (University of Arizona, 1976).
  3. J. E. Harvey, G. M. Callahan, “Wavefront Error Compensation Capabilities of Multi-Actuator Deformable Mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50 (1978).
  4. R. K. Tyson, D. M. Byrne, “The Effect of Wavefront Sensor Characteristics and Spatiotemporal Coupling on the Correcting Capability of a Deformable Mirror,” Proc. Soc. Photo-Opt. Instrum. Eng. 228, 21 (1980).
  5. R. K. Tyson, “Adaptive Optics Compensation of Thermal Distortion,” Proc. Soc. Photo-Opt. Instrum. Eng. 270, 142 (1981).
  6. R. K. Tyson, “Using the Deformable Mirror as a Spatial Filter: Application to Circular Beams,” Appl. Opt. 21, 787–793 (1982).
    [CrossRef] [PubMed]
  7. J. E. Pearson, S. Hansen, “Experimental Studies of a Deformable-Mirror Adaptive Optical System,” J. Opt. Soc. Am. 67, 325 (1977).
    [CrossRef]
  8. T. R. O’Meara, “The Multidither Principle in Adaptive Optics,” J. Opt. Soc. Am. 67, 306–333 (1977).
    [CrossRef]
  9. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  10. C. Lawson, R. Hansen, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1979).
  11. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).
  12. J. E. Harvey, G. M. Callahan, “Transfer Function Characterization of Deformable Mirrors,” J. Opt. Soc. Am. 67, 1367–1367 (1977).
  13. J. E. Harvey, Hughes Danburg Optical Systems, private communication.

1982 (1)

1981 (1)

R. K. Tyson, “Adaptive Optics Compensation of Thermal Distortion,” Proc. Soc. Photo-Opt. Instrum. Eng. 270, 142 (1981).

1980 (1)

R. K. Tyson, D. M. Byrne, “The Effect of Wavefront Sensor Characteristics and Spatiotemporal Coupling on the Correcting Capability of a Deformable Mirror,” Proc. Soc. Photo-Opt. Instrum. Eng. 228, 21 (1980).

1978 (1)

J. E. Harvey, G. M. Callahan, “Wavefront Error Compensation Capabilities of Multi-Actuator Deformable Mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50 (1978).

1977 (3)

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).

Byrne, D. M.

R. K. Tyson, D. M. Byrne, “The Effect of Wavefront Sensor Characteristics and Spatiotemporal Coupling on the Correcting Capability of a Deformable Mirror,” Proc. Soc. Photo-Opt. Instrum. Eng. 228, 21 (1980).

Callahan, G. M.

J. E. Harvey, G. M. Callahan, “Wavefront Error Compensation Capabilities of Multi-Actuator Deformable Mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50 (1978).

J. E. Harvey, G. M. Callahan, “Transfer Function Characterization of Deformable Mirrors,” J. Opt. Soc. Am. 67, 1367–1367 (1977).

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, Volume 7, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), pp. 245–340.

Gaskill, J. D.

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

Hansen, R.

C. Lawson, R. Hansen, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1979).

Hansen, S.

Harvey, J. E.

J. E. Harvey, G. M. Callahan, “Wavefront Error Compensation Capabilities of Multi-Actuator Deformable Mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50 (1978).

J. E. Harvey, G. M. Callahan, “Transfer Function Characterization of Deformable Mirrors,” J. Opt. Soc. Am. 67, 1367–1367 (1977).

J. E. Harvey, Hughes Danburg Optical Systems, private communication.

Lawson, C.

C. Lawson, R. Hansen, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1979).

O’Meara, T. R.

Pearson, J. E.

J. E. Pearson, S. Hansen, “Experimental Studies of a Deformable-Mirror Adaptive Optical System,” 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, Volume 7, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), pp. 245–340.

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, Volume 7, R. R. Shannon, J. C. Wyant, Eds. (Academic, New York, 1979), pp. 245–340.

Tyson, R. K.

R. K. Tyson, “Using the Deformable Mirror as a Spatial Filter: Application to Circular Beams,” Appl. Opt. 21, 787–793 (1982).
[CrossRef] [PubMed]

R. K. Tyson, “Adaptive Optics Compensation of Thermal Distortion,” Proc. Soc. Photo-Opt. Instrum. Eng. 270, 142 (1981).

R. K. Tyson, D. M. Byrne, “The Effect of Wavefront Sensor Characteristics and Spatiotemporal Coupling on the Correcting Capability of a Deformable Mirror,” Proc. Soc. Photo-Opt. Instrum. Eng. 228, 21 (1980).

Wagner, R. E.

R. E. Wagner, “Imagery Utilizing Multiple Focal Planes,” Ph.D. dissertation (University of Arizona, 1976).

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

Proc. Soc. Photo-Opt. Instrum. Eng. (3)

J. E. Harvey, G. M. Callahan, “Wavefront Error Compensation Capabilities of Multi-Actuator Deformable Mirrors,” Proc. Soc. Photo-Opt. Instrum. Eng. 141, 50 (1978).

R. K. Tyson, D. M. Byrne, “The Effect of Wavefront Sensor Characteristics and Spatiotemporal Coupling on the Correcting Capability of a Deformable Mirror,” Proc. Soc. Photo-Opt. Instrum. Eng. 228, 21 (1980).

R. K. Tyson, “Adaptive Optics Compensation of Thermal Distortion,” Proc. Soc. Photo-Opt. Instrum. Eng. 270, 142 (1981).

Other (6)

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

C. Lawson, R. Hansen, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1979).

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).

J. E. Harvey, Hughes Danburg Optical Systems, private communication.

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

R. E. Wagner, “Imagery Utilizing Multiple Focal Planes,” Ph.D. dissertation (University of Arizona, 1976).

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Figures (6)

Fig. 1
Fig. 1

Ideal low pass filter model of an adaptive optic with a cutoff frequency of the reciprocal of twice the actuator spacing T (attributed to Harvey and Callahan3).

Fig. 2
Fig. 2

H matrix is generally rectangular with the columns representing each actuator influence function and the rows representing points on the wavefront. The influence functions are assumed to be identical so that the columns have shifted versions of the same influence function. The S matrix is the system response matrix and the region affected by each actuator is indicated by a square with a width twice that of the influence function. The inner regions of the H and S matrices, indicated by the dashed lines, is negligibly affected by the edges of the array and is representative of the properties of matrices of infinite extent.

Fig. 3
Fig. 3

Orthogonalized Gaussian influence functions with a coupling at nearest neighbors of (a) 5%, (c) 15%, and (e) 50%. The frequency domain orthogonalized influence function with a coupling of (b) 5%, (d) 15%, and (f) 50%. Note that higher coupling, corresponding to a stiffer mirror surface, produces more zero crossings in the orthogonalized function and a more nearly rectangular spectrum.

Fig. 4
Fig. 4

Merit function vs coupling for a Gaussian influence function. Note that higher coupling, corresponding to a stiffer mirror surface, produces a better match to the ideal filter. As the coupling approaches 100%, the merit function goes to 1, and the spectrum of the orthogonalized influence function approaches the ideal low pass filter.

Fig. 5
Fig. 5

Spatial and frequency domain representations of the functional steps in adaptive optic modeling. The incident wavefront w(x) is shown in (a). The orthogonalized influence function m(x) is shown in (b). Part (c) represents the convolution of m′(x) and w(x). The effects of the finite actuator spacing are demonstrated in (d), and the least-squares fit wavefront is shown in (e). The residual wavefront that cannot be modeled by the adaptive element is shown in (f). This figure is meant to be qualitative in nature.

Fig. 6
Fig. 6

Surface contour representation of the system matrix S for twelve actuators with Gaussian influence functions and 37% coupling.

Equations (45)

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w m ( i ) = - a ( j ) m j ( i ) .
w m = H a ,
H ( i , j ) = m j ( i ) .
= w - w m .
σ 2 = t ,
a = ( H t H ) - 1 H t w .
w m = H ( H t H ) - 1 H t w .
S = H ( H t H ) - 1 H t ,
w m = S w .
m j ( i ) = m ( x i - x j ) ,
S = H H t ,
w m = H H t w ,
H ( i , j ) = m j ( i ) = m ( x i - x j ) .
w m ( x ) = m ( x ) * { [ m ( x ) * w ( x ) ] comb ( x / T ) } ,
W m ( ξ ) = M ( ξ ) [ M ( ξ ) W ( ξ ) * T comb ( T ξ ) ] ,
W m ( ξ ) = M ( ξ ) W ( ξ ) .
m ( x ) = sinc ( x T ) = sin ( π x T ) π x T .
m ( x ) = exp [ - ln ( c ) ( x T ) 2 ]
q = - 1 2 T + 1 2 T M ( ξ ) d ξ .
S δ ( x ) = m ( x ) * m ( x ) ,
u 0 = m m .
u = u 0 + f 1 u 1 u 0 + f 1 u 1 ,
u = u 0 ( x - T ) + u 0 ( x + T ) u 0 ( x - T ) + u 0 ( x + T ) ,
( u 0 + f 1 u 1 ) ( u 1 + f 1 u 0 ) = 0 ,
f 1 = - 1 1 ( 1 - 1 - 1 2 ) ,
m = u 0 + f 1 u 1 .
u = u 0 + f j u j u 0 + f j u j ,
u j = u 0 ( x - j T ) + u 0 ( x + j T ) u 0 ( x - j T ) + u 0 ( x + j T ) ,
f j = - 1 j ( 1 - 1 - j 2 ) ,
j = u 0 u j .
j = u 0 ( i ) u j ( i ) .
m ( x ) = m ( x ) + f 1 [ m ( x + T ) + m ( x - T ) ] + f 2 [ m ( x + 2 T ) + m ( x - 2 T ) ] +
a ( j ) = f k - j ,
f 0 = 1 , f - 1 = f 1 , f - 2 = f 2 , etc .
m = H a .
w m = H a ,
a = H t w ,
a ( j ) = H ( j , i ) w ( i ) ,
a ( j ) = m ( x j - x i ) w ( x i ) ,
a ( j ) = [ m ( x j - x i ) w ( x i ) d x i ] comb ( x / T ) ,
a ( j ) = [ m ( x ) * w ( x ) ] comb ( x / T ) .
w m ( i ) = H ( i , j ) a ( j ) ,
w m ( i ) = [ m ( x i - x j ) a ( x j ) d x j ] comb ( x Δ I ) ,
w m ( i ) = [ m ( x ) * a ( x ) ] comb ( x Δ I ) .
w m ( x ) = m ( x ) * { [ m ( x ) * w ( x ) ] comb ( x / T ) } ,

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