Abstract

The actuator influence functions of a typical deformable mirror are expanded in a Zernike polynomial decomposition. This expansion is then extended to a matrix formalism that describes the modal operation of the mirror. The size of the aperture over which the Zernikes are defined affects the accuracy of the expansion. The optimum size of this aperture is found by minimizing the variance of the wave-front error.

© 1993 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), Sec. 9.2 and App. VII.
  2. J. Y. Wang, D. E. Silva, “Wavefront interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  3. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981).
    [CrossRef]
  4. R. J. Noll, “Zernike polynomial and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  5. D. E. Novoseller, “Zernike-ordered adaptive correction of thermal blooming,” J. Opt. Soc. Am. A 5, 1937–1942 (1988).
    [CrossRef]
  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. R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991), and references therein.
  8. T. R. O'Meara, “Theory of multidither adaptive optical systems operating with zonal control of deformable mirrors,” J. Opt. Soc. Am. 67, 318–325 (1977).
    [CrossRef]
  9. K. E. Moore, G. N. Lawrence, “Zonal model of an adaptive mirror,” Appl. Opt. 29, 4622–4628 (1990).
    [CrossRef] [PubMed]
  10. J. E. Harvey, G. M. Callahan, “Wavefront error compensation capabilities of multi-actuator deformable mirrors,” in Adaptive Optical Components I, S. Holly, L. James, eds., Proc. Soc. Photo-Opt. Instrum. Eng.141, 50–57 (1978).
  11. E. S. Claflin, N. Bareket, “Configuring an electrostatic membrane mirror by least-squares fitting with analytically derived influence functions,” J. Opt. Soc. Am. A 3, 1833–1839 (1986).
    [CrossRef]

1990 (1)

1988 (1)

1986 (1)

1982 (1)

1981 (1)

1980 (1)

1977 (1)

1976 (1)

Bareket, N.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), Sec. 9.2 and App. VII.

Callahan, G. M.

J. E. Harvey, G. M. Callahan, “Wavefront error compensation capabilities of multi-actuator deformable mirrors,” in Adaptive Optical Components I, S. Holly, L. James, eds., Proc. Soc. Photo-Opt. Instrum. Eng.141, 50–57 (1978).

Claflin, E. S.

Harvey, J. E.

J. E. Harvey, G. M. Callahan, “Wavefront error compensation capabilities of multi-actuator deformable mirrors,” in Adaptive Optical Components I, S. Holly, L. James, eds., Proc. Soc. Photo-Opt. Instrum. Eng.141, 50–57 (1978).

Lawrence, G. N.

Mahajan, V. N.

Moore, K. E.

Noll, R. J.

Novoseller, D. E.

O'Meara, T. R.

Silva, D. E.

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, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991), and references therein.

Wang, J. Y.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), Sec. 9.2 and App. VII.

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (2)

Other (3)

J. E. Harvey, G. M. Callahan, “Wavefront error compensation capabilities of multi-actuator deformable mirrors,” in Adaptive Optical Components I, S. Holly, L. James, eds., Proc. Soc. Photo-Opt. Instrum. Eng.141, 50–57 (1978).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1980), Sec. 9.2 and App. VII.

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991), and references therein.

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Figures (9)

Fig. 1
Fig. 1

Arrangement of the 21-actuator deformable mirror. The total radius of the mirror is approximately four times the distance between actuators. We have also represented the Gaussian size of the influence function Ω, (where the influence function reaches a value of 1/e the maximum value). Three different aperture sizes have been plotted: solid curve, s = 1.0; dashed curve, s = 1.5; and dotted curve, s = 2.0.

Fig. 2
Fig. 2

Three-dimensional representation of the matrix relation ωm = a. Each slice of the matrix corresponds to the influence function of each actuator. Therefore the whole stack contains J slices.

Fig. 3
Fig. 3

Three-dimensional representation of the modal decomposition = Z C. Two stacks of a different number of slices are related by means of the matrix C. The transposed version of this matrix relation is also represented.

Fig. 4
Fig. 4

Gaussian-type influence functions as represented by a finite set of Zernike polynomials. The representation is more accurate as more polynomials are used, (a), (b), (c) show the restoration of the influence function using 45 (up to the eighth degree, n = 8), 66 (10th degree, n = 10), and 91 (12th degree, n = 12) Zernike polynomials, respectively; (d) shows the actual influence function corresponding to the central actuator. These figures correspond to a scaling factor s = 1. If the scaling factor increases, the influence function becomes wider and the fitting will be better.

Fig. 5
Fig. 5

Total variance of the whole actuator array σinf2 is represented as a function of the scale factor s for several decompositions including 6th- to 12th-degree polynomials. The parameter n is the maximum degree of the polynomial used in the expansion, which varies from 6 to 12.

Fig. 6
Fig. 6

When the circular aperture where the Zernikes are expanded is reduced, some portions of the outer influence functions are excluded. We have represented this decreasing in the characterization of the mirror in terms of the equivalent-area number of influence functions included inside the aperture.

Fig. 7
Fig. 7

Variance of the wave-front-error vector between the actual Zernikes and the restoration by the mirror σwf2 is given in this figure for the Zernike polynomials for the modal driving procedure. This figure also represents qualitatively the behavior of the variance when the mirror is driven in the zonal mode.

Fig. 8
Fig. 8

Value of the scaling factor at which the minimum variance is reached is plotted in this diagram as a function of the polynomial index. The different symbols denote the different method of calculating the fitting: diamonds are the zonal representation; pluses are the modal representation.

Fig. 9
Fig. 9

DM compared with several Zernike polynomials whose radial numbers n, azimuthal numbers m, polynomial indices k are noted for each row. The first column is the polynomial that the DM represents. The second column corresponds to modal driving. The zonal representation has a similar appearance and is not shown separately. The third column is obtained from the local value approximation. A scale factor s = 1.5 is used throughout. The values for σwf2 in each row are as follows from top to bottom: 0.1905 (modal), 0.1907 (zonal), 0.2085 (LVA); 0.1734 (modal), 0.1734 (zonal), 0.2118 (LVA); 0.2083 (modal), 0.2085 (zonal), 0.2477 (LVA); 0.1970 (modal), 0.1964 (zonal), 0.1260 (LVA); 0.1879 (modal), 0.1879 (zonal), 0.1876 (LVA).

Equations (27)

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ω m ( x , y ) = j = 1 J m j ( x , y ) a j ,
ω m = a ,
= ω ω m .
σ wf 2 = t ,
a = ( t ) 1 t ω ,
ω m = ( t ) 1 t ω .
ω m = t ω ,
m j ( x , y ) = k = 0 K 1 c jk Z k ( x , y ) ,
c jk = 1 π S m j ( x , y ) Z k ( x , y ) d x d y ,
= Z C ,
ω m = Z C a .
ω = Z I ,
ω m = Z C ( C t Z t Z C ) 1 C t Z t Z I = Z C ( C t C ) 1 C t I ,
a = ( C t C ) 1 C t I .
I = C ( t ) 1 t ω .
I = C a .
σ wf ( modal ) 2 = I t [ C ( C t C ) 1 C t ] I ,
σ wf ( zonal ) 2 = ω t [ ( t ) 1 t ] ω .
s = L r aperture ,
σ inf 2 = j = 1 J σ j 2 ,
j = m j k = 0 K 1 c jk Z k ,
σ j 2 = j t j .
c jk = 1 π δ ( x x j , y y j ) [ m ( x , y ) Z k ( x , y ) ] d x d y ,
Z k m j Z K ,
c jk = Z k ( x j , y j ) .
Z k ω m = j = 1 J c jk m j ,
Z { c jk } t .

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