Abstract

Full aperture optical system wave-front accuracy obtainable from single or. multiple subaperture wave-front measurements is analytically determined and illustrated with numerical examples. Insight gained from the analytic derivation and confirmed by the examples shows the effect of subaperture size, placement, and accuracy. The analysis also demonstrates that accurate full aperture aberrations (except tilt) can be estimated even with large uncertainty in relative subaperture tilt.

© 1984 Optical Society of America

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References

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  1. J. G. Thunen, O. Y. Kwon, “Full Aperture Testing With Subaperture,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 19 (1983).
  2. O. Y. Kwon, J. G. Thunen, “LODE: Subaperture Optical Testing,” LMSC D811929, 6Nov.1981.
  3. P. S. Maybeck, Stochastic Models, Estimation and Control, Vol. 1 (Academic, New York, 1979).

1983

J. G. Thunen, O. Y. Kwon, “Full Aperture Testing With Subaperture,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 19 (1983).

1981

O. Y. Kwon, J. G. Thunen, “LODE: Subaperture Optical Testing,” LMSC D811929, 6Nov.1981.

Kwon, O. Y.

J. G. Thunen, O. Y. Kwon, “Full Aperture Testing With Subaperture,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 19 (1983).

O. Y. Kwon, J. G. Thunen, “LODE: Subaperture Optical Testing,” LMSC D811929, 6Nov.1981.

Maybeck, P. S.

P. S. Maybeck, Stochastic Models, Estimation and Control, Vol. 1 (Academic, New York, 1979).

Thunen, J. G.

J. G. Thunen, O. Y. Kwon, “Full Aperture Testing With Subaperture,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 19 (1983).

O. Y. Kwon, J. G. Thunen, “LODE: Subaperture Optical Testing,” LMSC D811929, 6Nov.1981.

LMSC D811929

O. Y. Kwon, J. G. Thunen, “LODE: Subaperture Optical Testing,” LMSC D811929, 6Nov.1981.

Proc. Soc. Photo-Opt. Instrum. Eng.

J. G. Thunen, O. Y. Kwon, “Full Aperture Testing With Subaperture,” Proc. Soc. Photo-Opt. Instrum. Eng. 351, 19 (1983).

Other

P. S. Maybeck, Stochastic Models, Estimation and Control, Vol. 1 (Academic, New York, 1979).

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Figures (5)

Fig. 1
Fig. 1

Geometric relationship of a subaperture within the full aperture and definition of variables.

Fig. 2
Fig. 2

Placement of seven subapertures within the full aperture for the multiple subaperture optical test concept.

Fig. 3
Fig. 3

Effect of subaperture tilt measurement error on full aperture test accuracy.

Fig. 4
Fig. 4

Effect of subaperture tilt measurement error on full aperture test accuracy.

Fig. 5
Fig. 5

Full aperture wave-front estimation accuracy degradation as a function of subaperture radius for a single central subaperture.

Tables (10)

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Table I Zernike Mode Definitions Used for Sensitivity Matrix Calculations (Unit Disk Normalization)

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Table II Subaperture Zernike Mode Sensitivity Coefficients for Full Aperature Zernike Mode Aberrations

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Table III Subaperture Placement Parameters for the Multiple Subaperture Flat Optical Test Configuration

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Table IV Full Aperture Zernike Mode inverse Covariance Matrix

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Table V Symbolic Representation for the Nonzero Elements of the Matrix HTWH

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Table VI Symbolic Representation for the Full Aperture Zernike Coefficient Covariance Matrix σ A 2 = ( H T WH ) - 1

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Table VII Full Aperture Zernike Mode Estimation Accuracy as a Function of Subaperture Tilt Measurement Error (Ten-Full Aperture Modes Estimated Using Seven Subapertures with Ten Modes Each, σai = 0.006λ i > 3)

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Table VIII Full Aperture Zernike Mode Estimation Accuracy as a Function of Subaperture Tilt Measurement Error (Ten-Full Aperture Modes Estimated Using Seven Subapertures with Five Modes Each, σai = 0.006λ i > 3)

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Table IX Full Aperture Zernike Mode Estimation Accuracy as a Function of Subaperture Tilt Measurement Error (Ten-Full Aperture Modes Estimated Using Seven Subapertures with Ten Modes Each, σ a i = 0.006 λ / 10 i > 3)

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Table X (HTWT)−1 for the Single Central Subaperture

Equations (28)

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A = ( A 1 , A 2 , A 3 , A N ) T , full aperture Zernike coefficients , a = ( a 1 , a 2 , a 3 , a m ) T , subaperture Zernike coefficients ,
R ρ cos θ = h cos α + r ρ cos θ
R ρ sin θ = h sin α + r ρ sin θ ,
Z 1 = 2 ρ cos θ = 2 R · R ρ cos θ = 2 π ( h cos α + r ρ cos θ ) ,
Z 1 = 2 h R cos α + ( r R ) 2 ρ cos θ .
Z 2 = 2 ρ sin θ = 2 h R sin α + ( r R ) 2 ρ sin θ .
( R ρ ) 2 = ( R ρ ) 2 ( cos 2 θ + sin 2 θ ) = h 2 + r 2 ρ 2 + 2 h r ρ cos ( θ - α ) .
ρ 2 = ( h R ) 2 + ( r R ) 2 ρ 2 + ( h R ) ( r R ) ( cos α 2 ρ cos θ + sin α 2 ρ sin θ ) .
Z 3 = 3 ( 2 ρ 2 - 1 ) = 3 [ 2 ( h R ) 2 - 1 + ( r R ) 2 ] + ( r R ) 2 3 ( 2 ρ 2 - 1 ) + 2 3 ( h R ) ( r R ) ( cos α 2 ρ cos θ + sin α 2 ρ sin θ ) .
Z 3 = 3 [ 2 ( h R ) 2 - 1 + ( r R ) 2 ] + ( r R ) 2 Z 3 + 2 3 ( h R ) ( r R ) [ cos α Z 1 + sin α Z 2 ] .
S k A = S ( α k , h k ) A = a ( k ) ,
ɛ A = b - H A , b = [ a 1 a 2 a 3 a 7 ]             [ S 1 S 2 S 3 S 7 ] .
J = ɛ A T W ɛ A = ( b - H A ) T W ( b - H A ) .
J A = A [ ( b - H A ) T W ( b - H A ) ] = - 2 ( b - H A ) T WH = 0 ,
H T WH A = H T W b .
A = ( H T WH ) - 1 H T W b = Y b ,
Y = ( H T WH ) - 1 H T W .
W = ( σ a 2 ) - 1 ,
σ a 2 = [ σ a 2 ( 1 ) 0 0 0 σ a 2 ( 2 ) 0 0 0 0 σ a 2 ( 7 ) ]             Note : superscripts denote subaperture number .
σ A 2 = Y σ a 2 Y T = ( H T WH ) - 1 H T W σ a 2 W T H ( H T WH ) - 1 = ( H T WH ) - 1
σ a 2 ( i ) = [ σ t 2 0 σ t 2 σ 2 0 σ 2 ] ,
H T WH = k = 1 7 S T ( α k , h k ) σ a 2 S ( α k , h k ) .
cov ( A 4 ) = 1 C = ( 144 81 1 σ t 2 + 7 81 1 σ 2 ) - 1 = 81 σ t 2 σ 2 144 σ 2 + 7 σ t 2 .
[ σ A 2 ( 3 × 3 ) ] - 1 = H T WH 3 × 3 = [ 7 9 1 σ t 2 0 0 0 7 9 1 σ t 2 0 0 0 32 9 1 σ t 2 + 7 81 1 σ 2 ] .
cov ( A 3 ) = [ 32 9 1 σ t 2 + 7 81 1 σ 2 ] - 1 = 81 σ t 2 σ 2 32 ( 9 ) σ 2 + 7 σ t 2 .
[ cos ( A 3 ) ] 1 / 2 σ t 2 = 81 7 σ = 3.4 σ = 0.02 λ at ( σ = 0.006 λ ) .
[ cov ( A 3 ) ] 1 / 2 σ t = 0.006 λ = [ 32 9 1 ( 0.0003 λ ) 2 + 7 81 1 ( 0.006 λ ) 2 ] - 1 / 2 = 0.00016 λ ,
[ σ A i 2 σ a i 2 ] 1 / 2 = 1 10 { 2 ( R r ) 2 + 16 ( R r ) 2 [ ( R r ) 2 - 1 ] 2 + 3 ( R r ) 4 + 4 ( R r ) 6 + ( R r ) 8 } .

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