Abstract

A technique is described wherein the pupil phase profile is obtained from the image irradiance or point spread function. Results of numerical examples are shown that illustrate the procedure on spread functions with large aberrations and on irradiance measurements containing noise.

© 1977 Optical Society of America

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References

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  1. Richard A. Muller and Andrew Buffington, “Real-time correction of atmospherically degraded telescope images through image sharpening,” J. Opt. Soc. Am. 64, 1200–1210 (1974).
    [Crossref]
  2. Joseph W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 85.
  3. Harry C. Andrews, Computer Techniques in Image Processing (Academic, New York, 1970), p. 14.
  4. curfit is available from Cybernetic Service Company, 3508 Fifth Ave., Pittsburgh, Pa. 15213.

1974 (1)

Andrews, Harry C.

Harry C. Andrews, Computer Techniques in Image Processing (Academic, New York, 1970), p. 14.

Buffington, Andrew

Goodman, Joseph W.

Joseph W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 85.

Muller, Richard A.

J. Opt. Soc. Am. (1)

Other (3)

Joseph W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968), p. 85.

Harry C. Andrews, Computer Techniques in Image Processing (Academic, New York, 1970), p. 14.

curfit is available from Cybernetic Service Company, 3508 Fifth Ave., Pittsburgh, Pa. 15213.

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Figures (3)

FIG. 1
FIG. 1

Irradiance distribution of a focused wave front having 0.05 λ aberration on each of the first nine normalized Zernike polynomials. The wave front had an rms wave-front error of 0.15 λ with a 1.01. λ peak-to-peak variation.

FIG. 2
FIG. 2

Irradiance distribution of a focused wave front having 0.10 λ aberration on each of the first nine normalized Zernike polynomials. The wave front had an rms wave-front error of 0.30 λ with 2.02 λ peak-to-peak variation.

FIG. 3
FIG. 3

Irradiance distribution of focused wave front having no aberrations.

Tables (4)

Tables Icon

TABLE I Results of the least-squares fit algorithm for determining phase aberration coefficients from the irradiance pattern of Fig. 1. Coefficients are in units of λ.

Tables Icon

TABLE II Results of the least-squares fit algorithm for determining the phase aberration coefficients using the data shown in Fig. 1, except that a random noise term has been added to each point.

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TABLE III Results of the least-squares fit algorithm for determining phase aberration coefficients from the irradiance pattern of Fig. 2.

Tables Icon

TABLE IV Results of the least-squares fit algorithm for determining phase aberration coefficients from the irradiance pattern of Fig. 2. The starting point was 0.06 on all parameters.

Equations (9)

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U p ( x , y , a ) = A ( x , y ) exp ( i 2 π λ W ( x , y , a ) ) ,
U l ( x , y , a ) = exp [ ( i π / λ f ) ( x 2 + y 2 ) ] i λ f U p ( x , y , a ) × exp [ ( - i 2 π / λ f ) ( x x + y y ) ] d x d y .
I ( x , y , a ) = U l * U l ,
I m ( x , y ) = I ( x , y , a ) + ,
L ( a ) = C exp ( - x y 1 2 2 ( x , y , a ) / σ 2 ( x , y ) ) .
S = 1 2 i ( I M i - I i ( a ) ) 2 / σ i 2 ,
S a j = 0 ,             j = 1 ,     2 , , M
W = j a j F j ( x , y ) .
W ( x , y , a ) = 2 a 1 x + 2 a 2 y + 3 a 3 ( 2 r 2 - 1 ) + 6 a 4 ( x 2 - y 2 ) + 2 6 a 5 x y + 8 a 6 ( x 2 - 3 y 2 ) x + 8 a 7 ( y 2 - 3 x 2 ) y + 8 a 8 ( 3 r 2 - 2 ) x + 8 a 9 ( 3 r 2 - 2 ) y ,