Abstract

A method for obtaining the optical transfer function from the edge spread function is described. The method involves the inversion of a Fredholm integral equation of the first kind using sampling-theory concepts. Typical examples for both symmetric and asymmetric edge spread functions are studied and compared with the known solutions.

© 1965 Optical Society of America

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References

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  1. F. Scott, R. M. Scott, and R. V. Shack, Phot. Sci. Engr. 7, 345 (1963).
  2. R. Barakat and A. Houston (to be published).
  3. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, England, 1952), p. 188, 4th ed.
  4. R. Barakat, J. Opt. Soc. Am. 54, 920 (1964).
    [Crossref]
  5. It is interesting to note that S˜ is a Toeplitz matrix; that is, its elements (2.12) depend on the difference of m and n. Attempts were made to employ the theory of such matrices but unfortunately to no avail. The standard work on these forms is: U. Grenander and G. Szego, Toeplitz Forms and Their Applications (University of California Press, Berkeley, California, 1958).
  6. K. L. Nielsen, Methods in Numerical Analysis (The Macmillan Co., Inc., New York, 1956), Chap. 7.
  7. R. Barakat, J. Opt. Soc. Am. 54, 38 (1964).
    [Crossref]
  8. R. Barakat, J. Opt. Soc. Am. 52, 985 (1962).
    [Crossref]
  9. R. Barakat and M. V. Morello, J. Opt. Soc. Am. 52, 992 (1962).
    [Crossref]

1964 (2)

1963 (1)

F. Scott, R. M. Scott, and R. V. Shack, Phot. Sci. Engr. 7, 345 (1963).

1962 (2)

Barakat, R.

Grenander, U.

It is interesting to note that S˜ is a Toeplitz matrix; that is, its elements (2.12) depend on the difference of m and n. Attempts were made to employ the theory of such matrices but unfortunately to no avail. The standard work on these forms is: U. Grenander and G. Szego, Toeplitz Forms and Their Applications (University of California Press, Berkeley, California, 1958).

Houston, A.

R. Barakat and A. Houston (to be published).

Morello, M. V.

Nielsen, K. L.

K. L. Nielsen, Methods in Numerical Analysis (The Macmillan Co., Inc., New York, 1956), Chap. 7.

Scott, F.

F. Scott, R. M. Scott, and R. V. Shack, Phot. Sci. Engr. 7, 345 (1963).

Scott, R. M.

F. Scott, R. M. Scott, and R. V. Shack, Phot. Sci. Engr. 7, 345 (1963).

Shack, R. V.

F. Scott, R. M. Scott, and R. V. Shack, Phot. Sci. Engr. 7, 345 (1963).

Szego, G.

It is interesting to note that S˜ is a Toeplitz matrix; that is, its elements (2.12) depend on the difference of m and n. Attempts were made to employ the theory of such matrices but unfortunately to no avail. The standard work on these forms is: U. Grenander and G. Szego, Toeplitz Forms and Their Applications (University of California Press, Berkeley, California, 1958).

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, England, 1952), p. 188, 4th ed.

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, England, 1952), p. 188, 4th ed.

J. Opt. Soc. Am. (4)

Phot. Sci. Engr. (1)

F. Scott, R. M. Scott, and R. V. Shack, Phot. Sci. Engr. 7, 345 (1963).

Other (4)

R. Barakat and A. Houston (to be published).

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, England, 1952), p. 188, 4th ed.

It is interesting to note that S˜ is a Toeplitz matrix; that is, its elements (2.12) depend on the difference of m and n. Attempts were made to employ the theory of such matrices but unfortunately to no avail. The standard work on these forms is: U. Grenander and G. Szego, Toeplitz Forms and Their Applications (University of California Press, Berkeley, California, 1958).

K. L. Nielsen, Methods in Numerical Analysis (The Macmillan Co., Inc., New York, 1956), Chap. 7.

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Figures (5)

Fig. 1
Fig. 1

Comparison of exact transfer function (solid line) with the transfer function obtained from the edge spread function (dotted line) for an aberration-free system defocused W2 = 0.6λ.

Fig. 2
Fig. 2

Edge spread function in the paraxial plane for a system having third-order coma of amount W131 = 0.75λ in the azimuth ϕ = 45°. The dotted line represents the geometric edge.

Fig. 3
Fig. 3

Comparison of exact transfer-function modulus (–) with the transfer-function modulus obtained from the edge spread function with reference to geometric edge (– · –), and with reference to position marked by arrow (– – –). The system corresponds to one with third-order coma W131 = 0.75λ in the azimuth ϕ = 45°.

Fig. 4
Fig. 4

Comparison of exact transfer-function modulus (–) with the transfer-function modulus obtained from the randomly perturbed edge spread function with reference to geometric edge (– · –), and with reference to position marked by arrow (– – –). The system corresponds to one with third-order coma W131 = 0.75λ in the azimuth ϕ = 45°.

Fig. 5
Fig. 5

Comparison of exact transfer function phase with the transfer function phase obtained from the edge spread function. See text for description of curves.

Tables (1)

Tables Icon

Table I Comparison of transfer functions as obtained by inverting edge spread function with the exact value. The results refer to a circular aperture with optimum-balanced, fifth-order spherical aberration of amount W6 = 4λ. Thirty sampling points and 19Bn coefficients were employed. Inversion-1 refers to the use of five-place accuracy of E(v), inversion II to three-place accuracy of E(v).

Equations (24)

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0 ( v ) = { 0 v < 0 1 v > 0 ,
0 E ( v ) 1 ;             E ( - ) = 0 ;             E ( + ) = 1.
v = ( π / λ F ) z ,
Ω = ω / 2 λ F             ( - 2 ω 2 ) , .
E ( v ) = 1 2 + 1 2 π i - 2 2 T ( ω ) ω e i v ω d ω .
τ ( ν ) d E d v = 1 2 π - 2 2 T ( ω ) e i v ω d ω ,
T ( ω ) = n = - B n e ( i n π / 2 ) ω             ( - 2 ω 2 ) ,
( v ) = 1 2 π n - = B n [ Si ( 2 v + n π ) - Si ( - 2 v - n π ) ] = 1 π n = - B n Si ( 2 v + n π ) ,
P - t t sin x x d x = 2 Si ( t ) ,
P - t t cos x x d x = 0.
T ( ω ) = T r ( ω ) + i T i ( ω ) ,
T r ( ω ) = T r ( - ω ) = B 0 + n = 1 ( B n + B - n ) cos n π 2 ω ,
T i ( ω ) = - T i ( - ω ) = n = 1 ( B n - B - n ) sin n π 2 ω .
v m = m π / 2             m = 0 , ± 1 , ± 2 , ± 3 , .
( m π 2 ) = 1 π n = - B n S i [ π ( m + n ) ]             m = 0 , ± 1 , ± 2 , ,
S ˜ B ˜ = E ˜ ,
E ˜ = ( ( - 2 ) ( - 1 ) ( 0 ) ( + 1 ) ( + 2 ) ) ;             B ˜ = ( B - 2 B - 1 B 0 B + 1 B + 2 ) ;
S ˜ = ( $ - 4 $ - 3 $ - 2 $ - 1 $ 0 $ - 3 $ - 2 $ - 1 $ 0 $ 1 $ - 2 $ - 1 $ 0 $ 1 $ 2 $ - 1 $ 0 $ 1 $ 2 $ 3 $ 0 $ 1 $ 2 $ 3 $ 4 ) .
$ n + m ( 1 / π ) S i [ π ( n + m ) ] ,             n , m = 0 , ± 1 , ± 2 , .
T ( 0 ) = 1 = n = - B n ,
T ( 2 ) = 0 = n = - ( - 1 ) n B n .
W ( ρ ) = W 6 ( ρ 6 - 3 2 ρ 4 + 3 5 ρ 2 ) ,
E ( 0 ) = 1 2 + 1 π 0 2 T i ( ω ) ω d ω .
E ( v + Δ v ) = 1 2 + 1 2 π i - 2 2 T ( ω ) ω e i ( v + Δ v ) ω d ω = 1 2 + 1 2 π i - 2 2 T ( ω ) ω e i v ω d ω ,