Abstract

The transfer function is expressed as a trigonometric series whose coefficients are proportional to the sampled values of the edge response function. The series may be modified by means of added terms to take into account the known asymptotic behavior of the edge response function. Numerical results are given for pure defocusing.

© 1965 Optical Society of America

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References

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  1. R. Barakat, J. Opt. Soc. Am. 54, 920 (1964).
    [Crossref]
  2. The area under τ(x), which is the total energy, equals unity.
  3. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., London1959), pp. 369, 484.
  4. G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).
    [Crossref]
  5. See Ref. 1.
  6. M. Born and E. Wolf, Ref. 3, p. 483.
  7. M. Bora and E. Wolf, Ref. 3, p. 752.
  8. This formula is due to R. Barakat (private communication).
  9. E. C. Titchmarsh, The Theory of Functions (Oxford University Press, New York, 1939) 2nd ed., p. 434.
  10. G. H. Hardy, Divergent Series (Oxford University Press, New York, 1949).
  11. G. H. Hardy, Ref. 10, p. 101.
  12. R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).
    [Crossref]
  13. The first two expressions in (26) are obtained by summing the complex exponential, einu, according to the (C,1) definition. The remaining relationships in (26) can be found in R. Courant, Differential and Integral Calculus (Blackie & Son, Ltd., London and Glasgow, 1958),Vol. I, p. 446.
  14. R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 (1964).
    [Crossref]
  15. It is easily seen that (28) gives the values T1(0) = 1, and T1(1/2∊) = 0.

1964 (2)

1963 (1)

1955 (1)

Barakat, R.

Bora, M.

M. Bora and E. Wolf, Ref. 3, p. 752.

Born, M.

M. Born and E. Wolf, Ref. 3, p. 483.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., London1959), pp. 369, 484.

Courant, R.

The first two expressions in (26) are obtained by summing the complex exponential, einu, according to the (C,1) definition. The remaining relationships in (26) can be found in R. Courant, Differential and Integral Calculus (Blackie & Son, Ltd., London and Glasgow, 1958),Vol. I, p. 446.

Hardy, G. H.

G. H. Hardy, Divergent Series (Oxford University Press, New York, 1949).

G. H. Hardy, Ref. 10, p. 101.

Houston, A.

Titchmarsh, E. C.

E. C. Titchmarsh, The Theory of Functions (Oxford University Press, New York, 1939) 2nd ed., p. 434.

Toraldo di Francia, G.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., London1959), pp. 369, 484.

M. Born and E. Wolf, Ref. 3, p. 483.

M. Bora and E. Wolf, Ref. 3, p. 752.

J. Opt. Soc. Am. (4)

Other (11)

It is easily seen that (28) gives the values T1(0) = 1, and T1(1/2∊) = 0.

The first two expressions in (26) are obtained by summing the complex exponential, einu, according to the (C,1) definition. The remaining relationships in (26) can be found in R. Courant, Differential and Integral Calculus (Blackie & Son, Ltd., London and Glasgow, 1958),Vol. I, p. 446.

The area under τ(x), which is the total energy, equals unity.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Inc., London1959), pp. 369, 484.

See Ref. 1.

M. Born and E. Wolf, Ref. 3, p. 483.

M. Bora and E. Wolf, Ref. 3, p. 752.

This formula is due to R. Barakat (private communication).

E. C. Titchmarsh, The Theory of Functions (Oxford University Press, New York, 1939) 2nd ed., p. 434.

G. H. Hardy, Divergent Series (Oxford University Press, New York, 1949).

G. H. Hardy, Ref. 10, p. 101.

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Figures (5)

F. 1
F. 1

(a) Simulated line spread function. (b) Edge response function corresponding to (a).

F. 2
F. 2

Transfer functions for perfect systems with circular aperture: (———) 0.2 wavelength defocusing, (– – –) 1.0 wavelength defocusing, (–·–) 1.2 wavelength defocusing.

F. 3
F. 3

Error in modulation for transfer function calculated from edge response function, case 1, Table I: (– –) 0.2 wavelength defocusing, (– – –) 1.0 wavelength defocusing, (– –) 1.2 wavelength defocusing.

F. 4
F. 4

Error in modulation for transfer function calculated from edge response function, case II, Table I: (——) 0.2 wavelength defocusing, (– – –) 1.0 wavelength defocusing, (– –) 1.2 wavelength defocusing.

F. 5
F. 5

Error in modulation for transfer function calculated from edge response function, case III, Table I: (——) 0.2 wavelength defocusing, (– – –) 1.0 wavelength defocusing, (– –) 1.2 wavelength defocusing. (Note that the modulation error scale is 1/10 that of Figs. 3 and 4.)

Tables (1)

Tables Icon

Table I Assumed asymptotic behavior for E1(n∊) and E2(n∊) used in test calculations.

Equations (47)

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T ( f ) = τ ( x ) e i 2 π f x d x .
x i + 1 = x i = 1 / 2 f c =
τ ( x ) = n = τ ( n ) sinc π ( x n ) .
T ( f ) = { T 1 ( f ) + i T 2 ( f ) ( | f | f c ) 0 ( | f | > f c ) ,
T 1 ( f ) = τ e ( 0 ) + 2 n = 1 τ e ( n ) cos 2 π n f , T 2 ( f ) = 2 n = 1 τ 0 ( n ) sin 2 π n f ,
τ e ( n ) = [ τ ( n ) + τ ( n ) ] / 2 , τ 0 ( n ) = [ τ ( n ) τ ( n ) ] / 2 .
E ( x ) = x τ ( ξ ) d ξ ,
τ ( x ) = d E ( x ) / d x .
E ( x ) = 1 2 [ 1 + L i ( x ) ] , E ( x ) = 1 2 [ 1 L i ( x ) ] .
τ ( x ) = f c f c T ( f ) e i 2 π f x d f .
E ( x ) = T 1 ( 0 ) 2 + f c f c [ T 1 ( f ) sin 2 π f x 2 π f + T 2 ( f ) cos 2 π f x 2 π f ] d f .
E ( x ) = 1 2 + π n = τ ( n ) Si [ π ( x n ) ] .
Si ( x ) = 0 x sinc ( u ) d u
T ( f ) = d E ( x ) d x e i 2 π f x d x .
lim n S 1 + S 2 + + S n n = S ,
S ( x ) = x x a ( t ) d t
lim x 1 x x x S ( t ) d t = lim x x x ( x t ) x a ( t ) d t = S ,
a ( x ) d x S ( C , 1 ) .
T ( f ) = lim x x x ( x ξ ) x d E ( ξ ) d ξ e i 2 π f ξ d ξ = i 2 π f lim x x x ( x ξ ) x E ( ξ ) e i 2 π f ξ d ξ .
T ( f ) = i 2 π f S ( f ) ( C , 1 ) .
lim f 0 T ( f )
E ( x ) = n = E ( n ) sinc π ( x n ) .
n = sinc π ( ξ n ) .
n = 0 ( 1 ) n + 1 sin ( π ξ / ) π ( n ξ / ) + n = 1 ( 1 ) n sin ( π ξ / ) π ( n + ξ / ) .
| n = M N ( 1 ) n sin ( π ξ ξ ) | 1
T ( f ) = { n = i 2 π f E ( n ) e i 2 π f n ( | f | 1 / 2 = f c ) , 0 ( | f | > f c ) .
τ ( x ) = π n = [ cos π ( x / n ) π ( x / n ) sin π ( x / n ) π 2 ( x / n ) 2 ] E ( n ) = n = E ( n ) d sinc π ( x / n ) d π ,
E ( n ) = 1 2 + 2 0 1 / 2 [ T 1 ( f ) sin 2 π n f 2 π f + T 2 ( f ) cos 2 π n f 2 π f ] d f .
E ( n ) 1 + T 1 ( 0 ) 2 π 2 n T 1 ( 0 ) 24 π 4 ( n ) 3 T 2 ( 0 ) 8 π 3 ( n ) 2 , E ( n ) T 1 ( 0 ) 2 π 2 n T 1 ( 0 ) 24 π 4 ( n ) 3 T 2 ( 0 ) 8 π 3 ( n ) 2 .
E ( n ) 1 4 π 3 n + α ( n ) 2 + β ( n ) 3 , E ( n ) 4 π 3 n + α ( n ) 2 β ( n ) 3 .
E 1 ( n ) = [ E ( n ) + E ( n ) ] / 2 E 2 ( n ) = [ E ( n ) E ( n ) ] / 2 .
T 1 ( f ) = 4 π f n = 1 E 2 ( n ) sin 2 π n f ( C , 1 ) , T 2 ( f ) = 4 π f [ E 1 ( 0 ) 2 + n = 1 E 1 ( n ) cos 2 π n f ] ( C , 1 ) ,
E 1 ( n ) 1 2 + [ α / ( n ) 2 ] , E 2 ( n ) 1 2 [ 4 / π 3 n ] + [ β / ( n ) 3 ] .
n = N + 1 sin n u = cos ( N + 1 2 ) u 2 sin ( u / 2 ) ( C , 1 ) , n = N + 1 cos n u = sin ( N + 1 2 ) u 2 sin ( u / 2 ) ( C , 1 ) , n = 1 sin 2 π n u n = π B 1 ( u ) , n = 1 cos 2 π n u n 2 = π 2 B 2 ( u ) , n = 1 sin 2 π n u n 3 = 2 π 3 3 B 3 ( u ) ,
B 1 ( u ) = u 1 2 , B 2 ( u ) = u 2 u + 1 6 , B 3 ( u ) = u 2 ( 3 2 ) u 2 + ( 1 2 ) u .
e 0 = 8 / π e 1 = 16 2 / π + 4 3 π 4 ( β / ) , e 2 = 4 π 4 β , e 3 = ( 8 / 3 ) π 4 β , e 4 = 2 3 π 3 ( α / ) , e 5 = 4 π 3 α , e 6 = 4 π 3 α , e 7 = 4 / π 3 , e 8 = β / 3 e 9 = α / 3 .
T 1 ( f ) = 4 π f n = 1 N [ E 2 ( n ) + e 7 n + e 8 n 3 ] sin 2 π n f + e 0 f + e 1 f 2 + e 2 f 3 + e 3 f 4 + cos ( N + 1 2 ) 2 π f sinc π f , T 2 ( f ) = 4 π f { E 1 ( 0 ) 2 + n = 1 N [ E 1 ( n ) + e 9 n 2 ] cos 2 π n f } + e 4 f + e 5 f 2 + e 6 f 3 sin ( N + 1 2 ) 2 π f sinc π f .
α = [ E 1 ( N ) 1 2 ] ( N ) 2 , β = [ E 2 ( N ) 1 2 + 4 / π 3 N ] ( N ) 3 .
x = ( 2 / π ) υ 0 .
= ( 2 / π ) ,
n = υ 0 = 0 , 1 , 2 , , 12 .
1 2
1 2
1 2
1 2 4 / π 3 n
1 2 + [ α / ( n ) 2 ]
1 2 ( 4 / π 3 n ) + [ β / ( n ) 3 ]