Abstract

A description is given of a calculation procedure for obtaining the optical transfer function of an optical system from its design data for both axial and extra-axial object points. The first part of the calculation involves determining the wave aberration of the system to the fifth order (in the classical sense), with vignetting taken into account. The second part involves calculating the optical transfer function from this aberration by the methods of physical optics (diffraction theory). A high-speed computer is required but the particular program to be written from the equations depends upon the kind of computer to be used.

© 1963 Optical Society of America

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References

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  1. E. O’Neill, Selected Topics in Optics and Communication Theory (Itek Corp., Boston, 1958).
  2. H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 231, 91 (1955).
    [CrossRef]
  3. R. L. Lamberts, J. Opt. Soc. Am. 48, 490 (1958).
    [CrossRef]
  4. F. H. Perrin, J. Soc. Motion Picture Television Engrs. 69, 239 (1960). This paper contains an extensive bibliography.
  5. K. Miyamoto, J. Opt. Soc. Am. 48, 57, 567 (1958).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 483–489.
  7. H. H. Hopkins, Proc. Phys. Soc. (London) Ser. B 62, 22 (1949).
    [CrossRef]
  8. H. H. Hopkins, Proc. Phys. Soc. (London) Ser. B 69, 562 (1956).
    [CrossRef]
  9. R. Barakat, J. Opt. Soc. Am. 52, 985 (1962).
    [CrossRef]
  10. R. Barakat, M. Morello, J. Opt. Soc. Am. 52, 992 (1962).
    [CrossRef]

1962 (2)

1960 (1)

F. H. Perrin, J. Soc. Motion Picture Television Engrs. 69, 239 (1960). This paper contains an extensive bibliography.

1958 (2)

K. Miyamoto, J. Opt. Soc. Am. 48, 57, 567 (1958).
[CrossRef]

R. L. Lamberts, J. Opt. Soc. Am. 48, 490 (1958).
[CrossRef]

1956 (1)

H. H. Hopkins, Proc. Phys. Soc. (London) Ser. B 69, 562 (1956).
[CrossRef]

1955 (1)

H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 231, 91 (1955).
[CrossRef]

1949 (1)

H. H. Hopkins, Proc. Phys. Soc. (London) Ser. B 62, 22 (1949).
[CrossRef]

Barakat, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 483–489.

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. (London) Ser. B 69, 562 (1956).
[CrossRef]

H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 231, 91 (1955).
[CrossRef]

H. H. Hopkins, Proc. Phys. Soc. (London) Ser. B 62, 22 (1949).
[CrossRef]

Lamberts, R. L.

Miyamoto, K.

K. Miyamoto, J. Opt. Soc. Am. 48, 57, 567 (1958).
[CrossRef]

Morello, M.

O’Neill, E.

E. O’Neill, Selected Topics in Optics and Communication Theory (Itek Corp., Boston, 1958).

Perrin, F. H.

F. H. Perrin, J. Soc. Motion Picture Television Engrs. 69, 239 (1960). This paper contains an extensive bibliography.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 483–489.

J. Opt. Soc. Am. (4)

J. Soc. Motion Picture Television Engrs. (1)

F. H. Perrin, J. Soc. Motion Picture Television Engrs. 69, 239 (1960). This paper contains an extensive bibliography.

Proc. Phys. Soc. (London) Ser. B (2)

H. H. Hopkins, Proc. Phys. Soc. (London) Ser. B 62, 22 (1949).
[CrossRef]

H. H. Hopkins, Proc. Phys. Soc. (London) Ser. B 69, 562 (1956).
[CrossRef]

Proc. Roy. Soc. (London) Ser. A (1)

H. H. Hopkins, Proc. Roy. Soc. (London) Ser. A 231, 91 (1955).
[CrossRef]

Other (2)

E. O’Neill, Selected Topics in Optics and Communication Theory (Itek Corp., Boston, 1958).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 483–489.

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Figures (5)

Fig. 1
Fig. 1

Definition of wave aberration ϕ = Q ¯Q.

Fig. 2
Fig. 2

Typical vignetted aperture in exit pupil.

Fig. 3
Fig. 3

Modulation transfer curves of a certain 30 cm, f/5 lens for three image planes at different distances from the principal plane (zp = 0) and for an object point (A) on the axis and (B and C) 10° away. Values computed by the method of stationary phase are indicated in A by ○(zp = 0), △(zp = −0.7 mm), and □(zp = −1.5 mm). Upper scale of abscissas’ frequency in terms of s; lower scale in terms of R′.

Fig. 4
Fig. 4

Modulation transfer curves of a certain diffraction-limited lens for three image planes at different distances from the principal plane (zp = 0) and for an object point (A) on the axis and (B and C) 15° away. Upper scale of abscissas’ frequency in terms of s; lower scale in terms of R′.

Fig. 5
Fig. 5

Modulation transfer curves of a certain lens compared with experimental values (circles and triangles). A, object on axis, one image plane; B, object 15° from axis, image planes 0.15 and 0.30 mm from “principal” plane, meridional lines only. Upper scale of abscissas’ frequency in terms of s; lower scale in terms of R′.

Equations (24)

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y = y / a ,             u = ( x 2 + y 2 ) / a 2 ,
f = - k ϕ = f 1 y + f 2 u + f 3 y 2 + f 4 y u + f 5 u 2 + f 6 y 3 + f 7 y 2 u + f 8 y u 2 + f 9 u 3 ,
D ( s , t ) = 1 A S exp [ i f ( x + s 2 , y + t 2 ) - i f ( x - s 2 , y - t 2 ) ] d x d y .
s = λ R n sin α ,
x = x / a ,             y = y / a ,
D ( 0 , 0 ) = 1 A A d x d y = 1 ,
D ( 0 , t ) = D 1 + i D 2 , D = ( D 1 2 + D 2 2 ) 1 / 2 ,             θ = tan - 1 ( D 2 / D 1 ) ,
Q M = R - [ x 2 + ( y - y 0 ) 2 + f 2 ] 1 / 2
f 2 = R 2 - y 0 2 .
ϕ = Q ¯ Q = Q ¯ M - Q M
f = - k ϕ = - 2 π ϕ / λ .
p = ( π / λ ) ( a / R ) 2 z p
p ( y 0 / R ) 2 .
I = ( 2 / π ) { cos - 1 s / 2 - s / 2 [ 1 - ( ½ s ) 2 ] 1 / 2 } ,
I = S e i F ( x , y ) d x d y .
F ( x , y ) = F p + ( x - x p ) ( F x ) p + ( y - y p ) ( F y ) p ,
I p = A p exp { i [ F p + ( x - x p ) F x + ( y - y p ) F y ] } d x d y = exp ( i F p ) - - exp [ i ( F x x ¯ + F y y ¯ ) ] d x ¯ d y ¯ = exp ( i F p ) [ - exp ( i F x x ) d x ] [ - exp ( i F y y ) d y ] = exp ( i F p ) [ 2 0 cos ( F x x ) d x ] [ 2 0 cos ( F y y ) d y ] = 4 exp ( i F p ) sin ( F x ) F x sin ( F y ) F y = 4 2 exp ( i F p ) sinc ( F x ) sinc ( F y ) ,
sinc χ = ( sin χ ) / χ .
I = 4 2 exp ( i F p ) sinc ( F x ) p sinc ( F y ) p .
f ( x + s / 2 , y ) = f ( x , y ) + ( s / 2 ) 1 ! f x + ( s / 2 ) 2 2 ! f x x + ( s / 2 ) 3 3 ! f x x x + f ( x - s / 2 , y ) = f ( x , y ) - ( s / 2 ) 1 ! f x + ( s / 2 ) 2 2 ! f x x - ( s / 2 ) 3 3 ! f x x x +
f ( x + s / 2 , y ) - f ( x - s / 2 , y ) = s [ f x + ( s / 2 ) 2 3 ! f x x x + ( s / 2 ) 4 5 ! f x x x x x ] ,
F p = Z = s [ f x + ( s / 2 ) 2 3 ! f x x x + ( s / 2 ) 4 5 ! f x x x x x ] ( F x ) p = X = s [ f x x + ( s / 2 ) 2 3 ! f x x x x + ( s / 2 ) 4 5 ! f x x x x x x ] ( F y ) p = Y = s [ f x y + ( s / 2 ) 2 3 ! f x x x y + ( s / 2 ) 4 5 ! f x x x x x y ]
I = 4 2 e i Z sinc X sinc Y .
D ( s , 0 ) = 8 2 A cos Z sinc X sinc Y ,

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