Abstract

The notion that the optical contrast transfer function is a useful tool for assessing the performance of image-forming instruments has been accepted generally for some time and is now well established. This paper discusses one method of making the transition from ray-trace data to the evaluation of this important function. First, the light distribution in the point image is rigorously derived in terms of an integral over angular coordinates involving the eikonal function about a reference surface at infinity. Then, the ray-trace procedure is developed in the language of refraction and translation matrices culminating in matrix elements which are simply related to the eikonal coefficients of wave optics. Finally, the numerical evaluation of the contrast transfer function in amplitude and phase from these eikonal coefficients is presented, and the paper ends with an example showing the off-axis transfer function for line structures oriented at various azimuths. All calculations are carried out to fifth order in the eikonal coefficients, and emphasis is placed on the usefulness of this approach on relatively slow, low-capacity computing machines.

© 1963 Optical Society of America

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References

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  1. B. R. A. Nijboer, thesis, Groningen (1942).
  2. M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1959).
  3. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).
  4. R. K. Luneberg, Lecture Notes, Brown University, Providence (1944).
  5. E. Wolf, Proc. Roy Soc., A, 253, 349 (1959).
    [CrossRef]
  6. T. Smith, A Dictionary of Applied Physics, R. T. Glazebrook, ed. (Macmillan, London, 1923), Vol. IV, p. 287.
  7. W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, to be published).
  8. H. H. Hopkins, Proc. Phys. Soc., London B701002, 1162 (1957).
  9. J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge Univ. Press, New York, 1937).
  10. M. Herzberger, Strahlen Optik (Springer, Berlin, 1932).
  11. H. Osterberg, R. A. McDonald, “Symposium on Optical Image Evaluation,” NBS Circ. 526 (1954).
  12. R. Barakat, D. Lev, J. Opt. Soc. Am. 53, 324 (1963).
    [CrossRef]
  13. E. Marchand, R. Phillips, Appl. Opt. 2, 359 (1963).
    [CrossRef]

1963 (2)

1959 (1)

E. Wolf, Proc. Roy Soc., A, 253, 349 (1959).
[CrossRef]

1957 (1)

H. H. Hopkins, Proc. Phys. Soc., London B701002, 1162 (1957).

1954 (1)

H. Osterberg, R. A. McDonald, “Symposium on Optical Image Evaluation,” NBS Circ. 526 (1954).

Barakat, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1959).

Brouwer, W.

W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, to be published).

Herzberger, M.

M. Herzberger, Strahlen Optik (Springer, Berlin, 1932).

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc., London B701002, 1162 (1957).

Lev, D.

Luneberg, R. K.

R. K. Luneberg, Lecture Notes, Brown University, Providence (1944).

Marchand, E.

McDonald, R. A.

H. Osterberg, R. A. McDonald, “Symposium on Optical Image Evaluation,” NBS Circ. 526 (1954).

Nijboer, B. R. A.

B. R. A. Nijboer, thesis, Groningen (1942).

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

Osterberg, H.

H. Osterberg, R. A. McDonald, “Symposium on Optical Image Evaluation,” NBS Circ. 526 (1954).

Phillips, R.

Smith, T.

T. Smith, A Dictionary of Applied Physics, R. T. Glazebrook, ed. (Macmillan, London, 1923), Vol. IV, p. 287.

Synge, J. L.

J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge Univ. Press, New York, 1937).

Wolf, E.

E. Wolf, Proc. Roy Soc., A, 253, 349 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1959).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

NBS Circ. (1)

H. Osterberg, R. A. McDonald, “Symposium on Optical Image Evaluation,” NBS Circ. 526 (1954).

Proc. Phys. Soc., London (1)

H. H. Hopkins, Proc. Phys. Soc., London B701002, 1162 (1957).

Proc. Roy Soc., A (1)

E. Wolf, Proc. Roy Soc., A, 253, 349 (1959).
[CrossRef]

Other (8)

T. Smith, A Dictionary of Applied Physics, R. T. Glazebrook, ed. (Macmillan, London, 1923), Vol. IV, p. 287.

W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, to be published).

J. L. Synge, Geometrical Optics, An Introduction to Hamilton’s Method (Cambridge Univ. Press, New York, 1937).

M. Herzberger, Strahlen Optik (Springer, Berlin, 1932).

B. R. A. Nijboer, thesis, Groningen (1942).

M. Born, E. Wolf, Principles of Optics (Pergamon Press, London, 1959).

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley, Reading, Mass., 1963).

R. K. Luneberg, Lecture Notes, Brown University, Providence (1944).

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Figures (5)

Fig. 1
Fig. 1

(a) Coordinate systems. (b) Definition of angle–angle eikonal.

Fig. 2
Fig. 2

Wavefront configuration.

Fig. 3
Fig. 3

Rotation of coordinate axes for the computation of the contrast transfer function.

Fig. 4
Fig. 4

Rough sketch of normalized wave deformation.

Fig. 5
Fig. 5

Contrast transfer functions for wave deformation shown in Fig. 4 at several azimuth angles. Curve 1—on-axis; curve 2—off-axis ψ = 0; curve 3—off-axis ψ = π/4; curve 4—off-axis ψ = π/2.

Equations (41)

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a ( x , y ) = - + F ( v x , ν y ) exp [ 2 π i ( ν x x + ν y y ) ] d ν x d ν y ,
D ( ν x , ν y ) = - + s ( x , y ) exp [ - 2 π i ( ν x x + ν y y ) ] d x d y - + s ( x , y ) d x d y ,
D ( ν x , ν y ) = - + F ( μ x , μ y ) F * ( μ x - ν x , μ y - ν y ) d μ x d μ y - + F ( μ x , μ y ) d 2 μ x d μ y .
a ( P ) = Σ F ( A ) exp ( 2 π i λ W ) d σ ,
E L = - x , E M = - y ,
E x = - L , E y = - M .
A P ¯ = E ( L , M ) - P A ¯ + S R ¯ + ( A P ¯ - A R ¯ ) ,
S R ¯ = L ( x 0 + ξ ) + M ( y 0 + η ) ;
W = E ( L , M ) + L x 0 + M y 0 + ξ L + η M .
a ( P ) = Σ F ¯ ( L M ) exp 2 π i λ [ W 0 ( L , M ) + ξ L + η M ] d L d M
W 0 ( L , M ) = E ( L , M ) + L x 0 + M y 0 ,
ν x = L / λ , ν y = M / λ .
E L + x 0 + ξ = 0 , E M + y 0 + η = 0 ,
[ L i x i ] = [ 1 - A i 0 1 ] [ L i x i ] = R i [ L i x i ] and [ M i y i ] = [ 1 - A i 0 1 ] [ M i y i ] = R i [ M i y i ] ,
A i = n i cos φ i - n i cos φ i r i .
[ L i + 1 x i + 1 ] = [ 1 0 T i 1 ] [ L i x i ] = T [ L i x i ] and [ M i + 1 y i + 1 ] = [ L 0 T i 1 ] [ M i y i ] = T i [ M i y i ] ,
T i = t i / n i .
[ L x ] = T k R k T k - 1 R k - 1 T 2 R 2 T 1 R 1 T 0 [ L x ] ,
[ L x ] = [ B - A - D C ] [ L x ] and [ M y ] = [ B - A - D C ] [ M y ] .
u 1 = ½ ( x 2 + y 2 ) , u 2 = L x + M y , u 3 = ½ ( L 2 + M 2 ) .
( - B E u 1 - A ) x - ( B E u 2 + 1 ) L = 0 , ( D E u 1 + C + E u 2 ) x + ( D E u 2 + E u 3 ) L = 0.
E u 1 = - A B , E u 2 = - 1 B , E u 3 = D B , B C - A D = + 1.
E ( u 2 , u 3 ) = Σ i Σ j E i j u 2 i u 3 j E 00 + E 10 u 2 + E 01 u 3 + E 20 u 2 2 + E 11 u 2 u 3 + E 02 u 3 2 + E 30 u 2 3 + E 21 u 2 2 u 3 + E 12 u 2 u 3 2 + E 03 u 3 3 + .
E u 2 = - 1 B = E 10 + 2 E 20 u 2 + E 11 u 3 + 3 E 30 u 2 2 + 2 E 21 u 2 u 3 + E 12 u 3 2 + , E u 3 = D B = E 01 + E 11 u 2 + 2 E 02 u 3 + E 21 u 2 2 + 2 E 12 u 2 u 3 + 3 E 03 u 3 2 + .
u 2 = L x , u 3 = ½ L 2 ,
- 1 B = E 10 + ( 2 E 20 x ) L + ( ½ E 11 + 3 E 30 x 2 ) L 2 + ( E 21 x ) L 3 + ¼ E 12 L 4 , D B = E 01 + ( E 11 x ) L + ( E 02 + E 21 x 2 ) L 2 + ( E 12 x ) L 3 + ¾ E 03 L 4 .
β 0 = L / L m and γ 0 = M / M m .
D ( s , t ) = - F ( β 0 , γ 0 ) F * ( β 0 - s , γ 0 - t ) d β 0 d γ 0 - F 0 ( β 0 , γ 0 ) 2 d β 0 d γ 0 ,
F ( β 0 , γ 0 ) = exp [ 2 π i λ E ( β 0 , γ 0 ] , β 0 2 + γ 0 2 1 = 0 , β 0 2 + γ 0 2 > 1.
D ( s , ψ ) = - F ( β + s 2 , γ ) F * ( β - s 2 , γ ) d β d γ - F ( β , γ ) d 2 β d γ .
D ( s , ψ ) = 1 a exp [ i k s V ( β , γ ; ψ ) ] d β d γ ,
V ( β , α ; ψ ) = 1 s [ E ( β + s 2 , γ ; ψ ) - E ( β - s 2 , γ ; ψ ) ] = E β + 1 3 ! ( s 2 ) 2 3 E β 3 + 1 5 ! ( s 2 ) 4 5 E β 5 .
D ( s , ψ ) = 1 N Σ m Σ n e i Z sin X X sin Y Y ,
X = x k s V β , Y = y k s V α , Z = k s V ( β , γ ; ψ ) ,
D ( s , ψ ) = D ( s , ψ ) e i θ ( s , ψ ) ,
D ( s , ψ ) 2 = D R 2 ( s , ψ ) + D I 2 ( s , ψ ) , θ ( s , ψ ) = tan - 1 D I / D R ,
D R ( s , ψ ) = 1 N Σ m Σ n cos Z sin X X sin Y Y , D I ( s , ψ ) = 1 N Σ m Σ n sin Z sin X X sin Y Y .
E ( β , α ; ψ ) = k , l k + l = 0 6 A k l ( ψ ) β k γ l ,
x = E L , - x = E L , y = E M , - y = E M .
u ¯ 1 = ½ ( L 2 + M 2 ) , u ¯ 2 = L L + M M , u ¯ 3 = ½ ( L 2 + M 2 ) .
E u ¯ 1 = B A , E u ¯ 2 = - 1 A , E u ¯ 3 = C A , B C - A D = + 1.

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