Abstract

The calculation of image assessment critiera, e.g., the Strehl ratio, the point spread function, or the optical transfer function, involves the evaluation of an integral where the integrand is highly oscillatory over a large range of integration. Prefaced with a brief description of the well-known numerical quadrature methods adopted for the purpose, this paper presents a new quadrature technique that obviates the need for knowledge of derivatives of the argument of the exponential integrand. Some illustrative numerical results are presented.

© 1988 Optical Society of America

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References

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  1. R. Barakat, “The Calculation of Integrals Encountered in Optical Diffraction Theory,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer-Verlag, New York, 1980), p. 35.
    [CrossRef]
  2. P. J. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975).
  3. G. Black, E. H. Linfoot, “Spherical Aberration and the Information Content of Optical Images,” Proc. R. Soc. London Ser. A, 239, 522 (1957).
    [CrossRef]
  4. M. Kidger, “The Calculation of the Optical Transfer Function Using Gaussian Quadrature,” Opt. Acta 25, 665 (1978).
    [CrossRef]
  5. M. Plight, “The Rapid Calculation of the Optical Transfer Function for On Axis Systems Using the Orthogonal Properties of Tchebycheff Polynomials,” Opt. Acta 25, 849 (1978).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 464.
  7. E. C. Kintner, R. M. Sllitto, “A New Analytic Method for Computing the Optical Transfer Function,” Opt. Acta 23, 607 (1976).
    [CrossRef]
  8. M. De, L. N. Hazra, “Walsh Functions in Problems of Optical Imagery,” Opt. Acta 24, 221 (1977).
    [CrossRef]
  9. R. Barakat, “Application of the Sampling Theorem to Optical Diffraction Theory,” J. Opt. Soc. Am. 54, 920 (1964).
    [CrossRef]
  10. B. R. Frieden, “Image Evaluation by Use of the Sampling Theorem,” J. Opt. Soc. Am. 56, 1355 (1966).
    [CrossRef]
  11. E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, NJ, 1974).
  12. A. E. Siegman, “Quasi Fast Hankel Transform,” Opt. Lett. 1, 13 (1977).
    [CrossRef] [PubMed]
  13. S. Szapiel, “Point-Spread-Function Computation: Quasi-Digital Method,” J. Opt. Soc. Am. A 2, 3 (1985).
    [CrossRef]
  14. S. Szapiel, “Point-Spread Function Computation: Analytic End Correction in the Quasi-Digital Method,” J. Opt. Soc. Am. A 4, 625 (1987).
    [CrossRef]
  15. H. H. Hopkins, “The Numerical Evaluation of the Frequency Response of Optical Systems,” Proc. Phys. Soc. London Sect. B 70, 1002 (1957).
    [CrossRef]
  16. H. H. Hopkins, M. J. Yzuel, “The Computation of Diffraction Patterns in the Presence of Aberrations,” Opt. Acta 17, 157 (1970).
    [CrossRef]
  17. J. Macdonald, “The Calculation of the Optical Transfer Function,” Opt. Acta 8, 269 (1971).
    [CrossRef]
  18. A. Maréchal, “Étude des effects combines de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. Opt. 26, 257 (1947).
  19. L. N. Hazra, “A New Class of Optimum Amplitude Filters,” Opt. Commun. 21, 232 (1977).
    [CrossRef]
  20. P. K. Purkait, L. N. Hazra, M. De, “Walsh Functions in Lens Optimization II. Evaluation of Diffraction Based OTF for On Axis Imagery,” Opt. Acta 28, 389 (1981).
    [CrossRef]
  21. L. N. Hazra, “Extended Range Diffraction Based Merit Function for Lens Design Optimization,” Proc. Soc. Photo-Opt. Instrum. Eng. 655, 20 (1986).

1987 (1)

1986 (1)

L. N. Hazra, “Extended Range Diffraction Based Merit Function for Lens Design Optimization,” Proc. Soc. Photo-Opt. Instrum. Eng. 655, 20 (1986).

1985 (1)

1981 (1)

P. K. Purkait, L. N. Hazra, M. De, “Walsh Functions in Lens Optimization II. Evaluation of Diffraction Based OTF for On Axis Imagery,” Opt. Acta 28, 389 (1981).
[CrossRef]

1978 (2)

M. Kidger, “The Calculation of the Optical Transfer Function Using Gaussian Quadrature,” Opt. Acta 25, 665 (1978).
[CrossRef]

M. Plight, “The Rapid Calculation of the Optical Transfer Function for On Axis Systems Using the Orthogonal Properties of Tchebycheff Polynomials,” Opt. Acta 25, 849 (1978).
[CrossRef]

1977 (3)

M. De, L. N. Hazra, “Walsh Functions in Problems of Optical Imagery,” Opt. Acta 24, 221 (1977).
[CrossRef]

A. E. Siegman, “Quasi Fast Hankel Transform,” Opt. Lett. 1, 13 (1977).
[CrossRef] [PubMed]

L. N. Hazra, “A New Class of Optimum Amplitude Filters,” Opt. Commun. 21, 232 (1977).
[CrossRef]

1976 (1)

E. C. Kintner, R. M. Sllitto, “A New Analytic Method for Computing the Optical Transfer Function,” Opt. Acta 23, 607 (1976).
[CrossRef]

1971 (1)

J. Macdonald, “The Calculation of the Optical Transfer Function,” Opt. Acta 8, 269 (1971).
[CrossRef]

1970 (1)

H. H. Hopkins, M. J. Yzuel, “The Computation of Diffraction Patterns in the Presence of Aberrations,” Opt. Acta 17, 157 (1970).
[CrossRef]

1966 (1)

1964 (1)

1957 (2)

G. Black, E. H. Linfoot, “Spherical Aberration and the Information Content of Optical Images,” Proc. R. Soc. London Ser. A, 239, 522 (1957).
[CrossRef]

H. H. Hopkins, “The Numerical Evaluation of the Frequency Response of Optical Systems,” Proc. Phys. Soc. London Sect. B 70, 1002 (1957).
[CrossRef]

1947 (1)

A. Maréchal, “Étude des effects combines de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. Opt. 26, 257 (1947).

Barakat, R.

R. Barakat, “Application of the Sampling Theorem to Optical Diffraction Theory,” J. Opt. Soc. Am. 54, 920 (1964).
[CrossRef]

R. Barakat, “The Calculation of Integrals Encountered in Optical Diffraction Theory,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer-Verlag, New York, 1980), p. 35.
[CrossRef]

Black, G.

G. Black, E. H. Linfoot, “Spherical Aberration and the Information Content of Optical Images,” Proc. R. Soc. London Ser. A, 239, 522 (1957).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 464.

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, NJ, 1974).

Davis, P. J.

P. J. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975).

De, M.

P. K. Purkait, L. N. Hazra, M. De, “Walsh Functions in Lens Optimization II. Evaluation of Diffraction Based OTF for On Axis Imagery,” Opt. Acta 28, 389 (1981).
[CrossRef]

M. De, L. N. Hazra, “Walsh Functions in Problems of Optical Imagery,” Opt. Acta 24, 221 (1977).
[CrossRef]

Frieden, B. R.

Hazra, L. N.

L. N. Hazra, “Extended Range Diffraction Based Merit Function for Lens Design Optimization,” Proc. Soc. Photo-Opt. Instrum. Eng. 655, 20 (1986).

P. K. Purkait, L. N. Hazra, M. De, “Walsh Functions in Lens Optimization II. Evaluation of Diffraction Based OTF for On Axis Imagery,” Opt. Acta 28, 389 (1981).
[CrossRef]

L. N. Hazra, “A New Class of Optimum Amplitude Filters,” Opt. Commun. 21, 232 (1977).
[CrossRef]

M. De, L. N. Hazra, “Walsh Functions in Problems of Optical Imagery,” Opt. Acta 24, 221 (1977).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, M. J. Yzuel, “The Computation of Diffraction Patterns in the Presence of Aberrations,” Opt. Acta 17, 157 (1970).
[CrossRef]

H. H. Hopkins, “The Numerical Evaluation of the Frequency Response of Optical Systems,” Proc. Phys. Soc. London Sect. B 70, 1002 (1957).
[CrossRef]

Kidger, M.

M. Kidger, “The Calculation of the Optical Transfer Function Using Gaussian Quadrature,” Opt. Acta 25, 665 (1978).
[CrossRef]

Kintner, E. C.

E. C. Kintner, R. M. Sllitto, “A New Analytic Method for Computing the Optical Transfer Function,” Opt. Acta 23, 607 (1976).
[CrossRef]

Linfoot, E. H.

G. Black, E. H. Linfoot, “Spherical Aberration and the Information Content of Optical Images,” Proc. R. Soc. London Ser. A, 239, 522 (1957).
[CrossRef]

Macdonald, J.

J. Macdonald, “The Calculation of the Optical Transfer Function,” Opt. Acta 8, 269 (1971).
[CrossRef]

Maréchal, A.

A. Maréchal, “Étude des effects combines de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. Opt. 26, 257 (1947).

Plight, M.

M. Plight, “The Rapid Calculation of the Optical Transfer Function for On Axis Systems Using the Orthogonal Properties of Tchebycheff Polynomials,” Opt. Acta 25, 849 (1978).
[CrossRef]

Purkait, P. K.

P. K. Purkait, L. N. Hazra, M. De, “Walsh Functions in Lens Optimization II. Evaluation of Diffraction Based OTF for On Axis Imagery,” Opt. Acta 28, 389 (1981).
[CrossRef]

Rabinowitz, P.

P. J. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975).

Siegman, A. E.

Sllitto, R. M.

E. C. Kintner, R. M. Sllitto, “A New Analytic Method for Computing the Optical Transfer Function,” Opt. Acta 23, 607 (1976).
[CrossRef]

Szapiel, S.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 464.

Yzuel, M. J.

H. H. Hopkins, M. J. Yzuel, “The Computation of Diffraction Patterns in the Presence of Aberrations,” Opt. Acta 17, 157 (1970).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Acta (7)

H. H. Hopkins, M. J. Yzuel, “The Computation of Diffraction Patterns in the Presence of Aberrations,” Opt. Acta 17, 157 (1970).
[CrossRef]

J. Macdonald, “The Calculation of the Optical Transfer Function,” Opt. Acta 8, 269 (1971).
[CrossRef]

P. K. Purkait, L. N. Hazra, M. De, “Walsh Functions in Lens Optimization II. Evaluation of Diffraction Based OTF for On Axis Imagery,” Opt. Acta 28, 389 (1981).
[CrossRef]

E. C. Kintner, R. M. Sllitto, “A New Analytic Method for Computing the Optical Transfer Function,” Opt. Acta 23, 607 (1976).
[CrossRef]

M. De, L. N. Hazra, “Walsh Functions in Problems of Optical Imagery,” Opt. Acta 24, 221 (1977).
[CrossRef]

M. Kidger, “The Calculation of the Optical Transfer Function Using Gaussian Quadrature,” Opt. Acta 25, 665 (1978).
[CrossRef]

M. Plight, “The Rapid Calculation of the Optical Transfer Function for On Axis Systems Using the Orthogonal Properties of Tchebycheff Polynomials,” Opt. Acta 25, 849 (1978).
[CrossRef]

Opt. Commun. (1)

L. N. Hazra, “A New Class of Optimum Amplitude Filters,” Opt. Commun. 21, 232 (1977).
[CrossRef]

Opt. Lett. (1)

Proc. Phys. Soc. London Sect. B (1)

H. H. Hopkins, “The Numerical Evaluation of the Frequency Response of Optical Systems,” Proc. Phys. Soc. London Sect. B 70, 1002 (1957).
[CrossRef]

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

G. Black, E. H. Linfoot, “Spherical Aberration and the Information Content of Optical Images,” Proc. R. Soc. London Ser. A, 239, 522 (1957).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

L. N. Hazra, “Extended Range Diffraction Based Merit Function for Lens Design Optimization,” Proc. Soc. Photo-Opt. Instrum. Eng. 655, 20 (1986).

Rev. Opt. (1)

A. Maréchal, “Étude des effects combines de la diffraction et des aberrations géométriques sur l’image d’un point lumineux,” Rev. Opt. 26, 257 (1947).

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), p. 464.

R. Barakat, “The Calculation of Integrals Encountered in Optical Diffraction Theory,” in The Computer in Optical Research, B. R. Frieden, Ed. (Springer-Verlag, New York, 1980), p. 35.
[CrossRef]

P. J. Davis, P. Rabinowitz, Methods of Numerical Integration (Academic, New York, 1975).

E. O. Brigham, The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs, NJ, 1974).

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Figures (1)

Fig. 1
Fig. 1

p ¯ n is the average value of the function p(x,u) over the nth subinterval (xn,xn + ). For N equal subintervals of width 2 in the interval (a,b), xn = a + (2n − 1), n = 1, …,N.

Tables (5)

Tables Icon

Table I Strehl Ratios for β60 in 0(0.5) 4.0 with N = 1a

Tables Icon

Table II Strehl Ratios for β60 in 0(0.5) 4.0 with N = 2a

Tables Icon

Table III Strehl Ratios for β60 In 0(0.5) 4.0 with N = 4a

Tables Icon

Table IV Strehl Ratios for β60 in 0(0.5) 4.0 with N = 8a

Tables Icon

Table V Strehl Ratios for β60 in 0(0.5) 4.0 with N = 16a

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

F ( u ) = a b exp [ i p ( x , u ) ] d x .
F ( u ) = i = 1 N w i exp [ i p ( x i , u ) ]
p ( x , u ) = g ( x ) + u x ,
F ( u ) = a b exp [ i g ( x ) ] exp [ i u x ] d x = a b f ( x ) exp [ i u x ] d x ,
f ( x ) = exp [ i g ( x ) ] .
p ( x , u ) p ( x n , u ) + ( x - x n ) p ( x n , u ) ,
x n = a + ( 2 n - 1 ) ,
p ( x n , u ) = { d [ p ( x , u ) ] d x } x = x n .
F ( u ) = 2 n = 1 N sin { p ( x n , u ) } { p ( x n , u ) } exp [ i p ( x n , u ) ] .
p ¯ n = 1 2 x n - x n + p ( x , u ) d x .
F ( u ) = n = 1 N exp ( i p ¯ n ) x n - x n + exp { i [ p ( x , u ) - p ¯ n ] } d x .
F ( u ) = 2 n = 1 N [ 1 - 1 2 ( p n 2 ¯ - p ¯ n 2 ) ] exp ( i p ¯ n ) ,
p n 2 ¯ = 1 2 x n - x n + [ p ( x , u ) ] 2 d x .
F ( u ) = 2 n = 1 N exp ( i p ¯ n ) .
F ( 0 ) = 0 1 f ( r ) r d r ,
f ( r ) = exp [ i k W ( r ) ] .
I N ( 0 ) = F ( 0 ) / F 0 ( 0 ) 2 = F N ( 0 ) 2 ,
F N ( 0 ) = 2 0 1 exp [ i k W ( r ) ] r d r .
F N ( 0 ) = 0 1 exp [ i k W ( t ) ] d t .
F N ( 0 ) = 2 p = 1 N exp { i k W ( t p ) } sin { k W ( t p ) } { k W ( t p ) } } ,
W ( t p ) = [ d W ( t ) d t ] t = t p .
I N ( 0 ) = 4 2 p = 1 N q = 1 N sin { k W ( t p ) } { k W ( t p ) } sin { k W ( t q ) } { k W ( t q ) } × cos k [ W ( t p ) - W ( t q ) ] .
F N ( 0 ) = 2 p = 1 N exp { i k W ¯ p } [ 1 - ½ ( W p 2 ¯ - W ¯ p 2 ) ] ,
W ¯ p = 1 2 t p - t p + W ( t ) d t ,
W p 2 ¯ = 1 2 t p - t p + [ W ( t ) ] 2 d t .
I N ( 0 ) = 4 2 p = 1 N q = 1 N [ 1 - ½ ( W p 2 ¯ - W ¯ p 2 ) ] [ 1 - ½ ( W q 2 ¯ - W ¯ q 2 ) ] × cos [ k ( W ¯ p - W ¯ q ) ] .
I N ( 0 ) = 4 2 p = 1 N q = 1 N cos [ k ( W ¯ p - W ¯ q ) ] .
W 60 = ( 10 λ / π ) β 60 , W 40 = ( - 3 / 2 ) W 60 , W 20 = ( 3 / 5 ) W 60 .
W ( t p ) [ W ( t p + 1 ) - W ( t p ) ] / 2 .

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