Abstract

The interrelation of various image parameters such as point-spread function, line-spread function, edge response, flux passing through hole, flux passing through slit, and the spatial-frequency response is demonstrated. These image parameters are computed from the point-spread function in presence of aberrations and diffraction. The point-spread function is first calculated for varying amounts of spherical aberration, coma, astigmatism, and coma+astigmatism using the Zernike–Nijboer theory and then the other parameters are computed. The data are obtained for different focal positions. A discussion of method of computation of all these integrals on a Univac-1105 computer is included along with an analysis of errors.

© 1962 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. H. Selwyn, Phot. J. B88, 6, 46 (1948).
  2. O. H. Schade, Natl. Bur. Standards (U. S.) Circ. 526 (April29, 1954).
  3. R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, Rhode Island, 1944).
  4. P. M. Duffieux, “L’Integrale de Fourier et ses applications à l’optique,” Rennes (1946).
  5. A. Marechal, Étude des effets combines de la diffraction et des aberrations geometriques sur l’image d’un point lumineux (Editions de la Revue d’Optique Theorique et Instrumentale, Paris, France, 1948).
  6. W. H. Steel, “Étude des effets combines des aberrations et d’une obturation centrale de la pupille sur le contraste des image optique,” Rev. opt. 32, 4–26, 143–178, 269–306 (1953).
  7. O. H. Schade, RCA Rev. 9(1948).
  8. O. H. Schade, “Optical Image Evaluation,” Natl. Bur. Standards (U. S.), Circ. 526(April1954).
  9. “Image Evaluation Symposium,” Institute of Optics, University of Rochester (1955).
  10. Proc. Summer School on “Optical Image Assessment Using Frequency Response Techniques,” Imperial College, London (1958).
  11. E. L. O’Neill, Tech. Note 110, BUPRL (January1954).
  12. G. B. Parrent, thesis, Boston University (1955).
  13. H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).
  14. M. De, Proc. Roy. Soc. (London) 233A, 91 (1955).
  15. L. R. Baker, Proc. Phys. Soc. (London) BLXVIII, 871 (1955).
  16. F. D. Smith, J. Opt. Soc. Am. 45, 408 (1955).
  17. F. Zernike, Physica 5, 785 (1938).
    [Crossref]
  18. H. H. Hopkins, Proc. Roy. Soc. (London) A28 (1951).
  19. E. Wolf, Proc. Roy. Soc. (London) A230 (1955).
  20. A. Marechal and P. Croce, Compt. rend. 237, 607 (1953).
  21. E. O. O’Neill, IRE Trans. of the Profess. Group on Inf. Theory IT-2, No. 2 (1956).
  22. R. V. Shack, J. Research Natl. Bur. Standards,  56, 245 (1956).
    [Crossref]
  23. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), pp. 488–489.
  24. (x0,y0) are coordinates normalized in such a way that both the object point and its Gaussian image may be specified by the same ordered pair of numbers (x0,y0).
  25. B. R. A. Nijboer, “The Diffraction Theory of Aberrations,” thesis, University of Gronigen (1942).
  26. K. Nienhuis, “On the Influence of Diffraction on Image Formation in the Presence of Aberration,” thesis, University of Gronigen (1948).
  27. E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, England, 1955).

1956 (2)

E. O. O’Neill, IRE Trans. of the Profess. Group on Inf. Theory IT-2, No. 2 (1956).

R. V. Shack, J. Research Natl. Bur. Standards,  56, 245 (1956).
[Crossref]

1955 (4)

E. Wolf, Proc. Roy. Soc. (London) A230 (1955).

M. De, Proc. Roy. Soc. (London) 233A, 91 (1955).

L. R. Baker, Proc. Phys. Soc. (London) BLXVIII, 871 (1955).

F. D. Smith, J. Opt. Soc. Am. 45, 408 (1955).

1954 (2)

O. H. Schade, Natl. Bur. Standards (U. S.) Circ. 526 (April29, 1954).

O. H. Schade, “Optical Image Evaluation,” Natl. Bur. Standards (U. S.), Circ. 526(April1954).

1953 (3)

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

A. Marechal and P. Croce, Compt. rend. 237, 607 (1953).

W. H. Steel, “Étude des effets combines des aberrations et d’une obturation centrale de la pupille sur le contraste des image optique,” Rev. opt. 32, 4–26, 143–178, 269–306 (1953).

1951 (1)

H. H. Hopkins, Proc. Roy. Soc. (London) A28 (1951).

1948 (2)

E. H. Selwyn, Phot. J. B88, 6, 46 (1948).

O. H. Schade, RCA Rev. 9(1948).

1946 (1)

P. M. Duffieux, “L’Integrale de Fourier et ses applications à l’optique,” Rennes (1946).

1938 (1)

F. Zernike, Physica 5, 785 (1938).
[Crossref]

Baker, L. R.

L. R. Baker, Proc. Phys. Soc. (London) BLXVIII, 871 (1955).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), pp. 488–489.

Croce, P.

A. Marechal and P. Croce, Compt. rend. 237, 607 (1953).

De, M.

M. De, Proc. Roy. Soc. (London) 233A, 91 (1955).

Duffieux, P. M.

P. M. Duffieux, “L’Integrale de Fourier et ses applications à l’optique,” Rennes (1946).

Hopkins, H. H.

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

H. H. Hopkins, Proc. Roy. Soc. (London) A28 (1951).

Linfoot, E. H.

E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, England, 1955).

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, Rhode Island, 1944).

Marechal, A.

A. Marechal and P. Croce, Compt. rend. 237, 607 (1953).

A. Marechal, Étude des effets combines de la diffraction et des aberrations geometriques sur l’image d’un point lumineux (Editions de la Revue d’Optique Theorique et Instrumentale, Paris, France, 1948).

Nienhuis, K.

K. Nienhuis, “On the Influence of Diffraction on Image Formation in the Presence of Aberration,” thesis, University of Gronigen (1948).

Nijboer, B. R. A.

B. R. A. Nijboer, “The Diffraction Theory of Aberrations,” thesis, University of Gronigen (1942).

O’Neill, E. L.

E. L. O’Neill, Tech. Note 110, BUPRL (January1954).

O’Neill, E. O.

E. O. O’Neill, IRE Trans. of the Profess. Group on Inf. Theory IT-2, No. 2 (1956).

Parrent, G. B.

G. B. Parrent, thesis, Boston University (1955).

Schade, O. H.

O. H. Schade, Natl. Bur. Standards (U. S.) Circ. 526 (April29, 1954).

O. H. Schade, “Optical Image Evaluation,” Natl. Bur. Standards (U. S.), Circ. 526(April1954).

O. H. Schade, RCA Rev. 9(1948).

Selwyn, E. H.

E. H. Selwyn, Phot. J. B88, 6, 46 (1948).

Shack, R. V.

R. V. Shack, J. Research Natl. Bur. Standards,  56, 245 (1956).
[Crossref]

Smith, F. D.

F. D. Smith, J. Opt. Soc. Am. 45, 408 (1955).

Steel, W. H.

W. H. Steel, “Étude des effets combines des aberrations et d’une obturation centrale de la pupille sur le contraste des image optique,” Rev. opt. 32, 4–26, 143–178, 269–306 (1953).

Wolf, E.

E. Wolf, Proc. Roy. Soc. (London) A230 (1955).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), pp. 488–489.

Zernike, F.

F. Zernike, Physica 5, 785 (1938).
[Crossref]

Compt. rend. (1)

A. Marechal and P. Croce, Compt. rend. 237, 607 (1953).

IRE Trans. of the Profess. Group on Inf. Theory (1)

E. O. O’Neill, IRE Trans. of the Profess. Group on Inf. Theory IT-2, No. 2 (1956).

J. Opt. Soc. Am. (1)

F. D. Smith, J. Opt. Soc. Am. 45, 408 (1955).

J. Research Natl. Bur. Standards (1)

R. V. Shack, J. Research Natl. Bur. Standards,  56, 245 (1956).
[Crossref]

Natl. Bur. Standards (U. S.) Circ. 526 (1)

O. H. Schade, Natl. Bur. Standards (U. S.) Circ. 526 (April29, 1954).

Natl. Bur. Standards (U. S.), Circ. (1)

O. H. Schade, “Optical Image Evaluation,” Natl. Bur. Standards (U. S.), Circ. 526(April1954).

Phot. J. (1)

E. H. Selwyn, Phot. J. B88, 6, 46 (1948).

Physica (1)

F. Zernike, Physica 5, 785 (1938).
[Crossref]

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

L. R. Baker, Proc. Phys. Soc. (London) BLXVIII, 871 (1955).

Proc. Roy. Soc. (London) (4)

H. H. Hopkins, Proc. Roy. Soc. (London) A28 (1951).

E. Wolf, Proc. Roy. Soc. (London) A230 (1955).

H. H. Hopkins, Proc. Roy. Soc. (London) A217, 408 (1953).

M. De, Proc. Roy. Soc. (London) 233A, 91 (1955).

RCA Rev. (1)

O. H. Schade, RCA Rev. 9(1948).

Rennes (1)

P. M. Duffieux, “L’Integrale de Fourier et ses applications à l’optique,” Rennes (1946).

Rev. opt. (1)

W. H. Steel, “Étude des effets combines des aberrations et d’une obturation centrale de la pupille sur le contraste des image optique,” Rev. opt. 32, 4–26, 143–178, 269–306 (1953).

Other (11)

A. Marechal, Étude des effets combines de la diffraction et des aberrations geometriques sur l’image d’un point lumineux (Editions de la Revue d’Optique Theorique et Instrumentale, Paris, France, 1948).

R. K. Luneberg, Mathematical Theory of Optics (Brown University, Providence, Rhode Island, 1944).

“Image Evaluation Symposium,” Institute of Optics, University of Rochester (1955).

Proc. Summer School on “Optical Image Assessment Using Frequency Response Techniques,” Imperial College, London (1958).

E. L. O’Neill, Tech. Note 110, BUPRL (January1954).

G. B. Parrent, thesis, Boston University (1955).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), pp. 488–489.

(x0,y0) are coordinates normalized in such a way that both the object point and its Gaussian image may be specified by the same ordered pair of numbers (x0,y0).

B. R. A. Nijboer, “The Diffraction Theory of Aberrations,” thesis, University of Gronigen (1942).

K. Nienhuis, “On the Influence of Diffraction on Image Formation in the Presence of Aberration,” thesis, University of Gronigen (1948).

E. H. Linfoot, Recent Advances in Optics (Clarendon Press, Oxford, England, 1955).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

The coordinate system employed in the diffraction integral. The inclined ellipse represents the reference spherical wavefront of radius R passing through the center of the exit pupil. The Gaussian image point, a distance s from the optical axis, serves as the origin of the physical coordinates ρ′, ψ, z′, in terms of which the optical coordinates q, p are defined as p=πa2z′/R2λ; q=2π(a/R)(ρ′/λ).

Fig. 2
Fig. 2

The symmetric intensity distribution in the point image in the presence of primary spherical aberration of amounts 0, 1 4 λ, and 1 2 λ. The intensity is normalized to 1 at the center of the aberration-free image (A). The abscissa r is identical with q in the text.

Fig. 3
Fig. 3

The five one-dimensional functions derived from the point images of Fig. 2, normalized such that, for the aberration-free image, A(0)=E(∞)=Fs(∞)=Fc(∞)=F(0)=1. The abscissa r in 3-c is identical with q in the text. All abscissas are dimensionless quantities. Physical displacements may be found by multiplying by λ/2πNA, physical frequency by 2πNA/λ.

Fig. 4
Fig. 4

The five one-dimensional functions in the presence of primary astigmatism of three amounts (β22=3, 6, 10) in each of three focal planes ( p = 0 1 2 β 22, β22). The third number in the triplet gives the value of β22=2π times the number of wavelengths of astigmatism considered. The first number gives the value of p. p=0 gives sagittal (Gaussian) focal plane; p=β22 gives tangential focal plane; p = 1 2 β 22 gives plane of the geometrical circle. Normalization as in Fig. 3.

Fig. 5
Fig. 5

Line image (a), edge image (b), flux through hole (c), flux through slit (d), and frequency response (e) for three cases of primary coma. Center number in triplet gives value β31=2π times number of wavelengths of coma considered. First and third numbers in triplet as defined in Fig. 4. Normalization as in Fig. 3.

Fig. 6
Fig. 6

The five functions in the presence of combinations of primary astigmatism and coma. First number of triplet gives value of p; second gives β31; third gives β22. See also Figs. 4 and 5. Normalization as in Fig. 3.

Equations (21)

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

d i z 2 ( x 1 , y 1 ) = O ( x 0 , y 0 ) I z ( x 1 , y 1 ; x 0 , y 0 ) d x 0 d y 0 ,
i z ( x 1 , y 1 ) = - O ( x 0 , y 0 ) I z ( x 1 , y 1 ; x 0 , y 0 ) d x 0 d y 0
i z ( x 1 , y 1 ) = - O ( x 0 , y 0 ) I z ( x 1 - x 0 , y 1 - y 0 ) d x 0 d y 0 .
i z ( x 1 , y 1 ) = I z ( x 1 - x ¯ 0 , y 1 - y ¯ 0 ) .
A ( x 1 - x ¯ 0 ) A ( x ) = - I ( x , y ) d y .
E ( x 1 - x ¯ 0 ) E ( x 0 ) = - x 0 A ( x ) d x .
F c ( r ) = 0 r 0 2 π I ( r , ψ ) r d r d ψ .
F s ( 1 2 W ) = - - 1 2 W 1 2 W I ( x , y ) d x d y ,
F s ( 1 2 W ) = - 1 2 W 1 2 W A ( x ) d x = E ( 1 2 W ) - E ( - 1 2 W ) .
F ( ω ) = - A ( x ) e - 2 π i ω x d x ,
u ( q , ψ , p ) = 1 π 0 1 0 2 π exp { i [ p r 2 + q r cos ( ϕ - ψ ) - k V ] } r d r d ϕ ,
q = [ 2 π NA / λ ] ρ ;             p = [ π ( NA ) 2 / λ ] z .
x = q sin ψ ;             y = q cos ψ
( x , y ) = ( λ x / 2 π NA , λ y / 2 π NA ) .
A ( 0 ) = E ( ) = F c ( ) = F s ( ) = F ( 0 ) = 1 ,
I ( q , p ) = | 1 π 0 1 exp ( i β 40 r 4 - i p r 2 ) J 0 ( q r ) r d r | 2 .
x 0 x n y d y = h i = 0 n δ i y i - E ,
E = ( 3 / 80 ) y ( 4 ) h 5 + ( n - 4 ) ( 2 / 945 ) y ( 6 ) h 7
| 1 π 0 2 π 0 1 exp [ i q r cos ( ϕ - ψ ) - i β 31 ( r 3 - 2 3 r ) × cos ϕ - i β 22 r 2 cos 2 ϕ ] r d r d ϕ | 2 .
r 0 r 4 f ( r ) d r = 4 Δ r 90 ( 7 f 0 + 32 f 1 + 12 f 2 + 32 f 3 + 7 f 4 ) .
F s ( 1 2 W ) = E ( 1 2 W ) - E ( - 1 2 W ) .