Abstract

We describe an accurate technique for computing the diffraction point-spread function for optical systems. The approach is based on the combined method of ray tracing and diffraction, which implies that the computation is accomplished in a two-step procedure. First, ray tracing is employed to compute the wave-front error in a reference plane on the image side of the system and to determine the shape of the vignetted pupil. Next the Rayleigh–Sommerfeld diffraction theory, combined with the Kirchhoff approximation and the Stamnes–Spjelkavik–Pedersen method for numerical integration, is applied to compute the field in the region of the image. The method does not rely on small-angle approximations and works well for a pupil of general shape. Both scalar and electromagnetic computations are discussed and numerical results are presented.

© 1998 Optical Society of America

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References

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  1. H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
    [CrossRef]
  2. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, 1986), Sec. 12.6.
  3. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, 1986), Chap. 3, Sec. 6.3.2.
  4. R. Chander, “On tracing rays with specified endpoints,” Geophysics 41, 173–177 (1975).
  5. V. Cerveny and H. Hron, “The ray series method and dynamic ray tracing for 3-D inhomogeneous media,” Bull. Seism. Soc. Am. 70, 47–77 (1980).
  6. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).
  7. Equations (4.16a) and (4.18a) in Ref. 2.
  8. Equation (12.3b) in Ref. 2.
  9. H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. B 70, 1002–1005 (1957).
    [CrossRef]
  10. A. C. Ludwig, “Computation of radiation patterns involving double numerical integration,” IEEE Trans. Antennas Propag. AP-16, 767–769 (1968).
    [CrossRef]
  11. J. J. Stamnes, B. Spjelkavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 1331–1358 (1983).
    [CrossRef]
  12. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, 1986), Sec. 7.2.
  13. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peregrinus, London, 1976).
  14. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40.
  15. Equation (15.66a) in Ref. 2.
  16. Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  17. Ref. 2, Sec. 12.2.
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    [CrossRef]
  19. V. Dhayalan and J. J. Stamnes, “Focusing of electromagnetic waves in a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
    [CrossRef]
  20. J. J. Stamnes and V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
    [CrossRef]
  21. V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317–345 (1997).
    [CrossRef]
  22. V. Dhayalan and J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
    [CrossRef]
  23. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
    [CrossRef]
  24. A. Boivin and E. Wolf, “Electromagnetic fields in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
    [CrossRef]
  25. A. Boivin, J. Dow, and E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. A 57, 1171–1175 (1967).
    [CrossRef]
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    [CrossRef] [PubMed]
  29. H. Ling and S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  30. P. Török, P. Varga, Z. Laczic, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
    [CrossRef]
  31. P. Török, P. Varga, A. Konkol, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. II,” J. Opt. Soc. Am. A 13, 2232–2238 (1996).
    [CrossRef]
  32. D. G. Flagello, T. Milster, and A. E. Rosenbluth, “Theory of high-NA imaging in homogeneous thin films,” J. Opt. Soc. Am. A 13, 53–64 (1996).
    [CrossRef]
  33. S. H. Wiersma and T. D. Visser, “Defocusing of a converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
    [CrossRef]

1998 (1)

V. Dhayalan and J. J. Stamnes, “Focusing of electromagnetic waves in a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

1997 (2)

V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317–345 (1997).
[CrossRef]

V. Dhayalan and J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

1996 (4)

1995 (1)

1987 (1)

1986 (1)

1984 (1)

1983 (1)

J. J. Stamnes, B. Spjelkavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 1331–1358 (1983).
[CrossRef]

1981 (1)

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

1980 (1)

V. Cerveny and H. Hron, “The ray series method and dynamic ray tracing for 3-D inhomogeneous media,” Bull. Seism. Soc. Am. 70, 47–77 (1980).

1975 (1)

R. Chander, “On tracing rays with specified endpoints,” Geophysics 41, 173–177 (1975).

1974 (2)

A. Yoshida and T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system,” Optik (Stuttgart) 40, 322–331 (1974).

A. Yoshida and T. Asakura, “Electromagnetic field near the focus of a Gaussian beam,” Optik (Stuttgart) 41, 281–292 (1974).

1970 (1)

H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

1968 (1)

A. C. Ludwig, “Computation of radiation patterns involving double numerical integration,” IEEE Trans. Antennas Propag. AP-16, 767–769 (1968).
[CrossRef]

1967 (1)

A. Boivin, J. Dow, and E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. A 57, 1171–1175 (1967).
[CrossRef]

1965 (1)

A. Boivin and E. Wolf, “Electromagnetic fields in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

1957 (1)

H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. B 70, 1002–1005 (1957).
[CrossRef]

Asakura, T.

A. Yoshida and T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system,” Optik (Stuttgart) 40, 322–331 (1974).

A. Yoshida and T. Asakura, “Electromagnetic field near the focus of a Gaussian beam,” Optik (Stuttgart) 41, 281–292 (1974).

Barakat, R.

Boivin, A.

A. Boivin, J. Dow, and E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. A 57, 1171–1175 (1967).
[CrossRef]

A. Boivin and E. Wolf, “Electromagnetic fields in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Booker, G. R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40.

Cerveny, V.

V. Cerveny and H. Hron, “The ray series method and dynamic ray tracing for 3-D inhomogeneous media,” Bull. Seism. Soc. Am. 70, 47–77 (1980).

Chander, R.

R. Chander, “On tracing rays with specified endpoints,” Geophysics 41, 173–177 (1975).

Dhayalan, V.

V. Dhayalan and J. J. Stamnes, “Focusing of electromagnetic waves in a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317–345 (1997).
[CrossRef]

V. Dhayalan and J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

J. J. Stamnes and V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

Dow, J.

A. Boivin, J. Dow, and E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. A 57, 1171–1175 (1967).
[CrossRef]

Flagello, D. G.

Hopkins, H. H.

H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. B 70, 1002–1005 (1957).
[CrossRef]

Hron, H.

V. Cerveny and H. Hron, “The ray series method and dynamic ray tracing for 3-D inhomogeneous media,” Bull. Seism. Soc. Am. 70, 47–77 (1980).

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peregrinus, London, 1976).

Konkol, A.

Laczic, Z.

Lee, S. W.

Li, Y.

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Ling, H.

Ludwig, A. C.

A. C. Ludwig, “Computation of radiation patterns involving double numerical integration,” IEEE Trans. Antennas Propag. AP-16, 767–769 (1968).
[CrossRef]

Mansuripur, M.

Milster, T.

Pedersen, H. M.

J. J. Stamnes, B. Spjelkavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 1331–1358 (1983).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Rosenbluth, A. E.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 1331–1358 (1983).
[CrossRef]

Stamnes, J. J.

V. Dhayalan and J. J. Stamnes, “Focusing of electromagnetic waves in a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

V. Dhayalan and J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317–345 (1997).
[CrossRef]

J. J. Stamnes and V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 1331–1358 (1983).
[CrossRef]

Stavroudis, N.

N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).

Török, P.

Varga, P.

Visser, T. D.

Wiersma, S. H.

Wolf, E.

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

A. Boivin, J. Dow, and E. Wolf, “Energy flow in the neighborhood of the focus of a coherent beam,” J. Opt. Soc. Am. A 57, 1171–1175 (1967).
[CrossRef]

A. Boivin and E. Wolf, “Electromagnetic fields in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40.

Yoshida, A.

A. Yoshida and T. Asakura, “Electromagnetic field near the focus of a Gaussian beam,” Optik (Stuttgart) 41, 281–292 (1974).

A. Yoshida and T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system,” Optik (Stuttgart) 40, 322–331 (1974).

Yzuel, M. J.

H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

Appl. Opt. (1)

Bull. Seism. Soc. Am. (1)

V. Cerveny and H. Hron, “The ray series method and dynamic ray tracing for 3-D inhomogeneous media,” Bull. Seism. Soc. Am. 70, 47–77 (1980).

Geophysics (1)

R. Chander, “On tracing rays with specified endpoints,” Geophysics 41, 173–177 (1975).

IEEE Trans. Antennas Propag. (1)

A. C. Ludwig, “Computation of radiation patterns involving double numerical integration,” IEEE Trans. Antennas Propag. AP-16, 767–769 (1968).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Acta (2)

J. J. Stamnes, B. Spjelkavik, and H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 1331–1358 (1983).
[CrossRef]

H. H. Hopkins and M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[CrossRef]

Opt. Commun. (1)

Y. Li and E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Optik (Stuttgart) (2)

A. Yoshida and T. Asakura, “Electromagnetic field in the focal plane of a coherent beam from a wide-angular annular-aperture system,” Optik (Stuttgart) 40, 322–331 (1974).

A. Yoshida and T. Asakura, “Electromagnetic field near the focus of a Gaussian beam,” Optik (Stuttgart) 41, 281–292 (1974).

Phys. Rev. B (1)

A. Boivin and E. Wolf, “Electromagnetic fields in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Proc. Phys. Soc. B (1)

H. H. Hopkins, “The numerical evaluation of the frequency response of optical systems,” Proc. Phys. Soc. B 70, 1002–1005 (1957).
[CrossRef]

Proc. R. Soc. A (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

Pure Appl. Opt. (4)

V. Dhayalan and J. J. Stamnes, “Focusing of electromagnetic waves in a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

J. J. Stamnes and V. Dhayalan, “Focusing of electric-dipole waves,” Pure Appl. Opt. 5, 195–226 (1996).
[CrossRef]

V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317–345 (1997).
[CrossRef]

V. Dhayalan and J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchhoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

Other (10)

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, 1986), Sec. 7.2.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves (Peregrinus, London, 1976).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 40.

Equation (15.66a) in Ref. 2.

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, 1986), Sec. 12.6.

J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, 1986), Chap. 3, Sec. 6.3.2.

N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).

Equations (4.16a) and (4.18a) in Ref. 2.

Equation (12.3b) in Ref. 2.

Ref. 2, Sec. 12.2.

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Figures (14)

Fig. 1
Fig. 1

Diverging spherical wave from O transmitted through an optical system and producing a Gaussian image at O′.

Fig. 2
Fig. 2

Reflection and refraction at a curved interface. At the point at which the incident ray hits the surface, the incident wave is considered to behave locally as a plane wave and the curved interface is replaced by the local tangent plane.

Fig. 3
Fig. 3

Intensity contour map in a plane through the optical axis obtained through (a) on-axis focusing by a parabolic mirror and (b) the classical theory of focusing.

Fig. 4
Fig. 4

Intensity contour map in a plane normal to the optical axis obtained through (a) off-axis focusing by a parabolic mirror and (b) the classical theory of focusing.

Fig. 5
Fig. 5

Axial intensity obtained through on-axis focusing by a parabolic mirror.

Fig. 6
Fig. 6

Intensity contour map in a plane through the optical axis obtained through (a) on-axis focusing by a single plano-convex lens and (b) the classical theory of focusing.

Fig. 7
Fig. 7

Intensity contour map and spot diagram in a plane normal to the optical axis obtained through off-axis focusing by a single plano-convex lens.

Fig. 8
Fig. 8

Intensity contour map in a plane normal to the optical axis and passing through the tangential focus obtained through (a) off-axis focusing by an optimized doublet lens and (b) the classical theory of focusing.

Fig. 9
Fig. 9

Intensity contour map in a plane normal to the optical axis and passing through the central plane obtained through (a) off-axis focusing by an optimized doublet lens and (b) the classical theory of focusing.

Fig. 10
Fig. 10

(a) Focusing by a low-light-level photographic objective. (b) Transverse-aberration curves and a spot diagram in the image plane. (c) An intensity contour map in the image plane.

Fig. 11
Fig. 11

(a) Off-axis focusing by a spherical mirror having a hexagonal aperture with a central elliptical obscuration. (b) An intensity contour map in the image plane.

Fig. 12
Fig. 12

Axial intensity obtained through focusing by a single plano-convex lens with an 8.4λ aberration. (a) The result obtained by use of the CMRD. (b) The results of an independent calculation to check the accuracy.

Fig. 13
Fig. 13

On-axis electromagnetic focusing of a linearly polarized plane wave by a parabolic mirror. The electric-field amplitude squared of (a) the copolarized, (b) the longitudinal, and (c) the cross-polarized components.

Fig. 14
Fig. 14

On-axis electromagnetic focusing of a linearly polarized plane wave by an aspheric mirror giving a spherical aberration of half a wavelength at the edge of the aperture. The electric-field amplitude squared of (a) the copolarized and (b) the longitudinal components.

Equations (22)

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u x ,   y ,   0 ,   t = Re u x ,   y ,   0 exp - i ω t ,
u x ,   y ,   0 = u 0 x ,   y exp ik 0 ϕ x ,   y × exp - ikR 1 R 1 ,
R 1 = x - x 1 2 + y - y 1 2 + z 1 2 1 / 2 .
ϕ x ,   y = S x ,   y - S 0 .
u x 2 ,   y 2 ,   z 2 = - 1 2 π A   u x ,   y ,   0 z 2 exp ikR 2 R 2 d x d y ,
R 2 = x 2 - x 2 + y 2 - y 2 + z 2 2 1 / 2 .
u x 2 ,   y 2 ,   z 2 = A   g x ,   y exp if x ,   y d x d y ,
g x ,   y = 1 λ z 2 R 2 1 R 1 R 2   u 0 x ,   y ,
f x ,   y = k R 2 - R 1 + k 0 ϕ x ,   y - π 2 .
R 2 - R 1 = z 2 + z 1 R 2 + R 1 z 2 - z 1 + 1 z 2 + z 1 x 2 2 - x 1 2 + y 2 2 - y 1 2 - 2 x x 2 - x 1 - 2 y y 2 - y 1 .
E i r ,   t = Re E i r exp - i ω t ,
E i r = E 0 r exp ikr r .
E x ,   y ,   0 = E 1 x ,   y exp ik 0 ϕ x ,   y exp - ikR 1 R 1 ,
E i = A TE e ˆ TE + A TM e ˆ TM i , A TE = e ˆ TE · E i , A TM = e ˆ TM i · E i ,
E t = A TE T TE e ˆ TE + A TM T TM e ˆ TM t ,
T TE = 2   sin   θ t   cos   θ i sin θ i + θ t , T TM = 2   sin   θ t   cos   θ i sin θ i + θ t cos θ i + θ t ,
E r = - E i + 2 E i · n ˆ n ˆ ,
k r = k i - 2 k i · n ˆ n ˆ ,
E x 2 ,   y 2 ,   z 2 = - 1 2 π A E x ,   y ,   0 z 2 exp ikR 2 R 2 d x d y ,
E x 2 ,   y 2 ,   z 2 = A g x ,   y exp if x ,   y d x d y ,
g x ,   y = 1 λ E 1 x ,   y R 1 R 2 z 2 R 2 ,
f x ,   y = k R 2 - R 1 + k 0 ϕ x ,   y - π 2 ,

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