Abstract

Two mathematical innovations are presented that relate to calculating propagation of radiation through cylindrically symmetrical systems using Kirchhoff diffraction theory. The first innovation leads to an efficient means of computing Lommel functions of two arguments (u and ν), typically denoted by U n(u, ν) and V n(u, ν). This can accelerate computations involving Fresnel diffraction by circular apertures or lenses. The second innovation facilitates calculations of Kirchhoff diffraction integrals without recourse to the Fresnel approximation, yet with greatly improved efficiency like that characteristic of the latter approximation.

© 2001 Optical Society of America

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References

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  1. E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kresirunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).
  2. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).
  3. See, for example, W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 96.
  4. J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
    [CrossRef]
  5. W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970).
    [CrossRef]
  6. W. H. Steel, M. De, J. A. Bell, “Diffraction corrections in radiometry,” J. Opt. Soc. Am. 62, 1099–1103 (1972).
    [CrossRef]
  7. L. P. Boivin, “Diffraction corrections in radiometry: comparison of two different methods of calculation,” Appl. Opt. 14, 2002–2009 (1975).
    [CrossRef] [PubMed]
  8. L. P. Boivin, “Diffraction corrections in the radiometry of extended sources,” Appl. Opt. 15, 1204–1209 (1976).
    [CrossRef] [PubMed]
  9. E. L. Shirley, “Revised formulas for diffraction effects with point and extended sources,” Appl. Opt. 37, 6581–6590 (1998).
    [CrossRef]
  10. K. Schwarzschild, “Die Beugungsfigure im Fernrohr weit ausserhalb des Focus,” Sitzungsber. München Akad. Wiss., Math.-Phys. Kl. 28, 271–294 (1898).
  11. K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998).
    [CrossRef]
  12. In both innovations, exact or full calculations use exact formulas involving sums of Bessel functions [Eqs. (1), (2) and (31)]. However, the Bessel functions are numerically evaluated by sound and reasonably optimized methods.
  13. J. J. Stamnes, Waves in Focal Regions (Hilger, Boston, Mass., 1986).
  14. See, for example, F. W. J. Olver, Asymptotics and Special Functions (Peters, Wellesley, Mass., 1997), p. 238.
  15. See, for example, Ref. 14, p. 285.
  16. In practice the Faddeeva function is most helpful and has been calculated with software downloaded from http://gams.nist.gov ; G. P. M. Poppe, C. M. J. Wijers, “Algorithm 680: evaluation of the complex error function,” ACM (Assoc. Comput. Mach.) Trans. Math. Software 16, 47 (1990).
    [CrossRef]
  17. A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs, N.J., 1966).
  18. In this approximation the normal derivatives of the incident radiation field and its Green’s function are assumed to be ±ik times the respective quantities on the surface of an aperture, whichever sign is appropriate. Analogous approximations would be made in the case of diffraction by a lens.
  19. The combination of substitutions being made has the advantage that it leads to the correct answer in the limit of an infinite aperture.
  20. The various boundary conditions are discussed, for example, in J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  21. L. P. Boivin, “Reduction of diffraction errors in radiometry by means of toothed apertures,” Appl. Opt. 17, 3323–3328 (1978).
    [CrossRef] [PubMed]

1998

E. L. Shirley, “Revised formulas for diffraction effects with point and extended sources,” Appl. Opt. 37, 6581–6590 (1998).
[CrossRef]

K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998).
[CrossRef]

1978

1976

1975

1972

1970

W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970).
[CrossRef]

1956

J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
[CrossRef]

1898

K. Schwarzschild, “Die Beugungsfigure im Fernrohr weit ausserhalb des Focus,” Sitzungsber. München Akad. Wiss., Math.-Phys. Kl. 28, 271–294 (1898).

1885

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kresirunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).

Bell, J. A.

Blevin, W. R.

W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970).
[CrossRef]

Boivin, L. P.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

De, M.

Focke, J.

J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
[CrossRef]

Jackson, J. D.

The various boundary conditions are discussed, for example, in J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Lommel, E.

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kresirunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).

Mielenz, K. D.

K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998).
[CrossRef]

Olver, F. W. J.

See, for example, F. W. J. Olver, Asymptotics and Special Functions (Peters, Wellesley, Mass., 1997), p. 238.

Schwarzschild, K.

K. Schwarzschild, “Die Beugungsfigure im Fernrohr weit ausserhalb des Focus,” Sitzungsber. München Akad. Wiss., Math.-Phys. Kl. 28, 271–294 (1898).

Secrest, D.

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs, N.J., 1966).

Shirley, E. L.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Boston, Mass., 1986).

Steel, W. H.

Stroud, A. H.

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs, N.J., 1966).

Welford, W. T.

See, for example, W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 96.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

Abh. Bayer. Akad.

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kresirunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).

Appl. Opt.

J. Opt. Soc. Am.

J. Res. Natl. Inst. Stand. Technol.

K. D. Mielenz, “Algorithms for Fresnel diffraction at rectangular and circular apertures,” J. Res. Natl. Inst. Stand. Technol. 103, 497–509 (1998).
[CrossRef]

Metrologia

W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970).
[CrossRef]

Opt. Acta

J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
[CrossRef]

Sitzungsber. München Akad. Wiss., Math.-Phys. Kl.

K. Schwarzschild, “Die Beugungsfigure im Fernrohr weit ausserhalb des Focus,” Sitzungsber. München Akad. Wiss., Math.-Phys. Kl. 28, 271–294 (1898).

Other

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975).

See, for example, W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), p. 96.

In both innovations, exact or full calculations use exact formulas involving sums of Bessel functions [Eqs. (1), (2) and (31)]. However, the Bessel functions are numerically evaluated by sound and reasonably optimized methods.

J. J. Stamnes, Waves in Focal Regions (Hilger, Boston, Mass., 1986).

See, for example, F. W. J. Olver, Asymptotics and Special Functions (Peters, Wellesley, Mass., 1997), p. 238.

See, for example, Ref. 14, p. 285.

In practice the Faddeeva function is most helpful and has been calculated with software downloaded from http://gams.nist.gov ; G. P. M. Poppe, C. M. J. Wijers, “Algorithm 680: evaluation of the complex error function,” ACM (Assoc. Comput. Mach.) Trans. Math. Software 16, 47 (1990).
[CrossRef]

A. H. Stroud, D. Secrest, Gaussian Quadrature Formulas (Prentice-Hall, Englewood Cliffs, N.J., 1966).

In this approximation the normal derivatives of the incident radiation field and its Green’s function are assumed to be ±ik times the respective quantities on the surface of an aperture, whichever sign is appropriate. Analogous approximations would be made in the case of diffraction by a lens.

The combination of substitutions being made has the advantage that it leads to the correct answer in the limit of an infinite aperture.

The various boundary conditions are discussed, for example, in J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

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Figures (4)

Fig. 1
Fig. 1

Aperture or lens plane, plane of observation, and point of observation O are indicated for three cases. In each case, the combination of the incident wave and the lens or aperture causes there to be a circularly defined plane wave (top), divergent spherical wave (middle), or convergent spherical wave (bottom). In the convergent case, the focal plane and sign of u are indicated. The sign of u is ambiguous, because it depends on the manner in which the radiation field is defined in terms of Lommel functions. The ν = u cone, which defines the boundary between the geometrically illuminated and shadow regions, is indicated. In the top case, the sample O is in the shadow region; otherwise, it is in the illuminated region.

Fig. 2
Fig. 2

Required computation time for determining diffraction effects in seven geometries as a function of wavelength, in cases of exact evaluation of Lommel functions (Exact Calc.) and by the method developed in this paper [Eq. (14)].

Fig. 3
Fig. 3

Cylindrically symmetrical setup with six apertures (S, D, and A 1A 4), that we considered for examining diffraction effects. Transmission was computed for light incident as plane waves at various values of angle of incidence (θ), with transmission being the fraction of light entering aperture S that eventually passed through aperture D. The vertical scale is neither uniform nor consistent with the horizontal scale.

Fig. 4
Fig. 4

Amount of flux passing through aperture D for the case of a plane wave incident on S at an angle of incidence θ (compare with Fig. 3). The flux is normalized so that, in geometrical optics, one would have πr s 2 at θ = 0 mrad, where r s is the radius of aperture S. Results are plotted for the cases of none of the four A i apertures being present (solid curve), one of the four apertures being present (dashed curves), and all of the apertures being present (dotted curve). Each angle θ i is the angle between the optic axis and a line segment from the center of S to the edge of the respective aperture A i . Effects of the presence of each aperture A i are largest near the respective angle θ i , where there is one large peak resulting because of that aperture.

Tables (2)

Tables Icon

Table 1 Values of Vn (1000, ν) by Several Methods

Tables Icon

Table 2 Results for the Performance of Eqs. (46) and (31)

Equations (51)

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Unu, ν=s=0-1su/νn+2sJn+2sν,
Vnu, ν=s=0-1sν/un+2sJn+2sν.
Unu, ν=Vnν2/u, ν;  Vnu, ν=Unν2/u, ν.
Jmν2/πν cosν-mπ/2-π/4,
Vnu, νν/un2/πν1-ν2/u2-1×cosν-nπ/2-π/4.
Jmν2/πνcos ζ s=0-1sA2smν-2s-sin ζ s=0-1sA2s+1mν-2s+1,
ζ=ν-mπ/2-π/4
Asm=4m2-124m2-324m2-2s-12/s!8s.
Jmν2/πνcosν-mπ/2-π/4+m2/2ν,
Γnu, ν2/πνν/un expiν-nπ/2-π/4+n2/2ν,
V˜nu>ν, ν=Γnu, νs=0 αs expβs2.
αs expβs2=αsc/π-dh exp-ch2+sh1+i/2.
s=0S-1 αs expβs2=cπ-dh exp-ch2×1-αS expSh1+i/21-α exph1+i/2.
V˜nu>ν, ν=cπ Γnu, ν×-dh exp-ch21-α exph1+i/2.
-dh exp-ch21-α exph1+i/2=I1+I2,
I1=-dh exp-ch2-1Λ+h1+i/2,
I1=1+i2expicΛ2π erfc-1+i2c1/2Λ,
I2=-dh exp-ch211-α exph1+i/2+1Λ+h1+i/2,
limνu Re V˜nu>ν, ν=1-nJnν+cosν-nπ/2/2+Oν-3/2, 
V0ν, ν=J0ν+cosν/2,
V1ν, ν=sinν/2,
V2ν, ν=J0ν-cosν/2=-J2ν-cosν/2+Oν-3/2.
ψz1, r, Θ=m fmrexpimΘ,
ψz2, r, Θ=m gmrexpimΘ.
gms=k2πi0drrfmr×02πdΦ expimΦ+ikb2+r2+s2-2rs cos Φ1/2b2+r2+s2-2rs cos Φ1/2
gms=k22πi0drrfmrImk, b, r, s.
u=α+β+α-β/2,
ν=α+β-α-β/2,
Imk, b, r, s=02πdΦ expimΦ+iα-β cos Φ1/2α-β cos Φ1/2
Imk, b, r, s=02πdΦ expimΦ+iu2+ν2-2uνcos Φ1/2u2+ν2-2uνcos Φ1/2.
Imk, b, r, s=i l=0 jlνjlu+iylu2l+1×02πdΦ expimΘPlcos Φ.
02πdΦ expimΦPlcos Φ=πl-m-1!!l+m-1!!1+-1l+m1+δm02l+1l-m/2!l+m/2!.
α-β cos Φ1/2kb+r2+s2/2b-rs/bcos Φ
α-β cos Φ1/2kb
Imk, b, r, s2πi-mkb-1 expikb+r2+s2/2bJmkrs/b,
α-β cos Φ1/2=A-B cos Ψ.
Imk, b, r, s=02πdΨdΦdΨ×expimΦ+iA-iB cos Ψα-β cos Φ1/2.
dΦdΨ=2B sin Ψβ sin Φα-β cos Φ1/2,
Imk, b, r, s=02πdΨ2B sin Ψβ sin ΦexpiA+mΦ-B cos Ψ.
Φ-Ψ=-B2/βsin Ψ+,
LmΨ=2B sin Ψβ sin ΦexpimΦ-Ψ+B2βsin Ψ,
Imk, b, r, s=02πdΨLmΨexpiA+mΨ-B cos Ψ-mB2/βsin Ψ.
KmΨ=LmΨexpiBcosΨ-mB/β-cos Ψ-mB/βsin Ψ.
KmΨ=νexpiνΨKm,ν.
Imk, b, r, s=02πdΨKmΨexpiA+mΨ-B cosΨ-mB/β,
Imk, b, r, s=2π ν Km,νJν+mBexpiA-ν+m×π/2+ν+mmB/β.
dI/db=dI/dαdα/db,
d2I/db2=dI/dαd2α/db2+d2I/dα2dα/db2.
In=02πdΨKmΨexpiA+mΨ-B cosΨ-mB/β/A-B cos Ψn,
dI/dα=iI1-I2/2,
d2I/dα2=-I2-3iI3+3I4/4.

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