Abstract

The Fresnel approximation for off-axis illumination of a circular aperture is reexamined. The point-spread function for an aberration-free system can be expressed in terms of redefined optical coordinates. An improved expression is given for contours of constant intensity in the focal plane. The variation in axial width of the focal spot with angle of offset is discussed. The predictions are compared with exact calculations of the Rayleigh–Sommerfeld diffraction integral. Limitations for application in deconvolution of microscope images formed with objectives of finite tube length are discussed.

© 2004 Optical Society of America

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References

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  1. M. V. R. K. Murty, “On the theoretical limit of resolution,” J. Opt. Soc. Am. 47, 667–668 (1957).
    [CrossRef]
  2. V. A. Zverev, “Illumination distribution in the diffraction image of an off-axis point,” Sov. J. Opt. Technol. 53, 451–454 (1986).
  3. C. J. R. Sheppard, Z. Hegedus, “Resolution for off-axis illumination,” J. Opt. Soc. Am. A 15, 622–624 (1998).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, UK, 1975).
  5. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1980).
  6. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  7. C. J. R. Sheppard, “Imaging in optical systems of finite Fresnel number,” J. Opt. Soc. Am. A 3, 1428–1432 (1986).
    [CrossRef]
  8. S. F. Gibson, F. Lanni, “Diffraction by a circular aperture as a model for three-dimensional optical microscopy,” J. Opt. Soc. Am. A 6, 1357–1367 (1989).
    [CrossRef] [PubMed]
  9. C. J. R. Sheppard, P. P. Roberts, M. Gu, “Fresnel approximation for off-axis illumination of a circular aperture,” J. Opt. Soc. Am. A 10, 984–986 (1993).
    [CrossRef]
  10. C. J. R. Sheppard, M. Hrynevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” J. Opt. Soc. Am. A 9, 274–281 (1992).
    [CrossRef]
  11. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [CrossRef]
  12. C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
    [CrossRef]

1998

1993

1992

1989

1986

C. J. R. Sheppard, “Imaging in optical systems of finite Fresnel number,” J. Opt. Soc. Am. A 3, 1428–1432 (1986).
[CrossRef]

V. A. Zverev, “Illumination distribution in the diffraction image of an off-axis point,” Sov. J. Opt. Technol. 53, 451–454 (1986).

1984

1979

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

1957

Born, M.

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

Gibson, S. F.

Gu, M.

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Hegedus, Z.

Hrynevych, M.

Lanni, F.

Li, Y.

Murty, M. V. R. K.

Roberts, P. P.

Sheppard, C. J. R.

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1980).

Wolf, E.

Zverev, V. A.

V. A. Zverev, “Illumination distribution in the diffraction image of an off-axis point,” Sov. J. Opt. Technol. 53, 451–454 (1986).

Am. J. Phys.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

J. Mod. Opt.

C. J. R. Sheppard, M. Gu, “Imaging by a high aperture optical system,” J. Mod. Opt. 40, 1631–1651 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Sov. J. Opt. Technol.

V. A. Zverev, “Illumination distribution in the diffraction image of an off-axis point,” Sov. J. Opt. Technol. 53, 451–454 (1986).

Other

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

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, Cambridge, UK, 1980).

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Figures (4)

Fig. 1
Fig. 1

Contours of constant v for χ=60°, (a) calculated from the expression of Gibson and Lanni8 [IV, Eq. (15)], (b) calculated from Eq. (19) (V), and (c) as in the theory of Murty [VI, Eq. (20)].

Fig. 2
Fig. 2

Intensity distribution in the focal plane, for λ=633 nm, d=160 mm, χ=60°, and N=10, corresponding to a=1.423 mm, calculated by exact computation of Eq. (1) (I), and the contour for the first minimum (v=3.832) from Murty’s expression [VI, Eq. (20)].

Fig. 3
Fig. 3

Intensity along the line from the center of the aperture to the focal point for offset angles of 30° and 60° and N=10: (II) retaining the astigmatism term [Eq. (24)], (III) improved theory incorporating an extra defocus term dependent on the transverse coordinates of the focus and observation points [Eq. (25)], and (IV) theory of Gibson and Lanni [Eq. (26)]. II was shown to agree with the exact expression I for typical parameters.

Fig. 4
Fig. 4

Relative width of the intensity variation along the line from the center of the aperture to the focal point. The behavior for II and III is shown. The Fresnel number is assumed large. For comparison, IV predicts a constant value of unity.

Equations (33)

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U(P)=-iλAexp[-ik(r-s)]rszPs1+iksdS,
r=rF-aρxFrFcos θ+yFrFsin θ+a2ρ22rF-a2ρ22rFxFrFcos θ+yFrFsin θ2,
s=rP-aρxPrPcos θ+yPrPsin θ+a2ρ22rP-a2ρ22rPxPrPcos θ+yPrPsin θ2,
ξF,P=xF,PrF,P,ηF,P=yF,PrF,P,ζF,P=1rF,P
r-s=(rF-rP)-aρ[(ξF-ξP)cos θ+(ηF-ηP)sin θ]+a2ρ22ζF1-ξF2+ηF22-ζP1-ξP2+ηP22-a2ρ24{[ζF(ξF2-ηF2)-ζP(ξP2-ηP2)]cos 2θ+2(ξFηFζF-ξPηPζP)sin 2θ}.
r-s=(rF-rP)-aρ[(ξF-ξP)cos θ+(ηF-ηP)sin θ].
U(P)=-iNAζP02π01exp[-ik(r-s)]ρdρdθ,
N=a2/λrF.
U(P)=-iNAζP01J0(vρ)exp-12iuρ2ρdρ,
v=ka[(ξP-ξF)2+(ηP-ηF)2]1/2,
u=ka2(ωP-ωF),ωP,F=ζP,F[1-12(ξP,F2+ηP,F2)].
u=ka2(ζP-ζF).
U(P)=-iNAζP01J0(vρ)ρdρ=-iNAζPJ1(v)v,
v=ka[(ξP-sin χ)2+ηP2]1/2=karP[(xP-rP sin χ)2+yP2]1/2,
u=ka2ζP1-ξP2+ηP22-ζF1-sin2 χ2,
xP=xP-d tan χ,
rP=(xP2+2dxP tan χ+yP2+d2 sec2 χ)1/2,
rPd sec χ+xP sin χ+xP22d cos3 χ+yP22d cos χ.
v=kaxPd-tan χxP22d2cos2 χ+yP22d22×cos6 χ+yPd2cos2 χ1/2.
v=kad{xP2 cos6 χ+yP2 cos2 χ}1/2,
u=-2ka2xPd2sin χ cos2 χ(1-34sin2 χ),
u=-ka2xPd2sin χ cos2 χ.
ξP=ξF=ξ=sin χ,
ηP=ηF=0.
I3/I30=rFrP201J0πN2rFrP-1(sin2 χ)t×expiπNrFrP-11-sin2 χ2tdt2.
I2/I20=rFrP2sinπN2rFrP-11-sin2 χ2πN2rFrP-11-sin2 χ22,
I1/I10=rFrP2sinπN2rFrP-1πN2rFrP-12.
ξP=xP(d2+xP2+yP2)1/2,ηP=yP(d2+xP2+yP2)1/2,
ζP=1(d2+xP2+yP2)1/2.
ξF=Mxo(d2+M2(xo2+yo2)1/2,
ηF=Myo(d2+M2(xo2+yo2)1/2,
ζF=d-M2(zo-zs)d(d2+M2(xo2+yo2)1/2,
ζP-ζFM2zod(d2+M2(xo2+yo2)1/2-M2zsd(d2+xP2+yP2)1/2.

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