Abstract

Analytic expressions are given for the on-axis intensity predicted by the Rayleigh–Sommerfeld and Kirchhoff diffraction integrals for a scalar optical system of high numerical aperture and finite value of Fresnel number. A definition of the axial optical coordinate is introduced that is valid for finite values of Fresnel number, for high-aperture systems, and for observation points distant from the focus. The focal shift effect is reexamined. For the case when the focal shift is small, explicit expressions are given for the focal shift and the axial peak in intensity.

© 2003 Optical Society of America

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References

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  1. H. Osterberg, L. W. Shith, “Closed solutions of Rayleigh’s diffraction integral for axial points,” J. Opt. Soc. Am. 51, 1050–1054 (1961).
    [CrossRef]
  2. G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
    [CrossRef]
  3. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  4. P. Török, “Focusing of electromagnetic waves through a di-electric interface by lenses of finite Fresnel number,” J. Opt. Soc. Am. A 15, 3009–3015 (1998).
    [CrossRef]
  5. C. J. R. Sheppard, “Validity of the Debye approximation,” Opt. Lett. 25, 1660–1662 (2000).
    [CrossRef]
  6. Y. Li, “Dependence of the focal shift on Fresnel number and f number,” J. Opt. Soc. Am. 72, 770–774 (1982).
    [CrossRef]
  7. C. J. R. Sheppard, P. Török, “Dependence of focal shift on Fresnel number and angular aperture,” Opt. Lett. 23, 1803–1804 (1998).
    [CrossRef]
  8. W. Hsu, R. Barakat, “Stratton–Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 11, 623–629 (1994).
    [CrossRef]
  9. 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]
  10. C. J. R. Sheppard, M. Hrynevych, “Diffraction by a half-plane: a generalization of the Fresnel diffraction theory,” Opt. Lett. 16, 1060–1061 (1991).
    [CrossRef] [PubMed]
  11. C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  13. C. J. R. Sheppard, P. Török, “Effect of Fresnel number in focusing and imaging,” in Optics and Optoelectronics Theory, Devices and Applications, O. Nijhawan, A. Gupta, A. Musla, eds. (Naros, New Delhi, India, 1999), Chap. 106, pp. 635–649.

2000 (1)

1998 (2)

1994 (1)

1992 (1)

1991 (1)

1987 (1)

1982 (1)

1981 (1)

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

1961 (1)

1957 (1)

G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[CrossRef]

Barakat, R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Farnell, G.

G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[CrossRef]

Hrynevych, M.

Hsu, W.

Li, Y.

Y. Li, “Dependence of the focal shift on Fresnel number and f number,” J. Opt. Soc. Am. 72, 770–774 (1982).
[CrossRef]

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

Matthews, H. J.

Osterberg, H.

Sheppard, C. J. R.

Shith, L. W.

Török, P.

P. Török, “Focusing of electromagnetic waves through a di-electric interface by lenses of finite Fresnel number,” J. Opt. Soc. Am. A 15, 3009–3015 (1998).
[CrossRef]

C. J. R. Sheppard, P. Török, “Dependence of focal shift on Fresnel number and angular aperture,” Opt. Lett. 23, 1803–1804 (1998).
[CrossRef]

C. J. R. Sheppard, P. Török, “Effect of Fresnel number in focusing and imaging,” in Optics and Optoelectronics Theory, Devices and Applications, O. Nijhawan, A. Gupta, A. Musla, eds. (Naros, New Delhi, India, 1999), Chap. 106, pp. 635–649.

Wolf, E.

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

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Can. J. Phys. (1)

G. Farnell, “Calculated intensity and phase distribution in the image space of a microwave lens,” Can. J. Phys. 35, 777–783 (1957).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

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

Opt. Lett. (3)

Other (2)

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

C. J. R. Sheppard, P. Török, “Effect of Fresnel number in focusing and imaging,” in Optics and Optoelectronics Theory, Devices and Applications, O. Nijhawan, A. Gupta, A. Musla, eds. (Naros, New Delhi, India, 1999), Chap. 106, pp. 635–649.

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Figures (4)

Fig. 1
Fig. 1

Geometry of focusing with the Kirchhoff diffraction integral applied over a spherical surface. A spherical wave is incident on an aperture radius a. The intensity is observed at a point P at a distance z from the geometrical focus F.

Fig. 2
Fig. 2

Behavior of u/u0 with z/f, for different numerical apertures.

Fig. 3
Fig. 3

Behavior of z/f with u/2πN for different numerical apertures. Behavior according to the Li and Wolf theory is also shown.

Fig. 4
Fig. 4

Fractional focal shift as a function of Fresnel number N for different numerical apertures: (a) from analytic expression (17), (b) from approximate expression (9), (c) RSI, (d) RSII, (e) K.

Equations (46)

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U(z)=-ikf2 exp(-ikf)0αexp(ikr)rsin θdθ,
r=(f2+z2+2fz cos θ)1/2.
U(z)=-ikfzexp(-ikf)gf+zexp(ikr)dr,
U(z)=fz{exp[ik(g-f)]-exp(ikz)},
f=(d2+a2)1/2,
g=[(d+z)2+a2]1/2.
I(z)=2fz2 sin2k(g-f-z)2=2fz2 sin2k(f+z-g)2.
u=2k[f+z-g(z)],
I(z)=k(f+z-g)z2sin(u/4)(u/4)2.
πN=k(f-d)=2kf sin2(α/2),
u=4kz(f-d)f+z+g=8kzf sin2(α/2)f+z+g=4πNzf+z+g.
u04kz sin2(α/2)1+(z/f)cos2(α/2),
u02πNzf+z cos2(α/2),
z=u2πN[1-u/(2πN)sin2(α/2)][1-u/(2πN)]f
I=I0[1-u/(2πN)]2[1-u/(2πN)sin2(α/2)]2sin(u/4)u/42,
u(24/πN)cos2(α/2),
Δff-1cos2(α/2)+(π2N2/12)sec2(α/2),
II01+12 cos4(α/2)π2N2.
u=2πN1-tan(β/2)tan(α/2).
URSI(z)=f(d+z)gexp[ik(g-f)]g-f-exp(ikz)z.
URSI(z)=-f exp(ikz)1z-(d+z)exp(-iu/2)g(g-f).
cosδ2=cos α+cot βsin α-sin β=-sin α+sin βcos α-cos β.
URSI(z)=-exp(ikz)×sin β1-cotδ2cos βcos α+cos βexp(-iu/2),
I(z)=I0A(z)sin(u/4)u/42+B(z),
I0=[2k sin2(α/2)]2.
ARSI(z)=sin2(β/2)sin2(α/2)2 cos βcos α+cos β
BRSI(z)=sin βπN sin δcos α-cos δ cos βcos α+cos β2.
URSII(z)=-f exp(ikz)1z-d exp(-iu/2)f(g-f),
URSII(z)=-exp(ikz)sin β1-cotδ2×cos αcos α+cos βexp(-iu/2).
ARSII(z)=sin2(β/2)sin2(α/2)2 cos αcos α+cos β,
BRSII(z)=sin βπN sin δcos β-cos δ cos αcos α+cos β2.
UK(z)=-f exp(ikz)×1z-[f(d+z)+dg]exp(-iu/2)2fg(g-f),
UK(z)=-exp(ikz)sin β1-cotδ2exp(-iu/2).
AK(z)=sin2(β/2)sin2(α/2),
BK(z)=sin β tan(δ/2)2πN2.
URSI(ξp, ηp, ζp)=12πSU(ξ, η, ζ)nexp(ikg)gdS,
URSII(ξp, ηp, ζp)=-12πSU(ξ, η, ζ)nexp(ikg)gdS,
U(ξ, η, ζ)=exp(-ikr)r, r=[ξ2+η2+(ζ-f)2]1/2,
n=rζζ=0r=-frr,
n=-ζpgg.
URSI(0, 0, ζp)=-ζp2π0aexp(-ikr)r1ggexp(ikg)gϱdϱ,
URSII(0, 0, ζp)=f2π0aexp(ikg)g1rrexp(-ikr)rϱdϱ,
exp(-ikr)r1ggexp(ikg)gϱdϱ=-1gexp[ik(g-r)]g-r,
URSI(0, 0, ζp)=ζp2π1gexp[ik(g-r)]g-r0a.
URSII(0, 0, ζp)=f2π1rexp[ik(g-r)]g-r0a.
UK(0, 0, ζp)=14πζpg+frexp[ik(g-r)]g-r0a.

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