Abstract

Part II of this study is an application of the Rayleigh vector diffraction integrals to an investigation of the effect of focal shifts in converging spherical waves diffracted in systems of arbitrary relative aperture. The results are compared numerically with those obtained in Part I [J. Opt. Soc. Am. A 22, 68 (2005)] from the Kirchhoff vector diffraction theory. The effect of the numerical aperture (NA) on focal shifts can be considered in two regions: When NA0.5 the system behaves like an paraxial system, and the Fresnel number is the dominant factor. When 0.5<NA0.9 the absolute value of the relative focal shift decreases with increasing value of NA.

© 2005 Optical Society of America

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References

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  1. Y. Li, “Focal shifts in diffracted converging electromagnetic waves. I. Kirchhoff theory,” J. Opt. Soc. Am. A 22, 68 (2005).
    [CrossRef]
  2. R. K. Lunenberg, Mathematical Theory of Optics (University of California, Berkeley, California, 1964), Chap. 6.
  3. M. Klein, I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, New York, 1965).
  4. M. Born, E. Wolf, Principles of Optics, 9th ed. (Pergamon, New York, 2001), Sec. 8.11.
  5. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  6. Y. Li, “Focal shift in small-Fresnel-number focusing systems of different relative aperture,” J. Opt. Soc. Am. A 20, 234–239 (2003).
    [CrossRef]
  7. H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a subwavelength slit,” Opt. Exp. 11, 371–380 (2003).
    [CrossRef]
  8. H. F. Schouten, T. D. Visser, D. Lenstra, “Light transmission through a subwavelength slit: waveguide and optical vortices,” Phys. Rev. E 67, 036608-1–036608-4 (2003).
    [CrossRef]
  9. W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel number,” Opt. Commun. 119, 453–459 (1993).
    [CrossRef]
  10. H. Osterberg, L. W. Smith, “Close solution of Rayleigh diffraction integral for axial points,” J. Opt. Soc. Am. 51, 1050–1054 (1961).
    [CrossRef]
  11. C. J. R. Sheppard, P. Török, “Focal shift and axial coordinates for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20, 2156–2162 (2003).
    [CrossRef]

2005 (1)

2003 (4)

C. J. R. Sheppard, P. Török, “Focal shift and axial coordinates for high-aperture systems of finite Fresnel number,” J. Opt. Soc. Am. A 20, 2156–2162 (2003).
[CrossRef]

Y. Li, “Focal shift in small-Fresnel-number focusing systems of different relative aperture,” J. Opt. Soc. Am. A 20, 234–239 (2003).
[CrossRef]

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a subwavelength slit,” Opt. Exp. 11, 371–380 (2003).
[CrossRef]

H. F. Schouten, T. D. Visser, D. Lenstra, “Light transmission through a subwavelength slit: waveguide and optical vortices,” Phys. Rev. E 67, 036608-1–036608-4 (2003).
[CrossRef]

1993 (1)

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel number,” Opt. Commun. 119, 453–459 (1993).
[CrossRef]

1981 (1)

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

1961 (1)

Block, H.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a subwavelength slit,” Opt. Exp. 11, 371–380 (2003).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 9th ed. (Pergamon, New York, 2001), Sec. 8.11.

Gbur, G.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a subwavelength slit,” Opt. Exp. 11, 371–380 (2003).
[CrossRef]

Kay, I.

M. Klein, I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, New York, 1965).

Klein, M.

M. Klein, I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, New York, 1965).

Lenstra, D.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a subwavelength slit,” Opt. Exp. 11, 371–380 (2003).
[CrossRef]

H. F. Schouten, T. D. Visser, D. Lenstra, “Light transmission through a subwavelength slit: waveguide and optical vortices,” Phys. Rev. E 67, 036608-1–036608-4 (2003).
[CrossRef]

Li, Y.

Lunenberg, R. K.

R. K. Lunenberg, Mathematical Theory of Optics (University of California, Berkeley, California, 1964), Chap. 6.

Osterberg, H.

Schouten, H. F.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a subwavelength slit,” Opt. Exp. 11, 371–380 (2003).
[CrossRef]

H. F. Schouten, T. D. Visser, D. Lenstra, “Light transmission through a subwavelength slit: waveguide and optical vortices,” Phys. Rev. E 67, 036608-1–036608-4 (2003).
[CrossRef]

Sheppard, C. J. R.

Smith, L. W.

Török, P.

Visser, T. D.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a subwavelength slit,” Opt. Exp. 11, 371–380 (2003).
[CrossRef]

H. F. Schouten, T. D. Visser, D. Lenstra, “Light transmission through a subwavelength slit: waveguide and optical vortices,” Phys. Rev. E 67, 036608-1–036608-4 (2003).
[CrossRef]

Wang, W.

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel number,” Opt. Commun. 119, 453–459 (1993).
[CrossRef]

Wolf, E.

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel number,” Opt. Commun. 119, 453–459 (1993).
[CrossRef]

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, 9th ed. (Pergamon, New York, 2001), Sec. 8.11.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel number,” Opt. Commun. 119, 453–459 (1993).
[CrossRef]

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

Opt. Exp. (1)

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a subwavelength slit,” Opt. Exp. 11, 371–380 (2003).
[CrossRef]

Phys. Rev. E (1)

H. F. Schouten, T. D. Visser, D. Lenstra, “Light transmission through a subwavelength slit: waveguide and optical vortices,” Phys. Rev. E 67, 036608-1–036608-4 (2003).
[CrossRef]

Other (3)

R. K. Lunenberg, Mathematical Theory of Optics (University of California, Berkeley, California, 1964), Chap. 6.

M. Klein, I. Kay, Electromagnetic Theory and Geometrical Optics (Wiley, New York, 1965).

M. Born, E. Wolf, Principles of Optics, 9th ed. (Pergamon, New York, 2001), Sec. 8.11.

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Figures (4)

Fig. 1
Fig. 1

Geometry of vector diffraction at a circular aperture. (a) Definition of the coordinate systems and the positional vectors for aperture point Q and observation point P, (b) definition of characteristic distances.

Fig. 2
Fig. 2

Energy density distributions along the axis in systems of Fresnel number N=1 and NA from 0.1 to 0.9. Solid curves, systems of a10λ; dashed curves, systems of 10λ>a>λ. (a) Aplanatic systems, (b) parabolic mirrors.

Fig. 3
Fig. 3

Energy density distributions along the axis in aplanatic systems of Fresnel number N=0.5 and NA from 0.1 and 0.9. Solid curves, systems of a10λ; dashed curves for systems of 10λ>a>λ; dashed–dotted curves for systems of aλ.

Fig. 4
Fig. 4

Energy density distributions along the axis in aplanatic systems of Fresnel number N=1 and NA from 0.1 to 0.9, calculated by ignoring (ks)-1 in the d(s)=[1-i(ks)-1] term in the integrand of the diffraction integral. Here s is the distance between the aperture point Q and the point of observation point P. Solid curves, systems of a10λ; dashed curves, systems of 10λ>a>λ.

Tables (2)

Tables Icon

Table 1 Fractional Focal Shift Δf/f as a Function of the Fresnel Number N and NA for an Aplanatic System

Tables Icon

Table 2 Relative Excess of the Energy Density ΔwE/wE(z0) as a Function of the Fresnel Number N and NA for an Aplanatic System

Equations (55)

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Ex(r)=-12π (z=0)Ex(0)(ρ) G(r, ρ)z d2ρ,
Ey(r)=-12π (z=0)Ey(0)(ρ) G(r, ρ)z d2ρ,
Ez(r)=12π (z=0)Ex(0)(ρ) G(r, ρ)x+Ey(0)(ρ) G(r, ρ)yd2ρ,
ρ=iξ+jη,  ρ=|ρ|=ξ2+η2,
r=ix+jy+kz,  r=|r|=x2+y2+z2.
G(r, ρ)=exp(ik|r-ρ|)|r-ρ|=exp(iks)s  (s|r-ρ|),
G(r, ρ)q=ik-1s q-q0s G(r, ρ)=ikd(s) q-q0s                  exp(iks)s,
d(s)=1-1ik|r-ρ|=1-1iks.
Ex(r)=-ik2π (z=0)Ex(0)(ρ)d(s) zsexp(iks)s d2ρ,
Ey(r)=-ik2π (z=0)Ey(0)(ρ)d(s) zsexp(iks)s d2ρ,
Ez(r)=ik2π (z=0)x-ξs Ex(0)(ρ)+y-ηs Ey(0)(ρ)d(s) exp(iks)s d2ρ.
ξ=ρ cos ϕ,
η=ρ sin ϕ.
s=[(x-ξ)2+(y-η)2+z2]1/2=[r2+ρ2-2ρ(xcosϕ+ysinϕ)]1/2.
Ex(r)=-ik2π ρ=0ρ=aϕ=0ϕ=2πEx(0)(ρ, ϕ)d(s)×zsexp(iks)s ρdρdϕ,
Ey(r)=-ik2π ρ=0ρ=aϕ=0ϕ=2πEy(0)(ρ, ϕ)d(s)×zsexp(iks)s ρdρdϕ,
Ez(r)=ik2π ρ=0ρ=aϕ=0ϕ=2πEx(0)(ρ, ϕ) x-ρ cos ϕs+Ey(0)(ρ, ϕ) y-ρ sin ϕs×d(s)exp(iks)s ρdρdϕ,
Ei(g)=E(g)+E(g)=|E0|g0(θ)×cos2 ϕ cos θ+sin2 ϕ-cos ϕ sin ϕ(1-cos θ)cos ϕ sin θ,
g0(θ)=cos θ  (aplanatic system),
g0(θ)=11+cos θ  (parabolic mirror).
E(0)(ρ, ϕ)=iEx(0)(ρ, ϕ)+jEy(0)(ρ, ϕ)+kEz(0)(ρ, ϕ)=Ei(g) exp(ikΔ)l0/f=Ei(g) exp(ikΔ)cos Ω/cos θ  (ρ<a),
E(0)(ρ, ϕ)=0  (otherwise),
Δ=f-z0cos θ=z01cos Ω-1cos θ,
NA=a/f=sin Ω=sin θmax.
Ex(d)(r)=-ik|E0|2π ρ=0ρ=aϕ=0ϕ=2π                  cos2 ϕ cos θ+sin2 ϕcos Ω/cos θ×d(s) zsexp[ik(Δ+s)]s g0(θ)ρdρdϕ,
Ey(d)(r)=-ik|E0|2π ρ=0ρ=aϕ=0ϕ=2πcos ϕ sin ϕ(1-cos θ)cos Ω/cos θ×d(s) zsexp[ik(Δ+s)]s g0(θ)ρdρdϕ,
Ez(d)(r)=ik|E0|2π ρ=0ρ=aϕ=0ϕ=2π×cos2 ϕ cos θ+sin2 θcos Ω/cos θx-ρ cos ϕs-cos ϕ sin ϕ(1-cos θ)cos Ω/cos θy-ρ sin ϕs×d(s) exp[ik(Δ+s)]s g0(θ)ρdρdϕ.
s=z2+ρ2.
Ex(d)(z)=-ikz|E0|2π ρ=0ρ=aϕ=0ϕ=2π(cos2 ϕ sin θ+sin2 ϕ)d(s)×(cos θ)3cos Ωexp[ik(Δ+z2+ρ2)]z2+ρ2 ρdρdϕ,
Ey(d)(z)=-ikz|E0|2π ρ=0ρ=aϕ=0ϕ=2π[cos ϕ sin ϕ(1-cos θ)]d(s)×(cos θ)3cos Ωexp[ik(Δ+z2+ρ2)]z2+ρ2×ρdρdϕ,
Ez(d)(z)=-ik|E0|2π ρ=0ρ=aϕ=0ϕ=2π[(cos2 ϕ cos θ+sin2 θ)×cos ϕ-cos ϕ sin2 ϕ(1-cos θ)] (cos θ)3cos Ω×d(s)exp[ik(Δ+z2+ρ2)]z2+ρ2 ρdρdϕ.
Ey(d)(z)=Ez(d)(z)=0.
Ei(d)(0, 0, z)=iEx(d)(z)=-i ikz|E0|2 ρ=0ρ=a(1+cos θ) (cos θ)3cos Ω d(s)×exp[ik(Δ+z2+ρ2)]z2+ρ2 ρdρ.
WE(0, 0, z)=|Ei(d)(0, 0, z)|2=|E0|2πzλ ρ=0ρ=a(1+cos θ) (cos θ)3cos Ω d(s)×exp[ik(Δ+z2+ρ2)]z2+ρ2 ρdρ2.
ρ=z0 tan θ.
wE(z)=WE(0, 0, z)WE(0, 0, z0)=zz02θ=0θ=Ωb(θ)d(s) exp{iK[Δ/z0+(z/z0)2+tan2 θ]}(z/z0)2+tan2 θ dθθ=0θ=Ωb(θ)d(s) exp[iK(Δ/z0+1+tan2 θ)]1+tan2 θ dθ2,
b(θ)=(1+cos θ)tan θcos θ,
K=kz0=2πN cos Ωsin2 Ω=2πN 1-(NA)2(NA)2.
N=a2/λf.
b(θ)tan θcos θ.
Δff=zm-z0f=zm-z0z0cos Ω,
C(z, θ)[C(z, θ)+(z/z0)2C(z, θ)]+S(z, θ)[S(z, θ)+(z/z0)2S(z, θ)]=0,
C(z, θ)=θ=0θ=ΩB0(z, θ)cos Ψ(z, θ)+1Kτsin Ψ(z, θ)dθ,
S(z, θ)=θ=0θ=ΩB0(z, θ)sin Ψ(z, θ)-1Kτcos Ψ(z, θ)dθ,
C(z, θ)=-θ=0θ=ΩKB1(z, θ)+3K B3(z, θ)×sinΨ(z, θ)dθ-θ=0θ=ΩB2(z, θ)cos Ψ(z, θ)dθ,
S(z, θ)=θ=0θ=ΩKB1(z, θ)+3K B3(z, θ)×cos Ψ(z, θ)dθ-θ=0θ=ΩB2(z, θ)sin Ψ(z, θ)dθ,
Bp(z, θ)=b(θ)τ-(2+p) (p=0, 1, 2, 3),
Ψ(z, θ)=K(Δ/z0+τ).
ΔwEwE(z0)=(wE)max-1,
ks1,  d(s)1.
b(θ)=(1+cos θ)tan θcos θθ,
K=2πN cos Ωsin2 Ω2πN 1Ω2.
wE(z)wE(uN)=1-uN2πN2sin(uN/4)uN/42,
uN=2πNz-fz=2πNzf+z  (z=z-f).
tan(uN/4)uN/4=1-uN2πN.

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