Abstract

Starting with the vector formulation of the Kirchhoff diffraction theory, expressions for the total energy density distribution along the axis are presented without using any of the usual assumptions except the assumption made by Kirchhoff for the boundary conditions of a black screen. To make the Kirchhoff integral compatible with Maxwell’s equations, a line integral around the edge of the aperture is added in the analysis. The consequence of ignoring the contribution of this line integral to the axial field distribution is examined numerically. The focal shift effect is investigated for both aplanatic systems and parabolic mirrors having an arbitrary numerical aperture (NA) and finite value of the Fresnel number. The combined effects of the Fresnel number and NA on the focal shift are evaluated, and the validity of the results is carefully checked by comparing the wavelength with the system dimensions.

© 2005 Optical Society of America

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References

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  1. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  2. Y. Li, E. Wolf, “Focal shift in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  3. Y. Li, “Focal shift in small-Fresnel-number focusing systems of different relative aperture,” J. Opt. Soc. Am. A 20, 234–239 (2003).
    [CrossRef]
  4. J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, A. S. van de Nes, “Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
    [CrossRef]
  5. W. Hsu, R. Barakat, “Stratton–Chu vectorial diffraction of electromagnetic fields by aperture with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 11, 623–629 (1994).
    [CrossRef]
  6. J. Pu, B. Lu, S. Nemoto, “Three-dimensional intensity distribution of focused annular non-uniformly polarized beams,” J. Mod. Opt. 49, 1501–1513 (2002).
    [CrossRef]
  7. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
    [CrossRef]
  8. R. Barakat, “The intensity distribution and total illumination of aberration-free diffraction image,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961).
  9. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel number,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  10. T. D. Visser, S. H. Wiersma, “Diffraction of converging electromagnetic waves,” J. Opt. Soc. Am. A 9, 2034–2047 (1992).
    [CrossRef]
  11. M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, New York, 1999), Sec. 8.11.
  12. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–370 (1959).
    [CrossRef]
  13. V. S. Ignatovsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 4 (1920).
  14. V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 5 (1920).
  15. R. Barakat, “Diffraction of converging electromagnetic fields in the neighborhood of the focus of a parabolic mirror having a central obscuration,” Appl. Opt. 26, 3790–3795 (1987).
    [CrossRef] [PubMed]
  16. Y. Li, “Focal shift formulae,” Optik (Stuttgart) 69, 41–42 (1984).
  17. H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a sub-wavelength slit,” ACI Mater. J. 11, 371–380 (2003).
  18. 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]
  19. M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular aperture and complementary obstacles,” J. Appl. Phys. 26, 336–345 (1955).
    [CrossRef]
  20. G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
    [CrossRef]
  21. Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
    [CrossRef]
  22. W. Wang, E. Wolf, “Far-zone behavior of focused fields in systems with different Fresnel number,” Opt. Commun. 119, 453–459 (1993).
    [CrossRef]

2003 (4)

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a sub-wavelength slit,” ACI Mater. J. 11, 371–380 (2003).

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]

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

J. J. M. Braat, P. Dirksen, A. J. E. M. Janssen, A. S. van de Nes, “Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high-aperture optical system,” J. Opt. Soc. Am. A 20, 2281–2292 (2003).
[CrossRef]

2002 (1)

J. Pu, B. Lu, S. Nemoto, “Three-dimensional intensity distribution of focused annular non-uniformly polarized beams,” J. Mod. Opt. 49, 1501–1513 (2002).
[CrossRef]

1994 (1)

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]

1992 (1)

1987 (1)

1984 (2)

1983 (1)

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

1981 (2)

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

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

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–370 (1959).
[CrossRef]

1958 (1)

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[CrossRef]

1955 (1)

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular aperture and complementary obstacles,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

1954 (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

1920 (2)

V. S. Ignatovsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 4 (1920).

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 5 (1920).

Barakat, R.

Block, H.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a sub-wavelength slit,” ACI Mater. J. 11, 371–380 (2003).

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, New York, 1999), Sec. 8.11.

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

Braat, J. J. M.

Dirksen, P.

Ehrlich, M. J.

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular aperture and complementary obstacles,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

Farnell, G. W.

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[CrossRef]

Gbur, G.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a sub-wavelength slit,” ACI Mater. J. 11, 371–380 (2003).

Held, G.

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular aperture and complementary obstacles,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

Hsu, W.

Ignatovsky, V. S.

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 5 (1920).

V. S. Ignatovsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 4 (1920).

Janssen, A. J. E. M.

Lenstra, D.

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]

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a sub-wavelength slit,” ACI Mater. J. 11, 371–380 (2003).

Li, Y.

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

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel number,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[CrossRef]

Y. Li, “Focal shift formulae,” Optik (Stuttgart) 69, 41–42 (1984).

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

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

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

Lu, B.

J. Pu, B. Lu, S. Nemoto, “Three-dimensional intensity distribution of focused annular non-uniformly polarized beams,” J. Mod. Opt. 49, 1501–1513 (2002).
[CrossRef]

Nemoto, S.

J. Pu, B. Lu, S. Nemoto, “Three-dimensional intensity distribution of focused annular non-uniformly polarized beams,” J. Mod. Opt. 49, 1501–1513 (2002).
[CrossRef]

Platzer, H.

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Pu, J.

J. Pu, B. Lu, S. Nemoto, “Three-dimensional intensity distribution of focused annular non-uniformly polarized beams,” J. Mod. Opt. 49, 1501–1513 (2002).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–370 (1959).
[CrossRef]

Schouten, H. F.

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a sub-wavelength slit,” ACI Mater. J. 11, 371–380 (2003).

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]

Silver, S.

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular aperture and complementary obstacles,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

van de Nes, A. S.

Visser, T. D.

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]

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a sub-wavelength slit,” ACI Mater. J. 11, 371–380 (2003).

T. D. Visser, S. H. Wiersma, “Diffraction of converging electromagnetic waves,” J. Opt. Soc. Am. A 9, 2034–2047 (1992).
[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]

Wiersma, S. H.

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, “Three-dimensional intensity distribution near the focus in systems of different Fresnel number,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[CrossRef]

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

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

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–370 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, New York, 1999), Sec. 8.11.

ACI Mater. J. (1)

H. F. Schouten, T. D. Visser, G. Gbur, D. Lenstra, H. Block, “Creation and annihilation of phase singularities near a sub-wavelength slit,” ACI Mater. J. 11, 371–380 (2003).

Appl. Opt. (1)

Can. J. Phys. (1)

G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
[CrossRef]

J. Appl. Phys. (1)

M. J. Ehrlich, S. Silver, G. Held, “Studies of the diffraction of electromagnetic waves by circular aperture and complementary obstacles,” J. Appl. Phys. 26, 336–345 (1955).
[CrossRef]

J. Mod. Opt. (1)

J. Pu, B. Lu, S. Nemoto, “Three-dimensional intensity distribution of focused annular non-uniformly polarized beams,” J. Mod. Opt. 49, 1501–1513 (2002).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
[CrossRef]

Opt. Commun. (3)

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

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
[CrossRef]

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

Optik (Stuttgart) (1)

Y. Li, “Focal shift formulae,” Optik (Stuttgart) 69, 41–42 (1984).

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]

Proc. R. Soc. London, Ser. A (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–370 (1959).
[CrossRef]

Rep. Prog. Phys. (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

Trans. Opt. Inst. Petrograd (2)

V. S. Ignatovsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 4 (1920).

V. S. Ignatovsky, “Diffraction by a parabolic mirror having arbitrary opening,” Trans. Opt. Inst. Petrograd 1, paper 5 (1920).

Other (2)

R. Barakat, “The intensity distribution and total illumination of aberration-free diffraction image,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Pergamon, New York, 1999), Sec. 8.11.

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Figures (6)

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 and angles.

Fig. 2
Fig. 2

Geometrical illustration of the law of conservation of energy to determine the geometrical factor g0.

Fig. 3
Fig. 3

Focusing of a linearly polarized, monochromatic plane wave by a high-NA objective and bending of the E-vector when fields are transmitted through the aperture. The amount and direction of bending depend on the coordinates of the ray: (a) bending in the plane of azimuthal angle ϕ=0°, (b) bending in the plane of azimuthal angle ϕ=90°, (c) bending in the plane of an arbitrary azimuthal angle ϕ by the method of decomposing the E-vector into two components perpendicular and parallel to the plane of the azimuthal angle coordinate ϕ.

Fig. 4
Fig. 4

Energy density distributions along the optcal axis in systems that have Fresnel number N=1 and NA=0.1,0.3,0.9, for (a) aplanatic systems, (b) parabolic mirrors.

Fig. 5
Fig. 5

Energy density distributions along the optical axis for aplanatic systems of Fresnel number N=10 and NA=0.1,0.3,,0.9.

Fig. 6
Fig. 6

Energy density distributions along the optical axis for aplanatic systems of Fresnel number N=1 and (a) NA=0.1, (b) NA=0.5, (c) NA=0.9. See the text for explanations of curves C1, C2, and C3.

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(0) as a Function of the Fresnel Number N and NA for an Aplanatic System

Equations (60)

Equations on this page are rendered with MathJax. Learn more.

N=a2/λf,
NA=a/f,
E˜=Re[E(x)exp(-iωt)],
H˜=Re[H(x)exp(-iωt)],
EΣ(x)=Σ[ik(nˆ×HQ)G+(nˆ×EQ)×G+(nˆ  EQ)G]dσ,
HΣ(x)=Σ[ik(nˆ×EQ)G-(nˆ×HQ)×G+(nˆ  HQ)G]dσ,
G(ξ, x)=exp(ik|ξ-x|)4π|ξ-x|=exp(iks)4πs  (s|ξ-x|),
G(ξ, x)=qˆ(ik-1/s)G(ξ, x)=qˆikd(s)G(ξ, x),
d(s)=1-1/(iks).
E(x)=EΣ(x)+EΓ(x),
H(x)=HΣ(x)+HΓ(x),
EΓ(x)=1ik Γ(lˆ  HQ)Gdl,
HΓ(x)=-1ik Γ(lˆ  EQ)Gdl,
|E0|2δS0=|Ei|2δSi.
δS0=δSi cos θ.
g0(θ)=EiE0=cos θ.
g0(θ)=EiE0=11+cos θ.
Ei(g)(ϕ=0°)=|E0|g0(θ)cos θ0sin θ,
Ei(g)(ϕ=90°)=|E0|g0(θ)100.
|E|=|Ei|cos ϕ=|E0|g0(θ)cos ϕ,
|E|=|Ei|sin ϕ=|E0|g0(θ)sin ϕ,
E(g)=|E0|g0(θ)cos ϕcos ϕ cos θsin ϕ cos θsin θ.
E(g)=|E0|g0(θ)sin ϕsin ϕ-cos ϕ0.
Ei(g)=E(g)+E(g)=|E0|g0(θ)cos2 ϕcosθ+sin2 ϕ-cosϕsinϕ(1-cosθ)cosϕsinθ.
Ei(d)(x)=Σ[ik(nˆ × HQ)G+(nˆ × EQ)×G+(nˆ  EQ)G]dσ+1ik Γ(lˆ  HQ)Gdl,
HQ=nˆ × EQ.
Ei(d)(x)=ikΣ{nˆ×(nˆ×EQ)+d(s)[(nˆ × EQ)×qˆ+(nˆ  EQ)qˆ]}Gdσ+Γqˆd(s)[lˆ  (nˆ × EQ)]Gdl.
dσ=f2 sin θdθdϕ,
dl=adϕ,
nˆ=-sin θ cos ϕ-sin θ sin ϕcos θ,
lˆ=-sin ϕcos ϕ0,
qˆ=1τ sin θ cos ϕsin θ sin ϕ-z/f-cos θ,
τ=[(1+z/f)2-2(z/f)(1-cos θ)]1/2.
Ei(d)(z)=iEx(d)(z)+jEy(d)(z)+kEz(d)(z),
Ex(d)(z)=|E0|4 (NA)24 g0(Ω)[4+(1-cosΩ)cosΩ]×exp(iKτ)τ2 1-1iKτθ=Ω-iK0Ωχ(θ, z) (1+cos θ) g0(θ)×exp(iKτ)τsin θdθ,
Ey(d)(z)=0,
Ez(d)(z)=|E0|4NA4 g0(Ω)zf+cos Ω(1-cos Ω)cos Ω×exp(iKτ)τ2 1-1iKτθ=Ω,
K=kf=2πN(NA)2,
χ(θ, z)=1+1+(z/f)τ 1-1iKτ.
WE(z)=|Ei(d)(z)|2=|Ex(d)(z)|2+|Ez(d)(z)|2.
wE(z)=WE(z)WE(0)=|Ex(d)(z)|2+|Ez(d)(z)|2|Ex(d)(0)|2+|Ez(d)(0)|2.
Δff=zm-ff.
1-1iKτ1.
Ex(d)(z)=|E0|4 (NA)24 g0(Ω)[4+(1-cos Ω)cos Ω]×exp(iKτ)τ2θ=Ω-iK×0Ωχ(θ, z) exp(iKτ)τ g0(θ)×(1+cos θ)sin θdθ,
Ez(d)(z)=|E0|4NA4 g0(Ω)zf+cos Ω(1-cos Ω)cos Ω×exp(iKτ)τ2θ=Ω,
χ0(θ, z)=1+1+(z/f)τ.
wE(z)=1A(NA) 0Ωχ0(θ, z) exp(iKτ)τ g0(θ)×(1+cos θ)sin θdθ2,
A(NA)=20Ωg0(θ)(1+cos θ)sin θdθ=1615 1-58+38 1-(NA)2[1-(NA)2]34(aplanaticsystem),2[1-1-(NA)2](parabolicmirror).
C(θ, z)C(θ, z)+S(θ, z)S(θ, z)=0,
C(θ, z)=θ=0Ω χ0(θ, z)τcos(Kτ)g0(θ)×(1+cos θ)sin θdθ,
S(θ, z)=θ=0Ω χ0(θ, z)τsin(Kτ)g0(θ)×(1+cos θ) sin θdθ.
C(z, θ)=θ=0Ω1-[2χ0(θ, z)-1] (z/f)+cos θτ×cos(Kτ)-Kχ0(θ, z)[(z/f)+cosθ]sin(Kτ) ×g0(θ)τ2 (1+cos θ)sin θdθ, S(z, θ)
=θ=0Ω1-[2χ0(θ, z)-1] (z/f)+cos θτ×sin(Kτ)+Kχ0(θ, z)[(z/f)+cos θ]cos(Kτ)×g0(θ)τ2 (1+cos θ)sin θdθ,
ΔwEwE(z0)=(wE)max-1,
a/λ10,
f/λ10.
10>a/λ>1,
10>f/λ>1.
a/λ=N/NA,
f/λ=N/(NA)2.

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