Abstract

Recently, exact Kirchhoff solutions and the corresponding asymptotic solutions for the focusing of electromagnetic waves through a plane interface between two different dielectrics were reported. But the computation of exact results takes a long time because it requires the quadruple integration of a rapidly oscillating integrand. By using asymptotic techniques to perform two of the integrations, one can reduce the computing time dramatically. Therefore it is important to establish the accuracy and the range of validity of the asymptotic technique. To that end, we compare the exact and the asymptotic results for high-aperture, near-field focusing systems with a total distance from the aperture to the focal point of a few wavelengths and with a distance from the aperture to the interface as small as a fraction of a wavelength. The systems examined have f-numbers in the range from 0.6 to 0.9 and Fresnel numbers in the range from 0.4 to 3.5. Our results show that the accuracy of the asymptotic method increases with the aperture–interface distance when the aperture–focus distance is kept fixed and that it increases with the aperture–focus distance when the aperture–interface distance is kept fixed. To an accuracy of 7.8%, the asymptotic techniques are valid for aperture–interface distances as small as 0.5λ as long as the total distance from the aperture to the focal point exceeds 8λ. It is also shown that an accuracy of better than 1% can be obtained for the same aperture–interface distance of 0.5λ and for interface–observation-point distances as small as 0.1λ as long as the total distance from the aperture to the focal point exceeds 12λ. By use of the asymptotic technique the computing time is reduced by a factor of 103.

© 2000 Optical Society of America

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References

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  1. H. Ling, S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
    [CrossRef]
  2. J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Boston, 1986).
  3. T. D. Visser, S. H. Wiersma, “Defocusing of a converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
    [CrossRef]
  4. P. Török, P. Varga, Z. Laczic, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
    [CrossRef]
  5. P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
    [CrossRef]
  6. P. Török, P. Varga, G. Nemeth, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
    [CrossRef]
  7. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
    [CrossRef]
  8. D. G. Flagello, T. Milster, A. E. Rosenbluth, “Theory of high-NA imaging in homogeneous thin films,” J. Opt. Soc. Am. A 13, 53–64 (1996).
    [CrossRef]
  9. J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
    [CrossRef]
  10. J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [CrossRef]
  11. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  12. Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik (Stuttgart) 64, 207–218 (1983).
  13. Y. Li, H. Platzer, “An experimental investigation of diffraction patterns in low-Fresnel-number focusing systems,” Opt. Acta 30, 1621–1643 (1983).
    [CrossRef]
  14. Y. Li, “Dependence of the focal shift on Fresnel number and f-number,” J. Opt. Soc. Am. 72, 770–774 (1982).
    [CrossRef]
  15. V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
    [CrossRef]
  16. C. A. Taylor, B. J. Thompson, “Attempt to investigate experimentally the intensity distribution near the focus in the error-free diffraction patterns of circular and annular apertures,” J. Opt. Soc. Am. 48, 844–850 (1958).
    [CrossRef]
  17. G. W. Farnell, “Measured phase distribution in the image space of a microwave lens,” Can. J. Phys. 36, 935–943 (1958).
    [CrossRef]
  18. S. H. Wiersma, P. Török, T. D. Visser, P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
    [CrossRef]
  19. D. Jiang, J. J. Stamnes, “Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
    [CrossRef]
  20. J. J. Stamnes, D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
    [CrossRef]
  21. J. J. Stamnes, H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998).
    [CrossRef]
  22. H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. II. Numerical comparisons between exact, Debye, and Kirchoff theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
    [CrossRef]
  23. H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1292–1307 (1998).
    [CrossRef]
  24. V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
    [CrossRef]
  25. J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
    [CrossRef]

1998

D. Jiang, J. J. Stamnes, “Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

J. J. Stamnes, H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998).
[CrossRef]

H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. II. Numerical comparisons between exact, Debye, and Kirchoff theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
[CrossRef]

H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1292–1307 (1998).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

1997

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

S. H. Wiersma, P. Török, T. D. Visser, P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
[CrossRef]

1996

1995

1984

1983

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik (Stuttgart) 64, 207–218 (1983).

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

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

1982

1981

J. H. Erkkila, M. E. Rogers, “Diffracted fields in the focal volume of a converging wave,” J. Opt. Soc. Am. 71, 904–905 (1981).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchoff approximations,” Opt. Commun. 40, 81–85 (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]

1959

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[CrossRef]

1958

Booker, G. R.

Dhayalan, V.

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

Eide, H. A.

Erkkila, J. H.

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]

Flagello, D. G.

Jiang, D.

D. Jiang, J. J. Stamnes, “Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

Laczic, Z.

Lee, S. W.

Li, Y.

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik (Stuttgart) 64, 207–218 (1983).

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

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

Ling, H.

Milster, T.

Nemeth, G.

Pedersen, H. M.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[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]

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 A 253, 358–379 (1959).
[CrossRef]

Rogers, M. E.

Rosenbluth, A. E.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

Stamnes, J. J.

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

J. J. Stamnes, H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998).
[CrossRef]

D. Jiang, J. J. Stamnes, “Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1292–1307 (1998).
[CrossRef]

H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. II. Numerical comparisons between exact, Debye, and Kirchoff theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchoff approximations,” Opt. Commun. 40, 81–85 (1981).
[CrossRef]

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

Taylor, C. A.

Thompson, B. J.

Török, P.

Varga, P.

Visser, T. D.

Wiersma, S. H.

Wolf, E.

E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (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 A 253, 358–379 (1959).
[CrossRef]

Can. J. Phys.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. J. Stamnes, H. A. Eide, “Exact and approximate solutions for focusing of two-dimensional waves. I. Theory,” J. Opt. Soc. Am. A 15, 1285–1291 (1998).
[CrossRef]

H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. II. Numerical comparisons between exact, Debye, and Kirchoff theories,” J. Opt. Soc. Am. A 15, 1308–1319 (1998).
[CrossRef]

H. A. Eide, J. J. Stamnes, “Exact and approximate solutions for focusing of two-dimensional waves. III. Numerical comparisons between exact and Rayleigh–Sommerfeld theories,” J. Opt. Soc. Am. A 15, 1292–1307 (1998).
[CrossRef]

T. D. Visser, S. H. Wiersma, “Defocusing of a converging electromagnetic wave by a plane dielectric interface,” J. Opt. Soc. Am. A 13, 320–325 (1996).
[CrossRef]

P. Török, P. Varga, Z. Laczic, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: an integral representation,” J. Opt. Soc. Am. A 12, 325–332 (1995).
[CrossRef]

P. Török, P. Varga, G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive indices: structure of the electromagnetic field. I,” J. Opt. Soc. Am. A 12, 2136–2144 (1995).
[CrossRef]

P. Török, P. Varga, G. Nemeth, “Analytical solution of the diffraction integrals and interpretation of wave-front distortion when light is focused through a planar interface between materials of mismatched refractive indices,” J. Opt. Soc. Am. A 12, 2660–2671 (1995).
[CrossRef]

S. H. Wiersma, P. Török, T. D. Visser, P. Varga, “Comparison of different theories for focusing through a plane interface,” J. Opt. Soc. Am. A 14, 1482–1490 (1997).
[CrossRef]

H. Ling, S. W. Lee, “Focusing of electromagnetic waves through a dielectric interface,” J. Opt. Soc. Am. A 1, 965–973 (1984).
[CrossRef]

D. G. Flagello, T. Milster, A. E. Rosenbluth, “Theory of high-NA imaging in homogeneous thin films,” J. Opt. Soc. Am. A 13, 53–64 (1996).
[CrossRef]

Opt. Acta

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

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluation of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Opt. Commun.

J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchoff approximations,” Opt. Commun. 40, 81–85 (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]

Optik (Stuttgart)

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik (Stuttgart) 64, 207–218 (1983).

Proc. R. Soc. London A

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London A 253, 358–379 (1959).
[CrossRef]

Pure Appl. Opt.

D. Jiang, J. J. Stamnes, “Theoretical and experimental results for two-dimensional electromagnetic waves focused through an interface,” Pure Appl. Opt. 7, 627–641 (1998).
[CrossRef]

J. J. Stamnes, D. Jiang, “Focusing of two-dimensional electromagnetic waves through a plane interface,” Pure Appl. Opt. 7, 603–625 (1998).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electric-dipole waves in the Debye and Kirchoff approximations,” Pure Appl. Opt. 6, 347–372 (1997).
[CrossRef]

V. Dhayalan, J. J. Stamnes, “Focusing of electromagnetic waves into a dielectric slab. I. Exact and asymptotic results,” Pure Appl. Opt. 7, 33–52 (1998).
[CrossRef]

Other

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

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Figures (8)

Fig. 1
Fig. 1

In the aperture plane, z = 0, a current source radiates a converging electromagnetic wave that is focused through a plane dielectric interface at z = z 0. The permittivities in the two media are ∊0 for 0 < z < z 0 and ∊1 for z > z 0. The focus is at (0, 0, z f ). In the figure it is assumed that ∊1 is greater than ∊0.

Fig. 2
Fig. 2

Refraction through a plane dielectric interface. The main contribution to the field at the observation point (x, y, z) from the source point (x′, y′, 0) is due to the plane wave that is associated with the geometric ray that passes from the source point (x′, y′, 0) by means of the refraction point (x 0, y 0, z 0) to the observation point.

Fig. 3
Fig. 3

Plots of |F x |2 as a function of the axial observation distance from the aperture for various points in the aperture plane: (a) ρ′ = a and ϕ′ = π/4, (b) ρ′ = a/2 and ϕ′ = π/4, (c) ρ′ = a/10 and ϕ′ = π/4. Plots of |F x |2 as a function of the transverse observation distance for z = z f = 8λ for various points in the aperture plane: (d) ρ′ = a and ϕ′ = π/4, (e) ρ′ = a/2 and ϕ′ = π/4, (f) ρ′ = a/10 and ϕ′ = π/4. Here the aperture–interface distance is z 0 = 4λ, the aperture radius is a = 8λ, the aperture–focus distance is z f = 8λ, and the relative refractive index is n 2/n 1 = 1.5. The asymptotic and the exact results are shown as dashed and solid curves, respectively. All distances are in units of the wavelength.

Fig. 4
Fig. 4

Plots of |F x |2 as a function of the axial observation distance from the aperture for various points in the aperture plane: (a) ρ′ = a and ϕ′ = π/4, (b) ρ′ = a/2 and ϕ′ = π/4, (c) ρ′ = a/10 and ϕ′ = π/4. Plots of |F x |2 as a function of the transverse observation distance for z = 6λ for various points in the aperture plane: (d) ρ′ = a and ϕ′ = π/4, (e) ρ′ = a/2 and ϕ′ = π/4, (f) ρ′ = a/10 and ϕ′ = π/4. Here the aperture–interface distance is z 0 = 2λ, the aperture radius is a = 8λ, the aperture–focus distance is z f = 8λ, and the relative refractive index is n 2/n 1 = 1.5. The asymptotic and the exact results are shown as dashed and solid curves, respectively. All distances are in units of the wavelength.

Fig. 5
Fig. 5

Plots of |F x |2 as a function of the axial observation distance from the aperture for various points in the aperture plane: (a) ρ′ = a and ϕ′ = π/4, (b) ρ′ = a/2 and ϕ′ = π/4, (c) ρ′ = a/10 and ϕ′ = π/4. Plots of |F x |2 as a function of the transverse observation distance for z = z f = 8λ for various points in the aperture plane: (d) ρ′ = a and ϕ′ = π/4, (e) ρ′ = a/2 and ϕ′ = π/4, (f) ρ′ = a/10 and ϕ′ = π/4. Here the aperture–interface distance is z 0 = 0.5λ, the aperture radius is a = 8λ, the aperture–focus distance is z f = 8λ, and the relative refractive index is n 2/n 1 = 1.5. The asymptotic and the exact results are shown as dashed and solid curves, respectively. All distances are in units of the wavelength.

Fig. 6
Fig. 6

Exact and asymptotic results for the copolarized or x component of the focused electric field intensity |E x |2: (a) The observation points are along the z axis. (b) The observation points are transverse to the z axis for z 1 = 8λ. Here the aperture–interface distance is z 0 = 4λ, the aperture radius is a = 8λ, the aperture–focus distance is z f = 8λ, and the relative refractive index is n 2/n 1 = 1.5. The asymptotic and the exact results are shown as dashed and solid curves, respectively. All distances are in units of the wavelength.

Fig. 7
Fig. 7

Exact and asymptotic results for the copolarized component of the focused electric field intensity |E x |2: (a) The observation points are along the z axis. (b) The observation points are transverse to the z axis for z 1 = 8λ. Here the aperture–interface distance is z 0 = 2λ, the aperture radius is a = 8λ, the aperture–focus distance is z f = 8λ, and the relative refractive index is n 2/n 1 = 1.5. The asymptotic and the exact results are shown as dashed and solid curves, respectively. All distances are in units of the wavelength.

Fig. 8
Fig. 8

Exact and asymptotic results for the copolarized component of the focused electric field intensity |E x |2: (a) The observation points are along the z axis. (b) The observation points are transverse to the z axis for z 1 = 8λ. Here the aperture–interface distance is z 0 = 0.5λ, the aperture radius is a = 8λ, the aperture–focus distance is z f = 8λ, and the relative refractive index is n 2/n 1 = 1.5. The asymptotic and the exact results are shown as dashed and solid curves, respectively. All distances are in units of the wavelength.

Tables (1)

Tables Icon

Table 1 Maximum Inaccuracy of the Asymptotic Results for |Ex | 2 in the Focal Plane for Different Focusing Systems

Equations (29)

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

Er, t=ReETEr+ETMrexp-iωt,
Epr=A Ax, yexp-iϕx, y×Fpx, ydxdy,
Fpx, y=-fpkx, kyexpihkx, kydkxdky,
hkx, ky=kxx-x+kyy-y+kz0z0+kz1z-z0,
fTM=ωμ0πc2k0k1 TTMkxkt2k×kt×ez,
fTE=ωμ0πc2 TTEkykz0kt2kt×ez,
k=kt+kz1ez,  kt=kxex+kyey,
kzj=kj2-kt21/2, kj2=ωc2μjj+4πiσjω,  for j=0, 1.
TTE=2μ1kz0μ1kz0+μ0kz1,  TTM=k1k02k02μ1kz0k12μ0kz0+k02μ1kz1
ϕx, y=k0D1f+k1D2f-k0z0-k1zf-z0,
D1f=xf-x2+yf-y2+z021/2,  D2f=xf2+yf2+zf-z021/2.
Fpx, y -2πik0z0DD12fpkxs, kys×expik0D1+k1D2,
kxs=k0x0-xD1=k1x-x0D2,  kys=k0y0-yD1=k1y-y0D2.
x-x=ρ cos ϕ,  y-y=ρ sin ϕ,  x=ρ cos ϕ,  y=ρ sin ϕ.
Er=ETEr+ETMr=A Aρ, ϕexp-iϕρ, ϕFρ, ϕρdρdϕ,
Fρ, ϕ=k120BtJ0k1ρt+Dt2J1k1ρtk1ρtexpiψtkz1 tdt,
ψt=kz0z0+kz1z-z0,
kz0=k1kr-2-t21/2,  kz1=k11-t21/2,  kr=k1k0.
F=FH1+FH2+FI,
Fxρ, ϕFxTM+FxTE,
Relative Difference=|Exex|2-|Exasy|2|Exexmax|2,
ϕϕ+2π expit cosβ-ϕcos βdβ=2πi cos ϕJ1t.
ϕϕ+2π expit cosβ-ϕcos2 βdβ=πJ0t1+cos 2ϕ-cos 2ϕ 2J1tt.
ϕϕ+2π expit cosβ-ϕsin2 βdβ=πJ0t1-cos 2ϕ+cos 2ϕ 2J1tt.
ϕϕ+2π expit cosβ-ϕsin 2βdβ=-2π sin 2ϕ2 J1tt-J0t.
Bxt=2 ωμ0c2 kz1TTEkz0 sin2 ϕ+kz1k0k1 TTM cos2 ϕ,
Dxt=ωμ0c2 kz1TTEkz0-kz1k0k1 TTMcos 2ϕ,
Byt=-Dyt=ωμ0c2 kz1kz1k0k1 TTM-TTEkz0sin 2ϕ,
Bzt=0,  Dzt=-i ωμ0c2TTMkz1k0k1 ρt2 cos ϕ.

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