Abstract

Given a time-harmonic electric-current distribution over a finite planar aperture situated in front of a dielectric half-space, the transmitted field in the dielectric half-space is calculated by the plane-wave-spectrum technique. We concentrate on the current distribution that focuses the electromagnetic energy into a small spot in the dielectric medium, such as in hyperthermia applications. Numerical results are given to illustrate the variation of spot size and of focal-point electric-field intensity as functions of polarization, scanning, and the permittivity of the half-space. We find that (1) the maximum electric-field intensity is not at the focal point but rather at a point closer to the aperture; for a small aperture, this focal shift may be more than one wavelength; (2) when the dielectric half-space is introduced, the 3-dB spot size changes according to the ratio of the wavelength in the half-space to that in the original aperture medium; and (3) the electric-field intensity at the focal point is greatest when the permittivity of the half-space is less than that of the aperture medium, not when the two media are electrically matched.

© 1984 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Sec. 8.8.
  2. H. C. Minnett, B. MacA. Thomas, “Fields in the image space of symmetrical focusing reflectors,” Proc. IEE 115, 1419–1430 (1968).
  3. M. Landry, Y. Chassé, “Measurements of electromagnetic field intensity in focal region of wide-angle paraboloid reflector,” IEEE Trans. Antennas Propag. AP-19, 539–543 (1971).
    [CrossRef]
  4. C. C. Hung, R. Mittra, “Secondary pattern and focal region distribution of reflector antennas under wide-angle scanning,” IEEE Trans. Antennas Propag. AP-31, 756–763 (1983).
    [CrossRef]
  5. J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
    [CrossRef]
  6. J. J. Stamnes, “New methods for computing fields in focal regions,” Soc. Photo-Opt. Instrum. Eng. 358, 184–191 (1982).
  7. J. J. Stamnes, “The Luneburg apodization problem in the non-paraxial domain,” Opt. Commun. 38, 325–329 (1981).
    [CrossRef]
  8. J. J. Stamnes, “Focusing of a perfect wave and the Airy pattern formula,” Opt. Commun. 37, 311–314 (1981).
    [CrossRef]
  9. E. Wolf, Y. Li, “Conditions for the validity of the Debye integral representation of focused fields,” Opt. Commun. 39, 205–210 (1981).
    [CrossRef]
  10. J. J. Stamnes, B. Spjelkavik, “Focusing at small angular apertures in the Debye and Kirchhoff approximations,” Opt. Commun. 40, 81–85 (1981).
    [CrossRef]
  11. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  12. Y. Li, “Dependence of the focal shift on Fresnel number and f number,” J. Opt. Soc. Am. 72, 770–774 (1982).
    [CrossRef]
  13. M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
    [CrossRef]
  14. Y. Li, “Encircled energy of diffracted converging spherical waves,” J. Opt. Soc. Am. 73, 1101–1104 (1983).
    [CrossRef]
  15. H. Ling, S. W. Lee, W. Gee, “Frequency optimization of focused microwave hyperthermia applicators,” Proc. IEEE 72, 224–225 (1984).
    [CrossRef]
  16. J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
    [CrossRef]
  17. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), p. 130.
  18. M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), App. III.
  19. S. W. Lee, M. S. Sheshadri, V. Jamnejad, R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Tech. MTT-30, 12–19 (1982).
  20. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Sec. 5.5.
  21. G. C. Sherman, J. J. Stamnes, E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
    [CrossRef]
  22. W. Gee, S. W. Lee, N. Bong, C. Cain, R. Mittra, R. Magin, “Focused array hyperthermia applicator: theory and experiment,” IEEE Trans. Biomed. Eng. BME-31, 38–46 (1984).
    [CrossRef]
  23. A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
    [CrossRef]
  24. L. J. Ricardi, “Near-field characteristics of a linear array,” in Electromagnetic Theory and Antennas, E. C. Jordan, ed. (Pergamon, New York, 1962).
  25. J. W. Sherman, “Properties of focused apertures in the Fresnel region,” IRE Trans. Antennas Propag. AP-10, 399–408 (1962).
    [CrossRef]
  26. M. P. Bachynski, G. Bekefi, “Study of optical diffraction images at microwave frequencies,” J. Opt. Soc. Am. 47, 428–438 (1957).
    [CrossRef]

1984 (2)

H. Ling, S. W. Lee, W. Gee, “Frequency optimization of focused microwave hyperthermia applicators,” Proc. IEEE 72, 224–225 (1984).
[CrossRef]

W. Gee, S. W. Lee, N. Bong, C. Cain, R. Mittra, R. Magin, “Focused array hyperthermia applicator: theory and experiment,” IEEE Trans. Biomed. Eng. BME-31, 38–46 (1984).
[CrossRef]

1983 (2)

C. C. Hung, R. Mittra, “Secondary pattern and focal region distribution of reflector antennas under wide-angle scanning,” IEEE Trans. Antennas Propag. AP-31, 756–763 (1983).
[CrossRef]

Y. Li, “Encircled energy of diffracted converging spherical waves,” J. Opt. Soc. Am. 73, 1101–1104 (1983).
[CrossRef]

1982 (4)

S. W. Lee, M. S. Sheshadri, V. Jamnejad, R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Tech. MTT-30, 12–19 (1982).

J. J. Stamnes, “New methods for computing fields in focal regions,” Soc. Photo-Opt. Instrum. Eng. 358, 184–191 (1982).

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

M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
[CrossRef]

1981 (6)

J. J. Stamnes, “The Luneburg apodization problem in the non-paraxial domain,” Opt. Commun. 38, 325–329 (1981).
[CrossRef]

J. J. Stamnes, “Focusing of a perfect wave and the Airy pattern formula,” Opt. Commun. 37, 311–314 (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]

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

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

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
[CrossRef]

1976 (2)

J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
[CrossRef]

G. C. Sherman, J. J. Stamnes, E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

1971 (1)

M. Landry, Y. Chassé, “Measurements of electromagnetic field intensity in focal region of wide-angle paraboloid reflector,” IEEE Trans. Antennas Propag. AP-19, 539–543 (1971).
[CrossRef]

1968 (1)

H. C. Minnett, B. MacA. Thomas, “Fields in the image space of symmetrical focusing reflectors,” Proc. IEE 115, 1419–1430 (1968).

1965 (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

1962 (1)

J. W. Sherman, “Properties of focused apertures in the Fresnel region,” IRE Trans. Antennas Propag. AP-10, 399–408 (1962).
[CrossRef]

1957 (1)

Bachynski, M. P.

Bekefi, G.

Boivin, A.

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Bong, N.

W. Gee, S. W. Lee, N. Bong, C. Cain, R. Mittra, R. Magin, “Focused array hyperthermia applicator: theory and experiment,” IEEE Trans. Biomed. Eng. BME-31, 38–46 (1984).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Sec. 8.8.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), App. III.

Cain, C.

W. Gee, S. W. Lee, N. Bong, C. Cain, R. Mittra, R. Magin, “Focused array hyperthermia applicator: theory and experiment,” IEEE Trans. Biomed. Eng. BME-31, 38–46 (1984).
[CrossRef]

Chassé, Y.

M. Landry, Y. Chassé, “Measurements of electromagnetic field intensity in focal region of wide-angle paraboloid reflector,” IEEE Trans. Antennas Propag. AP-19, 539–543 (1971).
[CrossRef]

Felsen, L. B.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Sec. 5.5.

Gasper, J.

Gee, W.

H. Ling, S. W. Lee, W. Gee, “Frequency optimization of focused microwave hyperthermia applicators,” Proc. IEEE 72, 224–225 (1984).
[CrossRef]

W. Gee, S. W. Lee, N. Bong, C. Cain, R. Mittra, R. Magin, “Focused array hyperthermia applicator: theory and experiment,” IEEE Trans. Biomed. Eng. BME-31, 38–46 (1984).
[CrossRef]

Givens, M. P.

M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), p. 130.

Hung, C. C.

C. C. Hung, R. Mittra, “Secondary pattern and focal region distribution of reflector antennas under wide-angle scanning,” IEEE Trans. Antennas Propag. AP-31, 756–763 (1983).
[CrossRef]

Jamnejad, V.

S. W. Lee, M. S. Sheshadri, V. Jamnejad, R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Tech. MTT-30, 12–19 (1982).

Lalor, E.

G. C. Sherman, J. J. Stamnes, E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

Landry, M.

M. Landry, Y. Chassé, “Measurements of electromagnetic field intensity in focal region of wide-angle paraboloid reflector,” IEEE Trans. Antennas Propag. AP-19, 539–543 (1971).
[CrossRef]

Lee, S. W.

H. Ling, S. W. Lee, W. Gee, “Frequency optimization of focused microwave hyperthermia applicators,” Proc. IEEE 72, 224–225 (1984).
[CrossRef]

W. Gee, S. W. Lee, N. Bong, C. Cain, R. Mittra, R. Magin, “Focused array hyperthermia applicator: theory and experiment,” IEEE Trans. Biomed. Eng. BME-31, 38–46 (1984).
[CrossRef]

S. W. Lee, M. S. Sheshadri, V. Jamnejad, R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Tech. MTT-30, 12–19 (1982).

Li, Y.

Y. Li, “Encircled energy of diffracted converging spherical waves,” J. Opt. Soc. Am. 73, 1101–1104 (1983).
[CrossRef]

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]

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.

H. Ling, S. W. Lee, W. Gee, “Frequency optimization of focused microwave hyperthermia applicators,” Proc. IEEE 72, 224–225 (1984).
[CrossRef]

Magin, R.

W. Gee, S. W. Lee, N. Bong, C. Cain, R. Mittra, R. Magin, “Focused array hyperthermia applicator: theory and experiment,” IEEE Trans. Biomed. Eng. BME-31, 38–46 (1984).
[CrossRef]

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Sec. 5.5.

Minnett, H. C.

H. C. Minnett, B. MacA. Thomas, “Fields in the image space of symmetrical focusing reflectors,” Proc. IEE 115, 1419–1430 (1968).

Mittra, R.

W. Gee, S. W. Lee, N. Bong, C. Cain, R. Mittra, R. Magin, “Focused array hyperthermia applicator: theory and experiment,” IEEE Trans. Biomed. Eng. BME-31, 38–46 (1984).
[CrossRef]

C. C. Hung, R. Mittra, “Secondary pattern and focal region distribution of reflector antennas under wide-angle scanning,” IEEE Trans. Antennas Propag. AP-31, 756–763 (1983).
[CrossRef]

S. W. Lee, M. S. Sheshadri, V. Jamnejad, R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Tech. MTT-30, 12–19 (1982).

Ricardi, L. J.

L. J. Ricardi, “Near-field characteristics of a linear array,” in Electromagnetic Theory and Antennas, E. C. Jordan, ed. (Pergamon, New York, 1962).

Sherman, G. C.

G. C. Sherman, J. J. Stamnes, E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
[CrossRef]

Sherman, J. W.

J. W. Sherman, “Properties of focused apertures in the Fresnel region,” IRE Trans. Antennas Propag. AP-10, 399–408 (1962).
[CrossRef]

Sheshadri, M. S.

S. W. Lee, M. S. Sheshadri, V. Jamnejad, R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Tech. MTT-30, 12–19 (1982).

Spjelkavik, B.

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

Stamnes, J. J.

J. J. Stamnes, “New methods for computing fields in focal regions,” Soc. Photo-Opt. Instrum. Eng. 358, 184–191 (1982).

J. J. Stamnes, “The Luneburg apodization problem in the non-paraxial domain,” Opt. Commun. 38, 325–329 (1981).
[CrossRef]

J. J. Stamnes, “Focusing of a perfect wave and the Airy pattern formula,” Opt. Commun. 37, 311–314 (1981).
[CrossRef]

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

J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
[CrossRef]

J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
[CrossRef]

G. C. Sherman, J. J. Stamnes, E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

Thomas, B. MacA.

H. C. Minnett, B. MacA. Thomas, “Fields in the image space of symmetrical focusing reflectors,” Proc. IEE 115, 1419–1430 (1968).

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]

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

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), App. III.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Sec. 8.8.

IEEE Trans. Antennas Propag. (2)

M. Landry, Y. Chassé, “Measurements of electromagnetic field intensity in focal region of wide-angle paraboloid reflector,” IEEE Trans. Antennas Propag. AP-19, 539–543 (1971).
[CrossRef]

C. C. Hung, R. Mittra, “Secondary pattern and focal region distribution of reflector antennas under wide-angle scanning,” IEEE Trans. Antennas Propag. AP-31, 756–763 (1983).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

W. Gee, S. W. Lee, N. Bong, C. Cain, R. Mittra, R. Magin, “Focused array hyperthermia applicator: theory and experiment,” IEEE Trans. Biomed. Eng. BME-31, 38–46 (1984).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. W. Lee, M. S. Sheshadri, V. Jamnejad, R. Mittra, “Refraction at a curved dielectric interface: geometrical optics solution,” IEEE Trans. Microwave Theory Tech. MTT-30, 12–19 (1982).

IRE Trans. Antennas Propag. (1)

J. W. Sherman, “Properties of focused apertures in the Fresnel region,” IRE Trans. Antennas Propag. AP-10, 399–408 (1962).
[CrossRef]

J. Math. Phys. (1)

G. C. Sherman, J. J. Stamnes, E. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

J. Opt. Soc. Am. (5)

Opt. Commun. (6)

M. P. Givens, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 41, 145–148 (1982).
[CrossRef]

J. J. Stamnes, “The Luneburg apodization problem in the non-paraxial domain,” Opt. Commun. 38, 325–329 (1981).
[CrossRef]

J. J. Stamnes, “Focusing of a perfect wave and the Airy pattern formula,” Opt. Commun. 37, 311–314 (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]

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

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

Phys. Rev. B (1)

A. Boivin, E. Wolf, “Electromagnetic field in the neighborhood of the focus of a coherent beam,” Phys. Rev. B 138, 1561–1565 (1965).
[CrossRef]

Proc. IEE (1)

H. C. Minnett, B. MacA. Thomas, “Fields in the image space of symmetrical focusing reflectors,” Proc. IEE 115, 1419–1430 (1968).

Proc. IEEE (1)

H. Ling, S. W. Lee, W. Gee, “Frequency optimization of focused microwave hyperthermia applicators,” Proc. IEEE 72, 224–225 (1984).
[CrossRef]

Soc. Photo-Opt. Instrum. Eng. (1)

J. J. Stamnes, “New methods for computing fields in focal regions,” Soc. Photo-Opt. Instrum. Eng. 358, 184–191 (1982).

Other (5)

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961), p. 130.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), App. III.

M. Born, E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Sec. 8.8.

L. J. Ricardi, “Near-field characteristics of a linear array,” in Electromagnetic Theory and Antennas, E. C. Jordan, ed. (Pergamon, New York, 1962).

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), Sec. 5.5.

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Figures (14)

Fig. 1
Fig. 1

Focusing of electromagnetic waves through a dielectri interface. Aperture ∑ is in 1 and focal point Pf in 2.

Fig. 2
Fig. 2

(a) Determination of phase distribution of a perfectly focusing beam. (b) Wave fronts created by a perfectly focusing beam for 1 > 2, 1 = 2, and 1 < 2. When 1 = 2, the wave front is spherical.

Fig. 3
Fig. 3

Focusing in a single homogeneous medium (1 = 1).

Fig. 4
Fig. 4

(a) Illustration of geometry. (b) |E|2 contours on a decibel scale in the focal plane.

Fig. 5
Fig. 5

(a) Illustration of geometry. (b) |E|2 contours on a decibel scale in the E plane.

Fig. 6
Fig. 6

(a) Illustration of geometry. (b) |E|2 contours on a decibel scale in the H plane.

Fig. 7
Fig. 7

(a) |E|2 contours in the focal plane for a circularly polarized beam. The 3-dB spot has a diameter 0.7λ. (b) Surface plot of focal-plane pattern showing the Airy dark rings.

Fig. 8
Fig. 8

Focal-shift phenomenon. |E|2 is maximum at Pf′ instead of at the intended focus Pf.

Fig. 9
Fig. 9

Focal shift as a function of focal length f and of aperture diameter D.

Fig. 10
Fig. 10

Effect of scanning on focal-point intensity |E|2 along the E plane and the H plane.

Fig. 11
Fig. 11

Effect of scanning on 3-dB spot size.

Fig. 12
Fig. 12

Effect of focusing through a dielectric interface on 3-dB spot size.

Fig. 13
Fig. 13

Comparison of focal-plane patterns for 1 = 2 and 12.

Fig. 14
Fig. 14

Effect of focusing through a dielectric interface on focal-point intensity |E|2. The dashed line is the electric-field transmission coefficient of a normal incident plane wave.

Equations (54)

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

ϕ ( x 0 , y 0 ) = k 1 R 11 + k 2 R 21 - k 1 R 10 - k 2 R 20 ,
k 1 = ω μ 0 1 ,             k 2 = ω μ 0 2 ,
R 11 = [ ( x 11 - x 0 ) 2 + ( y 11 - y 0 ) 2 + d 2 ] 1 / 2 ,
R 21 = [ ( x f - x 11 ) 2 + ( y f - y 11 ) 2 + ( z f - d ) 2 ] 1 / 2 ,
R 10 = ( x 10 2 + y 10 2 + d 2 ) 1 / 2 ,
R 20 = [ ( x f - x 10 ) 2 + ( y f - y 10 ) 2 + ( z f - d ) 2 ] 1 / 2 ,
( x 0 - x 11 ) / ( y 0 - y 11 ) = ( x 0 - x f ) / ( y 0 - y f ) ,
k 1 [ ( x 0 - x 11 ) 2 + ( y 0 - y 11 ) 2 ] 1 / 2 / R 11 = k 2 [ ( x 11 - x f ) 2 + ( y 11 - y f ) 2 ] 1 / 2 / R 21 .
E x = - ψ 1 y ,             H x = 1 j k 1 Z 1 2 ψ 1 x z , E y = ψ 1 x ,             H y = 1 j k 1 Z 1 2 ψ 1 y z , E z = 0 ,             H z = 1 j k 1 Z 1 ( 2 z 2 + k 1 2 ) ψ 1 ,
E x = 1 j k 1 2 ψ ¯ 1 x z ,             H x = 1 Z 1 ψ ¯ 1 y , E y = 1 j k 1 2 ψ ¯ 1 y z ,             H y = - 1 Z 1 ψ ¯ 1 x , E z = 1 j k 1 ( 2 z 2 + k 1 2 ) ψ ¯ 1 ,             H z = 0.
[ ψ 1 ( x , y , z ) ψ ¯ 1 ( x , y , z ) ] = ( 1 2 π ) 2 - d α - d β exp [ - j ( α x + β y + γ 1 z ) ] [ Ψ ¯ 1 ( α , β ) Ψ ¯ 1 ( α , β ) ] ,
[ Ψ ¯ 1 ( α , β ) Ψ ¯ 1 ( α , β ) ] = Σ d x 0 d y 0 [ ψ 1 ( x 0 , y 0 , 0 ) ψ ¯ 1 ( x 0 , y 0 , 0 ) ] exp [ j ( α x 0 + β y 0 ) ] .
[ ψ 2 ( x , y , z ) ψ ¯ 2 ( x , y , z ) ] = ( 1 2 π ) 2 - d α - d β exp [ - j ( α x + β y + γ 2 z ) ] [ Ψ ¯ 2 ( α , β ) Ψ ¯ 2 ( α , β ) ] ,
Ψ ¯ 2 ( α , β ) = T ( α , β ) Ψ ¯ 1 ( α , β ) exp [ - j ( γ 1 - γ 2 ) d ] ,
Ψ ¯ 2 ( α , β ) = T ¯ ( α , β ) Ψ ¯ 1 ( α , β ) exp [ - j ( γ 1 - γ 2 ) d ] ,
T ( α , β ) = 2 1 + ( γ 2 / γ 1 ) ,
T ¯ ( α , β ) = 2 1 + ( 1 γ 2 / 2 γ 1 ) .
ψ 2 ( x , y , z ) = ( 1 2 π ) 2 Σ d x 0 d y 0 ψ 1 ( x 0 , y 0 , 0 ) F ,
F = - d α - d β T ( α , β ) exp [ - j f ( α , β ) ] ,
f ( α , β ) = α ( x - x 0 ) + β ( y - y 0 ) + γ 1 d + γ 2 ( z - d ) .
I = - d α - d β g ( α , β ) exp [ - j f ( α , β ; ω ) ] ,
α 1 = k 1 ( x 1 - x 0 ) / R 1 = k 2 ( x - x 1 ) / R 2 ,
β 1 = k 1 ( y 1 - y 0 ) / R 1 = k 2 ( y - y 1 ) / R 2 ,
R 1 = [ ( x 1 - x 0 ) 2 + ( y 1 - y 0 ) 2 + d 2 ] 1 / 2 ,
R 2 = [ ( x - x 1 ) 2 + ( y - y 1 ) 2 + ( z - d ) 2 ] 1 / 2 .
F ~ 2 π j ( k 1 d / R 1 2 ) ( DF ) T ( α 1 , β 1 ) exp [ - j ( k 1 R 1 + k 2 R 2 ) ] ,
DF = 1 [ 1 + ( R 2 / ρ ) ] 1 / 2 1 [ 1 + ( R 2 / ρ 2 ) ] 1 / 2 ,
ρ 1 = R 1 3 k 2 ( z - d ) 2 R 2 2 k 1 d 2 ,
ρ 2 = R 1 k 2 k 1 .
F = - d α - d β exp { - j [ α ( x - x 0 ) + β ( y - y 0 ) + γ z ] } ,
F ~ 2 π j k cos θ exp ( - j k R ) / R ,
R = [ ( x - x 0 ) 2 + ( y - y 0 ) 2 + z 2 ] 1 / 2 ,
cos θ = z / R .
ψ ( x , y , z ) = ( j k 2 π ) Σ d x 0 d y 0 ψ ( x 0 , y 0 , 0 ) ( z R ) e - i k R R .
R z ,             cos θ 1 ,
J ( x 0 , y 0 ) = - x ^ 2 A ( x 0 , y 0 ) exp [ j ϕ ( x 0 , y 0 ) ] δ ( z )
J ( x 0 , y 0 ) = ( 1 2 π ) 2 - d α - d β J ˜ ( α , β ) × exp [ - j ( α x 0 + β y 0 ) ] δ ( z ) x ^ ,
J ˜ ( α , β ) = Σ - 2 A ( x 0 , y 0 ) exp [ j ϕ ( x 0 , y 0 ) ] × exp [ j ( α x 0 + β y 0 ) ] d x 0 d y 0
H i ( x , y , z ) = ( j k 2 π ) Σ d x 0 d y 0 A ( x 0 , y 0 ) exp [ j ϕ ( x 0 , y 0 ) ] × e - j k R R [ 0 z / R - ( y - y 0 ) / R ] ,
E i ( x , y , z ) = Z ( j k 2 π ) Σ d x 0 d y 0 A ( x 0 , y 0 ) exp [ j ϕ ( x 0 , y 0 ) ] × e - j k R R 3 [ ( y - y 0 ) 2 + z 2 - ( y - y 0 ) ( x - x 0 ) - ( x - x 0 ) z ] ,
H t ( x , y , z ) = ( j k 1 2 π ) Σ d x 0 d y 0 A ( x 0 , y 0 ) exp [ j ϕ ( x 0 , y 0 ) ] × ( DF ) ( d R 1 2 ) exp [ - j ( k 1 R 1 + k 2 R 2 ) ] , × [ ( α β α 2 + β 2 ) [ T ( γ 2 γ 1 ) - T ¯ ] 1 ( α 2 + β 2 ) [ β 2 ( γ 2 γ 1 ) T + α 2 T ¯ ] - ( β γ 1 ) T ] α 1 , β 1 ,
E t ( x , y , z ) = ( j k 1 2 π ) Σ d x 0 d y 0 A ( x 0 , y 0 ) exp [ j ϕ ( x 0 , y 0 ) ] × ( DF ) ( d R 1 2 ) exp [ - j ( k 1 R 1 + k 2 R 2 ) ] , × [ ( 1 α 2 + β 2 ) [ T ¯ ( γ 2 ω 2 ) α 2 + T ( ω μ 0 γ 1 ) β 2 ] ( α β α 2 + β 2 ) [ T ¯ ( γ 2 ω 2 ) - T ( ω μ 0 γ 1 ) ] - T ¯ ( α ω 2 ) ] α 1 , β 1 ,
D = 16 λ 1 ,             f = 8 λ 1 ,             f / D = 0.5 ,
( 3 - dB spot / λ 1 ) = ( 3 - dB spot when 1 = 2 ) 1 / 2 .
T = | E t E i | = 2 1 + 2 / 1 .
T power = Re ( E t × H t * ) Re ( E i × H i * ) = ( 2 1 + 2 / 1 ) 2 2 1
f α = x - x 0 - [ d ( k 1 2 - α 2 - β 2 ) 1 / 2 + ( z - d ) ( k 2 2 - α 2 - β 2 ) 1 / 2 ] α = 0 ,
f β = y - y 0 - [ d ( k 1 2 - α 2 - β 2 ) 1 / 2 + ( z - d ) ( k 2 2 - α 2 - β 2 ) 1 / 2 ] β = 0.
( x - x 0 ) = ( 1 + k 1 R 2 k 2 R 1 ) ( x 1 - x 0 ) ,
( y - y 0 ) = ( 1 + k 1 R 2 k 2 R 1 ) ( y 1 - y 0 ) .
( x - x 0 ) / ( y - y 0 ) = ( x 1 - x 0 ) / ( y 1 - y 0 ) .
x - x 0 = ( x - x 1 ) + ( x 1 - x 0 )
y - y 0 = ( y - y 1 ) + ( y 1 - y 0 )
k 1 [ ( x 1 - x 0 ) 2 + ( y 1 - y 0 ) 2 ] 1 / 2 / R 1 = k 2 [ ( x - x 1 ) 2 + ( y - y 1 ) 2 ] 1 / 2 / R 2 .

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