Abstract

An analytical approximate solution of the electromagnetic field on a subwavelength elliptical hole in a thin perfectly conducting screen is presented. Illumination is a linear polarized, normally incident plane wave. A polynomial development method is used and allows one to obtain an easy-to-use analytical solution of the fields, which can be used to build analytical expressions of aperture fields for apertures in anisotropic structures.

© 2012 Optical Society of America

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References

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  1. Lord Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Philos. Mag. 43, 259–272 (1897).
    [CrossRef]
  2. H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
    [CrossRef]
  3. C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Phillips Res. Rep. 5, 321–332 (1950).
  4. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–668 (1998).
    [CrossRef]
  5. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
    [CrossRef]
  6. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
    [CrossRef]
  7. J.-B. Masson and G. Gallot, “Coupling between surface plasmons in subwavelength hole arrays,” Phys. Rev. B 73, 121401 (2006).
    [CrossRef]
  8. W. H. Eggimann, “Higher-order evaluation of electromagnetic diffraction by circular disks,” IRE Trans. Microwave Theory Tech. MIT-9, 408–418 (1961).
    [CrossRef]
  9. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
    [CrossRef]
  10. R. De Smedt and J. Van Bladel, “Magnetic polarizability of some small apertures,” IEEE Trans. Antennas Propag. 28, 703–707 (1980).
    [CrossRef]
  11. E. E. Okon and R. F. Harrington, “The polarizabilities of electrically small apertures of arbitrary shape,” IEEE Trans. Electromagn. Compat. EMC-23, 359–366 (1981).
    [CrossRef]
  12. E. Arvas and R. F. Harrington, “Computation of the magnetic polarizability of conducting disks and the electric polarizability of apertures,” IEEE Trans. Antennas Propag. 31, 719–725 (1983).
    [CrossRef]
  13. R. E. English and N. George, “Diffraction from a small square aperture: approximate aperture fields,” J. Opt. Soc. Am. A 5, 192–199 (1988).
    [CrossRef]
  14. P. Monk, Finite Element Method for Maxwell’s Equation(Oxford Science Publications, 2003).
  15. W. Yu, X. Yang, Y. Liu, and R. Mittra, Electromagnetic Simulation Techniques Based on the FDTD Method, Wiley Series in Microwave and Optical Engineering (Wiley, 2009).
  16. M. N. O. Sadiku, Numerical Techniques in Electromagnetism, 2nd ed. (CRC Press, 2000).
  17. D. J. Shin, A. Chavez-Pirson, S. H. Kim, S. T. Jung, and Y. H. Lee, “Diffraction by a subwavelength-sized aperture in a metal plane,” J. Opt. Soc. Am. A 18, 1477–1486 (2001).
    [CrossRef]
  18. K. Tanaka and M. Tanaka, “Analysis and numerical computation of diffraction of an optical field by a subwavelength-size aperture in a thick metallic screen by use of a volume integral equation,” Appl. Opt. 43, 1734–1746 (2004).
    [CrossRef]
  19. A. Sommerfeld, “Die Greensche Funktion der Schwingungs-gleichung,” Jahresber. Dtsch. Math. Ver. 21, 309–353 (1912).
  20. E. T. Copson, “Diffraction by a plane screen,” Proc. R. Soc. London. Ser. A 202, 277–284 (1950).
    [CrossRef]
  21. J. Boersma and E. Danick, “On the solution of an integral equation arising in potential problems for circular and elliptic disks,” SIAM J. Appl. Math. 53, 931–941 (1993).
    [CrossRef]
  22. C. J. Bouwkamp, “On integrals occurring in the theory of diffraction of electromagnetic waves by circular disk,” Proc. Kon. Med. Akad. Wetensch. 53, 654–661 (1950).
  23. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960).
  24. C. J. Bouwkamp, “On the diffraction of electromagnetic waves by small circular disks and holes,” Philips Res. Rep. 5, 401–422 (1950).
  25. P. Wolfe, “Eigenfunctions of the integral equation for the potential of the charge disk,” J. Math. Phys. 12, 1215–1218 (1971).
    [CrossRef]
  26. K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University, 2006).
  27. C. J. Bouwkamp, “On the evaluation of certain integrals occurring in the theory of freely vibrating circular disk and related problems,” Proc. Kon. Med. Akad. Wetensch. 52, 987–994 (1949).

2006 (2)

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef]

J.-B. Masson and G. Gallot, “Coupling between surface plasmons in subwavelength hole arrays,” Phys. Rev. B 73, 121401 (2006).
[CrossRef]

2004 (2)

2001 (1)

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–668 (1998).
[CrossRef]

1993 (1)

J. Boersma and E. Danick, “On the solution of an integral equation arising in potential problems for circular and elliptic disks,” SIAM J. Appl. Math. 53, 931–941 (1993).
[CrossRef]

1988 (1)

1983 (1)

E. Arvas and R. F. Harrington, “Computation of the magnetic polarizability of conducting disks and the electric polarizability of apertures,” IEEE Trans. Antennas Propag. 31, 719–725 (1983).
[CrossRef]

1981 (1)

E. E. Okon and R. F. Harrington, “The polarizabilities of electrically small apertures of arbitrary shape,” IEEE Trans. Electromagn. Compat. EMC-23, 359–366 (1981).
[CrossRef]

1980 (1)

R. De Smedt and J. Van Bladel, “Magnetic polarizability of some small apertures,” IEEE Trans. Antennas Propag. 28, 703–707 (1980).
[CrossRef]

1971 (1)

P. Wolfe, “Eigenfunctions of the integral equation for the potential of the charge disk,” J. Math. Phys. 12, 1215–1218 (1971).
[CrossRef]

1961 (1)

W. H. Eggimann, “Higher-order evaluation of electromagnetic diffraction by circular disks,” IRE Trans. Microwave Theory Tech. MIT-9, 408–418 (1961).
[CrossRef]

1954 (1)

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

1950 (4)

C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Phillips Res. Rep. 5, 321–332 (1950).

E. T. Copson, “Diffraction by a plane screen,” Proc. R. Soc. London. Ser. A 202, 277–284 (1950).
[CrossRef]

C. J. Bouwkamp, “On integrals occurring in the theory of diffraction of electromagnetic waves by circular disk,” Proc. Kon. Med. Akad. Wetensch. 53, 654–661 (1950).

C. J. Bouwkamp, “On the diffraction of electromagnetic waves by small circular disks and holes,” Philips Res. Rep. 5, 401–422 (1950).

1949 (1)

C. J. Bouwkamp, “On the evaluation of certain integrals occurring in the theory of freely vibrating circular disk and related problems,” Proc. Kon. Med. Akad. Wetensch. 52, 987–994 (1949).

1944 (1)

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

1912 (1)

A. Sommerfeld, “Die Greensche Funktion der Schwingungs-gleichung,” Jahresber. Dtsch. Math. Ver. 21, 309–353 (1912).

1897 (1)

Lord Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Philos. Mag. 43, 259–272 (1897).
[CrossRef]

Arvas, E.

E. Arvas and R. F. Harrington, “Computation of the magnetic polarizability of conducting disks and the electric polarizability of apertures,” IEEE Trans. Antennas Propag. 31, 719–725 (1983).
[CrossRef]

Bence, S. J.

K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University, 2006).

Bethe, H. A.

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

Boersma, J.

J. Boersma and E. Danick, “On the solution of an integral equation arising in potential problems for circular and elliptic disks,” SIAM J. Appl. Math. 53, 931–941 (1993).
[CrossRef]

Bouwkamp, C. J.

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

C. J. Bouwkamp, “On integrals occurring in the theory of diffraction of electromagnetic waves by circular disk,” Proc. Kon. Med. Akad. Wetensch. 53, 654–661 (1950).

C. J. Bouwkamp, “On the diffraction of electromagnetic waves by small circular disks and holes,” Philips Res. Rep. 5, 401–422 (1950).

C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Phillips Res. Rep. 5, 321–332 (1950).

C. J. Bouwkamp, “On the evaluation of certain integrals occurring in the theory of freely vibrating circular disk and related problems,” Proc. Kon. Med. Akad. Wetensch. 52, 987–994 (1949).

Chavez-Pirson, A.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960).

Copson, E. T.

E. T. Copson, “Diffraction by a plane screen,” Proc. R. Soc. London. Ser. A 202, 277–284 (1950).
[CrossRef]

Danick, E.

J. Boersma and E. Danick, “On the solution of an integral equation arising in potential problems for circular and elliptic disks,” SIAM J. Appl. Math. 53, 931–941 (1993).
[CrossRef]

De Smedt, R.

R. De Smedt and J. Van Bladel, “Magnetic polarizability of some small apertures,” IEEE Trans. Antennas Propag. 28, 703–707 (1980).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–668 (1998).
[CrossRef]

Eggimann, W. H.

W. H. Eggimann, “Higher-order evaluation of electromagnetic diffraction by circular disks,” IRE Trans. Microwave Theory Tech. MIT-9, 408–418 (1961).
[CrossRef]

English, R. E.

Gallot, G.

J.-B. Masson and G. Gallot, “Coupling between surface plasmons in subwavelength hole arrays,” Phys. Rev. B 73, 121401 (2006).
[CrossRef]

George, N.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–668 (1998).
[CrossRef]

Harrington, R. F.

E. Arvas and R. F. Harrington, “Computation of the magnetic polarizability of conducting disks and the electric polarizability of apertures,” IEEE Trans. Antennas Propag. 31, 719–725 (1983).
[CrossRef]

E. E. Okon and R. F. Harrington, “The polarizabilities of electrically small apertures of arbitrary shape,” IEEE Trans. Electromagn. Compat. EMC-23, 359–366 (1981).
[CrossRef]

Hobson, M. P.

K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University, 2006).

Jung, S. T.

Kim, S. H.

Lee, Y. H.

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–668 (1998).
[CrossRef]

Liu, Y.

W. Yu, X. Yang, Y. Liu, and R. Mittra, Electromagnetic Simulation Techniques Based on the FDTD Method, Wiley Series in Microwave and Optical Engineering (Wiley, 2009).

Masson, J.-B.

J.-B. Masson and G. Gallot, “Coupling between surface plasmons in subwavelength hole arrays,” Phys. Rev. B 73, 121401 (2006).
[CrossRef]

Mittra, R.

W. Yu, X. Yang, Y. Liu, and R. Mittra, Electromagnetic Simulation Techniques Based on the FDTD Method, Wiley Series in Microwave and Optical Engineering (Wiley, 2009).

Monk, P.

P. Monk, Finite Element Method for Maxwell’s Equation(Oxford Science Publications, 2003).

Okon, E. E.

E. E. Okon and R. F. Harrington, “The polarizabilities of electrically small apertures of arbitrary shape,” IEEE Trans. Electromagn. Compat. EMC-23, 359–366 (1981).
[CrossRef]

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Philos. Mag. 43, 259–272 (1897).
[CrossRef]

Riley, K. F.

K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University, 2006).

Sadiku, M. N. O.

M. N. O. Sadiku, Numerical Techniques in Electromagnetism, 2nd ed. (CRC Press, 2000).

Shin, D. J.

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, “Die Greensche Funktion der Schwingungs-gleichung,” Jahresber. Dtsch. Math. Ver. 21, 309–353 (1912).

Tanaka, K.

Tanaka, M.

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–668 (1998).
[CrossRef]

Van Bladel, J.

R. De Smedt and J. Van Bladel, “Magnetic polarizability of some small apertures,” IEEE Trans. Antennas Propag. 28, 703–707 (1980).
[CrossRef]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

Wolfe, P.

P. Wolfe, “Eigenfunctions of the integral equation for the potential of the charge disk,” J. Math. Phys. 12, 1215–1218 (1971).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–668 (1998).
[CrossRef]

Yang, X.

W. Yu, X. Yang, Y. Liu, and R. Mittra, Electromagnetic Simulation Techniques Based on the FDTD Method, Wiley Series in Microwave and Optical Engineering (Wiley, 2009).

Yu, W.

W. Yu, X. Yang, Y. Liu, and R. Mittra, Electromagnetic Simulation Techniques Based on the FDTD Method, Wiley Series in Microwave and Optical Engineering (Wiley, 2009).

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (2)

E. Arvas and R. F. Harrington, “Computation of the magnetic polarizability of conducting disks and the electric polarizability of apertures,” IEEE Trans. Antennas Propag. 31, 719–725 (1983).
[CrossRef]

R. De Smedt and J. Van Bladel, “Magnetic polarizability of some small apertures,” IEEE Trans. Antennas Propag. 28, 703–707 (1980).
[CrossRef]

IEEE Trans. Electromagn. Compat. (1)

E. E. Okon and R. F. Harrington, “The polarizabilities of electrically small apertures of arbitrary shape,” IEEE Trans. Electromagn. Compat. EMC-23, 359–366 (1981).
[CrossRef]

IRE Trans. Microwave Theory Tech. (1)

W. H. Eggimann, “Higher-order evaluation of electromagnetic diffraction by circular disks,” IRE Trans. Microwave Theory Tech. MIT-9, 408–418 (1961).
[CrossRef]

J. Math. Phys. (1)

P. Wolfe, “Eigenfunctions of the integral equation for the potential of the charge disk,” J. Math. Phys. 12, 1215–1218 (1971).
[CrossRef]

J. Opt. Soc. Am. A (2)

Jahresber. Dtsch. Math. Ver. (1)

A. Sommerfeld, “Die Greensche Funktion der Schwingungs-gleichung,” Jahresber. Dtsch. Math. Ver. 21, 309–353 (1912).

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–668 (1998).
[CrossRef]

Philips Res. Rep. (1)

C. J. Bouwkamp, “On the diffraction of electromagnetic waves by small circular disks and holes,” Philips Res. Rep. 5, 401–422 (1950).

Phillips Res. Rep. (1)

C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Phillips Res. Rep. 5, 321–332 (1950).

Philos. Mag. (1)

Lord Rayleigh, “On the passage of waves through apertures in plane screens, and allied problems,” Philos. Mag. 43, 259–272 (1897).
[CrossRef]

Phys. Rev. (1)

H. A. Bethe, “Theory of diffraction by small holes,” Phys. Rev. 66, 163–182 (1944).
[CrossRef]

Phys. Rev. B (1)

J.-B. Masson and G. Gallot, “Coupling between surface plasmons in subwavelength hole arrays,” Phys. Rev. B 73, 121401 (2006).
[CrossRef]

Proc. Kon. Med. Akad. Wetensch. (2)

C. J. Bouwkamp, “On integrals occurring in the theory of diffraction of electromagnetic waves by circular disk,” Proc. Kon. Med. Akad. Wetensch. 53, 654–661 (1950).

C. J. Bouwkamp, “On the evaluation of certain integrals occurring in the theory of freely vibrating circular disk and related problems,” Proc. Kon. Med. Akad. Wetensch. 52, 987–994 (1949).

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

E. T. Copson, “Diffraction by a plane screen,” Proc. R. Soc. London. Ser. A 202, 277–284 (1950).
[CrossRef]

Rep. Prog. Phys. (1)

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

Science (2)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef]

SIAM J. Appl. Math. (1)

J. Boersma and E. Danick, “On the solution of an integral equation arising in potential problems for circular and elliptic disks,” SIAM J. Appl. Math. 53, 931–941 (1993).
[CrossRef]

Other (5)

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960).

P. Monk, Finite Element Method for Maxwell’s Equation(Oxford Science Publications, 2003).

W. Yu, X. Yang, Y. Liu, and R. Mittra, Electromagnetic Simulation Techniques Based on the FDTD Method, Wiley Series in Microwave and Optical Engineering (Wiley, 2009).

M. N. O. Sadiku, Numerical Techniques in Electromagnetism, 2nd ed. (CRC Press, 2000).

K. F. Riley, M. P. Hobson, and S. J. Bence, Mathematical Methods for Physics and Engineering (Cambridge University, 2006).

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Figures (2)

Fig. 1.
Fig. 1.

Infinitely thin, perfectly conducting screen with elliptical hole of semimajor axis a and semiminor axis b . Both k b 1 and k a 1 . The electromagnetic plane wave is incident from z < 0 , ψ is the angle between the incident electric field, and the x axis. The transmitted field E⃗ t propagates in the z > 0 direction.

Fig. 2.
Fig. 2.

Evolution of the electric field in an elliptical hole ( a = 2 b ) with varying incident polarization. On the left E x and on the right E y , from top to bottom, ψ = ( 0 , π 4 , π 3 , π 2 ) . The plotted quantity is log | E | in order to increase the contrast of the patterns.

Tables (5)

Tables Icon

Table 1. G ( x , y ) = ε 0 2 π J ( x , y ) ρ d ρ d φ ( x x ) 2 + ( y y ) 2

Tables Icon

Table 2. H ( x , y ) = ε 0 2 π J ( x , y ) ( x x ) 2 + ( y y ) 2 d x d y

Tables Icon

Table 3. First-Order Development

Tables Icon

Table 4. Arbitrary Incident Electromagnetic Field

Tables Icon

Table 5. Arbitrary Incident Electromagnetic Field

Equations (57)

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

{ E⃗ t ( r⃗ ) = 1 ε 0 × F⃗ ( r⃗ ) H⃗ t ( r⃗ ) = 1 j μ 0 k c × E⃗ t ( r⃗ ) ,
F⃗ ( r⃗ ) = ε 0 2 π ellipse n⃗ × E⃗ t ( x , y , 0 ) e j k R R d x d y with R = ( x x ) 2 + ( y y ) 2 + z 2 ,
{ H x t ( r⃗ ) = H x i ( r⃗ ) H y t ( r⃗ ) = H y i ( r⃗ ) E z t ( r⃗ ) = E z i ( r⃗ ) .
{ x y 2 F x + k 2 F x = ε 0 E y i z x y 2 F y + k 2 F y = ε 0 E x i z F y x F x y = ε 0 E z i ,
{ J⃗ = J⃗ 0 + k J⃗ 1 + k 2 J⃗ 2 + k 3 J⃗ 3 + F⃗ = F⃗ 0 + k F⃗ 1 + k 2 F⃗ 2 + k 3 F⃗ 3 + .
J⃗ e j k r = J⃗ 0 + k ( J⃗ 1 + j r J⃗ 0 ) + k 2 ( J⃗ 2 + j r J⃗ 1 1 2 r 2 J⃗ 0 ) + k 3 ( J⃗ 3 + j r J⃗ 2 1 2 r 2 J⃗ 1 1 6 j r 3 J⃗ 0 ) + .
{ F⃗ 0 = ε 0 2 π ellipse J⃗ 0 d x d y R F⃗ 1 = ε 0 2 π ( ellipse J⃗ 1 d x d y R + j ellipse J⃗ 0 d x d y ) F⃗ 2 = ε 0 2 π ( ellipse J⃗ 2 d x d y R + j ellipse J⃗ 1 d x d y 1 2 ellipse J 0 R d x d y ) F⃗ 3 = ε 0 2 π ( ellipse J⃗ 3 d x d y R + j ellipse J⃗ 2 d x d y 1 2 ellipse J 1 R d x d y j 6 ellipse J 0 R 2 d x d y ) .
x y 2 F x 0 = x y 2 F y 0 = F y 0 x F x 0 y = 0 .
J⃗ 0 ( x , y ) = D x 0 ( x , y ) 1 x 2 a 2 y 2 b 2 e⃗ x + D y 0 ( x , y ) 1 x 2 a 2 y 2 b 2 e⃗ y ,
x D x 0 ( x , y ) + y D y 0 ( x , y ) = K 0 ( x , y ) ( 1 x 2 a 2 y 2 b 2 ) ,
x y 2 F x 1 = j ε 0 E i sin ( ψ ) , x y 2 F y 1 = j ε 0 E i cos ( ψ ) , F y 1 x = F x 1 y ,
x D x 1 ( x , y ) + y D y 1 ( x , y ) = K 1 ( x , y ) ( 1 x 2 a 2 y 2 b 2 ) ,
F⃗ 1 = ε 0 2 π ellipse J⃗ 1 d x d y R .
F x 1 = α 0 1 + α 1 1 x + α 2 1 y + α 3 1 x 2 + α 4 1 x y + α 5 1 y 2 , F y 1 = β 0 1 + β 1 1 y + β 2 1 x + β 3 1 y 2 + β 4 1 y x + β 5 1 x 2 ,
2 α 3 1 + 2 α 5 1 = j ε 0 E i sin ( ψ ) , 2 β 3 1 + 2 β 5 1 = j ε 0 E i cos ( ψ ) , β 2 1 = α 2 1 , 2 β 5 1 = α 4 1 , 2 α 5 1 = β 4 1 .
{ J x 1 ( x , y ) = ( η 0 1 + η 1 1 x + η 2 1 y + η 3 1 x 2 + η 4 1 x y + η 5 1 y 2 ) 1 x 2 a 2 y 2 b 2 J y 1 ( x , y ) = ( θ 0 1 + θ 1 1 y + θ 2 1 x + θ 3 1 y 2 + θ 4 1 y x + θ 5 1 x 2 ) 1 x 2 a 2 y 2 b 2 .
η 1 1 = θ 1 1 = 0 , η 2 1 = θ 2 1 , η 3 1 = η 0 1 a 2 , η 4 1 + θ 5 1 = θ 0 1 a 2 , η 5 1 + θ 4 1 = η 0 1 b 2 , θ 3 1 = θ 0 1 b 2 .
α 0 1 = π 2 g 0 η 0 1 + π 2 C 3 η 3 1 + π 2 C 6 η 5 1 α 1 1 = 0 α 2 1 = π 2 C 0 η 2 1 α 3 1 = π 2 C 1 η 3 1 + π 2 C 5 η 5 1 α 4 1 = π 2 C 7 η 4 1 α 5 1 = π 2 C 2 η 3 1 + π 2 C 4 η 5 1 β 0 1 = π 2 g 0 θ 0 1 + π 2 C 6 θ 3 1 + π 2 C 3 θ 5 1 β 1 1 = 0 , β 2 1 = π 2 C 1 θ 2 1 , β 3 1 = π 2 C 4 θ 3 1 + π 2 C 2 θ 5 1 , β 4 1 = π 2 C 7 θ 4 1 β 5 1 = π 2 C 5 θ 3 1 + π 2 C 1 θ 5 1 .
{ E x t ( 1 ) = θ 0 1 + θ 3 1 y 2 + θ 4 1 y x + θ 5 1 x 2 1 x 2 a 2 y 2 b 2 E y t ( 1 ) = η 0 1 + η 3 1 x 2 + η 4 1 x y + η 5 1 y 2 1 x 2 a 2 y 2 b 2 ,
x y 2 F x 3 + F x 1 = 0 , x y 2 F y 3 + F y 1 = 0 , F y 3 x = F x 3 y .
F x 3 ( x , y ) = ( i + j 4 ) α f ( i , j ) 3 x i y j and F y 3 ( x , y ) = ( i + j 4 ) β f ( i , j ) 3 y i x j ,
J x 3 ( x , y ) = ( i + j 4 ) η f ( i , j ) 3 x i y j 1 x 2 a 2 y 2 b 2 and J y 3 ( x , y ) = ( i + j 4 ) θ f ( i , j ) 3 y i x j 1 x 2 a 2 y 2 b 2 ,
η 1 3 = η 2 3 = η 6 3 = η 7 3 = η 8 3 = η 9 3 = 0 θ 1 3 = θ 2 3 = θ 6 3 = θ 7 3 = θ 8 3 = θ 9 3 = 0 .
{ E x t ( 3 ) = θ 0 3 + θ 3 3 y 2 + θ 4 3 y x + θ 5 3 x 2 + θ 10 3 y 4 + θ 11 3 y 3 x + θ 12 3 y 2 x 2 + θ 13 3 y x 3 + θ 14 3 y 4 1 x 2 a 2 y 2 b 2 E y t ( 3 ) = η 0 3 + η 3 3 x 2 + η 4 3 x y + η 5 3 y 2 + η 10 3 x 4 + η 11 3 x 3 y + η 12 3 x 2 y 2 + η 13 3 x y 3 + η 14 3 y 4 1 x 2 a 2 y 2 b 2 ,
{ E x t = k E x t ( 1 ) + k 3 E x t ( 3 ) E y t = k E y t ( 1 ) + k 3 E y t ( 3 ) .
A⃗ ( r⃗ ) = μ 0 4 π ellipse σ⃗ ( x , y ) e j k R R d x d y ,
{ E x ( x , y , 0 ) = E x i ( x , y , 0 ) E y ( x , y , 0 ) = E y i ( x , y , 0 ) H z ( x , y , 0 ) = H z i ( x , y , 0 ) ,
{ x y 2 A x + k 2 A x = μ 0 H y i z x y 2 A y + k 2 A y = μ 0 H x i z A y x A x y = μ 0 H z i .
{ F⃗ A⃗ ε 0 E⃗ i μ 0 H⃗ i σ⃗ n⃗ × E⃗ t .
x y 2 F x 0 = ε 0 E y i z , x y 2 F y 0 = ε 0 E x i z , F y 0 x F x 0 y = ε 0 E z i + ε 0 E z i x x + ε 0 E z i y y .
{ E x 0 = θ 0 0 + θ 1 0 y + θ 2 0 x + θ 3 0 y 2 + θ 4 0 y x + θ 5 0 x 2 1 x 2 a 2 y 2 b 2 E y 0 = η 0 0 + η 1 0 x + η 2 0 y + η 3 0 x 2 + η 4 0 x y + η 5 0 y 2 1 x 2 a 2 y 2 b 2 ,
x y 2 F x 1 = ε 0 2 E y i k z , x y 2 F y 1 = ε 0 2 E x i k z , F y 1 x F x 1 y = ε 0 E z i k + ε 0 2 E z i k x x + ε 0 2 E z i k y y .
η 0 1 = η 0 0 k , η 1 1 = 0 , η 2 1 = η 2 0 k , η 3 1 = η 3 0 k , η 4 1 = η 4 0 k , η 5 1 = η 5 0 k , θ 0 1 = θ 0 0 k , θ 1 1 = 0 , θ 2 1 = θ 2 0 k , θ 3 1 = θ 3 0 k , θ 4 1 = θ 4 0 k , θ 5 1 = θ 5 0 k ,
{ E x 1 = θ 0 1 + θ 1 1 y + θ 2 1 x + θ 3 1 y 2 + θ 4 1 y x + θ 5 1 x 2 1 x 2 a 2 y 2 b 2 E y 1 = η 0 1 + η 1 1 x + η 2 1 y + η 3 1 x 2 + η 4 1 x y + η 5 1 y 2 1 x 2 a 2 y 2 b 2 .
{ E x t ( 1 ) = 4 j 3 π 2 a 2 cos ψ x 2 cos ψ + x y sin ψ 2 y 2 cos ψ a 2 x 2 y 2 E i E y t ( 1 ) = 4 j 3 π 2 a 2 sin ψ 2 x 2 sin ψ + x y cos ψ y 2 sin ψ a 2 x 2 y 2 E i ,
{ E x t ( 1 ) = 4 j 3 π 2 a 2 x 2 2 y 2 a 2 x 2 y 2 E i E y t ( 1 ) = 4 j 3 π x y a 2 x 2 y 2 E i .
{ x = a ρ cos ( φ ) y = b ρ sin ( φ ) { x = a ρ cos ( φ ) y = b ρ sin ( φ ) ρ e i φ ρ e i φ = r e i θ .
F⃗ = ε 0 2 π ellipse J⃗ d x d y R
F⃗ = ε 0 2 π ellipse J⃗ R d x d y ,
G ( x , y ) = ε 0 2 π ellipse J ( x , y ) ρ d ρ d φ R ,
G ( x , y ) = ε 0 b 2 π circle J ( x , y ) ρ d ρ d φ r 1 p 2 sin 2 ( θ ) ,
g 0 = ε 0 b π 2 K ( p ) g ± 2 = 2 ε 0 b π 2 p 2 [ E ( p ) ( 1 1 2 p 2 ) K ( p ) ] g ± 4 = ε 0 b 3 π 2 p 4 [ ( 3 p 4 16 p 2 + 16 ) K ( p ) + ( 8 p 2 16 ) E ( p ) ] g ± 6 = ε 0 b 15 π 2 p 6 [ ( 15 p 6 158 p 4 + 384 p 2 256 ) K ( p ) + ( 46 p 4 256 p 2 + 256 ) ] g ± 8 = ε 0 b 105 π 2 p 8 [ ( 105 p 8 1856 p 6 + 8000 p 4 12288 p 2 + 6144 ) K ( p ) + ( 352 p 6 3776 p 4 + 9216 p 2 6144 ) E ( p ) ] ,
0 2 π 0 1 e i l θ r P n m ( 1 ρ 2 ) e i m φ 1 ρ 2 ρ d ρ d φ = 0 if | m + l | > n 0 2 π 0 1 e i l θ r P n m ( 1 ρ 2 ) e i m φ 1 ρ 2 ρ d ρ d φ = L m , n , l P n m + l ( 1 ρ 2 ) e i ( m + l ) φ if | m + l | n
L m , n , l = 2 l π Γ ( 1 2 n + 1 2 m + 1 2 ) Γ ( 1 2 n 1 2 m 1 2 l + 1 2 ) Γ ( 1 2 n 1 2 m + 1 2 ) Γ ( 1 2 n + 1 2 m + 1 2 l + 1 ) .
H ( x , y ) = ε 0 2 π ellipse J ( x , y ) ( x x ) 2 + ( y y ) 2 d x d y .
H ( 0 , 0 ) = ε 0 2 π ellipse J ( x , y ) x 2 + y 2 d x d y ,
{ H ( x , y ) x = ε 0 x 2 π J ( x , y ) ( x x ) 2 + ( y y ) 2 d x d y ε 0 2 π J ( x , y ) x ( x x ) 2 + ( y y ) 2 d x d y H ( x , y ) y = ε 0 y 2 π J ( x , y ) ( x x ) 2 + ( y y ) 2 d x d y ε 0 2 π J ( x , y ) y ( x x ) 2 + ( y y ) 2 d x d y ,
{ h 0 = 1 4 μ 0 a 2 b E ( p ) h 1 = μ 0 a 4 b 16 p 2 [ ( p 2 1 ) K ( p ) + ( p 2 + 1 ) E ( p ) ] h 2 = μ 0 b 3 a 2 16 p 2 [ ( 1 p 2 ) K ( p ) + ( 2 p 2 1 ) E ( p ) ] ,
η 1 3 = η 2 3 = η 6 3 = η 7 3 = η 8 3 = η 9 3 = 0 θ 1 3 = θ 2 3 = θ 6 3 = θ 7 3 = θ 8 3 = θ 9 3 = 0 .
η 10 3 = | S 1 D 2 D 3 D 4 D 5 D 6 D 7 D 8 D 9 D 10 | / ψ η 11 3 = | D 1 S 1 D 3 D 4 D 5 D 6 D 7 D 8 D 9 D 10 | / ψ η 12 3 = | D 1 D 2 S 3 D 4 D 5 D 6 D 7 D 8 D 9 D 10 | / ψ η 13 3 = | D 1 D 2 D 3 S 1 D 5 D 6 D 7 D 8 D 9 D 10 | / ψ η 14 3 = | D 1 D 2 D 3 D 4 S 1 D 6 D 7 D 8 D 9 D 10 | / ψ θ 10 3 = | D 1 D 2 D 3 D 4 D 5 S 1 D 7 D 8 D 9 D 10 | / ψ θ 11 3 = | D 1 D 2 D 3 D 4 D 5 D 6 S 1 D 8 D 9 D 10 | / ψ θ 12 3 = | D 1 D 2 D 3 D 4 D 5 D 6 D 7 S 1 D 9 D 10 | / ψ θ 13 3 = | D 1 D 2 D 3 D 4 D 5 D 6 D 7 D 8 S 1 D 10 | / ψ θ 14 3 = | D 1 D 2 D 3 D 4 D 5 D 6 D 7 D 8 D 9 S 1 | / ψ ,
D 1 = [ π 2 C 21 20 0 π 2 C 24 21 0 0 2 π 2 C 21 0 4 π 2 C 24 a 2 / b 2 0 ] T D 2 = [ 0 π 2 C 27 26 0 0 π 2 C 26 0 3 π 2 C 27 0 0 a 2 / b 2 ] T D 3 = [ π 2 C 31 29 0 π 2 C 30 31 0 0 2 π 2 C 31 0 4 π 2 C 30 1 0 ] T D 4 = [ 0 π 2 C 36 35 0 0 π 2 C 35 0 3 π 2 C 36 0 0 1 ] T D 5 = [ π 2 C 40 38 0 π 2 C 39 40 0 0 2 π 2 C 40 0 4 π 2 C 39 b 2 / a 2 0 ] T D 6 = [ 0 0 0 π 2 C 39 40 4 π 2 C 38 0 2 π 2 C 40 0 0 b 2 / a 2 ] T D 7 = [ 0 0 0 0 0 3 π 2 C 35 0 π 2 C 36 b 2 / a 2 0 ] T D 8 = [ 0 0 0 π 2 C 30 31 4 π 2 C 29 0 2 π 2 C 31 0 0 1 ] T D 9 = [ 0 0 0 0 0 3 π 2 C 26 0 π 2 C 27 1 0 ] T D 10 = [ 0 0 0 π 2 C 24 21 4 π 2 C 20 0 2 π 2 C 21 0 0 a 2 / b 2 ] T S 1 = [ ξ 1 1 ξ 2 1 ξ 3 1 ξ 4 1 ξ 5 1 ξ 6 1 ξ 7 1 ξ 7 1 0 0 ] T ψ = | D 1 D 2 D 3 D 4 D 5 D 6 D 7 D 8 D 9 D 10 | ,
δ 10 1 = 1 / 8 π 2 ( C 1 + C 8 ) η 3 1 + 1 / 8 π 2 ( C 5 + C 14 ) η 5 1 δ 11 1 = 1 / 6 π 2 ( C 7 + C 12 ) η 4 1 δ 12 1 = 1 / 4 π 2 ( C 2 + C 9 ) η 3 1 + 1 / 4 π 2 ( C 15 C 4 ) η 5 1 δ 13 1 = 1 / 2 π 2 C 11 η 4 1 δ 14 1 = 1 / 8 π 2 ( C 11 C 2 ) η 3 1 + 1 / 8 π 2 ( C 17 C 4 ) η 5 1 λ 10 1 = 1 / 8 π 2 ( C 17 C 4 ) θ 3 1 + 1 / 8 π 2 ( C 11 C 2 ) θ 5 1 λ 11 1 = 1 / 2 π 2 C 11 θ 4 1 λ 12 1 = 1 / 4 π 2 ( C 15 C 4 ) θ 3 1 + 1 / 4 π 2 ( C 9 C 2 ) θ 5 1 λ 13 1 = 1 / 6 π 2 ( C 12 C 7 ) θ 4 1 λ 14 1 = 1 / 8 π 2 ( C 14 C 5 ) θ 3 1 + 1 / 8 π 2 ( C 8 C 1 ) θ 5 1
C 1 0 = 1 / 2 ( g 0 C 1 ) C 0 0 , 1 = g 0 1 / 2 C 1 1 / 2 C 0 C 0 0 = 1 / 2 ( C 0 g 0 ) C 1 2 = 2 C 2 + 2 C C 3 10 = 1 / 2 ( C 3 C 10 ) C 3 13 = 1 / 2 ( C 13 C 3 ) C 4 5 = 2 C 5 + 2 C 4 C 6 16 = 1 / 2 ( C 6 C 16 ) C 6 19 = 1 / 2 ( C 19 C 6 ) C 10 13 , 3 = 1 / 2 C 13 + C 3 1 / 2 C 10 C 16 19 , 6 = 1 / 2 C 19 + C 6 1 / 2 C 16 C 21 20 = 12 C 20 + 2 C 21 C 24 21 = 2 C 21 + 12 C 24 C 22 23 = 2 C 23 2 C 22 C 27 26 = 6 C 26 + 6 C 27 C 31 29 = 12 C 29 + 2 C 31 C 30 31 = 2 C 31 + 12 C 30 C 32 33 = 2 C 33 2 C 32 C 36 35 = 6 C 35 + 6 C 36 C 40 38 = 12 C 38 + 2 C 40 C 39 40 = 2 C 40 + 12 C 39 C 41 42 = 2 C 42 2 C 41 .
ξ 1 = α 3 1 12 δ 10 1 2 δ 12 1 ξ 2 = α 4 1 6 δ 11 1 6 δ 13 1 ξ 3 = α 5 1 2 δ 12 1 12 δ 14 1 ξ 4 = β 3 1 2 λ 12 1 12 λ 10 1 ξ 5 = δ 11 1 4 λ 14 1 ξ 6 = 2 δ 12 1 3 λ 13 1 ξ 7 = 3 δ 13 1 2 λ 12 1 ξ 8 = 4 δ 14 1 λ 11 1
α 0 3 = η 0 3 π 2 g 0 + η 3 3 π 2 C 3 + η 5 3 π 2 C 6 + η 10 3 π 2 C 25 + η 12 3 π 2 C 34 + η 14 3 π 2 C 43 1 2 η 3 1 h 1 1 2 η 0 1 h 0 1 2 η 5 1 h 2 , α 1 3 = 1