Abstract

We consider electromagnetic scattering from penetrable cylinders of general cross section. After summarizing the basic T-matrix equations the low-frequency case is examined, which leads for nonmagnetic materials to the exact result T=iRR2 in the Rayleigh limit, satisfying both reciprocity and energy constraints. Here elements of R are given by integrals of regular wave functions over the cylinder surface. A “Rayleigh expansion” is then found that is convergent throughout the Rayleigh region and the lower end of the resonance region and requires no matrix inversion. For bodies of high aspect ratio, there is a problem with significance loss during numerical integration, due to large oscillatory terms. A class of surfaces has now been found for which these terms can be removed, however, enabling us to treat aspect ratios up to 1000:1. These methods are expected to apply also in three dimensions.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. K. Varadan and V. V. Varadan, eds., Acoustic, Electromagnetic, and Elastic Wave Scattering (Pergamon, 1980).
  2. P. C. Waterman, "New formulation of acoustic scattering," J. Acoust. Soc. Am. 45, 1417-1429 (1969).
    [CrossRef]
  3. R. L. Weaver and Y.-H. Pao, "Application of the transition matrix to a ribbon-shaped scatterer," J. Acoust. Soc. Am. 66, 1199-1206 (1979).
    [CrossRef]
  4. G. Kristensson and P. C. Waterman, "The T-matrix for acoustic and electromagnetic scattering by circular disks," J. Acoust. Soc. Am. 72, 1612-1625 (1982).
    [CrossRef]
  5. J. Björkberg and G. Kristensson, "Electromagnetic scattering by a perfectly conducting elliptic disk," Can. J. Phys. 65, 723-734 (1987).
    [CrossRef]
  6. S. Ström, "T matrix for electromagnetic scattering from an arbitrary number of scatterers with continuously varying electromagnetic properties," Phys. Rev. D 10, 2685-2690 (1974).
    [CrossRef]
  7. S. Ström and W. Zheng, "The null field approach to electromagnetic scattering from composite objects," IEEE Trans. Antennas Propag. 36, 376-382 (1988).
    [CrossRef]
  8. J. B. Schneider and I. C. Peden, "Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method," IEEE Trans. Antennas Propag. 36, 1317-1321 (1988).
    [CrossRef]
  9. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996).
    [CrossRef]
  10. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), Chap. 5.
  11. V. V. Varadan, A. Lakhtakia, and V. K. Varadan, "Comments on recent criticism of the T-matrix method," J. Acoust. Soc. Am. 84, 2280-2284 (1988).
    [CrossRef]
  12. M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, "T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database," J. Quant. Spectrosc. Radiat. Transf. 88, 357-406 (2004).
  13. M. I. Mishchenko and L. D. Travis, "T-matrix computations of light scattering by large spheroidal particles," Opt. Commun. 109, 16-21 (1994).
    [CrossRef]
  14. S. Havemann and A. J. Baran, "Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method," J. Quant. Spectrosc. Radiat. Transf. 89, 87-96 (2004).
    [CrossRef]
  15. A. Sarkissian, C. F. Gaumond, and L. R. Dragonette, "T-matrix implementation of forward scattering from rigid structures," J. Acoust. Soc. Am. 94, 3448-3453 (1993).
    [CrossRef]
  16. P. C. Waterman, "Surface fields and the T-matrix," J. Opt. Soc. Am. A 16, 2968-2977 (1999).
    [CrossRef]
  17. P. C. Waterman, "Symmetry, unitarity, and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971).
    [CrossRef]
  18. P. C. Waterman, "Matrix methods in potential theory and electromagnetic scattering," J. Appl. Phys. 50, 4550-4566 (1979).
    [CrossRef]
  19. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), p. 1372.
  20. A. S. Householder, The Theory of Matrices in Numerical Analysis (Blaisdell, 1964), p. 54.
  21. F. W. J. Olver, "Bessel functions of integer order," in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds., (U. S. Government Printing Offices, 1964) pp. 355-433.
  22. G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1962), pp. 145-150.

2004 (2)

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, "T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database," J. Quant. Spectrosc. Radiat. Transf. 88, 357-406 (2004).

S. Havemann and A. J. Baran, "Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method," J. Quant. Spectrosc. Radiat. Transf. 89, 87-96 (2004).
[CrossRef]

1999 (1)

1996 (1)

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996).
[CrossRef]

1994 (1)

M. I. Mishchenko and L. D. Travis, "T-matrix computations of light scattering by large spheroidal particles," Opt. Commun. 109, 16-21 (1994).
[CrossRef]

1993 (1)

A. Sarkissian, C. F. Gaumond, and L. R. Dragonette, "T-matrix implementation of forward scattering from rigid structures," J. Acoust. Soc. Am. 94, 3448-3453 (1993).
[CrossRef]

1988 (3)

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, "Comments on recent criticism of the T-matrix method," J. Acoust. Soc. Am. 84, 2280-2284 (1988).
[CrossRef]

S. Ström and W. Zheng, "The null field approach to electromagnetic scattering from composite objects," IEEE Trans. Antennas Propag. 36, 376-382 (1988).
[CrossRef]

J. B. Schneider and I. C. Peden, "Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method," IEEE Trans. Antennas Propag. 36, 1317-1321 (1988).
[CrossRef]

1987 (1)

J. Björkberg and G. Kristensson, "Electromagnetic scattering by a perfectly conducting elliptic disk," Can. J. Phys. 65, 723-734 (1987).
[CrossRef]

1982 (1)

G. Kristensson and P. C. Waterman, "The T-matrix for acoustic and electromagnetic scattering by circular disks," J. Acoust. Soc. Am. 72, 1612-1625 (1982).
[CrossRef]

1979 (2)

P. C. Waterman, "Matrix methods in potential theory and electromagnetic scattering," J. Appl. Phys. 50, 4550-4566 (1979).
[CrossRef]

R. L. Weaver and Y.-H. Pao, "Application of the transition matrix to a ribbon-shaped scatterer," J. Acoust. Soc. Am. 66, 1199-1206 (1979).
[CrossRef]

1974 (1)

S. Ström, "T matrix for electromagnetic scattering from an arbitrary number of scatterers with continuously varying electromagnetic properties," Phys. Rev. D 10, 2685-2690 (1974).
[CrossRef]

1971 (1)

P. C. Waterman, "Symmetry, unitarity, and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971).
[CrossRef]

1969 (1)

P. C. Waterman, "New formulation of acoustic scattering," J. Acoust. Soc. Am. 45, 1417-1429 (1969).
[CrossRef]

Babenko, V. A.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, "T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database," J. Quant. Spectrosc. Radiat. Transf. 88, 357-406 (2004).

Baran, A. J.

S. Havemann and A. J. Baran, "Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method," J. Quant. Spectrosc. Radiat. Transf. 89, 87-96 (2004).
[CrossRef]

Björkberg, J.

J. Björkberg and G. Kristensson, "Electromagnetic scattering by a perfectly conducting elliptic disk," Can. J. Phys. 65, 723-734 (1987).
[CrossRef]

Dragonette, L. R.

A. Sarkissian, C. F. Gaumond, and L. R. Dragonette, "T-matrix implementation of forward scattering from rigid structures," J. Acoust. Soc. Am. 94, 3448-3453 (1993).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), p. 1372.

Gaumond, C. F.

A. Sarkissian, C. F. Gaumond, and L. R. Dragonette, "T-matrix implementation of forward scattering from rigid structures," J. Acoust. Soc. Am. 94, 3448-3453 (1993).
[CrossRef]

Havemann, S.

S. Havemann and A. J. Baran, "Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method," J. Quant. Spectrosc. Radiat. Transf. 89, 87-96 (2004).
[CrossRef]

Householder, A. S.

A. S. Householder, The Theory of Matrices in Numerical Analysis (Blaisdell, 1964), p. 54.

Khlebtsov, N. G.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, "T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database," J. Quant. Spectrosc. Radiat. Transf. 88, 357-406 (2004).

Kristensson, G.

J. Björkberg and G. Kristensson, "Electromagnetic scattering by a perfectly conducting elliptic disk," Can. J. Phys. 65, 723-734 (1987).
[CrossRef]

G. Kristensson and P. C. Waterman, "The T-matrix for acoustic and electromagnetic scattering by circular disks," J. Acoust. Soc. Am. 72, 1612-1625 (1982).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), Chap. 5.

Lakhtakia, A.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, "Comments on recent criticism of the T-matrix method," J. Acoust. Soc. Am. 84, 2280-2284 (1988).
[CrossRef]

Mackowski, D. W.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, "T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database," J. Quant. Spectrosc. Radiat. Transf. 88, 357-406 (2004).

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996).
[CrossRef]

M. I. Mishchenko and L. D. Travis, "T-matrix computations of light scattering by large spheroidal particles," Opt. Commun. 109, 16-21 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), Chap. 5.

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), p. 1372.

Olver, F. W. J.

F. W. J. Olver, "Bessel functions of integer order," in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds., (U. S. Government Printing Offices, 1964) pp. 355-433.

Pao, Y.-H.

R. L. Weaver and Y.-H. Pao, "Application of the transition matrix to a ribbon-shaped scatterer," J. Acoust. Soc. Am. 66, 1199-1206 (1979).
[CrossRef]

Peden, I. C.

J. B. Schneider and I. C. Peden, "Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method," IEEE Trans. Antennas Propag. 36, 1317-1321 (1988).
[CrossRef]

Sarkissian, A.

A. Sarkissian, C. F. Gaumond, and L. R. Dragonette, "T-matrix implementation of forward scattering from rigid structures," J. Acoust. Soc. Am. 94, 3448-3453 (1993).
[CrossRef]

Schneider, J. B.

J. B. Schneider and I. C. Peden, "Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method," IEEE Trans. Antennas Propag. 36, 1317-1321 (1988).
[CrossRef]

Ström, S.

S. Ström and W. Zheng, "The null field approach to electromagnetic scattering from composite objects," IEEE Trans. Antennas Propag. 36, 376-382 (1988).
[CrossRef]

S. Ström, "T matrix for electromagnetic scattering from an arbitrary number of scatterers with continuously varying electromagnetic properties," Phys. Rev. D 10, 2685-2690 (1974).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996).
[CrossRef]

M. I. Mishchenko and L. D. Travis, "T-matrix computations of light scattering by large spheroidal particles," Opt. Commun. 109, 16-21 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), Chap. 5.

Varadan, V. K.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, "Comments on recent criticism of the T-matrix method," J. Acoust. Soc. Am. 84, 2280-2284 (1988).
[CrossRef]

V. K. Varadan and V. V. Varadan, eds., Acoustic, Electromagnetic, and Elastic Wave Scattering (Pergamon, 1980).

Varadan, V. V.

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, "Comments on recent criticism of the T-matrix method," J. Acoust. Soc. Am. 84, 2280-2284 (1988).
[CrossRef]

V. K. Varadan and V. V. Varadan, eds., Acoustic, Electromagnetic, and Elastic Wave Scattering (Pergamon, 1980).

Videen, G.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, "T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database," J. Quant. Spectrosc. Radiat. Transf. 88, 357-406 (2004).

Waterman, P. C.

P. C. Waterman, "Surface fields and the T-matrix," J. Opt. Soc. Am. A 16, 2968-2977 (1999).
[CrossRef]

G. Kristensson and P. C. Waterman, "The T-matrix for acoustic and electromagnetic scattering by circular disks," J. Acoust. Soc. Am. 72, 1612-1625 (1982).
[CrossRef]

P. C. Waterman, "Matrix methods in potential theory and electromagnetic scattering," J. Appl. Phys. 50, 4550-4566 (1979).
[CrossRef]

P. C. Waterman, "Symmetry, unitarity, and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971).
[CrossRef]

P. C. Waterman, "New formulation of acoustic scattering," J. Acoust. Soc. Am. 45, 1417-1429 (1969).
[CrossRef]

Watson, G. N.

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1962), pp. 145-150.

Weaver, R. L.

R. L. Weaver and Y.-H. Pao, "Application of the transition matrix to a ribbon-shaped scatterer," J. Acoust. Soc. Am. 66, 1199-1206 (1979).
[CrossRef]

Wriedt, T.

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, "T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database," J. Quant. Spectrosc. Radiat. Transf. 88, 357-406 (2004).

Zheng, W.

S. Ström and W. Zheng, "The null field approach to electromagnetic scattering from composite objects," IEEE Trans. Antennas Propag. 36, 376-382 (1988).
[CrossRef]

Can. J. Phys. (1)

J. Björkberg and G. Kristensson, "Electromagnetic scattering by a perfectly conducting elliptic disk," Can. J. Phys. 65, 723-734 (1987).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

S. Ström and W. Zheng, "The null field approach to electromagnetic scattering from composite objects," IEEE Trans. Antennas Propag. 36, 376-382 (1988).
[CrossRef]

J. B. Schneider and I. C. Peden, "Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method," IEEE Trans. Antennas Propag. 36, 1317-1321 (1988).
[CrossRef]

J. Acoust. Soc. Am. (5)

P. C. Waterman, "New formulation of acoustic scattering," J. Acoust. Soc. Am. 45, 1417-1429 (1969).
[CrossRef]

R. L. Weaver and Y.-H. Pao, "Application of the transition matrix to a ribbon-shaped scatterer," J. Acoust. Soc. Am. 66, 1199-1206 (1979).
[CrossRef]

G. Kristensson and P. C. Waterman, "The T-matrix for acoustic and electromagnetic scattering by circular disks," J. Acoust. Soc. Am. 72, 1612-1625 (1982).
[CrossRef]

V. V. Varadan, A. Lakhtakia, and V. K. Varadan, "Comments on recent criticism of the T-matrix method," J. Acoust. Soc. Am. 84, 2280-2284 (1988).
[CrossRef]

A. Sarkissian, C. F. Gaumond, and L. R. Dragonette, "T-matrix implementation of forward scattering from rigid structures," J. Acoust. Soc. Am. 94, 3448-3453 (1993).
[CrossRef]

J. Appl. Phys. (1)

P. C. Waterman, "Matrix methods in potential theory and electromagnetic scattering," J. Appl. Phys. 50, 4550-4566 (1979).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transf. (3)

M. I. Mishchenko, G. Videen, V. A. Babenko, N. G. Khlebtsov, and T. Wriedt, "T-matrix theory of electromagnetic scattering by particles and its applications: a comprehensive reference database," J. Quant. Spectrosc. Radiat. Transf. 88, 357-406 (2004).

S. Havemann and A. J. Baran, "Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method," J. Quant. Spectrosc. Radiat. Transf. 89, 87-96 (2004).
[CrossRef]

M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, "T-matrix computations of light scattering by nonspherical particles: a review," J. Quant. Spectrosc. Radiat. Transf. 55, 535-575 (1996).
[CrossRef]

Opt. Commun. (1)

M. I. Mishchenko and L. D. Travis, "T-matrix computations of light scattering by large spheroidal particles," Opt. Commun. 109, 16-21 (1994).
[CrossRef]

Phys. Rev. D (2)

S. Ström, "T matrix for electromagnetic scattering from an arbitrary number of scatterers with continuously varying electromagnetic properties," Phys. Rev. D 10, 2685-2690 (1974).
[CrossRef]

P. C. Waterman, "Symmetry, unitarity, and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971).
[CrossRef]

Other (6)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002), Chap. 5.

V. K. Varadan and V. V. Varadan, eds., Acoustic, Electromagnetic, and Elastic Wave Scattering (Pergamon, 1980).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), p. 1372.

A. S. Householder, The Theory of Matrices in Numerical Analysis (Blaisdell, 1964), p. 54.

F. W. J. Olver, "Bessel functions of integer order," in Handbook of Mathematical Functions, M.Abramowitz and I.A.Stegun, eds., (U. S. Government Printing Offices, 1964) pp. 355-433.

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, 1962), pp. 145-150.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Geometry of the penetrable cylinder.

Fig. 2
Fig. 2

Disparity in magnitude of the integrands of S 08 + and S 08 is shown versus ϑ for a 10:1 elliptic cylinder (see text). Real and imaginary parts of these quantities undergo sign changes near the minima seen here.

Fig. 3
Fig. 3

Geometry of the rectangular cylinder.

Fig. 4
Fig. 4

Double-convex lens, formed by the intersection of two circular cylinders.

Fig. 5
Fig. 5

Parabolic lens, formed by the intersection of two parabolic cylinders.

Fig. 6
Fig. 6

Bar plot showing the magnitudes of T-matrix elements on a logarithmic scale covering ten decades from unit amplitude (uppermost tick mark) down to the base plane (10:1 elliptic cylinder, q = 1 + i , δ = 2 1 2 , N = 8 ).

Fig. 7
Fig. 7

Plot of estimated number of significant figures remaining after truncation error (term-by-term comparison of elements for N = 8 , 10).

Fig. 8
Fig. 8

Significant figures of agreement for corresponding elements above and below the main diagonal in Fig. 6, testing reciprocity.

Fig. 9
Fig. 9

Significant figures of agreement between the left- and right-hand sides of Eq. (7c), checking energy balance for the example of Fig. 6.

Fig. 10
Fig. 10

Magnitudes of the T-matrix elements are shown for a 1000:1 elliptic cylinder. Scale is identical with that of Fig. 6 ( q = 1 + i , δ = 2 1 2 , N = 8 ).

Fig. 11
Fig. 11

Comparison of Eq. (12) with T using matrix inversion for a 10:1 rectangular cylinder shows four to six significant figure agreement throughout, checking the Rayleigh limit formula ( q = 1 + i , δ = 0.014 , N = 8 ).

Fig. 12
Fig. 12

Equation (13), using four terms in the summation, is compared with T (matrix inversion) for the cylinder of Fig. 11 at a higher frequency ( δ = 2 1 2 ) , showing five to more than ten significant figure agreement.

Tables (1)

Tables Icon

Table 1 Convergence Limit of the Rayleigh Expansion [Eq. (12)] for a 2:1 Rectangular Cylinder a

Equations (57)

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

ψ n ( k ρ ) = ϵ n 1 2 H n ( k ρ ) cos n ϑ , n = 0 , 1 , ,
χ n ( k ρ ) = ϵ n 1 2 2 ( 1 + μ ) 1 q n J n ( k ρ ) cos n ϑ , n = 0 , 1 , ,
E z inc ( k ρ ) = n = 0 a n χ n ( k ρ ) ,
E z sca ( k ρ ) = n = 0 f n ψ n ( k ρ ) ,
E z int ( k ρ ) = n = 0 α n χ n ( k ρ ) ,
Q m n = d σ [ μ χ m ( k ρ ) ψ n ( k ρ ) χ m ( k ρ ) ψ n ( k ρ ) ] , m , n = 0 , 1 , ,
Q = R + i S ,
a = i Q α ,
f = i R α ,
f = T a ,
Q T = R .
W m n = d σ μ χ m ( k * ρ ) χ n ( k ρ ) , m , n = 0 , 1 , .
1 2 Re ( α * W α ) = 1 4 α * ( W + W * ) α = 1 4 a * ( Q * ) 1 ( W + W * ) ( Q ) 1 a ,
Re T = T * T + 1 4 A ,
A = ( Q * ) 1 ( W + W * ) ( Q ) 1
S * S = I A .
Z = Q R
E z sca ( k ρ ) = m , n = 0 N a m T m n ψ n ( k ρ ) f ( ϑ ) ( 2 i π k ρ ) 1 2 exp ( i k ρ ) , k ρ 1 ,
f ( ϑ ) = m , n = 0 N a m T m n ϵ n 1 2 i n cos n ϑ ,
σ = 0 2 π d ϑ f ( ϑ ) 2 .
ϕ m ( k ρ ) ϕ n ( k ρ ) ϕ m ( k ρ ) ϕ n ( k ρ ) ,
R m n = ( q 2 1 ) r m n δ m + n + 2 , m , n = 0 , 1 , .
ϕ m ( k ρ ) ζ n ( k ρ ) ϕ m ( k ρ ) ζ n ( k ρ ) ,
S m n = I m n + s m n δ m n + 2 , m , n = 0 , 1 ,
S m n = I m n + s m n { δ m n + 2 , n m 1 ( δ ) for m + n even ( odd ) , otherwise } , m , n = 0 , 1 , .
Q 0 = Q i I .
T i [ R ( S I ) R ] R 2 .
T = i R R 2
T = ( I + p = 1 i p Q 0 p ) i R .
a x 2 + b y 2 + c x y + d x + e y + f = 0
( a cos 2 ϑ + b sin 2 ϑ + c cos ϑ sin ϑ ) ρ 2 ( ϑ ) + ( d cos ϑ + e sin ϑ ) ρ ( ϑ ) + f = 0 .
S = 0 .
ρ ( ϑ ) = x 0 cos ϑ + ( 1 x 0 2 sin 2 ϑ ) 1 2 ,
I ( x 0 ) = 0 π d ϑ cos 3 ϑ ρ ( ϑ ) ( Dirichlet case ) .
ρ ( ϑ ) = ( cos 2 ϑ + α 2 sin 2 ϑ ) 1 2 ,
Q = R + i S +
ρ ( ϑ ) = { ρ 1 ( ϑ ) , 0 ϑ tan 1 α ρ 2 ( ϑ ) , tan 1 α ϑ π 2 } ,
Q = R 1 ( pp ) + R 2 ( pp ) + i [ S 1 + ( pp ) S 1 ( npp ) + S 2 ( pp ) ] .
S m n = I m n + s m n { δ m n + 2 , n m δ n m , otherwise } , m , n = 0 , 1 , .
Q = R + i S + + .
Q = R ( pp ) + i [ S + + ( pp ) S ( npp ) ] .
d σ = d ϑ [ k ρ k ρ ( 1 k ρ ) k ρ ( ϑ ) ϑ ] .
J m ( z ) = J m + 1 ( z ) + ( m z ) J m ( z ) ,
N n ( z ) = N n 1 ( z ) ( n z ) N n ( z ) .
S m n = d m n 0 2 π d ϑ { μ k ρ J m ( q k ρ ) N n 1 ( k ρ ) cos m ϑ cos n ϑ + μ n J m ( q k ρ ) N n ( k ρ ) cos m ϑ [ ( 1 ρ ) ρ ( ϑ ) sin n ϑ cos n ϑ ] + q k ρ J m + 1 ( q k ρ ) N n ( k ρ ) cos m ϑ cos n ϑ m J m ( q k ρ ) N n ( k ρ ) [ cos m ϑ + ( 1 ρ ) ρ ( ϑ ) sin m ϑ ] cos n ϑ } ,
d m n = ( ϵ m ϵ n ) 1 2 2 ( 1 + μ ) 1 q m .
J m ( z ) = k = 0 ( 1 ) k ( z 2 ) 2 k + m k ! ( m + k ) ! ,
N n ( z ) = ( 1 π ) k = 0 n 1 ( n k 1 ) ! ( z 2 ) 2 k n k ! + ( 2 π ) ln ( z 2 ) J n ( z ) ( 1 π ) k = 0 ( 1 ) k [ ψ ( k + 1 ) + ψ ( n + k + 1 ) ] ( z 2 ) 2 k + n k ! ( n + k ) ! ,
ψ ( n ) = { γ for n = 1 ψ ( n 1 ) + 1 ( n 1 ) otherwise , }
J N sum ( m , n , q , z ) = ( 1 π ) k = 0 K s = 0 n 1 a ( m , k ) b ( n , s ) q 2 k + m ( z 2 ) 2 k + 2 s + m n + ( 1 π ) J m ( q z ) { 2 ln ( z 2 ) J n ( z ) k = 0 K [ ψ ( k + 1 ) + ψ ( n + k + 1 ) ] a ( n , k ) ( z 2 ) 2 k + n } .
a ( m , k ) = { 1 m ! for k = 0 a ( m , k 1 ) k ( m + k ) otherwise } ,
b ( n , s ) = { ( n 1 ) ! for s = 0 b ( n , s 1 ) s ( n s ) otherwise } ,
J N sum ( m , n , q , z ) = ( 1 π ) k = 0 n 2 s = 0 n 2 h [ ( 2 k + 2 s + m n + 1 2 ) ] × a ( m , k ) b ( n , s ) q 2 k + m ( z 2 ) 2 k + 2 s + m n .
J N ± ( m , n , q , z ) = { J N sum ± ( m , n , q , z ) if n > m J m ( q z ) N n ( z ) otherwise } .
J m ( z ) N n ( z ) = J n ( z ) N m ( z ) ( 1 π ) s = 0 ( n m 1 ) 2 c m n s ( 2 z ) n m 2 s ,
c m n s = ( m n + s + 1 ) s ( n s 1 ) ! s ! ( m + s ) ! .
Q 1 = [ I + i ( Q + ) 1 S ] 1 ( Q + ) 1 .

Metrics