Abstract

Numerical results are given for the backscattering cross section of a dielectric elliptical cylinder. Computations are carried out using the exact solutions in terms of Mathieu functions. Both polarizations of the incident wave are considered: one with the incident electric vector in the axial direction and the other with the incident magnetic vector in the axial direction. The parameters considered are: k02q2=0.004, 0.4, 4.0, 8.0, 12.0, 16.0, 20.0; 0≤qk0 cosξ0<6.5; 1/0=2.0; 0≤θ≤90°. k0 is the free-space wavenumber, q is the semifocal length, q cosξ0 is the length of the semimajor axis, θ is the angle of incidence with respect to the major axis, and 1/0 is the relative dielectric constant of the cylinder.

© 1965 Optical Society of America

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References

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  1. R. W. P. King and T. T. Wu, The Scattering and Diffraction of Waves (Hartford University Press, Cambridge, Massachusetts, 1959).
  2. C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
    [Crossref]
  3. H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).
  4. H. Hönl, A. W. Maue, and K. Westpfahl, “Theorie der Beugung,” Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1961), Vol. 25.
  5. C. Yeh, J. Math. Phys. 4, 65 (1963).
    [Crossref]
  6. For example, in the long-wavelength region, the Rayleigh method [ Rayleigh, Phil. Mag. 44, 28 (1897); A. F. Stevenson, J. Appl. Phys. 24, 1134 (1953); V. Twersky, J. Acoust. Soc. Am. 36, 1314 (1964)] is very useful; in the short wavelength region, the approximate treatment of diffraction problems by physical-optics techniques [J. B. Keller, J. Opt. Soc. Am. 52, 102 (1962); Y. M. Chen, J. Math. Phys. 5, 820 (1964)] is very successful.
    [Crossref]
  7. J. Meixner and F. W. Schäfke, Mathieu-Funktionen und Sphäroid-Funktionen (Springer-Verlag, Berlin, 1954).
    [Crossref]
  8. N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University Press, London, 1951).
  9. G. Blanch, J. Math. Phys. 25, 1 (1946).
  10. C. J. Bouwkamp, J. Math Phys. 26, 79 (1947).
  11. National Bureau of Standards, Tables Relating to Mathieu Functions (Columbia University Press, New York, 1951).
  12. G. Blanch and D. S. Clemm, Tables Relating to the Radial Mathieu Functions, Vol. 1, Functions of the First Kind (Aeronautical Research Laboratories, Office of Aerospace Research, U. S. Air Force, 1961).
  13. R. Barakat, A. Houston, and E. Levin, J. Math. Phys. 42, 142 (1963).

1963 (2)

C. Yeh, J. Math. Phys. 4, 65 (1963).
[Crossref]

R. Barakat, A. Houston, and E. Levin, J. Math. Phys. 42, 142 (1963).

1954 (1)

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
[Crossref]

1947 (1)

C. J. Bouwkamp, J. Math Phys. 26, 79 (1947).

1946 (1)

G. Blanch, J. Math. Phys. 25, 1 (1946).

1897 (1)

For example, in the long-wavelength region, the Rayleigh method [ Rayleigh, Phil. Mag. 44, 28 (1897); A. F. Stevenson, J. Appl. Phys. 24, 1134 (1953); V. Twersky, J. Acoust. Soc. Am. 36, 1314 (1964)] is very useful; in the short wavelength region, the approximate treatment of diffraction problems by physical-optics techniques [J. B. Keller, J. Opt. Soc. Am. 52, 102 (1962); Y. M. Chen, J. Math. Phys. 5, 820 (1964)] is very successful.
[Crossref]

Barakat, R.

R. Barakat, A. Houston, and E. Levin, J. Math. Phys. 42, 142 (1963).

Blanch, G.

G. Blanch, J. Math. Phys. 25, 1 (1946).

G. Blanch and D. S. Clemm, Tables Relating to the Radial Mathieu Functions, Vol. 1, Functions of the First Kind (Aeronautical Research Laboratories, Office of Aerospace Research, U. S. Air Force, 1961).

Bouwkamp, C. J.

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
[Crossref]

C. J. Bouwkamp, J. Math Phys. 26, 79 (1947).

Clemm, D. S.

G. Blanch and D. S. Clemm, Tables Relating to the Radial Mathieu Functions, Vol. 1, Functions of the First Kind (Aeronautical Research Laboratories, Office of Aerospace Research, U. S. Air Force, 1961).

Hönl, H.

H. Hönl, A. W. Maue, and K. Westpfahl, “Theorie der Beugung,” Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1961), Vol. 25.

Houston, A.

R. Barakat, A. Houston, and E. Levin, J. Math. Phys. 42, 142 (1963).

King, R. W. P.

R. W. P. King and T. T. Wu, The Scattering and Diffraction of Waves (Hartford University Press, Cambridge, Massachusetts, 1959).

Levin, E.

R. Barakat, A. Houston, and E. Levin, J. Math. Phys. 42, 142 (1963).

Maue, A. W.

H. Hönl, A. W. Maue, and K. Westpfahl, “Theorie der Beugung,” Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1961), Vol. 25.

McLachlan, N. W.

N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University Press, London, 1951).

Meixner, J.

J. Meixner and F. W. Schäfke, Mathieu-Funktionen und Sphäroid-Funktionen (Springer-Verlag, Berlin, 1954).
[Crossref]

Rayleigh,

For example, in the long-wavelength region, the Rayleigh method [ Rayleigh, Phil. Mag. 44, 28 (1897); A. F. Stevenson, J. Appl. Phys. 24, 1134 (1953); V. Twersky, J. Acoust. Soc. Am. 36, 1314 (1964)] is very useful; in the short wavelength region, the approximate treatment of diffraction problems by physical-optics techniques [J. B. Keller, J. Opt. Soc. Am. 52, 102 (1962); Y. M. Chen, J. Math. Phys. 5, 820 (1964)] is very successful.
[Crossref]

Schäfke, F. W.

J. Meixner and F. W. Schäfke, Mathieu-Funktionen und Sphäroid-Funktionen (Springer-Verlag, Berlin, 1954).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

Westpfahl, K.

H. Hönl, A. W. Maue, and K. Westpfahl, “Theorie der Beugung,” Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1961), Vol. 25.

Wu, T. T.

R. W. P. King and T. T. Wu, The Scattering and Diffraction of Waves (Hartford University Press, Cambridge, Massachusetts, 1959).

Yeh, C.

C. Yeh, J. Math. Phys. 4, 65 (1963).
[Crossref]

J. Math Phys. (1)

C. J. Bouwkamp, J. Math Phys. 26, 79 (1947).

J. Math. Phys. (3)

R. Barakat, A. Houston, and E. Levin, J. Math. Phys. 42, 142 (1963).

C. Yeh, J. Math. Phys. 4, 65 (1963).
[Crossref]

G. Blanch, J. Math. Phys. 25, 1 (1946).

Phil. Mag. (1)

For example, in the long-wavelength region, the Rayleigh method [ Rayleigh, Phil. Mag. 44, 28 (1897); A. F. Stevenson, J. Appl. Phys. 24, 1134 (1953); V. Twersky, J. Acoust. Soc. Am. 36, 1314 (1964)] is very useful; in the short wavelength region, the approximate treatment of diffraction problems by physical-optics techniques [J. B. Keller, J. Opt. Soc. Am. 52, 102 (1962); Y. M. Chen, J. Math. Phys. 5, 820 (1964)] is very successful.
[Crossref]

Rept. Progr. Phys. (1)

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
[Crossref]

Other (7)

H. C. van de Hulst, Light Scattering by Small Particles (John Wiley & Sons, Inc., New York, 1957).

H. Hönl, A. W. Maue, and K. Westpfahl, “Theorie der Beugung,” Handbuch der Physik, edited by S. Flügge (Springer-Verlag, Berlin, 1961), Vol. 25.

R. W. P. King and T. T. Wu, The Scattering and Diffraction of Waves (Hartford University Press, Cambridge, Massachusetts, 1959).

J. Meixner and F. W. Schäfke, Mathieu-Funktionen und Sphäroid-Funktionen (Springer-Verlag, Berlin, 1954).
[Crossref]

N. W. McLachlan, Theory and Application of Mathieu Functions (Oxford University Press, London, 1951).

National Bureau of Standards, Tables Relating to Mathieu Functions (Columbia University Press, New York, 1951).

G. Blanch and D. S. Clemm, Tables Relating to the Radial Mathieu Functions, Vol. 1, Functions of the First Kind (Aeronautical Research Laboratories, Office of Aerospace Research, U. S. Air Force, 1961).

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Figures (17)

Fig. 1
Fig. 1

Elliptical cylinder coordinates. F1 and F2 are the foci of the ellipse. The distance between foci is the focal distance 2q. ξ=const and η−const give, respectively, the locus of an ellipse and a hyperbola.

Fig. 2
Fig. 2

Backscattering cross section of an elliptical dielectric cylinder (1/0=2.0) for an incident E wave. Angle of incidence, θ=0°.

Fig. 3
Fig. 3

Backscattering cross section of an elliptical dielectric cylinder (1/0=2.0) for an incident E wave. Angle of incidence, θ=30°.

Fig. 4
Fig. 4

Backscattering cross section of an elliptical dielectric cylinder (1/0=2.0) for an incident E wave. Angle of incidence, θ=60°.

Fig. 5
Fig. 5

Backscattering cross section of an elliptical dielectric cylinder (1/0=2.0) for an incident E wave. Angle of incidence, θ=90°.

Fig. 6
Fig. 6

Backscattering cross section of an elliptical dielectric cylinder (1/0=2.0) for an incident H wave. Angle of incidence, θ=0°.

Fig. 7
Fig. 7

Backscattering cross section of an elliptical dielectric cylinder (1/0=2.0) for an incident H wave. Angle of incidence, θ=30°.

Fig. 8
Fig. 8

Backscattering cross section of an elliptical dielectric cylinder (1/0=2.0) for an incident H wave. Angle of incidence, θ=60°.

Fig. 9
Fig. 9

Backscattering cross section of an elliptical dielectric cylinder (1/0=2.0) for an incident H wave. Angle of incidence, θ=90°.

Fig. 10
Fig. 10

Backscattering cross section for an incident E wave as a function of semimajor axis for various angles of incidence. 1/0=2.0. k02q2/4=1.0.

Fig. 11
Fig. 11

Backscattering cross section for an incident E wave as a function of semimajor axis for various angles of incidence. 1/0=2.0. k02q2/4=3.0.

Fig. 12
Fig. 12

Backscattering cross section for an incident H wave as a function of semimajor axis for various angles of incidence. 1/0=2.0. k02q2/4=1.0.

Fig. 13
Fig. 13

Backscattering cross section for an incident H wave as a function of semimajor axis for various angles of incidence. 1/0=2.0. k02q2/4=4.0.

Fig. 14
Fig. 14

Variation of the backscattering cross section for an incident E wave as a function of the angle of incidence. 1/0=2.0, k02q2/4=0.1.

Fig. 15
Fig. 15

Variation of the backscattering cross section for an incident E wave as a function of the angle of incidence. 1/0=2.0, k02q2/4=4.0.

Fig. 16
Fig. 16

Variation of the backscattering cross section for an incident H wave as a function of the angle of incidence. 1/0=2.0, k02q2/4=0.1.

Fig. 17
Fig. 17

Variation of the backscattering cross section for an incident H wave as a function of the angle of incidence. 1/0=2.0, k02q2/4=4.0.

Equations (5)

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σ B = 2 π r ( P B s / S i ) ,
σ B E = lim r 2 π r ( E z B s 2 / E z i 2 ) ,
σ B H = lim r 2 π r ( H z B s 2 / H z i 2 ) ,
E z B s ( E wave ) = 2 ( - i ) ( 2 / π k 0 r ) 1 2 e i ( k 0 r + 1 4 π ) × [ m = 0 A m c e m ( η , γ 0 2 ) c e m ( θ , γ 0 2 ) + m = 1 B m s e m ( η , γ 0 2 ) s e m ( θ , γ 0 2 ) ] ,
H z B s ( H wave ) = 2 ( - i ) ( 2 / π k 0 r ) 1 2 e i ( k 0 r + 1 4 π ) × [ m = 0 U m c e m ( η , γ 0 2 ) c e m ( θ , γ 0 2 ) + m = 1 V m s e m ( η , γ 0 2 ) s e m ( θ , γ 0 2 ) ] ,