Abstract

The electromagnetic scattering by an arbitrarily oriented elliptical cylinder having different constitutive param eters than those of the background medium is treated in this work. The separation of variables method is used to solve this problem, but, due to the oblique incidence of the source fields, hybrid waves for the scattered and induced fields are generated, thus making the formulation complicated. Moreover, because of the different wave numbers between the scatterer and the background medium, the orthogonality relations for Mathieu functions do not hold, leading to more complicated systems, compared to those of normal incidence, which should be solved in order to obtain the solution for the scattered or induced fields. The validation of the results reveals the high accuracy of the implementation, even for electrically large scatterers. Both polarizations are considered and numerical results are given for various values of the parameters. The method is exact and can be used for reference as an alternative validation for future methods involving scattering problems.

© 2011 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  4. G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Scattering by an infinite elliptic metallic cylinder,” Electromagnetics 27, 159–182 (2007).
    [CrossRef]
  5. G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Electromagnetic scattering by an infinite elliptic dielectric cylinder with small eccentricity using perturbative analysis,” IEEE Trans. Antennas Propag. 58, 107–121 (2010).
    [CrossRef]
  6. A. R. Sebak, “Scattering from dielectric-coated impedance elliptic cylinder,” IEEE Trans. Antennas Propag. 48, 1574–1580(2000).
    [CrossRef]
  7. S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag. 45, 926–935 (1997).
    [CrossRef]
  8. S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinders,” IEEE Trans. Antennas Propag. 52, 189–196 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. J. L. Tsalamengas, “Exponentially converging Nyström methods applied to the integral-integrodifferential equations of oblique scattering/hybrid wave propagation in presence of composite dielectric cylinders of arbitrary cross section,” IEEE Trans. Antennas Propag. 55, 3239–3250 (2007).
    [CrossRef]
  18. O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915(1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. J. A. Roumeliotis, H. K. Manthopoulos, and V. K. Manthopoulos, “Electromagnetic scattering from an infinite circular metallic cylinder coated by an elliptic dielectric one,” IEEE Trans. Microwave Theory Tech. 41, 862–869 (1993).
    [CrossRef]
  24. A. L. V. Buren and J. E. Boisvert, “Accurate calculation of the modified Mathieu functions of integer order,” Q. Appl. Math. 65, 1–23 (2007).

2010

G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Electromagnetic scattering by an infinite elliptic dielectric cylinder with small eccentricity using perturbative analysis,” IEEE Trans. Antennas Propag. 58, 107–121 (2010).
[CrossRef]

M. Pastorino, Microwave Imaging (Wiley, 2010).
[CrossRef]

2009

R. Li, X. Han, and K. F. Ren, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E 79, 036602(2009).
[CrossRef]

S. C. Mao, Z. S. Wu, and H. Y. Li, “Three-dimensional scattering by an infinite homogeneous anisotropic elliptic cylinder in terms of Mathieu functions,” J. Opt. Soc. Am. A 26, 2282–2291(2009).
[CrossRef]

2008

2007

G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Scattering by an infinite elliptic metallic cylinder,” Electromagnetics 27, 159–182 (2007).
[CrossRef]

J. L. Tsalamengas, “Exponentially converging Nyström methods applied to the integral-integrodifferential equations of oblique scattering/hybrid wave propagation in presence of composite dielectric cylinders of arbitrary cross section,” IEEE Trans. Antennas Propag. 55, 3239–3250 (2007).
[CrossRef]

A. L. V. Buren and J. E. Boisvert, “Accurate calculation of the modified Mathieu functions of integer order,” Q. Appl. Math. 65, 1–23 (2007).

2005

N. L. Tsitsas, E. G. Alivizatos, H. T. Anastassiu, and D. I. Kaklamani, “Optimization of the method of auxiliary sources (MAS) for scattering by an infinite cylinder under oblique incidence,” Electromagnetics 25, 39–54 (2005).
[CrossRef]

2004

S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinders,” IEEE Trans. Antennas Propag. 52, 189–196 (2004).
[CrossRef]

2000

A. R. Sebak, “Scattering from dielectric-coated impedance elliptic cylinder,” IEEE Trans. Antennas Propag. 48, 1574–1580(2000).
[CrossRef]

1999

S. Caorsi, M. Pastorino, and M. Raffetto, “Analytic SAR computation in a multilayer elliptic cylinder for bioelectromagnetic applications,” Bioelectromagnetics 20, 365–371 (1999).
[CrossRef] [PubMed]

1998

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915(1998).
[CrossRef]

W. J. Byun, J. W. Yu, and N. H. Myung, “TM scattering from hollow and dielectric-filled semielliptic channels with arbitrary eccentricity in a perfectly conducting plane,” IEEE Trans. Microwave Theory Tech. 46, 1336–1339 (1998).
[CrossRef]

1997

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag. 45, 926–935 (1997).
[CrossRef]

1995

J. Yan, R. K. Gordon, and A. A. Kishk, “Electromagnetic scattering from impedance elliptic cylinders using finite difference method (oblique incidence),” Electromagnetics 15, 157–173(1995).
[CrossRef]

1993

J. A. Roumeliotis, H. K. Manthopoulos, and V. K. Manthopoulos, “Electromagnetic scattering from an infinite circular metallic cylinder coated by an elliptic dielectric one,” IEEE Trans. Microwave Theory Tech. 41, 862–869 (1993).
[CrossRef]

1991

A. Sebak and L. Shafai, “Generalized solutions for electromagnetic scattering by elliptical structures,” Comput. Phys. Commun. 68, 315–330 (1991).
[CrossRef]

1989

C. A. Balanis, Advanced Engineering Electromagnetics(Wiley, 1989).

1972

1966

1965

1964

1953

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

Alivizatos, E. G.

N. L. Tsitsas, E. G. Alivizatos, H. T. Anastassiu, and D. I. Kaklamani, “Optimization of the method of auxiliary sources (MAS) for scattering by an infinite cylinder under oblique incidence,” Electromagnetics 25, 39–54 (2005).
[CrossRef]

Anastassiu, H. T.

N. L. Tsitsas, E. G. Alivizatos, H. T. Anastassiu, and D. I. Kaklamani, “Optimization of the method of auxiliary sources (MAS) for scattering by an infinite cylinder under oblique incidence,” Electromagnetics 25, 39–54 (2005).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics(Wiley, 1989).

Boisvert, J. E.

A. L. V. Buren and J. E. Boisvert, “Accurate calculation of the modified Mathieu functions of integer order,” Q. Appl. Math. 65, 1–23 (2007).

Buren, A. L. V.

A. L. V. Buren and J. E. Boisvert, “Accurate calculation of the modified Mathieu functions of integer order,” Q. Appl. Math. 65, 1–23 (2007).

Byun, W. J.

W. J. Byun, J. W. Yu, and N. H. Myung, “TM scattering from hollow and dielectric-filled semielliptic channels with arbitrary eccentricity in a perfectly conducting plane,” IEEE Trans. Microwave Theory Tech. 46, 1336–1339 (1998).
[CrossRef]

Caorsi, S.

S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinders,” IEEE Trans. Antennas Propag. 52, 189–196 (2004).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Analytic SAR computation in a multilayer elliptic cylinder for bioelectromagnetic applications,” Bioelectromagnetics 20, 365–371 (1999).
[CrossRef] [PubMed]

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag. 45, 926–935 (1997).
[CrossRef]

Feshbach, H.

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

Gordon, R. K.

J. Yan, R. K. Gordon, and A. A. Kishk, “Electromagnetic scattering from impedance elliptic cylinders using finite difference method (oblique incidence),” Electromagnetics 15, 157–173(1995).
[CrossRef]

Han, X.

R. Li, X. Han, and K. F. Ren, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E 79, 036602(2009).
[CrossRef]

Kaklamani, D. I.

N. L. Tsitsas, E. G. Alivizatos, H. T. Anastassiu, and D. I. Kaklamani, “Optimization of the method of auxiliary sources (MAS) for scattering by an infinite cylinder under oblique incidence,” Electromagnetics 25, 39–54 (2005).
[CrossRef]

Kishk, A. A.

J. Yan, R. K. Gordon, and A. A. Kishk, “Electromagnetic scattering from impedance elliptic cylinders using finite difference method (oblique incidence),” Electromagnetics 15, 157–173(1995).
[CrossRef]

Li, H. Y.

Li, R.

R. Li, X. Han, and K. F. Ren, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E 79, 036602(2009).
[CrossRef]

Liou, K.-N.

Manthopoulos, H. K.

J. A. Roumeliotis, H. K. Manthopoulos, and V. K. Manthopoulos, “Electromagnetic scattering from an infinite circular metallic cylinder coated by an elliptic dielectric one,” IEEE Trans. Microwave Theory Tech. 41, 862–869 (1993).
[CrossRef]

Manthopoulos, V. K.

J. A. Roumeliotis, H. K. Manthopoulos, and V. K. Manthopoulos, “Electromagnetic scattering from an infinite circular metallic cylinder coated by an elliptic dielectric one,” IEEE Trans. Microwave Theory Tech. 41, 862–869 (1993).
[CrossRef]

Mao, S. C.

Martin, O. J. F.

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915(1998).
[CrossRef]

Morse, P. M.

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

Myung, N. H.

W. J. Byun, J. W. Yu, and N. H. Myung, “TM scattering from hollow and dielectric-filled semielliptic channels with arbitrary eccentricity in a perfectly conducting plane,” IEEE Trans. Microwave Theory Tech. 46, 1336–1339 (1998).
[CrossRef]

Pastorino, M.

M. Pastorino, Microwave Imaging (Wiley, 2010).
[CrossRef]

S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinders,” IEEE Trans. Antennas Propag. 52, 189–196 (2004).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Analytic SAR computation in a multilayer elliptic cylinder for bioelectromagnetic applications,” Bioelectromagnetics 20, 365–371 (1999).
[CrossRef] [PubMed]

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag. 45, 926–935 (1997).
[CrossRef]

Piller, N. B.

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915(1998).
[CrossRef]

Raffetto, M.

S. Caorsi, M. Pastorino, and M. Raffetto, “Analytic SAR computation in a multilayer elliptic cylinder for bioelectromagnetic applications,” Bioelectromagnetics 20, 365–371 (1999).
[CrossRef] [PubMed]

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag. 45, 926–935 (1997).
[CrossRef]

Ren, K. F.

R. Li, X. Han, and K. F. Ren, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E 79, 036602(2009).
[CrossRef]

Roumeliotis, J. A.

G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Electromagnetic scattering by an infinite elliptic dielectric cylinder with small eccentricity using perturbative analysis,” IEEE Trans. Antennas Propag. 58, 107–121 (2010).
[CrossRef]

G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Scattering by an infinite elliptic metallic cylinder,” Electromagnetics 27, 159–182 (2007).
[CrossRef]

J. A. Roumeliotis, H. K. Manthopoulos, and V. K. Manthopoulos, “Electromagnetic scattering from an infinite circular metallic cylinder coated by an elliptic dielectric one,” IEEE Trans. Microwave Theory Tech. 41, 862–869 (1993).
[CrossRef]

Rulf, B.

Savaidis, S. P.

G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Electromagnetic scattering by an infinite elliptic dielectric cylinder with small eccentricity using perturbative analysis,” IEEE Trans. Antennas Propag. 58, 107–121 (2010).
[CrossRef]

G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Scattering by an infinite elliptic metallic cylinder,” Electromagnetics 27, 159–182 (2007).
[CrossRef]

Sebak, A.

A. Sebak and L. Shafai, “Generalized solutions for electromagnetic scattering by elliptical structures,” Comput. Phys. Commun. 68, 315–330 (1991).
[CrossRef]

Sebak, A. R.

A. R. Sebak, “Scattering from dielectric-coated impedance elliptic cylinder,” IEEE Trans. Antennas Propag. 48, 1574–1580(2000).
[CrossRef]

Shafai, L.

A. Sebak and L. Shafai, “Generalized solutions for electromagnetic scattering by elliptical structures,” Comput. Phys. Commun. 68, 315–330 (1991).
[CrossRef]

Tsalamengas, J. L.

J. L. Tsalamengas, “Exponentially converging Nyström methods applied to the integral-integrodifferential equations of oblique scattering/hybrid wave propagation in presence of composite dielectric cylinders of arbitrary cross section,” IEEE Trans. Antennas Propag. 55, 3239–3250 (2007).
[CrossRef]

Tsitsas, N. L.

N. L. Tsitsas, E. G. Alivizatos, H. T. Anastassiu, and D. I. Kaklamani, “Optimization of the method of auxiliary sources (MAS) for scattering by an infinite cylinder under oblique incidence,” Electromagnetics 25, 39–54 (2005).
[CrossRef]

Tsogkas, G. D.

G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Electromagnetic scattering by an infinite elliptic dielectric cylinder with small eccentricity using perturbative analysis,” IEEE Trans. Antennas Propag. 58, 107–121 (2010).
[CrossRef]

G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Scattering by an infinite elliptic metallic cylinder,” Electromagnetics 27, 159–182 (2007).
[CrossRef]

Wu, Z. S.

Yan, J.

J. Yan, R. K. Gordon, and A. A. Kishk, “Electromagnetic scattering from impedance elliptic cylinders using finite difference method (oblique incidence),” Electromagnetics 15, 157–173(1995).
[CrossRef]

Yeh, C.

Yu, J. W.

W. J. Byun, J. W. Yu, and N. H. Myung, “TM scattering from hollow and dielectric-filled semielliptic channels with arbitrary eccentricity in a perfectly conducting plane,” IEEE Trans. Microwave Theory Tech. 46, 1336–1339 (1998).
[CrossRef]

Appl. Opt.

Bioelectromagnetics

S. Caorsi, M. Pastorino, and M. Raffetto, “Analytic SAR computation in a multilayer elliptic cylinder for bioelectromagnetic applications,” Bioelectromagnetics 20, 365–371 (1999).
[CrossRef] [PubMed]

Comput. Phys. Commun.

A. Sebak and L. Shafai, “Generalized solutions for electromagnetic scattering by elliptical structures,” Comput. Phys. Commun. 68, 315–330 (1991).
[CrossRef]

Electromagnetics

G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Scattering by an infinite elliptic metallic cylinder,” Electromagnetics 27, 159–182 (2007).
[CrossRef]

J. Yan, R. K. Gordon, and A. A. Kishk, “Electromagnetic scattering from impedance elliptic cylinders using finite difference method (oblique incidence),” Electromagnetics 15, 157–173(1995).
[CrossRef]

N. L. Tsitsas, E. G. Alivizatos, H. T. Anastassiu, and D. I. Kaklamani, “Optimization of the method of auxiliary sources (MAS) for scattering by an infinite cylinder under oblique incidence,” Electromagnetics 25, 39–54 (2005).
[CrossRef]

IEEE Trans. Antennas Propag.

J. L. Tsalamengas, “Exponentially converging Nyström methods applied to the integral-integrodifferential equations of oblique scattering/hybrid wave propagation in presence of composite dielectric cylinders of arbitrary cross section,” IEEE Trans. Antennas Propag. 55, 3239–3250 (2007).
[CrossRef]

G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Electromagnetic scattering by an infinite elliptic dielectric cylinder with small eccentricity using perturbative analysis,” IEEE Trans. Antennas Propag. 58, 107–121 (2010).
[CrossRef]

A. R. Sebak, “Scattering from dielectric-coated impedance elliptic cylinder,” IEEE Trans. Antennas Propag. 48, 1574–1580(2000).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by a multilayer elliptic cylinder under transverse-magnetic illumination: series solution in terms of Mathieu functions,” IEEE Trans. Antennas Propag. 45, 926–935 (1997).
[CrossRef]

S. Caorsi and M. Pastorino, “Scattering by multilayer isorefractive elliptic cylinders,” IEEE Trans. Antennas Propag. 52, 189–196 (2004).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

W. J. Byun, J. W. Yu, and N. H. Myung, “TM scattering from hollow and dielectric-filled semielliptic channels with arbitrary eccentricity in a perfectly conducting plane,” IEEE Trans. Microwave Theory Tech. 46, 1336–1339 (1998).
[CrossRef]

J. A. Roumeliotis, H. K. Manthopoulos, and V. K. Manthopoulos, “Electromagnetic scattering from an infinite circular metallic cylinder coated by an elliptic dielectric one,” IEEE Trans. Microwave Theory Tech. 41, 862–869 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. Rev. E

R. Li, X. Han, and K. F. Ren, “Generalized Debye series expansion of electromagnetic plane wave scattering by an infinite multilayered cylinder at oblique incidence,” Phys. Rev. E 79, 036602(2009).
[CrossRef]

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915(1998).
[CrossRef]

Q. Appl. Math.

A. L. V. Buren and J. E. Boisvert, “Accurate calculation of the modified Mathieu functions of integer order,” Q. Appl. Math. 65, 1–23 (2007).

Other

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

C. A. Balanis, Advanced Engineering Electromagnetics(Wiley, 1989).

M. Pastorino, Microwave Imaging (Wiley, 2010).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Geometry of the configuration.

Fig. 2
Fig. 2

Backscattering cross section versus the incident angle ψ. Solid curves, θ = 10 ° ; dashed curves, θ = 30 ° ; dashed–dotted curves, θ = 80 ° . The rest parameters are ϵ 2 / ϵ 1 = 2.54 , μ 2 / μ 1 = 1 , h = 0.6 , and k 1 c 0 = 10 π .

Fig. 3
Fig. 3

Forward scattering cross section versus the incident angle ψ. Solid curves, θ = 10 ° ; dashed curves, θ = 30 ° ; dashed–dotted curves, θ = 80 ° . The other values of the parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

Total scattering cross section versus the incident angle ψ. Solid curves, θ = 10 ° ; dashed curves, θ = 30 ° ; dashed–dotted curves, θ = 80 ° . The other values of the parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

Back scattering cross section in decibels versus the incident angle ψ, and for θ = 60 ° . Solid curves, k 1 c 0 = 2 π ; dashed curves, k 1 c 0 = 20 π . The other values of the parameters are the same as in Fig. 2.

Fig. 6
Fig. 6

Back scattering cross section versus the incident angle ψ, but for h = 0.8 . Solid curves, θ = 10 ° ; dashed curves, θ = 30 ° ; dashed–dotted curves, θ = 80 ° . The other values of the parameters are the same as in Fig. 2.

Fig. 7
Fig. 7

Back scattering cross section in decibels versus c 0 / λ 1 for ψ = 45 ° , θ = 10 ° , and h = 0.8 . The other values of the parameters are the same as in Fig. 2.

Fig. 8
Fig. 8

Back scattering cross section in decibels versus c 0 / λ 1 for ψ = 45 ° , θ = 70 ° , and h = 0.8 . The other values of the parameters are the same as in Fig. 2.

Tables (3)

Tables Icon

Table 1 Comparison with the Exact, Closed-Form Relations in [4, 5] Based on Asymptotic Analysis a

Tables Icon

Table 2 Comparison with the Highly Accurate Results from [17] a

Tables Icon

Table 3 Absolute Value of all Tangential Total Field Components on the Elliptic Boundary a

Equations (27)

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

E z i = exp ( j k 1 · r ) = exp [ j h 1 ( cosh η cos φ cos ψ + sinh η sin φ sin ψ ) ] exp ( j k 1 z cos θ ) ,
E z i = 8 π e j β z [ m = 0 j m S e m ( h 1 , cos ψ ) M m e ( h 1 ) J e m ( h 1 , cosh η ) S e m ( h 1 , cos φ ) + m = 1 j m S o m ( h 1 , cos ψ ) M m o ( h 1 ) J o m ( h 1 , cosh η ) S o m ( h 1 , cos φ ) ] ,
E z sc = e j β z [ m = 0 A m e H e m ( 2 ) ( h 1 , cosh η ) S e m ( h 1 , cos φ ) + m = 1 A m o H o m ( 2 ) ( h 1 , cosh η ) S o m ( h 1 , cos φ ) ] ,
E η sc = j β h η κ 1 2 E z sc η j ω μ 1 h η κ 1 2 H z sc φ ,
E φ sc = j β h η κ 1 2 E z sc φ + j ω μ 1 h η κ 1 2 H z sc η ,
H η sc = j ω ϵ 1 h η κ 1 2 E z sc φ j β h η κ 1 2 H z sc η ,
H φ sc = j ω ϵ 1 h η κ 1 2 E z sc η j β h η κ 1 2 H z sc φ ,
E z 2 = e j β z [ m = 0 C m e J e m ( h 2 , cosh η ) S e m ( h 2 , cos φ ) + m = 1 C m o J o m ( h 2 , cosh η ) S o m ( h 2 , cos φ ) ] ,
E z i + E z sc = E z 2 .
C i e = [ 8 π m = 0 j m S e m ( h 1 , cos ψ ) M m e ( h 1 ) J e m ( h 1 , cosh η 0 ) M m i e ( h 1 , h 2 ) + m = 0 A m e H e m ( h 1 , cosh η 0 ) M m i e ( h 1 , h 2 ) ] / [ J e i ( h 2 , cosh η 0 ) M i e ( h 2 ) ] , i = 0 , 1 , 2 ,
H z sc = H z 2 ,
F i e = m = 0 D m e H e m ( h 1 , cosh η 0 ) M m i e ( h 1 , h 2 ) / [ J e i ( h 2 , cosh η 0 ) M i e ( h 2 ) ] , i = 0 , 1 , 2 , ,
β κ 1 2 E z i φ β κ 1 2 E z sc φ + ω μ 1 κ 1 2 H z sc η = β κ 2 2 E z 2 φ + ω μ 2 κ 2 2 H z 2 η .
cos θ m = 1 H o m ( h 1 , cosh η 0 ) M m s o e ( h 1 ) A m o + cos θ ϵ 1 ϵ 2 μ 1 μ 2 m = 1 i = 1 A m o H o m ( h 1 , cosh η 0 ) M m i o ( h 1 , h 2 ) M i s o e ( h 2 , h 1 ) M i o ( h 2 ) + μ 1 ϵ 1 H e s ( h 1 , cosh η 0 ) M s e ( h 1 ) D s e μ 1 ϵ 1 ϵ 1 ϵ 2 m = 0 i = 0 D m e H e m ( h 1 , cosh η 0 ) J e i ( h 2 , cosh η 0 ) J e i ( h 2 , cosh η 0 ) M m i e ( h 1 , h 2 ) M i s e ( h 2 , h 1 ) M i e ( h 2 ) = cos θ 8 π m = 1 j m S o m ( h 1 , cos ψ ) J o m ( h 1 , cosh η 0 ) M m s o e ( h 1 ) M m o ( h 1 ) cos θ ϵ 1 ϵ 2 μ 1 μ 2 8 π m = 1 i = 1 j m S o m ( h 1 , cos ψ ) M m o ( h 1 ) J o m ( h 1 , cosh η 0 ) × M m i o ( h 1 , h 2 ) M i s o e ( h 2 , h 1 ) M i o ( h 2 ) , s = 0 , 1 , 2 , .
ω ϵ 1 κ 1 2 E z i η ω ϵ 1 κ 1 2 E z sc η β κ 1 2 H z sc φ = ω ϵ 2 κ 2 2 E z 2 η β κ 2 2 H z 2 φ .
RHS = ϵ 1 μ 1 8 π j s S o s ( h 1 , cos ψ ) J o s ( h 1 , cosh η 0 ) ϵ 1 μ 1 μ 1 μ 2 8 π m = 1 i = 1 j m S o m ( h 1 , cos ψ ) M m o ( h 1 ) J o m ( h 1 , cosh η 0 ) J o i ( h 2 , cosh η 0 ) J o i ( h 2 , cosh η 0 ) M m i o ( h 1 , h 2 ) M i s o ( h 2 , h 1 ) M i o ( h 2 ) , s = 1 , 2 , 3 ,
σ = lim ρ 2 π ρ | E sc | 2 | E i | 2 ,
H e o m ( h 1 , cosh η ) η 1 κ 1 a cosh η exp [ j ( κ 1 a cosh η ( 2 m + 1 ) π / 4 ) ] ,
H e o m ( h 1 , cosh η ) η 1 κ 1 a cosh η exp [ j ( κ 1 a cosh η ( 2 m + 1 ) π / 4 ) ] ( 1 2 j κ 1 a cosh η ) ,
k 1 σ b = 2 π sin θ [ | G E ( ψ + π ) | 2 + μ 1 ϵ 1 | G H ( ψ + π ) | 2 ] ,
k 1 σ f = 2 π sin θ [ | G E ( ψ ) | 2 + μ 1 ϵ 1 | G H ( ψ ) | 2 ] ,
[ G E ( φ ) G H ( φ ) ] = m = 0 [ A m e D m e ] j m S e m ( h 1 , cos φ ) + m = 1 [ A m o D m o ] j m S o m ( h 1 , cos φ ) .
k 1 Q t = 1 sin θ { m = 0 | A m e | 2 M m e ( h 1 ) + m = 1 | A m o | 2 M m o ( h 1 ) + μ 1 ϵ 1 [ m = 0 | D m e | 2 M m e ( h 1 ) + m = 1 | D m o | 2 M m o ( h 1 ) ] } .
0 2 π S e o m ( h , cos φ ) S e o v ( h , cos φ ) d φ = M m v e o ( h ) δ m v = δ m v 2 π n = 0 1 1 ε n B n e o ( h , m ) B n e o ( h , v )
0 2 π S e o m ( h 1 , cos φ ) S e o v ( h 2 , cos φ ) d φ = M m v e o ( h 1 , h 2 ) = 2 π n = 0 1 1 ε n B n e o ( h 1 , m ) B n e o ( h 2 , v )
0 2 π S e o m ( h , cos φ ) S o e v ( h , cos φ ) d φ = M m v e o o e ( h ) = π n = 1 n B n e o ( h , m ) B n o e ( h , v ) ,
0 2 π S e o m ( h 1 , cos φ ) S o e v ( h 2 , cos φ ) d φ = M m v e o o e ( h 1 , h 2 ) = π n = 1 n B n e o ( h 1 , m ) B n o e ( h 2 , v )

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