Abstract

An analytical, closed-form solution to the scattering problem from an infinite lossless or lossy elliptical cylinder coating a circular metal core is treated in this work. The problem is solved by expressing the electromagnetic field in both elliptical and circular wave functions, connected with one another by well-known expansion formulas. The procedure for solving the problem is cumbersome because of the nonexistence of orthogonality relations for Mathieu functions across the dielectric elliptical boundary. The solution obtained, which is free of Mathieu functions, is given in closed form, and it is valid for small values of the eccentricity h of the elliptical cylinder. Analytical expressions of the form S(h)=S(0)[1+g(2)h2+g(4)h4+O(h6)] are obtained, permitting an immediate calculation for the scattering cross sections. The proposed method is an alternative one, for small h, to the standard exact numerical solution obtained after the truncation of the system matrices, composed after the satisfaction of the boundary conditions. Both polarizations are considered for normal incidence. The results are validated against the exact solution, and numerical results are given for various values of the parameters.

© 2013 Optical Society of America

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  1. J. H. Richmond, “Scattering by a conducting elliptic cylinder with dielectric coating,” Radio Sci. 23, 1061–1066 (1988).
    [CrossRef]
  2. H. A. Ragheb and L. Shafai, “Electromagnetic scattering from a dielectric-coated elliptic cylinder,” Can. J. Phys. 66, 1115–1122 (1988).
    [CrossRef]
  3. A. K. Hamid and M. I. Hussein, “Electromagnetic scattering by a lossy dielectric-coated elliptic cylinder,” Can. J. Phys. 81, 771–778 (2003).
    [CrossRef]
  4. H. Ragheb, L. Shafai, and M. Hamid, “Plane-wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag. 39, 218–223 (1991).
    [CrossRef]
  5. A. Sebak, H. Ragheb, and L. Shafai, “Plane-wave scattering by dielectric elliptic cylinder coated with nonconfocal dielectric,” Radio Sci. 29, 1393–1401 (1994).
    [CrossRef]
  6. G. P. Zouros, “Oblique electromagnetic scattering from lossless or lossy composite elliptical dielectric cylinders,” J. Opt. Soc. Am. A 30, 196–205 (2013).
    [CrossRef]
  7. C. S. Kim and C. Yeh, “Scattering of an obliquely incident wave by a multilayered elliptical lossy dielectric cylinder,” Radio Sci. 26, 1165–1176 (1991).
    [CrossRef]
  8. S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by weakly lossy multilayer elliptic cylinders,” IEEE Trans. Antennas Propag. 46, 1750–1751 (1998).
    [CrossRef]
  9. S. Caorsi, M. Pastorino, and M. Raffetto, “EM field prediction inside lossy multilayer elliptic cylinders for biological-body modeling and numerical-procedure testing,” IEEE Trans. Biomed. Eng. 46, 1304–1309 (1999).
    [CrossRef]
  10. S. Caorsi, M. Pastorino, and M. Raffetto, “Radar cross section per unit length of a lossy multilayer elliptic cylinder,” Microw. Opt. Technol. Lett. 21, 380–384 (1999).
    [CrossRef]
  11. T.-K. Wu and L. L. Tsai, “Scattering by arbitrarily cross-sectioned layered lossy dielectric cylinders,” IEEE Trans. Antennas Propag. AP-25, 518–524 (1977).
  12. 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]
  13. G. P. Zouros and J. A. Roumeliotis, “Scattering by an infinite dielectric cylinder having an elliptic metal core: asymptotic solutions,” IEEE Trans. Antennas Propag. 58, 3299–3309 (2010).
    [CrossRef]
  14. 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]
  15. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  16. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).
  17. 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]
  18. G. C. Kokkorakis and J. A. Roumeliotis, “Power series expansions for Mathieu functions with small arguments,” Math. Comput. 70, 1221–1235 (2001).
    [CrossRef]
  19. G. D. Tsogkas, J. A. Roumeliotis, and S. P. Savaidis, “Scattering by an infinite elliptic metallic cylinder,” Electromagnetics 27, 159–182 (2007).
    [CrossRef]

2013 (1)

2010 (2)

G. P. Zouros and J. A. Roumeliotis, “Scattering by an infinite dielectric cylinder having an elliptic metal core: asymptotic solutions,” IEEE Trans. Antennas Propag. 58, 3299–3309 (2010).
[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]

2007 (2)

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, “Scattering by an infinite elliptic metallic cylinder,” Electromagnetics 27, 159–182 (2007).
[CrossRef]

2003 (1)

A. K. Hamid and M. I. Hussein, “Electromagnetic scattering by a lossy dielectric-coated elliptic cylinder,” Can. J. Phys. 81, 771–778 (2003).
[CrossRef]

2001 (1)

G. C. Kokkorakis and J. A. Roumeliotis, “Power series expansions for Mathieu functions with small arguments,” Math. Comput. 70, 1221–1235 (2001).
[CrossRef]

1999 (2)

S. Caorsi, M. Pastorino, and M. Raffetto, “EM field prediction inside lossy multilayer elliptic cylinders for biological-body modeling and numerical-procedure testing,” IEEE Trans. Biomed. Eng. 46, 1304–1309 (1999).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Radar cross section per unit length of a lossy multilayer elliptic cylinder,” Microw. Opt. Technol. Lett. 21, 380–384 (1999).
[CrossRef]

1998 (1)

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by weakly lossy multilayer elliptic cylinders,” IEEE Trans. Antennas Propag. 46, 1750–1751 (1998).
[CrossRef]

1994 (1)

A. Sebak, H. Ragheb, and L. Shafai, “Plane-wave scattering by dielectric elliptic cylinder coated with nonconfocal dielectric,” Radio Sci. 29, 1393–1401 (1994).
[CrossRef]

1993 (1)

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 (2)

C. S. Kim and C. Yeh, “Scattering of an obliquely incident wave by a multilayered elliptical lossy dielectric cylinder,” Radio Sci. 26, 1165–1176 (1991).
[CrossRef]

H. Ragheb, L. Shafai, and M. Hamid, “Plane-wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag. 39, 218–223 (1991).
[CrossRef]

1988 (2)

J. H. Richmond, “Scattering by a conducting elliptic cylinder with dielectric coating,” Radio Sci. 23, 1061–1066 (1988).
[CrossRef]

H. A. Ragheb and L. Shafai, “Electromagnetic scattering from a dielectric-coated elliptic cylinder,” Can. J. Phys. 66, 1115–1122 (1988).
[CrossRef]

1977 (1)

T.-K. Wu and L. L. Tsai, “Scattering by arbitrarily cross-sectioned layered lossy dielectric cylinders,” IEEE Trans. Antennas Propag. AP-25, 518–524 (1977).

Caorsi, S.

S. Caorsi, M. Pastorino, and M. Raffetto, “EM field prediction inside lossy multilayer elliptic cylinders for biological-body modeling and numerical-procedure testing,” IEEE Trans. Biomed. Eng. 46, 1304–1309 (1999).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Radar cross section per unit length of a lossy multilayer elliptic cylinder,” Microw. Opt. Technol. Lett. 21, 380–384 (1999).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by weakly lossy multilayer elliptic cylinders,” IEEE Trans. Antennas Propag. 46, 1750–1751 (1998).
[CrossRef]

Feshbach, H.

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

Hamid, A. K.

A. K. Hamid and M. I. Hussein, “Electromagnetic scattering by a lossy dielectric-coated elliptic cylinder,” Can. J. Phys. 81, 771–778 (2003).
[CrossRef]

Hamid, M.

H. Ragheb, L. Shafai, and M. Hamid, “Plane-wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag. 39, 218–223 (1991).
[CrossRef]

Hussein, M. I.

A. K. Hamid and M. I. Hussein, “Electromagnetic scattering by a lossy dielectric-coated elliptic cylinder,” Can. J. Phys. 81, 771–778 (2003).
[CrossRef]

Kim, C. S.

C. S. Kim and C. Yeh, “Scattering of an obliquely incident wave by a multilayered elliptical lossy dielectric cylinder,” Radio Sci. 26, 1165–1176 (1991).
[CrossRef]

Kokkorakis, G. C.

G. C. Kokkorakis and J. A. Roumeliotis, “Power series expansions for Mathieu functions with small arguments,” Math. Comput. 70, 1221–1235 (2001).
[CrossRef]

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]

Morse, P. M.

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

Pastorino, M.

S. Caorsi, M. Pastorino, and M. Raffetto, “Radar cross section per unit length of a lossy multilayer elliptic cylinder,” Microw. Opt. Technol. Lett. 21, 380–384 (1999).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “EM field prediction inside lossy multilayer elliptic cylinders for biological-body modeling and numerical-procedure testing,” IEEE Trans. Biomed. Eng. 46, 1304–1309 (1999).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by weakly lossy multilayer elliptic cylinders,” IEEE Trans. Antennas Propag. 46, 1750–1751 (1998).
[CrossRef]

Raffetto, M.

S. Caorsi, M. Pastorino, and M. Raffetto, “EM field prediction inside lossy multilayer elliptic cylinders for biological-body modeling and numerical-procedure testing,” IEEE Trans. Biomed. Eng. 46, 1304–1309 (1999).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Radar cross section per unit length of a lossy multilayer elliptic cylinder,” Microw. Opt. Technol. Lett. 21, 380–384 (1999).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by weakly lossy multilayer elliptic cylinders,” IEEE Trans. Antennas Propag. 46, 1750–1751 (1998).
[CrossRef]

Ragheb, H.

A. Sebak, H. Ragheb, and L. Shafai, “Plane-wave scattering by dielectric elliptic cylinder coated with nonconfocal dielectric,” Radio Sci. 29, 1393–1401 (1994).
[CrossRef]

H. Ragheb, L. Shafai, and M. Hamid, “Plane-wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag. 39, 218–223 (1991).
[CrossRef]

Ragheb, H. A.

H. A. Ragheb and L. Shafai, “Electromagnetic scattering from a dielectric-coated elliptic cylinder,” Can. J. Phys. 66, 1115–1122 (1988).
[CrossRef]

Richmond, J. H.

J. H. Richmond, “Scattering by a conducting elliptic cylinder with dielectric coating,” Radio Sci. 23, 1061–1066 (1988).
[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. P. Zouros and J. A. Roumeliotis, “Scattering by an infinite dielectric cylinder having an elliptic metal core: asymptotic solutions,” IEEE Trans. Antennas Propag. 58, 3299–3309 (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]

G. C. Kokkorakis and J. A. Roumeliotis, “Power series expansions for Mathieu functions with small arguments,” Math. Comput. 70, 1221–1235 (2001).
[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]

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, H. Ragheb, and L. Shafai, “Plane-wave scattering by dielectric elliptic cylinder coated with nonconfocal dielectric,” Radio Sci. 29, 1393–1401 (1994).
[CrossRef]

Shafai, L.

A. Sebak, H. Ragheb, and L. Shafai, “Plane-wave scattering by dielectric elliptic cylinder coated with nonconfocal dielectric,” Radio Sci. 29, 1393–1401 (1994).
[CrossRef]

H. Ragheb, L. Shafai, and M. Hamid, “Plane-wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag. 39, 218–223 (1991).
[CrossRef]

H. A. Ragheb and L. Shafai, “Electromagnetic scattering from a dielectric-coated elliptic cylinder,” Can. J. Phys. 66, 1115–1122 (1988).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Tsai, L. L.

T.-K. Wu and L. L. Tsai, “Scattering by arbitrarily cross-sectioned layered lossy dielectric cylinders,” IEEE Trans. Antennas Propag. AP-25, 518–524 (1977).

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]

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, T.-K.

T.-K. Wu and L. L. Tsai, “Scattering by arbitrarily cross-sectioned layered lossy dielectric cylinders,” IEEE Trans. Antennas Propag. AP-25, 518–524 (1977).

Yeh, C.

C. S. Kim and C. Yeh, “Scattering of an obliquely incident wave by a multilayered elliptical lossy dielectric cylinder,” Radio Sci. 26, 1165–1176 (1991).
[CrossRef]

Zouros, G. P.

G. P. Zouros, “Oblique electromagnetic scattering from lossless or lossy composite elliptical dielectric cylinders,” J. Opt. Soc. Am. A 30, 196–205 (2013).
[CrossRef]

G. P. Zouros and J. A. Roumeliotis, “Scattering by an infinite dielectric cylinder having an elliptic metal core: asymptotic solutions,” IEEE Trans. Antennas Propag. 58, 3299–3309 (2010).
[CrossRef]

Can. J. Phys. (2)

H. A. Ragheb and L. Shafai, “Electromagnetic scattering from a dielectric-coated elliptic cylinder,” Can. J. Phys. 66, 1115–1122 (1988).
[CrossRef]

A. K. Hamid and M. I. Hussein, “Electromagnetic scattering by a lossy dielectric-coated elliptic cylinder,” Can. J. Phys. 81, 771–778 (2003).
[CrossRef]

Electromagnetics (1)

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

IEEE Trans. Antennas Propag. (6)

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]

T.-K. Wu and L. L. Tsai, “Scattering by arbitrarily cross-sectioned layered lossy dielectric cylinders,” IEEE Trans. Antennas Propag. AP-25, 518–524 (1977).

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. P. Zouros and J. A. Roumeliotis, “Scattering by an infinite dielectric cylinder having an elliptic metal core: asymptotic solutions,” IEEE Trans. Antennas Propag. 58, 3299–3309 (2010).
[CrossRef]

H. Ragheb, L. Shafai, and M. Hamid, “Plane-wave scattering by a conducting elliptic cylinder coated by a nonconfocal dielectric,” IEEE Trans. Antennas Propag. 39, 218–223 (1991).
[CrossRef]

S. Caorsi, M. Pastorino, and M. Raffetto, “Electromagnetic scattering by weakly lossy multilayer elliptic cylinders,” IEEE Trans. Antennas Propag. 46, 1750–1751 (1998).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

S. Caorsi, M. Pastorino, and M. Raffetto, “EM field prediction inside lossy multilayer elliptic cylinders for biological-body modeling and numerical-procedure testing,” IEEE Trans. Biomed. Eng. 46, 1304–1309 (1999).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

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. A (1)

Math. Comput. (1)

G. C. Kokkorakis and J. A. Roumeliotis, “Power series expansions for Mathieu functions with small arguments,” Math. Comput. 70, 1221–1235 (2001).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

S. Caorsi, M. Pastorino, and M. Raffetto, “Radar cross section per unit length of a lossy multilayer elliptic cylinder,” Microw. Opt. Technol. Lett. 21, 380–384 (1999).
[CrossRef]

Radio Sci. (3)

C. S. Kim and C. Yeh, “Scattering of an obliquely incident wave by a multilayered elliptical lossy dielectric cylinder,” Radio Sci. 26, 1165–1176 (1991).
[CrossRef]

A. Sebak, H. Ragheb, and L. Shafai, “Plane-wave scattering by dielectric elliptic cylinder coated with nonconfocal dielectric,” Radio Sci. 29, 1393–1401 (1994).
[CrossRef]

J. H. Richmond, “Scattering by a conducting elliptic cylinder with dielectric coating,” Radio Sci. 23, 1061–1066 (1988).
[CrossRef]

Other (2)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

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

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Figures (11)

Fig. 1.
Fig. 1.

Geometry of the scatterer.

Fig. 2.
Fig. 2.

Back and forward-scattering cross section percentage error for ϵ1/ϵ2=2.54, μ1/μ2=1, R1/R2=0.8, x2=π, and ψ=30°. Solid curves, back and up to O(h2); dashed curves, back and up to O(h4); dashed–dotted curves, forward and up to O(h2); dotted curves: forward and up to O(h4). Black curves, left-hand axis, E-wave; gray curves, right-hand axis, H-wave.

Fig. 3.
Fig. 3.

Total scattering cross section percentage error for ϵ1/ϵ2=2.54, μ1/μ2=1, R1/R2=0.8, x2=π, and ψ=30°. Solid curves, up to O(h2); dashed curves, up to O(h4). Black curves, E-wave; gray curves, H-wave.

Fig. 4.
Fig. 4.

Back and forward-scattering cross section percentage error for ϵ1/ϵ2=4j10, μ1/μ2=1, R1/R2=0.2, x2=π, and ψ=30°. Solid curves, back and up to O(h2); dashed curves, back and up to O(h4); dashed–dotted curves, forward and up to O(h2); dotted curves, forward and up to O(h4). Black curves, left-hand axis, E-wave; gray curves, right-hand axis, H-wave.

Fig. 5.
Fig. 5.

Total scattering cross section percentage error for ϵ1/ϵ2=4j10, μ1/μ2=1, R1/R2=0.2, x2=π, and ψ=30°. Solid curves, up to O(h2); dashed curves, up to O(h4). Black curves, E-wave; gray curves, H-wave.

Fig. 6.
Fig. 6.

Backscattering, forward-scattering, and total scattering cross section percentage error when R1/R2=1 for ϵ1/ϵ2=2.54, μ1/μ2=1, x2=π, and ψ=45°. Solid curves, back and up to O(h4); dashed curves, forward and up to O(h4); dashed–dotted curves, total and up to O(h4). Black curves, left axis, E-wave; gray curves, right axis, H-wave.

Fig. 7.
Fig. 7.

Backscattering cross section for ϵ1/ϵ2=2.54jα, μ1/μ2=1, R1/R2=0.2, x2=π, and h=0.3. Solid curves, α=0; dashed curves, α=1; dashed–dotted curves, α=4. Black curves, left axis, E-wave; gray curves, right axis, H-wave.

Fig. 8.
Fig. 8.

Forward-scattering cross section for ϵ1/ϵ2=2.54jα, μ1/μ2=1, R1/R2=0.2, x2=π, and h=0.3. Solid curves, α=0; dashed curves, α=1; dashed–dotted curves, α=4. Black curves, left axis, E-wave; gray curves, right axis, H-wave.

Fig. 9.
Fig. 9.

Total scattering cross section for ϵ1/ϵ2=2.54jα, μ1/μ2=1, R1/R2=0.2, x2=π, and h=0.3. Solid curves, α=0; dashed curves, α=1; dashed–dotted curves, α=4. Black curves, left axis, E-wave; gray curves, right axis, H-wave.

Fig. 10.
Fig. 10.

Backscattering cross section in decibels for ϵ1/ϵ2=2.54, μ1/μ2=1, R1/R2=0.6, ψ=45°, and h=0.3. Black curve, E-wave; gray curve, H-wave.

Fig. 11.
Fig. 11.

Backscattering cross section percentage error for ϵ1/ϵ2=2.54, μ1/μ2=1, R1/R2=0.6, ψ=45°, and h=0.3. Black curve, E-wave; gray curve, H-wave.

Tables (3)

Tables Icon

Table 1. Values of k2σ(0), k2Qt(0), g(2) and g(4) for ϵ1/ϵ2=2.54, μ1/μ2=1, R1/R2=0.8, and x2=π

Tables Icon

Table 2. Values of k2σ(0), k2Qt(0), g(2) and g(4) for ϵ1/ϵ2=4j10, μ1/μ2=1, R1/R2=0.2, and x2=π

Tables Icon

Table 3. CPU Time (s) for Computation of k2σb using the Closed-Form Solution and the Exact Solution, for H-Wave Polarizationa

Equations (44)

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

Ezi=8π[m=0jmSem(h2,cosψ)Mme(h2)Jem(h2,coshξ)×Sem(h2,cosη)+m=1jmSom(h2,cosψ)Mmo(h2)×Jom(h2,coshξ)Som(h2,cosη)],
Ezsc=m=0PmHem(h2,coshξ)Sem(h2,cosη)+m=1QmHom(h2,coshξ)Som(h2,cosη),
EzI=i=0Ai[Ji(k1r)+qiNi(k1r)]cos(iφ)+i=1Ci[Ji(k1r)+qiNi(k1r)]sin(iφ),
Zi(k1r)cos(iφ)sin(iφ)=8π1/εi1/2m=01jmiBieo(h1,m)Mmeo(h1)×Zeom(h1,coshξ)Seom(h1,cosη),i01.
Ezi+Ezsc=EzI,1μ2(Ezi+Ezsc)ξ=1μ1EzIξ,
Ps=8πHes(h2,coshξ0)Mse(h2)×{i=0Ai1εim=0jmiBie(h1,m)Mmse(h1,h2)Mme(h1)×[Jem(h1,coshξ0)+qiNem(h1,coshξ0)]jsSes(h2,cosψ)Jes(h2,coshξ0)},s0,
i=0Ai1εim=0jmiBie(h1,m)Mmse(h1,h2)Mme(h1)×{μ2μ1[Jem(h1,coshξ0)+qiNem(h1,coshξ0)]Hes(h2,coshξ0)Hes(h2,coshξ0)[Jem(h1,coshξ0)+qiNem(h1,coshξ0)]}=jsSes(h2,cosψ)×[Jes(h2,coshξ0)Hes(h2,coshξ0)Hes(h2,coshξ0)Jes(h2,coshξ0)],s0.
i=01m=01aimsAicimsCi=bsds,s01.
aims(4)As4+aims(2)As2+aims(0)As+aims(+2)As+2+aims(+4)As+4=bs,i,m=s,s±2,s±4,s0.
aims(0)=Ds,s,s(0)+Ds,s,s(2)h2+[Ds,s,s(4)+Ds,s+2,s(4)+Ds,s2,s(4)(1δ0s)(1δ1s)]h4+O(h6),s0,aims(±2)=[Ds±2,s±2,s(2)+Ds±2,s,s(2)]h2+[Ds±2,s±2,s(4)+Ds±2,s,s(4)]h4+O(h6),s02,aims(±4)=[Ds±4,s±4,s(4)+Ds±4,s±2,s(4)+Ds±4,s,s(4)]h4+O(h6),s04,
bs=bs(0)+bs(2)h2+bs(4)h4+O(h6),s0.
As=As(0)+As(2)h2+As(4)h4+O(h6),
As(0)=bs(0)Ds,s,s(0),s0,
As(2)=1Ds,s,s(0){bs(2)bs(0)Ds,s,s(2)Ds,s,s(0)n=s±2bn(0)[Dn,s,s(2)+Dn,n,s(2)]Dn,n,n(0)},s0,
As(4)=1Ds,s,s(0){bs(4)bs(2)Ds,s,s(2)Ds,s,s(0)+bs(0)[Ds,s,s(2)]2[Ds,s,s(0)]2n=s±2bn(2)[Dn,s,s(2)+Dn,n,s(2)]Dn,n,n(0)+n=s±2bn(0)Dn,n,n(2)[Dn,s,s(2)+Dn,n,s(2)][Dn,n,n(0)]2+Ds,s,s(2)Ds,s,s(0)n=s±2bn(0)[Dn,s,s(2)+Dn,n,s(2)]Dn,n,n(0)+n=s,s+4bn(0)[Dn2,n2,s(2)+Dn2,s,s(2)]×[Dn,n2,n2(2)+Dn,n,n2(2)]/[Dn2,n2,n2(0)Dn,n,n(0)]+n=s,s4bn(0)[Dn+2,n+2,s(2)+Dn+2,s,s(2)]×[Dn,n+2,n+2(2)+Dn,n,n+2(2)]/[Dn+2,n+2,n+2(0)Dn,n,n(0)]n=s±2bn(0)[Dn,s,s(4)+Dn,n,s(4)]Dn,n,n(0)n=s,s+4bn(0)Dn,n2,s(4)Dn,n,n(0)n=s,s4bn(0)Dn,n+2,s(4)Dn,n,n(0)n=s,s±4bn(0)Dn,n,s(4)Dn,n,n(0)n=s±4bn(0)Dn,s,s(4)Dn,n,n(0)},s0.
Ps=Pims(0)+Pims(2)h2+Pims(4)h4+O(h6),i,m=s,s±2,s±4,s0.
k2σb=k2σb(0)[1+gb(2)h2+gb(4)h4+O(h6)],k2σf=k2σf(0)[1+gf(2)h2+gf(4)h4+O(h6)],k2Qt=k2Qt(0)[1+gt(2)h2+gt(4)h4+O(h6)],
Iuv()=Jeu()(x1)+qvNeu()(x1),Iuv()=Jeu()(x1)+qvNeu()(x1),=0,2,4,
Ls=Hes(0)(x2)Mse(2)(x2)+Hes(2)(x2)Mse(0)(x2),
Ds,s,s(0)={Bse(0)(x1,s)Msse(0)(x1,x2)[Hes(0)(x2)Iss(0)μ1/μ2Hes(0)(x2)Iss(0)]}/[μ1/μ2εsHes(0)(x2)Mse(0)(x1)],
Ds,s,s(2)={Bse(0)(x1,s)[Msse(0)(x1,x2)[μ1/μ2[x22Hes(2)(x2)Hes(0)(x2)Mse(0)(x1)Iss(0)Hes(0)(x2)[x12Hes(0)(x2)[Mse(0)(x1)Iss(2)Mse(2)(x1)Iss(0)]+x22Hes(2)(x2)Mse(0)(x1)Iss(0)]]+x12[Hes(0)(x2)]2[Mse(0)(x1)Iss(2)Mse(2)(x1)Iss(0)]]+Hes(0)(x2)Mse(0)(x1)Msse(2)(x1,x2)[Hes(0)(x2)Iss(0)μ1/μ2Hes(0)(x2)Iss(0)]]+Bse(2)(x1,s)Hes(0)(x2)Mse(0)(x1)Msse(0)(x1,x2)[Hes(0)(x2)Iss(0)μ1/μ2Hes(0)(x2)Iss(0)]}/{μ1/μ2εs[Hes(0)(x2)]2[Mse(0)(x1)]2},
Ds,s,s(4)=1εs{1Hes(0)(x2)Mse(0)(x1)μ1/μ2[Hes(0)(x2)Iss(0)Hes(0)(x2)Iss(0)μ1/μ2]Bse(4)(x1,s)Msse(0)(x1,x2)+Bse(2)(x1,s)μ1/μ2[Hes(0)(x2)]2[Mse(0)(x1)]2[[x12[Mse(0)(x1)Iss(2)Mse(2)(x1)Iss(0)][Hes(0)(x2)]2+[x22Hes(2)(x2)Hes(0)(x2)Mse(0)(x1)Iss(0)Hes(0)(x2)[Hes(0)(x2)[Mse(0)(x1)Iss(2)Mse(2)(x1)Iss(0)]x12+x22Hes(2)(x2)Mse(0)(x1)Iss(0)]]μ1/μ2]Msse(0)(x1,x2)+Hes(0)(x2)Mse(0)(x1)[Hes(0)(x2)Iss(0)Hes(0)(x2)Iss(0)μ1/μ2]Msse(2)(x1,x2)]+Bse(0)(x1,s)[Mse(0)(x1)]3[x22Hes(2)(x2)Hes(0)(x2)Iss(0)Msse(2)(x1,x2)[Mse(0)(x1)]2[Hes(0)(x2)]2+1μ1/μ2Hes(0)(x2)[Hes(0)(x2)Iss(0)Hes(0)(x2)Iss(0)μ1/μ2]Msse(4)(x1,x2)[Mse(0)(x1)]21μ1/μ2Hes(0)(x2)[[Hes(0)(x2)[Mse(0)(x1)Iss(2)Mse(2)(x1)Iss(0)]μ1/μ2+Hes(0)(x2)[Mse(2)(x1)Iss(0)Mse(0)(x1)Iss(2)]]x12+x22Hes(2)(x2)Mse(0)(x1)Iss(0)μ1/μ2]Msse(2)(x1,x2)Mse(0)(x1)+[1μ1/μ2Hes(0)(x2)[[Mse(2)(x1)]2Mse(0)(x1)Mse(4)(x1)][Hes(0)(x2)Iss(0)Hes(0)(x2)Iss(0)μ1/μ2]x14+1μ1/μ2[Hes(0)(x2)]2Mse(0)(x1)Mse(2)(x1)[[Hes(0)(x2)[Hes(0)(x2)Iss(2)x12+x22Hes(2)(x2)Iss(0)]x22Hes(2)(x2)Hes(0)(x2)Iss(0)]μ1/μ2x12[Hes(0)(x2)]2Iss(2)]x12+[Mse(0)(x1)]2[Hes(0)(x2)Iss(4)x14Hes(0)(x2)+Iss(4)x14μ1/μ2+x22[Hes(2)(x2)Hes(0)(x2)Hes(0)(x2)Hes(2)(x2)]Iss(2)x12[Hes(0)(x2)]2x24[Hes(0)(x2)]3[Hes(0)(x2)[Hes(2)(x2)]2Hes(0)(x2)Hes(2)(x2)Hes(2)(x2)+Hes(0)(x2)[Hes(0)(x2)Hes(4)(x2)Hes(4)(x2)Hes(0)(x2)]]Iss(0)]]Msse(0)(x1,x2)]},
Ds,s±2,s(4)=Bse(2)(x1,s±2)Ms±2,se(2)(x1,x2)εsμ1/μ2Hes(0)(x2)Ms±2e(0)(x1){Hes(0)(x2)Is±2,s(0)μ1/μ2Hes(0)(x2)Is±2,s(0)},
Ds±2,s±2,s(2)=1μ1/μ2εs±2Hes(0)(x2)Ms±2e(0)(x1){Hes(0)(x2)Is±2,s±2(0)μ1/μ2+Hes(0)(x2)Is±2,s±2(0)}Bs±2e(0)(x1,s±2)Ms±2,se(2)(x1,x2),
Ds±2,s,s(2)=1μ1/μ2εs±2Hes(0)(x2)Mse(0)(x1){Hes(0)(x2)Is,s±2(0)μ1/μ2Hes(0)(x2)Is,s±2(0)}Bs±2e(2)(x1,s)Msse(0)(x1,x2),
Ds±2,s±2,s(4)=1εs±2{Bs±2e(0)(x1,s±2)[Ms±2e(0)(x1)]2[Ms±2,se(2)(x1,x2)[Ms±2e(0)(x1)[x22Is±2,s±2(0)[Hes(2)(x2)Hes(0)(x2)Hes(0)(x2)Hes(2)(x2)]/[Hes(0)(x2)]2x12Hes(0)(x2)Is±2,s±2(2)Hes(0)(x2)+x12Is±2,s±2(2)μ1/μ2]+x12Ms±2e(2)(x1)[μ1/μ2Hes(0)(x2)Is±2,s±2(0)Hes(0)(x2)Is±2,s±2(0)]/[μ1/μ2Hes(0)(x2)]]+Ms±2e(0)(x1)Ms±2,se(4)(x1,x2)[Hes(0)(x2)Is±2,s±2(0)μ1/μ2Hes(0)(x2)Is±2,s±2(0)]/[μ1/μ2Hes(0)(x2)]]+Bs±2e(2)(x1,s±2)Ms±2,se(2)(x1,x2)[Hes(0)(x2)Is±2,s±2(0)μ1/μ2Hes(0)(x2)Is±2,s±2(0)]/[μ1/μ2Hes(0)(x2)Ms±2e(0)(x1)]},
Ds±2,s,s(4)=1εs±2{Bs±2e(2)(x1,s)[Msse(0)(x1,x2)[μ1/μ2[Hes(0)(x2)[x12Hes(0)(x2)[Mse(2)(x1)Is,s±2(0)Mse(0)(x1)Is,s±2(2)]x22Hes(2)(x2)Mse(0)(x1)Is,s±2(0)]+x22Hes(2)(x2)Hes(0)(x2)Mse(0)(x1)Is,s±2(0)]x12[Hes(0)(x2)]2[Mse(0)(x1)Is,s±2(2)Mse(2)(x1)Is,s±2(0)]]+Hes(0)(x2)Mse(0)(x1)Msse(2)(x1,x2)[μ1/μ2Hes(0)(x2)Is,s±2(0)Hes(0)(x2)Is,s±2(0)]]/[μ1/μ2[Hes(0)(x2)]2[Mse(0)(x1)]2]+Bs±2e(4)(x1,s)Msse(0)(x1,x2)[μ1/μ2Hes(0)(x2)Is,s±2(0)Hes(0)(x2)Is,s±2(0)]/[μ1/μ2Hes(0)(x2)Mse(0)(x1)]},
Ds±4,s±4,s(4)=1μ1/μ2εs±4Hes(0)(x2){Bs±4e(0)(x1,s±4)Ms±4,se(4)(x1,x2)[Hes(0)(x2)Is±4,s±4(0)μ1/μ2Hes(0)(x2)Is±4,s±4(0)]/[Ms±4e(0)(x1)]},
Ds±4,s±2,s(4)=1μ1/μ2εs±4Hes(0)(x2){Bs±4e(2)(x1,s±2)Ms±2,se(2)(x1,x2)[μ1/μ2Hes(0)(x2)Is±2,s±4(0)Hes(0)(x2)Is±2,s±4(0)]/[Ms±2e(0)(x1)]},
Ds±4,s,s(4)=1μ1/μ2εs±4Hes(0)(x2){Bs±4e(4)(x1,s)Msse(0)(x1,x2)[Hes(0)(x2)Is,s±4(0)μ1/μ2Hes(0)(x2)Is,s±4(0)]/[Mse(0)(x1)]},
Pims(0)=22[As(0)Iss(0)Bse(0)(x1,s)Msse(0)(x1,x2)Mse(0)(x1)εsjsJes(0)(x2)Ses(0)(ψ)]Hes(0)(x2)Mse(0)(x2),
Pims(2)=22π[Hes(0)(x2)]2[Mse(0)(x2)]2[As(0)Πs(0)+As2(0)Πs2(0)+As+2(0)Πs+2(0)+As(2)Πs(2)+Πs],
Πs(0)=1[Mse(0)(x1)]2εs{x22Bse(0)(x1,s)Mse(0)(x1)Msse(0)(x1,x2)Iss(0)Ls+x12Bse(0)(x1,s)Hes(0)(x2)Mse(0)(x2)×Msse(0)(x1,x2)[Mse(0)(x1)Iss(2)Mse(2)(x1)Iss(0)]+Bse(0)(x1,s)Hes(0)(x2)Mse(0)(x1)Mse(0)(x2)Msse(2)(x1,x2)Iss(0)+Bse(2)(x1,s)Hes(0)(x2)Mse(0)(x1)Mse(0)(x2)Msse(0)(x1,x2)Iss(0)},
Πs±2(0)=Hes(0)(x2)Mse(0)(x2)εs±2[Bs±2e(0)(x1,s±2)Ms±2,se(2)(x1,x2)Is±2,s±2(0)Ms±2e(0)(x1)Bs±2e(2)(x1,s)Msse(0)(x1,x2)Is,s±2(0)Mse(0)(x1)],
Πs(2)=1Mse(0)(x1)εsBse(0)(x1,s)Hes(0)(x2)Mse(0)(x2)Msse(0)(x1,x2)Iss(0),
Πs=jsx22Ses(0)(ψ)Jes(0)(x2)Lsjsx22Hes(0)(x2)Mse(0)(x2)[Ses(2)(ψ)Jes(0)(x2)+Ses(0)(ψ)Jes(2)(x2)].
Pims(4)=22π[Hes(0)(x2)]3[Mse(0)(x2)]3[As(0)Λs(0)+As2(0)Λs2(0)+As+2(0)Λs+2(0)+As4(0)Λs4(0)+As+4(0)Λs+4(0)+As(2)Λs(2)+As2(2)Λs2(2)+As+2(2)Λs+2(2)+As(4)Λs(4)+Λs],
Λs(0)=1εs{[Hes(0)(x2)]2[Mse(0)(x2)]2[[Mse(2)(x1)]2Mse(0)(x1)Mse(4)(x1)]Iss(0)Bse(0)(x1,s)Msse(0)(x1,x2)x14/[Mse(0)(x1)]3[Hes(0)(x2)]2[Mse(0)(x2)]2Mse(2)(x1)Iss(2)Bse(0)(x1,s)Msse(0)(x1,x2)x14[Mse(0)(x1)]2+[Hes(0)(x2)]2[Mse(0)(x2)]2Iss(4)Bse(0)(x1,s)Msse(0)(x1,x2)x14Mse(0)(x1)x22Hes(0)(x2)Mse(0)(x2)Ls×[Mse(0)(x1)Iss(2)Mse(2)(x1)Iss(0)]Bse(0)(x1,s)Msse(0)(x1,x2)x12/[Mse(0)(x1)]2+[Hes(0)(x2)]2[Mse(0)(x2)]2[Mse(0)(x1)Iss(2)Mse(2)(x1)Iss(0)][Bse(2)(x1,s)Msse(0)(x1,x2)x12+Bse(0)(x1,s)Msse(2)(x1,x2)x12]/[Mse(0)(x1)]2+x24[Hes(2)(x2)Mse(0)(x2)LsHes(0)(x2)[Hes(4)(x2)[Mse(0)(x2)]2+Hes(0)(x2)[Mse(0)(x2)Mse(4)(x2)[Mse(2)(x2)]2]]]Iss(0)Bse(0)(x1,s)Msse(0)(x1,x2)/Mse(0)(x1)x22Hes(0)(x2)Mse(0)(x2)LsIss(0)Bse(2)(x1,s)Msse(0)(x1,x2)/Mse(0)(x1)+[Hes(0)(x2)]2[Mse(0)(x2)]2Iss(0)Bse(4)(x1,s)Msse(0)(x1,x2)Mse(0)(x1)[Hes(0)(x2)]2[Mse(0)(x2)]2×n=s±2Ins(0)Bse(2)(x1,n)Mnse(2)(x1,x2)Mne(0)(x1)x22Hes(0)(x2)Mse(0)(x2)LsIss(0)Bse(0)(x1,s)Msse(2)(x1,x2)/Mse(0)(x1)+[Hes(0)(x2)]2[Mse(0)(x2)]2Iss(0)[Bse(2)(x1,s)Msse(2)(x1,x2)+Bse(0)(x1,s)Msse(4)(x1,x2)]Mse(0)(x1)},
Λs±2(0)=Hes(0)(x2)Mse(0)(x2)εs±2{Hes(0)(x2)Mse(0)(x2)[Mse(2)(x1)Is,s±2(0)Mse(0)(x1)Is,s±2(2)]Bs±2e(2)(x1,s)Msse(0)(x1,x2)x12/[Mse(0)(x1)]2Hes(0)(x2)Mse(0)(x2)Ms±2e(2)(x1)Is±2,s±2(0)Bs±2e(0)(x1,s±2)Ms±2,se(2)(x1,x2)x12/[Ms±2e(0)(x1)]2+Hes(0)(x2)Mse(0)(x2)Is±2,s±2(2)Bs±2e(0)(x1,s±2)Ms±2,se(2)(x1,x2)x12/Ms±2e(0)(x1)+x22LsIs,s±2(0)×Bs±2e(2)(x1,s)Msse(0)(x1,x2)/Mse(0)(x1)x22LsIs±2,s±2(0)Bs±2e(0)(x1,s±2)Ms±2,se(2)(x1,x2)/Ms±2e(0)(x1)Hes(0)(x2)Mse(0)(x2)Is,s±2(0)[Bs±2e(2)(x1,s)Msse(2)(x1,x2)+Bs±2e(4)(x1,s)Msse(0)(x1,x2)]Mse(0)(x1)+Hes(0)(x2)Mse(0)(x2)Is±2,s±2(0)[Bs±2e(0)(x1,s±2)Ms±2,se(4)(x1,x2)+Bs±2e(2)(x1,s±2)Ms±2,se(2)(x1,x2)]Ms±2e(0)(x1)},
Λs±4(0)=[Hes(0)(x2)]2[Mse(0)(x2)]2εs±4{Is,s±4(0)Bs±4e(4)(x1,s)Msse(0)(x1,x2)Mse(0)(x1)Is±2,s±4(0)Bs±4e(2)(x1,s±2)Ms±2,se(2)(x1,x2)Ms±2e(0)(x1)+Is±4,s±4(0)Bs±4e(0)(x1,s±4)Ms±4,se(4)(x1,x2)Ms±4e(0)(x1)},
Λs(2)=Hes(0)(x2)Mse(0)(x2)[Mse(0)(x1)]2εs{Hes(0)(x2)Mse(0)(x2)Mse(2)(x1)Iss(0)Bse(0)(x1,s)Msse(0)(x1,x2)x12+Hes(0)(x2)Mse(0)(x1)Mse(0)(x2)Iss(2)Bse(0)(x1,s)Msse(0)(x1,x2)x12x22Mse(0)(x1)LsIss(0)Bse(0)(x1,s)Msse(0)(x1,x2)+Hes(0)(x2)Mse(0)(x1)Mse(0)(x2)Iss(0)[Bse(2)(x1,s)Msse(0)(x1,x2)+Bse(0)(x1,s)Msse(2)(x1,x2)]},
Λs±2(2)=[Hes(0)(x2)]2[Mse(0)(x2)]2εs±2{Is±2,s±2(0)Bs±2e(0)(x1,s±2)Ms±2,se(2)(x1,x2)Ms±2e(0)(x1)Is,s±2(0)Bs±2e(2)(x1)sMsse(0)(x1,x2)Mse(0)(x1)},
Λs(4)=[Hes(0)(x2)]2[Mse(0)(x2)]2Iss(0)Bse(0)(x1,s)Msse(0)(x1,x2)Mse(0)(x1)εs,
Λs=js[Hes(0)(x2)]2Jes(4)(x2)[Mse(0)(x2)]2Ses(0)(ψ)x24+jsHes(0)(x2)Jes(2)(x2)Mse(0)(x2)LsSes(0)(ψ)x24jsJes(0)(x2)[Hes(2)(x2)Mse(0)(x2)LsHes(0)(x2)[Hes(4)(x2)[Mse(0)(x2)]2+Hes(0)(x2)[Mse(0)(x2)Mse(4)(x2)[Mse(2)(x2)]2]]]Ses(0)(ψ)x24+jsHes(0)(x2)Jes(0)(x2)Mse(0)(x2)LsSes(2)(ψ)x24js[Hes(0)(x2)]2[Mse(0)(x2)]2x24[Jes(2)(x2)Ses(2)(ψ)+Jes(0)(x2)Ses(4)(ψ)].

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