Abstract

It is shown analytically and numerically that a matched ε=μ reciprocal object with rotational symmetry will not produce any backscattering when illuminated along the axis of symmetry unless the body is invariant under a rotation by 180°. The purpose of this work is to generalize the monostatic theorem of Weston to arbitrary rotational symmetry, thereby providing a basic rule for scattering by complex bodies. The theory is illustrated by application to a few selected scatterers.

© 2005 Optical Society of America

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References

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  1. V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963).
    [CrossRef]
  2. C. Monzon, O. Kesler, “On the depolarization of bodies invariant under a rotation,” IEEE Trans. Antennas Propag. 49, 1868–1874 (2001).
    [CrossRef]
  3. K. S. Yee, A. H. Chang, “Scattering theorems with anisotropic surface boundary conditions for bodies of revolution,” IEEE Trans. Antennas Propag. 39, 1041–1043 (1991).
    [CrossRef]
  4. P. L. E. Uslenghi, “Scattering by an impedance sphere coated with a chiral layer,” Electromagnetics 10, 201–211 (1990).
    [CrossRef]
  5. P. L. E. Uslenghi, “Three theorems on zero backscattering,” IEEE Trans. Antennas Propag. 44, 269–270 (1996).
    [CrossRef]
  6. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  7. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
    [CrossRef] [PubMed]
  8. J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
    [CrossRef]
  9. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
    [CrossRef] [PubMed]
  10. G. Ruck, D. Barrick, W. Stuart, C. Kirchbaum, Radar Cross Section Handbook (Plenum, New York, 1970).
    [CrossRef]
  11. D. C. Jenn, Radar and Laser Cross Section Engineering (American Institute of Aeronautics and Astronautics, Washington, D.C., 1995).
  12. C. Monzon, “Radiation and scattering in homogeneous general biisotropic regions,” IEEE Trans. Antennas Propag. 38, 227–235 (1990).
    [CrossRef]
  13. P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
    [CrossRef]
  14. P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
    [CrossRef]
  15. L. N. Medgyesi-Mitschang, J. M. Putnam, M. B. Gedera, “Generalized method of moments for three-dimensional penetrable scatters,” J. Opt. Soc. Am. A 11, 1383–1398 (1994).
    [CrossRef]
  16. J. M. Putnam, M. B. Gedera, “CARLOS-3D: a general-purpose 3-D method of moments scattering code,” IEEE Antennas Propag. Mag. 35, 69–71 (1993).
    [CrossRef]
  17. C. Monzon, D. W. Forester, L. N. Medgyesi-Mitschang, “Scattering properties of an ideal homogeneous, causal ‘left handed’ sphere,” J. Opt. Soc. Am. A 21, 2311–2319 (2004).
    [CrossRef]

2004

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

C. Monzon, D. W. Forester, L. N. Medgyesi-Mitschang, “Scattering properties of an ideal homogeneous, causal ‘left handed’ sphere,” J. Opt. Soc. Am. A 21, 2311–2319 (2004).
[CrossRef]

2003

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

2001

C. Monzon, O. Kesler, “On the depolarization of bodies invariant under a rotation,” IEEE Trans. Antennas Propag. 49, 1868–1874 (2001).
[CrossRef]

2000

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

1999

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

1996

P. L. E. Uslenghi, “Three theorems on zero backscattering,” IEEE Trans. Antennas Propag. 44, 269–270 (1996).
[CrossRef]

1994

1993

J. M. Putnam, M. B. Gedera, “CARLOS-3D: a general-purpose 3-D method of moments scattering code,” IEEE Antennas Propag. Mag. 35, 69–71 (1993).
[CrossRef]

1991

K. S. Yee, A. H. Chang, “Scattering theorems with anisotropic surface boundary conditions for bodies of revolution,” IEEE Trans. Antennas Propag. 39, 1041–1043 (1991).
[CrossRef]

1990

P. L. E. Uslenghi, “Scattering by an impedance sphere coated with a chiral layer,” Electromagnetics 10, 201–211 (1990).
[CrossRef]

C. Monzon, “Radiation and scattering in homogeneous general biisotropic regions,” IEEE Trans. Antennas Propag. 38, 227–235 (1990).
[CrossRef]

1968

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

1963

V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963).
[CrossRef]

Barrick, D.

G. Ruck, D. Barrick, W. Stuart, C. Kirchbaum, Radar Cross Section Handbook (Plenum, New York, 1970).
[CrossRef]

Chang, A. H.

K. S. Yee, A. H. Chang, “Scattering theorems with anisotropic surface boundary conditions for bodies of revolution,” IEEE Trans. Antennas Propag. 39, 1041–1043 (1991).
[CrossRef]

Forester, D. W.

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

C. Monzon, D. W. Forester, L. N. Medgyesi-Mitschang, “Scattering properties of an ideal homogeneous, causal ‘left handed’ sphere,” J. Opt. Soc. Am. A 21, 2311–2319 (2004).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

Gedera, M. B.

L. N. Medgyesi-Mitschang, J. M. Putnam, M. B. Gedera, “Generalized method of moments for three-dimensional penetrable scatters,” J. Opt. Soc. Am. A 11, 1383–1398 (1994).
[CrossRef]

J. M. Putnam, M. B. Gedera, “CARLOS-3D: a general-purpose 3-D method of moments scattering code,” IEEE Antennas Propag. Mag. 35, 69–71 (1993).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Jenn, D. C.

D. C. Jenn, Radar and Laser Cross Section Engineering (American Institute of Aeronautics and Astronautics, Washington, D.C., 1995).

Kesler, O.

C. Monzon, O. Kesler, “On the depolarization of bodies invariant under a rotation,” IEEE Trans. Antennas Propag. 49, 1868–1874 (2001).
[CrossRef]

Kirchbaum, C.

G. Ruck, D. Barrick, W. Stuart, C. Kirchbaum, Radar Cross Section Handbook (Plenum, New York, 1970).
[CrossRef]

Loschialpo, P. F.

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

Medgyesi-Mitschang, L. N.

Monzon, C.

C. Monzon, D. W. Forester, L. N. Medgyesi-Mitschang, “Scattering properties of an ideal homogeneous, causal ‘left handed’ sphere,” J. Opt. Soc. Am. A 21, 2311–2319 (2004).
[CrossRef]

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

C. Monzon, O. Kesler, “On the depolarization of bodies invariant under a rotation,” IEEE Trans. Antennas Propag. 49, 1868–1874 (2001).
[CrossRef]

C. Monzon, “Radiation and scattering in homogeneous general biisotropic regions,” IEEE Trans. Antennas Propag. 38, 227–235 (1990).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Putnam, J. M.

L. N. Medgyesi-Mitschang, J. M. Putnam, M. B. Gedera, “Generalized method of moments for three-dimensional penetrable scatters,” J. Opt. Soc. Am. A 11, 1383–1398 (1994).
[CrossRef]

J. M. Putnam, M. B. Gedera, “CARLOS-3D: a general-purpose 3-D method of moments scattering code,” IEEE Antennas Propag. Mag. 35, 69–71 (1993).
[CrossRef]

Rachford, F. J.

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Ruck, G.

G. Ruck, D. Barrick, W. Stuart, C. Kirchbaum, Radar Cross Section Handbook (Plenum, New York, 1970).
[CrossRef]

Schelleng, J.

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Smith, D. L.

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

Smith, D. R.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

Stuart, W.

G. Ruck, D. Barrick, W. Stuart, C. Kirchbaum, Radar Cross Section Handbook (Plenum, New York, 1970).
[CrossRef]

Uslenghi, P. L. E.

P. L. E. Uslenghi, “Three theorems on zero backscattering,” IEEE Trans. Antennas Propag. 44, 269–270 (1996).
[CrossRef]

P. L. E. Uslenghi, “Scattering by an impedance sphere coated with a chiral layer,” Electromagnetics 10, 201–211 (1990).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Weston, V. H.

V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963).
[CrossRef]

Yee, K. S.

K. S. Yee, A. H. Chang, “Scattering theorems with anisotropic surface boundary conditions for bodies of revolution,” IEEE Trans. Antennas Propag. 39, 1041–1043 (1991).
[CrossRef]

Electromagnetics

P. L. E. Uslenghi, “Scattering by an impedance sphere coated with a chiral layer,” Electromagnetics 10, 201–211 (1990).
[CrossRef]

IEEE Antennas Propag. Mag.

J. M. Putnam, M. B. Gedera, “CARLOS-3D: a general-purpose 3-D method of moments scattering code,” IEEE Antennas Propag. Mag. 35, 69–71 (1993).
[CrossRef]

IEEE Trans. Antennas Propag.

C. Monzon, “Radiation and scattering in homogeneous general biisotropic regions,” IEEE Trans. Antennas Propag. 38, 227–235 (1990).
[CrossRef]

P. L. E. Uslenghi, “Three theorems on zero backscattering,” IEEE Trans. Antennas Propag. 44, 269–270 (1996).
[CrossRef]

V. H. Weston, “Theory of absorbers in scattering,” IEEE Trans. Antennas Propag. 11, 578–584 (1963).
[CrossRef]

C. Monzon, O. Kesler, “On the depolarization of bodies invariant under a rotation,” IEEE Trans. Antennas Propag. 49, 1868–1874 (2001).
[CrossRef]

K. S. Yee, A. H. Chang, “Scattering theorems with anisotropic surface boundary conditions for bodies of revolution,” IEEE Trans. Antennas Propag. 39, 1041–1043 (1991).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Phys. Rev. E

P. F. Loschialpo, D. L. Smith, D. W. Forester, F. J. Rachford, J. Schelleng, “Electromagnetic waves focused by a negative-index planar lens,” Phys. Rev. E 67, 026502 (2003).
[CrossRef]

P. F. Loschialpo, D. W. Forester, D. L. Smith, F. J. Rachford, J. Schelleng, C. Monzon, “Optical properties of an ideal homogeneous, causal ‘left handed’ material slab,” Phys. Rev. E 70, 036605 (2004).
[CrossRef]

Phys. Rev. Lett.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966 (2000).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef] [PubMed]

Sov. Phys. Usp.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Other

G. Ruck, D. Barrick, W. Stuart, C. Kirchbaum, Radar Cross Section Handbook (Plenum, New York, 1970).
[CrossRef]

D. C. Jenn, Radar and Laser Cross Section Engineering (American Institute of Aeronautics and Astronautics, Washington, D.C., 1995).

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Figures (13)

Fig. 1
Fig. 1

Layout of the monostatic problem. Here ψ denotes the least positive angular rotation about the z axis that leaves the scattering body invariant.

Fig. 2
Fig. 2

Canonical configuration depicting the axial incident and expected backscattered fields. The arrows indicate the scattering process. (a) Original scatterer, (b) scattering configuration after application of duality ( ε μ ) and performing a rotation of the plane of polarization so as to align it with the original configuration (a). Such a rotation does not change the scattering contribution according to Eq. (9). Enforcing the identity of scenarios (a) and (b) results in zero backscattering.

Fig. 3
Fig. 3

Two-dimensional plane-wave scattering by a left-handed cylinder: series solution. The plot on the left depicts the normalized bistatic cross section at 10 GHz . The homogeneous cylinder is of radius a = 1 cm and ε = μ = 1 . The surface on the right represents the magnitude of the time-harmonic electric field in a cross section of the near-field region 3 cm < x < 3 cm and 3 cm < y < 3 cm . Owing to the negative values of the constitutive parameters, a circular edge is visible and coincides with the contour of the cylinder. The arrow on the right indicates the direction of incidence ( ϕ = 0 ) .

Fig. 4
Fig. 4

Lorentzian frequency-dependent model F ( f ) = ε ( f ) = μ ( f ) for K = 3.88 , f 0 = 3.362 GHz and G = 3.362 × 10 3 GHz .

Fig. 5
Fig. 5

Tripole ( N = 3 ) scatterer. Coordinates (left), and CARLOS numerical model (right) with 2128 triangular surface patches. The cross section of the arms is a 0.5 × 0.5 square; each arm length is 1.1 , and the distance from the centroid of the scatterer to an arm end face is 1.53 .

Fig. 6
Fig. 6

Normalized bistatic cross section of the tripole scatterer at 2 GHz under axial plane-wave illumination with the E field directed along the u axis. The homogeneous body is characterized at 2 GHz by ε = μ = 5.482 j 0.004 . A clear backscattering ( θ s = ϕ s = 90 ° ) null is observed. The solid curve and the squares correspond to the principal E and H-plane Co-Pol bistatic cross section, respectively. The dotted curve and the crosses correspond to E and H-plane X-Pol, respectively.

Fig. 7
Fig. 7

Normalized bistatic cross section of the tripole scatterer at 3 GHz under axial plane-wave illumination with the E field directed along the u axis. The four curves correspond to the bistatic cross sections in the principal E and H planes and Co-Pol and X-Pol components as in Fig. 6. The homogeneous body is characterized at 3 GHz by ε = μ = 16.3 j 0.074 . A clear backscattering null is observed.

Fig. 8
Fig. 8

Normalized bistatic cross section of the tripole scatterer at 4 GHz under axial plane-wave illumination with the E field directed along the u axis. The four curves correspond to the bistatic cross sections in the principal E and H planes and for Co-Pol and X-Pol components as in Fig. 6. The homogeneous body is left-handed at 4 GHz since ε = μ = 6.054 j 0.021 . Within the confines of numerical error for this challenging configuration, a clear backscattering null is observed.

Fig. 9
Fig. 9

CARLOS numerical model of a five-arm scatterer ( N = 5 ) . The model employs 2494 triangular surface patches. The coordinates are the same as those presented in Fig. 5. The distance from the centroid of the object to the outermost point of an arm is 1.12 . The cross section of the arms are circles of diameter 0.24 . The object is roughly one free-space wavelength in physical size at 6 GHz .

Fig. 10
Fig. 10

Normalized bistatic cross section of the five-arm scatterer at 2 GHz under axial plane-wave illumination with the E field directed along the u axis. The four curves correspond to the bistatic cross sections in the principal E and H planes and for Co-Pol and X-Pol components as in Fig. 6. The homogeneous body is characterized at 2 GHz by ε = μ = 5.482 j 0.004 . A clear backscattering null is observed.

Fig. 11
Fig. 11

Normalized bistatic cross section of the five-arm scatterer at 3 GHz under axial plane-wave illumination with the E field directed along the u axis. The four curves correspond to the bistatic cross sections in the principal E and H planes and for Co-Pol and X-Pol components as in Fig. 6. The homogeneous body is characterized at 3 GHz by ε = μ = 16.3 j 0.074 . Within the confines of numerical error for this challenging configuration, a clear backscattering null is observed.

Fig. 12
Fig. 12

Normalized bistatic cross section of the five-arm scatterer at 4 GHz under axial-plane wave illumination with the E field directed along the u axis. The four curves correspond to the bistatic cross-sections in the principal E and H planes and for Co-Pol and X-Pol components as in Fig. 6. The homogeneous body is left-handed at 4 GHz since ε = μ = 6.054 j 0.021 . A clear backscattering null is observed.

Fig. 13
Fig. 13

Normalized bistatic cross section of the five-arm scatterer at 6 GHz under axial plane-wave illumination with the E field directed along the u axis. The four curves correspond to the bistatic cross sections in the principal E and H planes and for Co-Pol and X-Pol components as in Fig. 6. The homogeneous body is left-handed at 6 GHz since ε = μ = 0.317 j 0.001 . Within the confines of numerical error for this challenging configuration, a clear backscattering null is observed.

Equations (13)

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

[ E u s E v s ] = [ S u u S u v S v u S v v ] [ E u in E v in ] .
[ E u s E v s ] = [ S u u S u v S v u S v v ] [ E u in E v in ] .
L ̿ ( ψ ) [ E u s E v s ] = [ S u u S u v S v u S v v ] L ̿ ( ψ ) [ E u in E v in ] ,
L ̿ ( ψ ) = [ cos ( ψ ) sin ( ψ ) sin ( ψ ) cos ( ψ ) ] .
[ S u u S u v S v u S v v ] L ̿ ( ψ ) = L ̿ ( ψ ) [ S u u S u v S v u S v v ] .
S ̿ = S 0 [ cos ( β ) sin ( β ) sin ( β ) cos ( β ) ] .
S u u = S v v ,
S u v = S v u .
S u v = S v u .
S ̿ = S 0 1 ̿ .
S 0 = 0 .
H S H inc = E S E inc = S 0 .
F ( f ) = 1 + K 1 1 + j ( f G f o 2 ) ( f f o ) 2 .

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