Abstract

We studied the problem of diffraction of an electromagnetic plane wave by a perfectly conducting finite strip in a homogeneous bi-isotropic medium and obtained some improved results which were presented both mathematically and graphically. The problem was solved by using the Wiener-Hopf technique and Fourier transform. The scattered field in the far zone was determined by the method of steepest decent. The significance of present analysis was that it recovered the results when a strip was widened into a half plane.

© 2008 Optical Society of America

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  1. E. Beltrami, "Considerazioni idrodinamiche," Rend. Inst. Lombardo Acad. Sci. Lett.22,122 -131(1889). An English translation is available: Beltrami, E. "Considerations on hydrodynamics," Int. J. Fusion Energy 3,53- 57(1985). V. Trkal, "Paznámka hydrodynamice vazkych tekutin," C2d8 asopis pro Pe2d8 stování Mathematiky a Fysiky 48, 302 -311 1919. An English translation is available: V. Trkal, "A note on the hydrodynamics of viscous fluids," Czech J. Phys. 44, 97-1061994.
  2. S. Chandrasekhar, "Axisymmetric Magnetic Fields and Fluid Motions," Asrtophys. J,  124, 232 (1956).
    [CrossRef]
  3. A. Lakhtakia, "Viktor Trkal, Beltrami fields, and Trkalian flows," Czech. J. Phys. 44, 89 (1994).
    [CrossRef]
  4. C. F. Bohren, "Light scattering by an optically active sphere," Chem. Phys. Lett. 29, 458 (1974).
    [CrossRef]
  5. A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, Singapore, 1994).
    [CrossRef]
  6. A. Lakhtakia, "Time-dependent scalar Beltrami-Hertz potentials in free space," Int. J. Infrared Millim. Waves 15, 369-394 (1994).
    [CrossRef]
  7. A. Lakhtakia andW. S. Weiglhofer, "Covariances and invariances of the Beltrami-Maxwell postulates," IEE Proc. Sci. Meas. Technol. 142, 262-26 (1995).
    [CrossRef]
  8. V. V. Fisanov, "Distinctive features of edge fields in a chiral medium," Sov. J. Commun. Technol. Electronics 37, 93 (1992).
  9. S. Przezdziecki, "Field of a Point Source Within Perfectly Conducting Parallel-Plates in a Homogeneous Biisotropic Medium," Acta Physica Polonica A 83, 739 (1993).
  10. S. Asghar and A. Lakhtakia, "Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium," Int. J. Appl. Electromagn. Mater. 5, 181-188, (1994).
  11. J. P. McKelvey, "The case of the curious curl," Amer. J. Phys. 58, 306 (1990).
    [CrossRef]
  12. H. Zaghloul and O. Barajas, "Force- free magnetic fields," Amer. J. Phys. 58, 783 (1990).
    [CrossRef]
  13. V. K. Varadan, A. Lakhtakia, and V. V. Varadan, "A comment on the solutions of the equation.�?a = ka," J. Phys. A: Math. Gen. 20, 2649 (1987).
    [CrossRef]
  14. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer, Heidelberg, 1989).
  15. A. Lakhtakia, "Recent contributions to classical electromagnetic theory of chiral media," Speculat. Sci. Technol. 14, 2-17 (1991).
  16. B. D. H. Tellegen, Phillips Res. Rep. 3, 81 (1948).; errata: M. E. Van Valkenburg, ed., Circuit Theory: Foundations and Classical Contributions (Stroudsberg, PA: Dowden, Hutchinson and Ross, 1974).
  17. L. I. G. Chambers, "Propagation in a gyrational medium," Quart. J. Mech. Appl. Math. 9, 360 (1956), addendum: Quart. J. Mech. Appl. Math 11, 253-255 (1958).
    [CrossRef]
  18. J. C. Monzon, "Radiation and scattering in homogeneous general biisotropic regions," IEEE Trans. Antennas Propagat,  38, 227 (1990).
    [CrossRef]
  19. A. H. Sihvola and I. V. Lindell, "Theory of nonreciprocal and nonsymmetric uniform transmission lines," Microwave Opt. Technol. Lett. 4, 292 (1991).
  20. A. Lakhtakia and J. R. Diamond, "Reciprocity and the concept of the Brewster wavenumber," Int. J. Infrared Millim. Waves 12, 1167-1174 (1991).
    [CrossRef]
  21. A. Lakhtakia, "Plane wave scattering response of a unidirectionally conducting screen immersed in a biisotropic medium," Microwave Opt. Technol. Lett,  5, 163 (1992).
    [CrossRef]
  22. A. Lakhtakia and T. G. Mackay, "Infinite phase velocity as the boundary between positive and negative phase velocities," Microwave Opt Technol. Lett. 20, 165-166 (2004).
    [CrossRef]
  23. A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, "Negative phase velocity mediums," W. S. Weiglhofer and A. Lakhtakia (Eds.), Introduction to complex mediums for electromagnetics and optics, (SPIE Press. Bellingham, W. A, 2003).
  24. T. G. Mackay, "Plane waves with negative phase velocity in isotropic chiral mediums," Microwave Opt. Technol. Lett. 45, 120-121 (2005).
    [CrossRef]
  25. T. G. Mackay and A. Lakhtakia, "Plane waves with negative phase velocity in Faraday chiral mediums," Phys. Rev. E 69, 026602 (2004).
    [CrossRef]
  26. A. Lakhtakia and W. S. Weiglhofer, "Constraint on linear, homogeneous constitutive relations," Phys. Rev. E 50, 5017-5019 (1994).
    [CrossRef]
  27. A. Lakhtakia and B. Shanker, "Beltrami fields within continuous source regions, volume integral equations, scattering algorithms, and the extended Maxwell-Garnett model," Int. J. Appl. Electromagn. Mater. 4, 65-82 (1993).
  28. W. S. Weiglhofer, "Isotropie chiral media and scalar Hertz potential," J. Phys. A 21, 2249 (1988).
  29. B. Noble, Methods Based on the Wiener-Hopf Technique (Pergamon, London, 1958).
  30. S. Asghar, T. Hayat, and B. Asghar, "Cylindrical wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium," J. Mod. Opt. 3, 515-528 (1998).
    [CrossRef]
  31. S. Asghar and T. Hayat, "Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium," Int. J. Appl. Electromagn. Mechanics 9, 39-51 (1998).
  32. E. T. Copson, Asymptotic Expansions (Cambridge University Press, 1967).
  33. Mackay and Lakhtakia, "Electromagnetic fields in linear bianisotropic mediums, Prog. Opt. 51, 121-209 (2008).
    [CrossRef]
  34. F. I. Fedorov, Theory of Gyrotropy (Minsk: Nauka i Tehnika), 1976).

2008 (1)

Mackay and Lakhtakia, "Electromagnetic fields in linear bianisotropic mediums, Prog. Opt. 51, 121-209 (2008).
[CrossRef]

2005 (1)

T. G. Mackay, "Plane waves with negative phase velocity in isotropic chiral mediums," Microwave Opt. Technol. Lett. 45, 120-121 (2005).
[CrossRef]

2004 (2)

T. G. Mackay and A. Lakhtakia, "Plane waves with negative phase velocity in Faraday chiral mediums," Phys. Rev. E 69, 026602 (2004).
[CrossRef]

A. Lakhtakia and T. G. Mackay, "Infinite phase velocity as the boundary between positive and negative phase velocities," Microwave Opt Technol. Lett. 20, 165-166 (2004).
[CrossRef]

1998 (2)

S. Asghar, T. Hayat, and B. Asghar, "Cylindrical wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium," J. Mod. Opt. 3, 515-528 (1998).
[CrossRef]

S. Asghar and T. Hayat, "Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium," Int. J. Appl. Electromagn. Mechanics 9, 39-51 (1998).

1995 (1)

A. Lakhtakia andW. S. Weiglhofer, "Covariances and invariances of the Beltrami-Maxwell postulates," IEE Proc. Sci. Meas. Technol. 142, 262-26 (1995).
[CrossRef]

1994 (5)

A. Lakhtakia, "Time-dependent scalar Beltrami-Hertz potentials in free space," Int. J. Infrared Millim. Waves 15, 369-394 (1994).
[CrossRef]

E. Beltrami, "Considerazioni idrodinamiche," Rend. Inst. Lombardo Acad. Sci. Lett.22,122 -131(1889). An English translation is available: Beltrami, E. "Considerations on hydrodynamics," Int. J. Fusion Energy 3,53- 57(1985). V. Trkal, "Paznámka hydrodynamice vazkych tekutin," C2d8 asopis pro Pe2d8 stování Mathematiky a Fysiky 48, 302 -311 1919. An English translation is available: V. Trkal, "A note on the hydrodynamics of viscous fluids," Czech J. Phys. 44, 97-1061994.

A. Lakhtakia, "Viktor Trkal, Beltrami fields, and Trkalian flows," Czech. J. Phys. 44, 89 (1994).
[CrossRef]

S. Asghar and A. Lakhtakia, "Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium," Int. J. Appl. Electromagn. Mater. 5, 181-188, (1994).

A. Lakhtakia and W. S. Weiglhofer, "Constraint on linear, homogeneous constitutive relations," Phys. Rev. E 50, 5017-5019 (1994).
[CrossRef]

1993 (2)

A. Lakhtakia and B. Shanker, "Beltrami fields within continuous source regions, volume integral equations, scattering algorithms, and the extended Maxwell-Garnett model," Int. J. Appl. Electromagn. Mater. 4, 65-82 (1993).

S. Przezdziecki, "Field of a Point Source Within Perfectly Conducting Parallel-Plates in a Homogeneous Biisotropic Medium," Acta Physica Polonica A 83, 739 (1993).

1992 (2)

V. V. Fisanov, "Distinctive features of edge fields in a chiral medium," Sov. J. Commun. Technol. Electronics 37, 93 (1992).

A. Lakhtakia, "Plane wave scattering response of a unidirectionally conducting screen immersed in a biisotropic medium," Microwave Opt. Technol. Lett,  5, 163 (1992).
[CrossRef]

1991 (3)

A. Lakhtakia, "Recent contributions to classical electromagnetic theory of chiral media," Speculat. Sci. Technol. 14, 2-17 (1991).

A. H. Sihvola and I. V. Lindell, "Theory of nonreciprocal and nonsymmetric uniform transmission lines," Microwave Opt. Technol. Lett. 4, 292 (1991).

A. Lakhtakia and J. R. Diamond, "Reciprocity and the concept of the Brewster wavenumber," Int. J. Infrared Millim. Waves 12, 1167-1174 (1991).
[CrossRef]

1990 (3)

J. P. McKelvey, "The case of the curious curl," Amer. J. Phys. 58, 306 (1990).
[CrossRef]

H. Zaghloul and O. Barajas, "Force- free magnetic fields," Amer. J. Phys. 58, 783 (1990).
[CrossRef]

J. C. Monzon, "Radiation and scattering in homogeneous general biisotropic regions," IEEE Trans. Antennas Propagat,  38, 227 (1990).
[CrossRef]

1988 (1)

W. S. Weiglhofer, "Isotropie chiral media and scalar Hertz potential," J. Phys. A 21, 2249 (1988).

1987 (1)

V. K. Varadan, A. Lakhtakia, and V. V. Varadan, "A comment on the solutions of the equation.�?a = ka," J. Phys. A: Math. Gen. 20, 2649 (1987).
[CrossRef]

1974 (1)

C. F. Bohren, "Light scattering by an optically active sphere," Chem. Phys. Lett. 29, 458 (1974).
[CrossRef]

1958 (1)

L. I. G. Chambers, "Propagation in a gyrational medium," Quart. J. Mech. Appl. Math. 9, 360 (1956), addendum: Quart. J. Mech. Appl. Math 11, 253-255 (1958).
[CrossRef]

1956 (1)

S. Chandrasekhar, "Axisymmetric Magnetic Fields and Fluid Motions," Asrtophys. J,  124, 232 (1956).
[CrossRef]

Asghar, B.

S. Asghar, T. Hayat, and B. Asghar, "Cylindrical wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium," J. Mod. Opt. 3, 515-528 (1998).
[CrossRef]

Asghar, S.

S. Asghar, T. Hayat, and B. Asghar, "Cylindrical wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium," J. Mod. Opt. 3, 515-528 (1998).
[CrossRef]

S. Asghar and T. Hayat, "Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium," Int. J. Appl. Electromagn. Mechanics 9, 39-51 (1998).

S. Asghar and A. Lakhtakia, "Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium," Int. J. Appl. Electromagn. Mater. 5, 181-188, (1994).

Barajas, O.

H. Zaghloul and O. Barajas, "Force- free magnetic fields," Amer. J. Phys. 58, 783 (1990).
[CrossRef]

Beltrami, E.

E. Beltrami, "Considerazioni idrodinamiche," Rend. Inst. Lombardo Acad. Sci. Lett.22,122 -131(1889). An English translation is available: Beltrami, E. "Considerations on hydrodynamics," Int. J. Fusion Energy 3,53- 57(1985). V. Trkal, "Paznámka hydrodynamice vazkych tekutin," C2d8 asopis pro Pe2d8 stování Mathematiky a Fysiky 48, 302 -311 1919. An English translation is available: V. Trkal, "A note on the hydrodynamics of viscous fluids," Czech J. Phys. 44, 97-1061994.

Bohren, C. F.

C. F. Bohren, "Light scattering by an optically active sphere," Chem. Phys. Lett. 29, 458 (1974).
[CrossRef]

Chambers, L. I. G.

L. I. G. Chambers, "Propagation in a gyrational medium," Quart. J. Mech. Appl. Math. 9, 360 (1956), addendum: Quart. J. Mech. Appl. Math 11, 253-255 (1958).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, "Axisymmetric Magnetic Fields and Fluid Motions," Asrtophys. J,  124, 232 (1956).
[CrossRef]

Diamond, J. R.

A. Lakhtakia and J. R. Diamond, "Reciprocity and the concept of the Brewster wavenumber," Int. J. Infrared Millim. Waves 12, 1167-1174 (1991).
[CrossRef]

Fisanov, V. V.

V. V. Fisanov, "Distinctive features of edge fields in a chiral medium," Sov. J. Commun. Technol. Electronics 37, 93 (1992).

Hayat, T.

S. Asghar, T. Hayat, and B. Asghar, "Cylindrical wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium," J. Mod. Opt. 3, 515-528 (1998).
[CrossRef]

S. Asghar and T. Hayat, "Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium," Int. J. Appl. Electromagn. Mechanics 9, 39-51 (1998).

Lakhtakia, A.

T. G. Mackay and A. Lakhtakia, "Plane waves with negative phase velocity in Faraday chiral mediums," Phys. Rev. E 69, 026602 (2004).
[CrossRef]

A. Lakhtakia and T. G. Mackay, "Infinite phase velocity as the boundary between positive and negative phase velocities," Microwave Opt Technol. Lett. 20, 165-166 (2004).
[CrossRef]

A. Lakhtakia andW. S. Weiglhofer, "Covariances and invariances of the Beltrami-Maxwell postulates," IEE Proc. Sci. Meas. Technol. 142, 262-26 (1995).
[CrossRef]

A. Lakhtakia, "Viktor Trkal, Beltrami fields, and Trkalian flows," Czech. J. Phys. 44, 89 (1994).
[CrossRef]

A. Lakhtakia and W. S. Weiglhofer, "Constraint on linear, homogeneous constitutive relations," Phys. Rev. E 50, 5017-5019 (1994).
[CrossRef]

A. Lakhtakia, "Time-dependent scalar Beltrami-Hertz potentials in free space," Int. J. Infrared Millim. Waves 15, 369-394 (1994).
[CrossRef]

S. Asghar and A. Lakhtakia, "Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium," Int. J. Appl. Electromagn. Mater. 5, 181-188, (1994).

A. Lakhtakia and B. Shanker, "Beltrami fields within continuous source regions, volume integral equations, scattering algorithms, and the extended Maxwell-Garnett model," Int. J. Appl. Electromagn. Mater. 4, 65-82 (1993).

A. Lakhtakia, "Plane wave scattering response of a unidirectionally conducting screen immersed in a biisotropic medium," Microwave Opt. Technol. Lett,  5, 163 (1992).
[CrossRef]

A. Lakhtakia, "Recent contributions to classical electromagnetic theory of chiral media," Speculat. Sci. Technol. 14, 2-17 (1991).

A. Lakhtakia and J. R. Diamond, "Reciprocity and the concept of the Brewster wavenumber," Int. J. Infrared Millim. Waves 12, 1167-1174 (1991).
[CrossRef]

V. K. Varadan, A. Lakhtakia, and V. V. Varadan, "A comment on the solutions of the equation.�?a = ka," J. Phys. A: Math. Gen. 20, 2649 (1987).
[CrossRef]

Lindell, I. V.

A. H. Sihvola and I. V. Lindell, "Theory of nonreciprocal and nonsymmetric uniform transmission lines," Microwave Opt. Technol. Lett. 4, 292 (1991).

Mackay, T. G.

T. G. Mackay, "Plane waves with negative phase velocity in isotropic chiral mediums," Microwave Opt. Technol. Lett. 45, 120-121 (2005).
[CrossRef]

T. G. Mackay and A. Lakhtakia, "Plane waves with negative phase velocity in Faraday chiral mediums," Phys. Rev. E 69, 026602 (2004).
[CrossRef]

A. Lakhtakia and T. G. Mackay, "Infinite phase velocity as the boundary between positive and negative phase velocities," Microwave Opt Technol. Lett. 20, 165-166 (2004).
[CrossRef]

McKelvey, J. P.

J. P. McKelvey, "The case of the curious curl," Amer. J. Phys. 58, 306 (1990).
[CrossRef]

Monzon, J. C.

J. C. Monzon, "Radiation and scattering in homogeneous general biisotropic regions," IEEE Trans. Antennas Propagat,  38, 227 (1990).
[CrossRef]

Przezdziecki, S.

S. Przezdziecki, "Field of a Point Source Within Perfectly Conducting Parallel-Plates in a Homogeneous Biisotropic Medium," Acta Physica Polonica A 83, 739 (1993).

Shanker, B.

A. Lakhtakia and B. Shanker, "Beltrami fields within continuous source regions, volume integral equations, scattering algorithms, and the extended Maxwell-Garnett model," Int. J. Appl. Electromagn. Mater. 4, 65-82 (1993).

Sihvola, A. H.

A. H. Sihvola and I. V. Lindell, "Theory of nonreciprocal and nonsymmetric uniform transmission lines," Microwave Opt. Technol. Lett. 4, 292 (1991).

Varadan, V. K.

V. K. Varadan, A. Lakhtakia, and V. V. Varadan, "A comment on the solutions of the equation.�?a = ka," J. Phys. A: Math. Gen. 20, 2649 (1987).
[CrossRef]

Varadan, V. V.

V. K. Varadan, A. Lakhtakia, and V. V. Varadan, "A comment on the solutions of the equation.�?a = ka," J. Phys. A: Math. Gen. 20, 2649 (1987).
[CrossRef]

Weiglhofer, W. S.

A. Lakhtakia andW. S. Weiglhofer, "Covariances and invariances of the Beltrami-Maxwell postulates," IEE Proc. Sci. Meas. Technol. 142, 262-26 (1995).
[CrossRef]

A. Lakhtakia and W. S. Weiglhofer, "Constraint on linear, homogeneous constitutive relations," Phys. Rev. E 50, 5017-5019 (1994).
[CrossRef]

W. S. Weiglhofer, "Isotropie chiral media and scalar Hertz potential," J. Phys. A 21, 2249 (1988).

Zaghloul, H.

H. Zaghloul and O. Barajas, "Force- free magnetic fields," Amer. J. Phys. 58, 783 (1990).
[CrossRef]

Acta Physica Polonica A (1)

S. Przezdziecki, "Field of a Point Source Within Perfectly Conducting Parallel-Plates in a Homogeneous Biisotropic Medium," Acta Physica Polonica A 83, 739 (1993).

Amer. J. Phys. (2)

J. P. McKelvey, "The case of the curious curl," Amer. J. Phys. 58, 306 (1990).
[CrossRef]

H. Zaghloul and O. Barajas, "Force- free magnetic fields," Amer. J. Phys. 58, 783 (1990).
[CrossRef]

Asrtophys. J (1)

S. Chandrasekhar, "Axisymmetric Magnetic Fields and Fluid Motions," Asrtophys. J,  124, 232 (1956).
[CrossRef]

Chem. Phys. Lett. (1)

C. F. Bohren, "Light scattering by an optically active sphere," Chem. Phys. Lett. 29, 458 (1974).
[CrossRef]

Czech J. Phys. (1)

E. Beltrami, "Considerazioni idrodinamiche," Rend. Inst. Lombardo Acad. Sci. Lett.22,122 -131(1889). An English translation is available: Beltrami, E. "Considerations on hydrodynamics," Int. J. Fusion Energy 3,53- 57(1985). V. Trkal, "Paznámka hydrodynamice vazkych tekutin," C2d8 asopis pro Pe2d8 stování Mathematiky a Fysiky 48, 302 -311 1919. An English translation is available: V. Trkal, "A note on the hydrodynamics of viscous fluids," Czech J. Phys. 44, 97-1061994.

Czech. J. Phys. (1)

A. Lakhtakia, "Viktor Trkal, Beltrami fields, and Trkalian flows," Czech. J. Phys. 44, 89 (1994).
[CrossRef]

IEE Proc. Sci. Meas. Technol. (1)

A. Lakhtakia andW. S. Weiglhofer, "Covariances and invariances of the Beltrami-Maxwell postulates," IEE Proc. Sci. Meas. Technol. 142, 262-26 (1995).
[CrossRef]

IEEE Trans. Antennas Propagat (1)

J. C. Monzon, "Radiation and scattering in homogeneous general biisotropic regions," IEEE Trans. Antennas Propagat,  38, 227 (1990).
[CrossRef]

Int. J. Appl. Electromagn. Mater. (2)

A. Lakhtakia and B. Shanker, "Beltrami fields within continuous source regions, volume integral equations, scattering algorithms, and the extended Maxwell-Garnett model," Int. J. Appl. Electromagn. Mater. 4, 65-82 (1993).

S. Asghar and A. Lakhtakia, "Plane-wave diffraction by a perfectly conducting half-plane in a homogeneous bi-isotropic medium," Int. J. Appl. Electromagn. Mater. 5, 181-188, (1994).

Int. J. Appl. Electromagn. Mechanics (1)

S. Asghar and T. Hayat, "Plane wave diffraction by a perfectly conducting strip in a homogeneous biisotropic medium," Int. J. Appl. Electromagn. Mechanics 9, 39-51 (1998).

Int. J. Infrared Millim. Waves (2)

A. Lakhtakia and J. R. Diamond, "Reciprocity and the concept of the Brewster wavenumber," Int. J. Infrared Millim. Waves 12, 1167-1174 (1991).
[CrossRef]

A. Lakhtakia, "Time-dependent scalar Beltrami-Hertz potentials in free space," Int. J. Infrared Millim. Waves 15, 369-394 (1994).
[CrossRef]

J. Mod. Opt. (1)

S. Asghar, T. Hayat, and B. Asghar, "Cylindrical wave diffraction by a perfectly conducting strip in a homogeneous bi-isotropic medium," J. Mod. Opt. 3, 515-528 (1998).
[CrossRef]

J. Phys. A (1)

W. S. Weiglhofer, "Isotropie chiral media and scalar Hertz potential," J. Phys. A 21, 2249 (1988).

J. Phys. A: Math. Gen. (1)

V. K. Varadan, A. Lakhtakia, and V. V. Varadan, "A comment on the solutions of the equation.�?a = ka," J. Phys. A: Math. Gen. 20, 2649 (1987).
[CrossRef]

Microwave Opt Technol. Lett. (1)

A. Lakhtakia and T. G. Mackay, "Infinite phase velocity as the boundary between positive and negative phase velocities," Microwave Opt Technol. Lett. 20, 165-166 (2004).
[CrossRef]

Microwave Opt. Technol. Lett (1)

A. Lakhtakia, "Plane wave scattering response of a unidirectionally conducting screen immersed in a biisotropic medium," Microwave Opt. Technol. Lett,  5, 163 (1992).
[CrossRef]

Microwave Opt. Technol. Lett. (2)

A. H. Sihvola and I. V. Lindell, "Theory of nonreciprocal and nonsymmetric uniform transmission lines," Microwave Opt. Technol. Lett. 4, 292 (1991).

T. G. Mackay, "Plane waves with negative phase velocity in isotropic chiral mediums," Microwave Opt. Technol. Lett. 45, 120-121 (2005).
[CrossRef]

Phys. Rev. E (2)

T. G. Mackay and A. Lakhtakia, "Plane waves with negative phase velocity in Faraday chiral mediums," Phys. Rev. E 69, 026602 (2004).
[CrossRef]

A. Lakhtakia and W. S. Weiglhofer, "Constraint on linear, homogeneous constitutive relations," Phys. Rev. E 50, 5017-5019 (1994).
[CrossRef]

Prog. Opt. (1)

Mackay and Lakhtakia, "Electromagnetic fields in linear bianisotropic mediums, Prog. Opt. 51, 121-209 (2008).
[CrossRef]

Quart. J. Mech. Appl. Math (1)

L. I. G. Chambers, "Propagation in a gyrational medium," Quart. J. Mech. Appl. Math. 9, 360 (1956), addendum: Quart. J. Mech. Appl. Math 11, 253-255 (1958).
[CrossRef]

Sov. J. Commun. Technol. Electronics (1)

V. V. Fisanov, "Distinctive features of edge fields in a chiral medium," Sov. J. Commun. Technol. Electronics 37, 93 (1992).

Speculat. Sci. Technol. (1)

A. Lakhtakia, "Recent contributions to classical electromagnetic theory of chiral media," Speculat. Sci. Technol. 14, 2-17 (1991).

Other (7)

B. D. H. Tellegen, Phillips Res. Rep. 3, 81 (1948).; errata: M. E. Van Valkenburg, ed., Circuit Theory: Foundations and Classical Contributions (Stroudsberg, PA: Dowden, Hutchinson and Ross, 1974).

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Harmonic Electromagnetic Fields in Chiral Media (Springer, Heidelberg, 1989).

A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, Singapore, 1994).
[CrossRef]

B. Noble, Methods Based on the Wiener-Hopf Technique (Pergamon, London, 1958).

F. I. Fedorov, Theory of Gyrotropy (Minsk: Nauka i Tehnika), 1976).

E. T. Copson, Asymptotic Expansions (Cambridge University Press, 1967).

A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, "Negative phase velocity mediums," W. S. Weiglhofer and A. Lakhtakia (Eds.), Introduction to complex mediums for electromagnetics and optics, (SPIE Press. Bellingham, W. A, 2003).

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Figures (5)

Fig. 1.
Fig. 1.

Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ 1 at θ 0 = π 4 , k 1 xz = 1 , l=1.

Fig. 2.
Fig. 2.

Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ 1 at θ 0 = π 4 , k 1 xz = 1 , l=50.

Fig. 3.
Fig. 3.

Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ 1 at θ 0 = π 4 , k 1 xz = 1 , l=100.

Fig. 4.
Fig. 4.

Variation of the amplitude of separated field versus observation angle ϑ, for different values of δ 1 at θ 0 = π 4 , k 1 xz = 1 , l1022.

Fig. 5.
Fig. 5.

Variation of the amplitude of diffracted field in the half plane versus observation angle ϑ, for different values of δ 1 at θ 0 = π 4 , k 1xz =1.

Equations (129)

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D = ε E + ε α × E
B = μ H + μ β × H
× Q 1 = γ 1 Q 1 ,
× Q 2 = γ 2 Q 2 .
γ 1 = k ( 1 k 2 α β ) { 1 + k 2 ( α β ) 2 4 + k ( α + β ) 2 } ,
γ 2 = k ( 1 k 2 α β ) { 1 + k 2 ( α β ) 2 4 k ( α + β ) 2 } ,
Q 1 = η 1 η 1 + η 2 ( E + i η 2 H ) ,
Q 2 = i η 1 + η 2 ( E i η 1 H ) ,
η 1 = η 1 + k 2 ( α β ) 2 4 + k ( α β ) 2 ,
η 2 = η { 1 + k 2 ( α β ) 2 4 k ( α β ) 2 } ,
Q 1 = Q 1 t + y Q 1 y ,
Q 1 t = Q 1 x i + Q 1 z k .
Q 2 = Q 2 t + y Q 2 y .
i j k x y z Q 1 x Q 1 y Q 1 z = γ 1 ( Q 1 x i + Q 1 y j + Q 1 z k ) .
Q 1 x = 1 k 1 xz 2 [ ik y Q 1 y x γ 1 Q 1 y z ] ,
Q 1 z = 1 k 1 xz 2 [ ik y Q 1 y z + γ 1 Q 1 y x ] ,
k 1 xz 2 = γ 1 2 k y 2 .
Q 2 x = 1 k 2 xz 2 [ ik y Q 2 y x + γ 2 Q 2 y z ] ,
Q 2 z = 1 k 2 xz 2 [ ik y Q 2 y z γ 2 Q 2 y x ] ,
k 2 xz 2 = γ 2 2 k y 2 .
× Q 1 γ 1 Q 1 = S 1 ,
× Q 2 γ 2 Q 2 = S 2 ,
S 1 = η 1 η 1 + η 2 ( i γ 1 ω ε J ( 1 + α γ 1 ) K ) ,
S 2 = η 1 η 1 + η 2 ( i γ 2 ωμ K ( 1 + β γ 2 ) J ) .
1 + ω ε α η 2 = ( 1 k 2 α β ) ( 1 + α γ ) ,
1 ω ε α η 1 = ( 1 k 2 α β ) η 1 γ 2 ω μ ,
η 2 + ω μ β = ( 1 k 2 α β ) γ 1 ω ε ,
η 1 ω μ β = ( 1 k 2 α β ) η 1 ( 1 β γ 2 ) .
Q 1 y i η 2 Q 2 y = 0 , z = 0 , l x 0 ,
Q 1 x i η 2 Q 2 x = 0 , z = 0 , l x 0 .
1 k 1 xz 2 [ i k y Q 1 y x γ 1 Q 1 y z ] i η 2 1 k 2 xz 2 [ i k y Q 2 y x γ 2 Q 2 y z ] = 0 , z = 0 , l x 0 .
Q 1 y x δ Q 1 y z = 0 , z = 0 ± , l x 0 ,
δ = γ 2 k 1 xz 2 + γ 1 k 2 xz 2 i k y ( k 2 xz 2 k 1 xz 2 ) .
Q 1 y ( x , z + ) = Q 1 y ( x , z ) ; < x < l , x > 0 , z = 0 ,
Q 1 y ( x , z + ) z = Q 1 y ( x , z ) z ; < x < l , x > 0 , z = 0 .
Q 1 y ( x , 0 ) = O ( 1 ) and Q 1 y ( x , 0 ) z = O ( x 1 2 ) as x 0 + ,
Q 1 y ( x , 0 ) = O ( 1 ) and Q 1 y ( x , 0 ) z = O ( x + l ) 1 2 as x l .
Q 1 y ( x , z ) = Q 1 y inc ( x , z ) + Q 1 y sca ( x , z ) ,
Q 1 y inc ( x , y , z ) = exp [ i ( k y y + k 1 x x + k 1 z z ) ] ,
( 2 x 2 + 2 z 2 + k 1 xz 2 ) Q 1 y sca = 0 .
k 1 xz 2 = k 1 x 2 + k 1 z 2 = γ 1 2 k y 2 .
( x δ z ) Q 1 y inc + ( x δ z ) Q 1 y sca = 0 , z = 0 ± , l x 0 ,
Q 1 y sca ( x , z + ) = Q 1 y sca ( x , z ) ; < x < l , x > 0 , z = 0 ,
z Q 1 y sca ( x , z + ) = z Q 1 y sca ( x , z ) ; < x < l , x > 0 , z = 0 .
Ψ ¯ ( υ , z ) = 1 2 π Q 1 y sca ( x , z ) e i υ x d x = Ψ ¯ + ( υ , z ) + e i υ l Ψ ¯ ( υ , z ) + Ψ ¯ 1 ( υ , z ) ,
Ψ ¯ + ( υ , z ) = 1 2 π 0 Q 1 y sac ( x , z ) e i υ x d x ,
Ψ ¯ ( υ , z ) = 1 2 π l Q 1 y sac ( x , z ) e i υ ( x + l ) d x ,
Ψ ¯ 1 ( υ , z ) = 1 2 π l 0 Q 1 y sac ( x , z ) e i υ x d x .
Ψ ¯ 0 ( υ , 0 ) = i 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ] ,
Ψ ¯ 0 / ( υ , 0 ) = k 1 z 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ] .
( d 2 d z 2 + κ 2 ) Ψ ¯ ( υ , z ) = 0 ,
κ 2 = k 1 xz 2 υ 2 ,
Ψ ¯ 1 ( υ , 0 + ) = i υ δ [ Ψ ¯ 1 ( υ , 0 + ) + Ψ ¯ 0 ( υ , 0 ) ] Ψ ¯ 0 ( υ , 0 ) ,
Ψ ¯ 1 ( υ , 0 ) = i υ δ [ Ψ ¯ 1 ( υ , 0 ) + Ψ ¯ 0 ( υ , 0 ) ] Ψ ¯ 0 ( υ , 0 ) ,
Ψ ¯ ( υ , 0 + ) = Ψ ¯ ( υ , 0 ) = Ψ ¯ ( υ , 0 ) ,
Ψ ¯ + ( υ , 0 + ) = Ψ ¯ + ( υ , 0 ) = Ψ ¯ + ( υ , 0 ) ,
Ψ ¯ ( υ , 0 + ) = Ψ ¯ ( υ , 0 ) = Ψ ¯ ( υ , 0 ) ,
Ψ ¯ + ( υ , 0 + ) = Ψ ¯ + ( υ , 0 ) = Ψ ¯ + ( υ , 0 ) .
Ψ ¯ ( υ , z ) = { A ( υ ) e i κ z if z > 0 , C ( υ ) e i κ z if z < 0 .
Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 + ) = A ( υ ) ,
Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 ) = C ( υ ) ,
Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 + ) = i κ A ( υ ) ,
Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 ) = i κ C ( υ ) .
A ( υ ) = J 1 ( υ , 0 ) + J 1 ( υ , 0 ) i κ ,
C ( υ ) = J 1 ( υ , 0 ) + J 1 ( υ , 0 ) i κ ,
J 1 ( υ , 0 ) = 1 2 [ Ψ ¯ 1 ( υ , 0 + ) Ψ ¯ 1 ( υ , 0 ) ] ,
J 1 ( υ , 0 ) = 1 2 [ Ψ ¯ 1 ( υ , 0 + ) Ψ ¯ 1 ( υ , 0 ) ] .
Ψ ¯ + ( υ , 0 ) + e i v l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 + ) = i κ [ Ψ ¯ + ( υ , 0 ) + e i v l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 + ) ] ,
Ψ ¯ + ( υ , 0 ) + e i v l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 ) = i κ [ Ψ ¯ + ( υ , 0 ) + e i v l Ψ ¯ ( υ , 0 ) + Ψ ¯ 1 ( υ , 0 ) ] .
Ψ ¯ + ( υ , 0 ) + e ivl Ψ ¯ ( υ , 0 ) i κ L ( υ ) J 1 ( υ ) + k 1 z 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ] = 0 .
i υ Ψ ¯ + ( υ , 0 ) i υ e i υ l Ψ ¯ ( υ , 0 ) + δ L ( υ ) J 1 ( υ ) + k 1 x 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ] = 0 ,
L ( υ ) = ( 1 + υ δ κ ) .
L ( υ ) = ( 1 + υ δ κ ) = L + ( υ ) L ( υ ) ,
κ ( υ ) = κ + ( υ ) κ ( υ ) ,
A ( υ ) = 1 i κ L ( υ ) { Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + k 1 z 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ]
+ υ δ 1 κ L ( υ ) { Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) - k 1 z 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ] ,
C ( υ ) = 1 i κ L ( υ ) { Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + k 1 z 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ]
+ υ δ 1 κ L ( υ ) { Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) k 1 x 2 π ( k 1 x + υ ) [ 1 + exp [ i ( k 1 x + υ ) l ] ] ,
Ψ ¯ + ( υ , 0 ) + e i υ l Ψ ¯ ( υ , 0 ) + S ( υ ) J 1 ( υ ) = k 1 z 2 π ( k 1 x + υ ) [ 1 exp [ i ( k 1 x + υ ) l ] ] ,
i υ Ψ ¯ + ( υ , 0 ) i υ e i υ l Ψ ¯ ( υ , 0 ) + δ L + ( υ ) L ( υ ) J 1 ( υ ) = k 1 x 2 π ( k 1 x + υ ) [ 1 exp [ i ( k 1 x + υ ) l ] ] ,
S ( υ ) = i κ ( υ ) L ( υ ) = S + ( υ ) S ( υ ) ,
Ψ ¯ + ( υ , 0 ) = k 1 z S + ( υ ) 2 π [ G 1 ( υ ) + T ( υ ) C 1 ] ,
Ψ ¯ ( υ , 0 ) = k 1 z S ( υ ) 2 π [ G 2 ( υ ) + T ( υ ) C 2 ] ,
Ψ ¯ + ( υ , 0 ) = i L + ( υ ) 2 π υ [ G 1 ( υ ) + T ( υ ) C 1 ] ,
Ψ ¯ _ ( υ , 0 ) = i L _ ( υ ) 2 π υ [ G 2 ( υ ) T ( υ ) C 2 ] ,
S + ( υ ) = ( k 1 xz + υ ) 1 2 L + ( υ ) ,
S _ ( υ ) = e i π 2 ( k 1 xz υ ) 1 2 L _ ( υ ) ,
G 1 ( υ ) = 1 ( υ + k 1 x ) [ 1 S + ( υ ) 1 S + ( k 1 x ) ] e ilk 1 x R 1 ( υ ) ,
G 2 ( υ ) = e ilk 1 x ( υ k 1 x ) [ 1 S + ( υ ) 1 S + ( k 1 x ) ] R 2 ( υ ) ,
C 1 = S + ( k 1 xz ) [ G 2 ( k 1 xz ) + S + ( k 1 xz ) G 1 ( k 1 xz ) T ( k 1 xz ) 1 S + 2 ( k 1 xz ) T 2 ( k 1 xz ) ] ,
C 2 = S + ( k 1 xz ) [ G 1 ( k 1 xz ) + S + ( v ) G 2 ( k 1 xz ) T ( k 1 xz ) 1 S + 2 ( k 1 xz ) T 2 ( k 1 xz ) ] ,
G 1 ( υ ) = υ ( υ + k 1 x ) [ 1 L + ( υ ) 1 L + ( k 1 x ) ] e ilk 1 x R 1 ( υ ) ,
G 2 ( υ ) = e ilk 1 x ( υ - k 1 x ) [ υ L + ( υ ) + k 1 x L + ( k 1 x ) ] R 2 ( υ ) ,
C 1 = L + ( k 1 xz ) [ G 2 ( k 1 xz ) + L + ( k 1 xz ) G 1 ( k 1 xz ) T ( k 1 xz ) 1 L + 2 ( k 1 xz ) T 2 ( k 1 xz ) ] ,
C 2 = L + ( k 1 xz ) [ G 1 ( k 1 xz ) + L + ( k 1 xz ) G 2 ( k 1 xz ) T ( k 1 xz ) 1 L + 2 ( k 1 xz ) T 2 ( k 1 xz ) ] ,
R 1 , 2 ( υ ) = E 1 [ W 1 { i ( k 1 xz k 1 x ) l } W 1 { i ( k 1 xz + υ ) l } ] 2 π i ( υ ± k 1 x ) ,
T ( υ ) = 1 2 π i E 1 W 1 { i ( k 1 xz + υ ) l } ,
E 1 = 2 e i π 4 e ik 1 xz l ( l ) 1 2 ( i ) 1 h 1 ,
W n 1 2 ( p ) = 0 u n e u u + p du = Γ ( n + 1 ) e z 2 p 1 2 n 1 2 W 1 2 ( n + 1 ) , 1 2 n ( p ) ,
A ( υ ) C ( υ ) } = k 1 z sgn ( z ) 2 π i κ L ( υ ) { S + ( υ ) G 1 ( υ ) + S + ( υ ) T ( υ ) C 1 + e i υ l S ( υ ) × [ G 2 ( υ ) + T ( υ ) C 2 ] ( 1 e il ( k 1 x + υ ) ) ( k 1 x + υ ) }
+ υ δ 1 2 π κ L ( υ ) { L + ( υ ) G 1 ( υ ) + T ( υ ) L + ( v ) C 1 + e i υ l × [ ( L ( υ ) G 2 ( υ ) + T ( υ ) L + ( v ) C 2 ) ] ( 1 e il ( k 1 x + υ ) ) ( k 1 x + υ ) } ,
Q 1 y sca ( x , z ) = 1 2 π { A ( υ ) C ( υ ) } exp ( i κ z i υ x ) d υ ,
Q 1 y sca ( x , z ) = Ψ sep ( x , z ) + Ψ int ( x , z ) ,
Ψ sep ( x , z ) = k 1 z sgn ( z ) 2 π S + ( υ ) exp ( i κ z i υ x ) i κ L ( υ ) S + ( k 1 x ) ( k 1 x + υ ) d υ
+ k 1 z sgn ( z ) 2 π e il ( k 1 x + υ ) S ( υ ) exp ( i κ z i υ x ) i κ L ( υ ) S + ( k 1 x ) ( k 1 x + υ ) d υ
1 2 π δ 1 e il ( k 1 x + υ ) exp ( i κ z i υ x ) κ L ( υ ) ( k 1 x + υ ) d υ + 1 2 π L ( υ ) e il ( k 1 x + υ ) exp ( i κ z i υ x ) κ L ( υ ) ( k 1 x + υ ) L + ( k 1 x ) d υ
+ 1 2 π δ 1 exp ( i κ z i υ x ) κ L ( υ ) ( k 1 x + υ ) d υ ,
Ψ int ( x , z ) = k 1 z sgn ( z ) 2 π 1 ikL ( υ ) [ S + ( υ ) R 1 ( υ ) e ilk 1 x C 1 S + ( υ ) T ( υ )
+ S + ( υ ) e il υ R 2 ( υ ) C 2 T ( υ ) S + ( υ ) e il υ ] exp ( i κ z i υ x ) d υ
1 2 π δ 1 κ L ( υ ) [ T ( υ ) L + ( υ ) C 1 + T ( υ ) L _ ( υ ) C 2 L + ( υ ) R 1 ( υ ) e ilk 1 x
L _ ( υ ) R 2 ( υ ) e il υ ] exp ( i κ z i υ x ) d υ .
Q 1 y sca ( x , y ) = i k 1 xz 2 π ( π 2 k 1 xz r ) 1 2 { A ( k 1 xz cos ϑ ) C ( k 1 xz cos ϑ ) } sin ( ϑ ) exp ( i k 1 xz r + i π 4 ) ,
Q 1 y sca ( sep ) ( x , y ) = [ i k 1 z sgn ( z ) f 1 ( k 1 xz cos ϑ ) + g 1 ( k 1 xz cos ϑ ) ]
× 1 4 π k 1 xz ( 1 k 1 xz r ) 1 2 exp ( ik 1 xz r + i π 4 ) ,
Q 1 y sca ( int ) ( x , y ) = [ i k 1 z sgn ( z ) f 2 ( k 1 xz cos ϑ ) + g 2 ( k 1 xz cos ϑ ) ]
× 1 4 π k 1 xz ( 1 k 1 xz r ) 1 2 exp ( ik 1 xz r + i π 4 ) ,
f 1 ( k 1 xz cos ϑ ) = S + ( k 1 xz cos ϑ ) L ( k 1 xz cos ϑ ) S + ( k 1 x ) ( k 1 x k 1 xz cos ϑ )
e il ( k 1 x k 1 xz cos ϑ ) S + ( k 1 xz cos ϑ ) L ( k 1 xz cos ϑ ) S + ( k 1 x ) ( k 1 x k 1 xz cos ϑ ) ,
g 1 ( k 1 xz cos ϑ ) = 1 ( k 1 x k 1 xz cos ϑ ) [ ϑ 1 e il ( k 1 x k 1 xz cos ϑ ) L ( k 1 xz cos ϑ ) L + ( k 1 xz cos ϑ ) e il ( k 1 xz k 1 xz cos ϑ ) L ( k 1 xz cos ϑ ) L + ( k 1 x )
δ 1 L ( k 1 xz cos ϑ ) ] ,
f 2 ( k 1 x z cos ϑ ) = 1 L ( k 1 x z cos ϑ ) [ S + ( k 1 x z cos ϑ ) R 1 ( k 1 x z cos ϑ ) e i l k 1 x
+ S + ( k 1 x z cos ϑ ) e i l k 1 x z cos ϑ R 2 ( k 1 x z cos ϑ )
C 1 S + ( k 1 x z cos ϑ ) T ( k 1 x z cos ϑ )
C 2 T ( k 1 x z cos ϑ ) S + ( k 1 x z cos ϑ ) e i l k 1 x z cos ϑ ] ,
g 2 ( k 1 x z cos ϑ ) = 1 L ( k 1 x z cos ϑ ) [ L + ( k 1 x z cos ϑ ) R 1 ( k 1 x z cos ϑ ) e i l k 1 x .
+ L + ( k 1 x z cos ϑ ) R 2 ( k 1 x z cos ϑ ) e i l k 1 x z cos ϑ
T ( k 1 x z cos ϑ ) L + ( k 1 x z cos ϑ ) C 1
T ( k 1 x z cos ϑ ) L + ( k 1 x z cos ϑ ) C 2 ] .
A ( υ ) = 1 2 π [ k 1 z κ + ( υ ) L + ( υ ) i κ ( υ ) L ( υ ) ( k 1 x + υ ) κ + ( k 1 x ) L + ( k 1 x ) + δ 1 υ L + ( υ ) κ ( υ ) L ( υ ) ( k 1 x + υ ) L + ( k 1 x ) ] .

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