Abstract

The radiation by an extended linear source in the unbounded uniaxial dielectric medium with uniform current distribution is studied. The exact as well as approximate fields in the near and far zones are found analytically using the dyadic Green functions in the frequency domain. Elegant closed-form results are obtained when the Hertzian dipole is parallel and perpendicular to the optic axis of the uniaxial medium. When the dipole is parallel to the optic axis, only extraordinary waves are emitted. When the dipole is perpendicular to the optic axis, both ordinary and extraordinary waves are emitted; however, the radiations are suppressed along the optic axis and no extraordinary waves are emitted in a direction perpendicular to both the electric dipole and the optic axis. A comparison with the point dipole showed that the directivity of the radiation pattern can be controlled using the length of the Hertzian dipole.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Corrections

Aamir Hayat and Muhammad Faryad, "On the radiation from a Hertzian dipole of a finite length in the uniaxial dielectric medium: erratum," OSA Continuum 2, 2855-2855 (2019)
https://www.osapublishing.org/osac/abstract.cfm?uri=osac-2-10-2855

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References

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  1. H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw Hill, 1983), Ch. 10.
  2. H. C. Chen, “Dyadic green’s function and radiation in a uniaxially anisotropic medium,” Int. J. Electron. 35(5), 633–640 (1973).
    [Crossref]
  3. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994).
  4. K. Korzeb, M. Gajc, and D. A. Pawlak, “Compendium of natural hyperbolic materials,” Opt. Express 23(20), 25406–25424 (2015).
    [Crossref]
  5. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1970).
  6. K. G. Balmain, A. A. E. Luttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wirel. Propag. Lett. 1, 146–149 (2002).
    [Crossref]
  7. T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a sic superlens,” Science 313(5793), 1595 (2006).
    [Crossref]
  8. J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. USA 108(28), 11327–11331 (2011).
    [Crossref]
  9. Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
    [Crossref]
  10. U. Tumkur, L. Gu, J. K. Kitur, E. E. Narimanov, and M. A. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100(16), 161103 (2012).
    [Crossref]
  11. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
    [Crossref]
  12. H. Kuzmany, Solid State Spectroscopy (Springer-Verlag, 2009).
  13. A. Eroglu and J. Kyoon, “Far field radiation from an arbitrarily oriented hertzian dipole in the presence of a layered anisotropic medium,” IEEE Trans. Antennas Propag. 53(12), 3963–3973 (2005).
    [Crossref]
  14. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Radiation and canonical sources in uniaxial dielectric media,” Int. J. Electron. 65(6), 1171–1175 (1988).
    [Crossref]
  15. L. Alexeyeva, I. Kanymgaziyeva, and S. Sautbekov, “Generalized solutions of maxwell equations for crystals with electric and magnetic anisotropy,” J. Electromagn. Waves Appl. 28(16), 1974–1984 (2014).
    [Crossref]
  16. L. Tsang, E. Njoku, and J. A. Kong, “Microwave thermal emission from a stratified medium with nonuniform temperature distribution,” J. Appl. Phys. 46(12), 5127–5133 (1975).
    [Crossref]
  17. J. A. Kong, “Electromagnetic fields due to dipole antennas over stratified anisotropic media,” Geophysics 37(6), 985–996 (1972).
    [Crossref]
  18. Y. S. Kwon and J. J. H. Wang, “Computation of hertzian dipole radiation in stratified uniaxial anisotropic media,” Radio Sci. 21(6), 891–902 (1986).
    [Crossref]
  19. C. M. Tang, “Electromagnetic fields due to dipole antennas embedded in stratified anisotropic media,” IEEE Trans. Antennas Propag. 27(5), 665–670 (1979).
    [Crossref]
  20. S. Ali and S. Mahmoud, “Electromagnetic fields of buried sources in stratified anisotropic media,” IEEE Trans. Antennas Propag. 27(5), 671–678 (1979).
    [Crossref]
  21. N. Bellyustin and V. Dokuchaev, “Generation of electromagnetic waves by distributed currents in an anisotoipic medium,” Radiophys. Quantum Electron. 18(1), 10–17 (1975).
    [Crossref]
  22. T. N. C. Wang and T.-L. Wang, “Radiation resistance of small electric and magnetic antennas in a cold uniaxial plasma,” IEEE Trans. Antennas Propag. 20(6), 796–798 (1972).
    [Crossref]
  23. M. Faryad and A. Lakhtakia, Infinite-Space Dyadic Green Functions in Electromagnetism (Morgan & Claypool, 2018).
  24. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).
  25. A. Zangwill, Modern Electrodynamics (Cambridge, 2012), Ch. 20.
  26. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons, 1984).

2015 (1)

2014 (1)

L. Alexeyeva, I. Kanymgaziyeva, and S. Sautbekov, “Generalized solutions of maxwell equations for crystals with electric and magnetic anisotropy,” J. Electromagn. Waves Appl. 28(16), 1974–1984 (2014).
[Crossref]

2012 (2)

Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
[Crossref]

U. Tumkur, L. Gu, J. K. Kitur, E. E. Narimanov, and M. A. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100(16), 161103 (2012).
[Crossref]

2011 (1)

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. USA 108(28), 11327–11331 (2011).
[Crossref]

2007 (1)

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref]

2006 (1)

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a sic superlens,” Science 313(5793), 1595 (2006).
[Crossref]

2005 (1)

A. Eroglu and J. Kyoon, “Far field radiation from an arbitrarily oriented hertzian dipole in the presence of a layered anisotropic medium,” IEEE Trans. Antennas Propag. 53(12), 3963–3973 (2005).
[Crossref]

2002 (1)

K. G. Balmain, A. A. E. Luttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wirel. Propag. Lett. 1, 146–149 (2002).
[Crossref]

1988 (1)

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Radiation and canonical sources in uniaxial dielectric media,” Int. J. Electron. 65(6), 1171–1175 (1988).
[Crossref]

1986 (1)

Y. S. Kwon and J. J. H. Wang, “Computation of hertzian dipole radiation in stratified uniaxial anisotropic media,” Radio Sci. 21(6), 891–902 (1986).
[Crossref]

1979 (2)

C. M. Tang, “Electromagnetic fields due to dipole antennas embedded in stratified anisotropic media,” IEEE Trans. Antennas Propag. 27(5), 665–670 (1979).
[Crossref]

S. Ali and S. Mahmoud, “Electromagnetic fields of buried sources in stratified anisotropic media,” IEEE Trans. Antennas Propag. 27(5), 671–678 (1979).
[Crossref]

1975 (2)

N. Bellyustin and V. Dokuchaev, “Generation of electromagnetic waves by distributed currents in an anisotoipic medium,” Radiophys. Quantum Electron. 18(1), 10–17 (1975).
[Crossref]

L. Tsang, E. Njoku, and J. A. Kong, “Microwave thermal emission from a stratified medium with nonuniform temperature distribution,” J. Appl. Phys. 46(12), 5127–5133 (1975).
[Crossref]

1973 (1)

H. C. Chen, “Dyadic green’s function and radiation in a uniaxially anisotropic medium,” Int. J. Electron. 35(5), 633–640 (1973).
[Crossref]

1972 (2)

J. A. Kong, “Electromagnetic fields due to dipole antennas over stratified anisotropic media,” Geophysics 37(6), 985–996 (1972).
[Crossref]

T. N. C. Wang and T.-L. Wang, “Radiation resistance of small electric and magnetic antennas in a cold uniaxial plasma,” IEEE Trans. Antennas Propag. 20(6), 796–798 (1972).
[Crossref]

Alexeyeva, L.

L. Alexeyeva, I. Kanymgaziyeva, and S. Sautbekov, “Generalized solutions of maxwell equations for crystals with electric and magnetic anisotropy,” J. Electromagn. Waves Appl. 28(16), 1974–1984 (2014).
[Crossref]

Ali, S.

S. Ali and S. Mahmoud, “Electromagnetic fields of buried sources in stratified anisotropic media,” IEEE Trans. Antennas Propag. 27(5), 671–678 (1979).
[Crossref]

Balmain, K. G.

K. G. Balmain, A. A. E. Luttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wirel. Propag. Lett. 1, 146–149 (2002).
[Crossref]

Bartal, G.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. USA 108(28), 11327–11331 (2011).
[Crossref]

Bellyustin, N.

N. Bellyustin and V. Dokuchaev, “Generation of electromagnetic waves by distributed currents in an anisotoipic medium,” Radiophys. Quantum Electron. 18(1), 10–17 (1975).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1970).

Chen, H. C.

H. C. Chen, “Dyadic green’s function and radiation in a uniaxially anisotropic medium,” Int. J. Electron. 35(5), 633–640 (1973).
[Crossref]

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw Hill, 1983), Ch. 10.

Cortes, C. L.

Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
[Crossref]

Dokuchaev, V.

N. Bellyustin and V. Dokuchaev, “Generation of electromagnetic waves by distributed currents in an anisotoipic medium,” Radiophys. Quantum Electron. 18(1), 10–17 (1975).
[Crossref]

Eroglu, A.

A. Eroglu and J. Kyoon, “Far field radiation from an arbitrarily oriented hertzian dipole in the presence of a layered anisotropic medium,” IEEE Trans. Antennas Propag. 53(12), 3963–3973 (2005).
[Crossref]

Faryad, M.

M. Faryad and A. Lakhtakia, Infinite-Space Dyadic Green Functions in Electromagnetism (Morgan & Claypool, 2018).

Felsen, L. B.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994).

Gajc, M.

Gu, L.

U. Tumkur, L. Gu, J. K. Kitur, E. E. Narimanov, and M. A. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100(16), 161103 (2012).
[Crossref]

Guo, Y.

Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
[Crossref]

Hillenbrand, R.

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a sic superlens,” Science 313(5793), 1595 (2006).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Jacob, Z.

Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
[Crossref]

Kanymgaziyeva, I.

L. Alexeyeva, I. Kanymgaziyeva, and S. Sautbekov, “Generalized solutions of maxwell equations for crystals with electric and magnetic anisotropy,” J. Electromagn. Waves Appl. 28(16), 1974–1984 (2014).
[Crossref]

Kitur, J. K.

U. Tumkur, L. Gu, J. K. Kitur, E. E. Narimanov, and M. A. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100(16), 161103 (2012).
[Crossref]

Kong, J. A.

L. Tsang, E. Njoku, and J. A. Kong, “Microwave thermal emission from a stratified medium with nonuniform temperature distribution,” J. Appl. Phys. 46(12), 5127–5133 (1975).
[Crossref]

J. A. Kong, “Electromagnetic fields due to dipole antennas over stratified anisotropic media,” Geophysics 37(6), 985–996 (1972).
[Crossref]

Korobkin, D.

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a sic superlens,” Science 313(5793), 1595 (2006).
[Crossref]

Korzeb, K.

Kremer, P. C.

K. G. Balmain, A. A. E. Luttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wirel. Propag. Lett. 1, 146–149 (2002).
[Crossref]

Kuzmany, H.

H. Kuzmany, Solid State Spectroscopy (Springer-Verlag, 2009).

Kwon, Y. S.

Y. S. Kwon and J. J. H. Wang, “Computation of hertzian dipole radiation in stratified uniaxial anisotropic media,” Radio Sci. 21(6), 891–902 (1986).
[Crossref]

Kyoon, J.

A. Eroglu and J. Kyoon, “Far field radiation from an arbitrarily oriented hertzian dipole in the presence of a layered anisotropic medium,” IEEE Trans. Antennas Propag. 53(12), 3963–3973 (2005).
[Crossref]

Lakhtakia, A.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Radiation and canonical sources in uniaxial dielectric media,” Int. J. Electron. 65(6), 1171–1175 (1988).
[Crossref]

M. Faryad and A. Lakhtakia, Infinite-Space Dyadic Green Functions in Electromagnetism (Morgan & Claypool, 2018).

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref]

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref]

Luttgen, A. A. E.

K. G. Balmain, A. A. E. Luttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wirel. Propag. Lett. 1, 146–149 (2002).
[Crossref]

Mahmoud, S.

S. Ali and S. Mahmoud, “Electromagnetic fields of buried sources in stratified anisotropic media,” IEEE Trans. Antennas Propag. 27(5), 671–678 (1979).
[Crossref]

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994).

Molesky, S.

Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
[Crossref]

Narimanov, E. E.

U. Tumkur, L. Gu, J. K. Kitur, E. E. Narimanov, and M. A. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100(16), 161103 (2012).
[Crossref]

Njoku, E.

L. Tsang, E. Njoku, and J. A. Kong, “Microwave thermal emission from a stratified medium with nonuniform temperature distribution,” J. Appl. Phys. 46(12), 5127–5133 (1975).
[Crossref]

Noginov, M. A.

U. Tumkur, L. Gu, J. K. Kitur, E. E. Narimanov, and M. A. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100(16), 161103 (2012).
[Crossref]

Pawlak, D. A.

Sautbekov, S.

L. Alexeyeva, I. Kanymgaziyeva, and S. Sautbekov, “Generalized solutions of maxwell equations for crystals with electric and magnetic anisotropy,” J. Electromagn. Waves Appl. 28(16), 1974–1984 (2014).
[Crossref]

Shvets, G.

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a sic superlens,” Science 313(5793), 1595 (2006).
[Crossref]

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref]

Tang, C. M.

C. M. Tang, “Electromagnetic fields due to dipole antennas embedded in stratified anisotropic media,” IEEE Trans. Antennas Propag. 27(5), 665–670 (1979).
[Crossref]

Taubner, T.

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a sic superlens,” Science 313(5793), 1595 (2006).
[Crossref]

Tsang, L.

L. Tsang, E. Njoku, and J. A. Kong, “Microwave thermal emission from a stratified medium with nonuniform temperature distribution,” J. Appl. Phys. 46(12), 5127–5133 (1975).
[Crossref]

Tumkur, U.

U. Tumkur, L. Gu, J. K. Kitur, E. E. Narimanov, and M. A. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100(16), 161103 (2012).
[Crossref]

Urzhumov, Y.

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a sic superlens,” Science 313(5793), 1595 (2006).
[Crossref]

Varadan, V. K.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Radiation and canonical sources in uniaxial dielectric media,” Int. J. Electron. 65(6), 1171–1175 (1988).
[Crossref]

Varadan, V. V.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Radiation and canonical sources in uniaxial dielectric media,” Int. J. Electron. 65(6), 1171–1175 (1988).
[Crossref]

Wang, J. J. H.

Y. S. Kwon and J. J. H. Wang, “Computation of hertzian dipole radiation in stratified uniaxial anisotropic media,” Radio Sci. 21(6), 891–902 (1986).
[Crossref]

Wang, T. N. C.

T. N. C. Wang and T.-L. Wang, “Radiation resistance of small electric and magnetic antennas in a cold uniaxial plasma,” IEEE Trans. Antennas Propag. 20(6), 796–798 (1972).
[Crossref]

Wang, T.-L.

T. N. C. Wang and T.-L. Wang, “Radiation resistance of small electric and magnetic antennas in a cold uniaxial plasma,” IEEE Trans. Antennas Propag. 20(6), 796–798 (1972).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1970).

Xiong, Y.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref]

Yang, X.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. USA 108(28), 11327–11331 (2011).
[Crossref]

Yao, J.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. USA 108(28), 11327–11331 (2011).
[Crossref]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons, 1984).

Yin, X.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. USA 108(28), 11327–11331 (2011).
[Crossref]

Zangwill, A.

A. Zangwill, Modern Electrodynamics (Cambridge, 2012), Ch. 20.

Zhang, X.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. USA 108(28), 11327–11331 (2011).
[Crossref]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref]

Appl. Phys. Lett. (2)

Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
[Crossref]

U. Tumkur, L. Gu, J. K. Kitur, E. E. Narimanov, and M. A. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100(16), 161103 (2012).
[Crossref]

Geophysics (1)

J. A. Kong, “Electromagnetic fields due to dipole antennas over stratified anisotropic media,” Geophysics 37(6), 985–996 (1972).
[Crossref]

IEEE Antennas Wirel. Propag. Lett. (1)

K. G. Balmain, A. A. E. Luttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wirel. Propag. Lett. 1, 146–149 (2002).
[Crossref]

IEEE Trans. Antennas Propag. (4)

A. Eroglu and J. Kyoon, “Far field radiation from an arbitrarily oriented hertzian dipole in the presence of a layered anisotropic medium,” IEEE Trans. Antennas Propag. 53(12), 3963–3973 (2005).
[Crossref]

T. N. C. Wang and T.-L. Wang, “Radiation resistance of small electric and magnetic antennas in a cold uniaxial plasma,” IEEE Trans. Antennas Propag. 20(6), 796–798 (1972).
[Crossref]

C. M. Tang, “Electromagnetic fields due to dipole antennas embedded in stratified anisotropic media,” IEEE Trans. Antennas Propag. 27(5), 665–670 (1979).
[Crossref]

S. Ali and S. Mahmoud, “Electromagnetic fields of buried sources in stratified anisotropic media,” IEEE Trans. Antennas Propag. 27(5), 671–678 (1979).
[Crossref]

Int. J. Electron. (2)

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, “Radiation and canonical sources in uniaxial dielectric media,” Int. J. Electron. 65(6), 1171–1175 (1988).
[Crossref]

H. C. Chen, “Dyadic green’s function and radiation in a uniaxially anisotropic medium,” Int. J. Electron. 35(5), 633–640 (1973).
[Crossref]

J. Appl. Phys. (1)

L. Tsang, E. Njoku, and J. A. Kong, “Microwave thermal emission from a stratified medium with nonuniform temperature distribution,” J. Appl. Phys. 46(12), 5127–5133 (1975).
[Crossref]

J. Electromagn. Waves Appl. (1)

L. Alexeyeva, I. Kanymgaziyeva, and S. Sautbekov, “Generalized solutions of maxwell equations for crystals with electric and magnetic anisotropy,” J. Electromagn. Waves Appl. 28(16), 1974–1984 (2014).
[Crossref]

Opt. Express (1)

Proc. Natl. Acad. Sci. USA (1)

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. USA 108(28), 11327–11331 (2011).
[Crossref]

Radio Sci. (1)

Y. S. Kwon and J. J. H. Wang, “Computation of hertzian dipole radiation in stratified uniaxial anisotropic media,” Radio Sci. 21(6), 891–902 (1986).
[Crossref]

Radiophys. Quantum Electron. (1)

N. Bellyustin and V. Dokuchaev, “Generation of electromagnetic waves by distributed currents in an anisotoipic medium,” Radiophys. Quantum Electron. 18(1), 10–17 (1975).
[Crossref]

Science (2)

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref]

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a sic superlens,” Science 313(5793), 1595 (2006).
[Crossref]

Other (8)

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw Hill, 1983), Ch. 10.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1970).

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (IEEE, 1994).

H. Kuzmany, Solid State Spectroscopy (Springer-Verlag, 2009).

M. Faryad and A. Lakhtakia, Infinite-Space Dyadic Green Functions in Electromagnetism (Morgan & Claypool, 2018).

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

A. Zangwill, Modern Electrodynamics (Cambridge, 2012), Ch. 20.

A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (John Wiley & Sons, 1984).

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Figures (8)

Fig. 1.
Fig. 1. Schematic showing a dipole (thick line) oriented parallel to the optic axis $\hat {\mathbf {c}}=\hat {\mathbf {{x}}}$. The field point P is located at position vector $\mathbf {r}$ with respect to the origin.
Fig. 2.
Fig. 2. Schematic showing a dipole (thick vertical line) oriented parallel to the optic axis $\hat {\mathbf {c}}=\hat {\mathbf {{z}}}$. The field point P is located at a position vector $\mathbf {r}$ with respect to the center of the dipole.
Fig. 3.
Fig. 3. Schematic showing a dipole (thick vertical line) oriented perpendicular to the optic axis $\hat {\mathbf {c}}=\hat {\mathbf {{x}}}$. The field point P is located at a position vector $\mathbf {r}$ with respect to the center of the dipole.
Fig. 4.
Fig. 4. Far-field radiation pattern of a Hertzian dipole given by Eq. (50), which is oriented parallel to the optic axis ($z$ axis) and lying in a uniaxial medium (rutile) with $p_0=1/\omega$, $\varepsilon _a=8.427$, $\varepsilon _b=6.843$, $\mu _b=1$ , $\lambda _{o}=0.584\,\mu$m [26], and (left) $L=0.1 \lambda _{o}$, (right) $L=0.2 \lambda _{o}$. The plot is given for $0\leq \theta \leq \pi$ and $\pi /2\leq \phi \leq 3\pi /2$.
Fig. 5.
Fig. 5. Far-field radiation pattern of ordinary waves given by Eq. (76) when the dipole is oriented along $z$ axis and the optic axis along $x$ axis for a uniaxial medium (rutile) with $\varepsilon _a=8.427$, $\varepsilon _b=6.843$, $\mu _b=1$ , $\lambda _{o}=0.584\,\mu$m [26], and (left) $L=0.1 \lambda _{o}$, (right) $L=0.2 \lambda _{o}$. The plot is given for $0\leq \theta \leq \pi$ and $\pi /2\leq \phi \leq 3\pi /2$. The pattern is symmetric about $xz$ plane.
Fig. 6.
Fig. 6. Same as Fig. 5 except that the pattern is that of extraordinary waves given by Eq. (77).
Fig. 7.
Fig. 7. Far-field radiation pattern of a point dipole given by Eq. (85), when it is oriented parallel to the optic axis ($z$ axis) and lying in a uniaxial medium (rutile) with $p_0=1/\omega$, $\varepsilon _a=8.427$, $\varepsilon _b=6.843$, $\mu _b=1$ , $\lambda _{o}=0.584\,\mu$m. The plot is given for $0\leq \theta \leq \pi$ and $\pi /2\leq \phi \leq 3\pi /2$.
Fig. 8.
Fig. 8. Far-field radiation pattern of (left) ordinary waves given by Eq. (86) and (right) extraordinary waves given by Eq. (87), when the point dipole is oriented along $z$ axis and the optic axis along $x$ axis for a uniaxial medium (rutile) with $p_0=1/\omega$, $\varepsilon _a=8.427$, $\varepsilon _b=6.843$, $\mu _b=1$ , $\lambda _{o}=0.584\,\mu$m [26]. The plot is given for $0\leq \theta \leq \pi$ and $\pi /2\leq \phi \leq 3\pi /2$.

Equations (107)

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× H ( r ) + i ω D ( r ) = J e ( r ) ,
× E ( r ) i ω B ( r ) = 0 ,
D ( r ) = ρ e ( r ) ,
B ( r ) = 0 ,
D ( r ) = ε _ _ E ( r ) , B ( r ) = μ o μ b H ( r ) ,
ε _ _ = ε o ε _ _ r = ε o [ ε b I _ _ + ( ε a ε b ) c ^ c ^ ] ,
E ( r ) = V G _ _ e e ( R ) J e ( r ) d 3 r ,
H ( r ) = V G _ _ m e ( R ) J e ( r ) d 3 r ,
R = r r .
G _ _ e e ( R ) = i ω μ o μ b { g e ( R ) ε a ε _ _ r 1 [ 1 1 i k o n o R e 1 ( k o n o R e ) 2 ] g e ( R ) [ 1 3 i k o n o R e 3 ( k o n o R e ) 2 ] ε a 2 ( ε _ _ r 1 R ) ( ε _ _ r 1 R ) R e 2 + 1 ε b [ ε b g o ( R ) ε a g e ( R ) ] K _ _ ( R ) + R g o ( R ) R e g e ( R ) i k o n o ( R × c ^ ) ( R × c ^ ) [ I _ _ c ^ c ^ 2 K _ _ ( R ) ] } ,
G _ _ m e ( R ) = ε a ε b ( 1 i k o n o R e ) g e ( R ) ( R × c ^ ) [ R × ( R × c ^ ) ] R e 2 ( R × c ^ ) ( R × c ^ ) + [ g e ( R ) g o ( R ) ] ( R c ^ ) [ c ^ × ( R × c ^ ) ] ( R × c ^ ) + ( R × c ^ ) [ c ^ × ( R × c ^ ) ] [ ( R × c ^ ) ( R × c ^ ) ] 2 ( 1 i k o n o R ) g o ( R ) [ R × ( R × c ^ ) ] ( R × c ^ ) R 2 [ ( R × c ^ ) ( R × c ^ ) ] ,
K _ _ ( R ) = ( R × c ^ ) ( R × c ^ ) ( R × c ^ ) ( R × c ^ ) ,
ε _ _ r 1 = 1 ε b I _ _ ( 1 ε b 1 ε a ) c ^ c ^ ,
n o = ε b μ b , k o = ω μ o ε o .
g o ( R ) = exp ( i k o n o R ) 4 π R and g e ( R ) = exp ( i k o n o R e ) 4 π R e
R e = ε a ε b ( R × c ^ ) ( R × c ^ ) + ( R c ^ ) 2 .
G _ _ e e ( R ) i ε a 4 π ω ε o ε b ( 3 ε a ( ε _ _ r 1 R ) ( ε _ _ r 1 R ) R e 5 ε _ _ r 1 R e 3 )
G _ _ m e ( R ) 0
G _ _ e e ( R ) i ω μ o μ b { g e ( R ) ε a ε _ _ r 1 g e ( R ) ε a 2 ( ε _ _ r 1 R ) ( ε _ _ r 1 R ) R e 2 + 1 ε b [ ε b g o ( R ) ε a g e ( R ) ] K _ _ ( R ) } .
ε _ _ r 1 ε a ( ε _ _ r 1 R ) ( ε _ _ r 1 R ) R e 2 1 ε b K _ _ ( R ) = [ R × ( R × c ^ ) ] [ R × ( R × c ^ ) ] ε b R e 2 ( R × c ^ ) ( R × c ^ ) ,
G _ _ e e ( R ) = i ω μ o μ b { g e ( R ) ε a [ R × ( R × c ^ ) ] [ R × ( R × c ^ ) ] ε b R e 2 ( R × c ^ ) ( R × c ^ ) + g o ( R ) K _ _ ( R ) } .
G _ _ m e ( R ) i k o n o { g o ( R ) [ R × ( R × c ^ ) ] ( R × c ^ ) R ( R × c ^ ) ( R × c ^ ) ε a ε b g e ( R ) ( R × c ^ ) [ R × ( R × c ^ ) ] R e [ ( R × c ^ ) ( R × c ^ ) ] }
J e ( r ) = { i ω p o δ ( y ) δ ( z ) x ^ , | x | L , 0 , | x | > L .
E ( r ) = ε d p o 4 π ε o L L [ 3 ε a ( ε _ _ r 1 R x ) ( ε _ _ r 1 R x ) x ^ R e x 5 ε _ _ r 1 x ^ R e x 3 ] d x ,
ε d = ε a / ε b ,
R e x = ε d ( y 2 + z 2 ) + ( x x ) 2 ,
R x = ( x x ) x ^ + y y ^ + z z ^ .
ε _ _ r 1 R x = 1 ε b ( y y ^ + z z ^ ) + 1 ε a ( x x ) x ^ ,
ε _ _ r 1 x ^ = 1 ε a x ^ .
E ( r ) = ε d p o 4 π ε o ε a { 3 ε d ( y y ^ + z z ^ ) L L ( x x ) [ ε d ( y 2 + z 2 ) + ( x x ) 2 ] 5 2 d x + 3 x ^ L L ( x x ) 2 [ ε d ( y 2 + z 2 ) + ( x x ) 2 ] 5 2 d x x ^ L L 1 [ ε d ( y 2 + z 2 ) + ( x x ) 2 ] 3 2 d x }
E ( r ) = ε d p o 4 π ε o ε b { y y ^ + z z ^ ( 1 [ ε d ( y 2 + z 2 ) + ( x L ) 2 ] 3 2 1 [ ε d ( y 2 + z 2 ) + ( x + L ) 2 ] 3 2 ) + x ^ ε d ( L x [ ε d ( y 2 + z 2 ) + ( x L ) 2 ] 3 2 + L + x [ ε d ( y 2 + z 2 ) + ( x + L ) 2 ] 3 2 ) } .
E ( r ) = ω 2 μ o μ b p o { L L g e ( R x ) [ ε a ε _ _ r 1 x ^ ε a 2 ( ε _ _ r 1 R x ) ( ε _ _ r 1 R x ) x ^ R e x 2 ] d x } ,
E ( r ) = ω 2 μ o μ b p o 4 π [ x ^ L L exp ( i k o n o R e x ) R e x d x ε d ( y y ^ + z z ^ ) L L ( x x ) exp ( i k o n o R e x ) R e x 3 d x x ^ L L ( x x ) 2 exp ( i k o n o R e x ) R e x 3 d x ] ,
R e x r e x r e x
r e = x 2 + ε d ( y 2 + z 2 ) .
E ( r ) ω 2 μ o μ b p o 4 π exp ( i k o n o r e ) [ ( I 1 r e I 3 r e 3 ) x ^ ε d ( y y ^ + z z ^ r e 3 ) I 2 ] ,
x = r sin θ cos ϕ , y = r sin θ sin ϕ , z = r cos θ ,
r e = r sin 2 θ cos 2 ϕ + ε d ( sin 2 θ sin 2 ϕ + cos 2 θ ) = r Φ ( θ , ϕ ) .
E ( r ) = k o μ b p o ε d 2 π ε o n o r exp ( i k o n o r Φ ) Φ 2 [ ( sin 2 θ sin 2 ϕ + cos 2 θ ) sin θ cos ϕ x ^ sin θ sin ϕ y ^ cos θ z ^ ] sin ( k o L n o sin θ cos ϕ Φ ) .
H ( r ) = k o n o ω p o ε d L L g e ( R x ) ( R x × x ^ ) [ R x × ( R x × x ^ ) ] x ^ R e x ( R x × x ^ ) ( R x × x ^ ) d x .
H ( r ) = k o n o ω p o ε d 4 π ( y z ^ + z y ^ ) L L exp ( i k o n o R e x ) R e x 2 d x ,
R x × x ^ = y z ^ + z y ^ , [ R x × ( R x × x ^ ) ] x ^ = ( y 2 + z 2 ) .
H ( r ) = k o n o ω p o ε d 4 π exp ( i k o n o r e ) r e 2 ( y z ^ + z y ^ ) I 1 .
H ( r ) = k o p o c ε d 2 π r exp ( i k o n o r Φ ) Φ sin θ cos ϕ ( cos θ y ^ sin θ sin ϕ z ^ ) sin ( k o L n o sin θ cos ϕ Φ ) .
d P d Ω = 1 2 r ^ R e ( E × H ) r 2 .
d P d Ω = k o 2 μ b p o 2 c ε d 2 8 π 2 ε o n o ( sin 2 θ sin 2 ϕ + cos 2 θ ) Φ 3 sin 2 θ cos 2 ϕ sin 2 ( k o L n o sin θ cos ϕ Φ ) .
E ( r ) = k o μ b p o ε d 2 π ε o n o r exp ( i k o n o r Θ ) Θ 2 [ sin 2 θ cos θ x ^ sin θ cos ϕ y ^ sin θ sin ϕ z ^ ] sin ( k o L n o cos θ Θ ) ,
Θ = cos 2 θ + ε d sin 2 θ ,
H ( r ) = k o p o c ε d 2 π r exp ( i k o n o r Θ ) Θ cos θ ( sin θ cos ϕ z ^ + sin θ sin ϕ y ^ ) sin ( k o L n o cos θ Θ ) .
d P d Ω = k o 2 μ b p o 2 c ε d 2 8 π 2 ε o n o sin 2 θ Θ 3 cos 2 θ sin 2 ( k o L n o cos θ Θ ) .
J e ( r ) = { i ω p o δ ( x ) δ ( y ) z ^ , | z | L , 0 , | z | > L .
E ( r ) = ε d p o 4 π ε o L L [ 3 ε a ( ε _ _ r 1 R z ) ( ε _ _ r 1 R z ) z ^ R e z 5 ε _ _ r 1 z ^ R e z 3 ] d z ,
R z = x x ^ + y y ^ + ( z z ) z ^
R e z = x 2 + ε d y 2 + ε d ( z z ) 2 .
E ( r ) = 3 ε d 2 p o 4 π ε o ε a { ( x x ^ + ε d y y ^ ) L L ( z z ) [ x 2 + ε d y 2 + ε d ( z z ) 2 ] 5 2 d z + z ^ ε d L L ( z z ) 2 [ x 2 + ε d y 2 + ε d ( z z ) 2 ] 5 2 d z z ^ 3 L L 1 [ x 2 + ε d y 2 + ε d ( z z ) 2 ] 3 2 d z } ,
E ( r ) = 3 ε d 2 p o 4 π ε o ε a [ ( x x ^ + ε d y y ^ ) { 1 [ x 2 + ε d y 2 + ε d ( z L ) 2 ] 3 2 1 [ x 2 + ε d y 2 + ε d ( z + L ) 2 ] 3 2 } + z ^ ε d ( x 2 + ε d y 2 ) { ( z L ) 2 [ x 2 + ε d y 2 + ε d ( z L ) 2 ] 3 2 ( z + L ) 2 [ x 2 + ε d y 2 + ε d ( z + L ) 2 ] 3 2 } + z ^ 3 ( x 2 + ε d y 2 ) { ( z L ) [ x 2 + ε d y 2 + ε d ( z L ) 2 ] 1 2 ( z + L ) [ x 2 + ε d y 2 + ε d ( z + L ) 2 ] 1 2 } ] .
E ( r ) = ω 2 μ o μ b p o 4 π { ε d L L [ ( x 2 y y ^ x y 2 x ^ ) ( z z ) + x 2 z ^ ( z z ) 2 x x ^ ( z z ) 3 ] y 2 + ( z z ) 2 × exp ( i k o n o R e z ) R e z 3 d z + L L ( y 2 z ^ y ( z z ) y ^ y 2 + ( z z ) 2 ) exp ( i k o n o R z ) R z d z } .
R e z = [ r e 2 + ε d z ( z 2 z ) ] 1 2 ,
R e z r e ε d z z r e
R z = [ r 2 + z ( z 2 z ) ] 1 2
R z r z z r
y 2 + ( z z ) 2 y 2 + z 2
E ( r ) ω 2 μ o μ b p o 4 π { ε d exp ( i k o n o r e ) r e 3 ( y 2 + z 2 ) [ ( x 2 y y ^ x y 2 x ^ ) I 4 + x 2 z ^ I 5 x x ^ I 6 ] + exp ( i k o n o r ) r ( y 2 + z 2 ) ( y 2 z ^ I 7 y y ^ I 8 ) } ,
E = E o + E e ,
E o ( r ) = k o μ b p o 2 π ε o n o r ( y ^ + tan θ sin ϕ z ^ ) exp ( i k o n o r ) sin θ sin ϕ ( sin 2 θ sin 2 ϕ + cos 2 θ ) sin ( k o n o L cos θ )
E e ( r ) = k o μ b p o 2 π ε o n o r [ ( sin 2 θ sin 2 ϕ + cos 2 θ ) x ^ + sin 2 θ cos ϕ sin ϕ y ^ + sin θ cos θ cos ϕ z ^ ] exp ( i k o n o r Φ ) sin θ cos ϕ Φ 2 ( sin 2 θ sin 2 ϕ + cos 2 θ ) sin ( ε d k o n o L cos θ Φ )
H ( r ) = k o n o ω p o L L { g o ( R z ) [ R z × ( R z × x ^ ) ] ( R z × x ^ ) z ^ R z ( R z × x ^ ) ( R z × x ^ ) ε d g e ( R z ) ( R z × x ^ ) [ R z × ( R z × x ^ ) ] z ^ R e z ( R z × x ^ ) ( R z × x ^ ) } d z .
H ( r ) = k o n o ω p o 4 π { L L exp ( i k o n o R z ) R z 2 [ x y 2 y ^ + y 3 x ^ x y ( z z ) z ^ + y ( z z ) 2 x ^ ] y 2 + ( z z ) 2 d z ε d L L exp ( i k o n o R e z ) R e z 2 [ x y ( z z ) z ^ + x ( z z ) 2 y ^ ] y 2 + ( z z ) 2 d z } .
H ( r ) k o n o ω p o 4 π { exp ( i k o n o r ) r 2 ( y 2 + z 2 ) [ ( y 3 x ^ x y 2 y ^ ) I 7 x y z ^ I 8 + y x ^ I 9 ] ε d exp ( i k o n o r e ) r e 2 ( y 2 + z 2 ) ( x y z ^ I 4 + x y ^ I 5 ) } ,
H = H o + H e ,
H o ( r ) = k o p o c 2 π r [ ( sin 2 θ sin 2 ϕ + cos 2 θ ) x ^ sin 2 θ cos ϕ sin ϕ y ^ sin θ cos θ cos ϕ z ^ ] exp ( i k o n o r ) sin θ sin ϕ cos θ ( sin 2 θ sin 2 ϕ + cos 2 θ ) sin ( k o n o L cos θ ) ,
H e ( r ) = k o p o c 2 π r ( cos θ y ^ + sin θ sin ϕ z ^ ) exp ( i k o n o r Φ ) sin θ cos ϕ Φ ( sin 2 θ sin 2 ϕ + cos 2 θ ) sin ( ε d k o n o L cos θ Φ ) .
r ^ ( E e × H o ) = 0 , r ^ ( E o × H e ) = 0 ,
d P d Ω = d P o d Ω + d P e d Ω ,
d P o d Ω = 1 2 r ^ R e ( E o × H o ) r 2
d P e d Ω = 1 2 r ^ R e ( E e × H e ) r 2 .
d P o d Ω = k o 2 μ b p o 2 c 8 π 2 ε o n o sin 2 θ sin 2 ϕ cos 2 θ ( sin 2 θ sin 2 ϕ + cos 2 θ ) sin 2 ( k o n o L cos θ ) .
d P e d Ω = k o 2 μ b p o 2 c 8 π 2 ε o n o sin 2 θ cos 2 ϕ Φ 3 ( sin 2 θ sin 2 ϕ + cos 2 θ ) sin 2 ( ε d k o L n o cos θ Φ ) .
d P d Ω = k o 2 μ b p o 2 c 8 π 2 ε o n o sin 2 θ ( sin 2 θ sin 2 ϕ + cos 2 θ ) [ cos 2 ϕ Φ 3 sin 2 ( ε d k o L n o cos θ Φ ) + sin 2 ϕ cos 2 θ sin 2 ( k o n o L cos θ ) ] .
G _ _ e e ( R ) = i ω μ o μ b [ ( ε a ε _ _ r 1 R ^ R ^ ) + ε b ε a 2 ε b ( I _ _ c ^ c ^ ) ] g o ( R )
G _ _ m e ( R ) = i k o n o ( ε d + 1 ) 2 g o ( R ) c ^ × I _ _ .
E ( r ) = k o 2 μ b p o L 4 π ε o ( ε d + 1 ) exp ( i k o n o r ) r z ^ .
H ( r ) = k o 2 n o p o c L 4 π ( ε d + 1 ) exp ( i k o n o r ) r y ^ .
d P d Ω = k o 4 μ b n o p o 2 c L 2 32 π 2 ε o ( ε d + 1 ) 2 .
J e ( r ) = i ω p o δ ( x ) δ ( y ) δ ( z ) z ^ .
d P d Ω = k o 4 μ b n o p o 2 c ε d 2 32 π 2 ε o Θ 5 sin 2 θ
d P o d Ω = k o 4 μ b n o p o 2 c 32 π 2 ε o sin 2 θ sin 2 ϕ sin 2 θ sin 2 ϕ + cos 2 θ .
d P e d Ω = k o 4 μ b n o p o 2 c ε d 2 32 π 2 ε o Φ 5 sin 2 θ cos 2 ϕ cos 2 θ sin 2 θ sin 2 ϕ + cos 2 θ .
d P d Ω = k o 4 μ b n o p o 2 c 32 π 2 ε o ( ε d 2 cos 2 θ cos 2 ϕ Φ 5 + sin 2 ϕ ) sin 2 θ sin 2 θ sin 2 ϕ + cos 2 θ .
I 1 = L L exp ( i s x x ) d x = 2 sin ( s L x ) s x ,
s = k o n o r e .
I 1 = 2 Φ k o n o sin θ cos ϕ sin ( k o n o L sin θ cos ϕ Φ ) .
I 2 = L L ( x x ) exp ( i s x x ) d x = 2 i s L x cos ( L s x ) + 2 ( i + s x 2 ) sin ( L s x ) s 2 x 2 .
I 2 = 2 r Φ k o n o sin ( k o n o L sin θ cos ϕ Φ ) .
I 3 = L L ( x x ) 2 exp ( i s x x ) d x = i s 3 x 3 { exp ( i s L x ) [ 2 2 i s ( L x ) x + s 2 ( L x ) 2 x 2 ) ] exp ( i s L x ) [ 2 + 2 i s ( L + x ) x + s 2 ( L + x ) 2 x 2 ] } .
I 3 = 2 r 2 Φ sin θ cos ϕ k o n o sin ( k o n o L sin θ cos ϕ Φ ) .
I 4 = L L ( z z ) exp ( i s ε d z z ) d z = 2 i ε d s L z cos ( ε d L s z ) + 2 ( i + ε d s z 2 ) sin ( ε d L s z ) ε d 2 s 2 z 2 .
I 4 = 2 r Φ ε d k o n o sin ( ε d k o n o L cos θ Φ ) .
I 5 = L L ( z z ) 2 exp ( i s ε d z z ) d z = i s 3 z 3 ε d 3 { exp ( i s L z ε d ) [ 2 2 i s ( L z ) z ε d + s 2 ( L z ) 2 z 2 ε d 2 ] exp ( i s L z ε d ) [ 2 + 2 i s ( L + z ) z ε d + s 2 ( L + z ) 2 z 2 ε d 2 ] } .
I 5 = 2 r 2 Φ cos θ ε d k o n o sin ( ε d k o n o L cos θ Φ ) .
I 6 = L L ( z z ) 3 exp ( i s ε d z z ) d z = 1 s 4 z 4 ε d 4 { exp ( i s L z ε d ) [ 6 + i 6 s ( L z ) z ε d 3 s 2 ( L z ) 2 z 2 ε d 2 i s 3 ( L z ) 3 z 3 ε d 3 ] exp ( i s L z ε d ) [ 6 i 6 s ( L + z ) z ε d 3 s 2 ( L + z ) 2 z 2 ε d 2 + i s 3 ( L + z ) 3 z 3 ε d 3 ] } .
I 6 = 2 r 3 Φ cos 2 θ ε d k o n o sin ( ε d k o n o L cos θ Φ ) .
I 7 = L L exp ( i τ z z ) d z
τ = k o n o r .
I 7 = 2 sin ( k o n o L cos θ ) k o n o cos θ .
I 8 = L L ( z z ) exp ( i τ z z ) d z = 2 r sin ( k o n o L cos θ ) k o n o .
I 9 = L L ( z z ) 2 exp ( i τ z z ) d z = 2 r 2 cos θ sin ( k o n o L cos θ ) k o n o .

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