Abstract

The reflection and transmission of full-vector X waves normally incident on planar half-spaces and slabs are studied. For this purpose, X waves are expanded in terms of weighted vector Bessel beams; this new decomposition and reconstruction method offers a more lucid and intuitive interpretation of the physical phenomena observed upon the reflection or transmission of X waves when compared to the conventional plane-wave decomposition technique. Using the Bessel beam expansion approach, we have characterized changes in the field shape and the intensity distribution of the transmitted and reflected full-vector X waves. We have also identified a novel longitudinal shift, which is observed when a full-vector X wave is transmitted through a dielectric slab under frustrated total reflection condition. The results of our studies presented here are valuable in understanding the behavior of full-vector X waves when they are utilized in practical applications in electromagnetics, optics, and photonics, such as trap and tweezer setups, optical lithography, and immaterial probing.

© 2012 Optical Society of America

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    [CrossRef]
  4. A. M. Shaarawi and I. M. Besieris, “Superluminal tunneling of an electromagnetic X wave through a planar slab,” Phys. Rev. E 62, 7415–7421 (2000).
    [CrossRef]
  5. A. M. Shaarawi and I. M. Besieris, “Ultra-fast multiple tunnelling of electromagnetic X-waves,” J. Phys. A 33, 8559–8576 (2000).
    [CrossRef]
  6. A. M. Shaarawi, B. H. Tawfik, and I. M. Besieris, “Superluminal advanced transmission of X waves undergoing frustrated total internal reflection: the evanescent fields and the Goos–Hänchen effect,” Phys. Rev. E 66, 046626 (2002).
    [CrossRef]
  7. A. M. Shaarawi, I. M. Besieris, A. M. Attiya, and E. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air–dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–249 (2001).
    [CrossRef]
  8. A. M. Attiya, E. A. El-Diwany, and A. M. Shaarwai, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” in 17th National Radio Science Conference, NRSC’2000 (IEEE, 2000), pp. B11.1–B11.12.
  9. M. A. Salem and H. Bağcı, “Energy flow characteristics of vector X-waves,” Opt. Express 19, 8526–8532 (2011).
    [CrossRef]
  10. B. Zhang and B.-I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. 103, 243901 (2009).
    [CrossRef]
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  14. L. A. Ambrosio and H. E. Hernández-Figueroa, “Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime,” Opt. Express 18, 24287–24292 (2010).
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  15. Z. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
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  16. T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
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    [CrossRef]
  18. F. I. Fedorov, “On the theory of total reflection,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955).
  19. C. Imbert, “Calculation and experimental proof of transverse shift by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
    [CrossRef]
  20. E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell equations,” Physica A 252, 586–610 (1998).
    [CrossRef]
  21. M. Zamboni-Rached, E. Recami, and F. Fontana, “Superluminal localized solutions to maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
    [CrossRef]
  22. M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
    [CrossRef]
  23. M. Zamboni-Rached, F. Fontana, and E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
    [CrossRef]
  24. A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
    [CrossRef]
  25. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Sixth Edition: A Comprehensive Guide (Academic, 2005).
  26. S. Ruschin and A. Leizer, “Evanescent Bessel beams,” J. Opt. Soc. Am. A 15, 1139–1143 (1998).
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  27. W. B. Williams and J. B. Pendry, “Generating Bessel beams by use of localized modes,” J. Opt. Soc. Am. A 22, 992–997 (2005).
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  28. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006).
    [CrossRef]
  29. J. J. Miret and C. J. Zapata-Rodríguez, “Diffraction-free propagation of subwavelength light beams in layered media,” J. Opt. Soc. Am. B 27, 1435–1445 (2010).
    [CrossRef]
  30. G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Generation of enhanced evanescent Bessel beam using band-edge resonance,” J. Appl. Phys. 108, 074304 (2010).
    [CrossRef]
  31. G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Evanescent Bessel beam generation through filtering highly focused cylindrical vector beams with a defect mode one-dimensional photonic crystal,” Opt. Commun. 283, 2272–2276 (2010).
    [CrossRef]
  32. A. V. Novitsky and L. M. Barkovsky, “Total internal reflection of vector Bessel beams: Imbert–Fedorov shift and intensity transformation,” J. Opt. A 10, 075006 (2008).
    [CrossRef]
  33. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
    [CrossRef]
  34. P. Einziger and L. Felsen, “Evanescent waves and complex rays,” IEEE Trans. Antennas Propag. 30, 594–605 (1982).
    [CrossRef]
  35. M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express 18, 25482–25493 (2010).
    [CrossRef]
  36. M. A. Salem and H. Bağcı, “Enhancing propagation characteristics of truncated localized waves in silica,” in 2011 IEEE International Symposium on Antennas and Propagation (APSURSI) (IEEE, 2011), pp. 500–503.
  37. M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
    [CrossRef]
  38. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).
  39. A. V. Novitsky and D. V. Novitsky, “Change of the size of vector Bessel beam rings under reflection,” Opt. Commun. 281, 2727–2734 (2008).
    [CrossRef]
  40. P. Hillion, “Electromagnetic tunneling Bessel beams,” Opt. Commun. 153, 199–201 (1998).
    [CrossRef]
  41. D. Mugnai, “Passage of a Bessel beam through a classically forbidden region,” Opt. Commun. 188, 17–24 (2001).
    [CrossRef]
  42. D. Mugnai, “Bessel beam through a dielectric slab at oblique incidence: the case of total reflection,” Opt. Commun. 207, 95–99 (2002).
    [CrossRef]

2011 (3)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Z. Zheng, B.-F. Zhang, H. Chen, J. Ding, and H.-T. Wang, “Optical trapping with focused Airy beams,” Appl. Opt. 50, 43–49 (2011).
[CrossRef]

M. A. Salem and H. Bağcı, “Energy flow characteristics of vector X-waves,” Opt. Express 19, 8526–8532 (2011).
[CrossRef]

2010 (7)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4, 103–106 (2010).
[CrossRef]

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[CrossRef]

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Generation of enhanced evanescent Bessel beam using band-edge resonance,” J. Appl. Phys. 108, 074304 (2010).
[CrossRef]

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Evanescent Bessel beam generation through filtering highly focused cylindrical vector beams with a defect mode one-dimensional photonic crystal,” Opt. Commun. 283, 2272–2276 (2010).
[CrossRef]

J. J. Miret and C. J. Zapata-Rodríguez, “Diffraction-free propagation of subwavelength light beams in layered media,” J. Opt. Soc. Am. B 27, 1435–1445 (2010).
[CrossRef]

L. A. Ambrosio and H. E. Hernández-Figueroa, “Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime,” Opt. Express 18, 24287–24292 (2010).
[CrossRef]

M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express 18, 25482–25493 (2010).
[CrossRef]

2009 (1)

B. Zhang and B.-I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. 103, 243901 (2009).
[CrossRef]

2008 (2)

A. V. Novitsky and L. M. Barkovsky, “Total internal reflection of vector Bessel beams: Imbert–Fedorov shift and intensity transformation,” J. Opt. A 10, 075006 (2008).
[CrossRef]

A. V. Novitsky and D. V. Novitsky, “Change of the size of vector Bessel beam rings under reflection,” Opt. Commun. 281, 2727–2734 (2008).
[CrossRef]

2006 (1)

2005 (1)

2004 (1)

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[CrossRef]

2003 (2)

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[CrossRef]

M. Zamboni-Rached, F. Fontana, and E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
[CrossRef]

2002 (3)

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

A. M. Shaarawi, B. H. Tawfik, and I. M. Besieris, “Superluminal advanced transmission of X waves undergoing frustrated total internal reflection: the evanescent fields and the Goos–Hänchen effect,” Phys. Rev. E 66, 046626 (2002).
[CrossRef]

D. Mugnai, “Bessel beam through a dielectric slab at oblique incidence: the case of total reflection,” Opt. Commun. 207, 95–99 (2002).
[CrossRef]

2001 (4)

D. Mugnai, “Passage of a Bessel beam through a classically forbidden region,” Opt. Commun. 188, 17–24 (2001).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, and E. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air–dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–249 (2001).
[CrossRef]

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,” Prog. Electromagn. Res. 30, 191–211 (2001).
[CrossRef]

M. Zamboni-Rached, E. Recami, and F. Fontana, “Superluminal localized solutions to maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
[CrossRef]

2000 (2)

A. M. Shaarawi and I. M. Besieris, “Superluminal tunneling of an electromagnetic X wave through a planar slab,” Phys. Rev. E 62, 7415–7421 (2000).
[CrossRef]

A. M. Shaarawi and I. M. Besieris, “Ultra-fast multiple tunnelling of electromagnetic X-waves,” J. Phys. A 33, 8559–8576 (2000).
[CrossRef]

1998 (3)

E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell equations,” Physica A 252, 586–610 (1998).
[CrossRef]

S. Ruschin and A. Leizer, “Evanescent Bessel beams,” J. Opt. Soc. Am. A 15, 1139–1143 (1998).
[CrossRef]

P. Hillion, “Electromagnetic tunneling Bessel beams,” Opt. Commun. 153, 199–201 (1998).
[CrossRef]

1992 (1)

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[CrossRef]

1982 (1)

P. Einziger and L. Felsen, “Evanescent waves and complex rays,” IEEE Trans. Antennas Propag. 30, 594–605 (1982).
[CrossRef]

1972 (1)

C. Imbert, “Calculation and experimental proof of transverse shift by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

1955 (1)

F. I. Fedorov, “On the theory of total reflection,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955).

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

Ambrosio, L. A.

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Sixth Edition: A Comprehensive Guide (Academic, 2005).

Attiya, A. M.

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, and E. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air–dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–249 (2001).
[CrossRef]

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,” Prog. Electromagn. Res. 30, 191–211 (2001).
[CrossRef]

A. M. Attiya, E. A. El-Diwany, and A. M. Shaarwai, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” in 17th National Radio Science Conference, NRSC’2000 (IEEE, 2000), pp. B11.1–B11.12.

Bagci, H.

M. A. Salem and H. Bağcı, “Energy flow characteristics of vector X-waves,” Opt. Express 19, 8526–8532 (2011).
[CrossRef]

M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express 18, 25482–25493 (2010).
[CrossRef]

M. A. Salem and H. Bağcı, “Enhancing propagation characteristics of truncated localized waves in silica,” in 2011 IEEE International Symposium on Antennas and Propagation (APSURSI) (IEEE, 2011), pp. 500–503.

M. A. Salem and H. Bağcı, “Detecting electromagnetic cloaks using backward-propagating waves,” in Proceedings of the XXXth URSI General Assembly (IEEE, 2011), pp. 1–3.

Barkovsky, L. M.

A. V. Novitsky and L. M. Barkovsky, “Total internal reflection of vector Bessel beams: Imbert–Fedorov shift and intensity transformation,” J. Opt. A 10, 075006 (2008).
[CrossRef]

Besieris, I. M.

A. M. Shaarawi, B. H. Tawfik, and I. M. Besieris, “Superluminal advanced transmission of X waves undergoing frustrated total internal reflection: the evanescent fields and the Goos–Hänchen effect,” Phys. Rev. E 66, 046626 (2002).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, and E. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air–dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–249 (2001).
[CrossRef]

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,” Prog. Electromagn. Res. 30, 191–211 (2001).
[CrossRef]

A. M. Shaarawi and I. M. Besieris, “Superluminal tunneling of an electromagnetic X wave through a planar slab,” Phys. Rev. E 62, 7415–7421 (2000).
[CrossRef]

A. M. Shaarawi and I. M. Besieris, “Ultra-fast multiple tunnelling of electromagnetic X-waves,” J. Phys. A 33, 8559–8576 (2000).
[CrossRef]

Betzig, E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Chen, H.

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4, 103–106 (2010).
[CrossRef]

Christodoulides, D. N.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4, 103–106 (2010).
[CrossRef]

Ciattoni, A.

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[CrossRef]

Conti, C.

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[CrossRef]

Davidson, M. W.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett. 7, 684–685 (1971).
[CrossRef]

Ding, J.

Einziger, P.

P. Einziger and L. Felsen, “Evanescent waves and complex rays,” IEEE Trans. Antennas Propag. 30, 594–605 (1982).
[CrossRef]

El-Diwany, E.

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, and E. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air–dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–249 (2001).
[CrossRef]

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,” Prog. Electromagn. Res. 30, 191–211 (2001).
[CrossRef]

El-Diwany, E. A.

A. M. Attiya, E. A. El-Diwany, and A. M. Shaarwai, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” in 17th National Radio Science Conference, NRSC’2000 (IEEE, 2000), pp. B11.1–B11.12.

Fahrbach, F. O.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[CrossRef]

Fedorov, F. I.

F. I. Fedorov, “On the theory of total reflection,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955).

Felsen, L.

P. Einziger and L. Felsen, “Evanescent waves and complex rays,” IEEE Trans. Antennas Propag. 30, 594–605 (1982).
[CrossRef]

Fontana, F.

M. Zamboni-Rached, F. Fontana, and E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
[CrossRef]

M. Zamboni-Rached, E. Recami, and F. Fontana, “Superluminal localized solutions to maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
[CrossRef]

Galbraith, C. G.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Galbraith, J. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Gao, L.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Greenleaf, J. F.

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[CrossRef]

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Hernández-Figueroa, H. E.

L. A. Ambrosio and H. E. Hernández-Figueroa, “Gradient forces on double-negative particles in optical tweezers using Bessel beams in the ray optics regime,” Opt. Express 18, 24287–24292 (2010).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

Hillion, P.

P. Hillion, “Electromagnetic tunneling Bessel beams,” Opt. Commun. 153, 199–201 (1998).
[CrossRef]

Imbert, C.

C. Imbert, “Calculation and experimental proof of transverse shift by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[CrossRef]

Leizer, A.

Lu, J.-Y.

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves—exact solutions to free-space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[CrossRef]

Lu, Y.

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Evanescent Bessel beam generation through filtering highly focused cylindrical vector beams with a defect mode one-dimensional photonic crystal,” Opt. Commun. 283, 2272–2276 (2010).
[CrossRef]

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Generation of enhanced evanescent Bessel beam using band-edge resonance,” J. Appl. Phys. 108, 074304 (2010).
[CrossRef]

Milkie, D. E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Ming, H.

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Evanescent Bessel beam generation through filtering highly focused cylindrical vector beams with a defect mode one-dimensional photonic crystal,” Opt. Commun. 283, 2272–2276 (2010).
[CrossRef]

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Generation of enhanced evanescent Bessel beam using band-edge resonance,” J. Appl. Phys. 108, 074304 (2010).
[CrossRef]

Miret, J. J.

Mugnai, D.

D. Mugnai, “Bessel beam through a dielectric slab at oblique incidence: the case of total reflection,” Opt. Commun. 207, 95–99 (2002).
[CrossRef]

D. Mugnai, “Passage of a Bessel beam through a classically forbidden region,” Opt. Commun. 188, 17–24 (2001).
[CrossRef]

Nóbrega, K. Z.

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

Novitsky, A. V.

A. V. Novitsky and D. V. Novitsky, “Change of the size of vector Bessel beam rings under reflection,” Opt. Commun. 281, 2727–2734 (2008).
[CrossRef]

A. V. Novitsky and L. M. Barkovsky, “Total internal reflection of vector Bessel beams: Imbert–Fedorov shift and intensity transformation,” J. Opt. A 10, 075006 (2008).
[CrossRef]

Novitsky, D. V.

A. V. Novitsky and D. V. Novitsky, “Change of the size of vector Bessel beam rings under reflection,” Opt. Commun. 281, 2727–2734 (2008).
[CrossRef]

Pendry, J. B.

Planchon, T. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Porto, P. D.

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[CrossRef]

Recami, E.

M. Zamboni-Rached, F. Fontana, and E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

M. Zamboni-Rached, E. Recami, and F. Fontana, “Superluminal localized solutions to maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
[CrossRef]

E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell equations,” Physica A 252, 586–610 (1998).
[CrossRef]

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4, 103–106 (2010).
[CrossRef]

Rohrbach, A.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[CrossRef]

Rui, G.

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Generation of enhanced evanescent Bessel beam using band-edge resonance,” J. Appl. Phys. 108, 074304 (2010).
[CrossRef]

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Evanescent Bessel beam generation through filtering highly focused cylindrical vector beams with a defect mode one-dimensional photonic crystal,” Opt. Commun. 283, 2272–2276 (2010).
[CrossRef]

Ruschin, S.

Salem, M. A.

M. A. Salem and H. Bağcı, “Energy flow characteristics of vector X-waves,” Opt. Express 19, 8526–8532 (2011).
[CrossRef]

M. A. Salem and H. Bağcı, “On the propagation of truncated localized waves in dispersive silica,” Opt. Express 18, 25482–25493 (2010).
[CrossRef]

M. A. Salem and H. Bağcı, “Detecting electromagnetic cloaks using backward-propagating waves,” in Proceedings of the XXXth URSI General Assembly (IEEE, 2011), pp. 1–3.

M. A. Salem and H. Bağcı, “Enhancing propagation characteristics of truncated localized waves in silica,” in 2011 IEEE International Symposium on Antennas and Propagation (APSURSI) (IEEE, 2011), pp. 500–503.

Shaarawi, A. M.

A. M. Shaarawi, B. H. Tawfik, and I. M. Besieris, “Superluminal advanced transmission of X waves undergoing frustrated total internal reflection: the evanescent fields and the Goos–Hänchen effect,” Phys. Rev. E 66, 046626 (2002).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, and E. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air–dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–249 (2001).
[CrossRef]

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,” Prog. Electromagn. Res. 30, 191–211 (2001).
[CrossRef]

A. M. Shaarawi and I. M. Besieris, “Superluminal tunneling of an electromagnetic X wave through a planar slab,” Phys. Rev. E 62, 7415–7421 (2000).
[CrossRef]

A. M. Shaarawi and I. M. Besieris, “Ultra-fast multiple tunnelling of electromagnetic X-waves,” J. Phys. A 33, 8559–8576 (2000).
[CrossRef]

Shaarwai, A. M.

A. M. Attiya, E. A. El-Diwany, and A. M. Shaarwai, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” in 17th National Radio Science Conference, NRSC’2000 (IEEE, 2000), pp. B11.1–B11.12.

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[CrossRef]

Tawfik, B. H.

A. M. Shaarawi, B. H. Tawfik, and I. M. Besieris, “Superluminal advanced transmission of X waves undergoing frustrated total internal reflection: the evanescent fields and the Goos–Hänchen effect,” Phys. Rev. E 66, 046626 (2002).
[CrossRef]

Wang, H.-T.

Wang, P.

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Generation of enhanced evanescent Bessel beam using band-edge resonance,” J. Appl. Phys. 108, 074304 (2010).
[CrossRef]

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Evanescent Bessel beam generation through filtering highly focused cylindrical vector beams with a defect mode one-dimensional photonic crystal,” Opt. Commun. 283, 2272–2276 (2010).
[CrossRef]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Sixth Edition: A Comprehensive Guide (Academic, 2005).

Williams, W. B.

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4, 103–106 (2010).
[CrossRef]

Wu, B.-I.

B. Zhang and B.-I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. 103, 243901 (2009).
[CrossRef]

Zamboni-Rached, M.

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[CrossRef]

M. Zamboni-Rached, F. Fontana, and E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

M. Zamboni-Rached, E. Recami, and F. Fontana, “Superluminal localized solutions to maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
[CrossRef]

Zapata-Rodríguez, C. J.

Zhan, Q.

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Evanescent Bessel beam generation through filtering highly focused cylindrical vector beams with a defect mode one-dimensional photonic crystal,” Opt. Commun. 283, 2272–2276 (2010).
[CrossRef]

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Generation of enhanced evanescent Bessel beam using band-edge resonance,” J. Appl. Phys. 108, 074304 (2010).
[CrossRef]

Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006).
[CrossRef]

Zhang, B.

B. Zhang and B.-I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. 103, 243901 (2009).
[CrossRef]

Zhang, B.-F.

Zheng, Z.

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[CrossRef]

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G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Generation of enhanced evanescent Bessel beam using band-edge resonance,” J. Appl. Phys. 108, 074304 (2010).
[CrossRef]

J. Opt. A (1)

A. V. Novitsky and L. M. Barkovsky, “Total internal reflection of vector Bessel beams: Imbert–Fedorov shift and intensity transformation,” J. Opt. A 10, 075006 (2008).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

J. Phys. A (1)

A. M. Shaarawi and I. M. Besieris, “Ultra-fast multiple tunnelling of electromagnetic X-waves,” J. Phys. A 33, 8559–8576 (2000).
[CrossRef]

Nat. Methods (1)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8, 417–423 (2011).
[CrossRef]

Nat. Photon. (2)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4, 103–106 (2010).
[CrossRef]

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon. 4, 780–785 (2010).
[CrossRef]

Opt. Commun. (6)

G. Rui, Y. Lu, P. Wang, H. Ming, and Q. Zhan, “Evanescent Bessel beam generation through filtering highly focused cylindrical vector beams with a defect mode one-dimensional photonic crystal,” Opt. Commun. 283, 2272–2276 (2010).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[CrossRef]

A. V. Novitsky and D. V. Novitsky, “Change of the size of vector Bessel beam rings under reflection,” Opt. Commun. 281, 2727–2734 (2008).
[CrossRef]

P. Hillion, “Electromagnetic tunneling Bessel beams,” Opt. Commun. 153, 199–201 (1998).
[CrossRef]

D. Mugnai, “Passage of a Bessel beam through a classically forbidden region,” Opt. Commun. 188, 17–24 (2001).
[CrossRef]

D. Mugnai, “Bessel beam through a dielectric slab at oblique incidence: the case of total reflection,” Opt. Commun. 207, 95–99 (2002).
[CrossRef]

Opt. Express (3)

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[CrossRef]

Phys. Rev. E (6)

M. Zamboni-Rached, E. Recami, and F. Fontana, “Superluminal localized solutions to maxwell equations propagating along a normal-sized waveguide,” Phys. Rev. E 64, 066603 (2001).
[CrossRef]

M. Zamboni-Rached, K. Z. Nóbrega, E. Recami, and H. E. Hernández-Figueroa, “Superluminal X-shaped beams propagating without distortion along a coaxial guide,” Phys. Rev. E 66, 046617 (2002).
[CrossRef]

M. Zamboni-Rached, F. Fontana, and E. Recami, “Superluminal localized solutions to Maxwell equations propagating along a waveguide: the finite-energy case,” Phys. Rev. E 67, 036620 (2003).
[CrossRef]

A. Ciattoni, C. Conti, and P. D. Porto, “Vector electromagnetic X waves,” Phys. Rev. E 69, 036608 (2004).
[CrossRef]

A. M. Shaarawi, B. H. Tawfik, and I. M. Besieris, “Superluminal advanced transmission of X waves undergoing frustrated total internal reflection: the evanescent fields and the Goos–Hänchen effect,” Phys. Rev. E 66, 046626 (2002).
[CrossRef]

A. M. Shaarawi and I. M. Besieris, “Superluminal tunneling of an electromagnetic X wave through a planar slab,” Phys. Rev. E 62, 7415–7421 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

B. Zhang and B.-I. Wu, “Electromagnetic detection of a perfect invisibility cloak,” Phys. Rev. Lett. 103, 243901 (2009).
[CrossRef]

Physica A (1)

E. Recami, “On localized “X-shaped” superluminal solutions to Maxwell equations,” Physica A 252, 586–610 (1998).
[CrossRef]

Prog. Electromagn. Res. (2)

A. M. Attiya, E. El-Diwany, A. M. Shaarawi, and I. M. Besieris, “Reflection and transmission of X-waves in the presence of planarly layered media: the pulsed plane wave representation,” Prog. Electromagn. Res. 30, 191–211 (2001).
[CrossRef]

A. M. Shaarawi, I. M. Besieris, A. M. Attiya, and E. El-Diwany, “Reflection and transmission of an electromagnetic X-wave incident on a planar air–dielectric interface: spectral analysis,” Prog. Electromagn. Res. 30, 213–249 (2001).
[CrossRef]

Other (6)

A. M. Attiya, E. A. El-Diwany, and A. M. Shaarwai, “Transmission and reflection of TE electromagnetic X-wave normally incident on a lossy dispersive half-space,” in 17th National Radio Science Conference, NRSC’2000 (IEEE, 2000), pp. B11.1–B11.12.

H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves (Wiley, 2008).

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Sixth Edition: A Comprehensive Guide (Academic, 2005).

M. A. Salem and H. Bağcı, “Detecting electromagnetic cloaks using backward-propagating waves,” in Proceedings of the XXXth URSI General Assembly (IEEE, 2011), pp. 1–3.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

M. A. Salem and H. Bağcı, “Enhancing propagation characteristics of truncated localized waves in silica,” in 2011 IEEE International Symposium on Antennas and Propagation (APSURSI) (IEEE, 2011), pp. 500–503.

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Figures (11)

Fig. 1.
Fig. 1.

The reflection and transmission of | E ϕ | of a zero-order ordinary full-vector X wave with azimuthal dependence n = 1 and parameters α = 2 × 1 0 13 s , q = 0.8 , A e = 1 / ϵ 0 , and A h = i / μ 0 normally incident from free space on a planar interface of a dielectric half-space with the constant medium parameters ϵ 2 = 3.9 and μ 2 = 1 , at different time instances, where its centroid is (a)  0.5 ps away from the interface (incident), (b) just at the interface, and (c)  0.5 ps away from the interface (reflected). Note that the ζ scale is different for each medium (difference not shown in figure).

Fig. 2.
Fig. 2.

Amplitudes of the TE and TM components of the incident E ϕ at the centroid of a zero-order ordinary full-vector X wave with azimuthal dependence n = 1 and parameters α = 2 × 1 0 13 s , q = 0.8 , A e = 1 / ϵ 0 , and A h = i / μ 0 .

Fig. 3.
Fig. 3.

Amplitudes of the TE and TM components of the incident E ϕ in the ρ ζ plane of a zero-order ordinary full-vector X wave with azimuthal dependence n = 1 and parameters α = 2 × 1 0 13 s , q = 0.8 , A e = 1 / ϵ 0 , and A h = i / μ 0 .

Fig. 4.
Fig. 4.

Reflection and transmission of | E ϕ | of a zero-order ordinary full-vector X wave with azimuthal dependence n = 1 and parameters α = 2 × 1 0 13 s , q = 0.8 , A e = 1 / ϵ 0 , and A h = i / μ 0 , normally incident from a homogeneous medium with the constant characteristics ϵ 1 = 3.9 and μ 1 = 1 onto a planar interface of free space, at different time instances, where its centroid is (a)  0.5 ps away from the interface (incident), (b) just at the interface, and (c)  0.5 ps away from the interface (reflected). Note that the ζ scale is different for each medium (difference not shown in figure).

Fig. 5.
Fig. 5.

Amplitudes of the incident, the reflected, and the evanescent transmitted E z at the centroid of a zero-order ordinary full-vector X wave with azimuthal dependence n = 1 and parameters α = 2 × 1 0 13 s , q = 0.8 , A e = 1 / ϵ 0 , and A h = i / μ 0 at the interface between the two media M 1 and M 2 .

Fig. 6.
Fig. 6.

Reflection and transmission of | E ϕ | of an ordinary full-vector X wave of order m = 28 and with azimuthal dependence n = 1 , and parameters α = 1 0 14 s , q = 0.8 , A e = 1 / ϵ 0 , and A h = i / μ 0 , normally incident from free space onto a planar interface of a homogeneous fused silica half-space, at different time instances, where its centroid is (a)  10 fs away from the interface (incident), (b) just at the interface, and (c)  10 fs away from the interface (reflected).

Fig. 7.
Fig. 7.

Plot of (a) the reflected and (b) the transmitted | E ϕ | of the full-vector X wave depicted in Fig. 6 at the time instance t = 1 ps .

Fig. 8.
Fig. 8.

Reflection and transmission of | E ϕ | of a zero-order ordinary full-vector X wave with azimuthal dependence n = 1 and parameters α = 2 × 1 0 13 s , q = 0.8 , A e = 1 / ϵ 0 , and A h = i / μ 0 , normally incident from free space on a planar dielectric slab with the constant medium parameters ϵ 2 = 3.9 and μ 2 = 1 and of thickness L = 0.1 mm at different time instances, where its centroid is (a)  5 ps away from the interface (incident), (b) just at the interface, and (c)  5 ps away from the interface (reflected).

Fig. 9.
Fig. 9.

Plot of (a) the multiple reflected and (b) the multiple transmitted | E ϕ | of the full-vector X wave depicted in Fig. 8 at the time instance t = 5 ps .

Fig. 10.
Fig. 10.

Reflection and transmission of | E ϕ | of a zero-order ordinary full-vector X wave with azimuthal dependence n = 1 and parameters α = 2 × 1 0 13 s , q = 0.8 , A e = 1 / ϵ 0 , and A h = i / μ 0 , normally incident from a homogeneous medium with the constant characteristics ϵ 1 = 3.9 and μ 1 through free-space gap of thickness L = 0.1 mm at different time instances, where its centroid is (a)  5 ps away from the interface (incident), (b) just at the interface, and (c)  5 ps away from the interface (reflected).

Fig. 11.
Fig. 11.

Plot of the TE- and TM-polarized components and the full vector of (a) the incident, (b) the reflected, and (c) the transmitted | E ϕ | of the full-vector X wave depicted in Fig. 10 at ρ = 0 ; (a), (b)  z = 0 and (c)  z = L at different time instances showing the advanced reflection and transmission in addition to the shift in the z direction.

Equations (44)

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Ψ ˜ n ( k ρ , k z , ω ) = Ω ( ω ) e i n ϕ δ ( k z [ ω V ] ) δ ( k ρ 2 [ ω 2 c 2 k z 2 ] ) ,
Ψ ( ρ , ϕ , z , t ) = 0 d k z d ω 0 d k ρ Ψ ˜ n ( k ρ , k z , ω ) J n ( k ρ ρ ) e i ( k z z ω t ) ,
Ψ ( ρ , ϕ , ζ ) = e i n ϕ 0 d ω Ω ( ω ) J n ( ω ρ 1 c 2 1 V 2 ) e i ω ζ ,
E = ( · Π e ) 1 c 2 2 t 2 Π e μ 0 × ( t Π h ) ,
H = ϵ 0 × ( t Π e ) + ( · Π h ) 1 c 2 2 t 2 Π h .
( E ( ρ , ϕ , z , t ) H ( ρ , ϕ , z , t ) ) = ( E ( ρ ) H ( ρ ) ) e i n ϕ e i ( k z z ω t ) .
( d 2 d ρ 2 + n ρ d d ρ + [ k ρ 2 ( n ρ ) 2 ] ) ( E z ( ρ ) H z ( ρ ) ) = 0.
E z = A ˜ h J n ( k ρ ρ ) e i n ϕ e i ( k z z ω t ) , H z = A ˜ e J n ( k ρ ρ ) e i n ϕ e i ( k z z ω t ) .
E ρ = e i n ϕ k ρ 2 e i ( k z z ω t ) ( n μ k ρ A e ˜ + i k z A h ˜ ρ ) J n ( k ρ ρ ) , E ϕ = e i n ϕ k ρ 2 e i ( k z z ω t ) ( n k z ρ A h ˜ i μ k A e ˜ ρ ) J n ( k ρ ρ ) , H ρ = e i n ϕ k ρ 2 e i ( k z z ω t ) ( n ϵ k ρ A h ˜ + i k z A e ˜ ρ ) J n ( k ρ ρ ) , H ϕ = e i n ϕ k ρ 2 e i ( k z z ω t ) ( n k z ρ A e ˜ + i ϵ k A h ˜ ρ ) J n ( k ρ ρ ) ,
z ^ × E i + z ^ × E r = z ^ × E t ,
z ^ × H i + z ^ × H r = z ^ × H t ,
γ e = A e ˜ r / A e ˜ , γ h = A h ˜ r / A h ˜ ,
τ e = A e ˜ t / A e ˜ , τ h = A h ˜ t / A h ˜ ,
μ 1 ( 1 + γ e ) = μ 2 τ e , k z 1 ( 1 γ e ) = k z 2 τ e , ϵ 1 ( 1 + γ h ) = ϵ 2 τ h , k z 1 ( 1 γ h ) = k z 2 τ h .
γ e = μ 2 k z 1 μ 1 k z 2 μ 2 k z 1 + μ 1 k z 2 , γ h = ϵ 2 k z 1 ϵ 1 k z 2 ϵ 2 k z 1 + ϵ 1 k z 2 ,
τ e = 2 μ 1 k z 1 μ 2 k z 1 + μ 1 k z 2 , τ h = 2 ϵ 1 k z 1 ϵ 2 k z 1 + ϵ 1 k z 2 .
γ e = μ 2 q μ 1 q 2 + n 2 2 n 1 2 μ 2 q + μ 1 q 2 + n 2 2 n 1 2 , γ h = ϵ 2 q ϵ 1 q 2 + n 2 2 n 1 2 ϵ 2 q + ϵ 1 q 2 + n 2 2 n 1 2 ,
τ e = 2 μ 1 q μ 2 q + μ 1 q 2 + n 2 2 n 1 2 , τ h = 2 ϵ 1 q ϵ 2 q + ϵ 1 q 2 + n 2 2 n 1 2 .
Ψ ( ρ , ϕ , ζ ) = 2 e i ϕ 1 + η 1 λ ρ 1 + η ,
γ e = μ 2 q i μ 1 n 1 2 ( q 2 + n 2 2 ) μ 2 q + i μ 1 n 1 2 ( q 2 + n 2 2 ) , γ h = ϵ 2 q i ϵ 1 n 1 2 ( q 2 + n 2 2 ) ϵ 2 q + i ϵ 1 n 1 2 ( q 2 + n 2 2 ) ,
τ e = 2 μ 1 q μ 2 q + i μ 1 n 1 2 ( q 2 + n 2 2 ) , τ h = 2 ϵ 1 q ϵ 2 q + i ϵ 1 n 1 2 ( q 2 + n 2 2 ) .
g e = 2 arctan ( μ 1 n 1 2 ( q 2 + n 2 2 ) μ 2 q ) , g h = 2 arctan ( ϵ 1 n 1 2 ( q 2 + n 2 2 ) ϵ 2 q ) .
Ψ ( ρ , ϕ , ζ ) = e i n ϕ 0 d ω Ω ( ω ) J n ( n ( ω ) ω ρ 1 c 2 1 V 2 ) e i ω ζ ,
z ^ × E i + z ^ × E r = z ^ × E + + z ^ × E ,
z ^ × H i + z ^ × H r = z ^ × H + + z ^ × H ,
z ^ × E + + z ^ × E = z ^ × E t ,
z ^ × H + + z ^ × H = z ^ × H t .
γ ˜ e = A e ˜ r / A e ˜ , γ ˜ h = A h ˜ r / A h ˜ ,
a e = A e ˜ + / A e ˜ , a h = A h ˜ + / A h ˜ ,
b e = A e ˜ / A e ˜ , b h = A h ˜ / A h ˜ ,
τ ˜ e = A e ˜ t / A e ˜ , τ ˜ h = A h ˜ t / A h ˜ ,
[ 1 γ ˜ e / h ] = 1 τ e / h [ 1 γ e / h γ e / h 1 ] [ a e / h b e / h ] , [ e i k z 2 L 0 0 e i k z 2 L ] [ a e / h b e / h ] = 1 1 γ e / h [ 1 γ e / h γ e / h 1 ] [ τ e / h 0 ] .
γ ˜ e ( ω ) = Γ e ( ω ) [ 1 e 2 i ω c q 2 + n 2 2 n 1 2 L ] ( μ 1 2 [ q 2 + n 2 2 n 1 2 ] μ 2 q 2 ) , γ ˜ h ( ω ) = Γ h ( ω ) [ 1 e 2 i ω c q 2 + n 2 2 n 1 2 L ] ( ϵ 1 2 [ q 2 + n 2 2 n 1 2 ] ϵ 2 2 q 2 ) ,
a e ( ω ) = 2 Γ e ( ω ) μ 1 q ( μ 2 q + μ 1 q 2 + n 2 2 n 1 2 ) , a h ( ω ) = 2 Γ h ( ω ) ϵ 1 q ( ϵ 2 q + ϵ 1 q 2 + n 2 2 n 1 2 ) ,
b e ( ω ) = e 2 i ω c q 2 + n 2 2 n 1 2 L a e ( ω ) , b h ( ω ) = e 2 i ω c q 2 + n 2 2 n 1 2 L a h ( ω ) ,
τ ˜ e ( ω ) = 4 Γ e ( ω ) e i ω c ( q q 2 + n 2 2 n 1 2 ) L μ 1 μ 2 q q 2 + n 2 2 n 1 2 , τ ˜ h ( ω ) = 4 Γ h ( ω ) e i ω c ( q q 2 + n 2 2 n 1 2 ) L ϵ 1 ϵ 2 q q 2 + n 2 2 n 1 2 ,
Γ e ( ω ) = [ e 2 i ( ω / c ) q 2 + n 2 2 n 1 2 L ( μ 1 q 2 + n 2 2 n 1 2 μ 2 q ) 2 ( μ 1 q 2 + n 2 2 n 1 2 + μ 2 q ) 2 ] 1 ,
Γ h ( ω ) = [ e 2 i ( ω / c ) q 2 + n 2 2 n 1 2 L ( ϵ 1 q 2 + n 2 2 n 1 2 ϵ 2 q ) 2 ( ϵ 1 q 2 + n 2 2 n 1 2 + ϵ 2 q ) 2 ] 1 ,
g ˜ e / h = π arctan ( P e / h Q e / h coth ( ω | V 2 | L ) ) ,
t ˜ e / h = ω V 1 L + arctan ( Q e / h P e / h tanh ( ω | V 2 | L ) ) ,
D e / h g = g ˜ e / h ω ,
D e / h t = t ˜ e / h ω L V 1 ,
D e / h g = L | V 2 | P e / h Q e / h sinh 2 ( ω | V 2 | L ) { Q e / h 2 + P e / h 2 coth 2 ( ω | V 2 | L ) } ,
D e / h t = L | V 2 | P e / h Q e / h cosh 2 ( ω | V 2 | L ) { P e / h 2 + Q e / h 2 tanh 2 ( ω | V 2 | L ) } .

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