Abstract

We investigate the behavior of full-vector electromagnetic Bessel beams obliquely incident at an interface between two electrically different media. We employ a Fourier transform domain representation of Bessel beams to determine their behavior upon reflection and transmission. This transform, which is geometric in nature, consists of elliptical support curves with complex weighting associated with them. The behavior of the scattered field at an interface is highly complex, owing to its full-vector nature; nevertheless, this behavior has a straightforward representation in the transform domain geometry. The analysis shows that the reflected field forms a different vector Bessel beam, but in general, the transmitted field cannot be represented as a Bessel beam. Nevertheless, using this approach, we demonstrate a method to propagate a Bessel beam in the refractive medium by launching a non-Bessel beam at the interface. Several interesting phenomena related to the behavior of Bessel beams are illustrated, such as polarized reflection at Brewster’s angle incidence, and the Goos–Hänchen and Imbert–Federov shifts in the case of total reflection.

© 2013 Optical Society of America

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  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]
  2. F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nat. Commun. 3, 632 (2012).
    [CrossRef]
  3. B. Yalizay, T. Ersoy, B. Soylu, and S. Akturk, “Fabrication of nanometer-size structures in metal thin films using femtosecond laser Bessel beams,” Appl. Phys. Lett. 100, 031104 (2012).
    [CrossRef]
  4. R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100, 061126 (2012).
    [CrossRef]
  5. H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab on Chip 12, 4816–4820 (2012).
    [CrossRef]
  6. J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
    [CrossRef]
  7. I. A. Litvin, A. Dudley, and A. Forbes, “Poynting vector and orbital angular momentum density of superpositions of Bessel beams,” Opt. Express 19, 16760 (2011).
    [CrossRef]
  8. D. Mugnai and I. Mochi, “Superluminal X-wave propagation: energy localization and velocity,” Phys. Rev. E 73, 016606 (2006).
    [CrossRef]
  9. D. Mugnai, “Propagation of Bessel beams from a dielectric to a conducting medium,” Appl. Opt. 50, 2654–2658 (2011).
    [CrossRef]
  10. D. Mugnai, “Bessel beam through a dielectric slab at oblique incidence: the case of total reflection,” Opt. Commun. 207, 95–99 (2002).
    [CrossRef]
  11. A. Aiello and J. P. Woerdman, “Goos–Hänchen and Imbert–Fedorov shifts of a nondiffracting Bessel beam,” Opt. Lett. 36, 543–545 (2011).
    [CrossRef]
  12. A. V. Novitsky and L. M. Barkovsky, “Total internal reflection of vector Bessel beams: Imbert–-Fedorov shift and intensity transformation,” J. Opt. Pure Appl. Opt. 10, 075006 (2008).
    [CrossRef]
  13. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  14. S. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
    [CrossRef]
  15. M. Salem, A. Kamel, and E. Niver, “Microwave Bessel beams generation using guided modes,” IEEE Trans. Antennas Propag. 59, 2241–2247 (2011).
    [CrossRef]
  16. M. A. Salem and H. Bağci, “Reflection and transmission of normally incident full-vector X waves on planar interfaces,” J. Opt. Soc. Am. A 29, 139–152 (2012).
    [CrossRef]
  17. E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002), pp. 589–601.

2012

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nat. Commun. 3, 632 (2012).
[CrossRef]

B. Yalizay, T. Ersoy, B. Soylu, and S. Akturk, “Fabrication of nanometer-size structures in metal thin films using femtosecond laser Bessel beams,” Appl. Phys. Lett. 100, 031104 (2012).
[CrossRef]

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100, 061126 (2012).
[CrossRef]

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab on Chip 12, 4816–4820 (2012).
[CrossRef]

M. A. Salem and H. Bağci, “Reflection and transmission of normally incident full-vector X waves on planar interfaces,” J. Opt. Soc. Am. A 29, 139–152 (2012).
[CrossRef]

2011

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]

M. Salem, A. Kamel, and E. Niver, “Microwave Bessel beams generation using guided modes,” IEEE Trans. Antennas Propag. 59, 2241–2247 (2011).
[CrossRef]

A. Aiello and J. P. Woerdman, “Goos–Hänchen and Imbert–Fedorov shifts of a nondiffracting Bessel beam,” Opt. Lett. 36, 543–545 (2011).
[CrossRef]

D. Mugnai, “Propagation of Bessel beams from a dielectric to a conducting medium,” Appl. Opt. 50, 2654–2658 (2011).
[CrossRef]

I. A. Litvin, A. Dudley, and A. Forbes, “Poynting vector and orbital angular momentum density of superpositions of Bessel beams,” Opt. Express 19, 16760 (2011).
[CrossRef]

2008

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

2006

D. Mugnai and I. Mochi, “Superluminal X-wave propagation: energy localization and velocity,” Phys. Rev. E 73, 016606 (2006).
[CrossRef]

2002

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

1993

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[CrossRef]

1991

S. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

1987

Aiello, A.

Akturk, S.

B. Yalizay, T. Ersoy, B. Soylu, and S. Akturk, “Fabrication of nanometer-size structures in metal thin films using femtosecond laser Bessel beams,” Appl. Phys. Lett. 100, 031104 (2012).
[CrossRef]

Arita, Y.

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab on Chip 12, 4816–4820 (2012).
[CrossRef]

Bagci, H.

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. Pure Appl. Opt. 10, 075006 (2008).
[CrossRef]

Bergmann, G.

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab on Chip 12, 4816–4820 (2012).
[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]

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]

Deacon, K. S.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100, 061126 (2012).
[CrossRef]

Dholakia, K.

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab on Chip 12, 4816–4820 (2012).
[CrossRef]

Dudley, A.

Durnin, J.

Ersoy, T.

B. Yalizay, T. Ersoy, B. Soylu, and S. Akturk, “Fabrication of nanometer-size structures in metal thin films using femtosecond laser Bessel beams,” Appl. Phys. Lett. 100, 031104 (2012).
[CrossRef]

Fahrbach, F. O.

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nat. Commun. 3, 632 (2012).
[CrossRef]

Forbes, A.

Friberg, A. T.

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[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]

Gunn-Moore, F. J.

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab on Chip 12, 4816–4820 (2012).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002), pp. 589–601.

Heisterkamp, A.

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab on Chip 12, 4816–4820 (2012).
[CrossRef]

Kamel, A.

M. Salem, A. Kamel, and E. Niver, “Microwave Bessel beams generation using guided modes,” IEEE Trans. Antennas Propag. 59, 2241–2247 (2011).
[CrossRef]

Litvin, I. A.

Marchington, R. F.

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab on Chip 12, 4816–4820 (2012).
[CrossRef]

Meyers, R. E.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100, 061126 (2012).
[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]

Mishra, S.

S. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

Mochi, I.

D. Mugnai and I. Mochi, “Superluminal X-wave propagation: energy localization and velocity,” Phys. Rev. E 73, 016606 (2006).
[CrossRef]

Mugnai, D.

D. Mugnai, “Propagation of Bessel beams from a dielectric to a conducting medium,” Appl. Opt. 50, 2654–2658 (2011).
[CrossRef]

D. Mugnai and I. Mochi, “Superluminal X-wave propagation: energy localization and velocity,” Phys. Rev. E 73, 016606 (2006).
[CrossRef]

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

Niver, E.

M. Salem, A. Kamel, and E. Niver, “Microwave Bessel beams generation using guided modes,” IEEE Trans. Antennas Propag. 59, 2241–2247 (2011).
[CrossRef]

Novitsky, A. V.

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

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]

Praveen, B. B.

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab on Chip 12, 4816–4820 (2012).
[CrossRef]

Rendall, H. A.

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab on Chip 12, 4816–4820 (2012).
[CrossRef]

Rohrbach, A.

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nat. Commun. 3, 632 (2012).
[CrossRef]

Salem, M.

M. Salem, A. Kamel, and E. Niver, “Microwave Bessel beams generation using guided modes,” IEEE Trans. Antennas Propag. 59, 2241–2247 (2011).
[CrossRef]

Salem, M. A.

Shih, Y.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100, 061126 (2012).
[CrossRef]

Soylu, B.

B. Yalizay, T. Ersoy, B. Soylu, and S. Akturk, “Fabrication of nanometer-size structures in metal thin films using femtosecond laser Bessel beams,” Appl. Phys. Lett. 100, 031104 (2012).
[CrossRef]

Tunick, A. D.

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100, 061126 (2012).
[CrossRef]

Turunen, J.

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[CrossRef]

Woerdman, J. P.

Yalizay, B.

B. Yalizay, T. Ersoy, B. Soylu, and S. Akturk, “Fabrication of nanometer-size structures in metal thin films using femtosecond laser Bessel beams,” Appl. Phys. Lett. 100, 031104 (2012).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

B. Yalizay, T. Ersoy, B. Soylu, and S. Akturk, “Fabrication of nanometer-size structures in metal thin films using femtosecond laser Bessel beams,” Appl. Phys. Lett. 100, 031104 (2012).
[CrossRef]

R. E. Meyers, K. S. Deacon, A. D. Tunick, and Y. Shih, “Virtual ghost imaging through turbulence and obscurants using Bessel beam illumination,” Appl. Phys. Lett. 100, 061126 (2012).
[CrossRef]

IEEE Trans. Antennas Propag.

M. Salem, A. Kamel, and E. Niver, “Microwave Bessel beams generation using guided modes,” IEEE Trans. Antennas Propag. 59, 2241–2247 (2011).
[CrossRef]

J. Opt. Pure Appl. Opt.

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

J. Opt. Soc. Am. A

Lab on Chip

H. A. Rendall, R. F. Marchington, B. B. Praveen, G. Bergmann, Y. Arita, A. Heisterkamp, F. J. Gunn-Moore, and K. Dholakia, “High-throughput optical injection of mammalian cells using a Bessel light beam,” Lab on Chip 12, 4816–4820 (2012).
[CrossRef]

Nat. Commun.

F. O. Fahrbach and A. Rohrbach, “Propagation stability of self-reconstructing Bessel beams enables contrast-enhanced imaging in thick media,” Nat. Commun. 3, 632 (2012).
[CrossRef]

Nat. Methods

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]

Opt. Commun.

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

S. Mishra, “A vector wave analysis of a Bessel beam,” Opt. Commun. 85, 159–161 (1991).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. E

D. Mugnai and I. Mochi, “Superluminal X-wave propagation: energy localization and velocity,” Phys. Rev. E 73, 016606 (2006).
[CrossRef]

Pure Appl. Opt.

J. Turunen and A. T. Friberg, “Self-imaging and propagation-invariance in electromagnetic fields,” Pure Appl. Opt. 2, 51–60 (1993).
[CrossRef]

Other

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002), pp. 589–601.

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Figures (5)

Fig. 1.
Fig. 1.

Illustration of an obliquely incident Bessel beam at different angles of incidence on a planar interface between two dielectric half-spaces (left), and the corresponding spectral support representation in the spatial Fourier domain (right), as given by Eq. (5).

Fig. 2.
Fig. 2.

Illustration of the projection of the elliptical support curves on the hemispheres of incidence, reflection, and refraction in the kxkz plane as given by Eq. (6). The incident Bessel beam has a cone angle ζ=π/6 and is incident at angle θ=π/4 onto a half-space with χ=2k. The figure shows the deformation in the support curve of the refracted field resulting in a diverging refracted beam.

Fig. 3.
Fig. 3.

Bessel beam incident at Brewster’s angle θ=π/2.7433 onto a dielectric half-space with ϵχ=3.9ϵk, μχ=μk, and Ae=iAh=1. The reflection and refraction of the Ex component in the xz plane are shown in (a), and the cross section of its reflected field is shown in (b). The reflection and refraction of the Ey component in the xz plane are shown in (c), and the cross section of its reflected field is shown in (d). The figure clearly shows that the reflected field is mostly polarized in the y direction. Note that the field values are normalized with respect to their respective incident fields.

Fig. 4.
Fig. 4.

(a) Illustration of the projection of the elliptical support curves on the hemispheres of incidence, reflection, and refraction in the kxkz plane as given by Eq. (6). The support curve of the incident field is deformed such that the refracted field forms a Bessel beam with cone angle ζ=π/6 and refraction angle ϑ=π/4 through a half-space with χ=2k. Note that for this cone angle and material, ϑ=π/4 is the maximum possible refraction angle for a complete Bessel beam, since greater angles would drive the incident spectrum into the evanescence region. (b) Bessel beam refracted at angle ϑ=π/6 into a medium with ϵχ=3.9ϵk and μχ=μk. The incident and reflected fields do not form a Bessel beam; however, the incident field is converging at the interface.

Fig. 5.
Fig. 5.

Comparison between the scattered fields upon incidence under total reflection condition. In (a), (b), and (c), the refraction medium is a right-handed material with ϵχ=ϵk/3.9 and μχ=μk, and in (d), (e), and (f), the refraction medium is a left-handed metamaterial with ϵχ=ϵk/3.9 and μχ=μk. In (a), a positive lateral GH shift along the x axis in the reflected Ex is visible in the xz plane, in contrast to a negative GH lateral shift for the same component in (d). The lateral and rotational GH shifts for Ex are visible in (b) and (c), while these shifts are mirrored in (e) and (f). The change in the transverse field structure due to the IF shift is visible in (b), (c), (e), and (f).

Equations (17)

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

ψn(ρ,ϕ,z,t)=AJn(kρρ)einϕei[kzzωt].
E(r)=μ×[ickΠh(r)]+[·Πe(r)]+k2Πe(r),H(r)=ϵ×[ickΠe(r)]+[·Πh(r)]+k2Πh(r),
Πe(r)=Aek2sin2(ζ)Jn(ksin(ζ)ρ)ei[kcos(ζ)z+nϕ]z^,Πh(r)=Ahk2sin2(ζ)Jn(ksin(ζ)ρ)ei[kcos(ζ)z+nϕ]z^,
Eρ=csc2(ζ)k[nμρAe+icos(ζ)Ahρ]Jn(ksin(ζ)ρ),Eϕ=csc2(ζ)k[nρcos(ζ)AhiμAeρ]Jn(ksin(ζ)ρ),Ez=AeJn(ksin(ζ)ρ),Hρ=csc2(ζ)k[nϵρAh+icos(ζ)Aeρ]Jn(ksin(ζ)ρ),Hϕ=csc2(ζ)k[nρcos(ζ)Ae+iϵAhρ]Jn(ksin(ζ)ρ),Hz=AhJn(ksin(ζ)ρ),
x=xcos(θ)+zsin(θ),ρ=x2+y2,y=y,ϕ=arctan(y/x),
kx(σ)=k[sin(ζ)cos(θ)cos(σ)+cos(ζ)sin(θ)],ky(σ)=ksin(ζ)sin(σ),
kz(σ)=k2[kx2(σ)+ky2(σ)],
sin(ϑ)=nknχ2+[nχ2nk2]tan2(ζ)sin(θ),
[ExHx](x,y)={[EρHρ](ρ,ϕ)cos(ϕ)[EϕHϕ](ρ,ϕ)sin(ϕ)}cos(θ)[EzHz](ρ,ϕ)sin(θ),[EyHy](x,y)=[EρHρ](ρ,ϕ)sin(ϕ)+[EϕHϕ](ρ,ϕ)cos(ϕ).
E˜xi(k)+E˜xr(k)=E˜xt(k),H˜xi(k)+H˜xr(k)=H˜xt(k),E˜yi(k)+E˜yr(k)=E˜yt(k),H˜yi(k)+H˜yr(k)=H˜yt(k),
E˜xr=[G˜ekxkzkG˜hkyμkω]ei[kxx+kyykzkz],H˜xr=[G˜hkxkzk+G˜ekyϵkω]ei[kxx+kyykzkz],E˜yr=[G˜ekykzk+G˜hkxμkω]ei[kxx+kyykzkz],H˜yr=[G˜hkykzkG˜ekxϵkω]ei[kxx+kyykzkz],E˜xt=[T˜ekxkzχ+T˜hkyμχω]ei[kxx+kyy+kzχz],H˜xt=[T˜hkxkzχ+T˜ekyϵχω]ei[kxx+kyy+kzχz],E˜yt=[T˜ekykzχ+T˜hkxμχω]ei[kxx+kyy+kzχz],H˜yt=[T˜hkykzχ+T˜ekxϵχω]ei[kxx+kyy+kzχz],
G˜e={kzχωHi+ϵχE+i}/De,G˜h={kzχωEi+μχH+i}/Dh,T˜e={kzkωHi+ϵkE+i}/De,T˜h={kzkωEi+μkH+i}/Dh,
E˜zr=G˜e[kx2+ky2]ei[kxx+kyykzkz],H˜zr=G˜h[kx2+ky2]ei[kxx+kyykzkz],E˜zt=T˜e[kx2+ky2]ei[kxx+kyy+kzχz],H˜zt=T˜h[kx2+ky2]ei[kxx+kyy+kzχz].
tan(θB)=nχ2+[nχ2nk2]tan2(ζ)nk.
E˜xi=[IekxkzkIhkyμkω]ei[kxx+kyy+kzkz],H˜xi=[Ihkxkzk+Iekyϵkω]ei[kxx+kyy+kzkz],E˜yi=[Iekykzk+Ihkxμkω]ei[kxx+kyy+kzkz],H˜yi=[IhkykzkIekxϵkω]ei[kxx+kyy+kzkz],
Ie={kzkωHt+ϵkE+t}/D˜e,Ih={kzkωEt+μkH+t}/D˜h,
Δxe=kxϵkϵχ[k2χ2]kzk|kzχ|[ϵχ2kzk2ϵkkzχ2],Δxh=kxμkμχ[k2χ2]kzk|kzχ|[μχ2kzk2μk2kzχ2].

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