Abstract

Radiation emitted by an electric dipole consists of traveling and evanescent plane waves. Usually, only the traveling waves are observable by a measurement in the far field, since the evanescent waves die out over a length of approximately a wavelength from the source. We show that when the radiation is passed through an interface with a medium with an index of refraction larger than the index of refraction of the embedding medium of the dipole, a portion of the evanescent waves are converted into traveling waves, and they become observable in the far field. The same conclusion holds when the waves pass through a layer of finite thickness. Waves that are transmitted under an angle larger than the so-called anti-critical angle θac(1) are shown to originate in evanescent dipole waves. In this fashion, part of the evanescent spectrum of the radiation becomes amenable to observation in the far field. We also show that in many situations the power in the far field coming from evanescent waves greatly exceeds the power originating in traveling waves.

© 2004 Optical Society of America

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References

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  1. D. W. Pohl, “Scanning near-field optical microscopy,” in Advances in Optical and Electron Microscopy, T. Mulvey, C. J. R. Sheppard, eds. (Academic, San Diego, Calif., 1991), p. 243.
  2. D. W. Pohl, D. Courjon, (eds.,) Near Field Optics, Proceedings of the NATO Advanced Research Workshop on Near Field Optics, Series E, Applied Sciences, Vol. 242 (Kluwer, Dordrecht, The Netherlands, 1993).
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    [CrossRef]
  4. M. A. Paesler, P. J. Moyer, Near Field Optics, Theory, Instrumentation, and Applications (Wiley, New York, 1996).
  5. M. Ohtsu, ed., Near-Field Nano/Atom Optics and Technology (Springer, Berlin, 1998).
  6. K. T. V. Grattan, B. T. Meggitt, eds., Optical Fiber Sensor Technology, Fundamentals (Kluwer, Dordrecht, The Netherlands, 2000).
  7. D. Courjon, Near-Field Microscopy and Near-Field Optics (World Scientific, Singapore, 2003).
  8. L. Novotny, D. W. Pohl, P. Regli, “Light propagation through nanometer-sized structures: the two-dimensional-aperture scanning near-field optical microscope,” J. Opt. Soc. Am. A 11, 1768–1779 (1994).
    [CrossRef]
  9. H. Heinzelmann, B. Hecht, L. Novotny, D. W. Pohl, “Forbidden light scanning near-field optical microscopy,” J. Microsc. 177, 115–118 (1995).
    [CrossRef]
  10. D. Van Labeke, F. Baida, D. Barchiesi, D. Courjon, “A theoretical model for the inverse scanning tunneling optical microscope (ISTOM),” Opt. Commun. 114, 470–480 (1995).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. B. Hecht, D. W. Pohl, H. Heinzelmann, “Tunnel near-field optical microscopy: TNOM-2,” in Photons and Local Probes, O. Marti, R. Möller, eds. (Kluwer, Dordrecht, The Netherlands, 1995), p. 93–107.
  15. B. Hecht, H. Bielefeldt, D. W. Pohl, “Influence of detection conditions on near-field optical imaging,” J. Appl. Phys. 84, 5873–5882 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  21. H. F. Arnoldus, J. T. Foley, “Traveling and evanescentparts of the optical near field,” J. Mod. Opt. 50, 1883–1901 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. W. Lukosz, R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67, 1615–1619 (1977).
    [CrossRef]
  31. W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers,” Phys. Rev. B 22, 3030–3038 (1980).
    [CrossRef]

2003 (2)

H. F. Arnoldus, J. T. Foley, “Spatial separation of the traveling and evanescent parts of dipole radiation,” Opt. Lett. 28, 1299–1301 (2003).
[CrossRef] [PubMed]

H. F. Arnoldus, J. T. Foley, “Traveling and evanescentparts of the optical near field,” J. Mod. Opt. 50, 1883–1901 (2003).
[CrossRef]

1998 (1)

B. Hecht, H. Bielefeldt, D. W. Pohl, “Influence of detection conditions on near-field optical imaging,” J. Appl. Phys. 84, 5873–5882 (1998).
[CrossRef]

1995 (3)

D. Van Labeke, D. Barchiesi, F. Baida, “Optical characterization of nanosources used in scanning near-field optical microscopy,” J. Opt. Soc. Am. A 12, 695–703 (1995).
[CrossRef]

H. Heinzelmann, B. Hecht, L. Novotny, D. W. Pohl, “Forbidden light scanning near-field optical microscopy,” J. Microsc. 177, 115–118 (1995).
[CrossRef]

D. Van Labeke, F. Baida, D. Barchiesi, D. Courjon, “A theoretical model for the inverse scanning tunneling optical microscope (ISTOM),” Opt. Commun. 114, 470–480 (1995).
[CrossRef]

1994 (2)

1992 (1)

1987 (1)

1986 (1)

O. Keller, “Screened electromagnetic propagators in nonlocal metal optics,” Phys. Rev. B 34, 3883–3899 (1986).
[CrossRef]

1984 (1)

1981 (1)

J. E. Sipe, “The dipole antenna problem in surface physics: a new approach,” Surf. Sci. 105, 489–504 (1981).
[CrossRef]

1980 (1)

W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers,” Phys. Rev. B 22, 3030–3038 (1980).
[CrossRef]

1977 (2)

1976 (2)

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
[CrossRef]

1974 (1)

K. H. Drexhage, “Interaction of light with monomolecular dye layers,” Prog. Opt. XII, 163–232 (1974).
[CrossRef]

1972 (1)

Arnoldus, H. F.

H. F. Arnoldus, J. T. Foley, “Spatial separation of the traveling and evanescent parts of dipole radiation,” Opt. Lett. 28, 1299–1301 (2003).
[CrossRef] [PubMed]

H. F. Arnoldus, J. T. Foley, “Traveling and evanescentparts of the optical near field,” J. Mod. Opt. 50, 1883–1901 (2003).
[CrossRef]

Baida, F.

D. Van Labeke, D. Barchiesi, F. Baida, “Optical characterization of nanosources used in scanning near-field optical microscopy,” J. Opt. Soc. Am. A 12, 695–703 (1995).
[CrossRef]

D. Van Labeke, F. Baida, D. Barchiesi, D. Courjon, “A theoretical model for the inverse scanning tunneling optical microscope (ISTOM),” Opt. Commun. 114, 470–480 (1995).
[CrossRef]

Bainier, C.

D. Courjon, C. Bainier, “Near field microscopy and near field optics,” Rep. Prog. Phys. 57, 989–1028 (1994).
[CrossRef]

Barchiesi, D.

D. Van Labeke, F. Baida, D. Barchiesi, D. Courjon, “A theoretical model for the inverse scanning tunneling optical microscope (ISTOM),” Opt. Commun. 114, 470–480 (1995).
[CrossRef]

D. Van Labeke, D. Barchiesi, F. Baida, “Optical characterization of nanosources used in scanning near-field optical microscopy,” J. Opt. Soc. Am. A 12, 695–703 (1995).
[CrossRef]

Bielefeldt, H.

B. Hecht, H. Bielefeldt, D. W. Pohl, “Influence of detection conditions on near-field optical imaging,” J. Appl. Phys. 84, 5873–5882 (1998).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), App. III, p. 890.

Carniglia, C. K.

Courjon, D.

D. Van Labeke, F. Baida, D. Barchiesi, D. Courjon, “A theoretical model for the inverse scanning tunneling optical microscope (ISTOM),” Opt. Commun. 114, 470–480 (1995).
[CrossRef]

D. Courjon, C. Bainier, “Near field microscopy and near field optics,” Rep. Prog. Phys. 57, 989–1028 (1994).
[CrossRef]

D. Courjon, Near-Field Microscopy and Near-Field Optics (World Scientific, Singapore, 2003).

Depasse, F.

Drexhage, K. H.

K. H. Drexhage, “Interaction of light with monomolecular dye layers,” Prog. Opt. XII, 163–232 (1974).
[CrossRef]

C. K. Carniglia, L. Mandel, K. H. Drexhage, “Absorption and emission of evanescent photons,” J. Opt. Soc. Am. 62, 479–486 (1972).
[CrossRef]

Foley, J. T.

H. F. Arnoldus, J. T. Foley, “Traveling and evanescentparts of the optical near field,” J. Mod. Opt. 50, 1883–1901 (2003).
[CrossRef]

H. F. Arnoldus, J. T. Foley, “Spatial separation of the traveling and evanescent parts of dipole radiation,” Opt. Lett. 28, 1299–1301 (2003).
[CrossRef] [PubMed]

Gasper, J.

Girard, C.

Hecht, B.

B. Hecht, H. Bielefeldt, D. W. Pohl, “Influence of detection conditions on near-field optical imaging,” J. Appl. Phys. 84, 5873–5882 (1998).
[CrossRef]

H. Heinzelmann, B. Hecht, L. Novotny, D. W. Pohl, “Forbidden light scanning near-field optical microscopy,” J. Microsc. 177, 115–118 (1995).
[CrossRef]

B. Hecht, “Forbidden light scanning near-field optical microscopy,” doctoral thesis (University of Basel, Basel, Switzerland, 1996).

B. Hecht, D. W. Pohl, H. Heinzelmann, “Tunnel near-field optical microscopy: TNOM-2,” in Photons and Local Probes, O. Marti, R. Möller, eds. (Kluwer, Dordrecht, The Netherlands, 1995), p. 93–107.

Heinzelmann, H.

H. Heinzelmann, B. Hecht, L. Novotny, D. W. Pohl, “Forbidden light scanning near-field optical microscopy,” J. Microsc. 177, 115–118 (1995).
[CrossRef]

B. Hecht, D. W. Pohl, H. Heinzelmann, “Tunnel near-field optical microscopy: TNOM-2,” in Photons and Local Probes, O. Marti, R. Möller, eds. (Kluwer, Dordrecht, The Netherlands, 1995), p. 93–107.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 396.

Keller, O.

O. Keller, “Screened electromagnetic propagators in nonlocal metal optics,” Phys. Rev. B 34, 3883–3899 (1986).
[CrossRef]

Kunz, R. E.

Lalor, É.

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

Lukosz, W.

Mandel, L.

C. K. Carniglia, L. Mandel, K. H. Drexhage, “Absorption and emission of evanescent photons,” J. Opt. Soc. Am. 62, 479–486 (1972).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 3.2.4.

Massey, G. A.

Moyer, P. J.

M. A. Paesler, P. J. Moyer, Near Field Optics, Theory, Instrumentation, and Applications (Wiley, New York, 1996).

Novotny, L.

Paesler, M. A.

M. A. Paesler, P. J. Moyer, Near Field Optics, Theory, Instrumentation, and Applications (Wiley, New York, 1996).

Pohl, D. W.

B. Hecht, H. Bielefeldt, D. W. Pohl, “Influence of detection conditions on near-field optical imaging,” J. Appl. Phys. 84, 5873–5882 (1998).
[CrossRef]

H. Heinzelmann, B. Hecht, L. Novotny, D. W. Pohl, “Forbidden light scanning near-field optical microscopy,” J. Microsc. 177, 115–118 (1995).
[CrossRef]

L. Novotny, D. W. Pohl, P. Regli, “Light propagation through nanometer-sized structures: the two-dimensional-aperture scanning near-field optical microscope,” J. Opt. Soc. Am. A 11, 1768–1779 (1994).
[CrossRef]

D. W. Pohl, “Scanning near-field optical microscopy,” in Advances in Optical and Electron Microscopy, T. Mulvey, C. J. R. Sheppard, eds. (Academic, San Diego, Calif., 1991), p. 243.

B. Hecht, D. W. Pohl, H. Heinzelmann, “Tunnel near-field optical microscopy: TNOM-2,” in Photons and Local Probes, O. Marti, R. Möller, eds. (Kluwer, Dordrecht, The Netherlands, 1995), p. 93–107.

Regli, P.

Sherman, G. C.

J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
[CrossRef]

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

Sipe, J. E.

J. E. Sipe, “New Green function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
[CrossRef]

J. E. Sipe, “The dipole antenna problem in surface physics: a new approach,” Surf. Sci. 105, 489–504 (1981).
[CrossRef]

Stamnes, J. J.

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

J. Gasper, G. C. Sherman, J. J. Stamnes, “Reflection and refraction of an arbitrary electromagnetic wave at a plane interface,” J. Opt. Soc. Am. 66, 955–961 (1976).
[CrossRef]

Van Labeke, D.

D. Van Labeke, F. Baida, D. Barchiesi, D. Courjon, “A theoretical model for the inverse scanning tunneling optical microscope (ISTOM),” Opt. Commun. 114, 470–480 (1995).
[CrossRef]

D. Van Labeke, D. Barchiesi, F. Baida, “Optical characterization of nanosources used in scanning near-field optical microscopy,” J. Opt. Soc. Am. A 12, 695–703 (1995).
[CrossRef]

Vigoureux, J. M.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 3.2.4.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), App. III, p. 890.

Appl. Opt. (2)

J. Appl. Phys. (1)

B. Hecht, H. Bielefeldt, D. W. Pohl, “Influence of detection conditions on near-field optical imaging,” J. Appl. Phys. 84, 5873–5882 (1998).
[CrossRef]

J. Math. Phys. (1)

G. C. Sherman, J. J. Stamnes, É. Lalor, “Asymptotic approximations to angular-spectrum representations,” J. Math. Phys. 17, 760–776 (1976).
[CrossRef]

J. Microsc. (1)

H. Heinzelmann, B. Hecht, L. Novotny, D. W. Pohl, “Forbidden light scanning near-field optical microscopy,” J. Microsc. 177, 115–118 (1995).
[CrossRef]

J. Mod. Opt. (1)

H. F. Arnoldus, J. T. Foley, “Traveling and evanescentparts of the optical near field,” J. Mod. Opt. 50, 1883–1901 (2003).
[CrossRef]

J. Opt. Soc. Am. (4)

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

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

Opt. Commun. (1)

D. Van Labeke, F. Baida, D. Barchiesi, D. Courjon, “A theoretical model for the inverse scanning tunneling optical microscope (ISTOM),” Opt. Commun. 114, 470–480 (1995).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (2)

W. Lukosz, “Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers,” Phys. Rev. B 22, 3030–3038 (1980).
[CrossRef]

O. Keller, “Screened electromagnetic propagators in nonlocal metal optics,” Phys. Rev. B 34, 3883–3899 (1986).
[CrossRef]

Prog. Opt. (1)

K. H. Drexhage, “Interaction of light with monomolecular dye layers,” Prog. Opt. XII, 163–232 (1974).
[CrossRef]

Rep. Prog. Phys. (1)

D. Courjon, C. Bainier, “Near field microscopy and near field optics,” Rep. Prog. Phys. 57, 989–1028 (1994).
[CrossRef]

Surf. Sci. (1)

J. E. Sipe, “The dipole antenna problem in surface physics: a new approach,” Surf. Sci. 105, 489–504 (1981).
[CrossRef]

Other (11)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 3.2.4.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), App. III, p. 890.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 396.

M. A. Paesler, P. J. Moyer, Near Field Optics, Theory, Instrumentation, and Applications (Wiley, New York, 1996).

M. Ohtsu, ed., Near-Field Nano/Atom Optics and Technology (Springer, Berlin, 1998).

K. T. V. Grattan, B. T. Meggitt, eds., Optical Fiber Sensor Technology, Fundamentals (Kluwer, Dordrecht, The Netherlands, 2000).

D. Courjon, Near-Field Microscopy and Near-Field Optics (World Scientific, Singapore, 2003).

D. W. Pohl, “Scanning near-field optical microscopy,” in Advances in Optical and Electron Microscopy, T. Mulvey, C. J. R. Sheppard, eds. (Academic, San Diego, Calif., 1991), p. 243.

D. W. Pohl, D. Courjon, (eds.,) Near Field Optics, Proceedings of the NATO Advanced Research Workshop on Near Field Optics, Series E, Applied Sciences, Vol. 242 (Kluwer, Dordrecht, The Netherlands, 1993).

B. Hecht, “Forbidden light scanning near-field optical microscopy,” doctoral thesis (University of Basel, Basel, Switzerland, 1996).

B. Hecht, D. W. Pohl, H. Heinzelmann, “Tunnel near-field optical microscopy: TNOM-2,” in Photons and Local Probes, O. Marti, R. Möller, eds. (Kluwer, Dordrecht, The Netherlands, 1995), p. 93–107.

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Figures (9)

Fig. 1
Fig. 1

An electric dipole, with dipole moment d, in a medium with index of refraction n1 is located a distance H above the xy plane and on the z axis. The region -L<z<0 contains a dielectric material with index of refraction n2, and the region z<-L is filled with a material with index of refraction n3.

Fig. 2
Fig. 2

Angular spectrum source (dipole) waves with wave vectors k emanate from the plane z=H. (a) A traveling dipole wave, for z>H, travels in the positive z direction and for z<H in the negative z direction. When the dipole wave is traveling, so is the specular wave, represented by wave vector kr. (b) An evanescent dipole wave decays in the positive z direction for z>H and in the negative z direction for z<H. In this case, the reflected wave (r) is also evanescent, and each wave travels along the xy plane with wave vector k. The waves in the layer can be either traveling or evanescent, and also the transmitted wave can be either traveling or evanescent, traveling or decaying away from the boundary z=-L.

Fig. 3
Fig. 3

Polar diagram of the intensity distribution A(θ, ϕ). The dashed line indicates the xy plane, and the vertical axis is the z axis. The graph represents A(θ, ϕ) as the distance to the origin, given the polar angle θ. The parameters are n1=1.41, n2=n3=1, and h=4π. The orientation of the dipole is taken as the spherical unit vector e1=-(ex+iey)/2, representing a dipole with a dipole moment that rotates counterclockwise in the xy plane. For this case, A(θ, ϕ) has no ϕ dependence.

Fig. 4
Fig. 4

Radiation pattern A(θ, ϕ) for n1=1, n2=n3=1.41, h=4π, and u=e1. The anti-critical angle is θac(1)=45°, indicated by the dashed line. The fraction of evanescent power for this case is f=0.56%.

Fig. 5
Fig. 5

Polar diagram of the intensity distribution B(θ) for n1=1, n2=n3=4.47, for which θac(1)=30°, and h=0. The dipole unit vector here is u=ez, and f=95%. We see that almost all power in z<-L comes from evanescent waves, and it is interesting to notice that the emission of traveling waves in z>H is negligible.

Fig. 6
Fig. 6

Graphs of Ptr/(n3Po), Pev/(n3Po) and the corresponding fraction f, as a function of n1/n3 for a dipole along the z axis.

Fig. 7
Fig. 7

Same as Fig. 6 but for a dipole in the xy plane. Of interest here is that the evanescent power can exceed the power of a free dipole in medium n3.

Fig. 8
Fig. 8

Same as Fig. 7 but for h=0.00314. Already for this small value of h, we see that Pev/(n3Po) falls to zero for n1/n30.

Fig. 9
Fig. 9

Graph of the intensity distribution A(θ, ϕ), illustrating the effect of the second critical angle θac(2). The parameters are n1=1, n2=1.73, n3=2, h=0, l=2π and u=e1. Here we have θac(1)=30°, θac(2)=60°, and f=75%. We see that owing to the layer thickness, almost no power is emitted in the range θac(2)<θt<π/2.

Equations (50)

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

sin θc(2)=n2/n1.
sin θc(1)=n3/n1.
Es(r)=i8π2on12d2k1β [ko2n12d-(d  k)k]×exp[ik(r-Hez)],zH,
β=ko2n12-k2 fork<kon1ik2-ko2n12 fork>kon1.
Bs(r)=-iω ×Es(r),
α=k/ko.
νi=ni2-α2,i=1, 2, 3,
k=k+koν1ez,z>Hk-koν1ez,z<H.
kr=k+koν1ez
kt=k-koν3ez.
es=1kk×ez,
ep,a=1kaka×es,
ep,t=-1n3 [ν3(k/k)+αez].
ko2n12d-(d  k)k=ko2n12σ=s,p(d  eσ)eσ,
Es(r)=iko8π2od2kσ=s,pexp(iν1h)ν1 (deσ)eσ×exp(ik  r),z<H.
Es(r)=iko8π2od2kσ=s,pexp(-iν1h)ν1 (d  eσ,r)eσ,r×exp(ikrr),z>H.
Er(r)=iko8π2od2kσ=s,pexp(iν1h)ν1 Rσ(α)×(d  eσ)eσ,rexp(ikrr),z>0,
E(r)=iko8π2od2k1ν1exp[i(kr+ν1koz)]×σ=s,p[exp(-iν1h)d  eσ,r+exp(iν1h)Rσ(α)d  eσ],z>H.
E(r)=iko8π2od2k1ν1exp[i(kr-ν3koz+ν1h)]×σ=s,pTσ(α)(d  eσ)eσ,t,z<-L,
d2k1koνi f(k)exp[i(kr+koνi|z|)]
-2πir f(k,o)exp(inikor),i=1or3,
E(r)=ko24πorexp[in1(kor-h cos θ)]×[(d  eϕ)eϕ+(d  eθ)eθ]+ko24πorexp[in1(kor+h cos θ)][Rs(αo)×(d  eϕ)eϕ-Rp(αo)(d  eθ+2 sin θd  ez)eθ],
Rs(αo)(d  eϕ)eϕ-Rp(αo)(d  eθ+2 sin θd  ez)eθ
=-Rs(αo)(d˜  eϕ)eϕ+Rp(αo)(d˜  eθ)eθ,
d˜=d-d,
E(r)=-ko2n3cos θ4πorexp[i(kon3r+hν1,o)] 1ν1,o×[Ts(αo)(d  eϕ)eϕ-1n1 Tp(αo)×(ν1,od  eρ+n3sin θd  ez)eθ].
ν1,o=n12-n32sin θ.
B(r)=nicrˆ×E(r),i=1or3.
S(r)=12μoReE(r)×B(r)*,
dPdΩ=PoA(θ, ϕ),
Po=ω412πoc3d  d*.
A(θ, ϕ)=3n18π |eθu+eθu˜Rp(n1sin θ)×exp(2in1h cos θ)|2+3n18π |eϕu×[1+Rs(n1sin θ)exp(2in1h cos θ)]|2,
A(θ, ϕ)=3n338πcos2 θTs(n3sin θ)ν1,o2|eϕu|2+1n12Tp(n3sin θ)ν1,o2×|ν1,oueρ+n3sin θ uez|2×exp[-2h Im(n12-n32sin2 θ)1/2].
sin θac(1)=n1/n3,
B(θ)=sin θ02πdϕA(θ, ϕ),
dPdθ=PoB(θ).
B(θ)=38n1sin θ(1-|uz|2)×[|1+Rs(n1sin θ)exp(2in1h cos θ)|2+cos2 θ|1-Rp(n1sin θ)exp(2in1h cos θ)|2]+34n1sin3 θ|uz|2×|1+Rp(n1sin θ)exp(2in1h cos θ)|2,
B(θ)=3n338n12sin θ cos2 θ|n12-n32sin2 θ| [(1-|uz|2)×(n12|Ts(n3sin θ)|2+|n12-n32sin2 θ||Tp(n3sin θ)|2)+2|uz|2n32sin2 θ|Tp(n3sin θ)|2]×exp[-2h Im(n12-n32sin2 θ)1/2],
Ptr=Poπ-θac(1)πdθB(θ),
Pev=Poπ/2π-θac(1)dθB(θ).
f=PevPev+Ptr×100%.
Pev/(n3Po)=121-r1(1+r)2 {(1-|uz|2)×[3-(1-r2)(1-r)-rη(r)]+2|uz|2[1-r2-3r+η(r)]},
η(r)=3r1-r2ln1+1-r2r.
sin θac(2)=n2/n3.
Λs=(ν1+ν2)(ν2+ν3)+(ν1-ν2)(ν2-ν3)×exp(2iν2l),
Λp=(n22ν1+n12ν2)(n32ν2+n22ν3)+(n22ν1-n12ν2)(n32ν2-n22ν3)exp(2iν2l),
Rs(α)=1Λs [(ν1-ν2)(ν2+ν3)+(ν1+ν2)(ν2-ν3)×exp(2iν2l)],
Rp(α)=1Λp [(n22ν1-n12ν2)(n32ν2+n22ν3)+(n22ν1+n12ν2)(n32ν2-n22ν3)exp(2iν2l)],
Ts(α)=4ν1ν2Λsexp[i(ν2-ν3)l],
Tp(α)=4ν1ν2n1n22n3Λpexp[i(ν2-ν3)l].

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