Abstract

Longitudinal and transverse shifts of a light beam at total internal reflection was experimentally studied by far-field measurements on the reflected field. We propose to use a scanning tunneling optical microscope (STOM) to study these shifts in transmission, and we present a theoretical model of this proposed experiment to obtain a numerical estimation of these shifts. We study the reflection and the transmission of a three-dimensional polarized incident beam. We verify the validity of our formalism by studying the Goos–Hanchen shift in reflection and by comparing our results with published ones. Then we calculate STOM images of the transmitted field distribution. On the images the well-known Goos–Hanchen shift is easily observed. But we also encounter a smaller shift, perpendicular to the plane of incidence. This transverse shift was also observed in reflection by Imbert and Levy [Nouv. Rev. Opt. 6, 285 (1975)]. We study the variations of the two shifts versus various parameters such as the angle of incidence, the optical index, and the incident polarization. Then we discuss the feasibility of the near-field observation of these shifts.

© 2000 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  42. J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  48. D. Van Labeke, D. Barchiesi, “Probes for scanning tunneling optical microscopy: a theoretical comparison,” J. Opt. Soc. Am. A 10, 2193–2201 (1993).
    [CrossRef]

1998

L. I. Perez, F. Ciocci, “Nonspecular first-order effects in Kretschmann’s configuration,” J. Mod. Opt. 45, 2487–2502 (1998).
[CrossRef]

D. Van Labeke, F. I. Baida, J. M. Vigoureux, “A theoretical study of near-field detection and excitation of surface plasmons,” Ultramicroscopy 71, 351–359 (1998).
[CrossRef]

J. J. Greffet, M. Nieto-Vesperinas, “Field theory for generalized bidirectional reflectivity: derivation of Helmholtz’s reciprocity principle and Kirchhoff’s law,” J. Opt. Soc. Am. A 15, 2735–2744 (1998).
[CrossRef]

1997

N. Bonomo, R. A. Depine, “Nonspecular reflection of ordinary and extraordinary beams in uniaxial media,” J. Opt. Soc. Am. A 14, 3402–3409 (1997).
[CrossRef]

M. A. Porras, “Nonspecular reflection of general light beams at a dielectric interface,” Opt. Commun. 135, 369–377 (1997).
[CrossRef]

1996

1995

Z. Bouchal, M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

R. A. Depine, N. E. Bonomo, “Spatial modifications of Gaussian beams reflected at anisotropic–uniaxial interfaces,” J. Mod. Opt. 42, 2401–2412 (1995).
[CrossRef]

P. Dawson, K. W. Smith, F. de Fornel, J.-P. Goudonnet, “Imaging of surface plasmon launch and propagation using a photon scanning tunneling microscope,” Ultramicroscopy 57, 287–292 (1995).
[CrossRef]

A. Madrazo, M. Nieto-Vesperinas, “Detection of subwavelength Goos–Hanchen shifts from near-field intensities: a numerical simulation,” Opt. Lett. 20, 2445–2447 (1995).
[CrossRef]

1994

G. Y. Leng, D. F. Gu, X. Y. Zang, “Nonspecular longitudinal shift of the beam reflected from an interface containing an absorbing medium,” Opt. Eng. 33, 2612–2616 (1994).
[CrossRef]

1993

1992

D. Van Labeke, D. Barchiesi, “Scanning-tunneling optical microscopy: a theoretical macroscopic approach,” J. Opt. Soc. Am. A 9, 732–738 (1992).
[CrossRef]

J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
[CrossRef]

J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

F. Bretenaker, A. L. Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hanchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992).
[CrossRef] [PubMed]

1990

1989

1988

1987

1986

1985

1984

1977

J. P. Hugonin, R. Petit, “Etude générale des déplacements à la réflexion totale,” J. Opt. (Paris) 8, 73–87 (1977).
[CrossRef]

O. Costa de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3553–3562 (1977).
[CrossRef]

J. J. Cowan, B. Anicin, “Longitudinal and transverse displacements of a bounded microwave beam at total internal reflection,” J. Opt. Soc. Am. 67, 1307–1314 (1977).
[CrossRef]

1975

Y. Levy, C. Imbert, “Amplification des déplacements à la réflexion totale,” Opt. Commun. 13, 43–47 (1975).
[CrossRef]

C. Imbert, Y. Levy, “Déplacement d’un faisceau lumineux par réflexion totale: filtrage des états de polarisation et amplification,” Nouv. Rev. Opt. 6, 285–296 (1975).
[CrossRef]

1973

O. Costa de Beauregard, C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D 7, 3555–3563 (1973).
[CrossRef]

1972

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

1971

1947

F. Goos, H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. (Leipzig) 1, 333–345 (1947).
[CrossRef]

Anicin, B.

Baida, F. I.

D. Van Labeke, F. I. Baida, J. M. Vigoureux, “A theoretical study of near-field detection and excitation of surface plasmons,” Ultramicroscopy 71, 351–359 (1998).
[CrossRef]

Barchiesi, D.

D. Van Labeke, D. Barchiesi, “Probes for scanning tunneling optical microscopy: a theoretical comparison,” J. Opt. Soc. Am. A 10, 2193–2201 (1993).
[CrossRef]

D. Van Labeke, D. Barchiesi, “Scanning-tunneling optical microscopy: a theoretical macroscopic approach,” J. Opt. Soc. Am. A 9, 732–738 (1992).
[CrossRef]

D. Van Labeke, D. Barchiesi, “Theoretical problems in scanning near-field optical microscopy,” in Near Field Optics, D. W. Pohl, D. Courjon, eds., Vol. 242 of NATO Advanced Science Institute Series (Kluwer Academic, Dordrecht, The Netherlands, 1993), pp. 157–178.

Baylard, C.

J. J. Greffet, C. Baylard, “Nonspecular reflection from a lossy dielectric,” Opt. Lett. 18, 1129–1131 (1993).
[CrossRef] [PubMed]

J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

Bielefeldt, H.

B. Hecht, L. Novotny, H. Bielefeldt, Y. Inouye, D. W. Pohl, “Local excitation, scattering, and interferences of surface plasmons,” Phys. Rev. Lett. 77, 1889–1892 (1996).
[CrossRef] [PubMed]

Bonomo, N.

Bonomo, N. E.

R. A. Depine, N. E. Bonomo, “Spatial modifications of Gaussian beams reflected at anisotropic–uniaxial interfaces,” J. Mod. Opt. 42, 2401–2412 (1995).
[CrossRef]

Bouchal, Z.

Z. Bouchal, M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Bretenaker, F.

L. Dutriaux, A. L. Floch, F. Bretenaker, “Measurement of the transverse displacement at total reflection by helicoidal laser eigenstates,” Europhys. Lett. 24, 345–349 (1993).
[CrossRef]

F. Bretenaker, A. L. Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hanchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992).
[CrossRef] [PubMed]

Chan, C. C.

Ciocci, F.

L. I. Perez, F. Ciocci, “Nonspecular first-order effects in Kretschmann’s configuration,” J. Mod. Opt. 45, 2487–2502 (1998).
[CrossRef]

Costa de Beauregard, O.

O. Costa de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3553–3562 (1977).
[CrossRef]

O. Costa de Beauregard, C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D 7, 3555–3563 (1973).
[CrossRef]

Courjon, D.

D. Courjon, K. Sarayeddine, M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

Cowan, J. J.

Dawson, P.

P. Dawson, K. W. Smith, F. de Fornel, J.-P. Goudonnet, “Imaging of surface plasmon launch and propagation using a photon scanning tunneling microscope,” Ultramicroscopy 57, 287–292 (1995).
[CrossRef]

de Fornel, F.

P. Dawson, K. W. Smith, F. de Fornel, J.-P. Goudonnet, “Imaging of surface plasmon launch and propagation using a photon scanning tunneling microscope,” Ultramicroscopy 57, 287–292 (1995).
[CrossRef]

Depine, R. A.

N. Bonomo, R. A. Depine, “Nonspecular reflection of ordinary and extraordinary beams in uniaxial media,” J. Opt. Soc. Am. A 14, 3402–3409 (1997).
[CrossRef]

R. A. Depine, N. E. Bonomo, “Spatial modifications of Gaussian beams reflected at anisotropic–uniaxial interfaces,” J. Mod. Opt. 42, 2401–2412 (1995).
[CrossRef]

Dutriaux, L.

L. Dutriaux, A. L. Floch, F. Bretenaker, “Measurement of the transverse displacement at total reflection by helicoidal laser eigenstates,” Europhys. Lett. 24, 345–349 (1993).
[CrossRef]

F. Bretenaker, A. L. Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hanchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992).
[CrossRef] [PubMed]

Fainman, Y.

Falco, F.

Fillard, J. P.

J. P. Fillard, Near-Field Optics and Nanoscopy (World Scientific, Singapore, 1996).

Floch, A. L.

L. Dutriaux, A. L. Floch, F. Bretenaker, “Measurement of the transverse displacement at total reflection by helicoidal laser eigenstates,” Europhys. Lett. 24, 345–349 (1993).
[CrossRef]

F. Bretenaker, A. L. Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hanchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992).
[CrossRef] [PubMed]

Goos, F.

F. Goos, H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. (Leipzig) 1, 333–345 (1947).
[CrossRef]

Goudonnet, J.-P.

P. Dawson, K. W. Smith, F. de Fornel, J.-P. Goudonnet, “Imaging of surface plasmon launch and propagation using a photon scanning tunneling microscope,” Ultramicroscopy 57, 287–292 (1995).
[CrossRef]

Greffet, J. J.

Gu, D. F.

G. Y. Leng, D. F. Gu, X. Y. Zang, “Nonspecular longitudinal shift of the beam reflected from an interface containing an absorbing medium,” Opt. Eng. 33, 2612–2616 (1994).
[CrossRef]

Hall, D.

Hanchen, H.

F. Goos, H. Hanchen, “Ein neuer and fundamentaler Versuch zur total Reflection,” Ann. Phys. (Leipzig) 1, 333–345 (1947).
[CrossRef]

Hecht, B.

B. Hecht, L. Novotny, H. Bielefeldt, Y. Inouye, D. W. Pohl, “Local excitation, scattering, and interferences of surface plasmons,” Phys. Rev. Lett. 77, 1889–1892 (1996).
[CrossRef] [PubMed]

Horowitz, B. R.

Hugonin, J. P.

J. P. Hugonin, R. Petit, “Etude générale des déplacements à la réflexion totale,” J. Opt. (Paris) 8, 73–87 (1977).
[CrossRef]

Imbert, C.

O. Costa de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3553–3562 (1977).
[CrossRef]

Y. Levy, C. Imbert, “Amplification des déplacements à la réflexion totale,” Opt. Commun. 13, 43–47 (1975).
[CrossRef]

C. Imbert, Y. Levy, “Déplacement d’un faisceau lumineux par réflexion totale: filtrage des états de polarisation et amplification,” Nouv. Rev. Opt. 6, 285–296 (1975).
[CrossRef]

O. Costa de Beauregard, C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D 7, 3555–3563 (1973).
[CrossRef]

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

Inouye, Y.

B. Hecht, L. Novotny, H. Bielefeldt, Y. Inouye, D. W. Pohl, “Local excitation, scattering, and interferences of surface plasmons,” Phys. Rev. Lett. 77, 1889–1892 (1996).
[CrossRef] [PubMed]

Kou, E. F. Y.

Kou, F. Y.

Lekner, J.

J. Lekner, Theory of Reflection of Electromagnetic and Particle Waves (Nijhoff, Dordrecht, The Netherlands, 1987).

Leng, G. Y.

G. Y. Leng, D. F. Gu, X. Y. Zang, “Nonspecular longitudinal shift of the beam reflected from an interface containing an absorbing medium,” Opt. Eng. 33, 2612–2616 (1994).
[CrossRef]

Levy, Y.

O. Costa de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3553–3562 (1977).
[CrossRef]

C. Imbert, Y. Levy, “Déplacement d’un faisceau lumineux par réflexion totale: filtrage des états de polarisation et amplification,” Nouv. Rev. Opt. 6, 285–296 (1975).
[CrossRef]

Y. Levy, C. Imbert, “Amplification des déplacements à la réflexion totale,” Opt. Commun. 13, 43–47 (1975).
[CrossRef]

Lin, L.

Madrazo, A.

Moyer, P. J.

M. A. Paesler, P. J. Moyer, Near-Field Optics (Wiley, New York, 1996).

Mukunda, M.

Nasalski, W.

Nieto-Vesperinas, M.

Novotny, L.

B. Hecht, L. Novotny, H. Bielefeldt, Y. Inouye, D. W. Pohl, “Local excitation, scattering, and interferences of surface plasmons,” Phys. Rev. Lett. 77, 1889–1892 (1996).
[CrossRef] [PubMed]

Olivik, M.

Z. Bouchal, M. Olivik, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Paesler, M. A.

M. A. Paesler, P. J. Moyer, Near-Field Optics (Wiley, New York, 1996).

Pedrotti, F. L.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice Hall, London, 1993), Chap. 22.

Pedrotti, L. S.

F. L. Pedrotti, L. S. Pedrotti, Introduction to Optics, 2nd ed. (Prentice Hall, London, 1993), Chap. 22.

Perez, L. I.

L. I. Perez, F. Ciocci, “Nonspecular first-order effects in Kretschmann’s configuration,” J. Mod. Opt. 45, 2487–2502 (1998).
[CrossRef]

Petit, R.

J. P. Hugonin, R. Petit, “Etude générale des déplacements à la réflexion totale,” J. Opt. (Paris) 8, 73–87 (1977).
[CrossRef]

Pohl, D. W.

B. Hecht, L. Novotny, H. Bielefeldt, Y. Inouye, D. W. Pohl, “Local excitation, scattering, and interferences of surface plasmons,” Phys. Rev. Lett. 77, 1889–1892 (1996).
[CrossRef] [PubMed]

Porras, M. A.

M. A. Porras, “Nonspecular reflection of general light beams at a dielectric interface,” Opt. Commun. 135, 369–377 (1997).
[CrossRef]

Saleh, B.

B. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 3.

Sarayeddine, K.

D. Courjon, K. Sarayeddine, M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

Shamir, J.

Simon, R.

Smith, K. W.

P. Dawson, K. W. Smith, F. de Fornel, J.-P. Goudonnet, “Imaging of surface plasmon launch and propagation using a photon scanning tunneling microscope,” Ultramicroscopy 57, 287–292 (1995).
[CrossRef]

Spajer, M.

D. Courjon, K. Sarayeddine, M. Spajer, “Scanning tunneling optical microscopy,” Opt. Commun. 71, 23–28 (1989).
[CrossRef]

Sudarshan, E. C. G.

Tamir, T.

Teich, M. C.

B. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 3.

Van Labeke, D.

D. Van Labeke, F. I. Baida, J. M. Vigoureux, “A theoretical study of near-field detection and excitation of surface plasmons,” Ultramicroscopy 71, 351–359 (1998).
[CrossRef]

D. Van Labeke, D. Barchiesi, “Probes for scanning tunneling optical microscopy: a theoretical comparison,” J. Opt. Soc. Am. A 10, 2193–2201 (1993).
[CrossRef]

D. Van Labeke, D. Barchiesi, “Scanning-tunneling optical microscopy: a theoretical macroscopic approach,” J. Opt. Soc. Am. A 9, 732–738 (1992).
[CrossRef]

D. Van Labeke, D. Barchiesi, “Theoretical problems in scanning near-field optical microscopy,” in Near Field Optics, D. W. Pohl, D. Courjon, eds., Vol. 242 of NATO Advanced Science Institute Series (Kluwer Academic, Dordrecht, The Netherlands, 1993), pp. 157–178.

Vigoureux, J. M.

D. Van Labeke, F. I. Baida, J. M. Vigoureux, “A theoretical study of near-field detection and excitation of surface plasmons,” Ultramicroscopy 71, 351–359 (1998).
[CrossRef]

J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
[CrossRef]

Zang, X. Y.

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Figures (13)

Fig. 1
Fig. 1

Experimental scheme of the device.

Fig. 2
Fig. 2

Longitudinal displacement of the reflected beam. Normalized curves Δr/λ=f(θ-θc). Curve (a), TE; curve (b), TM; and curve (c), circular polarization. The width of the circularly polarized incident beam a is equal to 1000λ (λ=632.8 nm is the wavelength of the incident light); the refractive index of the incident medium is n1=1.94.

Fig. 3
Fig. 3

Variations of the longitudinal shift of the reflected beam versus n1. Normalized curves Δr/aλ=f(n1). a is ∼3.5 µm, and the wavelength of the incident light is λ=632.8 nm. Curve (a), TE; curve (b), TM; and curve (c), circularly polarized beam. The angle of incidence is θc=arcsin(1/n1).

Fig. 4
Fig. 4

STOM images of the intensity distribution of the transmitted field in the case of (a) TE polarization and (b) TM polarization. The incident beam is characterized by its wavelength λ=632.8 nm, and a=3.5 µm. The mean angle of incidence is the critical angle [θc=arcsin(1/n1)].

Fig. 5
Fig. 5

Contour plots of the intensity distribution of the incident beam (solid curves) and of the transmitted field (dashed curves) in the case of right circular polarization. The wavelength of the incident beam is λ=632.8 nm, and a=3.5 µm. The mean angle of incidence is the critical angle [θc=arcsin(1/n1), where n1=1.94]. We clearly see a longitudinal shift (along the X direction) of ∼510 nm and a transverse shift (along the Y direction) of ∼74 nm.

Fig. 6
Fig. 6

Longitudinal shift of the transmitted intensity distribution. Normalized curves Δt/λ=f(θ-θc). Curve (a), TE; curve (b), TM; and curve (c), circular polarization. a=1000λ (λ=632.8 nm is the wavelength of the incident light). The glass index is n1=1.94.

Fig. 7
Fig. 7

Variations of the longitudinal shift of the transmitted intensity distribution versus refractive index n1. Normalized curves Δt/aλ=f(n1). The angle of incidence is θc=arcsin(1/n1), a is equal to 3.5 µm, and the wavelength of the incident light is λ=632.8 nm. Curve (a), TE; curve (b), TM; and curve (c), circularly polarized beam.

Fig. 8
Fig. 8

Longitudinal displacement of the reflected beam and of the transmitted evanescent field versus γ. Curves (a) and (b) correspond to the transmitted and to the reflected beams, respectively, in the case of a linear polarization [Eq. (14)]. The same variations [curves (c) and (d)] are obtained in the case of an elliptic polarization with Eq. 15.

Fig. 9
Fig. 9

Variations of the transverse displacement in transmission δt with respect to the angle of incidence (θ): curve (a) for n1=1.92, curve (b) for n1=1.75, and curve (c) for n1=1.5. a=1000λ (λ=632.8 nm is the wavelength of the incident light).

Fig. 10
Fig. 10

Comparison of the variations of the transverse displacement versus the polarization of the incident beam. Curves (a) and (b) correspond to transmission and reflection cases, respectively, of a linear incident polarization. γ=0 corresponds to TM polarization and γ=90° to TE polarization. Curves (c) for transmission and (d) for reflection are performed in the case of a right elliptic polarization. γ=45° corresponds to a circular polarization. The incidence angle is θ=θc+2° and n1=1.94.

Fig. 11
Fig. 11

Variations of the transverse displacement of the reflected (dashed curve) and transmitted (solid curve) fields in the case of a linearly polarized incident beam with respect to the angle of incidence (θ) for n1=2.5. a=1000λ (λ=632.8 nm is the wavelength of the incident light). The polarization is described by Eq. (14) with γ=45°.

Fig. 12
Fig. 12

Normalized cross sections made on Fig. 4. They show that the longitudinal shift can easily be measured.

Fig. 13
Fig. 13

Contour plots of intensity distribution of the transmitted field in the case of a right circular polarization (solid curve) and in the case of a left circular polarization (dashed curve).

Equations (28)

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Einc=Ainc(u, v)exp[+i(ux+vy+w1z)]dudv.
w1=n12 ω2c2-u2-v21/2.
Axinc(u, v)=αx A0(u, v),
Ayinc(u, v)=αy A0(u, v).
A0(u, v)=πa2A0 exp[-a2(u2+v2)/4].
uAxinc+vAyinc+w1Azinc=0.
Einc(X, Y, Z)=Ainc(U, V)exp[+i(UX+VY+W1Z)]dUdV(Z<0).
W1=n12 ω2c2-U2-V21/2.
AXinc(U, V)=A0(U cos θ-W1 sin θ, V)×αx-V sin θWαy,
AYinc(U, V)=A0(U cos θ-W1 sin θ, V)×αycos θ+U sin θW1,
AZinc(U, V)=-A0(U cos θ-W1 sin θ, V)×Uαx+V cos θαyW1.
Et(X, Y, Z)=T(U, V)Ainc(U, V)×exp[+i(UX+VY+W2Z)]dUdV
forZ>0,
Er(X,Y,Z)=R(U, V)Ainc(U, V)×exp[+i(UX+VY-W1Z)]dUdV
forZ<0.
W2=n22 ω2c2-U2-V21/2.
T(U, V)XX=tpU2+tsV2U2+V2,
T(U, V)YY=tpV2+tsU2U2+V2,
T(U, V)XY=T(U, V)YX=UV(ts-tp)U2+V2,
T(U, V)XZ=T(U, V)YZ=0,
R(U, V)XX=-rpU2+rsV2U2+V2,
R(U, V)YY=-rpV2+rsU2U2+V2,
R(U, V)XY=R(U, V)YX=UV(rs+rp)U2+V2,
R(U, V)XZ=R(U, V)YZ=0.
αx=cos(γ),
αy=sin(γ).
αx=cos(γ),
αy=±i sin(γ).

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