Abstract

It is predicted that the Goos–Hänchen effect can be resonantly enhanced by placing a metallic quantum well (ultrathin film) at the dielectric–vacuum (air) interface. We study the enhancement of the phenomenon, as it appears in frustrated total internal reflection with p-polarized light, both theoretically and numerically. Starting from boundary conditions for the electromagnetic field, which in a self-consistent manner take into account the quantum-well dynamics, we derive new expressions for the amplitude reflection and transmission coefficients of light, and from these the stationary phase approximation to the Goos–Hänchen shifts is obtained. It is shown that large peaks appear in the Goos–Hänchen shift below the critical angle in reflection, and these are located at the minima for the energy reflection coefficient. Both positive and negative shifts may occur, and the number of peaks depends on the gap width. To determine the accuracy of the simple stationary phase approximation, we carry out a rigorous stationary energy-transport calculation of the Goos–Hänchen shift. Although the overall agreement between the two approaches is good, the stationary phase approach mostly overestimates the peak heights. For a Gaussian incident beam, the resonance displacement of the reflected beam can be as large as the Gaussian width parameter. It is suggested that the possible relation between the Goos–Hänchen effect and the optical tunneling phenomenon in the two-prism configuration should be reinvestigated by depositing quantum wells on the glass–vacuum interfaces to obtain a better spatial photon localization.

© 2002 Optical Society of America

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References

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  1. Ph. Balcou, L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
    [CrossRef]
  2. E. Pollak, W. H. Miller, “New physical interpretation for time in scattering theory,” Phys. Rev. Lett. 53, 115–118 (1984).
    [CrossRef]
  3. M. Büttiker, R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, 1739–1742 (1982).
    [CrossRef]
  4. A. A. Stahlhofen, “Photonic tunneling time in frustrated total internal reflection,” Phys. Rev. A 62, 012112 (2000).
    [CrossRef]
  5. O. Keller, “Relation between spatial confinement of light and optical tunneling,” Phys. Rev. A 60, 1652–1671 (1999).
    [CrossRef]
  6. O. Keller, “Optical tunneling: a fingerprint of the lack of photon localizability,” J. Opt. Soc. Am. B 18, 206–217 (2001).
    [CrossRef]
  7. J. Broe, O. Keller, “Superluminal interactions in near-field optics,” J. Microsc. 202, 286–293 (2001).
    [CrossRef] [PubMed]
  8. O. Keller, “On the quantum physical relation between photon tunnelling and near-field optics,” J. Microsc. 202, 261–272 (2001).
    [CrossRef] [PubMed]
  9. F. Von Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).
    [CrossRef]
  10. F. Falco, T. Tamir, “Improved analysis of nonspecular phenomena in beams reflected from stratified media,” J. Opt. Soc. Am. A 7, 185–190 (1990).
    [CrossRef]
  11. S. L. Chuang, “Lateral shift of an optical beam due to leaky surface-plasmon excitations,” J. Opt. Soc. Am. A 3, 593–599 (1986).
    [CrossRef]
  12. W. Nasalski, “Longitudinal and transverse effects of nonspecular reflection,” J. Opt. Soc. Am. A 13, 172–181 (1996).
    [CrossRef]
  13. C. W. Hsue, T. Tamir, “Lateral displacement and distortion of beams incident upon a transmitting-layer configuration,” J. Opt. Soc. Am. A 2, 978–987 (1985).
    [CrossRef]
  14. C. W. Hsue, T. Tamir, “Lateral beam displacements in transmitting layered structures,” Opt. Commun. 49, 383–387 (1984).
    [CrossRef]
  15. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
    [CrossRef]
  16. H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik (Stuttgart) 32, 116–137 (1970).
  17. J. J. Greffet, C. Baylard, “Nonspecular reflection from a lossy dielectric,” Opt. Lett. 18, 1129–1131 (1993).
    [CrossRef] [PubMed]
  18. J. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
    [CrossRef]
  19. S. Zhang, T. Tamir, “Spatial modification of Gaussian beams diffracted by reflection gratings,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
    [CrossRef]
  20. A. K. Ghatak, M. R. Shenoy, I. C. Goyal, K. Thyagarajan, “Beam propagation under frustrated total reflection,” Opt. Commun. 56, 313–317 (1986).
    [CrossRef]
  21. B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61, 586–594 (1971).
    [CrossRef]
  22. H. M. Lai, F. C. Cheng, W. K. Tang, “Goos–Hänchen effect around and off the critical angle,” J. Opt. Soc. Am. A 3, 550–557 (1986).
    [CrossRef]
  23. O. Keller, “Sheet-model description of the linear optical response of quantum wells,” J. Opt. Soc. Am. B 12, 987–996 (1995).
    [CrossRef]
  24. A. Madrazo, M. Nieto-Vesperinas, “Detection of subwavelength Goos–Hänchen shifts from near-field intensities: a numerical simulation,” Opt. Lett. 20, 2445–2447 (1995).
    [CrossRef]
  25. F. I. Baida, D. Van Labeke, “Numerical study of the displacement of a three-dimensional Gaussian beam transmitted at total internal reflection: near-field applications,” J. Opt. Soc. Am. A 17, 858–866 (2000).
  26. P. J. Feibelman, “Surface electromagnetic fields,” Prog. Surf. Sci. 12, 287 (1982).
    [CrossRef]
  27. O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
    [CrossRef]

2001 (3)

O. Keller, “Optical tunneling: a fingerprint of the lack of photon localizability,” J. Opt. Soc. Am. B 18, 206–217 (2001).
[CrossRef]

J. Broe, O. Keller, “Superluminal interactions in near-field optics,” J. Microsc. 202, 286–293 (2001).
[CrossRef] [PubMed]

O. Keller, “On the quantum physical relation between photon tunnelling and near-field optics,” J. Microsc. 202, 261–272 (2001).
[CrossRef] [PubMed]

2000 (2)

1999 (1)

O. Keller, “Relation between spatial confinement of light and optical tunneling,” Phys. Rev. A 60, 1652–1671 (1999).
[CrossRef]

1997 (1)

Ph. Balcou, L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

1996 (2)

W. Nasalski, “Longitudinal and transverse effects of nonspecular reflection,” J. Opt. Soc. Am. A 13, 172–181 (1996).
[CrossRef]

O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
[CrossRef]

1995 (2)

1993 (1)

1992 (1)

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

1990 (1)

1989 (1)

1986 (4)

1985 (1)

1984 (2)

C. W. Hsue, T. Tamir, “Lateral beam displacements in transmitting layered structures,” Opt. Commun. 49, 383–387 (1984).
[CrossRef]

E. Pollak, W. H. Miller, “New physical interpretation for time in scattering theory,” Phys. Rev. Lett. 53, 115–118 (1984).
[CrossRef]

1982 (2)

M. Büttiker, R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, 1739–1742 (1982).
[CrossRef]

P. J. Feibelman, “Surface electromagnetic fields,” Prog. Surf. Sci. 12, 287 (1982).
[CrossRef]

1971 (1)

1970 (1)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik (Stuttgart) 32, 116–137 (1970).

1947 (1)

F. Von Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).
[CrossRef]

Baida, F. I.

Balcou, Ph.

Ph. Balcou, L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

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]

Broe, J.

J. Broe, O. Keller, “Superluminal interactions in near-field optics,” J. Microsc. 202, 286–293 (2001).
[CrossRef] [PubMed]

Büttiker, M.

M. Büttiker, R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, 1739–1742 (1982).
[CrossRef]

Cheng, F. C.

Chuang, S. L.

Dutriaux, L.

Ph. Balcou, L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

Falco, F.

Feibelman, P. J.

P. J. Feibelman, “Surface electromagnetic fields,” Prog. Surf. Sci. 12, 287 (1982).
[CrossRef]

Ghatak, A. K.

A. K. Ghatak, M. R. Shenoy, I. C. Goyal, K. Thyagarajan, “Beam propagation under frustrated total reflection,” Opt. Commun. 56, 313–317 (1986).
[CrossRef]

Goyal, I. C.

A. K. Ghatak, M. R. Shenoy, I. C. Goyal, K. Thyagarajan, “Beam propagation under frustrated total reflection,” Opt. Commun. 56, 313–317 (1986).
[CrossRef]

Greffet, J. J.

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]

Hänchen, H.

F. Von Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).
[CrossRef]

Horowitz, B. R.

Hsue, C. W.

C. W. Hsue, T. Tamir, “Lateral displacement and distortion of beams incident upon a transmitting-layer configuration,” J. Opt. Soc. Am. A 2, 978–987 (1985).
[CrossRef]

C. W. Hsue, T. Tamir, “Lateral beam displacements in transmitting layered structures,” Opt. Commun. 49, 383–387 (1984).
[CrossRef]

Keller, O.

O. Keller, “Optical tunneling: a fingerprint of the lack of photon localizability,” J. Opt. Soc. Am. B 18, 206–217 (2001).
[CrossRef]

J. Broe, O. Keller, “Superluminal interactions in near-field optics,” J. Microsc. 202, 286–293 (2001).
[CrossRef] [PubMed]

O. Keller, “On the quantum physical relation between photon tunnelling and near-field optics,” J. Microsc. 202, 261–272 (2001).
[CrossRef] [PubMed]

O. Keller, “Relation between spatial confinement of light and optical tunneling,” Phys. Rev. A 60, 1652–1671 (1999).
[CrossRef]

O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
[CrossRef]

O. Keller, “Sheet-model description of the linear optical response of quantum wells,” J. Opt. Soc. Am. B 12, 987–996 (1995).
[CrossRef]

Lai, H. M.

Landauer, R.

M. Büttiker, R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, 1739–1742 (1982).
[CrossRef]

Lotsch, H. K. V.

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik (Stuttgart) 32, 116–137 (1970).

Madrazo, A.

Miller, W. H.

E. Pollak, W. H. Miller, “New physical interpretation for time in scattering theory,” Phys. Rev. Lett. 53, 115–118 (1984).
[CrossRef]

Nasalski, W.

Nieto-Vesperinas, M.

Pollak, E.

E. Pollak, W. H. Miller, “New physical interpretation for time in scattering theory,” Phys. Rev. Lett. 53, 115–118 (1984).
[CrossRef]

Shenoy, M. R.

A. K. Ghatak, M. R. Shenoy, I. C. Goyal, K. Thyagarajan, “Beam propagation under frustrated total reflection,” Opt. Commun. 56, 313–317 (1986).
[CrossRef]

Stahlhofen, A. A.

A. A. Stahlhofen, “Photonic tunneling time in frustrated total internal reflection,” Phys. Rev. A 62, 012112 (2000).
[CrossRef]

Tamir, T.

Tang, W. K.

Thyagarajan, K.

A. K. Ghatak, M. R. Shenoy, I. C. Goyal, K. Thyagarajan, “Beam propagation under frustrated total reflection,” Opt. Commun. 56, 313–317 (1986).
[CrossRef]

Van Labeke, D.

Von Goos, F.

F. Von Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).
[CrossRef]

Zhang, S.

Ann. Phys. (Leipzig) (1)

F. Von Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).
[CrossRef]

J. Microsc. (2)

J. Broe, O. Keller, “Superluminal interactions in near-field optics,” J. Microsc. 202, 286–293 (2001).
[CrossRef] [PubMed]

O. Keller, “On the quantum physical relation between photon tunnelling and near-field optics,” J. Microsc. 202, 261–272 (2001).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

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

Opt. Commun. (3)

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

A. K. Ghatak, M. R. Shenoy, I. C. Goyal, K. Thyagarajan, “Beam propagation under frustrated total reflection,” Opt. Commun. 56, 313–317 (1986).
[CrossRef]

C. W. Hsue, T. Tamir, “Lateral beam displacements in transmitting layered structures,” Opt. Commun. 49, 383–387 (1984).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (1)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik (Stuttgart) 32, 116–137 (1970).

Phys. Rep. (1)

O. Keller, “Local fields in the electrodynamics of mesoscopic media,” Phys. Rep. 268, 85–262 (1996).
[CrossRef]

Phys. Rev. A (2)

A. A. Stahlhofen, “Photonic tunneling time in frustrated total internal reflection,” Phys. Rev. A 62, 012112 (2000).
[CrossRef]

O. Keller, “Relation between spatial confinement of light and optical tunneling,” Phys. Rev. A 60, 1652–1671 (1999).
[CrossRef]

Phys. Rev. Lett. (3)

Ph. Balcou, L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. 78, 851–854 (1997).
[CrossRef]

E. Pollak, W. H. Miller, “New physical interpretation for time in scattering theory,” Phys. Rev. Lett. 53, 115–118 (1984).
[CrossRef]

M. Büttiker, R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. 49, 1739–1742 (1982).
[CrossRef]

Prog. Surf. Sci. (1)

P. J. Feibelman, “Surface electromagnetic fields,” Prog. Surf. Sci. 12, 287 (1982).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Configuration used to study the Goos–Hänchen effect in this paper: two prisms both with a relative dielectric constant and placed a distance d apart from each other in vacuum. On each prism a quantum well (ultrathin film) (QW) is deposited. A monochromatic beam with a Gaussian profile is incident on the left prism–vacuum interface at a mean angle of incidence θi. The Goos–Hänchen shifts obtained in reflection (Δr) and transmission (Δt) are 0, in general, different, as indicated.

Fig. 2
Fig. 2

Geometrical meaning of the different terms entering the analytical expressions given for the Goos–Hänchen shifts Δrsp and Δtsp in Eqs. (5). The zero points of the phases for the modes constituting the incident and reflected beams are arbitrarily chosen at the point (0, zi) and for the modes in the transmitted beam are chosen at (0, z=zt). It is indicated that Δrsp and Δtsp are, in general, different.

Fig. 3
Fig. 3

Goos–Hänchen shift Δrsp as a function of the angle of incidence θi. The peaks in Δrsp are those expected to occur at θi values where |r|0. Both positive and negative peak values appear, and the different peak values (in micrometers) are given. The distance between the prisms is d=3 μm.

Fig. 4
Fig. 4

Goos–Hänchen shifts Δr (solid curve) and Δt (dotted curve) together with Δ0 (dashed curve) obtained in the absence of the quantum wells for a gap of d=1 μm as a function of the angle of incidence θi. The peak values are written on the figure close to the respective peaks. Away from the resonances, Δr and Δt are almost equal. As expected, the (numerical) peak heights never exceed the beam parameter (here a=1 mm) substantially.

Fig. 5
Fig. 5

Goos–Hänchen shifts Δr (solid curve), Δt (dotted curve), and Δ0 (dashed curve) for a gap of d=3 μm as a function of θi. When the gap is increased, more peaks occur in reflection (compare with Fig. 4), but now the peaks take both positive and negative values. The numerical values of the peaks in this case as well are bounded by the beam width (parameter a=1 mm).

Fig. 6
Fig. 6

With reference to the left y axis are shown |r|2 (dotted curve), |t|2 (dashed curve), and the sum of the the two, |r|2+|t|2 (dotted–dashed curve), which is not equal to unity because of the finite value of τ. The curve for |t|2 is never close to zero, and therefore no peaks occur in transmission. The curve for |r|2 is for specific values of θi close to zero. The solid curve shows Δr from Fig. 5 (values on the right y axis), and it is seen that the peaks in Δr occur for the values of θi where |r|20.

Fig. 7
Fig. 7

Goos–Hänchen shift Δr as a function of θi for different values of the relaxation time [τ=280 fs (solid curves), τ=28 fs (dashed curves), and τ=2.8 fs (dotted curves)]. The gap width used is d=1 μm. The peak width increases with decreasing τ, while the peak value (given in micrometers close to the peaks) decreases.

Fig. 8
Fig. 8

Goos–Hänchen shifts Δr and Δrsp for τ=28 fs (dashed curve) and τ=280 fs (solid curve) around the first peak in Fig. 7. Large differences occur between Δr and Δrsp in the peaks, particularly for large values of τ.

Fig. 9
Fig. 9

Closeup of the left peak in Fig. 4 together with Δr for different beam widths [a=0.1 mm (dotted curve), a=1 mm (dotted–dashed curve), and a=2 mm (dashed curve)]. For comparison, Δrsp (solid curve), which is independent of the beam width, is also shown. As expected, the best agreement between the stationary phase and stationary energy-transport approaches is obtained for large beam width. The difference between Δr and Δrsp for a=0.1 mm is approximately 1 order of magnitude.

Equations (60)

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Ei(x, z)=-e(q)A(q; zi)exp[iq(z-zi)]×exp(iq x)dq,
Er(x, z)=-100-1e(q)r(q; zi)A(q; zi)×exp[-iq(z-zi)]exp(iq x)dq,
Et(x, z)=-e(q)t(q; zt)A(q; zi)exp[iq(z-zt)]×exp(iq x)dq,
r(q; zi)=|r(q)|exp[iϕr(q; zi)],
t(q; zt)=|t(q)|exp[iϕt(q; zt)],
Δrsp=-ϕr(q; zi)qq0-2zitan θi,
Δtsp=-ϕt(q; zt)qq0-(d-zt+zi)tan θi,
Δrsp=-ϕr(q; 0)qq0,
Δtsp=-ϕt(q; d)qq0,
Js(z; q)=J0s(q)δ(z)+Jds(q)δ(z-d),
J0s=S0E(z=0+; q),
Jds=S0E(z=d-; q).
S0=Sx00Sz.
Ex(z=0+)-Ex(z=0-)=q0ω SzEz(z=0+),
Ez(z=0+)-Ez(z=0-)=q0ω SxEx(z=0+),
Ex(z=d+)-Ex(z=d-)=q0ω SzEz(z=d-),
Ez(z=d+)-Ez(z=d-)=q0ω SxEx(z=d-).
r(q; 0)ExrExi={[(q0)2(1+βz)2-q2(1+βx)2]×exp(-iq0d)-[(q0)2(1-βz)2-q2(1-βx)2]exp(iq0d)}D-1,
t(q;d)ExtExi=4q0q(1-βxβz)D-1,
D=[q0(1+βz)+q(1+βx)]2exp(-iq0d)-[q0(1-βz)-q(1-βx)]2exp(iq0d),
βx=q00ω Sx,βz=q20ωq0 Sz
Δrsp=Δtsp=-ϕD(q; zi)qq0,
σ(z; ω)=n(z)e2τme(1-iωτ) U,
n(z)=meπ2i(εF-εi)Θ(εF-εi)|Ψi(z)|2.
meπ2i[εF(dQW)-εi]Θ(εF(dQW)-εi)=ZN+dQW,
S0=S0U=QWσ(z; ω)dz.
S0=223π23τe2me1/2EF3/2dQW1-iωτ,
Δrsp(q)=-11+tan2 ϕr(q) tan ϕr(q)q,
r(q)=αR+iαI+(βR+iβI)(q-q0),
Δrsp(q)=αIβR-αRβI|r(q)|2.
αRβR+αIβI=0.
Δrsp(q)=αIβRβR2+βI2|r(q)|2,
r0(q)=(q-qzero)×[βR0+iβI0+(γR0+iγI0)(q-qzero)],
Δr0,sp(q)=(βI0γR0-βR0γI0) (q-qzero)2|r0(q)|2
βR0γR0+βI0γI0=0.
Δr0,sp(qzero)Δrsp(qmin)=γR0βI0βRαIαR2+αI2βR2+βI2,
Ei(x, z; ω)=-q-qA(q)exp[i(qx+qz)]dq,
q=+ωc02-(q)21/2
q=+i(q)2-ωc021/2
q=qcos θi+qsin θi
q=qcos θi-qsin θi,
q=±ωc02-q21/2
Ei(x, z; ω)=--(ω/c0)+-(ω/c0)(ω/c0)cos θi+(ω/c0)cos θi(ω/c0)+(ω/c0) q-qA(q)×exp(iqx)exp(iqz)dq,
Ei(x, z;ω)=c1+-(ω/c0)cos θi (ω/c)0ωc02-q21/2-qAqcos θi-ωc02-q21/2sin θi×exp(iqx)expiωc02-q21/2zcos θi+qsin θiωc02-q21/2dq +(ω/c0)cos θi(ω/c0)+c2ωc02-q21/2qAqcos θi+ωc02-q21/2sin θi×exp(iqx)exp-iωc02-q21/2zcos θi-qsin θiωc02-q21/2dq,
c1:Im(q)=tan θi[Re(q)]2-ωc02cos2 θi1/2for Re(q)<-ωc0cos θi,
c2:Im(q)=tan θi[Re(q)]2-ωc02cos2 θi1/2for Re(q)>ωc0cos θi.
Er(q, ω)=r(q)100-1Ei(q, ω),
Et(q, ω)=t(q)Ei(q, ω).
S(x, z; ω)=12μ0Re[E(x, z; ω)×B*(x, z; ω)].
Sη(x, z; ω)=12μ0Re[Eη(x, z; ω)×Bη*(x, z; ω)],
η=r, t,
|Sr(Δr, z=0-; ω)|=max[|Sr(x, z=0-; ω)|],
|St(Δt, z=d+; ω)|=max[|St(x, z=d+; ω)|].
A(q)=A0exp-a24 (q)2,
Ei(x, 0; ω)
=2πa A0nωc01-c0nωa214-(x)2a2+xia2×exp-(x)2a2.
0exp(-β2x2)cos(αx)dx=π2βexp-α24β2,
β>0.
0x2nexp(-β2x2)cos(αx)dx
=(-1)nπ(2β)2n+1 H2nα2βexp-α24β2,β>0,

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