Abstract

Past studies have shown that beams reflected by a single dielectric interface exhibit lateral and focal shifts under total-reflection conditions or angular shifts if a partial reflection regime is maintained. We investigate these effects for Gaussian beams incident upon multilayered media by using an analysis that treats the three beam-shifting phenomena in a unified manner. This approach reveals a novel fourth effect that manifests itself as an expansion or a reduction of the beam waist. All the four nonspecular phenomena are evaluated for typical layered configurations, and simple approximate relations are derived. The results show that the reflected beam fields may be considerably different from those predicted by geometrical optics if incidence occurs at an angle around which the reflectance function varies rapidly.

© 1986 Optical Society of America

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References

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  1. F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Physik 1(6), 333–345 (1947).
    [CrossRef]
  2. H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik 32, 116–137, 189–204 (1970); Optik 32, 299–319, 553–569 (1971).
  3. B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,”J. Opt. Soc. Am. 61, 586–594 (1971).
    [CrossRef]
  4. T. Tamir, H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,”J. Opt. Soc. Am. 61, 1397–1413 (1971).
    [CrossRef]
  5. Y. Levy, C. Imbert, “Amplification of the Goos–Hänchen shift by the interposition of thin films,”C. R. Acad. Sci. Paris 275, 723–725 (1972).
  6. J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
    [CrossRef]
  7. Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).
  8. O. C. de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3533–3562 (1977).
    [CrossRef]
  9. M. McGuirk, C. K. Carniglia, “An angular spectrum representation approach to the Goos–Hänchen shift,”J. Opt. Soc. Am. 67, 103–107 (1977).
    [CrossRef]
  10. C. K. Carniglia, K. R. Brownstein, “Focal shift and ray mode for total internal reflection,”J. Opt. Soc. Am. 67, 121–122 (1977).
    [CrossRef]
  11. I. A. White, A. W. Snyder, C. Pask, “Directional change of beams undergoing partial reflection,”J. Opt. Soc. Am. 67, 703–705 (1977).
    [CrossRef]
  12. W. J. Wild, C. L. Giles, “Goos–Hänchen shifts from absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
    [CrossRef]
  13. A. Puri, D. N. Pattanayal, J. L. Birman, “Resonance effects on total internal reflection and lateral (Goos–Hänchen) beam displacement at the interface between nonlocal and local dielectric,” Phys. Rev. B 28, 5877–5886 (1983).
    [CrossRef]
  14. V. Shah, T. Tamir, “Absorption and lateral shift of beams incident upon lossy multilayered media,”J. Opt. Soc. Am. 73, 37–44 (1983).
    [CrossRef]
  15. P. Mazur, B. Djafari-Rouhani, “Effect of surface polaritons on the lateral displacement of a light beam at a dielectric interface,” Phys. Rev. B 30, 6759–6762 (1984).
    [CrossRef]
  16. C. W. Hsue, T. Tamir, “Lateral beam displacements in transmitting structures,” Opt. Commun. 49, 383–387 (1984).
    [CrossRef]
  17. C. W. Hsue, T. Tamir, “Lateral displacement and distortion of beams incident upon a transmitting-layer configuration,” J. Opt. Soc. Am. A 2, 978–988 (1985).
    [CrossRef]
  18. C. C. Chan, T. Tamir, “Angular shift of a Gaussian beam reflected near the Brewster angle,” Opt. Lett. 10, 378–380 (1985).
    [CrossRef] [PubMed]
  19. C. W. Hsue, “The scattering of Gaussian beams incident on a transmitting layer configuration,” Ph.D. dissertion (Polytechnic Institute of New York, Brooklyn, N.Y., 1985).
  20. R. P. Riesz, R. Simon, “Reflection of a Gaussian beam from a dielectric slab,” J. Opt. Soc. Am. A 2, 1809–1817 (1985).
    [CrossRef]

1985 (3)

1984 (2)

P. Mazur, B. Djafari-Rouhani, “Effect of surface polaritons on the lateral displacement of a light beam at a dielectric interface,” Phys. Rev. B 30, 6759–6762 (1984).
[CrossRef]

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

1983 (2)

A. Puri, D. N. Pattanayal, J. L. Birman, “Resonance effects on total internal reflection and lateral (Goos–Hänchen) beam displacement at the interface between nonlocal and local dielectric,” Phys. Rev. B 28, 5877–5886 (1983).
[CrossRef]

V. Shah, T. Tamir, “Absorption and lateral shift of beams incident upon lossy multilayered media,”J. Opt. Soc. Am. 73, 37–44 (1983).
[CrossRef]

1982 (1)

W. J. Wild, C. L. Giles, “Goos–Hänchen shifts from absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
[CrossRef]

1977 (4)

1974 (1)

Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

1973 (1)

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

1972 (1)

Y. Levy, C. Imbert, “Amplification of the Goos–Hänchen shift by the interposition of thin films,”C. R. Acad. Sci. Paris 275, 723–725 (1972).

1971 (2)

1970 (1)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik 32, 116–137, 189–204 (1970); Optik 32, 299–319, 553–569 (1971).

1947 (1)

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Physik 1(6), 333–345 (1947).
[CrossRef]

Antar, Y. M.

Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

Bertoni, H. L.

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

T. Tamir, H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,”J. Opt. Soc. Am. 61, 1397–1413 (1971).
[CrossRef]

Birman, J. L.

A. Puri, D. N. Pattanayal, J. L. Birman, “Resonance effects on total internal reflection and lateral (Goos–Hänchen) beam displacement at the interface between nonlocal and local dielectric,” Phys. Rev. B 28, 5877–5886 (1983).
[CrossRef]

Boerner, W. M.

Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

Brownstein, K. R.

Carniglia, C. K.

Chan, C. C.

de Beauregard, O. C.

O. C. de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3533–3562 (1977).
[CrossRef]

Djafari-Rouhani, B.

P. Mazur, B. Djafari-Rouhani, “Effect of surface polaritons on the lateral displacement of a light beam at a dielectric interface,” Phys. Rev. B 30, 6759–6762 (1984).
[CrossRef]

Felsen, L. B.

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

Giles, C. L.

W. J. Wild, C. L. Giles, “Goos–Hänchen shifts from absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
[CrossRef]

Goos, F.

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Physik 1(6), 333–345 (1947).
[CrossRef]

Hänchen, H.

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Physik 1(6), 333–345 (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, 978–988 (1985).
[CrossRef]

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

C. W. Hsue, “The scattering of Gaussian beams incident on a transmitting layer configuration,” Ph.D. dissertion (Polytechnic Institute of New York, Brooklyn, N.Y., 1985).

Imbert, C.

O. C. de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3533–3562 (1977).
[CrossRef]

Y. Levy, C. Imbert, “Amplification of the Goos–Hänchen shift by the interposition of thin films,”C. R. Acad. Sci. Paris 275, 723–725 (1972).

Levy, Y.

O. C. de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3533–3562 (1977).
[CrossRef]

Y. Levy, C. Imbert, “Amplification of the Goos–Hänchen shift by the interposition of thin films,”C. R. Acad. Sci. Paris 275, 723–725 (1972).

Lotsch, H. K. V.

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik 32, 116–137, 189–204 (1970); Optik 32, 299–319, 553–569 (1971).

Mazur, P.

P. Mazur, B. Djafari-Rouhani, “Effect of surface polaritons on the lateral displacement of a light beam at a dielectric interface,” Phys. Rev. B 30, 6759–6762 (1984).
[CrossRef]

McGuirk, M.

Pask, C.

Pattanayal, D. N.

A. Puri, D. N. Pattanayal, J. L. Birman, “Resonance effects on total internal reflection and lateral (Goos–Hänchen) beam displacement at the interface between nonlocal and local dielectric,” Phys. Rev. B 28, 5877–5886 (1983).
[CrossRef]

Puri, A.

A. Puri, D. N. Pattanayal, J. L. Birman, “Resonance effects on total internal reflection and lateral (Goos–Hänchen) beam displacement at the interface between nonlocal and local dielectric,” Phys. Rev. B 28, 5877–5886 (1983).
[CrossRef]

Ra, J. W.

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

Riesz, R. P.

Shah, V.

Simon, R.

Snyder, A. W.

Tamir, T.

White, I. A.

Wild, W. J.

W. J. Wild, C. L. Giles, “Goos–Hänchen shifts from absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
[CrossRef]

Ann. Physik (1)

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Physik 1(6), 333–345 (1947).
[CrossRef]

C. R. Acad. Sci. Paris (1)

Y. Levy, C. Imbert, “Amplification of the Goos–Hänchen shift by the interposition of thin films,”C. R. Acad. Sci. Paris 275, 723–725 (1972).

Can. J. Phys. (1)

Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).

J. Opt. Soc. Am. (6)

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

Opt. Commun. (1)

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

Opt. Lett. (1)

Optik (1)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect,” Optik 32, 116–137, 189–204 (1970); Optik 32, 299–319, 553–569 (1971).

Phys. Rev. A (1)

W. J. Wild, C. L. Giles, “Goos–Hänchen shifts from absorbing media,” Phys. Rev. A 25, 2099–2101 (1982).
[CrossRef]

Phys. Rev. B (2)

A. Puri, D. N. Pattanayal, J. L. Birman, “Resonance effects on total internal reflection and lateral (Goos–Hänchen) beam displacement at the interface between nonlocal and local dielectric,” Phys. Rev. B 28, 5877–5886 (1983).
[CrossRef]

P. Mazur, B. Djafari-Rouhani, “Effect of surface polaritons on the lateral displacement of a light beam at a dielectric interface,” Phys. Rev. B 30, 6759–6762 (1984).
[CrossRef]

Phys. Rev. D (1)

O. C. de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3533–3562 (1977).
[CrossRef]

SIAM J. Appl. Math. (1)

J. W. Ra, H. L. Bertoni, L. B. Felsen, “Reflection and transmission of beams at a dielectric interface,” SIAM J. Appl. Math. 24, 396–413 (1973).
[CrossRef]

Other (1)

C. W. Hsue, “The scattering of Gaussian beams incident on a transmitting layer configuration,” Ph.D. dissertion (Polytechnic Institute of New York, Brooklyn, N.Y., 1985).

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Figures (10)

Fig. 1
Fig. 1

Geometry of the reflecting structure. The coordinates of the incident and the reflected beams are given by (xi, zi) and (xr, zr), respectively.

Fig. 2
Fig. 2

Outline of the geometrical-optics reflected beam (shown by dashed lines) and the actual reflected beam (shown by solid lines). Usually L′ < w, but L′ is shown larger here for clarity.

Fig. 3
Fig. 3

Typical pole–null pairs in the complex κ = κ′ + plane for three situations: (a) totally reflecting structures, (b) partially reflecting structures with the null below the real axis, and (c) partially reflecting structures with the null above the real axis.

Fig. 4
Fig. 4

Universal curve for L′/Lm versus v = δ′/κp for totally reflecting configurations, as given by Eq. (38).

Fig. 5
Fig. 5

Universal curve for F′/Fm versus v = δ′/κp for totally reflecting configurations. The solid line is given by Eq. (42), whereas the dashed line shows a more accurate curve for a specific case found through Eq. (39).

Fig. 6
Fig. 6

Variation of |r(θ)| versus θ for perpendicularly polarized waves incident upon a dielectric slab (a) from below or (b) from above. The dashed curves show the approximate variation of |r(θ)| if Eq. (22) is used in the neighborhood of the third minimum.

Fig. 7
Fig. 7

Variation of L = L′ + iL″ versus δ′ for a beam incident as shown in Fig. 6(a). The value δ′ = 0 corresponds to θi = 45.25°.

Fig. 8
Fig. 8

Variation of F = F′ + iF″ versus δ′ for the same situation as in Fig. 7.

Fig. 9
Fig. 9

Variation of L = L′ + iL″ versus δ′ for a beam incident as shown in Fig. 6(b). The value δ′ = 0 corresponds to θi = 60.44°.

Fig. 10
Fig. 10

Variation of F = F′ + iF″ versus δ′ for the same situation as in Fig. 9.

Equations (49)

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E i = ( w / w i ) exp [ - ( x i / w i ) 2 + i k i z i ] ,
w i 2 = w 2 + i ( 2 z i / k i )
E i = k i w 2 π - exp [ - ( k i w s / 2 ) 2 - i k i ( s x i - c z i ) ] d s ,
s = sin ( θ - θ i ) ,
c = cos ( θ - θ i ) ,
c = 1 - ( s 2 / 2 ) ,
E r = k i w 2 π - r ( θ ) exp [ - ( k i w s / 2 ) 2 + i k i ( s x r + c z r ) ] d s ,
r ( θ ) exp [ ln r ( θ ) r ( θ i ) exp [ s r d r d s 0 + s 2 2 d d s 0 ( 1 r d r d s ) ] ,
L = L + i L = i k i r d r d s 0 ,
F = F + i F = - i k i d d s 0 ( 1 r d r d s ) .
E r = k i w 2 π r ( θ i ) exp ( i k i z r ) - exp [ ( k i w r s / 2 ) 2 + i k i s ( x r - L ) ] d s ,
w f 2 = w 2 + i ( 2 / k i ) ( z r - F ) .
E r = ( w / w f ) r ( θ i ) exp [ - ( x r - L ) 2 / w f 2 + i k i z r ] .
L = i k i r d r d θ ,
F = - d L d θ i ,
w m 2 = Re ( w f 2 ) z r = F = w 2 + ( 2 F / k i ) = w 2 ( 1 + μ ) ,
μ = 2 F k i w 2 .
E r 2 = ( w / w f ) r ( θ i ) 2 exp { - 2 Re [ ( x r - L ) / w f }
k i w 2 ( 1 + μ ) ( x r - L ) = 2 L ( z r - F ) .
tan α = ( x r - L ) / ( z r - F ) .
α = 2 L k i w 2 ( 1 + μ ) = 2 L k i w m 2 .
r ( θ ) = R 0 ( κ - κ n ) / ( κ - κ p ) ,
κ = n i sin θ ,
L λ = i 2 π ( κ n - κ p ) cos θ i ( κ i - κ n ) ( κ i - κ p ) ,
κ i = n i sin θ i .
F = L ( tan θ i - i k i L 2 κ i - κ n - κ p κ n - κ p ) .
κ a = κ a + i κ a = 1 2 ( κ p + κ n ) ,
Δ = Δ + i Δ = 1 2 ( κ p - κ n ) ,
δ = κ i - κ a = δ - i κ a ,
L λ = i π Δ Δ 2 - δ 2 cos θ i ,
F = L ( tan θ i + i 2 π n i L λ δ Δ ) .
Δ = κ a = 0             and             Δ = κ p = - κ n > 0.
L λ = κ p ( κ p ) 2 + ( δ ) 2 cos θ i π .
L m λ = cos θ 0 κ p ,
κ p = n i sin θ 0 .
v = δ / κ p ,
L cos θ 0 / L m cos θ i = ( 1 + v 2 ) - 1 .
L / L m = ( 1 + v 2 ) - 1 ,
F λ = sin θ i κ p ( 1 + v 2 ) π [ 1 + 2 v κ p ( 1 + v 2 ) n i cos 2 θ i sin θ i ] .
F λ = 2 n i v π [ cos θ i κ p ( 1 + v 2 ) ] 2 .
F m λ = 3 3 / 2 n i 8 π ( cos θ f κ p ) 2
F F m = 16 3 3 / 2 v ( 1 + v 2 ) 2 3.08 v ( 1 + v 2 ) 2 .
F m / L m ( 3 3 / 2 / 8 κ p ) n i cos θ 0 = 2.041 n i ( L m / λ ) .
L λ ( κ n - κ p ) cos θ i 2 π κ n κ p .
r ( θ ) = R e i ϕ ,
k i L = d ϕ d θ i ,
α = 2 ( k i w m ) 2 × 1 R d R d θ i .
k i F = - d L d θ i ,
μ = 2 ( k i w ) 2 [ ( 1 2 d R d θ i ) 2 - 1 R d 2 R d θ i 2 ] .

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