Abstract

The field resulting from a beam incident upon a planar dielectric layer placed between media having different refractive indices is examined by using both an approximate approach and a highly accurate Gaussian-beam analysis. We find that, if the incidence angle is adjusted to couple energy to a leaky wave that may be guided by the layer, both the reflected and the transmitted beams may undergo large lateral displacements. The transmitted beam is then always shifted in a forward direction, but the displacement of the reflected beam can be either forward or backward. We show that these effects are produced by distortion of the beam profiles caused by the leaky-wave coupling mechanism, which accounts for lateral shifts of the beam axes. The possible presence of absorption loss in the layer is also considered, and we find that the reflected beam may be effectively suppressed under certain critical combinations of the physical parameters involved.

© 1985 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Goos, H. Haenchen, “Ein neuer und fundamentaler Ver-such zur Totalreflexion,” Ann. Physik 1(6), 333–345 (1947).
    [CrossRef]
  2. H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Haenchen effect,” Optik 32, 116–137, 189–204 (1970);Optik 32, 299–319, 553–569 (1971).
  3. T. Tamir, H. L. Bertoni, “Lateral displacement of optical beams at multilayered and periodic structures,” J. Opt. Soc. Am. 61, 1397–1413 (1971).
    [CrossRef]
  4. 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, 3553–3562 (1977).
    [CrossRef]
  5. V. Shah, T. Tamir, “Absorption and lateral shift of beams incident upon lossy multilayered media,” J. Opt. Soc. Am. 73, 37–44 (1983).
    [CrossRef]
  6. C. W. Hsue, T. Tamir, “Lateral beam displacements in transmitting layered structures,” Opt. Commun. 49, 383–387 (1984).
    [CrossRef]
  7. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1980), pp. 101–107.
  8. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 1–49.
  9. C. W. Hsue, T. Tamir, “Evolution of TE surface and leaky waves guided by an asymmetric layer configuration,” J. Opt. Soc. Am. A 1, 923–931 (1984).
    [CrossRef]
  10. K. Ogusu, M. Miyagi, S. Nishida, “Leaky TE modes on an asymmetric three-layered slab waveguide,” J. Opt. Soc. Am. 70, 48–52 (1980).
    [CrossRef]
  11. V. Shah, T. Tamir, “Anomalous absorption by multilayered media,” Opt. Commun. 37, 383–387 (1981).
    [CrossRef]
  12. H. Blok, J. M. van Splunter, H. G. Janssen, “Leaky-wave modes and their role in the numerical evaluation of the field excited by a line source in a non-symmetric, inhomogeneously-layered waveguide,” Appl. Sci. Res. 41, 223–236 (1984).
    [CrossRef]
  13. L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), pp. 455–462.
  14. See the appendix on p. 43 of Ref. 5.
  15. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 297.
  16. E. S. Birger, L. A. Vainshtein, Sov. Phys. Tech. Phys. 18, 1405–1411 (1974).
  17. L. A. Vainshtein, Sov. Phys. Usp. 19, 189–205 (1976).
    [CrossRef]

1984 (3)

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

C. W. Hsue, T. Tamir, “Evolution of TE surface and leaky waves guided by an asymmetric layer configuration,” J. Opt. Soc. Am. A 1, 923–931 (1984).
[CrossRef]

H. Blok, J. M. van Splunter, H. G. Janssen, “Leaky-wave modes and their role in the numerical evaluation of the field excited by a line source in a non-symmetric, inhomogeneously-layered waveguide,” Appl. Sci. Res. 41, 223–236 (1984).
[CrossRef]

1983 (1)

1981 (1)

V. Shah, T. Tamir, “Anomalous absorption by multilayered media,” Opt. Commun. 37, 383–387 (1981).
[CrossRef]

1980 (1)

1977 (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, 3553–3562 (1977).
[CrossRef]

1976 (1)

L. A. Vainshtein, Sov. Phys. Usp. 19, 189–205 (1976).
[CrossRef]

1974 (1)

E. S. Birger, L. A. Vainshtein, Sov. Phys. Tech. Phys. 18, 1405–1411 (1974).

1971 (1)

1970 (1)

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

1947 (1)

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

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 297.

Bertoni, H. L.

Birger, E. S.

E. S. Birger, L. A. Vainshtein, Sov. Phys. Tech. Phys. 18, 1405–1411 (1974).

Blok, H.

H. Blok, J. M. van Splunter, H. G. Janssen, “Leaky-wave modes and their role in the numerical evaluation of the field excited by a line source in a non-symmetric, inhomogeneously-layered waveguide,” Appl. Sci. Res. 41, 223–236 (1984).
[CrossRef]

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1980), pp. 101–107.

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, 3553–3562 (1977).
[CrossRef]

Felsen, L. B.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), pp. 455–462.

Goos, F.

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

Haenchen, H.

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

Hsue, C. W.

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

C. W. Hsue, T. Tamir, “Evolution of TE surface and leaky waves guided by an asymmetric layer configuration,” J. Opt. Soc. Am. A 1, 923–931 (1984).
[CrossRef]

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, 3553–3562 (1977).
[CrossRef]

Janssen, H. G.

H. Blok, J. M. van Splunter, H. G. Janssen, “Leaky-wave modes and their role in the numerical evaluation of the field excited by a line source in a non-symmetric, inhomogeneously-layered waveguide,” Appl. Sci. Res. 41, 223–236 (1984).
[CrossRef]

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, 3553–3562 (1977).
[CrossRef]

Lotsch, H. K. V.

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

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 1–49.

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), pp. 455–462.

Miyagi, M.

Nishida, S.

Ogusu, K.

Shah, V.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 297.

Tamir, T.

Vainshtein, L. A.

L. A. Vainshtein, Sov. Phys. Usp. 19, 189–205 (1976).
[CrossRef]

E. S. Birger, L. A. Vainshtein, Sov. Phys. Tech. Phys. 18, 1405–1411 (1974).

van Splunter, J. M.

H. Blok, J. M. van Splunter, H. G. Janssen, “Leaky-wave modes and their role in the numerical evaluation of the field excited by a line source in a non-symmetric, inhomogeneously-layered waveguide,” Appl. Sci. Res. 41, 223–236 (1984).
[CrossRef]

Ann. Physik (1)

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

Appl. Sci. Res. (1)

H. Blok, J. M. van Splunter, H. G. Janssen, “Leaky-wave modes and their role in the numerical evaluation of the field excited by a line source in a non-symmetric, inhomogeneously-layered waveguide,” Appl. Sci. Res. 41, 223–236 (1984).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Commun. (2)

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

V. Shah, T. Tamir, “Anomalous absorption by multilayered media,” Opt. Commun. 37, 383–387 (1981).
[CrossRef]

Optik (1)

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

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, 3553–3562 (1977).
[CrossRef]

Sov. Phys. Tech. Phys. (1)

E. S. Birger, L. A. Vainshtein, Sov. Phys. Tech. Phys. 18, 1405–1411 (1974).

Sov. Phys. Usp. (1)

L. A. Vainshtein, Sov. Phys. Usp. 19, 189–205 (1976).
[CrossRef]

Other (5)

L. B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, Englewood Cliffs, N.J., 1973), pp. 455–462.

See the appendix on p. 43 of Ref. 5.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), p. 297.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1980), pp. 101–107.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), pp. 1–49.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1

Geometry of a beam incident upon a transmitting dielectric layer.

Fig. 2
Fig. 2

Locations of poles (shown by crosses) and nulls (shown by circles) for a beam incident, as in Fig. 1, upon a typical lossless configuration, (a) Complex τ0 = τ0′ + 0″ plane, (b) Complex τ2 = τ2′ + 2″ plane, (c) Uppermost sheet of the κ = κ ′ + iκ″ plane; on this sheet, both τ0′ and τ2′ are positive.

Fig. 3
Fig. 3

Location of poles and nulls for the same situation, as in Fig. 2 except that now the beam is incident from the lower (substrate) region.

Fig. 4
Fig. 4

Profiles of beam intensity P = |F|2 for the incident, reflected, and transmitted beams, Pi,Pr, and Pt, respectively, in a typical situation with (a) incidence from the upper region, and (b) incidence from the lower region. The cases are shown for the TE15 mode, with κp = 0.875 + 0.055i and κn = 0.866 ± 0.017i.

Fig. 5
Fig. 5

Loci of poles and nulls for a typical layer geometry, for the TE18 mode and varying values of the substrate index n2.

Fig. 6
Fig. 6

Beam profiles for the situation shown in Fig. 5.

Fig. 7
Fig. 7

Loci of poles and nulls for a typical layer geometry for varying values of the loss factor ν1 in the layer medium, (a) Incidence from the upper region and (b) incidence from the lower region.

Fig. 8
Fig. 8

Beam profiles for the situation shown in Fig. 7(a).

Fig. 9
Fig. 9

Beam profiles for the situation shown in Fig. 7(b).

Equations (46)

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

F r ( x , z ) = 1 2 π A ( κ ) r ( κ ) exp [ i k ( κ x + τ 0 z ) ] d κ
F t ( x , z ) = 1 2 π A ( κ ) t ( κ ) exp [ i k ( κ x + τ 1 h ) i k τ 2 ( z + h ) ] d κ .
τ j = ( n j 2 κ 2 ) 1 / 2 ,
κ i = n 0 sin θ 0 .
D r = D r + i D r = i r ( κ i ) k r ( κ i ) ,
D t = D t + i D t = i t ( κ i ) k t ( κ i ) .
r ( κ ) = N ( κ ) D ( κ ) = N 0 ( κ κ n 1 ) ( κ κ n 2 ) ( κ κ n m ) D 0 ( κ κ p 1 ) ( κ κ p 2 ) ( κ κ p m ) ,
exp = [ i k κ p x + i k ( n 0 2 κ p 2 ) 1 / 2 z ]
r ( κ ) = R κ κ n κ κ p ,
D r = i ( κ n κ p ) k ( κ i κ n ) ( κ i κ p ) .
D r κ n κ p k κ n κ p .
t ( κ ) = T κ p κ κ p ,
D t = i k ( κ p κ i ) .
D t 1 k κ p .
F i ( x , 0 ) = exp [ ( x / w x ) 2 + i k κ i x ] ,
A ( κ ) = F i ( x , 0 ) exp ( i k κ x ) d ( k x ) = k w x π exp [ ( k w x / 2 ) 2 ( κ κ i ) 2 ] .
F r ( x , 0 ) = R k w x 2 π κ κ n κ κ p × exp [ ( k w x / 2 ) 2 ( κ κ i ) 2 + i k κ x ] d κ .
F r ( x , 0 ) = R [ 1 + i π 2 k w x ( κ p κ n ) exp ( γ 2 ) erfc ( γ ) ] × exp [ ( x / w x ) 2 + i k κ i x ] ,
γ = x / w x + i ( k w x / 2 ) ( κ i κ p ) .
F t ( x , h ) = T k w x 2 π κ p κ κ p exp [ ( k w x / 2 ) 2 ( κ κ i ) 2 + i k ( κ x + τ 1 h ) ] d κ .
τ 1 = n 1 cos θ 1 ( κ κ i ) tan θ 1 ( κ κ i ) 2 / ( 2 n 1 cos 3 θ 1 ) ,
n 0 sin θ 0 = n 1 sin θ 1 .
F t ( x , h ) = i π 2 T k w x κ x exp ( γ 1 2 ) erfc ( γ 1 ) × exp [ ( x 1 / w 1 ) 2 + i ( k κ i x + ϕ 1 ) ] ,
γ 1 = ( x 1 / w 1 ) + i ( k w 1 / 2 ) ( κ i κ p ) .
x 1 = x h tan θ 1 ,
w 1 = w x [ 1 + i 2 h / ( k n 1 w x 2 cos 3 θ 1 ) ] 1 / 2 ,
ϕ 1 = k n 1 h cos θ 1 .
π exp ( γ 1 2 ) erfc ( γ 1 ) 1 / γ 1 .
γ 1 = ( k w 1 κ p / 2 ) [ 1 ( 2 x 1 / k w 1 2 κ p ) ] ( k w 1 κ p / 2 ) exp [ ( 2 x 1 / k w 1 2 κ p ) ] .
F t ( x , h ) i T κ p w x κ p w 1 exp [ ( x 1 1 / k κ p w 1 ) 2 + i ( k κ i x + ϕ 1 ) ] .
n 1 = n 1 ( 1 + i ν 1 ) ,
Δ κ n , p i n 1 2 ν 1 κ n , p ,
r ( κ ) = N ( κ ) D ( κ ) = ( s τ 2 u τ 0 ) i ( τ 1 2 s u τ 0 τ 2 ) tan ( k τ 1 h ) / τ 1 ( s τ 2 + u τ 0 ) i ( τ 1 2 + s u τ 0 τ 2 ) tan ( k τ 1 h ) / τ 1 ,
u = s = 1 for TE incidence , u = ( n 1 / n 0 ) 2 and s = ( n 1 / n 2 ) 2 for TM incidence .
t ( κ ) = 2 υ D ( κ ) cos ( k τ 1 h ) ,
υ = { τ 0 for TE incidence u τ 2 for TM incidence
D ( κ p ) = ( s τ 2 p + u τ 0 p ) i ( τ 1 p 2 + s u τ 0 p τ 2 p ) × tan ( k τ 1 p h ) / τ 1 p = 0 .
D * ( κ p ) = ( s τ 2 p * + u τ 0 p * ) + i ( τ 1 p * 2 + s u τ 0 p * τ 2 p * ) × tan ( k τ 1 p * h ) / τ 1 p * = 0 .
N ( κ n ) = ( s τ 2 n u τ 0 n ) i ( τ 1 n 2 + s u τ 0 n τ 2 n ) × tan ( k τ 1 n / h ) / τ 1 n = 0 ;
N * ( κ n ) = [ s ( τ 2 n * ) + u τ 0 n * ] i [ τ 1 n * 2 + s u τ 0 n * ( τ 2 n * ) ] × tan ( k τ 1 n * h ) / τ 1 n * = 0 .
f ( τ 0 , τ 1 , τ 2 ) = ( s τ 2 ± u τ 0 ) i ( τ 1 2 ± s u τ 0 τ 2 ) × tan ( k τ 1 h ) / τ 1 = 0 ,
Δ f = f τ 0 Δ τ 0 + f τ 1 Δ τ 1 + f τ 2 Δ τ 2 + f n 1 Δ n 1 = 0.
κ Δ κ = τ 0 Δ τ 0 = n 1 Δ n 1 τ 1 Δ τ 1 = τ 2 Δ τ 2 .
Δ κ p , n = n 1 Δ n 1 κ k h i [ ( s ¯ τ 2 2 ± ū τ 0 2 ) ( 1 2 τ 2 2 / n 1 2 ) ] k h + i [ s ¯ ( n 1 2 n 2 2 ) ± ū ( n 1 2 n 0 2 ) ] ,
s ¯ = s / τ 2 ( τ 1 2 s 2 τ 2 2 ) ,
ū = u / τ 0 ( τ 1 2 u 2 τ 0 2 ) .

Metrics