Abstract

The net displacement of the intensity peak in relatively narrow Gaussian beams reflected at a dielectric interface is shown to be produced by a combination of four distinct nonspecular effects, namely, lateral, focal, and angular shifts and a modification of the beam-waist magnitude. We also find that these effects cause the axis of the reflected beam to follow a trajectory that depends on the beam width and on the distance from the waist of the geometric-optical reflected beams. To determine all the details of this nonspecular behavior, we derive a new expression for the reflected field; in contrast to previously reported results, this expression also holds for small beam widths and for incidence angles equal or close to the critical angle of total reflection. Our derivation yields accurate results for the four distinct nonspecular effects and provides a consistent explanation of the available experimental data on the net displacement of the beam peak.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 1, 333–345 (1947).
    [Crossref]
  2. H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect, Parts I–IV” 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. Sci. Am. 61, 586–594 (1971).
    [Crossref]
  4. J. J. Cowan, B. Anicin, “Longitudinal and transverse displacements of a bounded microwave beam at a total internal reflection,” J. Opt. Soc. Am. 67, 1307–1314 (1977).
    [Crossref]
  5. 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]
  6. Y. M. Antar, W. M. Boerner, “Gaussian beam interaction with a planar dielectric interface,” Can. J. Phys. 52, 962–972 (1974).
  7. I. A. White, A. W. Snyder, C. Pask, “Directional change of beams undergoing partial reflection,” J. Opt. Soc. Am. 67, 703–705 (1977).
    [Crossref]
  8. R. P. Riesz, R. Simon, “Reflection of a Gaussian beam from a dielectric slab,” J. Opt. Soc. Am. A 2, 1809–1817 (1985).
    [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, 703–705 (1977).
    [Crossref]
  11. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
    [Crossref]
  12. C. C. Chan, T. Tamir, “Beam phenomena at and near critical incidence upon a dielectric interface,” J. Opt. Soc. Am. A 4, 655–663 (1987).
    [Crossref]
  13. 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]
  14. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980), p. 1064.

1987 (1)

1986 (2)

1985 (1)

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]

1971 (1)

B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Sci. Am. 61, 586–594 (1971).
[Crossref]

1970 (1)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos–Hänchen effect, Parts I–IV” 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. Phys. 1, 333–345 (1947).
[Crossref]

Anicin, B.

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]

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.

Cheng, F. C.

Cowan, J. J.

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]

Goos, F.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980), p. 1064.

Hänchen, H.

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

Horowitz, B. R.

B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Sci. Am. 61, 586–594 (1971).
[Crossref]

Lai, H. M.

Lotsch, H. K. V.

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

McGuirk, M.

Pask, C.

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.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980), p. 1064.

Simon, R.

Snyder, A. W.

Tamir, T.

Tang, W. K.

White, I. A.

Ann. Phys. (1)

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

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. Sci. Am. (1)

B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Sci. Am. 61, 586–594 (1971).
[Crossref]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (4)

Optik (1)

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

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)

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980), p. 1064.

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 (11)

Fig. 1
Fig. 1

Geometry of the incident and reflected beams, showing the geometric-optical reflected-beam coordinates (xr, zr) and the actual reflected-beam coordinates (xm, zm).

Fig. 2
Fig. 2

Variation of |r(θ)| for θi = 41.12° and n = 1.491. The solid curve marked NTL refers to the derivation in this paper, whereas the short-dashed curve marked LCT refers to the expansion used by Lai et al.13 The exact result is shown by the long-dashed curve.

Fig. 3
Fig. 3

Variation of L′ versus θi, for TM polarization, n = 1.491, kw = 60.425, and zr = 0. The long-dashed curve marked CT refers to the derivation of Chan and Tamir,12 whereas the solid curve marked NTL refers to that obtained in the present study.

Fig. 4
Fig. 4

Variation of F′ versus θi, for the same parameters and notation as in Fig. 3.

Fig. 5
Fig. 5

Variation of α versus θi, for the same parameters and notation as in Fig. 3.

Fig. 6
Fig. 6

Variation of μ versus θi, for the same parameters and notation as in Fig. 3.

Fig. 7
Fig. 7

Variation of S versus θi, for the same parameters and notation as in Fig. 3. The short-dashed curve marked LCT refers to the derivation used by Lai et al.13

Fig. 8
Fig. 8

Variation of S versus θi, for TM and TE polarizations, with n = 1.471, kw = 60, and zr = 60 cm. The experimental points are those obtained by Cowan and Anicin.4

Fig. 9
Fig. 9

Variation of S versus θi, for TM polarization and n = 1.491, with kw and zr chosen to give the best fit to the experimental data.

Fig. 10
Fig. 10

Variation of F′/w and α versus kw for n = 1.491, z = 60 cm, and θi = θc.

Fig. 11
Fig. 11

Variation of S/w and L′/w versus kw for n = 1.491, z = 60 cm, and θi = θc.

Equations (62)

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

G i = G i ( x i , z i ) = ( w / w i ) exp [ ( x i / w i ) 2 + i k z i ] ,
w i 2 = w 2 + i ( 2 z i / k ) .
k w 1 ,
G i = k w 2 π 1 / 2 exp [ ( kws / 2 ) 2 + i k ( s x i + c z i ) ] d s ,
s = sin ( θ θ i ) ,
c = cos ( θ θ i ) ,
c = 1 s 2 / 2
G r = G r ( x r , z r ) = k w 2 π 1 / 2 r ( θ ) exp [ ( kws / 2 ) 2 + i k ( s x r + c z r ) ] d s ,
r ( θ ) = cos θ m ( sin 2 θ c sin 2 θ ) 1 / 2 cos + m ( sin 2 θ c sin 2 θ ) 1 / 2 ,
m = { 1 for normal ( TE ) polarization n 2 for parallel ( TM ) polarization .
sin θ c = 1 / n ,
Im [ ( sin 2 θ c sin 2 θ ) 1 / 2 ] 0 ,
G g = G g ( x r , z r ) = ( w / w r ) r ( θ i ) exp [ ( x r / w r ) 2 + i k z r ] ,
w r 2 = w 2 + i ( 2 z r / k ) .
G r = A r ( x r , z r ) ( w / w r ) r ( θ i ) exp [ ( x r L ) 2 / w f 2 + i k z r ] ,
w f 2 = w 2 + i 2 ( z r F ) / k ,
S = S ( z r ) = L + ( z r F ) tan α ,
α tan α = 2 L / [ ( 1 + μ ) ( k w 2 ) ] ,
μ = 2 F / ( k w 2 ) .
w m 2 = w 2 ( 1 + μ ) ,
u = u ( θ , θ ± ) = [ sin ( θ c θ ± ) s ± ] 1 / 2 ,
r ( θ ) = ν = 0 N [ R ( ν ) ( θ ± ) ( u u ± ) ν ] + Δ r ,
R ( ν ) ( θ ± ) = 1 ν ! d ν r ( θ ) d u ν | θ = θ ± ,
u ± = u ( θ ± , θ ± ) = sin 1 / 2 ( θ c θ ± ) ,
Δ r R ( N + 1 ) ( θ ± ) ( u u ± ) N + 1 .
| R ( N + 1 ) ( θ c ) ( 2 / k w r ) N + 1 | < 0.01 | ν = 1 N R ( ν ) ( θ c ) ( 2 / k w r ) ν | .
G r = k w 2 π 1 / 2 R ( ν ) ( θ ± ) ( u u ± ) ν × exp [ ( kws / 2 ) 2 + i k ( s x r + c z r ) ] d s + Δ G r ,
( Δ G r ) = 0 .
r ( θ ) = R i + ν = 0 N ρ ν ( θ ± ) u ν with R i = r ( θ i ) ,
ρ 0 ( θ ± ) = R ( 0 ) ( θ ± ) R ( 0 ) ( θ i ) R ( 1 ) ( θ ± ) u ± + R ( 2 ) ( θ ± ) u ± 2 R ( 3 ) ( θ ± ) u ± 3 , ρ 1 ( θ ± ) = R ( 1 ) ( θ ± ) 2 R ( 2 ) ( θ ± ) u ± + 3 R ( 3 ) ( θ ± ) u ± 2 , ρ 2 ( θ ± ) = R ( 2 ) ( θ ± ) 3 R ( 3 ) ( θ ± ) u ± , ρ 3 ( θ ± ) = R ( 3 ) ( θ ± ) .
υ = k w r [ sin ( θ c θ i ) s ] / 2 1 / 2 ,
u ( θ , θ ± ) = η ( θ ± ) ( υ 2 1 / 2 / k w r ) 1 / 2 ,
η ( θ ± ) = [ cos ( θ c θ ± ) / cos ( θ c θ i ) ] 1 / 2 .
0 exp ( i β υ υ 2 / 2 ) υ p 1 d υ = exp ( β 2 / 4 ) Γ ( p ) D p ( i β ) ,
G r = ( w / w r ) exp [ ( x r / w r ) 2 + i k z r ] [ R i + g ( x r , z r ) ] ,
g ( x r , z r ) = ν = 0 N [ c ν ( + ) D 1 ν / 2 ( + i β ) + c ν ( ) D 1 ν / 2 ( i β ) ] ,
β = β ( x r , z r ) = 2 1 / 2 [ ( x r , w r ) + i ( k w r / 2 ) sin ( θ c θ i ) ] ,
c ν ( ± ) = i ± ν / 2 σ ν ρ ν ( θ ± ) η ν ( θ ± ) ,
σ ν = ( k w ) ν / 2 Γ ( ν / 2 + 1 ) ( 2 π ) 1 / 2 ( i 2 1 / 2 w / w r ) ν / 2 exp ( β 2 / 4 ) .
x ¯ = x r / w r , x ¯ s = S / w r ,
R i + g ( x r , z r ) = exp [ ln ( R i + g 0 ) + g 1 ( x ¯ x ¯ s ) + g 2 ( x ¯ x ¯ s ) 2 ] ,
g 0 = g ( S , z r ) ,
g 1 = d g ( S , z r ) d x ¯ = 2 1 / 2 R 1 + g 0 d g ( S , z r ) d β ,
g 2 = 1 2 d 2 g ( S , z r ) d x ¯ 2 = 1 2 g 1 2 + 1 R 1 + g 0 d 2 g ( S , z r ) d β 2 .
d d β [ exp ( β 2 / 4 ) D p ( ± i β ) ] = i exp ( β 2 / 4 ) D p + 1 ( ± i β ) ,
d q g ( S , z r ) d β q = ( i ) q ν = 0 N [ c ν ( + ) D 1 ν / 2 + q ( + i β ) + ( ) q c ν ( ) D 1 ν / 2 + q ( i β ) ] .
A r ( x r , z r ) = ( 1 + g 0 / R i ) exp { g 1 x ¯ s + g 2 x ¯ s 2 + ( g 1 2 g 2 x ¯ s ) 2 / [ 4 ( 1 g 2 ) ] } ,
L / w = g 1 2 g 2 x ¯ s 2 ( 1 g 2 ) ( w r / w ) ,
F / w = i g 2 2 ( 1 g 2 ) ( k w ) ( w r / w ) 2 .
R ( 0 ) ( θ ) R ( 0 ) ( θ c ) + R ( 1 ) ( θ c ) u i with u i = u ( θ i , θ i ) ,
ρ 0 ( θ ± ) ρ 0 ( θ c ) R ( 1 ) ( θ c ) u i , ρ 1 ( θ ± ) ρ 1 ( θ c ) R ( 1 ) ( θ c ) ,
c ν ( + ) i ν c ν ( ) ( ν = 0 , 1 ) .
D p ( β ) = Γ ( p + 1 ) ( 2 π ) 1 / 2 [ exp ( i p π / 2 ) D p 1 ( i β ) + exp ( i p π / 2 ) D p 1 ( i β ) ] ,
g ( x r , z r ) = R ( 1 ) ( θ c ) [ u i + 2 1 / 4 ( k w ) 1 / 2 exp ( i π / 4 + β 2 / 4 ) D 1 / 2 ( β ) + i 1 / 2 i 1 / 2 2 5 / 4 ( k w ) 1 / 2 u i exp ( β 2 / 4 ) D 3 / 2 ( i β ) ,
S = w d 1 2 d 2 x ¯ 2 2 ( 1 d 2 ) ,
d 1 ( g 1 w r / w ) + ( g 1 w r / w ) 2 w r w r w r 2 w r 2 ,
d 2 = g 2 + g 2 2 w r w r w r 2 w r 2 ,
L / w ( k w ) 1 / 2 | 1 + 2 i ( z r / w ) ( k w ) 1 | 1 / 2 ,
F / w ( k w ) 1 / 2 | 1 + 2 i ( z r / w ) ( k w ) 1 | 3 / 2 ,
α ( L / w ) ( k w ) 1 ,
μ ( F / w ) ( k w ) 1 .
S / w ( k w ) 1 / 2 + a 1 ( k w ) 1 + a 2 ( z r / w ) ( k w ) 3 / 2 ,

Metrics