Abstract

A general expression for the Goos–Hänchen lateral shift of a Gaussian wave beam, applicable for incident angle θ equal to, near, or larger than the critical angle θc, is obtained. It yields finite and smooth results around θc and coincides with Artmann’s classical result otherwise. Another expression for the lateral shift of a different beam-field distribution is also obtained and shown to be quite similar to that of the Gaussian beam. Comparisons with an existing theoretical expression and with available experimental data (for θθc) are made. Discussion of a different expression for θ away from θc obtained through an energy method is also presented.

© 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, 333–346 (1947); “Neumessung des Strahlversetzungseffektes bei Totalreflexion,” Ann. Physik 5, 251–252 (1949).
    [Crossref]
  2. K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Physik 2, 87–102 (1948).
    [Crossref]
  3. R. H. Renard, “Total reflection: a new evaluation of the Goos–Hänchen Shift,”J. Opt. Soc. Am. 54, 1190–1197 (1964). Strictly speaking, the analytic expression obtained in this paper applies only to incident angles close to the critical angle (see the discussion in Section 6).
    [Crossref]
  4. J. L. Agudin, “Time delay of scattering processes,” Phys. Rev. 171, 1385–1387 (1968).
    [Crossref]
  5. B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,”J. Opt. Soc. Am. 61, 586–594 (1971).
    [Crossref]
  6. K. W. Chiu, J. J. Quinn, “On the Goos–Hänchen effect: a simple example of time delay scattering process,” Am. J. Phys. 40, 1847–1851 (1972).
    [Crossref]
  7. M. McGuirk, C. K. Carniglia, “An angular spectrum representation approach to the Goos–Hänchen Shift,”J. Opt. Soc. Am. 67, 103–107 (1977).
    [Crossref]
  8. See, for example, H. Kogelnik, H. P. Weber, “Rays, stored energy, and power flow in dielectric waveguides,” J. Opt. Soc. Am. 64, 174–185 (1974).
    [Crossref]
  9. W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966), pp. 323–332.
  10. The width σ defined here is related to Horowitz’s and Tamir’s w by σ=w/2.
  11. J. J. Cowan, B. Aničin, “Longitudinal and transverse displacements of a bounded microwave beam at total internal reflection,”J. Opt. Soc. Am. 67, 1307–1314 (1977).
    [Crossref]
  12. Similar measurement of the shifts for one incident angle was carried out earlier by V. Akylas, J. Kaur, T. M. Knasel, “Longitudinal shift of a microwave beam at total internal reflection,” Appl. Opt. 13, 742–743 (1974).
    [Crossref] [PubMed]
  13. J. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 337, Eq. (3.4621).
  14. The indication made by Horowitz and Tamir that the shift is nearly constant for θ≃ θc is misleading.
  15. The reason for taking such a value is for convenience of comparison with Cowan’s and Aničin’s experimental data in Section 5.
  16. Note that our S(θ) is related to the Cowan–Aničin X(θ) by S(θ) = X(θ)cos θ.
  17. Cowan and Aničin essentially took kσ=43/2≐30. Judging from their experimental data, we feel that kσ= 43 reflects the real situation better.

1977 (2)

1974 (2)

1972 (1)

K. W. Chiu, J. J. Quinn, “On the Goos–Hänchen effect: a simple example of time delay scattering process,” Am. J. Phys. 40, 1847–1851 (1972).
[Crossref]

1971 (1)

1968 (1)

J. L. Agudin, “Time delay of scattering processes,” Phys. Rev. 171, 1385–1387 (1968).
[Crossref]

1964 (1)

1948 (1)

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Physik 2, 87–102 (1948).
[Crossref]

1947 (1)

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Physik 1, 333–346 (1947); “Neumessung des Strahlversetzungseffektes bei Totalreflexion,” Ann. Physik 5, 251–252 (1949).
[Crossref]

Agudin, J. L.

J. L. Agudin, “Time delay of scattering processes,” Phys. Rev. 171, 1385–1387 (1968).
[Crossref]

Akylas, V.

Anicin, B.

Artmann, K.

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Physik 2, 87–102 (1948).
[Crossref]

Carniglia, C. K.

Chiu, K. W.

K. W. Chiu, J. J. Quinn, “On the Goos–Hänchen effect: a simple example of time delay scattering process,” Am. J. Phys. 40, 1847–1851 (1972).
[Crossref]

Cowan, J. J.

Goos, F.

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Physik 1, 333–346 (1947); “Neumessung des Strahlversetzungseffektes bei Totalreflexion,” Ann. Physik 5, 251–252 (1949).
[Crossref]

Gradshteyn, J. S.

J. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 337, Eq. (3.4621).

Hänchen, H.

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Physik 1, 333–346 (1947); “Neumessung des Strahlversetzungseffektes bei Totalreflexion,” Ann. Physik 5, 251–252 (1949).
[Crossref]

Horowitz, B. R.

Kaur, J.

Knasel, T. M.

Kogelnik, H.

Magnus, W.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966), pp. 323–332.

McGuirk, M.

Oberhettinger, F.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966), pp. 323–332.

Quinn, J. J.

K. W. Chiu, J. J. Quinn, “On the Goos–Hänchen effect: a simple example of time delay scattering process,” Am. J. Phys. 40, 1847–1851 (1972).
[Crossref]

Renard, R. H.

Ryzhik, I. M.

J. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 337, Eq. (3.4621).

Soni, R. P.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966), pp. 323–332.

Tamir, T.

Weber, H. P.

Am. J. Phys. (1)

K. W. Chiu, J. J. Quinn, “On the Goos–Hänchen effect: a simple example of time delay scattering process,” Am. J. Phys. 40, 1847–1851 (1972).
[Crossref]

Ann. Physik (2)

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Physik 1, 333–346 (1947); “Neumessung des Strahlversetzungseffektes bei Totalreflexion,” Ann. Physik 5, 251–252 (1949).
[Crossref]

K. Artmann, “Berechnung der Seitenversetzung des totalreflektierten Strahles,” Ann. Physik 2, 87–102 (1948).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. (5)

Phys. Rev. (1)

J. L. Agudin, “Time delay of scattering processes,” Phys. Rev. 171, 1385–1387 (1968).
[Crossref]

Other (7)

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer-Verlag, New York, 1966), pp. 323–332.

The width σ defined here is related to Horowitz’s and Tamir’s w by σ=w/2.

J. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 337, Eq. (3.4621).

The indication made by Horowitz and Tamir that the shift is nearly constant for θ≃ θc is misleading.

The reason for taking such a value is for convenience of comparison with Cowan’s and Aničin’s experimental data in Section 5.

Note that our S(θ) is related to the Cowan–Aničin X(θ) by S(θ) = X(θ)cos θ.

Cowan and Aničin essentially took kσ=43/2≐30. Judging from their experimental data, we feel that kσ= 43 reflects the real situation better.

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Figures (5)

Fig. 1
Fig. 1

The usual XZ coordinate system for Eq. (6), the incident XZ′ coordinate system, and the reflection XZ″ coordinate system.

Fig. 2
Fig. 2

Universal lateral shift as a function of incident angle. Solid line, according to Eq. (24) for a Gaussian beam; dashed line, according to Eq. (40) for a single-slit beam.

Fig. 3
Fig. 3

Lateral shift, in units of k−1 = λ/2π, versus the incident angle for the TE case and for = 103, 102, and 10, taking n = 1.491.

Fig. 4
Fig. 4

Comparison of various theoretical results with the experimental data of Cowan and Aničin. Δ, the TM case; ○, the TE case.

Fig. 5
Fig. 5

A wave beam totally reflected from a dielectric–vacuum interface. The dashed line and the dotted line indicate, respectively, the boundary of the strip of width S and that of the edge layer of width λ. (The drawing is not to scale.)

Equations (62)

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δ ( sin θ - sin θ c ) sin θ θ - θ c 1 ,
S HT ( θ ) = α ( k σ ) 1 / 2 2 k × Re { e i π / 4 exp ( γ 0 2 / 4 ) D - 1 / 2 ( γ 0 ) 1 + α [ ( - δ ) 1 / 2 - e i π / 4 exp ( γ 0 2 / 4 ) D 1 / 2 ( γ 0 ) / ( k σ ) 1 / 2 ] }
γ 0 i k σ δ .
D ν ( γ 0 ) e γ 0 ² / ( 2 ν π ) 1 / 2 [ 1 / Γ ( 1 / 2 - ν / 2 ) - 2 1 / 2 γ 0 / Γ ( - ν / 2 ) ] ,
F ( x , z ) = - F ˜ ( k x ) exp [ i ( k x x + k z z - ω t ) d k x ,
k = ( k x 2 + k z 2 ) 1 / 2 = n ω / c
R ( k x ) = ( k 2 - k x 2 ) 1 / 2 - m ( k 2 sin 2 θ c - k x 2 ) 1 / 2 ( k 2 - k x 2 ) 1 / 2 + m ( k 2 sin 2 θ c - k x 2 ) 1 / 2 ,
m = 1             or             m = n 2 ,
k x = k x cos θ + k z sin θ , k z = k x sin θ - k z cos θ ,
k z = ( k 2 - k z 2 ) 1 / 2 .
F = - R ( k x ) F ˜ ( k x ) exp [ i ( k x x + k z z - ω t ) ] d k x .
k σ 1 ,
k x / k 1
R ( k x ) = exp ( - i ϕ ) exp ( - i ϕ 0 ) exp ( - i ϕ k x k x ) ,
S A = 2 m × cos 2 θ c × sin θ k ( sin 2 θ - sin 2 θ c ) 1 / 2 [ cos 2 θ + m 2 ( sin 2 θ - sin 2 θ c ) ] .
R ( k x ) = [ A + B ( - Δ - k x / k ) 1 / 2 + C k x / k + E ( - Δ - k x / k ) 1 / 2 k x / k ] / G ,
A = cos 2 θ - m 2 ( sin 2 θ - sin 2 θ c ) , B = - 2 m cos θ ( sin 2 θ ) 1 / 2 , C = - 2 m 2 cos 2 θ c sin 2 θ / G , E = 2 m sin θ ( sin 2 θ ) 1 / 2 [ m 2 ( cos 2 θ + cos 2 θ c ) - cos 2 θ ] / G , G = cos 2 θ + m 2 ( sin 2 θ - sin 2 θ c ) , Δ = ( sin 2 θ - sin 2 θ c ) / sin 2 θ .
tan θ k σ ,
G ( n 4 - 1 ) sin 2 θ / k σ ,
2 ( n 2 + 1 ) ( n 2 - 1 ) 1 / 2 k σ .
F ˜ ( k x ) = σ F 0 exp ( - k x 2 σ 2 / 2 ) ,
- d τ × τ 1 / 2 exp ( - a τ 2 + i b τ ) ,
F ( x , z ) = ( 2 π ) 1 / 2 ( F 0 / G ) exp ( - x 2 / 2 σ 2 ) f ( x ) × exp [ i ( k z - ω t ) ] ,
f ( x ) = A + B e i π / 4 × e β 2 / 4 × D 1 / 2 ( β ) / ( k σ ) 1 / 2 + i C x / k σ 2 + i E e i π / 4 × e β 2 / 4 × [ D 1 / 2 ( β ) x / σ + D - 1 / 2 ( β ) / 2 ] / ( k σ ) 3 / 2 , β = i k σ Δ - x / σ .
exp ( - x 2 / 2 σ 2 ) f ( x ) = exp ( - x 2 2 σ 2 + ln f )
ln f = a o + a 1 x / σ + a 2 x 2 / 2 σ 2
exp ( - x 2 / 2 σ 2 ) f ( x ) exp { - [ ( 1 - a 2 ) x 2 - 2 σ a 1 x ] / 2 σ 2 } .
S ( θ ) = σ Re ( a 1 ) / [ 1 - Re ( a 2 ) ] .
a 1 = e i π / 4 × e β 0 2 / 4 [ - B D - 1 / 2 ( β 0 ) + i 2 E D 1 / 2 ( β 0 ) / k σ ] + i 2 C / ( k σ ) 1 / 2 2 [ ( k σ ) 1 / 2 A + B e i π / 4 × e β 0 2 / 4 × D 1 / 2 ( β 0 ) ] ,
a 2 = - B e i π / 4 × e β 0 2 / 4 × D - 3 / 2 ( β 0 ) / [ 4 A ( k σ ) 1 / 2 ] ,
β 0 = i k σ Δ .
p + q ( θ - θ c ) ,
S ( θ ) = 1.16 × S ( θ c ) Re { e i π / 4 × e β 0 2 / 4 × D - 1 / 2 ( β 0 ) } ,
S ( θ c ) = 1.22 × m ( tan θ c ) 1 / 2 k - 1 ( k σ ) 1 / 2
S m / S ( θ c ) = 1.185
θ m = θ c + 0.765 / k σ ,
D ν ( β 0 ) = e - β 0 2 / 4 × β 0 ν [ 1 + ( β 0 - 2 - order terms ) ]
S ( θ ) = B ( 2 C Δ - A ) - 2 A E Δ 2 k ( A 2 + B 2 Δ ) Δ 1 / 2 .
A 2 + B 2 Δ = [ cos 2 θ - m 2 ( sin 2 θ c - sin 2 θ ) ] 2
2 m ( sin 2 θ ) 1 / 2 cos 2 θ c [ cos 2 θ - m 2 ( sin 2 θ c - sin 2 θ ) ] / cos θ .
F ˜ ( k x ) = { F 0 / k for k x k 0 otherwise ,
F ( x , z ) = 2 k F 0 ( sin k x k x ) exp [ i ( k z - ω t ) ] ,
Δ θ - θ c .
F = ( 2 F 0 / G ) exp [ i ( k z - ω t ) ] g ( k x ) ,
g ( ξ ) = A sin ξ / ξ + 1 / 2 B exp ( - i Δ ξ / ) J ( ξ )
J ( 2 3 / 2 ) - 1 Δ - Δ + ( - η ) 1 / 2 exp ( i k x η ) d η .
exp ( i k x η ) = 1 + i k x η .
J = 1 3 ( 1 - Δ ) 3 / 2 - ξ 5 ( 1 + Δ ) 5 / 2 + i [ 1 3 ( 1 + Δ ) 3 / 2 - ξ 5 ( 1 - Δ ) 5 / 2 ] ,
J = ξ 5 [ ( Δ - 1 ) 5 / 2 - ( Δ + 1 ) 5 / 2 ] + i 3 [ ( Δ + 1 ) 3 / 2 - ( Δ - 1 ) 3 / 2 ] ,
I g 2 ,
d I d x = 0
S ( θ c ) = 6 m ( 2 tan θ c ) 1 / 2 / ( 5 k 1 / 2 ) .
S ( θ ) = S ( θ c ) × h ( Δ / ) ,
h ( a ) = { ( 1 + a ) 3 / 2 ( 1 - 2 a / 3 ) for a < 1 ( a - 1 ) 3 / 2 ( 2 a / 3 + 1 ) - ( a + 1 ) 3 / 2 ( 2 a / 3 - 1 ) for a 1 .
k σ = 2 π n σ / λ 0 = 43.
R = N / D ,
N = k 2 - k x 2 + m 2 ( k 2 sin 2 θ c - k x 2 ) - 2 m ( k 2 - k x 2 ) 1 / 2 ( k 2 sin 2 θ c - k x 2 ) 1 / 2 , D = k 2 - k x 2 - m 2 ( k 2 sin 2 θ c - k x 2 ) .
( k 2 sin 2 θ c - k x 2 ) 1 / 2 = k ( sin 2 θ ) 1 / 2 ( - Δ - k x / k ) 1 / 2 ,
N / k 2 = A - ( m 2 + 1 ) sin 2 θ ( k x / k ) + B [ 1 - tan θ ( k x / k ) ] ( - Δ - k x / k ) 1 / 2
D / k 2 = G + ( m 2 - 1 ) sin 2 θ ( k x / k ) ,
( - Δ - k x / k ) 1 / 2
[ - Δ - k x / k - ( k x / k ) 2 cot 2 θ ] 1 / 2 .

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