Abstract

The lateral (Goos–Hänchen) shift of a gaussian light beam incident from a denser medium upon the interface to a rarer medium is investigated by means of a rigorous integral representation comprising a continuous plane-wave spectrum. By applying a Fresnel approximation to that integral, we derive the lateral displacement for angles of incidence that are arbitrarily close to the critical angle of total reflection. Our results show that, in general, the lateral displacement is a function of the beam width, as well as the incidence angle; the classical expression appears as a limit case which holds only for large beam widths and for incidence angles that are not too close to the critical angle. An analysis of our expression for the beam shift reveals that, as the incidence angle approaches the critical angle of total reflection, the beam shift approaches a constant value that is strongly dependent on the beam width, in contrast to the classical expression, which predicts an infinitely large displacement; we also find that the maximum lateral displacement occurs at an angle that is slightly larger than the critical angle. Numerical results are presented in terms of normalized curves that are applicable to a wide range of realistic beams.

© 1971 Optical Society of America

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References

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  1. F. Goos and H. Hänchen, Ann. Physik (6)  1, 333 (1947).
    [Crossref]
  2. H. K. V. Lotsch, J. Opt. Soc. Am. 58, 551 (1968).
    [Crossref]
  3. H. K. V. Lotsch, “Die Strahlversetzung bei Totalreflexion: Der Goos–Hänchen Effekt”; Dr. Ing. dissertation, Technical University of Aachen, Germany, 1970.Also, Optik 32, 116, 189, 299, 553 (1970/71), in English.
  4. T. Tamir and A. A. Oliner, J. Opt. Soc. Am. 59, 942 (1969).
  5. H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1073 (1964).
    [Crossref]
  6. L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Ch. I, pp. 100–117.
  7. K. Artmann, Ann. Physik (6)  8, 270 (1951).
  8. H. Wolter, Z. Naturforsch. 5a, 143 (1950).
  9. Although Er0 does contain the geometric-optics field, it is not, strictly speaking, a result derivable solely from geometric optics because it also contains the effects of the free-space diffraction of the beam. If, however, we take the incident beam, with its freespace diffraction, and geometrically reflect it at z=0 (i.e., replace z by − z) and multiply the resultant field by the reflectance evaluated at the angle of beam incidence, we obtain Er0, as is subsequently shown. It is in this sense, therefore, that we shall henceforth refer to Er0 as the geometric-optics field.
  10. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics(Springer, New York, 1966), Ch. VIII, p. 331.
  11. K. Artmann, Ann. Physik (6)  2, 87 (1948).
    [Crossref]
  12. Reference 10, p. 324.
  13. Reference 10, p. 326.
  14. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 337, Eq. (3.462-1).
  15. Reference 14, p. 1066, Eq. (9.248-1).

1969 (1)

1968 (1)

1964 (1)

1951 (1)

K. Artmann, Ann. Physik (6)  8, 270 (1951).

1950 (1)

H. Wolter, Z. Naturforsch. 5a, 143 (1950).

1948 (1)

K. Artmann, Ann. Physik (6)  2, 87 (1948).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, Ann. Physik (6)  1, 333 (1947).
[Crossref]

Artmann, K.

K. Artmann, Ann. Physik (6)  8, 270 (1951).

K. Artmann, Ann. Physik (6)  2, 87 (1948).
[Crossref]

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Ch. I, pp. 100–117.

Goos, F.

F. Goos and H. Hänchen, Ann. Physik (6)  1, 333 (1947).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 337, Eq. (3.462-1).

Hänchen, H.

F. Goos and H. Hänchen, Ann. Physik (6)  1, 333 (1947).
[Crossref]

Lotsch, H. K. V.

H. K. V. Lotsch, J. Opt. Soc. Am. 58, 551 (1968).
[Crossref]

H. K. V. Lotsch, “Die Strahlversetzung bei Totalreflexion: Der Goos–Hänchen Effekt”; Dr. Ing. dissertation, Technical University of Aachen, Germany, 1970.Also, Optik 32, 116, 189, 299, 553 (1970/71), in English.

Magnus, W.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics(Springer, New York, 1966), Ch. VIII, p. 331.

Oberhettinger, F.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics(Springer, New York, 1966), Ch. VIII, p. 331.

Oliner, A. A.

Osterberg, H.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 337, Eq. (3.462-1).

Smith, L. W.

Soni, R. P.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics(Springer, New York, 1966), Ch. VIII, p. 331.

Tamir, T.

Wolter, H.

H. Wolter, Z. Naturforsch. 5a, 143 (1950).

Ann. Physik (3)

F. Goos and H. Hänchen, Ann. Physik (6)  1, 333 (1947).
[Crossref]

K. Artmann, Ann. Physik (6)  8, 270 (1951).

K. Artmann, Ann. Physik (6)  2, 87 (1948).
[Crossref]

J. Opt. Soc. Am. (3)

Z. Naturforsch. (1)

H. Wolter, Z. Naturforsch. 5a, 143 (1950).

Other (8)

Although Er0 does contain the geometric-optics field, it is not, strictly speaking, a result derivable solely from geometric optics because it also contains the effects of the free-space diffraction of the beam. If, however, we take the incident beam, with its freespace diffraction, and geometrically reflect it at z=0 (i.e., replace z by − z) and multiply the resultant field by the reflectance evaluated at the angle of beam incidence, we obtain Er0, as is subsequently shown. It is in this sense, therefore, that we shall henceforth refer to Er0 as the geometric-optics field.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics(Springer, New York, 1966), Ch. VIII, p. 331.

L. M. Brekhovskikh, Waves in Layered Media (Academic, New York, 1960), Ch. I, pp. 100–117.

H. K. V. Lotsch, “Die Strahlversetzung bei Totalreflexion: Der Goos–Hänchen Effekt”; Dr. Ing. dissertation, Technical University of Aachen, Germany, 1970.Also, Optik 32, 116, 189, 299, 553 (1970/71), in English.

Reference 10, p. 324.

Reference 10, p. 326.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), p. 337, Eq. (3.462-1).

Reference 14, p. 1066, Eq. (9.248-1).

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Figures (5)

F. 1
F. 1

Lateral displacement of actual reflected beam (solid lines) relative to geometrically reflected beam (dashed lines). The upper medium is the denser medium.

F. 2
F. 2

Geometry of the reflected-beam problem. showing the various coordinate systems.

F. 3
F. 3

Lateral displacement as a function of incidence angle for two different beam widths, for n = 1.94 (fused-quartz–air interface) and normal polarization. The appropriate classical result D, is shown by the dashed curve. The displacements for parallel polarization may be found, to an excellent approximation for the values shown, by merely multiplying the values of D0 and D0 by m.

F. 4
F. 4

Normalized critical displacement D c / ( w λ 0 ) 1 2 vs refractive index n, for parallel polarization (solid curve) and normal polarization (dashed curve).

F. 5
F. 5

Relative displacement D/Dc vs the incidence parameter k w δ / 2, for incidence angles larger than θc (solid curve) and incidence angles smaller than θc (dashed curve).

Equations (71)

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n = k / k 0 > 1 .
E w ( x i , 0 ) = exp [ ( x i / w ) 2 ] / π 1 2 w ,
k w 1
E a p ( x , h ) = [ E w ( x i , 0 ) exp ( i k z i ) ] z = h = exp [ ( x cos θ / w ) 2 + ikx sin θ ] π 1 2 w .
E inc ( x , z ) = 1 2 π + Φ ( k x ) exp { i [ k x x + k z ( z + h ) ] } d k x ,
k x 2 + k z 2 = k 2 = ( 2 π / λ ) 2 ,
Φ ( k x ) = + E a p ( x , h ) exp ( i k x x ) d x ,
Φ ( k x ) = exp ( { [ ( k x k sin θ ) / 2 cos θ ] w } 2 ) cos θ .
E inc ( x , z ) = 1 2 π cos θ + exp [ ( k x k sin θ 2 cos θ w ) 2 ] × exp { i [ k x x + k z ( z + h ) ] } d k x .
E ref 1 ( x , z ) = 1 2 π cos θ + Γ ( k x ) exp [ ( k x k sin θ 2 cos θ w ) 2 ] × exp { i [ k x x k z ( z h ) ] } d k x ,
Γ ( k x ) = ( k 2 k x 2 ) 1 2 m ( k 0 2 k x 2 ) 1 2 ( k 2 k x 2 ) 1 2 + m ( k 0 2 k x 2 ) 1 2 .
m = { n 2 for polarisation parallel to the plan of incidence , 1 for polarisation normal to the plan of incidence ,
k z k cos θ ( k x k sin θ ) tan θ [ ( k x k sin θ ) 2 / 2 k cos 3 θ ] .
E inc { exp [ ( x i / w i ) 2 ] / π 1 2 w i } exp ( i k z i ) ,
w i 2 = w 2 + i ( 2 / k ) ( z i x i tan θ ) .
sin θ c = 1 / n .
Γ ( k x ) = Γ ( k sin θ ) [ 1 + Γ c ( k x ) ] ,
Γ c ( k x ) = [ Γ ( k x ) Γ ( k sin θ ) ] / Γ ( k sin θ ) ] ,
E ref 1 ( x , z ) = E r 0 ( x , z ) [ 1 + e c ( x , z ) ] ,
E r 0 ( x , z ) = Γ ( k sin θ ) 2 π cos θ + exp [ ( k x k sin θ 2 cos θ w ) 2 ] × exp { i [ k x x k z ( z h ) ] } d k x ;
e c ( x , z ) = Γ ( k sin θ ) 2 π cos θ E r 0 ( x , z ) × + Γ c ( k x ) exp [ ( k x k sin θ 2 cos θ w ) 2 ] × exp { i [ k x x k z ( z h ) ] } d k x .
E r 0 ( x , z ) = Γ ( k sin θ ) E inc ( x , z ) .
E r 0 Γ ( k sin θ ) exp [ ( x r / w r ) 2 ] π 1 2 w r exp ( i k z r ) ,
w r 2 = w 2 + i ( 2 / k ) ( z r x r tan θ ) .
e c A ( θ ) [ ( δ ) 1 2 2 1 4 e i π / 4 exp ( γ 2 / 4 ) ( k w r ) 1 2 D 1 2 ( γ ) ] ,
A ( θ ) = 4 m cos 2 θ c × sin θ cos 1 2 θ ( sin θ + sin θ c ) 1 2 [ cos 2 θ + m 2 ( sin 2 θ + sin 2 θ c ) ] ,
γ = 2 1 2 ( i k w r δ 2 x r w r ) ,
δ = ( sin θ sin θ c ) sec θ θ θ c ,
w r w
E ref 1 Γ ( k sin θ ) π 1 2 w × exp [ ( x r / w ) 2 + ln ( 1 + e c ) ] exp ( i k z r ) .
ln ( 1 + e c ) = a 0 + a 1 ( x r / w ) + a 2 ( x r / w ) 2 + ,
a n = ( 1 / n ! ) [ d n / d ( x r / w ) n ] [ ln ( 1 + e c ) ] γ = γ 0
γ 0 = γ | ( x r / w ) = 0 = ikw δ / 2 .
( x r w ) 2 + ln ( 1 + e c ) = 1 a 2 w 2 × [ x r a 1 w 2 ( 1 a 2 ) ] 2 + a 0 + a 1 2 4 ( 1 a 2 ) .
D ¯ cos θ = a 1 w / 2 ( 1 a 2 ) ,
w ¯ 2 = w 2 / ( 1 a 2 ) ,
f = a 0 + a 1 2 / 4 ( 1 a 2 ) ,
E ref 1 Γ ( k sin θ ) π 1 2 w ¯ exp [ ( x r D ¯ cos θ w ¯ ) 2 ] × exp ( i k z r + f ) ,
D A ( θ ) 2 5 / 4 cos θ Re { ( w k ) 1 2 e i π / 4 exp ( γ 0 2 / 4 ) D 1 2 ( γ 0 ) 1 + A ( θ ) [ ( δ ) 1 2 ( 2 1 4 e i π / 4 / ( k w ) 1 2 ) exp ( γ 0 2 / 4 ) D 1 2 ( γ 0 ) ] } ,
D = m λ 0 ( n 2 1 ) π n 3 × tan θ ( sin 2 θ sin 2 θ c ) 1 2 [ cos 2 θ + m 2 ( sin 2 θ sin 2 θ c ) ] .
| γ 0 | = | k w δ / 2 | 1
exp ( γ 2 / 4 ) D ν ( γ ) γ ν , for | γ | 1 and | arg γ | < 3 π / 4 .
| γ 0 | = | k w δ / 2 | 1 .
exp ( γ 2 4 ) D ν ( γ ) D ν ( 0 ) = Γ ( 1 2 ) 2 ν / 2 Γ [ ( 1 ν ) / 2 ] , for | γ | 1 .
D Γ ( 1 4 ) A ( θ ) 4 cos θ Re [ ( w π k ) 1 2 × e i π / 4 1 A ( θ ) / ( k w ) 1 2 { [ ( 2 π ) 1 2 / Γ ( 1 4 ) ] e i π / 4 ( k w δ ) 1 2 ] } ] .
D D c = Γ ( 1 4 ) m ( tan θ c ) 1 2 2 3 2 n 1 2 π cos θ c ( w λ c ) 1 2 ,
D A ( θ ) 2 5 / 4 cos θ Re [ ( w k ) 1 2 e i π / 4 exp ( γ 0 2 / 4 ) D 1 2 ( γ 0 ) ] .
( d / d δ ) D = 0 ,
Re [ ( 1 i ) w 3 2 exp ( γ 0 2 / 4 ) D 3 2 ( γ 0 ) ] = 0 ,
( d / d γ ) [ exp ( γ 2 / 4 ) D ν ( γ ) ] = ν exp ( γ 2 / 4 ) D ν 1 ( γ ) .
γ 0 = ikw δ / 2 i 0.77 .
σ = ( k x k sin θ ) / k cos θ ,
k x = k ( 1 σ cot θ ) sin θ .
k z k cos θ [ 1 + σ tan θ ( σ 2 / 2 ) sec 2 θ ] .
E inc k 2 π exp { i k [ x sin θ + ( z + h ) cos ] } × exp [ ( k w i σ 2 ) 2 ] × exp { i k [ z + h ] sin θ x cos θ ] σ } d σ ,
w i 2 = w 2 + i 2 ( z + h ) / k cos θ .
x i r = x cos θ ( z ± h ) sin θ ,
z i r = x sin θ ( z ± h ) cos θ ,
τ = σ δ = ( sin θ c k x / k ) sec θ ,
k x = k [ sin θ ( τ + δ ) cos θ ] .
k z k cos θ [ 1 + ( τ + δ ) tan θ ( τ + δ ) 2 / 2 cos 2 θ ] .
Γ c ( k x ) = A ( θ ) [ τ 1 2 ( δ ) 1 2 ] + O { [ τ 1 2 ( δ ) 1 2 ] 2 } ,
e c k A ( θ ) 2 π 1 2 E r 0 exp [ ( k w r δ 2 ) 2 i k δ x r ] × exp ( i k z r ) [ I 1 ( δ ) 1 2 I 2 ] ,
I 1 = + τ 1 2 exp [ ( k w r τ 2 ) 2 ikb τ ] d τ ,
I 2 = + exp [ ( k w r τ 2 ) 2 ikb τ ] d τ ,
b = x r ( i k δ / 2 ) w r 2 .
I 1 = 0 + τ 1 2 exp [ ( k w r τ 2 ) 2 ikb τ ] d τ + i 0 + τ 1 2 exp [ ( k w r τ 2 ) 2 + ikb τ ] d τ ,
I 1 = [ ( k w r ) 2 2 ] 3 2 Γ ( 3 2 ) exp ( γ 2 4 ) × [ D 3 2 ( i γ ) + i D 3 2 ( i γ ) ] ,
D ν ( γ ) = Γ ( ν + 1 ) ( 2 π ) 1 2 [ e i ν π / 2 D ν 1 ( i γ ) + e i ν π / 2 D ν 1 ( i γ ) ]
I 1 = 2 5 / 4 π ( k w r ) 3 2 e i π / 4 exp ( γ 2 / 4 ) D 1 2 ( γ ) .
I 2 = ( 2 π / k w r ) exp [ ( b / w r ) 2 ] .