Abstract

The reflection of a beam of light at a plane interface is treated using the angular spectrum representation. The Goos-Hänchen shift is found to be proportional to the first derivative of the phase of the reflectance. The second derivative of the phase gives rise to a shift of the reflected beam along its direction of propagation. This new shift, called a focal shift, is different from the extra propagation distance of the beam predicted on the basis of a ray model for total internal reflection. Expressions are presented for the Goos-Hänchen and focal shifts for both s and p polarization.

© 1977 Optical Society of America

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References

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  1. F. Goos and H. Hänchen, Ann. Phys. (Leipz.) 1, 333 (1947).
    [CrossRef]
  2. K. Artmann, Ann. Physik (6) 2, 87 (1948).
    [CrossRef]
  3. B. R. Horowitz and T. Tamir, J. Opt. Soc. Am. 61, 586 (1971).
    [CrossRef]
  4. M. McGuirk and C. K. Carniglia, J. Opt. Soc. Am. 65, 1168 (1975).
  5. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 74.
  6. H. Kogelnik and H. P. Weber, J. Opt. Soc. Am. 64, 174 (1974).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 48 ff.
  8. E. Lalor, J. Opt. Soc. Am. 58, 1235 (1968).
    [CrossRef]
  9. W. H. Carter, Opt. Acta 21, 871 (1974).
    [CrossRef]
  10. M. Green, P. Kirkby, and R. S. Timsit, Phys. Lett. A 45, 259 (1973).
    [CrossRef]
  11. H. K. V. Lotsch, J. Opt. Soc. Am. 58, 551 (1968).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 49.
  13. R. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964).
    [CrossRef]
  14. I. Newton, Optiks (Dover, New York, 1952).

1975 (1)

M. McGuirk and C. K. Carniglia, J. Opt. Soc. Am. 65, 1168 (1975).

1974 (2)

1973 (1)

M. Green, P. Kirkby, and R. S. Timsit, Phys. Lett. A 45, 259 (1973).
[CrossRef]

1971 (1)

1968 (2)

1964 (1)

1948 (1)

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

1947 (1)

F. Goos and H. Hänchen, Ann. Phys. (Leipz.) 1, 333 (1947).
[CrossRef]

Artmann, K.

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

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 49.

Burke, J. J.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 74.

Carniglia, C. K.

M. McGuirk and C. K. Carniglia, J. Opt. Soc. Am. 65, 1168 (1975).

Carter, W. H.

W. H. Carter, Opt. Acta 21, 871 (1974).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 48 ff.

Goos, F.

F. Goos and H. Hänchen, Ann. Phys. (Leipz.) 1, 333 (1947).
[CrossRef]

Green, M.

M. Green, P. Kirkby, and R. S. Timsit, Phys. Lett. A 45, 259 (1973).
[CrossRef]

Hänchen, H.

F. Goos and H. Hänchen, Ann. Phys. (Leipz.) 1, 333 (1947).
[CrossRef]

Horowitz, B. R.

Kapany, N. S.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 74.

Kirkby, P.

M. Green, P. Kirkby, and R. S. Timsit, Phys. Lett. A 45, 259 (1973).
[CrossRef]

Kogelnik, H.

Lalor, E.

Lotsch, H. K. V.

McGuirk, M.

M. McGuirk and C. K. Carniglia, J. Opt. Soc. Am. 65, 1168 (1975).

Newton, I.

I. Newton, Optiks (Dover, New York, 1952).

Renard, R. H.

Tamir, T.

Timsit, R. S.

M. Green, P. Kirkby, and R. S. Timsit, Phys. Lett. A 45, 259 (1973).
[CrossRef]

Weber, H. P.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 49.

Ann. Phys. (Leipz.) (1)

F. Goos and H. Hänchen, Ann. Phys. (Leipz.) 1, 333 (1947).
[CrossRef]

Ann. Physik (6) (1)

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

J. Opt. Soc. Am. (6)

Opt. Acta (1)

W. H. Carter, Opt. Acta 21, 871 (1974).
[CrossRef]

Phys. Lett. A (1)

M. Green, P. Kirkby, and R. S. Timsit, Phys. Lett. A 45, 259 (1973).
[CrossRef]

Other (4)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 48 ff.

N. S. Kapany and J. J. Burke, Optical Waveguides (Academic, New York, 1972), p. 74.

I. Newton, Optiks (Dover, New York, 1952).

M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 49.

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Figures (2)

FIG. 1
FIG. 1

Reflection at a boundary B between two optical regions. The x, z axis is used for the incident beam. The reflected beam is expressed in terms of the x′, z′ axis. The y and y′ axes are directed out of and into the page, respectively. The angle of incidence and reflection of the beam is θ0. Region 2 is the low-index region in the case of total internal reflection.

FIG. 2
FIG. 2

Ray model of total internal reflection illustrating the incident beam, ideal reflected beam, and the shifted (actual) reflected beam. The angle of reflection is θ0. The incident beam travels along the dashed path PQR as if it were reflected at a depth Z. This gives rise to the GH shift D.

Equations (49)

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k = n ω / c ,
e i ( k x x + k z z ) ,
k z = ( k 2 - k x 2 ) 1 / 2 .
α = arcsin ( k x / k ) .
A 0 ( k x ) = 1 2 π E ( x , 0 ) e - i k x x d x .
E ( x , z ) = A 0 ( k x ) e i ( k x x + k z z ) d k x .
A ( k x , z ) = A 0 ( k x ) e i k z z .
k x A 0 ( k x ) 2 d k x = 0.
A 0 ( k x ) max [ A 0 ( k x ) ] ,
k x / k > sin α 0 .
A 0 ( k x ) e i ( k x x + k z z )
A r ( k x ) e i ( k x x + k z z ) ,
k x = k x ,
k z = k z .
A r ( k x ) = r ( k x ) A 0 ( k x ) e i k z z O .
E r ( x , z ) = A r ( k x ) e i ( k x x + k z z ) d k x
E r ( x , z ) = r ( k x ) A ( k x , z 0 ) e i ( k x x + k z z ) d k x .
E I ( x , z ) = A ( k x , z 0 ) e i ( k x x + k z z ) d k x .
θ c = arcsin ( 1 / n ) .
θ 0 > θ c + α 0 .
θ 0 < 90 - α 0 .
r ( k x ) = e i δ ( k x ) ,
δ ( k x ) = δ 0 + k x D + k x 2 F / 2 k + ,
δ 0 = δ ( 0 ) ,
D = d δ d k x | k x = 0 ,
F = k d 2 δ d k x 2 | k x = 0 .
E r ( x , z ) = e i δ 0 A ( k x , z 0 ) e i [ k x ( x + D ) + k z z ] d k x .
E r ( x , z ) = e i δ 0 E I ( x + D , z ) .
θ = θ 0 - arcsin ( k x / k ) .
D = - λ 2 π d δ d θ | θ = θ 0 ,
δ s ( θ ) = - 2 tan - 1 [ ( n 2 sin 2 θ - 1 ) 1 / 2 / n cos θ ] + π .
D s = ( λ / π ) n sin θ 0 / ( n 2 sin 2 θ 0 - 1 ) 1 / 2 .
δ p ( θ ) = - 2 tan - 1 [ n ( n 2 sin 2 θ - 1 ) 1 / 2 / cos θ ] + π ,
D p = D s / [ ( n 2 + 1 ) sin 2 θ 0 - 1 ] .
k x 2 / 2 k = k - k z - O ( k x 2 / k 3 ) .
δ ( k x ) = δ 0 + k x D + ( k - k z ) F .
E r ( x , z ) = e i ( δ 0 + k F ) A ( k x , z 0 ) e i [ k x ( x + D ) + k z ( z - F ) ] d k x
E r ( x , z ) = e i ( δ 0 + k F ) E I ( x + D , z - F ) .
F = 1 k d 2 δ d θ 2 | θ = θ 0 .
F s = ( λ / π ) n cos θ 0 / ( n 2 sin 2 θ 0 - 1 ) 3 / 2 .
F s = D s cot θ 0 / ( n 2 sin 2 θ 0 - 1 ) .
F p = ( 1 + f ) F s / [ ( n 2 + 1 ) sin 2 θ 0 - 1 ]
F p = ( 1 + f ) D p cot θ 0 / ( n 2 sin 2 θ 0 - 1 ) ,
f = 2 ( n 2 sin 2 θ 0 - 1 ) ( n 2 + 1 ) sin 2 θ 0 / [ ( n 2 + 1 ) sin 2 θ 0 - 1 ] .
D s ( λ / π ) / ( n 2 sin 2 θ 0 - 1 ) 1 / 2
D p n 2 D s .
F s ( λ / π ) ( n 2 - 1 ) 1 / 2 / ( n 2 sin 2 θ 0 - 1 ) 3 / 2
F p n 2 F s .
F r = - D cot θ 0 .