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.

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  1. F. Goos and H. Hänchen, Ann. Physik (6) 1, 333 (1947).
  2. H. K. V. Lotsch, J. Opt. Soc. Am. 58, 551 (1968).
  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).
  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 s 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 hence forth refer 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).
  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).

Artmann, K.

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

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

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).

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).

Lotsch, H. K. V.

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

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.

T. Tamir and A. A. Oliner, J. Opt. Soc. Am. 59, 942 (1969).

Osterberg, H.

H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1073 (1964).

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.

H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1073 (1964).

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.

T. Tamir and A. A. Oliner, J. Opt. Soc. Am. 59, 942 (1969).

Wolter, H.

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

Other

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

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

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.

T. Tamir and A. A. Oliner, J. Opt. Soc. Am. 59, 942 (1969).

H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1073 (1964).

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

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

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

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 s 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 hence forth refer 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.

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

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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