Abstract

The reflection and refraction of a beam of light is treated. Approximate solutions of Maxwell’s equations are used to describe the electromagnetic field of the beam being limited in the transverse direction. The point of departure is the classical paper by Schaefer and Pich. The laws of reflection and refraction are derived. Fresnel’s formulas and their corrections are presented for both polarizations. The case of total reflection is investigated for <i>E</i> polarization in greater detail. The electromagnetic fields and the time-average Poynting vectors are explicitly derived for both the optically dense and less-dense media. The flow of energy at total reflection is studied extensively. It is shown that, due to the flow of energy in the less-dense medium, the center of gravity of the reflected beam is displaced, as was suggested by v. Fragstein. This leads to a shift of the totally reflected beam with respect to a geometrically reflected beam, as was experimentally demonstrated by Goos and (Lindberg-) Haenchen. New expressions for this “Goos-Haenchen shift” are derived. These expressions reduce to the classical formulas deduced by Artmann, v. Fragstein, Wolter, and Maecker if the angle of incidence is only slightly larger than the critical angle of total reflection. In consistency with Renard’s viewpoint they, however, predict vanishing shifts in the limit of grazing incidence.

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  1. G. Quincke, Sitzber. d. k. Ges. . Berlin, p. 294 (1865); Pogg. Ann. 127, 1 and 199 (1865).
  2. W. Voigt, Nachr. Akad. Wiss. Goettingen, II. Math. Physik. Kl., p. 49 (1884); Ann. Physik (9) 67, 185 (1899); Ann. Physik (4) 34, 798 (1911).
  3. E. Ketteler, Ann. Physik (9) 67, 879 (1899).
  4. P. Drude, The Theory of Optics (Dover Publications, Inc., New York 1959), originally published in German (1900).
  5. A. Eichenwald, Ann. d. Moskauer Kais. Ingenieur-Hochschule, p. 15, April 1908; J. Russ. Phys.-Chem. Soc. 41, phys. part, 131 (1909).
  6. E. E. Hall, Phys. Rev. 15, 73 (1902).
  7. W. von Ignatowsky and E. Oettinger, Ann. Physik (4) 25, 99 (1908); (4) 26, 1031 (1908); (4) 37, 901 (1912); and (4) 37, 911 (1912).
  8. C. Schaefer and G. Gross, Ann. Physik (4) 32, 648 (1910).
  9. W. Voigt, Ann. Physik (9) 67, 185 (1899).
  10. A. Eichenwald, Ann. Physik (4) 35, 1037 (1911); W. Voigt, Ann. Physik (4) 34, 797 (1911), and (4) 36, 866 (1911).
  11. E. Funk, "Reflexion und Brechung optischer Kugelwellen und das Problem der Totalreflexion"; dissertation at E.T.H. Zürich (1921). This investigation was suggested by H. Weyl.
  12. J. Picht, Ann. Physik (5) 3, 433 (1929).
  13. P. Debye, Ann. Physik (4) 30, 755 (1909).
  14. J. Picht, Ann. Physik (4) 77, 685 (1925).
  15. J. Picht, Z. Physik 39, 933 (1926).
  16. F. Noether, Ann. Physik (5) 11, 141 (1931).
  17. C. Schaefer and R. Pich, Ann. Physik (5) 30, 245 (1937).
  18. C. Schaefer, Einfuehrung in die theoretische Physik, Vol 3, Part 1 (Walter de Gruyter Verlag, Berlin, 1932 or 1949), Ch. 7.
  19. K. Artmann, Ann. Physik (6) 8, 270 (1951).
  20. H. Arzelies, Ann. Phys. (Paris) (12) 1, 5 (1946); and (12) 2, 517 (1947).
  21. H. Arzelies, Rev. Opt. 27, 205 (1948).
  22. F. I. Fedorov, Dokl. Akad. Nauk. SSSR 105, 465 (1955).
  23. H. Maecker, Ann. Physik (6) 4, 409 (1949).
  24. H. Maecker, Ann. Physik (6) 10, 115 (1952).
  25. H. Maecker, Ann. Physik (6) 10, 153 (1952)
  26. G. Lehmann and H. Maecker, Ann. Physik (6) 10, 161 (1952).
  27. H. Ott, Ann. Physik (5) 41, 443 (1942).
  28. H. Ott, Ann. Physik (6) 4, 432 (1949).
  29. J. Pavageau, Compt. Rend. 263B, 276 (1966).
  30. J. Pavageau and S. Tergiman, Compt. Rend. 263B, 352 (1966).
  31. J. Pavageau and S. Tergiman, Compt. Rend. 263B, 856 (1966).
  32. J. Picht, Z. Physik 40, 521 (1926).
  33. J. Picht, Z. Physik 58, 667 (1929).
  34. J. Picht, Ann. Physik (5) 19, 913 (1934).
  35. J. Picht, Zur Theorie der Totalreflexion, Abhandlungen d. deutschen Akad. Wiss. Berlin (Akademie-Verlag, Berlin, 1956).
  36. B. P. Sandford, "Frustrated Total Reflection and its Application to the Measurement of the Refractive Index and Thickness of Thin Transparent Films"; M.S. thesis, Victoria University College, Wellington, New Zealand (1957).
  37. C. Schaefer and C. von Fragstein, Ann. Physik (6) 6, 39 (1949).
  38. H. Schilling, Ann. Physik (7) 16, 122 (1965).
  39. A. Schoch, Ergeb. Exakt. Naturwiss. 23, 127 (1950).
  40. O. von Schmidt, Ann. Physik (5) 19, 891 (1934).
  41. H. Wolter, Z. Naturforsch. 5a, 139 (1950).
  42. H. Wolter, Ann. Physik (6) 7, 341 (1950).
  43. H. Wolter, Z. Naturforsch. 5a, 276 (1950).
  44. P. Acloque and C. Guillemet, Compt. Rend. 250, 4328 (1960).
  45. H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1078 (1964).
  46. F. Goos and H. Haenchen, Ann. Physik (5) 43, 383 (1943).
  47. F. Goos and H. Haenchen, Ann. Physik (6) 1, 333 (1947).
  48. F. Goos and H. Lindberg-Haenchen, Ann. Physik (6) 5, 251 (1949).
  49. K. Artmann, Ann. Physik (6) 2, 87 (1948).
  50. C. von Fragstein, Ann. Physik (6) 4, 271 (1949).
  51. H. Wolter, Z. Naturforsch. 5a, 143 (1950).
  52. R. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964). It should be pointed out that Artmann's exact expression49 for H polarization isd11=(λ1/π) sin φ1[sin2φ1-n2]-½×[n2(1-n2)/(n4 COS2φ1+sin2φ1-n2)]and not d11= (λl/π) sinφ1[sin2φ1-n2]-1/2, as was stated by Renard. Artmann explicitly stating that he considers only angles close to the critical angle of total reflection, approximated the expression between the brackets by n-2.
  53. K. Artmann, Ann. Physik (6) 7, 209 (1950).
  54. H. Hora, Optik 17, 409 (1960).
  55. A. Schoch, Acustica 2, 18 (1952).
  56. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959), pp. 557–558.
  57. H. A. Lorentz, Abhandlungen über theoretische Physik (B. G. Teubner, Leipzig, Berlin, 1907), pp. 415–442.
  58. K. Artmann, Ann. Physik (6) 8, 285 (1951).
  59. K. Artmann, Ann. Physik (6) 15, 1 (1954).
  60. C. von Fragstein and C. Schaefer, Ann. Physik (6) 12, 84 (1953).
  61. C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill Book Co., New York, 1965), pp. 8–9.
  62. It should be noted that all the quantities δ are defined for E polarization, but the indicator 1 is omitted for simplicity. The reader is cautioned that the quantities φ and Ψ used by Joos to denote (δr)11 and (δr)1 respectively, should have reversed signs. G. Joos, Theoretical Physics (Hafner Publishing Company, New York, 1950), 3rd ed., pp. 354–355; also Lehrbuch der theoretischen Physik (Akademische Verlagsgesellschaft Geest and Portig K.-G., Leipzig, 1956), p. 317.
  63. W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill Book Co., New York, 1950), p. 443.
  64. W. N. Hansen, Spectrochim. Acta 21, 815 (1965).
  65. W. N. Hansen, T. Kuwana, and R. A. Osteryoung, Anal. Chem. 6638, 1810 (1966).
  66. T. Hirschfeld, Can. Spectry. 11, 112 (1966).
  67. P. A. Wilks and T. Hirschfeld, Appl. Spectry. Rev. 1, 99 (1967).
  68. H. Schilling, Ann. Physik (7) 20, (1968).
  69. Note added to the galley proof. Bloembergen and Lee71 reported the observation of a strong generation of second harmonics near the critical angle of total reflection, pointing out the significance of phase matching. The aforementioned phenomenon, predicted in 1962 by Bloembergen and Pershan72 from a nonlinear viewpoint was expected by this author on the basis of the linear theory presented: The Goos-Haenchen shift may be interpreted by saying that the beam of light is reflected by a virtual plane parallel to the interface and a little distance inside the optically less-dense medium, as mentioned in Sec. 5. This distance, depending on the angle of incidence, has its maximum value for some angle near the critical angle of total reflection, decreasing rapidly as the angle of incidence increases.51 That is to say, the light penetrates deepest into the less-dense medium in the vicinity of the critical angle. Therefore, if the less-dense medium is nonlinear, the maximum interaction between light and the nonlinear medium occurs at an angle of incidence close to the critical angle of total reflection. This interpretation of the Goos-Haenchen shift was, in a way, experimentally verified by Bloembergen and Lee.71
  70. H. K. V. Lotsch, Optik 26, 112 and 181 (1967/1968).
  71. N. Bloembergen and C. H. Lee, Phys. Rev. Letters 19, 835 (1967).
  72. N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962).

Acloque, P.

P. Acloque and C. Guillemet, Compt. Rend. 250, 4328 (1960).

Artmann, K.

K. Artmann, Ann. Physik (6) 15, 1 (1954).

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

K. Artmann, Ann. Physik (6) 7, 209 (1950).

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

K. Artmann, Ann. Physik (6) 8, 285 (1951).

Arzelies, H.

H. Arzelies, Ann. Phys. (Paris) (12) 1, 5 (1946); and (12) 2, 517 (1947).

H. Arzelies, Rev. Opt. 27, 205 (1948).

Bloembergen, N.

N. Bloembergen and C. H. Lee, Phys. Rev. Letters 19, 835 (1967).

N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962).

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959), pp. 557–558.

Debye, P.

P. Debye, Ann. Physik (4) 30, 755 (1909).

Drude, P.

P. Drude, The Theory of Optics (Dover Publications, Inc., New York 1959), originally published in German (1900).

Eichenwald, A.

A. Eichenwald, Ann. d. Moskauer Kais. Ingenieur-Hochschule, p. 15, April 1908; J. Russ. Phys.-Chem. Soc. 41, phys. part, 131 (1909).

A. Eichenwald, Ann. Physik (4) 35, 1037 (1911); W. Voigt, Ann. Physik (4) 34, 797 (1911), and (4) 36, 866 (1911).

Fedorov, F. I.

F. I. Fedorov, Dokl. Akad. Nauk. SSSR 105, 465 (1955).

Funk, E.

E. Funk, "Reflexion und Brechung optischer Kugelwellen und das Problem der Totalreflexion"; dissertation at E.T.H. Zürich (1921). This investigation was suggested by H. Weyl.

Goos, F.

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

F. Goos and H. Lindberg-Haenchen, Ann. Physik (6) 5, 251 (1949).

F. Goos and H. Haenchen, Ann. Physik (5) 43, 383 (1943).

Gross, G.

C. Schaefer and G. Gross, Ann. Physik (4) 32, 648 (1910).

Guillemet, C.

P. Acloque and C. Guillemet, Compt. Rend. 250, 4328 (1960).

Haenchen, H.

F. Goos and H. Haenchen, Ann. Physik (5) 43, 383 (1943).

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

Hall, E. E.

E. E. Hall, Phys. Rev. 15, 73 (1902).

Hansen, W. N.

W. N. Hansen, T. Kuwana, and R. A. Osteryoung, Anal. Chem. 6638, 1810 (1966).

W. N. Hansen, Spectrochim. Acta 21, 815 (1965).

Hirschfeld, T.

T. Hirschfeld, Can. Spectry. 11, 112 (1966).

P. A. Wilks and T. Hirschfeld, Appl. Spectry. Rev. 1, 99 (1967).

Hora, H.

H. Hora, Optik 17, 409 (1960).

Ketteler, E.

E. Ketteler, Ann. Physik (9) 67, 879 (1899).

Kuwana, T.

W. N. Hansen, T. Kuwana, and R. A. Osteryoung, Anal. Chem. 6638, 1810 (1966).

Lee, C. H.

N. Bloembergen and C. H. Lee, Phys. Rev. Letters 19, 835 (1967).

Lehmann, G.

G. Lehmann and H. Maecker, Ann. Physik (6) 10, 161 (1952).

Lindberg-Haenchen, H.

F. Goos and H. Lindberg-Haenchen, Ann. Physik (6) 5, 251 (1949).

Lorentz, H. A.

H. A. Lorentz, Abhandlungen über theoretische Physik (B. G. Teubner, Leipzig, Berlin, 1907), pp. 415–442.

Lotsch, H. K. V.

H. K. V. Lotsch, Optik 26, 112 and 181 (1967/1968).

Maecker, H.

H. Maecker, Ann. Physik (6) 10, 115 (1952).

H. Maecker, Ann. Physik (6) 4, 409 (1949).

H. Maecker, Ann. Physik (6) 10, 153 (1952)

G. Lehmann and H. Maecker, Ann. Physik (6) 10, 161 (1952).

Noether, F.

F. Noether, Ann. Physik (5) 11, 141 (1931).

Oettinger, E.

W. von Ignatowsky and E. Oettinger, Ann. Physik (4) 25, 99 (1908); (4) 26, 1031 (1908); (4) 37, 901 (1912); and (4) 37, 911 (1912).

Osterberg, H.

H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1078 (1964).

Osteryoung, R. A.

W. N. Hansen, T. Kuwana, and R. A. Osteryoung, Anal. Chem. 6638, 1810 (1966).

Ott, H.

H. Ott, Ann. Physik (6) 4, 432 (1949).

H. Ott, Ann. Physik (5) 41, 443 (1942).

Papas, C. H.

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill Book Co., New York, 1965), pp. 8–9.

Pavageau, J.

J. Pavageau and S. Tergiman, Compt. Rend. 263B, 856 (1966).

J. Pavageau and S. Tergiman, Compt. Rend. 263B, 352 (1966).

J. Pavageau, Compt. Rend. 263B, 276 (1966).

Pershan, P. S.

N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962).

Pich, R.

C. Schaefer and R. Pich, Ann. Physik (5) 30, 245 (1937).

Picht, J.

J. Picht, Ann. Physik (5) 19, 913 (1934).

J. Picht, Zur Theorie der Totalreflexion, Abhandlungen d. deutschen Akad. Wiss. Berlin (Akademie-Verlag, Berlin, 1956).

J. Picht, Ann. Physik (5) 3, 433 (1929).

J. Picht, Ann. Physik (4) 77, 685 (1925).

J. Picht, Z. Physik 39, 933 (1926).

J. Picht, Z. Physik 58, 667 (1929).

J. Picht, Z. Physik 40, 521 (1926).

Quincke, G.

G. Quincke, Sitzber. d. k. Ges. . Berlin, p. 294 (1865); Pogg. Ann. 127, 1 and 199 (1865).

Renard, R. H.

R. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964). It should be pointed out that Artmann's exact expression49 for H polarization isd11=(λ1/π) sin φ1[sin2φ1-n2]-½×[n2(1-n2)/(n4 COS2φ1+sin2φ1-n2)]and not d11= (λl/π) sinφ1[sin2φ1-n2]-1/2, as was stated by Renard. Artmann explicitly stating that he considers only angles close to the critical angle of total reflection, approximated the expression between the brackets by n-2.

Sandford, B. P.

B. P. Sandford, "Frustrated Total Reflection and its Application to the Measurement of the Refractive Index and Thickness of Thin Transparent Films"; M.S. thesis, Victoria University College, Wellington, New Zealand (1957).

Schaefer, C.

C. Schaefer, Einfuehrung in die theoretische Physik, Vol 3, Part 1 (Walter de Gruyter Verlag, Berlin, 1932 or 1949), Ch. 7.

C. Schaefer and C. von Fragstein, Ann. Physik (6) 6, 39 (1949).

C. von Fragstein and C. Schaefer, Ann. Physik (6) 12, 84 (1953).

C. Schaefer and R. Pich, Ann. Physik (5) 30, 245 (1937).

C. Schaefer and G. Gross, Ann. Physik (4) 32, 648 (1910).

Schilling, H.

H. Schilling, Ann. Physik (7) 16, 122 (1965).

H. Schilling, Ann. Physik (7) 20, (1968).

Schoch, A.

A. Schoch, Ergeb. Exakt. Naturwiss. 23, 127 (1950).

A. Schoch, Acustica 2, 18 (1952).

Smith, L. W.

H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1078 (1964).

Smythe, W. R.

W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill Book Co., New York, 1950), p. 443.

Tergiman, S.

J. Pavageau and S. Tergiman, Compt. Rend. 263B, 856 (1966).

J. Pavageau and S. Tergiman, Compt. Rend. 263B, 352 (1966).

Voigt, W.

W. Voigt, Ann. Physik (9) 67, 185 (1899).

W. Voigt, Nachr. Akad. Wiss. Goettingen, II. Math. Physik. Kl., p. 49 (1884); Ann. Physik (9) 67, 185 (1899); Ann. Physik (4) 34, 798 (1911).

von Fragstein, C.

C. von Fragstein and C. Schaefer, Ann. Physik (6) 12, 84 (1953).

C. Schaefer and C. von Fragstein, Ann. Physik (6) 6, 39 (1949).

C. von Fragstein, Ann. Physik (6) 4, 271 (1949).

von Ignatowsky, W.

W. von Ignatowsky and E. Oettinger, Ann. Physik (4) 25, 99 (1908); (4) 26, 1031 (1908); (4) 37, 901 (1912); and (4) 37, 911 (1912).

von Schmidt, O.

O. von Schmidt, Ann. Physik (5) 19, 891 (1934).

Wilks, P. A.

P. A. Wilks and T. Hirschfeld, Appl. Spectry. Rev. 1, 99 (1967).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959), pp. 557–558.

Wolter, H.

H. Wolter, Ann. Physik (6) 7, 341 (1950).

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

H. Wolter, Z. Naturforsch. 5a, 139 (1950).

H. Wolter, Z. Naturforsch. 5a, 276 (1950).

Other (72)

G. Quincke, Sitzber. d. k. Ges. . Berlin, p. 294 (1865); Pogg. Ann. 127, 1 and 199 (1865).

W. Voigt, Nachr. Akad. Wiss. Goettingen, II. Math. Physik. Kl., p. 49 (1884); Ann. Physik (9) 67, 185 (1899); Ann. Physik (4) 34, 798 (1911).

E. Ketteler, Ann. Physik (9) 67, 879 (1899).

P. Drude, The Theory of Optics (Dover Publications, Inc., New York 1959), originally published in German (1900).

A. Eichenwald, Ann. d. Moskauer Kais. Ingenieur-Hochschule, p. 15, April 1908; J. Russ. Phys.-Chem. Soc. 41, phys. part, 131 (1909).

E. E. Hall, Phys. Rev. 15, 73 (1902).

W. von Ignatowsky and E. Oettinger, Ann. Physik (4) 25, 99 (1908); (4) 26, 1031 (1908); (4) 37, 901 (1912); and (4) 37, 911 (1912).

C. Schaefer and G. Gross, Ann. Physik (4) 32, 648 (1910).

W. Voigt, Ann. Physik (9) 67, 185 (1899).

A. Eichenwald, Ann. Physik (4) 35, 1037 (1911); W. Voigt, Ann. Physik (4) 34, 797 (1911), and (4) 36, 866 (1911).

E. Funk, "Reflexion und Brechung optischer Kugelwellen und das Problem der Totalreflexion"; dissertation at E.T.H. Zürich (1921). This investigation was suggested by H. Weyl.

J. Picht, Ann. Physik (5) 3, 433 (1929).

P. Debye, Ann. Physik (4) 30, 755 (1909).

J. Picht, Ann. Physik (4) 77, 685 (1925).

J. Picht, Z. Physik 39, 933 (1926).

F. Noether, Ann. Physik (5) 11, 141 (1931).

C. Schaefer and R. Pich, Ann. Physik (5) 30, 245 (1937).

C. Schaefer, Einfuehrung in die theoretische Physik, Vol 3, Part 1 (Walter de Gruyter Verlag, Berlin, 1932 or 1949), Ch. 7.

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

H. Arzelies, Ann. Phys. (Paris) (12) 1, 5 (1946); and (12) 2, 517 (1947).

H. Arzelies, Rev. Opt. 27, 205 (1948).

F. I. Fedorov, Dokl. Akad. Nauk. SSSR 105, 465 (1955).

H. Maecker, Ann. Physik (6) 4, 409 (1949).

H. Maecker, Ann. Physik (6) 10, 115 (1952).

H. Maecker, Ann. Physik (6) 10, 153 (1952)

G. Lehmann and H. Maecker, Ann. Physik (6) 10, 161 (1952).

H. Ott, Ann. Physik (5) 41, 443 (1942).

H. Ott, Ann. Physik (6) 4, 432 (1949).

J. Pavageau, Compt. Rend. 263B, 276 (1966).

J. Pavageau and S. Tergiman, Compt. Rend. 263B, 352 (1966).

J. Pavageau and S. Tergiman, Compt. Rend. 263B, 856 (1966).

J. Picht, Z. Physik 40, 521 (1926).

J. Picht, Z. Physik 58, 667 (1929).

J. Picht, Ann. Physik (5) 19, 913 (1934).

J. Picht, Zur Theorie der Totalreflexion, Abhandlungen d. deutschen Akad. Wiss. Berlin (Akademie-Verlag, Berlin, 1956).

B. P. Sandford, "Frustrated Total Reflection and its Application to the Measurement of the Refractive Index and Thickness of Thin Transparent Films"; M.S. thesis, Victoria University College, Wellington, New Zealand (1957).

C. Schaefer and C. von Fragstein, Ann. Physik (6) 6, 39 (1949).

H. Schilling, Ann. Physik (7) 16, 122 (1965).

A. Schoch, Ergeb. Exakt. Naturwiss. 23, 127 (1950).

O. von Schmidt, Ann. Physik (5) 19, 891 (1934).

H. Wolter, Z. Naturforsch. 5a, 139 (1950).

H. Wolter, Ann. Physik (6) 7, 341 (1950).

H. Wolter, Z. Naturforsch. 5a, 276 (1950).

P. Acloque and C. Guillemet, Compt. Rend. 250, 4328 (1960).

H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 54, 1078 (1964).

F. Goos and H. Haenchen, Ann. Physik (5) 43, 383 (1943).

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

F. Goos and H. Lindberg-Haenchen, Ann. Physik (6) 5, 251 (1949).

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

C. von Fragstein, Ann. Physik (6) 4, 271 (1949).

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

R. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964). It should be pointed out that Artmann's exact expression49 for H polarization isd11=(λ1/π) sin φ1[sin2φ1-n2]-½×[n2(1-n2)/(n4 COS2φ1+sin2φ1-n2)]and not d11= (λl/π) sinφ1[sin2φ1-n2]-1/2, as was stated by Renard. Artmann explicitly stating that he considers only angles close to the critical angle of total reflection, approximated the expression between the brackets by n-2.

K. Artmann, Ann. Physik (6) 7, 209 (1950).

H. Hora, Optik 17, 409 (1960).

A. Schoch, Acustica 2, 18 (1952).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959), pp. 557–558.

H. A. Lorentz, Abhandlungen über theoretische Physik (B. G. Teubner, Leipzig, Berlin, 1907), pp. 415–442.

K. Artmann, Ann. Physik (6) 8, 285 (1951).

K. Artmann, Ann. Physik (6) 15, 1 (1954).

C. von Fragstein and C. Schaefer, Ann. Physik (6) 12, 84 (1953).

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill Book Co., New York, 1965), pp. 8–9.

It should be noted that all the quantities δ are defined for E polarization, but the indicator 1 is omitted for simplicity. The reader is cautioned that the quantities φ and Ψ used by Joos to denote (δr)11 and (δr)1 respectively, should have reversed signs. G. Joos, Theoretical Physics (Hafner Publishing Company, New York, 1950), 3rd ed., pp. 354–355; also Lehrbuch der theoretischen Physik (Akademische Verlagsgesellschaft Geest and Portig K.-G., Leipzig, 1956), p. 317.

W. R. Smythe, Static and Dynamic Electricity (McGraw-Hill Book Co., New York, 1950), p. 443.

W. N. Hansen, Spectrochim. Acta 21, 815 (1965).

W. N. Hansen, T. Kuwana, and R. A. Osteryoung, Anal. Chem. 6638, 1810 (1966).

T. Hirschfeld, Can. Spectry. 11, 112 (1966).

P. A. Wilks and T. Hirschfeld, Appl. Spectry. Rev. 1, 99 (1967).

H. Schilling, Ann. Physik (7) 20, (1968).

Note added to the galley proof. Bloembergen and Lee71 reported the observation of a strong generation of second harmonics near the critical angle of total reflection, pointing out the significance of phase matching. The aforementioned phenomenon, predicted in 1962 by Bloembergen and Pershan72 from a nonlinear viewpoint was expected by this author on the basis of the linear theory presented: The Goos-Haenchen shift may be interpreted by saying that the beam of light is reflected by a virtual plane parallel to the interface and a little distance inside the optically less-dense medium, as mentioned in Sec. 5. This distance, depending on the angle of incidence, has its maximum value for some angle near the critical angle of total reflection, decreasing rapidly as the angle of incidence increases.51 That is to say, the light penetrates deepest into the less-dense medium in the vicinity of the critical angle. Therefore, if the less-dense medium is nonlinear, the maximum interaction between light and the nonlinear medium occurs at an angle of incidence close to the critical angle of total reflection. This interpretation of the Goos-Haenchen shift was, in a way, experimentally verified by Bloembergen and Lee.71

H. K. V. Lotsch, Optik 26, 112 and 181 (1967/1968).

N. Bloembergen and C. H. Lee, Phys. Rev. Letters 19, 835 (1967).

N. Bloembergen and P. S. Pershan, Phys. Rev. 128, 606 (1962).

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