Abstract

A quantitative analysis of internal conical refraction in biaxial crystals, for which an extensive knowledge of ray geometry is not required, is presented. Except for the wave incident along an optic axis, the various waves in the incident wave bundle are shown to couple to modes of definite wave velocity and polarization in the crystal, and the dependence of these quantities on wave direction is found, using a first-order expansion in the angle between an optic axis and the wave vector. The Poynting vector for each mode is determined. From the dependence of the Poynting vector on the wave direction, the shape of the cone is determined. Our mathematical expansion permits this to be done in a simple manner. In particular, this analysis shows that rays with the same polarization do not lie along radii of the cone, a result not previously given. Pecularities associated with the wave incident along the optic axis, and reasons for not observing them, are discussed. Use of such an explicit analysis aids in clarifying the nature of internal conical refraction and its consequences.

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  1. W. R. Hamilton, Mathematical Papers, Vol. I, A. W. Conway and J. W. Synge, Eds. (Cambridge University Press, London, 1931).
  2. J. C. Poggendorf, Pogg. Ann 48, 461 (1839).
  3. W. V. Haidinger, Pogg. Ann 96, 469 (1855).
  4. W. Voigt, Z. Physik 6, 818 (1905).
  5. W. Voigt, Z. Physik 6, 672 (1905).
  6. W. Voigt, Ann. Physik 18, 645 (1905).
  7. R. N. Ditchburn, Light (Wiley-Interscience Inc., New York, 1963), 2nd ed., pp. 616 ff.
  8. Principles of Optics, M. Born and E. Wolf, Eds. (Pergamon Press, Ltd., Oxford, 1964), 2nd ed., pp. 686 ff.
  9. A. Bramley, Appl. Phys. Letters 5, 210 (1964).
  10. R. P. Burns, Appl. Opt. 3, 1505 (1964).
  11. P. C. Waterman, Phys. Rev. 113, 1240 (1959).
  12. H. J. McSkimin and W. L. Bond, J. Acoust. Soc. Am. 39, 499 (1966).
  13. See Ref. 8, pp. 9 ff., 33.
  14. This is true as long as the angular spread of the incident wave bundle is not too large. To see this, assume, for simplicity, that the crystal surface is normal to the optic axis. Outside the crystal, the wave normal and ray directions coincide. In the crystal, any wave whose normal makes an angle θ to the optic axis results from the refraction of an incident wave whose inclination to the optic axis is θ´=θ22+θυn2(cosø±l)]e½ according to Snell's law. For small angles θ´=θυ2e-1, independent of θ and ø. The azimuthal angle ø is not changed by refraction. Hence, refraction at small angles does not change the shape of the incident wave bundle. This argument can be generalized to cases in which the surface is not normal to the optic axis.
  15. W. Haas and R. Johannes, Appl. Opt. 5, 1066 (1966).

Bond, W. L.

H. J. McSkimin and W. L. Bond, J. Acoust. Soc. Am. 39, 499 (1966).

Bramley, A.

A. Bramley, Appl. Phys. Letters 5, 210 (1964).

Burns, R. P.

R. P. Burns, Appl. Opt. 3, 1505 (1964).

Ditchburn, R. N.

R. N. Ditchburn, Light (Wiley-Interscience Inc., New York, 1963), 2nd ed., pp. 616 ff.

Haas, W.

W. Haas and R. Johannes, Appl. Opt. 5, 1066 (1966).

Haidinger, W. V.

W. V. Haidinger, Pogg. Ann 96, 469 (1855).

Hamilton, W. R.

W. R. Hamilton, Mathematical Papers, Vol. I, A. W. Conway and J. W. Synge, Eds. (Cambridge University Press, London, 1931).

Johannes, R.

W. Haas and R. Johannes, Appl. Opt. 5, 1066 (1966).

McSkimin, H. J.

H. J. McSkimin and W. L. Bond, J. Acoust. Soc. Am. 39, 499 (1966).

Poggendorf, J. C.

J. C. Poggendorf, Pogg. Ann 48, 461 (1839).

Voigt, W.

W. Voigt, Ann. Physik 18, 645 (1905).

W. Voigt, Z. Physik 6, 818 (1905).

W. Voigt, Z. Physik 6, 672 (1905).

Waterman, P. C.

P. C. Waterman, Phys. Rev. 113, 1240 (1959).

Other (15)

W. R. Hamilton, Mathematical Papers, Vol. I, A. W. Conway and J. W. Synge, Eds. (Cambridge University Press, London, 1931).

J. C. Poggendorf, Pogg. Ann 48, 461 (1839).

W. V. Haidinger, Pogg. Ann 96, 469 (1855).

W. Voigt, Z. Physik 6, 818 (1905).

W. Voigt, Z. Physik 6, 672 (1905).

W. Voigt, Ann. Physik 18, 645 (1905).

R. N. Ditchburn, Light (Wiley-Interscience Inc., New York, 1963), 2nd ed., pp. 616 ff.

Principles of Optics, M. Born and E. Wolf, Eds. (Pergamon Press, Ltd., Oxford, 1964), 2nd ed., pp. 686 ff.

A. Bramley, Appl. Phys. Letters 5, 210 (1964).

R. P. Burns, Appl. Opt. 3, 1505 (1964).

P. C. Waterman, Phys. Rev. 113, 1240 (1959).

H. J. McSkimin and W. L. Bond, J. Acoust. Soc. Am. 39, 499 (1966).

See Ref. 8, pp. 9 ff., 33.

This is true as long as the angular spread of the incident wave bundle is not too large. To see this, assume, for simplicity, that the crystal surface is normal to the optic axis. Outside the crystal, the wave normal and ray directions coincide. In the crystal, any wave whose normal makes an angle θ to the optic axis results from the refraction of an incident wave whose inclination to the optic axis is θ´=θ22+θυn2(cosø±l)]e½ according to Snell's law. For small angles θ´=θυ2e-1, independent of θ and ø. The azimuthal angle ø is not changed by refraction. Hence, refraction at small angles does not change the shape of the incident wave bundle. This argument can be generalized to cases in which the surface is not normal to the optic axis.

W. Haas and R. Johannes, Appl. Opt. 5, 1066 (1966).

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