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.

© 1969 Optical Society of America

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References

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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 0 to the optic axis results from the refraction of an incident wave whose inclination to the optic axis is θ′=θ[υ22+θυn2(cosϕ±1)]e−12 according to Snell’s law. For small angles θ′ = θυ2e−1, independent of θ and ϕ. The azimuthal angle ϕ 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).

1966 (2)

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

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

1964 (2)

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

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

1959 (1)

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

1905 (3)

W. Voigt, Z. Physik 6, 818 (1905).

W. Voigt, Z. Physik 6, 672 (1905).

W. Voigt, Ann. Physik 18, 645 (1905).

1855 (1)

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

1839 (1)

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

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.

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, Z. Physik 6, 818 (1905).

W. Voigt, Z. Physik 6, 672 (1905).

W. Voigt, Ann. Physik 18, 645 (1905).

Waterman, P. C.

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

Ann. Physik (1)

W. Voigt, Ann. Physik 18, 645 (1905).

Appl. Opt. (2)

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

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

Appl. Phys. Letters (1)

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

J. Acoust. Soc. Am. (1)

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

Phys. Rev. (1)

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

Pogg. Ann (2)

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

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

Z. Physik (2)

W. Voigt, Z. Physik 6, 818 (1905).

W. Voigt, Z. Physik 6, 672 (1905).

Other (5)

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

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.

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 0 to the optic axis results from the refraction of an incident wave whose inclination to the optic axis is θ′=θ[υ22+θυn2(cosϕ±1)]e−12 according to Snell’s law. For small angles θ′ = θυ2e−1, independent of θ and ϕ. The azimuthal angle ϕ 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.

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Figures (4)

F. 1
F. 1

Transformation of coordinates from the principal axes to a coordinate set fixed by the chosen optic axis. The axes x1 and x3 are, respectively, the principal axes associated with the fastest and slowest principal velocities. The direction of the intermediate-velocity principal axis, x2, is normal to the plane of the paper. In the transformed coordinates, the z axis is oriented along the chosen optic axis, and the y axis is oriented along the x2 principal axis. The angle γ between the x3 and z axes is equal to tan 1 [ ( υ 1 2 υ 2 2 ) / ( υ 2 2 υ 3 2 ) ] 1 2, where υ1, υ2, and υ3 are the principal velocities associated, respectively, with the x1, x2, and x3 axes.

F. 2
F. 2

Definition of the angles describing the orientation of a propagation vector S with respect to the coordinate system fixed by the optic axis. The angle θ is the inclination angle of the propagation vector with respect to the chosen optic axis. The angle ϕ is the azimuthal angle of the propagation vector with respect to the x axis in the plane normal to the optic axis.

F. 3
F. 3

Cone of internal conical refraction. The line OA is in the direction of the optic axis, and OC is the axis of the cone. The angle χ, which is defined in the text, is the vertex angle of the cone. The circle ABP is the circle of conical refraction; B is the point diametrically opposite the point A where the optic axis intercepts the circle; P is a typical point on the circle.

F. 4
F. 4

Variation of the ray direction in the crystal with the direction of the incident ray. Rear view of cone base. Positions of points A, B, C, and P are shown in Fig. 3. The angle ρ between the radius vector to a given point P and the intermediate-velocity principal axis is linearly related to the azimuthal angle ϕ of the propagation vector, as shown in the text. For fixed ϕ, the ray moves along the direction PP′, making an angle ρ′ to the intermediate-velocity principal axis, as the inclination angle θ of the propagation vector is varied.

Tables (1)

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Table I Relation between the incident and resultant beams in a biaxial crystal.

Equations (53)

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2 E i ( / x i ) ( E ) = c 2 ( 2 D i / t 2 ) ,
E i = ( υ i 2 / c 2 ) D i i = 1 , 2 , 3 ,
υ i 2 2 D i j = 1 3 υ j 2 2 D j x i x j = 2 D i t 2 ( i = 1 , 2 , 3 ) .
P = ( c / 8 π ) R ( E × H ) ,
υ 2 = υ 2 2 + θ υ n 2 ( cos ϕ ± 1 ) ,
υ n 2 = [ ( υ 1 2 υ 2 2 ) ( υ 2 2 υ 3 2 ) ] 1 2
D y / D x = sin ϕ / ( cos ϕ ± 1 ) .
Mode I , D y / D x = tan ϕ / 2 Mode II , D y / D x = cot ϕ / 2 .
P x ( θ , ϕ ) = ( D 2 / 8 π c 2 ) { υ 2 υ n 2 cos 2 ϕ / 2 + θ υ 3 cos 2 ϕ } P y ( θ , ϕ ) = ( D 2 / 8 π c 2 ) { υ 2 υ n 2 cos ϕ / 2 sin ϕ / 2 + θ υ 3 sin ϕ } P z ( θ , ϕ ) = ( D 2 / 8 π c 2 ) υ 2 3 ,
P x ( θ , ϕ ) = ( D 2 / 8 π c 2 ) { υ 2 υ n 2 sin 2 ϕ / 2 θ υ 3 cos 2 ϕ } P y ( θ , ϕ ) = ( D 2 / 8 π c 2 ) { υ 2 υ n 2 cos ϕ / 2 sin ϕ / 2 + θ υ 3 sin ϕ } P z ( θ , ϕ ) = ( D 2 / 8 π c 2 ) υ 2 3 .
[ P x ( D 2 / 16 π c 2 ) υ 2 υ n 2 ] 2 + P y 2 = ( D 2 / 16 π c 2 ) υ 2 υ n 2
tan ρ = ± sin ρ tan ρ .
x 1 = [ υ 1 2 υ 2 2 υ 1 2 υ 3 2 ] 1 2 x 2 = 0 x 3 = [ υ 2 2 υ 3 2 υ 1 2 υ 3 2 ] 1 2
x = α x 1 + β x 3 y = x 2 z = β x 1 + α x 3 ,
x 1 = α x β z x 2 = y x 3 = β x + α z ,
α = [ υ 2 2 υ 3 2 υ 1 2 υ 3 2 ] 1 2 β = [ υ 1 2 υ 2 2 υ 1 2 υ 3 2 ] 1 2 .
S x = sin θ cos ϕ S y = sin θ sin ϕ S z = cos θ .
S x = θ cos ϕ S y = θ sin ϕ S z = 1 1 2 θ 2 .
D i = S i υ 2 υ i 2 j = 1 3 υ j 2 S j D j ( i = 1 , 2 , 3 ) .
i = 1 3 S i 2 υ 2 υ i 2 = 0 .
( α θ cos ϕ β [ 1 1 2 θ 2 ] ) 2 υ 2 υ 1 2 + θ 2 sin 2 ϕ υ 2 υ 2 2 + ( β θ cos ϕ + α [ 1 1 2 θ 2 ] ) 2 υ 2 υ 3 2 = 0 .
υ 4 υ 2 [ υ 2 2 ( 1 θ 2 + θ 2 cos 2 ϕ ) + ( υ 3 2 α 2 + υ 1 2 β 2 ) cos 2 ϕ + ( α 2 υ 1 2 + β 2 υ 3 2 ) ( 1 θ 2 ) + 2 α β ( υ 1 2 υ 3 2 ) θ cos ϕ + θ 2 sin 2 ϕ ( υ 1 2 + υ 3 2 ) ] + { υ 2 2 [ α 2 υ 3 2 + β 2 υ 1 2 ] θ 2 cos 2 ϕ + υ 2 2 ( β 2 υ 3 2 + α 2 υ 1 2 ) ( 1 θ 2 ) + 2 ( υ 1 2 υ 3 2 ) υ 2 2 α β θ cos ϕ + θ 2 sin 2 ϕ υ 1 2 υ 3 2 } = 0 .
α 2 υ 1 2 + β 2 υ 3 2 = υ 2 2 β 2 υ 1 2 + α 2 υ 3 2 = υ 1 2 υ 2 2 + υ 3 2 .
υ n 2 = ( υ 1 2 υ 3 2 ) α β = [ ( υ 1 2 υ 2 2 ) ( υ 2 2 υ 3 2 ) ] 1 2 ;
υ 4 2 υ 2 [ υ 2 2 ( 1 θ 2 ) + υ n 2 θ cos ϕ + θ 2 / 2 ( υ 1 2 + υ 3 2 ) ] + υ 2 2 ( υ 1 2 + υ 3 2 υ 2 2 ) θ 2 cos 2 ϕ + υ 2 4 ( 1 θ 2 ) + 2 υ 2 2 υ n 2 θ cos ϕ + υ 1 2 υ 3 2 θ 2 sin 2 ϕ = 0 .
υ 2 = υ 2 2 + θ υ n 2 [ cos ϕ ± 1 ] .
j = 1 3 υ j 2 S j D j L 2 .
L 2 = θ υ 2 2 ( D x cos ϕ + D y sin ϕ ) + ( υ 1 2 + υ 3 2 υ 2 2 ) S z D z υ n 2 [ D x S z + S x D z ] .
D z = 1 / S z [ S x D x + S y D y ] = θ [ D x cos ϕ + D y sin ϕ ] .
L 2 = θ [ 2 υ 2 2 ( υ 1 2 + υ 3 2 ) ] [ D x cos ϕ + D y sin ϕ ] υ n 2 [ D x θ 2 cos ϕ ( D x cos ϕ D y sin ϕ ) ] .
D y = D 2 = θ sin ϕ θ υ n 2 os ϕ ± 1 ] υ n 2 D x 0 ( θ )
D y / D x = sin ϕ / ( cos ϕ ± 1 ) + 0 ( θ ) ,
S × H = ( υ / c ) D .
( υ / c ) D x = θ sin ϕ H z H y ( υ / c ) D y = H x θ cos ϕ H z .
H z = θ ( cos ϕ H x + sin ϕ H y ) .
H x = ( υ / c ) D y H y = ( υ / c ) D x .
c 2 E x = υ 2 2 D x υ n 2 D z c 2 E y = υ 2 2 D y c 2 E z = υ n 2 D x + ( υ 1 2 + υ 3 2 υ 2 2 ) D z .
P x = ( 8 π c ) 2 { υ 2 υ n 2 | D x | 2 + θ υ 2 [ ( υ n 4 / 2 υ 2 2 ) ( cos ϕ ± 1 ) | D x | 2 + ( υ 1 2 + υ 3 2 ) ( | D x | 2 cos ϕ + D x D y * sin ϕ ] υ 2 2 cos ϕ ( | D x | 2 + | D y | 2 ) 2 υ 2 2 sin ϕ D x D y * ] } , P y = ( 8 π c ) 2 { υ 2 υ n 2 D x D y * + θ υ 2 [ ( υ n 4 / 2 υ 2 2 ) ( cos ϕ ± 1 ) D x D y * + ( υ 1 2 + υ 3 2 ) ( D x D y * cos ϕ + | D y | 2 sin ϕ ) 2 υ 2 2 cos ϕ D x D y * υ 2 2 ( | D y | 2 | D x | 2 ) sin ϕ ] } , P z = ( 8 π c ) 2 { υ 2 3 ( | D x | 2 + | D y | 2 ) + θ [ ( υ 2 υ n 2 / 2 ) ( cos ϕ ± 1 ) ( | D x | + | D y | 2 ) + υ 2 υ n 2 ( | D x | 2 cos ϕ + D x D y * sin ϕ ) ] } ,
Mode I D x = D cos ϕ / 2 Mode II D x = D sin ϕ / 2 D y = D sin ϕ / 2 D y = D cos ϕ / 2 .
tan Ψ = P x / P y .
ρ = 2 Ψ 90 °
tan ρ = tan ( 2 Ψ 90 ° ) = cot 2 Ψ
P x / P y = cot ϕ / 2 = tan ( 90 ° ϕ / 2 ) .
Ψ = 90 ° ϕ / 2 ,
tan ρ = cot ( 180 ° ϕ ) = cot ϕ .
P x / P y = tan ϕ / 2 ,
Ψ = ϕ / 2 ,
tan ρ = cot ( ϕ ) = cot ϕ .
tan ρ = cot ϕ .
tan ρ = [ P x ( θ , ϕ ) P x ( 0 , ϕ ) ] / [ P y θ , ϕ ) P y ( 0 , ϕ ) ] = cos 2 ϕ / sin ϕ .
tan ρ = cos 2 ϕ / sin ϕ .
tan ρ = ± cos ϕ tan ρ .
tan ρ = sin ρ tan ρ .