## Abstract

Formulas have been derived relating the direction vector of the extraordinary ray in a uniaxial anisotropic medium to the ordinary ray. With these formulas one can trace rays through uniaxial crystals. Tracing rays into the crystal involves finding the ordinary ray by using the usual ray-tracing formula for isotropic media, then by using the derived equations to find the extraordinary ray. To trace an extraordinary ray out of the crystal, one first constructs the fictive ordinary ray corresponding to it through its point of contact with the refracting surface. Using the ray-tracing formula for isotropic media one traces this ray through the surface.

© 1962 Optical Society of America

### References

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1. See, for example, John Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958), Appendix B, Interferometers by J. Dyson.
2. P. Kaemmerer, Neues. Jahrb. Mineral. Geol. Beilage Bd. 20, 159–320 (1900).
3. F. C. A. Pockels, Lehrbuch der Kristalloptik (Leipzig and Berlin, 1904).
4. A. W. Conway and J. L. Synge, The Mathematical Papers of Sir William Rowan Hamilton (Cambridge University Press, New York, 1931), Vol. I, p. 336 ff.
5. A. A. Lebedev, Rev. opt. 9, 385 (1930).
6. See any standard text such as Jenkins and White, Fundamentals of Optics (McGraw-Hill, New York, 1950), Chap. 25. See also L. Fletcher, The Optical Indicatrix and the Transmission of Light in Crystals (Oxford University Press, London, 1892).
7. That (6) represents an ellipsoid of revolution with the axis of revolution coincident with A can be seen as follows. First consider the special case where A=(1,0,0). Let X=(x,y,z). Then (6) becomesne2[(x-r)2+y2+z2]+(n02-ne2)(x-r)2=s2,which can be written in the standard form of an ellipsoid of revolutionn02(x-r)2+ne2(y2+z2)=s2.The general case where A is arbitrary reduces to the special case upon application of a rotation of the coordinate system about the point (r,0,0) which maps A into (1,0,0).

#### 1930 (1)

A. A. Lebedev, Rev. opt. 9, 385 (1930).

#### 1900 (1)

P. Kaemmerer, Neues. Jahrb. Mineral. Geol. Beilage Bd. 20, 159–320 (1900).

#### Conway, A. W.

A. W. Conway and J. L. Synge, The Mathematical Papers of Sir William Rowan Hamilton (Cambridge University Press, New York, 1931), Vol. I, p. 336 ff.

#### Dyson, J.

See, for example, John Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958), Appendix B, Interferometers by J. Dyson.

#### Jenkins,

See any standard text such as Jenkins and White, Fundamentals of Optics (McGraw-Hill, New York, 1950), Chap. 25. See also L. Fletcher, The Optical Indicatrix and the Transmission of Light in Crystals (Oxford University Press, London, 1892).

#### Kaemmerer, P.

P. Kaemmerer, Neues. Jahrb. Mineral. Geol. Beilage Bd. 20, 159–320 (1900).

#### Lebedev, A. A.

A. A. Lebedev, Rev. opt. 9, 385 (1930).

#### Pockels, F. C. A.

F. C. A. Pockels, Lehrbuch der Kristalloptik (Leipzig and Berlin, 1904).

#### Strong, John

See, for example, John Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958), Appendix B, Interferometers by J. Dyson.

#### Synge, J. L.

A. W. Conway and J. L. Synge, The Mathematical Papers of Sir William Rowan Hamilton (Cambridge University Press, New York, 1931), Vol. I, p. 336 ff.

#### White,

See any standard text such as Jenkins and White, Fundamentals of Optics (McGraw-Hill, New York, 1950), Chap. 25. See also L. Fletcher, The Optical Indicatrix and the Transmission of Light in Crystals (Oxford University Press, London, 1892).

#### Neues. Jahrb. Mineral. Geol. Beilage Bd. (1)

P. Kaemmerer, Neues. Jahrb. Mineral. Geol. Beilage Bd. 20, 159–320 (1900).

#### Rev. opt. (1)

A. A. Lebedev, Rev. opt. 9, 385 (1930).

#### Other (5)

See any standard text such as Jenkins and White, Fundamentals of Optics (McGraw-Hill, New York, 1950), Chap. 25. See also L. Fletcher, The Optical Indicatrix and the Transmission of Light in Crystals (Oxford University Press, London, 1892).

That (6) represents an ellipsoid of revolution with the axis of revolution coincident with A can be seen as follows. First consider the special case where A=(1,0,0). Let X=(x,y,z). Then (6) becomesne2[(x-r)2+y2+z2]+(n02-ne2)(x-r)2=s2,which can be written in the standard form of an ellipsoid of revolutionn02(x-r)2+ne2(y2+z2)=s2.The general case where A is arbitrary reduces to the special case upon application of a rotation of the coordinate system about the point (r,0,0) which maps A into (1,0,0).

F. C. A. Pockels, Lehrbuch der Kristalloptik (Leipzig and Berlin, 1904).

A. W. Conway and J. L. Synge, The Mathematical Papers of Sir William Rowan Hamilton (Cambridge University Press, New York, 1931), Vol. I, p. 336 ff.

See, for example, John Strong, Concepts of Classical Optics (Freeman, San Francisco, 1958), Appendix B, Interferometers by J. Dyson.

### Cited By

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### Figures (5)

Fig. 1

Application of Huygens’ principle to ordinary ray.

Fig. 2

Spherical and ellipsoidal wavelets in a uniaxial crystal.

Fig. 3

Application of Huygens’ principle to extraordinary ray.

Fig. 4

The N, P, Q coordinate system. N is perpendicular to refracting surface at point of incidence. P lies in plane of incidence. Q is perpendicular to plane of incidence. S0 is a unit vector in the direction of the ordinary ray.

Fig. 5

Ordinary ray and wave surface in the plane of incidence. Q vector perpendicular to illustration.

### Equations (39)

$Q = S 0 × N [ 1 - ( N · S 0 ) 2 ] 1 2 ,$
$P = N × Q , Q = P × N , N = Q × P .$
$X · S 0 = 0.$
$F ≡ ( X - r P ) 2 - s 2 / n 0 2 = 0 ,$
$S 0 = ( n 0 / s ) ( X 0 - r P ) .$
$s = - n 0 r ( S 0 · P ) .$
$G ≡ n e 2 ( X - r P ) 2 + ( n 0 2 - n e 2 ) [ ( X - r P ) · A ] 2 - s 2 = 0 ,$
$X · B = 0.$
$B · Q = 0.$
$b B = n e 2 ( X e - r P ) + ( n 0 2 - n e 2 ) [ ( X e - r P ) · A ] A .$
$t S e = X e - r P ,$
$b B = t [ n e 2 S e + ( n 0 2 - n e 2 ) ( S e · A ) A ] .$
$t 2 [ n e 2 + ( n 0 2 - n e 2 ) ( S e · A ) 2 ] = n 0 2 r 2 ( S 0 · P ) 2 .$
$n e 2 ( S e · Q ) + ( n 0 2 - n e 2 ) ( S e · A ) ( A · Q ) = 0.$
$b ( B · P ) = t [ n e 2 ( S e · P ) + ( n 0 2 - n e 2 ) ( S e · A ) ( A · P ) ] .$
$b B · ( X e - r P ) = n e 2 ( X e - r P ) 2 + ( n 0 2 - n e 2 ) [ ( X e - r P ) · A ] 2 ,$
$- b ( B · P ) = n 0 2 r ( S 0 · P ) 2 .$
$n e 2 ( S e · P ) + ( n 0 2 - n e 2 ) ( S e · A ) ( A · P ) = - ( r / t ) n 0 2 ( S 0 · P ) 2 .$
$n e 2 + ( n 0 2 - n e 2 ) ( S e · A ) 2 = k 2 n 0 2 ( S 0 · P ) 2 .$
$R = n e 2 S e + ( n 0 2 - n e 2 ) ( S e · A ) A .$
$R · Q = 0 ,$
$R · P = - k n 0 2 ( S 0 · P ) 2 .$
$R 2 = n 0 2 [ k 2 ( S 0 · P ) 2 ( n 0 2 + n e 2 ) - n e 2 ] .$
$R = ( R · N ) N + ( R · P ) P ,$
$R 2 = ( R · N ) 2 + ( R · P ) 2 .$
$( R · N ) 2 = R 2 - ( R · P ) 2 = n 0 4 { k 2 ( S 0 · P ) 2 [ ( S 0 · N ) 2 + μ 2 ] - μ 2 }$
$R = - k n 0 2 ( S 0 · P ) 2 P + n 0 2 { k 2 ( S 0 · P ) 2 [ ( S 0 · N ) 2 + μ 2 ] - μ 2 } 1 2 N .$
$S e = [ R - ( 1 - μ 2 ) ( R · A ) A ] / n e 2 .$
$n e 4 = R 2 - ( 1 - μ 4 ) ( R · A ) 2 .$
$k 2 ( S 0 · P ) 2 - μ 2 = ( 1 - μ 2 ) [ - k ( S 0 · P ) 2 ( P · A ) + ( N · A ) { k 2 ( S 0 · P ) 2 [ ( S 0 · N ) 2 + μ 2 ] - μ 2 } 1 2 ] 2 .$
$k ( S 0 · P ) = - μ / [ μ 2 + ( S 0 · N ) 2 - U 2 ] 1 2$
$U 2 [ 1 - ( 1 - μ 2 ) ( N · A ) 2 ] - 2 ( 1 - μ 2 ) ( S 0 · P ) ( P · A ) ( N · A ) U + ( 1 - μ 2 ) [ 1 - ( S 0 · P ) 2 ( P · A ) 2 ] - ( S 0 · N ) 2 = 0 ,$
$U = ( 1 - μ 2 ) ( S 0 · P ) ( P · A ) ( N · A ) + σ 1 - ( 1 - μ 2 ) ( N · A ) 2 ,$
$σ 2 = ( 1 - μ 2 ) 2 ( N · A ) 2 + ( S 0 · N ) 2 - ( 1 - μ 2 ) [ 1 + ( N · A ) 2 ( S 0 · N ) 2 - ( P · A ) 2 ( S 0 · P ) 2 ] .$
$R = n 0 2 μ [ ( S 0 · P ) P + U N ] [ μ 2 + ( S 0 · N ) 2 - U 2 ] 1 2 .$
$B = ( S 0 · P ) P + U N [ ( S 0 · P ) 2 + U 2 ] 1 2 .$
$( S 0 · P ) = μ 2 ( S e · P ) + ( 1 - μ 2 ) ( A · S e ) ( A · P ) [ μ 2 + ( 1 - μ 2 ) ( A · S e ) 2 ] 1 2 .$
$ne2[(x-r)2+y2+z2]+(n02-ne2)(x-r)2=s2,$
$n02(x-r)2+ne2(y2+z2)=s2.$