.Formation by Crystalline Media. 

 General Considerations. 



249 



The ordinary light in uniaxial crystals is propagated in 

 precisely the same way as in ordinary isotropic substances, 

 so that the laws for image-formation by this light are well 

 known and need no further discussion. It is the extra- 

 ordinary images in these crystals and both images in biaxials 

 which follow more complex laws, — due to the more compli- 

 cated shape of the wave-front and the fact that the rays are 

 not necessarily at right angles to the waves. In investi- 

 gating the laws for the image-formation in such cases, there- 

 fore, it is necessary to take into account the curvatures of 

 the incident wave as well as the nature of the refracting 

 surface and the change of velocity on refraction. The image 

 is in general astigmatic, so that the investigation involves 

 the determination of the positions and directions of two 

 focal lines. 



As an example of the method which must be employed, 

 consider the formation of the image of a point-source within 

 a uniaxial crystal by the extraordinary light refracted into 

 air through a plane surface which is normal to the optic 

 axis of the crystal. Such an image is intended to be seen 

 by the eye or a low-power microscope, so that the pencil of 

 rays coming from object or image-point is of small angular 

 aperture. For this reason the standard approximations of 

 the " first order " discussion of image-formation are justified 

 and will be assumed without comment. The procedure here 

 followed is not the same as that of Stokes, although it 

 amounts to the same thing in the end. 



As is well known *, this extraordinary wave in a uniaxial 

 crystal travels out from a point-source in the form of a 

 spheroid whose axis of revolution is the optic axis of the 

 costal. The following are geometrical and optical properties 

 of such a wave-front. 



The principal indices of refraction are n x equal to V/a, 

 and n 2 equal to Y/c, (1), where Y is the velocity of light in 

 air (or vacuum), a is the velocity of the extraordinary wave 

 along the axis, and c its velocity at right angles to the axis ; 

 a is also the velocity of the ordinary wave. 



If a Wave has travelled a distance x along the axis it has 

 .also travelled a distance y at right angles to the axis such 

 that y — cx\a^ (2), so that the semi-axes of the wave-surface 



* Drude, ' Theory of Optics,' Properties of Transparent Crystals, 

 from (55) setting b equal to c. Further relations are quoted or are 

 deduced very directly from results in the same chapter ; they will be 

 referred to by the number of the equation which represents them. 



