Huygens Theory of Double Refraction when the Optic Axis is parallel to the refracting surface and lies in the plane of incidence for the case of normal and oblique incidence:

Introduction:
The Huygens theory of double refraction explains the phenomenon in which a ray of light passing through certain crystals splits into two separate rays, each with a different velocity and direction. This phenomenon occurs when the optic axis of the crystal is parallel to the refracting surface and lies in the plane of incidence. The theory was formulated by Christiaan Huygens in the 17th century and has been widely accepted in understanding double refraction.

Normal Incidence:
1. When a ray of light travels through a crystal with its optic axis parallel to the refracting surface and lying in the plane of incidence, the incident wavefront is divided into two new wavefronts.
2. One wavefront behaves as if the crystal were isotropic and follows the laws of ordinary refraction.
3. The other wavefront, known as the extraordinary wavefront, undergoes anisotropic refraction and follows a different set of laws.
4. The extraordinary wavefront experiences a change in velocity and direction, resulting in double refraction.

Oblique Incidence:
1. When light is incident obliquely on the crystal with the optic axis parallel to the refracting surface and lying in the plane of incidence, a similar phenomenon occurs.
2. The incident wavefront is divided into an ordinary wavefront and an extraordinary wavefront.
3. The ordinary wavefront follows the laws of ordinary refraction and behaves as if the crystal were isotropic.
4. The extraordinary wavefront experiences anisotropic refraction and follows different laws.
5. The extraordinary ray undergoes a change in velocity and direction, resulting in double refraction.

Conclusion:
The Huygens theory of double refraction explains the behavior of light when passing through crystals with an optic axis parallel to the refracting surface and lying in the plane of incidence. It describes the division of the incident wavefront into ordinary and extraordinary wavefronts, each following different laws of refraction. This theory provides a fundamental understanding of the phenomenon of double refraction and is widely used in the field of optics.