1. A very narrow laser beam is shone into a sphere of solid glass that has uniform refractive index n>1 and radius ρ . The beam makes an angle α  in air with the normal to the surface of the sphere.

Sketch the path of the beam in the circular cross-section of the sphere defined by the normal and the beam direction and show all reflected and transmitted rays. Using Snell's Law at each reflection, explain why the beam cannot become a whispering gallery ray propagating around the inner surface of the sphere. 

2.  An asymmetric, multimode, step-profile slab waveguide has a core index of 1.5. The cladding index above the core has value 1.49 and the cladding index below the core has value 1.48. Rays propagate in the core in the x-z plane and make an angle θz relative to the z-axis along the centre of the waveguide.

(a) Determine the range of (positive) values of θz in degrees for which only bound rays will propagate along the waveguide.

(b) How many different types of rays can propagate along the waveguide?

Sketch a typical path for each type of ray.

(c) Determine the values of θz in degrees that delineate between the 3 types of ray paths.