1. A very narrow laser beam in air is shone into a sphere of solid glass that has a 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.  describe why the beam cannot become a bound whispering gallery ray propagating around the inside the glass-air surface of the sphere.

2. An asymmetric, multimode, step-profile slab waveguide has a core index of 1.49. The cladding index above the core has value 1.47 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 θ_z relative to the z-axis along the centre of the waveguide. 

(a) Determine the range of (positive) values of θ_z 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 θ_z in degrees that delineate between the 3 types of ray paths.

3. An asymmetric, multimode slab waveguide has a core width 2ρ, a uniform cladding index n_cl in the regions above and below the core, and a variable refractive-index profile n_co(x) in the core defined by:

n_co(x)= n_co[1-2Δ(x/ρ)²]^1/2  for 0≤x≤ ρ

n_co(x)= n_co for - ρ ≤x≤0

Where n_co is a constant, Δ is the relative index difference with standard definition and the x-axis is orthogonal to the z-axis of symmetry of the waveguide. Assume that all ray paths are periodic and start at x=z=0. The initial angle the ray path makes with the z-axis at x=0 is θ_z, i.e. dx/dz=tanθ_z at z=0.

(a) Sketch, approximately, the shape of a periodic bound ray path along the waveguide. 

(b) Substitute the above core profile into the second-order differential equation for ray    paths as given in lectures and solve the resulting equation to determine the curved bound ray paths for 0≤x≤ ρ as a function of z, θ_z and σ. What is the maximum permissible value of θ_z for a curved bound ray path?

(c) Use the path equation derived in (a) to determine an analytical expression for the bound ray half-period z_p, i.e. the distance between successive crossing points of the ray path on the z-axis.

4. A weakly guiding, step-profile slab waveguide has a core half-width of 4 microns. The core index is 1.47 and the cladding index is 1.46. Determine the range of source wavelength values in nanometres for which the waveguide will support: (i) a single mode; (ii) two modes; (iii) three modes; (iv) more than three modes.

5. Consider a fibre whose core index is 1.5 and whose cladding index is 1.485.  The core radius is 100 mm.  At what bending radius does a ray traveling along the fibre axis strike the cladding at the critical angle in the bend?

6. A weakly guiding optical fibre propagates just the first three bound modes. The three modes have scalar transverse electric fields ψ₁(x,y), ψ₂(x,y) and ψ₃(x,y) with respective propagation constants β₁, β₂ and β₃ respectively. At the beginning of the fibre, a source on the end-face excites the three modes with electric field amplitudes a₁, a₂ and a₃, respectively. 

(a) prepare down an algebraic expression for the total bound electric field at any position along the fibre.

(b) Starting with the expression for the total field in (a), use lecture notes to determine an integral expression for the total guided mode power in the fibre in terms of ψ₁(x,y), ψ₂(x,y) and ψ₃(x,y).

(c) Confirm that the resulting expression in (b) is equal to the sum of the powers carried by each of the three modes.

7. A multimode, step-profile fibre has a core radius of 40 microns, a core index of 1.51 and a cladding index of 1.49. 

(a) If the fibre end-face is excited by a source with a wavelength of 900nm, what is the maximum number of bound modes that can be excited?

(b) What is the minimum number of bound modes that can be excited regardless of the wavelength of the source?

(c) If the end-face of the fibre in (a) is now illuminated simultaneously by 2 sources that emit light at wavelengths of 800 and 1,000nm, respectively, what is the total number of different bound modal states that can be excited in the fibre?