A weakly guiding optical fibre propagates just two bound modes with scalar transverse electric fields ψ₁(x,y) and ψ₂(x,y), and propagation constants β1 and β2, respectively. At the beginning of the fibre, a source excites both modes with amplitudes a1 and a2, respectively. 

(a) Write down an algebraic expression for the total bound electric field anywhere along the fibre.

(b) This field repeats itself periodically along the fibre with a longitudinal period zb. By examining the expression for the accumulated phase difference between the 2 modes as they propagate determine zb in terms of the propagation constants of the two modes.

(c) Using the expression for the total field in (a), determine the total guided power in the fibre and confirm that it is equal to the sum of the powers in the 2 modes.