Q1. The transfer Matrix for a lossless optical coupler is:

where Ei's and Eo's are the input and output electric fields at different ports and "k" in the transfer matrix is the coupling coefficient.

(a) Assuming k = 20 cm^-1, find the shortest length of the coupler that couples 30% of the power from port 1 to port 2.

(b) Show that when the signal is applied to any one of the input ports, the "transmitted" signal and the "coupled signal" always have a 90-degree phase difference.

(c) Show that the above device, a 3dB coupler with a fiber loop of length Δl at the output, works as a reflecting mirror. Find the reflection coefficient as a function of the length of fiber loop Δl. (Hint: Use the property that the transfer matrix in coupled-mode analysis can be cascaded.)

Q2. For a single mode slab waveguide with a grating shown in the figure,

(a) Calculate the coupling coefficient, k, for the grating, (b) plot the power reflection curve versus Δβ/|k| with Δβ = β - (π/Λ), (c) find the "photonic bandgap" in terms of |k|.

Q3. (a) Describe how a Δβ coupler work. What are its advantages compared with a single-section directional coupler.

(b) Show the switching diagram (Kl vs. Δl) with detailed illustration.

(c) Consider an input with a(0) = 1, b(0) = 0, Δl = π/2, and Kl = π, where l is the total length of the Δβ coupler. Find the output powers, |a(l)|² and |b(l)|².