Effect of the coupling phase on locking of longitudinally coupled semiconductor lasers

The success of the very stable and reliable widely tuneable coupled cavity laser (CCL) reported in [1, 2] was due to the coupling element being based on a multi-mode (MMI) waveguide [2]. Such couplers bring about transmission properties with well-defined and engineerable coupling phase [3]. Rate-equation theory [4] for two identical longitudinally coupled Fabry-Pérot semiconductor lasers predicts W = 2k √ 1+α 2 √(P 1 -P 2 ) 2 /4P 1 P 2 + cos 2 (φ c - Arctanα), (1) where κ and φ c are the coupling rate and phase, respectively, P j (j=1, 2) are the laser intensities after coupling and a is the linewidth enhancement parameter. In lowest order of coupling and when both lasers operate not too close to their (uncoupled) thresholds, the P j are in good approximation given by P j = P j 0 {1+k √P 0 k / P 0 j B j cos(φ c + (-1) k Arcsin (ω 21 /W)}, (j, k = 1, 2; k ≠ j), where P 0 j are the intensities before coupling, b j ≈ 2σ c (p j + 1) / (p j Γ) with P j the pump strength of laser j, σ c ≡ sgn[cos(Φ c - Arctanα)] and ω 21 ≡ ω 2 - ω 1 is the uncoupled detuning frequency between laser 2 and 1.