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.