Theory of microscopic semiconductor lasers with external optical feedback: supplement

The properties of microscopic semiconductor lasers with external optical feedback are theoretically analysed. The size-dependence of the critical feedback level, at which the laser first becomes unstable, is clarified, showing how the dominant indicator of feedback stability is the gain of the laser, irrespective of size. The impact of increased spontaneous emission β-factors and over-damped operation is evaluated, exposing a diminished phase sensitivity of microscopic lasers, and a trade-off between modulation bandwidth and feedback stability is identified.


LANG-KOBAYASHI EQUATIONS
The fundamental model of semiconductor lasers with weak external optical feedback is the Lang-Kobayashi equations, first presented in Ref. 1. These consist of a conventional field equation supplemented by an external feedback term and a rate equation for the carrier density in the active region. In this work, we apply a generalised travelling-wave form of the Lang-Kobayashi equations due to the much improved computational speed and flexibility of this approach. The general properties of these two models are essentially equivalent, particularly in the small-signal regime, from which the critical feedback level (Eq. 1 in the main text) is derived. The travellingwave form of the Lang-Kobayashi model is based on the work of Ref. 2 and was first presented in Ref. 3

. The field equation is in this form
where A + (t) is the complex amplitude of the field envelope in the laser cavity and N(t) is the carrier density in the active region. The other parameters are the laser diode round-trip time τ in , the confinement factor Γ, the group velocity v g , the differential gain g N , the linewidth enhancement factor, α, the round-trip time of the external cavity, τ D , and the feedback fraction κ = r 3 (1 − |r 2 | 2 )/|r 2 |, where r 3 is the amplitude reflectivity of the external mirror, while r 2 is the amplitude reflectivity of the laser mirror facing towards the external mirror. The parameters ω and N s are expansion parameters corresponding to the stationary solution of the laser without external feedback, and are found by solving the oscillation condition which ensures that the field replicates itself after a single round-trip. Finally, the term τ in F L (t) represents an integrated noise contribution from spontaneous emission over the round-trip τ in , with F L being a Langevin noise source [3]. Eq. (S1) can be efficiently evolved in time through an iterative process, and due to the assumption of weak feedback, the external field only needs to be stored for a time τ D . The field equation is complemented by a rate equation for the carrier density [4]: Here R p is the pump rate, τ s is an effective carrier lifetime, g(N) = g N (N − N 0 ) is the gain, with differential gain g N and N 0 being the transparency carrier density, while the photon density is related to the field amplitude through the equation N p (t) = σ s |A + (t)| 2 /V p , where V p = V c /Γ is the photon volume and σ s is given by [2] σ s = 2 0 nn ḡ hω where 0 is the vacuum permittivity, n is the material refractive index, n g is the group index, g th is the threshold gain at the frequency ω, α i represents internal losses, and r 1 is the reflectivity of the mirror facing away from the external mirror. In order to adapt Eq. (S3) to the iterative framework of Eq. (S1) for convenient time stepping, a Taylor expansion is utilised to write the time evolution of N(t) as which is justified as the carrier density varies slowly compared to the round-trip time, τ in , which is in the range of 100s of fs to few ps.
To generate the phase diagrams in figure S1, a 500 ns time-domain trajectory is calculated for each value of the external reflectivity, r 3 , and the delay time, τ D . The output is then characterised using the relative intensity noise (RIN), defined as where δP(t) 2 and P(t) are respectively the variance and mean of the output power, P(t). The output power is calculated from the field amplitude as is the distributed mirror loss coefficient and L is the cavity length. This results in two distinct regimes, corresponding to stable (continuous-wave) and unstable (varying periodic, quasi-periodic and chaotic dynamics), which are the blue and yellow regions in the phase diagrams. The initial condition is taken as the minimum linewidth mode, as that is the most stable operation point [5]. The details of how to calculate this mode are given in Ref. 6.
The parameter values used in the calculations are listed in Table S1. The threshold gain is calculated from the photon lifetime as which in turn is used to calculate the solitary threshold carrier density, assuming a logarithmic relation between the carrier density and the material gain: The threshold pump rate is then determined as R p,th = N s /τ s (S10) Finally, the non-linear gain suppression is included as g(N, N p ) = 1 1 + NL N P g 0 log N N 0 (S11) but this is generally negligible in the cases here due to the choice of pump rate. Figure S1 shows calculated stability phase diagrams for well-known commercial lasers, specifically an in-plane edge emitting Fabry-Perot (FP) laser and a vertical cavity surface emitting laser (VCSEL), as well as a photonic crystal laser (Ref 7). These are calculated using the previously described computational model, with the parameters listed in table S1. The stability boundary expressed by Eq. 1 of the main text is indicated by the vertical red line in each figure, and it is clear that it provides an excellent estimate of the worst-case (i.e. minimum required) feedback level for instabilities to occur. The lasers in figure S1(b) are chosen to cover a wide range of devices, to illustrate the applicability of Eq. 1 of the main text. The approximations used in deriving Eq. 1 of the main text are (see e.g. Refs. 8 and 6) ω 2 R κ 2 /τ 2 in and ω 2 R γκ/τ in , where ω R is the relaxation oscillation angular frequency, τ in = 2L/v g is the laser roundtrip time, and κ = r 3 /r 2 (1 − |r 2 | 2 ) is the conventional feedback fraction [1]. Due to the size-scaling of both γ and ω R , combined with the explicit dependence on the cavity length through τ in , the size-dependence of these approximations is complicated to predict. However, numerical investigations indicate that the approximations are only invalidated for unusual combinations of cavity length and mirror reflectivity (e.g. L 1 mm and r 2 0.9) together with large pump rates (> 10 times the threshold rate). Thus, the equation is in practice widely applicable, even for the case of microscopic lasers, and as such may be used to predict their feedback properties. As such, the main takeaway here is that the analytical expression for the critical feedback level provides an excellent estimate of the worst-case onset of instability, validating its use also for analysis of microscopic lasers.

SMALL-SIGNAL STABILITY BOUNDARIES
One can also, more efficiently, obtain these phase diagrams by determining the instability boundaries in the small-signal regime. By taking the Laplace transform of the linearised dynamical equations, and searching for zeros of the determinant crossing the real axis, one can calculate the boundary of stability. This approach is used to phenomenologically illustrate the impact of overdamping in figure 3 of the main text, and is based on the work of Refs. 8 and 6. The stability boundary is given by the equation where y is the principal value of ωτ D , ω R is the relaxation oscillation frequency, γ is the relaxation oscillation damping rate and Ω is given by the solutions to Given the coupled nature of these transcendental equations, determining the stability boundary is a complicated numerical problem. One approach to creating the phase diagram is simply to sweep r 3 and τ D , and for each combination calculate the first N solutions to Eq. (S13), where N is the number of 2π multiples spanned by ω R τ D . Then, for each of these solutions check the equality in Eq. (S12), and assign the stability based on this check. This is the approach used to generate figure 3 in the main text, in accordance with Ref. 6, and it clearly agrees well with the full numerical simulations of figure S1. For (a), γ and ω R have their natural values, while in (b) γ is scaled by a factor of 5, and in (c) a factor of 10.