1. Introduction

Semiconductor lasers with external optical feedback have been extensively studied over the preceding decades [1–6] due to the complexity of the underlying non-linear dynamics and the severe requirements on optical isolation imposed by the dynamical instabilities arising from the feedback. With the importance of microscopic lasers for optical interconnects in next-generation computer chips [7], and the absence of practically realisable and CMOS-compatible optical isolators, it is of crucial importance to understand what the requirements are for such microscopic on-chip lasers to be stable towards unintentional optical feedback. Studies are beginning to emerge [8–12], but most previous work on microscopic semiconductor lasers with feedback has been based on specific cases with narrow focus on single parameters [9,10,13]. However, when a laser is scaled down, several of the properties defining the feedback response change, and it is important to obtain a complete understanding of the consequences of down-scaling the device size.

In this work, we derive general results on the feedback properties of lasers, when the device shrinks into the microscopic regime. We focus on the critical feedback level, which is defined here as the value of the external mirror amplitude reflectivity, $r_3$, at which dynamical instabilities first arise, and is thus the most important feedback-related quantity for practical applications of microscopic lasers [4]. In particular, it is demonstrated that small lasers are not intrinsically less stable than their macroscopic counterparts, although a straight-forward application of the general expression for the critical feedback level would suggest this.

Various feedback-resistant lasers have recently been proposed and demonstrated based on this expression with a view to applications in optical interconnects. Examples include lasers relying on increased damping rates and small $\alpha$-factors [6] or high internal quality factors (corresponding to low threshold gain) [16,17]. Other ways of reducing the gain, e.g. slow light [18], may also be viable for realising highly stable lasers, while it was also shown that operation deep in the over-damped regime through the Fano laser geometry yields extraordinary stability [11]. These examples all utilise different ways of achieving highly stable devices, and can generally be explained in terms of the theory outlined in this work, which concludes by giving design rules for obtaining feedback resistant microscopic lasers and demonstrating unavoidable trade-offs in device design.

2. Size-scaling of the critical feedback level

The critical feedback level is given by [3]

In practice, Eq. (1) is viable for a wide range of semiconductor lasers with many different parameter combinations, and the applicability is explored further in the supplementary by dynamical simulations of the full Lang-Kobayashi equations. Figure 1(a) provide a realistic application example, showing a set of CPUs interconnected by an optical waveguide on a micro-chip. In this case, back-reflections from the interface between the waveguide and the detector provides external optical feedback to the laser, with a delay time defined by the waveguide length. Depending on the length of the waveguide and the properties of the laser, the level of feedback necessary to initiate dynamical instabilities varies, with the worst-case being given by Eq. (1). This variation is illustrated on a logarithmic scale in Fig. 1(b), showing the critical feedback level as a function of the mirror reflectivity and cavity length of the source laser. Indicated with red crosses are three example lasers: the photonic crystal (PhC) laser of Ref. [14] and text book examples of vertical cavity surface emitting (VCSEL) and edge-emitting Fabry-Perot (FP) lasers from Ref. [15] currently used in optical communication applications. Interestingly, these have very similar critical feedback levels, despite the fact that they have vastly different sizes and mirror configurations.

This similarity is unintuitive and non-trivial as when inspecting Eq. (1), one observes that the critical feedback level scales linearly with the cavity length, from which the straight-forward conclusion is that decreasing the cavity length reduces the stability of the laser. In reality, however, the situation is more complicated for two reasons. Firstly, the cavity length and mirror reflectivity are constrained by the maximum available gain, and linked through the round-trip condition for the field amplitude, which, in the limit of weak feedback, is

Here, $R_x$ are the power reflectivities of the left and right mirrors, $\alpha _i$ represents internal losses and $\Gamma g_{th}$ is the modal threshold gain. From this condition, one can express, e.g., the mirror reflectivity as function of the cavity length and substitute into Eq. (1), yielding

For decreasing cavity length, the expression converges to

Here, $\tau _s$ is the carrier lifetime, $g_N$ is the differential gain, $N_p$ is the photon density, $\Gamma$ is the confinement factor, $\beta$ is the spontaneous emission factor, $R_{sp}$ is the total rate of spontaneous emission, and $a_p$ is the derivative of the gain with respect to the photon density, accounting for non-linear gain suppression [3]. In particular, one notes the dependence on the spontaneous emission $\beta$-factor, which has been viewed as a defining characteristic of microscopic lasers due to the possibility of achieving near-unity values, leading to threshold-less lasing [21,22]. In terms of feedback stability, however, the impact of the increased $\beta$-factor is simply a small increase of the damping rate, which drops off rapidly as the laser is biased beyond the threshold, due to the inverse proportionality to the photon density in the second term of Eq. (5). As such, the value of $\beta$ itself is of limited importance for the feedback properties, except near the lasing threshold, as in e.g. Reference [10].

 

Fig. 1. (a) Schematic of on-chip optical interconnect between CPUs. Interfaces, at e.g. the detector, lead to unintentional external optical feedback to the laser. (b) Calculated critical feedback level (dB, color scale) as function of cavity length and mirror reflectivity. Inset crosses indicate the locations of different example lasers. "PhC" is the photonic crystal laser of Ref. [14], while the "VCSEL" and "FP" (Fabry-Perot) labels correspond to textbook examples from Ref. [15]. The white area corresponds to devices with unrealistically large threshold gain ($\Gamma g_{th} > 200$ cm$^{-1}$).

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The size scaling of the critical feedback level is thus determined by the available modal gain, linking $r_2$ and $L$, and the size-scaling of the damping rate. As demonstrated in Fig. 2, the analysis is surprisingly simple. The figure shows the calculated critical feedback level as function of the cavity length, $L$, for different constant values of the net modal gain at threshold, as indicated by the legend. These curves are obtained by variation of the mirror reflectivity, $r_2$, to pin the threshold gain. Clearly, the critical feedback level is determined almost exclusively by the net modal gain at which the laser operates, essentially irrespective of the size of the device. There is only a weak size-scaling from small variations in the damping rate owing to the change in system dimensions, but in the range from conventional edge-emitting lasers ($L > 300 \ \mu$m) to microscopic lasers ($L \simeq 1 \mu$m), lasers operating with equivalent net gain are practically equally sensitive to external feedback. This is also evident from Fig. 1(b), where the most stable lasers are in the top right corner, as these are the devices with the lowest modal threshold gain. The importance of the gain also explains the practical equivalence of the critical feedback levels of the three example lasers in Fig. 1(b), despite their vastly different size. The difference in mirror reflectivities means that in practice they operate at similar modal gains, and thus have similar critical feedback levels.

 

Fig. 2. Calculated critical feedback level from Eq. (1) as function of the laser cavity length for different values of the modal threshold gain.

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3. Impact of over-damped operation

It has previously been pointed out [23] that microscopic lasers tend to operate as class A [24] lasers in the over-damped, relaxation-oscillation free regime. This is an important factor to keep in mind when evaluating the feedback properties of micro-lasers, as over-damping is beneficial for feedback stability due to the fact that dynamic instabilities are driven by relaxation oscillations becoming un-damped [3]. Figure 3 illustrates the effect of over-damped operation, by showing stability phase diagrams for the photonic crystal laser of Ref. [14], for different values of the damping rate, $\gamma$, with constant relaxation oscillation frequency, $\omega _R$, so that the ratio $\omega _R/\gamma$ is changed. Here the blue region represents stable continuous-wave output, while the yellow region represents dynamically varying output. The phase diagrams are calculated using a small-signal analysis as in Ref. [3], with the detailed methodology presented in the Supplement 1. It should be noted that the properties demonstrated here are general to the Lang-Kobayashi model, and thus also apply to e.g. macroscopic lasers in the over-damped regime, but they are emphasized here due to the general tendency of microscopic lasers to be over-damped.

 

Fig. 3. Calculated stability diagrams for the photonic crystal laser of Ref. [14], but with different values of the damping rate, $\gamma$, in order to change the ratio $\omega _R/\gamma$. Blue is stable continuous-wave output and yellow is unstable operation, illustrating both the shift of the critical feedback level and the flattening of the stability boundary. The vertical red lines are the critical feedback levels calculated from Eq. (1).

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The increase of $\gamma$ has two effects: the first is a linear increase of the critical feedback level, according to Eq. (1). This can be understood simply from a qualitative consideration of the time evolution of the field amplitude, $|E(t)|$, described by the evolution equation

The physical explanation for the flattening is that the critical feedback level of Eq. (1) corresponds to the case where the external feedback is phase matched to the internal relaxation oscillation frequency, $\omega _R$, of the laser. Over-damped operation then suppresses or even eliminates this phase matching dependence due to the absence of relaxation oscillations, as shown in Fig. 3. Furthermore, the lower boundary of the delay time for which instabilities arise scales with the absolute value of $\omega _R$, due to the necessity of having at least half a period of relaxation oscillations within the delay time for phase matching. Thus, the stability of microscopic lasers is in general less sensitive to fine variations in the phase of the external feedback due to their over-damped operation.

The stability is uniquely determined by Eq. (1) in the case where only a single stationary solution of the oscillation condition exists. The number of solutions is quantified by the $C$-parameter [2,3],

4. Bandwidth trade-off in designing feedback resistant lasers

As has been demonstrated in the previous sections, the straight-forward way to design a laser with a high resistance to external feedback is to minimize the modal threshold gain, cf. Figure 2. However, due to the intimate relation between the damping rate, relaxation oscillation frequency and photon lifetime, doing so is not without consequences for the modulation bandwidth. For lasers obeying a conventional rate equation description (as assumed for validity of the Lang-Kobayashi model), the maximum 3 dB intensity modulation bandwidth of a given laser by variation of the pump power is given approximately by [15]

5. Conclusion

The simple and general conclusion of this work is that the most important design rule for developing feedback-resistant microscopic semiconductor lasers is that the lasers in question must operate with the lowest possible modal threshold gain. Designing microscopic lasers with low operational gain is also desirable due to the requirements of high power efficiency and low energy consumption [7] for data transmission in on-chip interconnects. However, reducing the gain limits the maximum available intensity modulation bandwidth, so that in practice there is a trade-off between feedback stability, energy consumption and speed. Additionally, the intrinsic operation in the over-damped regime of microscopic lasers reduces the sensitivity to the phase of the external optical feedback. These considerations are crucial for the design of micro-scale lasers to be employed for chip-scale data communication, given the absence of integrated optical isolators.