# Subharmonic combs in short-cavity swept lasers

A Stroganov, A V Kovalev, E A Viktorov

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A Stroganov, A V Kovalev, E A Viktorov. Subharmonic combs in short-cavity swept lasers. 2022 European Semiconductor Laser Workshop (ESLW), Sep 2022, Neuchatel, Switzerland. hal-04041600

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# Subharmonic combs in short-cavity swept lasers

A. Stroganov,¹,* A. V. Kovalev,¹ and E. A. Viktorov¹

¹ ITMO University, Birzhevaya Liniya 14, 199034 Saint Petersburg, Russia

*stroganov@itmo.ru

**Abstract:** We analyze dynamics of a short-cavity swept semiconductor laser source and show possibility of several distinct regimes which indicate existence of subharmonic resonances in the system and may be justified by the bifurcation structure of the steady state solutions. Observed periodic regimes yield optical frequency combs in spectral domain.

## 1. Introduction

Today there is a number of crucial technologies which rely on coherent swept sources, such as optical coherence tomography (OCT), LiDAR or distributed fiber sensing. This fact created the necessity and required basis for considerable advancements of the sources and detailed studies of possible dynamics of such systems. At the same time, there is general trend on making optical systems more compact or even fully integrating them on a chip. Typical presence of Kerr nonlinearity in such miniaturized systems enables multistability and hysteresis effects due to folded steady state solutions of formed cavities. Nonlinearity also leads to a variety of comb generation approaches, including recent demonstrations of subharmonic combs in ring resonators, where several solitons form controlled dissipative time crystal states [1]. And while nonlinear properties and non-negligible effects in fibers or integrated photonics materials are known and expected, it was also recently shown that short-cavity semiconductor lasers with intracavity spectral filtering may behave as nonlinear resonators even in absence of any nonlinear medium [2]. This fact opens new interesting prospects for further analysis of dynamics and applications of such configurations. Delay Differential Equations (DDEs) were shown to be an efficient tool to model the behaviour of swept systems and were able to predict and describe main experimental results for both long and short cavity semiconductor lasers [3].

## 2. Model description

We study the dynamics of a short-cavity swept source using the model from Ref [2]. The set of two DDEs describes the evolution of the complex electrical field amplitude $\tilde{E}$ and cumulative saturable gain G as

$$ \gamma^{-1} \frac{d\tilde{E}}{dt} + (1 + i\Delta)\tilde{E} = \sqrt{\kappa} e^{\frac{1-i\alpha}{2}} G \tilde{E}(t-1), \quad (1) $$

$$ \eta^{-1} \frac{dG}{dt} = J - G - (e^G - 1) |\tilde{E}(t-1)|^2, \quad (2) $$

where t is time normalized such that the cavity round trip time is equal to 1. Coefficient $\gamma$ is bandwidth of the intracavity spectral filtering multiplied by the round trip time, and $\eta$ is the ratio of the cavity round trip time and the carrier density relaxation time. The attenuation factor $\kappa$ describes linear intensity losses per round trip, $\alpha$ is the linewidth enhancement factor, and J is the pump parameter. The tuning is described via time-dependent $\Delta = \Delta(t)$, which defines the detuning between the central frequency of the narrowband tunable filter and the reference frequency, which coincides with the frequency of one of the laser modes. For convenience of further analysis, substitution $\tilde{E} = E e^{-i \int_0^t \Delta(x) dx}$ is used to transform these equations to

$$ \gamma^{-1} \frac{dE}{dt} + E = \sqrt{\kappa} e^{\frac{1-i\alpha}{2}} G - i\phi(t) E (t-1), \quad (3) $$

$$ \eta^{-1} \frac{dG}{dt} = J - G - (e^G - 1) |E(t-1)|^2, \quad (4) $$

where $\phi(t) = \int_{t-1}^t \Delta(x) dx$.
### 3. Results and discussion

In our analysis, linear tuning is defined by $\phi(t) = \varepsilon * t$, where $\varepsilon$ is a control parameter corresponding to the tuning speed. By sweeping $\varepsilon$ in wide range of values, several distinct behaviours were observed: fundamental, with $2\pi$ periodicity; subharmonics with $2\pi \cdot n$, n=2,3,4,5; and regions with non-periodic and chaotic behaviours. It can be illustrated that appearance of these subharmonic resonances in the system is governed by $2\pi$-periodicity and hysteretic structure of steady state solutions and their corresponding bifurcation points. All periodic regimes produce optical frequency combs in spectral domain, with the mode spacing directly defined by the tuning speed and corresponding dynamical regime, may which allow generation of combs with defined characteristics (see Figure 1).

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Intensity plotted against tuning parameter $\phi$ (a-d) and corresponding spectra (e-f) for various regimes: $\varepsilon = 0.10$ (fundamental), $\varepsilon = 0.24$ (fundamental), $\varepsilon = 0.28$ (subharmonic 1/2), and $\varepsilon = 0.32$ (subharmonic 1/3); other parameters are $\gamma = 0.25$, $\eta = 1.0$, $\kappa = 0.35$, $\alpha = 5$, and $J = 10$.</figcaption></figure>

### 4. Conclusion

Short-cavity semiconductor lasers demonstrate asymmetric steady state with foldover and hysteretic properties. We report presence of subharmonic periodic regimes in such lasers in case of linear frequency tuning. These regimes correspond to frequency combs in spectral domain with subharmonic repetition rates defined by the tuning speed parameter. This underlines a new feature in possible dynamical behaviours of compact swept sources.

### References

1. H. Taheri, A. B. Matsko, L. Maleki, and K. Sacha, "All-optical dissipative discrete time crystals," Nat. Commun. **13**, 848 (2022).

2. A. V. Kovalev, P. S. Dmitriev, A. G. Vladimirov, A. Pimenov, G. Huyet, and E. A. Viktorov, "Bifurcation structure of a swept-source laser," Phys. Rev. E **101**, 012212 (2020).

3. S. Slepneva, B. Kelleher, B. O'Shaughnessy, S. P. Hegarty, A. G. Vladimirov, and G. Huyet, "Dynamics of fourier domain mode-locked lasers," Opt. Express **21**, 19240-19251 (2013).