Mode-locked lasers, also called comb lasers [1] or laser frequency combs [2], have gained considerable interest, such as in frequency metrology [3], applied spectroscopy [4], or ranging [5]. Of particular interest are dense frequency combs, i.e., with small mode spacing through low, sub-GHz, repetition rates, for instance, to resolve the typically GHz-wide absorption lines of atmospheric trace gases [6] or to extract thermodynamic properties (temperature, density, and pressure) of molecular gas samples from pressure broadening and line shifting [7]. Other applications lie in microwave photonics, where optical comb lines can be used as a local oscillator, enabling data to be upconverted or downconverted across different RF bands by switching the carrier frequency [8]. Further potential applications of low repetition rate frequency combs could be enabled by miniaturization through photonic integration with electrically pumped semiconductor optical amplifiers. Examples are the upscaling of quantum optical systems [9,10,11,12] or compact LIDAR sensors [13]. In all instances, maintaining a high level of frequency stability is essential.

To address the named drawbacks, there has been a push to develop frequency combs based on integrated photonics, which offers compact, chip-based solutions that can be scaled toward more complex systems. An example is generating Kerr-combs in micro-ring resonators [14,15]. These combs can provide extremely wide spectra; however, the repetition rates are high, typically above 50 GHz and the conversion efficiency is typically only at the percent level [16]. Monolithic mode-locked diode lasers provide the unique advantage of direct electrical pumping, eliminating the need for optical pumping and thus providing higher optical power and overall efficiencies [17]. Electrical pumping and monolithic integration of the laser cavity strongly decreases the susceptibility to acoustic perturbations and simplifies the overall system, including thermal management. However, monolithic mode-locked diode lasers share a drawback with Kerr frequency combs in that they also exhibit high GHz repetition rates due to their short cavity length. A fundamental limitation of monolithic diode lasers is the broader intrinsic linewidth of the comb frequencies [18,19], which is due to the short cavity length as well, in combination with a relatively higher intrinsic waveguide loss in diode laser amplifiers.

An interesting solution for lowering the repetition rate and the intrinsic linewidth of diode lasers is to increase the cavity length by heterogeneous integration [1] and hybrid integration [20]. These techniques involve the use of extended cavities such as that made of low-loss Si₃N₄ feedback circuits and integrated saturable absorbers to generate mode-locked pulses. Although such feedback circuits can provide long roundtrip lengths in a chip-size format, there is a fundamental stability problem. Lowering the repetition rates to the sub-GHz range is prone to chaotic fluctuations [21,22] or harmonic mode-locking [23] with several pulses per roundtrip and the latter increasing the repetition rate again. The reasons are the short gain lifetimes and recovery times that are typically around 1 ns in semiconductor optical amplifiers (SOA) [24]. If the roundtrip time of the pulses in the laser resonator becomes longer than the upper state lifetime, amplified spontaneous emission between pulses destabilizes mode-locking [25]. This limitation makes it difficult to achieve the desired sub-GHz repetition rates with diode lasers using extended cavities, whether using bulk optics or integrated photonics.

To open the path toward sub-GHz repetition rate mode-locking with integrated diode lasers, here we explore the alternative approach of Fourier domain mode-locking (FDML) [26]. With FDML, the laser cavity does not contain a saturable absorber such that the output is quasi-continuous. Instead, the frequency of the laser is modulated synchronously with the roundtrip time. Essentially, the output power is constant while the optical frequency is periodically chirped with the roundtrip rate, which generates a comb spectrum. FDML of a diode laser with an integrated feedback circuit has been suggested and numerically analyzed by Heck et al. in [27]. As no pulses are formed, stimulated emission continuously saturates the gain, which suppresses amplified spontaneous emissions independent of the cavity roundtrip time. We note that FDML is actually related to frequency modulation (FM) mode-locking as both can generate a quasi-continuous, periodically chirped output. However, there are some differences in their working principles and laser parameters. With FDML, the output is generated with an intracavity spectral filter, with the filter frequency periodically varied [26]. In an FM mode-locked laser, it is the cavity length that is varied via phase modulation; however, there is no spectral filter and one aims at low dispersion [28]. Our laser contains a narrowband intracavity filter with high dispersion, which is essential for self mode-locking. Another difference is that FM mode-locking can also generate short pulses, with a proper amount of dispersion and gain or loss variation [29] or via additional nonlinear effects [30]. The property of FDML being intrinsically absorber-free is also of practical relevance. For the telecom wavelength range, saturable absorbers integrated with semiconductor amplifiers are available within multi-project wafer (MPW) runs; however, other wavelength ranges require dedicated fabrication involving much higher costs. These fundamental differences and practical aspects make FDML more universal, as sub-GHz generation can presumably be obtained in many wavelength ranges due to the wide availability of semiconductor amplifiers.

In recent experiments, FDML based on diode lasers has been successfully demonstrated by using spectral feedback filtering [31,32]. We note that spectral filtering, such as with an intracavity grating, is required with standard (pulsed) mode-locking to restrict the destabilizing influence of intracavity dispersion. In contrast, to obtain stable FDML, sharper spectral filtering is required. The qualitative explanation is that the filter-induced bandwidth reduction and the associated, steep filter dispersion counteract ultrashort pulse formation and impose quasi-continuous oscillation. In more detail [33] detuned loading introduced by the filter [34] enables amplification of relaxation oscillation (RO) if the RO frequency is in resonance with the mode spacing frequency. The RO then generates multiple sidebands in the laser cavity modes, such that single-frequency oscillation is suppressed.

The lowest repetition rates reported with this approach with diode lasers so far are 255 MHz [32] and 360 MHz [31]. However, for the former, bulk optics was used for cavity length extension and spectral filtering (a lens-coupled Bragg fiber), which is not suitable for chip integration. In the other case (360 MHz), the laser was fully integrated with a SiN waveguide delay line for cavity length extension, while a Bragg waveguide served for spectral feedback filtering. The observed average repetition rate matched the extended cavity length. However, the authors also noted signs of dynamic instability due to independent locking of subgroups of modes, which we address to an insufficiently narrow Bragg filtering. In contrast, stable FDML has been observed in hybrid integrated lasers enabled by the sharper spectral filtering via micro-ring resonators [35,36]. The repetition rates of these lasers were, however, about an order of magnitude higher than in [31,32], i.e., 5 GHz and 2.5 GHz, respectively.