Characterization of the synchronization regimes of a self-injected two-frequency laser

Summary form only given. Frequency-shifted feedback was recently used to synchronize the beat note between the two modes of a dual-frequency laser with the radiofrequency emitted by a local microwave oscillator. This technique allows one to obtain an optically-carried RF clock signal that can find applications in microwave photonics. Two regimes of synchronization are then found [1]. When the frequency difference Δν between the two oscillators is smaller than a frequency f_A determined by the feedback amount, the two oscillators are phase-locked, leading to a constant relative phase φ. Then, a regime of frequency locking without phase locking appears. This regime, also called bounded phase [2,3], extends the frequency synchronization beyond f_A up to a frequency f_B. Finally, when Δν is larger than f_B, the two oscillators are not synchronized, resulting in an unbounded drifting phase. Here, we experimentally characterize the three regimes and compare the results to numerical simulations. Our dual-frequency Nd:YAG laser emitting at 1064 nm, is detailed in [1]. The laser cavity contains two quarter-wave plates in order to enforce dual-frequency operation with an adjustable beating frequency Qy.Qx between two linearly polarized x and y modes. We choose Qy. Qx around 200 MHz. The feedback arm contains a Bragg cell driven at fAO and a quarter-wave plate that flips the x polarization into a y polarization after reflection on the feedback mirror. The optical frequency of the reinjected beam is thus shifted by 2fAO. Hence the laser dynamics is governed by a third frequency Qx + 2fAO, and ´Q is equal to Qy - Qx - 2fAO. The figure shows the so-called phasor plots [3] and the corresponding spectra. It evidences the three regimes, i.e., locked (in which the relative phase I between the beat note and the local oscillator is fixed), bounded (in which I oscillates in a bounded interval) and unbounde- (in which I grows indefinitely). By using a Lang-Kobayashi-type model, the phase evolution can be studied mathematically. In particular, using asymptotic methods, the transition between the locked and the bounded regime is found to be associated to a bifurcation. Conversely, the transition from the bounded to the unbounded regime is not a bifurcation, as formerly stated [2]. This is highlighted by the power spectra (e) and (f) which are similar. However, by measuring the phase noise spectra, we have found experimentally that the bounded regime has the same quality, in terms of spectral purity as the phase locked regime. In both cases the long-term phase stability of the local oscillator is reported on the beat note. Conversely, in the unbounded regime, the phase stability of the beat note is essentially that of the freerunning laser. This is confirmed by numerical simulations and demonstrates that bounded-phase regime truly enlarges the locking range. We are now extending our studies to injected semiconductor lasers, where boundedphase is also observed [3].