IntroductionThe interaction of short optical pulses with laser cavity modes is important in, for example, formation of mode-locked pulse trains, optical clock recovery, and external optical feedback [1]. In a dispersive nonlinear medium such as a semiconductor, a treatment of the optical field is required which goes beyond discrete modal frequencies. A description of coherent interactions requires resolution of the fast oscillating carrier wave on the femtosecond timescale, while the return to equilibrium may take place over nanoseconds due to a long cavity lifetime or relaxation oscillation period. In inhomogeneous structures such as multi-section or DFB lasers, sub-wavelength to millimeter spatial scales must also be described.The spatio-temporal dynamics of the electric field may be calculated from Maxwell's equations using a finite-difference time-domain (FDTD) method. Coupling a Lorentzian resonance allows an approximate model of the optical gain [2]. Realistic microscopic models of the semiconductor gain based on the semiconductor Bloch equations [3] are at present too computationally intensive for simulation over such a range of temporal scales.We consider a class of recent experiments in which a short optical pulse is injected into a semiconductor laser diode, allowing a study of the pulse-cavity interactions on time-scales shorter than the cavity roundtrip time. In addition to the expected pulse broadening and relaxation oscillations, new phenomena such as stable, long-lived 'dark pulses' were observed [4,5].
Numerical MethodsWe have coupled an FDTD calculation of the electric field with multiple Lorentzian resonances which approximate the spectral dependence of the semiconductor gain. Figure 1 shows that four resonances provide a good fit to the semiconductor gain spectrum [6]. We found that spatially resolving the carrier lifetime and incorporating carrier diffusion was essential in achieving a physically realistic number of lasing modes.
Results and discussionDuring the propagation of a short pulse through a population-inverted semiconductor a region of depleted gain is left behind the injected pulse. For a laser under CW operation this region of depleted gain can evolve into a long lived 'dark pulse'. Figure 2 shows the simulated output from the laser facet following the injection of an optical pulse with a central frequency of 193 THz, resonant with the CW emission wavelength. The initial perturbation induces relaxation oscillations (period ~ 50 ps) which persist for ~400 ps. A bright pulse is emitted from the facet every ~7 ps, after each cavity round-trip. The amplitude of successive bright pulses decays on the timescale of a few hundred ps. A dark pulse forms after the first few round trips, and is the dominant feature after ~200 ps. On a longer time scale, this feature oscillates (with ~500 ps period) between a predominantly bright and dark pulse superimposed on the CW emission. All of this behavior is in good qualitative agreement with experimental results [4]. Inset (A) of Figure 2 shows the ...