Frequency Comb Generation by Bloch Gain Induced Giant Kerr Nonlinearity

Optical nonlinearities are known to coherently couple amplitude and phase of light, which can result in the formation of periodic waveforms. Such waveforms are referred to as optical frequency combs. Here we show that Bloch gain—a nonclassical phenomenon that was first predicted in the 1930s—can play an essential role in comb formation. We develop a self-consistent theoretical model that considers all aspects of comb dynamics: band structure, electron transport, and cavity dynamics. In quantum cascade lasers, Bloch gain gives rise to a giant Kerr nonlinearity, which enables frequency modulated combs and serves as the physical origin of the linewidth enhancement factor. Bloch gain also triggers the formation of solitonlike structures in ring resonators, paving the way toward electrically driven Kerr combs.

Figure 1

Illustration of different systems and their optical susceptibilities. The complex susceptibility of a QCL can be represented as a sum of both Bloch and harmonic contributions.

Figure 2

Bloch gain in QCLs. (a) Unsaturated optical gain from Eq. (1) represented as the sum of the harmonic Lorentzian contribution and dispersive Bloch gain. (b)–(c) Effect of saturation on real ${\chi }_R$ and imaginary part ${\chi }_I$ of the optical susceptibility, whether the Bloch gain is included or not. Red dots indicate values at gain peak, and arrows show the direction of the increasing intensity. Bloch gain induces asymmetric gain, redshift, and nonzero values of ${\chi }_R$ at the gain peak. (d) Saturated gain clamped to the threshold for three values of the current density $J(\mathrm{kA}/{\mathrm{cm}}^2)$. (e) Spectrally calculated LEF for the same current densities as in (d). Values at the gain peak are indicated with dots. (f) LIV characteristic of the QCL design [30] operating in continuous wave at room temperature. (g) Dependence of LEF and factor $b$ from Eq. (2) on the current density. A linearlike dependence of both values is observed.

Figure 3

FM combs in a Fabry-Pérot cavity. (a) Temporal evolution of the intensity. Excluding Bloch gain results in an unlocked state (bottom). Including Bloch gain leads to a stable FM comb (top). The waveforms in the last round trip are shown on the right. (b) Evolution of the intensity spectrum with the increasing current density $J$. Near threshold, the spectrum consists of a single mode and broadens with the increase of $J$. The Bloch gain induced Kerr nonlinearity forms a locked FM comb over the entire range of $J$, as is seen from the autocorrelation value equal to unity (blue line on the left). Neglecting the Bloch contribution leads to unlocked states with the autocorrelation value smaller than unity (red line on the left). The four spectra on the right are taken at $2.5\mathrm{kA}/{\mathrm{cm}}^2$ and $4.5\mathrm{kA}/{\mathrm{cm}}^2$, indicated by the white dashed lines.

Figure 4

Spatial patterns in a monolithic ring laser comb. (a) Schematic of a ring cavity laser. (b) Intensity spectra of a ring QCL. Bloch gain leads to a multimode instability through phase turbulence [25]. A ${\mathrm{sech}}^2$ envelope is fitted to the spectrum. Elimination of Bloch gain yields single-mode operation. (c) Temporal evolution of the intensity shows an initial turbulent regime that forms a frequency comb after 510 000 round trips.