Coherent coupling of transverse modes in quantum cascade lasers

Quantum cascade lasers (QCLs) boast a combination of nonlinear dynamic properties that is unique among solid-state lasers. On one hand, an ultrafast (~ 1 ps) electron relaxation time in excited subbands and an even faster relaxation of the intersubband coherence make QCLs the only solid-state lasers belonging to the group of class-A lasers, in which population inversion and polarization adiabatically follow the dynamics of the electromagnetic field. On the other hand, a very large dipole moment of the intersubband transitions in combination with high intracavity intensity gives rise to a large Rabi frequency of the order of 1 terahertz. Therefore, one can expect that nonlinear effects originating from spectral and spatial hole burning that are only marginal or non-existent in other types of semiconductor lasers will be ubiquitous in QCLs. In this project, in collaboration with F. Capasso group at Harvard, we investigate the nonlinear coupling between the transverse modes of quantum cascade lasers. We present evidence for stable phase coherence of multiple transverse modes over a large range of injection currents. We explain the phase coherence by a four-wave mixing interaction originating from the strong optical nonlinearity of the gain transition. The phase-locking conditions predicted by theory are supported by spectral data and both near- and far-field mode measurements.

Results and experimental figures are from:

[1] N. Yu, L. Diehl, E. Cubukcu, D. Bour, S. Corzine, G. Höfler, A. K. Wojcik, K.B. Crozier, A. Belyanin, and F. Capasso, Coherent Coupling of Multiple Transverse Modes in a Quantum Cascade Laser, Phys. Rev. Lett., 102, 013901 (2009).

FIG. 1 (a) SEM image of the facet of a QCL with a 12-mm-wide active region, and simulations showing from top to bottom the electric field of TM₀₀, TM₀₁, and TM₀₂ modes for the device; the arrows in the figure indicate the polarization of the field. (b) Measured far-field intensity distributions of the device (open circles) and their fits (solid lines) at driving currents of 1.7, 1.5, 1.35, 1.2, 1, 0.75, 0.70, and 0.68 A (from top to bottom). The curves are shifted vertically for clarity. Clearly, above threshold current of ~ 1 A, the radiation pattern becomes asymmetric indicating coherent coupling between transverse modes.

FIG. 2. (a) Spectra of a QCL with a 12-mm-wide active region. (b) Fourier transform of the spectra. The laser is driven by pulsed current I=1.0 A (upper panel) and 0.75 A (lower panel). The spectrum reveals three combs with different periodicities corresponding to three transverse modes. The differences between mode frequencies are non-equidistant below threshold (at 0.75 A) and become locked into equidistant relationship above phase-locking threshold (1 A).

We focus our analysis on a buried heterostructure QCL with an active region of 12 mm, lasing on three transverse modes at a wavelength of 7 mm. The three interacting transverse modes are TM₀₀, TM₀₁ and TM₀₂, shown in Fig.1a. The phase locking between these modes is achieved through four-wave mixing of the type TM₀₀+TM₀₂®TM₀₁+TM₀₁ and therefore it has to have a constant 2F_TM01-F_TM00-F_TM02 phase difference in the phase-locked regime. Our analysis shows that the stable phase locking is possible when the above phase difference is equal to zero (or multiple of 2p), the amplitudes of TM₀₀ and TM₀₂ modes are close to each other, and frequencies of the hot modes are locked into the condition. These locking conditions are in excellent agreement with observed modal amplitudes, phases, and frequencies as reported in [1] and shown in Figs. 1-3. The stability of the solutions is confirmed by solving the equations for a random set of arbitrary initial conditions, as shown in Fig. 4. Our numerical results reported here confirm the above locking conditions and show that they are reached independently of the initial conditions. This shows that mode locking process is robust and stable.

Fig. 3. a) Experimental measurements of relative mode amplitudes. b) Experimental measurements of modal phase differences. c) Numerical results of modal amplitudes. Red triangles: TM₀₁ mode; black squares: TM₀₂ mode; blue circles: TM₀₀ mode.

(a) (b)

Fig. 4. a) Numerical solutions for the modal amplitudes with a random set of initial conditions. The solutions converge to steady state confirming the stability of the solutions. b) The calculations of phase difference F_TM00+F_TM02 -2F_TM01 confirm the stability of the mode locking; the same initial conditions as in a) were used here.