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Physical Review B

Saturation and self-lensing in self-assembled quantum dots with constant-wave driving

A. Tierno, T. Ackemann, T. Maggipinto, Massimo Brambilla

Published Jul 22, 2009• 6 citations
Publisher

Abstract

The nonlinear optical response of self-assembled quantum dots to cw driving is analyzed via numerical simulations of a spatially resolved rate equation model. The saturation is shown to follow a behavior in between the one for a dominantly, homogeneously, and inhomogeneously broadened medium. Self-lensing is suggested to probe the refractive-index nonlinearities and to open a complementary way of characterizing phase-amplitude coupling ($\ensuremath{\alpha}$ factor) in quantum-dots samples. For conservative assumptions on current samples the minimum focal length is predicted to be $\ifmmode\pm\else\textpm\fi{}1.7\text{ }\text{mm}$ for an input beam with $15\text{ }\ensuremath{\mu}\text{m}$ radius at a detuning of 1.1 inhomogeneous linewidths from gain center.

DOI
10.1103/physrevb.80.035314

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Sep 1, 2004•1 citation
The nonlinear optical response of self-assembled quantum dots to cw driving is analyzed via numerical simulations of a spatially resolved rate equation model. The saturation is shown to follow a behavior in between the one for a dominantly, homogeneously, and inhomogeneously broadened medium. Self-lensing is suggested to probe the refractive-index nonlinearities and to open a complementary way of characterizing phase-amplitude coupling (\ensuremathα\ensuremath{\alpha}\ensuremathα factor) in quantum-dots samples. For conservative assumptions on current samples the minimum focal length is predicted to be \ifmmode±\else\textpm\fi1.7 mm\ifmmode\pm\else\textpm\fi{}1.7\text{ }\text{mm}\ifmmode±\else\textpm\fi1.7 mm for an input beam with 15 \ensuremathμm15\text{ }\ensuremath{\mu}\text{m}15 \ensuremathμm radius at a detuning of 1.1 inhomogeneous linewidths from gain center.
We analyze the strength of the nonlinear refractive index change associated with the saturation of the ground state absorption or gain of self-assembled quantum dot samples under cw optical injection. The model is based on a two-level framework but takes into account the strong coupling of quantum dots to the wetting layer by thermal excitations and Auger processes. Due to the inhomogeneous broadening and limited density the effect turns out to be weak though it should be experimentally detectable using self-lensing, the shift of Fabry-Perot resonances or interferometric schemes.
Spectral broadening and cross relaxation of an aggregate of highly carrier-injected quantum dots have been studied experimentally by evaluating the gain saturation and optical-wave mixing characteristics of traveling-wave laser amplifiers that include self-assembled 0.98-µm InGaAs dots. Measured signal-gain spectra at room temperature showed that the gain decreases over the entire gain spectra when an intense and narrow-linewidth saturating signal is injected at a higher-order subband transition of the quantum dot. Differential-gain spectrum was calculated from the saturated and unsaturated signal gain spectra, and revealed asymmetric gain saturation with respect to the wavelength of the saturating signal. These results suggest that cross relaxation is so fast that the dot aggregate has a homogeneously broadened gain spectrum. Self-saturation and cross-saturation characteristics, and the detuning characteristics of highly nondegenerate four-wave mixing were also investigated. The latter revealed that the characteristic times of two cross-relaxation processes are about 1 ps and 100 fs.
The nonlinear optical phase shift in self-assembled InAs quantum dots (QDs) under resonant excitation in a ground-state transition was measured by a unique two-color pump∕probe ellipsometric polarization analysis. This ellipsometric analysis makes use of the large optical birefringence of SK-QD [(SK) — Stranski-Krastanov] originating from the asymmetric structure. A phase shift of 0.5π rad was obtained at an input pump pulse energy density of 30pJ∕μm2, a detuning of 11meV, and a time delay of 20ps in a 1mm long waveguide having QDs with a peak wavelength of 1290nm, a volume density of 4×1015cm−3, and inhomogeneous broadening of 35meV. Analysis revealed that the phase shift is mainly attributed to the absorption saturation for TE-polarized light, though other mechanisms also could contribute at higher pumping. The calculation, based on the two-level approximation, revealed that the minimum energy density for π shift is 240fJ∕μm2, calculated under ideal conditions.
We numerically investigate the effect of inhomogeneous broadening caused by quantum-dot (QD) size fluctuations on the gain and phase recovery of QD semiconductor optical amplifiers (SOAs). We establish 1088 coupled rate equations to simulate the carrier dynamics of the inhomogeneously broadened QD ensembles as inhomogeneous broadening increases. When all the QD ensembles are identical and inhomogeneous broadening becomes zero, eight coupled rate equations are solved for the homogeneous QDs. The gain and phase recovery responses are calculated when an ultrashort pump pulse is injected into a QD SOA. As the inhomogeneous broadening increases, the slow component of the phase recovery at the QD ground state increases due to the enlarged contribution from the slow phase recovery of carrier reservoirs such as the QD excited states. By separately calculating the gain and phase recovery responses of the homogeneous QDs with different sizes, we identify how increasing inhomogeneous broadening affects the enlarged slow phase recovery components from carrier reservoirs. We also demonstrate that the effect of inhomogeneous broadening on the temporal variation of the α-factor is more significant compared to the injection pump power.
The dynamics of ultrashort pulses propagating in a quantum dot amplifier is determined by a complex nonlinear coupling and dynamic interplay of light fields and carriers in the spatially inhomogeneous quantum dot ensemble. Computational modeling shows that in spite of the large complexity the strong localization of the carrier inversion and the low amplitude phase coupling may allow the amplification and transmission of ultrahort light pulses with minimum deterioration of the pulse properties (e.g. pulse shape, duration). The theoretical description is based on spatially resolved Quantum Dot Maxwell-Bloch equations that describe the spatio-temporal light field and inter-/intra-level carrier dynamics in each quantum dot of a typical quantum dot ensemble. In particular, this includes spontaneous luminescence, counterpropagation of amplified spontaneous emission and induced recombination as well as carrier diffusion in the wetting layer of the laser. Intradot scattering via emission and absorption of phonons, as well as the scattering with the carriers and phonons of the surrounding wetting layer are dynamically included on a mesoscopic level. Spatial fluctuations in size and energy levels of the quantum dots and irregularities in the spatial distribution of the quantum dots in the active layer are simulated via statistical methods. Simulation results of the nonlinear pulse propagation in quantum dot optical amplifiers allow visualization and interpretation of fundamental nonlinear processes such as selective depletion and re-filling of quantum dot energy levels leading to a complex gain and index dynamics that affect the amplitude and phase of a propagating light pulse. Computational modelling thus may lay the foundation for an optimization and tayloring of pulse properties.