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International Conference on Transparent Optical Networks

Nonlinear phase-shifts in self-assembled quantum dot samples under CW driving

T. Ackemann, A. Tierno, Sylvain Barbay, Robert Kuszelewicz, Massimo Brambilla

Published Jun 1, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.1109/icton.2010.5549301

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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Sep 1, 2004•1 citation
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.
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.
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.
The effects of the hydrogenic impurity on the electron-related non-linear optical processes in a InAs/GaAs dome-shaped quantum dot with a wetting layer under applied electric fields are studied within the density-matrix formalism. The one-electron energy levels and wave functions are calculated using the effective mass approximation and the finite element method. The non-linear optical absorption, relative refractive index change and non-linear optical rectification associated with interlevel transitions are calculated under a strong probe field excitation for both in-plane and z-polarisation of the incident light. According to our results as the electric field increases the absorption and dispersion peaks decrease and exhibit red shift. Hydrogenic impurity located at the origin induces a blue shift in the optical responses. For the optical absorption coefficient the peaks magnitude is enhanced by the impurity presence independent of the electric field strengths, whereas the non-linear optical rectification is larger in the case with impurity only for zero applied electric field.
I recent years, significant attention has been given to nonlinear optical phenomena, such as nonlinear optical absorption and dispersion, derived from a driven quantum transition of semiconductor nanostructures, such as semiconductor quantum wells, quantum disks and mainly quantum dots. An important issue that is widely studied is the effect of probe electromagnetic field intensity on the total (nonlinear) absorption coefficient and on the total (nonlinear) index of refraction. Some of the phenomena that are commonly found for large electromagnetic field intensities are the creation of a strong dip in the absorption spectrum near or at resonance (the so-called bleaching effect), negative absorption near or at resonance that leads to optical gain and the change of the slope of the total index of refraction near or at resonance from negative to positive that changes the behavior of the system from fast to slow light. Unfortunately, these phenomena do not exist and are artifacts of a common abuse of the theoretical methodology for the study of the total absorption coefficient and the total index of refraction. The main error is that the total absorption coefficient and the total index of refraction was initially derived from density matrix equations using perturbation theory and includes first-order (linear) as well as third-order (nonlinear) terms. This method gives correct results for small electromagnetic field intensities and it leads to saturation of absorption and dispersion in this regime. However, for larger intensities it gives the misleading results mentioned above due to either failure of the order of perturbation theory considered in the derivation of the total absorption coefficient and the total index of refraction or even failure of perturbation theory at all in the treatment of the problem. In this work, we revisit the problem of the total absorption coefficient and the total index of refraction in a symmetric semiconductor quantum dot structure (we choose the example of a core-shell quantum dot) under a strong probe field excitation. We use the two-level model, solve the relevant density matrix equations under the rotating wave approximation and under steady state conditions, and obtain the correct form of the nonlinear optical susceptibility that is then used for the derivation of the formulae of the total absorption coefficient and the total index of refraction under the interaction with a strong probe field. Actually, the commonly used perturbation theory formulae can be obtained by our formulae using a series expansion and keeping terms up to second order in electric field amplitude. We also derive the electronic structure of the quantum dot system by solving numerically the time-independent Schrodinger equation, under the effective mass approximation, using the potential morphing method and obtain the energy difference and the electric dipole matrix element for the relevant transitions of the quantum dot system. Then, for the specific quantum dot system we compare the results of the total absorption coefficient and the total index of refraction for the two methodologies (the perturbation result and our result) for different electromagnetic field intensities.
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.