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IEEE Journal of Quantum Electronics

Two-level description of gain and mixing susceptibilities in amplifying semiconductor materials

Bruno J. Thedrez, A. Jones, Robert Frey

Published Aug 1, 1988• 18 citations
Publisher

Abstract

Optical-intensity-dependent gain and mixing susceptibilities of amplifying semiconductor materials have been calculated using the density-matrix approach. The model, which takes intraband relaxation and carrier-injection rate into account, allows for numerical computation of these susceptibilities, whatever the medium saturation is. As a simplification, an equivalent two-level system is derived which represents the gain and nearly degenerate four-wave mixing susceptibilities. Such an equivalence, which leads to the same field-induced polarization as the conventional rate equation model, also connects the two-level (or rate equation) parameters to the real characteristics of the semiconductor material.>

DOI
10.1109/3.7076

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Optical-intensity-dependent gain and mixing susceptibilities of amplifying semiconductor materials have been calculated using the density-matrix approach. The model, which takes intraband relaxation and carrier-injection rate into account, allows for numerical computation of these susceptibilities, whatever the medium saturation is. As a simplification, an equivalent two-level system is derived which represents the gain and nearly degenerate four-wave mixing susceptibilities. Such an equivalence, which leads to the same field-induced polarization as the conventional rate equation model, also connects the two-level (or rate equation) parameters to the real characteristics of the semiconductor material.>
Saturation properties of absorption, dispersion, and mixing susceptibilities in semiconductors have been described using the density matrix approach. This model, which takes into account both intraband thermalization and interband relaxation including Auger recombination, allows for numerical computation of the susceptibilities whatever the temperature and laser intensity are. As a simplification, an equivalent two-level system is derived which represents very well absorption and mixing susceptibilities for any laser wavelength, dispersion being correctly described by this model only when the pump frequency is close to or smaller than the band-gap energy. Such an equivalence is very easy to handle and allows for great simplification when the susceptibilities are introduced into the propagation equations.
Quantum dot (QD)-based semiconductor optical amplifiers offer unique properties compared with conventional devices based on bulk or quantum well material. Due to the bandfilling properties of QDs and the existence of a nearby reservoir of carriers in the form of a wetting layer, QD semiconductor optical amplifiers may be operated in regimes of high linearity, i.e. with a high saturation power, but can also show strong and fast nonlinearities by breaking the equilibrium between discrete dot states and the continuum of wetting layer states. In this paper, we analyse the interplay of these two carrier populations in terms of a simple rate equation model. Based on the steady-state and small-signal properties of the model, we analyse and discuss the optical modulation response and the four-wave mixing properties of QD semiconductor optical amplifiers, in particular emphasizing the role of ultrafast gain dynamics.
The theory of four-wave mixing (FWM) in the quantum dot (QD) semiconductor optical amplifiers (SOAs) is discussed by combining the QD rate equations system, the quantum-mechanical density-matrix theory, and the pulse propagation in QD SOAs including the three region of QD structure ground state (GS), excited state (ES), and wetting layer. Also, relations for differential gain, gain integral, and nonlinear susceptibility of both pump, probe, and signal pulses were discussed. Gain and differential gain have been calculated for QD structure. FWM efficiency and its components [spectral hole burning (SHB), carrier heating, and carrier density pulsation] are calculated. It is found that inclusion of ES in the formulas and in the calculations is essential since it works as a carrier reservoir for GS. It is found that QD SOA with enough capture time from ES to GS will reduce the SHB component, and so it is suitable for telecommunication applications that require symmetric conversion and independent detuning.
The intracavity absorption (or gain) coefficient of two-level atoms embedded inside a Fabry–Perot cavity has been demonstrated to be well represented by an equivalent two-level system. The parameters of this new two-level system are simply related to that of the medium and to the cavity characteristics. When associated with well-known results of fluorescence spectra and nearly degenerate four-wave mixing in two-level media, this equivalence provides a universal description of intracavity resonance fluorescence and relaxation oscillations in any nonlinear absorptive, amplifying, or dispersive Fabry–Perot cavities.
We consider an intrinsic semiconductor medium subject to an injection current, an arbitrarily intense electromagnetic wave, and one or two non-saturating waves. We assume that the saturating wave intensity is constant throughout the interaction region and ignore transverse variations. We further assume that the carrier–carrier scattering is sufficiently fast that the carriers are described approximately by Fermi-Dirac distributions. We find that the probe-wave propagation is described by Beer’s law or coupled-mode equations. These equations are determined by a Fourier expansion technique highly developed for two-level atoms. The absorption and coupling coefficients experience asymmetric dips generated by pump scattering off carrier-density pulsations induced by the pump–probe interference. These effects are reminiscent of coherent-dip phenomena well-known in two-level systems but differ significantly in form.
A calculation of degenerate four-wave mixing in a homogeneously broadened two-level saturable absorber is presented. An analytical formula for the local electric dipole polarization responsible for degenerate four-wave mixing is derived for arbitrary electric-field strengths. The overall efficiency of the medium is calculated by numerically solving the coupled wave equations with the use of the phase-matching electric dipole polarization. Solutions are presented when the laser is tuned on and off resonance for various input laser intensities and linear absorption coefficients. Optimal conditions for maximum efficiency are discussed.