Comparative Study of Transfer Matrix Formalism vs Single-Mode Model for Semiconductor Microcavities

Abstract

An optical semiconductor micrcavity consisting of two distributed bragg reflectors (DBRs) and a quantum well between, can be modeled using a transfer matrix approach, which solves the propagation through the DBR mirrors and the cavity segment in between the mirrors. Such an approach is easy to use if the interband polarization of the quantum well PQW is a given function of time or frequency, which includes the case of linear optical response, where PQW is given in terms of the linear susceptibility and the electric field at the position of the quantum well, EQW. In many cases of practical interest, the quantum well response is a nonlinear function of EQW, in which case the transfer matrix approach becomes impractical. In such cases, a time differential equation for PQW, which is of the form i¯ h dPQW(t) dt= F[PQW(t),EQW(t)] where F is a nonlinear function of PQW, is solved via time-stepping from earlier to later times. To obtain the electric field EQW needed as input to the PQW solution, a commonly used phenomenological approach utilizes the single-mode equation i¯ h dEQW(t) dt= hωcEQW(t)−ΩPQW(t) + S(t) with the source term S(t) being defined by S(t) = ¯ htcE+ inp(t) and corresponding constants that are defined in section 5 of this thesis. However, apart from containing phenomenological parameters, the simple source term entering the single-mode equation does not account for propagation, retardation, and pulse filtering effects of the incident light field traversing the DBR mirror. In this thesis, an alternate approach is presented along with evidence of its validity using a bounded convolution integral instead. The integral is used to determine the electric field as a function of time and therefore can be used to determine the time derivative of the polarization. The integral being EQW(t) =Z t −∞ [A(t−t0)E+ inp(t0) + B(t−t0)PQW(t0)]dt0. We show in the final sections that it is adequate to use this bounded integral to resolve pulses in the time domain. Evidence of that is done using a gaussian pulse and linear response. This method could then be used in conjunction with a time stepping algorithm to resolve nonlinear responses.