Spontaneous decay rates in one-dimensional waveguides

It is well-known that the spontaneous decay rates of emitting sources such as atoms or quantum wells can be largely modified in optical micro-cavities. − 4 Controlled spontaneous-emission plays a key role in a new generation of micro and nano-optical devices. Highperformance micro-cavity lasers, for instance, have been already experimentally demonstrated. 6 Spontaneous decay rates modifications are also expected to impact the performance of optical waveguide amplifiers as the guided mode area radii of these devices become smaller than ≈ 1μm. 8 In order to characterize one dimensional nanoand micro-optical devices one key decomposition of the total decay rate (τ 0 ) into two components was introduced in Refs. [3,4]. The decay rate was divided into the guided modes (τ g ) and into the radiated modes (τ r ), where the total decay rate is τ −1 0 = τ −1 g + τ −1 r . In contrast to large devices, where just the total decay rate must be considered for their characterization, the modeling of nanoand micro-optical devices requires the measure of both components of the decay rate. The classical method to determine the spontaneous decay rate of an emitting source embedded in a one-dimensional waveguide consists in the measure of the exponential decay rate of the Amplified Spontaneous Emission (ASE) output power when the pump source is switched off. The spontaneous decay rate is given by the exponential decay coefficient of the ASE output power. Two natural questions arise here:(1) How can τg and τr be measured? (2) What is actually measured when using the classical method to determine the spontaneous decay rate? In this Letter we answer these two questions. We show that the classical method to measure τ0 in one dimensional waveguides gives τr if used in long length waveguides and actually τ0 in short waveguides (assuming no reflections at the waveguide ends). We also show how these measures are modified in lossy mediums, i. e., when a background loss coefficient is incorporated into the rate and propagation equations. Then we show how these ideas are useful on devices of practical interests. Three cases are considered: Erbium Doped Waveguide and Fiber Amplifiers (EDWAs and EDFAs) and Semiconductor Optical Amplifiers (SOAs). Theory: decay rate measures and background loss influence. We employ the analytical solution for the longitudinal z dependence of the rate equations presented in Ref. [8] to investigate the measure of the spontaneous decay rate in one dimensional waveguides. The analytical solution presented there is valid only for waveguides in which the excited state population of the emitting source, N2(z), is constant along the fiber. Since the measure of the decay rate is performed when N2 → 0 along z, this approximation is valid when measuring τ0. The rate equation is: