Basic “Small Signal” Modulation Response

Most predictions of direct modulation response behavior of laser diodes are derived from a small signal analysis of the spatially averaged rate equations (1.19) and (1.20). This approach involves the assumption that the laser diode is driven by a “small” sinusoidal current at frequency ω, superimposed on a DC bias current: J(t) = J₀ + j(ω) exp(iωt). The photon and electron density variables, n and p, are assumed to similarly consist of a “steady state” part, and a “small” time-varying part: n(t) = n₀ + n(t); p(t) = p₀ + p(t). Furthermore, the “small” time-dependent part is assumed to be sinusoidally varying in time, at the same frequency as the modulating current, i.e., n(t) = n(ω) exp(iωt), p(t) = p(ω) exp(iωt), where n(ω), p(ω) are complex quantities in general, thus incorporating the relative phase shifts between the drive current and the electron and phot on responses. As for what constitutes a “small” signal, one examines (1.19) and (1.20), and observes that the difficulty which prevents a simple analytic solution originates from the product term “np”, present in both equations. A well known mathematical technique for obtaining an approximate solution is to first solve the equations in the “steady state”, assuming no time variations in J(t); i.e., j(ω) = 0 consequently there would be no time variations in n and p either — n(ω) = p(ω) = 0. One then simply solves for n₀ and p₀ as a function of J₀. It turns out that the solutions thus obtained are incredibly simple, namely, n₀ = 1 and p₀ = J₀ — 1 if J₀ > 1 whereas, n₀ = J and p₀ = 0 if J₀ < 1. A straight forward physical interpretation of these simple results is that the quantity J₀ = 1 represents the lasing threshold current of the laser. Thus, in the steady state when no modulation current is applied, the relation between the optical output power from the laser (which is proportional to P0) to the input current J₀, is simply: P₀ = 0 if J₀ < 1; P₀ = J₀ — 1 if J₀ > 1.¹ There is a “knee” in the optical output power vs. input current relationship. Above this knee the output optical power is a strictly linear function of input current — in principle, assuming absence of any device imperfections. Thus from the above simple considerations the direct modulation characteristic of the laser should be strictly linear and free of distortions.

It turns out that this conclusion is only valid for modulation at “low” frequencies, which is to be expected since the conclusion is drawn from a steady state solution of the rate equations. Chapter 3 will examine the full frequency dependence of various modulation distortions. It will be shown in Chap. 3 that, apart from those induced by device imperfections. The product term n(t)p(t) in the rate equations (1.19) and (1.20), due to fundamental stimulated emission responsible for the laser action, is the fundamental source of nonlinear distortions in directly modulated laser diodes. This thus establishes a fundamental lower limit to the amount of distortion generated in direct modulation of the laser diode which cannot be removed by means of clever device design. Consequently all ultra-linear fiber-optic transmitter which employs directly modulated laser diode must use some form of electrical distort ion-compensation techniques to correct for the laser modulation distortion.