A global antiphase theorem

Abstract

Abstract. We determine the conditions under which the global rate equations lead to an antiphase theorem for a laser oscillating on N modes. The global rate equations are N integro-differential equations for the modal intensities coupled to a single differential equation for the space-dependent population inversion. The antiphase theorem states that the total lasing intensity is characterized by a single relaxation oscillation frequency. It holds in the limit of frequency-independent linear gain and loss of the oscillating modes. Additional constraints apply, depending on the nature of the pumping. A quantitative formulation of these conditions is derived.Keywords: Global rate equations, antiphase theorem, Lyapunov theorem, end-pumping, filling factorIn this letter we consider the global rate equations describing the dynamics of a homogeneously broadened solid state laser in which there are no phase-dependent interactions. In dimensionless form, these equations can be written aswhere f p (z) ≡ |φ p (z)| 2 and φ p (z) is a cavity eigenmode, I(p, t) is the intensity of mode p normalized to its saturation value, p = 1, . . . , N where N is the number of modes above threshold, time and time constants have been scaled to the population inversion decay time, κ p is the decay rate (or loss rate) of mode p, γ p is the linear gain of mode p in units of the gain of the first mode to lase, J (z, t) is the population inversion and w(z) is the longitudinal pumping profile. The cavity length is L.There exist very few results which rely only on these N + 1 rate equations and do not require a modal expansion of the population inversion. Among them is the Lyapunov theorem proved by Ghiner et al [2,3]. This theorem can be stated in terms of the steady state solutionsĪ p andJ (z) of equations (1) and (2) as follows. The set of modes for which (i) I p (t) = 0 and J (z, t) −J (z) = 0 for 0 z L and 0 t < ∞ and (ii)J (z) > 0 andĪ p > 0 for 0 z L, is characterized by the Lyapunov functionThe proof is straightforward and does not require more conditions than expressed in the theorem. The Lyapunov function verifies the equationUnder the conditions of the theorem, the function F has the two necessary properties to be a Lyapunov function:The theorem is of limited interest since the set of modes that appears in equation (3) is restricted to the oscillating modes because of the assumption I p = 0 for ∀t (including t = ∞ and thereforeĪ p = 0). Another restriction is that the population inversion in and out of steady state may not vanish anywhere in the laser cavity and must be positive everywhere in steady state:J (z) > 0 and J (z, t) = 0 both for ∀z. This means that the theorem does not apply to partially filled cavities for which w = J = 0 in the empty sections. Despite these limitations, this theorem is important since it suggests that there are properties that can be analysed at the level of the global rate equations (1), (2) which do not need a modal expansion of the population inversion as in the Tang, Statz and deMars (TSD) rate equati...

Global rate equation description of a laser

“…Another reason is the unrealistic property that the number of relaxation oscillation frequencies and damping rates increases without limit as the number of moments which is retained in the hierarchy of evolution equations increases. Finally, in a recent paper [19], we have proved that a general antiphase theorem can be derived from equations (1,2). In this context, it appeared clearly that assessing the role of the filling factor is not a problem if the theory is developed without any specification of the longitudinal pump profile and the results are expressed in terms of averages of the pump.…”

“…which is an alternative way to express the antiphase theorem [19]. In the same limit, the TSD model yields the frequencies…”

“…In the simple case of constant pump w 1 = w 2 = w and with the definitions γ 1 = 1, γ 2 ≡ γ introduced without loss of generality, equation (18) becomes (20) while equation (19) leads to the solution (6). The simultaneous solution of these equations is…”