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Cited by 5 publications
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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...
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...
We report on recent experimental results of the dynamics of a laser-diode-pumped free-running Nd:YVO 4 laser operating in a two-mode regime. We observe intrinsic quasiperiodic and chaotic oscillations as well as a locking of pulsation frequencies. We perform an asymptotic analysis of model rate equations in which an intensity-dependent cross-gain ͑i.e., nonlinear gain͒ mechanism of direct mode-mode coupling is introduced in addition to the coupling mechanism via cross saturation of population inversions. We show that the intrinsic instability originates from the nonlinear gain mechanism. The observed locking of pulsation frequencies is successfully reproduced by simulations based on the proposed rate equations.
We apply the multiple time scale method to perform a nonlinear analysis of the Tang, Statz, and deMars rate equations, which describe an N-mode Fabry-Perot laser in which all modes have an equal gain and a large loss rate. In addition to the two relaxation oscillation frequencies ⍀ L and ⍀ R known from the linearized analysis, we find the four frequencies 2⍀ L , ⍀ R Ϯ⍀ L , and 2⍀ R . The signature of antiphased dynamics for the new frequencies is that there are no relaxation oscillations in the total intensity at ⍀ R Ϯ⍀ L . The laser steady state is shown to be stable, being characterized by damping rates derived explicitly. Relations among these damping rates are obtained. We also study the role played by the initial condition in governing the manifestation of the antiphase dynamics and the relative magnitude of the modal intensity power spectrum peak heights at the two main frequencies ⍀ L and ⍀ R . Finally, we deal with the resonant case ⍀ R ϭ2⍀ L . In this case, inphased dynamics is shown to appear at 2⍀ R , instead of at ⍀ R .
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