The master equation for passive modelocking Closing a half-century challenge

. Passive modelocking (PML) of lasers is pivotal in modern science and industry. Here, solving a half-century-long challenge, we present the first master equation (ME) describing PML on all time scales, from Q-switched to fundamental and harmonic modelocking, valid for both slow and fast saturable absorption, short and long cavities. The proposed ME should become the workhorse for analytical and numerical studies of ultrafast lasers in both the photonics engineering and laser physics communities.


The Problem
Passive modelocking (PML) plays a central role in the science and technology of ultrafast lasers and optical frequency combs [1][2][3][4].Surprisingly, the quest for a rigorous and comprehensive PML theoretical framework has remained mostly unsuccessful for almost fifty years after H. A. Haus seminal contributions [5,6].
PML is achieved by material or effective saturatingloss devices, categorized as "fast" or "slow" depending on how the pulse duration compares to the absorber recovery time.Typically, resonant absorbers like semiconductor saturable absorber mirrors (SESAM) and carbon nanotubes are slow, whereas artificial absorbers, as in Kerrlens or nonlinear polarization rotation modelocking, are fast, showing an instantaneous response.The main PML modeling problem lies in the need to cover the extremely different timescales that affect gain and absorption.The problem is less pronounced in the PML with a fast saturable absorber, but even then, the original Master Equation (ME) by Haus and related models are of limited validity yielding stable pulses only in a narrow range of pump values close to the lasing threshold [7].Phenomenological inclusion of a saturating quintic nonlinearity, in models for fiber lasers typically neglecting gain dynamics, overcomes such limitation [8], featuring however a challenging direct quantitative comparison with experiments [9,10].
The ME consists of one partial-differential equation for the electric field complex amplitude A(T, t), evolving in slow-time T counting the number of roundtrips, and resolved by the fast-time coordinate t over one cavity roundtrip, coupled to the dynamical equations for the gain G and absorption Q.Only recently has the characterization of the dynamics of G and Q on both slow and fast time scales been seriously tackled by Hausen et al. [11] and Nizette and Vladimirov [12], whose proposals partially solve the problem.Both approaches work around the Vladimirov-Turaev time-delay model for semiconductor lasers [13] and treat the field A as a periodic function of t.The approach in [11] incorporates slow-time dynamical information on G and Q via Dirichlet boundary conditions.The solution is not fully consistent but works if pulses stay away from the (mathematical) boundaries in t.However, since pulses drift [14], see Fig. 1 (b) and (c), an ad-hoc counter-advection term is added to the ME, whose coefficient is computed after every cavity roundtrip [11].The artificial term would be trivial if the whole system of equations obeyed periodic boundary conditions.However, with hard (mathematical) boundaries as in this case, it makes analytical treatments involved and can lead to spurious instabilities, e.g. in the presence of noise [15].The main limitation of the approach in [12] instead is that it is only applicable to lasers with a cavity roundtrip time T R much shorter than the gain recovery time T G and comparable to or longer than the absorber recovery time T Q , thus invalid for VECSELs and external cavity laser diodes.

The Solution
Here we present a universal ME for PML, rigorously derived from the classic Haus model [6] with the methodology in [16], which solves this half-century-long challenge.Denoting a generic T R -average as ⟨•⟩, the ME reads with G net ≡ G − Q − 1 the net gain, including alphafactors for semiconductor materials, D a complex coefficient describing finite gain bandwidth and group velocity dispersion, K the Kerr nonlinearity coefficient, Ḡ ≡ ⟨G⟩ and Q ≡ ⟨Q⟩ the average gain and absorption, G 0 and Q 0 their respective unsaturated values, ℓ and s the linear cavity loss and saturation parameters.Finally, g ≡ G − Ḡ and q ≡ Q − Q, describing gain and absorption dynamic saturation, are explicit functions of the light intensity I ≡ |A| 2 .The ME (1) is valid for any evolution time scale of the material variables, from solid-state and semiconductor to fiber lasers, with slow or fast absorbers and long or short cavities.The ME captures all pulsing dynamics occurring in the same laser on extremely different time frames (from ms to ps and below): Q-switched modelocking, fundamental and harmonic modelocking, and subthreshold localized structures [17]; see Fig. 1.Our formalism allows analytical prediction of Q-switching and self-modelocking instabilities, growth rates, frequency and bandwidth, and also captures the leading-edge instability [13].Further, the ME can easily incorporate quantum coherence effects [16] and associated spontaneous modelocking phenomena.The ME obeys exact periodic boundary conditions in t for all variables, simplifying analytical investigations and warranting reliable, easy and fast numerical integration with the split-step Fourier algorithm.All these features should make ME (1) the workhorse for analytical and numerical studies of ultrafast lasers in both the photonics engineering and laser physics communities.
This work is part of Project PID2020-120056GB-C22, funded by MCIN/AEI/10.13039/501100011033,and Project EP/W002868/1, funded by EPSRC.AMP acknowledges support from the Royal Academy of Engineering (Research Fellowship Scheme).

Figure 1 .
Figure 1.Examples of PML regimes with a slow absorber from ME (1) for fixed parameters but the pump G 0 , which is lower in (a,b) than in (c,d).(a,b): Q-switched modelocking with a period of ≃ 100 roundtrips; the inset in (a) shows two pulses at ensuing cavity roundtrips.(c,d): Harmonic modelocking transition from one to two pulses per roundtrip; (d): final roundtrip (red, continuous) and initial roundtrip (blue, dashed) pulses of the evolution in (c).(e): Average intensity ⟨I⟩ of subthreshold fundamental modelocked pulses (FML) vs pump G 0 .