# Statistical physical theory of mode-locking laser generation with a frequency comb

###### Abstract

A study of the Mode-locking lasing pulse formation in closed cavities is presented within a statistical mechanical framework where the onset of laser coincides with a thermodynamic phase transition driven by the optical power pumped into the system. Electromagnetic modes are represented by classical degrees of freedom of a Hamiltonian model at equilibrium in an effective ensemble corresponding to the stationary laser regime. By means of optimized Monte Carlo numerical simulations, the system properties are analyzed varying mode interaction dilution, gain profile and number of modes. Novel properties of the resulting mode-locking laser phase are presented, not observable by previous approaches based on mean-field approximations. For strong dilution of the nonlinear interaction network, power condensation occurs as the whole optical intensity is taken by a few electromagnetic modes, whose number does not depend on the size of the system. For all reported cases laser thresholds, intensity spectra, phase waves and ultra-fast electromagnetic pulses are computed.

## I Introduction

In multimode lasers with many cavity modes, nonlinear interactions originate among modes. One notable mechanism inducing interaction is saturable absorption, i.e., the progressive depletion of low power tails of the light pulse traveling through the cavity at each roundtrip. This causes the consequent amplification of very short pulses composed by modes with locked phases, a phenomenon called mode-locking Haus (2000, 1984). Mode-locking (ML) derives from the nonlinear synchronization constraint on the oscillations of interacting modes. Given any quadruplet of modes this is expressed by the frequency matching condition (FMC):

(1) |

where is the single mode line-width. Phase-locking occurs at the ML lasing threshold and corresponds to some long-range order in the set of modes in the cavity.

We adopt a statistical mechanical approach to describe the optical properties of stimulated light emission from cavities with a large number of modes. In this approach the generation of a multimode ML lasing regime from a fluorescent continuous wave (CW) regime as the optical power in the cavity is increased can be characterized as a thermodynamic phase transition between a disordered phase and a phase with long-range order . The stationary laser system can be treated as a thermodynamic system at equilibrium in a thermal bath whose effective temperature is proportional to the inverse squared power pumped into the cavity Antenucci et al. (2014a); Marruzzo and Leuzzi (2015). Since the first attempt by Gordon and Fischer in the early 00’s Gordon and Fischer (2002), this approach has been performed in a mean-field fully connected approximation corresponding in the optical language to the so-called narrow-band approximation, see also Refs. [Angelani et al., 2006; Leuzzi et al., 2009; Conti and Leuzzi, 2011; Antenucci et al., 2014a]. This consists in choosing mode frequencies in a narrow band-width around the central frequency of the cavity. So narrow that the frequency interspacing between resonant modes is less than the linewidth of each mode. In this way Eq. (1) is practically always satisfied and, therefore, actually irrelevant in determining lasing properties.

In the present work we introduce frequency dependent populations of modes, considering gain profiles and the effect of nontrivial frequency matching on the mode couplings. This analysis requires to go beyond the limits of validity of mean-field theory and it is carried out by means of optimized Monte Carlo (MC) simulations running on GPU’s. An exhaustive numerical analysis accounting for the fluctuations induced by these new ingredients reveals that, depending on the optical system properties, on the cavity topology, and on the relative gain-to-nonlinearity strength, different thermodynamic-like phases occur. Such regimes range from a ferromagnetic-like one, where all mode phases are aligned, to a phase-wave one, where phases of modes at nearby frequencies are strongly correlated, though not equal to each other. The ferromagnetic behavior occurs in the low finesse limit of the narrowband approximation. Non-trivial phase locking occurs, instead, at high finesse. In the latter case we show how, distributing the frequencies according to an optical frequency comb Udem et al. (2002); Baltus̃ka et al. (2003); Schliesser et al. (2006), intensity spectra and pulse phase delay observed in ultra-short pulses are reproduced Brabec and Krausz (2000).

As it will be taken up in the following, previous studies based on mean-field theory are exact only in the narrow band-width case. In this paper we go beyond the mean-field approximation, accounting also for situations in which different modes exhibit different frequencies

Our study introduces two essential and new ingredients. The first one is the FMC, yielding mode interaction networks that are no longer described by mean-field theory, in which non-trivial multimode emission spectra and mode phase correlations above threshold occur. The second ingredient is a random dilution of the interacting network, modeling possible topological disorder in arbitrary cavity structures, as, e.g., multi cavity channels not exactly equal to each other. We will show that, as far as it is not too strong, the latter kind of dilution does not alter at all the laser transition properties. Below a certain dilution point, however, in the lasing phase the whole optical power condenses into a small set of connected modes, scaling independently of the number of modes.

## Ii The Model

Expanding the electromagnetic field in the complete base of normal modes Murray Sargent III, Marlan O’Scully and Willis E. Lamb (1978)

(2) |

the equilibrium dynamics of the time-dependent complex amplitudes is given by the Hamiltonian Gordon and Fischer (2002)

(3) |

where and are chosen as real numbers, neglecting
dispersion and Kerr-lens effect. The physical meaning of the
coefficients comes from the equivalence of the Hamiltonian dynamical
equation with the Haus master equation Haus (2000):
is the net gain profile, is the self-amplitude modulation (SAM)
coefficient. The ML sum runs over a subset of quadruplets such that
for each element the FMC holds. The latter
implies that in the non-linear term of Eq. (3) three
non-equivalent orderings of quadruplets contribute to the sum, each
one consisting of eight equivalent index permutations ^{1}^{1}1The
three orderings inequivalent with respect to the FMC are
, and
. Given a quadruplet the
equivalent permutations are (i): , (ii): , (iii): and their combinations.. The Hamiltonian is symmetrized with respect to these orderings. The
coupling strength in Eq. (3) is taken as , where
is the number of quadruplets, making the Hamiltonian
extensive.

The total optical energy stored in the system is and it is kept constant in the dynamics by external power pumping. Eq. (3) is a direct generalization of the Hamiltonian studied in Ref. Gordon and Fischer (2002) and can be seen as the ordered limit of the random laser theory analyzed in Refs. Leuzzi et al. (2009); Conti and Leuzzi (2011); Antenucci et al. (2014a). From the point of view of statistical mechanics the driven optical system composed by the cavity, the amplifying medium and the optical power pumped into the system can be described by Eq. (3), considering it as the Hamiltonian of a system at equilibrium with an effective thermal bath. The role of the inverse temperature is played by the pumping rate squared: . Here is the inverse heat bath temperature, regulating spontaneous emission. It is usually represented as white noise in a Langevin dynamics Gordon and Fischer (2002, 2003); Gat et al. (2004); Angelani et al. (2006); Leuzzi et al. (2009); Conti and Leuzzi (2011); Antenucci et al. (2014a); Marruzzo and Leuzzi (2015).

## Iii Mode interaction network

Thermodynamic phases are determined by the interaction network, as well. In the following we will undergo the analysis of networks with a varying degree of dilution. This will be expressed as number of quadruplets vs. number of modes . We will discuss data for , .

Two essentially different types of topologies will be investigated, depending on the frequency bandwidth being narrow or finite. Both topologies can be further diluted upon homogeneously randomly removing quadruplets. The “Narrow Band-width Topology” (NBT) is low finesse, i.e., , and the role of frequencies is irrelevant. The fully connected instance, consisting in interacting quadruplets, corresponds to a closed Fabry-Perot-like cavity where all longitudinal modes are localized in the same spatial region. Possible random diluted NBT’s correspond to more complicated geometries, including multi-channels set-ups. For finite bandwidth, instead, we will work in the high-finesse limit, , with sets of equispaced frequencies Bellini and Hansch (2000); Diddams et al. (2000); Udem et al. (2002); Baltus̃ka et al. (2003); Schliesser et al. (2006). We will term this a “Frequency Comb Topology” (FCT). In this case the list of quadruplets is extracted from those nontrivially satisfying Eq. 1: modes are not all equivalent to each other and mean-field theory does not hold.

## Iv Numerical Simulations and Data Analysis

We performed
extensive Monte Carlo simulations of equilibrium dynamics by means of
the exchange MC Hukushima and Nemoto (1996) algorithm and the synchronous,
fully parallel MC Peretto (1984); Mahmoudi and Saad (2014); Metz and Theumann (2008, 2009). The
latter, indeed, remarkably turns out to reproduce a reliable dynamics
in the present model Antenucci
et al. (2014b). In the NBT, system sizes
from to have been simulated for random dilutions of
, . ^{2}^{2}2We also tested the
network-to-network fluctuations over different numbers of network
realizations finding that the fluctuations of any obserables over the distribution
of topologies are one order of magnitude lower than thermal
fluctuations, ,
already in the worst case of small size .. For the FCT, we
simulated systems of size with number of frequencies
in each case, and and upon
applying the FMC filter.

The gain is taken as Gaussian with varying mean square displacement. We checked thermal equilibration, i.e., the onset of the pumped stationary regime, by looking at the energy relaxation and at the symmetry of the distribution of complex amplitude values deep in the lasing phase. In the following we present our results about (I) laser thresholds identification, (II) intensity spectra, and (III) phase waves, electromagnetic pulses and their correlations.

### (I) Laser threshold

The estimate of the laser threshold is obtained from the finite size scaling (FSS) analysis of the behavior of the energy vs. pumping rate, as shown in Fig. 1 for the FCT (for the NBT the energy behavior is the same). For low the system is in an incoherent continuous wave regime with uncorrelated phases and zero energy per mode. As increases a phase transition occurs indicated by a discontinuity in the energy. For the NBT the critical point is analytically known Antenucci et al. (2014a) and pointed out to as an arrow in Fig. 1. For , the FSS of the discontinuity point is compatible with the fully connected analytical limit, as reported in Tab. 1. The critical thresholds for the FCT case, estimate by FSS for and for are reported in Tab. 1. The CW/ML laser phase transition is first order: in the inset of Fig. 1 both the spinodal and the critical points are displayed, e.g., for in a FCT. Spinodal points occur both in NBT and in FCT. In Fig. 2 the average mode magnitudes are plotted. This is for randomly independently oscillating amplitudes and it discontinuously increases at the ML lasing threshold indicating intensity mode-locking. For , tends to in the NBT and to in the FCT case.

band | Narrow | Band | Frequency | Comb | ||
---|---|---|---|---|---|---|

Exact | ||||||

#### Power condensation

As the dilution is strong, i.e., , each mode interacts in a number of quadruplets. Above threshold the whole power turns out to be taken by a small number of connected modes and the probability to find a configuration with energy equipartition is negligible in the thermodynamic limit. In the mean field approximation one can prove that in order to display power condensation it must be Antenucci et al. (2014b) as confirmed by numerical simulations. In the following we focus on more connected networks.

### (II) Intensity spectra

In Fig. 3 we show two instances of the spectra vs. in FCT systems with Gaussian gain profiles with different variances. In the left panel the mean square displacement of the gain profile in the wavelength dominion is large () in comparison to the spectral free range, whereas in the right panel it is of the same order of magnitude (). In the first case, below the CW spectrum is flat and suddenly sharpens at the ML threshold . To underline this, spectra are shown right below and above in Fig. 3. In the small case the spectra appears already narrower in the CW regime, following , as displayed in the right panel of Fig. 3 for the lowest simulated pumping rate. At though, their narrowing qualitatively changes and becomes progressively independent of as increases, eventually taking the same spectral shape of the previous case.

We show in Fig. 3 the cumulative detections of very many pulses, as in data acquisition from ultra-fast ML lasers. In the MC dynamics used in simulation, though, each MC step corresponds to a pulse generation. Within our approach it is, then, possible to look at the dynamics at much shorter time intervals, where the mode amplitude and intensity profile in fluctuates from pulse to pulse. This is connected to changes in the spectral phase delay of the electromagnetic pulse. Different spectral dynamics are reported in Video of Sup. Mat. SuM (2014) (see details in Sec. VI).

### (III) Electromagnetic pulses and phase delay

In terms of slow complex amplitudes, cf. Eq (2), , , the electromagnetic pulse is

The time operatively labels a single MC step in our simulations, i.e., the interval between two pulses. In Fig. 4 we show at four different times in the dynamics. In the NBT, in the ML regime all modes acquire same modulus and phase. In a FCT, instead, at a non-trivial phase-locking occurs, such that the mode phases exhibit a linear dependence on the mode frequencies: , as shown in the bottom insets of Fig. 4. The pulse is, thus, unchirped Haus (2000). The spectral phase delay, or group delay, of the optical pulse does not depend on the frequency of mode . It changes, though, with time , from pulse to pulse. Within our approach we thus find the typical spectral phase frequency profile at each given pulse and its pulse-to-pulse dynamics, cf. Video in Sup. Mat. SuM (2014) (see details in Sec. VI).

#### Phase waves lifetime

Let us define the time average over an equilibrated set of data () on a time window : . In the FCT, after a time , the average global phase correlation function , defined as

(4) | |||||

(5) |

is observed to decay to zero. This is at difference with the NBT where, in the high power regime, is finite also for . For the FCT, the distribution of correlation times as the optical power varies across the lasing threshold is sharply peaked around its logarithmic average below threshold, cf. Fig. 5. For increasing the distribution tends to a flat curve.

#### Vanishing two-mode correlators

A related phenomenon is that the average over of two-mode phase correlations , cf. Eq. (5), vanishes, as shown in Fig. 6, implying a zero ensemble average. This occurs though modes with frequencies , are correlated at each time , cf. bottom insets of Fig. 4, and when . In Fig. 6 we show as function of for and . In the top panel, at shorter time window , one clearly observes that is completely uncorrelated independently of for . As the pumping increases above the threshold, displays a non-trivial behavior as a function of . Above threshold, thus, the global phase correlation function , cf. Eq. (4), becomes larger the higher the pumping. In other words, the correlation time grows with and overcomes : . In the bottom panel of Fig. 6 we consider, instead, a time window larger than the average correlation time for most of the simulated pumping values , cf. Fig. 5: it can be observed that for practically almost all , but the smallest ones for large power. The oscillations displayed by in Fig. 6 in the ML laser regime are due to the fact that different phase delays are involved in the thermal average. Indeed, cf. bottom insets of Fig. 4, the slope of changes with time .

The origin of the vanishing of two-mode correlators is reminiscent of symmetry conservation in gauge lattice theories Kogut (1979) and will be discussed elsewhere Antenucci et al. (2014b). We just mention that the main difference in the lasing regime for the two topologies is that in the NBT the global symmetry is spontaneously broken, whereas in the FCT it is conserved across the threshold.

## V Conclusions

We present the first statistical mechanical approach to the study of real-world ultrashort mode-locked multimode lasers in closed optical cavities, including possible degrees of topological disorder. In previous approaches, statistical mechanical systems with distinct resonances have been studied in the mean-field approximation, see, e. g., Refs. Gordon and Fischer (2003); Gat et al. (2004); Rosen et al. (2010). The key point is, though, that the mean field solution is exact only in the narrow band-width limit. When describing inhomogeneous topologies, such as the Frequency Comb Topology of equi-spaced well-refined resonances, the only thing that the mean-field theory can account for is a shift in the pumping threshold resulting from the dilution (in fact, just a modification of the coupling constant). The nature of the predicted mode-locked regime remains, indeed, identical to the one predicted assuming narrow band-width. This limit basically lies in the very definition of the mean-field method: since the fluctuations of the mode degrees of freedom are neglected a many-body problem is actually reduced to a one-body problem, in which all modes exhibit a common average phase and a common average intensity. The inhomogeneity in frequency dependence of the interaction network in more realistic cases, in which modes with near-by frequencies have stronger coupling, is simply neglected by construction. Because of the mean-field assumption, previous approaches have not, and could not have, accounted for the main properties here reported: phase waves, non-equipartition threshold and vanishing two-mode correlators.

Our approach, going beyond mean-field theory, with Monte Carlo simulations of equilibrium dynamics, allows to reproduce and study the onset of the lasing regime and the behavior of emission spectra and laser pulses and relative group phase delays at any supplied power. The existence of metastable lasing regimes marked by spinodal points in the energy behavior, cf. inset of Fig. 1, accounts for the onset of optical bistability Gibbs (1985); Baas et al. (2004). The phenomenon of power condensation for extreme dilution of mode interaction and the vanishing of the equal time two-mode phase correlations for long times are properties that can be experimentally tested. Furthermore, this kind of approach opens the way to further analyze the carrier-envelop offset phase behavior, and the tolerance to disorder in the coupling SAM coefficient. The latter analysis is useful, e.g., for stabilized micro resonator in chip-based devices Lee et al. (2012); Saha et al. (2013) in which technical precision undergoes size constraints and controlling material damage is a true challenge. Eventually, including open cavity terms Viviescas and Hackenbroich (2003); Hackenbroich et al. (2003) and strong disorder in the nonlinear coupling Angelani et al. (2006); Leuzzi et al. (2009); Conti and Leuzzi (2011); Antenucci et al. (2014a), our approach can be applied to the study of random lasers Lawandy et al. (1994); Cao et al. (1998); Cao (2005); Wiersma (2008); Ghofraniha et al. (2015).

## Acknowledgements

The authors would like to thank Claudio Conti, Andrea Crisanti and Giorgio Parisi for stimulating discussions. The research leading to these results has received funding from the Italian Ministry of Education, University and Research under the Basic Research Investigation Fund (FIRB/2008) program/CINECA grant code RBFR08M3P4 and under the PRIN2010 program, grant code 2010HXAW77-008 and from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA grant agreement n¡ 290038, NETADIS project.

## Vi Supplemental Material

### Spectra dynamics [Video 1]

In the file Video1_spectral_dynamics.mpg a video is shown for the dynamics of spectra for a FCT system with modes and frequencies. The gain wavelength profile is taken uniform, i.e. . The finite size threshold for this specific system is . Dynamic sequences at three pumping rate are reported: in the incoherent CW regime (), slightly above the lasing threshold () and for high pumping (). Each MC step corresponds to the interval between two pulsed emissions in the mode-locked lasing regime. In the video each single frame is averaged over 10 subsequent Monte Carlo steps and frames are shown at intervals of 100 Monte Carlo steps.

### Lasing pulse and phase delay dynamics [Video 2]

In the file Video2_phase_delay_dynamycs.mpg a video is shown for the stationary dynamics of the relationship of lasing pulses in a simulation of a FCT system of modes with an initial random dilution of and a final number of interacting quadruplets . The lasing system is at optical power , right above the lasing threshold . The initial distribution of the gain among frequencies is taken as uniform. The interval between each frame is 100 Monte Carlo steps. It can be observed that at each time , corresponding to a pulsed emission, is approximately linear, yielding a well defined phase delay independent from . As the dynamics runs, though, changes, progressively taking a broad interval of values.

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