The Spontaneous Cascading Mechanism Behind Critical Phenomena in Self-Coupled Lasers

TL;DR

This paper investigates a novel self-coupled laser system that exhibits spontaneous cascading leading to abrupt phase transitions, revealing a universal mechanism behind critical phenomena in lasers through experimental and theoretical analysis.

Contribution

It introduces and analyzes a new nonlinear self-coupled laser system demonstrating spontaneous cascading and mixed-order transitions, expanding understanding of abrupt laser phenomena.

Findings

Observation of long-lived cascading plateau at critical point

Critical branching factor equals one at transition

Universal cascading mechanism suggested for abrupt transitions

Abstract

The basic physics of lasers is characterized by a second-order continuous phase transition at the critical lasing threshold. Nevertheless, laser bistability with abrupt transitions has been reported in some laser systems, but its underlying mechanism has never been explored. Here we study experimentally and theoretically a novel nonlinearly self-coupled laser system. We show both experimentally and theoretically that this system experiences spontaneous cascading that yields an abrupt mixed-order transition. At the critical point, a long-lived cascading plateau is observed, characterized by a critical branching factor equal to one. When deviating from criticality, the branching factor departs monotonically from one. The critical scaling close to and at the critical point resembles similar phenomena observed recently in other interdependent systems, suggesting a common universal cascading origin for abrupt transitions. Our results shed light on the cascading mechanism of abrupt transitions in laser systems, which can be utilized for future research and applications.

Equations

$$I_{i+1}=I_0+f(P_i),$$

$$|P_{i+1}-P_i|<{ϵ}_P,$$

$$\begin{array}{c}P-P_c\sim (I-I_c{)}^{\beta }\end{array}$$

$$\begin{array}{c}\tau \propto (I-I_c^{-}{)}^{-\zeta },\end{array}$$

$$\begin{array}{c}\frac{dn}{dt}=KnN-{\gamma }_{c}n+b{\gamma }_{2}N\\ \frac{dN}{dt}=\frac{I}{q}-KnN-{\gamma }_{2}N.\end{array}$$

$$\begin{array}{c}\frac{dI}{dt}=I_0-I+f(n)\end{array}$$

$$\begin{array}{c}{\gamma }_c\cdot n=I_0-\frac{{\gamma }_c{\gamma }_2}{K}+\alpha \cdot \tanh (qn)\end{array}$$

Full text

⋆⋆footnotetext: These authors contributed equally to this work.

Since the early days of laser research, phase transitions and critical phenomena in lasers have attracted significant attention. It has been recognized that the transition above threshold in lasers exhibits characteristics akin to a non-equilibrium phase transition. In most laser systems, the transition between spontaneous emission and coherent stimulated emission occurs smoothly, resembling the continuous state change observed in a second-order phase transition Scott et al. (1975); DeGiorgio and Scully (1970); Graham and Haken (1970); Grossmann and Richter (1971); Dohm (1972). However, under specific configurations, abrupt and discontinuous changes in laser intensity have been observed, indicating a first-order abrupt phase transition. This abrupt transition type was first reported in gas lasers Lisitsyn and Chebotaev (1968) and semiconductor lasers Lasher (1964); Nathan et al. (1965) by incorporating elements such as saturable absorbers into the laser cavity. Moreover, a hysteresis effect observed in the laser output intensity versus pump input characteristics was interpreted as analogous to a liquid-gas transition near the critical point Kazantsev and Surdutovich (1970). Since then, the concept of bistability at the origin of the hysteresis loop observed in lasers has garnered significant attention in developing bistable optical devices, which are crucial for advancing all-optical computing Hesselink et al. (2008), in semiconductor lasers Nathan et al. (1965); Kawaguchi (1982, 1987); Harari et al. (2018) and in intensity-coupled lasers Dutta et al. (1984); Oudar and Kuszelewicz (1984).

Rosen et al. Rosen et al. (2010) further investigated abrupt transitions in lasers by measuring critical exponents near the transition. These abrupt transitions provide a theoretical basis for understanding laser instabilities Kröger (2015) and optical turbulence Ikeda et al. (1980), and opened new applications in chaos communications, random number generation, and reservoir computing Sciamanna and Shore (2015); Ohtsubo (2017). Despite these findings, the underlying mechanisms driving such abrupt transitions in laser systems remain unclear. Understanding the conditions and the mechanisms that govern these sharp changes could have significant implications for understanding phase transitions and, particularly, designing more efficient and tunable lasers with precise control over their emission properties.

In this Letter, we investigate both theoretically and experimentally a self-coupled laser, a system that incorporates a computer-controlled intensity feedback on the pump. By selecting a nonlinear coupling function, we demonstrate that it can exhibit an abrupt transition. Our investigation reveals a critical slowing down of the laser’s transient time as the system approaches the transition point, characterized by critical exponents. This observation is explained here in terms of a spontaneous chain reaction of critical cascade changes within the system. The feedback mechanism operates slowly at criticality with a critical branching factor equal exactly to one, ultimately leading to a cascade of small changes that yield a sudden, abrupt laser switch-on as well as switch-off. This is similar to the cascading phenomena observed in the extensively studied framework of interdependent networks theory Buldyrev et al. (2010); Zhou et al. (2014) that has recently been identified in several physical systems, including interdependent superconductors Bonamassa et al. (2023); Gross et al. (2025) and interdependent ferromagnetic networks Gross et al. (2024). In these systems, dependencies between nodes in different networks allow failure propagation, eventually leading to a cascading failure and mixed-order transition with a branching factor of one at criticality. Our experimental and theoretical analysis of the underlying mechanism of the self-coupled laser system includes measurements of the critical exponents, which are found to be identical to those observed in interdependent networks. This result proves that self-coupled lasers belong to the mixed-order universality class of interdependent systems Bonamassa et al. (2025). Furthermore, this shows that the universality class of interdependent systems is broader and is determined by the underlying cascade mechanism of the system and not the network structure, which is absent in self-coupled lasers.

Self-coupled laser - experiments - The experimental setup consists of a temperature-controlled, electrically pumped commercial laser diode (For more details see See Supplemental Material at [URL will be inserted by publisher] ). Self-coupling is implemented by dynamically adjusting the injection current based on the measured laser output power. More specifically, a computer processes the optical signal and updates the injection current according to a predefined feedback function $f$, such that

[TABLE]

where $P_{i}$ is the laser output power measured at injection current $I_{i}$, $I_{0}$ is the initially set current, and $i$ denotes the iteration step. The system is considered to have reached steady state when the output power converges within a specified tolerance, i.e.,

[TABLE]

where $\epsilon_{P}=10^{-4}$mW is an arbitrary convergence threshold. Compared to all other laser systems with optoelectronic feedback described in the literature Lin and Liu (2003); Abarbanel et al. (2001); Ghalib et al. (2012); Satyan et al. (2009), the advantage of this method is the unique ability to choose the function $f(P)$ that couples the photodetected laser output $P$ with the injection current $I_{0}$.

When nonlinear coupling is introduced, an abrupt transition occurs, as illustrated in Fig. 1(a). This behavior contrasts with the shown case of linear coupling, $f(P)=\alpha P$, where the laser transition remains continuous and manifests only as an increase in slope efficiency with increasing $\alpha$. With nonlinear coupling, the laser turns on abruptly at the critical input intensity $I_{c}^{+}$ and switches off abruptly at $I_{c}^{-}$, forming a hysteresis loop as shown in Fig. 1(b). This loop can be accurately traced experimentally by increasing/decreasing the injection current $I_{0}$ in small increments of 10$\mu$A. The unstable solution branch shown in Fig. 1(b) is reconstructed using the method detailed in See Supplemental Material at [URL will be inserted by publisher] .

The broadness of the hysteresis, $I_{c}^{+}-I_{c}^{-}$ is found to depend on the coupling strength $\alpha$. This leads to the phase diagram ($I$,$\alpha$) presented in Fig. 1(c) for two values of temperature, $T$=10°C and $T$=7°C. In this phase diagram, the critical currents, $I_{c}^{-}(\alpha)$ and $I_{c}^{+}(\alpha)$, appear as straight lines separating the lasing and non-lasing phases. Bistability starts at the tricritical point for which $\alpha=\alpha_{c}$. For $\alpha<\alpha_{c}$, $I_{c}^{-}=I_{c}^{+}$ and the transition is continuous, while for $\alpha>\alpha_{c}$ the transition is abrupt with bistability characterized by $I_{c}^{+}>I_{c}^{-}$.

Critical behavior - experiments - Mixed-order phase transitions are a unique class of phase transitions, displaying both an abrupt change similar to first-order transitions and scaling laws and critical exponents near the critical point similar to continuous second-order transitions Mukamel (2024); Boccaletti et al. (2016); Gross et al. (2022); Alert et al. (2017); Korbel et al. (2025). To classify the nature of the transition and its universality class, we examine the critical behavior near the critical points. We find that the laser power near both critical points $I_{c}^{-}$ and $I_{c}^{+}$ follows a power-law:

[TABLE]

where the exponent $\beta=0.52$ near $I_{c}^{-}$ (see inset of Fig. 1(b)) and $\beta=0.50$ near $I_{c}^{+}$ See Supplemental Material at [URL will be inserted by publisher] , thus confirming the expected square-root scaling predicted for interdependent networks Buldyrev et al. (2010), and similar to the results of Rosen et al. Rosen et al. (2010). Interestingly, the inset of Fig. 1(b) shows that logarithmic plots for different values of $\alpha$ at fixed temperature collapse onto a single line, indicating that the proportionality coefficient in Eq. (1) is independent of $\alpha$.

Plateau - experiments - Critical exponents are universal descriptors that tell us how the system behaves near the transition, but not why and how the system transitions. To elucidate the mechanism underlying this self-coupled laser mixed-order transition, we investigate the dynamics of the laser in the vicinity of the critical point. To this end, we prepare the system in a non-equilibrium state and monitor its relaxation toward equilibrium. We first consider the non-lasing regime, $I_{0}<I_{c}^{-}$, and apply a brief perturbation $\delta I$ such that $I_{0}+\delta I>I_{c}^{-}$, momentarily driving the system into the lasing state. The optical output power $P$ is monitored and the transient injection current $I=I_{0}+f(P)$ is iteratively updated until the system returns to its non-lasing steady state (Fig. 2(a)). As $I_{0}$ approaches the lower critical threshold, $I_{c}^{-}$, the relaxation exhibits a pronounced slowdown, forming a characteristic plateau. The duration $\tau$ of this plateau is measured and found to follow a power law scaling near the transition:

[TABLE]

with a critical exponent $\zeta$ = 0.50 (see Fig. 2(b)). A similar critical slowing down is observed when probing the system from above the upper threshold $I_{c}^{+}$ See Supplemental Material at [URL will be inserted by publisher] . In this case, a brief negative perturbation $I_{0}-\delta I$ temporarily turns the laser off. As $I_{0}$ approaches $I_{c}^{+}$, the relaxation back to the lasing state develops a plateau whose duration scales as a power law with the same exponent $\zeta$.

Branching factor - experiments - The emergence of the plateau, along with its scaling properties, signals a progressive transformation that governs the slowing down of the laser dynamics near the critical point. The quantity that best probes this cascading mechanism is the branching factor, $\eta$. The branching factor measures the average number of secondary events triggered during the cascade by a single primary event. This control parameter has been used to model e.g. epidemic or forest fire spread, information cascades, neural avalanches, to determine whether an event dies out or leads to a large cascade. When $\eta$ reaches unity, the system undergoes a phase transition. In the coupled-laser system, the branching factor can be obtained by measuring $\eta=\langle\frac{P_{i+1}-P_{i}}{P_{i}-P_{i-1}}\rangle$, where $P_{i}$ is the transient laser output power at step $i$ and $\langle...\rangle$ stands for time averaging over the plateau duration. As the initial injection current, $I_{0}$, is reduced stepwise toward the transition $I_{c}^{-}$, the branching factor $\eta$ is found to increase from below one (lasing state) to one, at the critical point, when the laser stops emitting. Beyond that point, the branching factor becomes larger than one, as illustrated in Fig. 2(c). Measurement methods are detailed in See Supplemental Material at [URL will be inserted by publisher] .

Self-coupled laser - theory - To elucidate the mechanisms underlying the above experimentally observed abrupt transitions and cascading phenomena, we develop a theoretical framework based on a simplified analytical model of the self-coupled laser. This model describes a single-mode laser in terms of three key quantities: the number of photons $n$, the number of excited atoms $N$, and the injection current $I$. As will be shown, this model successfully captures the essential features of the experimentally observed phenomena.

In the absence of coupling, their time evolution is governed by the standard set of coupled rate equations for semiconductor lasers Agrawal (2002):

[TABLE]

Here, $\gamma_{c}$ is the photon loss rate, $\gamma_{2}$ the atomic decay rate, $K$ the stimulated-transition coefficient, $q$ the carrier charge, and $b$ the spontaneous-emission factor. Numerical results use $\gamma_{c}=7.479\,\mathrm{s}^{-1}$, $\gamma_{2}=4.291\,\mathrm{s}^{-1}$, $q=1$, $K=1.5$, and $b=10^{-3}$, while the analytical treatment assumes $b\approx 0$.

An additional rate equation

[TABLE]

captures the feedback coupling between $I$ and $n$, where $I_{0}$ denotes the initial current injected at $t=0$. In the steady state, the pump current becomes $I=I_{0}+f(n)$, reflecting the feedback from the laser output, given the proportionality between the photon density $n$ and the power $P$.

We begin by introducing the linear coupling function, $f(n)=\alpha n$. Below the critical threshold, the trivial solution $n=0$ is the stable solution, while above the threshold, a new non-zero steady-state solution emerges: $n=\frac{I-\gamma_{c}\gamma_{2}/K}{\gamma_{c}-\alpha}$. Despite the coupling, the transition remains continuous but exhibits a steeper linear slope of $1/(\gamma_{c}-\alpha)$, compared to $1/\gamma_{c}$ in the absence of coupling ($\alpha=0$), as illustrated in Fig. 1(d).

Next, we examine the system under a nonlinear coupling of the form, $f(n)=\alpha\cdot\tanh(qn)$.The coupled equations 3 and 4 are integrated numerically using the Euler method until transients vanish, and the steady state is recorded. This time, the self-coupled laser exhibits an abrupt jump (Fig. 1(d)), closely reproducing the experimental observations (Fig. 1(a)). The steady-state solution of the nonlinearly-coupled laser satisfies the following equation:

[TABLE]

The multiplicity of solutions found - 2 stable and 1 unstable - indicates the presence of a bistable regime, as confirmed by numerically solving the laser equations (Fig. 1(e)). The resulting hysteresis loop between critical points, $I_{c}^{-}$ and $I_{c}^{+}$, is characterized by the coexistence of two stable states: an active (lasing) state and an inactive (non-lasing) state. By systematically varying the parameters $\alpha$ and $I$, we qualitatively reproduce the experimental phase diagram, as shown in Fig. 1(f).

Critical behavior - theory - To further validate our simplified theoretical model, we now examine the critical exponents and scaling behavior near the abrupt transition. The critical exponent $\beta$, characterizing the growth of the optical power above the critical injection current $I_{c}^{-}$, is extracted by numerically solving Eq. (5). The best fit of $P-P_{c}$ versus $I_{0}-I_{c}^{-}$ yields a slope of $0.50$ as shown in the inset of Fig. 1(e), in excellent agreement with the experimentally determined value $\beta=0.52$. This result is further supported by an independent analytical derivation, which yields $\beta=0.5$ and confirms the independence with $\alpha$ of the proportionality coefficient See Supplemental Material at [URL will be inserted by publisher] .

Plateau - theory - Next, the dynamics of the self-coupled laser are investigated theoretically by solving the coupled equations Eqs. (3) and (4). The system is perturbed below $I_{c}^{-}$ and the time evolution of the laser output returning to the steady state is computed. As the initial condition approaches the critical point, a pronounced critical slowdown is observed, characterized by the emergence of a plateau of length $\tau$. Figure 2(e) shows $\tau$ as a function of $I_{c}^{-}-I$ on a double logarithmic scale, confirming a power-law scaling with an exponent $\zeta=0.52$, in close agreement with the experimentally measured value.

Branching factor - theory - The branching factor is defined as the average number of photons generated per photon from the previous time step. We find that $\eta>1$ above the critical injection current and $\eta<1$ below threshold, while at the critical point $\eta\to 1$ from both directions (Fig. 2(f)), in agreement with the experiment. At this stage of the transition, each photon reproduces, on average, exactly one photon—revealing the hallmark of a self-sustained critical process. The theory thus uncovers, at the microscopic scale, the cascading dynamics that manifest experimentally as the abrupt phase transition.

Discussion - In summary, we identify a universal microscopic cascading mechanism that drives the macroscopic abrupt phase transition observed experimentally in a nonlinearly self-coupled laser. This transition is marked by a long-lived transient plateau near criticality and a branching factor converging to unity, and is characterized by novel critical exponents that capture the scaling of the dynamics. Our findings place this coupled laser system within the broader class of physical systems exhibiting mixed-order transitions—featuring both discontinuities and critical behavior Bonamassa et al. (2025); Mukamel (2024); Boccaletti et al. (2016); Gross et al. (2022); Alert et al. (2017); Korbel et al. (2025)—akin to those identified in the theory of interdependent networks Buldyrev et al. (2010); Zhou et al. (2014). Furthermore, these results demonstrate that the universality class of interdependent systems is broader than previously thought, being dictated by the underlying cascade mechanism rather than by network structure, which is entirely absent in self-coupled lasers.

In See Supplemental Material at [URL will be inserted by publisher] , we explore an alternative nonlinear coupling function that gives rise to double bistability, further enriching the pathways for controlling phase transitions in self-coupled systems. Together, our results not only open new avenues for fundamental research into critical phase transitions and cascading phenomena, but also point toward practical applications in the design of optical switches, photonic memory elements, and dynamically reconfigurable laser systems.

Acknowledgments - S.H. and P.S. thank Prof. M. Segev for initiating this collaboration. P.S. acknowledges the precious help of his lab manager, Leonid Wolfson. This work was supported by The Israel Science Foundation grants 3777/25, 1698/22, 2630/20, 189/19; the United States–Israel Binational Science Foundation NSF/BSF Grants 2015694, 2019740 and 2020245, the BSF Grant 2022158; the Israel Ministry of Innovation, Science $\&$ Technology, Grant No. 01017980; The Science Minister-Smart Mobility, Grant No. 1001706769; EU H2020 Project OMINO, Grant No. 101086321 and the Israeli VATAT project on power-grid. B.G. acknowledges the support of the Fulbright Postdoctoral Fellowship Program. C.X. acknowledges the Excellence Fellowship for international postdoctoral researchers funded by the Israel Academy of Sciences and Humanities and the Council for Higher Education of Israel. J.W. has been supported by the Excel@BINA scholarship from the Nanocenter at Bar Ilan University.