Physics Informed Machine Learning for Laser Dynamics: Numerical Modeling of Solitary Semiconductor Lasers With and Without Delayed Optical Feedback

Department of Electrical and Computer Engineering, Princeton University December 12, 2025

Abstract

Growing machine learning workloads are increasingly constrained by the physical limits of conventional electronic hardware. In particular, the energy and latency costs associated with data movement now dominate many applications. This challenge has motivated major architectural innovations, such as Google’s TPUs and Nvidia’s novel Tesla architecture, designed to reduce data movement and communication overhead, and has renewed interest in physical computing substrates capable of performing useful transformations directly in hardware. Photonic Reservoir Computing (RC) is one such direction: it leverages the large bandwidth and nonlinear dynamics of optical systems to process information at extremely high speeds with significant energy reduction. This work develops a physics-informed study of a solitary semiconductor laser, both with and without delayed optical feedback, as a framework for photonic reservoir computing. Building on the Lang–Kobayashi rate equations, a modular simulation framework was constructed that reproduces the laser’s transition from stable continuous wave operation to more complex oscillatory and chaotic regimes. The model allows systematic exploration of spontaneous emission, linear lasing, and saturation, and it provides a way to quantify how feedback strength, delay time, and phase reshape the underlying attractor structure.

This connects laser dynamics to practical requirements imposed by reservoir computing. This work outlines how this single laser configuration fits into our lab’s broader roadmap: beginning with a mirror-based delay line, then moving toward mutually coupled lasers in which each device acts as both a nonlinear node and a feedback element. In that context, parameters such as cavity length, coupling strength, and linewidth enhancement factor become design variables rather than fixed device properties. The resulting simulation and analysis pipeline provides a reproducible foundation for bridging semiconductor laser physics with future photonic hardware.

1 Introduction and Motivation

1.1 Limits of Classical Hardware

Deep learning now underpins performance in domains ranging from language modeling and autonomous driving to drug discovery and climate prediction. Modern models routinely contain hundreds of millions to trillions of parameters, and training them can require sustained exaflops of computation across large clusters of GPUs or specialized accelerators.

Yet despite steady architectural refinements, several physical and architectural bottlenecks increasingly limit the performance of conventional digital hardware:

• Memory wall and data movement. For many workloads, the dominant cost is no longer arithmetic but the movement of data between memory and compute units. Fetching a single 32-bit value from DRAM can consume orders of magnitude more energy than performing a floating point operation on it, forcing designers to adopt deeper memory hierarchies with increasing complexity.

• Interconnect bottlenecks. With clock frequencies plateauing in the low-GHz regime, communication between chips and even across racks often becomes the limiting factor in large-scale training. Distributing a model across many devices introduces synchronization overheads that directly impact latency and throughput.

• Thermal constraints. Since Dennard scaling has long since broken down, shrinking transistors no longer guarantees reduced power density. As a result, both data center and edge deployments face strict thermal limits that cap compute density.

• Specialization versus flexibility. While GPUs and TPUs deliver remarkable performance, they remain fundamentally general-purpose numeric engines. Many learning tasks, especially those involving temporal structure or repetitive inference, could in principle benefit from tighter coupling between the underlying physics and the algorithm.

These considerations have motivated a broader search for physical computing systems, whose intrinsic dynamics or material properties can implement useful machine learning primitives without the overhead of repeatedly simulating them in digital logic.

1.2 Photonic and Neuromorphic Computing

Photonic computing has emerged as a promising candidate in this search. Light propagates without resistive heating, and photonic integrated circuits can exploit wavelength-division multiplexing to route many independent channels through a single waveguide. Moreover, semiconductor lasers, modulators, and other nonlinear optical elements naturally provide the nonlinear transformations required for computation.

Several characteristics make photonic systems particularly attractive:

1. High bandwidth. Optical carriers support THz-scale bandwidths, and even modest integrated modulators and detectors routinely operate at tens of GHz—well beyond typical digital clock speeds.

2. Low latency feedback. On-chip optical paths can form sub-nanosecond feedback loops, enabling dynamical systems that evolve orders of magnitude faster than their electronic counterparts.

3. Massive parallelism. Wavelength and spatial multiplexing allow many signals to coexist within a compact footprint, providing opportunities for highly parallel computation.

However, photonics is not a direct replacement for digital logic. Building fully programmable, large-scale neural networks in optics remains challenging, especially when precise weight control is required. Instead, many recent efforts adopt a neuromorphic perspective: leveraging the natural dynamics of optical devices as analog processors while keeping training and control simple enough to be practical.

1.3 Reservoir Computing Framework

A reservoir is a fixed recurrent dynamical system that nonlinearly embeds input time series into a high-dimensional state space. Only the readout layer is trained, typically via linear regression, so the internal structure of the reservoir can remain unchanged (First Order Reduced and Controlled Error (FORCE), originally introduced by Sussillo and Abbott explores a method for training reservoirs online using a fast update rule rather than large offline gradient descent training).

Several properties make RC particularly compatible with physical systems:

• Minimal training overhead. Because only the readout parameters are optimized, hardware designs can focus on producing rich internal dynamics rather than tunable weights.

• Robustness to imperfections. In contrast to conventional recurrent neural networks, small perturbations in the physical reservoir’s internal structure need not degrade performance, provided the overall dynamics retain sufficient richness and memory.

• Natural alignment with physical dynamics. Many physical systems—including lasers with feedback, mechanical oscillators, and analog electronic circuits—already exhibit nonlinear dynamics with fading memory. RC offers a way to convert these natural behaviors into computational resources.

Time-delay reservoirs form an especially practical architecture. A single nonlinear node is embedded in a feedback loop of duration $\tau_f$, and by sampling the system along this loop, one obtains an effective “network of $N$ virtual nodes.” This approach has been demonstrated experimentally using optoelectronic loops, semiconductor lasers with optical feedback, and integrated photonic delay lines.

1.4 Research Strategy: Single to Coupled Lasers

This project fits into a broader effort within our lab to develop photonic reservoirs on chip. The simulation and modeling roadmap proceeds in stages:

• Stage 1: Single laser with mirror-based feedback. A solitary semiconductor laser whose output is reflected by a virtual mirror forms the simplest delay-based reservoir. The reflected field acts as a memory term, and the resulting dynamics can already be surprisingly rich.

• Stage 2: Replace the mirror with a second laser. Introducing a second laser in place of the mirror produces a mutually coupled system in which each laser provides both nonlinearity and feedback to the other. Even a two-laser configuration yields significantly higher-dimensional dynamics.

• Stage 3: Fabrication-aware reservoir design. Ultimately, we aim to design and fabricate networks of coupled lasers and waveguides whose physical parameters (cavity lengths, coupling coefficients, linewidth enhancement factors, and more) are tuned for specific ML tasks.

A prerequisite for this roadmap is a detailed understanding of the behavior of a single laser with and without feedback. This work provides that foundation by characterizing the solitary device’s dynamical regimes and by framing the results in a way that directly connects to machine learning workflows.

2 Literature Review

2.1 Lang–Kobayashi Model and Laser Chaos

The Lang–Kobayashi rate equations, introduced in 1980, remain the standard theoretical framework for describing semiconductor lasers subject to external optical feedback [1]. By extending basic photon–carrier rate equations with a delayed feedback term, the model captures a wide range of experimentally observed behaviors. These include the emergence of external cavity modes, low-frequency fluctuations, and the transition to coherence collapse despite the model’s relatively compact form.

A detailed classification of these dynamical regimes is provided in Ohtsubo’s monograph [2], which applies tools from bifurcation theory and numerical continuation to map out the stability boundaries of the system. Depending on the feedback strength, delay time, and feedback phase, the Lang–Kobayashi model can exhibit stable continuous-wave solutions, periodic or quasi-periodic oscillations, and fully developed chaos. These regimes have been studied across a variety of laser types, and experimental results align closely with the predictions of the model.

2.2 Reservoir Computing with Photonic Delay Systems

The idea of using dynamical systems with delay as reservoirs for machine learning tasks emerged from early experiments with optoelectronic feedback loops and photonic delay architectures [3, 6]. In these systems, a single nonlinear node—often realized with a Mach–Zehnder modulator or a semiconductor laser—is driven by an input signal as well as a delayed version of its own output. Sampling the resulting time trace at multiple points within the delay produces a set of virtual nodes whose collective state serves as the reservoir.

Early work by Soriano et al. [3] showed that delay-coupled photonic systems naturally generate high-dimensional transient dynamics, and that these dynamics can be systematically harnessed for computation. Their experiments demonstrated that even a single nonlinear node with delayed feedback can emulate a large recurrent network, effectively establishing the conceptual foundation for photonic reservoir computing.

Van der Sande et al. [4] extended this insight by examining what it takes to move such systems from a conceptual model to a practical information-processing platform. They analyzed how design choices such as feedback strength, input scaling, and sampling rate affect achievable data rates, robustness to noise, and overall dynamical stability. This work clarified the engineering constraints that govern real photonic reservoirs.

Hermans et al. [6] then reframed these dynamical systems within a modern machine learning framework. By showing how nonlinear response, fading memory, and state separation relate to learnability, they connected physical system parameters to standard ML notions of function approximation and generalization. This perspective helped explain why certain photonic reservoirs excel at complex temporal tasks and provided a principled way to evaluate and compare physical instantiations.

When the nonlinear node is itself a semiconductor laser, the reservoir inherits the rich dynamical repertoire associated with delayed optical feedback. Harkhoe and Van der Sande [9] quantify the computational capacity of such lasers using a suite of standardized benchmark tasks. Their findings reinforce an idea that appears repeatedly across the literature: operating near the so-called “edge of chaos” often yields the best trade-off between expressiveness and reproducibility.

2.3 Recent Photonic Reservoir Architectures

Beyond bulk fiber loops, a diverse set of photonic reservoir architectures has emerged in recent years. A few examples illustrate the range of possibilities:

• Integrated delay-based reservoirs. Microring resonators and integrated photonic feedback loops provide compact, stable delay elements with low loss. Such platforms also support multiplexing across wavelengths or spatial modes, enabling parallel processing within a small footprint [13].

• Spin lasers and quantum dot reservoirs. Devices based on spin polarization or quantum dot gain media introduce internal degrees of freedom such as polarization dynamics or multi-state carrier populations that can enrich reservoir states and support multi-channel operation [11].

• Mutually coupled laser networks. Instead of relying on a single nonlinear element, some architectures use multiple lasers that are mutually coupled, often with additional delayed paths. These networks combine spatial and temporal complexity, significantly expanding the dimensionality of the reservoir [12].

These developments point toward a growing need for modeling frameworks that can address both isolated single-node systems and more intricate networks of coupled photonic devices. The single-laser characterization presented in this work is meant to serve exactly that purpose: it provides the baseline understanding required before extending the simulations to multiple interacting lasers.

3 Prior Work and Scope

3.1 Existing Simulation Tools

Before this project, our lab maintained several numerical scripts implementing solitary semiconductor laser models based on standard rate equations. These tools were primarily intended for device-level characterization. In practice, they were used to estimate threshold currents and slope efficiencies from simulated PI curves, study how variations in material parameters such as differential gain, gain saturation, and linewidth enhancement factor affect steady-state lasing behavior, and provide qualitative guidance for choosing bias points in optical communication or sensing experiments.

Although these scripts were useful for exploratory analysis, they had limitations that became clear when investigating feedback dynamics. Most importantly, optical feedback was either absent or treated in highly simplified, quasi-static ways; as a result, the tools could not reproduce many of the dynamical phenomena reported in the literature. In addition, advanced diagnostics such as phase-space reconstruction, transient removal, and systematic parameter sweeps were not automated. Lastly, the outputs were not structured with machine learning in mind, making it difficult to directly integrate them into RC workflows.

The lab’s earlier tools were well-suited for basic device characterization but not for studying the nonlinear dynamical regimes that are essential for reservoir computing.

3.2 Contributions to Laser Modeling

The work presented here builds on this foundation but extends it in several directions to support both physics analysis and machine learning applications. The main contributions are as follows:

1. Unified ODE/DDE modeling. The solitary and feedback laser models were refactored into a set of modular simulation functions. The solitary dynamics are solved using a standard adaptive Runge–Kutta method, while the delayed feedback system is handled using a specialized DDE solver. Both approaches use consistent units (nanosecond time base, ns$^{-1}$ rates) and clearly defined transient and sampling windows. This unified structure makes it straightforward to compare solitary and feedback dynamics under identical conditions.

2. Systematic parameter sweeps. The framework supports sweeps over injection current, feedback strength, delay time, and feedback phase. These ranges were chosen based on experimentally realistic values. The results are stored in structured data formats to facilitate reproducibility and downstream analysis.

3. Physics-informed ML features. For each operating point, the code computes a feature vector containing quantities that are both meaningful to laser physicists and relevant to RC tasks: normalized current, slope efficiency, carrier sensitivity, variance of the intensity, and dimensionless delay. These features enable surrogate modeling, assist in hyperparameter searches, and provide an interpretable link between device physics and ML performance.

4. Documentation and reproducibility. All simulations, plots, and analysis routines are documented in the accompanying simulation codebase. Figures appearing in this paper are generated directly from that framework, and the overall structure is designed to scale naturally to multi-laser systems.

These elements form a simulation and analysis pipeline that extends beyond basic laser modeling. The goal is not only to reproduce known physical phenomena but also to present the results in a way that makes them immediately useful for exploring reservoir computing architectures. The rest of this paper covers the physical and ML-relevant insights obtained from the simulations, while leaving implementation details to the accompanying code.

4 Theoretical Framework and Numerical Methods

4.1 Solitary Laser Model

To model the solitary semiconductor laser, we adopt a standard single-mode rate-equation framework with gain saturation. Writing the complex intracavity field as

$$E(t) = x(t) + i\,y(t)$$

and defining its intensity as

$$r^2 = x^2 + y^2 = |E|^2,$$

the equations of motion take the form

$$\dot{x}(t) = \frac{1}{2}(G-\gamma)x - \alpha y, \tag{1}$$ $$\dot{y}(t) = \frac{1}{2}(G-\gamma)y + \alpha x, \tag{2}$$ $$\dot{N}(t) = \frac{I}{e} - \gamma_e N - G(x^2+y^2). \tag{3}$$

Here $N(t)$ denotes the carrier density, $I$ the injection current, $e$ the elementary charge, $\alpha$ the linewidth enhancement factor, and $\gamma$ and $\gamma_e$ the photon and carrier decay rates, respectively. The material gain is modeled as

$$G = \frac{g\,(N - N_0)}{1 + s(x^2+y^2)}, \tag{4}$$

where $g$ is the differential gain, $N_0$ the transparency carrier density, and $s$ the gain saturation parameter.

The model parameters are chosen to match the characteristics of devices used in our lab:

• $\alpha = 3.0$,

• $\tau_p = 2.02\ \mathrm{ps}$ ($\gamma = 1/\tau_p$),

• $\tau_N = 1.54\ \mathrm{ns}$ ($\gamma_e = 1/\tau_N$),

• $g$ and $s$ adjusted to reproduce experimental threshold and slope efficiency.

These equations capture the essential interplay between the optical field and the carrier population. In the absence of feedback, they produce the familiar behavior of solitary semiconductor lasers: a transition from spontaneous emission to coherent lasing, followed by a steady-state intensity governed by gain saturation.

4.2 Delayed Self-Feedback: Lang–Kobayashi Extension

To study the effects of delayed optical feedback, we extend the solitary laser model using the Lang–Kobayashi formalism. The delayed field re-enters the cavity after a round-trip time $\tau_f$, with a feedback strength $k_f$ and phase $\phi_f$. The resulting delay differential equations (DDEs) are:

$$\dot{x}(t) = \frac{1}{2}(G-\gamma)x - \alpha y + k_f\Big[x(t-\tau_f)\cos(\phi_f) - y(t-\tau_f)\sin(\phi_f)\Big], \tag{5}$$ $$\dot{y}(t) = \frac{1}{2}(G-\gamma)y + \alpha x + k_f\Big[y(t-\tau_f)\cos(\phi_f) + x(t-\tau_f)\sin(\phi_f)\Big], \tag{6}$$ $$\dot{N}(t) = \frac{I}{e} - \gamma_e N - G(x^2+y^2). \tag{7}$$

The terms $F_x$, $F_y$, and $F_N$ denote the solitary dynamics defined previously. In compact complex notation, the equations correspond to

$$\frac{dE}{dt} = \frac{1}{2}(1 - i\alpha)\,[G-\gamma]\,E + k_f\,E(t-\tau_f)\,e^{-i\omega_0 \tau_f}, \tag{8}$$

but the real-valued representation is more convenient for the solvers used in this work.

Adding feedback fundamentally changes the structure of the equations: instead of a three-dimensional ODE, the system becomes an infinite-dimensional DDE, and its solution depends on the full history of the state over the interval $[t-\tau_f, t]$. This shift is responsible for the dramatic increase in dynamical richness observed later in the numerical results.

4.3 Numerical Integration and Transient Handling

For solitary laser simulations, we integrate the ODE using an adaptive time-step scheme via solve_ivp. Each simulation runs for a total time $T_{\mathrm{tot}}$, of which the initial $T_{\mathrm{trans}}$ is discarded as transient. The remaining trajectory, of length $T_{\mathrm{sample}}$, is sampled uniformly with a time step $\Delta t$ to generate the time series for $x(t)$, $y(t)$, and $N(t)$.

The delayed feedback case requires a solver capable of handling DDEs. For this, we use jitcdde, which compiles the system into efficient C code and maintains the state history required to resolve the delayed terms. The initial history is set to a small perturbation around the steady-state solution of the solitary laser. As in the ODE case, we discard an initial transient window before sampling the trajectory. The sampling rate is chosen to adequately resolve both the internal relaxation oscillations and the external cavity frequency $1/\tau_f$.

4.4 Parameter Sweeps and Data Structures

The primary control parameter for the solitary laser is the injection current $I$, which we sweep from below threshold into the saturation regime. For each $I$, we compute summary statistics:

• mean intensity $\langle |E|^2 \rangle$,

• mean carrier density $\langle N \rangle$,

• standard deviations of intensity and carrier density,

• numerical estimates of slope efficiency via finite-difference derivatives.

For the feedback system, we perform nested sweeps over injection current $I$ and feedback strength $k_f$, and selectively vary feedback delay $\tau_f$ and phase $\phi_f$ for representative operating points. Simulation results are stored in Python dictionaries keyed by $(I, k_f, \tau_f, \phi_f)$, and later flattened into .npz and HDF5 archives for ease of use.

This structured format allows consistent downstream analysis, including computation of features for machine learning applications.

4.5 Physics-Informed Feature Engineering

To interface the simulations with reservoir computing workflows, we extract a set of physics-informed features for each operating point:

• raw parameters: $I$, $k_f$, $\tau_f$, $\phi_f$;

• normalized current: $I_{\mathrm{norm}} = (I - I_{\mathrm{th}})/I_{\mathrm{th}}$;

• slope efficiency estimate: $\eta_s = d\langle |E|^2 \rangle/dI$;

• carrier sensitivity: $S_N = d\langle N \rangle/dI$;

• intensity variance:

$$\sigma^2_{|E|^2} = \left\langle \left(|E|^2 - \langle |E|^2 \rangle\right)^2 \right\rangle;$$

• dimensionless delay: $\tau_f/\tau_p$ (for feedback cases).

These features are intentionally chosen to bridge physical and computational perspectives. They can support surrogate modeling, guide hyperparameter searches for RC tasks, and provide interpretable indicators of when the laser operates in regimes of high dynamical richness.

5 Numerical Results: Laser Dynamics and Regimes

5.1 Solitary Laser Dynamics

Figure 1 captures the fundamental input–output behavior of a solitary semiconductor laser. For currents below the threshold ($I_{\mathrm{th}} \approx 17.35\ \mathrm{mA}$), the output intensity remains effectively zero, because spontaneous emission cannot accumulate into a coherent field. The curve exhibits an abrupt turn-on at threshold, after which intensity increases linearly with current—reflecting the textbook slope efficiency of the device. The fitted line demonstrates that the model behaves ideally in this region, with $R^2 \approx 1$, meaning the simulated laser cleanly transitions into a stable CW lasing regime without distortions or relaxation oscillations large enough to influence the steady-state average.

The shaded region on the far right marks the onset of gain saturation, where the linearity begins to break down. This occurs because as current increases, carrier density becomes fully clamped near its threshold value, and additional current primarily increases internal losses rather than supporting proportionally larger stimulated emission. As a result, intensity begins to deviate from the linear trend.

Physical interpretation:

• Below threshold: Carrier recombination is dominated by spontaneous emission $\to$ negligible output.

• At threshold: The system reaches the point where gain matches cavity loss, leading to coherent field build-up.

• Above threshold (linear region): Carriers are clamped, stimulated emission dominates, and intensity scales linearly with current.

• Approaching saturation: The differential slope decreases because the device can no longer convert injected current into photons with the same efficiency.

This plot confirms that the solitary laser behaves as expected: clean threshold behavior, linear slope efficiency, and high-power saturation.

Figure 1: Power–current (PI) curve of the solitary laser. The sharp transition at $I_{\mathrm{th}}$ marks the onset of stimulated emission. The nearly linear region above threshold corresponds to typical operating conditions, while the slow roll-off reflects gain saturation.

Time-domain simulations, shown in Fig. 2, reveal the transient dynamics across these regimes. At $0.8\times I_{\mathrm{th}}$, the intensity remains at the numerical noise floor ($10^{-28}$ to $10^{-25}$), as spontaneous emission fails to build into coherent oscillation. At threshold ($1.0\times I_{\mathrm{th}}$), the system attempts to ignite, showing small transient spikes that collapse back to noise, reflecting the critical balance between gain and loss. Above threshold ($1.5\times I_{\mathrm{th}}$ and $2.5\times I_{\mathrm{th}}$), the laser exhibits characteristic relaxation oscillations: the intensity rises rapidly, overshoots its steady-state value, and then settles into a stable continuous-wave equilibrium. The carrier density mirrors this behavior, stabilizing at a clamped value slightly above the transparency level.

Figure 2: Transient dynamics of the solitary laser. The panels show the evolution of intensity and carrier density for currents below, at, and above threshold. Note the transition from noise-dominated behavior to stable relaxation oscillations as current increases.

Phase-space trajectories in the $(N, E^2)$ plane (Fig. 3) trace the evolution of the complex electric field ($E = x + iy$) at different injection currents. Each trajectory shows how the field spirals (or collapses) toward its steady state. Importantly, the color encodes the instantaneous intensity, allowing us to see how amplitude and phase evolve together.

• $0.8\times I_{\mathrm{th}}$ (below threshold): The trajectory collapses directly toward the origin. The field decays because the gain is insufficient to overcome cavity losses. There is no oscillatory behavior—just exponential decay.

• $1.0\times I_{\mathrm{th}}$ (at threshold): The system is marginally stable. The trajectory slowly converges toward zero amplitude because any spontaneously generated field is not amplified enough to escape decay.

• $1.5\times I_{\mathrm{th}}$ (above threshold): The trajectory begins at a small, noise-driven field and moves outward before decaying into a non-zero fixed point. This reflects the transient relaxation oscillation behavior: gain initially overshoots, intensity rises too fast, then the system self-corrects and settles.

• $2.5\times I_{\mathrm{th}}$ (well above threshold): The system reaches a strong, stable steady-state field with a much larger amplitude. Because the relaxation oscillations are strongly damped in this solitary-laser configuration, the trajectory is almost monotonic and does not form a visible spiral.

Physical interpretation: These phase-space trajectories show that the solitary laser is globally stable and always converges to a single fixed point determined by the injection current. Below threshold, the stable point is the origin. Above threshold, the stable point shifts to a non-zero field amplitude. The lack of loops or chaos here is expected—without feedback, a solitary semiconductor laser naturally behaves like a damped nonlinear oscillator that always relaxes to a fixed equilibrium.

Figure 3: Phase-space trajectory of the solitary laser. All trajectories converge to a stable fixed point, indicating low-dimensional dynamics in the absence of feedback.

5.2 Feedback-Induced Dynamics

Figure 4 directly contrasts the solitary laser (no feedback) with versions of the same laser subject to optical self-feedback of increasing strength ($k_f = 0.1$ and $k_f = 0.5$). The comparison illustrates how feedback modifies the laser’s core dynamical properties.

PI Curves (top row): The no-feedback case shows the same clean threshold and linear slope described earlier. When feedback is introduced, the average intensity remains similar in magnitude, but the error-bars (temporal variability) increase substantially. This reflects the onset of feedback-induced modulation: the laser intensity begins to oscillate or fluctuate due to the delayed reinjection of light back into the cavity.

$E^2$ vs $N$ relationships (middle row): Without feedback, the relationship is almost monotonic and collapses to a single equilibrium point at higher carrier density—corresponding to the clamped $N_{\mathrm{th}}$. With feedback, the same region expands into a significantly larger spread. This spread corresponds to quasi-periodic or chaotic-like excursions in the $E^2$–$N$ phase space due to the delay term. In other words, feedback breaks the single steady-state attraction point and replaces it with a time-dependent orbit.

Time Series of $E^2$ and $N$ (bottom row): This is where the effect becomes clearest. In the no-feedback case, the intensity rises quickly and then flattens into a smooth, constant value, while the carrier density clamps and remains fixed. With feedback, however, the system never settles. Both intensity and carrier density exhibit continuous oscillations whose amplitude depends on feedback strength. For small feedback ($k_f = 0.1$), the oscillations are mild; for stronger feedback ($k_f = 0.5$), they grow in magnitude and move the laser into a more unstable dynamical regime.

Physical interpretation: Adding delayed self-feedback fundamentally changes the nature of the solitary laser by destroying the fixed point that normally attracts the trajectories. Instead, it introduces periodic or quasi-periodic oscillations into both optical intensity and carrier dynamics, which can lead to coherence collapse or chaotic regimes for sufficiently large feedback.

The comparison thus demonstrates the classical transition from a stable CW laser ($k_f = 0$) to a modulated or chaotic laser ($k_f > 0$), which is consistent with Lang–Kobayashi theory.

Figure 4: Comparison of solitary (left) and delayed feedback (right) intensity dynamics at the same injection current. Feedback prevents convergence to a fixed point and produces persistent oscillations or irregular dynamics.

The transition to chaos becomes even more evident in the frequency domain. Although not pictured here, the power spectral density evolves from one dominated by discrete peaks (relaxation oscillations and external cavity modes) to a broadband spectrum as the feedback strength increases. This broadband structure is consistent with deterministic chaos, as documented extensively in the literature [2, 3].

In phase space, feedback disrupts the simple spiral-to-fixed-point behavior of the solitary system. Instead of collapsing onto a single orbit, trajectories expand into multi-loop or space-filling attractors whose shape depends strongly on $(I, k_f, \tau_f, \phi_f)$. These extended structures provide the high-dimensional state representations that are useful in reservoir computing.

5.3 Feedback Parameter Space

To map out how feedback modifies the laser’s dynamics, we perform a sweep over injection current $I$ and feedback strength $k_f$. For each pair $(I, k_f)$, we compute the standard deviation of the intensity time series as a simple proxy for dynamical richness. The results are shown in Fig. 5.

Figure 5: Parameter-space summary over injection current and feedback strength. Dark regions indicate nearly steady-state dynamics; brighter regions correspond to persistent or chaotic fluctuations. The intermediate band is a candidate “edge of chaos” region.

Three qualitatively distinct regions emerge:

• Stable region. For small $k_f$ and moderate currents, feedback induces only shallow modulation around the solitary fixed point. Dynamics remain low-dimensional.

• Edge of chaos region. At intermediate $k_f$, the laser displays strong but bounded fluctuations. These dynamics are complex enough to encode useful temporal structure while remaining reproducible—an appealing combination for reservoir computing [9, 10].

• Strongly chaotic region. Larger feedback strengths or particular combinations of parameters lead to large excursions in intensity. While interesting from a dynamical-systems perspective, such regimes may not be practical for physical hardware due to clipping, thermal concerns, or poor repeatability.

Additional sweeps over delay time $\tau_f$ and phase $\phi_f$ reveal resonances between the external cavity period and intrinsic laser timescales. These effects, well documented in [10], highlight the importance of careful delay engineering when designing physical RC systems. Even small changes in delay can dramatically reshape the attractor topology and memory properties of the reservoir.

6 Machine Learning and Roadmap

6.1 Laser Dynamics and ML Requirements

From a machine learning perspective, a semiconductor laser with delayed feedback functions as a nonlinear dynamical system that maps an input current waveform $I(t)$ to an optical intensity output $|E(t)|^2$. When used as a reservoir, the input sequence modulates the laser, and the resulting optical waveform or a sampled version of it constitutes the reservoir state. For this arrangement to be practically useful, several properties must hold:

• Nonlinear response. The system must respond nonlinearly to modulations in $I(t)$. Without sufficient nonlinearity, the reservoir cannot perform the transformations required for tasks such as parity, XOR-like operations, or chaotic sequence prediction.

• Finite memory. A useful reservoir exhibits “fading memory”: the influence of an input perturbation persists for a finite time before decaying. This property is essential for tasks involving temporal dependencies, such as speech recognition or time-series forecasting.

• High-dimensional state space. Distinct input histories should lead to distinguishable internal states. In delay-based systems, this richness emerges from the interaction between internal laser dynamics and the delayed reinjection term.

• Reproducibility and robustness. Although nonlinear and sometimes sensitive to initial conditions, a physical reservoir must behave reproducibly for a given parameter setting and input sequence. Otherwise, training a readout becomes unstable or ineffective.

The simulations in Section 5 illustrate how these properties depend on the choice of $I$, $k_f$, $\tau_f$, and $\phi_f$. In particular, the intermediate band highlighted in Fig. 5 often provides an appealing balance: the dynamics are rich enough to support nonlinear processing yet structured enough to remain usable for training. This is consistent with the broader “edge of chaos” intuition found in both photonic and non-photonic reservoir computing literature.

6.2 Physics-Informed Features

A key goal of this project is to connect numerical laser simulations with machine learning workflows in a way that maintains interpretability. The physics-informed features introduced in Section 4 serve several roles:

• Surrogate modeling and design space exploration. Full delay differential equation simulations are computationally expensive. By training surrogate models on the derived features, one can quickly approximate summary statistics such as intensity variance or approximate Lyapunov exponents. This accelerates exploration of large parameter spaces.

• Hyperparameter guidance for RC tasks. When evaluating the reservoir on benchmark tasks (e.g., Mackey–Glass prediction, NARMA-10, spoken digits), these features can be correlated with task performance. Doing so helps identify task-agnostic indicators of good operating regimes. For instance, moderate normalized current and intermediate feedback strength tend to perform well for prediction tasks—a trend also observed in [9].

• Transferability to coupled laser architectures. Because the features are grounded in physical quantities rather than the specifics of a single laser configuration, they extend naturally to systems with multiple mutually coupled lasers. In such systems, quantities like slope efficiency or intensity variance can be computed per laser, and coupling terms simply add new parameters to the feature vector.

These features therefore form a bridge between the detailed physics of semiconductor lasers and the practical considerations of ML model selection and hyperparameter tuning.

6.3 Future Coupled Architectures

The longer-term vision of our lab is to move beyond single-laser reservoirs and develop networks of optically coupled lasers whose dynamics can be engineered for specialized computational tasks. In broad terms:

• Single mirror-based feedback. The system studied here—one laser with external-cavity feedback—implements a temporal reservoir using a single nonlinear element. Its memory arises from the delay line, and its nonlinearity arises from the semiconductor gain dynamics.

• Mutually coupled lasers. Replacing the passive mirror with a second laser introduces an additional layer of dynamical interaction: each laser processes the incoming field while simultaneously serving as a feedback element for the other. Even a pair of such lasers can produce dynamics that are far richer than those achievable with a single node.

• Integrated photonic networks. On-chip implementations could extend this idea to small arrays of lasers, waveguides, and microring resonators. Fabrication parameters—cavity lengths, coupling gaps, and waveguide lengths, among others—then become controllable design variables. By tuning these values, one could shape the reservoir’s memory depth, nonlinearity, and spectral bandwidth.

The single-laser results in this work provide a reference point for these more complex architectures. Understanding how feedback modifies the behavior of one device is a necessary step toward predicting how multiple devices will interact. Moreover, the simulation and feature-extraction pipeline developed here can be adapted to multi-laser systems by adding coupling terms and expanding the parameter space.

Overall, the intent is to progress from fundamental dynamical understanding to task-aware reservoir design, guided by both physical intuition and machine learning performance metrics.

7 Discussion and Future Directions

7.1 Energy Efficiency Trade-offs

The simulations presented here highlight a central tension that often appears in photonic reservoir computing: regimes that are most energy efficient are not always the most computationally expressive. Biasing the laser deep in the linear regime yields high slope efficiency and stable operation, but the resulting dynamics are close to linear and therefore limited in computational capability. In contrast, operating near threshold or under intermediate feedback conditions introduces strong nonlinearities and richer dynamics, but at the cost of increased sensitivity and reduced power efficiency.

Quantifying this trade-off more rigorously is an important direction for future work. A natural next step is to combine the simulations with energy models that estimate photons per operation or effective energy per MAC. Doing so would help situate photonic reservoirs relative to traditional accelerators in terms of energy consumption, particularly for inference tasks where energy per output is critical. Understanding this balance will likely become even more important if fabricated coupled laser networks are used for dedicated ML workloads.

7.2 Task-Level Benchmarks

While the present study focuses on characterizing laser dynamics, the next stage is to connect these dynamics directly to task-level performance. This involves implementing reservoir computing tasks using the simulated time series as reservoir states, applying virtual-node sampling, and training linear readout layers. Typical benchmarks include chaotic time-series prediction (e.g., Mackey–Glass), nonlinear autoregressive moving average tasks (e.g., NARMA-10), and simple classification problems.

By evaluating performance across a broad sweep of $(I, k_f, \tau_f, \phi_f)$, it becomes possible to test whether the “edge of chaos” regions identified in Fig. 5 indeed correspond to optimal performance. Prior work [9, 6] strongly suggests this connection, but validating it using a model calibrated to the lab’s actual device parameters would provide more direct guidance for experimental implementations.

7.3 Fabrication-Aware Design

The long-term objective is to move toward fabrication-aware design of photonic reservoirs. The single-laser results serve as the first step in this process, offering a way to map from physical parameters to dynamical properties and then to ML-relevant behavior. Extending this approach to coupled laser systems requires introducing inter-device coupling terms and accounting for wavelength detuning, coupling strength, and phase relationships.

Once a coupled-laser simulation framework is in place, one could begin exploring how geometric and material parameters—cavity lengths, waveguide delays, coupling gaps, mirror reflectivities—affect both the dynamical richness and practical usability of the resulting reservoir. Ideally, this would lead to a workflow that allows researchers to specify desired computational properties (e.g., memory depth or nonlinear mixing strength) and obtain a set of device-level design recommendations.

Ultimately, this direction points toward a more integrated perspective on photonic reservoir computing, where device physics, dynamical systems theory, and machine learning performance criteria inform one another. The work in this paper provides the foundation for that progression by establishing a reproducible bridge between numerical laser modeling and ML-oriented analysis.

Acknowledgements

I would like to thank everyone who supported this project. I am grateful to Professor Claire Gmachl for her guidance, steady encouragement, and for providing an environment in which independent research could develop productively. I also thank Godwin Duan, whose earlier Lang–Kobayashi simulation framework formed the basis for the modeling work here and whose advice on numerical methods and parameter choices was extremely helpful. I appreciate the many conversations with Chixuan Chen on semiconductor laser physics, feedback configurations, and the connection between simulation and experiment, all of which informed the direction of this work. Finally, I am thankful to the broader lab community for their time, ideas, and the collaborative atmosphere that contributed to the completion of this project.

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