Lang-Kobayashi Model Research Articles

Retrieval of the Dielectric Properties of a Resonant Material in the Terahertz Region via Self-Detection Near Field Optical Microscopy

We present a numerical and analytical study of the self-detection scattering type near field optical microscopy (SD s-SNOM), a recently demonstrated technique based on a combination of self-mixing interferometry and scattering near- field microscopy. This scheme, which exploits a terahertz (THz) quantum cascade laser as both a laser source and detector, allows to investigate the optical properties of resonant materials in the THz range with resolution far beyond the diffraction limit. Our study, developed by using a modified version of the Lang-Kobayashi model, is focused on the weak feedback regime (Acket parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$C ≈ 10^{-1}$</tex-math></inline-formula> ), where we derive an approximated method for the retrieval of the scattering coefficient of the SD s-SNOM configuration applied to a sample of Cesium Bromide (CsBr). These results were used in turn to derive the dielectric permittivity of the sample, reporting a good accuracy in the estimation of its phonon resonances.

Cascading of all-optical logic gates based on semiconductor laser phase-locking

Optical information processing is an essential part of contemporary photonics, and it remains a challenge for researchers around the world. The presented work investigates numerically a maximum number of all-optical logic gates in a cascade. The functionality of the all-optical logic gate is based on the injection locking technique with semiconductor lasers. The possibility of cascading a sequence of AND and OR gates, providing a high-quality final signal, is demonstrated for a planar vertical-cavity surface-emitting laser using the traditional Lang–Kobayashi model. It is shown that the maximum number of cascaded gates is limited to three for repetition rates less than 1 Gbit/s.

Calculation of the steady states in dynamic semiconductor laser models

We discuss numerical challenges in calculating stable and unstable steady states of widely used dynamic semiconductor laser models. Knowledge of these states is valuable when analyzing laser dynamics and different properties of the lasing states. The example simulations and analysis mainly rely on 1(time)+1(space)-dimensional traveling-wave models, where the steady state defining conditions are formulated as a system of nonlinear algebraic equations. The performed steady state calculations reveal limitations of the Lang-Kobayashi model, explain nontrivial bias threshold relations in lasers with several electrical contacts, or predict and explain transient dynamics when simulating such lasers.

Self-Pulsations in Terahertz Quantum Cascade Lasers under Strong Optical Feedback: The Effect of Multiple Reflections in the External Cavity.

We have recently reported the self-pulsation phenomenon under strong optical feedback in terahertz (THz) quantum cascade lasers (QCLs). One important issue, however, we left open: the effect of multiple round trips in the external cavity on the laser response to feedback. Our current analysis also casts additional light on the phenomenon of self-pulsations. Using only one external cavity round trip (ECRT) in the model has been the common approach following the seminal paper by Lang-Kobayashi in 1980. However, the conditions under which the Lang-Kobayashi model, in its original single-ECRT formulation, is applicable has been rarely explored. In this work, we investigate the self-pulsation phenomenon under multiple ECRTs. We found that the self-pulsation waveform changes when considering more than one ECRT. This we attribute to the combined effect of the extended external cavity length and the frequency modulation of the pulsation frequency by the optical feedback. Our findings add to the understanding of the optical feedback dynamics under multiple ECRTs and provide a pathway for selecting the appropriate numerical model to study the optical feedback dynamics in THz QCLs and semiconductor lasers in general.

Impact of FBG feedback phase on laser dynamics

Fiber Bragg gratings (FBGs) have been advantageously used to improve the chaotic properties of semiconductor lasers. Though these components are known to be highly sensitive to environmental conditions, feedback phase fluctuations are often neglected. In this work, we experimentally demonstrate that the small variations of the propagation time induced by a simple thermal tuning of the FBG are sufficient to induce significant changes of the laser behavior. We report periodic stability enhancements linked with phase variations and highlight that both phase variation and phase offsets play an important role. Last, we show a good qualitative agreement with simulations based on an expanded version of the Lang-Kobayashi model.

Dynamics of a semiconductor laser with feedback and modulation: experiments and model comparison.

We study experimentally and numerically the dynamics of a semiconductor laser near threshold, subject to optical feedback and sinusoidal current modulation. The laser operates in the low frequency fluctuation (LFF) regime where, without modulation, the intensity shows sudden spikes at irregular times. Under particular modulation conditions the spikes lock to the modulation and their timing becomes highly regular. While the modulated LFF dynamics has received a lot of attention, an in-depth comparison with the predictions of the Lang-Kobayashi (LK) model has not yet been performed. Here we use the LK model to simulate the laser dynamics and use the Fano factor to quantify the regularity of the timing of the spikes. The Fano factor is calculated by counting the number of spikes in successive segments of the intensity time-series and keeps information about temporal order in the spike sequence that is lost when the analysis is based on the distribution of inter-spike intervals. Here we compare the spike timing regularity in experimental and in simulated spike sequences as a function of the modulation amplitude and frequency and find a good qualitative agreement. We find that in both experiments and simulation for appropriate conditions the spike timing can be highly regular, as revealed by very small values of the Fano factor.

Experimental and Numerical Study of Locking of Low-Frequency Fluctuations of a Semiconductor Laser with Optical Feedback

We study the output of a semiconductor laser with optical feedback operated in the low-frequency fluctuations (LFFs) regime and subject to weak sinusoidal current modulation. In the LFF regime, the laser intensity exhibits abrupt drops, after which it recovers gradually. Without modulation, the drops occur at irregular times, while, with weak modulation, they can lock to the external modulation and they can occur, depending on the parameters, every two or every three modulation cycles. Here, we characterize experimentally the locking regions and use the well-known Lang–Kobayashi model to simulate the intensity dynamics. We analyze the effects of several parameters and find that the simulations are in good qualitative agreement with the experimental observations.

Challenge-Response Authentication Scheme With Chaotic Lasers

In this paper, we numerically evaluate a hardware-based method for authentication using a pair of twin chaotic lasers. This method belongs to the PUF (Physically Unclonable Function) class; however, it does not require storing the response of the hardware in a database, because both users own the same hardware (one of the twin chaotic lasers), producing, on the fly, the same response, i.e., the same chaotic waveform, when subject to the same stimulus (injection from a third laser) in the same working conditions (local injection, pump current). A bit sequence can be easily obtained from the chaotic waveforms by electronic processing, and authentication consists in comparing the sequences produced by the twin lasers. This scheme is proposed above all for authentication of a client in an unsecure environment to a server in a secure environment, but it can be used also for peer-to-peer authentication. Both the case of open- and close-loop chaotic lasers are considered. Simulations are based on the Lang-Kobayashi model.

Locking Phenomena in Semiconductor Lasers near Threshold with Optical Feedback and Sinusoidal Current Modulation

The dynamics of semiconductor lasers with optical feedback and current modulation has been extensively studied, and it is, by now, well known that the interplay of modulation and feedback can produce a rich variety of nonlinear phenomena. Near threshold, in the so-called low frequency fluctuations regime, the intensity emitted by the laser, without modulation, exhibits feedback-induced spikes, which occur at irregular times. When the laser current is sinusoidally modulated, under appropriate conditions, the spikes lock to the modulation and become periodic. In previous works, we studied experimentally the locked behavior and found sub-harmonic locking (regular spike timing such that a spike is emitted every two or three modulation cycles), but we did not find spikes with regular timing, emitted every modulation cycle. To understand why 1:1 regular locking was not observed, here, we perform simulations of the well-known Lang–Kobayashi model. We find a good qualitative agreement with the experiments: with small modulation amplitudes, we find wide parameter regions in which the spikes are sub-harmonically locked to the modulation, while 1:1 locking occurs at much higher modulation amplitudes.

Effect of Shockley-Read-Hall recombination on the static and dynamical characteristics of epitaxial quantum-dot lasers on silicon

We semianalytically and numerically investigate the static and dynamical characteristics of quantum-dot (QD) lasers directly grown on silicon by considering the Shockley-Read-Hall (SRH) recombination. The static characteristics are studied through small-signal analysis, including the ${\ensuremath{\alpha}}_{H}$ factor, damping factor, and modulation dynamics. In addition, the feedback dynamics are analyzed through improved corpuscular rate equations based on the classical Lang-Kobayashi (LK) model with time series, bifurcation diagrams, and phase portraits. We find that a smaller ${\ensuremath{\alpha}}_{H}$ factor but larger damping factor are obtained by decreasing the nonradiative recombination lifetime. On top of that, in both the short- and long-external-cavity regimes, any decrease of the SRH recombination lifetime obliterates significantly chaotic regions and shifts the first Hopf bifurcation point to higher feedback values. Overall, this work provides insights into the understanding of QD laser physics, hence highlighting the influence of the SRH lifetime on the reflection sensitivity of epitaxial QD lasers on silicon. These results are qualitatively consistent with recent experiments and are therefore helpful for designing feedback-resistant lasers for future photonic integrated circuits operating without optical isolation.

Stable-unstable switching dynamics in semiconductor lasers with external cavities

Optical feedback in a semiconductor laser was recently used for photonic microwave square-wave generation by invoking quasiperiodicity for repeated switching between a stable stage of constant intensity and an unstable stage of intensity oscillations. Based on the Lang-Kobayashi model for a laser with external-cavity feedback, we investigate such stable-unstable switching dynamics. First, the model is verified to produce the stable-unstable switching dynamics as the feedback strength and delay are adjusted. The model agrees with former experiments in spite of previous doubts. Then, the analytical Hopf boundary for the minimum linewidth mode is derived to be matching the boundaries of the switching regions. Moreover, the trajectory in the plane of instantaneous frequency and gain is analyzed during the stable stage, where the laser is found to follow an ellipse in visiting different external cavity modes. Most importantly, an analytical derivation yields the switching period $\ensuremath{\tau}+{\ensuremath{\tau}}_{\ensuremath{\varepsilon}}$ that is slightly expanded from the feedback delay time $\ensuremath{\tau}$, where the expansion time ${\ensuremath{\tau}}_{\ensuremath{\varepsilon}}$ is found to be inversely proportional to the feedback strength. Such a switching period expansion explains the generation of a shifted frequency component below $1/\ensuremath{\tau}$ in quasiperiodic laser dynamics. The results generally contribute to understanding the switching dynamics in photonic microwave generation.

Optimized properties of chaos from a laser diode

We perform an experimental parametric study of the chaos generated by a laser diode subjected to phase-conjugate feedback. In addition to the typical figure of merit, i.e., chaos bandwidth, the corresponding spectral flatness and permutation entropy at delay is analyzed. Our experimental observations reveal that the chaos can be generated with a bandwidth of ≈29GHz, a spectral flatness up to 0.75, and a permutation entropy at delay of up to 0.99. These optimized performances are maintained over a large range of parameters and have not been achieved in the conventional optical feedback configuration. Interestingly, reducing the pump current reduces the chaos bandwidth while keeping the spectral flatness and the permutation entropy at delay the same as observed for increased pump current. Our experimental findings are consistent with the presented numerical simulations produced using the Lang-Kobayashi model.

Analysis of doppler induced optical feedback interferometry for point-of-use refractometry

The paper presents a flow-refractometry based on the Doppler Effect. The presented work provides an in-depth analysis of the design and principle of the Doppler based refractometry. The simulative study is performed by solving the modified Lang-Kobayashi model numerically, and the results of the study are verified. The paper also presents a modification in the design to enhance the measuring capability of the Doppler based refractometry. The modification in the design improves the measuring range of the refractometry by increasing the amount of reflection available for the interference. The proposed method can measure a wide range of the refractive index. The proposed design of the refractometry reports a standard deviation of the order of 10−5 for the refractive index ranging from 1.3 to 2.5. The design and precision of the refractometer also qualify the requirement of the point-of-care devices based on the refractive index variations developed for COVID-19 diagnosis.

Simulation of Monolithically Integrated Semiconductor Laser Subject to Random Feedback and Mutual Injection

Aiming at the problems of limited bandwidth and periodicity of chaotic laser, a monolithically integrated chaotic semiconductor laser subject to random feedback and mutual injection is proposed. The chaotic laser consists of two distributed feedback laser diodes (DFB-LD), two semiconductor optical amplifiers (SOA) and a passive optical waveguide. Optical injection of two coupled DFB-LDs achieves chaotic signal shaping and bandwidth enhancement. Random grating along the passive waveguide provides stochastic feedback to suppress the time delay signature (TDS). A theoretical model is established based on the Lang-Kobayashi model to explore the dynamics of the proposed chaotic laser. A chaotic signal with TDS of 0.06 and bandwidth of 13.12 GHz is obtained by simulation. Furthermore, the effects of parameters including feedback ratio, mutual injection ratio, and detuning frequency on the TDS and bandwidth of chaotic signal are investigated. The results show that the broadband chaotic laser signal with TDS suppression can be obtained by controlling parameters reasonably.

Suppression of Coherence Collapse in Semiconductor Fano Lasers.

We show that semiconductor Fano lasers strongly suppress dynamic instabilities induced by external optical feedback. A comparison with conventional Fabry-Perot lasers shows orders of magnitude improvement in feedback stability and in many cases even total suppression of coherence collapse, which is of major importance for applications in integrated photonics. The laser dynamics are analyzed using a generalization of the Lang-Kobayashi model for semiconductor lasers with external feedback, and an analytical expression for the critical feedback level is derived.

Efficient and Optical Feedback Tolerant Hybrid Laser Design for Silicon Photonics Applications

We present a design of a tunable hybrid laser based on III-V Reflective Semiconductor Optical Amplifier (RSOA) and Silicon Photonics (SiPh) external mirror that represents a good trade off between high Wall-Plug Efficiency (WPE) and high tolerance to optical feedback caused by unwanted back reflections from the rest of the SiPh chip. The sensitivity to optical feedback of different configurations, an important issue in many SiPh applications, is evaluated through the calculation of the critical feedback level based on an effective Lang-Kobayashi model and the main results are validated through numerical simulations of laser dynamics. We conclude that hybrid lasers with long effective external cavities typically designed to reduce the laser linewidth can be also exploited to improve the tolerance to spurious optical feedback.

Assessing Security of Chaos Communications Against Eavesdropping by Pump Detection

In this paper, we numerically evaluate a method that could be tried by an eavesdropper to force the standard two-laser scheme of chaos private transmission. This method is based on the detection of the voltage arising across a semiconductor laser junction and uses two lasers: one is synchronized to the transmitter (thus, reproducing chaos and rejecting the message), the other is selected and trimmed to copy the transmitter (i.e., reproducing both chaos and the message). The message is then obtained by difference and chaos cancellation. By assuming the usual parameter mismatch, which is expected to be achieved by an eavesdropper, it is shown that the method does not represent a substantial threat to the privacy of the transmission system. The scheme is also briefly considered as an alternative, but less performant, scheme for chaotic cryptography. The simulations are based on the Lang-Kobayashi model.

Інтерферометрія зворотного зв’язку в лазерному резонаторі. Параметрична модель.

The article is devoted to an alternative view on the explanation of the mechanism of action of devices based on laser feedback interferometry (LFI-model), which is caused by contradictions between the theory of existing models and practical results. A view other than the Lang-Kobayashi model (LK model) and the parametric LFI model (P model) are proposed. Due to the theory, which is based on the parametric properties of the LFI-model, the level of advantage of LFI technology over traditional technologies is quantitatively substantiated. The article gives an example of calculating the useful signal power. It is calculated that at a distance of 500 meters, the signal according to the P-model is 34 dB higher than the signal calculated by traditional models. Thus, from the traditional models follows the inverse square dependence of the signal-to-noise ratio (S / N ratio, hereinafter - SNR) on the distance to the target l_t, viz.: SNR ~ l_t^{-2}. In practice, the SNR is much higher. Within the P-model, another dependence of SNR on l_t, is theoretically proved and experimentally confirmed, viz.: SNR ~ (l_t * ln l_t^2)^{-1/2}. Traditional models do not consider the presence of a useful signal in the pump current, while, in fact, its power is more than 10 times greater than the radiation power in the resonator. The P-model eliminates contradictions between theoretical models and practical results.

Experimental characterization of the transition to coherence collapse in a semiconductor laser with optical feedback.

Semiconductor lasers with time-delayed optical feedback display a wide range of dynamical regimes, which have found various practical applications. They also provide excellent testbeds for data analysis tools for characterizing complex signals. Recently, several of us have analyzed experimental intensity time-traces and quantitatively identified the onset of different dynamical regimes, as the laser current increases. Specifically, we identified the onset of low-frequency fluctuations (LFFs), where the laser intensity displays abrupt dropouts, and the onset of coherence collapse (CC), where the intensity fluctuations are highly irregular. Here we map these regimes when both, the laser current and the feedback strength vary. We show that the shape of the distribution of intensity fluctuations (characterized by the standard deviation, the skewness, and the kurtosis) allows to distinguish among noise, LFFs and CC, and to quantitatively determine (in spite of the gradual nature of the transitions) the boundaries of the three regimes. Ordinal analysis of the inter-dropout time intervals consistently identifies the three regimes occurring in the same parameter regions as the analysis of the intensity distribution. Simulations of the well-known time-delayed Lang-Kobayashi model are in good qualitative agreement with the observations.

Chaotic bursting in semiconductor lasers.

We investigate the dynamic mechanisms for low frequency fluctuations in semiconductor lasers subjected to delayed optical feedback, using the Lang-Kobayashi model. This system of delay differential equations displays pronounced envelope dynamics, ranging from erratic, so called low frequency fluctuations to regular pulse packages, if the time scales of fast oscillations and envelope dynamics are well separated. We investigate the parameter regions where low frequency fluctuations occur and compute their Lyapunov spectra. Using the geometric singular perturbation theory, we study this intermittent chaotic behavior and characterize these solutions as bursting slow-fast oscillations.