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Communications in Theoretical Physics

Determination of optical properties of a single-mode class-A laser amplifier using the noise fluxes generated by cavity Langevin force

J. Jahanpanah, Davood Hashemi dastjerdi, A A Rahdar

Published Feb 10, 2025• 0 citations
Publisher

Abstract

Abstract The noise feature of a single-mode class-A laser amplifier is investigated by solving the Maxwell–Bloch equations of motion in the presence of the fluctuation force of cavity Langevin. The aim is to calculate the simultaneous fluctuations that are superimposed on the amplitude and phase of the cavity electric field, as well as the atomic population inversion. The correlation function of these fluctuations yields the amplitude, phase, and spontaneous emission noise fluxes, respectively. The amplitude and spontaneous emission noise fluxes exhibit the Lorentzian profiles in both the below-threshold state and the injection-locking region of the above-threshold state. While noise is typically viewed negatively in science and engineering, this research highlights its positive role as a valuable tool for measuring the optical properties of a laser amplifier. For instance, the degree of first-order temporal coherence (DFOTC) is derived by taking the Fourier transform of the amplitude noise flux. The damping rate of DFOTC is associated with the coherence time of the light emitted by the laser amplifier. Furthermore, the uncertainty relation between noise bandwidth and coherence time is confirmed. Finally, it is demonstrated that the input pumping noise flux, together with the output amplitude and spontaneous emission noise fluxes, satisfy the principle of flux conservation.

DOI
10.1088/1572-9494/adb413

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Abstract The noise feature of a single-mode class-A laser amplifier is investigated by solving the Maxwell–Bloch equations of motion in the presence of the fluctuation force of cavity Langevin. The aim is to calculate the simultaneous fluctuations that are superimposed on the amplitude and phase of the cavity electric field, as well as the atomic population inversion. The correlation function of these fluctuations yields the amplitude, phase, and spontaneous emission noise fluxes, respectively. The amplitude and spontaneous emission noise fluxes exhibit the Lorentzian profiles in both the below-threshold state and the injection-locking region of the above-threshold state. While noise is typically viewed negatively in science and engineering, this research highlights its positive role as a valuable tool for measuring the optical properties of a laser amplifier. For instance, the degree of first-order temporal coherence (DFOTC) is derived by taking the Fourier transform of the amplitude noise flux. The damping rate of DFOTC is associated with the coherence time of the light emitted by the laser amplifier. Furthermore, the uncertainty relation between noise bandwidth and coherence time is confirmed. Finally, it is demonstrated that the input pumping noise flux, together with the output amplitude and spontaneous emission noise fluxes, satisfy the principle of flux conservation.
The noise feature of a single mode class-A laser is investigated in the presence (amplifier) and absence (free-running) of an input signal. The Maxwell-Bloch equations of motion have been solved after adding the cavity Langevin force to calculate fluctuations that imposed to the atomic population inversion and the amplitude and phase of cavity electric field. The correlation function of these fluctuations is then used to derive the spontaneous emission, amplitude, and phase noise fluxes in the below and above-threshold states. The bandwidth of noise fluxes is not only adjusted by the amplitude and frequency detuning of input signal, but also by the laser pumping and cavity damping rates. On the other hand, the degree of first-order temporal coherence (DFOTC) is turned out as the correlation function of the amplitude fluctuation so that its Fourier transform led to the amplitude noise flux. The coherence time plays a dual role in order that it is equal to the damping rate invers of DFOTC and at the same time has an uncertainty relation with the bandwidth of amplitude noise flux. Finally, the flux conservation requires a balance between the input pumping noise flux and the output amplitude and spontaneous emission noise fluxes.
The fluctuation effect of laser pumping rate on the output noise fluxes of class-A lasers is investigated. The method is based on the role of cavity Langevin force as a fluctuating force in the absence of the atomic population inversion and dipole moment Langevin forces. The temporal fluctuations induced to the phase and amplitude of the cavity electric field and the atomic population inversion are calculated in both below and above threshold states. Our aim is to derive correlation functions for the fluctuating variables of the cavity electric field and the atomic population inversion to determine the noise fluxes emerging from the cavity mirrors and measured by an optical detector and those radiated in the form of a spontaneous emission in all spatial directions. We introduce a heuristic conservation relation that connects the noise flux generated by the laser pumping system with those distributed among the laser variables. Finally, the results are confirmed by demonstrating the energy conservation law.
It has been customary to ignore the key role of the cavity Langevin force on the laser noise fluxes. Here, the noise fluxes of the laser pump and spontaneous and stimulated emission (laser) radiation are calculated for class-B lasers by considering the cavity Langevin force alone. They have a symmetrical double-peak profile around the cavity resonance frequency, which is in complete agreement with the experimental results of the heterodyne gain profile of class-B laser amplifiers. This agreement indicates the lasers undertake the same mechanism in amplifying an input signal as superimposing noise fluxes on their variables. Meanwhile, the noise flux profiles of the cavity Langevin force are consistent with those produced by two other Langevin forces, associated with atomic population inversion and the atomic dipole moment. Finally, the noise fluxes generated by the cavity Langevin force are confirmed by demonstrating the flux conservation law.
The class-C lasers are well recognized to have very narrow atomic transition bandwidths. Therefore, a minor mismatch between the cavity resonance and atomic transition frequencies (CAF) is normally led to the large modifications in the gain and noise profiles. The motion of class-C lasers is described by the three variables of cavity electric field, atomic population inversion, and dipole moment of atoms that are coupled to each other by the optical Maxwell-Bloch equations. The fluctuating features of these three variables are completely elaborated by solving their equations of motion in the presence of the cavity Langevin force. The results peculiarly indicate that the noise profile of the cavity electric field exactly mimics the amplification profile of an input signal by the laser in both below and above threshold states. The both gain and noise profiles simultaneously tend to infinity at the normalized pumping rates and CAF mismatches determined by the laser stability theory. The noise flux profiles are finally confirmed by illustrating the flux conservation.
A theoretical study is presented of the gain and the noise spectra of laser amplifiers at pumping rates both above and below the oscillation threshold. The theory is valid for both class-A and class-B laser systems, where the decay rate of the collective atomic dipole moment greatly exceeds the decay rates of the atomic population inversion and the laser cavity. The gain and the noise properties are shown to be strongly influenced by a four-wave mixing of the signal and image frequencies via a modulation of the population inversion. Particular attention is paid to the residual noise flux that persists in the laser output above threshold and to the typical characteristics of a second-order phase transition displayed by the gain in the threshold region. The results for the above-threshold laser amplifier are compared with experimental measurements of both gain and noise for the argon-ion (class-A) and CO2{\mathrm{CO}}_{2}CO2​ (class-B) lasers. The signal-to-noise ratios for both heterodyne and direct-detection measurements are evaluated, and the conditions are determined under which optical amplification can lead to an enhancement of sensitivity.
The random nature of spontaneous emission leads to unavoidable fluctuations in a laser's output. This is often included through random Langevin forces in laser rate equations, but this approach falls short for nanolasers. In this paper, we show that the laser quantum noise can be quantitatively computed for a very broad class of lasers by starting from simple and intuitive rate equations and merely assuming that the number of photons and excited electrons only takes discrete values. While the approach has seen previous success, we here derive it rigorously from an open quantum system master equation, whereas it was previously introduced only on phenomenological grounds. We further show that in the many-photon limit, the model simplifies to Langevin equations. We perform an extensive comparison of different approaches for computing quantum noise in lasers, identifying the best approach for different system sizes, ranging from nanolasers to macroscopic lasers, and different levels of excitation, i.e., cavity photon number. In particular, we show that below the laser threshold, stochastic fluctuations in the numerical solution to the Langevin equations can drive populations to unphysical negative values, requiring the introduction of population bounds, which in turn skew the noise statistics, leading to inaccuracies. The Laser Markov Chain model, on the other hand, is accurate for all pump values and laser sizes when collective emitter effects are excluded.