Using population rate equations, we have calculated the intrinsic intensity fluctuations expected in the output of 3-level and 4-level CW laser oscillators. Such fluctuations stem from the quantum shot noise implicit in a restriction of the atomic and photon population-operator eigenvalues to discrete integral values (in the manner of Shimoda, Takahasi, and Townes). The intrinsic fluctuations below threshold are typical of a Gaussian distribution ofe-mfield amplitudes (Bose-Einstein distribution of photons) and above threshold of an amplitude-stabilizede-mfield (Poisson distribution of photons). The noise properties of 3-level and 4-level lasers are qualitatively similar, although for a given output power the intensity fluctuations of 3-level oscillators are greater than (or equal to) those in 4-level oscillators, principally as a consequence of the increased spontaneous emission associated with the larger upper-state occupation necessary to achieve a net gain in 3-level systems.
Using population rate equations, we have calculated the intrinsic intensity fluctuations expected in the output of 3-level and 4-level CW laser oscillators. Such fluctuations stem from the quantum shot noise implicit in a restriction of the atomic and photon population-operator eigenvalues to discrete integral values (in the manner of Shimoda, Takahasi, and Townes). The intrinsic fluctuations below threshold are typical of a Gaussian distribution ofe-mfield amplitudes (Bose-Einstein distribution of photons) and above threshold of an amplitude-stabilizede-mfield (Poisson distribution of photons). The noise properties of 3-level and 4-level lasers are qualitatively similar, although for a given output power the intensity fluctuations of 3-level oscillators are greater than (or equal to) those in 4-level oscillators, principally as a consequence of the increased spontaneous emission associated with the larger upper-state occupation necessary to achieve a net gain in 3-level systems.
Using linearized rate equations, we have calculated the intensity fluctuations expected in the output of four-level cw laser oscillators. Intrinsic quantum effects have been included by restricting the eigenvalues of photon and atomic population operators to discrete integral values and by using equivalent noise sources. The total output fluctuations increase monotonically with output power; the relative fluctuations decrease monotonically, and at high output levels inversely as the output power squared. The low-frequency fluctuations differ in that the relative fluctuations increase with output below threshold (noise amplifier region) but decrease sharply above threshold (dynamic saturation region). Curves typical of gas, solid, and diode lasers are illustrated. The theory agrees well with preliminary experimental results.
The quantum noise properties of atomic and semiconductor lasers are compared. The intensity fluctuation spectrum of the output field is calculated for the case of regularly, i.e., sub-Poissonian, pumped lasers. The effects of spontaneous emission on the output amplitude fluctuations may be suppressed in regularly pumped atomic lasers by requiring the lifetime of the lower lasing level to be short in comparison with that of the upper laser level. The output amplitude fluctuations then tend to zero at the laser frequency. In regularly pumped semiconductor lasers, net stimulated emission of photons dominates over spontaneous emission because of the combination of pumping and rapid intraband thermalization due to Coulomb scattering.
The basic equations governing noise phenomena are derived from first principles and applied to examples in optical communications. Quantum noise arises from two sources, the momentum fluctuations of electrons at optical frequencies and the uncertainty-related fluctuations of the electromagnetic field. Shot noise results from the beating of the noise sources with the signal field. In high-gain amplifiers, the spontaneous-emission noise dominates shot noise and results in a noise figure of at least 3 dB. It is shown explicitly how, at high power, both the laser field and the laser noise source become classical. The increase in noise in lasers with open cavities is not due to enhanced spontaneous emission as once thought, but to single-pass amplification. The noise fields and spontaneous currents have Gaussian distributions, while nonlasing modes have exponential photon-number distributions. Low-frequency intensity fluctuations arise from the electric current driving the laser and can be sub-Poissonian, in contrast to shot noise, which has a Poissonian distribution. The calculational tools are a wave equation for the field operator and a rate equation for the carrier-number operator, each containing spontaneous current noise sources. The correlation functions of these sources are determined by the fluctuation-dissipation theorem. [S0034-6861(96)00503-X]
The intensity fluctuations of a three-level laser are known to drop below shot noise in the presence of depletion of the ground state. We study the fluctuations of the laser output as a function of the parameter $\ensuremath{\varepsilon},$ defined as the fraction of atoms needed for the laser to operate. For a sufficiently small number of active atoms, the value of $\ensuremath{\varepsilon},$ and thereby the ground-state depletion, can be appreciable even for modest pumping. This suggests that the intensity fluctuations in the laser output would decrease below shot noise as the number of active atoms is reduced. A microscopic approach that uses a quantum-trajectory method and a macroscopic approach using semiclassical rate equations both show, however, that rather than decreasing, the intensity fluctuations actually increase with $\ensuremath{\varepsilon}.$ We find that the fluctuations of the output are determined by the dependence of the cycle time of the atoms on the number of photons in the laser mode.
The amplitude and phase noise spectra of the internal and external fields of a laser oscillator are calculated using the quantum mechanical Langevin equations. The outgoing field spectra are demonstrated to be different from the internal field spectra. The outgoing photon flux\hat{N}from a laser oscillator operating at far above the threshold has a spectral density equal to the classical shot noise level,2\langle\hat{N}\rangle. On the other hand, the internal photon number spectrum is Lorentzian, and the variance is equal to the average photon number,\langle\hat{N}\rangle. The difference stems from the quantum mechanical interference between the transmitted internal field and the reflected zero-point fluctuation at the output coupling mirror. The relation between the present analysis and the previous noise theory of a laser is also discussed.
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.