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Applied Physics B

The influence of resonator outcoupling and losses on the photon statistics of the LASER

Marc Eichhorn

Published Sep 8, 2022• 1 citation
Publisher

Abstract

Abstract An extension to the quantum-mechanical laser master equation for the density operator is derived to incorporate the beam-splitter effect caused by typical dielectric laser output couplers. This effect gives rise to a significant change in the photon statistical distribution of the part of the laser light reflected back into the resonator and, therefore, may have an influence on the total laser output photon statistics. Different cases without and with additional intra-cavity losses were discussed and their influence on the expected laser photon statistics was deduced. As a result, it was found that the well-known Poisson distribution of laser light is in most cases the result of additional losses or absorption, which act uncorrelatedly on single photons. In a laser with negligible additional losses where outcoupling is dominated by the beam-splitter effect, the photon statistics reveal to be mainly non-Poisson. A Poisson distribution would only occur for very low outcoupling rates, i.e., high finesse cavities. It is found that in the limit of strong outcoupling even the distribution of thermal light can result.

DOI
10.1007/s00340-022-07897-6

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Abstract An extension to the quantum-mechanical laser master equation for the density operator is derived to incorporate the beam-splitter effect caused by typical dielectric laser output couplers. This effect gives rise to a significant change in the photon statistical distribution of the part of the laser light reflected back into the resonator and, therefore, may have an influence on the total laser output photon statistics. Different cases without and with additional intra-cavity losses were discussed and their influence on the expected laser photon statistics was deduced. As a result, it was found that the well-known Poisson distribution of laser light is in most cases the result of additional losses or absorption, which act uncorrelatedly on single photons. In a laser with negligible additional losses where outcoupling is dominated by the beam-splitter effect, the photon statistics reveal to be mainly non-Poisson. A Poisson distribution would only occur for very low outcoupling rates, i.e., high finesse cavities. It is found that in the limit of strong outcoupling even the distribution of thermal light can result.
So far quantum-mechanical laser theories used approaches based on different but equivalent formulations either relying on a density matrix description or on the Fokker-Planck equation [1], [2]. In all these descriptions an important approximation and assumption have been used since: The outcoupling of photons out of the laser resonator was modeled by a loss term in analogy to what an equivalent intra-cavity absorption of photons would cause to the laser quantum state, i.e., an uncorrelated loss of single photons, thereby incorporating the finite cavity Q-factor into the description of the laser. Although this procedure is consistent in terms of energy and average photon number it lacks the fact that a typical dielectric laser output mirror is not just a uncorrelated photon extracting device like an absorbing or scattering object. As the output coupler is a beam splitter it has a complex action on the incoming intra-cavity photon state. The photon state, therefore, reflected back into the laser resonator is not just shifted to a lower average photon number but is a rather complex combination of various number states that are binomially distributed. The goal of this paper is to investigate the consequences of these findings on the laser photon statistics of continuous-wave lasers, now described by the new master equation including the beam-splitter effect [3]. Therefrom, the detailed balance of probability fluxes (Fig. 1) obtains as and the results with the common laser theory based on absorption-like outcoupling used so far is compared.
The interaction is considered between NNN stationary two-level atoms and a radiation field described by a single cavity mode. The state vector for the complete system of atoms plus radiation is expressed as a linear superposition of states constructed from a product of photon states in the nnn representation and products of Pauli spin eigenstates describing all combinations of atoms in the lower and upper energy levels. Equations for the corresponding probability amplitudes are derived by substituting this superposition into the Schr\"odinger equation. The resulting equations are combined into bilinear form and phenomenological damping contributions are added. After the neglect of certain of the bilinear quantities, a master equation is derived which governs the probability Pmn{{P}_{m}}^{n}Pm​n of having mmm atoms in the lower level and nnn photons in the mode. This master equation takes account of multiple single-quantum absorption and emission processes but not of simultaneous multiple processes involving two or more atoms at a time. The equation which governs the expected number of photons 〈n〉〈n〉〈n〉 derived from the master equation bears a close resemblance to a rate equation. The effect of radiation loss from the cavity is incorporated into the master equation. Numerical calculations for a QQQ-spoiled laser show that the statistics of the number of photons in the mode bear a qualitative resemblance to Poisson statistics.
The statistics of the beam of excited atoms that pumps a micromaser influence the photon number distribution of the cavity field. Of the cases studied so far, regular injection statistics minimize the number fluctuations and Poisson statistics maximize them. Here we extend these studies in two directions. Previous master equations describing sub-Poissonian injection statistics were derived by using standard reservoir theory. Recently, corrections to these equations have been found that involve the commutator of the gain and loss operators. Describing the system by a discrete map, we can avoid the approximations inherent in reservoir theory. The results are compared to those of the master equation. Differences are found to be small for large photon number but increasingly important for decreasing photon number and pump beam intensity, indicating the breakdown of reservoir theory in this regime. We give an approximate expression for the photon number distribution for weak damping and discuss its connection with semiclassical theory. Finally, we generalize our treatment for super-Poissonian pump fluctuations.
The photon counting statistics of the fluorescence of a single atom driven by a resonant coherent field has been calculated and the results checked experimentally. The authors examine the effect of detuning the coherent field and/or the effect of an additional thermal field on the photon counting statistics. They work with a master equation that they have used previously and derive exact results for the probability distribution function P(n, tau ) for the emitted photons and for the normalised second factorial moment Q. The subPoissonian character of the photon statistics, for which Q<0, is most pronounced for weak thermal fields or weak detuned coherent fields. For short counting times tau the statistic is dominated by any spontaneous emission that occurs, as expected. Increasing numbers of black-body photons tend to increase the Q values, as might be expected-especially in the case of an initially excited atom-but no simple rule emerges. The recent experiments in the microwave region on Rydberg atoms interacting with thermal radiation in (low-Q) cavities offers the possibility of observing thermal-field effects on the photon counting statistics.
A quantum statistical analysis of a two-photon laser is presented in a generalization of the one-photon laser. A two-photon loss mechanism is assumed by taking detailed-balance considerations into account. The expressions for the photon statistical distribution exhibit the exact influence of the Doppler parameter and other relaxation parameters. In addition, the photon statistics can be studied as a function of the detuning. Threshold conditions for the laser action are noticed. The peak photon number and half-width of the photon distributions are obtained. It is found that the photon distribution can be written as a ratio of the one-photon laser-photon distribution to the Poisson distribution for photons.
We derive the photon-number probability distribution and the resulting degree of second-order optical coherence for light emission from a uniformly distributed active species within a multiple-light-scattering medium. This is obtained from a master equation describing the probability distribution for photons in the vicinity of position r, traveling with a wave vector k, related, in turn, to a coarse-grained average of the optical Wigner coherence function. Using a simple model for isotropic, spatially uncorrelated scatterers, this reduces to a generalization of the master equation of a conventional laser in which the medium behaves like a random collection of low-quality factor cavities that are coupled by photon diffusion between a given cavity and its neighbors. Laserlike coherence, on average, is obtained in the random laser above a specific pumping threshold. Photon-number statistics above and below the lasing threshold are computed by first assuming that the atomic response to the local electromagnetic fields is nearly instantaneous. Corrections to this simple model, arising from nonadiabatic atomic dynamics, are then estimated. The dependence of the photon statistics on scatterer density, gain concentration, and position within a sample reveal that, on average, increase of the scattering strength (decrease of the photon transport mean free path) in the medium leads to a sharper peak in the local photon-number distribution, characteristic of increased local coherence in the optical field. We also evaluate the coherence of the output field at points outside the random-laser medium. This is a weighted average of radiation emitted at different positions in the sample, exhibiting varying degrees of coherence due to variations in the local pumping intensity.