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Physical Review A

Dynamics of laser buildup from quantum noise

F. T. Arecchi, W. Gadomski, R. Meucci, José Antonio Roversi

Published Apr 1, 1989• 55 citations
Publisher

Abstract

We measure the statistical distribution of buildup times for the field of a single-mode ${\mathrm{CO}}_{2}$ laser when cavity losses have rapidly changed from a large to a small value. The ratio between the average buildup time and its statistical spread is shown to be a sensitive test of the quantum fluctuations in the off state. This method operates as a ``statistical microscope,'' permitting the accurate determination of a small mean photon number in the infrared region by optical amplification, and a discrimination of the initial mean photon number from noise contributions along the amplification process. Following the buildup, the laser pulse decays on two time scales. The faster one, due to the radiation coupling, can be fitted by a Toda oscillator model; the slower one, due to the collisional dynamics of the excited population, provides a quantitative evaluation of two distinct population decay rates, one within the rotational manifold and the other one out of the vibrational band.

DOI
10.1103/physreva.39.4004

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We measure the statistical distribution of buildup times for the field of a single-mode CO2{\mathrm{CO}}_{2}CO2​ laser when cavity losses have rapidly changed from a large to a small value. The ratio between the average buildup time and its statistical spread is shown to be a sensitive test of the quantum fluctuations in the off state. This method operates as a ``statistical microscope,'' permitting the accurate determination of a small mean photon number in the infrared region by optical amplification, and a discrimination of the initial mean photon number from noise contributions along the amplification process. Following the buildup, the laser pulse decays on two time scales. The faster one, due to the radiation coupling, can be fitted by a Toda oscillator model; the slower one, due to the collisional dynamics of the excited population, provides a quantitative evaluation of two distinct population decay rates, one within the rotational manifold and the other one out of the vibrational band.
We introduce a "statistical microscope" which allows the accurate determination of a small number of infrared photons inside a laser cavity by optical (rather than electron) multiplication. For this purpose, we induce a loss switch in a CO 2 laser and then measure the statistical distribution of passage times through a threshold value within the linear regime. Our approach is model independent, since the measured cumulants provide both average values and associated error bars. Furthermore, our method discriminates the initial photon number from noise contributions along the amplifying path, thus it does not depend on the statistical features of the amplifying process.
The transient statistical properties of a single-mode laser field are investigated both experimentally and theoretically. The physical system is a QQQ-switched He-Ne gas laser oscillating on the 6328-\AA{} transition. The measurements are performed by photon-counting techniques, using the so-called linear method which enables resolving times to be in the nsec region. The effects of the photomultiplier and of the attenuators on the statistics are taken into account. The measurements are interpreted in terms of the available laser theories as well as in terms of a simple phenomenological model which attributes the statistical spread to the initial photon distribution, neglecting the stochastic forces in the course of the buildup. Finally, some of the approximations of the theories are discussed, and the sensitivity of a transient measurement is compared with that of a stationary-intensity-correlation measurement.
A short-cavity dye laser has been constructed to study fluctuations arising from quantum noise and mode coupling in multimode systems. Typical dye lasers (i.e., long-cavity) have been found to have cw dynamics that are dominated by coherent wave mixing effects.1 The quantum noise in these lasers also manifests itself as a variance in the buildup times of the laser light intensity. Buildup times of singlemode long-cavity lasers have been studied.2 The relative importance of quantum noise is enhanced in short cavities making the turn-on transient the dominant feature in the laser light fluctuations. We have measured the turn-on transient of individual modes from a multimode short-cavity dye laser following sudden turn-on of the pump power. The shortness of the cavity has the effect of making the decay times of the intracavity field and the molecular population similar. The observed buildup and rise times are compared with those predicted by a semiclassical multimode laser theory with stochastic noise added for several pump rates. Moreover, semiregular oscillations of the total intensity have been observed to occur in short bursts separating intervals of nearly constant cw laser output.
In this chapter, laser rate equations were used to derive another set of simpler rate equations that couple cavity photon number and the population of excited active atoms as the dynamical variables describing the laser. These equations were used to explore the buildup of laser oscillations from noise in a laser cavity and the remarkable growth of cavity photon number as the laser crosses threshold of oscillation. These modified rate equations were used to derive a Master equation for the cavity photon number probabilities. The steady-state solution of this equation was used to discuss equally remarkable transformation of photon statistics of laser light near threshold. The effect of spontaneous emission on setting the ultimate limit on laser linewidth was also discussed.
Buildup of laser oscillation is expressed in terms of the probabilities p(n)(t) that there are n photons in the laser cavity. These probabilities obey a coupled set of difference-differential equations of motion. The method and results of numerical integration of these equations are given here for the initial condition that no photons exist (radiation field vacuum). Corresponding equations of motion for the moments of the photon distribution are derived, and their time dependences are given. A movie showing the time development of the p(n) from an initial vacuum is given on the corners of the November 1970 issue of Applied Optics and can be seen by flipping through the pages.
The description of the statistical dynamics of quantum oscillators is formulated in terms of the Wigner distribution, analogous to the more commonly used PPP distribution, with explicit formulas being obtained for its time evolution and for average values. This formulation is desirable because, e.g., the Wigner distribution always exists whereas the PPP distribution does not. The formalism is applied to the process of parametric amplification in a single mode, which may be considered as the degenerate form of the well-known two-mode case. This degeneracy gives rise to significantly different properties; for example, the PPP distribution for the single mode of interest evolves from a circularly symmetric two-dimensional Gaussian into an elliptically symmetric form and ceases to exist after a finite time, even for amplification in the presence of losses. This is contrary to the two-mode case. The corresponding Wigner distribution is found to exist as a well-behaved function for all time as expected, regardless of the amount of losses, and is used to calculate average values of various quantities of interest. It is found, e.g., that in the lossless case the average number of photons in the signal mode always becomes infinite as $t\ensuremath{\rightarrow}\ensuremath{\infty}$. This is in contrast to the corresponding classical result for the lossless case which allows the signal to decay rather than to grow with time, depending on the relative phase between the signal and the pump. Field fluctuations are discussed and found to have some unusual properties. The combination of frequency up-conversion with single-mode amplification is also described briefly. The effect of the quantization of the pump oscillator is considered in an Appendix.