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Physical Review A

Linewidth of four-level microcavity lasers

Leno M. Pedrotti, Michael Sokol, Perry R. Rice

Published Mar 1, 1999• 10 citations
Publisher

Abstract

We use standard quantum laser theory to examine the linewidth of lasers with a four-level atomic gain medium as a function of the spontaneous emission rate into the lasing mode and the decay rates from the upper and lower lasing levels. We find that the laser linewidth ceases to exhibit the customary decrease with increased photon number when a critical photon number, proportional to the ratio of the decay rate from the lower lasing level to the rate of spontaneous emission into the lasing mode, is reached. This critical photon number would decrease as the size of the optical cavity is decreased. In addition we show that, in general, the laser linewidth depends critically on the ratio of the overall decay rates from the upper and lower lasing levels.

DOI
10.1103/physreva.59.2295

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We use standard quantum laser theory to examine the linewidth of lasers with a four-level atomic gain medium as a function of the spontaneous emission rate into the lasing mode and the decay rates from the upper and lower lasing levels. We find that the laser linewidth ceases to exhibit the customary decrease with increased photon number when a critical photon number, proportional to the ratio of the decay rate from the lower lasing level to the rate of spontaneous emission into the lasing mode, is reached. This critical photon number would decrease as the size of the optical cavity is decreased. In addition we show that, in general, the laser linewidth depends critically on the ratio of the overall decay rates from the upper and lower lasing levels.
We review the physics underlying the process of spontaneous emission, with a special focus on spontaneous emission into a resonator mode. We define the mode volume, verify the fundamental modal dimensions, present the spectral mode profile, the coherence time, the Q-factor, the Füchtbauer-Ladenburg equation, and the Purcell factor, and discuss their influence on different types of lasers. We obtain the relation between peak emission cross section, radiative lifetime, and emission linewidth. By interpreting spontaneous emission as stimulated emission driven by vacuum fluctuations, we derive the spontaneous-emission rate into a resonator mode and establish physical expressions for the fractions of spontaneous emission and total decay from the upper laser level into this mode. Furthermore, we discuss coupling of the atomic system with the coherent field inside a lasing resonator mode, resulting in the formation of a Mollow triplet, and demonstrate that it leads to a reduction of the spontaneous-emission rate by a factor of 2.
Essentially closed-form formulae are given for the linewidth of inhomogeneously broadened lasers, with arbitrary cavity detuning and pumping statistics (Poissonian or quiet pumps). The so-called `bad-cavity' mode of operation is treated. The general formula is obtained from a semiclassical theory first derived from a linearized symmetrically ordered quantum theory in the limit of a large number of atoms. This semiclassical theory is most easily explained in terms of independent quantum jumps performed by emitting and absorbing atoms. The effect of inhomogeneous broadening on linewidth is found to be negligible for some pumping schemes, but when the atoms are independently pumped the linewidth is proportional (at some constant power level) to the gain-medium spectral width. We consider, in particular, an intermediate situation in which the rate at which atoms are promoted from the lower to the upper level (pump rate) is proportional to the lower-state population for each atom class. For that particular model, pump fluctuations do not affect the linewidth, even when the cavity is detuned from the centre of the atomic transition frequency distribution.
We calculate the output spectrum of a single-atom laser in a microcavity across a wide range of operating conditions. We considered both three-level and four-level atomic level structures. We used a numerical routine to calculate spectra that is more efficient than others used previously. We found that the linewidth of a single-atom laser generally scales as the inverse of the photon number and that there is no pump value at which an abrupt change occurs that might locate a lasing threshold. For a three-level gain atom we found vacuum–Rabi splitting similar to that found by Loffler et al. [Phys. Rev. A 55, 3923 (1997)] and used quantum trajectory theory to obtain a new interpretation of the results. For a four-level gain atom the vacuum–Rabi structure can appear at a small nonzero pump level and is maintained for large pumps, even when the intracavity photon number is larger than unity and the laser is on. We use the quantum trajectory approach to explain these results.
We study degenerate two-photon micromasers pumped by cascade three-level atoms, where each atomic energy level has a different linewidth. Our model allows us to recover the ordinary two-photon micromaser situation when these widths vanish. In the limit of large linewidths we obtain a two-photon laser model. We show that trapping states are extremely sensitive to atomic decay, even when the decay time is small in comparison to the transit time through the cavity. Unlike the one-photon micromaser, they are sensitive to the linewidth of the upper, as well as the lower energy level. We have also studied the effects of atomic decay on the starting time of micromasers and on the noise. We found that there is a significant difference between the behavior of two-photon micromasers and lasers as we change the ratio of the widths of the upper and lower levels.
The theory of a laser system with a ladder-type configuration is studied in detail based on the quantum Langevin approach. By using an external field to link the lower lasing level with another atomic level, whose decay rate is much larger, laser intensity significantly increases and the quantum-limited linewidth can be quenched. We also discuss the spectrum of fluctuations of the output field, and the result shows that the fluctuations at low frequencies can be much suppressed too. On the other hand, this quenching approach can realize a laser output between two atomic levels, whose decay rates do not satisfy the usual lasing condition that the decay rate of the lower lasing level should be larger than that of the upper lasing level. It will be very useful to realize a laser output with the wavelength we want. This quenching approach has been widely used in the absorption spectrum of the ytterbium optical lattice clock and in the laser cooling approach for calcium atoms. Here we apply it in the stimulated emission of lasers.
Usually, the cavity is considered an intrinsic part of laser design to enable coherent emission. For different types of cavities, it is assumed that the light coherence is achieved by different ways. We show that regardless of the type of cavity, the lasing condition is universal and is determined by the ratio of the width of the atomic spectrum to the product of the number of atoms and the spontaneous radiation rate in the laser structure. We demonstrate that cavity does not play a crucial role in lasing since it merely decreases the threshold by increasing the photon emission rate thanks to the Purcell effect. A threshold reduction can be achieved in a cavity-free structure by tuning the local density of states of the electromagnetic field. This paves the way for the design of laser devices based on cavity-free systems.