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Optics Communications

Laser linewidth and operator orderings

Ning Lü

Published Nov 1, 1990• 6 citations
Publisher

Abstract

We demonstrate explicitly that, after including the saturation effects of a laser diffusion coefficients,(i) the phase diffusion coefficients given by the normal-ordering P and the antinormal-ordering Q functions become equal, implying that the laser linewidth is independent of the operator orderings of different quasiprobability functions; and (ii) the intensity diffusion coefficients given by the two functions are still different, but leading to the same photon-number variance.

DOI
10.1016/0030-4018(90)90487-e

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We demonstrate explicitly that, after including the saturation effects of a laser diffusion coefficients,(i) the phase diffusion coefficients given by the normal-ordering P and the antinormal-ordering Q functions become equal, implying that the laser linewidth is independent of the operator orderings of different quasiprobability functions; and (ii) the intensity diffusion coefficients given by the two functions are still different, but leading to the same photon-number variance.
We study the linewidth of a one-photon laser through the approach of the two-time correlation function. Our numerical calculations cover the regions of below, near, and above threshold. Well below threshold, the laser linewidth decreases linearly with increasing linear gain. Near and above threshold, in comparison with the approximate linewidth given by the phase-diffusion coefficient at the mean photon number (which is inversely proportional to the laser intensity---the Schawlow-Townes formula), we find that the laser intensity fluctuations increase the laser linewidth and that the Schawlow-Townes formula is valid only when the mean photon number is large and the photon-number variance is much smaller than the square of the mean photon number. Near threshold, the laser linewidth can be twice as large as that given by the phase-diffusion coefficient at the mean photon number (i.e., twice as large as that given by the Schawlow-Townes formula).
Starting from the field master equation for a laser oscillating simultaneously in two modes, we probe the dependence of its fundamental phase-diffusion linewidth on the degree of coupling between the two modes. We find that the heterodyned intrinsic linewidth shows the usual decrease when the output power of each mode increases with an increase in the gain, while the nonlinear self-saturation and cross-coupling coefficients are held constant. The linewidth also decreases with increase in the cross-coupling between the modes for constant gain when the output power of each mode is kept constant by decreasing the self-saturation coefficient. The linewidth, however, increases with increasing cross-coupling when (a) the gain and the self-saturation coefficient are held constant and hence the output power in each mode decreases and (b) the gain is increased keeping the output power and the self-saturation of each mode constant.
A quantum theory of the spectral width of a laser with a saturable absorber is presented. We perform a density matrix calculation yielding the off-diagonal element pn,n+1(t ) which is proportional to the average value of the electric field (E(t )) associated with the laser field having a large photon number n. The general linewidth theory is applied to any general laser which has saturable absorber features for a portion of the building of the laser field. We have found that the saturable absorber action contribution to the linewidth of such a laser can be substantial.
In this chapter, we present a theory of the laser based on the Heisenberg–Langevin approach. This is a different, but completely equivalent approach to the density operator approach discussed in the previous chapter. In general, the density operator approach is better suited to study the photon statistics of the radiation field whereas the Heisenberg–Langevin approach has certain calculational advantages in the determination of phase diffusion coefficients, and consequently laser linewidth. In Section 12.1, a simple approach to determine laser linewidth based on a linear theory is presented. This analysis is especially interesting and useful in that it includes atomic memory effects, something that is difficult to do within a density matrix theory. In Sections 12.2–12.4, we consider the complete nonlinear theory of the laser and rederive all the important quantities related to the quantum statistical properties of the radiation field. A simple Langevin treatment of the laser linewidth including atomic memory effects† The full nonlinear quantum theory of the laser discussed in the previous chapter yields most of the interesting quantum statistical properties of the radiation field. In many problems of interest, however, we do not need such an elaborate treatment. For example, as we saw in the previous chapter, the natural linewidth of the laser can be determined from a linearized theory of the laser. That is, the full nonlinear theory serves to determine the amplitude of the field but the phase fluctuations about this operating point are described by a linear theory.
The problem of the quantum-limited or intrinsic linewidth of a good-cavity laser is revisited. Starting from the Scully-Lamb master equation, we present a fully analytical treatment to determine the correlation function and the spectrum of the cavity field at steady state. For this purpose, we develop an analytical approximation method that implicitly incorporates the microscopic fluctuations of both the phase and intensity of the field, and, in addition, takes full account of the saturation of the nonlinear gain. Our main result is a simple formula for the quantum-limited linewidth that is valid from near to far above threshold and also includes the presence of thermal photons. Close to the threshold, the linewidth is twice as large as predicted by the standard phase-diffusion treatment neglecting intensity fluctuations, and even 50% above threshold the increase is still considerable. In general, quantum fluctuations of the intensity are present and continue to influence the linewidth as long as the photon-number distribution is not strictly Poissonian. This inherent relationship is displayed by a formula relating the linewidth and the Mandel Q parameter. More than 100% above treshold the linewidth is found to be smaller than predicted by the standard treatment, since the simple phase-diffusion model increasingly overestimates the rate of phase fluctuations by neglecting gain saturation. In the limit of a very large mean photon number the expected perfectly coherent classical field is obtained.