Excess noise for coherent radiation propagating through amplifying random media

The coherent radiation emitted by a laser has a noise spectral density P equal to the time-averaged photocurrent Ī. This noise is called photon shot noise, by analogy with electronic shot noise in vacuum tubes. If the radiation is passed through an amplifying medium, P increases more than Ī because of the excess noise due to spontaneous emission [1]. For an ideal linear amplifier, the (squared) signal-to-noise ratio Ī/P drops by a factor of two as one increases the gain. One says that the amplifier has a noise figure of 2. This is a lower bound on the excess noise for a linear amplifier [2]. Most calculations of the excess noise assume that the amplification occurs in a single propagating mode. (Recent examples include work by Loudon and his group [3,4].) The minimal noise figure of 2 refers to this case. Generalisation to amplification in a multi-mode waveguide is straightforward if there is no scattering between the modes. The recent interest in amplifying random media [5] calls for an extension of the theory of excess noise to include inter-mode scattering. Here we present such an extension. Our central result is an expression for the probability distribution of the photocount in terms of the transmission and reflection matrices t and r of the multi-mode waveguide. (The noise power P is determined by the variance of this distribution.) Single-mode results in the literature are recovered for scalar t and r. In the absence of any incident radiation our expression reduces to the known photocount distribution for amplified spontaneous emission [6]. We find that inter-mode scattering strongly increases the excess noise, resulting in a noise figure that is much larger than 2. We present explicit calculations for two types of geometries, waveguide and cavity, distinguishing between photodetection in transmission and in reflection. We also discuss the parallel with absorbing media. We use the method of random-matrix theory [7] to obtain the required information on the statistical properties of the transmission and reflection matrices of an ensemble of random media. Simple analytical results follow if the number of modes N is large (i.e. for high-dimensional matrices). Close to the laser threshold, the noise figure F exhibits large sample-to-sample fluctuations, such that the ensemble average diverges. We compute for arbitrary N ≥ 2 the distribution p(F) of F in the ensemble of disordered cavities, and show that F = N is the most probable value. This is the generalisation to multi-mode random media of the single-mode result F = 2 in the literature.