Noise in semiconductor lasers

Abstract
It has been demonstrated experimentally and theoretically from first principles that for semiconductor lasers the complex light-field amplitude obeys a Van der Pol equation with a noisy driving term. This noise source is mainly due to the fluctuations of the atomic dipole moments, while the population fluctuations do not strongly influence the noise properties of the light amplitude and phase. In laser junctions, however, the population fluctuations can, at least in principle, be measured via the current noise. Starting with the quantum mechanical Langevin equations for the operators of the light mode, the atomic dipole moments, and the population of an individualkstate, one can derive an equation of motion for total population of the conduction bandC. This equation can be split into a mean equation (rate equation) and an equation for the superimposed fluctuations. The rate equation is\dot{\bar{N}}_{C} = R_{P} - R_{SP} + \bar{n}E_{VC} - (\bar{n} + 1)E_{CV}, whereR_{P}, R_{SP}, are the rates of pump and spontaneous emission.E_CV(\bar{n} + 1)is the emission rate into the separately treated laser mode,\bar{n}E_{VC}is the corresponding absorption rate, and\bar{n}is the mean number of photons in the laser mode. Because of fast electron-electron relaxation one has an electron gas with a fluctuating quasi-Fermi level rather than electrons with individually fluctuating population. The equation of the fluctuating Fermi level can be solved below and above threshold. One gets the following result:\langle\deltaN^{2}\rangle = (T_{C}/2)(R_{P} + R_{SP} + (\bar{n} + 1) E_{CV} + \bar{n}E_{VC}) + k(\bar{n}). Far below threshold this expression is reduced to\langle\deltaN^{2}\rangle = (T_{C.0}/2)(R_{P} - R_{SP}), whereT_{C.0}^{-1} = (d/dN_{C})(R_{SP} - R_{P}). This result is known to be valid for degenerate semiconductors. The laser action shortens the relaxation time toT_{C}^{-1} = T_{C.0}^{-1} + (d/dN_{C}) (E_{CV} - E_{VC}). The transition rates due to the coherent light field contribute in the same way to the shot noise as the rates of pump and spontaneous emission,k(\bar{n})is due to the light-field fluctuations. The main contribution is proportional to the intensity noise of the laser light; an additional contribution has its origin in the correlation between the fluctutions in light field and in population. Below threshold the photons are in a Bose distribution, i.e.,\langle\Deltan^{2}\rangle = \bar{n}(\bar{n} + 1). Therefore below but near threshold the noise due to the laser photons increases quadratically and becomes constant above threshold, because the intensity noise of the light field itself becomes constant. The junction current noise spectrum at low frequencies is given byS_{I}(\omega \approx 0) = 2eI + 4e^{2}\bar{n}E_{VC} + 4e^{2}/T_{C}\cdot k(\bar{n}). The first term, besides the usual shot noise2eI, appears because the noise is given by the sum of emission and absorption rates, while the currentIis given by the difference. The term due to the light-field fluctuations leads to an increase of the noise quadratically inInear threshold and it yields an approximately constant amount of noise above threshold.

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