Introduction

At first sight, it may very well seem that the two subjects linked in the title have little in common. What do semiconductor lasers have to do with behavior normally associated with fully developed hydrodynamic turbulence? In order to make the connection, we begin by reviewing the salient features of semiconductor lasers. In many ways, they are like two level lasers in that the coherent light output is associated with the in phase transitions of an electron from a higher to lower energy state. In semiconductors, the lower energy state is the valence band from which sea electrons are removed leaving behind positively charged holes. The higher energy state is the conduction band. The quantum of energy released corresponds to an excited electron in the conduction band combining with a hole in the lower band below the bandgap. Bandgaps, or forbidden energy zones are features of the energy spectrum of an electron in periodic potentials introduced in this case by the periodic nature of the semiconductor lattice.

However, there are two important ways in which the semiconductor laser differs from and is more complicated than the traditional two-level laser model. First, there is a continuum of bandgaps parameterized by the electron momentum ${\bf k}$ and the laser output is a weighted sum of contributions from polarizations corresponding to electron-hole pairs at each momentum value. In this feature, the semiconductor laser resembles an inhomogeneosly broadened two level laser. Second, electrons and holes interact with each other via Coulomb forces. Although this interaction is screened by the presence of many electrons and holes, it is nonetheless sufficiently strong to lead to a nonlinear coupling between electrons and holes at different momenta. The net effect of these collisions is a redistribution of carriers (the common name for both electrons and holes) across the momentum spectrum. In fact it is the fastest ($\approx 100$ fs.) process (for electric field pulses of duration greater than picoseconds) and because of this, the gas of carriers essentially relaxes to a distribution corresponding to an equilibrium of this collision process. This equilibrium state is commonly taken to be that of thermodynamic equilibrium for fermion gases, the Fermi-Dirac distribution characterized by two parameters, the chemical potential $\mu$ and temperature $T$, slightly modified by the presence of broadband pumping and damping.

But the Fermi-Dirac distribution is not the only equilibrium of the collision process. There are other stationary solutions, called finite flux equilibria, for which there is a finite and constant flux of carriers and energy across a given spectral window. The Fermi-Dirac solution has zero flux of both quantities. It is the aim of this letter to suggest that these finite flux equilibria are more relevant to situations in which energy and carriers are added in one region of the spectrum, redistributed via collision processes to another region where they are absorbed. Moreover, it may be advantageous to pump the laser in this way because such a strategy may partially overcome the deleterious effects of Pauli blocking. The Pauli exclusion principle means that two electrons with the same energy and spin cannot occupy the same state at a given momentum. This leads to inefficiency because the pumping is effectively multiplied by a factor $(1-n_s({\bf k})), s=e,h$ for electrons and holes respectively, denoting the probability of not finding electron (hole) in a certain ${\bf k}$ (used to denote both momentum and spin) state. But, near the momentum value corresponding to the lasing frequency $\omega _L$, $n_s({\bf k})$ is large ( $n_e({\bf k})+n_h({\bf k})$ must exceed unity) and Pauli blocking significant. Therefore, pumping the laser in a window about $\omega _0>\omega _L$ in such a way that one balances the savings gained by lessening the Pauli blocking (because the carriers density $n_s({\bf k})$ decreases with $k=\vert{\bf k}\vert$) with the extra input energy required (because $k$ is larger), and then using the finite flux solution to transport carriers (and energy) back to lasing frequency, seems an option worth considering. The aim of this letter is to demonstrate, using the simplest possible model, that this alternative is viable. More detailed results using more sophisticated (but far more complicated) models will be given later.

These finite flux equilibria are the analogies of the Kolmogorov spectra associated with fully developed, high Reynolds number hydrodynamic turbulence and the wave turbulence of surface gravity waves on the sea. In the former context, energy is essentially added at large scales (by stirring or some instability mechanism), is dissipated at small (Kolmogorov and smaller) scales of the order of less than the inverse three quarter power of the Reynolds number. It cascades via nonlinear interactions from the large scales to the small scales through a window of transparency (the inertial range in which neither forcing nor damping is important) by the constant energy flux Kolmogorov solution. Indeed, for hydrodynamic turbulence, the analogue to the Fermi-Dirac distribution, the Rayleigh-Jeans spectrum of equipatitions, is irrelevant altogether. The weak turbulence of surface gravity waves is the classical analogue of the case of weakly interacting fermions. The mechanism for energy and carrier density (particle number) transfer is "energy" and "momentum" conserving binary collisions satisfying the "four wave resonance" conditions

$${\bf k}+{\bf k_1}={\bf k_2}+{\bf k_3}, \ \ \ \ \omega ({\bf k})+\omega ({\bf k_1})=\omega ({\bf k_2})+\omega ({\bf k_3}).$$ (1)

In the semiconductor context, $\hbar \omega ({\bf k})=\hbar \omega _{\rm gap} +\epsilon _e({\bf k})+\epsilon _h({\bf k})$ (which can be well approximated by $\alpha +\beta k^2$) where $\hbar \omega _{\rm gap}=\epsilon _{\rm gap}$ corresponds to the minimum bandgap and $\epsilon _e({\bf k})$, $\epsilon _h({\bf k})$ are electron and hole energies. In each case, there is also a simple relation $E({\bf k})=\omega n({\bf k})$ between the spectral energy density $E({\bf k})$ and carrier (particle number) density $n({\bf k})$. As a consequence of conservation of both energy and carriers, it can be argued (schematically shown in Figure 1 and described in its caption), that the flux energy (and some carriers) from intermediate momentum scales (around $k_0$ say) at which it is injected, to higher momenta (where it is converted into heat) must be accompanied by the flux of carriers and some energy from $k_0$ to lower momenta at which it will be absorbed by the laser. It is the latter solution that we plan to exploit.