The near- and far-field photoluminescence spectra of a gated two-dimensionalelectron gas have been measured. Spatial fluctuations in the electron densityare found to be manifested as spatial fluctuations in the emission amplitude ofthe negatively charged exciton (X$^{-}$) and in the peak energy of the neutralexciton (X). Consequently, the far-field X$^{-}$ spectrum is a homogoneouslybroadened Lorentzian, while the X lineshape exhibits substantial inhomogenousbroadening. We present a novel and simple technique to extract the electrondensity from the X$^{-}$ spectrum. We show that width of the far-field X lineis proportional to the electron density fluctuations; hence it can be used tocharacterize the inhomogeneity in the electron density.
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