Effect of atomic line shape on spatial coherence in mirrorless lasers.

Abstract

The statistical properties of the mirrorless-laser output radiation field, E(r,t), and its spectrum, E(r,\ensuremath{\nu}), are studied. The study is based on the analysis of the cross-spectral density integrated over a band of frequencies of width B; ${\mathit{M}}^{\mathit{B}}$(r,r,'\ensuremath{\nu})\ensuremath{\equiv}${\mathcal{F}}_{\ensuremath{\nu}\mathrm{\ensuremath{-}}\mathit{B}/2}^{\ensuremath{\nu}+\mathit{B}/2}$d\ensuremath{\nu}'' ${\mathcal{F}}_{\ensuremath{\nu}\mathrm{\ensuremath{-}}\mathit{B}/2}^{\ensuremath{\nu}+\mathit{B}/2}$d\ensuremath{\nu}'〈E\ifmmode \tilde{}\else \~{}\fi{}(r,\ensuremath{\nu}')E${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}^{\mathrm{*}}$ and of the power spectrum ${\mathrm{\ensuremath{\Lambda}}}^{\mathit{B}}$(r,\ensuremath{\nu})\ensuremath{\equiv}${\mathit{M}}^{\mathit{B}}$(r,r,\ensuremath{\nu}). The relation between the spectral properties and the spatial correlations is characterized by the ratio between measurements in a narrow band near the atomic line center and measurements in a band of frequencies much larger than the atomic linewidth, namely by the ratios a(r)\ensuremath{\equiv}${\mathrm{\ensuremath{\Lambda}}}_{1}^{\mathit{B}}$\ensuremath{\ll}\ensuremath{\Gamma}(r,0)/${\mathrm{\ensuremath{\Lambda}}}_{2}^{\mathit{B}}$\ensuremath{\gg}\ensuremath{\Gamma}(r,0) and b(r,r')\ensuremath{\equiv}${\mathit{M}}_{1}^{\mathit{B}}$\ensuremath{\ll}\ensuremath{\Gamma}(r,r',0)/${\mathit{M}}_{2}^{\mathit{B}}$\ensuremath{\gg}\ensuremath{\Gamma}(r,r',0 ). It is shown that, for systems with small spatial variations in the atomic population-inversion density, both the ratios a and b are independent of the observation points r,r' and their value equals the ratio between the bandwidth ${\mathit{B}}_{1}$ and the gain-narrowed atomic linewidth. For systems with a strong spatial variation in the population inversion, it is shown that the ratio b depends on the observation points r,r' and is larger than the ratio between the bandwidth ${\mathit{B}}_{1}$ and the gain-narrowed atomic linewidth. This effect is demonstrated by a detailed numerical calculation for a mirrorless laser with a Lorenzian atomic line shape.