Photon distribution function for long-distance propagation of partially coherent beams through the turbulent atmosphere

The photon density operator function is used to calculate light beam propagation through turbulent atmosphere. A kinetic equation for the photon distribution function is derived and solved using the method of characteristics. Optical wave correlations are described in terms of photon trajectories that depend on fluctuations of the refractive index. It is shown that both linear and quadratic disturbances produce sizable effects for long-distance propagation. The quadratic terms are shown to suppress the correlation of waves with different wave vectors. We examine the intensity fluctuations of partially coherent beams (beams whose initial spatial coherence is partially destroyed). Our calculations show that it is possible to significantly reduce the intensity fluctuations by using a partially coherent beam. The physical mechanism responsible for this pronounced reduction is similar to that of the Hanbury-Braun–Twiss effect.

Figure 1

The dependence of beam radius on the coherence length parameter $(r_1∕r_0{)}^2$ for $q_0={10}^7{\mathrm{m}}^{-1},(l_0∕2\pi )={10}^{-3}\mathrm{m}$. There is a wide range of $(r_1∕r_0{)}^2$ for which the beam radius is almost constant.

Figure 2

Dependence of scintillation index on turbulence strength for weak fluctuations; $q_0$ and $l_0$ are as in Fig. 1. Decreasing of scintillation index with the decrease of source coherence is seen (similar to the results of Korotkova et al. [21]).

Figure 3

Plots ${\sigma }^2\left(C_n^2\right)$ for different initial coherence $(r_1∕r_0{)}^2$; $q_0$ and $l_0$ are as in Fig. 1. Left-side curves calculated with employing Eqs. (48, 49) exhibit negligible effect of partial coherence (merged curves), that is in contrast to results shown in Fig. 2. Two very different parameters ${\rho }_0^2$ are used in these cases. Black squares show ${\sigma }^2$ obtained with the assumption of straight trajectories in Eq. (36).

Figure 4

The same as in Fig. 3 but for longer distance. The upper curve corresponds to coherent beam. It approaches asymptotically to the value ${\sigma }^2=1$.