Long-exposure imaging through weak turbulence

Effects of weak turbulence on images of coherent sources or coherently illuminated objects taken with exposure times much greater than the turbulence's time constant are examined using the extended Fresnel principle. Two cases are considered. One in which the turbulent medium fills the region between the imaging system and the object and the other in which the turbulence occurs as a phase screen directly before the object. Assuming the Kolmogoroff spectrum for the index of refraction fluctuations and specifying a modulated Gaussian form to describe the object, a closed form result is reached which illustrates the effect of the turbulence on the image in a conceptually simple manner. A loss of resolution is found, manifesting itself as an effective reduction of the lens size. A simple relation is derived that relates the effective lens size to the actual lens size and the coherence length ρ _o, of a spherical wave propagating through the turbulent medium. This relation agrees well with the empirically known fact that increasing the size of the primary lens of a telescope beyond approximately 10 cm does little to improve resolution. Turbulence is also found to cause a coherent object to seem incoherent from the standpoint of the viewer if the resolution spot size of the imaging system is larger than ρ _o. Approximations upon which these results are based are supported by numerical calculations for particular objects. Finally, the implications of these discoveries are compared with papers that demonstrate superresolution for short- and long-exposure imaging.