  When waves propagate through a complex medium like the turbulent atmosphere
the wave field becomes incoherent and the wave intensity forms a complex
speckle pattern. In this paper we study a speckle memory effect in the
frequency domain and some of its consequences. This effect means that certain
properties of the speckle pattern produced by wave transmission through a
randomly scattering medium is preserved when shifting the frequency of the
illumination. The speckle memory effect is characterized via a detailed novel
analysis of the fourth-order moment of the random paraxial Green's function at
four different frequencies. We arrive at a precise characterization of the
frequency memory effect and what governs the strength of the memory. As an
application we quantify the statistical stability of time-reversal wave
refocusing through a randomly scattering medium in the paraxial or beam regime.
Time reversal refers to the situation when a transmitted wave field is recorded
on a time-reversal mirror then time reversed and sent back into the complex
medium. The reemitted wave field then refocuses at the original source point.
We compute the mean of the refocused wave and identify a novel quantitative
description of its variance in terms of the radius of the time-reversal mirror,
the size of its elements, the source bandwidth and the statistics of the random
medium fluctuations.
