We present a family of circular Bessel probability density functions that are capable of describing the intensity, amplitude, and field statistics of waves in any random medium, with only the assumption of circularity. The well-known zero-mean circular Gaussian statistics break down in the Anderson localization and the weakly scattering regimes, where the field can no longer be regarded as the sum of a multitude of independent random phasors. We find that in such scenarios circular Bessel statistics apply because the field can be modeled as a random phasor sum with a random number of contributing phasors. The validity of our density functions is verified through numerical simulations of electromagnetic waves propagating in 2D random media. Having a set of density functions that work in all scattering regimes provides a framework for modeling wave propagation in random media, facilitating random media characterization, imaging in and through scatter, and random laser design.
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