Derivation of statistics of real delay time from statistics of imaginary delay time using spectroscopic technique in weakly disordered optical media

Delay time is defined as a time that a wave spent in a scattering medium before it escapes, and this can be derived by the energy derivative of the phase of the scattering wave. Considering the complex reflection amplitude R=|R|exp(i{\theta}) of a light wave, real delay time {\tau}_r (i.e., sojourn or Wigner delay time), which is the energy derivative of the real phase ({\tau}_r =d{\theta}/cdk), and complex delay time {\tau}_i , which is the energy derivative of the reflection coefficient ({\tau}_i=d{\theta}_i/cdk, |R|=r^1/2=exp(-{\theta}_i)), have the same statistical form and a mirror image with a shift in time in weak disorder and short length regime. Real delay time statistics obtained from the reflection coefficient can be attributed to the strong correlation between the reflection coefficient and its phase in this regime. The range of the scattering parameters for this validation, as well as application of this method in characterization of weakly disordered media is also discussed.