On the time-domain response of Cole - Cole dielectrics

We investigate time-domain electromagnetic pulse propagation in a dispersive lossy dielectric half-space whose properties are described in the frequency-domain by the Cole - Cole model ε(ω) = ε _∞ + (ε_s - ε _∞)/(1 + (iωτ)α), 0 < α < 1. With asymptotic techniques we calculate the small-depth impulse response and determine it is infinitely smooth at the wavefront. This result contrasts the case of the Debye medium (α = l) in which the wavefront supports discontinuities that decay exponentially with depth. Then, with asymptotic and numerical methods we investigate the large-depth impulse response. We find that while the saddle-point method accurately predicts the space-time location of the peak of the response it is of limited applicability in the approximation of such response for times past the arrival of the peak. Significantly, we find the peak of the response for 0 < α < 1 arrives earlier than in the case of α = 1. Our asymptotic results are validated with independent results obtained numericall.