Pulse propagation in a resonant medium: Application to sound waves in superfluid 3He-B.

Abstract

We give a description of the propagation of pulsed signals in a resonant medium based both on a time-domain analytical solution of the problem and on experimental observations in superfluid $^{3}\mathrm{He}$. These signals propagate according to a wave equation involving the unperturbed velocity ${c}_{0}$ and coupled to an internal mode characterized by a resonance frequency ${\ensuremath{\omega}}_{m}$, an oscillator strength \ensuremath{\lambda}, and a lifetime \ensuremath{\tau}. This mode crossing situation may exhibit a region of anomalous dispersion for which d\ensuremath{\omega}/dkl0; this leads to well-known difficulties in the application of the concept of group velocity to wave packets with center frequency in the neighborhood of ${\ensuremath{\omega}}_{m}$. We tackle this problem as follows. Using the slowly varying envelope approximation, we transform the Fourier integral representing the general wave propagation solution at location x and retarded local time t'=t-x/${c}_{0}$ into a time-convolution integral for the wave envelope. We then split this time integral into a short-local-time part and a long-time part.Depending on circumstances, the short-time part is either the signal itself (far from ${\ensuremath{\omega}}_{m}$ or if it varies rapidly with respect to \ensuremath{\tau}) or a precursory motion which is akin to Sommerfeld's precursor and which has already been discussed from another point of view by Crisp and others. The envelope of these precursors behaves as (t'/\ensuremath{\Lambda}x${)}^{n/2}$${J}_{n}$(2 \ensuremath{\surd}\ensuremath{\Lambda}xt' ), with \ensuremath{\Lambda}=\ensuremath{\lambda}${\ensuremath{\omega}}_{0}^{2}$/2${c}_{0}$, n being the order of the first nonzero derivative of the envelope at the origin (x=0). As the flight path in the resonant medium increases, it decays as a power of ${x}^{\mathrm{\ensuremath{-}}1/2}$ only and its pseudofrequency \ensuremath{\Lambda}x/2\ensuremath{\pi} increases. This part of the response, which we call a resonant precursor of order n, arises from the free response of the Lorentz oscillators to their own field and decays in time as exp(-t'/\ensuremath{\tau}). Characteristic wiggling patterns of this kind have been observed in sound propagation measurements in superfluid $^{3}\mathrm{He}$, described below in detail, and, more recently on an optical system using ultrashort laser pulses by Rothenberg, Grischkowsky, and Balant.The long-time part of the convolution integral describes the gradual distortion of the envelope as it propagates. This leads to the concept of complex group velocity, already introduced by Johnson, and shows its general applicability: pulses smooth on time scale \ensuremath{\tau} propagate with the classical group velocity as long as causality is preserved. This result, already stated for Gaussian pulses by Garrett and McCumber, is compared critically to the early and conflicting predictions on signal propagation velocity made by Brillouin and by Baerwald. Thus, for signals which are not very short, the received envelope is made up of two parts, precursors, which travel with velocity ${c}_{0}$, and a delayed signal which is damped exponentially, and travels with the classical group velocity and, close to resonance, suffers distortion governed by the size of dA/dt with respect to A. Computer simulations illustrate these different concepts.