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By R et al Kingslake

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Also, find X(t - t') that relates the polarization P(t) to £(t'). (To handle the pole at w = 0, replace w by w + i71 and let 71- O. ) Suppose the electrons are subjected to a pulsed electric field, which vanishes as the time t - +00. Notice that J(t) vanishes as t - +00, but P(t) does not. (b) The electrons in this problem have an average velocity v described by dv e v -=-£(t)--. ' By directly integrating this equation, for general E(t), show that when the electrons respond to a field pulse, lim P(t) = neAx , ,-+00 where Ax is the displacement of the electron gas in response to the pulse.

An array of oscillators, each with a dipole moment with time dependence exp[ -i(wl - W2)t], will generate an electromagnetic wave at frequency (WI W2), as one may appreciate by inserting the nonlinear dipole moment density into the Maxwell equations; we shall see how this occurs in Chap. 4, for an important example. 21 e). Thus, we also get resonant enhancement if the frequency (WI - W2) lies close to the resonance frequency of the oscillator. 21 d) shows a resonance when (WI + (2) lies near Woo We have here an example of a three wave interaction: the wave of frequency WI "mixes" with that of frequency W2 to, among other things, produce a third output wave at (WI - W2)' If anyone of the waves, including the output wave, is in resonance with the physical system (the oscillator in this case) that provides the nonlinearity responsible for their interaction, we obtain resonance enhancement.

1 We have seen that free electrons are characterized by a frequency dependent conductivity a(w) ne 2 = -1/ m 'r 1 . -lW and they contribute a term to the frequency dependent susceptibility of the form ne 2 X(w) 1 = --;;; w(w + i/,r) . 34 2. Linear Dielectric Response of Matter (a) Find expressions for u(t - t'), the "real time conductivity" which relates the current density J(t) to the past behavior of the electric field £(t). Also, find X(t - t') that relates the polarization P(t) to £(t'). (To handle the pole at w = 0, replace w by w + i71 and let 71- O.

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