lecture16 - 2.57 Nano-to-Macro Transport Processes Fall...

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2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 16 Review of last lecture 2 Q 1->2 1 1 2 1 2 Q 2->1 z vEf τ 1 2 JJG 1 v x > 0 In Lecture 14, we have discussed the energy exchange between two points. We have spent time on calculating the transmissivity for cases in the right two figures. The 12 velocity v 1 is the group velocity, not the phase velocity determined by v = ω / k (derived x tk x .) from constant phase Φ= x Consider a plane traveling along x-direction A e i ( t kx ) . The velocity v = / k is NOT the speed of signal or energy propagation. We see that the x plane wave represented by above equation extends from minus infinite to plus infinite in both time and space. There is no start or finish and it does not represent any meaningful signal. In practice, a signal has a starting point and an ending point in time. Let’s suppose that a harmonic signal at frequency ω o is generated during a time period [0,t o ], as shown in the following figure (a). t t ω o t o o t 2 π Amplitude Frequency (a) (b) (c) Such a finite-time harmonic signal can be decomposed through Fourier series into the summation of truly plane waves with time extending from minus infinite to plus infinite, as shown in figure (b). The frequencies of these plane waves are centered around ω o and their amplitudes decay as frequency moves away from ω o , as illustrated in figure (c). 2.57 Fall 2004 – Lecture 16 1
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One can better understand these pictures by actually carrying out the Fourier expansion ( a sin 2 π n t ). Because each of the plane waves in such a series expansion is at a n T frequency slight different from the central frequency ω o , it also has a corresponding wavevector that is different from k o , as determined by the dispersion relation between ω( k). The subsequent propagation of the signal can be obtained from tracing the spatial evolution of all these Fourier components as a function of time. For simplicity, let’s consider that the signal is an electromagnetic wave. We pick up only two of the Fourier components and consider their superposition, one at frequency ω ο −∆ω/2 and another at frequency ω ο + ∆ω/2 propagating along positive x-direction [figure (a)]. The superposition of these two waves gives the electric fields as y ( , Ex t ) = a e x p { i ( ω ) t −( k o k x } + a e x { i ( + ) t ( k } o ) o o ) = 2c o s ( ∆ω t kx ) e x ⎡− i ( k ) a o o ω o - ∆ω ω o + ∆ω (b) (a) (c) The electric field is schematically shown in above figures. There appears to be two waves, one is the carrier wave at central frequency ω ο ( term exp i ( k ) ), another is the o o modulation of the carrier wave by a wave at frequency ∆ω (the amplitude term a t y ( , o ( ∆ω •− ) in ) ) . The latter one changes much slower compared with the former one.
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lecture16 - 2.57 Nano-to-Macro Transport Processes Fall...

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