lecture13 - 2.57 Nano-to-Macro Transport Processes Fall...

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2.57 Nano-to-Macro Transport Processes Fall 2004 Lecture 13 Review of previous lectures 1. Energy transport between two points Q 1->2 1 2 Q 2->1 x 2. Plane waves & their interface reflection We are interested in the wave energy at points 1 and 2 on two sides of the interface. Transmission wave Reflection wave Energy barrier u wave 1 2 Incoming x 3. Oblique incidence of an electromagnetic wave onto an interface k i E i H i k r E t k t θ i θ r θ t n 1 n 2 E r z x x Assuming flat interface, we have θ = . In last lecture, we have derived i r 2.57 Fall 2004 – Lecture 13 1
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E // r n 2 cos θ i + n 1 cos E // t 2 n 1 cos i = = , r // E // i n 2 cos + n 1 cos t , t // = E // i = n 2 cos + n 1 cos i t i t which are known as the Fresnel coefficients of reflection and transmission. For normal incidence ( i = 0 ), similarity exists between case 2 and case 3 (see the following table). Electron propagation across a barrier Normal incidence onto an interface Ψ = k 1 k 2 E // r n 2 + n 1 r r = = = r // Ψ i k 1 + k 2 E // i n 2 + n 1 2 k 1 2 n 1 = t = t // n 2 + n 1 k 1 + k 2 2 2 2 S rz S r B kk 2 1 = , = = R // r // R =− J / J = = r i S iz S i , A k 1 + k 2 * = , n 2 cos 2 t J t Re ( k 2 ) 2 T // S tz = Re t // T = = t S n 1 cos i , J i Re () k 1 * Discussions 1) Critical & total internal reflection A) n 1 <n 2 The Snell law is applied, n 1 sin = n 2 sin . i t n 1 <n 2 n 2 x Note: The Snell law indicates the momentum conservation, or wavevector k x 1 = k x 2 on the interface. B) n 1 >n 2 Because the maximum angle of the refracted wave is θ t =90 o , there exists an angle of incidence above which no real solution for θ t exists. This critical angle happens when, according to the Snell law, n 1 sin = n 2 90 sin o or sin = n 2 c c n 1 Above this angle, the reflectivity equals one, i.e., all the incident energy is reflected (T=0, R=1). 2.57 Fall 2004 – Lecture 13 2
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For an electromagnetic wave incident above the critical angle, the Snell law gives, sin θ = n 1 sin i > 1 , t n 2 and thus, 2 cos t = 1 sin t = i n 1 sin i 2 1 = ai . n 2 In the wave function of the transmitted wave, the imaginary cos t leads to an exponential decay wave E // t exp i ω t nx sin + nz cos t 2 t 2 ⎟⎥ c o 2 nz a E = // t exp i t sin t 2 c o c o , which is similar to the encountered evanescent wave.
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lecture13 - 2.57 Nano-to-Macro Transport Processes Fall...

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