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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 9

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online