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This article explains the concept of Snell's Law and Total Internal Reflection in the context of bending light. It also discusses anomalies in refraction and the physics of negative index. Includes diagrams and equations.
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Total Internal reflection (TIR) Bend away so much it comes back !!! n2 qt = 900 n2 n1 n1 qC qC q > qC Critical Angle Total Internal Reflection sinqC/sin(900) = n2/n1 < 1
The world according to Carp qc qc www.seafriends.org.nz/ phgraph/water.htm Snell’s circle
TIR: Fiber Optic cables and diamonds Pictures courtesy Joseph F. Alward, Physics, University of the Pacific
m,e < 0 n = -me Snell : bend away from normal Anomalous material: bend towards normal Anomalous refraction n < 0 Then qt must be -negative Another anomaly f large l large too Predicted by Victor Veselago Observed by Smith, Pendry
The physics of negative index Nanoellipsoid • LC of split ring gives m < 0 near resonance Review, Shalaev (Purdue) • Wires give e < 0 • Combo gives n < 0
Hot off the press! Metamaterial Expts – Duke Univ. http://www.bbc.co.uk/news/science-environment-20265623
Electronic Analogs (source) Photons in free space: w = ck Electrons in a solid: w = Ak2 Graphene Electrons: w = vk (drain) VS VD VG1 VG2 Graphene (2010 Physics Nobel) PN junction switch in graphene Lensless Focusing Total Internal Reflection
Diffraction Limiting neighboring spherical waves prevents their coalescence into a new planar wavefront Neighboring delayed spherical wavefronts create new bent planar wavefront Light bends through this slit (Diffraction)
Can we translate these pictures into equations?
Any rt. Handed triad (Diff. in book pg. 294) Er ej(wrt+br. r ) Et ej(wtt-bt. r ) Ei ej(wit-bi. r ) Deriving Snell’s Law from Maxwell’s eqns Match E║, especially phase, all along boundary wi = wr = wt = w
Er ej(wrt+br. r ) Et ej(wtt-bt. r ) Ei ej(wit-bi. r ) E polarized ║ to plane Match Ecomponent ║interface wi = wr = wt = w
Er ej(wt+br. r ) Et ej(wt-bt. r ) Ei ej(wt-bi. r ) Deriving Snell’s Law x qr qt z qi From b.r part of phase n1w/c x sinqi = n1w/c x sinqr = n2w/c x sinqt
x qr qt z qi Er ej(wt+br. r ) Et ej(wt-bt. r ) Ei ej(wt-bi. r ) Match E component ║interface Eicosqi - Ercosqr = Etcosqt
Hr ej(wt+br. r ) Ht ej(wt-bt. r ) Hr ej(wt-bi. r ) Match H component ║ interface x qr qt z qi (Ei+ Er)(e1/m1) = Et (e2/m2)
R║ = |Er/Ei|2 = _____________ Z1cosqi – Z2cosqt 2 [ ] Z1cosqi + Z2cosqt Normal incidence R = [ _____ ]2 =G║2 Z = (m/e) Z1-Z2 Z1+Z2 Reflection 1/(me) = c/n + Snell’s law Use
What about other two BCs? There is no H to match What if we try matching eE ? e1(Eisinqi + Ersinqr) = e2Etsinqt e1(Ei+ Er)sinqi = e2Etsinqt (Ei+ Er)(e1/m1) = Et (e2/m2) n1sinqi = n2sinqt 4th equation redundant Simply reinforces Snell’s Law !
Z1cosqt - Z2cosqi R = [ _____________ ]2 Z1cosqt + Z2cosqi E polarized to plane x Hr Er Et 2 qr qt Ht z qi Ei Match E component ║interface Match H component ║interface Hi Interchange E and H ie, Z 1/Z
sin2(qi – qt) tan2(qi – qt) = _________ = _________ sin2(qi + qt) tan2(qi + qt) Z1cosqt - Z2cosqi Z2cosqt – Z1cosqi R = [ _____________ ]2 R║ = [ _____________ ]2 Z1cosqt + Z2cosqi Z2cosqt + Z1cosqi Summary of Reflectivity
