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Understanding Snell's Law and Total Internal Reflection

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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Understanding Snell's Law and Total Internal Reflection

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  1. Back to Snell’s Law

  2. 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

  3. The world according to Carp qc qc www.seafriends.org.nz/ phgraph/water.htm Snell’s circle

  4. TIR: Fiber Optic cables and diamonds Pictures courtesy Joseph F. Alward, Physics, University of the Pacific

  5. Laws are meant to be broken…

  6. Bending Light: Optical Camouflage

  7. 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

  8. 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

  9. Hot off the press! Metamaterial Expts – Duke Univ. http://www.bbc.co.uk/news/science-environment-20265623

  10. 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

  11. 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)

  12. Can we translate these pictures into equations?

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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)

  18. 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

  19. 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 !

  20. 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

  21. 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

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