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Reflection and refraction

Reflection and refraction. Optics, Eugene Hecht, Chpt. 4. Notation. Start with propagating waves: E = E 0 cos(kx - w t) and B = B 0 cos(kx - w t) Use complex amplitudes (as in ac circuits): E 0 cos(kx - w t) = (1/2) (E 0 expi(kx - w t) + c.c.) drop (1/2) and c.c. part

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Reflection and refraction

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  1. Reflection and refraction Optics, Eugene Hecht, Chpt. 4

  2. Notation • Start with propagating waves: • E = E0 cos(kx - wt) and B = B0 cos(kx - wt) • Use complex amplitudes (as in ac circuits): • E0 cos(kx - wt) = (1/2) (E0 expi(kx - wt) + c.c.) • drop (1/2) and c.c. part • E = E0 e i(kx - wt) and B = B0 e i(kx - wt) Three waves, Ei, Er, Et • Define reflection and transmission coefficients • Er = r Ei, Et = t Ei • Reflected and transmitted power -- Er2, Et2 • Er2 = r2 Ei2, Et2 = t2 Ei2 • Reflected power R = r2, transmitted power T = t2 1 r n1 n2 t r2 + t2 = 1

  3. Snell’s law • Momentum parallel to surface is conserved • no boundary to bounce off • ki sin qi = kr sin qr = kt sin qt • ni sin qi = nr sin qr = nt sin qt • Law of reflection: • ni = nr --> qi = qr • Law of refraction • ni sin qi = nt sin qt qr qi ki kr n1 n2 kt qt

  4. Total internal reflection • From high index to low index nt > ni • Maximum value of sin qt = 1 • Snell’s law: sin qimax = ni / nt < 1 • Critical angle:sin qcritical = ni / nt • Larger angles: • cannot satisfy Snell’s law • no transmission • total internal reflection • Evanescent wave on surface • k-vector: kevan = ni ki sin qi > ki nt • wavelength: levan = li / sin qi < lt • sub-wavelength in medium nt qr qi ki kr ni nt kt qt

  5. S and P polarizations • General case of reflection and refraction at boundary • Different results for different polarizations • S-polarization • Electric field polarized perpendicular to incidence plane • parallel to boundary surface • P-polarization • Electric field polarized in incidence plane • component of E-field perpendicular to boundary surface Boundary

  6. E is normal to plane of incidence Eperpendicular, S-polarization E is parallel to surface • No space charge -- Ei + Er = Et Two components of B • Perpendicular to surface • No magnetic monopoles • Bi sin qi + Br sin qr = Bt sin qt • Parallel to surface • mi = mr = mt -- most materials • -Bi cos qi + Br cos qr = -Bt cos qt Need second equation for E • B is related to E by B = E/v = nE/c • Perpendicular B’s • niEi sin qi + nrEr sin qr = ntEt sin qt • use Snell’s law -- same as E-field equation • Parallel B’s • - niEi cos qi + nrEr cos qr = - ntEt cos qt • use Snell’s law: • rperpendicular = (ni cos qi - nt cos qt) / (ni cos qi + nt cos qt) • tperpendicular = (2ni cos qi ) / (ni cos qi + nt cos qt)

  7. E is in plane of incidence Eparallel, P-polarization Two components of E • Parallel to surface • No space charge • Ei cos qi + - Er cos qr = Et cos qt • Perpendicular to surface • Space charge attenuates Et • ni2Ei sin qi + nr2Er sin qr = nt2Et sin qt • use Snell’s law • niEi + nrEr = ntEt • B is parallel to surface • Bi + Br = Bt • B is related to E by B = E/v = nE/c • same as perpendicular E • rparallel = (nt cos qi - ni cos qt) / (nt cos qi + ni cos qt) • tparallel = (2ni cos qi ) / (nt cos qi + ni cos qt)

  8. Normal incidence Perpendicular • qi = qr = qt = 0 • rnormal = - rparallel = rperpendicular • sign difference comes from definition • either E or B must flip sign on reflection • symmetry property -- propagation reversed • Energy flow must reverse: S = e0 c E X B • rnormal = (nt - ni) / (ni + nt) • tnormal = (2ni) / (ni + nt) Special cases • Low to high index • ni < nt --rnormal > 0 (positive) • High to low index • ni > nt --rnormal < 0 (negative) • tnormal > 1 ??? Energy flow: S = n e0 c2 E2 = n Svacuum • (nrr2 + ntt2)/ni = 1 = R2 + T2 Parallel

  9. Energy flow -- non-normal incidence • General case • energy into boundary surface = energy out • A ni cos qi = A nr r2 cos qr + A nt t2 cos qt • Reference to input energy • 1 = r2 + t2 (nt cos qt / ni cos qi) = R + T • T = t2 (nt cos qt / ni cos qi)

  10. Reflectivity vs angle Case of external reflection: low to high index, nt > ni • rperpendicular = (ni cos qi - nt cos qt) / (ni cos qi + nt cos qt) • tperpendicular = (2ni cos qi ) / (ni cos qi + nt cos qt) • rparallel = (nt cos qi - ni cos qt) / (nt cos qi + ni cos qt) • tparallel = (2ni cos qi ) / (nt cos qi + ni cos qt) • Transmissions similar for both polarizations Reflections: • Note rperpendicular always negative • nt cos qt > ni cos qi • rparallel goes to zero, changes sign • nt cos qi = ni cos qt 1 r ni , air nt , glass t

  11. 1 r ni , glass nt , air t Reflectivity vs angle Case of internal reflection: high to low index, ni > nt • rperpendicular = (ni cos qi - nt cos qt) / (ni cos qi + nt cos qt) • tperpendicular = (2ni cos qi ) / (ni cos qi + nt cos qt) • rparallel = (nt cos qi - ni cos qt) / (nt cos qi + ni cos qt) • tparallel = (2ni cos qi ) / (nt cos qi + ni cos qt) • Transmissions similar for both polarizations Reflections: • Note rperpendicular always positive • nt cos qt < ni cos qi • rparallel goes to zero, changes sign • nt cos qi = ni cos qt • Both cases: r --> 1 above critical angle

  12. i r ni , air nt , glass t i r ni , glass nt , air t Polarization (Brewster) angle • Reflection --> 0 for one polarization • rparallel goes to zero • rparallel = (nt cos qi - ni cos qt) / (nt cos qi + ni cos qt) • tparallel = (2ni cos qi ) / (nt cos qi + ni cos qt) • rparallel = 0 when nt cos qi = ni cos qt • Snell’s law gives: tan qi = tan qBrewster = nt / ni • rparallel --> 0 • tparallel --> ni / nt

  13. i r ni , glass nt , air t Phase shifts Perpendicular • rperpendicular = (ni cos qi - nt cos qt) / (ni cos qi + nt cos qt) • tperpendicular = (2ni cos qi ) / (ni cos qi + nt cos qt) • rparallel = (nt cos qi - ni cos qt) / (nt cos qi + ni cos qt) • tparallel = (2ni cos qi ) / (nt cos qi + ni cos qt) Phase shifts • Both tperpendicular and tparallel always in phase • rperpendicular always p phase shift • rparallel starts out with 0 phase • switches to p beyond Brewster angle • Above critical angle nt < ni, • both rperpendicular and rparallel have phase shifts Parallel

  14. Phase for total internal reflection Reflection coefficients • Reflectivities • rperpendicular = (ni cos qi - nt cos qt) / (ni cos qi + nt cos qt) • rparallel = (nt cos qi - ni cos qt) / (nt cos qi + ni cos qt) • Replacement for cos qt from Snell’s law • Complex reflection coefficients

  15. Internal reflection i r ni , glass nt , air t Summary • Transmission -- nothing unusual • Critical angle: • internal reflection = high to low index • total internal reflection, evanescent wave • Brewster angle: • P-polarization • no reflection, both internal & external reflection Phase shifts Reflectivity Differential phase total internal reflection

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