1 / 15

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

dobry
Download Presentation

Reflection and refraction

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related