Post on 05-Feb-2016
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Reflection and refraction
Optics, Eugene Hecht, Chpt. 4
Notation
• Start with propagating waves:– E = E0 cos(kx - t) and B = B0 cos(kx - t)
• Use complex amplitudes (as in ac circuits): – E0 cos(kx - t) = (1/2) (E0 expi(kx - t) + c.c.) – drop (1/2) and c.c. part
• E = E0 e i(kx - t) and B = B0 e i(kx - t)
Three waves, Ei, Er, Et
• Define reflection and transmission coefficients• Er = r Ei, Et = t Ei
• Reflected and transmitted power -- Er2, Et
2
• Er2 = r2 Ei
2, Et2 = t2 Ei
2
• Reflected power R = r2, transmitted power T = t2
1r
t
r2 + t2 = 1
n1
n2
Snell’s law• Momentum parallel to surface is conserved
– no boundary to bounce off
– ki sin i = kr sin r = kt sin t
– ni sin i = nr sin r = nt sin t
• Law of reflection:– ni = nr --> i = r
• Law of refraction– ni sin i = nt sin t
ki kr
kt
n1
n2
i r
t
Total internal reflection• From high index to low index nt > ni
• Maximum value of sin t = 1
• Snell’s law: sin imax = ni / nt < 1
• Critical angle: sin critical = ni / nt
• Larger angles: – cannot satisfy Snell’s law– no transmission– total internal reflection
• Evanescent wave on surface– k-vector: kevan = ni ki sin i > ki nt
– wavelength: evan = i / sin i < t
– sub-wavelength in medium nt
ki kr
kt
ni
nt
i r
t
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
E is normal to plane of incidenceEperpendicular, S-polarizationE is parallel to surface• No space charge -- Ei + Er = Et
Two components of B• Perpendicular to surface
– No magnetic monopoles– Bi sin i + Br sin r = Bt sin t
• Parallel to surface– i = r = t -- most materials– -Bi cos i + Br cos r = -Bt cos t
Need second equation for E• B is related to E by B = E/v = nE/c• Perpendicular B’s
– niEi sin i + nrEr sin r = ntEt sin t
– use Snell’s law -- same as E-field equation• Parallel B’s
– - niEi cos i + nrEr cos r = - ntEt cos t
– use Snell’s law: • rperpendicular = (ni cos i - nt cos t) / (ni cos i + nt cos t)• tperpendicular = (2ni cos i ) / (ni cos i + nt cos t)
E is in plane of incidenceEparallel, P-polarizationTwo components of E• Parallel to surface
– No space charge – Ei cos i + - Er cos r = Et cos t
• Perpendicular to surface– Space charge attenuates Et
– ni2Ei sin i + nr
2Er sin r = nt2Et sin t
– 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 i - ni cos t) / (nt cos i + ni cos t)• tparallel = (2ni cos i ) / (nt cos i + ni cos t)
Normal incidence• i = r = t = 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 = 0 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 0 c2 E2 = n Svacuum
• (nrr2 + ntt2)/ni = 1 = R2 + T2
Perpendicular
Parallel
Energy flow -- non-normal incidence
• General case– energy into boundary surface = energy out
– A ni cos i = A nr r2 cos r + A nt t2 cos t
• Reference to input energy– 1 = r2 + t2 (nt cos t / ni cos i) = R + T
• T = t2 (nt cos t / ni cos i)
Reflectivity vs angleCase of external reflection: low to high index, nt > ni
– rperpendicular = (ni cos i - nt cos t) / (ni cos i + nt cos t)– tperpendicular = (2ni cos i ) / (ni cos i + nt cos t)– rparallel = (nt cos i - ni cos t) / (nt cos i + ni cos t)– tparallel = (2ni cos i ) / (nt cos i + ni cos t)
• Transmissions similar for both polarizationsReflections:• Note rperpendicular always negative
– nt cos t > ni cos i
• rparallel goes to zero, changes sign– nt cos i = ni cos t
1r
t
ni , air
nt , glass
Reflectivity vs angleCase of internal reflection: high to low index, ni > nt
– rperpendicular = (ni cos i - nt cos t) / (ni cos i + nt cos t)– tperpendicular = (2ni cos i ) / (ni cos i + nt cos t)– rparallel = (nt cos i - ni cos t) / (nt cos i + ni cos t)– tparallel = (2ni cos i ) / (nt cos i + ni cos t)
• Transmissions similar for both polarizationsReflections:• Note rperpendicular always positive
– nt cos t < ni cos i
• rparallel goes to zero, changes sign– nt cos i = ni cos t
• Both cases: r --> 1 above critical angle
1r
tnt , air
ni , glass
Polarization (Brewster) angle• Reflection --> 0 for one polarization• rparallel goes to zero
– rparallel = (nt cos i - ni cos t) / (nt cos i + ni cos t)– tparallel = (2ni cos i ) / (nt cos i + ni cos t)
• rparallel = 0 when nt cos i = ni cos t
• Snell’s law gives: tan i = tan Brewster = nt / ni
– rparallel --> 0– tparallel --> ni / nt
i r
t
ni , air nt , glass
i r
t
ni , glass nt , air
Phase shifts• rperpendicular = (ni cos i - nt cos t) / (ni cos i + nt cos t)• tperpendicular = (2ni cos i ) / (ni cos i + nt cos t)• rparallel = (nt cos i - ni cos t) / (nt cos i + ni cos t)• tparallel = (2ni cos i ) / (nt cos i + ni cos t)Phase shifts• Both tperpendicular and tparallel always in phase• rperpendicular always phase shift• rparallel starts out with 0 phase
– switches to beyond Brewster angle• Above critical angle nt < ni,
– both rperpendicular and rparallel have phase shifts
i r
t
ni , glass nt , air
Perpendicular
Parallel
Phase for total internal reflection• Reflectivities
– rperpendicular = (ni cos i - nt cos t) / (ni cos i + nt cos t)– rparallel = (nt cos i - ni cos t) / (nt cos i + ni cos t)
• Replacement for cos t from Snell’s law
• Complex reflection coefficients
)(sin1cos 2
2
it
it n
n
222
222
sincos
sincos
tiiiti
tiiitiparallel
nin
ninr
22
22
sincos
sincos
tiii
tiiilarperpendicu
ni
nir
Reflection coefficients
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
i r
t
ni , glass nt , air
Internal reflection
Reflectivity
Phase shifts
Differential phasetotal internal reflection