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 qi qr 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 qi qr 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 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
Perpendicular
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. 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 1 r t
nt , air
ni , glass

12. 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 i r t
ni , air
nt , glass i r t
ni , glass
nt , air

13. Phase shifts
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
Perpendicular
Parallel i r t
ni , glass
nt , air

14. Phase for total internal reflection
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
Reflection coefficients

15. total internal reflection
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
Phase shifts
Reflectivity
Differential phase
total internal reflection