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

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