Chapter 23: Fresnel equations. Recall basic laws of optics Law of reflection: ii normal n1n1 n2n2 rr...
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Transcript of Chapter 23: Fresnel equations. Recall basic laws of optics Law of reflection: ii normal n1n1 n2n2 rr...
Recall basic laws of optics
Law of reflection: ri
qi
normal
n1
n2
qr
qt
Law of refraction“Snell’s Law”:
1
2
sin
sin
n
n
t
i
Easy to derive on the basis of:Huygens’ principle: every point on a wavefront may be regarded as a
secondary source of wavelets
Fermat’s principle: the path a beam of light takes between two points is the one which is traversed in the least time
Incident, reflected, refracted, and normal in same plane
E and B are harmonic
)sin(
)sin(
0
0
t
t
rkBB
rkEE
Also, at any specified point in time and space,
cBE where c is the velocity of the propagating wave,
m/s 10998.21 8
00
c
We’ll also determine the fraction of the light reflected vs. transmitted
and the change in the phase upon reflection
external reflection,
Incidence angle, qi
1.0
.5
0
0° 30° 60° 90°
R
TR
T
qi
Let’s start with polarization…light is a 3-D vector field
linear polarization circular polarization
y
x
z
Plane of incidence: formed by and k and the normal of the interface plane
…and consider it relative to a plane interface
k
normal
kBE
TE: transverse electrics: senkrecht polarized
(E-field sticks in and out of the plane)
Polarization modes (= confusing nomenclature!)
TM: transverse magneticp: plane polarized
(E-field in the plane)E
M
M
E
E Eperpendicular ( ), horizontal parallel ( || ), vertical
always relative to plane of incidence
y
x x
y
Plane waves with k along z directionoscillating electric field
Any polarization state can be described as linear combination of these two:
“complex amplitude” contains all polarization info
yeExeE yx
tkziy
tkzix ˆˆ 00
E
tkziiy
ix eyeExeE yx ˆˆ 00E
At the boundary point:
phases of the three waves must be equal:
true for any boundary point and time, so let’s take 0r
ttt tri
tri or
hence, the frequencies are equal
and if we now consider 0t
)()()( ttt ttrrii rkrkrk
rkrkrk tri
which means all three propagation vectors lie in the same plane
rkrkrk tri
focus on first two terms:
rkrk ri
rrii rkrk sinsin
incident and reflected beams travel in same medium; same l
ri kk
hence we arrive at the law of reflection:
ri
Reflection
rkrkrk tri
now the last two terms:
rkrk tr
ttrr rkrk sinsin
reflected and transmitted beams travel in different media (same frequencies; different wavelengths!):
cnvk rr // 1
which leads to the law of refraction:
tr nn sinsin 21
cnvk tt // 2
Refraction
Boundary conditions from Maxwell’s eqns
yE
yE
yE
0
0
0
ˆ
ˆ
ˆ
tt
rr
ii
E
E
E
for both electric and magnetic fields, components parallel to boundary plane must be continuous as boundary is passed
complex field amplitudes
electric fields:
TE waves
continuity requires:
tri EEE
parallel to boundary plane
Boundary conditions from Maxwell’s eqnsfor both electric and magnetic fields, components parallel to boundary plane must be continuous as boundary is passed
)(
)(
)(
)ˆsinˆcos(
)ˆsinˆcos(
)ˆsinˆcos(
tittt
tirrr
tiiii
t
r
i
eBB
eBB
eBB
rk
rk
rk
zxB
zxB
zxB
magnetic fields:
coscoscos tri BBB continuity requires:
same analysis can be performed for TM waves
TE waves
TE waves TM waves
n2
tri EEE
ttrrii BBB coscoscos
tri BBB
ttrrii EEE coscoscos
BvBE nc
Summary of boundary conditions
n1
iE
iE
rE
rE
tE
tE
tB
tB
iB
iB
rB
rB
amplitudes are related:
TE waves TM waves
tri EEE
tttiriiii EnEnEn coscoscos
ttriii EnEnEn
ttirii EEE coscoscos
tnn
i
tnn
i
i
rTE
i
t
i
t
E
Er
coscos
coscos
tinn
tinn
i
rTM
i
t
i
t
E
Er
coscos
coscos
For reflection: eliminate Et , separate Ei and Er , and take ratio:
Get all in terms of E and apply law of reflection (qi = qr):
Apply law of refraction and let :ttii nn sinsin
Fresnel equations
ii
iiTE
n
nr
22
22
sincos
sincos
ii
iiTM
nn
nnr
222
222
sincos
sincos
i
t
n
nn
ii
i
i
tTE
nE
Et
22 sincos
cos2
For transmission: eliminate Er , separate Ei and Et , take ratio…
And together:
ii
i
i
tTM
nn
n
E
Et
222 sincos
cos2
TE waves TM waves
Fresnel equations
TETE rt 1 TETM rtn 1
internal reflection:
External and internal reflections
external reflection: 21 nn
21 nn
11
2 n
nn
11
2 n
nn
occur when
n = n2/n1 = 1.5
External reflections (i.e. air-glass)
normal grazing
- at normal and grazing incidence, coefficients have same magnitude- negative values of r indicate phase change- fraction of power in reflected wave = reflectance =
- fraction of power transmitted wave = transmittance =
2
2
i
r
i
r
E
Er
P
PR
2
cos
costn
P
PT
i
t
i
t
RTM = 0(here, reflected light TE polarized;RTE = 15%)
22
1
1
n
nrR
at normal : 4%
Not
e: R
+T =
1
http://www.ray-ban.com/clarity/index.html?lang=uk
Glare
- incident angle where RTM = 0 is:
- both and reach values of unity before q=90°
total internal reflection
p
2TETE rR 2
TMTM rR
1
211 sin)(sinn
nnc
Internal reflections (i.e. glass-air)
n = n2/n1 = 1.5
total internal reflection
Conservation of energy
tri PPP
1TR
it’s always true that
and
in terms of irradiance (I, W/m2)
ttrrii AIAIAI using laws of reflection and refraction, you can deduce
2
2
0
0 rE
E
I
I
P
PR
i
r
i
r
i
r
and2
cos
costnT
i
t
Summary:Reflectance and Transmittance for anAir-to-Glass Interface
Perpendicular polarization
Incidence angle, qi
1.0
.5
00° 30° 60° 90°
R
T
Parallel polarization
Incidence angle, qi
1.0
.5
00° 30° 60° 90°
R
T
Summary:Reflectance and Transmittance for aGlass-to-Air Interface
Perpendicular polarization
Incidence angle, qi
1.0
.5
00° 30° 60° 90°
R
T
Parallel polarization
Incidence angle, qi
1.0
.5
00° 30° 60° 90°
R
T
at qp, TM is perfectly transmitted with no reflection
Brewster’s angle for internal and external reflections
at Brewster’s angle, “s skips and p plunges”s-polarized light (TE) skips off the surface; p-polarized light (TM) plunges in
http://www.brewstersociety.com/cbs_sundaymorning_09.html
Brewster’s other angles: the kaleidoscope
Phase changes upon reflection
-recall the negative reflection coefficients
-indicates that sometimes electric field vector reverses direction upon reflection:
-p phase shiftexternal reflection: all angles for TE and at for TM internal reflection: more complex…
ErEr )(
0)(
0 titii eEreEreEr rkrk
p
Phase changes upon reflection: internal
in the region , r is complexc
222
222
22
22
sincos
sincos
sincos
sincos
nin
ninr
ni
nir
TM
TE
reflection coefficients in polar form: ierr
f phase shift on reflection
Phase changes upon reflection: internaldepending on angle of incidence, - < < p f p
cos
sin
2tan
cos
sin
2tan
2
2222
n
nn TMTE
Exploiting the phase differencecircular polarization
-consists of equal amplitude components of TE and TM linear polarized light, with phases that differ by ±p/2
-can be created by internal reflections in a Fresnel rhomb
each reflection produces a π/4 phase delay
http://www.halbo.com/fr_rhmb.htm
Summary of phase shifts on reflection
TE mode TM mode
airglass
external reflection
TE mode TM mode
airglass
internal reflection