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7/31/2019 Fresnel Reflection Lecture13
1/19
13. Fresnel's Equations for Reflection
and RefractionIncident, transmitted, and reflected beams
Boundary conditions: tangential fields are continuous
Reflection andtransmission
coefficients
The "Fresnel Equations"
Brewster's Angle
Total internal reflection
Power reflectanceand transmittance
Augustin Fresnel1788-1827
7/31/2019 Fresnel Reflection Lecture13
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Posing the problem
What happens when light, propagating in auniform medium, encounters a smooth interfacewhich is the boundary of another medium with adifferent refractive index?
k-vector of the
incident light
boundarynincident
ntransmitted
This is really two problems, not just one.
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Definitions: Planes of Incidence, the
interface and S and P polarizationsS polarization is the perpendicular polarization, and it sticks upout of the plane of incidence
Plane of the interface (y=0, the xzplane) (perpendicular to screen)
P polarization is the parallel polarization, and it lies parallelto the plane of incidence.
Plane of incidence (z=0)is the plane thatcontains the incident
and reflected k-vectors.
x
y
z
I R
T
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ni
nt
ik
rk
tk
i r
t
EiBi
Er
Br
Et
Bt
Interface
Fresnel EquationsWe would like to compute the fraction of a light wave reflected andtransmitted by a flat interface between two media with different refrac-tive indices. Fresnel was the first to do this calculation (early 1800s).
x
y
zBeam geometry forlight with its electric
field sticking up out ofthe plane of incidence(i.e., out of the page)
We treat the case of S polarization first:
the xz plane (y = 0)
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ni
nt
ik
rk
tk
i r
t
EiBi
Er
Br
Et
Bt
Interface
Boundary Condition for the Electric
Field at an Interface: s polarizationx
y
z
In other words,
The Tangential Electric Field is Continuous
So: Ei(y = 0) +Er(y = 0) = Et(y = 0)
The component ofthe E-field that lies inthe xz plane iscontinuous as you
move across theplane of the interface.
Here, all E-fields arein the z-direction,
which is in the planeof the interface.
(Were not explicitly writingthe x, z, and t dependence,
but it is still there.)
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Boundary Condition for the Magnetic
Field at an Interface: s polarization
ni
nt
ik
rk
tk
i r
t
Ei
Bi
ErBr
Et
Bt
Interface
x
y
z
ii
*It's really the tangentialB/, but we're using i t0
Bi(y = 0) cosi +Br(y = 0) cosr = Bt(y = 0) cost
The Tangential Magnetic Field* is Continuous
In other words,
The total B-field in theplane of the interface iscontinuous.
Here, all B-fields are inthe xy-plane, so we takethe x-components:
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Reflection and Transmission for
Perpendicularly Polarized Light
Ignoring the rapidly varying parts of the light wave and keeping
only the complex amplitudes:
0 0 0
0 0 0
cos( ) cos( ) cos( )
i r t
i i r r t t
E E E
B B B
0 0 0 0
0 0 0 0
:
( ) cos( ) ( ) cos( )
Substituting for using
t i r t
i r i i t r i t
E E E E
n E E n E E
0 0 0( ) cos( ) cos( ) i r i i t t t n E E n E
0 0/( / ) / .
But and
i rB E c n nE c Substituting into the second equation:
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Reflection & Transmission Coefficients
for Perpendicularly Polarized Light
0 0 0 0
0 0
( ) cos( ) ( ) cos( ) :
cos( ) cos( ) cos( ) cos( )
i r i i t r i t
r i i t t i i i t t
n E E n E E
E n n E n n
Rearranging yields
0 0/ 2 cos( ) / cos( ) cos( )t i i i i i t t t E E n n n
0 0/ , ist iE EAnalogously, the transmission coefficient,
0 0/ cos( ) cos( ) / cos( ) cos( )r i i i t t i i t t r E E n n n n
0 0/r iE ESolving for yields the reflection coefficient :
These equations are called the Fresnel Equations for
perpendicularly polarized (s-polarized) light.
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ni
nt
ik
rk
tk
i r
t
EiBi Er
Br
EtBt
Interface
Fresnel EquationsParallel electric field
x
y
z
Beam geometryfor light with itselectric fieldparallel to theplane of incidence(i.e., in the page)
Note that the reflected magnetic field must point into the screen toachieve for the reflected wave. The x with a circle
around it means into the screen.
E B k
Note that Hechtuses a differentnotation for the
reflected field,which is confusing!
Ours is better!
This leads to adifference in the
signs of someequations...
Now, the case of P polarization:
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Reflection & Transmission Coefficients
for Parallel Polarized Light
These equations are called the Fresnel Equations for
parallel polarized (p-polarized) light.
|| 0 0/ cos( ) cos( ) / cos( ) cos( )r i i t t i i t t ir E E n n n n
|| 0 0/ 2 cos( ) / cos( ) cos( )t i i i i t t it E E n n n
Solving forE0r/ E0i yields the reflection coefficient, r||:
Analogously, the transmission coefficient, t|| = E0t/ E0i, is
For parallel polarized light, B0i B0r = B0t
and E0icos(i) +E0rcos(r) = E0tcos(t)
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To summarize
||
cos( ) cos( )
cos( ) cos( )
i t t i
i t t i
n nr
n n
||
2 cos( )
cos( ) cos( )
i i
i t t i
nt
n n
2 cos( )
cos( ) cos( )
i i
i i t t
nt
n n
cos( ) cos( )
cos( ) cos( )
i i t t
i i t t
n nr
n n
s-polarized light: p-polarized light:
And, for both polarizations: sin( ) sin( )i i t t n n
planeofincidenceincidentwave
transmitted wave
interface
planeofincidenceincidentwave
transmitted wave
interface
E-field vectors are red.k vectors are black.
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Reflection Coefficients for an
Air-to-Glass Interface
Incidence angle, i
Reflection
coefficient,
r
1.0
.5
0
-.5
-1.0
r||
r
0 30 60 90
The two polarizations areindistinguishable at = 0
Total reflection at = 90for both polarizations.
nair 1 < nglass 1.5
Brewsters angle
r||=0!Zero reflection for parallel
polarization at 56.3
Brewster's angleThe value of this angledepends on the value ofthe ratio ni/nt:
Brewster = tan-1(nt/ni)
Sir David Brewster1781 - 1868
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Incidence angle, i
Reflection
coefficient,
r
1.0
.5
0
-.5
-1.0
r||
r
0 30 60 90
Brewstersangle
Total internalreflection
Critical
angle
Criticalangle
Total internal reflection
above the "critical angle"
crit
sin-1
(nt/ni) 41.8for glass-to-air
nglass > nair
(The sine in Snell's Law
can't be greater than one!)
Reflection Coefficients for a
Glass-to-Air Interface
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Reflectance (R)
R Reflected Power / Incident Power r r
i i
I A
I A
Because the angle of incidence = the angle of reflection,the beam area doesnt change on reflection.
Also, n is the same for both incident and reflected beams.
A = Area
20 0
02
c
I n E
iwi nint
r wi
2R rSo: since
2
0 2
2
0
r
i
Er
E
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Transmittance (T)
t t
i i
I A
I A A = Area
20 0
02
cI n E
cos( )cos( )
t t t
i i i
A wA w
t
iwi
wt
nint
If the beamhas width wi:
20 020
0 2
220 0 0
0
2
2
t tt t tt t t t t
i i i i ii i ii i
cn E
n E wI A w n wT t
cI A w n wn E w
n E
The beam expands (or contracts) in one dimension on refraction.
since
2
0 2
2
0
t
i
Et
E
2cos
cos
t t
i i
nT t
n
T Transmitted Power / Incident Power
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Reflectance and Transmittance for an
Air-to-Glass Interface
Note that it is NOT true that: r+ t= 1.
But, it is ALWAYS true that: R + T = 1
Perpendicular polarization
Incidence angle, i
1.0
.5
00 30 60 90
R
T
Parallel polarization
Incidence angle, i
1.0
.5
00 30 60 90
R
T
Brewstersangle
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Perpendicular polarization
Incidence angle, i
1.0
.5
00 30 60 90
R
T
Reflectance and Transmittance for a
Glass-to-Air Interface
Parallel polarization
Incidence angle, i
1.0
.5
00 30 60 90
R
T
Note that the critical angle is the same for both polarizations.
And still, R + T = 1
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Reflection at normal incidence,i= 0
2
t i
t i
n nR
n n
2
4t i
t i
n nT
n n
When i = 0, the Fresnelequations reduce to:
For an air-glass interface (ni = 1 and nt= 1.5),
R = 4% and T= 96%
The values are the same, whicheverdirection the light travels, from air toglass or from glass to air.
This 4% value has big implicationsfor photography.
lens flare
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Windows look like mirrors at night (when youre in the brightly lit room)
One-way mirrors (used by police to interrogate bad guys) are just
partial reflectors (actually, with a very thin aluminum coating).
Disneyland puts ghouls next to you in the haunted house using partialreflectors (also aluminum-coated).
Optical fibers use total internal reflection.
Where youve seen Fresnels Equations in action
R = 100% R = 90%Laser medium
0% reflection!
0% reflection!
Some lasers use Brewsters angle components to avoid reflective losses: