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20. Fresnel equations 20. Fresnel equations
• EM Waves at an Interface
• Fresnel Equations: Reflection and Transmission Coefficients
• Brewster’s Angle
• Total Internal Reflection• Evanescent Waves
• The Complex Refractive Index• Reflection from Metals
2 2
2 2
2 2 2
2 2 2
cos sin:cos sin
cos sin:c
:
os sin
r
r
E nTE rE n
E
r reflection coefficient
n nTM rE n n
θ θ
θ θ
θ θ
θ θ
− −= =
+ −
− −= =
+ −
2 2
2 2 2
2cos:cos sin
2 cos:cos sin
:
t
t
t transE
TE tE nE nTM tE
mission coeffi e
n
ci nt
n
θ
θ θθ
θ θ
= =+ −
= =+ −
n2
θE
Et
Erθr
θt
n1
We will derive the Fresnel equations We will derive the Fresnel equations
1
2
nnn ≡
EM Waves at an InterfaceEM Waves at an Interface
r
TE mode
EM Waves at an InterfaceEM Waves at an Interface
( )( )( )
exp
exp
:
:
:
( ), ,
exp
i oi i i
r or r r
t ot t t
E E i k r t
E E i k r t
E E
Incident beam
Reflected beam
Transmitted beam
At the boundary between the two media the x y plane
i k r t
all waves must exist simultaneouslya
ω
ω
ω
⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤− ⎦
−
= ⋅⎣
rr r r
rr r r
rr r r
( ) ( ) ( )
,
exp exp exp
oi i i or r r ot t t
i r t
nd .Therefore, for all t and for all r on the inter
the tangential component must be equal on both sides of the interface
n E n E n E
face
n E i k r t n E i k r t n E i k r tω ω ω⎡ ⎤ ⎡ ⎤ ⎡× ⋅ − + × ⋅ − = × ⋅ −⎣ ⎦ ⎣
×
⎦
× + = ×
r
r r rr r rr r r) )
)
)
r r r) )
( ) ( ) ( ) : Phase matching at the boundary!
,:
i i r r t t
the only way that this can be true over the entire interfAssuming that the wave amplitudes are cons
k r
ace and for al
t k r t k r
tl t is i
t
antf
ω ω ω⇒ ⋅ − = ⋅ − = ⋅ −
⎤⎣ ⎦
r r rr r r
EM Waves at an InterfaceEM Waves at an Interface
( ) ( ) ( )
(Frequency does not change at the boundar
0,
y!)
(Phases on the boundary does not chan
0
)
,
ge!
i r t
i t
t
r
i r
i i r r t t
At r this results int t t
At t this results in
k r k r k
S
r
k r t k r t k r t
ω ω ωω ω ω
ω ω ω
== =
⇒
=
= =
⋅ ⋅ = ⋅⇒ =
⋅ − = ⋅ − = ⋅ −
r r rr
r
r r
r r rr r r
( ) ( ) ( )
,
0
.
,
i r i t t r
i i
i r t
ubtracting any pair of these factors results in
k k r k k r k k r
This equation k r constant is the equation for a plane perpendicular to k rConsequently the above relation implies t
k k a
a
nd k
h t
ar
− ⋅ = − ⋅ = − ⋅ =
⋅ = ⋅
r r
r r r r r rr r
r
r
r
rr r
. e coplanar in the plane of incidence
EM Waves at an InterfaceEM Waves at an Interface
co
0,
,
sin si
ns
n
,
sin
ta
s
n
n
t
i
i r i i r r
i
ii r
t
r i ri
r
At t
Considering the relation for the incident and reflected beams
k r k r k r k r
Since the incident and reflected beams are in t
k r k r
he same mediumnk k
k r
c
θ θ
ω θ θ θ θ
=
⋅ = ⋅ ⇒ =
= = ⇒ = ⇒
⋅ = ⋅ =
=
⋅ =
r rr r
r r rr r r
: law of reflection
sin sin : law of
,
sin s n
re
i
,
i t i i t t
i ti ii tt t
Considering the relation for the incident and transmitted beams
k r k r k r k r
But the incident and transmitted
n
beams are in different median nk nk
c c
θ
θ
θ
ω ω θ =
⋅ = ⋅ ⇒ =
= = ⇒
r rr r
fraction
Development of the Fresnel EquationsDevelopment of the Fresnel Equations
cos co
' ,
s co
:
si r t
i i r r t t
E E EB B B
From Maxwell s EM field theorywe have the boundary conditions at the interface
Th tangential
components of both E and B are equal onboth sides o
e above co
f the i
nditions imply that th
for the T
e
E case
θ θ θ+ =
− =
r r
0
cos cos
.,
c
:
os
.
i i r r t t
i t
i r t
We have alsoassumed that as is true formost dielectric materia
nterface
E E EB B B
For the TM mode
lsμ μ μ
θ θ θ− + = −+ =
≅ ≅
TE-case
TM-case
Development of the Fresnel EquationsDevelopment of the Fresnel Equations
1
1 1
1
2
2
1
:
cos cos
:
cos cos c
v
s
c
o
os
i i r
i r t
i
r t t
i
i r r t t
cRecall that E B Bn
Let n refractive index of incident mediumn refractive index of refracting me
For the TM m
diu
ode
E
For the TE mode
E E En E n E
E En
n
E n
n
c
m
E
EB
θ θ θ
θ θ θ
=⎛ ⎞= = ⇒⎜
−
⎟⎝ ⎠
==
=
=
=
−
−
+
+
+
2r tE n E=
TE-case
TM-case
n1
n2
n1
n2
Development of the Fresnel EquationsDevelopment of the Fresnel Equations
2
1
cos cos:cos cos
cos cos:co
:
sin sin
cos 1
s cos
i tr
i i t
i tr
i
t
i
t
t
i
t
Eliminating E
nnn
n
n n
nETE case rE n
nETM case rE
from each set of equationsand solving for the reflection coefficient we obtain
where
We know that
n
θ θθ θ
θ θθ θ
θ θ
θ
−= =
+
−=
=
=
=
+
=
−2
2 2 22
sinsin 1 sinit in nnθθ θ= − = −
TE-case
TM-case
n1
n2
n1
n2
Now we have derived the Fresnel EquationsNow we have derived the Fresnel Equations
TE-case
TM-case
n1
n2
n1
n2
2 2
2 2
2 2 2
2 2 2
cos sin:
cos sin
cos sin:
cos
:
:
s
in
i ir
i i i
i ir
i i i
transmission coefficient t
TE
reflection coefficienSubst
c
ts r
nETE case rE n
n nETM case rE n
as
ituting we obtain the Fresnel equations for
For the
e
n
θ θ
θ θ
θ θ
θ θ
− −= =
+ −
− −= =
+ −
2 2
2 2 2
2cos:cos sin
2 co
:
s:
1: 1
cos sin
t i
i i i
t i
i i i
EtE n
E nTM case tE n
TE t rT nt r
n
M
θ
θ θ
θ
θ θ
=
= =+ −
= =+
+= +
−
1
2
nnn ≡
These mean just the boundary conditions
Power : Reflectance(R) and Transmittance(T)Power : Reflectance(R) and Transmittance(T).
1
,,
:
:
tr
i i
i r t
R and T are the ratios of reflected and transmitThe quantitiesThe ratiosrespectively to
ted powers
PPR TP P
R T
r and t are ratios of electric field amplitudes
From conservation of ener
the incident power
P P P
We can
gy
= =
= += + ⇒
21 0
20 00
:
cos cos
cos cos cos
1
cos
1 c22
i i i r r r t t t
i i r r
i i r r t
i i r r t
i
t
t
t t
iexpress the power in each of the fie
n terms of the product of an irradiance and areaP I A P I A P I A
I
lds
I A I A I A
But n c
I I
I n c
I A I I A
E
A
E
θ θ θ
ε
θ θ θ
ε
=⇒=+
=+
⇒
= = =
+
= 2 21 0 0 2 0 0
2 2 2 20 2 0 0
2 220 0
2 20 0
02 2 2 20 1 0 0 0
1 1os cos cos2 2
cos cos 1cos
cos coscos
co
s
s
co
i
r t t t
i i
r r t t
r t t r t t
i i i i
i
i i
i
n cE n cE
E n E E E
E ER r T n
n R TE
E
n E E
nE
E
θ ε θ ε θ
θ θθ
θ θθ θ
θ⎛ ⎞ ⎛ ⎞
=
= +
⎛ ⎞⇒ = + = + = +⎜ ⎟
⎝ ⎠
= = =⎜ ⎟ ⎜⎝ ⎝
⇒⎠ ⎠
2t⎟ 2
2
coscos*
coscos
*
tnttnT
rrrR
i
t
i
t⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
==
θθ
θθ
20-2. External and Internal Reflection20-2. External and Internal Reflection
1 2
1
2
1
1 22
:
1:
1
Internal reflection
External reflect
n n
nnn
i n
n
on
n
n
n
< = >
⇒ =>
⇒
<
2 2
,
sin i
For internal reflection
n may be an im total inteaginary rnal refnum lecber .tionθ− ⇒
( ) 1( ) :
0 tan p
pp
Note for polarizing angBrewster's angle for the Tle
r wh
M se
nen
ca
θ
θ
θ −=⇒ =
External reflection
r
transmission
t
θi
n=1.50
critical angle : θi = θc
1
2sinnnnc =≡θ
No reflection of TM mode
Derivation of Brewster’s AngleDerivation of Brewster’s Angle
( )
( )
4
2 2 2
2 2 2
2 2 2
4 2 2 2
2 22 22
2 2
2
cos sin0
c
( ) :
cos sin
cos sin 0
1 1 4 1 sin sin1 1 4cos sin2cos 2cos
1 1 4si
os si
n 4si
n
p
p p
p p
p p
p pp
p
p
p
p
p
p
p
Brewster's angle for polarizing angle for the TM case
n n
n n
n
n nr
n n
θ
θ θ
θ θ
θ θ
θ θθ
θ θ
θ θ
θ θ
θ
⇒ = −
− + =
± − −± −⇒ = =
± − +=
− −= =
+ −
( )42
1
2
2
22
2
1 1 2sinn2cos 2cos
sin s
1.5
cin
tos
ancos
0, 56.31
pp
p
pp
p
p
p
p
p
For n
n n n
θθ
θ θ
θθ
θθ
θ
θ −=
± −=
= ⇒ ⇒
= =
=
°
Exist both in external and internal
externalreflectioninternal
reflection
θp θp
θc
R
Total Internal Reflection (TIR) Total Internal Reflection (TIR)
2
1
1 1
total internal refl
sin , (TE and TM) .
, ,
* 1
:
ect
.
i
1
)
on(c
c
nFor internal reflectio
For n for both cases
called
TE ca
r is a complex numberR rr
r
For
n
s
n
R
n
TI
θ θ
θ θ
− == =
⇒⇒ =
= <
=
>
2 2
2 2
2 2 2
2 2 2
cos sin:
cos sin
cos sin:
cos sin
i ir
i i i
i ir
i i i
nEe rE n
n nETM cas
i
i
i
ie r
E n n
θ θ
θ θ
θ θ
θ θ
− −= =
+ −
− −= =
+ −
internalreflection
R
r
TIR
θc
20-3. Phase changes on External Reflection20-3. Phase changes on External Reflection
( ) ( )( )0
0
, ,,
.
:
exp
ex
180 ( ) 00
p
0
xp
0
e
r i
i
i
the phase shift is for r
For
When r is a real number as it always is for external reflectionthen the phase shift is fora
r
nd
E r E
r E i k r t
r E i k r t
r
i
π
π
π
ω
ω
° = <
<
= −
⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅
°
⎣ ⎦
>
+−
r r
r r
externalreflection
TE TMπ π
r
수직입사인경우TE 경우만π-phase 변화,즉, 위상반전이일어나는가?
(Ex) Why two r’s are opposite in their signs at θi 0 ?(Ex) Why two r’s are opposite in their signs at θi 0 ?
Transmission coeff.
Reflection coeff.
Brewster angleonly for TM pol.
rTM > 0
rTE < 0
The opposite signs in r’s correspond completelyto the phase shift of π after reflection in both cases.
E3 is opposite to E1
E3 is opposite to E1
r
t
TE
TM
Phase Shifts for Internal ReflectionPhase Shifts for Internal Reflection
2
. :
0 0 18
cos sin sit
( )
ancos si
0
n
0
ii i
c
c
i
TIR case r is comWhen then r is a real number and the phase shift will be andWhen then and for both the TE and TM cases has the form
a ib i er e ea
p
ib
for r for rx
e
le
i
αα φ
α
θθ θ
α α αα α
θ
−− −
+
− −= = = = = ⇒
< ° > <
=+ +
≥°
2 2
2 2
2 2
2
ncos
cos sin:
cos si
( ).
n
cos sin
tan tan
i ir
i i i
i i
is the phase shift for total internal
ba
i nETE case rE i n
reflection TI
a b n
R
αα
θ θ
θ θ
θ
α
φ α
φ
θ
=
− −= =
⇒
=
+ −
= = −
=
2 21
2 21
2
2 2sin2 cos
sin
2 tancos
sin2 tan
c
:
os
iTE
iTM
i n
A similar analysis for the TM case gives
n
nn
θ
θφ
θ
θ
θ
φθ
φ
−
−
⎛ ⎞−⎜ ⎟=⎜ ⎟⎝ ⎠
⎛ ⎞−
−
⎜ ⎟=⎜ ⎟⎝
⎛ =⎟⎝ ⎠
⎠
⎞⎜ r
Internal reflection
TIR
Phase Shifts for Internal ReflectionPhase Shifts for Internal Reflection
π π
TM TE
2 2 2 21 1
2
sin sin2 tan 2 tan
cos cosi i
TM TE
n nn
θ θφ φ
θ θ− −⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
Phase Shifts for Internal ReflectionPhase Shifts for Internal Reflection
'p
'p c
2 21
c2
c
2 21
180 ( ) <0 <
sin2 tan <
cos
0 <
sin2 tan
cos
TM
i
TE i
nn
n
π θ θφ θ θ θ
θθ θ
θ
θ θ
φ θθ
−
−
⎧⎪⎪⎪⎪= θ θ
⎧⎪⎪ ⎞⎨ ⎟⎪ ⎟⎪ ⎠⎩
c
c
0 <:
0 >TM TEθ θ
φ φθ θ
⎧=− ⎨
>⎩
o
o
45 53 1.5TM TE iNote near when nφ φ θ− = = =o o
Phase Shifts for Internal Reflection:The Fresnel Rhomb
Phase Shifts for Internal Reflection:The Fresnel Rhomb
45 53 1.5TM TE iNote at when nφ φ θ− = = =o o
Linearly polarized light (45o)
CircularlyPolarized
light
20-5. Evanescent Waves at an Interface20-5. Evanescent Waves at an Interface
( )( )( )
( )
: exp
: exp
: exp
exp
si
:
n
i oi i i
r or r r
t ot t t
t ot t t
t t t
Incident beam E E i k r t
Reflected beam E E i k r t
Transmitted beam E E
For the transmitted
i k
bea
r t
E E i k r
k
m
t
r k
ω
ω
ω
ω
θ
⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦⎡ ⎤= ⋅ −⎣ ⎦
⎡ ⎤= ⋅ −⎣ ⎦
⋅ =
rr r r
rr r r
rr r r
r r
r r )( ) ( )( )
2
22
cos
sin cos
sin, cos 1 sin 1
, :
sin
sinc
( )
o s 1it
t t
t t t
it t
iWhen n
x k y x x yy
k x y
Butn
th
i a purely imaginary numbe
total internal reflection
r
en
n
θ
θ
θθ
θ
θθ
θ
θ
+ ⋅ +
= +
=
= −
>
−
⇒
− =
) ) )
Evanescent Waves at an InterfaceEvanescent Waves at an Interface( )
2
2 2
sin
sin 1
,:
sin
:
sin sin21 1
:
tt t
it
i
i
t
t
iwith an TIR condition n
i y
For the transmitted beamwe can write the phase factor as
k r k xn
Defining the coefficient
kn n
We can write the transmitted wa s
n
v
E
e a
θ
α
θ θπαλ
θ
θ⎛ ⎞⋅ = +⎜ ⎟
⎜ ⎟⎝ ⎠
=
−
− −
=
=
>
r r
( )0sin ep
.
xx pe t tt y
amplitude will decay rapidThe evanescent waveas it penetrates into the lower refractive ind
ly
k xE i t
ex med
n
ium
θ ω α⎡ ⎤⎛ ⎞− −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
Note that the incident and reflection wavesform a standing wave in x direction
n2
n1
n1 > n2 h
( )0sinexp expt tt t
k xE E i t yn
θ ω α⎡ ⎤⎛ ⎞= − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
x
y
2
2
1 1sin2 1
t ot
i
E Ee
h
n
λα θπ
⎛ ⎞= ⇒ = =⎟⎝ ⎠
−⎜Penetration depth:
Frustrated TIRFrustrated TIR
Tp = fraction of intensity transmitted across gap
Zhu et al., “Variable Transmission Output Coupler and Tuner for Ring Laser Systems,”Appl. Opt. 24, 3611-3614 (1985).
d
n1=n2=1.5171.65
d/λ
Frustrated Total Internal ReflectanceFrustrated Total Internal Reflectance
Zhu et al., “Variable Transmission Output Coupler and Tuner for Ring Laser Systems,”Appl. Opt. 24, 3611-3614 (1985).
Pellin-Broca prism
20-6. Complex Refractive Index20-6. Complex Refractive Index
0
2 2 2
0
( )
1 2
1
I I
R I
R R
For a material with conductivity
n i n n i n
n i n
n
n i σε ω
σ
σε ω
⎛ ⎞= + = +
⎛ ⎞= + = − +⎜ ⎟
⎝
⎝ ⎠
⎠
⎜ ⎟
%
%
2 2
0 02
2
02
22 4 2
0 0
2
02
1 1 42
2
:
1 22
1 02 2
:
1 1 42
2 I
R I R I RI
I I II
I
Solving for the real and imaginary components we obtain
n n n n nn
n n nn
From the quadratic solution we obta n
n
i
n
σ σε ω ε ω
σ σε ω ε ω
σε ω
σε ω
− = = ⇒ =
⎛ ⎞ ⎛ ⎞⇒
⎛ ⎞+ +
− = ⇒ − − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞± + ⎜ ⎟
⎝⎜⎝=⇒
⎟⎠⎠=
.IWe need to take the positive root because n is a real number
Complex Refractive IndexComplex Refractive Index
2
0
0 0 00
, ,
2 2
,
2 2I I R I
With this approximation the complex refractive index for m
For most metals with light in the microwave region
n n
etals becomes
n
Substituting our expression for the complex refractive index
nσε ω
σ εω
σ σ σε ω ε ω ε ω
>>
= = =≅ ⇒
( ) ( ) ( )
( ) ( )
0 0
0
ˆex
ˆex ˆexp
p
.
exp
p R k
k
I
R I
k
back intoour expression for the electric field we obtain
E E i k r t E i
ni u r tc
n i n u r tc
The first exponential term is oscillatory The
n uc
E
E r
ω
ω ω
ω ω⎧ ⎫⎡ ⎤⎡ ⎤= ⋅ − = + ⋅ −⎨ ⎬⎢ ⎥
⎧ ⎫⎡ ⎤⋅ −⎨ ⎬⎢ ⎥⎣⎡ ⎤− ⋅⎢ ⎥⎣ ⎦
⎣ ⎦ ⎣ ⎦⎩
⎩
⎭
=⎦⎭
rr r rr r
r r r
/ .RThe second exponent
M wave propagates with aial has a real argument
velocity of(abso
n crbed).
Complex Refractive IndexComplex Refractive Index
( )* *0 0
0
..
ˆ2exp
ˆ2exp
I k
I k
The second term leads to absorption of the beam in metals due to inducinga current in the medium This causes the irradiance to decrease as the wavepropagates through the medium
n u rI EE E E
c
n uI I
ω
ω
⋅⎡ ⎤≡ = −⎢ ⎥
⎣ ⎦
= −
rr r r r
( ) ( )0 ˆexp
2 4:
k
I I
rI u r
c
The absorption coefficient is defined n ncω πα
α
λ
⋅⎡ ⎤= − ⋅⎡
= =
⎤⎢ ⎥ ⎣ ⎦⎣ ⎦
rr
( ) ( )0 ˆex ˆexpp Ik kRn uni u r
crt
cE E ω ω⎧ ⎫⎡ ⎤⋅ −⎨ ⎬⎢ ⎥
⎡ ⎤− ⋅=⎣ ⎦⎩ ⎢ ⎥⎣ ⎦⎭
rr r r
20-7. Reflection from Metals20-7. Reflection from Metals
2 2
2 2
2 2 2
2 2 2
cos sin:
cos sin
cos sin:
cos sin
:
i ir
i i i
i ir
i i i
Reflection from metals is analyzed
ETE case rE
nETM cas
by substituting the complex refractive index n in the Fresnel equation
e rE n
Sub
n
n
n
n
sti u
s
t t
θ θ
θ θ
θ θ
θ θ
− −= =
+ −
− −= =
+ −
% %
%
%
%
%
%
( ) ( )( ) ( )
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
2 2 2
2 2 2
2 2 2 2 2
2 2 2 2 2
:
cos sin 2:
cos sin 2
2 cos sin 2:
2 cos sin 2
i R I i R Ir
i i R I i R I
R I R I i R I i R Ir
i R I R I i R I i R I
R Iing we obtain
n n i n nErE n n i n n
n n i n n n n i n nErE n n i n n
TE case
T
n n
n n iM case
n n
i n
θ θ
θ θ
θ θ
θ θ
− − − += =
+ − − +
⎡ ⎤− + − − − +⎣ ⎦= =⎡ ⎤− + + − − +
= +
⎣ ⎦
%
Reflection from Metals at normal incidence (θi=0)Reflection from Metals at normal incidence (θi=0)
( ) ( )( ) ( )
( ) ( )( ) ( )
( )( )
( )( )
2 2 2
2 2 2
22 2
22 2
, 0 :
cos sin 2
cos sin 2
1 2 1
1 2 1
:
&
1
1
i
i R I i R Ir
i i R
R I
R I
I i R I
R I R I R I
R I R I R I
At normal incidence
n n i n nErE n n
TE TM cases
n i
i n n
n n i n n n i n
n n i n n n i n
power ref
nr
n i
T ehe
n
l c
θ
θ θ
θ θ
= °
− − − += =
+ − − +
− − + − −= =
+ − +
=−
+ −
− −+
∴
( )( )
( )( )
( )( )
2 2
2 2*
2 2
2 2
1 1 1 21 1 1 2
11
R I
R
R I R I R R I
R I
I
R I R R I
is given by
n i n n i n n n nR r rn i n n i n
tance R
n nR
n n
n n n⎡ ⎤ ⎡
− +=
⎤− − − + ⎛ ⎞− + += = =⎢ ⎥ ⎢ ⎥ ⎜ ⎟+ − + + + + +⎝ ⎠⎣ ⎦ ⎣
+
⎦
+
Reflection from MetalsReflection from Metals
At normal incidence(from Hecht, page 113)
Reflectance
θiλ
visible
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