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 n
TE r
E n
E
r reflection coefficient
n n
TM r
E n n
θ θ
θ θ
θ θ
θ θ
− −
= =
+ −
− −
= =
+ −
2 2
2 2 2
2 cos :
cos sin
2 cos :
cos sin
:
t
t
t trans TE t E
E n
E n
TM t
E
mission coeffi e
n ci nt
n
θ
θ θ
θ
θ θ
= =
+ −
= =
+ −
n
2E θ
E
tE
rθ
rθ
tn
1We will derive the Fresnel equations We will derive the Fresnel equations
1 2
n
n ≡ n
EM Waves at an Interface EM Waves at an Interface
r
TE mode
EM Waves at an Interface EM 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 simultaneously a
ω ω
ω
⎡ ⎤
= ⎣ ⋅ − ⎦
⎡ ⎤
= ⎣ ⋅ − ⎦
⎡ − ⎤⎦
−
= ⎣ ⋅
r r r r
r r r r
r r 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⋅ −ωt ⎤⎦ + ×n E ⎡⎣i k r⋅ −ω t ⎤ = ×n E ⎡i k r⋅ −ωt
×
⎦
× + = ×
r
r r r
r r r r 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 interf Assuming that the wave amplitudes are cons
k r
ace and for al
t k r t k r
t
l t is i t
ant
f
ω ω ω
⇒ ⋅ − = ⋅ − = ⋅ −
⎣ ⎤⎦
r r r r r r
EM Waves at an Interface EM 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 in
t 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 ω
ω ω ω
ω ω ω
=
= =
⇒
=
= =
⋅ ⋅ = ⋅
⇒ =
⋅ − = ⋅ − = ⋅ −
r r r r
r
r r
r r r r 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 r Consequently the above relation implies t
k k a
a nd k
h t ar
− ⋅ = − ⋅ = − ⋅ =
⋅ = ⋅
r r
r r r r r r r r
r
r
r
r r r
. e coplanar in the plane of incidence
EM Waves at an Interface EM Waves at an Interface
co 0,
,
sin si
ns
n
, sin
ta
s n
n t
i
i r i i r r
i
i
i r
t
r i r
i 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 medium k k n
k r
c
θ θ
ω θ θ θ θ
=
⋅ = ⋅ ⇒ =
= = ⇒ = ⇒
⋅ = ⋅ =
=
⋅ =
r r r r
r r r r r r
: law of reflection
sin sin : law of ,
sin s n
re i
,
i t i i t t
i t
i i
i t t 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 media
n n
k k n
c c
θ
θ θ
ω ω θ =
⋅ = ⋅ ⇒ =
= = ⇒
r r r r
fraction
Development of the Fresnel Equations Development of the Fresnel Equations
cos co
' ,
s co
:
s
i r t
i i r r t t
E E E
B B B
From Maxwell s EM field theory
we have the boundary conditions at the interface
Th tangential
components of both E and B are equal on both 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 also assumed that as is true for most dielectric materia
nterface
E E E
B B B
For the TM mode
ls
μ μ μ
θ θ θ
− + = −
+ =
≅ ≅
TE-case
TM-case
Development of the Fresnel Equations Development 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
Recall that E B c B n
Let n refractive index of incident medium n refractive index of refracting me
For the TM m
diu
ode
E
For the TE mode
E E E
n E n E
E E
n
n
E n
n
c
m
E
B E
θ θ θ
θ θ θ
⎛ ⎞ =
= = ⎜ ⇒
−
⎝ ⎠⎟
=
=
=
=
=
−
− +
+
+
2
r t
E = n E
TE-case
TM-case
n1n2
n1
n2
Development of the Fresnel Equations Development of the Fresnel Equations
2 1
cos cos
: cos cos
cos cos
: co
:
sin sin
cos 1
s cos
i t
r
i i t
i t
r i t
i
t
t
i
t
Eliminating E
n n n
n
n n
n TE case r E
E n
n TM case r E
E
from each set of equations
and solving for the reflection coefficient we obtain
where
We know that
n
θ θ
θ θ
θ θ
θ θ
θ θ
θ
= = −
+
= −
=
=
=
+
=
− 2 sin22 2 2
sin t n 1 i n sin i
n
θ = − θ = − θ
TE-case
TM-case
n1n2
n1
n2
Now we have derived the Fresnel Equations Now we have derived the Fresnel Equations
TE-case
TM-case
n1n2
n1
n2
2 2
2 2
2 2 2
2 2 2
cos sin
:
cos sin
cos sin
:
cos
:
:
sin
i i
r
i i i
i i
r
i i i
transmission coefficient t TE
reflection coefficien Subst
c
ts r E n
TE case r
E n
n n
TM case r E
E 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
2 cos :
cos sin
2 co
:
: s
1
: 1
cos sin
t i
i i i
t i
i i i
t E
E n
E n
TM case t
E n
TE t r
T nt r
n
M
θ
θ θ
θ
θ θ
=
= =
+ −
= =
+
+
= +
−
1 2
n n ≡ n
These mean just the boundary conditions
Power : Reflectance(R) and Transmittance(T) Power : Reflectance(R) and Transmittance(T)
.
1
, ,
: :
r t
i i
i r t
R and T are the ratios of reflected and transmit The quantities
The ratios respectively to
ted powers
P
R P T
P 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
= =
= +
= + ⇒
2 1 0 2
0 0 0
:
cos cos
cos cos cos
1
cos
1 c
2 2
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
i
express the power in each of the fie
n terms of the product of an irradiance and area
P 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
=
⇒
= +
=
+
⇒
= = =
+
= 1 0 02 2 0 02
2 2 2 2
0 2 0 0
2 2
0 2 0
2 2
0 0
0
2 2 2 2
0 1 0 0 0
1 1
os cos cos
2 2
cos cos
1
cos
cos cos
cos
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 E
R r T n
n R T
E
E
n E E
E n
E
θ ε θ ε θ
θ θ
θ
θ θ
θ θ
θ
⎛ ⎞ ⎛ ⎞
=
= +
⎛ ⎞
⇒ = + = + ⎜ ⎟ = +
⎝ ⎠
= = ⎜ ⎟ = ⎜
⎝ ⎝
⇒ ⎠ ⎠
t2
⎟ 2
2
cos
* cos cos
cos
*
t n
tt n
T
r rr R
i t i
t ⎟⎟
⎠
⎜⎜ ⎞
⎝
=⎛
⎟⎟⎠
⎜⎜ ⎞
⎝
=⎛
=
=
θ θ θ
θ
20-2. External and Internal Reflection 20-2. External and Internal Reflection
1 2
1
2 1
1 2
2
:
1 :
1
Internal reflection External reflect
n n
n n i n n
n on
n n
n
< = >
⇒ =
>
⇒
<
2 2
, sin i
For internal reflection
n − θ may be an imaginary number ⇒ total internal reflection.
( )
1( ) :
0 tan
p
p p
Note Brewster's angle for polarizing angle for the T
r wh
M se
n en
ca θ
θ
θ = −
⇒ =
External reflection r
transmission t
θ
in=1.50
critical angle : θi = θc
1
sin
2n n n
c
≡ =
θ
No reflection of TM mode
Derivation of Brewster’s Angle Derivation of Brewster’s Angle
( )
( )
4
2 2 2
2 2 2
2 2 2
4 2 2 2
2 2
2 2
2
2 2
2
cos sin
0
c
( ) :
cos sin
cos sin 0
1 1 4 1 sin sin 1 1 4 cos sin
2 cos 2 cos
1 1 4 si
os si
n 4 si
n
p
p p
p p
p p
p p
p
p p
p
p
p p
p
Brewster's angle for polarizing angle for the TM case
n n
n n
n
n n
r
n n
θ
θ θ
θ θ
θ θ
θ θ
θ θ θ
θ θ
θ θ
θ
⇒ = −
− + =
± − −
± −
⇒ = =
± − +
=
− −
= = + −
( )
4 2
1 2 2
2 2
2
1 1 2 sin n
2 cos 2 cos
sin s
1.5 c
in t
os an
cos
0, 56.31
p p
p
p
p p
p
p
p
p
For n
n n n
θ θ
θ θ
θ θ
θ θ θ
θ −
=
= ± −
= ⇒ ⇒
= =
=
°
Exist both in external and internal
external reflection internal
reflection
θ
pθ
pθ
cR
Total Internal Reflection (TIR)
Total Internal Reflection (TIR)
2 1
1 1
total internal refl
sin , (TE and TM) .
, ,
* 1
:
ect
.
i
1
)
c on(
c
For internal reflectio n
For n for both cases
called
TE ca
r is a complex number R 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 i
r
i i i
i i
r
i i i
E n e r
E n
n n
TM cas E
i i
i i
e r E n n
θ θ
θ θ
θ θ
θ θ
− −
= =
+ −
− −
= =
+ −
internal reflection
R
r
TIR
θ
c20-3. Phase changes on External Reflection 20-3. Phase changes on External Reflection
( )
( )
( )
0
0
, ,
,
. :
exp
ex
180 ( ) 0
0
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 reflection then the phase shift is for
a 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
external reflection
TE TM
π π
r
수직입사인 경우
TE 경우만π-phase 변화,
즉, 위상 반전이
일어나는가?
(Ex) Why two r’s are opposite in their signs at θi Æ 0 ? (Ex) Why two r’s are opposite in their signs at θi Æ 0 ?
Transmission coeff.
Reflection coeff.
Brewster angle only for TM pol.
rTM > 0
rTE < 0
Î The opposite signs in r’s correspond completely to the phase shift of π after reflection in both cases.
Î E3 is opposite to E1
Î E3 is opposite to E1
r t
TE
TM
Phase Shifts for Internal Reflection Phase Shifts for Internal Reflection
2
.
:
0 0 18
cos sin si
t
( )
cos si an
0
n
0
i
i i
c c
i
TIR case r is com
When then r is a real number and the phase shift will be and
When then and for both the TE and TM cases has the form
a ib i e
r e e
a
p
ib
for r for r
x
e
le
i
α α φ
α
θ θ θ
α α α
α α
θ
− − −
+
− −
= = = = = ⇒
< ° > <
+ + =
≥
°
2 2
2 2
2 2
2
n cos
cos sin
:
cos si
( ).
n
cos sin
tan tan
i i
r
i i i
i i
is the phase shift for total internal b a
i n
TE case r E
E i n
reflection TI
a b n
R α
α
θ θ
θ θ
θ
α
φ α φ
θ
=
− −
= =
⇒
=
+ −
= = −
=
2 2
1
2 2
1
2
2 2
sin
2 cos
sin
2 tan
cos
2 tan sin
c
:
os
i TE
i TM
i n
A similar analysis for the TM case gives n
n n
θ
φ θ
θ
θ
θ
φ θ
φ
−
−
⎛ − ⎞
⎜ ⎟
= ⎜⎝ ⎟⎠
⎛ − ⎞
−
⎜ ⎟
= ⎜⎝ ⎟
⎛ ⎟ =
⎝ ⎠
⎠
⎜ ⎞ r
Internal reflection
TIR
Phase Shifts for Internal Reflection Phase Shifts for Internal Reflection
π π
TM TE
2 2 2 2
1 1
2
sin sin
2 tan 2 tan
cos cos
i i
TM TE
n n
n
θ θ
φ φ
θ θ
−
⎛ − ⎞
−⎛ − ⎞
⎜ ⎟ ⎜ ⎟
= =
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
Phase Shifts for Internal Reflection Phase Shifts for Internal Reflection
' p '
p c
2 2
1
2 c
c
2 2
1
180 ( ) <
0 <
2 tan sin <
cos
0 <
2 tan sin
cos
TM
i
TE i
n n
n
π θ θ
φ θ θ θ
θ θ θ
θ
θ θ
φ θ
θ
−
−
⎧⎪
⎪⎪⎪
= ⎨ <
⎪ ⎛ − ⎞
⎪ ⎜ ⎟
⎪ ⎜ ⎟
⎪ ⎝ ⎠
⎩
= ⎛ −
⎜⎜
⎝
o
o
o
>θ θc
⎧⎪⎪ ⎞
⎨ ⎟
⎪ ⎟
⎪ ⎠
⎩
c c
0 <
: 0 >
TM TE
φ φ θ θ
θ θ
− ⎧= ⎨ ⎩ >
o o
45 TM TE i 53 1.5 Note φ −φ = o near θ = o when n =
Phase Shifts for Internal Reflection:
The Fresnel Rhomb
Phase Shifts for Internal Reflection:
The Fresnel Rhomb
TM TE 45 i 53 1.5 Note φ −φ = o at θ = o when n =
Linearly polarized light (45
o)
Circularly Polarized
light
20-5. Evanescent Waves at an Interface 20-5. 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
⎡ ⎤
= ⎣ ⋅ − ⎦
⎡ ⎤
= ⎣ ⋅ − ⎦
⎡ ⎤
= ⎣ ⋅ − ⎦
⎡ ⎤
= ⎣ ⋅ − ⎦
⋅ =
r
r r r
r r r r
r
r r r
r r
r r
(
)) ( )
( )
2
2 2
cos
sin cos
, cos 1 sin 1 sin
, :
sin c sin
( )
os t i 1
t t
t t t
i
t t
When i n
x k y x x yy
k x y
But n
th
i a purely imaginary numbe
total internal reflection
r en
n
θ θ
θ θ
θ
θ θ
θ
θ
+ ⋅ +
= +
=
= −
>
−
⇒
− =
) ) )
Evanescent Waves at an Interface Evanescent Waves at an Interface
( )
2
2 2
sin
sin 1
, :
sin
:
sin 2 sin
1 1
:
t
t t
i t
i
i t
t
with an TIR condition i n
i y For the transmitted beam
we can write the phase factor as
k r k x n
Defining the coefficient
k n n
We can write the transmitted wa s n
v E
e a θ
α
θ π θ
α λ
θ
⎛ θ ⎞
⋅ = ⎜⎜⎝ + ⎟⎟⎠
=
−
− −
=
=
>
r r
( )
0
sin e
p
. x
x p
e t t
t y
amplitude will decay rapid The evanescent wave
as it penetrates into the lower refractive ind
ly
E i k x t
ex med n
ium
θ ω α
⎡ ⎛⎜ − ⎞⎟⎤ −
⎢ ⎝ ⎠⎥
⎣ ⎦
Note that the incident and reflection waves form a standing wave in x direction n2
n1
n1 > n2 h
( )
0
exp t sin t exp
t t
E E i k x t y
n
θ ω α
⎡ ⎛ ⎞⎤
= ⎢⎣ ⎜⎝ − ⎟⎠⎥⎦ −
x y
2 2
1 1
2 sin 1
t ot
i
E E
e h
n λ
α π θ
=⎛ ⎞⎟ ⇒ = =
⎝ ⎠ −
⎜
Penetration depth:
Frustrated TIR Frustrated 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=1.517 1.65
d/λ
Frustrated Total Internal Reflectance Frustrated 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 Index 20-6. 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 σ
ε ω
σ
σ ε ω
⎛ ⎞
= + = +
⎛ ⎞
= + ⎜ ⎟ = − +
⎝
⎝ ⎠
⎠
⎜ ⎟
%
%
2 2
0 0
2
2
2 0
2
2 4 2
0 0
2
2 0
1 1 4
2 2
:
1 2
2
1 0
2 2
:
1 1 4
2
2 I
R I R I R
I
I I I
I
I
Solving for the real and imaginary components we obtain
n n n n n
n
n n n
n
From the quadratic solution we obta n
n i
n
σ σ
ε ω ε ω
σ σ
ε ω ε ω
σ ε ω
σ ε ω
− = = ⇒ =
⎛ ⎞ ⎛ ⎞
⇒
⎛ ⎞
+ +
− = ⇒ − − =
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎛ ⎞
± + ⎜ ⎟
⎝ ⎜
= ⎝
⇒
⎟⎠
= ⎠
I .
We need to take the positive root because n is a real number
Complex Refractive Index Complex Refractive Index
2
0
0 0 0
0
, ,
2 2
,
2 2
I 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 n
σ ε ω
σ ε
ω
σ σ σ
ε ω ε ω ε ω
>>
= = =
≅ ⇒
( ) ( ) ( )
( ) ( )
0 0
0
ex ˆ
ex ˆ exp ˆ
p
. exp
p R k
k
I
R I
k
back into our expression for the electric field we obtain
E E i k r t E i
i n u r t
c
n i n u r t
c
The first exponential term is oscillatory The
n u
c
E
E r
ω
ω ω
ω ⎧ ⎡ ω ⎤⎫
⎡ ⎤
= ⋅ − = ⎨ ⎢ + ⋅ − ⎥⎬
⎧ ⎡ ⋅ − ⎤⎫
⎨ ⎢⎣ ⎥⎬
⎡− ⋅ ⎤
⎢ ⎥
⎣ ⎦
⎣ ⎦ ⎩ ⎣ ⎦
⎩
⎭
= ⎦⎭
r r r r r r
r r r
R / . The second exponent
M wave propagates with a ial has a real argument
velocity of (abso
n c rbed).
Complex Refractive Index Complex Refractive Index
( )
* *
0 0
0
.
. 2 ˆ
exp 2 ˆ
exp
I k
I k
The second term leads to absorption of the beam in metals due to inducing a current in the medium This causes the irradiance to decrease as the wave
propagates through the medium
n u r
I EE E E
c
n u
I I
ω ω
⋅
⎡ ⎤
≡ = ⎢− ⎥
⎣ ⎦
= −
r r r r r
( )
0 exp(
ˆ)
2 4
:
k
I I
r I u r
c
The absorption coefficient is defined n n c
ω π
α α
λ
⋅
⎡ ⎤
= ⎡− ⋅
= =
⎢ ⎥ ⎣ ⎤⎦
⎣ ⎦
r r
( ) ( )
0 exp R ˆk exp nI ˆk
n u
i u r
c t r
E = E ⎧⎨ ω⎡⎢ ⋅ − ⎤⎥⎫⎬ ⎡− cω ⋅ ⎤
⎣ ⎦
⎩ ⎭ ⎢⎣ ⎥⎦
r r r r
20-7. Reflection from Metals 20-7. Reflection from Metals
2 2
2 2
2 2 2
2 2 2
cos sin
:
cos sin
cos sin
:
cos sin
:
i i
r
i i i
i i
r
i i i
Reflection from metals is analyzed
TE case r E E
E n TM cas
by substituting the complex refractive index n in the Fresnel equation
e r E 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 I
r
i i R I i R I
R I R I i R I i R I
r
i R I R I i R I i R I
R I
ing we obtain
n n i n n
r E
E n n i n n
n n i n n n n i n n
r E
E n n i n n
TE case
T
n n
n n i
M case
n n i n
θ θ
θ θ
θ θ
θ θ
− − − +
= =
+ − − +
⎡ − + ⎤ − − − +
⎣ ⎦
= =
⎡ − + ⎤ + − − +
= +
⎣ ⎦
%