fresnel equations
TRANSCRIPT
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23. Fresnel Equations
• EM Waves at boundaries
• Fresnel Equations:
Reflection and Transmission Coefficients
• Brewster’s Angle
• Total Internal Reflection (TIR)
• Evanescent Waves
• The Complex Refractive Index
• Reflection from Metals
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2 2
2 2
2 2 2
2 2 2
cos sin
cos sin
cos sin
c
:
os sin
r
TE
rTM
E n r
E n
E n n r
r reflection coef
E n n
ficient
θ θ
θ θ
θ θ
θ θ
= =
= =
2 2
2 2 2
2cos
cos sin
2 cos
cos s
:
in
tTE
tTM
t transmission coefficie
E t
E n
E n t
E
n
n
t
n
θ
θ θ
θ
θ θ
= =
= =
n
θ
E
E
t
E
r
θ
r
θ
t
n
We will derive the Fresnel equations
1
2
n
n
n
nn
incident
d transmitte =≡
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EM Waves at an Interface
( )
( )
( )
: exp
: exp
: exp
oi
or
i i i
r r
t t
r
t ot
k
k
k
Incident beam E i r t
Reflected beam E i r t
Tran
E
E
smitted beam E i t E r
ω
ω
ω
⎡ ⎤= ⋅ −⎣ ⎦
⎡ ⎤= ⋅ −⎣ ⎦
⎡ ⎤= ⋅ −⎣ ⎦
rr r r
r r
r
r
r
r
r
r
1 0
1 0
2 0
i
r
t
k n k
k n k
k n k
=
=
=
r
r
r
r
TE mode
n1
n2
iθ
n1
n2
TM mode
Note the definition of the positive E-field directions in both cases.
n
n
i E r
r E r
t E r
ik r
r k r
t k r
oi E r
or E
r
ot E r
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EM Waves at an Interface
( ), ,
points
,
the tangential component must be equal on both sides of the inter
At the boundary between the two media the x y plane all waves must exist simultaneously
and .
Therefore for all time t and fo
f
all
ac
boundary r on th te
e
r e in
−
r,rface
( )
( )
( )
exp
exp
exp
:
:
:
i oi i
r or
i
r
t ot t
r
t
E E i r t
E E i r
Incident beam
Reflected beam
Transmitted bea
k
k
k m
t
E E i r t
ω
ω
ω
⎡ ⎤= ⋅ −⎣ ⎦
⎡ ⎤= ⋅ −⎣ ⎦
⎡ ⎤= ⋅ −⎣ ⎦
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 interfac
k r t k r t k r t
e and for all t is i
Assuming that the wave amplitudes are constan
f
t
ω ω ω ⇒ ⋅ − = ⋅ − = ⋅ −r r rr r r
( ) ( ) ( )exp exp expoi i i or r r ot t
i r t
t n E i k r t n E i k r t n E i k
n E n E n
r t
E
ω ω ω ⎡ ⎤ ⎡ ⎤ ⎡ ⎤× ⋅ − + ×
× +
⋅ − = × ⋅ −⎣ ⎦ ⎣ ⎦ ⎣ ⎦
× = ×r r rr r r
r r r ) )
r r r ) ) )
)n
n
i E r
r E r
t E r
ik
r
r k
r
t k r
oi E r
or E r
ot E r
n̂
r r
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EM Waves at an Interface
Normal
x
ik r
r k r
t k r
1 0n k
2 0n k
r r
1 0
1 0
2 0
i
r
t
k n k
k n k
k n k
=
=
=
r
r
r
( ) ( ) ( )Phase matching condition:
i i r r t t k r t k r t k r t ω ω ω ⋅ − = ⋅ − = ⋅ −r r rr r r
(Frequency does not change at the boundary!)
0,
t
i t
i r
r
At r this results in
t t t ω ω ω
ω ω ω
=
= =
⇒ = =
r
, ,
, ,
,
and .
, .i r
i r t
i r
t
t
k r constant
the equation for a plane perpendicular to k
k k and k are coplanar in the plane of incidence
r ⇒ ⋅
⇒
=→
r r
r
r
r
r r
(Phases on the boundary does not change
,
!)
0
i r t
At t this
k r
results i
k k
n
r r ⋅ ==
=⇒ ⋅ ⋅r r rr r r
n
n
i E r
r E r
t E
r
ik r
r k r
t k
r
r r
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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
ii 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
nk k
k r
c
θ θ
ω θ θ θ θ
=
⋅ = ⋅ ⇒ =
= = ⇒ = ⇒
⋅ = ⋅ =
=
⋅ =
r rr r
r r rr r r
: law of reflection
,
sin sin
sin sin : law of refractio
,
n
i t i
i i t
i t t
i t t t i
Considering the relation for the incident and transmitted beams
k r k r k r k r
But the incident and transmitted beams are in different media
n nk k
c c
n n
θ θ
ω θ
ω θ =
⋅ = ⋅ ⇒ =
= = ⇒
r rr r
iθ
r θ
t θ
Normal
x
ik r
r k r
t k r
1 0n k
2 0n k
r r
n
θ
ι
θ
r
θ
t
n
x
i E r
r E
r
t E
r
ik r
r k r
t k r
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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
For the TM mod
B B
e
ls
θ θ θ
μ μ μ
+ =
− + = −
≅ ≅
TE-case
TM-case
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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
c Recall that E B B
n
Let n refractive index of incident mediumn refractive index of refracting me
For the TM m
diu
For the TE mod
nE B
c
ode
E E
e
E E E
n E n E E
n
m
n
E
n E E
θ θ
θ θ θ
θ
=⎛ ⎞
= = ⇒⎜ ⎟⎝ ⎠
==
−
+
+
=
+
−
=
=
2r t n E = −
TE-case
TM-case
n1
n2
n1
n2
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Development of the Fresnel Equations
2
1
sin sin
cos cos:cos cos
cos cos:
cos co
:
s
cos
i t r T
t
i t
E
i i t
i t r TM
i i t
from each set of e
n E TE case r E n
n E TM ca
quations
and solving for the refle
Eliminating
ction coefficient we obtain
wher
E
nne
We know t
se r E
n
a
n
t
n
n
h
θ
θ θ
θ
θ
θ θ
θ θ
θ
θ −= =+
− += =
=
=
+
22 2 2
2
sin1 sin 1 sini
t t in n n
n
θ θ θ = − = − = −
TE-case
TM-case
n1
n2
n1
n2
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TE-case
TM-case
n1
n2
n1
n2
Now we have derived the Fresnel Equations
2 2
2 2
2 2 2
2 2 2
cos sin
cos si
:
:
:
:
n
cos sin
cos sin
i ir TE
i i i
i ir TM
i i i
Substituting we obtain t reflection coefficiehe Fresnel equa nts r
n E r
E n
n n E r E n
transmission c
tions for
TE case
TM case
For the
T
oeffici n t
n
e t
θ θ
θ θ
θ θ
θ θ
− −= =
+ −
− + −= =+ −
2 2
2 2 2
2coscos sin
2 cos
: 1
:
cos sin
1
:
:
TE TE
TM T
t iTE
i i i
t iTM
i
M
i i
E case
TM ca
E t E
TE t r
TM n
n
E nt
E n
t
nse
r
θ θ θ
θ
θ θ
= =+ −
= =
+
= −
+ −
=
1
2n
nn ≡
These just mean the boundary conditions.
:
:
i r t
i r t
For the TE case E E E
For the TM mode B B B
+ =
− + = −
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Power : Reflectance (R) and Transmittance (T)
.
1
,
,
:
:
t r
i i
i r t
R and T are the ratios of reflected and transmit
The quantities
The ratios
respectively to
ted powers
PP R 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 00
:
cos cos
cos cos cos
1
cos
1c
22
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
=⇒
=
+
=
+
⇒
= = =
+
= 2 2
1 0 0 2 0 0
2 2 2 2
0 2 0 0
2 22
0 02 2
0 0
0
2 2 2 2
0 1 0 0 0
1 1os 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
n E
E
θ ε θ ε θ
θ θ
θ
θ θ
θ θ
θ
⎛ ⎞ ⎛ ⎞
=
= +
⎛ ⎞⇒ = + = + = +⎜ ⎟
⎝ ⎠
= = =⎜ ⎟ ⎜⎝ ⎝ ⇒ ⎠ ⎠
2
t ⎟
2
2
cos
cos*cos
cos
*
t ntt nT
r rr R
i
t
i
t ⎟⎟ ⎠
⎞
⎜⎜⎝
⎛
=⎟⎟ ⎠
⎞
⎜⎜⎝
⎛
=
==
θ
θ
θ
θ
2
2
coscos _
cos cos
out out out out out
in in in in in
n E I Power ratio
I n E
θ θ
θ θ
⎛ ⎞= = ⎜ ⎟
⎜ ⎟⎝ ⎠
A
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23-2. External and Internal Reflection
( )
2 1
2 2
2 1
,
/ 1 sin 0 are always real
TE TM
n n n nr
n n
θ
⇒
⇒
>
= > ⇒ − ≥
⇒
rTE
t
n=1.50
Brewster’s angle (or, polarizing angle)(No reflection of TM mode)( ) 1
:
tan0TM p p
f No or tht e TM case
n
e
r whenθ θ θ −=⇒ = =
External Reflection
, 0TE TM r >
, 0TE TM r <
, 0TE TM
t >2 2
2
2 2
2
2
2 2 2
cos sin
c
cos sin
co os sins sin
i ii
TM
i
i
TE
i ii
n nr
n
nr
nn
θ θ
θ
θ θ
θ θ θ
− + −=
+ −
− −=
+ −
, If 0 then there are no phase changes after reflection.TE TM r ⇒ >
,
, , ,
If 0 then there are always ( 180 ) phase changes.
TE TM
TE TM TE TE M
i
TM T
r
er r r π
π ⇒ < =→ = =−
o
[ ]2 1/ 1n n n= >
rTM
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( )2 2
,
,
, ,
, If sin 0, =1,
=1
(- ~ ) phase change may occur after reflect
ion
!TE T TE TM
TE TM
i i
TE TM TE
M
TM
BUT r are compn r
r
r r e e
lex
φ φ
θ
φ π π
⇒ − <
→
→ = =
→ +
( )1tan
: 0
p
TM
No Brewster's angle n
for the TM
e
case
t
r
θ −=
=
critical angle
Internal Reflection
2 2
2
2 2
2
2
2 2 2
cos sin
c
cos sin
co os sins sin
i ii
TM
i
i
TE
i ii
n nr
n
nr
nn
θ θ
θ
θ θ
θ θ θ
− + −=
+ −
− −=
+ −
, 0TE TM
r >
, 0TE TM
r <
Total internal reflection (TIR) when θ > θ c
TIR region
( ) ( )21
2 2
2 1
2 2
/ 1
sin 0, , sin 0
n n n
n o
n n
r nθ θ
⇒ = <
⇒ − > − <
>
( )2 2
,
2 1
If sin 0, =1
sin ( / )
TE TM
c
n r
n n n
θ
θ
⇒ − =
→ = =
( )2 2
,
,
,
If sin 0, are always real
If 0 then there are no phase changes after reflection.
If 0 then there are ( 180 ) phase changes.
TE TM
TE TM
TE TM
n r
r
r
θ
π
⇒ − >
→ >
→ < = o
[ ]2 1/ 1n n n= <
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Derivation of Brewster’s Angle
( )
4 2 2 2
2 2 2
4 2 2 2
1
2 2 2
2 2 2 2
c
( ) :
cos sin
cos s
os sin0
cos sin
1.50, 56.
in
( 1) os si
tan
n 0
31
p
p p
p p
TM p
p p
p
p
p p
p p
Brewster's angle for polarizing ang
n nr
n n
For n
le
n n
n n
n n c
n
θ
θ θ
θ θ
θ θ θ
θ
θ
θ
θ
θ
θ
−
⇒ = −
− +
− + −= =
⎡ ⎤= − − =⎣
+
= =
=
⎦
−
⇒
°
externalreflectioninternal
reflection
θ p θ p
θ c
R
External & Internal reflections, but TM-polarization only
1: sin nnc <=θ
or nnn p 11 : tan <>=θ
TE & TM polarizations, but Internal reflection only
Brewster ‘s angle :
Critical angle :
TETM
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Total Internal Reflection (TIR)
2
1
1 , ,
and for both (TE and TM) cases.
sin t
: 1
1
otal internal r
* 1
eflection( )
cos
r TE
i
c
n Internal reflection n
n
r R r
r is a complex
call
numb
eFor n TI d
e
r
E r E
r
Rθ θ −
⇒
= <
= = =
≥
=
=
⇒
=
2 2
2 2
2 2 2
2 2 2
sincos sin
cos sin
cos sin
i i
i i
i ir TM
i i i
i nn
n n E r
E n
i
i n
i
θ θ θ θ
θ θ
θ θ
− −+ −
− + −= =
+ −
internalreflection
R
r
θ c
Complex value
R = 1
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rTEn=1.50
, 0TE TM
r >
, 0TE TM r <
, 0TE TM
t >
23-3. Phase changes on reflection
,
,
,
18
,
,0
) 0
0
.0 (
TE T
TE TM
TE TM
M the phase
r is always a real numb
the
er for ex
phase sh
shift is fo
ift is
ternal reflection
then
an for r
r
d
r
π
° >
° = <
TE TM
Phase shift after External Reflection
For TE case, π phase shift for all incident angles For TM case, π phase shift for θ < θp
No phase shift for θ > θp
External Reflection External Reflection
External reflection
rTM
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cθ θ >:
Complex value
In TIR region
Phase shift after Internal Reflection Internal reflection
1
0 for sin is complex in TIR region where
TE TE
TE c
TE c
i i
TE TE
r nr
r r e eφ φ
θ θ θ θ
−
⇒ > < =⇒ >
→ = =
For TM case, no phase shift for θ < θp
π phase shift for θp < θ < θc
TM(θ) phase shift for θ > θc
For TE case, no phase shift for θ < θ
c
TE(θ) phase shift for θ > θc
TIR TIR
1
0 for tan 0 for
TM
TM p
TM p c
i
TM TM TM TM
r nr
r r e r φ
θ θ θ θ θ
φ π
−
⇒ > < =⇒ < < <
→ = − = → =
is complex in TIR region where
TM TM
TM c
i i
TM TM
r
r r e eφ φ
θ θ ⇒ >
→ = =
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2
:
cos sin sintan
cos sin2
( )
s
co
ii i
i
c TI When then and for both the TE and TM cases has the form
a ib i e br e e
a ib i
R case r is complex
e a
α α φ
α
α α α φ α α
α
θ θ
α α
−−
+
− −= = = =
≥
== = −= ⇒+ +
2 2
2 2
2 2
2
1
2 2
2
cos sin:
cos sin
cos sin
sintan t
( ).
sin2tan
co
an2 cos
s
i ir TE
i i i
i i
iT
i
E
TE
i
is the phase shift on total internal reflection TIR
n
i n E TE case r
E i n
a b n
n
A simi
φ
θ θ
θ θ
θ θ
θ φ
θ
θ φ α
θ
−
− −= =
+ −
= = −−⎛ ⎞
⎛ ⎞−⎜ ⎟= −⎜ ⎟
⇒ = − =⎜⎝ ⎠
⎝ ⎠
⎟
2 2
1
2
sin2tan
cos
:
i
TM
i
lar analysis for the TM case giv
n
es
nθ φ π
θ
− ⎛ ⎞−⎜ ⎟= −⎜ ⎟
⎝ ⎠
Internal reflection
: i cθ θ >
: i cθ θ >
TIR(Complex r )
Phase shifts on total Internal Reflection for both TE- and TM-cases
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2 2
1
2
2 2
1
sin2tan
cos
sin2tan
cos
i
TM
i
TE
n
n
n
θ φ π
θ
θ φ
θ
−
−
⎛ ⎞−⎜ ⎟= −⎜ ⎟
⎝ ⎠⎛ ⎞−⎜ ⎟= −⎜ ⎟⎝ ⎠
Internal reflection
Complex value
Therefore, after TIR is ………..,TE TM r
,TE TM r
,TE TM φ
2 2
2 2
2 2 2
2 2 2
cos sin
cos sin
cos sin
cos sin
i ir TE
i i i
i ir TM
i i i
n E r
E n
n n
i
i
in n
i
E r
E
θ θ
θ θ
θ θ
θ θ
− −= =+ −
− + −= =
+ −
( )cincident θ θ >caseTIR For
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Internal reflectionTIR(Complex r )
Summary of Phase Shifts on Internal Reflection
'
p
'
p c
2 2
1c2
0 <
( 180 ) <
sin2 tan <cos
TM
i nn
θ θ
φ π θ θ θ
θ π θ θ θ
−
⎧⎪⎪⎪⎪
= = <⎨⎪
⎛ ⎞−⎪ ⎜ ⎟−⎪ ⎜ ⎟
⎪ ⎝ ⎠⎩
o
o
p
p c
c
0 <
<
0
TM TE
θ θ
φ φ φ π θ θ θ
θ θ
⎧=⎪Δ = − = <⎨⎪
> <⎩
o
o
c
2 2
1c
0 <
sin2 tan >cos
TE i n
θ θ
φ θ θ θ θ
−
⎧⎪⎪ ⎛ ⎞=
−⎨ ⎜ ⎟−⎪ ⎜ ⎟⎪ ⎝ ⎠⎩
o
φ Δ
TM φ
TE φ
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Fresnel Rhomb
Linearly polarized light (45o)
Circularly
Polarizedlight
3 53 1.5
4
After two consequentive TIRs,
3
2
2
TM TE i
TM TE
TM TE
Note near when n
Quarter wave retarder
π φ φ θ
π
φ φ π
φ φ φ
− = = =
→
→ − =
→ Δ = − =
→ −
o
φ ΔTM φ
TE φ
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Linearlypolarized light
(45o)
CircularlyPolarized
light
69 ???2
2
TM TE i
TM TE
Note near when n
Quarter wave retarder
π φ φ θ
π φ φ φ
− = = =
→ Δ = − =
→ −
oφ ΔTM φ
TE φ
Quarter-wave retardation after TIR
n
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23-5. 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
( )
os 1it
t t
t t t
it t
iWhen n
x k z x x zz
k x z
But n
th
i a purely imaginary numbe
total internal reflection
r
en
n
θ
θ
θ θ
θ
θ θ
θ
θ
+ ⋅ +
= +
=
= −
− −
⇒
=
>
) ) )
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Evanescent Waves at an Interface
( )
2
2 2
sin
sin1
,
:
sin
:
sin sin21 1
:
t t t
it
i
i
t
t
iwith an TIR condition n
i z
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 θ π α
λ
θ
θ ⎛ ⎞⋅ = +⎜ ⎟
⎜ ⎟⎝ ⎠
=
−
− −
=
=
>
r r
( )0
sine p
.
xx pe t t t z
amplitude will decay rapid The evanescent wave
as it penetrates into the lower refractive ind
ly
k x E i 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
sinexp expt t
t t
k x E E i t z
n
θ ω α
⎡ ⎤⎛ ⎞= − −⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦
x
z
2
2
1
1
sin2 1
t ot
i
E E
e
h
n
λ
α θ π
⎛ ⎞= ⇒ = =⎟
⎝ ⎠ −
⎜Penetration depth:
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Frustrated TIR
Tp = fraction of intensitytransmitted across gap
Zhu et al., “Variable Transmission OutputCoupler and Tuner for Ring Laser Systems,” Appl. Opt. 24, 3610-3614 (1985).
d
n1=n2=1.517
1.65
d/λ
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Frustrated Total Internal Reflectance
Zhu et al., “Variable Transmission OutputCoupler and Tuner for Ring Laser Systems,” Appl. Opt. 24, 3610-3614 (1985).
Pellin-Broca prism
d = 1 ~ λ: changing the reflectanceRotation: changing the wavelength resonant at θB
d
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23-6. Complex Refractive Index
2 2 2
0
0
( ) :
1
1
2 R I R I
R I For a material with conductivity
n i n n i n
n i n i n
n
σ
ε ω σ
σ
ε ω
⎛
⎛ ⎞
= + = −
⎞= + = +⎜ ⎟
⎝ ⎠
+⎜ ⎟⎝ ⎠%
%
2
0
2
0
2 2
2 4 2
0 0
2
0
2
022
:
1 2
1 02 2
:
1 1
2
1 1 42
42
2 2
R
I
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 nn
From the quadratic solution we obt
n
n
n
ain
n
σ
ε ω
σ σ
ε ω ε ω
σ
ε
σ
ε
σ
ω
ω
ε ω
− = = ⇒
⎛ ⎞ ⎛ ⎞⇒ − = ⇒ − − =⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
⎛ ⎞± +
=
⎛ ⎞+ + ⎜ ⎟
⎝ ⎜⎝
=⇒
⎟ ⎠
=
. I We need to take the positive root because n is a real number
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Complex Refractive Index
( )
( ) ( )
( )
0
0
0
exp
ˆexp
eˆ xexp p Rk
R k
I
I
Substituting our expression for the complex refractive index back into
our expression for the electric field we obtain
E E i k r t
E i n i n u r t c
E n
i u r t c
n
cω
ω
ω
ω
ω
⎡ ⎤= ⋅ −⎣
⎧ ⎫⎡ ⎤⋅ − −
⎦
⎧ ⎫⎡ ⎤= + ⋅ −⎨ ⎬⎢ ⎥
⎨ ⎬⎢ ⎥⎣ ⎦
⎣ ⎦⎭
⎩
⎩
=⎭
rr
r
r r
r r
r
( )
.
.
ˆ
/
k
R
The first exponential term is oscillatory
The EM wave propagates w
u r
The second exponential has a r
ith a ve
eal arg
locity of
ument (absor .
n c
bed)
⎡ ⎤⋅⎢ ⎥⎣ ⎦
r
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Complex Refractive Index
( )* *
0 0
0
.
.
ˆ2exp
ˆ2exp
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
ˆexp
2 4:
k
I I
r I u r
c
The absorption coefficient is defined n n
c
ω π α
α
λ
⋅⎡ ⎤= − ⋅⎡
= =
⎤⎢ ⎥ ⎣ ⎦⎣ ⎦
rr
( ) ( )0ˆex ˆexp p I
k k R n
un
i u r c
r t c
E E ω ω ⎧ ⎫⎡ ⎤
⋅ −⎨ ⎬⎢ ⎥⎡ ⎤
− ⋅=⎣ ⎦⎩
⎢ ⎥⎣ ⎦⎭
rr r r
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23-7. Reflection from Metals
2 2
2 2
2 2 2
2 2 2
cos sin:
cos sin
cos sin:
cos sin
:
i ir TE
i i i
i ir TM
i i i
Reflection from metals is analyzed
E TE case r
E
n E TM cas
by substituting the complex refractive index n i
e r E n
S
n the Fresnel
n
n
equa
ub
ti
n
n
n
o s
s
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
R I
I i R I
TE case
TM case
tituting we obtain
n n i n n E r
E n n i n n
n n i n n n n i n n E r
E n n i n n n n
n n i
i n n
n
θ θ
θ θ
θ θ
θ θ
− − − += =
+ − − +
⎡ ⎤− − + + − − +⎣ ⎦= =⎡ ⎤− + + − +
=
⎣
+
−⎦
%
Reflectance
θ i
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Reflection from Metals at normal incidence (θi=0)
( )
( )
( )( )
( )( )
( )
( )
2 2
2 2
*
2 2
2 2
1 1 1 2
1 1
1
1
1
1
1
2
R I
R I R I R R I
R I R I R R
R I
R I
I
R I
The is given by
R r r
n i n n i n n n n
n i n n i n n n n
power refl
n i n
r n i
ectance R
n
n n
R n n
∴
=⎡ ⎤ ⎡ ⎤− − − + ⎛ ⎞− + +
= =⎢ ⎥ ⎢ ⎥ ⎜ ⎟+ − + + + + +⎝ ⎠
− +
= +
− −
= + −
⎦
+
⎣ ⎣ ⎦
2 2
2 2
2 2 2
2 2 2
cos sin 1
1cos sin
cos sin 1
1cos sin
i i
TE
i i
i i
TM
i i
nr
n
n n
n
n
nr
nn n
θ θ
θ θ
θ θ
θ θ
− − −= =
++ −
− + − −= =
++ −
%
%
% % %
%
%
% %
%
, 0 :i At normal incidence θ = ° At normal incidence(from Hecht, page 113)
visible