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ENE 451Fundamental of Optical Engineering
Lecture 7
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Reflection at Plane Dielectric Interface
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Boundary conditions:E and T must be continuous.
Region 1:
Reflection at Plane Dielectric Interface
ˆ ˆx y z
i xi zi
r xe ye ze
r x z
11
1 1
2
sin , cos
xi zi xr zii x i z i x i zi r
i r
xi zi
r xr zr
E E e E e
n
I I
r x z
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Region 2:
As we know that E is continuous.
Reflection at Plane Dielectric Interface
22
2 2
2
sin , cos
xt zti x i zt
t
xt zt
E E e
n
I I
xi xr xti x i x i xi r tE e E e E e
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Therefore,
Reflection at Plane Dielectric Interface
1
2
1 2
1 2
sin
sin
sin sin
sin sin
xi xr
xt
I
I
I I
n I n I
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Total internal reflection
1 2 1 2
2
1
1 2
1
If and sin
sin
sin c
n n n I n
nI
n
nI I
n
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For I ≥ Ic implies “no refracted wave”. This allows light to propagate with no loss.
This result does not depend on polarization and the wave.
For I < Ic I, there will be a reflected wave and the refracted
The ratio of Er or Et to Ei depends on the polarization (direction of Ei).
Total internal reflection
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Total internal reflection
Plane of incidence is defined by the propagation vector of incident wave and normal to the plane of the interface.
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Total internal reflection
Consider 2 cases: (i) Ei normal to plane of incidence.
This is called ‘s-polarization’ or ‘perpendicular polarization’.
ˆi ix yE E e
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Total internal reflection
(ii) Ei in plane of incidence.
This is called ‘p-polarization’ or ‘perpendicular polarization’.
ˆ ˆi ix x iz zE E e E e
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1
2
3
4
Fresnel’s equations
2
2
tan
tan
2cos sin
sin cos( )
sin
sin
2cos sin
sin
Reflectivity =
rp
i p
tp
i p
rs
i s
ts
i s
r
i
I IEr
E I I
E I It
E I I I I
I IEr
E I I
E I It
E I I
Er R
E
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For normal incidence
Fresnel’s equations
1 2
1
2
1
2
1
2
2 1
2 1
tanFrom #1: and R sin sin
tan
0 :
p
p
p
I Ir n I n I
I I
nI I I
n
nI InI I
rnI I I In
n nr
n n
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We can also show from #3 that
Fresnel’s equations2
2 2 1
2 1
At , 1
p p
c p s
n nR r
n n
I I R R
2
2 1
2 1
At normal incidence I 0,
s
s p
n nR
n n
R R
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The angle that makes no light reflection for p-polarization.
Maximum polarization occurs at this angle for reflected light.
Light reflected at any other angle but Brewster’s is partially linearly polarized.
Brewster’s angle
90
tan 90
0: no reflection for p-polarization
B
p
I I
R
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Note: There is no Brewster’s angle for s-polarization.
Brewster’s angle
1 2
2
2
2
1
1 2
1
sin sin
sin 90
cos
tan
tan
B
B
B
B
B
n I n I
n I
n I
nI
n
nI
n
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Calculate Brewster’s angle for light traveling from a medium of refractive index 1.81 into a medium of index 1.52.
Example
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What is the angle of incidence for complete polarization to occur on reflection at the boundary between water (n=4/3) and glass (n=1.589) assuming the light comes from (a) water and (b) glass.
Example
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Unpolarized light Linearly polarized light Partially linearly polarized light Circular polarization Elliptical polarization
Polarization of light
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Birefringent median can transform polarization.
Birefringent media have different refractive indices for orthogonal polarizations.
Polarization of light
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Let us consider
Polarization of light
0
0
0
0
0
ˆ-Suppose
ˆAfter length L:
ˆ ˆ-Suppose
ˆ ˆAfter length L: 2
22,
ˆ ˆ2
x
yx
x yx
inc x
i Linc x
inc x y
i Li Linc x y
yxx y
i L Li Lout x y
E E e
E E e e
E E e e
EE e e e e
nn
EE e e e e
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The output polarization is the same as incident one.
This is called “Full-wave”.
Polarization of light
0
Define phase retardation
If 2 ; 0, 1, 2,...
then 1
ˆ ˆ2
x y
x
x y
i L L
i Lout x y
L L
n n
e
EE e e e
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This is called “Half-wave”.
Polarization of light
0
If 2 1 ; 0, 1, 2,...
then 1
ˆ ˆ2
x y
x
i L L
i Lout x y
n n
e
EE e e e
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This is called “quarter-wave”.
Polarization of light
/ 2
0
1If 2 ; 0, 1, 2,...
2
then
ˆ ˆ2
x y
x
i L L i
i Lout x y
n n
e e i
EE e e ie
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After leaving a crystal
Consider real part of an electric field by assuming
Polarization of light
0
0
ˆ ˆ2
ˆ ˆRe( ) cos sin2
xi Lx y
x y
EE e e ie e
EE ze ze
1xi Le
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Polarization of light
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Polarization of light
0 ˆ ˆ ; 12
i zx y
EE e iKe e K
Elliptrical polarization
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These are components which transform polarization.
Wave plates