fundamental of optical engineering lecture 7. boundary conditions:e and t must be continuous. ...
TRANSCRIPT
ENE 451Fundamental of Optical Engineering
Lecture 7
Reflection at Plane Dielectric Interface
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
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
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
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
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
Total internal reflection
Plane of incidence is defined by the propagation vector of incident wave and normal to the plane of the interface.
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
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
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
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
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
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
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
Calculate Brewster’s angle for light traveling from a medium of refractive index 1.81 into a medium of index 1.52.
Example
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
Unpolarized light Linearly polarized light Partially linearly polarized light Circular polarization Elliptical polarization
Polarization of light
Birefringent median can transform polarization.
Birefringent media have different refractive indices for orthogonal polarizations.
Polarization of light
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
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
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
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
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
Polarization of light
Polarization of light
0 ˆ ˆ ; 12
i zx y
EE e iKe e K
Elliptrical polarization
These are components which transform polarization.
Wave plates