1 optical properties of materials … reflection … refraction (snell’s law) … index of...
TRANSCRIPT
1
Optical Properties of Materials
0
1
1
0
0
1
1
0
00
sin
sin
sin
sin
v
vv
cn
n
n
… reflection
… refraction (Snell’s law)
… index of refraction
Index of refraction
Absorption
2
Maxwell’s Equations
HμμHμB
EεεEεD
Ej
DDr
t
BEE
r
r
r
0
0
div
rot
… electric field … magnetic field … electric displacement field … magnetic induction … current density … electric charge density … electrical conductivity … permittivity … permeability
Materials equations
3
Maxwell’s Equations
0div0
divdiv
rot
0
0
E
EεD
t
HμE
r
r
0div
rot 0
H
Et
EεH r
… no free charge
t
Eσ
t
EεεμμE
r
E
r
EEE
t
E
t
Eε
t
H
t
HμE
rr
r
r
2
2
002
2
2
2
0
2
2
0
0
rotrot
divgradrotrot
rot
rotrotrot
… wave equation
The Wave Equation
4
rrμεv
cn
2
22
t
Eσ
t
EεεμμE
r
Err
2
2
002
2
Et
E
Eit
E
22
2
Ekr
E
22
2
00
1
c
trkiEE
exp0
2
2220
2
ck
EσiωωεεμEk rr
2
002
ωε
σμinkk
ωε
σμin
c
ωk
ωσμiμωc
nk
r
r
r
0
220
2
0
22
22
02
2
22
ωε
σiεεεn
k
kμr
0
220
2~~~1
5
Refraction and Absorption
k … wave vector
… angular frequency
c … velocity of light
n … index of refraction
… electrical conductivity
Complex permittivity:permittivity and losses
Complex index of refraction:refraction and absorption
nn
inin
in
ininn
inn
i
2;
2
~~
2~
~
~
02
221
01
22
212
2222
0
6
Amplitude and Intensity of the Propagating Wave
rkrkntirkntiEI
EEEI
rkrkntiEE
inrktiEE
inkk
rktiEE
000
2
0
2
absorption
0
wavegpropagatin
00
00
0
0
2expexpexp
expexp
exp
exp
7
Relationship between Dielectric and Optical Constants
210
12
2222
~~
2~
iin
inninn
00
02
221
42
2
nn
n
n
122
212
11
2
0
21
2
122
212
11
2
0
21
2
2
1
2
1
n
* dielectric constant = permittivity
Insulator
8
nn
nn2
200 0;024
0 … non-conducting material
… no absorption, no losses
… the index of refraction is a real quantity
9
Penetration Depth
zc
IzkII
zknzktinzktiEI
EEEI
zknzktiEE
2exp2exp
2expexpexp
expexp
000
000
2
0
2
absorption
0
wavegpropagatin
00
4421
2
12exp
1:
00
0
cczz
c
Ie
zc
II
Ie
Iz
ee
e
… dependent on frequency (wavelength) and absorption
0
04
cnz
n
e
10
Penetration Depth and Absorption(Examples)
* absorption = damping
11
Reflection and Transmission
𝜃i 𝜃r
𝜃t
1
2
vexp
2exp
exp
0
0
0
rstiEE
rstiEE
rktiEE
111
211
sinsinsin
sin
v
θ
v
θ
v
θ
θs
v
s
v
s
v
s
tri
x
(t)x
(r)x
(i)x
121
2
11
22
2
1
sin
sin
sinsin
nn
n
με
με
v
v
θ
θ
θθ
t
i
ri
Reflection:
Transmission:(Snell’s law)
Same amplitude and phase of wave at the point “0”
12
Electric and Magnetic Field
The original wave:
11
)(
1)(
1||)(
1)(
||)()(
||)(
v
cossin
v
sincos
sincos
iii
i
ii
iz
iiy
ii
ix
ii
iz
iiy
ii
ix
zxt
rst
eAHeAHeAH
eAEeAEeAE
iii
iii
𝐼 𝑅
𝑇
EsH
The vectors of the electric and magnetic fields are perpendicular to the propagation direction of the wave.
𝜃i𝜃r
EsHE
𝒔𝑯
𝑬
13
Electric and Magnetic FieldThe transmitted wave:
22
222
cossin
sincos
sincos
v
θzθxtω
v
rstωτ
eεθTHeεTHeεθTH
eθTEeTEeθTE
tt(t)
t
iτt
(t)z
iτ||
(t)y
iτt
(t)x
iτt||
(t)z
iτ(t)y
iτt||
(t)x
ttt
ttt
The reflected wave:
11
)(
1)(
1||)(
1)(
||)()(
||)(
v
cossin
v
sincos
sincos
rrr
r
ir
rz
iry
ir
rx
ir
rz
iry
ir
rx
zxt
rst
eRHeRHeRH
eREeREeRE
rrr
rrr
14
Fresnel Equations… are obtained from the boundary conditions: Tangential components of
and have to be continuous at the interface (surface).
)t(y
)r(y
)i(y
)t(x
)r(x
)i(x
)t(y
)r(y
)i(y
)t(x
)r(x
)i(x
HHHHHH
EEEEEE
TRA
TRA
TRA
TRA
2||||1
t2i1
t||i||||
coscos
coscos
15
Fresnel Coefficients
Ann
nnR
Ann
nnR
Ann
nT
Ann
nT
ti
ti
ti
ti
ti
i
ti
i
coscos
coscos
coscos
coscos
coscos
cos2
coscos
cos2
21
21
||12
12||
21
1
||12
1||
ti
ti
ti
ti
ti
i
ti
i
nn
nnr
nn
nnr
nn
nt
nn
nt
coscos
coscos
coscos
coscos
coscos
cos2
coscos
cos2
21
21
12
12||
21
1
12
1||
it
ti
nnn
nn
221
22
2
21
sin1
cos
sinsin
Snell
ii
ii
inn
i
inn
i
ii
i
inn
i
i
nnn
nnnr
nnn
nnnr
nnn
nt
nnn
nt
221
221
221
221
221
222
221
222
||
221
221
1
221
222
1||
sincos
sincos
sincos
sincos
sincos
cos2
sincos
cos2
2
1
2
1
2
1
16
Index of Refraction(Experimental Examples)
17
Materials with different refractive indices are very important for complex optical systems
18
Transmission and Reflection
0
2
0
2
0
2
||
t
r
tII
rII
EEEI
R
T
R
Brewster angle – complete polarization of reflected electromagnetic wave (polarization of light)
Vacuum Glass (n=1,5)
RR
RRP
RRR
||
||
||21
Vacuum Glass: n=1.5
19
Transmission and Reflection
Vacuum Germanium (n=5,3)
Vacuum Germanium: n=5,3
20
Optical Reflection
Glass (n=1,5) Vacuum
Total internal reflection
Glass (n=1,5) Vacuum
21
Total Internal Reflection
1
2
2
1
2
1
21
21
arcsin
1sinsin
1sinsin
sinsin
n
n
n
n
n
n
nn
nn
c
tc
ti
ti
n1
n2
c
Glass (n = 1,5): c = 41,8°
Water (n = 2): c = 30°
22
Transmission and Reflectionwith Complex Index of Refraction
23
Transmission and Reflectionwith an Incident Angle of 0°
ii
ii
inn
i
inn
i
ii
i
inn
i
i
nnn
nnnr
nnn
nnnr
nnn
nt
nnn
nt
221
221
221
221
221
222
221
222
||
221
221
1
221
222
1||
sincos
sincos
sincos
sincos
sincos
cos2
sincos
cos2
2
1
2
1
2
1
21
21
12
12||
21
1
12
1||
22
0sin1cos0
nn
nnr
nn
nnr
nn
nt
nn
nt
iii
rr
tt
||
||
2
21
21
nn
nnR
2
1
1
n
nRInterface material - vacuum:
24
Table 11.2Refractive index and absorption index of some materials with nm
n
4
… absorption index… absorption coefficient… index of refraction… wavelength
25
Transmission and Reflection with Complex Index of Refraction
Copper
n = 0.14
k = 3.35
R = 95.6 %
Vacuum Copper (n=0.14-3.35i)
26
Transmission and Reflection with Complex Index of Refraction
Sodium
n = 0.048
k = 1.86
R = 95.8 %
Vacuum Sodium (n=0.048-1.86i)
27
Transmission and Reflection with Complex Index of Refraction
Gallium
n = 3.69
k = 5.43
R = 71.3 %
Vacuum Gallium (n=3.69-5.43i)
28
Transmission and Reflectionwith Complex Index of Refraction
Cobalt
n = 2.0
k = 4.0
R = 68.0 %
Vacuum Cobalt (n=2.0-4.0i)
29
30
Reflection with Complex Index of Refraction
22
222
1
1
11
11
1
1
n
n
inin
inin
n
nR
Influence of absorption (weakening, damping)
on the reflection
31
Reflection with Complex Index of Refraction
Total external reflection vanishes
32
Reflectivity as function of Refractive Index and Absorption
Reflectivity increases with increasing index of refraction and
an increasing absorption index
Fig. 11.2Reflectivity as function of absorption and refractive index
33
Refractive Index as function of Wavelength
Color of Materials
(Sphalerite)
(Rutile)
Material
Fig. 11.5Refractive index as function of absorption index and absorption coefficient as function of wavelength for Si (a), KCl (b) and Cu (c).
34
Reflection and Transmissionof a Thin Film
3322
332223
3322
2223
2211
221112
2211
1112
coscos
coscos
coscos
cos2
coscos
coscos
coscos
cos2
nn
nnr
nn
nt
nn
nnr
nn
nt
Fresnel coefficients at the interfaces:
22
11
33
22312
22312
22312
2312
cos
cos
11
rtn
n
err
errr
err
ettt
i
i
i
i
RT
cos
2cos0 nttnkk Phase shift:
35
Reflection and Transmissionof a Thin Film
Constant wavelength (monochromatic radiation)
Thickness of the film is ten times of the wavelength
Reflection
Vacuum Glass (n = 1.5, t = 6 μm) Vacuum, λ = 600 nm
Angle of incidence (degree)
Inte
nsi
ty (
%)
36
Reflection and Transmissionof a Thin Film
Constant wavelength (monochromatic radiation)
Thickness of the film is two times of the wavelength
Reflection
Vacuum Glass (n = 1.5, t = 1.2 μm) Vacuum, λ = 600 nm
Angle of incidence (degree)
Inte
nsi
ty (
%)
37
Reflection and Transmissionof a Thin Film
Constant wavelength (monochromatic radiation)
Thickness of the film is 40 times of the wavelength
Reflection
Vacuum Glass (n = 1.5, t = 24 μm) Vacuum, λ = 600 nm
Angle of incidence (degree)
Inte
nsi
ty (
%)
38
Reflection and Transmissionof a Thin Film
Different wavelengths (polychromatic radiation)
Thickness of film is 1.2 m
Different “Colors” are reflected and transmitted differently.
Vacuum Glass (n = 1.5, t = 1.2 μm) Vacuum, λ = 300-600 nm
Angle of incidence (degree)
Inte
nsi
ty (
%)