Download Complex index of refraction

Optical Properties of Materials
Index of refraction
 0   0
sin  0 n1

sin 1 n0
n
c
v
… reflection
… refraction
(Snell’s law)
… index of refraction
sin  0 v1

sin 1 v0
Absorption
1
Maxwell’s Equations




B

E

rot
E



r
t

 
  D  div D  
r
Materials equations


j  E



D  εE  ε 0 ε r E



B  μH  μ0 μ r H
¶
¶D
´ H = rot H =
+j
¶r
¶t
¶
× B = div B = 0
¶r
𝐸 … electric field
𝐻 … magnetic field
𝐷 … electric displacement field
𝐵 … magnetic induction
𝑗 … current density
𝜌 … electric charge density
𝜎 … electrical conductivity
𝜖 … permittivity
𝜇 … permeability
2
Maxwell’s Equations


H
rot E    0 μr
t


div D   0 εr div E  

  0  div E  0 … no free charge



E
rot H   0 εr
 E
t

div H  0


H
rotrot E    0 μr rot
t



2
H
 E
E
rot
  0 εr 2  
t
t
t


 2E
rotrot E  graddiv
E  2

r
0



2
2


 E
 E
E 

 … wave equation
 2   rotrot E  μ0 μr  ε0 εr 2  σ
r
t
t 

3
The Wave Equation



2


 E
 E
E 


 2  E  μ0 μr  ε0 εr 2  σ
r
t
t 


 
 
E  E0 exp i k  r  t
2




2
 k E    0 μ r ε 0 ε r ω  iω σ E
2
n2 2
k  2 ω  iμ0 μr ωσ
c
ω2  2
μσ
2
k  2  n  i r 
c 
ε0 ω 
2

μσ
k 2  k02  n 2  i r 
ε0 ω 



 E
2
 k E
2
r
2
c
1
 0 0
c2
n  2  ε r μr
v
2
2
 2  
2
k0  
  2
c
  
2



E
 iE
t

2

 E
2
  E
t 2
k 2 ~2 ~ ~
σ
μr  1  2  n  ε  ε  ε  i
k0
ε0 ω
4
Refraction and Absorption

 0
n~  n  i
~    i
2
n~ 2  n  i   n 2  2i   2
n~ 2  ~   1  i 2

 0

1  n 2   2 ;  2 
 2n
 0
n 2  2in   2   1  i
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
5
Amplitude and Intensity
of the Propagating Wave

 
E  E0 exp i t  k  r
k  k0 n  i 




 
 
E  E0 exp i t  k0  r n  i 
 
 
E  E0 exp i t  nk0  r  exp  k0  r
 
 

propagating wave
I  E  E  E

absorption
2
I  E0
2

  


 
 
 
exp i t  nk0  r  exp  i t  nk0  r  exp  2k0  r

6
Relationship between
Dielectric and Optical Constants
1  n 2   2

2 
 2n
 0
  2n 0  4n 0
2
n~ 2  n  i   n 2   2  2in

n~ 2  ~  1  i
  1  i 2
 0
2




1


   1   12  12   22   1
n 2    12  
2
 0 




2




1


   1   12  12   22   1
 2    12  
2
 0 








7
* dielectric constant = permittivity
Insulator
 0
  4n 0  2n 0    0;  2  0
… non-conducting material
  n2  n  
… the index of refraction is a real
quantity
… no absorption, no losses
8
Penetration Depth
E  E0 expi t  k0nz  exp k0 z 
 
propagating wave
absorption
I  E  E  E
2
I  E0 expi t  k0nz  exp i t  k0nz  exp 2k0 z 
2
 2 
I  I 0 exp 2k0 z   I 0 exp 
z
c


1
ze : I  I 0
e
 2  1
I  I 0 exp  
z   I0
c  e

2
c
c

ze  1  ze 


c
2 4 4

40 n
cn 0
ze 


… dependent on frequency
(wavelength) and absorption
9
Penetration Depth and Absorption
(Examples)
𝑊  𝑧e
𝑘𝜅
* absorption = damping
10
Reflection and Transmission

 
E  E0 exp i t  k  r
1
𝜃i
2

  2   
E  E0 exp i  t 
s  r 

  
 
  s  r 
E  E0 exp i  t 

v 
 
𝜃r
𝜃t
Same amplitude and phase of
wave at the point “0”
s (i)
s (r)
s (t)
x
x

 x
v1
v1
v2
s x  sin θ
sin θi sin θr sin θt


v1
v1
v1
Reflection:
Transmission:
(Snell’s law)
sin θi  sin θr
sin θi v1
ε μ
n
  2 2  2  n12
sin θt v2
ε1 μ1 n1
11
Electric and Magnetic Field

 
H  sE
The vectors of the electric and magnetic
fields are perpendicular to the propagation
direction of the wave.
   
EH s E
𝑬
𝐼
𝒔
𝜃i 𝜃r
𝑅
𝑇
𝑯
The original wave:
E x(i )   A|| cos  i e i i
E y(i )   A e i i
H x(i )   A cos  i 1 e i i
E z(i )  A|| sin  i e i i
H y(i )   A|| 1 e i i
 s (i )  r 


    t  x sin  i  z cos  i 
 i    t 


v1 
v1



H z(i )  A sin  i 1 e i i
12
Electric and Magnetic Field
The transmitted wave:
E x(t)  T|| cos θt e  iτ t
 iτ t
E (t)

T
e
y

E z(t)  T|| sin θt e  iτ t
iτ t
H x(t)  T cos θt ε2 e iτ t H (t)
y  T|| ε 2 e
 
 s (t)  r 
 x sin θt  z cos θt 
  ω t 

τ t  ω t 
v2 
v2



H z(t)  T sin θt ε2 e iτ t
The reflected wave:
E x( r )   R|| cos  r e i r
E y( r )  R e i r
E z( r )  R|| sin  r e i r
H x( r )   R cos  r 1 e i r H y( r )   R|| 1 e i r
 s ( r )  r 


    t  x sin  r  z cos  r 
 r    t 


v1 
v1



H z( r )  R sin  r 1 e i r
13
Fresnel Equations
… are obtained from the boundary conditions: Tangential components of 𝐸
and 𝐻 have to be continuous at the interface (surface).
Ex( i )  Ex( r )  Ex( t )
H x( i )  H x( r )  H x( t )
Ey( i )  Ey( r )  Ey( t )
H y( i )  H y( r )  H y( t )
A  R cos
||
||
i
 T|| cos  t
A  R  T
 A  R  1 cosi  T  2 cos t
1 A||  R||    2 T
14
Fresnel Coefficients
T|| 
2n1 cos i
A||
n2 cos i  n1 cos t
t|| 
2n1 cos i
n2 cos i  n1 cos t
T 
2n1 cos i
A
n1 cos i  n2 cos t
t 
2n1 cos i
n1 cos i  n2 cos t
R|| 
n2 cos i  n1 cos t
A||
n2 cos i  n1 cos t
r|| 
n2 cos i  n1 cos  t
n2 cos i  n1 cos  t
r 
n1 cos i  n2 cos  t
n1 cos i  n2 cos t
Snell
n cos i  n2 cos t
R  1
A
n1 cos i  n2 cos t
n1 sin  i  n2 sin  t
cos  t 
t|| 
2n1 cos i
1
n22  n12 sin 2  i
n2
n2 cos i  n1 n22  n12 sin 2 i
n
t 
2
n2 cos i  n1 n22  n12 sin 2 i
n
r|| 
2
n2 cos i 
n1
n2
n22
 n12 sin 2  i
r 
2n1 cos i
n1 cos i  n22  n12 sin 2  i
n1 cos i  n22  n12 sin 2 i
n1 cos  i  n22  n12 sin 2 i
15
Index of Refraction
(Experimental Examples)
16
Materials with different
refractive indices are very
important for complex optical
systems
17
Transmission and Reflection
I  E E  E
R  Ir I0  r
T  It I0  t
Vacuum  Glass: n=1.5
2
2
2
R ||  0
Brewster angle – complete
polarization of reflected
electromagnetic wave
(polarization of light)
R 
P
1
2
R ||  R  
R ||  R 
R ||  R 
Vacuum  Glass (n=1,5)
18
Transmission and Reflection
Vacuum  Germanium: n=5,3
Vacuum  Germanium (n=5,3)
19
Optical Reflection
Glass (n=1,5)  Vacuum
Total internal
reflection
Glass (n=1,5)  Vacuum
20
Total Internal Reflection
n1  n2
n1 sin  i  n2 sin  t
n1
sin  i  sin  t  1
n2
n1
sin  c  sin  t  1
n2
 c  arcsin
n2
c
n2
n1
n1
Glass (n = 1,5): c = 41,8°
Water (n = 2): c = 30°
21
Transmission and Reflection
with Complex Index of Refraction
22
Transmission and Reflection
with an Incident Angle of 0°
t|| 
2n1 cos i
n2 cos i  n1 n22  n12 sin 2 i
n
t 
2
n2 cos i  n1 n22  n12 sin 2 i
n
r|| 
2
n2 cos i 
n1
n2
n22
 n12 sin 2  i
r 
2n1 cos i
n1 cos i  n22  n12 sin 2  i
n1 cos i  n22  n12 sin 2 i
n1 cos  i  n22  n12 sin 2 i
i  0 cos i  1 sin i  0
2n1
n2  n1
n n
r||  2 1
n2  n1
t|| 
n n
R 1 2
n1  n2
2
2n1
n1  n2
n n
r  1 2
n1  n2
t 
t||  t
r||  r
n 1
Interface material - vacuum: R 
n 1
2
23
Table 11.2
Refractive index 𝑛 and absorption index 𝜅
of some materials with 𝜆 = 589 nm


4n
𝜅… absorption index
𝛼… absorption coefficient
𝑛… index of refraction
… wavelength
24
Transmission and Reflection
with Complex Index of Refraction
Vacuum  Copper (n=0.14-3.35i)
Copper
n = 0.14
k = 3.35
R = 95.6 %
25
Transmission and Reflection
with Complex Index of Refraction
Vacuum  Sodium (n=0.048-1.86i)
Sodium
n = 0.048
k = 1.86
R = 95.8 %
26
Transmission and Reflection
with Complex Index of Refraction
Vacuum  Gallium (n=3.69-5.43i)
Gallium
n = 3.69
k = 5.43
R = 71.3 %
27
Transmission and Reflection
with Complex Index of Refraction
Vacuum  Cobalt (n=2.0-4.0i)
Cobalt
n = 2.0
k = 4.0
R = 68.0 %
28
29
Reflection with Complex Index of Refraction

n  1
n  i  1  n  i  1 n  12   2
R 


n  i  1  n  i  1 n  12   2
n  1
2
Influence of absorption
(weakening, damping)
on the reflection
30
Reflection with Complex Index of Refraction
Total external reflection vanishes
31
Reflectivity as function of Refractive Index
and Absorption
Reflectivity increases with
increasing index of refraction and
an increasing absorption index
Fig. 11.2
Reflectivity as function of absorption and
refractive index
32
Refractive Index as function of Wavelength
Material
(Sphalerite)
Color of
Materials
(Rutile)
Fig. 11.5
Refractive index as function of absorption index and absorption coefficient as
function of wavelength for Si (a), KCl (b) and Cu (c).
33
Reflection and Transmission
of a Thin Film
Fresnel coefficients at the
interfaces:
t
t12t23ei
1  r12 r23e 2i
T 
r
r12  r23e 2i
1  r12 r23e 2i
n3 cos 3 2
t
n1 cos 1
Phase shift:   k  nk0t cos  
R r
2


t12

2n1 cos 1
n1 cos 1  n2 cos  2
r12 
n1 cos 1  n2 cos  2
n1 cos 1  n2 cos  2

t23

2n2 cos  2
n2 cos  2  n3 cos 3

r23

n2 cos  2  n3 cos 3
n2 cos  2  n3 cos 3
2
nt cos 
34
Reflection and Transmission
of a Thin Film
Intensity (%)
Vacuum  Glass (n = 1.5, t = 6 μm)  Vacuum, λ = 600 nm
Constant wavelength
(monochromatic
radiation)
Reflection
Thickness of the film
is ten times of the
wavelength
Angle of incidence (degree)
35
Reflection and Transmission
of a Thin Film
Intensity (%)
Vacuum  Glass (n = 1.5, t = 1.2 μm)  Vacuum, λ = 600 nm
Constant wavelength
(monochromatic
radiation)
Reflection
Thickness of the film
is two times of the
wavelength
Angle of incidence (degree)
36
Reflection and Transmission
of a Thin Film
Vacuum  Glass (n = 1.5, t = 24 μm)  Vacuum, λ = 600 nm
Intensity (%)
Constant wavelength
(monochromatic
radiation)
Reflection
Thickness of the film
is 40 times of the
wavelength
Angle of incidence (degree)
37
Reflection and Transmission
of a Thin Film
Vacuum  Glass (n = 1.5, t = 1.2 μm)  Vacuum, λ = 300-600 nm
Different
wavelengths
(polychromatic
radiation)
Intensity (%)
Thickness of film
is 1.2 m
Different “Colors”
are reflected and
transmitted
differently.
Angle of incidence (degree)
38