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Optoelectronics
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2016. 11. 17.
Changhee Lee
School of Electrical and Computer Engineering
Seoul National Univ.
Chapter 6. Optics of Solids
Part 4 – Propagation of light in crystals
Changhee Lee, SNU, Korea
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https://en.wikipedia.org/wiki/Plasmon
Propagation of light in crystals
Birefringence is the optical property of a material having a refractive index that depends on the
polarization and propagation direction of light. These optically anisotropic materials are said to be
birefringent (or birefractive). Crystals with non-cubic crystal structures are often birefringent, as
are plastics under mechanical stress.
A calcite crystal laid upon a
graph paper with blue lines
showing the double
refraction
https://en.wikipedia.org/wiki
/Birefringence
A model to illustrate the anisotropic polarizability of a crystal
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Optoelectronics
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333231
232221
131211
333231
232221
131211
tensordielectric
,
χ
χ)(1
Eχ)E(1D χEP
o
oo
z
y
x
o
z
y
x
E
E
E
P
P
P
ε
ε
Propagation of light in crystals
For ordinary nonabsorbing crystals the tensor is symmetric so there always exists a set of coordinate
axes, called principal axes, such that the tensor assumes the diagonal form
33
22
11
00
00
00
χ
The tensor assumes the diagonal form where three ’s are known as the principal susceptibilies.
constants dielectric principal ,1 ,1 ,1 333322221111 KKK
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2
2
2
2
2
2
2
2 ~~
)(tc
N
tc
N
Eχ
EE
Propagation of light in crystals
2
2
2
2
2
1)(
ttco
PEE
zzyxyyzxxz
yzzyyzxxxy
xzzxyyxxzy
Ec
Ec
kkEkkEkk
Ec
EkkEc
kkEkk
Ec
EkkEkkEc
kk
33
22
11
2
2
2
222
2
2
2
222
2
2
2
222
)(
)(
)(
For monochromatic plane wave of the usual form )( tie rk
χEEEEkk
χEEEkk
2
2
2
22
2
2
2
2
-)(
)(
cck
cc
Changhee Lee, SNU, Korea
Optoelectronics
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2016. 2nd Semester
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Propagation of light in crystals
axis in the points if velocity phase ,1 0, If
axis in the points if velocity phase ,1 0, If
)(
)(
axis) the toe transversis field (
33
33
22
22
2
2
2
22
2
2
2
22
2
2
2
2
33
22
33
22
11
zK
c
kK
cckE
yK
c
kK
cckE
Ec
Ec
k
Ec
Ec
k
xEEc
Ec
x
y
zz
yy
xxx
E
E
E
Consider a wave propagating in the direction of one of the principal axes, say the x axis. In this case
of the usual form 0 , zyx kkkk
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Optoelectronics
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2016. 2nd Semester
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Propagation of light in crystals
In the principal coordinate systems,
33
22
11
00
00
00
χ
33333
22222
11111
1
1
1
Kn
Kn
Kn
Introduce the three principal indices of refraction
0
)(
)(
)(
222
3
222
2
222
1
y
y
x
yxyzxz
zyzxxy
zxyxzy
E
E
E
kknc
kkkk
kkkknc
kk
kkkkkknc
0)1(-)(2
22 EχEEkk
ck
can be written as
Changhee Lee, SNU, Korea
Optoelectronics
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Propagation of light in crystals
ellipse)an of(equation 1)/()/(
circle) a of(equation )(
0)()()(
2
1
2
2
2
2
2
3
22
2222
2
22
1
222
3
cn
k
cn
k
nc
kk
kkknc
knc
kknc
yx
yx
yxxyyx
Consider any one of the coordinate planes, say the xy plane. In this plane 0zk
Similar equations are obtained for the xz and the yz planes. The intercept of the k surface with each
coordinate plane consists of one circle and one ellipse.
In order for a nontrivial solution 0) det(
0
)(
)(
)(
222
3
222
2
222
1
yxyzxz
zyzxxy
zxyxzy
kknc
kkkk
kkkknc
kk
kkkkkknc
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Optoelectronics
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2016. 2nd Semester
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Propagation of light in crystals
For any given direction of the wave vector k,
there are two possible values for k, i.e., two
values of the phase velocity corresponding to two
mutually orthogonal directions of polarization.
A light wave of arbitrary polarization can always
be resolved into two orthogonally polarized
waves, Thus, when unpolarized light, or light of
arbitrary polarization propagates through a
crystal, it can be considered to consist of two
independent waves that are polarized
orthogonally with respect to each other and
travelling with different phase velocities.
For propagation in the direction of an optic axis,
there is only one value of k. Thus, two
orthogonally polarized waves propagate in this
direction with the same phase velocity.
Saleh and Teich, pp.214-215
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Optoelectronics
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2016. 2nd Semester
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Propagation of light in crystals
Biaxial crystal Uniaxial positive crystal
no < ne
Uniaxial negative crystal
no > ne
Isotropic crystal, n=n1=n2=n3 example, cubic crystal
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• Uniaxial crystal: There is a single direction governing the optical anisotropy whereas all
directions perpendicular to it are optically equivalent. Thus rotating the material around this axis
does not change its optical behavior. This special direction is known as the optic axis of the
material.
• A ray with a linear polarization direction perpendicular to the optic axis is called an ordinary ray.
• A ray with a linear polarization in the direction of the optic axis is called an extraordinary ray.
• The ordinary ray will always experience a refractive index of no, whereas the refractive index of
the extraordinary ray will be in between no and ne, depending on the ray direction as described by
the index ellipsoid. The magnitude of the difference is quantified by the birefringence: Dn=ne – no
• The propagation (as well as reflection coefficient) of the ordinary ray is simply described by no as
if there were no birefringence involved. However the extraordinary ray, as its name suggests,
propagates unlike any wave in a homogenous optical material. Its refraction (and reflection) at a
surface can be understood using the effective refractive index (a value in between no and ne).
• Biaxial crystals are characterized by three refractive indices corresponding to three principal axes
of the crystal. For most ray directions, both polarizations would be classified as extraordinary rays
but with different effective refractive indices.
Propagation of light in crystals
https://en.wikipedia.org/wiki/Birefringence
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Optoelectronics
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Propagation of light in crystals
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https://en.wikipedia.org/wiki/Bravais_lattice
a = b = c
α = β = γ = 90°
a ≠ b ≠ c
α ≠ β ≠ γ ≠ 90°
a ≠ c
α = γ = 90°, β ≠ 90°
a ≠ b ≠ c
α = β = γ = 90°
a = b ≠ c
α = β = γ = 90°
a = b = c
α = β = γ ≠ 90°
a = b
α = β = 90°, γ = 120°
Crystals in three dimensional space
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Propagation of light in crystals
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Propagation of light in crystals
0)()(
)()()(
2222
2
222222
3
22
22222
3
222
2
222
1
zyxzxzyxyx
zyyxzxzy
kkknc
kkkkknc
kk
kkkknc
kknc
kknc
In general, we can expand the determinant as follows:
0
)(
)()(
2222222222222222
222
3
222
2
222
1
22222
3
22222
2
22222
12
2
222
1
2
3
222
3
2
2
222
2
2
1
2
2
22
3
2
2
2
1
3
2
2
zyxzyxzyzyyxxzzy
yxxzzyyzxzzyxyzxyx
xzzyyx
kkkkkkkkkkkkkkkk
kknkknkknkkkknkkkknkkkknc
kknnkknnkknnc
nnnc
222222
22222222
zyxzyx
zyyxxzzy
kkkkkk
kkkkkkkk
][][)(][][][)(
)(
2
3
2
2
22222
12
22
3
2
2
2222222
12
2
222
3
222
2
222
1
22222
3
22222
2
22222
12
2
nnkkkknc
nnkkkkkknc
kknkknkknkkkknkkkknkkkknc
zyxxzyzxyx
yxxzzyyzxzzyxyzxyx
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2016. 2nd Semester
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Propagation of light in crystals
(1) Isotropic crystal, n1=n2=n3=n
0)()( 222
2
2
2
1
2
2
3
2
1
2
2
3
2
2
2
2
2
22
2
1
22
2
3
22
2
22
2
2
zyx
zyxxzzyyxkkk
nn
k
nn
k
nn
k
n
kk
n
kk
n
kk
cc
2
3
2
2
2
12
2
)()equation Previous( nnnc
)degenerate(doubly 0)(
0)())((2)(
2
2
2
2
2
2
22
2
22
2
2
nc
kn
k
c
n
k
n
k
cc
(2) Uniaxial crystal, n1=n2=no, n3=ne
)( ),( ,0)()(
0)()(
2
222
2
2
2
2
2
22
2
2
2
2
2
2
2
2
2
22
2
2
2
22
2
2
2
22
2
22
2
22
2
22
2
2
cnk
cn
k
n
kk
cn
k
cn
k
n
kk
n
k
n
kk
n
k
n
kk
n
kk
n
kk
cc
o
o
z
e
yx
oo
z
e
yx
o
z
e
yx
oo
zy
o
zx
e
yx
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Optoelectronics
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2016. 2nd Semester
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Alternative form
0)()(
)()()(
2222
2
222222
3
22
22222
3
222
2
222
1
zyxzxzyxyx
zyyxzxzy
kkknc
kkkkknc
kk
kkkknc
kknc
kknc
We define effective refractive index:
0)()()(
)()()(
22222
2
2222
3
2222
1
22
222
3
222
2
222
1
zyxzxyxzy
zyx
kkkknc
kkknc
kkknc
kk
kknc
kknc
kknc
0)()()()(
)()()()()(
22
2
22
1
222
1
22
3
2
22
3
22
2
222
3
22
2
22
1
knc
knc
kknc
knc
k
knc
knc
kknc
knc
knc
zy
x
1
)()()( 2
3
2
2
2
2
2
2
2
1
2
2
n
ck
k
nc
k
k
nc
k
k zyx
22
3
2
2
2
2
2
2
2
1
2
21
nnn
s
nn
s
nn
s zyx
kcn
),,(),,( zyxzyx ssskkkk
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Propagation of light in uniaxial crystals
Saleh and Teich, p.218
2
2
2
2
2
sincos
)(
1
eo nnn
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The Poynting vector and the ray velocity
SkES
Dk
PED 0,D
toparallenot is
o
cos
vty ray veloci
u
o
HES
DHk
HEk
2
2
2
2
22
2
22
2
2
vcos
-)(
)(
Dc
ED
ck
ck
c
o
o
o
o
DE
DDDE
DEEkk
DEkk
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Propagation of light in crystals
Saleh and Teich, p.219
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Propagation of light in uniaxial crystals
Saleh and Teich, p.219
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Phase-velocity surface
0
/
/
/
22242
3
22242
2
22242
1
2
yxyzxz
zyzxxy
zxyxzy
vvcvnvvkv
vvvvcvnvv
vvvvvvcvn
v
vk
2
4
2
1
2
2
2
2
2
3
2222
vvv
vvv
plane, For the
cnn
n
c
xy
yx
yx
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The Poynting vector and the ray velocity
If the coordinate axes are principal axes of the crystal,
, ,2
333
2
222
2
111 n
DDE
n
DDE
n
DDE zz
zo
yy
yoxx
xo
0)(
0)(
0)(
22
2
3
2
22
2
2
2
22
2
1
2
yxzyzyxzx
zyzzxyxyx
zxzyxyzyx
uun
cDuuDuuD
uuDuun
cDuuD
uuDuuDuun
cD
0
/
/
/
222
3
2
222
2
2
222
1
2
yxyzxz
zyzxxy
zxyxzy
uuncuuuu
uuuuncuu
uuuuuunc
ellipse)(an circle) (a 222
1
22
22
3
222 cunun
n
cuu yxyx
The equations of the intercepts in the xy plane are obtained by setting 0zu
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6.8 Double refraction (birefringence)
boundary)(at rkrk o
21
11
sinsin
sinsin
kk
kk
o
o
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6.8 Double refraction (birefringence)
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6.8 Double refraction (birefringence)
eee
ooo
o
o
n
n
nknk
kk
sin)(sinfor which at (TM)on polarizati parallel of ary waveextraordinAn
sinsinfor which at (TE)on polarizati orthogonal of aveordinary wAn
crystal, uniaxial afor wavesrefracted Two
sin)(sin ,)( medium, canisotropian In
sinsin condition matching-phase
1
1
1
1
Saleh and Teich, p.221
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6.8 Double refraction (birefringence)
Saleh and Teich, p.222
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6.8 Double refraction (birefringence)
Saleh and Teich, p.222
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Polarizing prisms
Suppose we have a negative uniaxial crystal, such
as calcite, and the internal angle of incidence is such
that
In this case we have total internal reflection for the
ordinary wave but not for the extraordinary wave.
Thus, the refracted wave is completely polarized.
oe nn sin
1
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Polarizing prisms
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6.9 Optical activity
Optical activity = The ability to rotate the plane of polarization of light passing through the medium.
When a beam of linearly polarized light is passed through an optically active medium, the light
emerges with its plane of polarization turned through an angle that is proportional to the length of the
path of the light through the medium.
Specific rotatory power = the amount of rotation per unit length of travel.
If the sense of rotation of the plane of polarization is to the right, as a right-handed screw pointing in
the direction of propagation, the substance is called dextrorotatory or right-handed. If the rotation is to
the left, the substance is called levorotatory or left-handed.
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6.9 Optical activity
)(power rotatory specific
)(2
)(
RR
RRRR
nn
lnn
c
lnn
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6.10 Faraday rotation in solids
constant)Verdet ( VVBl
The presence of the field causes the dielectric to become optically active. This phenomenon was
discovered in 1845 by Michael Faraday.
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6.11 Other magneto-optic and electro-optic effects
Kerr electro-optic effect: When an optically
isotropic substance is placed in a strong
electric field, it becomes doubly refracting,
discovered in 1875 by J. Kerr.
- 2
|| oλKEnn
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6.11 Other magneto-optic and electro-optic effects
Cotton-Mouton effect: Magnetic analogue of the Kerr electro-optic effect.
Pockels effect: When certain kinds of birefringent crystals are placed in an electric field, their
indices of refraction are altered by the presence of the field.
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6.12 Nonlinear Optics (NLO)