Nonlinear Notes 1
Optical Effect we experience typically in nature are linear
Absorption
A =
cl
Raman scattering
I
c
Linearly proportional: Light interacts with matter in proportion to its concentration
1961 Franken at Univ of Michican Ruby La
ser focused on a crystal 1st nonlinear
observation
Transitted light consisted of both wavelenths both 649.3 and 347.15 nm
2nd Harmonic optical effect
Today
Since advent of powerful lasers many nonlinear effects used.
Sum and frequency difference mi
xing
Kerr electroptical effect
third Harmoic generation
Four wave mixing
Stimulate Raman Scattering
Two photon absorption
Nonlinear effects normally only appear in the presence of large electric or magnetic
fields.
Why?
Light interacts with matter in
ducing oscillating dipole moments in the medium. These
can be modulated by the vibrations and
the oscillating dipoles
may emit at frequencies
that are the same or are modulated by the vibrations of the molecules
Thes depns on the polarizability of molecu
les at least to a first approximation, since to
generate the oscillating dipoles the electron could needed to slosh back and forth. This is
generally a harmonic type oscillation which leads to an oscillating field with similar
characteristics and the inco
ming EM.
However at high levels of the incoming irradiance, the dipoles no longer responds
linearly with the field and the dipoles are anharmonic
That is if the response is linear the dipole follow Hookes Law and the potential energy
curve is parabolic
But at high enough driving forces the dipole restoring force no longer increases in a
linear way with the displacement and Hookes law is violated the potential energy curve
strays from a perfect parabola!
How it strays depends on the molecular structure
of the material itself.
Generally speaking there are two broad categories for molecular structures
Those with a center of symmetry isotropic
Those without a center of symmetry anisotropic
For the centrosymmetric case, the dipole restoring potenti
al energy curve may look like
For the noncentrosymmetric case, the dipole resoring potential energy curve may look
like
In a concentrated or condensed medium especially each dipole is SUSCEPTIBLE to the
local fields of its neighbors as well as t
he external filed of the light wave
Remember in Raman scattering, the polarizability of the molecul
e was important and one
wrote that the Polarization =
P =
E
However now this is no longer the case when the dipole of the neighbors and nearest
neighbors
influence each other especially with an intense driving force. Intense EM so
here then more terms are needed in the Polarization expression.
P =
1
E +
2
E:E +
3
E:E:E + ………
where
n
symbolizes the susceptibility tensor
1
=
the polarizabilit
y
2
= the hyperpolarizability in microscopic systems, the
3
is referred to as the 3rd order
susceptibility.
If the field intensity is low, then the electric field is purely linear and the Polarization
induced is linear and the material exhibits all of
the ordinary optical properties.
If the field strength is higher than the induced polarization becomes nonlinear then the
Polarization is distorted from the original EM field
see figs
Fourier analysis of the Polarizaton waves provide u
s a way to see what optical effects are
associated wth the optical properties of the materials
Consider the nonlinear response of the noncentrosymmetric media to a high powered
laser light source at
1
Fourier analysis of the Polarization reveals that it is three superimposed waveforms
1) sinusoidal wave with twice the frequency of the electric field
2) a negative dc component
3) a sinusoidal wave with the same frequency as the electric field
These are 2nd harmonic generation, optical rectification, and normal refraction
If two monochromatic waves with frequencies
1
and
2
are now incident on a non

centrosymmetric crystal
The resulting polarization wave induced in the material has now t
he following
components
1) component at the fundamental
1
2) component at the fundamental
2
3) 2nd harmonic at 2
1
4) 2nd harmonic at 2
2
5) the combination sum
1
+
2
6) the combination difference
1

2
7) a dc component
Generally speaking, the sec
ond order nonlinear effect involves wave

mixing processes in
which waves of differing frequencies merge to produce others. How does this happen?
Light waves are not supposed to exchange energy with each other and they don't in
vacuum, in the medium
The
nonlinear polarization wave itself couples the energy from one wave to another.
This effect is called optical parametric interaction.
How large is this effect?
Depends on the relative phase velocities of the polarization wave as the light waves move
th
rough the medium.
Most of what we will be looking at are plane waves which is a wave whose phase is the
same at a given instant at all points in each plane perpendicular to some direction of
propagation
E(z,t) = E
0
exp(

i(wt ± kz) k is the wave
vector =
/c
kz is the phase factor and is constant for a given t and z no matter what the values of x
and y are.
Since the electric field initiates a dipole moment which then reradiates
The Polarization wave of oscillating dipoles creates the secondary wave, bu
t the
secondary induced wave is produced proportionally to the acceleration of the elctrons the
lags it by 90
o
. The sum of this with the original wave produces a refracted wave with a
slightly different phase factor that slightly lags the original EM.
Also the 2nd Harmonic wave has a frequency that is twice the original EM.
Overlap of
these beams
can add intensity to the beams but in general the index of
refraction is different at different frequencies and so the wave propagation speed is
different.
To allow more overlap of the waves, one needs the medium to have the same index of
refraction at the 2nd Harmonic and the
original frequency and to get intense 2nd
harmonic generation one needs the material to be birefrigent
Birefringent the index of
refraction depends on the direction and polarization of the
propagating light.
Thus if a polarized light passes through at just the correct angle, the phase velocites of the
induced polarization and the 2nd harmonic wave can be made equal.
Other co
nsiderations for SHG Materials
Nonlinear optical effects can take place in all phases of matter.
Historically inorganic crystals have proven to be a very convenient medium.
Crystals are grouped into seven general types:
Triclinic, Monoclinic, Orthorhom
bic, Trigonal, Tetragonal, Hexagonal, Cubic
Cubic is the most symmetric, triclinic is the least.
Additionally, the fit into one of the 32 point groups
Based on Symmetry Considerations it is
IMPOSSIBLE
for centrosymmetric (isotropic)
materials to have
any nonlinear phenomenon associated with the even order
susceptibilities. Only the odd order susceptibilities will contribute.
So second harmonic generation is not possible
for in a centrosymmetric medium or
nonlinear phenomenon associated with the hyper
polarizability are not possible in a
centrosymmetric medium.
For noncentrosymmetric media both even and odd order susceptibility terms may
contribute.
A consideration of the points groups shows that 11 of them have controsymmetric
symmetry, so only 21
of them could show the effect .
Additional symmetry considerations reduce this to 18.
Classic crystals
KDP

potassium dihydrogen phosphate
KD
*
P

deuterated KDP
KTP KTiOPO
4
pretty high
2
used to double cw light to get green laser pointers
LiNbO
3
has low threshold damage
BANANNA BaNa
2
Nb
5
O
15
BBO beta barium borate
LBO Lithium triborate
Organic NLOs
Poly diacetylenes with high
2
Often we might be interested in nonlinear effects in gases and liquids. These are
isotropic media sin
ce the molecules do not have a fixed position relative to the incoming
light.
Second order effects are impossible here so we must go to the 3rd order effects.
How many susceptibility terms are there that could possibly contribute to the induced
polari
zation?
Turns out
1st order there are 9 terms
2nd order there are 27 terms
3rd order there are 81 terms
How do these come about?
Lets go back and consider the Polarization and rewrite it carefully
P
a
=
b
1
ab
E
b
+
b
g
2
abg
b
g
d
3
abgd
E
b
E
g
E
d
+ higher order
The sums are over the coordinates in the system such as x, y, z.
Writing out a few of these terms
Writing the Polarization in the x direction from the contribution from the 1st order
sus
ceptibility gives
P
x
=
xx
E
x
+
xy
E
y
+
xz
E
z
P
y
=
yx
E
x
+
yy
E
y
+
yz
E
z
P
x
=
zx
E
x
+
zy
E
y
+
zz
E
z
From this one can see how the 9 susceptibility terms contribute. It depends on the
polarization direction of the induced Polarization and also the polari
zation of the incident
electromagnetic field.
Can be thought of as a 3 by 3 matrix of the susceptibility terms.
What about the 2nd order susceptibility?
Write out the polarization in the x direction.
P
x
=
2
xxx
E
x
E
x
+
2
xxy
E
x
E
y
+
2
xxz
E
x
E
z
+
2
xyx
E
y
E
x
+
2
xyy
E
y
E
y
+
2
xyz
E
y
E
z
+
2
xzx
E
z
E
x
+
2
xzy
E
z
E
y
+
2
xzz
E
z
E
z
The P
x
and P
y
terms would be written similarly. One can see that this would give a total
of 27 terms 9 2nd order susceptibility terms for each direction of the Polarizatio
n.
This then can be thought of as a 3 by 3 by 3 matix
Or three of the 3 by 3 matrices of the first order susceptibility
Continuing with this then the 3rd order susceptibility can be thought of as
three 3 by 3 by 3 matrices for a total of 27 x 3

81
terms.
Now when writing out the Polarization for the CARS it is written as
P
3
CARS
(
3
, z) = D
3
abgd
(

3
,
1
1
,

2
)
1
2
E
2
exp [i(2k
1

k
2
)z]
One sees that the subscripts on the 3rd order susceptibility correspond to the polarization
directions of the induced Polarization wave at
3
, the polarization of the incoming wave
at electric fi
eld E
1
, the polarization of the 2nd incoming electric field at
1
and the
polarization of the 2nd incoming electric field at E
2
Note also that the sum of the Energies corresponding to

3,
1,
1

2
where the negative
on the
means the energy is negati
ve, must sum to be equal to zero.
For the isotropic medium only a few of the third order susceptibility terms are nonzero.
In particular these are
1111
=
xxxx
=
yyyy
=
zzzz
1122
=
xxyy
=
etc.
1212
=
xyxy
= etc.
1221
=
xyyx
= etc.
1111
=
1
122
+
1212
+
1221
Thus different Combinations of the the 3rd order susceptibility can be accessed using
different experimental geometries in CARS
D
3
CARS
(polarization directions)
P
cars
E
1
E
1
E
2
3rd order susceptibility term
Now this i
s just the induced Polarization
We really want the Electromagnetic field produced by the induced polarization
Need to solve Maxwell's Eqns. to get the new electric field emitted at
3
due to the
induced Polarization from the fields at
1
E
1
E
2
Come up w
ith the nonlinear Raman gain eqn.
d
2
E/dz
2
+ 2ik
3
dE/dz =

4
3
2
/c
2
CARS
E
1
2
E
2
exp (i
kz)
usually neglect the 2nd order term and assuming z = L and E
(
3
) = 0 at z =0
This can be solved to give
2
(w3)
/dz
2
+ 2ik
3
dE
(
w3
)
/dz =

4
3
2
/ c
2
cars
E
w1
2
E
w2
2
exp i
kz
k
= 2
k
1

k
2

k
3
k
3
= n
3
3
And the time averaged intensity of the antistokes wave at z can be determined from this
Neglecting the 2nd order term and using I
(w3)
= n
3
c/(8
)
E
(w3)
2
one can get:
I
(w3)
= 256
4
3
2
/ (n
1
2
n
2
n
3
c
4
)
cars
2
I
1
2
I
2
L
2
[(sin (
kL/2)) /
kL/2)]
2
L is the coherence length or the interaction length
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