Nonlinear Optics
Basic concept of optical nonlinearity
The polarization density in the presence of an electric field:
P∝E
Here we assume that the medium is isotropic. When the field is strong, the displacement of
electrons from their equilibrium point may deviate from linearity:
P = ε 0 χE + 2dE 2 + 4 χ ( 3) E 3
(1)
The coefficients describe the first order, second order and third order nonlinearity.
If the medium has an inversion symmetry, the second order nonlinearity, characterized by d,
is zero. Why?
Physical pictures: When the potential as a function of displacement deviates from a perfect
parabola, the electrons driven by an electric field may create a polarization. For example: a
DC polarization can be created by an electric in an asymmetric potential. An asymmetric
shape in the polarization-electric field relation results in a distorted polarization which, in
turn, generates a wave which contains the harmonic waves whose frequencies are two times
the fundamental.
The polarization created by the radiation then acts as a source for the generation of new
waves.
P = ε 0 χE + 2dE 2 + 4 χ ( 3) E 3
Second order nonlinearity
P ( 2 ) = 2dE 2 = 2d ( E1e jω1t + E 2 e jω2t + cc ) 2
ω1=ω2
•Second harmonic generation
•Optical rectification
•
•Electrooptic effect
ω1=0
ω1≠ω2
•Frequency conversion
•Optical parametric process
P = ε 0 χE + 2dE 2 + 4 χ ( 3) E 3
Third order nonlinearity
P ( 3) = 4 χ ( E1e jω1t + E2 e jω 2t + E3e jω3t + cc ) 3
ω1=ω2 =ω3
•Third harmonic generation
•Degenerate four-wave mixing
ω1=ω2
ω1≠ω2 ≠ω3
•Optical phase conjugation
•Optical Kerr effect
•Self-phase modulation
•Self- focusing
•Raman gain
•Four-wave mixing
I
In general, the polarization density and electric field is related by a tensor. For example,
( 3)
Pi = ε 0 χ ij E j + 2d ijk E j E k + 4 χ ijkl
E j E k El + .......
(1.1)
where each of the i,j,k,l are can be x, y, or z. The coordinate axis does not need to be one
that diagonalizes the dielectric tensor.
See Yariv’s.
Nonlinear coefficients- the anharmonic oscillator model
Consider the bound electron model with an electron in a anharmonic potential well driven by
electromagnetic wave of frequency ω. The displacement of electron from equilibrium follows
the following relations:
eE jωt
(e + cc)
2m
Assuming that the solution of this equation has the following form:
1
X = (q1e jωt + q 2 e j 2ωt + q 3 e j 3ωt + cc)
2
..
.
x + γ x + ω 02 + Dx 2 =
(1.2)
Solve for the q’s
eE
1
q1 = ( ) 2
m (ω 0 − ω 2 ) + jωγ
(1.3)
q2 =
− De 2 E 02
[
(1.4)
]
2
2m 2 (ω 02 − ω 2 ) + jωγ (ω 02 − ω 2 + 2 jωγ )
The nonlinear coefficients are larger in the vicinity of resonance lines..
Nonlinear wave equation
From Lecture 2 Eq. (7), the Helmholtz equation with a source is given by
1 ∂2E
∂2P
=
µ
0
c02 ∂t 2
∂t 2
The polarization has a linear and nonlinear
components.
∇2 E −
P = ε 0 χE + PNL
Four-wave mixing :
beam deflection by an oscillating grating
(3)
The linear part can be combined with the other
time derivative term to give rise to a speed of
light c =c0/n. Eq. (2) can be rewritten as
1 ∂2E
∇ E − 2 2 = −S
c ∂t
∂ 2 PNL
∂t 2
k3
k4
k1
2
S = −µ0
(2)
(4)
The directions and frequencies of the k’
Second order nonlinearity
Consider the case of two sinusoidal electric fields in a nonlinear medium
2
P ( 2 ) = 2 d ( Eω1 e j 2ω1t + Eω 2 e jω 2t + cc ) 2
k2
(5)
In general, the two electric fields can be in different direction and the polarization is related
to the individual fields through a tensor. The mixing of the field results in four nonlinear
polarization densities oscillating at 2ω1, 2ω2, ω1+ω2 , and ω1-ω2 , representing the sources for
second harmonic generation of ω1 and ω2 , sum and difference frequency generation. Note
the for second harmonic generation, only one wave is needed.
Pockels effect
A special case is when one of the waves is a DC electric field with ω2=0, the polarization
density is given by
P ( 2 ) = 2dE 2 = 2d EDC Eω1 e jω1t
(6)
This term contribute to a change in the refractive index
2d
E DC
(7)
nε 0
The change of refractive index by applied voltage is the Pockels effect, which has wide
applications in optics as optical shutter or modulators.
∆n =
Frequency conversion
2
2
P ( 2 ) = 2 d ( Eω1 e j 2ω1t + Eω 2 e j 2ω 2t + 2 Eω1 Eω* 2 e j (ω1 −ω 2 ) t + 2 E ω1 Eω 2 e j (ω1 +ω 2 ) t
(8)
Thus the nonlinear medium can be used to mix two optical waves of difference frequencies
and generate a third wave at the sum and difference frequencies. Although the waves produce
polarization densities at various frequencies, not all of them are generated in a meaningful
way unless the following conditions are simultaneously satisfied.
Frequency matching : ω 3 = ω1 + ω 2
Phase matching :
k 3 = k1 + k 2
(9)
(10)
Physical meaning: Using propagation delay of SHG in one dimension as an example. The
waves must have the same speed (phase velocity) to ensure constructive interference along the
path.
Coupled-wave theory with second-order nonlinearity
∇2 E −
1 ∂2E
= −S
c 2 ∂t 2
∂ 2 PNL
∂2 2
S = −µ 0
=
−
2
d
µ
E
0
∂t 2
∂t 2
(11)
From (4)
The field is a superposition of three waves
1
1
E=
( E q exp( jω q t ) + cc) =
E q exp( jω q t )
2
q =1, 2 , 3 2
q = ±1, ±2 , ±3
(12)
where E-q is defined to be Eq *
By substitution the electric filed into (11), and carrying out the time derivative
S=
1
µ0d
(ω q + ω r ) 2 E q E r exp[( jω q + ω r )t ]
2
q , r = ±1, ±2 , ±3
The Hemlholtz equations with sources are
(∇ 2 + k s2 ) E s = − S s
s = 1,2,3
(13)
where Ss is the component of S with a frequency, and ks=nsω/c. Among the frequencies in the
three equation, the interaction takes place only when
ω 3 = ω1 + ω 2
(14)
(∇ 2 + k12 ) E1 = −2 µ 0ω12 dE3 E 2*
(15-1)
*
1
(∇ + k ) E 2 = −2 µ 0ω dE 3 E
(15-2)
(∇ 2 + k 32 ) E 3 = −2 µ 0ω 32 dE1 E 2
(15-3)
2
2
2
2
2
With this constraint,
This is the Helmholtz equations for three-wave mixing.
Special case: Collinear plane waves
In the normalized form
E q = (2π ω q )1 / 2 a q exp(− jk q z )
q = 1,2,3
(16)
The coupled mode equations are, from (15) and (16) and using slow varying amplitude
approximation.
da1
= − jga 3 a 2* exp(− j∆kz )
dz
da 2
= − jga3 a1* exp(− j∆kz )
dz
da 3
= − jga1 a 2* exp( j∆kz )
dz
(17)
(18)
(19)
where g 2 = 2 ω1ω 2ω 3η 3 d 2 and η = ( µ 0 / ε 0 )1 / 2 , and
mismatch of the three waves.
∆k = k 3 − k 2 − k1 is the phase
For an input of two frequencies, the wave of the third frequency cannot build up unless the
phase mismatch is zero. Second harmonic generation
Angle tuning
The electric displacement vector and the electric field are related through a dielectric tensor
Dk = ε kl El
(1)
where k and l refer to the Cartesian coordinates (x,y,z). Using the principal axes as the
coordinate system, the dielectric tensor is diagonal. The index ellipsoid is described by
x12 x 22 x33
+
+
=1
(2)
n12 n22 n33
where 1 / n1 ,1 / n2 , and 1 / n3 are the principal values. The index ellipsoid of an isotropic
medium is a sphere.
A wave propagating in the z-direction and linearly polarized in the x-direction travels with
phase velocity c0/n1.
What about a wave traveling in the z-direction and an electric field on the z=0 plane and
making a 45-degree angle with the x-axis?
In an uniaxial crystal,
1
cos 2 (θ ) sin 2 θ
=
+
n 2 (θ )
n02
ne2
The ray with index n0 is the ordinary wave and n(θ) the extraordinary wave.
Walk-off
OA
o and e
OA
no>ne
o
e
ne
no
ne
n(θ)
no
no