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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:
2 ( 3 ) 3
P = ε 0 χ E + 2 dE + 4 χ E (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.
2 ( 3 ) 3
P =ε 0 χ E + 2 dE + 4 χ E
Second order nonlinearity
2 1 2
( 2 ) 2
P 2 dE 2 d ( Ee^1 Ee^2 cc )
j t j t
ω ω
ω 1 = ω 2 •Second harmonic generation •Optical rectification
ω 1 =0 •Electrooptic effect
ω 1 ≠ω 2 •Frequency conversion •Optical parametric process
I
2 ( 3 ) 3
P =ε 0 χ E + 2 dE + 4 χ E
Third order nonlinearity
3 1 2 3
P (^3 )= 4 χ( Eej^ ω^1 t + Eej ω^2 t + Eej^ ω^3 t + cc )
ω 1 = ω 2 •Optical Kerr effect •Self-phase modulation •Self- focusing
ω 1 = ω 2 = ω 3 •Third harmonic generation •Degenerate four-wave mixing •Optical phase conjugation
ω 1 ≠ω 2 ≠ω 3 •Raman gain •Four-wave mixing
In general, the polarization density and electric field is related by a tensor. For example,
Pi = ε 0 χ ijEj + 2 dijkEjEk + 4 χ ijkl (^3 ) EjEkEl + ....... (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
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 2 ω 1, 2 ω 2, ω 1+ ω 2 , and ω 1 - ω 2 , representing the sources for
second harmonic generation of ω 1 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 ω 2 =0 , the polarization
density is given by
(6)
This term contribute to a change in the refractive index
n E^ DC
d n 0
The change of refractive index by applied voltage is the Pockels effect, which has wide applications in optics as optical shutter or modulators.
Frequency conversion
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 = ω 1 + ω 2 (9)
Phase matching : k (^) 3 k 1 k 2
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
j t P dE d EDC E e 1 1 2 2
( 2 ) 2 ω = = ω
P (^2 ) d E^2 ej^^2 t E^2 ej^2 t E E * ej ( ) t E E ej (^12 )^ t 1 2
1 2 1 2
2 2
1 2 ( 1 2 2 ω ω ω ω
ω ω ω ω
ω ω
ω ω = + + −^ + +
2 2
2 2 0
2 0
2
2 2
2
E
t
d
t
P
S
S
t
E
c
E
NL
μ μ
From (4) The field is a superposition of three waves
= =± ±±
1 , 2 , 3 1 , 2 , 3
exp( ) 2
( exp( ) ) 2
q
q q q
E Eq j ω q t cc E j ω t (12)
where E-q is defined to be Eq * By substitution the electric filed into (11), and carrying out the time derivative
=± ± ±
, 1 , 2 , 3
2 2 0 ( ) exp[( )]
q r
S μ d ω q ω r EqEr j ω q ω r t
The Hemlholtz equations with sources are
( ∇ 2 + k (^) s^2 ) Es =− Ss s = 1 , 2 , 3 (13)
where Ss is the component of S with a frequency , and ks=ns ω /c. Among the frequencies in the
three equation, the interaction takes place only when
With this constraint,
This is the Helmholtz equations for three-wave mixing.
Special case: Collinear plane waves
In the normalized form
E q = ( 2 π ω q )^1 /^2 aq exp(− jkqz ) q = 1 , 2 , 3 (16)
The coupled mode equations are, from (15) and (16) and using slow varying amplitude approximation.
(^1) jga 3 a * 2 exp( j kz ) dz
da = − −∆ (17)
(^2) jga 3 a 1 *exp( j kz ) dz
da = − − ∆ (18)
(^3) jga 1 a * 2 exp( j kz ) dz
da = − ∆ (19)
3 2
2 1 0 1
2 1
(∇ 2 + k ) E =− 2 μ ω dE E (15-1)
3 1
2 2 0 2
2 2
(∇ 2 + k ) E =− 2 μ ω dE E (15-2)
1 2
2 3 0 3
2 3
(∇ 2 + k ) E =− 2 μ ω dE E (15-3)
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