Non-linear optics

1. Nonlinear Susceptibility of a Classical
Anharmonic Oscillator
Lorentz model will be considered by allowing the possibility of a
nonlinearity in the restoring force exerted on the electron. The analysis differ
depending upon whether or not the medium possesses inversion symmetry.
• First we treat the case of a non-centrosymmetric medium
• Second we find that such a medium can give rise to a second-order optical
nonlinearity.
• Third we treat the case of a medium that possesses a center of symmetry
and find that the lowest-order nonlinearity that can occur in this case is a
third-order nonlinear susceptibility.

2. 1.4.1 Noncentrosymmetric Media
the equation of motion of the electron position 𝐱 to be of the form:
……1
is Electric field
-e is the electric charge of the electron
-2m𝛾x is the damping force
The restoring force is: 𝐹𝑟𝑒𝑠𝑡𝑜𝑟𝑖𝑛𝑔= -m𝜔𝑜
2
𝑥-ma 𝑥2
……………2
where a is a parameter that characterizes the strength of the nonlinearity.
To understand the nature of this form of the restoring force from the potential
energy function as
……….3

3. The first term in eq. 3 corresponds to a harmonic potential and the second term corresponds to
an anharmonic correction term, as in Fig. 1. The present model can describe only non-
centrosymmetric media because we have assumed that the potential energy function U(𝑥) of
Eq. 3. contains both even and odd powers of 𝑥; for a centrosymmetric medium only even
powers of 𝑥 could appear, because the potential function U(𝑥) must possess the symmetry
U(𝑥) = U(−𝑥).
Fig. 1. Potential energy function for
a noncentrosymmetric medium.
Assume that the applied optical field is of the form
………….4
where E1 = E(ω1) and E2 = E(ω2)
No general solution to Eq. 1 for an applied field of the form (4) is known If the applied field is sufficiently weak, the nonlinear
term a𝑥2
will be much smaller than the linear term 𝜔𝑜
2
𝑥 for any displacement 𝑥 that can be induced by the field. So Eq. 1. can be
solved using Rayleigh–Schrödinger perturbation theory in quantum mechanics. By replacing 𝐸(t) in eq.1 by 𝜆𝐸(t), where λ is a
parameter that between 0 to 1 . Then eq. 1 can be as
………5

4. The terms lead respectively to the equations
………..6
the lowest-order contribution 𝑥(1) is governed by the same equation as that of the conventional (i.e., linear) Lorentz model.
Its steady-state solution is given by
……..7
Where the amplitude 𝑥(1)
(𝜔𝑗) have the form
𝑥(1)(𝜔𝑗) = -
𝑒
𝑚
𝐸𝑗
𝐷(𝜔𝐽)
………………… 8
Where 𝐷(𝜔𝐽)=𝜔𝑜
2- 𝜔𝑗
2
-2i𝜔𝑗𝛾 ………….9
By squaring the term 𝑥(1)(t) and substituted in to 6 to obtain the lowest-order correction term 𝑥(2)
To determine the response at frequency 2𝜔1 so we solve
……..10
And …………11

5. Substitution of Eq. 11 into Eq. 10 leads to the result
……..12
The amplitudes of the responses at the other frequencies are
…..13a
…..13b
……13c
……13d
The linear susceptibility is defined through the relation 𝑃 1
𝜔𝑗 = 𝜖𝑜𝜒 1
𝜔𝑗 E(𝜔𝑗) ……..14
The linear contribution to the polarization is given 𝑃 1
𝜔𝑗 = -Ne𝑥(1)
(𝜔𝑗) ……..15
Where N is the density of the atom. Using 7 and 8 the linear susceptibility is
--------16a
Where D(𝜔𝑗) = 2𝜔𝑜 (𝜔𝑗 − 𝜔𝑜 − 𝑖𝛾) so: ……..16b

6. the Eq, 16b shows explicitly the real and imaginary parts of the susceptibility.
The nonlinear susceptibilities are calculated in an analogous manner, the non-linear susceptibility describing
second-harmonic generation is defined by the relation :
𝑃 2 2𝜔1 = 𝜖𝑜𝜒 2 2𝜔1, 𝜔1, 𝜔1 𝐸(𝜔1)2 ………..17
𝑃 2 2𝜔1 is the amplitude of the component of the nonlinear polarization oscillating at frequency 2𝜔1
𝑃 2 2𝜔1 =-Ne𝜒(2) 2𝜔1 ………………………..18
From 12 𝜒 2 2𝜔1, 𝜔1, 𝜔1 =
𝑁
𝑒3
𝑚2 𝑎
𝜖𝑜𝐷 2𝜔1 𝐷2 𝜔1
…………………19
Or in terms of the product of linear susceptibilities as
𝜒 2
2𝜔1, 𝜔1, 𝜔1 =
𝜖𝑜
2 𝑚 𝑎
𝑁2𝑒3 𝜒(1)
2𝜔1 [𝜒(1)
𝜔1 ]2
……………20
FIG.2 Linear optical response as predicted by the Lorentz model of the atom.
The imaginary part of 𝜒(1) gives the atomic absorption profile, and the real
part represents a contribution to the real part of the refractive index. Note that
the full-width at half maximum of Im 𝜒 1
is equal to 2γ . The vertical axis is
plotted in normalized units. To obtain the numerical value of 𝜒(1) , the value
on the vertical axis should be multiplied by Ne/(2 𝜖𝑜 mγ )

7. A crucial conclusion : [ The second-order susceptibility is proportional to the product of three linear susceptibilities].
This dependence is the reasons why one conventionally expresses the second-order susceptibility as a function of three
frequencies, each of which is the argument of one of the linear susceptibilities appearing on the right-hand side of this equation.
Nonlinear susceptibility for second-harmonic generation of 𝜔2 field is obtained from Eqs. 19 and 20 by substitution 𝜔1→ 𝜔2
The nonlinear susceptibility describing sum-frequency generation obtained by means a similar calculation.
𝑃(2)(𝜔1 + 𝜔2) = 2𝜖𝑜𝜒(2) (𝜔1 + 𝜔2 , 𝜔1, 𝜔2)E(𝜔1)𝐸( 𝜔2) …….21
and
𝑃(2)(𝜔1 + 𝜔2) = −𝑁𝑒𝑥(2) (𝜔1 + 𝜔2) ……….22
The nonlinear susceptibility is ………….23
in terms of the product of linear susceptibilities as
……………….24
From comparing of Eqs.19 and 23 , as 𝜔2 approaches 𝜔1 , 𝜒(2) (𝜔1 + 𝜔2 , 𝜔1, 𝜔2) approaches 𝜒(2) (2𝜔1, 𝜔1, 𝜔1)
For difference-frequency generation
…………….25

8. For optical rectification of the 𝜔1 fied :
……………26
The conclusion : [the lowest-order nonlinear contribution to the polarization of a noncentrosymmetric material is second
order in the applied field strength].
1.4.2 Miller’s Rule
An empirical rule due to Miller (Miller, 1964; see also Garrett and Robinson, 1966 ).
Miller noted that the quantity is nearly constant for all non-centrosymmetric crystals :
𝜒(2) (𝜔1+ 𝜔2 , 𝜔1, 𝜔2)
𝜒(1) (𝜔1+ 𝜔2) 𝜒(1)( 𝜔1) 𝜒(1)(𝜔2)
……………..27
this quantity will be nearly constant only if [
𝑚𝑎𝜖𝑜
2
𝑁2𝑒3 ……..28 ] is nearly constant
N is the atomic number density ~ 1022
𝑐𝑚−3
, the parameters m and e are fundamental constants, the size of the nonlinear
coefficient a can be estimated the linear and nonlinear contributions to the restoring force is given by Eq. 2.
The displacement 𝑥 of the electron from its equilibrium position is approximately equal to the size of the atom.

9. This distance is of the order of the separation between atoms, that is, of the lattice constant d. This leads to m𝜔𝑜
2𝑑 = 𝑚𝑎𝑑2
Or
𝑎 =
𝜔𝑜
2
𝑑
………..29
𝜔𝑜 and d are roughly the same for most solids, the quantity a would also be expected to be the same for all materials.
Using the nonlinear coefficient a given by Eq. 29 to estimate of the size of the second-order susceptibility under highly
nonresonant conditions. By replace D(𝜔) by 𝜔𝑜
2
in the denominator of Eq.23 and set N equal to 1/𝑑3 and set a from 29
Than
𝜒(2) =
𝑒3
𝜖𝑜𝑚2𝜔𝑜
4𝑑4 ………….30
Using the typical values 𝜔𝑜= 1× 1016 𝑟𝑎𝑑
𝑠
, d = 0.3nm, e = 1 × 1016𝐶, 𝑚 = 9.1 × 10−31𝐾𝑔 than
𝜒(2) ≅6.9 × 10−12 m/V

10. 1.4.3 Centrosymmetric Media
• For the case of a centrosymmetric medium he electronic restoring force is
• …………31
where b is a parameter that characterizes the strength of
the nonlinearity. This restoring force corresponds
to the potential energy function:
…….32
Fig.3 Potential energy function for a
centrosymmetric medium
This potential function is illustrated in the Fig. 3
is the lowest-order correction term to the parabolic
potential described by the term
A special cases is the isotropic material (as well as being centrosymmetric).
Examples of such materials are glasses and liquids. In such a case, the restoring force have the form
…………33
where r is the vector displacement of the electron from its equilibrium position
The equation of motion for the electron displacement from equilibrium is thus
………34

11. The applied field is ………35
Or
…..36
˜
Replace 𝐸(t ) in Eq. 34 by λ 𝐸(t ), where λ is a parameter that characterizes the strength of the perturbation
and equal to unity at the end of the calculation.
A solution to Eq. 34 having the form of a power series in the parameter λ:
……….37
insert Eq. 37 into the equation of motion 34 and require that the terms proportional to λ𝑛
vanish separately for each value of n.
……..38a
……..38b
….. ……..38c
for n = 1, 2, and 3, respectively
The steady-state solution for 38a is …….39a
Where ……..39b
With 𝐷 𝜔𝑛 is given by 𝐷 𝜔𝑛 =𝜔𝑜
2
− 𝜔𝑛
2
− 2𝑖𝜔𝑛𝛾, Since the polarization at frequency 𝜔𝑛 is given by
………40
The Cartesian components of the polarization as …….41

12. The linear susceptibility is ………42
With 𝜒(1)(𝜔𝑛) given by eq. 16a
Where 𝛿𝑖𝑗 is the Kronecker delta, which is defined such that 𝛿𝑖𝑗 =1 for i=j and 𝛿𝑖𝑗=0 for i≠j
The second-order response of the system is described by Eq. 38b. Since this equation is damped but not driven, its steady-
state solution vanishes, that is, 𝒓(2) = 0
To calculate the third-order response, substitute the expression for 𝒓 1 (𝑡) in eq. 39a in eq. 38c which give:
……….43
Because of the summation over m, n, and p, the right-hand side of this equation contains many different frequencies. We
denote one of these frequencies by ωq = ωm+ωn+ωp. The solution to Eq. (43) can then be written in the form
……….44
substitute Eq. 44) into Eq. 43 and find 𝑟 3
(𝜔𝑞) is given by:
…….45
Since the coefficient of 𝑟 3 (𝜔𝑞) on the left side is just 𝐷 𝜔𝑛 than : ………46

13. The amplitude of the polarization component oscillating at frequency ωq is given in terms of this amplitude by
𝑷 3
𝜔𝑞 = −𝑁𝑒𝑟 3
(𝜔𝑞) ………47
Next recall the definition of the third-order nonlinear susceptibility given by the following equation:
…….48
Than ………49
Since this equation contains a summation over the dummy variables m, n, and p, there is more than one possible choice
for the expression for the nonlinear susceptibility. One choice for this expression for the susceptibility is,
………..50
It is conventional to define nonlinear susceptibilities in a manner that displays this symmetry, which is known as intrinsic
permutation symmetry. Since there are six possible permutations of the orders in which Ej(ωm), Ek(ωn), and El(ωp) may
be taken, the third-order susceptibility can be defined to be one-sixth of the sum of the six expressions analogous to Eq.
50 with the input fields taken in all possible orders. The resulting form for the nonlinear susceptibility is given by,
……….51
This expression can be rewritten in terms of the linear susceptibilities at the four different frequencies ωq , ωm , ωn , and
ωp by using Eq. 16a to eliminate the resonance denominator factors D(ω).

14. Thereby obtain
……..52
The value of the constant b that appears in this result to estimate the value of the constant a appears in expressions
of 𝜒(2)
when the displacement 𝑥 becomes comparable to the atomic dimension d, that is, when 𝑚𝜔𝑜
2
d = mb𝑑3
, which
implies that 𝑏 =
𝜔𝑜
2
𝑑2 ……..53
Using this expression for b, we can now estimate the value of the nonlinear susceptibility. For the case of nonresonant
excitation, D(ω) is approximately equal to 𝜔𝑜
2
, and hence from Eq. 51 we obtain
………54
……..55

15. Properties of the Nonlinear
Susceptibility
The aim of this section is:
- Study some of the formal symmetry properties of the nonlinear
susceptibility
- why it is important that we understand these symmetry
properties.

16. Tmutual interaction of three waves of frequencies 𝜔1, 𝜔2 𝑎𝑛𝑑 𝜔3 = 𝜔1 + 𝜔2
interact in a lossless second-order nonlinear optical medium.
the nonlinear polarizations 𝑃(𝜔𝑖) influencing each of them
we therefore need to determine the six tensors quantities
and six additional tensors in which each frequency is replaced by its negative
these 12 tensors thus consists of 27 Cartesian components, as many as 324 different (complex) numbers need to be
specified in order to describe the interaction.
from symmetry considerations that relate the various components of 𝜒(2)
a number of restrictions is considered
Hence far fewer than 324 numbers are needed to describe the nonlinear coupling.
So these formal properties of the nonlinear susceptibility will be discussed .

17. 1.5.1 Reality of the Fields