quantum-mechanical theory of the nonlinear optical
TRANSCRIPT
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Quantum-Mechanical
Theory of the Nonlinear
Optical Susceptibility
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Contents:
3.1.Introduction
3.2. SchrΓΆdinger Calculation of Nonlinear Optical Susceptibility
3.3. Density Matrix Formulation of Quantum Mechanics
3.4. Perturbation Solution of the Density Matrix Equation of Motion
3.5. Density Matrix Calculation of the Linear Susceptibility
3.6. Density Matrix Calculation of the Second-Order Susceptibility
3.7. Density Matrix Calculation of the Third-Order Susceptibility
3.8. Electromagnetically Induced Transparency
3.9. Local-Field Corrections to the Nonlinear Optical Susceptibility
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In this chapter, we use the laws of quantum mechanics to derive explicit expressions for the nonlinear optical susceptibility.
β’ (1) these expressions display the functional form of the nonlinear
optical susceptibility and hence show how the susceptibility
depends on material parameters such as dipole transition moments
and atomic energy levels
β’ (2) these expressions display the internal symmetries of the
susceptibility
β’ (3) these expressions can be used to make predictions of the
numerical values of the nonlinear susceptibilities.
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The idea behind resonance enhancement of the nonlinear optical susceptibility is shown schematically in Fig. 3.1.1 for the case of third-harmonic generation.
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The idea behind resonance enhancement of the nonlinear optical susceptibility is shown schematically in Fig. 3.1.1 for the case of third-harmonic generation.
third-harmonic generation in terms of the virtual levels introduced in Chapter 1
real atomic levels, indicated by solid horizontal lines
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Three possible strategies for enhancing the efficiency of third-harmonic generation through the technique of resonance enhancement are illustrated in Fig. 3.1.2.
the one-photon
transition is nearly
resonant
the two-photon
transition is nearly
resonant
the three-photon
transition is nearly
resonant
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SchrΓΆdinger Equation Calculation of the Nonlinear Optical Susceptibility
Hamiltonian for a free atom
the time-dependent
SchrΓΆdinger equation
At first a review on quantum mechanics:
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Energy Eigenstates:
For the case in which no external field is applied to the atom, the Hamiltonian π» is simply equal to π»0 , and SchrΓΆdingerβs equation possesses solutions in the form of energy eigenstates. These states have the form:
By substituting this form into
the SchrΓΆdinger equation
n is a label used to distinguish the various
solutions. For future convenience, we assume
that these solutions are chosen in such a
manner that they constitute a complete,
orthonormal set satisfying the condition
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Perturbation Solution to SchrΓΆdingerβs Equation:
For the general case in which the atom is exposed to an electromagnetic field, SchrΓΆdingerβs equation, we replace the Hamiltonian by :
0 β€Ξ» β€ 1 corresponds to the actual physical situation
We now seek a solution to SchrΓΆdingerβs equation in the form of a power
series in Ξ»:
By substituting this form into
the SchrΓΆdinger equation
1
2
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answer1
2
an
swer
gives the probability amplitude that, to Nth order in the perturbation, the atom is in energy
eigenstatel at time t.
If is substituted into ,we find that the probability amplitudes
obey the system of Equations :
2
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To simplify this equation, we multiply each side from the left by ππβ and we integrate
the resulting equation over all space. Then through use of the orthonormality condition,
we obtain the equation:
once the probability amplitudes of order Nβ1 are determined, the amplitudes of the
next higher order(N)can be obtained by straightforward time integration. In particular,
we find that:
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β’ To determine the first-order amplitudes ππ(1)(π‘) we set ππ
0in equal to
Ξ΄ππ, corresponding to an atom known to be in state gin zeroth order.
β’ We represent the optical field ΰ·©E(t) as a discrete sum of (positive and negative)
frequency components as:
Through use of & we can then replace π½ππ(πβ²) by:
(electric dipole transition moment)
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:
N=1
N=2
N=3
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Linear Susceptibility
β’ According to the rules of quantum mechanics, the expectation value of the electric dipole moment is given by:
By substituting π(π) & π(π) we find that:
the population decay rate of the upper level m
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we formally replace Οπ by - Οπ in the second term, in which case the expression
becomes:
the number density of atoms
.
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β’ Note that if g denotes the ground state, it is impossible for the second term to
become resonant, which is why it is called the antiresonant contribution.
resonant antiresonant
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Second-Order Susceptibility
By substituting π(π) , π(π) & π(π) we find that:
As in the case of the linear susceptibility, this equation can be rendered more by
βΟπa transparent by replacing Οπ by βΟπin the second term and by replacing Οπ
and Οπ by βΟπ in the third term:
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.
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denotes the intrinsic permutation operator. This operator tells us to average the
expression that follows it over both permutations of the frequencies Οp and Οq of
the applied fields.
A
A
B
B
C
C
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For the case of highly nonresonant excitation, such that the resonance frequencies
Οmg and Οng can be taken to be real quantities, the expression for Ο(2) can be
simplified still further. In particular, under such circumstances above equation can
be expressed as:
full permutation operator, denoted six permutations:
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Third-Order Susceptibility:
By substituting π(π) , π(π), π(π) & π(π) we find that:
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Since the expression is summed over all positive and negative values of Οπ, Οπ&Οπ,
we can replace these quantities by their negatives in those expressions where the
complex conjugate of a field amplitude appears. We thereby obtain the expression:
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.
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1
1
2
2
3
3
4
4
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Third-Harmonic Generation in Alkali Metal Vapors
As an example of the use of we next calculate the nonlinear
optical susceptibility describing third-harmonic generation in a vapor of
sodium atoms.
We assume that the incident radiation is linearly polarized in the z
direction. Consequently, the nonlinear polarization will have only a z
component, and we can suppress the tensor nature of the nonlinear
interaction. If we represent the applied field as:
we find that the nonlinear polarization can be represented as:
.
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Energy-level diagram
of the sodium atom.
The third-harmonic
generation process.
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can become fully resonant. This term becomes fully resonant when Ο is nearly equal to Οmg, 2Ο is nearly equal to Οng,and 3Ο is nearly equal to ΟΞ½g.
According to the selection rule:
&
since the ground state is ansstate
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Density Matrix Formulation of Quantum Mechanics
We use this formalism because it is capable of treating effects, such as collisional
broadening of the atomic resonances, that cannot be treated by the simple
theoretical formalism based on the atomic wave function. We need to be able to
treat such effects for a number of related reasons.
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Let us begin by reviewing how the density matrix formalism follows
from the basic laws of quantum mechanics:
We can hence represent the wavefunction of states as
the time-independent SchrΓΆdinger equationthe time-dependent SchrΓΆdinger equation
β’ which are assumed to be orthonormal in
that they obey the relation
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&
multiply each side from
the left by π’πβ (π) and
integrate over all space:
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In Dirac
notation
The expectation value π΄ can be expressed in terms of the probability amplitudes πΆππ (π‘) by:
We will back to π¨ by
the density matrix
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Under such circumstances, where the precise state of the system is unknown, the density
matrix formalism can be used to describe the system in a statistical sense. Let us denote by p(s)
the probability that the system is in the state s.
In terms of p(s), we define the elements of the density matrix of the system by:
This relation can also be written symbolically as
denotes an ensemble average,
that is, an average over all of the
possible states of the system.
Are understood to run over all of the energy eigenstates of the system.
The elements of the density matrix have the following physical interpretation: The diagonal
elements πππ give the probability that the system is in energy eigenstaten.
The off-diagonal elements have a somewhat more abstract interpretation:πππ gives the
βcoherenceβ between levels n and m, in the sense that πππ will be nonzero only if the
system is in a coherent superposition of energy eigenstate n and m.
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Back to:
the expectation value for the case in which the exact state of the system is not known
is obtained by averaging π¨ over all possible states of the system, to yield:
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In order to determine how any expectation value evolves in time, it is thus necessary only to
determine how the density matrix itself evolves in time. By direct time differentiation of :
we find that:
assume that p(s) does
not vary in time
0
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describes how the density matrix evolves in time as the result of interactions that are included in
the Hamiltonian. However, there are certain interactions (such as those resulting from collisions
between atoms) that cannot conveniently be included in a Hamiltonian description. Such
interactions can lead to a change in the state of the system, and hence to a nonvanishing value of
dp(s)/dt.
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We include such effects in the formalism by adding phenomenological damping terms to
There is more than one way to model such decay processes. We shall often model such
processes by taking the density matrix equations to have the form:
a phenomenological damping term, which indicates that Οππ relaxes to its equilibrium value
Οππ(ππ)
at rate Ξ³ππ. Since Ξ³ππ is a decay rate, we assume that Ξ³ππ=Ξ³ππ. In addition, we
make the physical assumption that:
the rate per atom at which population
decays from level m to level n
the damping rate of the Οππ coherence
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under quite general conditions the off-diagonal elements can be represented as:
total decay rates of population out of levels n and m
the dipole dephasing rate
due to processes (such as
elastic collisions)
Lets see why depends upon the population decay rates in the manner indicated?
note that if level n has life time Οn =1
Πn, the probability to be in level n must decay as:
Thus, the coherence between the two levels must vary as
But since the ensemble average of πΆπβπΆπ is just Οππ, whose damping rate is denoted Ξ³ππ, it
follows that:
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Example: Two-Level Atom
and thus the density matrix describing the atom
is the two-by-two matrix given explicitly by:
The matrix representation of the dipole moment operator is:
where
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Perturbation Solution of the Density Matrix Equation of Motion
Lets back to
Remember that:
In general, this equation cannot be solved exactly for physical systems of interest, and for this
reason it is useful to develop a perturbative technique for solving it.
Hamiltonian of the free atom
We assume that the states n represent the energy eigenfunctions unof the unperturbed
Hamiltonian π»0and thus satisfy the equation
As a consequence, the matrix representation of π»0 is diagonal that is:
The commutator can thus be expanded as
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we replace π½ππ in by Ξ»π½ππ, where Ξ» is a parameter ranging between zero and one that
characterizes the strength of the perturbation. The value Ξ»=1 is taken to represent the actual
physical situation. We now seek a solution to in the form of a power series in Ξ» that is:
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Separated to:
We take the steady-state solution to this equation to be
where
Now that Οππ(0)
is known, can be integrated. To do so, we make a change of variables by
representing Οππ(1)
as:
? Next slideβ¦
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This equation can be
integrated to give
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Density Matrix Calculation of the Linear Susceptibility
Remember that:
We represent the applied field as
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By replacing in :
can be
evaluated
explicitly
as:
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We next use this result to calculate the expectation value of the induced dipole moment:
Decompose π(π‘) into its frequency components according to
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Rewrite
Interchange the dummy indices n and m in the second summation so that the two summations
can be recombined as:
use the fact that
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first make the simplifying assumption that all of the population is in one level (typically the
ground state), which we denote as level a. Mathematically, this assumption can be stated as:
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Lorentzian line shape with
a linewidth (full width at
half maximum) equal to
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Linear Response Theory
The summation over n includes all of the magnetic sublevels of the
atomic excited states. However, on average only one-third of the aβn
transitions will have their dipole transition moments parallel to the
polarization vector of the incident field, and hence only one-third of
these transitions contribute effectively to the susceptibility.
Lets define oscillator strength of the aβn transition:
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Let us next see how to calculate the refractive index and absorption coefficient.
linear dielectric constant linear susceptibility If assumed
The significance of the refractive index n(Ο) is that the propagation of a plane wave through
the material system is described by:
Hence, the intensity π° = πππΊπ ΰ·©π¬(π, π)π of this wave varies with position in the medium
according to: ?
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We define the atomic polarizability πΈ(π)(π) as the coefficient relating the induced dipole
moment π(π) and the applied field E(Ο):
The susceptibility and polarizability are related (when local-field corrections can be ignored)
through:
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the absorption cross section π
Remember:
Remember from slide 40:
Next we calculate the maximum values that the polarizability and absorption cross section can
attain. We consider the case of resonant excitation (Ο=Οna)of some excited level n. We find,
through use of and dropping the nonresonant contribution, that the polarizability is
purely imaginary and is given by: Next slide
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the state associated with level n
that is excited by the incident light.
1/3 no longer appears because we are now
considering a particular state of the upper level and
are no longer summing over n.
When is Max?
When it is min
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Density Matrix Calculation of the Second-Order Susceptibility
The applied optical field is expressed as:
a
b
Substituted
in . b
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Substituted in . a
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We have given the complicated expression in curly braces the label π²πππ because it
appears in many subsequent equations.
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We next calculate the expectation value of the atomic dipole moment:
polarization oscillating:
We define the nonlinear susceptibility through the equation:
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If we now perform the summation over the dummy indices l, m , and n and retain only those
terms in which both factors in the denominator are resonant, we find that the nonlinear
susceptibility is given by:
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We represent the density operator as
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In the numerators
of terms (π2β² ) and (π1)
are identical and that
their denominators can
be combined as follows:
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The same procedure can be performed on terms
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Density Matrix Calculation of the Third-Order Susceptibility
We also represent the electric field as:
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The nonlinear polarization oscillating at frequency Οp + Οq + Οr is given by
By combining through ,we find that the third-order susceptibility is given by:
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Next slide
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Double-sided Feynman diagrams
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Double-sided Feynman diagrams
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Double-sided Feynman diagrams
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Double-sided Feynman diagrams
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Double-sided Feynman diagrams
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Double-sided Feynman diagrams
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Double-sided Feynman diagrams
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Double-sided Feynman diagrams
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Electromagnetically Induced Transparency
Typical situation for observing electromagnetically induced transparency (Fig. 3.8.1) :
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It should be obey SchrΓΆdingerβs equation in the form:
where in the rotating-wave and electric-dipole approximations we can express
the interaction energy as:
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We turn this result into three separate equations by the usual procedure of
multiplying successively by π’πβ , π’π
β ,and π’πβ and integrating the resulting equation
over all space. Assuming the quantities π’π to be orthonormal, we obtain:
We next introduce the explicit forms of the matrix elements of V:
Also, we measure energies relative to that of the ground state a so that:
In addition, we introduce the Rabi frequencies
Equations thus become: Next slide
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We next note that because these equations must be valid for arbitrary values of the parameter Ξ»,
the coefficients of each power of Ξ» must satisfy the equations separately. In particular, the
portions of that are independent of Ξ» are given by:
We take the solution to these equations to be the one corresponding to the assumed initial
conditions that is,
for all times. Next, we note that the portions of that are linear in Ξ» are given by:
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We can thus find the steady state solution to these equations by setting the time derivatives to
zero:
We solve these equations algebraically to find that
The physical quantity of primary interest is the induced dipole moment, which can be
determined as follows:
We thus find that the dipole moment amplitude is given by
and consequently that the polarization is given by
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which is purely imaginary!
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We now calculate the response leading to sum-frequency generation. We express the wavefunction
in the interaction picture as:
We now separate this expression into four equations by the usual method of multiplying
successively by π’πβ , π’π
β , π’πβ, and π’π
β and integrating over all Space, we find that
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We next represent the matrix elements of the interaction Hamiltonian as
and introduce the detuning factors as
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We next need to find the simultaneous, steady-state solutions to Eqs. (3.8.32c) and (3.8.32d).
We set the time derivatives to zero to obtain
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The induced dipole moment at the sum frequency is now calculated as
We thus find that the complex amplitude of the induced dipole moment is give by
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Local-Field Corrections to the Nonlinear Optical Susceptibility
β’ Local-Field Effects in Linear Optics:
we represent the dipole moment induced in a typical molecule as:
the local field that is, the effective
electric field that acts on the molecule.
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The field, which we identify as the Lorentz local field, is given by
By definition, the polarization of the material is given by
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Local-Field Effects in Nonlinear Optics:
the polarization now has both linear and nonlinear contributions:
We represent the linear contribution as
Remember:
to express the factor NΞ± that appears in the
resulting expression in terms of the linear
dielectric constant. We thereby obtain
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Next we consider the displacement vector
the nonlinear source polarization defined by
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This result shows how local-field effects are incorporated into the wave equation.
We now apply the results given by and to the case of second-order nonlinear
interactions. We define the nonlinear susceptibility by means of the equation
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where
The nonlinear polarization (i.e., the second-order contribution to the dipole moment
per unit volume) can be represented as
we find that the nonlinear
susceptibility can be represented as Next slide
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where
gives the local-field enhancement factor for the second-order susceptibility.
This result is readily generalized to higher-order nonlinear interaction:
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We assume that the total polarization (including both linear and nonlinear contributions)
at the third-harmonic frequency is given by
Using:
This equation is now solved
algebraically for P(3Ο) to obtain
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We can identify the first and second terms of this expression as the linear and
third-order polarizations, which we represent as
where (in agreement with the unusual Lorentz Lorenz law) the linear susceptibility is
given by:
and where the third-order susceptibility is given by