en>MathsPoetry: simpler image ? revert if you disagree

2013-05-10T09:27:20Z

simpler image ? revert if you disagree

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{{Orphan|date=October 2013}}

In [[optics]], '''Miller's Rule''' is an empirical rule which gives an estimate of the order of magnitude of the nonlinear coefﬁcient.

More formally, it states that the coefficient of the second order electric susceptibility response (<math>\chi_{\text{2}}</math>) is proportional to the product of the first-order susceptibilities (<math>\chi_{\text{1}}</math>) at the three frequencies which <math>\chi_{\text{2}}</math> is dependant upon.<ref>{{cite book

| last = Boyd
| first = Robert
| title = Nonlinear Optics
| publisher = Academic Press
| year = 2008
| isbn = 0123694701 }}</ref> The proportionality coefficient is known as Miller's coefficient <math>\delta</math>.

== Definition ==
The first order susceptibility response is given by:

:<math>\chi_{\text{1}}(\omega)=\frac{Nq^2}{m\varepsilon_0} \frac{1}{\omega_\mathrm{0}^2-\omega^2-\tfrac{i\omega}{\tau}} </math>

where:
* <math>\omega</math> is the frequency of oscillation of the [[electric field]];
* <math>\chi_{\textrm{1}}</math> is the first order electric susceptibility, as a function of <math>\omega</math>;
* ''N'' is the number density of oscillating charge carriers ([[electron]]s);
* ''q'' is the [[fundamental charge]];
* ''m'' is the mass of the oscillating charges, the electron mass;
* <math>\varepsilon_0</math> is the [[Vacuum permittivity|electric permittivity of free space]];
* ''i'' is the imaginary unit;
* <math>\tau</math> is the free carrier relaxation time;

For simplicity, we can define <math>D(\omega)</math>, and hence rewrite <math>\chi_{\text{1}}</math>:

:<math>D(\omega)=\frac{1}{\omega_\mathrm{0}^2-\omega^2-\tfrac{i\omega}{\tau}} </math>
:<math>\chi_{\text{1}}(\omega)=\frac{Nq^2}{\varepsilon_0 m} \frac{1}{D(\omega)} </math>

The second order susceptibility response is given by:

:<math>\chi_{\text{2}}(2\omega)=\frac{Nq^3\zeta_2}{\varepsilon_0m^2} \frac{1}{D(2\omega)D(\omega)^2} </math>
where <math>\zeta_2</math> is the first [[anharmonicity]] coefficient.
It is easy to show that we can thus express <math>\chi_{\text{2}}</math> in terms of a product of <math>\chi_{\text{1}}</math>

:<math>\chi_{\text{2}}(2\omega)=\frac{\varepsilon_0^2 m \zeta_2}{N^2q^3} \chi_{\text{1}}(\omega)\chi_{\text{1}}(\omega)\chi_{\text{1}}(2\omega)</math>

The constant of proportionality between <math>\chi_{\text{2}}</math> and the product of <math>\chi_{\text{1}}</math> at three different frequencies is Miller's coefficient:

:<math> \delta=\frac{\varepsilon_0^2 m \zeta_2}{N^2q^3} </math>

== References ==
{{Reflist}}

[[Category:Optics]]

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en>MathsPoetry: simpler image ? revert if you disagree