Miller's Rule (optics)

In optics, Miller's Rule is an empirical rule which gives an estimate of the order of magnitude of the nonlinear coefﬁcient.^[1]

More formally, it states that the coefficient of the second order electric susceptibility response ( $\chi _{\text{2}}$ ) is proportional to the product of the first-order susceptibilities ( $\chi _{\text{1}}$ ) at the three frequencies which $\chi _{\text{2}}$ is dependent upon.^[2] The proportionality coefficient is known as Miller's coefficient $\delta$ .

Definition

The first order susceptibility response is given by:

\chi _{\text{1}}(\omega )={\frac {Nq^{2}}{m\varepsilon _{0}}}{\frac {1}{\omega _{\mathrm {0} }^{2}-\omega ^{2}-{\tfrac {i\omega }{\tau }}}}

where:

$\omega$ is the frequency of oscillation of the electric field;
$\chi _{\textrm {1}}$ is the first order electric susceptibility, as a function of $\omega$ ;
N is the number density of oscillating charge carriers (electrons);
q is the fundamental charge;
m is the mass of the oscillating charges, the electron mass;
$\varepsilon _{0}$ is the electric permittivity of free space;
i is the imaginary unit;
$\tau$ is the free carrier relaxation time;

For simplicity, we can define $D(\omega )$ , and hence rewrite $\chi _{\text{1}}$ :

D(\omega )=\omega _{\mathrm {0} }^{2}-\omega ^{2}-{\tfrac {i\omega }{\tau }}

\chi _{\text{1}}(\omega )={\frac {Nq^{2}}{\varepsilon _{0}m}}{\frac {1}{D(\omega )}}

The second order susceptibility response is given by:

\chi _{\text{2}}(2\omega )={\frac {Nq^{3}\zeta _{2}}{\varepsilon _{0}m^{2}}}{\frac {1}{D(2\omega )D(\omega )^{2}}}

where $\zeta _{2}$ is the first anharmonicity coefficient. It is easy to show that we can thus express $\chi _{\text{2}}$ in terms of a product of $\chi _{\text{1}}$

\chi _{\text{2}}(2\omega )={\frac {\varepsilon _{0}^{2}m\zeta _{2}}{N^{2}q^{3}}}\chi _{\text{1}}(\omega )\chi _{\text{1}}(\omega )\chi _{\text{1}}(2\omega )

The constant of proportionality between $\chi _{\text{2}}$ and the product of $\chi _{\text{1}}$ at three different frequencies is Miller's coefficient:

\delta ={\frac {\varepsilon _{0}^{2}m\zeta _{2}}{N^{2}q^{3}}}

References

↑ Miller, R. C. (1964). "Optical second harmonic generation in piezoelectric crystals". Applied Physics Letters. 5 (1): 17-19.
↑ Boyd, Robert (2008). Nonlinear Optics. Academic Press. ISBN 0123694701.

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