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 (${\displaystyle \chi _{\text{2}}}$) is proportional to the product of the first-order susceptibilities (${\displaystyle \chi _{\text{1}}}$) at the three frequencies which ${\displaystyle \chi _{\text{2}}}$ is dependent upon.^[2] The proportionality coefficient is known as Miller's coefficient ${\displaystyle \delta }$.

The first order susceptibility response is given by:

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

where:

• ${\displaystyle \omega }$ is the frequency of oscillation of the electric field;
• ${\displaystyle \chi _{1}}$ is the first order electric susceptibility, as a function of ${\displaystyle \omega }$;
• 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;
• ${\displaystyle \varepsilon _{0}}$ is the electric permittivity of free space;
• i is the imaginary unit;
• ${\displaystyle \tau }$ is the free carrier relaxation time;

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

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

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

The second order susceptibility response is given by:

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

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

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

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

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