Miller's rule (optics) explained

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

Definition

The first order susceptibility response is given by:$$\backslash chi\_1(\backslash omega)\; =\; \backslash frac\; \backslash frac$$

where:

\omega is the frequency of oscillation of the electric field;\chi _{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 _{1} :$$D(\backslash omega)\; =\; \backslash omega\_0^2\; -\; \backslash omega^2\; -\; \backslash tfrac$$$$\backslash chi\_1(\backslash omega)\; =\; \backslash frac\; \backslash frac$$

The second order susceptibility response is given by:$$\backslash chi\_2(2\backslash omega)\; =\; \backslash frac\; \backslash frac$$where

\zeta _{2} is the first anharmonicity coefficient.It is easy to show that we can thus express \chi _{2} in terms of a product of \chi _{1} $$\backslash chi\_2(2\backslash omega)\; =\; \backslash frac\; \backslash chi\_1(\backslash omega)\; \backslash chi\_1(\backslash omega)\; \backslash chi\_1(2\backslash omega)$$

The constant of proportionality between

\chi _{2} and the product of \chi _{1} at three different frequencies is Miller's coefficient:$$\backslash delta\; =\; \backslash frac$$

Notes and References

1. Miller . R. C. . Optical second harmonic generation in piezoelectric crystals . Applied Physics Letters . 5 . 1 . 17–19 . 1964 . 10.1063/1.1754022 .
2. Book: Boyd , Robert . Nonlinear Optics . Academic Press . 2008 . 978-0123694706 .