Physics: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 ([math]\displaystyle{ \chi_{\text{2}} }[/math]) is proportional to the product of the first-order susceptibilities ([math]\displaystyle{ \chi_{\text{1}} }[/math]) at the three frequencies which [math]\displaystyle{ \chi_{\text{2}} }[/math] is dependent upon.[2] The proportionality coefficient is known as Miller's coefficient [math]\displaystyle{ \delta }[/math].
Definition
The first order susceptibility response is given by: [math]\displaystyle{ \chi_1(\omega) = \frac{Nq^2}{m\varepsilon_0} \frac{1}{\omega_0^2 - \omega^2 - \tfrac{i\omega}{\tau}} }[/math]
where:
- [math]\displaystyle{ \omega }[/math] is the frequency of oscillation of the electric field;
- [math]\displaystyle{ \chi_1 }[/math] is the first order electric susceptibility, as a function of [math]\displaystyle{ \omega }[/math];
- 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;
- [math]\displaystyle{ \varepsilon_0 }[/math] is the electric permittivity of free space;
- i is the imaginary unit;
- [math]\displaystyle{ \tau }[/math] is the free carrier relaxation time;
For simplicity, we can define [math]\displaystyle{ D(\omega) }[/math], and hence rewrite [math]\displaystyle{ \chi_1 }[/math]: [math]\displaystyle{ D(\omega) = \omega_0^2 - \omega^2 - \tfrac{i\omega}{\tau} }[/math] [math]\displaystyle{ \chi_1(\omega) = \frac{Nq^2}{\varepsilon_0 m} \frac{1}{D(\omega)} }[/math]
The second order susceptibility response is given by: [math]\displaystyle{ \chi_2(2\omega) = \frac{Nq^3\zeta_2}{\varepsilon_0 m^2} \frac{1}{D(2\omega) D(\omega)^2} }[/math] where [math]\displaystyle{ \zeta_2 }[/math] is the first anharmonicity coefficient. It is easy to show that we can thus express [math]\displaystyle{ \chi_2 }[/math] in terms of a product of [math]\displaystyle{ \chi_1 }[/math] [math]\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) }[/math]
The constant of proportionality between [math]\displaystyle{ \chi_2 }[/math] and the product of [math]\displaystyle{ \chi_1 }[/math] at three different frequencies is Miller's coefficient: [math]\displaystyle{ \delta = \frac{\varepsilon_0^2 m \zeta_2}{N^2 q^3} }[/math]
References
- ↑ Miller, R. C. (1964). "Optical second harmonic generation in piezoelectric crystals". Applied Physics Letters 5 (1): 17–19. doi:10.1063/1.1754022.
- ↑ Boyd, Robert (2008). Nonlinear Optics. Academic Press. ISBN 978-0123694706.
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