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 () is proportional to the product of the first-order susceptibilities () at the three frequencies which is dependent upon.[2] The proportionality coefficient is known as Miller's coefficient .

Definition

The first order susceptibility response is given by:

where:

  • is the frequency of oscillation of the electric field;
  • is the first order electric susceptibility, as a function of ;
  • 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;
  • is the electric permittivity of free space;
  • i is the imaginary unit;
  • is the free carrier relaxation time;

For simplicity, we can define , and hence rewrite :

The second order susceptibility response is given by:

where is the first anharmonicity coefficient. It is easy to show that we can thus express in terms of a product of

The constant of proportionality between and the product of at three different frequencies is Miller's coefficient:

References

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