Re: harmonics, anharmonic ...

[John Denker <jsd@MONMOUTH.COM>]

At 08:21 AM 7/8/99 +0100, Ludwik Kowalski wrote:
> The way in which "nonlinearity" is responsible for the appearance
> of harmonics is clear in electronics. A diode, for example, will
> turn the pure harmonic signal into a rectified signal which is the
> sum many frequencies (plus d.c.). What is nonlinear here? The
> relation between V and I.

We are agreeing on everything so far.

> But what do we mean by the "nonlinearity" when we refer to
> atoms doubling a laser frequency ? What versus what is nonlinear?
> Is the term nonlinearity appropriate in this case?

As you know:
 *) In a harmonic oscillator, the mass experiences a force linearly related
to its displacement.  Examples include
    --) a particle moving in a parabolic potential well, and
    --) almost anything, in the limit of small amplitude.
 *) In most crystals, the atoms are held in position by forces that are
harmonic to a good approximation, for any reasonable level of applied
optical excitation.

To answer your question:
  *) In contrast, in an anharmonic oscillator, such as the rather unusual
crystals used for nonlinear optics, the force is a nonlinear function of
displacement.

At the next level of detail:  A typical nonlinear optics crystal is lithium
niobate.  It has the perovskite crystal structure.  It is a ferroelectric.
It is being operated not too far from its Curie temperature, so it is
relatively easy to move the atoms.  That's right, atoms.  Usually one
thinks of optical phenomena as affecting only the electrons in a crystal,
while the ions are just spectators, but the real world is more complicated
than that.

As a quick and dirty model, imagine lithium ions in little boxes with steep
walls and flat floors -- an exceedingly anharmonic potential.  Right at the
Curie temperature the floor is flat to fourth order, and there is no
parabolic piece to the potential whatsoever.  Just below Tc there is a
slight hump in the floor, macroscopic ferroelectric effects, and two pits
that are parabolic to second order, while the fourth-order terms are huge.
Just above Tc there is only one pit, a small parabolic term, and again huge
fourth-order terms.

Does that answer your question?

Cheers --- jsd