sum frequency generation of light

[Jamie]

Hi,

I'm trying to understand second harmonic generation / frequency doubling
of coherent light in non-linear crystals, which is a specific case of
sum frequency generation, where two phase matched input lights of
variable frequencies, combine in matter to create a higher frequency
output light. I am wondering about the quantum mechanics of the two
phase matched light beams and how they create a higher frequency output
light ie. like in this image:

http://comp.chem.tohoku.ac.jp/eng/img/sfgscheme.jpg

If the two input light beams with different frequencies are able to
create an output light beam with a summed frequency, how is this
described with quantum mechanics, or atomic/molecular electron orbitals?

cheers,
Jamie

 

[Martin Brown]

Hi,

I'm trying to understand second harmonic generation / frequency doubling
of coherent light in non-linear crystals, which is a specific case of
sum frequency generation, where two phase matched input lights of
variable frequencies, combine in matter to create a higher frequency
output light. I am wondering about the quantum mechanics of the two
phase matched light beams and how they create a higher frequency output
light ie. like in this image:

http://comp.chem.tohoku.ac.jp/eng/img/sfgscheme.jpg

If the two input light beams with different frequencies are able to
create an output light beam with a summed frequency, how is this
described with quantum mechanics, or atomic/molecular electron orbitals?

Try

http://www.scitopics.com/Sum_Frequency_Generation_SFG_Spectroscopy.html

or

http://iopscience.iop.org/0953-8984/17/8/002/

I think you may need to be a member of IOP to read this latter link, but
the abstract should give you keywords to use in a wider Google search.
Might also be worth a look on ADS abstracts too...

It is certainly not worth $33 to read it!

 

[Jamie]

IIRC the usual method is to define a second-order dielectric
susceptibility operator, and then hit it with two copies of the bra and
ket vectors to compute the expectation value.

We classical types just measure the second-order susceptibility, compute
the output beam strength using the slowly-varying envelope
approximation, and put the shot noise in by hand afterwards.

Hi,

It sounds like quantum mechanics/electron orbitals aren't able to
explain this phenomenon.

From your explanation I guess the two input light beams are polarizing
the matter (at three discrete frequencies f1, f2, f1+f2), and the second
order susceptibility is the polarization that occurs at the summed
frequency of the two beams, which creates the new higher frequency EM
field starting with a new electric field.

So if you have three in-phase input beams of variable frequencies, there
would be three discrete summed frequencies (f1+f2, f1+f3, f1+f2+f3)?

cheers,
Jamie

 

[Jamie]

Try

http://www.scitopics.com/Sum_Frequency_Generation_SFG_Spectroscopy.html

Hi,

That link makes sum frequency generation sound like a quantum process.
Maybe there is always a classical sum frequency output, but as the
spectroscope frequency is shifted there are peaks based on
atomic/molecular quantum orbitals. It sounds like it is two separate
phenomenon.

cheers,
Jamie

 

[Jamie]

Not at all. It's well understood quantum-mechanically, and if I reach
back far enough into the archives, I probably still understand the
quantum bit well enough to do calculations, though I don't usually need
a lot of quantum for what I do. Nico Bloembergen's book, published in
(iirc) 1963, has all that stuff in it.

Quantum mechanics is a helpful shortcut for some things, interestingly
enough, e.g. the sum and difference frequency mixing in an acousto-optic
cell. Conservation of energy and momentum tells you that the light beam
that gets bent along the acoustic propagation direction has been
upshifted, because each photon has notionally absorbed a phonon; light
diffracted backwards (against the acoustic propagation direction) has
been downshifted, again because notionally each photon has produced a
phonon by stimulated emission.

Hi,

Does that mean that every time light passes through a crystal that there
are material vibrations in the crystal called phonons? Or are there
only phonons in special cases of light interacting with crystals such as
sum frequency generation?

cheers,
Jamie

 

[Tim Williams]

Does that mean that every time light passes through a crystal that there
are material vibrations in the crystal called phonons? Or are there
only phonons in special cases of light interacting with crystals such as
sum frequency generation?

Phonons are present all the time -- they are the quantum mechanical
vibrations of a crystal. If you imagine a crystal as a ball-and-spring
lattice, you can imagine waves traveling through it. The quantum picture
says these vibrations are quantized in energy levels. These vibrations
contain energy, so this represents a bulk material property which contains
energy, throughout the space of the crystal. If you touch a crystal to
another, it's going to transfer this shaking, until both are shaking with
about as much energy (note, not the same velocity or frequency, because
that's material dependent). Starting to sound like the transfer of heat,
right? Great!

Phonons are also acoustic vibrations. Whereas thermal vibrations are
randomly distributed, acoustic vibrations are coherent (or generally much
more so). Although the frequencies are fairly low, the energy levels are
fairly high, since the mass is a lot higher than the effective mass of a
photon (MHz acoustic phonons might be closer in energy to ~THz optical
photons, but you'd have to check the equations to see if that's the right
order of magnitude).

Last piece of the puzzle, photons can interact with phonons because
phonons cause changes in density, and therefore permittivity, etc. This
modulates the wavelength at least, and can lead to exchanges in energy by
coherent or incoherent (scattering) processes.

Finally, I'm not big on the whole acousto-optical statistical quantum
mechanical systems thing, so I leave lots of room to be very wrong here,
and will let others continue...

Tim

 

[Jamie]

You have to get your head around a couple of things to understand QM,
assuming that that's your intention. The first is that a photon, or a
phonon, is an elementary excitation of a normal mode of some system,
i.e. the electromagnetic field in a laser cavity or an elastic wave in
some solid, respectively. The usual classical differential equations
continue to apply, but the excitations are quantized. (There are higher
order corrections that lead to all sorts of field theory that I've never
studied, but at ordinary energies and field strengths, this is the case.)

A system containing bits of vacuum and dielectric will have certain
electromagnetic normal modes, which when quantized give you photons.
It'll also have acoustic normal modes, which when quantized give you
phonons. But the fact that they have names shouldn't make you think that
they qualify as _things_ in the same way that electrons do, for instance.

The second is linearity. The basic equations of mathematical physics are
all linear, i.e. if you have two forcing functions, the resulting
response is the sum of the responses to the two functions separately.

All _materials_ become nonlinear eventually, at some level, but the
vacuum doesn't, or not until you get to ridiculous energy densities, and
you need quantum field theory to handle that.

Coupling between photons and phonons is called the acousto-optic effect,
or at short acoustic wavelengths, Brillouin scattering. It's normally a
very weak effect, much weaker than optical absorption, say.

Optical frequency doubling and four-wave mixing don't require phonons to
exist at all--all they need is a nonlinear electric susceptibility term.
You apply one or more pump beams to the crystal, which sets up a
dielectric polarization at 2f and DC, or f1+-f2, and that polarization
in turn starts radiating. Since the phase of the polarization is
controlled by the phase(s) of the pump beam(s) at each point in space,
its direction is also controlled.

You need to pick crystal orientations and beam k vectors appropriately
to get the radiation from each peak to add coherently, which doesn't
happen by accident, which is why there isn't a whole lot of second
harmonic generation in most optical systems.

Hi,

Thanks for all the information, I agree a difficulty in understanding
photons/phonons is they don't qualify as material things as you say!

Sum frequency generation is a linear process by definition, so I don't
see how the "phonons" could be thought of as quantized for that case
where two frequencies add up.


cheers,
Jamie

 

[Jamie]

Sum frequency generation is bilinear, not linear linear--i.e. it's
linear in each of the two input amplitudes, like an analogue multiplier
chip. If it were purely linear, there wouldn't be any cross products
generated.

Hi,

I think the two input frequencies can still create an arbitrary output
frequency though.

cheers,
Jamie

 

[Jamie]

I think you aren't actually interested in understanding.

Hi,

Well I was trying to imply that if there can be an arbitrary frequency
generated by the two input frequencies, then that would indicate that
the intermediate phonon would not be quantized perhaps.

cheers,
Jamie