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4-wave mixing too awesome for amateurs?

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hboy007

Sat Aug 13 2011, 11:45AM

Registered Member #1667 Joined: Sat Aug 30 2008, 09:57PM
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Posts: 374

1313232498 1667 FT0 4wave Convolution Teaser

(the fourier transform of a delta function yields unity in the frequency domain so it does not affect the rest of a product, see below)

Four-wave mixing is perhaps one of the most jaw-dropping methods in non-linear optics. But what is this, you might ask. Let me try to explain.

Non-linear optics and green laser pointers
If you take a medium without point inversion symmetry (centrosymmetry), e.g. a KD*P crystal, the electrons inside cannot be moved as easily in one way as in the other. When the electron motion is coupled to an alternating external field, a sinusoidal electric force cannot move the electrons in a sinusoidal manner, in fact sinus halfwaves are buldged in one direction. Since the motion of the oscillating electron is still periodic with the period duration of the incident wave, only oscillations with integer multiples of the base frequency can contribute to the deformation of the path.
The first thing to be considered is of course the second harmonic of the incident wave and a mixing of the first and second harmonic usually describes the assumed motion of the electron. Now that's what makes your 1064nm laser transform into juicy green 532nm and is called "second harmonic generation" or "frequency doubling".

making it work: phase matching
We've neglected the fact that green light is generated throughout the whole slab of KDP crystal and waves generated from any position in the crystal must interfere constructively with the rest of the green beam. This translates into the condition that both 1064nm and 532nm beams must be phase-matched. So we call this the phase matching condition which is guaranteed by picking a non-linear crystal that is also birifingent so we can dial the refractive index by rotating the crystal. Unfortunately, the refractive index depends on the lattice constants of the crystal which have the habit of drifting away as the crystal heats up. That's what makes that damn green laser pointers so unreliable.

higher order phenomena do cool stuff
Let's get back to 4-wave mixing. I've told you the story of second order non-linear optics effects. The next order of non-linear interaction basically brings the interaction of three waves that generate a fourth frequency. I forgot to mention that the process above can be considered as "light mixing with itself" where the difference frequency vanishes and the sum frequency is just twice the incident frequency, so the implication of a third wave is just the next step to do.

putting it all together
The output frequency is connected to all three waves by a nasty 81-component tensor which we don't have to touch for the moment. More important is the fact that all the input waves must be present to generate an output wave. This enables us to overlay two fourier images and get the product of them at each point by 4-wave mixing. Let's assemble something cool:

from fourier optics or basic photography experience we know that where the f-stop is mounted, the coverage radius on the film or sensor is not directly affected (except for the outer edges) when putting objects into the aperture or closing the blades, however, images start getting blurry again when the f-stop is closed. This is known as the diffraction limit and ultimately performs low pass filtering on the image. So somewhere near the lens plane the image must exist in a frequency-domain form. Actually it is the fourier transform of the image.
Have you seen images like these before? Or have you ever bought a digital camera that says "digital image stabilizer"? Well, then you know at least one application of (de)convolution of an image. Convolution works like replacing every pixel in an image with a stamp that has the brightness of the pixel. When you take pictures at night without a tripod, all the lights are smeared in longer exposures. The way they are smeared defines the stamp pattern. This is called the blur kernel. The cool thing is that the image can to some extent be recovered by guessing the kernel and applying the inverse operation.

conclusion
The convolution theorem states that a convolution performed in real space is equivalent to multiplying the fourier representations of both the image and the blur kernel or its inverse (and therein lies the rub of deconvolution). Now all the forward and back-transforming of images consumes a noticeable amount of processing power when done on portable devices because the amount of data processed is large even when using fast algorithms.

The cool thing is that an optical system can already deliver the fourier transform of an image and third harmonic non-linear optics can do the multiplication. The same piece of optics then does the inverse transform and there you have it. Now have a look at the image I quoted from the paper above. It looks way cooler now that you know what it means, doesn't it? How come amateurs have only made it to holography and lasers? Why not do more of the fancy non-linear stuff?

Dr. ISOTOP

Sat Aug 13 2011, 04:14PM

Registered Member #2919 Joined: Fri Jun 11 2010, 06:30PM
Location: Cambridge, MA
Posts: 652

Convolutions and Fourier transforms are cool and all, but what sort of hardware does it take to do all this?
If its anything like SHG then its *very* difficult to achieve without the right setup.

hboy007

Sat Aug 13 2011, 05:14PM

Registered Member #1667 Joined: Sat Aug 30 2008, 09:57PM
Location:
Posts: 374

I suppose the fourier transform setup is quite nasty to build and operate, but there are numerous applications of 4-wave mixing in materials analysis and spectroscopy. Some are covered in the paper I referred to.
Sorry, just got excited about the convolution process so I had to share this with you.

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