### Wave Mixing

August 20, 2018

One social construct of my undergraduate days was the college "mixer" in which male and female students would interact socially. This was an innocent time before people could swipe left and right, and when hookups were a perversion rather than a recreation. A college mixer was a place at which a scientist could meet non-scientists and attempt to discover why someone would spend his/her life randomly splashing paint onto canvas.[1]

Mixed drinks and radio mixing boards were likewise a part of my youth, but the more important mixing was scientific and technical. As a novice chemist I learned the difference between a mixture of substances and the reaction product of substances that is a chemical compound. I also made use of a common laboratory mixer, the hot plate with integral magnetic stirrer.

Laboratory hot plate with magnetic stirrer.

The boiling points of aqueous solutions are near the boiling point of water, 100 °C.

The Curie temperature of the most common bar magnet bar magnet, Alnico 5, is above 500 °C, so the stirring magnet is largely unaffected by the temperature.

(Wikimedia Commons image by Ruhrfisch.)

One mixer employed in electrical engineering is the frequency mixer, an electronic circuit that combines two frequencies to produce the sum and difference of these frequencies. This process is magically done by a nonlinear circuit element, such as a diode. This magic, exposed to be a simple trick by mathematical analysis (shown here), enables superheterodyne receivers by mixing a local oscillator with a radio signal to produce a more easily amplified and detected intermediate frequency.

One frequency mixer that I've used often to bring high frequency signals into an easily measurable range is a digital mixer created with a D-type flip-flop. The two square wave inputs F1 and F2 to this circuit produce an output ΔF that's the difference of F1 and F2 when these frequencies are within 50% of each other. It will even give a difference related to harmonics of F1, giving a modulus output (F2 mod F1).

Digital mixer circuit. Left, the usual abstract representation of a signal mixer. Middle, a realization of a digital mixer using logic gates to create a D-type flip-flop. In a D-type flip flop, the output, Q, gets the value of D on the leading-edge (low-to-high) transition of the square wave input at CLK. Right, a practical implementation of a digital mixer using half of a CD4013 integrated circuit. (Created using Inkscape. Click for larger image.)

Signal mixing isn't limited to radio frequency signals. At lower frequency, audio signals at high intensity, about the sound level of a home belt sander or a hand-held electric drill, will produce a non-linear response in the human ear.[2]

At much higher frequency, a non-linear optical material, such as lithium niobate, will allow four-wave mixing in which high intensity light from three lasers at different frequencies will produce light at multiple frequencies. The process is most efficient when there is phase matching of the optical signals, which is generally achieved by selecting certain propagation directions in the non-linear crystal.

If the frequencies of the three light signals are f1, f2, and f3, the resulting frequencies produced will be
f1 ± f2 ± f3
In the end, you'll have twelve different wavelengths of light, three of which are the starting laser signals.

I've had optical physics colleagues who worked on four-wave mixing, and it's one of those topic area in which the devil is in the details; specifically the phase matching and the available optical crystals. To transcend these problems, a team of physicists from Sandia National Laboratories (Albuquerque, New Mexico) and the Friedrich Schiller University Jena (Jena, Germany) has accomplished such frequency synthesis with a gallium arsenide-based dielectric metamaterial. Their results are published in an open access paper in Nature Communications.[3-4].

Phase-matching of the fundamental and generated light frequencies is needed when using non-linear optical crystals for frequency synthesis since the nonlinear optical processes are weak.[3] Gallium arsenide has large nonlinear susceptibility coefficients, and it has low optical loss below the bandgap, but it can't be used since it's a dispersive material.[3] That's why frequency synthesis is relegated to birefringent materials such as lithium niobate and β-barium borate that have much smaller nonlinear susceptibility.[3] A technique called quasi-phase matching can be used for other materials, but it will function over just a small range of frequencies.[3]

Metamaterials consist of nanoscale repeating structures of dimension smaller than a wavelength, and these interact with electromagnetic waves such a light differently than conventional materials.[4] The research team fabricated a gallium arsenide metasurface as an array of cylinders about 500 nanometers tall with a diameter of about 400 nanometers spaced about 840 nanometers apart from each another.[4] The laser infrared wavelengths f1 and f2 mixed to create eleven color frequencies that included f1 + f1, f1 + f2, f2 + f2, f1 + f1 + f2, and f1 + f2 + f2.[3-4]

Illustration of the light output of the Sandia National Laboratories metamaterial optical mixer. The infrared light intensity is about ten times stronger than that of the red light. (Sandia Labs image by Michael Vittitow, modified)[4]

At this time, the conversion efficiency of this unique optical metamixer is low, but the efficiency might be improved by stacking multiple layers of metamaterial.[4] The optical mixing was only obtained through use of femtosecond lasers that produce short, but powerful, light pulses.[4] Says Sandia physicist and paper author, Polina Vabishchevich,
"With this tiny device and two laser pulses we were able to generate 11 new colors at the same time, which is so cool... We don't need to change angles or match phases."[4]

There may be some applications for such a plethora of wavelengths in fiberoptic communication, atmospheric research, remote sensing, and quantum optics.[4] This process would be useful in the creation of intense laser light at wavelengths not generated by today's lasers.[4] This research was funded by the US Department of Energy's Office of Science.[4]

### References:

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