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It's pretty straight-forward to visualise second-order non-linear optical mixing processes in which two waves of frequencies $\omega_{1}$ and $\omega_{2}$ mix up to generate new waves of different frequencies.

For example the sum-frequency generation: enter image description here

Or the difference-frequency generation: enter image description here

But my question is, what new waves could the two ($\omega_{1}$ and $\omega_{2}$) waves generate by third-order non-linear optical mixing processes?

Would it be possible to have a two-wave input and a two-wave output? For example, $\omega_{1}$ and $\omega_{2}$ input; $\omega_{1}+\omega_{2}$ and $\omega_{1}+2\omega_{2}$ output?

The only examples of third order processes that I found on my references involve an input of three waves, as in: enter image description here

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I like Brandon's very physically intuitive answer: mine is a little drier. It is simply that three waves $E_j(t);\,j=1,2,3$ mix through $n^{th}$ order nonlinearity by way of $n^{th}$ power term $\left(\sum_{j=1}^3 E_j(t) e^{-i\,\omega_j\,t} + E_j(t)^* e^{i\,\omega_j\,t}\right)^n$ in the Taylor series for the input to output transfer function. So in the $n^{th}$ order term we get frequencies

$$|\sum_j a_{n,j} \omega_j|\qquad(1)$$

where:

$$\sum_j |a_{n,j}| = n;\quad a_{n,j} \in \{0,1,2,\cdots n\}\qquad(2)$$

So you need, as you can guess, at least third order to get terms where all three frequencies are together in the sum. From (1) you'll get:

$$\begin{array}{c} |\omega_1 \pm 2\omega_2|\\ |\omega_1 \pm 2\omega_3|\\ |2 \omega_1 \pm \omega_2|\\ |2 \omega_1 \pm \omega_3|\\ |2 \omega_2 \pm \omega_3|\\ |\omega_1 \pm \omega_2\pm\omega_3| \end{array}$$

the last term is the only third order one where all three frequencies combine. You need to go to four wave mixing and higher order to get more "interesting" linear combinations of frequencies of the form being $|2 \omega_1 \pm \omega_2\pm\omega_3|$ and fifth order to get $|3 \omega_1 \pm \omega_2\pm\omega_3|$.

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Why not imagine the third-order process as a two-stage second-order process like so: two-stage third-order process

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