Before we delve into the microscopic physical processes and mathematical relationships that enable nonlinear interactions in some materials, it is worthwhile to simply see what they look like.

The simplest example of a nonlinear process is the addition and subtraction of the colors of two light beams incident on an appropriate material. For example, when two red (800 nm) photons hit a barium borate (BBO) crystal at just the right angle, their energies can add up to produce one violet (400 nm) photon. This process is known as **second harmonic generation** (SHG).

The incident photons don’t have to originate from the same light beam, nor come in from the same angle. In fact, when photons are incident from different angles on a nonlinear crystal, their different ways they interact are resolved in specific dimensions, giving rise to patterns like this:

In this example, three IR beams labeled 1, 2, and 3 (ω_{1} = ω_{2} = ω_{3} = 7,700 cm^{-1}) are incident on a KTP crystal from different directions, and the above pattern is seen at the output. To understand this pattern, we can start with the “triangle” formed by the three white-looking spots. Even though ω_{1,2,3} are invisible, the beams resulting from the doubling (15,400 cm^{-1} orange) and tripling (23,100 cm^{-1}, violet) of this frequency can be seen overlapping at the triangle edges. The spots appear white where the orange and violet colors overlap, but the individual components can be seen “leaking” around the edge.

The bright orange spots at the midpoints along the sides of the triangle represent the process whereby one photon from one beam interacts with one photon from a second beam. For example, the lowermost orange spot represents the *sum frequency* of beams 1 and 2.

The violet spots on either side of the orange spots also represent interactions between two beams, but, in that case, the “weight” of one beam is larger. For example, the leftmost violet spot along the “1-2” side represents the process 2ω_{1} + ω_{3}, whereby two photons from beam 1 add up their energy with one photon from beam 3. Another three-photon interaction is shown by the violet spot in the middle of triangle, ω_{1}+ω_{2}+ω_{3}, which has equal contribution from all three beams.

Finally, the dim orange spots outside of our “defined” triangle originate from processes involving *difference frequency generation. *For example, the spot labelled 4ω_{1} – 2ω_{3} involves the subtractive interaction of the fourth harmonic of beam 1 with the second harmonic of beam 3. Note that, as a 6-photon process, this output beam is much less probable, and thus dimmer, than the central ones.

One may reasonably ask: why don’t we observe nonlinear processes in our daily life? A simple answer is that, even if we encounter objects and materials that are good hosts for multi-photon interactions, the probability of those interactions is much smaller than that of a single-photon process. In order to observe processes such as the ones we saw above, one must focus extremely large amounts of light on the material, which requires a laser. As we will see in the next sections, the electric field from a laser beam can be intense enough to distort the normally linear response of the material and enable multi-photon processes.