As I sat on the vibrating bus with my head leaning against its vibrating window, trying to read text on my cellphone, I noticed that the visual OTF induced by the vibration changed over time as the relative phase of components of the vibration changed. It occurred to me that this may provide a feasible way to measure oscillations that are too fast to measure directly or even rectify.
My vibrating head, vibrating because of the vibrating window, reduced the text to mostly just a blur — the convolution of the true text image with the path my eye was taking through each cycle of the vibration. If the path were an ellipse, I'd have seen just a blur and little more; but in fact there were two or three copies of the text, perfectly clear but overlapping, at spots in the vibration path where my eye had temporarily come to rest. But these copies moved as the vibration changed.
Suppose you have an oscillation of some unknown frequency around 1 MHz vibrating a mirror which is directing a focused laser beam at the wall. Without the vibration, the beam would draw a point; with the vibration, it draws a line, which is brighter at the ends than in the middle, because the beam spends more time at the ends. In particular there are singularities of maximum brightness at the ends of the line, like caustics, because the beam actually becomes stationary there. You can see these phenomena with your eyes even though the oscillations are four orders of magnitude too fast for your eyes to see them. They allow you to measure the amplitude of the vibration, at least if you have calibrated the mirror, but not its frequency. They give you some information about the shape of the waveform — how much time it spends at each height — but not in what order.
Now, maybe this is not the best example, because in real life, you could vibrate the mirror in the other axis with a lower-frequency signal, say 100 kHz, and observe the Lissajous pattern; you could adjust the frequency of the reference signal until the Lissajous pattern was stable, although this might be very challenging to do by hand — to get the Lissajous pattern to be shifting at less than 50 Hz, your harmonic frequency needs to be within 50 Hz of the unknown signal. But let’s suppose you have only one dimension to work with, and that the unknown signal is very spectrally pure.
If you add a 100-kHz reference signal to the displacement in the same axis, the pattern of bright and dark in the line projected on the wall will change. If the unknown frequency is a precise multiple of the reference, it will produce a stable pattern of bright and dark — in particular, at the points where the sum wave has zero derivative, you have more of those bright singularities that previously appeared only at the ends of the line. If the reference signal has a small enough amplitude, there will be 20 of them, but as the reference signal amplitude increases relative to the higher-frequency unknown signal, more and more of these singularities will disappear.
If the harmonic relation is imperfect, this will manifest as a continuous phase shift between the reference frequency and the unknown frequency, with the bright spots moving around; just as with a Lissajous figure, the speed of this phase shift tells you the precise difference in frequency.
(This is related to CCD oscilloscope, which concerns a different and much simpler way to measure fast signals with slow sensors.)