Can an electromagnetic wave be pulsated at the same frequency of the wave itself? This is a thought experiment that's been bugging me for some time and I haven't been able to find any literature on this topic. I am not a physicist, please bear with me :).
Suppose I have a 500nm laser. Suppose I have a way to turn the laser on and off at a frequency of c/500nm. What will I see if I shine that laser at my wall?
Now suppose that I have a way to turn the laser on and off at a higher frequency, will any light be emitted from the laser at all?
 A: It's not that trivial. When you turn a laser on, you start with spontaneous emission, which is amplified by the gain medium. After few round-trips in the cavity, the amplified mode becomes dominant over spontaneous emission. So in the extreme case, where you turn the laser on and off with the frequency of its light, it wouldn't be laser.
In reality, this process is more complex since you have to consider all the dynamics. Take a semiconductor laser, you need to bring carriers to the active medium (p-n junction, quantum well) and get them out again. So you have the electronic problem of a capacity that is charged and discharged. In addition, the electromagnetic mode that builds up and causes stimulated emission will change your occupation statistics of the carriers, as the generated photons consume electron hole pairs.
Though not an expert in these fields, I'm guessing there are similar things to consider for other types of lasers.
A: What you describe is a radio wave. The electrons in the antenna rod are accelerated synchronously and this time they all emit photons together. With a receiver one can observe this pulsed electromagnetic radiation.
For a weak source and / or large distances it is possible to reconstruct the wave frequency even from the remaining single photons of such a very weak signal or to distinguish in astronomy between different sources.

Now suppose that I have a way to turn the laser on and off at a higher frequency, will any light be emitted from the laser at all?

Let’s get back to the radio source. Driving the antenna with higher frequency reduces the radiated power of the desired frequency and increases the reactive power. But still you get a modulated EM radiation.
In a setup with a continuous wave laser - if it is possible to place a shutter in front of the laser and the shutter can be driven at a higher frequency than the frequency of the laser color - you will again get a modulated EM radiation. This frequency can be measured (using again a shutter with a little different frequency).
