Trying to understand small increments of light. From a particle perspective, a photon is the smallest increment, dependent on frequency as E=hf. From Maxwell's wave perspective, light propagates as a result of the energy passing between the E field and the B field. The rate at which this occurs gives the frequency. So from a wave perspective, how short can a wave of light be? Less than one wavelength? If I have a source, a radio transmitter, that emits 10 1/4 lambda of light, and 10 lambda is received/filtered/reflected, can the 1/4 thats left when the filter is removed still propagate?
Think some more about classical E&M before you worry about photons. Nope, that's not the way an EM wave works. The energy does not pass from E to B and back again. E and B are in phase. At the crest of the wave, E and B are both maximum simultaneously. Ninety degrees later, at the node, they are both zero simultaneously. The energy density is E^{2} + B^{2}. That means there is plenty of energy at the crest, shared by E and B in equal amounts, and none at all at the node. A short wave pulse cannot be a pure sine wave. If you cut off a sine wave into a short pulse, the wave will no longer be pure, it will contain high frequencies. Frequencies so high that their wavelength is comparable to the pulse length.
Example, an extremely low frequency em wave, 0.1hz, if you take a 2.5 second burst is it not photons present in a 1/4 wavelength of a sinusoidal charge/magnetic field?
I'm thinking about it in terms of a particle accelerator. If I use the 1/4 lambda above can I aim that at a proton and get it to accelerate due to the charge wave?
Yes, just as the shape of water waves will change if you do something to disturb their propagation. This is one of the reasons that Bill_K is advising you to understand classical E&M before you try thinking in terms of photons.
Alright. So can I read any math that will describe this? the limit of the existence of the wave? The other effect I was interested in is the faraday effect... how sharply/irregularly can I twist a linearly polarised wave and keep it as propagating light?
Bill_K's reply in post #2 seems intuitive enough: Try this: Draw a sine wave with one full frequency cycle. Now remove/erase as much of the sine wave as you want to represent a shorter pulse. When you're done erasing, draw/add vertical extension(s) to the curve that remains to connect to the zero 'node' axis. Now do you see why fractions of frequency pulse result as Bill_K described?
If you want to learn more about these disturbances people keep referencing: http://hyperphysics.phy-astr.gsu.edu/hbase/math/fft.html#c1 Of course, it depends on what filter you put on the signal, but in my experience* it tends towards a sinc function. *I have no experience.