I found this thread, and so this site, whilst searching google for an answer to my question, exciting times!
So you split a polychromatic beam, put it back together again and the only time every wavelength is constructive is when the beams have travelled EXACTLY the same distance.
If your source light has a defined range of wavelengths, e.g. exactly 400 - 700 nm wont there be other points where all is constructive between EXACTLY the same distance and infinity?
Especially seeing as energy is quantized, and so there aren’t infinite wavelengths hiding within any small part of the spectrum.
My small head can’t not imagine the phases starting all over again at some point.
The quantization of energy only applies within a given frequency: specifically, the energy in a light wave with frequency f (or wavelength c/f) comes in multiples of hf, where h is Planck’s constant. There’s nothing preventing f from taking on arbitrary values within a continuum of frequencies, unless your experimental setup has filtered out all but finitely many frequencies.
If only finitely many wavelengths (c/f1, c/f2, c/f3, for example) showed up in your polychromatic beam, you could have one of the split beams travel an extra distance L = LCM(c/f1, c/f2, c/f3) and get constructive interference of all the wavelengths. As soon as you allow the wavelengths within the beam to take on all values from the continuum 400 - 700 nm, the least common multiple is no longer well-defined, let alone finite.
Actually even in that case the LCM might not be defined. We’re easily deceived into thinking that wavelengths are commensurate when they’re all reported as an integer number of nanometers, but in fact we only get integers due to the limited precision of our measurements. Taking any two real numbers from the interval [400, 700], it’s overwhelmingly improbable that they would happen to be commensurate; for most pairs of real numbers the LCM is not defined.
On the other hand, if the ratio of the two wavelengths is irrational, you won’t ever get an exact correspondence, but you will get a great many approximate correspondences, some closer than others.