Lets say we have a double slit setup where a plane wave hits 2 narrow slits and travels down a small channel. One of the channels has a dielectric which will slow down the light passing through it. The dielectric's refractive index is continuous so that all the light is transmitted and none is reflected. The light passing through the dielectric gets a phase delay of $2\pi n$ where n is a positive integer greater than the distance between the slits divided by the wavelength of light used plus 1. If we send through one photon at a time, will there be no interference pattern? If we sent one wave front through the double slit, the waves coming out of either slit wouldn't be able to interfere with each other due to them never overlapping due to their phase delay. I suppose it could be possible that there could be multiple photons in one wave front and that it could also be possible that one photon's wave function could be spread out over multiple wave fronts. I guess that kind of complicates things. I'm guessing that if we send a continuous plane wave through we'll see a diffraction pattern as normal. Though if we knew how many photons were entering both slits combined over time and the amount of photons hitting our screen over time, we may be able to determine how many of the photons passed through each slit. I don't know if that knowledge would interfere with a diffraction pattern's formation though. Any help would be appreciated.
It appears that you're thinking of a photon as if it were a tiny particle-- but it's not. It is a wave, extended in width, height, and length. Its length depends on the coherence length of the source. Actually, the coherence length of the source is the effective length of the photons. When each individual photon goes through the double slit interferometer, it is a wave going through both slits. But when it is detected at downstream, it's found to be at a random point location with a likelihood proportional to the intensity of the wave.
In the experiment you described, you are delaying the wave at one slit by some amount. As long as that amount is less than the coherence length of the source (i.e., the length of the wave packet), it is still possible to get unterference where the two parts of the wave overlap.