I am not a physicist. I am just curious. Does photon distance increase when light spreads out? When they leave the flashlight, they are close, but as the light spreads, do they leave gaps in between where there is no light?
The term photon describes an elementary particle from the table of elementary particles . They are called "particles" because they interact at a point, with a quantum mechanical probability describing the interaction.
Electromagnetic radiation emerges from a confluence of photons, the wave function of each photon in a superposition ( note this is not an interaction) with the others, so that the squared total amplitude give the classical wave. This image may help understand this.
Does photon distance increase when light spreads out?
For a definition of "distance" quantum mechanically, yes.
When they leave the flashlight, they are close, but as the light spreads, do they leave gaps in between where there is no light?
You have to define "no light". For very large attenuation single photons can be detected, the eye will not see any "light" . In quantum mechanical terms the probability of a photon in superposition with an another photon becomes very small, when the intensity of the light falls at very large distances, but the "gaps" are not perceivable by the human eye. A photon particle detector would find gaps in interaction from the incoming beam .
The photons, as they move away from the source, are spreading as an expanding sphere. So, if you consider just one spherical pulse, then, yes, with distance, the photons will be so far apart that some places, there will be no photons.
However, source of light generally is not a single/short pulse. It is a continuous source. And the light will spread in all possible directions.
Now let us breakdown the continuous light into tiny spherical pulses. Each individual pulse will have spatial gaps, but the other pulses will not have same spatial gaps. So, given a point, you may not see light all the time, but you will see some light some times. Therefore, yes, at large enough distances, there will be time gaps all the places. It is hard to say there will be space gaps given sufficient amount of time.
If the beam is focused (or partly focused), they will converge and become closer up to the focal point (or area). Beyond that or if the beam is not focused they will diverge, separation will increase. In the direction of the beam the velocity of light being constant they remain equally separated. Unless they are traveling through a time changing medium, which can reduce their speed by different amounts, refract or reflect them by different amounts.
Light are quanta, and quantum mechanics can be a bit weird, if you've not met it before...
"Are there gaps?" No.
OK, I'll have a go at this it is tricky to explain and to understand. (No-one has an intuitive feel for it as it is outside normal human experience.)
A single photon is naturally spread out and does so in the same way as a wave.
Never-the-less when it interacts all the energy it has will be involved in the interaction at one location (as a classical particle).
So (and hopefully) clearly this photon thing is not simply a wave. It is something that behaves sometimes like familiar waves and sometimes like familiar particles ("wave-particle duality") but is not actually either of those things.
Look up the Double-Slit experiment if you are curious, you will see that sometimes it behaves in spooky ways we never normally experience.
Upshot is, although if you put a detector in a diverged beam it's possible for light to pass by with-out interacting, but that doesn't mean there was a gap, because there can be less than 100% probability of interacting at a position when quantum mechanics is considered.
It's been shown with quantum teleportation experiments, that entanglement and therefore the distribution distance of a wave function can be macroscopic, (kilometers); To my knowledge no-one has yet put a limit on how far they can extend, distance is probably irrelevant to this phenomena. Thus the interaction probability does not vary as you move a detector across within a diverged light beam.