Speed of light in double-slit experiment

Question

Given the fact that the photons arrive at different places on the screen, does the speed of each photon is different ?

Answer 1

No, the light rays all travel at the speed of light $c$ but because they travel different distances they arrive at slightly different times. And indeed it is the difference in the light travel time that causes the interference pattern.

If we take the phase of the light at the slits as our reference, then the phase change once the light reaches the screen is given by:

$$ \phi = 2\pi\frac{\ell}{\lambda} $$

where $\lambda$ is the wavelength of the light and $\ell$ is the distance travelled by the light from the slits to the screen. The distance $\ell$ is related to the travel time $t$ by:

$$ \ell = ct $$

So we can rewrite our phase equation as:

$$\begin{align} \phi &= 2\pi\frac{ct}{\lambda} \\ &= 2\pi \nu t \end{align}$$

where $\nu$ is the frequency of the light and we're using the fact that $\nu=c/\lambda$.

So if you have two light rays (one from each slit) with travel times $t_1$ and $t_2$ then the difference between their phases will be:

$$ \Delta\phi_{12} = 2\pi \nu (t_1 - t_2) $$

When $\Delta\phi_{12}$ is a multiple of $2\pi$ we get constructive interference and a bright spot, and when $\Delta\phi_{12}$ is a multiple of $2\pi$ plus an extra $\pi$ we get a dark spot.

So the light travel times are different and it's precisely this difference that causes interference.

Response to comment:

Have a read through my answer to What is the relation between electromagnetic wave and photon?. It is rarely useful to describe interference by considering photons because interference is a wave phenomenon. It can be done but you would need to resort to a full quantum field theory calculation. If you really want to do this see for example Young's Double Slit Experiment in Quantum Field Theory by Masakatsu Kenmoku and Kenji Kume.