Yes, there can be an interference pattern on the screen if you shine light through two slits and then reflect it off a mirror. This phenomenon is known as "double-slit interference."
When coherent light, such as a laser beam or light from a single-source, passes through two closely spaced slits, it diffracts and creates two separate sources of waves. These waves then spread out and overlap with each other, resulting in an interference pattern on a screen placed in the path of the diffracted light.
If the diffracted light from the double slits is then reflected off a mirror, it undergoes an additional phase change. The reflected light will interfere with itself when it combines with the light from the other slit on the screen. This interference can create a pattern of alternating bright and dark regions on the screen, known as an interference pattern or fringe pattern.
The specific pattern depends on the wavelength of the light, the distance between the slits, and the distance between the slits and the screen. The interference pattern arises from constructive and destructive interference of the waves, resulting in regions where the waves reinforce each other (bright fringes) and regions where they cancel each other out (dark fringes).
It's worth noting that the visibility and characteristics of the interference pattern depend on factors such as the coherence of the light source, the size of the slits, and any potential disturbances or imperfections in the setup. However, in ideal conditions, shining light through two slits and reflecting it off a mirror can indeed produce an interference pattern on the screen.