Interference pattern with photons in superposition

In the enormous amount of scientific literature it can be quite challenging to find the experiments you are looking for. For example with the following idea I came up with during my search for some more understanding about how all those quantum concepts fundamentally work.

This is the idea: After photons went through a beam splitter, they are in superposition of path A and path B. Then they go through a double slit. What probability function will be measured on the screen?

I expect it to be an interference pattern, with half the intensity of what it would have been if there was no beam splitter and just one detector.

Now I wonder, has this experiment been done and are that indeed the results?

If you know the answer to this, please let me know!