How to calculate Phase shift for fixed amount of beam steering ?

Hey guys!

I am trying to understand the impact of phase shift on beam steering angle. I have read theory using Array factors and trying to find a correlation between final plot and phase shift. But still I couldn't get the gist.
I specifically want to know the formula for relation between phase shift and the change in beam steering angle. How to decide the value of phase shift? Or you guys could advice me any book for this specific problem. It will be a great help.

I found this image but couldn't understand it.

Its a simple sinus calculation.
The incoming wave has to reach each element at the same time (traveling the same distance)
Instead of thinking of an outgoing wave it might be easier for you if you visualize it as an incoming wave.
Say the wave is coming in from the left at 20 degrees and hits two antennas that are offset some distance from each other.
The phase shift should delay the left most antenna signal so that it arrives at the same time at the right most signal.
The right most signal has traveled longer as the antenna is offset some distance.
if the distance converted to wave length is a quarter wave length then you need to delay it 90 degres.