Hello ,I have learned in class that by performing beam steering we decrease the effective area of the array and thus reduce the directivity.
In the array factor shown bellow.
beta=kdsin(theta)
so if we scan by 45 degrees we got an exponent coefficient.
but we take the absolute value of the array factor so this phase exponent is 1.
is there any mathematical proof where we could see that our directivity will reduce as the steering angle goe from 0 to 90 degree direction?
Thanks.

 

These concepts are complicated, here is a good place to start:

P. Hannan, "The element-gain paradox for a phased-array antenna," in IEEE Transactions on Antennas and Propagation, vol. 12, no. 4, pp. 423-433, July 1964,

You can get the pdf here:
https://www.semanticscholar.org/paper/The-element-gain-paradox-for-a-phased-array-antenna-Hannan