azimuth extent for range migration algorithm
I'm going to assume that you are operating in Stripmap mode with no squinting i.e. broadside mode. The aperture size is given by the real antenna azimuth beamwidth (in radians) times the range to the target. What you use as the beamwidth is really a design question - you can use 3dB point or the null-null width. The beamwidth will also be affected by the elevation beampattern somewhat.

You can see near range targets will have a smaller synthetic aperture and thus have smaller coherent gain ( fewer azimuth pulses that can be coherently added). Farther range targets will have a larger coherent gain because more pulses can be coherently added. Most SAR systems have a relatively small range swath widths and so the change in the aperture length is pretty small. The beamwidth is what limits the extent of the azimuth Doppler freqency and thus determines the azimuth resolution.

There are a couple of approaches to handling larger range swath widths. The problem with large range swath widths has more to do with handling the change in range curvatures over the range swath rather than the change in the aperture length. One of the better books to discuss this is by Cumming and Wong:"Digital Processing of Synthetic Aperture Radar Data"

Note that in practical processing systems - if the synthetic aperture at your range of interest contains N pulse, you tend to process much more than N pulses so that you get more azimuth data at the output of your processing. If you only process N pulses - then you get 1 point in azimuth, while if you process 2N pulses you can process N+1 synthetic apertures (1->N,2->N+1, ...,N+1->2N) and you get more effective processing.

In Spotlight mode you steer the antenna beam at one spot on the ground throughout the data collection. This allows a higher-resolution image to be formed but over a smaller area.