Generally, it's said that for incoherent laser beam combining, the intensity scales with N (where N = # of lasers), whereas the intensity scales with $N^2$ for coherent combining.
Why? Because in coherent combining, the amplitudes add up, and the intensity scales with the square of the amplitude.
However, things seema bit more nuanced and I feel like I've read some confusing info while delving deeper into the theory, which I'll elaborate on below. Imagine two methods of coherent combining:
Method A --> "Side-by-side", or "tiled", combining. N lasers are placed in an array with fill factor 1, all in phase.
What I don't understand stems from the following paper: https://www.ll.mit.edu/news/Fan_LaserBeamCombining.pdf states that:
Another way to look at the nonideality of CBC systems is to recognize that, for the ideal phasing of an array of N elements, the on-axis far-field intensity will be N times higher than for the same array with no fixed phase relations (incoherent) among the elements . This can be simply understood by recognizing that the radiance of the incoherent array is at best that of a single element and that an ideally phased system will have a radiance of N times this amount.
This suggests that if the far-field intensity is a factor N higher for incoherent combining (compared to a single source), it should be $N*N=N^2$ higher for coherent combining. Directly after, it reads
As a note, it has been often stated that the on-axis intensity of CBC systems scales as $N^2$ (e.g., see ), which appears to be at odds with the stated brightness scaling. The on-axis intensity does scale as $N^2$, but only if the emitting aperture grows proportional to N. If instead the aperture size is constrained to a fixed size, then the on-axis intensity scales only as N.
For method A (tiled), I can see how the emitting aperture grows (due to the array) and that the intensity scales indeed with $N^2$. However, for method B (filled-aperture), one of the characteristics is that the aperture size remains constant.
Then, does this indeed mean that using this coherent combining method, you'll really only get a factor N rather than $N^2$ higher intensity?
(If so, what happened to the square of the amplitude? Is there any way to increase the area to get the $N^2$ dependency?)
Some considerations that I felt fit not in the question, but might be useful:
For Method A, I'd say the total beam power scales with N (lasers). Due to the reduced divergence, the final on-axis intensity then scales with $N^2$. (One factor N due to the nr of lasers, one due to the reduced divergence)
For Method B, I'd say the total beam intensity scales with $N^2$ (lasers, constructive interference) and the divergence is unaffected, resulting also in an on-axis intensity of $N^2$, rather than the N suggested above. Extra: If the intensity ($W/m^2$) scales with $N^2$ increases, and the beam area remains the same, I'd still expect an $N^2$ dependency for the power as well, even though you can't 'make' extra power?