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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.
$$I_{array,incoherent}=N*I_{single}$$
$$I_{array,coherent}=N^2*I_{single}$$
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.

Tiled combining

Method B --> "Filled-aperture" combining. N lasers are combined using 50/50 beamsplitters and using the 100% constructive interference from one output when the phase is just right. Filled-aperture combining

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 [3]. 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 [4]), 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?

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  • $\begingroup$ The total power scales exactly the same for both coherent and incoherent sources. Where would the additional energy magically come from in the coherent system? If you put ten times 10W in, all that can ever come out is 100W. $\endgroup$ – CuriousOne Aug 3 '16 at 12:31
  • $\begingroup$ Good point, I've mixed up some terms. Will edit them, but then still the question remains: If combining coherent, the intensity scales with N^2 (or not). Since in Method B the beam area remains the same, the power would have had to increase, since intensity=power/area? $\endgroup$ – SvB Aug 3 '16 at 12:43
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    $\begingroup$ The coherent sources will simply combine into a different spatial/angular pattern than the incoherent ones and the max. intensity of that pattern will increase in the coherent case, even though the total power will stay the same. One can't make energy with interference, one can only redistribute it differently. $\endgroup$ – CuriousOne Aug 3 '16 at 12:46
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Asking the question, trying to explain my thoughts and the direction CuriousOne pushed me in with his comments make me think I understand it now, so I'll try to go ahead and answer my own question:

YES, both method A and method B will increase the power of the final beam with a factor N. ONLY method A will increase the final intensity of the beam with a factor $N^2$: One factor due to the power, one factor due to the reduced divergence.
Where I thought methods A and B were different, but with similar end effects, I think I was wrong: Method A is better if you're looking for high intensity.

As to my confusion on how the intensity of a beam can scale with $N^2$ whereas the power can not (even when the beam stays the same size): I had lost sight of the second leaving the beamsplitter:

Imagine a 1W laser hitting a 50/50 beam splitter. Each side has 0.5W output. Coherently combining a second 1W laser will result in a single 2W output and one "0W" output. So yes, from 0.5W to 2W is indeed a factor $N^2$, but no power was generated since this power was 'taken' from the other output. While the power of that specific beam increased by a factor $N^2$, the power of the system increased with only a factor N.

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