Thanks for those images. You probably didn't need us to tell you that the second plate appeared woefully underexposed at 200' as compared to your image at 100'.
Here's the theory, which assumes that the camera is not Rayleigh limited at the 200' zoom.
If you were only adding illumination at the camera, and you wanted the camera to show the plate as brightly at 200' as it had when the plate was at 100', then you would need 16x the illumination at the camera. That's probably unaffordable, or unsightly, or both.
Let's mix it up a bit: what illumination is required to achieve equivalent brightness at the camera, depending upon where you place the illumination? Let's call the baseline the 1x case, with the camera and illumination co-located and the plate at 100' distance. Now consider the camera and plate separated by 200':
(case 1) Illumination at camera: 16x illumination required
(case 2) Illumination at 100' from plate: 4x illumination required
(case 3) Illumination at 50' from plate: 1x illumination required
From your comment, it seems that you have placed "another one near the target." I take this to mean that you are fulfilling the conditions of (case 3), plus a little bit more (mostly meaningless) illumination back at the camera, yet your experience doesn't agree with my theory. Is that correct?
...because I was going to recommend an experiment in which you move that extra illumination to within 50' of the plate, temporarily if need be, then have someone drive through the detection zone and see how it looked. I was going to say, ... "and you will see that this case with 1x illumination at 50' from the plate, with the plate 200' from the camera, should look about as bright as your baseline 100' case which had 1x illumination only at the camera."
But it seems as if you have done that, and it certainly does not match my claims. Is that correct? And you're pretty sure this illuminator is pointing at the plate, right? And the closer illumination is of comparable power to the one that was co-located with the camera to give you that acceptable plate at 100', right?
So now, I'm wondering, could you also have been using some auxillary illumination closer to the plate for that 100' baseline case? I mean, is there something I'm not understanding correctly about your setup that gave the acceptable results at 100'?
Otherwise, the last thing I'd like to verify is that the camera is not Rayleigh limited at the 200' zoom. I'd like to calculate whether or not your camera's lens is capable of resolving the numbers at that distance, regardless of the illumination.
Thanks for providing the lens info: f/1.4 5-50. Absent any other info, I'll assume you're using max zoom at 200'. The given f/1.4 would be for the 5 zoom. 50 zoom would yield f/14, at best. But at 100', to keep the same linear field of view, the lens would probably have been at f/7.
Going back to Airy Disc formula,
@ Infrared: 0.850 um x f/14 = 12 um equivalent pixel resolution
@ Infrared: 0.850 um x f/7 = 6 um equivalent pixel resolution
At 100', you resolved the plate with an effective pixel linear dimension of 6 um.
At 200', you could not resolve the plate with an effective pixel linear dimension of 12 um. 2x worse linear resolution is 4x worse area resolution. Now let's zoom in on the first & second plates to see if that sort of difference might tell the story...
The 100' plate (left) appears fairly clear and perhaps even over resolved.
The 200' plate (right) does appear more coarse, but also appears under illuminated. Looking closely, you can almost resolve some characters. For example, is the 2nd character an M? Is the 4th an R? Is the 5th a D? It's teasing us from the edge of ledgibility.
Bottom line: all this theory hasn't given us much insight. I'd say, move all of those detached illuminators up & bury that plate in brightness. If it still looks like the above comparison, Rayleigh is to blame. Otherwise, let's just blame it on anisoplanaticism.