Olsen's ribbon mikes often used horn-effect to tweak-up the top octave.
"PZM"s are often used on 2-surface and sometimes 3-surface "corners" to give support and some directivity in free space. These are conical horns.
By eye, these horns are ~6" across the mouth, without flange. They do nothing below 400Hz. Of course many mike flaws are above 400Hz. And specifically the flaws caused by finite diaphragm diameter and stick-mike case length.
I've seriously sketched large horn mikes for semi-fixed installation, 2 feet or more across the mouth. Polar plots may be estimated from similar speaker horn plots. Exponential is not really a virtue, the way I saw it, though I did not consider damping of the diaphragm. (At least he spells "damping" properly.) The Simpsons appear to be conical, at least near the mouth. If the flare cut-off must be within the audio band, conical has mild effects and exponential leverages itself into a catastrophe just below cutoff.
> the initial image montage is lots and lots of pictures of the SAME microphone...
Further down the page is shown two mikes in a case. The lighting is too casual to be PhotoChop.
> I would guess that any 'reverse-horn' would have SERIOUS issues in the time domain...
An infinite horn is a perfect acoustic transformer.
Getting "infinite-enough" is a practical problem.
A physical diaphragm is generally an inexact match to horn impedance. The mismatch can be used to good effect, but that causes other effects.
If you want an unfair experiment, remember Reciprocity Principle. Steal the horns and drivers off the top of a PA stack and wire to a mike input. Directivity is identical. Frequency response is complicated... there is an exact answer but I think you need the current-response (not voltage response) when worked as a speaker, to plot the response as a microphone. Whatever, you will want to EQ it to taste. Usable bass response may be much better as mike than as speaker. Accept that mikes are held to a higher standard than speakers are, so flaws which are accepted in speakers (like an early JBL2440's 11KHz ring/dip, and most horns' half-wave-mouth directivity bump) will be embarassing as a mike.
> When I try to imagine what happens with the incoming spheric wave traveling into the horn, I see a lot of reflections...
For "sane" mike technique, the wavefront should be plane for all practical purpose.
If you put a 6" mike at 6" from a small source, yeah, it'll be spherical.
Either way: I don't think reflections are an issue on a simple shape like this. If you trace all the rays it does not give "reflections" but a mild broadening, and only at the very top of the bandwidth implied by throat dimensions. That may be what the center spear is for.
Or eyeball it. Take his shape and mirrorize the inside. Put a bright orange disk at the throat. Stand on stage. What do you see? I believe you will only see the disk, no reflections in the walls.
I may be wrong on that.