I've done this before. The requirement was for a 1µs. Phosphor LEDs are variable. Some are fast and others are slow.
Attached is a schematic of the test set-up I used to drive LEDs with 1µs pulses of hundreds of amps, hence the low value current limiting resistor. It's possible to overdrive LEDs by a factor of 10, for short pulse lengths. Note transformers are used to drive the MOSFET and measure the current, to avoid large currents passing through the grounds of the test equipment.
I'm working on digesting all the idea's that have been suggested here. Zero999, I'm just making sure my flea brain is understanding the diagram, you use an inductor to drive the LEDs rather then using a Flash Capacitor.
I appreciate the lead on the LED you were testing. Vela mentions they spent a lot of time trying to find the right LED's for the job. I suspect this gets back to the issue of how long the phosphor stays excited. I'm fairly confident they are using white LED's, the images of the unit show the LED's and the yellowish tint characteristic of white LED's is clearly visible. This would simplify the circuit, as we wouldn't need to worry about white balancing the RGB LED's. Since LED's MFG's aren't typically concerned about the switching time of a White LED, I can image there is a lot of variation, possibly even from lot to lot.
I've studied his schematics and everything he's doing makes sense. I am assuming the length of the pulse is being controlled by the software on the MC.
Microcontroller control only really makes sense if the micro triggers both the chain of events you want to photograph as well as the flash.
If you are say pulling the trigger on a gun with a string it makes much more sense to just keep everything analogue and trigger on sound or a light gate, no micro necessary.
As far as triggers go, I was planning on using an IR optical sensor. I've put one together before, and have had fairly good success with it. As the bullet passes by the "shadow" creates a signal. The trouble with using this directly is that the duration of this pulse is dependent on bullet velocity and bullet length. Fast short bullets would create pulses that would likely be far less then 500ns, while long slow bullets would create a pulse that would last far longer then 500ns. (These type of sensors are commonly used in high end chronographs)
So we would want the MC to trigger on the pulse, with a programmable delay (just a few microseconds) before sending the trigger for the flash. By having a short delay, we can dial in the position of the bullet in the frame without physically moving the set up. I know I've tried to use an Ardiuno to create a short 500ns square wave but I seem to remember that it struggled. I don't think the code was fast enough to switch the I/O pin that quickly. It will be another problem I'll need to work through.
To Marco's comment of using low voltage and high amps, I was thinking on that last night. A 10s (37v) Lipo battery rated at 4000mAh could provide a pretty good jolt of energy across an LED. With the duration being so low, I don't think there would be any lasting damage to the Lipo and it would be good for quite some time. There may be other reasons that this wouldn't work, but I was brainstorming.