To get an easily measurable photoacoustic effect, you might need a rather powerful light pulse. I'd recommend having a look at LIDAR light sources, they're typcally 900nm (not quite 850), are cheap, put out tens of watts and have ns rise and fall times.
A quick perusing of the internets turned up the TPGAD1S09H which is affordable and in stock. While normally driven with a dedicated GaN FET switch and bias/pulse shaping network, in the interests of simplicity, you may be able to use a chunky MOSFET gate driver instead as it'll already have additional logic to sharpen up your control signal's rising and falling edges, the UCC27614DSGR is one such module (although any fast low-side driver could likely be substituted)

In the diagram above, above all else make the red loop path as compact as you possibly can otherwise you're going to have trouble getting decent rise and fall times.
Cb should be sized so that it only holds enough energy for one pulse in order to minimise the risk of cooking the laser should your CPU decide to just leave the thing permanently on - note also that EN is tied to the driver's local supply? We're exploiting EN's under-voltage lockout capability as an additional safety measure.
Rb should be small enough to charge up Cb between pulses but be large enough to act as a fault current limiter (start with 50mA, that laser can't dissipate that much average power, it is pretty small). Ri should be set to limit peak laser current to something sensible (couple amps in this case), start with half an ohm and go from there.
Needless to say, simulate everything as best you can before turning anything on! High power in small spaces can release much magic smoke very fast...