No, No, No, Not classical FTIR. Pain in the neck to process.
Your on the right track though.
The easy way is to use a HENE laser or these days, EDCL diode laser, and the Unknown laser into a Michelson Interferometer with the beams one above another. The interferometer is built with prisms and cube beam split ters, not mirrors, because they are cheap and self aligning. One prism o f the interferometer is moving on a flexure, a linear bearing, or in case o f one famous publication, a HO scale flat car and track. All you need is two photodiodes and two TTL counters at the output of the interferometer, the ratio of the known wavelength to the unknown wavelength gives you the d ata. With care in the design of the counters, you can even get it to read out directly in nanometers. Used, they used to be a few hundred Dollars, bu t a quick check on Ebay shows that the price is now outrageous.
The other way is a interferometer built of a reference plate and a t ilted wedge. You shine the reference laser into it using a flip mirror an d measure the fringes with a linear CCD. The reference laser beam is moved and the beam to be measured is routed into the system. The ratio of the di stances from fringe peak to peak gives you the data to solve for the wavel ength. I have one of those setting in the garage, but I don't have the ISA card for it, nor the software. I can route the signal into a storage scope , but have never bothered to go further. It is just two pieces of coated, very accurately made glass, and a tiny single mode HENE laser in the base. That is the preferred way for a pulsed laser.
One day I was out visiting a friend, and was told to go next door and get any gear I could use as the lab's occupant had passed away. Her last reque st was seeing that her gear was used for Education, otherwise her employer was tossing it. So I have a Wavemeter, but their IT department took the PC s before I got there. :-(
Steve