Below I suggested it would be easy for EDM or ECM. Have I been misled?
Compared to say machining aluminum or steel you will be practicing some patience.
Not necessarily a bad thing to have in the world of metal working. ("increased feed speed is not a substitute for bad planning")
You can also cut tungsten with water jets loaded with abrasives, I've seen a 2" thick block cut like that, it was quite impressive. But that won't get you a surface. You'd need to start with a sheet of the right thickness.
Tungsten foil/sheet isn't that hard to come by.
Cost a small fortune too.
There's also the issue of electrode potentials: sucking a coulomb off your anode at 0.5 volts takes only half a joule, but sucking off the same coulomb at 2 volts takes 2 joules. I don't understand standard electrode potentials well enough to predict what voltage you need for efficient tungsten ECM, but on first glance at the reference table, it doesn't look particularly bleak. The tungsten entries are in the middle of the table, near things like lead, tin, and iron, rather than way down on the upper east side with hoity-toity mercury and iridium, much less out in the ethereal districts where gold and ozone dwell.
Fascinating processes by the way, and the precision that you can reach with EDM is insane.
I think the density ratio for tungsten and steel (19:8, just over 2:1) is actually more even than the atomic mass ratio (23:7, just over 3:1). So I'd think that even on a volume-per-coulomb basis ECM would cut tungsten faster, assuming comparable faradaic efficiencies. Do you have any idea about the electrode-potential question? I'm totally lost.
That's a question worth settling, from memory the speed is not so much a function of the material as it is of the current density that you can achieve with the gear you have. Thousands of Amps will get you some pretty good speeds, on the order of a cubic cm of material removed per minute for a 25KW machine.
I also wonder what would happen if you tried to plasma cut it, that should work as well.
If there is one thing I really miss from life in Canada then it was the fully equipped machine shop, that was such a great thing to have. You could go from idea to a pretty decent physical implementation fast than you could have ever mail ordered the parts and there is so much satisfaction in working metal. The plasma torch I had there was a relatively small one (12 KW) but it still cut anything an everything I ever threw at it with pretty impressive speed. But I never tried it on Tungsten. Missed chance! I suspect it wouldn't work all that well because Tungsten isn't the best conductor and probably would backsplatter all over the place from the compressed air dispersing the metal that would melt rather than evaporate so you get a puddle rather than a stream of gas.
Electrolytic processes are really interesting and complicated, and 200+ years after Davy used them to revolutionize chemistry, I think they're still underused. I'm preetty sure you can produce Fresnel reflectors for a given wavelength, for example, by anodizing aluminum foil to about a quarter-wavelength depth, and across a wide range of visible wavelengths by anodizing it to about 3 microns depth, with the electropolishing effect inherently eliminating the small asperities that make it so slow to produce lenses and mirrors by grinding.
Plasma cutting should work, but should be slower on W than on iron; WP says that at room temperature its specific heat is 24.27 J/mol/K, but at 183.84 g/mol, that's only 0.13 J/g/K. For iron the same figures are 25.10 J/mol/K and 55.845 g/mol, so 0.45 J/g/K. So tungsten should heat up three times as fast with the same power output at room temperature; but tungsten's heat of fusion is four times higher, and I think that's actually the dominant component of plasma-cutting energy consumption. But I think you have roughly four orders of magnitude more knowledge about plasma cutting than I do.
Has anybody tried that?
Reading I've done since then explains that, when anodizing, the anodized layer grows up out of the surface roughly as far as the surface gets depressed, and the anodized layer has a higher refractive index than air, so, for many purposes, you don't need to etch nearly as much metal. (The anodized layer is thin enough that it would be adequate in many applications that normally require a first-surface mirror, though not all.)
It's well known that if you're instead anodizing silicon you can fabricate a rugate filter by varying the anodizing current to vary the density of the anodized layer, that when anodizing titanium you get brilliant iridescence due to the high refractive index of the oxide and consequent strong single-layer dichroic filtering effect, and that there's a negative feedback on the current from the oxide layer thickness in all of these cases, requiring higher voltage to get a thicker oxide; this should also tend to even out small surface asperities, and is not the same effect as anodic leveling. The possibility of including such dichroic filters on the surface of the holographic reflective optics, with wavelength response tailored at a submillimeter or even submicron X-Y resolution, could be interesting for fabricating a wide variety of optical systems.
If you end up trying it, I'd love to hear about the results! If I'm not dead by then. Though I'm sure you have a long list of ideas of your own that you can't wait to try, like everyone else capable of trying this sort of thing.
All this seems to me like a crucial enabling technology for feedback control of machine tools, because light waves don't distort or change their dimensions much when the temperature in the room changes or there's a side load on your gantry.