The choice isn't so broad once you put "transparent"...
Pure polystyrene is quite brittle. Flexible if really thin, never soft. Shock-proofing additives (acrylonitrile and butadiene) lose the transparency.
Pmma and PC are the very transparent polymers (better than glass), but their softening point resembles Pet.
PE, PP are all hazy. PVA less so.
The best extrusion candidate for me is
PMP, poly(methylpentene). It's very soft and flexible, much more transparent than other polyolefines,
but it resists boiling water for sterilization. Not exactly cheap. You've already seen it at syringes: nicely transparent and shock-proof.
http://en.wikipedia.org/wiki/Polymethylpentenehttp://de.wikipedia.org/wiki/Polymethylpenten injection at 270°C, ouch...
links to datasheets there.
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The other option would be - if acceptable for your application - to polymerize or polycondensate in situ. You could
mix your optical additives first, then polymerize. MMA does it at room temperature with the UV fraction of fluorescent lamps or sunlight (hence work under red light or filament bulbs, and mind the nasty volatiles). Not the extrusion you planned, a bit brittle, but transparent and cold. MMA sells as a glue for PMMA but is useable alone, in moderate thickness.
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What about
laminating your optical additives between sheets of transparent plastic like PC? This must work at a lesser temperature than extrusion. Or
include the additive in a much thinner glue layer (MMA or standard glue) between two mechanical sheets?