Can a broken light filter shift the wavelength of light exiting it? I've measured the wavelength of light emitted from a red Helium–Neon laser passing through a red intensity filter (indirectly, by fitting a function to a single slit diffraction pattern, then forcing the fit to the theoretical values). It came out around 735nm, whereas the theoretical value for He-Ne is 632.8nm (looked it up online, I don't have manufacturer specs).
The light cannot receive any energy from the filter so the wavelength can only be shifted up. The filter is visibly worn out and cracked, and it does a poor job filtering the intensity.
There is nothing wrong with the fitting method as it works well with a green laser. And I tried fitting with 3 different data sets for the red one, to no avail.
What phenomena shifts exiting light wavelength in general? (apart from relativistic corrections)
 A: No, there is no way to convert a beam of HeNe laser to a laser beam of another wavelength with a glass filter. There are some phenomena, like fluorescence or Raman scattering, that will produce light of different wavelength, but it will not follow the path of the laser beam. Also, it will be weaker (much weaker in case of Raman scattering) and have a much broader wavelength distribution.
Therefore, there must be a mistake in your wavelength calculations. You didn't describe how you measured it exactly (in particular, the part about forcing the fit to the theoretical values). You could put a description in another question so that people here could check if your method is correct.
A: Well you did not say just what sort of Red filter you are using.   It could be a multi-layer interference filter (unlikely by the sound of it), or it could be an ordinary absorptive glass filter, such as the Schott sharp cutoff glasses.
A typical such glass, would be the RG645 type, which has a nominal 50% transmission at 645 nm,(for 3 mm thick)  beyond the He-Ne wavelength.  If you actually observed the 632.8 nm output through a mono-chromator, you would find, that the laser is indeed quite attenuated.
BUT !!   it is very well known that these glasses are strongly fluorescent, and your 735nm output sounds like a typical value that RG 645 would fluoresce at.
These glasses demonstrate the folly of believing that the Beer's law of absorption, also applies to energy transmission.    The absorption of the input wavelength photons may follow from Beer's law versus thickness, but those absorbed photons, refuse to stay dead, and are simply re-incarnated at some lower frequency, and retransmitted.  Of course the fluorescence emission is isotropic, so the forward beam is further attenuated by that scattering.  In the end, absorbed energy would heat the glass, so the re-emission could be at thermal wavelengths; but much of the energy will escape by radiation at some frequency.
