How to smooth the spectrum of a light source? Could somebody please tell me if there's a reasonably cheap substance or device can I use to smooth the spectrum of a light source?
For example, if the spectrum has spikes as in the blue graph below, is it possible to smear in terms of wavelength (not spatially, as in ordinary diffusers) so that it resembles the red graph more?

The actual shape of the smoothing (convolution kernel) is not really important, but it is preferable that most of the energy is transmitted.
I've searched for "spectrum flattening" filters, but they instead attenuate certain wavelengths rather than spread them out.
I suppose it's possible to rapidly move the light source forward and back so that the Doppler effect would cause a similar shift, but the speeds required would be tremendous!
 A: This might not be feasible for your setup, but you could try rapidly rotating your light source, which would Doppler broaden your spectral lines. You're correct in that the speed would need to be a substantial with respect to the speed of light. As an example, if your frequency is 500nm let's say. If you'd like it to spead out on the order of a single nanometer, you would need to solve:
$$ f = f_{0} \left( 1 + \frac{v}{c} \right)$$
for v, the velocity, given that $f_0 = 500$nm, and $f = 501$nm. If you did so the velocity would have to be 600,000 m/s.  
Alternatively, you could try heating your source up so that the atoms/molecules on average have higher velocities which will broaden your spectral lines (since some will be traveling away from you and some will be traveling toward you).

Assuming that the thermal energy of an atom at temperature T with mass M is equally divided among each degree of freedom it has (see: Equipartition theorem), the root mean square velocity relates to the temperature in the following way:
$$ v_{rms} = \sqrt{\frac{3RT}{M}} $$
where M is the mass of a mole of ideal gas particles. The question is then, in order to see a spreading of 1nm, you would need some of the particles to reach the aforementioned velocity, and at what temperature is this reached for the particular gas you're interested in using?
I personally think the easiest thing to do is to do this numerically, in fact there are plenty of python packages which could assist you in convolving your spectral lines with a known smoothing kernal. You could use a Gaussian kernel (effectively a point-spread-function). If you're hell bent on it being a physical spreading of your emission lines I think you have your work cut out for you (unless I'm forgetting something here).
A: When you scatter light off of a material there is a photon-phonon interaction which will shift the photon frequency depending on the phonon energy (Raman scattering, Brillouin scattering). The effect is quite small, however.
How much broadening do you need? Rayleigh scattering through a warm, high density gas will probably go quite far in messing up the spectrum for you :)
A: The phosphors lining the glass tube of a fluorescent light do a pretty good job of smearing the atomic mercury line emission spectrum into something closer to black body radiation.  There could be a phosphor mix that accomplishes what you want...
