Does light influence transparency? Let's consider this thought experiment:
There is a sheet of a (more or less) transparent material. On one side of the sheet there are two light sources illuminating one spot on the sheet. The sources have the same spectrum, let's say a black body spectrum of the same temperature. The first source is pulsed on/off with a given frequency. The second source is either always on or always off.
On the other side of the material, close to the illuminated spot, there is a detector that is only sensitive to pulsed light of the first source and is not affected by the constant light of the second source.
Is it possible that the detected intensity of the pulsed light is influenced by the constant light due to material effects in the sheet? (disregard any effects within the detector)
If so, which material properties are relevant to this?
Does this situation change if one or both light sources are monochromatic?
And, if both are monochromatic, if they have the same or different wavelengths?
(Disclaimer: I'm not a physicist, i'm an electrical engineer. Thus my concepts and my choice of words might be awkward.)
 A: Yes, for example if you use high intensities on a saturable absorber. What you describe is a possible lock-in method to measure this incoherent non-linear effect. The material absorption decreases as the power received increases. At high intensity (high number of arriving photons per second per area), the electrons of the saturable absorber (SA) are excited to higher energy levels and do no have the time to relax back to the ground state, resulting in the depletion of the ground state and consequent decrease of absorption. Materials that behave like this include semicondictors, media with absorbing dopant ions, carbon nanotubes and graphene. The relaxation time of the excited state needs to be somehow long, but fast SA also exist for application in ultrashort laser pulses. As they are mainly used in pulsed applications (light with broad spectra) they should work ok also with incoherent light (depending on their absorption spectrum and the light spectrum).
This is one example, another could be achieved using a medium with high Kerr effect. Here the index of refraction of the material changes with light intensity. The result could be to focus your beam.
