There are several basic issues with your scene, the first one I want to address is your Prism. It is not detailed enough, so there is not a large enough number of angles to bounce the light. Add a Bevel modifier, and a Subdivision modifier to your mesh, and then fix your material to be just a Glass BSDF node with an IOR of 1.44-1.90 (depends on personal preference, this is the realistic range of glass).
The prism is the most important part of the scene, so take your time. This should be something like what you end up with:
Now model the light beams. I have the white one coming in, and curving slightly out the other side (artistic license, not technically correct). I also have them overlapping, so that in the material I can fade from one beam into the next. Like so:
Now work on the materials, I set mine up like this:
Notice that you may need to change mapping values for both materials. Also, I unwrapped the Spectrum mesh and am using the UV map to map the Color Ramp just to give some more control.
Now that the basic scene is setup, it is just a case of tweaking lighting and the prism until you have the desired result.