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The dispersion of white light beam while passing through a triangular prism is well known. Considering the reversibility of optical path, it should be possible to reverse the experiment.

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  1. How can one implement the reverse experiment of lights' dispersion in experiment?

  2. What are the necessary and sufficient conditions of the incident light beam(s) in order to obtain a white light beam after passing through a dispersive prism? For example, can one remove some of light beams with specific wavelengths or adjust their intensity but still obtains white light beam?

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It is indeed possible. This was a famous experiment by Isaac Newton (published in 1672). Place a lens of focal length $f$ a distance $2f$ from the first prism. Add a second identical prism $2f$ past the lens and rotate it round until white light emerges.

The lens is required to bring the rays back together. It creates an image of the exit of prism one on the entrance of prism two.

The figure below shows how Newton did it. White light enters at O, it is split and recombined by the second prism. The third prism splits the light again. This experiment demonstrated that white light is made from lots of separate wavelengths. Before this it was thought that coloured light was a modification of white light (eg a red filter adds something to white light to make the light red).

In answer to your second question, if you block any of the beams you will not get white light, or at least the same white light. You will get a mixture of the remaining colours.

Newton's version from Opticks part 1

The image is from Opticks Part one and its the last experiment (PROP. XI. PROB. VI.) which you can read here

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