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How frequency remains constant though the medium changes?

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When we think of light, we can describe it as an electromagnetic wave or as a flux of particles - photons. The latter description is more fundamental: If you could have a light source with sensitive enough intensity knob, then after just turning it on (minimum intensity), you'd be sending out photons one by one. I believe that answers to your deep questions lie therein. Behold:

Energy of one light quantum (one photon) can be written E=hf,where h is a universal (Planck's) constant, E is energy and f is frequency. We cannot divide photon in pieces, so its energy must stay constant and frequency goes the same way. Devices that appear to divide photons (or change photons' frequency) actually first swallow-destroy the incoming photons and then emit other photons at a different frequency. Frequency of light does not ever change, as long as you can be sure that the photons are the same as the photons at the beginning.

Wavelength L is, on the other hand, tied with energy through its speed, E=hf=hv/L. Atoms of materials, even gases like air, impede the flow of photons - photons bounce off of the atoms (elastic collisions) or are swallowed and re-emitted by the atoms (inelastic collisions). Like I wrote above, a photon swallowed and re-emitted is a different photon. So, it is not part of the original light stream. The Snell's laws speak only about the part of light (photons) that experienced only elastic collisions in a material.

So, in passing from one material to another, light changes wavelength proportionally to the change of speed, so that the ratio v/L=fv/L=f remains constant. But does that mean that it changes color? That depends, how you define color! As color is usually defined via wavelength (i.e. visible light wavelengths in the range 300-700 nm), then indeed, color changes on the interface of two optical materials with different indexes of refraction (like air-glass, air-water, etc).

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