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Refraction at an interface never causes a phase change—but a reflection can, depending on the indexes of refraction on the two sides of the interface

I have seen many answers why the phase doesn't change during refraction but most of the answerer describe the boundary condition. Can someone enlighten me on how these two topics are related or provide some link to read?

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Light is composed of electric and magnetic fields that are perpendicular to each other. The tangential component of electric and magnetic fields are continuous across boundaries. If we use these continuity relations as well as the equations relating the speed of light $c$ and the index of refraction $n$: $$\frac{E}{B} = c = \frac{c_0}{n}$$ a very useful set of equations known as Fresnel's equations can be derived. A phase shift of $\pi$ is represented by a flipping of sign of the electric field across the boundary.

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I often answer this question with the analogous situation of a travelling wave on a string. If the end of the string is tied down, the reflected wave flips phase. This is analogous to an interface where the 'outer' material has greater index of refraction.
If the end of the string is free to move, the reflected wave holds phase. This is analogous to an interface where the 'outer' material has a lower index of refraction.
See this exact discussion here

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