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There are many applets on the internet which demonstrate Huygens Principle or reflection/refraction geometrically. This is one such example: https://www.olympus-lifescience.com/zh/microscope-resource/primer/java/reflection/huygens/

In all of these applets it appears that the wave is polarized so that the E field is going "out of the screen". This seems crucial to the argument as this guarantees that the surface is 'activated' by the wave crest (or any other part of the wave ) in a staggered way to get the desired effect.

Does Huygens Principle of reflection take polarization into account, or must one assume that the wave is polarized so that the E field points out of the screen? Lastly if it does take it into account, how?

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In practice one should always take into account the polarization of light. For instance, the amplitude of light parallel-polarized to the incident plane is zero when the incident angle satisfies $$ \theta_\text{incident} = \arctan \left( \frac{n_2}{n_1} \right ) $$ where $n_1$ is the refractive index of the incident medium and $n_2$ is the refractive index of the other medium. The angle $\theta$ is called Brewster's Angle. The full treatment of polarized light through interfaces is given by Fresnel equations, which follows from applying electromagnetic theory to light.

I believe in most demonstrations of Huygen's Principle polarization is discarded for pedagogical reasons, since by that time you only want to explain the wave nature of light, which works the same way for scalar waves as for vector waves, but the former doesn't have the complications arising from polarization.

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