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I am solving exercises where I occasionally need to decompose the radiance of an EM wave in its components, but I don't understand how it's done. In this context, reflectance ($R$) and transmissivity ($T$), defined as power ratios derived from Fresnel's laws, should always verify that $R+T=1$ due to the conservation of energy. In the case I'm having trouble with, where the electric field is neither $p-$ nor $s-$polarized, but at an angle $\theta$ with the incidence plane of a wave that interacts with an interphase between two dielectric materials, my professor decomposes R as:

$$R = R_s \sin^2 \alpha + R_p \cos^2 \alpha$$

If $R_s$ and $R_p$ could be seen as components of a vector (which I guess is not possible), there would need to be a $^2$ exponent in $R_s$, $R_p$ and $R$. But this doesn't seem to be the case, and I am confused as to why this happens. Does this have anything to do with the requirement that $R+T = 1$, or is this a mistake on my professor's behalf?

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$R_s$ and $R_p$ are indeed the squared moduli of the reflection coefficients of the fields $r_s$ and $r_p$. In fact, by calculating the reflected field first using $r_s$ and $r_p$ first and afterwards calculating the reflected power (squaring $\vec E$) and using $$ R_s = \vert r_s\vert^2 $$ and $$ R_p = \vert r_p\vert^2 $$ you arrive at $$E_{refl}^2=E^2 (R_s \sin^2\theta + R_p\cos^2\theta) $$ what leads directly to the formula you cited.

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  • $\begingroup$ Thank you very much! My book didn't point at the relation between $r$ and $R$, and I therefore didn't think they were so obviously related. $\endgroup$ – mar Dec 26 '19 at 18:43

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