When light encounters a boundary between two dielectric media, part of the wave is reflected and part is transmitted.
The detailed behavior depends on the angle of incidence, the refractive indices of the two media, and the polarization of the incident field.
In optics, the most basic quantities describing this process are the Fresnel amplitude reflection coefficients, usually denoted \(r_s\) and \(r_p\).
These are amplitude ratios, not power ratios. Their squared magnitudes give the reflected power fractions.
The subscript \(s\) refers to polarization perpendicular to the plane of incidence, while \(p\) refers to polarization parallel to that plane.
For a single interface between media of refractive indices \(n_1\) and \(n_2\), the incidence and transmission angles are related by Snell’s law:
\[
n_1\sin\theta_i=n_2\sin\theta_t.
\]
Once the transmitted angle \(\theta_t\) is known, the Fresnel amplitude reflection coefficients are
\[
r_s=\frac{n_1\cos\theta_i-n_2\cos\theta_t}{n_1\cos\theta_i+n_2\cos\theta_t},
\qquad
r_p=\frac{n_2\cos\theta_i-n_1\cos\theta_t}{n_2\cos\theta_i+n_1\cos\theta_t}.
\]
These formulas tell you how the reflected electric-field amplitude compares to the incident amplitude for each polarization component.
The associated reflected power fractions are
\[
R_s=|r_s|^2,
\qquad
R_p=|r_p|^2.
\]
One of the most important special angles in polarization optics is Brewster’s angle.
It is the angle of incidence for which the parallel reflection coefficient vanishes:
\[
r_p=0.
\]
Setting the numerator of \(r_p\) equal to zero leads to Brewster’s law:
\[
\theta_B=\tan^{-1}\!\left(\frac{n_2}{n_1}\right).
\]
This means that if light travels from medium \(n_1\) into medium \(n_2\) at angle \(\theta_B\), the reflected \(p\)-polarized component is completely absent.
The reflected beam is then purely \(s\)-polarized.
This is the physical basis of polarization by reflection and one of the reasons polarized sunglasses are so effective at reducing glare from water, glass, roads, and other reflective surfaces.
For the sample case of air to glass,
\(n_1=1\) and \(n_2=1.5\).
The Brewster angle is
\[
\theta_B=\tan^{-1}(1.5)\approx 56.3^\circ.
\]
If the incident angle is chosen to be approximately \(56.3^\circ\), then the Fresnel coefficient for \(p\)-polarized light becomes
\[
r_p=0.
\]
Consequently,
\[
R_p=|r_p|^2=0,
\]
so the \(p\)-polarized reflected power also vanishes.
The reflected \(s\)-component, however, generally remains nonzero.
That is why the reflected beam does not disappear entirely; instead, it becomes polarized.
A useful geometric fact is that at Brewster’s angle, the reflected and refracted rays are perpendicular:
\[
\theta_B+\theta_t=90^\circ.
\]
This relationship follows directly from combining Snell’s law with the Brewster condition \(r_p=0\).
It gives a very intuitive geometric picture of why the parallel component fails to reflect at that special angle.
There is another important angle that can appear in related problems: the critical angle.
If \(n_1>n_2\), then
\[
\theta_c=\sin^{-1}\!\left(\frac{n_2}{n_1}\right)
\]
marks the start of total internal reflection.
Above this angle, no real transmitted angle exists. In that regime, the Fresnel reflection coefficients become complex phase factors, although their magnitudes remain 1 for lossless media.
This calculator reports that situation explicitly and distinguishes it from the ordinary Brewster-angle regime.
It is important to remember that \(r_s\) and \(r_p\) are amplitude coefficients, not intensity coefficients.
Their signs and phases matter in wave interference and multilayer optics, while \(R_s\) and \(R_p\) describe reflected power.
At a more advanced university level, one studies how these coefficients behave in thin films, dielectric stacks, metallic reflections, and absorbing media with complex refractive indices.
But for a single transparent interface, Brewster’s law and the Fresnel formulas already capture the central physics very well.
\[
r_s=\frac{n_1\cos\theta_i-n_2\cos\theta_t}{n_1\cos\theta_i+n_2\cos\theta_t},
\qquad
r_p=\frac{n_2\cos\theta_i-n_1\cos\theta_t}{n_2\cos\theta_i+n_1\cos\theta_t},
\]
\[
R_s=|r_s|^2,
\qquad
R_p=|r_p|^2,
\qquad
\theta_B=\tan^{-1}\!\left(\frac{n_2}{n_1}\right).
\]
Together, these formulas let you determine the reflected amplitudes, the reflected power fractions, the Brewster angle, and the special condition under which the reflected \(p\)-polarized light vanishes.
