Brewster’s angle is the special angle of incidence at which reflected light from a dielectric interface becomes perfectly linearly polarized.
More precisely, the reflected parallel or p-polarized component disappears, while the perpendicular or s-polarized component remains.
This is why reflected glare from water, glass, and road surfaces can become strongly polarized, and it is also why polarized sunglasses are so effective at reducing certain types of reflection.
Consider light traveling from a medium of refractive index \(n_1\) into a second medium of index \(n_2\).
The Brewster angle \(\theta_B\) is
\[
\theta_B = \tan^{-1}\!\left(\frac{n_2}{n_1}\right).
\]
This formula applies for a simple interface between lossless dielectric media.
For the common case of air to glass, where \(n_1=1.00\) and \(n_2=1.50\), the calculation is
\[
\theta_B = \tan^{-1}(1.50) \approx 56.3^\circ.
\]
That means when light hits a glass surface at about \(56.3^\circ\) relative to the normal, the reflected beam contains no \(p\)-polarized component.
The reflected light is then entirely \(s\)-polarized.
This is the physical basis of “polarization by reflection.”
The deeper reason Brewster’s angle occurs can be seen from the geometry of reflection and refraction.
At Brewster’s angle, the reflected and refracted rays are perpendicular.
In other words,
\[
\theta_B + \theta_t = 90^\circ,
\]
where \(\theta_t\) is the transmitted or refracted angle.
Combined with Snell’s law,
\[
n_1 \sin\theta_i = n_2 \sin\theta_t,
\]
this leads directly to the tangent formula for \(\theta_B\).
It is one of the most elegant results in geometrical and wave optics because it links polarization, refraction, and reflection in a single simple expression.
To go beyond the Brewster angle itself, it is useful to examine the Fresnel reflectance formulas.
For an incident angle \(\theta_i\), the reflected power fractions for the two linear polarization components are
\[
R_s = \left(\frac{n_1\cos\theta_i-n_2\cos\theta_t}{n_1\cos\theta_i+n_2\cos\theta_t}\right)^2,
\qquad
R_p = \left(\frac{n_2\cos\theta_i-n_1\cos\theta_t}{n_2\cos\theta_i+n_1\cos\theta_t}\right)^2.
\]
Here \(R_s\) is the reflectance for the perpendicular polarization, and \(R_p\) is the reflectance for the parallel polarization.
At Brewster’s angle, the numerator of the \(R_p\) expression becomes zero, so
\[
R_p = 0.
\]
This does not mean that all reflection disappears. Only the \(p\)-component disappears.
The \(s\)-component generally remains nonzero, which is why reflected light at Brewster’s angle is polarized rather than absent.
If the incident light is unpolarized, the reflected beam becomes partially reduced in intensity but strongly polarized.
For the sample case air to glass, the Brewster angle is about \(56.3^\circ\).
Near this angle, \(R_p\) drops to zero while \(R_s\) remains positive.
That is why reflected glare from windows and water surfaces often has a dominant linear polarization direction.
Polarized sunglasses are designed to block one linear polarization more strongly than the other, which significantly reduces this kind of glare.
Another important related idea is the critical angle.
If \(n_1 > n_2\), there may be an angle above which total internal reflection occurs:
\[
\theta_c = \sin^{-1}\!\left(\frac{n_2}{n_1}\right).
\]
Brewster’s angle and the critical angle are different concepts.
Brewster’s angle refers to the disappearance of the reflected \(p\)-component at an interface.
The critical angle refers to the loss of any real transmitted ray when light travels from a higher-index medium into a lower-index one.
In advanced optics, both ideas appear together in multilayer coatings, laser cavities, fiber optics, and polarization control.
At a more advanced university level, one studies multiple reflections, absorbing materials, complex refractive indices, and thin-film interference.
For metals, the simple Brewster-angle formula usually does not apply in the same clean way because absorption changes the Fresnel coefficients.
But for ordinary transparent dielectric media, Brewster’s law remains one of the most useful polarization results in classical optics.
\[
\theta_B = \tan^{-1}\!\left(\frac{n_2}{n_1}\right),
\qquad
n_1\sin\theta_i = n_2\sin\theta_t,
\qquad
R_p=0 \text{ at } \theta_i=\theta_B.
\]
Together, these relations explain when reflected light becomes fully polarized, how the angle depends on the refractive indices, and why glare from glass or water can be strongly polarization-dependent.
