10.4 Brewster's angle

Definition of Brewster's angle

Brewster's angle is the angle of incidence at which p-polarized light is perfectly transmitted through a transparent dielectric surface, with zero reflection. This makes it one of the cleanest results you'll encounter in boundary electrodynamics: a specific, calculable angle where one polarization component simply refuses to reflect.

When unpolarized light hits a surface at Brewster's angle, the reflected beam comes back perfectly s-polarized. The physical reason is striking: at Brewster's angle, the reflected and refracted rays are exactly perpendicular to each other. Since the reflected wave would need to be driven by oscillating dipoles in the refracted medium, and those dipoles can't radiate along their own oscillation axis, the p-component has no mechanism to reflect.

Brewster's angle is named after Sir David Brewster, who first described the phenomenon experimentally in 1812.

Relationship to polarization

Recall the convention for labeling polarization at an interface:

• s-polarization (from German senkrecht, perpendicular): the electric field oscillates perpendicular to the plane of incidence.
• p-polarization (parallel): the electric field oscillates within the plane of incidence.

At Brewster's angle, the reflected light is purely s-polarized. The refracted (transmitted) light is partially polarized, carrying all of the p-component and most of the s-component. The transmitted beam is not fully polarized because s-polarized light still partially transmits at this angle.

Snell's law and refractive indices

Snell's law governs refraction at the boundary between two isotropic media:

$n_1 \sin \theta_1 = n_2 \sin \theta_2$

where $\theta_1$ is the angle of incidence, $\theta_2$ is the angle of refraction, and $n_1$, $n_2$ are the refractive indices of the respective media. This relationship is essential to the Brewster's angle derivation because the perpendicularity condition between reflected and refracted rays translates directly into a constraint via Snell's law.

Derivation of Brewster's angle

The derivation proceeds from the Fresnel equations by finding the incidence angle that zeros out the p-polarized reflection coefficient.

Fresnel equations

The Fresnel equations give the amplitude reflection and transmission coefficients for each polarization at a planar dielectric interface:

s-polarization:

$r_s = \frac{n_1 \cos \theta_1 - n_2 \cos \theta_2}{n_1 \cos \theta_1 + n_2 \cos \theta_2}$

$t_s = \frac{2n_1 \cos \theta_1}{n_1 \cos \theta_1 + n_2 \cos \theta_2}$

p-polarization:

$r_p = \frac{n_2 \cos \theta_1 - n_1 \cos \theta_2}{n_2 \cos \theta_1 + n_1 \cos \theta_2}$

$t_p = \frac{2n_1 \cos \theta_1}{n_2 \cos \theta_1 + n_1 \cos \theta_2}$

These coefficients are ratios of electric field amplitudes (reflected or transmitted) to the incident amplitude. The power reflectance and transmittance are $R = |r|^2$ and $T = 1 - R$ (for lossless media), though note that the transmittance expression requires a correction factor of $\frac{n_2 \cos\theta_2}{n_1 \cos\theta_1}$ when written in terms of $|t|^2$.

Conditions for zero reflection: step-by-step

1. Set $r_p = 0$: This requires the numerator of the p-reflection coefficient to vanish: $n_2 \cos \theta_1 = n_1 \cos \theta_2$

2. Apply Snell's law to eliminate $\theta_2$. From $n_1 \sin\theta_1 = n_2 \sin\theta_2$, you can write $\cos\theta_2 = \sqrt{1 - \frac{n_1^2}{n_2^2}\sin^2\theta_1}$. Substituting and simplifying (or using the geometric argument below) yields the result.

3. Geometric shortcut: The condition $r_p = 0$ is equivalent to requiring that the reflected and refracted rays be perpendicular, i.e., $\theta_1 + \theta_2 = 90°$. Substituting $\theta_2 = 90° - \theta_1$ into Snell's law:

$n_1 \sin\theta_B = n_2 \sin(90° - \theta_B) = n_2 \cos\theta_B$

1. Solve for Brewster's angle: $\tan \theta_B = \frac{n_2}{n_1}$

This is Brewster's law. The perpendicularity condition ($\theta_1 + \theta_2 = 90°$) is the physical heart of the result: the reflected ray direction coincides with the oscillation axis of the p-polarized dipoles in the second medium, so they cannot radiate in that direction.

Properties of Brewster's angle

Dependence on materials

Brewster's angle depends only on the ratio of refractive indices:

$\theta_B = \arctan\left(\frac{n_2}{n_1}\right)$

Some concrete values:

Note that Brewster's angle exists for both external reflection ($n_2 > n_1$) and internal reflection ($n_1 > n_2$). For the internal case, $\theta_B < 45°$.

Wavelength independence

Brewster's angle is independent of wavelength as long as the refractive indices themselves don't vary significantly with wavelength (i.e., the materials are non-dispersive). In practice, most dielectrics have weak enough dispersion that $\theta_B$ shifts by less than a degree across the visible spectrum. This makes Brewster-angle devices effective over broad bandwidths.

Brewster's angle vs. critical angle

These two special angles arise from different physics and should not be confused:

• Brewster's angle ($\theta_B = \arctan(n_2/n_1)$): the angle at which p-polarized reflection vanishes. It exists for light going from either medium into the other.
• Critical angle ($\theta_c = \arcsin(n_2/n_1)$, with $n_1 > n_2$): the angle above which total internal reflection occurs. It only exists when light travels from a denser to a rarer medium.

For a glass-to-air interface ($n_1 = 1.5$, $n_2 = 1.0$): $\theta_B \approx 33.7°$ and $\theta_c \approx 41.8°$. Brewster's angle is always less than the critical angle when both exist, which you can verify from the inequality $\arctan(x) < \arcsin(x)$ for $0 < x < 1$.

Applications of Brewster's angle

Polarizing filters and Brewster windows

Brewster-angle plate polarizers use a stack of dielectric plates, each oriented at Brewster's angle. At every interface, s-polarized light partially reflects while p-polarized light fully transmits. After several plates, the transmitted beam is highly p-polarized. These are simple, broadband, and handle high laser powers well.

Brewster windows are standard in gas laser cavities (e.g., He-Ne, CO$_2$). The windows at each end of the gain tube are tilted at Brewster's angle so that p-polarized light passes with zero reflection loss. This preferentially amplifies the p-polarization, producing a linearly polarized output beam without needing an intracavity polarizer.

Glare reduction

Light reflecting off horizontal surfaces (water, roads, glass) near Brewster's angle is predominantly s-polarized. Polarized sunglasses are oriented to block this s-polarized glare while transmitting the p-component. The effect is strongest when the viewing angle is close to Brewster's angle for the reflecting surface.

Brewster angle microscopy

Brewster angle microscopy (BAM) is a surface-sensitive technique for imaging thin films at interfaces (e.g., Langmuir monolayers on water). A p-polarized laser hits the bare water surface at Brewster's angle, producing zero reflection from the clean interface. Any thin film present changes the local refractive index, causing detectable reflected light. This gives high-contrast images of film morphology without fluorescent labels.

Experimental verification

Measuring Brewster's angle

A typical measurement setup:

1. Direct a laser beam through a linear polarizer set to transmit p-polarization.
2. Aim the beam at the dielectric sample mounted on a rotation stage.
3. Place a photodetector to capture the specularly reflected beam.
4. Rotate the sample through a range of incidence angles, recording reflected intensity at each angle.
5. Identify Brewster's angle as the incidence angle where reflected intensity reaches its minimum.

For clean dielectric surfaces, the reflected p-polarized intensity should drop nearly to zero. Any residual signal usually indicates surface contamination, roughness, or slight polarization misalignment.

Determining refractive indices

Once $\theta_B$ is measured, you can extract the refractive index ratio directly:

$\frac{n_2}{n_1} = \tan \theta_B$

If one index is known (e.g., $n_1 = 1$ for air), the other follows immediately. This is a practical, non-destructive method for characterizing the optical properties of substrates and thin films.

Polarization state analysis

To confirm the Fresnel equation predictions, you can place a rotating analyzer (a second polarizer) in the reflected beam path. At Brewster's angle, the reflected light should be purely s-polarized, meaning the analyzer will show a clean $\cos^2\phi$ dependence with a null when aligned to block s-polarization. Deviations from this ideal behavior reveal information about surface quality or absorption.

Limitations and special cases

Absorbing media and complex refractive indices

The standard Brewster's angle derivation assumes lossless dielectrics with real refractive indices. For absorbing media (metals, semiconductors, doped materials), the refractive index becomes complex: $\tilde{n} = n + ik$, where $k$ is the extinction coefficient.

In this case, $r_p$ never reaches exactly zero. Instead, there's a pseudo-Brewster angle where the p-reflectance reaches a minimum. For metals, this minimum can still be substantial. Measuring the pseudo-Brewster angle and the minimum reflectance together provides both $n$ and $k$ of the absorbing material.

Anisotropic materials

In anisotropic media (crystals, liquid crystals), the refractive index depends on the propagation direction and polarization state. This means:

• There may be multiple Brewster angles depending on the crystal orientation relative to the interface.
• The ordinary and extraordinary rays experience different refractive indices, so the simple $\tan\theta_B = n_2/n_1$ formula doesn't directly apply.
• Proper treatment requires the 4×4 transfer matrix method (Berreman formalism) or similar approaches that account for the full dielectric tensor.

Non-planar interfaces

The Brewster's angle derivation assumes a perfectly flat, infinite planar boundary. On curved surfaces (lenses, fibers), the local incidence angle varies from point to point, so the Brewster condition is satisfied only along specific curves on the surface. Modeling these situations requires ray tracing or full-wave numerical methods like FDTD. For surfaces with curvature much larger than the beam diameter, the planar approximation remains reasonable locally.