Why does the p-polarized reflection go to zero at the Brewster angle?

At the Brewster angle, the reflected and refracted rays are 90 degrees apart. For p-polarized light, the electric field oscillates in the plane of incidence. When the refracted ray is perpendicular to the reflected ray, the direction of the oscillating dipole induced in the second medium that would normally re-radiate as a reflected wave points exactly along the propagation direction of the reflected ray. An oscillating dipole cannot radiate along its own axis, so the reflection for that polarization vanishes.

Does this mean light reflected off a lake or window is completely polarized?

At the specific Brewster angle for the air-water or air-glass interface, the reflected light is 100% s-polarized (perpendicular to the plane). At other angles, the reflection contains a mix of s- and p-polarizations. Polarized sunglasses use this principle, with lenses oriented to block the dominant horizontal (s-polarized) glare reflected from flat surfaces like water or roads.

Can the Brewster angle exist for light going from a higher index to a lower index material (e.g., glass to air)?

Yes, Brewster's law (tan θ_B = n₂/n₁) still applies. However, when n₁ > n₂ (e.g., glass to air), the calculated θ_B is greater than 45 degrees. It's crucial to note that this angle is measured within the denser medium (glass). For angles of incidence in the denser medium greater than the critical angle, total internal reflection occurs, which supersedes the Brewster condition.

What are the main limitations of this simulator's model?

The model assumes ideal conditions: perfectly smooth interfaces, isotropic and homogeneous dielectric materials with no absorption or scattering, and perfectly monochromatic plane waves. Real-world surfaces have roughness, materials can be absorbing (like metals, where Brewster's angle is complex), and light sources have a finite bandwidth, which can slightly blur the sharp minimum in the reflectance curve.

Brewster Angle

The Brewster Angle simulator visualizes a fundamental phenomenon in polarization optics: the angle of incidence at which light with a particular polarization is perfectly transmitted through a dielectric interface. At the heart of the model is Brewster's law, expressed as tan θ_B = n₂/n₁, where θ_B is the Brewster angle, and n₁ and n₂ are the refractive indices of the incident and transmitting media, respectively. The core physical principle is that when the incident angle equals θ_B, the reflected and refracted rays are perpendicular to each other (θ_i + θ_t = 90°). This geometric condition forces the reflected light's electric field oscillation direction (for p-polarized light, parallel to the plane of incidence) to align with the direction of propagation of the refracted ray, making reflection impossible. Consequently, the reflectance for p-polarized light, R_p, drops to zero at this angle. The simulator plots the Fresnel equations for reflectance, R_s and R_p, as functions of the incident angle, allowing users to observe the distinct minimum in R_p and the non-zero behavior of R_s (s-polarized, perpendicular to the plane). Simplifications include assuming perfectly smooth, planar interfaces between ideal, non-absorbing dielectric materials and perfectly monochromatic, plane wave light. By interacting with the model, students can explore how changing the refractive index ratio shifts the Brewster angle, verify the perpendicularity condition, and gain an intuitive understanding of polarization by reflection, a principle applied in glare-reducing sunglasses and laser cavity windows.

Who it's for: Undergraduate physics and engineering students studying electromagnetism, optics, or photonics, particularly when covering polarization, Fresnel equations, and boundary conditions for electromagnetic waves.

Key terms

Brewster's Angle
Polarization
Fresnel Equations
Reflectance
Refractive Index
p-polarization
s-polarization
Angle of Incidence

Live graphs

How it works

At a dielectric boundary, reflectivity depends on polarization. For p-polarization (electric field in the plane of incidence), there is an incidence angle θ_B — Brewster angle — where the reflected wave vanishes and the reflected and refracted directions are perpendicular. Then tan θ_B = n₂/n₁. The reflected light at Brewster is purely s-polarized (electric field perpendicular to the plane of incidence).

Key equations

tan θ_B = n₂ / n₁ · θᵢ + θₜ = 90° at Brewster

Fresnel (power): R_s, R_p from r_s, r_p at the interface

Frequently asked questions

Why does the p-polarized reflection go to zero at the Brewster angle?: At the Brewster angle, the reflected and refracted rays are 90 degrees apart. For p-polarized light, the electric field oscillates in the plane of incidence. When the refracted ray is perpendicular to the reflected ray, the direction of the oscillating dipole induced in the second medium that would normally re-radiate as a reflected wave points exactly along the propagation direction of the reflected ray. An oscillating dipole cannot radiate along its own axis, so the reflection for that polarization vanishes.
Does this mean light reflected off a lake or window is completely polarized?: At the specific Brewster angle for the air-water or air-glass interface, the reflected light is 100% s-polarized (perpendicular to the plane). At other angles, the reflection contains a mix of s- and p-polarizations. Polarized sunglasses use this principle, with lenses oriented to block the dominant horizontal (s-polarized) glare reflected from flat surfaces like water or roads.
Can the Brewster angle exist for light going from a higher index to a lower index material (e.g., glass to air)?: Yes, Brewster's law (tan θ_B = n₂/n₁) still applies. However, when n₁ > n₂ (e.g., glass to air), the calculated θ_B is greater than 45 degrees. It's crucial to note that this angle is measured within the denser medium (glass). For angles of incidence in the denser medium greater than the critical angle, total internal reflection occurs, which supersedes the Brewster condition.
What are the main limitations of this simulator's model?: The model assumes ideal conditions: perfectly smooth interfaces, isotropic and homogeneous dielectric materials with no absorption or scattering, and perfectly monochromatic plane waves. Real-world surfaces have roughness, materials can be absorbing (like metals, where Brewster's angle is complex), and light sources have a finite bandwidth, which can slightly blur the sharp minimum in the reflectance curve.

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Frequently asked questions

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