Brewster Angle Calculator
The incidence angle at which p-polarized light is not reflected at all (zero reflectance) — θB = arctan(n₂/n₁). Also inverse mode: determine the refractive index from a measured Brewster angle.
Mode 1 — from refractive index to angle
θB = arctan(n₂ / n₁)
Brewster angle θB
–°
Refraction angle θt (= 90° − θB)
–°
θB (radians)
–
p reflectance (theoretical)
0
s reflectance at θB
–
The reflected and refracted rays are always perpendicular (90°) exactly at the Brewster angle. Angles are measured from the normal line (dashed line).
Mode 2 — from measured angle to refractive index
n₂ = n₁ × tan(θB)
Refractive index of destination medium
n₂ = –
Quick examples (computed when the page opens)
| From → to | θB | Refraction angle |
|---|
Worked examples
Air → crown glass. n₁ = 1.0003, n₂ = 1.52.
θB = arctan(1.52 / 1.0003) = arctan(1.51954) = 56.65°.
Refraction angle θt = 90° − 56.65° = 33.35°.
Air → water. n₁ = 1.0003, n₂ = 1.333.
θB = arctan(1.333 / 1.0003) = 53.11°; refraction angle 36.89°.
Inverse. Measured Brewster angle 56.66° in air:
n₂ = 1.0003 × tan(56.66°) = 1.5205 ≈ crown glass.
Honest limitation: only the p component's reflection (polarization parallel to the plane of incidence) becomes zero at the Brewster angle. The s component still reflects, and unpolarized light is only partially affected. The Brewster angle differs from the critical angle (onset of total internal reflection) — for refraction, the critical angle, and TIR use the angle-of-refraction tool.
Frequently asked questions (FAQ)
What is the Brewster angle?
The Brewster angle (θ_B) is the incidence angle at which p-polarized light is not reflected at all — its reflectance is zero. The formula is θ_B = arctan(n2/n1), where n1 is the refractive index of the source medium and n2 of the destination medium. At this angle the reflected light becomes fully linearly polarized (only the s component).
What is the Brewster angle from air to glass?
With n_air = 1.0003 and n_crown_glass = 1.52: θ_B = arctan(1.52/1.0003) = 56.65°. The refraction angle is 90° − 56.65° = 33.35°, so the reflected and refracted rays are perpendicular.
What is the Brewster angle from air to water?
With n_air = 1.0003 and n_water = 1.333: θ_B = arctan(1.333/1.0003) = 53.11°. The refraction angle is 90° − 53.11° = 36.89°.
Does the Brewster angle also exist from inside glass (internal)?
Yes. From crown glass to air: θ_B = arctan(1.0003/1.52) = 33.35°. This differs from the critical angle for total internal reflection (41.15° for crown glass to air) — the internal Brewster angle is always smaller than the critical angle, so it can be reached before TIR occurs.
Why can p-polarized reflection be zero at the Brewster angle?
At the Brewster angle, the reflected and refracted rays form exactly a 90° angle. In the microscopic model, electrons in the second medium oscillate along the refracted ray — i.e., perpendicular to the reflected ray's direction of travel — so they cannot radiate a wave toward the reflected ray for the p component. This zero is theoretical for an ideal surface; real surfaces approach zero.
Is unpolarized light affected too?
Yes, partially. Unpolarized light consists of equal p and s components. At the Brewster angle the p component is not reflected at all, but the s component still reflects — for example 15.65% for air to crown glass, or 7.81% for air to water. The result is that the reflected light becomes linearly polarized (pure s). The s component is never zero at the Brewster angle.
How do I measure the refractive index from the Brewster angle?
Measure the incidence angle at which p-polarized reflection vanishes, then compute n2 = n1 × tan(θ_B). Example: measured θ_B of 56.66° in air → n2 = 1.0003 × tan(56.66°) = 1.5205 (crown glass). Another example: 53.12° → n2 = 1.3332 (water).
What is the difference between the Brewster angle and the critical angle?
The Brewster angle is about polarization: at that angle p reflectance becomes zero, and it exists for every pair of media. The critical angle is about refraction: the incidence angle at which refraction reaches 90° and total internal reflection occurs — it only exists when light travels from a denser to a rarer medium. The critical angle and TIR calculator is available in the angle-of-refraction tool.
Can s-polarization also be zero at some angle?
No. From the Fresnel equations, the s reflection coefficient is never zero as long as n1 ≠ n2 (only trivially zero when both media have the same refractive index, which means there is no boundary). Only the p component has a zero-reflection angle.
Where is the incidence angle measured from?
As with all geometric optics laws, the incidence, reflection, and refraction angles are all measured from the normal line (the line perpendicular to the surface), not from the surface itself. This calculator uses that convention.
Free tool by NRTechStudio