Brewster Angle Calculator

Calculator inputs

Calculation mode

Angle unit

Decimal precision

Incident-medium preset

Transmitted-medium preset

Known Brewster angle

Incident index n₁

Use the refractive index at the selected wavelength.

Transmitted index n₂

Real-valued indices suit ideal transparent materials.

Incident-medium name

Transmitted-medium name

Wavelength

Optional analysis angle

Compare Fresnel values away from Brewster incidence.

Graph scale

Comparison materials

Hold Ctrl or Command to select several materials.

Refractive index can change with wavelength, temperature, composition, and material quality.

Angle sweep table

Reflectance values ignore absorption.

Incidence angle	Refraction angle	p-reflectance	s-reflectance	Status

Material and interface comparison

Incident medium	Compared medium	n₂	Brewster angle	Critical angle	Direction

Formula used

θ_B = tan⁻¹(n₂ / n₁)

At this angle, the reflected and refracted rays are perpendicular.

θ_t = 90° − θ_B

n₁ sin θ_B = n₂ sin θ_t

How to use this calculator

Select the calculation mode.
Choose materials or enter custom refractive indices.
Select degrees or radians for reverse calculations.
Choose output precision and optional wavelength information.
Select comparison materials for multiple-interface results.
Press Calculate to view results, graphs, and tables.
Export the calculation as CSV or PDF.

Worked examples

Interface	n₁	n₂	Approximate Brewster angle	Interpretation
Air to water	1.000293	1.333	53.12°	Useful for glare reduction from water.
Air to crown glass	1.000293	1.520	56.65°	Common optical-window approximation.
Air to diamond	1.000293	2.417	67.49°	Higher index produces a larger angle.
Water to crown glass	1.333	1.520	48.75°	The smaller index ratio lowers the angle.
Crown glass to air	1.520	1.000293	33.36°	Total internal reflection can occur later.

Understanding Brewster angle

Brewster angle is a special incidence angle. P-polarised reflection ideally falls to zero. The reflected beam becomes strongly s-polarised.

Snell’s law still controls refraction. The reflected and refracted rays become perpendicular. Their two angles total exactly ninety degrees.

Real materials may absorb light. Roughness and coatings also change reflection. Complex refractive indices require advanced optical modelling.

Brewster angle versus critical angle

Brewster angle concerns polarisation and minimum p-reflection. Critical angle concerns total internal reflection. They describe different interface conditions entirely.

Practical applications

Applications include polarising filters and laser windows. It also supports microscopy and telecommunications. Glare reduction remains a familiar everyday example.

Frequently asked questions

What is Brewster angle?

It is the incidence angle where ideal p-polarised reflection becomes zero.

Which refractive index goes first?

Use n₁ for the incident medium and n₂ for the transmitted medium.

Can Brewster angle exist from glass to air?

Yes. The formula remains valid before total internal reflection begins.

Why is p-polarised reflectance zero?

The reflected dipole radiation cannot radiate along its oscillation direction.

Does s-polarised reflection also vanish?

No. S-polarised reflectance usually remains significant at Brewster incidence.

Does wavelength affect the answer?

Yes. Refractive index often changes with wavelength through material dispersion.

What happens when both indices are equal?

The ideal interface produces no reflection for any incidence angle.

Is Brewster angle the critical angle?

No. Brewster angle minimises p-reflection. Critical angle starts total reflection.

Can this calculator handle absorbing materials?

It uses real indices. Absorbing materials need complex refractive-index calculations.

Why might measurements differ from the result?

Surface roughness, coatings, absorption, temperature, and wavelength can shift observations.