Brewster Angle Calculator - Polarizing Angle and Refraction

A Brewster angle calculator solves the polarizing angle theta_B, the refraction angle theta_2, and the Fresnel p-polarized reflectance for any pair of refractive indices n1 and n2. It applies Brewster's law theta_B = arctan(n2 / n1) and the Fresnel equations for Rp so you can read the angle at which reflected light is fully s-polarized without setting up a laser bench.

• • Set up a Brewster window for a gas laser: Pick the tilt angle that eliminates p-polarized reflection so the cavity oscillates in one linear polarization.
• • Compute the angle for polarized sunglasses and camera filters: Confirm why glare off roads and water is s-polarized above the Brewster angle and how polarizing filters block it.
• • Solve an introductory optics homework problem: Plug in n1 and n2 for glass, water, or diamond and read theta_B plus theta_2.
• • Plan a holographic recording setup: Tilt the reference beam to Brewster's angle so the back of the holographic film does not reflect into the hologram.

Brewster's angle is named after the Scottish physicist Sir David Brewster, who reported in 1815 that the polarizing angle of a transparent surface equals arctan(n2/n1). The same geometry explains why polarized sunglasses work against road glare and why laser cavities prefer a tilted window to remove reflective loss for the working polarization.

When you move from reflection at a single surface to refraction through a lens, Lensmakers Equation Calculator turns refractive index and surface radii into focal length and diopter power using the same n-driven geometry.

The calculator applies Brewster's law, derives the refraction angle from Snell's law, and evaluates Fresnel reflectance at the user-selected incidence. Rp goes to zero for the p polarization when the angle equals theta_B.

• n1: Refractive index of the incident medium. Air is 1.000, water is 1.333.
• n2: Refractive index of the transmitted medium. Crown glass is about 1.5, sapphire about 1.77, diamond about 2.417.
• incidence angle theta_1: Angle from the surface normal to the incoming ray, in degrees. The Fresnel equations evaluate Rp at this angle.
• theta_B: Brewster angle output, in degrees, equal to atan(n2 / n1) times 180 divided by pi.
• theta_2: Refraction angle at theta_B, equal to 90 minus theta_B. The reflected and refracted rays are mutually perpendicular at this point.
• Rp and Rs: Power reflectance for the p polarization (in the plane of incidence) and the s polarization (perpendicular to the plane). At theta_B, Rp goes to zero and Rs stays positive.

The derivation starts from the Fresnel equation for the p-polarized amplitude reflectance and asks when it equals zero. Combining R_p = 0 with Snell's law gives tan(theta_B) = n2 / n1, which is Brewster's law. The refraction angle follows from theta_B + theta_2 = 90 degrees, and the result is real whether n2 is larger or smaller than n1 as long as the incidence stays below the critical angle.

According to Wikipedia - Brewster's angle, Brewster's law states that theta_B = arctan(n2 / n1), and the reflected and refracted rays are mutually perpendicular at that point.

According to OpenStax University Physics - Polarization, R_p goes to zero when the reflected and refracted rays are at right angles, giving tan(theta_B) = n2 / n1 below the critical angle.

For a curved reflector the law of reflection still holds at the surface, and Mirror Equation Calculator uses that geometry together with object and image distances to find the focal length of a concave or convex mirror.

Four ideas come up every time you apply Brewster's law to a real surface. Understanding them keeps the formula, the orthogonality, and the Fresnel reflectance from producing the wrong sign or magnitude.

The same surface that polarizes light at Brewster's angle also limits the rays that pass through the optical system, and Aperture Area Calculator converts f-number and focal length into entrance-pupil area using the same pi * r^2 form.

Four quick steps take you from a material pair (such as glass in air) to the polarizing angle, the refraction angle, and the p-polarized Fresnel reflectance at any incidence angle you choose.

1. 1 Enter the incident medium index n1: Type the refractive index of the medium the light travels in before the surface. Use 1.000 for air, 1.333 for water, or the value from a glass data sheet for a submerged interface.
2. 2 Enter the transmitted medium index n2: Type the refractive index the light enters through. Use 1.500 for crown glass, 1.770 for sapphire, or 2.417 for diamond at the sodium D line.
3. 3 Set the incidence angle: Default the incidence angle to the calculated Brewster angle so Rp reads zero, or enter the angle from your optical layout to read the actual reflectance.
4. 4 Read theta_B, theta_2, Rp, and Rs: Use theta_B and theta_2 as the headline outputs, then Rp and Rs to confirm which polarization survives the reflection.

For a helium-neon laser with a glass window in air, enter n1 = 1.000 and n2 = 1.500. Leave the incidence angle at 56.31 degrees and read theta_B = 56.31 degrees, theta_2 = 33.69 degrees, and Rs = 0.1486.

Because Brewster's law is just an arctangent of the index ratio, Arctan Calculator is the right tool when you want to check the ratio step alone or work in radians rather than degrees.

The Brewster angle calculator gives you the polarizing angle, the refraction angle, and the Fresnel reflectance for both polarizations in a single pass.

• • Read theta_B without solving arctan by hand: Type n1 and n2 and read the polarizing angle to two decimals.
• • Verify the orthogonality condition in one click: The refraction angle is always 90 minus theta_B, so you can confirm the rays are perpendicular for any material pair.
• • Compare Rp and Rs at the same incidence: The Rp and Rs outputs show which polarization survives the reflection and how much light is lost, the trade-off a Brewster window makes in a laser cavity.
• • Try common material pairs quickly: Use the default 1.0 and 1.5 inputs for a glass-air interface or change n2 to 1.333 for water and 2.417 for diamond without leaving the page.
• • Plan polarization optics and laser setups: Read the tilt angle for a Brewster window, the orientation for polarized sunglasses, or the reference-beam angle for a holographic plate from the same inputs.

Reflected and refracted waves at Brewster's angle still satisfy the wave equation, and Harmonic Wave Equation Calculator resolves the wavelength, frequency, and wave number that the Brewster condition preserves across the interface.

Five inputs and conditions move the Brewster angle the most, plus three limitations to keep in mind before quoting a window tilt.

• • The calculator uses the lossless dielectric Fresnel equations. It does not include magnetic permeability, complex refractive index for absorbing media, or anisotropic crystals.
• • Brewster's law assumes an idealized flat interface. Real surfaces with roughness or thin-film coatings shift the effective Brewster angle and change the reflectance minimum.
• • When n1 > n2, the Brewster angle theta_B = arctan(n2 / n1) always lies below the critical angle arcsin(n2 / n1), so the result is real as long as the entered incidence stays sub-critical too.

According to RP Photonics - Brewster Windows, gas lasers with low round-trip gain often use windows tilted at the Brewster angle to eliminate p-polarized reflection and force single-polarization oscillation.

Dispersion shifts the Brewster angle with wavelength the same way it shifts Bragg diffraction peaks, so Braggs Law Calculator gives the matching wavelength, d spacing, and diffraction order that move with n(lambda).