Use this Brewster’s angle calculator to find the polarization angle at an interface from two refractive indices, or solve for a missing refractive index if you already know the angle. You can also compute the refraction angle at Brewster incidence as a quick self-check.
Advanced options
How to use our Brewster's Angle Calculator
- Select what you want to solve for: Brewster angle, refractive index n2, or refractive index n1.
- Enter refractive index n1 (incident medium), which is the material the light starts in (example: air).
- Enter refractive index n2 (transmitted medium), which is the material the light enters (example: glass).
- Pick an angle unit for results (degrees or radians). This unit also applies to the Brewster angle input when you solve for a refractive index.
- If solving for n1 or n2, open Advanced options and enter the Brewster angle value.
- (Optional) In Advanced options, set "Also compute refraction angle" to Yes to calculate the transmitted angle using Snell's law and see the sum-of-angles check.
- (Optional) Choose a rounding setting to control how many decimal places are shown.
- Click Calculate. If you see an error, change the exact field named in the message (for example, a refractive index must be greater than 0).
Definitions
Refractive index (n): A number that tells how fast light travels in a material compared to vacuum; it is unitless and must be greater than 0 in this calculator. [1]
Incident medium (n1): The first material the light is traveling in before it reaches the surface.
Transmitted medium (n2): The second material the light enters after crossing the surface.
Brewster angle (theta_B): The incident angle where reflected light with parallel polarization (p-polarized) is minimized (idealized as zero) for a simple dielectric interface. [2]
Refraction angle (theta_T): The angle of the transmitted ray in medium 2, measured from the surface normal; related to the incident angle by Snell's law. [4]
Surface normal: An imaginary line perpendicular to the surface; angles here are measured from this line (not from the surface).
Methodology
Inputs and units
All angles in the math below are handled in radians internally. If you choose degrees, the calculator converts degrees to radians for trig functions and converts results back to degrees for display.
radians = degrees * (pi / 180)
degrees = radians * (180 / pi)
Mode 1: Solve for Brewster angle
When you enter n1 and n2, Brewster's law gives the Brewster angle (angle of incidence). [2]
theta_B = arctan(n2 / n1)
The calculator blocks invalid refractive indices (n1 less than or equal to 0, or n2 less than or equal to 0) because the formula would not describe a physical refractive index. [1]
Mode 2: Solve for n2 (transmitted medium)
If you know the Brewster angle and n1, rearrange the same relationship to solve for n2.
n2 = n1 * tan(theta_B)
If theta_B is too close to 90 degrees (or pi/2 radians), tan(theta_B) becomes extremely large and the result is not usable, so the calculator shows an error instead of a misleading number.
Mode 3: Solve for n1 (incident medium)
If you know the Brewster angle and n2, solve for n1.
n1 = n2 / tan(theta_B)
If theta_B is too close to 0, tan(theta_B) is too close to 0 and division would fail, so the calculator shows an error.
Optional: Refraction angle at Brewster incidence (Snell check)
If you turn on the check, the calculator uses Snell's law to compute the transmitted (refracted) angle when the incident angle equals the Brewster angle. [4]
theta_T = arcsin( (n1 / n2) * sin(theta_B) )
Domain rule: arcsin(x) is only real when x is between -1 and 1. If the value is slightly outside that range due to rounding, the calculator clamps it into [-1, 1] for display and shows a warning; if it is meaningfully outside the range, theta_T is shown as N/A with a warning.
90 degree (pi/2) perpendicular-ray self-check
At Brewster incidence for the standard non-magnetic dielectric case, the reflected and refracted rays are perpendicular, so the angles (measured from the normal) add to a right angle. [2]
theta_B + theta_T = 90 degrees
theta_B + theta_T = pi / 2 radians
The calculator reports the sum in your selected unit so you can quickly catch swapped n1 and n2, a wrong angle unit, or an impossible Snell setup.
Rounding and display
Rounding only changes what is shown. Internally, calculations use full floating-point precision, then the final outputs are rounded to your chosen number of decimal places. Refractive index outputs are shown with up to 6 decimals (trailing zeros are trimmed) to stay readable while still being useful for lab work.