The Brewster's angle calculator above returns the angle of incidence at which light reflected from a surface is completely plane-polarised. At that one angle, the component of the incoming light polarised in the plane of incidence is not reflected at all, so everything that comes back off the surface vibrates in a single direction. Enter the refractive indices of the two media and the calculator returns the angle; enter a measured angle instead and it returns the index of the reflecting material.
Arb Digital builds free tools that show the working rather than just the answer. Alongside the polarising angle this page reports the refraction angle inside the second medium, the fraction of the other polarisation that still gets reflected, the reflectance at normal incidence for comparison, and the critical angle for total internal reflection when the geometry allows one. Those four numbers are what turn a textbook angle into something you can use at a bench or behind a camera.
What This Brewster's Angle Calculator Does
Light striking a surface can be split into two polarisations. The p-polarised component vibrates in the plane containing the incoming ray and the surface normal; the s-polarised component vibrates perpendicular to it. The two reflect with different strengths at every angle, and at one particular angle the p-polarised reflection falls to zero. That angle is Brewster's angle, and its tangent is the ratio of the two refractive indices.
Because the p-polarised reflection vanishes there, whatever light does come back is purely s-polarised. It is not merely partially polarised, as it is at every other angle; at Brewster's angle it is complete. This is why a polarising filter can eliminate glare from water or glass so effectively, and why the effect is strongest at a specific viewing angle rather than at all of them.
The calculator also reports the residual s-polarised reflectance, computed from the Fresnel equations rather than approximated, because that number tells you how much glare survives even at the ideal angle. For an air-to-glass interface it is around sixteen per cent, far above the roughly four per cent you get at normal incidence. The polarisation is perfect at Brewster's angle; the reflection is not weak.
How to Use It
- Enter the medium the light is travelling in first. That is n₁, and it is air at 1.0003 for almost every practical case. Using exactly 1 changes the answer by a few hundredths of a degree.
- Set n₂ from the preset list or type it in. Refractive index varies with wavelength, so a value quoted for green light will be slightly low for blue and slightly high for red.
- Measure angles from the normal. Brewster's angle is measured from the perpendicular to the surface, so a value of 56 degrees means the light is arriving at a shallow 34 degrees above the glass.
- Switch to solving for n₂ if you have measured the angle. Rotating a polariser while changing the viewing angle until the reflection disappears is a genuine way to measure a refractive index.
- Check the critical angle field. It only exists when light is travelling from the denser medium into the lighter one, and the calculator says so explicitly when it does not.
The Formula: How Brewster's Angle Is Calculated
Section 1.7 of OpenStax University Physics Volume 3, on polarization, gives Brewster's law as tanθb = n₂/n₁, where n₁ is the index of the medium the light starts in and n₂ is the index of the surface it reflects from. The same section works two examples, obtaining 53.1 degrees for light reflecting from water and 56.7 degrees for light reflecting from crown glass, both from air.
Work the default values. With n₁ = 1 and n₂ = 1.52, the ratio is 1.52 and its arctangent is 56.66 degrees. Snell's law then gives the refraction angle: sinθt = sin(56.66°) ÷ 1.52 = 0.8355 ÷ 1.52 = 0.5497, so θt = 33.34 degrees. Notice that 56.66 plus 33.34 is exactly ninety, which is the defining geometric property of Brewster's angle and a fast way to sanity-check any result.
The residual reflection comes from the Fresnel equation for s-polarised light. With cosθi = 0.5494 and cosθt = 0.8355, the amplitude coefficient is (1 × 0.5494 − 1.52 × 0.8355) ÷ (1 × 0.5494 + 1.52 × 0.8355) = −0.7206 ÷ 1.8194 = −0.3960. Squaring gives a reflectance of 0.157, so about 15.7 per cent of the s-polarised light bounces off. Refractive index values themselves come from measurement, and section 1.3 of the same textbook, on refraction, sets out the underlying relationship between index, speed and bending angle.
Why the Reflection Is Polarised at All
The usual explanation runs through the electrons in the second medium. The incoming wave drives them into oscillation, and an oscillating charge radiates, which is what the reflected beam actually is. But a dipole radiates nothing along its own axis of oscillation. At Brewster's angle the refracted ray happens to point in exactly the direction the p-polarised dipoles are vibrating, and the reflected ray is perpendicular to the refracted ray, so it lies along that dead axis. No p-polarised light can be radiated in that direction, and only the s component survives.
This is why the ninety-degree relationship between the reflected and refracted rays is not a coincidence but the mechanism itself. It also explains why the effect requires a dielectric. A metal surface has free electrons rather than bound dipoles, its refractive index is complex, and there is no angle at which the p-polarised reflection falls to zero. Metals have a pseudo-Brewster angle where the reflection reaches a minimum, but it never vanishes and the reflected light is never fully polarised.
The dependence on wavelength follows too. Refractive index changes across the spectrum, so the Brewster angle for blue light differs slightly from the one for red. Over the visible range for common glass the shift is only a few tenths of a degree, small enough to ignore for photography but not for a laser window where a fraction of a per cent of loss per pass compounds.
Polarising Filters, Glare and the Angle That Actually Works
A circular polariser on a camera works because reflections from water, wet roads, foliage and glass are partially polarised, and strongly so near Brewster's angle. Water has an index of about 1.333, giving a polarising angle near 53 degrees from the vertical, which is why glare on a lake vanishes most completely when the camera is tilted roughly a third of the way down from horizontal rather than pointed straight at the surface.
Sunglasses exploit the same effect with a fixed orientation. Light reflected from horizontal surfaces is polarised horizontally, so the lenses are made to block the horizontal component, which removes road and water glare while leaving most of the scene intact. Tilt your head ninety degrees while wearing them and the glare comes back, which is a two-second demonstration of the whole principle.
The limitation worth knowing is that Brewster's angle removes glare from dielectrics only. Reflections from bare metal, from a chrome bumper or a polished aluminium panel, stay unpolarised and a polariser will barely touch them. Anyone who has tried to shoot a car and found the paint glare disappearing while the trim glare stayed has met this directly.
Brewster Windows and Laser Cavities
Gas lasers frequently seal their tubes with windows tilted to Brewster's angle rather than mounted perpendicular to the beam. A perpendicular window reflects about four per cent of the light at each surface, and in a cavity where the beam passes back and forth hundreds of times, that loss is fatal. A Brewster window transmits the p-polarised component with essentially zero reflection loss, at the cost of throwing away the s component entirely.
The side effect is that the laser output becomes linearly polarised, which is usually welcome and occasionally a nuisance. Any downstream optics that behave differently for the two polarisations, including beamsplitters and many diffraction gratings, will interact with that fixed orientation. Our diffraction grating calculator covers the angular part of that problem, and the Snell's law calculator handles the refraction angles at each tilted surface.
The same idea appears in anti-glare display coatings, in ellipsometry for measuring thin films, and in the polarising beamsplitters used in optical instruments. In each case the practical value is the same: a specific angle where one polarisation is transmitted without loss and the other is separated cleanly.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Measuring the angle from the surface — Brewster's angle is measured from the normal, so it looks like a shallow grazing angle when you are standing beside the surface.
- Getting n₁ and n₂ the wrong way round — the ratio inverts, so an air-to-glass angle of 56.7 degrees becomes 33.3 degrees for glass to air. Both are real, and they are complements of each other.
- Expecting the reflection to disappear — only the p-polarised half vanishes. The s-polarised half is still there, and for glass it is stronger at Brewster's angle than at normal incidence.
- Applying it to metal surfaces — metals have a complex refractive index and no true Brewster's angle, so a polariser will not clear glare from chrome or polished steel.
- Ignoring wavelength — refractive index varies across the spectrum, so a single Brewster angle is only exact for one colour.
Related Free Tools From Arb Digital
Refraction and reflection angles at any incidence are covered by the Snell's law calculator, and image formation by the thin lens equation calculator. For interference and periodic structures, use the diffraction grating calculator or the Bragg's law calculator. Wavelength and energy questions carry on with the wavelength calculator and the photon energy calculator. Angle unit conversions are handled by the angle converter. The full free online tools hub lists everything Arb Digital publishes.
Frequently Asked Questions
It is the angle of incidence at which light reflected from a surface is completely plane-polarised. Its tangent equals the refractive index of the reflecting medium divided by that of the medium the light is travelling in, so for air to crown glass it is about 56.7 degrees.
From the normal, the line perpendicular to the surface. An angle of 56.7 degrees therefore describes light arriving at a shallow 33.3 degrees above the surface itself, which is why glare on water disappears when you look down at it obliquely rather than steeply.
No. Only the component polarised in the plane of incidence disappears. The perpendicular component still reflects, and for an air-to-glass surface it is about sixteen per cent, which is stronger than the four per cent reflected at normal incidence.
The reflected and refracted rays are exactly ninety degrees apart at Brewster's angle. The electrons driven by the refracted wave vibrate along the direction the reflected ray would travel, and an oscillating dipole radiates nothing along its own axis, so that polarisation cannot be reflected.
Not properly. Metals have a complex refractive index because they absorb light, so the reflection for the parallel polarisation reaches a minimum but never falls to zero. Polarising filters therefore remove glare from water and glass far better than from chrome or polished steel.
Brewster's angle is about polarisation and exists for any pair of media. The critical angle is about total internal reflection and only exists when light travels from a denser medium into a lighter one. The critical angle is always the larger of the two when both exist.
Yes. Rotate a polariser to block the reflection and vary the viewing angle until the reflected light disappears entirely. The tangent of that angle gives the index of the reflecting material directly, provided the surface is a clean dielectric rather than a metal or a coating.
This tool is provided for educational and estimating use. It assumes clean, non-absorbing dielectric media and a single wavelength, and does not model coatings, absorption or surface roughness.
