Brewster’s Angle Calculator
Brewster's Angle: the One Angle Where Reflected Light Is Perfectly Polarised
Hit water at 53 degrees from the normal and the light that bounces off is completely polarised, horizontally, every time. That is the whole reason polarising sunglasses kill the glare off a lake and off a wet road. tanθB = n₂/n₁ gives the angle; this page also gives how much light actually comes back, and a ray diagram at your own angle.
Brewster's angle, with Fresnel reflectances
Air to water, n₁ = 1.000 and n₂ = 1.333, with the ray diagram drawn at 53°
Brewster’s angle, and the geometry that proves it
- θB
- Brewster’s angle, also called the polarising angle, measured from the normal. At this one angle the reflected light has no p component at all, so it is completely polarised perpendicular to the plane of incidence — horizontally, for a horizontal surface
- n₁, n₂
- the indices before and after the boundary. Only the ratio matters, so air-to-water and water-to-something-1.33-times-denser give the same angle. For air, n₁ = 1 and the formula collapses to the form most textbooks print, tanθB = n
- θB + θt = 90°
- the geometric statement of the same thing, and the easier one to remember: at Brewster’s angle the reflected and refracted rays are at right angles to each other. The diagram draws that arc when you are within a degree of the angle. It is also the proof — put θt = 90 − θB into Snell’s law and the tangent falls out in two lines
- s and p
- the two polarisations, from the German senkrecht (perpendicular) and parallel, relative to the plane of incidence. Only p goes to zero at Brewster’s angle; s reflects more strongly there than at normal incidence, not less, which is why glare is bright as well as polarised
- Rs, Rp
- the Fresnel reflectances — the fractions of the intensity reflected, which is why each is the square of an amplitude coefficient. At normal incidence they are equal and reduce to ((n₁−n₂)/(n₁+n₂))2, which is 2.04% for air and water: a window reflects about 4% per surface and that is where the figure comes from
- degree of polarisation
- (Rs − Rp)/(Rs + Rp), which is 0 when the reflection treats both polarisations alike and 1 at Brewster’s angle. It stays high over a wide range of angles, which is why sunglasses work over a whole afternoon rather than for one minute
Worked example
Air to water, n₁ = 1.000 and n₂ = 1.333, with the ray diagram drawn at 53°
THE ANGLE ITSELF. tanθB = n₂/n₁ = 1.333/1.000 = 1.333, so θB = arctan(1.333) = 53.12° from the normal. Not from the surface: from the surface it would be 36.88°, and that is the answer people write down when they are thinking about the water rather than the normal.
WHERE THAT FORMULA COMES FROM, IN THREE LINES. Brewster's condition is that the reflected and refracted rays are perpendicular, so θB + θt = 90°, so θt = 90° − θB. Put that into Snell's law: n₁sinθB = n₂sin(90° − θB) = n₂cosθB. Divide both sides by n₁cosθB and you have tanθB = n₂/n₁. The whole law is Snell's law plus one right angle.
CHECK THE RIGHT ANGLE NUMERICALLY. At 53.12°, sinθt = (1.000/1.333) sin 53.12° = 0.7500 × 0.7999 = 0.5999, so θt = 36.88°, and 53.12 + 36.88 = 90.00° exactly. That is the sum row above, and it is the fastest way to tell whether you have the right angle.
WHY p VANISHES. The reflected light is radiated by electrons in the water driven along the direction of the refracted ray's electric field. An oscillating charge radiates nothing along its own axis of oscillation. For p polarisation at Brewster's angle, that axis points exactly along the reflected ray, so there is nothing to emit in that direction. The reflected light has only the s component left, and it is perfectly polarised.
HOW MUCH ACTUALLY COMES BACK. Fresnel at 53.12° gives Rp = 0 and Rs = 7.83%, so unpolarised sunlight reflects 3.91% — all of it horizontally polarised. Compare the normal-incidence figure, ((1 − 1.333)/(1 + 1.333))2 = 2.04%. So glare is both brighter and purer at Brewster's angle than straight on.
AND WHY THE SUNGLASSES WORK. That 3.91% is horizontally polarised, and it sits on top of the light coming out of the water, which is not polarised. A filter with a vertical axis removes essentially all of the glare and only half of the rest, so the contrast between them improves by a factor of about two. The Cornell lecture says exactly this: sunglasses transmit only vertically polarised light “to cut glare due to reflections from water”. Tip your head 90° and the glare comes straight back.
Brewster’s angle for some common boundaries, light arriving from air
| Second medium | n₂ | θ_B = arctan(n₂) | Reflectance there, unpolarised | Where n₂ comes from |
|---|---|---|---|---|
| Air (no boundary) | 1.000 | 45.00° | 0 | exact by definition |
| Water | 1.333 | 53.12° | 3.91% | common tabulation, not sourced here |
| Glass, as the Cornell example uses it | 1.5 | 56.31° | 5.08% | stated in that lecture |
| Window glass, typical | 1.52 | 56.66° | 5.29% | common tabulation, not sourced here |
| Diamond | 2.417 | 67.52° | 20.5% | common tabulation, not sourced here |
Why one particular angle turns reflection into a polariser
Reflection is not neutral. Light whose electric field lies in the plane of incidence — p polarisation — and light whose field is perpendicular to it — s — reflect by different amounts at every angle except straight on. At one particular angle the p reflection falls all the way to zero, and everything that comes back is s. That angle is Brewster’s, and tanθB = n₂/n₁.
Why it happens, which is better than remembering it. The reflected beam is not bounced; it is re-radiated by electrons in the second medium being driven back and forth by the refracted wave. An oscillating charge radiates sideways and nothing at all along its own axis of oscillation — the same dipole pattern that decides how a radio aerial works. For p polarisation those electrons oscillate in the plane of the diagram, perpendicular to the refracted ray, so when the reflected direction lines up with that axis there is nothing to send that way. The alignment happens exactly when the reflected and refracted rays are 90° apart, and that condition plus Snell’s law gives the tangent in two lines.
The sunglasses, and the road. Glare off a horizontal surface — a lake, a wet road, a car bonnet, a shop window — is a reflection at a fairly large angle of incidence, so it is strongly s polarised, and for a horizontal surface s means horizontal. A polarising filter with a vertical transmission axis therefore removes nearly all of the glare and only half of the ordinary unpolarised light from the scene, so what you lose in brightness you more than gain in contrast. The Cornell lecture states the application directly: sunglasses “transmit only vertically polarized light to cut glare due to reflections from water”. Tilt your head through ninety degrees and the glare comes straight back, which is the cheapest demonstration of polarisation there is.
How sharp is it? Not very, and that is the useful part: the degree-of-polarisation row stays high for a good fifteen degrees either side of the exact angle, so sunglasses work through an afternoon rather than for an instant. What is sharp is the p reflectance itself, which really does reach exactly zero, and that is what makes a Brewster window useful: a plate set at θB in a laser cavity passes p polarisation with no reflection loss at all, which is why many gas lasers emit polarised light with no polariser in them.
Brewster and critical are different angles. If you set the first medium denser than the second, the page gives you both. The critical angle is where refraction stops; Brewster’s is where the reflection is purest, and it is always the smaller of the two. Past the critical angle the reflection is total and unpolarised, which is the opposite of Brewster’s behaviour, so the two must not be confused — for water to air they are 36.88° and 48.75°.
What the syllabus documents say. This batch searched the whole CBSE Class XII curriculum for 2025-26 and Brewster’s law appears nowhere in it, and it is not in the NCERT Chapter 10 copy read either — that chapter’s polarisation section runs from transverse waves through polaroids to Malus’s law and stops. Those are statements about two particular documents, not about any paper or any other board, and this batch did not read a syllabus for any other board. It is recorded because claiming examinable status this page cannot evidence would be worse than admitting the gap. The physics is standard and the derivation above is complete either way. See also Malus’s law, which is what happens after this light meets a filter, and Snell’s law and the critical angle, which is half of the derivation.
Frequently asked questions
Is Brewster’s angle measured from the surface or from the normal?
From the normal, like every other angle in ray optics. For air to water that is 53.12° from the normal, which is 36.88° from the water surface — and if you measure from the surface you will get the refraction angle by accident, because the two add to 90° at exactly this angle. That coincidence makes the mistake unusually hard to spot. The diagram on this page draws the normal explicitly and puts the grey arc between the normal and the incident ray so there is nothing to guess.
Does ALL the light reflect at Brewster’s angle, since it is perfectly polarised?
No — and this is the most common misreading. Perfectly polarised is not the same as completely reflected. At Brewster’s angle for air and water only about 7.8% of the s component reflects and none of the p does, so unpolarised sunlight loses about 3.9% of its energy to the reflection. The other 96% goes into the water. What is special is the purity of the small amount that comes back, not its size. If you want all of it back you need total internal reflection, which is a different angle, only happens going from dense to rare, and gives you light that is not polarised at all.
Why do polarising sunglasses have a vertical axis?
Because the glare they exist to remove is horizontally polarised. Glare comes off horizontal surfaces — water, wet roads, car bonnets — and the s direction for a horizontal surface is horizontal. A vertical filter blocks the whole of a horizontally polarised beam and only half of ordinary unpolarised light, so the glare drops far more than the scene does. This is also why they can make LCD car dashboards and phone screens go black at certain angles: those screens emit polarised light, and if its plane happens to be horizontal your glasses remove it completely.
What is the difference between Brewster’s angle and the critical angle?
They answer different questions. The critical angle is the angle beyond which no light gets through at all: it only exists going from a denser medium to a rarer one, and past it the reflection is total and unpolarised. Brewster’s angle is where the reflection is purest: it exists in both directions, and most of the light still goes through. For water to air they are 36.88° and 48.75°, and Brewster’s is always the smaller, so as you tilt a ray over inside water you meet the polarising angle well before you lose the refracted ray. Set n₁ above n₂ above and the page gives you both.
Why doesn’t a polarising filter cut glare off a metal surface?
Because the dipole argument fails for a conductor. In glass or water the reflected light is re-radiated by bound electrons, and at one angle the p component has nowhere to go. In a metal the electrons are free, the response is almost entirely out of phase and the two polarisations reflect by nearly the same amount at every angle — there is no angle at which Rp reaches zero. So glare off chrome or off still water behave completely differently, and a photographer’s polariser that kills the reflection in a pond does almost nothing to the shine on a car’s bumper. This page’s Fresnel formulas assume a transparent, non-absorbing second medium and do not apply to metals.
Related calculators
References
- Cornell University, Physics Department / CLASSE, P3330 Experimental Optics, Lecture 3 (I. Bazarov). Gives the Fresnel amplitude reflection coefficients in the compact form r⊥ = (1 − nm)/(1 + nm) and r∥ = (m − n)/(m + n), with n = nt/ni and m = cosθt/cosθi. Those are algebraically identical to the long forms this page implements: divide the usual numerator and denominator through by nicosθi and you get them, which is the check that this page’s Fresnel arithmetic is that source’s Fresnel arithmetic. For the air-to-glass case, with nair ≈ 1 and nglass ≈ 1.5, it states “Zero reflection for parallel polarization at Brewster’s angle (56.3° for these values of ni and nt)” — and arctan(1.5) = 56.3099°, which is the independent check that the arctangent on this page is the right arctangent. For polarisation in use it says sunglasses “transmit only vertically polarized light to cut glare due to reflections from water”. What this source does NOT contain, and is therefore not attributed to it: the formula tanθB = n2/n1, and any statement that the reflected and refracted rays are 90° apart. Both are derived from the geometry in the worked example on this page instead. Several of the slides’ fractions were damaged in extraction and are read from the layout, so they are reported as a reading rather than a verbatim quotation.
- NCERT, Physics, Class XII, Part II, Chapter 10, Wave Optics (same caveat about the hosted copy). The chapter’s sections are 10.1 Introduction, 10.2 Huygens Principle, 10.3 Refraction and Reflection of Plane Waves using Huygens Principle, 10.4 Coherent and Incoherent Addition of Waves, 10.5 Interference of Light Waves and Young’s Experiment, 10.6 Diffraction and 10.7 Polarisation. Section 10.7 gives Equation (10.18), I = I₀cos2θ, and names it: “This is known as Malus’ law.” It is explicit about the thing that catches people out — “where I₀ is the intensity of the polarized light after passing through P₁”, not the intensity of the unpolarised beam that arrived — and states separately that “the intensity coming out of a single polaroid is half of the incident intensity”. Brewster’s law does not appear anywhere in the chapter. Section 10.7 runs from transverse waves through polaroids to Malus’ law and its applications and stops; there is no tan iB = n. The chapter was searched for the word. Figures are absent from the extracted copy and several equations are garbled, so only the statements quoted above are relied on.
- Central Board of Secondary Education, Physics (Code No. 042), Classes XI–XII, Senior Secondary Curriculum 2025-26, read from cbseacademic.nic.in. Unit VI Optics, Chapter 9 Ray Optics lists the lens maker’s formula by name, alongside the thin lens formula, magnification, power of a lens, thin lenses in contact and refraction through a prism. Chapter 10 Wave Optics lists Huygens’ principle, the proof of the laws of reflection and refraction, coherent sources, interference, “Young’s double slit experiment and expression for fringe width (No derivation final expression only)” and “diffraction due to a single slit, width of central maxima (qualitative treatment only)” — and then stops. “Polarisation”, “Malus” and “Brewster” appear nowhere in the Optics unit. The whole document was searched: the only occurrence of the word at all is “Dielectrics and electric polarization” in Unit I, which is electrostatics and a different thing. This is reported because it is load-bearing: on that document the lens maker’s formula is named examinable content and the two polarisation laws are not.
- The refractive indices on these pages, and how far each can be trusted. n = 1.5 for glass is the figure the Cornell lecture above uses for its own worked Brewster angle, so it is sourced. n = 1.333 for water at visible wavelengths is the figure in common tabulation and this batch did not reach a primary standards publication for it; it is consistent with the Cornell example to the precision that example is quoted at. Real glass is not one number: crown glass, flint glass and polycarbonate run from roughly 1.5 to 1.6 and every one of them disperses, so the index you should use is the one the question gives you. Water’s index falls slightly as it warms and rises towards the blue end of the spectrum, which is why a Brewster angle quoted to four figures is false precision. The indices used in the comparison tables are marked in the table itself with where each came from, and the ones that could not be sourced say so rather than being dressed up.