Brewster’s Angle Calculator

Brewster’s Angle Calculator for any two media. Find the polarizing angle from both indices, or one index from a measured angle, with Fresnel reflectance and a pile-of-plates count.

How the Brewster’s Angle Calculator Works

The Brewster’s Angle Calculator has three modes in its Solve For menu. The default finds the Brewster angle θB from two refractive indices. The other two work backward from an angle you measured.

In the default mode, n₁ is the index of the medium the light travels in. n₂ is the index of the surface it strikes. The reverse modes return n₂ when you know the incident medium, or n₁ when you know the surface.

Each index box has a material menu with thirteen presets. They run from ice and water through acrylic, glass, sapphire and diamond, up to infrared ZnSe and germanium. You can also type any index from 0.01 to 100.

Three folded sections sit under the main inputs. Wavelength moves N-BK7, F2, fused silica and sapphire to a laser line such as 532 nm or 1064 nm. Your Incidence Angle adds the reflectance at the angle you actually work at.

Brewster Window takes a beam diameter and a plate thickness in mm, cm or inches. It returns the window size and beam shift for a plate set at θB. The tool covers clear, non-absorbing materials, so metals are outside its range.

Reading the Result Cards

For the default air to glass case (n₁ = 1, n₂ = 1.5), the Brewster’s Angle Calculator returns 56.31°. At that angle the reflected and refracted rays sit exactly 90° apart. The hero also shows the angle in radians.

The first card gives the critical angle for total internal reflection, 41.81° for glass. It also shows the refraction angle inside the glass, 33.69°, and which side of the surface can trap light.

The second card shows how much light still reflects at θB. All of it is s-polarized. For glass that is 14.79% of the s-light, or 7.40% of an unpolarized beam, compared with 4% at normal incidence.

The third card counts the plates a pile-of-plates polarizer needs to reach 90% polarization. Glass needs 52 plates. The card also shows that 5 plates reach 46.47% and 20 plates reach 77.64%.

The fourth card covers your own incidence angle. At 45° on glass, p-light reflects 0.8466% and s-light reflects 9.201%. The reflected glare is already 83.15% polarized, well short of θB.

Below the cards sit the step-by-step working, a ray diagram and a reflectance chart. The Brewster window table and a material table follow. Results update as you type, and the Copy, Share and Print buttons carry every value.

Brewster’s Angle Formula

The Brewster’s angle formula is also called the polarizing angle formula or Brewster’s law. It links the angle to the ratio of the two refractive indices. The angle is measured from the surface normal, not from the surface.

$$\tan\theta_B = \frac{n_2}{n_1}$$

$$\theta_B = \arctan\left(\frac{n_2}{n_1}\right)$$

Here n₁ is the index of the medium the light comes from, and n₂ is the index of the medium it hits. Swap the two and you get the internal Brewster angle. It is always 90° minus the external one, so glass gives 56.31° from air and 33.69° from inside.

Deriving Brewster’s Law From Snell’s Law

Snell’s law, the first law of refraction, says n₁ sin θi = n₂ sin θt. The law of reflection makes the reflected angle equal the incident angle. At Brewster’s angle, the reflected and refracted rays also meet at a right angle, so θB + θt = 90°.

$$n_1 \sin\theta_B = n_2 \sin\left(90^\circ – \theta_B\right) = n_2 \cos\theta_B$$

Dividing both sides by n₁ cos θB gives tan θB = n₂ ÷ n₁. The reason lies with the electrons in medium 2. They oscillate along the field of the refracted p-light, which at θB points straight along the reflected ray.

An oscillating electron sends no light along its own line of motion. So at θB, no p-light can leave in the reflected direction. Only the s-light, whose electrons oscillate across the page, reflects.

Angle of Refraction at Brewster’s Angle

The general angle of refraction formula is θt = arcsin(n₁ sin θi ÷ n₂). At Brewster’s angle it reduces to θt = 90° − θB. The calculator shows this value on its first card for any pair of media.

For air to water, the refracted ray travels at 36.88° from the normal. For air to diamond, it travels at just 22.48°, because diamond bends light much more strongly than water or glass.

Worked Example: Brewster Angle for Water

Light in air (n₁ = 1.000) striking calm water (n₂ = 1.333) gives tan θB = 1.333. So θB = arctan(1.333) = 53.12°. At that angle water reflects 7.83% of s-light and no p-light.

That works out to 3.91% of the incoming sunlight. It is close to double the 2.04% the water reflects when you look straight down at it, but all of it is now s-polarized.

A ray hitting water at 53.12° from the normal is 36.88° above the horizon. So lake glare is most strongly polarized when the sun, or your line of sight, sits about 37° above the water. Polarized sunglasses and camera filters cut the most glare there.

Brewster Angle Table by Material

The table lists the Brewster’s Angle Calculator presets with light coming from air. “θB in” is the angle for light arriving from air. “θB out” is the internal angle for light leaving the material into air.

Visible materials use their catalog index near 589 nm. ZnSe and germanium use the 10.6 µm CO2 laser line, which is the wavelength these two infrared materials are normally used at.

The glass indices come straight from the maker’s own data sheets. SCHOTT lists N-BK7 at nd = 1.51680 and F2 at nd = 1.62004, both measured at 587.6 nm.

Coherent’s optical materials data sheet gives ZnSe 2.4028 at 10.6 µm. Crystran’s germanium data runs from 4.0034 at 9.72 µm to 4.0026 at 11.04 µm.

Covestro rates its Makrolon polycarbonate grades between 1.584 and 1.587 under ISO 489. Water at 20 °C is 1.3330 at the sodium line, the figure behind the NIST and IAPWS water index.

Sapphire is birefringent, so its Brewster angle depends on how the crystal is cut. The preset uses the ordinary-ray index of 1.768. Light polarized along the c-axis sees about 1.760, which moves θB from 60.51° to 60.40°.

Brewster Angle by Wavelength

Every refractive index changes with wavelength, so the polarizing angle does too. The Brewster’s Angle Calculator handles this with the Sellmeier equation. It gives the index at any wavelength from six fitted constants.

$$n^2(\lambda) = 1 + \frac{B_1 \lambda^2}{\lambda^2 – C_1} + \frac{B_2 \lambda^2}{\lambda^2 – C_2} + \frac{B_3 \lambda^2}{\lambda^2 – C_3}$$

Here λ is in micrometers. The constants for N-BK7 and F2 come from SCHOTT’s data sheets. Fused silica uses Malitson’s 1965 measurements, and sapphire uses Malitson and Dodge’s fit for the ordinary ray.

Each set of constants reproduces its catalog index. N-BK7 comes out at 1.51680 and fused silica at 1.45846 at 587.6 nm. Pick a laser line in the Wavelength section and these four presets update on their own.

The angle falls as the wavelength grows, because each index falls toward the infrared. F2 flint moves the most, by 0.88° from 405 nm to 1550 nm. Fused silica moves only 0.47° over the same range.

For most windows that shift sits inside the broad p-reflectance dip. It matters more when you set up a low-loss laser cavity, or when you measure an index from θB with a laser.

Water, ice, acrylic, polycarbonate and diamond keep their catalog values when you change the wavelength. ZnSe and germanium stay at 10.6 µm. For these, type the supplier’s index for your line into the box as a custom value.

Critical Angle vs Brewster Angle

Both angles come from the same index ratio, but they mean different things. Brewster’s angle is where p-light stops reflecting, and it exists for light crossing a surface in either direction.

The critical angle is where all light starts reflecting. It only exists when light goes from a denser medium into a less dense one, such as from glass or water out into air.

Inside the dense medium, sin θc = n₂ ÷ n₁ while tan θB = n₂ ÷ n₁. The tangent of an angle is always larger than its sine, so θB always lands below θc. The Brewster’s Angle Calculator marks both on its chart when n₁ is larger than n₂.

Past θc, the incidence card switches to total internal reflection. It then reports the phase shift between s and p light. That shift is what a Fresnel rhomb uses to turn linear polarization into circular polarization.

Brewster’s Angle Diagram

The Ray Diagram panel in the Brewster’s Angle Calculator draws the incident, reflected and refracted beams at θB. Dots mark s-polarization, with the electric field pointing out of the page. Short ticks across a ray mark p-polarization.

The incident ray carries both dots and ticks. The reflected ray carries dots only. The refracted ray carries every tick plus the s-light that did not reflect. A right-angle marker shows the 90° gap between the two outgoing rays.

The Reflectance Chart

The Reflectance vs Angle chart plots Rs and Rp from 0° to 90°. Rs climbs steadily toward 100% at grazing incidence. Rp falls to zero at θB and then rises again toward the same 100%.

Your own incidence angle appears as two dots on the curves. The table under the chart lists both curves every 10°, plus extra rows for θB and for the angle you entered.

The dip is broad, and that matters in practice. For air to glass, p-reflectance stays under 0.1% from 53.1° to 59.1°. It stays under 0.01% from 55.3° to 57.2°.

So a window set a degree off θB still loses very little p-light. The same broad dip makes the exact minimum hard to find by eye, which is the main problem in a lab measurement.

Measuring Brewster’s Angle in an Experiment

A common lab setup sends a laser through a polarizer set to p-polarization. The beam hits a flat sample on a rotation stage. As the stage turns, the reflected spot dims to a minimum and then brightens again.

The stage reading at the minimum is θB. With unpolarized light, you can instead view the reflected beam through a second polarizer. Turn the sample until that polarizer blocks the glare completely.

Because the minimum is broad, bracket it. Note two angles on either side where the spot looks equally dim, then average them. This gives a steadier reading than chasing the darkest point.

Then switch the Brewster’s Angle Calculator to Index of Medium 2 and enter that angle with n₁ = 1. Used this way, it works as a Brewster’s angle calculator for physics labs, turning one reading into a refractive index.

A reading of 56.5° gives n₂ = tan 56.5° = 1.511. The result is sensitive near 56°. A reading of 56.0° gives 1.483, while a reading of 57.0° gives 1.540 for the same sample.

That means a ±0.5° error shifts the index by about ±0.03. It is enough to blur the difference between acrylic and crown glass, so take several readings and average them.

Set the Wavelength section to your laser before you compare results. SCHOTT lists N-BK7 at 1.51509 for a 632.8 nm helium-neon laser. That gives 56.57°, not the 56.60° you get at the catalog line.

Where the Polarizing Angle Matters

Brewster’s angle turns up wherever someone needs to control glare, polarize a beam or pass light through a window with little loss. The calculator’s cards and tables are built around these jobs.

Polarized Sunglasses and Camera Filters

Glare from water, wet roads and car windows arrives near Brewster’s angle. Much of it is s-polarized, with the field running horizontally. Polarized lenses have a vertical transmission axis that blocks that field.

The same effect helps when you glass water with binoculars, since cutting glare shows what is under the surface. The Binoculars Range Calculator handles the distance side of that job.

Brewster Windows in Lasers

Gas lasers such as helium-neon tubes seal the tube with windows tilted to θB. The RP Photonics Encyclopedia entry on Brewster windows, by Dr. Rüdiger Paschotta, explains that they lose far less p-light than an anti-reflection coating.

The same entry notes an s-loss of roughly 15% per glass surface. That loss is what forces the laser to emit p-polarized light, with no separate polarizer needed inside the cavity.

A tilted window also needs a larger face. A round beam of diameter D lands on the window as an ellipse with a long axis of D ÷ cos θB. That is 1.80 D for glass and 2.60 D for ZnSe.

The Aperture Area Calculator sizes that clear aperture. A window too small for the tilt clips the beam in one direction. That coarsens the limit you would get from the Angular Resolution Calculator.

A tilted plate also shifts the beam sideways. The beam crosses the plate at the refraction angle θt, and the size of the shift depends on the plate thickness t.

$$d = \frac{t \sin\left(\theta_B – \theta_t\right)}{\cos\theta_t}, \qquad \theta_t = 90^\circ – \theta_B$$

The Brewster Window section works these out for you. A 2 mm helium-neon beam through a 5 mm glass plate needs a 3.606 mm window face. The beam shifts sideways by 2.311 mm and travels 6.009 mm inside the plate.

Laser tubes often tilt their two windows in opposite directions, so the two shifts cancel. For a CO2 laser, a 10 mm beam through 5 mm of ZnSe at 67.41° needs a 26.03 mm face and shifts by 3.817 mm.

Pile-of-Plates Polarizers

A stack of thin plates at θB passes all the p-light. Each surface strips a share of the s-light. With m plates, and counting light bounced between them, the beam reaches this degree of polarization.

$$P = \frac{m R_s}{1 + (m – 1) R_s}, \qquad R_s = \left(\frac{n^2 – 1}{n^2 + 1}\right)^2$$

Five glass plates reach only 46.47%, and twenty reach 77.64%. Reaching 90% takes 52 glass plates, which is why glass piles are mostly used as a teaching demo.

High-index infrared materials do far better. ZnSe needs 10 plates and germanium only 3. Both work in the 8 to 14 µm thermal band, where the Blackbody Radiation Calculator puts a 300 K object’s peak near 9.7 µm.

X-Ray Polarizers

The same electron rule works for X-rays, where every refractive index sits just below 1. A crystal that scatters X-rays through exactly 90° reflects only one polarization. A Bragg angle of 45° therefore acts like a Brewster angle.

The Bragg’s Law Calculator finds the d-spacing and wavelength that put a crystal on that 45° condition. It is the X-ray version of tilting a glass plate to θB.

The Optics Calculators hub collects the related tools. HyperPhysics at Georgia State University also has a short Brewster angle and polarization by reflection page for students.

Brewster’s Angle FAQs

These answers cover the questions people ask most about Brewster’s angle. They run from its basic meaning to wavelength, metals and the units used for polarization.

What is meant by Brewster’s angle?

Brewster’s angle is the angle of incidence at which reflected light is completely polarized. At that angle p-light passes into the second medium with zero reflection. Only s-light bounces off the surface.

The angle is named after the Scottish physicist Sir David Brewster, who published tan θB = n₂ ÷ n₁ in 1815. It depends on both media, not one. Glass gives 56.31° from air but only 48.37° under water.

Is there a Brewster’s angle calculator app?

You do not need a separate app. The Brewster’s Angle Calculator runs in any phone or desktop browser. On a phone, the extra sections stay folded until you open them, so the main inputs fit on one screen.

Inputs are processed in your browser. The Share button sends the full result list to a messaging app. Print / PDF saves it as a page for a lab notebook or report.

Does Brewster’s angle change with wavelength?

Yes, because the refractive index changes with wavelength. For N-BK7 the shift is about 0.15° between a 632.8 nm helium-neon laser and a 1064 nm Nd:YAG laser.

The shift grows over wider ranges. F2 flint moves 0.88° between 405 nm violet and 1550 nm telecom light. The Wavelength section applies these shifts to N-BK7, F2, fused silica and sapphire.

Is there a Brewster angle for metals?

Metals absorb light, so their refractive index is complex. Their p-reflectance never falls all the way to zero. It only dips to a minimum at what is called the pseudo-Brewster angle.

The Brewster’s Angle Calculator works with clear, non-absorbing materials only. Use it for glass, plastics, liquids and crystals rather than polished metal or strongly absorbing films.

What are the two types of polarisation in reflection?

Reflection splits light into s-polarization and p-polarization. In s-light the field is perpendicular to the plane of incidence. The s comes from the German word senkrecht, meaning perpendicular.

In p-light the field lies in that plane. Each type has its own reflectance curve. Brewster’s angle is where the p-curve touches zero while the s-curve keeps rising.

Optics also teaches three polarization states of light, which are linear, circular and elliptical. Light reflected at θB is linear. A Fresnel rhomb working past the critical angle can turn it circular.

What are the four types of polarization?

“Four types of polarization” usually refers to dielectric polarization in materials, not to light. The four are electronic, ionic, orientational and space-charge polarization.

Electronic polarization is the one tied to optics. It sets the refractive index at visible frequencies, and so it sets Brewster’s angle for glass, water and plastics.

What is the SI unit of polarization?

Dielectric polarization P is dipole moment per unit volume. Its SI unit is the coulomb per square meter (C/m²). Molecular dipole moments are often given in debye, where 1 D is about 3.336 × 10⁻³⁰ C·m.

The degree of polarization of a light beam is different. It is a plain percentage with no unit, and it is what the calculator’s cards report for reflected and transmitted light.