Is the brewster angle measured from the surface or normal?

It is measured from the normal, which is perpendicular to the surface. When going from air into glass it will be about 56.7 degrees relative to the normal but about 33.3 degrees relative to the glass surface itself. If a material gives an angle that is measured from the surface please subtract 90 degrees from that angle before entering it here.

Why is reflected light at Brewster's angle completely polarized?

The reflection is actually a re-emission that happens when the incident wave makes electrons in the surface oscillate. An oscillating charge does not radiate along its axis of motion. At Brewster's angle, the direction that p-polarized light wants to be reflected into exactly matches this axis. So only s-polarized components remain, and the reflected light is purely s-polarized.

Does the brewster angle mean that no light is reflected?

No. Only the p-polarized half of light is cancelled out. The s-polarized half continues to be reflected. When light enters crown glass from air, this value is about 15.6%, so for normal non-polarized light about 7.8% will be reflected at the surface. What's special about this angle is the purity of the reflected light, not the amount of reflection.

How is the brewer's angle different from the critical angle?

The Brewster angle uses the arctangent of the ratio of indices of refraction and exists at all transparent interfaces. The critical angle uses the arcsine and only exists when light goes from a higher index medium to a lower index one. For any two given media, the arctangent is always less than the arcsine so that Brewster's angle occurs first and long before total reflection begins.

Can you measure the index of refraction by determining the polarization angle?

Yes. This is a standard method of measurement with a typical experimental setup. You rotate the sample until the reflection visible through a polarizing filter disappears, read off the angle and multiply the tangent of that angle by the refractive index of the surrounding medium. If you enter the angle and leave the second refractive index blank, then a tool will calculate this value backwards.

Why is the transmitted light ray barely polarized?

Since a surface does not reflect the p-component and only reflects a very small part of the s-component, the polarization strength of the continuing light is only a few percent. When glass panes are stacked on top of each other, this small portion is removed again at each surface so that by simply stacking about ten glass panes, the polarization strength of the transmitted light beam can be increased to over 90%.

Brewster Angle Calculator

The surface you are looking at

Refractive index of the incident medium (n1)

Refractive index of the reflecting medium (n2)

Brewster's angle (from the normal)

More results

Brewster's angle

deg (°)

Refraction angle at this angle: deg (°)
Reflected share of unpolarized light: %
Reflectance of s-polarized light: %

Polarization of the transmitted beam: %
Reflectance head-on, for comparison: %
Index ratio n2/n1

At this angle the p-polarized reflectance is exactly zero, so everything that bounces off is s-polarized: the light waves all vibrate parallel to the surface. Turn a polarizing filter to block that direction and the glare disappears. Only

7.82%

of an ordinary unpolarized beam reflects here, the rest goes through.

The reflected ray and the refracted ray leave at exactly 90 degrees to each other, which is the geometric signature of Brewster's angle. The refracted ray runs off at

33.35 deg (°)

from the normal. Measuring the angle up from the surface instead of down from the normal gives that same number, which is the mix-up worth watching for.

Reflectance curve

Show how reflectance varies with the angle of incidence

The p-polarized curve drops to exactly zero at Brewster's angle. That dip is the whole effect.

Common materials

Show Brewster's angle for common materials

Brewster's angle and surface reflectance for everyday materials, with your own pair added.

Brewster's angle and surface reflectance for common materials in air, with your own surface added
Material (from air)	Refractive index	Brewster's angle	s-polarized reflectance	Reflectance head-on
Water	1.333	53.11 deg	7.81%	2.03%
Ice	1.31	52.64 deg	6.94%	1.80%
Acrylic (PMMA)	1.49	56.12 deg	14.34%	3.87%
Crown glass	1.52	56.65 deg	15.65%	4.25%
Flint glass	1.62	58.31 deg	20.07%	5.59%
Polycarbonate	1.585	57.74 deg	18.52%	5.11%
Sapphire	1.77	60.53 deg	26.61%	7.72%
Silicon (infrared)	3.42	73.70 deg	70.96%	29.97%
Diamond	2.417	67.52 deg	50.06%	17.19%
Your surface	1.52	56.65 deg	15.65%	4.25%

Good to know

Brewster's law is a single line:

tan(θ_B) = n₂ ÷ n₁

. The angle is measured from the normal, the line perpendicular to the surface, not from the surface itself.

This angle is why a polarizing filter clears the reflection off a shop window or a lake, why laser tubes are capped with windows tilted to exactly this angle, and why ellipsometers measure film thickness by hunting for the reflectance minimum. Metals and other absorbing surfaces never reach a true zero, only a shallow dip called the pseudo-Brewster angle.

If you tilt a pane of glass and reduce the reflections that are visible through a polarizing filter until they almost disappear, then you have found the Brewster angle. This is the incident angle on a transparent surface where p-polarization is not reflected at all by the surface, so that all reflected light becomes pure s-polarization.

This calculator computes the results based on the refractive index of the materials on either side of a surface and can also perform calculations in reverse. If any one of the three values is entered, the other two are computed. This allows measured polarization angles to be converted into unknown refractive indices.

It shows not only the angle but also the actual effect that the surface has on light. You can determine the amount of reflected light, the degree of polarization of both reflected and transmitted light, as well as how many sheets of glass stacked together are needed to make a practical polarizer.

What is the Brewster Angle?

At a surface there are two different types of polarization that are independent of each other: s-polarization, whose oscillation is parallel to the surface and perpendicular to the plane of incidence, and p-polarization, which oscillates in the plane of incidence.

The effects on the light are different and this difference changes with increasing angle. At a particular angle the reflection rate for p-polarization is exactly zero. This angle is called the Brewster's angle, also known as the polarization angle.

Reflection occurs when electrons oscillating at the surface re-emit radiation. However, oscillating charges do not radiate along their axis of motion. At Brewster's angle, the direction in which reflected p-polarized light must travel exactly matches this non-radiating axis, and thus no p-polarization is produced.

Formula for the Brewster angle:

Brewster's law relates one angle to two indices of refraction, that's all it is.

\tan\theta_B = \frac{n_2}{n_1} \qquad\Longrightarrow\qquad \theta_B = \arctan\!\left(\frac{n_2}{n_1}\right)

where n1 is the index of refraction of the medium that light is traveling through and n2 is the index of refraction of the material that the light hits. The angle is measured with respect to the normal, which is a line perpendicular to the surface.

By rearranging the formula, unknown materials can be identified by using the polarization angle calculated from this equation.

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

Since only the ratio of n2 to n1 appears, the Brewster angle for a surface does not change if both indices are simultaneously increased by a factor of two. It is helpful to remember that this ratio equals the tangent of the angle, as it provides an easy way to check validity.

Why the reflected and refracted rays make a 90-degree angle with each other.

Combining the Brewster's law and Snell's law leads to a simple result. Snell's law is n1*sin(theta_B) = n2*sin(theta_t), and Brewster's law is n2 = n1*tan(theta_B). This results in:

\sin\theta_t = \frac{n_1 \sin\theta_B}{n_2} = \frac{\sin\theta_B}{\tan\theta_B} = \cos\theta_B

If sine and cosine are equal, that means that the sum of their two angles is a right angle.

\theta_B + \theta_t = 90^\circ

So the reflected and refracted rays are perpendicular to each other when they leave the surface. This is a way of quickly determining the Brewster angle on a ray diagram, and it also means that the angle of refraction at the Brewster angle is simply 90 degrees minus the Brewster angle itself.

This resulting angle of refraction has another interesting consequence: the Brewster's Angle also applies to light traveling in the opposite direction from Medium 2 into Medium 1. This is because arctan(n1/n2) is the complement of arctan(n2/n1).

Example: From air to crown glass.

We use air with a refractive index of n equals 1.0003 and crown glass with a refractive index of n equals 1.52. We divide the refractive indices, then take the arctangent.

\theta_B = \arctan\!\left(\frac{1.52}{1.0003}\right) = \arctan(1.51954) \approx 56.65^\circ

The refracted ray will travel at an angle that is the difference between 90 and 56.65, making a 33.35 degree angle with the normal. When the polarizing filter is rotated to block s-polarization, reflection from this glass will be almost completely suppressed.

The reflection is not complete but nearly so since there is still an s-component in the reflected light. At Brewster's angle this reflection is about 15.6%. Also only about 7.8% of unpolarized light is reflected because only half of natural light is s-polarized.

How much light is reflected at the Brewster angle?

The reflection for p-polarization is by definition zero. The reflection for s-polarization is given by the Fresnel equation and when we substitute the geometric relationship of 90 degrees it can be simplified to a simpler form.

R_s = \cos^2(2\theta_B) = \left(\frac{n_2^2 - n_1^2}{n_2^2 + n_1^2}\right)^2 \qquad R_p = 0

Since the unpolarized light beam consists of equal parts of both components, only half of the s-component is actually reflected.

R_{\text{unpol}} = \frac{R_s + R_p}{2} = \frac{R_s}{2}

At the surface where air meets glass this is about 7.8%. In contrast a light ray hitting that same surface head on has about 4.3%. The Brewster angle is not the angle at which total reflection becomes less, but rather the angle at which reflected light becomes fully polarized.

Brewster's angle for common materials

Most transparent materials in air are within a narrow range because the refractive index once it exceeds 1.3 causes the arctangent function to rapidly have a lower rate of change.

Material (in air)	Refractive index	Brewster's angle	s-polarized reflectance
Water	1.333	53.11 degrees	7.81%
Ice	1.31	52.64 degrees	6.94%
Acrylic (PMMA)	1.49	56.12 degrees	14.34%
Crown glass	1.52	56.65 degrees	15.65%
Polycarbonate	1.585	57.74 degrees	18.52%
Flint glass	1.62	58.31 degrees	20.07%
Sapphire	1.77	60.53 degrees	26.61%
Diamond	2.417	67.52 degrees	50.06%
Silicon (infrared)	3.42	73.70 degrees	70.96%

For this reason, photographers need only know one rule of thumb: If you're shooting a subject at an angle of about 55 degrees to the normal, be it water, glass or glossy paint, then using a polarizing filter will get you close to Brewster's Angle conditions.

Reflection and transmission in stacked glass plates to produce polarized light

The reflected ray at the Brewster angle is fully polarized but weak. The transmitted ray is strong but has little polarization because only a very small part of the s-component can be removed from a single surface while the p-component is not removed at all.

A single crown glass plate has two Brewster surfaces and the degree of polarization of transmitted light is about 17%. As more plates are stacked on top of each other, the degree of polarization increases further since each surface makes a small additional contribution to reducing the s-component.

T_s = (1 - R_s)^{2m} \qquad P = \frac{1 - T_s}{1 + T_s} \qquad \frac{I_{\text{out}}}{I_{\text{in}}} = \frac{1 + T_s}{2}

Here m is the number of glass plates and 2m is the number of surfaces. With ten overlapping crown glass plates, the degree of polarization reaches about 94%, while still allowing through about 52% of the light. This is close to the value of 50% defined by half of the p-polarization that is not affected.

This stacked system is one of the oldest polarizers used in optics and still today it is used for wavelengths where film polarizers are not available such as in the ultraviolet or infrared regions.

Brewster's angle and critical angle

When light passes from a medium with higher refractive index to one with lower refractive index, such as from glass into air, there is both a Brewster's angle and a critical angle at the interface. These are not the same thing and never coincide.

\theta_B = \arctan\!\left(\frac{n_2}{n_1}\right) \qquad \theta_c = \arcsin\!\left(\frac{n_2}{n_1}\right) \qquad \theta_B < \theta_c

Since the arctangent is always less than the arcsine for the same ratio, the polarization angle reaches its limit first. For light going from crown glass to air, the Brewster's angle is 33.35 degrees while total internal reflection begins at 41.15 degrees.

In the normal case, when light passes from a medium with lower refractive index to one with higher refractive index, there is no critical angle but only the Brewster's Angle.

How to use this calculator:

In most cases you will leave the index of refraction for the incident material at its value for air and enter the index of refraction for the material being examined. The angles are left blank. The screen displays the Brewster's angle, the corresponding angle of refraction, and the fraction of light reflected.

To identify a material, enter the measured polarization angle and leave the second refractive index blank. The calculation tool will calculate backwards using Brewster's law and output the refractive index. This is the basic principle of the Brewster Angle Refractometer.

If you open the reflection curve, you can see the attenuation of p-polarization in both media. With the multilayer option, you can design a size of a multilayer polarizer.

Here are some limitations to keep in mind.

The Brewster law assumes that both media are transparent and non-magnetic. Since metals and other absorbing surfaces have a complex index of refraction, the reflection rate for p-polarized light does not actually drop to zero but reaches a flatter minimum. This minimum is called pseudo-Brewster angle and requires full complex Fresnel analysis.

The index of refraction also changes with wavelength so the Brewster angle for red light is slightly different from that for blue. For normal glasses this change is small and negligible for photography but can be important in precision optics.

Thin-film coatings also change the situation. As stacked anti-reflection layers have an inherent layer structure, the formulas for a single surface no longer apply.

This tool is for learning and quick estimation purposes. For precision optics work you should use the refractive index measured for the exact wavelength, temperature, and polarization.

Frequently asked questions

Is the brewster angle measured from the surface or normal?: It is measured from the normal, which is perpendicular to the surface. When going from air into glass it will be about 56.7 degrees relative to the normal but about 33.3 degrees relative to the glass surface itself. If a material gives an angle that is measured from the surface please subtract 90 degrees from that angle before entering it here.
Why is reflected light at Brewster's angle completely polarized?: The reflection is actually a re-emission that happens when the incident wave makes electrons in the surface oscillate. An oscillating charge does not radiate along its axis of motion. At Brewster's angle, the direction that p-polarized light wants to be reflected into exactly matches this axis. So only s-polarized components remain, and the reflected light is purely s-polarized.
Does the brewster angle mean that no light is reflected?: No. Only the p-polarized half of light is cancelled out. The s-polarized half continues to be reflected. When light enters crown glass from air, this value is about 15.6%, so for normal non-polarized light about 7.8% will be reflected at the surface. What's special about this angle is the purity of the reflected light, not the amount of reflection.
How is the brewer's angle different from the critical angle?: The Brewster angle uses the arctangent of the ratio of indices of refraction and exists at all transparent interfaces. The critical angle uses the arcsine and only exists when light goes from a higher index medium to a lower index one. For any two given media, the arctangent is always less than the arcsine so that Brewster's angle occurs first and long before total reflection begins.
Can you measure the index of refraction by determining the polarization angle?: Yes. This is a standard method of measurement with a typical experimental setup. You rotate the sample until the reflection visible through a polarizing filter disappears, read off the angle and multiply the tangent of that angle by the refractive index of the surrounding medium. If you enter the angle and leave the second refractive index blank, then a tool will calculate this value backwards.
Why is the transmitted light ray barely polarized?: Since a surface does not reflect the p-component and only reflects a very small part of the s-component, the polarization strength of the continuing light is only a few percent. When glass panes are stacked on top of each other, this small portion is removed again at each surface so that by simply stacking about ten glass panes, the polarization strength of the transmitted light beam can be increased to over 90%.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Brewster's angle
Derivation of Brewster's law, the dipole-radiation explanation, and the pseudo-Brewster angle for absorbing media.
HyperPhysics: Polarization by Reflection
Georgia State University reference on Brewster's angle and polarization at a dielectric surface.
RP Photonics Encyclopedia: Brewster's Angle
Practical treatment of Brewster windows, Brewster-angle plates, and laser applications.
Encyclopedia Britannica: Brewster's law
Short historical and physical account of the law David Brewster published in 1811.