# Brewster angle

The Brewster angle (after Sir David Brewster , 1781–1868) or polarization angle is a quantity of optics . It specifies the angle at which, of the light incident on the interface between two dielectric media , only the portions polarized perpendicular to the plane of incidence (based on the electric field component) are reflected. The reflected light is then linearly polarized.

When it is reflected at the interface between two media, at least one of which is not an ideal dielectric, the light is only partially polarized. In contrast to the ideal case, a part of the part polarized parallel to the plane of incidence is also reflected here, but this has a minimum in a completely analogous manner at the so-called pseudo-Brewster angle.

## Phenomenological description

Location of Brewster's angle when light is reflected at the interface between two ideal dielectrics (k 1 = k 2 = 0)
Representation of the Brewster angle ${\ displaystyle \ theta _ {\ mathrm {B}}}$

An electromagnetic wave, e.g. B. visible light, which does not strike an interface between two media perpendicularly, is partly reflected and partly refracted into the second medium . The reflected or the refracted portion is at least partially polarized. This can be observed using a polarization filter , for example . If the filter is rotated around the direction of propagation of the reflected light at a constant angle of incidence , a minimum brightness can be determined when the transmission axis of the filter lies in the plane of incidence of the wave. The depth of this brightness minimum depends on the angle of incidence selected. At a certain angle of incidence, the minimum brightness is zero, so only light polarized perpendicular to the plane of incidence is reflected. This angle of incidence is called the Brewster angle. Its value depends on the refractive indices of the media between which the transition takes place. It is therefore dependent on the material of the media and the frequency of the electromagnetic wave.

The observations by the Fresnel formulas described, by means of which the reflectivity as a function of angle of incidence and the refractive indices , for the respective polarization of the incident light can be calculated (see figure). ${\ displaystyle \ theta _ {\ mathrm {1}}}$ ${\ displaystyle n_ {1}}$${\ displaystyle n_ {2}}$

## Physical basics

An unpolarized wave can be split into two mutually perpendicular linearly polarized components. The vectors of the electric field strength are perpendicular or parallel to the plane of incidence (perpendicular or parallel linearly polarized component, also called s or p polarization). The incoming light beam excites the electrons of the material to vibrate ( displacement polarization ). This creates a collection of atomic dipoles ( Hertzian dipoles ) that oscillate in the direction of polarization and thereby emit secondary waves that only interfere constructively in the direction of the reflected and the refracted beam .

In the case of the vertically polarized component, the incoming and outgoing directions are perpendicular to the direction of oscillation of the dipoles and thus in a direction in which a Hertzian dipole radiates at its maximum. The polarization is retained. The situation is different with the excitation by the parallel polarized component, here there is an angle between the direction of oscillation of the Hertzian dipoles and the direction of the reflected wave, which varies from the angle of incidence . The radiation intensity of the dipole ( ) also varies with the angle of excitation , so both polarization components are emitted with different intensities and the reflected light is partially polarized. In the case (the direction of the dipole axis is identical to the direction of the reflection), there is no radiation of the dipoles in the direction of reflection for the parallel polarized portion and the reflected light is completely perpendicularly linearly polarized. ${\ displaystyle \ theta _ {1}}$${\ displaystyle \ alpha = 90 ^ {\ circ} - (\ theta _ {1} + \ theta _ {2})}$${\ displaystyle I}$${\ displaystyle I \ sim \ sin ^ {2} \ alpha}$${\ displaystyle \ alpha = 0}$

## Brewster's law

The angle of incidence (Brewster's angle) at which the reflected light is completely perpendicularly polarized at the transition between any two non-magnetizable media can be derived from Snell's law of refraction and the radiation characteristics of the dipoles ( forced oscillation ) described above. In the following it is assumed that the jet hits another material from the air. As a good approximation, one can assume a refractive index of for air . The material has the refractive index . ${\ displaystyle \ theta _ {B}}$${\ displaystyle n_ {1} = 1}$${\ displaystyle n_ {2}}$

${\ displaystyle n_ {1} \ sin \ left (\ theta _ {1} \ right) = n_ {2} \ sin \ left (\ theta _ {2} \ right)}$

For a ray incident at Brewster's angle, the refracted ray is perpendicular to the reflected ray:

${\ displaystyle \ theta _ {1} = \ theta _ {\ mathrm {B}}}$
${\ displaystyle \ theta _ {2} = 90 ^ {\ circ} - \ theta _ {\ mathrm {B}}}$

together with the law of reflection (angle of incidence equals angle of reflection ) one obtains by inserting ${\ displaystyle \ theta _ {1}}$${\ displaystyle \ theta _ {\ mathrm {a}}}$

${\ displaystyle n_ {1} \ sin \ left (\ theta _ {\ mathrm {B}} \ right) = n_ {2} \ sin \ left (90 ^ {\ circ} - \ theta _ {\ mathrm {B }} \ right) = n_ {2} \ cos \ left (\ theta _ {\ mathrm {B}} \ right)}$

and after changing

${\ displaystyle \ theta _ {\ mathrm {B}} = \ arctan \ left ({\ frac {n_ {2}} {n_ {1}}} \ right)}$( Brewster's law ).

## Example - reflection on glass

Scheme of a polarizer based on the Brewster angle. To increase the degree of polarization, several individual polarizers are placed one behind the other.

From the previous section it can be seen that a glass plate can be used as a polarizer by irradiating it at Brewster's angle.

With the angles mentioned above and the law of refraction, the following applies:

${\ displaystyle \ theta _ {\ mathrm {a}} +90 ^ {\ circ} + \ theta _ {2} = 180 ^ {\ circ}; \ quad \ theta _ {2} = 90 ^ {\ circ} - \ theta _ {1}}$
${\ displaystyle \ theta _ {1} = \ theta _ {\ mathrm {a}} \,}$

Assuming in the materials used air ( ) and ordinary glass ( soda-lime glass , , Note: no quartz glass ( ), there and in accordance ) if they were ideal, non-absorbent materials ( dielectrics ), applies the Brewster described above Law: ${\ displaystyle n_ {1} = 1}$${\ displaystyle n_ {2} = 1 {,} 55}$${\ displaystyle \ mathrm {SiO} _ {2}}$${\ displaystyle n_ {2} = 1 {,} 46}$${\ displaystyle \ theta _ {\ mathrm {B, \ mathrm {Air-SiO_ {2}}}}}$${\ displaystyle 55 {,} 59 ^ {\ circ}}$

${\ displaystyle \ theta _ {\ mathrm {B}} = \ arctan \ left ({\ frac {n_ {2}} {n_ {1}}} \ right)}$

A Brewster angle results for the transition from air to glass:

${\ displaystyle \ theta _ {\ mathrm {B, \ mathrm {Air-Glass}}} = \ arctan \ left ({\ frac {n_ {2}} {n_ {1}}} \ right) = \ arctan \ left ({\ frac {1 {,} 55} {1 {,} 0}} \ right) = 57 {,} 17 ^ {\ circ}}$

According to the Fresnel equations , the following degrees of reflection ( or ) result for the perpendicular or parallel portion of the incident light : ${\ displaystyle R_ {s}}$${\ displaystyle R_ {p}}$

${\ displaystyle R_ {s} = r_ {s} ^ {2} = \ left ({\ frac {n_ {1} \ cos (\ theta _ {1})) - n_ {2} \ cos (\ theta _ { 2})} {n_ {1} \ cos (\ theta _ {1}) + n_ {2} \ cos (\ theta _ {2})}} \ right) ^ {2} = \ left ({\ frac {\ cos (\ theta _ {B, \ mathrm {air-glass}}) - {\ sqrt {{\ tilde {n}} ^ {2} - \ sin ^ {2} \ theta _ {B, \ mathrm {Air-glass}}}}} {\ cos (\ theta _ {B, \ mathrm {air-glass}}) + {\ sqrt {{\ tilde {n}} ^ {2} - \ sin ^ {2 } \ theta _ {B, \ mathrm {air-glass}}}}}} \ right) ^ {2} = 0 {,} 1699 = 16 {,} 99 \, \%}$
${\ displaystyle R_ {p} = r_ {p} ^ {2} = \ left ({\ frac {n_ {2} \ cos (\ theta _ {1})) - n_ {1} \ cos (\ theta _ { 2})} {n_ {2} \ cos (\ theta _ {1}) + n_ {1} \ cos (\ theta _ {2})}} \ right) ^ {2} = \ left ({\ frac {{\ tilde {n}} ^ {2} \ cos (\ theta _ {B, \ mathrm {air-glass}}) - {\ sqrt {{\ tilde {n}} ^ {2} - \ sin ^ {2} \ theta _ {B, \ mathrm {air-glass}}}}} {{\ tilde {n}} ^ {2} \ cos (\ theta _ {B, \ mathrm {air-glass}}) + {\ sqrt {{\ tilde {n}} ^ {2} - \ sin ^ {2} \ theta _ {B, \ mathrm {Air-Glass}}}}}} \ right) ^ {2} = 0 {,} 0 = 0 {,} 0 \, \%}$

with and${\ displaystyle {\ tilde {n}} = {\ frac {n_ {2}} {n_ {1}}}}$${\ displaystyle \ theta _ {1} = \ theta _ {B, \ mathrm {air-glass}}}$

The reflected light is accordingly linearly polarized completely perpendicular to the plane of incidence.

In the case of irradiation with unpolarized light (all polarizations equally represented), the degree of reflection can be determined using the arithmetic mean of the two components, the following applies:

${\ displaystyle R = {\ frac {R_ {s} + R_ {p}} {2}}}$

For the described transition from unpolarized light, only 8.5% of the incident intensity (at Brewster's angle) is therefore reflected.

## literature

• F. Pedrotti, L. Pedrotti, W. Bausch, H. Schmidt: Optics for engineers: Fundamentals . 2nd Edition. Springer, Berlin 2001, ISBN 3-540-67379-2 .