3.3.3 Fresnel equations

Snell’s law describes how the angles of incidence and transmission are related through the refractive indices of the respective media.

But what about the intensity of transmitted and reflected light?

We are looking at a picture of the Montana glacier parks.

White mountain tops are reflected in a perfectly still lake Mcdonald.

It is a beautiful example of this video’s subject.

Not only do we see the mountain tops reflected in the water, we can also find the each and every pebble lying on the bottom of the lake.

A fraction of the light, reflected by the mountain tops, is reflected off the surface of the water.

Another part is transmitted into the water and reflects off the rocks and pebbles.

The Fresnel equations describe the intensity of the reflected and transmitted fractions.

In this video we will learn which properties of light and matter affect the intensity of the transmitted and reflected light waves.

We will introduce the concept of light polarization.

Then we will discuss the fresnel equations and, finally, we will apply them to improve our solar cells.

Let's look into the parameters that affect the fractions of light that are transmitted and reflected.

This figure shows a light beam incident on the interface of two media.

The refractive indices of both media, that describe how easily light propagates through a material, play an important role.

The angle of incidence is also an important parameter.

With the refractive indices and the angle of incidence we can calculate the angle of transmission, according to Snell’s law.

However, these are not the only important properties.

The polarization of the incident light also affects the reflection and transmission.

As we recall, electromagnetic waves propagate through an oscillation in an electric field and an oscillation in a magnetic field.

The two fields are perpendicular to each other.

The polarization of light refers to the plane of the electric field by convention.

When the plane of the electric field of an incident light wave is parallel to the plane of incidence, the light wave is said to be P-polarized.

When the plane of the electric field is perpendicular to the plane of incidence however, the light wave is said to be S-polarized.

The S stands for senkrecht, which is the German word for perpendicular.

When we calculate the intensity of reflected and transmitted light, the polarization will be an important factor, so keep this concept in mind as we move forward.

A fraction of the intensity of an incident light beam is reflected at the interface of two media.

This fraction is called the reflectivity and is denoted by a capital R.

The fraction of light intensity that is transmitted is called the transmissivity, and is denoted by a capital T.

Since the full intensity of the incident light beam is either reflected or transmitted, the sum of the reflectivity and transmissivity is one.

Augustin-Jean Fresnel derived equations for the reflection coefficient and the transmission coefficient, from which the reflectivity and transmissivity can be calculated.

For convenience's sake, we will focus our discussion of the fresnel equations on the reflection.

A full derivation and description of the fresnel coefficients, and the transmissivity, is provided in the text file following this video.

The reflection coefficient is defined as the ratio of the electric field strength of the reflected wave, to the electric field strength of the incident wave.

The P-polarized reflection coefficient is a function of the product of n1 and the cosine of the angle of transmission and the product of the n2 and the cosine of the angle of incidence.

As we can see, the reflection coefficient of S-polarized light, so the light perpendicular to the plane of incidence, is slightly different. n1 is multiplied by the angle of incidence rather than the angle of transmission and the reverse is true for n2.

The intensity of an electric field is proportional to the square of the electric field strength.

The reflection coefficient is the fraction of the reflected electric field strength and the reflectivity is the fraction of the reflected light intensity.

We can therefore calculate the reflectivity by taking the square of the reflection coefficient.

The reflection coefficients are not the same for S- and P-polarized light.

As a consequence, both polarizations show different reflective behaviour.

This graph shows the reflectivity as a function of the angle of incidence.

The red line represents S-polarized light and the blue line P-polarized light.

The grey line is simply an average of the two.

This graph holds true only in the case of light going to a material with a higher refractive index, so if n1 is smaller than n2.

There is a significant difference in reflectivity between the two polarizations.

The point where the difference is greatest, is where none of the P-polarized light is reflected.

This happens when the angle of incidence equals the so called Brewster angle.

The Brewster angle is equal to the inverse tangent of n2 over n1.

The brewster angle finds a use in optical material characterization techniques, like ellipsometry.

The reflective behaviour of light is a lot different when we move from a material with a high refractive index to a material with a lower one, so when n1 is greater than n2.

As we encountered in the discussion of Snell’s law, moving towards a low refractive index material gives rise to total internal reflection.

We observe the same difference between S- and P-polarized light.

From the critical angle onwards however all light is fully reflected, regardless of the polarization.

Furthermore, we can observe in the graph that in the case of normal incidence there is no difference between light polarizations In this special case, of normal incidence, the Fresnel equations can be simplified.

This illustration shows the same plane of incidence, but now for normal incidence.

When the angle of incidence is zero degrees, the angles of reflection and transmission become zero degrees as well.

Consequently, all the cosines of the angles can be struck from the Fresnel equations.

The reflectivity of S- and P-polarized light, therefore both become a function of only the refractive indices.

The reflectivity under normal incidence is often used to compare the surface reflectivity of different materials.

We have seen however that for most angles there is a difference between S- and P-polarized reflectivity.

For these angles we have to calculate the total reflectivity.

The sources that enlighten our daily lives, from light bulbs to our favourite life giving star, emit unpolarized light.

This means the light contains equal amount of S- and P-polarized electromagnetic waves.

In order to calculate the total reflectivity of an incident beam, we therefore simply sum the contribution of both polarizations and divide it by two.

The Fresnel equations can be used to choose materials specifically to minimize or maximize the reflectivity.

At the front interface of a solar cell, for example, we want the reflectivity to be minimal.

If we consider the case of normal incidence, the reflectivity and transmissivity are described by these relations Let’s assume the materials to be air and silicon, with a refractive index of 1 and 4,3 respectively.

About 39 percent of light moving from air to silicon will be reflected.

This means that about 61 percent will be transmitted into the solar cell.

Now we introduce an additional layer with a refractive index, denoted by n1, between that of air and silicon.

Light is now reflected at two interfaces , each with a smaller refractive index mismatch than that of air and silicon.

The effective, or total transmissivity is equal to the product of the transmissivity of all interfaces.

The effective reflectivity however is calculated by finding what fraction of the initial light reaches a certain interface and then multiplying it by the reflectivity at that interface.

Summing the contributions of all interfaces will give the effective reflectivity.

This yields the shown equations.

Here the reflectivity is a function of the refractive indices involved and the transmissivity is equal to 1 minus the reflectivity.

If n1 equals 2.07, 77% of the light intensity incident on the additional layer will be transmitted into the silicon layer.

A comparison shows that the introduction of the additional layer, with a refractive index between that of air and silicon, has reduced the total reflection.

While 39% of the light intensity is reflected of the air-silicon interface, the total reflection with the additional layer amounts to 23%.

A film that is introduced at the front of the solar cell, to minimize front reflection by decreasing the refractive index mismatch, is called a Rayleigh film.

The effectiveness of a Rayleigh film depends strongly on its refractive index.

In this graph the refractive index of the Rayleigh film is plotted against the total reflectivity.

The blue line is the total reflectivity, in the case of no Rayleigh film.

As we can see n1 has an optimum value.

This optimum value is given by the geometric mean of the refractive indices of the surrounding media.

In the case of silicon and air, the optimum refractive index for the Rayleigh film is 2.1.

In summary, we defined the difference between S- and P-polarization.

We discussed the Fresnel equations and used them to calculate the reflectivity and transmissivity.

We found that the difference in reflectivity between polarizations is greatest at the Brewster angle.

For normal incidence the reflectivity and transmissivity are the same for both polarizations, and therefore the equations simplify.

We also saw that natural light sources emit unpolarized light, which contains equal amount of S- and P-polarized light.

Finally, we used the Fresnel coefficient to calculate how a Rayleigh film can decrease the reflection at the front of a solar cell.