Refractive Index

Imagine you are going for a run along a smooth dirt path, and you approach a waist-deep river. You need to cross the river and don't want to slow your run, so you decide to press forward through it. As you enter the water, you try to maintain the same speed as before, but quickly realize that the water is slowing you down. Finally, making it to the other side of the river, you pick up the same speed as before and continue with your run. In the same way that the speed of your run decreased as you ran through the water, optics tells us that the propagation speed of light decreases as it travels through different materials. Every material has a refractive index that gives the ratio between the speed of light in the vacuum and the speed of light in the material. The refractive index allows us to determine the path a light beam will take as it travels through the material. Let's learn more about the refractive index in optics!

Fig. 1 - Water slows down a runner like different materials slow down the propagation speed of light.

Definition of Refractive Index

When light is traveling through a vacuum, or empty space, the light's speed of propagation is simply the speed of light, \(3.00\times10^8\mathrm{\frac{m}{s}}.\) Light travels slower when it goes through a medium such as air, glass, or water. A light beam passing from one medium to another at an incident angle will experience reflection and refraction. Some of the incident light will be reflected off of the surface of the medium at the same angle as the incident angle with respect to the surface normal, while the rest will be transmitted at a refracted angle. The normal is an imaginary line perpendicular to the boundary between both media. In the image below, a light ray experiencing reflection and refraction as it passes from medium \(1\) to medium \(2,\) appears in light green. The thick blue line depicts the boundary between both media whereas the skinny blue line perpendicular to the surface represents the normal.

Fig. 2 - A light beam is reflected and refracted as it passes from one medium to another.

Every material has an index of refraction that gives the ratio between the speed of light in a vacuum and the speed of light in the material. This helps us to determine the refracted angle.

A light beam traveling at an angle from a material that has a lower refractive index to one with a higher refractive index will have a refraction angle that bends toward the normal. The refraction angle bends away from the normal when it travels from a higher refractive index to a lower one.

Formula for Refractive Index

The refractive index, \(n,\) is dimensionless since it's a ratio. It has the formula \[n=\frac{c}{v},\] where \(c\) is the speed of light in vacuum and \(v\) is the speed of light in the medium. Both quantities have units of meters per second, \(\mathrm{\frac{m}{s}}.\) In a vacuum, the refractive index is unity, and all other media have a refractive index that is greater than one. The index of refraction for air is \(n_\mathrm{air}=1.0003,\) so we generally round to a few significant figures and take it to be \(n_{\mathrm{air}}\approx 1.000.\) The table below shows the refractive index for various media to four significant figures.

The law of refraction, Snell's law, uses the refractive index to determine the refracted angle. Snell's law has the formula

\[n_1\sin\theta_1=n_2\sin\theta_2,\]

where \(n_1\) and \(n_2\) are the indices of refraction for two media, \(\theta_1\) is the incident angle, and \(\theta_2\) is the refracted angle.

Critical Angle of the Index of Refraction

For light traveling from a medium of a higher index of refraction to a lower one, there is a critical angle of incidence. At the critical angle, the refracted light beam skims the surface of the medium, making the refracted angle a right angle with respect to the normal. When the incident light hits the second medium at any angle greater than the critical angle, the light is totally internally reflected, so that there is no transmitted (refracted) light.

We calculate the critical angle using the law of refraction. As mentioned above, at the critical angle the refracted beam is tangent to the surface of the second medium so that the refraction angle is \(90^\circ.\) Thus, \(\sin\theta_1=\sin\theta_\mathrm{crit}\) and \(\sin\theta_2=\sin(90^\circ)=1\) at the critical angle. Substituting these into the law of refraction gives us:

\[\begin{align*}n_1\sin\theta_1&=n_2\sin\theta_2\\[8pt]\frac{n_2}{n_1}&=\frac{\sin\theta_1}{\sin\theta_2}\\[8pt]\frac{n_2}{n_1}&=\frac{\sin\theta_\mathrm{crit}}{1}\\[8pt]\sin\theta_\mathrm{crit}&=\frac{n_2}{n_1}.\end{align*}\]

Since \(\sin\theta_\mathrm{crit}\) is equal to or less than one, this shows that the refractive index of the first medium must be greater than that of the second for total internal reflection to occur.

Measurements of Refractive Index

A common device that measures the refractive index of a material is a refractometer. A refractometer works by measuring the refraction angle and using it to calculate the refractive index. Refractometers contain a prism upon which we place a sample of the material. As light shines through the material, the refractometer measures the refraction angle and outputs the refractive index of the material.

A common use for refractometers is to find the concentration of a liquid. A hand-held salinity refractometer measures the amount of salt in salt water by measuring the refraction angle as light passes through it. The more salt there is in the water, the greater the refraction angle is. After calibrating the refractometer, we place a few drops of salt water on the prism and cover it with a cover plate. As light shines through it, the refractometer measures the refraction index and outputs the salinity in parts per thousand (ppt). Beekeepers also use hand-held refractometers in a similar way to determine how much water is in honey.

Fig. 3 - A hand-held refractometer uses refraction to measure the concentraion of a liquid.

Examples of the Refractive Index

Now let's do some practice problems for the refractive index!