## Refractive Index Explained: Snell's Law & Maxwell's Equations

The refractive index is the factor by which an electromagnetic wave slows down inside of a material relative to the speed of light in vacuum \(c_0 =\)299,792,458 m/s. This means the speed \(v\) of an electromagnetic wave inside of a medium with refractive index \(n\) is

\(v = c_0/n\)

Eq. (1)

For example, if the refractive index of a material is 2.0, then the speed of the wave is half of the speed of light in vacuum.

The refractive index is an important parameter in optics and electromagnetics. One common use of the refractive index is calculating the angle of a refracted wave. Suppose a wave propagating in a medium with refractive index \(n_1\) encounters a flat interface into a second medium that has a refractive index \(n_2\). If that wave encounters that interface at an angle \(\theta_1\), the angle \(\theta_2\) of the wave on the other side of the interface (i.e. refracted wave) is calculated using Snell’s law.

\(n_1 \sin \theta_1 = n_2 \sin \theta_2\)

Eq. (2)

Snell’s law and refraction is the key mechanism behind operation of lenses, light guiding in optical fibers, appearance of rainbows, and much more. The above discussion is usually how refractive index is defined and described, but this gives little insight about how materials exhibit a refractive index due to atomic scale mechanisms, how refractive index is related to the more fundamental electromagnetic parameters (permittivity and permeability), and what it means to have a complex refractive index.

### How does a refractive index happen?

At the atomic scale, the electromagnetic fields put mechanical forces on the charges (i.e. electrons and protons) that displace them. However, the negatively charged electrons are bound to the atoms due to the electrostatic attraction to the positively charged protons. The electrostatic attraction makes the system act like a mass on a spring where the applied wave stretches the spring and the electrostatic attraction is the restoring force of the spring. Like a mass on a spring, the displacement of the electrons away from their equilibrium positions is a resonant phenomenon.

When charges displace, they accelerate or decelerate and radiate tiny secondary waves. The secondary waves are at the same frequency as the applied wave, but they are out of phase due to the resonance, so the secondary waves interfere with the applied wave. If the interference between the applied wave and all the secondary waves is averaged, and we pretend the overall wave is still just the applied wave, it looks as if the applied wave has slowed down. Interestingly, the individual electromagnetic waves always travel at the speed of light in vacuum.

### How does refractive index relate to permittivity and permeability?

James Clerk Maxwell did not invent or discover what today we call Maxwell’s equations. Instead, he unified the equations from other researchers. This let him for the first time derive the electromagnetic wave equation.

\(\nabla^2 \vec{E} + \omega^2 \mu \varepsilon \vec{E} = 0\)

Eq. (3)

In this equation, the del operator \(\nabla\) is just a three-dimensional spatial derivative, \(\omega\) is the angular frequency, \(\mu\) is the permeability of the medium, \(\varepsilon\) is the permittivity of the medium, and \(\vec{E}\) is the electric field intensity. A long time before Maxwell, the general wave equation was known.

\(\nabla^2 \psi + \left( \frac{\omega}{v} \right)^2 \psi = 0\)

Eq. (4)

In this equation, \(v\) is the speed of the wave and \(\psi\) is the disturbance. The disturbance could be anything from displacement of a point on a string, to pressure, an electric or magnetic field, or other things. Maxwell compared equations (3) and (4) and realized that \(\omega^2\mu\varepsilon = \left( \frac{\omega}{v} \right)^2\). This let Maxwell derive an equation to calculate the speed \(v\) of an electromagnetic wave from the fundamental electromagnetic parameters.

\(v = 1/\sqrt{\mu \varepsilon}\)

Eq. (5)

Very often, the permittivity is written as \(\varepsilon = \varepsilon_0 \varepsilon_{\textrm{r}}\), where the permittivity of vacuum is \(\varepsilon_0 = 8.8541878128 \times 10^{-12}\) F/m. The term \(\varepsilon_{\textrm{r}}\) is the relative permittivity, also called dielectric constant. The relative permittivity has no units and is the more convenient term to communicate permittivity through because it is generally in the range \(1 \leq \varepsilon_{\textrm{r}} \leq 20\). Similarly, the permeability can be written as \(\mu = \mu_0 \mu_{\textrm{r}}\) where the vacuum permeability is \(\mu_0 = 1.25663706212 \times 10^{-6} \) H/m. The term \( \mu_{\textrm{r}} \) is the relative permeability and is also a more convenient term to communicate permeability. In vacuum, \( \mu = \mu_0 \) and \( \varepsilon = \varepsilon_0 \) and the speed of light is

\(c_0 = 1/\sqrt{\mu_0 \varepsilon_0}\)

Eq. (6)

Now, Eq. (5) can be expanded to

\( v = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \frac{1}{\sqrt{\mu_{\textrm{r}} \varepsilon_{\textrm{r}}}} \)

Eq. (7)

From Eq. (6), it is clear that the \(1/\sqrt{\mu_0 \varepsilon_0}\) term in Eq. (7) is the speed of light \(c_0\). This lets Eq. (7) be written as

\( v = \frac{c_0}{\sqrt{\mu_{\textrm{r}} \varepsilon_{\textrm{r}}}} \)

Eq. (8)

Inspecting Eq. (8) shows that the term \(\sqrt{\mu_{\textrm{r}} \varepsilon_{\textrm{r}}}\) is the factor by which the wave slows down relative to the speed of light in vacuum. This was the definition of refractive index, so

\(n = \sqrt{\mu_{\textrm{r}} \varepsilon_{\textrm{r}}}\)

Eq. (9)

Equation (9) is how the refractive index is related to the permeability and permittivity of a medium.

### How is refractive index a complex number?

We live in the real world where numbers are always just purely real. At some point, however, mathematicians discovered we can greatly simplify the math involved when analyzing systems operating a single frequency by using complex numbers. This is commonly called time-harmonic, or frequency-domain, analysis.

When we analyze electromagnetics or optics in the frequency domain, most quantities become complex in order to convey phase and amplitude. So what is a complex refractive index? Let’s experiment with the math to find out.

A wave propagating in the \(+ z\) direction can be written in the frequency-domain as \(E(z) = E_0 \exp \left( -k k_0 n z \right)\) where \(k_0 = 2 \pi / \lambda_0 \) is the vacuum wave number and \(\lambda_0\) is the vacuum wavelength. A complex refractive index can be written in terms of its real and imaginary parts as

\(n = n_{\textrm{o}} – j \kappa\)

Eq. (10)

The real part \(n_{\textrm{o}}\) is called the ordinary refractive index and the imaginary part \(\kappa\) is called the extinction coefficient. Putting this into the frequency-domain equation for a wave leads to

\(E(z) = \exp \left( -j k_0 n_{\textrm{o}} z \right) \exp \left( -k_0 \kappa z \right)\)

Eq. (11)

Physical meaning about the real and imaginary parts can be obtained by inspecting the right side of Eq. (11). The ordinary refractive index acts much like how we thought refractive index behaved before we considered it as a complex quantity. The ordinary refractive index determines the speed of a wave, or how fast it oscillates as a function of position. The extinction coefficient determines how quickly a wave decays with distance. A wave decays when a medium has absorption loss, so the extinction coefficient quantifies loss. A negative extinction coefficient describes gain.