Phase velocity and refractive index

The phase velocity is related to the permittivity and permeability as follows:

$latex \displaystyle v_p = \frac{1}{\sqrt{\epsilon \mu}} $

For our purposes, we’re mostly interested in nonmagnetic materials whose permeability is equal to the permeability of vacuum:

$latex \displaystyle \mu = \mu_0 = 4 \pi \times 10^{-7} H / m $

in SI units.  This isn’t true for all materials, and we’ll note exceptions as they come along.  We are very interested in dielectric materials whose permittivity differs from the permittivity of vacuum, however.  We typically write the permittivity as the product of a relative permittivity and the vacuum permittivity:

$latex \displaystyle \epsilon = \epsilon_r \epsilon_0$

where the vacuum permittivity is:

$latex \displaystyle \epsilon_0 = 8.854 \times 10^{-12} F / m $

 The relative permittivity $latex \epsilon_r$ depends on the material you’re in, and may be a function of position, time, and frequency.  To understand the relative permittivity, we need to describe material polarization.  Briefly, the atoms and molecules composing a material have both positive charges (protons in the nucleus) and negative charges (electrons orbiting in ‘shells’ or clouds), which are pulled apart in space if an electric field is applied.  This charge separation is called ‘polarization’.  The relative permittivity is a measure of how strongly a material’s atoms or molecules polarize (i.e. how much charge separation is achieved) in response to an applied electric field.  As an example, the relative permittivity of water for visible light is 1.8.  The relative permittivity of vacuum is always 1.

As an electromagnetic wave travels through a material, the electric field of the wave separates the positive and negative charges by a tiny distance (polarizes the material).  Those polarized atoms are electric dipoles, and the ends of the dipoles oscillate as the electromagnetic wave passes [visualization?].  Electromagnetic theory tells us that oscillating dipoles emit electromagnetic waves.  Therefore, the electromagnetic wave is strongly coupled to the atomic dipoles in the material, as the wave creates the dipoles via polarization, and the dipoles emit electromagnetic waves.

When an electromagnetic wave (such as light) traveling in vacuum encounters a dielectric object, let’s say a block of glass, we say the light is incident from vacuum onto glass.  As we’ll see later, some of the light will reflect from the face of the glass, and some will penetrate into the glass.  The portion of the electromagnetic wave that enters the glass is replaced by a new electromagnetic wave traveling at a lower speed, because the permittivity of the glass is higher than that of vacuum.  It is somewhat intuitive that light will travel more slowly as the permittivity increases, because the material polarizes more strongly in response to the wave.

A little more information here, click if curious.

 How does the wave get ‘replaced’ inside the glass?  Well, the incoming electromagnetic wave induces polarization in the atoms composing the glass, which in turn emit two new electromagnetic waves: one that exactly cancels the incoming wave inside the glass, and another that travels at the slower speed (the phase velocity calculated using the glass’s relative permittivity).  This is called the Ewald-Oseen extinction theorem, and is derived in Born and Wolf.  When I get enough time I’ll put together a little Green’s function-based derivation for normally incident light.

The phase velocity of an electromagnetic wave (such as light) in vacuum is a constant independent of frequency:

 $latex \displaystyle v_p^{vac}  = \frac{1}{\sqrt{\epsilon_0 \mu_0}} \equiv c = 3.0 \times 10^{8} m/s $

In optics, it is more common to use the refractive index rather than the relative permittivity, which for nonmagnetic materials may be defined as

$latex \displaystyle n = \sqrt{\epsilon_r}$

Using these new symbols, we see that the phase velocity of light in a material may be written as

$latex \displaystyle  v_p = \frac{c}{n} $

In summary: when a time-harmonic electromagnetic wave enters a material, its phase velocity is reduced by a factor called the refractive index. What else changes about the wave?  We saw in the last section that

$latex \displaystyle \lambda \nu = \frac{c}{n}$

Since the right hand side of this equation changes when going from vacuum into a dielectric material (such as glass), it follows that the left hand side must change as well.  It turns out that the frequency $latex \nu$ of the wave does not change, but the wavelength$latex \lambda$ does.  Logically, since the right hand side of the equation is reduced by a factor $latex n$, and the frequency $latex \nu$ is unchanged, the wavelength $latex \lambda$ must be reduced by a factor $latex n$:

$latex \displaystyle \lambda = \frac{\lambda_0}{n}$

where $latex \lambda_0$ is the free-space wavelength, i.e. the wavelength of the wave in vacuum, whereas $latex \lambda$ is the actual wavelength of the wave in whatever material it’s passing through.