# Plasma Optics (IV)

To start this section, we need the wave equation expression, which we have derived before (The Wave Equation):

where u is wave function and c is the speed of light in medium, defined by

Meanwhile, we know that the relationship between refractive index n has connection with the speed of light in medium by this expression:

We have the speed of light, dielectric function and magnetic permeability in free space, $c_0, \epsilon_0, \mu_0$, respectively. For nonmagnetic material, the value of $\mu=\mu_0$, so that we have the simplified expression:

From this equation, we can say that negative dielectric function will give imaginary value of refractive index. In the other way around, positive dielectric function will give positive refractive index. We hope to come back to this section in the next section.

Let us come back to the expression of the wave equation and wave function. We can get the dispersion relation for electromagnetic waves by inserting equation wave function u into wave equation,

The last equation is dispersion relation and it tells us two important things:

• If the dielectric function, $\epsilon$ is real and positive, the wavevector k is real and positive.
• If the dielectric function, $\epsilon$ is real and negative, the wavevector k is imaginary.

Ok, now we can apply these conditions to the wave function. As a note, absorption is negligible in the following case

First condition is when the dielectric function, $\epsilon$ is real and positive, we will have real and positive value of wavevector k. Let’s say that the value of the k is 1, then

The result of plotting the wave function with real and positive k as a function of distance is shown below:

Figure 1 shows us how light will propagate through the material (grey colored). Since light can propagate through material, it will become transparent. This happens when the dielectric function is positive, i.e. incoming light has wavelength smaller than the plasma wavelength of the respected material (see Plasma Optics (III)), as we expected. Since there is no absorption from the material (from the assumption), the amplitude of light propagating inside the material has the same value of the light propagate outside of the material, and it will look transparent in the infinity. This will be different if we assume that the material has ability to absorb the light, where it will be only reaching certain distance.

The second condition is when the dielectric function, $\epsilon$ is real and negative, we will have imaginary value of wavevector k. Let’s say that the value of the k is $\sqrt{-1}$, then

The result of plotting the wave function whose the value of k is imaginary as a function of distance is shown below:

Figure 2 shows us how light will be reflected through the material (grey colored). Physically speaking from the figure 2, the light is not reflected, but rather absorbed. Therefore the material becomes solid. This happens when the dielectric function is negative, i.e. incoming light has wavelength higher than the plasma wavelength of the respected material (see Plasma Optics (III)), as we expected. We can say that with the negative value of dielectric function, the wave of light is damped directly in the surface of the material. This is not the same with the first condition (with the assumption where the material has absorption coefficient), the light can propagate to some certain distance before it is vanished.

How big is the influence of the absorption of the material to the light propagation? We will discuss this in the next section.