# Numerical Aperture on The Fiber Optic (1)

Yesterday, I re-arranged my bookshelf containing my notes I had made during Master course in NTNU. I felt nostalgic with these notes and they brought me back to the time when I tried to write them as neat as possible so that they can be useful (readable) for me in the future. I thought that it must be better to document them digitally to prevent these notes to be permanently damaged when they are just in the form of paper.

In the following note, I will write the study on the numerical aperture of fiber optic. Fiber optic is a cable made of glass (or plastic) threads (fibers), used for transmitting data in guided mode (for example: internet connection) utilizing light waves. We can see how it looks like in figure 1. I find the concept of fiber optic is very interesting because it uses simple optical phenomenon in our daily life. Surprisingly, the basic knowledge for explaining the guided-light has been attained in junior high school. Unfortunately, what we learnt during that time was only simple phenomenon without realizing there is an advanced technology depending on it.

I will divide this explanation into three sections: first we deal with the Snell’s law. This is quite basic of refraction phenomenon, but I think it is very important to grasp the fundamental part. Next, we will move to the total internal reflection, where we will make use of the trait of Snell’s law to control the light. Finally, the last part will deal with fiber optics, by utilizing total internal reflection property.

# 1. Snell’s law

The basic law governing guided transmitted light in fiber optic is Snell’s law. This law deals with light coming from point P one media (with refraction index 1, $n_{1}$) with certain angle relative to the normal axis ${\theta_1}$, to point Q another media (with refraction index 2, $n_{2}$) where the light will have change its angle relative to the normal axis ${\theta_2}$ (figure 2). Alteration of light direction (refraction) is begun at the interface between two different media. We can meet with this phenomenon daily by observing how the straw in the glass of water looks bent or the strange form of stair in the swimming pool.

How does direction of light change? Roughly speaking because of the atomic structure composing the media. The mass of the atom will create certain density in that media leading to the specified interaction between light and the atom and it will determine speed of light penetrating in that media. We can say that refraction characteristics of material are varying from one to another. Therefore, in figure 2, propagation of light with speed of ${v_1}$ in media ${n_1}$ changes its speed to ${v_2}$ in media ${n_2}$. Since there is an alteration of speed meaning there is a slightly change in the direction of light, relative to the normal axis, and in this case is media with ${\theta_1}$ to ${\theta_2}$.

To get rough qualitative measurement of the angle in each media, we can use this rule of thumb (based on figure 2: the incidence angle in medium 1 is larger than the refraction angle in medium 2 (${\theta_1} > {\theta_2}$) shows that light propagate faster in medium 1 than in medium 2 (${v_1} > {v_2}$). Since light propagate faster in medium 1 compared to medium 2, we can conclude that refraction index in medium 1 is smaller than medium 2 (${n_1} < {n_2}$). The denser the material is, the higher is the value of refractive index and the light will propagate slower depicted with the smaller angle of refraction (it will move closer to the normal axis).

For the case of refraction index in medium 1 is larger than medium 2 (${n_1} > {n_2}$), the result will show exactly in the reverse way: light will propagate slower in medium 1 than in medium 2 (${v_1} < {v_2}$) due to the fact that medium 1 is denser than medium 2 and this will lead to the incidence angle in medium 1 is smaller than the refraction angle in medium 2 (${\theta_1} < {\theta_2}$).

Based on the relationship we have established so far, this part will be closed with the formula between angle of incidence ${\theta_1}$ and refraction ${\theta_2}$, speed v and refraction index n for medium 1 and 2,

$\frac{\sin \theta_1}{\sin \theta_2}=\frac{v_1}{v_2}=\frac{n_2}{n_1}$