Numerical Aperture on The Fiber Optic (3)

3. Fiber optic: application of total internal reflection

The first and second part should be enough to provide illustration of how the fiber optic works: first, it depends heavily on the Snell’s law in governing how the light behaves after passing the interface of two materials with different refractive index number. Second, total internal reflection phenomenon is established owing to the fact that light coming at certain angle, defined as critical angle, is able to have reflection instead of refraction, as the light comes from the denser medium to less denser medium (n_{1}>n_{2}). Mathematical expression for the first and second part are described in here and here, respectively.

Now, we can go back to the figure 1. First, we need to realize that there are three different refractive index, as depicted below:

Figure 5. Fiber optic with three different refractive indexes
Figure 5. Fiber optic with three different refractive indexes

Refractive index for air (where, later, the source of light-laser- comes from), core and cladding of fiber optic are designed as it is so that light with certain range of angle, can be guided (controlled) from one point to another. To get an idea of the value of critical angle need to be fulfilled, we can start first inside the fiber optic, in the interface between core and cladding. We can start describe it as it in the figure 4 in the right picture. To have total internal reflection means that refractive index of core must be larger than refractive index of cladding (n_{2}>n_{3}).

Figure 6. Total internal reflection in the interface between core and cladding
Figure 6. Total internal reflection in the interface between core and cladding

Then we follow the same procedure as in the second part,

tir

Next, we can determine the correlation between air and core of the fiber optic (figure 7). Since total internal reflection in this boundary is not needed (or precisely, avoided), we can proceed with the refractive index of air smaller than core of the fiber optic (n_{1}<n_{2}).

Figure 7. Incident light and its refraction in the interface between air and core of fiber optic
Figure 7. Incident light and its refraction in the interface between air and core of fiber optic

Applying Snell’s law, we can calculate the incident angle required to achieve total internal reflection,

n_1 \sin \theta_1 = n_2 \sin \theta_4

We do not know \sin \theta_4 but we do know the value of \sin \theta_2 or \sin \theta_{c(i)}. Based on the figure 7, we can figure the value of \theta_4 from ordinary trigonometry case,

Figure 8. Trigonometry case to find out the value of Θ4
Figure 8. Trigonometry case to find out the value of Θ4

Mathematically, $\sin \theta_{4}$ is equal to $\sin \left( \frac{\pi}{2}-\theta_2 \right)$. To re-define $\sin \theta_{4}$, trigonometry identity of $\sin (\frac{\pi}{2}-\theta)=\cos \theta$ can be utilized. Then the continuation of the last equation is,

NA

There it is, we find already the incident angle required to conduct total internal reflection in the fiber optic. {\textit{Numerical aperture (NA)} is defined as the qualitative limitation of receiving angle of the fiber optic with respect to the incident angle of light. Therefore, the numerical aperture for fiber optic can be defined as follows:

NAresult

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