Numerical Aperture on The Fiber Optic (2)

2. Total Internal Reflection

As it has been discussed in the first part that, the incident light coming from one medium to another as it strikes the boundary with certain angle, will be refracted in definite manner according to the n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2}. Let’s just start this section by increasing incident angle of light in figure 2, first part. This is what will happen if the incident angle is increased:

Figure 3. Refraction phenomenon, with n1 is smaller than n2
Figure 3. Refraction phenomenon, with n1 is smaller than n2

As we know from the behavior described in the first part, we observed the light being refracted in increasing manner when the angle of incident light is increased as well. It is less likely that we can notice something intriguing in here.

How about if medium one has larger value than medium two (n_{1}>n_{2})? We should have light will have lower velocity in medium 1 than in medium 2 (v_{1} < v_{2}) and thus angle of light with respect to the normal in medium 1 is smaller than in medium 2 (\theta_{1}<\theta_{2}). As we increase the angle of incident in medium 1, we will this result:

Figure 4. Refraction phenomenon evolves to total internal reflection, with n1 larger than n2
Figure 4. Refraction phenomenon evolves to total internal reflection, with n1 larger than n2

Total internal reflection is depicted in the right illustration of figure 2. In this phenomenon, the medium boundary acts as a reflector, instead of refractor under two conditions:

  • Light propagates from medium with higher refractive index to the medium with lower refractive index (n_{1}>n_{2}).
  • The angle of incident light has to satisfy minimum of certain angle, i.e. critical angle \theta_{c}.

Total internal reflection can be considered when the refracted light is in parallel direction with the interface (figure 2, middle illustration), i.e. \theta_{2}=90^{\circ}. Using Snell’s law equation described earlier in the first part and two conditions stated above, we can quantify the required incident angle so that it can trigger total internal reflection:

n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2}

n_{1} \sin \theta_{c}=n_{2} \sin 90^{\circ}

n_{1} \sin \theta_{c}=n_{2}

\sin \theta_c=\frac{n_2}{n_1}

\theta_c=\sin^{-1} \frac{n_2}{n_1}

Total internal reflection is occured when the incident angle fulfills the critical angle whose value is determined by the medium 1 and 2, as it is expressed in the last formula of the above equations, called the critical angle equation:

\theta_c=\sin^{-1} \frac{n_2}{n_1}

This equation establishes the requirement for total internal reflection to take place: medium 2 must have refractive index smaller than medium 1. If it happened in opposite way, there will be no solution for this equation, meaning that total internal reflection does not take place.

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