Question

In a step index fiber, the index of refraction undergoes a discontinuity (jump) at the core-cladding boundary, as shown in the figure. Infrared light with wavelength \(1550 \mathrm{nm}\) propagates through such a fiber by total internal reflection at the corecladding boundary. The index of refraction of the core for the infrared light is \(n_{\text {core }}=1.48\). If the maximum angle, \(\alpha_{\max },\) at which light can enter the fiber with no light lost into the cladding is \(\alpha_{\max }=14.033^{\circ},\) calculate the percent difference between the index of refraction of the core and the index of refraction of the cladding.

Short answer

Answer: The percent difference between the index of refraction of the core and the index of refraction of the cladding is approximately \(0.95\%\).

Step-by-step solution

Find the critical angle for total internal reflection

Using Snell's Law, we can write: $$n_{\text {core}} \times \sin \theta_c = n_{\text {clad}} \times \sin 90^\circ$$ Since \(\sin 90^\circ = 1\), we can rewrite the equation as: $$\theta_c = \sin^{-1}\left(\frac{n_{\text {clad}}}{n_{\text {core}}}\right)$$

Calculate the critical angle using the maximum entrance angle

Now, we can use the given maximum angle (\(\alpha_{\max}\)) to find the critical angle (\(\theta_c\)). The relationship between \(\alpha_{\max}\) and \(\theta_c\) is: $$\alpha_{\max} + \theta_c = 90^\circ$$ We can now solve for the critical angle: $$\theta_c = 90^\circ - \alpha_{\max} = 90^\circ - 14.033^\circ = 75.967^\circ$$

Calculate the index of refraction for the cladding

Using the critical angle (\(\theta_c\)) and the index of refraction of the core (\(n_{\text {core }}\)), we can now find the index of refraction for the cladding (\(n_{\text {clad }}\)): $$n_{\text{clad}} = n_{\text{core}} \times \sin\theta_c = 1.48 \times \sin75.967^\circ \approx 1.466$$

Calculate the percent difference between the index of refraction of the core and the cladding

Finally, we will find the percent difference between the index of refraction of the core (\(n_{\text {core }}\)) and the index of refraction of the cladding (\(n_{\text {clad}}\)): $$\text{Percent Difference} = \frac{n_{\text{core}} - n_{\text{clad}}}{n_{\text{core}}} \times 100 = \frac{1.48 - 1.466}{1.48} \times 100 \approx 0.95 \%$$ Therefore, the percent difference between the index of refraction of the core and the index of refraction of the cladding is approximately \(0.95\%\).