A step index fibre has a relative refractive index difference of 0.86% Estimate the critical angle at the core-cladding interface

Answer

To find the critical angle at the core-cladding interface of a step index fiber with a given relative refractive index difference of 0.86%, we can follow these steps:

### Step 1: Understand the Relative Refractive Index Difference
The relative refractive index difference (Δ) is given by the formula:
\[ \Delta = \frac{n_1 - n_2}{n_1} \times 100 \]
where \(n_1\) is the refractive index of the core and \(n_2\) is the refractive index of the cladding.

### Step 2: Convert the Relative Refractive Index Difference to a Decimal
Given that Δ = 0.86%, we can express this as:
\[ \Delta = 0.0086 \]

### Step 3: Rearrange the Formula to Find \(n_2\)
From the relative refractive index difference formula, we can rearrange it to find \(n_2\):
\[ n_1 - n_2 = \Delta \cdot n_1 \]
This can be rewritten as:
\[ n_2 = n_1 - \Delta \cdot n_1 = n_1(1 - \Delta) \]
Substituting \(\Delta\):
\[ n_2 = n_1(1 - 0.0086) = n_1 \cdot 0.9914 \]

### Step 4: Use Snell's Law to Find the Critical Angle
The critical angle (\(\theta_c\)) at the core-cladding interface can be calculated using Snell's law:
\[ \sin(\theta_c) = \frac{n_2}{n_1} \]
Substituting \(n_2 = n_1 \cdot 0.9914\):
\[ \sin(\theta_c) = \frac{n_1 \cdot 0.9914}{n_1} = 0.9914 \]

### Step 5: Calculate the Critical Angle
To find the critical angle, we take the inverse sine:
\[ \theta_c = \sin^{-1}(0.9914) \]
Using a calculator:
\[ \theta_c \approx 82.48^\circ \]

### Final Answer
The critical angle at the core-cladding interface is approximately \(82.48^\circ\).

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