AIIMS 2005 Physics Optical Fibres MCQ Question

To determine the maximum acceptance angle at the air-core interface of an optical fiber, we first need to understand the refractive indices involved and the concept of total internal reflection.

Key Concepts

1. Refractive Indices:

   • Let $n_1$ be the refractive index of the core (the central part of the fiber).
   • Let $n_2$ be the refractive index of the cladding (the outer part surrounding the core).
   • The air surrounding the fiber has a refractive index of approximately $n_0 = 1$.

2. Total Internal Reflection:

   • Light is guided through the core of the fiber due to total internal reflection, which occurs when light travels from a medium of higher refractive index (core) to a medium of lower refractive index (cladding).
   • The critical angle $\theta_c$ for total internal reflection can be calculated using Snell's law.

Calculation of the Maximum Acceptance Angle

The critical angle $\theta_c$ is given by the formula:

$$\sin(\theta_c) = \frac{n_2}{n_1}$$

Here, $n_2 < n_1$ ensures that light can be totally internally reflected at this angle. Rearranging gives:

$$\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)$$

The maximum acceptance angle $\theta_{max}$ at the air-core interface can be related to the critical angle. The acceptance angle is the angle at which light can enter the fiber and still be guided through it. This angle is given as:

$$\sin(\theta_{max}) = \frac{n_2}{n_1}$$

Solution Using the Correct Answer

To find the expression for $\theta_{max}$:

1. We have established that the maximum acceptance angle corresponds to the critical angle when light enters from air into the core.
2. The maximum acceptance angle can be calculated as:

$$\theta_{max} = \sin^{-1}\left(\frac{n_2}{n_1}\right)$$

However, we want the angle specifically at the air-core interface. Here, the incidence angle when light enters from air is taken into account, and we know:

$$\sin(\theta_{max}) = \sqrt{n_1^2 - n_2^2}$$

Thus, the correct answer is: B) $\sin^{-1} \sqrt{n_1^2 - n_2^2}$

Why Other Options Are Incorrect

• Option A: $\sin^{-1}(μ₀μₙ₁)$

  • This expression is unrelated to the acceptance angle in optical fibers. It does not correctly relate to the refractive indices involved.

• Option C: $\tan^{-1}(n₂/n₁)$

  • The tangent function does not apply to finding the acceptance angle directly in this context. The correct relationship involves the sine function, as shown earlier.

• Option D: $\tan^{-1}(n₁/n₂)$

  • While it is a valid mathematical expression, it does not relate to the acceptance angle in optical fibers. The acceptance angle is derived from the sine of the critical angle, not the tangent.

Conclusion

Thus, the maximum acceptance angle at the air-core interface of an optical fiber, based on the refractive indices of the core and cladding, is correctly expressed by option B), which accounts for the necessary conditions for total internal reflection and the relationship between the refractive indices.