Optical Fiber Acceptance Angle Calculation

This solution calculates the maximum angle of acceptance for an optical fiber based on the given refractive index values.

Fiber Parameters Identification

• Core refractive index, $n_1 = 1.5$.
• Core/cladding index difference, $\Delta n = n_1 - n_2 = 0.01$.

Numerical Aperture (NA) Calculation

The Numerical Aperture (NA) determines the light-gathering ability of the fiber. A common approximation, especially for small index differences, is used here:

$ NA \approx n_1 \sqrt{2 \Delta n} $

Substitute the given values:

$ NA \approx 1.5 \times \sqrt{2 \times 0.01} $

$ NA \approx 1.5 \times \sqrt{0.02} $

$ NA \approx 1.5 \times 0.14142 $

$ NA \approx 0.21213 $

Acceptance Angle Calculation

The maximum angle of acceptance ($\theta_{max}$) is the maximum angle at which light entering the fiber core from the surrounding medium (usually air, with refractive index $n_{air} \approx 1$) will be propagated through total internal reflection. The relationship is:

$ NA = n_{air} \sin(\theta_{max}) $

Assuming the surrounding medium is air ($n_{air} \approx 1$):

$ NA = \sin(\theta_{max}) $

$ 0.21213 = \sin(\theta_{max}) $

Solving for $\theta_{max}$:

$ \theta_{max} = \arcsin(0.21213) $

$ \theta_{max} \approx 12.25^\circ $

Conclusion

The calculated maximum angle of acceptance is approximately $12.25^\circ$. This value is closest to option B, $12.1^\circ$.