Optical Fibre Demonstration with a Sharp Bend
A school demonstration uses a step-index plastic optical fibre with core refractive index 1.50 and cladding refractive index 1.48. The fibre is 2.0 m long and is coupled to a laser pointer whose beam enters the flat end at angles up to 25° from the fibre axis. The student bends the fibre into a loop of radius 3 cm; light leaks at the bend and the far end dims. She calculates the critical angle as sin⁻¹(1.48/1.50)=80.6° and concludes that any ray inside the core will be totally internally reflected, so the bend should not matter. The fibre end is cut with a rough blade, and the cladding is accidentally scratched. She can change the launch angle, use a longer wavelength, or make a larger loop.
1. Calculate the fibre's numerical aperture and the maximum launch angle in air for guided rays. Does the 25° beam satisfy it?
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For a step-index fibre, NA = √(n₁²−n₂²) = √(1.50²−1.48²) = √(2.25−2.1904) = √0.0596 = 0.244. The maximum launch angle in air is θ_a = sin⁻¹(NA) = sin⁻¹(0.244) = 14.1°. The laser pointer launches rays up to 25°, which is larger than 14.1°, so many rays do not satisfy the guidance condition. Those rays refract into the cladding at the input end and are lost. The critical angle of 80.6° is the angle inside the core at the core–cladding boundary, not the external launch angle; it cannot be compared directly with 25°. This alone explains part of the dimming.
2. The student says TIR at 80.6° means bending cannot cause loss. Explain why a 3 cm bend can still make light leak.
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In a straight fibre, a guided ray strikes the side wall at an angle greater than the critical angle, so it is totally internally reflected. In a bend, the outer wall is curved and the ray's angle of incidence at that wall changes. A ray that was safely above the critical angle in the straight section can fall below it at the outside of the bend and refract into the cladding. The bend also converts guided modes into leaky or radiation modes. The effect grows as the bend radius decreases, so a 3 cm loop is much lossier than a gentle bend. A rough cut end and a scratched cladding add scattering and absorption, so the far end dims even if the bend alone is not the only cause.
3. Which remedies would you choose and which would you avoid? Consider launch angle, loop radius, wavelength, and end preparation.
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Reduce the launch angle below about 14° by using a lens or a smaller aperture, so that the input rays satisfy the numerical-aperture condition. Make the loop radius much larger than 3 cm, or keep the fibre straight, to avoid bend-induced loss. Re-cut and polish the end squarely, and avoid scratching the cladding; a rough end scatters light before it can be guided. A longer wavelength would generally increase bend loss in this plastic step-index fibre, so it is not the remedy to choose. The priority is proper launch, a gentle bend and a clean end; after that, compare the output power quantitatively rather than judging only by brightness.