The Optical Fiber Lighthouse · AI Generator #GEN-E31D8F14
Physics · optics
Problem
A lighthouse uses an unusual optical fiber system to amplify its beam. The lighthouse sits on a cliff 100 m above sea level. Light from the lighthouse lamp enters a horizontal optical fiber of length L=50m and core refractive index n1=1.46, surrounded by cladding with refractive index n2=1.44. The fiber has a circular cross-section with core radius a=25μm and is positioned so that its exit end is at the cliff edge.
A ship's captain observes that when the lighthouse beam exits the fiber, it forms a cone of light. The captain measures that this cone has a half-angle of θ=4.8° in air (refractive index n0=1.00).
Due to weather conditions, the lighthouse keeper notices that only light rays that travel the fiber by undergoing exactly 250 total internal reflections along the entire fiber length make it to the exit with sufficient intensity to be useful.
(a) What is the numerical aperture (NA) of this optical fiber?
(b) What is the path length traveled by the useful light rays inside the fiber core?
(c) At what angle α (measured from the fiber axis) do these useful rays propagate inside the fiber core?