The Fiber Optic Periscope · AI Generator #GEN-2081BAE1
Physics · optics
Problem
A submarine engineer designs a simple periscope using a bent optical fiber. The fiber has a circular cross-section with core radius r=50μm and is made of glass with refractive index n1=1.46. The fiber is surrounded by a cladding material with refractive index n2=1.44 and is bent into a U-shape with radius of curvature R=2.0cm at the bend. Light enters the straight section of the fiber at various angles.
For the periscope to work effectively, light must successfully propagate through both the straight section and around the curved bend without escaping. In the straight section, total internal reflection occurs at the core-cladding interface. However, when the fiber bends, the geometry changes and some light rays that were previously trapped may escape.
(a) What is the maximum acceptance angle θmax (measured from the fiber axis) for light entering the straight section of the fiber in air?
(b) For a light ray entering at angle θmax that successfully propagates through the straight section, what is the angle α this ray makes with the fiber axis when it reaches the beginning of the curved section?
(c) When this ray travels around the curved section, at what radial distance rcritical from the center of curvature does total internal reflection just begin to fail? Express your answer in terms of the bend radius R.