A medical endoscope uses a flexible optical fiber whose core has index n1=1.48 and whose cladding has index n2=1.44. The light entering the fiber is confined to a half-angle of divergence of 8∘ with respect to the fiber axis.
When the fiber bends, rays follow circular arcs inside the core. Excessive curvature can cause some rays to strike the core–cladding interface at angles smaller than the critical angle, allowing light to escape.
Assume the worst-case ray is launched at the maximum entrance divergence angle and then follows a circular path of radius R centered on the axis of curvature.
(a) Derive an expression for the largest angle θint that this worst-case ray can make with the normal to the core–cladding interface in a straight section of fiber.
(b) Establish a relationship between R and θint by considering the geometry of the bent path and the cylindrical core of radius rc=62μm. Clearly state any approximations you make.
(c) Using the requirement that total internal reflection be maintained at every point along the bend, calculate the minimum allowable bend radius Rmin for the fiber.
(d) Discuss briefly one practical implication of your result for the design or handling of medical endoscopes.