A designer wishes to build an underwater optical communication cable. The glass core (index nc=1.62) is coated with a transparent polymer cladding of unknown index np. The cable will operate while fully submerged in seawater (nw=1.33). To guarantee that no guided ray escapes, every internal reflection must satisfy two separate conditions:
1. Total internal reflection at the core–polymer boundary.
2. Total internal reflection at the polymer–water boundary, so that no light leaks from the cladding into the water.
Guided rays may strike the core–polymer interface at angles as small as θmin=30∘ with respect to the normal.
Neglect absorption and assume planar interfaces for parts (a)–(d).
(a) Write an inequality that relates nc, np, and θmin ensuring total internal reflection at the core–polymer interface.
(b) Write a second inequality that must be satisfied at the polymer–water interface for all possible rays emerging from the core into the polymer.
(c) Combine the two inequalities to determine the largest allowable value of np that guarantees confinement of the guided rays.
(d) The polymer manufacturer offers materials with indices 1.40, 1.46, and 1.52. Which of these meet the design specification? Justify your choice quantitatively.