The Lighthouse Lens Paradox · AI Generator #GEN-7D4B1139
Physics · optics
Problem
A coastal lighthouse uses a Fresnel lens system to focus its beam. The lighthouse keeper notices something peculiar: when fog rolls in with tiny water droplets of radius r=2.0μm, the lighthouse beam appears to "bend backward" at a specific angle.
The lighthouse emits light at wavelength λ=589nm (sodium D-line). The fog droplets have refractive index n=1.33 and are spherical. When the keeper measures the scattered light intensity, she finds that there is a sharp minimum in the backward direction at angle θ=165° from the forward direction.
This phenomenon occurs because the fog droplets act as tiny optical cavities that support whispering gallery modes - electromagnetic waves that circulate around the droplet's circumference by total internal reflection, similar to acoustic whispering galleries in circular buildings.
For a whispering gallery mode to exist, the optical path length around the droplet's circumference must equal an integer multiple of the wavelength inside the droplet. The intensity minimum occurs when destructive interference happens between the direct scattered light and light that has completed exactly one circulation around the droplet.
(a) Find the effective circumference path length that light travels when it circulates once around the droplet at the critical angle for total internal reflection.
(b) Determine how many wavelengths fit around this optical path, and verify this explains the observed scattering minimum at θ=165°.