eigenray

An eigenray is a ray path through an optical system that arrives at a predetermined point or detector with a specific set of angular properties. Unlike a general ray that might scatter or miss entirely, an eigenray is one of a discrete set of solutions to the ray equations that satisfy both the starting point (the source) and the ending point (the target) simultaneously. The term comes from the German eigen, meaning characteristic or proper, borrowed from eigenvalue mathematics where solutions satisfy particular boundary conditions.

In imaging systems, eigenrays matter because they represent the actual paths along which light couples into a detector or optical element. A single source can connect to a single target through multiple eigenrays if the geometry permits, each with a different internal angle or number of reflections. In fiber optics and waveguides, eigenrays are the discrete bound modes that propagate without loss; in geometrical optics through complex systems (lenses, mirrors, prisms), they are the ray paths that satisfy both Fermat's principle and the boundary constraints of the system.

Eigenrays versus general rays

A free ray launched from a source in any direction will eventually hit something or escape to infinity. An eigenray is constrained: it must land exactly at the target. This constraint drastically reduces the number of allowed paths. In a laser cavity or resonator, the eigenrays are the standing-wave modes; each one represents a stable path the light repeatedly takes. In a more general optical imaging system, eigenrays describe which ray paths actually contribute light to the image formed at the detector plane.

Computing eigenrays requires solving the ray equations as a boundary-value problem rather than an initial-value problem. This is more demanding numerically than tracing a single ray from source to wherever it goes. The reward is that eigenrays tell you which paths matter physically: they are the ones that carry energy efficiently between source and target, and they determine the throughput and aberration signature of the system.

In atmospheric and underwater optics, eigenrays take on a different meaning: they are the ray solutions through a stratified medium (layers of different refractive index) that connect a source at one depth to a receiver at another. The number and angle of eigenrays depend on the sound or light speed profile, and oceanographers and sonar engineers use eigenray theory to predict signal propagation in variable conditions. Here too the term emphasizes that only certain discrete ray angles solve the full problem of reaching the target through a structured medium.