Numerical Aperture - Fiber Optics

Fiber Optics

A multi-mode optical fiber will only propagate light that enters the fiber within a certain cone, known as the acceptance cone of the fiber. The half-angle of this cone is called the acceptance angle, θ_max. For step-index multimode fiber, the acceptance angle is determined only by the indices of refraction of the core and the cladding:

where n_core is the refractive index of the fiber core, and n_clad is the refractive index of the cladding. While the core will accept light at higher numerical apertures (higher angles), those rays will not totally reflect off the core–cladding interface, and so will not be transmitted to the other end of the fiber.

When a light ray is incident from a medium of refractive index n to the core of index n_core at the maximum acceptance angle, Snell's law at the medium–core interface gives

From the geometry of the above figure we have:

where is the critical angle for total internal reflection.

Substituting cos θ_c for sin θ_r in Snell's law we get:

By squaring both sides

Solving, we find the formula stated above:

This has the same form as the numerical aperture in other optical systems, so it has become common to define the NA of any type of fiber to be

where n_core is the refractive index along the central axis of the fiber. Note that when this definition is used, the connection between the NA and the acceptance angle of the fiber becomes only an approximation. In particular, manufacturers often quote "NA" for single-mode fiber based on this formula, even though the acceptance angle for single-mode fiber is quite different and cannot be determined from the indices of refraction alone.

The number of bound modes, the mode volume, is related to the normalized frequency and thus to the NA.

In multimode fibers, the term equilibrium numerical aperture is sometimes used. This refers to the numerical aperture with respect to the extreme exit angle of a ray emerging from a fiber in which equilibrium mode distribution has been established.