page_title: Numerical aperture
section_title: Fiber optics
hierarchical_section_title: Numerical aperture / Fiber optics
context_page_description: In optics, the numerical aperture of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the property that it is constant for a beam as it goes from one material to another, provided there is no refractive power at the interface. The exact definition of the term varies slightly between different areas of optics. Numerical aperture is commonly used in microscopy to describe the acceptance cone of an objective, and in fiber optics, in which it describes the range of angles within which light that is incident on the fiber will be transmitted along it.
context_section_description: A multi-mode optical fiber will only propagate light that enters the fiber within a certain range of angles, known as the acceptance cone of the fiber. The half-angle of this cone is called the acceptance angle, θₘₐₓ. For step-index multimode fiber in a given medium, the acceptance angle is determined only by the indices of refraction of the core, the cladding, and the medium:
  where n is the refractive index of the medium around the fiber, ncore is the refractive index of the fiber core, and nclad is the refractive index of the cladding. While the core will accept light at 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. The derivation of this formula is given below.
  When a light ray is incident from a medium of refractive index n to the core of index ncore 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 θc for sin θᵣ 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 (NA) in other optical systems, so it has become common to define the NA of any type of fiber to be
  where ncore 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.
caption: A multi-mode fiber of index n1 with cladding of index n2.