Numerical Aperture (NA), a measure of the acceptance angle of a fiber, is a dimensionless quantity. For applications, it is most commonly expressed as
is the maximum 1/2 acceptance angle of the fiber,
is the index of refraction of the material outside of the fiber; this material is typically air, making it equal to approximately 1.0, as shown in the figure below.
Figure: A ray at the maximum 1/2 acceptance angle propagates into a fiber.
Numerical Aperture can also be defined in terms of the index of refraction of the fiber core and the cladding. Due to Snell’s law, there is a critical angle above which all of the light at a fiber-cladding interface will be totally internally reflected. In turn, this means that there is a maximum acceptance angle at which light can enter the fiber. Following Snell’s law, the maximum acceptance angle can be determined:
is the index of refraction of the fiber core,
is the index of refraction of the cladding,
is the critical angle for total internal reflection,
is the maximum 1/2 acceptance angle.
This is the common way numerical aperture is defined for optical fibers. It is also important to note that these equations assume that a Gaussian beam is being outputted from the fiber.