Numerical Aperture (NA), a measure of the acceptance angle of a fiber, is a dimensionless quantity. For applications, it is most commonly expressed as
where θa is the maximum 1/2 acceptance angle of the fiber, and ni 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 cladding. Due to Snell’s law, there is a critical angle above which all of the light at a fiber-cladding interface will experience total internal reflection. 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:
nf is the index of refraction of the fiber core, nc is the index of refraction of the cladding,
θc is the critical angle for total internal reflection, and
θa 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.