Numerical aperture
2024-6-10
Definition:
Abbreviation: NA The sine of the maximum angle of incidence for some optical devices, or the sine of the acceptance angle for a waveguide or fiber.
Numerical aperture (NA) has two different meanings, one for fiber optics and one for imaging fibers.
Numerical aperture of a fiber or waveguide
In a ray diagram, a beam can be viewed as propagating in air and incident on the core of a step-index fiber (perpendicularly cleaved) with a large mode area.
Figure 1: The incident light is scattered and then totally reflected at the core-cladding interface. This only happens if the angle of incidence is not very large.
The numerical aperture (NA) of a fiber is the sine of the maximum angle of incidence such that the transmitted light beam can propagate in the core. The NA is determined by the difference in refractive index between the core and the cladding, more precisely by the following relationship:
This formula is obtained by propagating the transmitted light beam at the core-cladding interface at the total reflection angle. Where n0 is the refractive index of the medium near the fiber, which is approximately 1 if it is air.
Similarly, the NA of other waveguides can be defined.
The propagation angle determined by the numerical aperture can be converted into the maximum transverse spatial frequency of the light wave, which is equal to the numerical aperture divided by the vacuum wavelength. (For the angular frequency or transverse wave vector component, the maximum value is equal to 2π times this value.)
In single-mode fiber and few-mode fiber, the details of wave propagation need to be taken into account, and the above relationship can only give an approximate result, while the result is relatively accurate in the case of multimode fiber.
When the core area is very small (such as single-mode fiber), the wave characteristics of the beam become obvious and the ray diagram is inaccurate. (The beam divergence angle cannot be ignored.)
The above formula can still be used to define NA. When the refractive index shape is not rectangular, that is, non-step refractive index fiber, this concept becomes problematic.
A large numerical aperture usually corresponds to a large beam divergence angle of the fundamental mode at the end of the fiber, but the beam divergence angle also depends on the core diameter.
For non-step-index fibers (fibers where the core index is not constant), the effective numerical aperture can be defined based on the equivalent step-index cross section, which gives similar mode properties. The numerical aperture can also be calculated based on the refractive index at the maximum point in the core.
Other methods are based on the far-field shape of the light emitted from the fiber, usually taking the sine of the angle at which the light intensity decays to 5% of the maximum value.
In any case, the specific definition of the numerical aperture should be indicated when using the numerical aperture value, because different definitions will give different results.
The NA of single-mode fiber is usually on the order of 0.1, but can vary from 0.05 to 0.4. Multimode fibers usually have larger numerical apertures, such as 0.3. Photonic crystal fibers have very large numerical apertures.
Abbreviation: NA The sine of the maximum angle of incidence for some optical devices, or the sine of the acceptance angle for a waveguide or fiber.
Numerical aperture (NA) has two different meanings, one for fiber optics and one for imaging fibers.
Numerical aperture of a fiber or waveguide
In a ray diagram, a beam can be viewed as propagating in air and incident on the core of a step-index fiber (perpendicularly cleaved) with a large mode area.
Figure 1: The incident light is scattered and then totally reflected at the core-cladding interface. This only happens if the angle of incidence is not very large.
The numerical aperture (NA) of a fiber is the sine of the maximum angle of incidence such that the transmitted light beam can propagate in the core. The NA is determined by the difference in refractive index between the core and the cladding, more precisely by the following relationship:
This formula is obtained by propagating the transmitted light beam at the core-cladding interface at the total reflection angle. Where n0 is the refractive index of the medium near the fiber, which is approximately 1 if it is air.
Similarly, the NA of other waveguides can be defined.
The propagation angle determined by the numerical aperture can be converted into the maximum transverse spatial frequency of the light wave, which is equal to the numerical aperture divided by the vacuum wavelength. (For the angular frequency or transverse wave vector component, the maximum value is equal to 2π times this value.)
In single-mode fiber and few-mode fiber, the details of wave propagation need to be taken into account, and the above relationship can only give an approximate result, while the result is relatively accurate in the case of multimode fiber.
When the core area is very small (such as single-mode fiber), the wave characteristics of the beam become obvious and the ray diagram is inaccurate. (The beam divergence angle cannot be ignored.)
The above formula can still be used to define NA. When the refractive index shape is not rectangular, that is, non-step refractive index fiber, this concept becomes problematic.
A large numerical aperture usually corresponds to a large beam divergence angle of the fundamental mode at the end of the fiber, but the beam divergence angle also depends on the core diameter.
For non-step-index fibers (fibers where the core index is not constant), the effective numerical aperture can be defined based on the equivalent step-index cross section, which gives similar mode properties. The numerical aperture can also be calculated based on the refractive index at the maximum point in the core.
Other methods are based on the far-field shape of the light emitted from the fiber, usually taking the sine of the angle at which the light intensity decays to 5% of the maximum value.
In any case, the specific definition of the numerical aperture should be indicated when using the numerical aperture value, because different definitions will give different results.
The NA of single-mode fiber is usually on the order of 0.1, but can vary from 0.05 to 0.4. Multimode fibers usually have larger numerical apertures, such as 0.3. Photonic crystal fibers have very large numerical apertures.
