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Numerical Aperture

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Acronym: NA

Definition: sine of the maximum angle of an incident beam of some optical device, or the sine of the acceptance angle of a waveguide or fiber

German: numerische Apertur

The term numerical aperture (NA) is used with two different meanings, depending on the context, which may be fiber optics or imaging optics.

Numerical Aperture of an Optical Fiber or Waveguide

In a ray picture, a light beam can be considered which propagates in air and hits the core of a (perpendicularly cut) step-index fiber with large mode area.

acceptance angle of a fiber

Figure 1: An incident light ray is first refracted and then undergoes total internal reflection at the core–cladding interface. However, that works only if the incidence angle is not too large.

The numerical aperture (NA) of the fiber is the sine of the maximum angle of an incident ray with respect to the fiber axis, so that the transmitted beam is guided in the core. The NA is determined by the refractive index difference between core and cladding, more precisely by the relation

numerical aperture

which can be derived from the requirement that the transmitted beam at the core/cladding interface propagates with the critical angle for total internal reflection. Here, n0 is the refractive index of the medium around the fiber, which is close to 1 in case of air.

In a similar way, the NA can also be defined for other kinds of waveguides.

The limitation for the propagation angle by the numerical aperture translates into a maximum transverse spatial frequency of light, which is the numerical aperture divided by the vacuum wavelength. (For angular spatial frequencies or transverse wave vector components, the limit is 2π times that value.) Note that for single-mode fibers and few-mode fibers, where the detailed wave propagation needs to be taken into account, that rule gives only a rough estimate, whereas it is fairly accurate for highly multimode fibers.

Calculation of the NA of a Fiber

Core index:
Cladding index:
Numerical aperture: calc

After you have modified some inputs, click the "calc" button to recalculate the output.

It is assumed that the external medium is air (n = 1), and that the index contrast is small.

For small core areas (e.g. for single-mode fibers), the wave nature of the beams becomes essential, and the ray picture becomes invalid. (The beam divergence can then no longer be ignored.) The above equation may still be used to define the NA via the refractive indices. The concept becomes questionable for non-rectangular refractive index profiles, i.e., for non-step-index fibers.

A high NA usually relates to a large beam divergence for the fundamental mode exiting the fiber end, but this beam divergence also depends on the core diameter. For fibers other than step-index fibers (where the core does not have a constant refractive index), an effective numerical aperture may be defined based on an equivalent step-index profile, which leads to similar mode properties. Alternatively, one may calculate an NA from the maximum refractive index in the core. Still other methods are based on the far field profile of the light exiting a fiber, usually taking the sine of the angle at which the intensity decays to 5% of its maximum value. In any case, the precise underlying definition should be given when quoting such values, as different definitions can lead to different results.

For a single-mode fiber, the NA is typically of the order of 0.1, but can vary roughly between 0.05 and 0.4. Multimode fibers typically have a higher numerical aperture of e.g. 0.3. Very high values are possible for photonic crystal fibers.

A higher NA has the following consequences:

Numerical Aperture of a Lens

The NA of a lens (or a microscope objective, which is a combination of lenses) is defined as the sine of the angle of the marginal ray coming from the focal point, multiplied by the refractive index of the medium from which the input beam comes. The NA of a lens (and not its focal length) is what limits the size of a beam waist which can be formed with that lens. Therefore, high-NA lenses are required e.g. for players and recorders of data storage media such as CDs, DVDs and Blu-ray Discs.

In a microscope, the NA limits the image resolution obtainable. It can be increased by using some immersion oil with large refractive index between sample and objective lens, so that the NA can be greater than 1.

Lenses with high NA are also required for collimating laser beams which originate from small apertures. For example, this is the case for low-power single-mode laser diodes. When a lens with too low NA is used, the resulting collimated beam can be distorted (aberrated) or even truncated.

Bibliography

[1]Article on numerical aperture and resolution, http://microscopy.fsu.edu/primer/anatomy/numaperture.html, of the Optical Microscopy Primer

See also: acceptance angle in fiber optics, fibers, single-mode fibers, waveguides, fiber core, V number, focal length, lenses

Categories: fibers and other waveguides, general optics

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