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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

Categories: fiber optics and waveguides, general optics

Formula symbol: NA

Units: (dimensionless)

How to cite the article; suggest additional literature

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

Enter input values with units, where appropriate. 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 or Objective

The NA of a lens (or a microscope or photographic objective, which is a combination of lenses) is essentially a quantitative measure for the range of angles which the lens can handle. As an example, consider a microscope objective collecting light from a small point of an observed object. Using a ray optics picture, the NA is the sine of the angle of the marginal ray (i.e., the ray with maximum angle against that axis which can still get into the objective and contribute to the observed image) multiplied with the refractive index of the medium in which the objective is working. The NA (usually referring to the object side of the objective) does not only determine how bright the observed image can be, but more importantly sets a limit to the obtained spatial resolution: the finest resolvable details have a size of approximately λ / (2 NA), assuming that the objective does not produce excessive image aberrations. On the other hand, high NA is associated with a small depth of field: only objects within a small range of distances from the objective can be seen with a sharp image.

Similarly, the NA of a lens used for focusing a laser beam, for example, determines the minimum possible beam radius in the focus (beam waist) to λ / (pi NA), assuming a collimated Gaussian beam (having diffraction-limited beam quality) with correct input beam radius hits the lens. In such cases, the numerical aperture is not defined via a marginal ray, but based on the maximum beam divergence half-angle which the lens can handle without truncation or excessive aberrations.

High-NA lenses are required for various applications:

In photography, it is not common to specify the numerical aperture of an objective. Instead, one often uses the so-called f number (formula symbol N), which is the focal length divided by the diameter of the entrance pupil. The f number can often be increased by manually or automatically reducing some internal aperture size, e.g. in order to avoid excessive exposure or to increase the depth of field. If the objective is focused at infinite distance, the numerical aperture is approximately 1 / (2 N), assuming that a photograph objective is used in air (as normal).

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

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