# V Number

Definition: a normalized frequency parameter, which determines the number of modes of a step-index fiber

Category: fiber optics and waveguides

Formula symbol: *V*

Units: (dimensionless number)

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Author: Dr. Rüdiger Paschotta

The *V number* is a dimensionless parameter which is often used in the context of step-index fibers.
It is defined as

where λ is the vacuum wavelength, *a* is the radius of the fiber core, and NA is the numerical aperture.
Of course, the *V* number should not be confused with some velocity *v*, e.g. the phase velocity of light, and also not with the Abbe number, which is also sometimes called V-number.

The *V* number can be interpreted as a kind of normalized optical frequency.
(It is proportional to the optical frequency, but rescaled depending on waveguide properties.)
It is relevant for various essential properties of a fiber:

- For
*V*values below ≈ 2.405, a fiber supports only one mode per polarization direction (→*single-mode fibers*). - Multimode fibers can have much higher
*V*numbers. For large values, the number of supported modes of a step-index fiber (including polarization multiplicity) can be calculated approximately as

- The
*V*number determines the fraction of the optical power in a certain mode which is confined to the fiber core. For single-mode fibers, that fraction is low for low*V*values (e.g. below 1), and reaches ≈ 90% near the single-mode cut-off at*V*≈ 2.405. - There is also the so-called Marcuse equation for estimating the mode radius of a step-index fiber from the
*V*number; see the article on mode radius. - A low
*V*number makes a fiber sensitive to micro-bend losses and to absorption losses in the cladding. However, a high*V*number may increase scattering losses in the core or at the core–cladding interface.

For certain types of photonic crystal fibers, an *effective V number* can be defined, where *n*_{cladding} is replaced with an effective cladding index.
The same equations as for step-index fibers can then be used for calculating quantities such as the single-mode cut-off, mode radius and splice losses.

### Bibliography

[1] | A. W. Snyder and J. D. Love, Optical Waveguide Theory, Chapman and Hall, London (1983) |

See also: fibers, step-index fibers, fiber core, numerical aperture, single-mode fibers, multimode fibers, Abbe number

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