# V Number

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

Category: fiber optics and waveguides

Units: (dimensionless number)

Formula symbol: <$V$>

Author: Dr. Rüdiger Paschotta

Cite the article using its DOI: https://doi.org/10.61835/8ty

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

where <$\lambda$> 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 <$n_\rm{cladding}$> 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

## Questions and Comments from Users

2020-10-21

In the case of Photonic crystal fibers (PCFs) do we use the same concept of V-number for figuring out the number of modes?

The author's answer:

Strictly speaking, the numerical aperture and <$V$> number are not defined for a photonic crystal fiber. Further, the formation of modes is influenced by other physical aspects, e.g. photonic bandgap effects. At most, you can for some of those fibers get a rough estimate of the number of modes, based on an intelligent guess how to assign a reasonable <$V$> number to such a fiber.

2022-12-01

What if the fiber is not cylindrical? How to calculate the V-number of a fiber with irregular core shape, for example, a rectangular shape?

The author's answer:

The V number is not defined for such cases – only for ordinary step-index fibers.

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2020-06-14

What is the relation between V number and power flow in cladding?

The author's answer:

Tentatively, for fibers with low V number a larger fraction of the total optical power propagates in the fiber cladding. The numerical value, however, depends on the details, not just the V number.