 Encyclopedia … combined with a great Buyer's Guide!

# V Number

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

Units: (dimensionless number)

Formula symbol: <$V$>

Author:

Get citation code:

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

$$V = \frac{{2\pi }}{\lambda }a\;{\rm{NA}} = \frac{{2\pi }}{\lambda }a\sqrt {n{{_{{\rm{core}}}^{}}^2} - n{{_{{\rm{cladding}}}^{}}^2}}$$

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.

## Calculation of the NA and V Number of a Fiber

 Wavelength: Core index: Cladding index: Core radius: Numerical aperture: calc V number: calc

Enter input values with units, where appropriate. After you have modified some values, click a “calc” button to recalculate the field left of it.

It is assumed that the external medium is air (n = 1).

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
$$M \approx \frac{{{V^2}}}{2}$$
• 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

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

## Questions and Comments from Users

2020-06-14

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

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.

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?

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? Share this with your friends and colleagues, e.g. via social media:   These sharing buttons are implemented in a privacy-friendly way!