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# V Number

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

Formula symbol: V

Units: (dimensionless number)

Author:

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.

## 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 • 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 ncladding 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) If you like this article, share it with your friends and colleagues, e.g. via social media: