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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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URL: https://www.rp-photonics.com/v_number.html

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

V number

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
number of modes versus V number
  • 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.

Questions and Comments from Users

2020-06-14

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

Answer from the author:

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?

Answer from the author:

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.

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Bibliography

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

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See also: fibers, step-index fibers, fiber core, numerical aperture, single-mode fibers, multimode fibers, Abbe number
and other articles in the category fiber optics and waveguides

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