# V Number

Author: the photonics expert Dr. Rüdiger Paschotta

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: <$V$>

The *V number* is a dimensionless parameter which is often used in the context of step-index fibers (but normally not usable for other kinds of refractive index profiles). It is defined as

where <$\lambda$> is the vacuum wavelength, <$r_{\rm co}$> 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 a high <$V$> number, the number of guided modes can be estimated as <$V^2 / 4$>, when counting only one polarization direction, or twice that number for both polarization directions.

The formula for the estimated number of guided modes can be generalized for arbitrary index profiles, where effectively we use an average value of <$V^2$> in the fiber core:

$$M \approx \frac{\pi}{\lambda^2} \int {\left(n^2 - n_{\rm{cl}}^2\right) \: {\rm d}A}$$This is shown in a case study:

## Case Study: Number of Modes of a Highly Multimode Fiber

We seek a simple equation for estimating for the number of modes of a highly multimode fiber, which can be applied to fiber designs with arbitrary shapes of the refractive index profile. Instead of applying complicated mathematics, we build a hypothesis and subject that to multiple numerical tests.

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.

## More to Learn

Encyclopedia articles:

### Bibliography

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

## 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.