Glossary entry

V number

Dimensionless parameter determining the number of guided modes in a step-index optical fiber. Single-mode operation requires V < 2.405.

The V number (also called the V parameter or normalized frequency) of an optical fiber is defined as

V \;=\; \frac{2\pi a}{\lambda_0} \sqrt{n_\text{core}^2 - n_\text{clad}^2} \;=\; \frac{2\pi a}{\lambda_0} \cdot \text{NA},

where $a$ is the core radius, $\lambda_0$ is the free-space wavelength, $n_\text{core}$ and $n_\text{clad}$ are the core and cladding refractive indices, and NA is the numerical aperture.

Single-mode operation requires $V < 2.405$ (the first zero of the Bessel function $J_0$ ). Above this cutoff, additional transverse modes begin to propagate. For large $V$ , the total number of supported modes is approximately $V^2/2$ .

Typical values:

Fiber	Wavelength	$V$
SMF-28	1550 nm	2.27
SMF-28	1310 nm	2.68
HI-1060	1064 nm	2.31
50 / 125 μm graded-index MMF	850 nm	$\sim$ 30
62.5 / 125 μm MMF	850 nm	$\sim$ 50

The single-mode cutoff wavelength is $\lambda_c = 2\pi \, a \, \text{NA} \, / \, 2.405$ . Operating a fiber below $\lambda_c$ produces multi-mode propagation. The slight exception for SMF-28 at 1310 nm ( $V = 2.68$ ) is accommodated by the LP $_{11}$ mode being effectively cut off through additional design parameters.

For photonic integrated circuit waveguides, an analogous parameter exists but the analysis is geometry-specific (rectangular cross-section, step-index in two directions); the simple V-number formula does not apply directly.