V number

The V number, or normalized frequency, of an optical fiber combines the core radius $a$, the free-space wavelength $\lambda$ and the numerical aperture into one dimensionless number,

$$V=\frac{2\pi a}{\lambda }\sqrt{n_{\text{core}}^2-n_{\text{clad}}^2}=\frac{2\pi a}{\lambda }\text{NA},$$

which decides how many transverse modes the core guides. A step-index fiber guides a single mode, LP₀₁, when $V<2.405$, the first zero of the Bessel function $J_0$; above that value the LP₁₁ mode is guided as well, and a large core guides about $V^2\mathrm{/}2$ modes. Standard single-mode fiber has $V\approx 2.0$ at 1550 nm, and a 50 µm graded-index multimode core with an NA of 0.20 has $V=37$ at 850 nm. Because $V$ is inversely proportional to wavelength, every fiber becomes multimode at a short enough wavelength; the wavelength at which $V=2.405$ is its cutoff wavelength, ${\lambda }_c=2\pi a\text{NA}\mathrm{/}2.405$. The Fiber Numerical Aperture and V Number Calculator gives $V$, the guided LP modes and the cutoff for any fiber.

Mode cutoffs

In a weakly guiding step-index fiber, the guided modes are the linearly polarized LP${}_{lm}$ modes, and each appears at a fixed value of $V$: the LP${}_{0m}$ modes at the zeros of $J_1$ and the LP${}_{lm}$ modes with $l\ge 1$ at the zeros of $J_{l-1}$.

Each LP${}_{lm}$ mode with $l\ge 1$ has two orientations, and every spatial mode two polarizations. The fundamental LP₀₁ mode has no cutoff, so a step-index core always guides at least one mode. The second and third rows are the three-mode and six-mode fibers used for mode-division multiplexing, described in the few-mode fiber entry.

Number of modes in multimode fiber

For large $V$, a step-index core guides about $V^2\mathrm{/}2$ modes, counting both polarizations, and a graded-index core with a parabolic profile about $V^2\mathrm{/}4$:

The modes travel at different group velocities, which is the modal dispersion that limits the bandwidth of multimode links; the graded-index profile exists to equalize those velocities.

Standard single-mode fiber

Standard single-mode fiber (ITU-T G.652) has a core about 8.2 µm in diameter and a datasheet NA of 0.14. Taken at face value, those figures give $V=2.75$ at 1310 nm, above cutoff. The datasheet NA is measured from the far-field angle at 1% of the peak power, which overstates the NA of the equivalent step-index core; the core-cladding index difference of about 0.36% corresponds to $\text{NA}\approx 0.123$, which gives

and a theoretical cutoff near 1.3 µm. The LP₁₁ mode is weakly guided at wavelengths just short of its cutoff wavelength and is lost in bends within a few meters of fiber, so the cutoff measured on a cabled length is shorter: G.652 specifies a cable cutoff of 1260 nm or less, and the fiber is effectively single mode at 1310 nm.

V and the mode field diameter

As $V$ falls, the fundamental mode spreads further into the cladding. Marcuse's fit for a step-index core gives the mode radius $w$ as

$$\frac{w}{a}\approx 0.65+\frac{1.619}{V^{3\mathrm{/}2}}+\frac{2.879}{V^6},$$

accurate to about 1% for $1.2<V<2.4$. The mode field diameter $2w$ is 1.10 times the core diameter at $V=2.405$, 1.27 times at $V=2.0$ and 1.78 times at $V=1.5$; in the Gaussian approximation the share of power inside the core falls from 81% to 71% to 47% over the same range. A mode that extends far into the cladding is easily lost at bends, so the bend loss of a single-mode fiber rises toward longer wavelengths, where $V$ is smaller. Single-mode fibers are therefore designed with $V$ between about 2 and 2.4 over their operating band, as large as single-mode operation allows, and few-mode fibers keep $V$ away from the cutoff of their highest mode.

The calculator starts from the step-index equivalent of standard single-mode fiber, 8.2 µm and NA 0.123, the values behind the table above; with the datasheet NA of 0.14 it gives 9.2 µm at 1550 nm.

Waveguides other than fibers

A symmetric slab waveguide of thickness $d$ has an analogous parameter, $V=(\pi d\mathrm{/}\lambda )\sqrt{n_{\text{core}}^2-n_{\text{clad}}^2}$, and guides one TE and one TM mode when $V<\pi \mathrm{/}2$, that is, when $d<\lambda \mathrm{/}(2\sqrt{n_{\text{core}}^2-n_{\text{clad}}^2})$. For silicon in silica at 1550 nm this limit is 245 nm, which is why silicon photonics uses silicon layers about 220 nm thick. The Waveguide Mode Explorer solves the slab exactly. Rectangular channel waveguides have no single cutoff formula; their modes are found with a numerical solver or approximately with the effective index method.

Common questions

What V number makes a fiber single mode?

A step-index fiber is single mode when $V<2.405$. Graded-index and other profiles have their own cutoff values, higher than 2.405 for a graded core.

How many modes does a multimode fiber carry?

About $V^2\mathrm{/}2$ for a step-index core and $V^2\mathrm{/}4$ for a parabolic graded-index core, counting both polarizations: roughly 340 modes in 50 µm graded-index fiber at 850 nm and 1,000 in 62.5 µm fiber.

How do you find the cutoff wavelength from the V number?

At a wavelength where the fiber has a V number $V$, the cutoff wavelength is ${\lambda }_c=\lambda V\mathrm{/}2.405$. A fiber with $V=2.0$ at 1550 nm has ${\lambda }_c=1289$ nm.

Why is standard single-mode fiber single mode at 1310 nm when its NA of 0.14 gives V = 2.75?

The quoted NA comes from a far-field measurement that overstates the step-index equivalent, which is about 0.123 and gives $V\approx 2.4$ at 1310 nm. The weakly guided LP₁₁ mode is also stripped by bends in a cabled fiber, so the specified cable cutoff is 1260 nm or less.

References: D. Gloge, "Weakly guiding fibers," Applied Optics 10, 2252 (1971). D. Marcuse, "Loss analysis of single-mode fiber splices," Bell System Technical Journal 56, 703 (1977). A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983). ITU-T Recommendation G.652, Characteristics of a single-mode optical fibre and cable (2016).