Fibre V-Number (Normalised Frequency)

Everything about how a step-index fibre guides collapses into one dimensionless number. Gloge showed in 1971 that for a weakly guiding fibre — one where $n_1 \approx n_2$, which covers essentially all doped silica — the mode structure depends only on $V = 2\pi a\,\mathrm{NA}/\lambda$. Below $V = 2.405$, the first zero of the Bessel function $J_0$, exactly one spatial mode propagates and the fibre is single-mode. Above it, a second mode group appears, then a third, and the count climbs.

The 2.405 is not a rule of thumb. It falls out of matching the Bessel-function field inside the core to the decaying modified-Bessel field in the cladding, and the cutoff of the $\mathrm{LP}_{11}$ mode lands exactly on $J_0$'s first root. It is one of the tidiest results in guided optics.

The thing readers most often get wrong is that single-mode is a property of a WAVELENGTH BAND, not of a spool. $V$ is inversely proportional to $\lambda$, so the same fibre is single-mode at long wavelengths and multimode at short ones. A standard telecom fibre with an 8.2 µm core and NA 0.13 has $V = 2.16$ at 1550 nm — single-mode, as sold. At 1064 nm the same fibre gives $V = 3.15$, and at 633 nm it gives $V = 5.29$ and carries something like fourteen modes. Sending a red alignment laser down a single-mode telecom fibre and expecting a clean Gaussian out the far end is a disappointment waiting to happen, and it is not the fibre's fault.

Set $V = 2.405$ and solve for wavelength and you get the CUTOFF WAVELENGTH — the shortest wavelength at which the fibre is still single-mode. Single-mode operation means working above it. Two honesty notes on that number. The theoretical cutoff computed here is not what a datasheet quotes: the measured "cable cutoff" is shorter, usually by tens of nanometres, because bends and the finite length of real deployed cable strip the second mode before it can travel far. And this whole analysis is step-index. A graded-index fibre has a different cutoff condition, and for the usual parabolic profile the effective $V$ is smaller by roughly $\sqrt2$.

Because $V$ depends on the PRODUCT $a \cdot \mathrm{NA}$, a designer can hold the same modal behaviour with a small core and a large NA or a large core and a small NA. Telecom fibre takes the second route — a low NA around 0.12 buys the largest single-mode core, which makes splicing and launching easier. Large-mode-area fibres for high-power lasers push it further, down toward NA 0.06 to keep a 20 or 25 µm core effectively single-mode, because a wide mode is what keeps the intensity in the glass below the nonlinear thresholds. The price is bend sensitivity: a weakly guided mode leaks out of a curve a high-NA fibre would not notice.

One last correction to a common mental picture. The guided field is not confined to the core. Near cutoff a substantial fraction of the power travels in the cladding, which is why the mode field diameter of a single-mode fibre is always somewhat larger than its physical core, and why bend loss climbs sharply as $V$ falls toward 1.