Number of Modes in a Step-Index Fibre

Once a fibre is well past cutoff, counting its modes one at a time by solving Bessel eigenvalue equations stops being worth the effort, and a beautifully simple asymptotic result takes over. Gloge's argument is essentially geometric: the guided modes fill a region of phase space whose area is proportional to $V^2$, each mode occupies a fixed cell of that space, and the count comes out at $M \approx V^2/2$.

The estimate is good where estimates of this kind are good — when there are many modes. At $V = 20$ it gives 200 and it is close. At $V = 5$ it gives 12.5 and it is loose, because an argument that counts modes by area has nothing to say when the count is small enough to list. And below $V = 2.405$ it is meaningless: the fibre carries exactly one spatial mode, and the formula's answer of 2.9 is not a mode count but a signal that you are in the wrong regime.

Three qualifications, and none of them is fine print. First, THIS IS THE WEAKLY GUIDING APPROXIMATION. Gloge's whole derivation assumes $n_1 \approx n_2$, which is true of doped silica where the indices differ in the third decimal place, and false of plastic fibre, air-clad designs and anything with a large index step, where the exact vector treatment is needed instead. Second, it is a STEP-INDEX result: a graded-index fibre of the same $V$ carries roughly half as many, $V^2/4$ for the usual parabolic profile, which is one of the reasons graded-index fibre displaced step-index for data links. Third, these are spatial modes, and each supports two polarisation states, so the count of propagating states is twice this.

What the number is actually for is dispersion. More modes means more spread in path length, which means a pulse launched sharp arrives smeared, which means less bandwidth over a given distance. That is the whole reason multimode fibre is confined to buildings and campuses while long-haul telecom is single-mode. A graded-index profile is a partial fix: the outer rays travel through lower-index glass and go faster, which very nearly cancels their longer path, and it buys a large factor of bandwidth over step-index without giving up the easy launch.

Run backwards, $V = \sqrt{2M}$ is a rough design tool. Because it is a square root, doubling the modal capacity needs only a 41% larger core — which is part of why multimode core diameters cluster so tightly around the standard 50 and 62.5 µm rather than spreading across a wide range.