7.6.11 Modes of the fibre and the single-mode condition

The modes of the round core are found as in the circular metal guide of Section 7.4, with Bessel functions. Inside the core the fields vary as $J_{l} (u r / a) cos l ϕ$ , outside as $K_{l} (w r / a) cos l ϕ$ , where $K_{l}$ is the modified Bessel function of the second kind, which decays roughly as $e^{- w r / a} / r$ . The parameters are the radial versions of $u$ and $w$ ,

u w u^{2} + w^{2} = a k_{0}^{2} n_{1}^{2} - τ^{2}, = a τ^{2} - k_{0}^{2} n_{2}^{2}, = V^{2}, (7.6.36)

so $J_{l}$ plays the role of the slab's cosine and $K_{l}$ that of its exponential.

The exact modes of the rod are hybrid: they have both $E_{z}$ and $H_{z}$ (they are called HE and EH modes), except the circularly symmetric TE $_{0 m}$ and TM $_{0 m}$ .
When $Δ ≪ 1$ the boundary barely distinguishes TE from TM (the factor $n_{2}^{2} / n_{1}^{2}$ of the slab is almost $1$ ), and the exact modes group into nearly degenerate sets, the linearly polarised LP $_{l m}$ modes, whose transverse field points in one fixed direction.
The lowest, LP $_{01}$ (the HE $_{11}$ mode), has no cutoff, like TE $_{0}$ of the slab.

LP modes of the weakly guiding fibre

u \frac{J _{l - 1} ( u )}{J _{l} ( u )} = - w \frac{K _{l - 1} ( w )}{K _{l} ( w )}, LP_{01} : u \frac{J _{1} ( u )}{J _{0} ( u )} = w \frac{K _{1} ( w )}{K _{0} ( w )} . (7.6.37)

LP $_{l m}$ is cut off at the $m$ -th zero of $J_{l - 1} (V)$ , the root $V = 0$ counted only for $l = 0$ (so LP $_{01}$ has no cutoff and LP $_{02}$ starts at $3.832$ ). The second mode, LP $_{11}$ , starts where $J_{0} (V) = 0$ , so the fibre is single-mode for

V < 2.405, λ_{0} > λ_{c} = \frac{2 π a NA}{2.405} . (7.6.38)

For $V ≫ 1$ the fibre carries about $V^{2} /2$ modes (counting both polarisations).

In plain words

A fibre is a round slab. Its modes are named by two numbers, the number of field periods around the axis ( $l$ ) and across the radius ( $m$ ). Make the core thin enough, or the index step small enough, that $V < 2.405$ , and only the fundamental mode survives.

The tool below shows the single-mode fibre of Problem 7.6.3 at $1550 nm$ . Drag the launch ray out of the acceptance cone, then shorten $λ_{0}$ until LP $_{11}$ appears at $V = 2.405$ , below $λ_{c} = 1257 nm$ .

InteractiveLoading simulation…

Figure 7.6.10 shows $b = w^{2} / V^{2} = (n_{eff}^{2} - n_{2}^{2}) / (n_{1}^{2} - n_{2}^{2})$ of the first six LP modes, computed from Eq. (7.6.37). The single-mode range $V < 2.405$ is shaded.

The normalised propagation constant b of the LP modes of a step-index fibre against V (computed from Eq. (7.6.37)). The cutoffs lie at the zeros of J_{l-1}: 2.405 for LP_{11}, 3.832 for LP_{21} and LP_{02}, 5.136 for LP_{31} and 5.520 for LP_{12}. Green: the single-mode range. — **Figure 7.6.10.** The normalised propagation constant $b$ of the LP modes of a step-index fibre against $V$ (computed from Eq. (7.6.37)). The cutoffs lie at the zeros of $J_{l - 1}$ : $2.405$ for LP $_{11}$ , $3.832$ for LP $_{21}$ and LP $_{02}$ , $5.136$ for LP $_{31}$ and $5.520$ for LP $_{12}$ . Green: the single-mode range.

Single-mode fibre

Above the cutoff wavelength $λ_{c}$ only LP $_{01}$ propagates, in two orthogonal polarisations. All the light then travels with one $τ$ , and the modal dispersion of the next subsection disappears. Standard telecom fibre has $λ_{c} < 1260 nm$ , so it is single-mode in both telecom windows, at $1310$ and at $1550 nm$ . Its core is $8$ to $9 μ m$ across, and it can be that large only because the NA is small: with the NA of a multimode fibre a single-mode core would have a radius of about $1.3 μ m$ , far too small to couple light into (Problems 7.6.3 and 7.6.4).