Optical Fiber: LP Modes, Dispersion and Pulse Broadening

An interactive view of the weakly guiding step-index optical fibre. The $\mathrm{LP}_{lm}$ modes solve $U J_{l-1}(U)/J_l(U) = -W K_{l-1}(W)/K_l(W)$ with $V^2 = U^2 + W^2$ and normalised index $b = 1 - U^2/V^2 \in (0,1)$; $\mathrm{LP}_{01}$ has no cutoff while $\mathrm{LP}_{11}$ (the single-mode limit) cuts off at the first zero of $J_0$, $V = 2.40483$, and $\mathrm{LP}_{21}/\mathrm{LP}_{02}$ near the first nonzero zero of $J_1$. The dispersion panel draws the universal $b$-$V$ curves with the single-mode region shaded and the operating point marked; the cross-section panel shows the $|E|^2$ pattern of the selected mode (the $\cos(l\phi)$ azimuthal lobes and the $J_l$ radial structure inside the core, $K_l$ evanescent tail outside); the pulse panel shows a chirp-free Gaussian broadening as $T(z) = T_0\sqrt{1 + (z/L_D)^2}$, $L_D = T_0^2/|\beta_2|$, with the energy conserved. The single-mode condition $V \lt 2.40483$ (the first zero of $J_0$), the universal $b$-$V$ dispersion curves, and the dispersion-limited Gaussian pulse broadening are the physical content.