Question 1 of 6: Group Index, Dispersion Parameter, and a Step-Index Fiber at 1350 nm

Question 1: Group Index, Dispersion Parameter, and a Step-Index Fiber at 1350 nm (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Part (a): phase velocity $v_p=c/n$, group velocity $v_g=c/N$. Part (b): $\lambda=1350$ nm, core diameter $2a=20\ \mu\text{m}$, $n_1=1.465$, $N_1=1.474$, $n_2=1.462$, $N_2=1.466$, total intramodal dispersion $D=20\ \text{ps}/(\text{nm}\cdot\text{km})$, source spectral width $\Delta\lambda=15$ nm, bit rate $B=100$ Mbit/s in RZ format.

Find. (a) A derivation of $N=n-\lambda\,dn/d\lambda$ from $\beta=n\omega/c$ and an expression for $d^2\beta/d\omega^2$ in terms of $N$. (b)(i) Whether the fiber is single- or multimode, and the mode count if multimode. (ii) The maximum fiber length for 100 Mbit/s RZ transmission. (c) The physical origins of dispersion in a fiber.

Approach. Part (a) is a direct chain-rule substitution of $\lambda=2\pi c/\omega$ into $\beta(\omega)=n(\lambda(\omega))\,\omega/c$. Part (b)(i) uses the normalized frequency $V=2\pi a\,\mathrm{NA}/\lambda$ against the single-mode cutoff $V_c=2.405$; part (b)(ii) combines the intermodal spread set by the two group indices with the intramodal (chromatic) spread from the given coefficient and source linewidth, then applies the RZ pulse-broadening criterion.

Part (a) — group index and the dispersion parameter

Since $\lambda=2\pi c/\omega$, the propagation constant $\beta=n(\lambda)\,\omega/c$ is a function of $\omega$ through both the explicit $\omega$ and the implicit $\lambda(\omega)$ dependence of $n$: $$\frac{d\beta}{d\omega}=\frac{n}{c}+\frac{\omega}{c}\frac{dn}{d\lambda}\frac{d\lambda}{d\omega}.$$ From $\lambda=2\pi c/\omega$, $\dfrac{d\lambda}{d\omega}=-\dfrac{2\pi c}{\omega^2}=-\dfrac{\lambda}{\omega}$, so $$\frac{d\beta}{d\omega}=\frac{n}{c}-\frac{\omega}{c}\frac{dn}{d\lambda}\frac{\lambda}{\omega} =\frac{1}{c}\left(n-\lambda\frac{dn}{d\lambda}\right)\equiv\frac{N}{c}.$$ This identifies $\boxed{N=n-\lambda\,dn/d\lambda}$ and confirms $v_g=d\omega/d\beta=c/N$, exactly the definition given.

Differentiating $d\beta/d\omega=N(\lambda(\omega))/c$ once more with respect to $\omega$, using the same $d\lambda/d\omega=-\lambda/\omega$: $$\frac{d^2\beta}{d\omega^2}=\frac{1}{c}\frac{dN}{d\lambda}\frac{d\lambda}{d\omega} =-\frac{\lambda}{c\,\omega}\frac{dN}{d\lambda}.$$ Substituting $\omega=2\pi c/\lambda$ gives the dispersion parameter entirely in terms of the group index $N$: $$\boxed{\frac{d^2\beta}{d\omega^2}=-\frac{\lambda^2}{2\pi c^2}\frac{dN}{d\lambda}}.$$ This is the standard group-velocity-dispersion coefficient $\beta_2$; it is directly proportional to how fast the group index itself varies with wavelength — the same quantity a fiber's manufacturer reports as the chromatic dispersion coefficient $D=(1/c)\,dN/d\lambda=-(2\pi c/\lambda^2)\beta_2$, used numerically in part (b)(ii).

Part (b)(i) — single-mode or multimode?

1. Numerical aperture. $$\mathrm{NA}=\sqrt{n_1^2-n_2^2}=\sqrt{1.465^2-1.462^2}=0.0937.$$
2. Normalized frequency (V-number). With core radius $a=10\ \mu\text{m}$, $$V=\frac{2\pi a\,\mathrm{NA}}{\lambda}=\frac{2\pi(10\times10^{-6})(0.0937)}{1350\times10^{-9}}=4.36.$$ Since $V=4.36>V_c=2.405$, the fiber is multimode at 1350 nm.
3. Mode count. For a step-index fiber the total number of guided modes (both polarizations, all azimuthal orders) is approximated by $$M\approx\frac{V^2}{2}=\frac{4.36^2}{2}\approx\boxed{9\text{--}10\ \text{modes}}.$$

Part (b)(ii) — maximum RZ link length at 100 Mbit/s

1. Intermodal spread per unit length. Part (b)(i) showed the fiber is multimode, so the dominant broadening is the delay difference between the fastest and slowest guided modes. The question supplies the core and cladding group indices for exactly this purpose (Kasap's step-index estimate): $$\frac{\Delta\tau_{inter}}{L}\approx\frac{N_1-N_2}{c}=\frac{1.474-1.466}{2.998\times10^{8}\ \text{m/s}}=2.67\times10^{-11}\ \text{s/m}=26.7\ \frac{\text{ns}}{\text{km}}.$$
2. Intramodal (chromatic) spread per unit length. From the stated total intramodal dispersion coefficient and the source linewidth, $$\frac{\Delta\tau_{intra}}{L}=D\,\Delta\lambda=\left(20\ \frac{\text{ps}}{\text{nm}\cdot\text{km}}\right)(15\ \text{nm})=300\ \frac{\text{ps}}{\text{km}}=0.300\ \frac{\text{ns}}{\text{km}}.$$
3. Total spread. The two mechanisms are independent, so they combine in quadrature: $$\frac{\Delta\tau}{L}=\sqrt{26.7^2+0.300^2}\ \frac{\text{ns}}{\text{km}}=26.7\ \frac{\text{ns}}{\text{km}}.$$ Intermodal dispersion outweighs the chromatic term by a factor of about 89, so the chromatic contribution is negligible here.
4. Bit period and RZ criterion. $$T_b=\frac{1}{B}=\frac{1}{100\times10^6}=10\ \text{ns}.$$ An RZ pulse occupies only the first half of its bit slot, so the tolerable dispersion-induced spread is taken as half the bit period (the stricter criterion than NRZ's full $T_b$): $$\Delta t_{\max}=\frac{T_b}{2}=5\ \text{ns}=5000\ \text{ps}.$$
5. Maximum length. $$L_{\max}=\frac{\Delta t_{\max}}{\Delta\tau/L}=\frac{5\ \text{ns}}{26.7\ \text{ns/km}}=0.187\ \text{km}\approx\boxed{190\ \text{m}}.$$

This is why a multimode step-index fiber is a short-haul medium. If the modal term were ignored, chromatic dispersion alone would allow $5000/300=16.7$ km, which overstates the reach by nearly two orders of magnitude. With only about nine guided modes ($V=4.36$), the $(N_1-N_2)/c$ spread is an upper-bound estimate, so the true reach may be somewhat longer. It is still a few hundred metres, not kilometres.

Part (c) — sources of dispersion in an optical fiber

Three physically distinct mechanisms broaden a pulse as it propagates. Intermodal (modal) dispersion occurs only in multimode fiber: different mode groups travel at different group velocities (equivalently, different effective ray angles), so a pulse launched into many modes arrives smeared over a delay set by $N_1\Delta/c$ per unit length, where $\Delta=(n_1-n_2)/n_1$; it is eliminated entirely in single-mode fiber. Chromatic (intramodal) dispersion is present even in a single mode, because the source has a finite spectral width $\Delta\lambda$ and the fiber's group index $N(\lambda)$ varies with wavelength; it splits into material dispersion (the glass's own $dN/d\lambda$) and waveguide dispersion (the mode's effective index changes with $\lambda$ through the core/cladding confinement, even for a hypothetically non-dispersive glass) — the two can partially cancel near 1310 nm in standard silica fiber. Polarization-mode dispersion (PMD) arises in single-mode fiber from residual core ellipticity or stress birefringence, which splits the nominally degenerate two polarization states into slightly different group velocities; it grows only as $\sqrt{L}$ (a random-mode-coupling process) and is usually the smallest term except in very-high-bit-rate long-haul links.