Question 7 of 7: 100 km Fiber Link — Single-Mode Check, Dispersion Limit, DCF Sizing and Power Budget

Question 7: 100 km Fiber Link — Single-Mode Check, Dispersion Limit, DCF Sizing and Power Budget (20 marks)

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.

Find. (a) single-mode? (b) dispersion-limit failure, (c) minimum DCF length, (d) minimum transmitter power (dBm and W), (e) how $D$ is engineered.

Approach. Check the normalized frequency $V$ against the single-mode cutoff; compare the accumulated chromatic dispersion against the bit period (and the standard $B\Delta\tau\le0.2$ design rule); size the negative-dispersion DCF to bring the total back under budget; then close the power budget including the extra DCF loss.

1. Part (a) — single-mode operation. With core radius $a=5\ \mu\text{m}$, $$\mathrm{NA}=\sqrt{n_1^2-n_2^2}=\sqrt{1.465^2-1.461^2}=0.1082,$$ $$V=\frac{2\pi a\,\mathrm{NA}}{\lambda}=\frac{2\pi(5\times10^{-6})(0.1082)}{1550\times10^{-9}} =\boxed{2.193}.$$ Since $V=2.193 < V_c=2.405$, the fiber is operating single mode at 1550 nm.
2. Part (b) — dispersion-limit check.
   Link budget: 100 km main span plus dispersion-compensating fiber (DCF) before the receiver.
   The bit period at 2.5 Gb/s is $T_b=1/B=400\ \text{ps}$. The accumulated chromatic dispersion over the full 100 km span, using the source's own $0.5\ \text{nm}$ linewidth, is $$\Delta\tau=D\,\Delta\lambda\,L=(20)(0.5)(100)=\boxed{1000\ \text{ps}}.$$ This already exceeds the entire bit slot $T_b=400\ \text{ps}$, so pulses spread into their neighbours long before any receiver-design margin is even considered. Applying the standard NRZ dispersion-limit rule $B\Delta\tau\le0.2$, the allowed spreading is only $$\Delta\tau_{\text{max}}=\frac{0.2}{B}=80\ \text{ps},$$ so the actual dispersion is $12.5\times$ over budget: the link cannot operate at 2.5 Gb/s over the full 100 km without dispersion compensation.
3. Part (c) — minimum DCF length. Full compensation is not required, only enough negative dispersion to bring the total accumulated $D\cdot L$ product back under the $\Delta\tau_{\text{max}}/\Delta\lambda$ ceiling: $$D\,L_{\text{main}}+D_{\text{DCF}}\,L_{\text{DCF}}\le\frac{\Delta\tau_{\text{max}}}{\Delta\lambda},$$ $$(20)(100)+(-100)\,L_{\text{DCF}}=\frac{80}{0.5}=160\ \text{ps/nm},$$ $$L_{\text{DCF}}=\frac{2000-160}{100}=\boxed{18.4\ \text{km}}.$$
4. Part (d) — minimum transmitter power. The main span's loss budget is $$L_{\text{main}}=\alpha L+6(0.05)+2(0.2)=(0.2)(100)+0.3+0.4=20.7\ \text{dB}.$$ The 18.4 km of DCF adds its own loss (0.5 dB/km) plus its stated splice and connector: $$L_{\text{DCF}}^{\text{loss}}=(0.5)(18.4)+0.05+0.2=9.45\ \text{dB},$$ $$L_{\text{total}}=20.7+9.45=30.15\ \text{dB}.$$ The receiver sensitivity in dBm is $$P_{\text{Rx,sens}}=10\log_{10}\!\left(\frac{20\ \mu\text{W}}{1\ \text{mW}}\right) =-16.99\ \text{dBm},$$ so the minimum transmitter power that closes the budget is $$P_{\text{Tx,min}}=P_{\text{Rx,sens}}+L_{\text{total}} =-16.99+30.15=\boxed{13.16\ \text{dBm}} =\boxed{20.7\ \text{mW}\approx0.0207\ \text{W}}.$$
5. Part (e) — engineering the dispersion coefficient. The chromatic dispersion coefficient $D(\lambda)=D_{\text{material}}(\lambda)+D_{\text{waveguide}}(\lambda)$ is the sum of a material term (fixed by the glass composition, essentially un-tunable for a given host glass) and a waveguide term that depends on the core radius, the index difference $n_1-n_2$, and how much of the mode's power extends into the cladding at the wavelength of interest. Because the waveguide term can be pushed positive or negative by shaping the refractive-index profile — a smaller core with higher $\Delta n$, a triangular or depressed-cladding ("W") profile, segmented-core profiles, etc. — fiber designers dial in a specific total $D(\lambda)$: shifting the material zero-dispersion wavelength out to 1550 nm (dispersion-shifted fiber), flattening $D$ across the whole C-band (dispersion-flattened fiber), or, as used for the DCF in part (c), engineering a large negative waveguide term to build a fiber whose $D$ is strongly negative for exactly this compensation role.