Dispersion Compensation
Calculates chromatic dispersion limits and DCF (dispersion-compensating fiber) requirements: the rms pulse broadening of a span, the 4Bσ ≤ 1 bit-rate limit and power penalty, the DCF length that cancels the dispersion at one wavelength, and the residual dispersion that a slope mismatch leaves across the band.
Span without compensation (at 1550 nm)
Accumulated dispersion
1360 ps/nm
(D + S(λ − λ₀)) · L
RMS broadening
681 ps
σ_b = √(σ_D² + λ²|D_acc|/4πc), added to the optimum input pulse; 4Bσ_b = 27.2
Dispersion-limited bit rate
0.367 Gb/s
below 10 Gb/s; 4Bσ ≤ 1, σ the total output width with the optimum input pulse (√(λ²|D_acc|/2πc) for a narrow source)
Dispersion penalty
eye closed
With DCF (dispersion nulled at λ₀ = 1550 nm)
DCF length
13.6 km
Lc = −D·L / DDCF
Residual slope
1.92 ps/nm²
59 % of the span slope compensated
Usable band (≤ 1 dB)
1534.8–1565.2 nm
λ₀ ± 15.2 nm
Residual at 1550 nm
0 ps/nm
Sres · (λ − λ₀)
RMS broadening
0.339 ps
σ_b = √(σ_D² + λ²|D_acc|/4πc), optimum input pulse
Dispersion penalty
4.00e-4 dB
Residual Dispersion after DCF vs Wavelength
The DCF cancels the span's dispersion only at λ₀. Elsewhere the residual is Sres·(λ − λ₀); the dashed lines mark where the penalty reaches 1 dB at 10 Gb/s (±29.1 ps/nm).
Model
σD = √((Daccσλ)² + ½ (Saccσλ²)²), σb = √(σD² + λ²|Dacc|/4πc), δ = −5 log₁₀(1 − (4Bσb)²) dB
Lc = −D·L / DDCF, Sres = S·L + SDCF·Lc, Dacc(λ) = Sres(λ − λ₀)
Agrawal, Fiber-Optic Communication Systems, §2.4.2–2.4.3 and §5.4.4: for a broad source (σλ much larger than the modulation bandwidth, ≈ λ²B/c, 0.08 nm at 10 Gb/s and 1550 nm) σb reduces to σD; for a narrow one the transmitted Gaussian pulse is taken at its optimum width, which adds λ²|Dacc|/4πc to σb² for the penalty. The bit-rate limit uses the total output width σ instead, √(λ²|Dacc|/2πc) = √(|β2|L) for a narrow source. D varies linearly with λ. A DCF compensates the whole band only when its relative slope SDCF/DDCFequals the span's S/D.
Textbook approximation. Not included: the coherent β₃ term in the usable band; the curvature of D(λ), which is large for real DCFs away from their design band; PMD, nonlinearity and DCF loss.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Linear dispersion model D(λ) = D + S(λ − λ₀), Gaussian source of any width launched as the optimum Gaussian pulse (coherent β₃ term at λ₀ left out). Not included: dispersion curvature, PMD, nonlinear effects and DCF loss.
- Agrawal G. P. (2021). Fiber-Optic Communication Systems, 5th ed., §2.4.3 and §5.4.4. Wiley.
- NPTEL, Optical Communication, Module 12, “System degradation and power penalty”. Tabulated power-penalty values used as a check.