Tool · Fiber and telecom

Dispersion and Pulse Broadening Calculator

How much does a pulse spread in a length of fiber, and how far can a link run before dispersion limits its bit rate? The calculator takes the fiber dispersion at the operating wavelength, entered directly or computed from the zero-dispersion wavelength and slope with the ITU-T G.652 formula, converts it to β₂ and β₃, and reports the dispersion length, the broadening or compression of a chirped Gaussian pulse, the accumulated dispersion, the length of compensating fiber that cancels it, and the dispersion-limited bit rate and length. Background: chromatic dispersion, group-velocity dispersion, and dispersion compensation.

Fiber dispersion

Wavelength λ (nm)

Zero-dispersion wavelength λ₀ (nm)

Zero-dispersion slope S₀ (ps/(nm²·km))

Dispersion D (ps/(nm·km))

Dispersion slope S (ps/(nm²·km))

D and S at the wavelength above; the D(λ) plot extends them as a straight line

Pulse and link

Input pulse FWHM (ps)

Gaussian intensity; the 1/e half-width is T₀ = T_FWHM/(2√ln 2) = T_FWHM/1.665

Initial chirp C

input field exp[−(1 + iC)T²/(2T₀²)]; the pulse first compresses when Cβ₂ < 0 (frequency chirp)

Fiber length L (km)

Bit rate B (Gb/s)

Source

Source RMS width σ_λ (nm)

Compensating fiber

D_DCF (ps/(nm·km))

S_DCF (ps/(nm²·km))

at the operating wavelength; the defaults are a slope-matched fiber for standard single-mode fiber

Presets

Learn with it

Three short experiments. Each one sets the inputs, says where to look, and asks for a prediction before it shows the result.

Readouts

Pulse intensity in and out

inputafter L

Pulse width versus distance

with chirp Cunchirped

Dispersion D(λ), 1200–1700 nm

Checked against

These checks run in your browser on every load. They compare the conversions and closed forms with textbook and standards values and with a numerical propagation of a chirped Gaussian pulse done here by discrete Fourier transform.

Check	Expected	Computed	Tolerance

Expected values come from the ITU-T G.652 dispersion formula and the CWDM4 MSA dispersion limits, from the Gaussian-pulse results in Agrawal, Nonlinear Fiber Optics (section 3.2) and Fiber-Optic Communication Systems (section 2.4), and from hand evaluation of the same formulas. The numerical rows propagate the chirped Gaussian field through the fiber transfer function exp(iβ₂ω²z/2) on a 512-point grid and measure its RMS width, peak and energy, which tests the closed form without assuming it. Tolerances are relative to the expected value; where a row lists several values, the tolerances follow in the same order.

The model

The dispersion parameter $D$ , in ps/(nm·km), is entered with its slope $S = dD/d\lambda$ , or computed from the zero-dispersion wavelength $\lambda_0$ and the slope there, $S_0$ , with the form ITU-T G.652 uses for standard single-mode fiber:

D(\lambda) = \frac{S_0}{4}\left(\lambda - \frac{\lambda_0^4}{\lambda^3}\right)

The group-velocity dispersion and the third-order dispersion follow from $D$ and $S$ :

\beta_2 = -\frac{D\lambda^2}{2\pi c}

S = \left(\frac{2\pi c}{\lambda^2}\right)^{2}\beta_3 + \frac{4\pi c}{\lambda^3}\,\beta_2

The input is a Gaussian pulse whose intensity has a full width at half maximum $T_\mathrm{FWHM} = 2\sqrt{\ln 2}\,T_0 \approx 1.665\,T_0$ , where $T_0$ is the half-width at 1/e intensity, and whose field carries a linear chirp $C$ : $U(0,T) = \exp[-(1+iC)T^2/2T_0^2]$ . With the dispersion length $L_D = T_0^2/|\beta_2|$ and $\xi = \beta_2 z/T_0^2$ , the pulse stays Gaussian and its width after a distance $z$ is

\frac{T_1}{T_0} = \left[(1 + C\xi)^2 + \xi^2\right]^{1/2}

while its peak falls by the same factor. When $C\beta_2 < 0$ the pulse first compresses, reaching its narrowest point

z_\mathrm{min} = \frac{|C|}{1+C^2}\,L_D

T_\mathrm{min} = \frac{T_0}{(1+C^2)^{1/2}}

and it regains its input width at $2z_\mathrm{min}$ . The accumulated dispersion of a length $L$ is $DL$ in ps/nm, and a compensating fiber of dispersion $D_\mathrm{DCF}$ and slope $S_\mathrm{DCF}$ cancels it in

L_\mathrm{DCF} = -\frac{DL}{D_\mathrm{DCF}}

leaving a residual slope $SL + S_\mathrm{DCF}L_\mathrm{DCF}$ in ps/nm², which vanishes when the two fibers have the same relative dispersion slope $S/D$ . The dispersion-limited bit rate uses the criteria of Agrawal's Fiber-Optic Communication Systems (section 2.4.3), which keep the RMS width of the broadened pulse within a quarter of the bit slot. For a source whose spectrum is narrow compared with the signal bandwidth, with the input width chosen to minimise the output width,

B\,(|\beta_2|L)^{1/2} \le \tfrac{1}{4}

and for a source of RMS spectral width $\sigma_\lambda$ that dominates the pulse spectrum,

B L |D| \sigma_\lambda \le \tfrac{1}{4}

The first criterion does not use the pulse width set in the calculator, because it assumes the optimal input width. Both are rules for direct detection without equalization; receivers with electronic equalization, and coherent receivers whose DSP inverts the dispersion, operate far beyond them.

The model is linear: self-phase modulation and the other Kerr effects, polarization-mode dispersion, and fiber loss (which scales the pulse without changing its shape) are left out. The broadening uses $\beta_2$ only; third-order dispersion is reported, with its length scale $L_D' = T_0^3/|\beta_3|$ , and the readout panel says when $L_D'$ is shorter than $L_D$ , which happens close to the zero-dispersion wavelength, where the pulse broadens asymmetrically and the Gaussian result underestimates the width. In direct entry the D(λ) plot extends the set $D$ and $S$ as a straight line.

The fiber presets use published values. Standard single-mode fiber takes $\lambda_0 = 1310$ nm and $S_0 = 0.092$ ps/(nm²·km), within the G.652 range of 1300–1324 nm, and its maximum slope, which gives 17.5 ps/(nm·km) at 1550 nm, inside the 18.0 ps/(nm·km) maximum of Corning SMF-28e. The G.655 preset is Corning LEAF, 4 ps/(nm·km) at 1550 nm and 10 at 1625 nm, hence a slope of 0.08 ps/(nm²·km). The G.653 preset puts $\lambda_0$ at the nominal 1550 nm with $S_0 = 0.085$ ps/(nm²·km). The compensating fiber has $D = -100$ ps/(nm·km), the value of single-clad designs, with a slope of −0.34 ps/(nm²·km) that gives a relative dispersion slope of 0.0034 nm⁻¹, inside the 0.0028–0.0042 nm⁻¹ that OFS quotes for its slope-matched fiber. The O-band preset is the worst case of the 100G CWDM4 MSA: 1264.5 nm on fiber with $\lambda_0 = 1324$ nm and $S_0 = 0.093$ ps/(nm²·km) over 2 km, for which the MSA quotes −11.9 ps/nm.

Worked example

The default preset is standard single-mode fiber at 1550 nm with $\lambda_0 = 1310$ nm and $S_0 = 0.092$ ps/(nm²·km). Since $1310^4/1550^3 = 790.8$ nm, the G.652 formula gives $D = 0.023 \times 759.2 = 17.46$ ps/(nm·km) and $S = 0.0582$ ps/(nm²·km), so $\beta_2 = -22.27\ \mathrm{ps^2/km}$ and $\beta_3 = 0.131\ \mathrm{ps^3/km}$ . An unchirped 25 ps pulse has $T_0 = 15.01$ ps and $L_D = 225.4/22.27 = 10.12$ km. After 50 km, $\xi = -4.940$ and $T_1/T_0 = (1 + 24.40)^{1/2} = 5.040$ : the pulse leaves 126.0 ps wide with 0.198 of its input peak. The accumulated dispersion is 873.0 ps/nm, which 8.73 km of fiber at −100 ps/(nm·km) cancels; the slopes then add to $2.910 - 2.968 = -0.058\ \mathrm{ps/nm^2}$ . For a narrow-linewidth source the dispersion-limited bit rate over 50 km is $1/(4 \times 33.37\ \mathrm{ps}) = 7.49$ Gb/s, and at 10 Gb/s the limit is 28.1 km. The third-order length $L_D'$ is 25,800 km, so leaving $\beta_3$ out changes nothing visible here. These are the values the readout panel shows.

Related: the link budget explorer adds up the loss and penalties of a link, and the glossary entries on frequency chirp and the zero-dispersion wavelength cover the two inputs that most change the result.

References: G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic Press, 2013), sections 3.2 and 3.3. G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2010), sections 2.3 and 2.4. ITU-T Recommendation G.652, Characteristics of a single-mode optical fibre and cable (2016). ITU-T Recommendation G.653, Characteristics of a dispersion-shifted, single-mode optical fibre and cable (2010). Corning SMF-28e optical fiber product information PI1344 (2007). Corning LEAF optical fiber product information PI1107 (2014). 100G CWDM4 MSA Technical Specifications, revision 1.1 (2015). I. P. Kaminow and T. L. Koch, eds., Optical Fiber Telecommunications IIIA (Academic Press, 1997). OFS, “Chromatic dispersion compensation,” IEEE 802.3cs task force presentation (May 2019).