Photonics Essentials: Chapter 9 Problems

Source

Thomas P. Pearsall, Photonics Essentials: An Introduction with Experiments (McGraw-Hill, 2003), Chapter 9, Optical Fibers and Optical Fiber Amplifiers, Problems 9.1–9.4, printed pages 222–224.

The calculations follow the chapter’s simple step-index and transform-limited pulse models. Catalog values are rounded measurements and need not be mutually exact inputs to those models.

Problem 9.1: Corning SMF-28 estimates

The normalized frequency is

\[V=\frac{2\pi a}{\lambda}\mathrm{NA}.\]

Using \(a=8.2/2=4.1\ \mu\mathrm m\) and the NA quoted at \(1310\ \mathrm{nm}\),

\[V_{1310} =\frac{2\pi(4.1)(0.14)}{1.310} =\boxed{2.75}.\]

For weak guidance,

\[\Delta=\frac{n_1-n_2}{n_1} \approx\frac{\mathrm{NA}^2}{2n_1^2}.\]

Taking the stated effective group index as the requested estimate for \(n_1\),

\[\boxed{\Delta\approx \frac{0.14^2}{2(1.4677)^2} =4.55\times10^{-3}=0.455\%}.\]

Equivalently, \(n_1-n_2\approx0.00668\).

An ideal step-index fiber becomes single mode at \(V=2.405\), giving

\[\lambda_c =\frac{2\pi a\,\mathrm{NA}}{2.405} =\boxed{1.50\ \mu\mathrm m}.\]

This literal result conflicts with the data sheet’s identification of the fiber as single-mode at \(1310\ \mathrm{nm}\). It also predicts \(V_{1310}>2.405\). The reason is that nominal core diameter, measured NA, mode-field diameter, and effective group index cannot be combined as exact parameters of a single ideal step-index profile. The calculation is a useful consistency check, not a replacement for the manufacturer’s measured cable cutoff.

Problem 9.2: Material-dispersion distance

Bit periods

\[T_b=\frac1B.\]

Bit rate	2.5 Gbit/s	10 Gbit/s	40 Gbit/s
\(T_b\)	400 ps	100 ps	25 ps
Allowed 50% broadening	200 ps	50 ps	12.5 ps

Dispersion near 1550 nm

A visual linear fit to Figure 9.10 near \(1550\ \mathrm{nm}\) gives approximately

\[\boxed{ M(\lambda)\approx 17+0.07[\lambda(\mathrm{nm})-1550] \ \mathrm{\frac{ps}{nm\,km}} }.\]

The values are graph estimates; using a different pair of well-read points can change the intercept by about \(1\ \mathrm{ps/(nm\,km)}\).

Modulation linewidth

The chapter uses \(\Delta f\approx2/T_b=2B\). Since \(f_0=c/\lambda_0\),

\[\Delta\lambda_{\mathrm{mod}} \approx\lambda_0\frac{\Delta f}{f_0} =\frac{2\lambda_0^2B}{c}.\]

Following the book, add this linearly to the zero-modulation linewidth \(0.3\ \mathrm{nm}\):

Bit rate	\(\Delta\lambda_{\mathrm{mod}}\)	\(\Delta\lambda_{\mathrm{total}}\)
2.5 Gbit/s	0.0401 nm	0.340 nm
10 Gbit/s	0.160 nm	0.460 nm
40 Gbit/s	0.641 nm	0.941 nm

Distance limit

Material-dispersion broadening is

\[\Delta t_M=L\,M\,\Delta\lambda.\]

Overlap begins under the problem’s criterion when \(\Delta t_M=0.5T_b\). Therefore,

\[L_{\max}=\frac{0.5T_b}{M\Delta\lambda}.\]

Using \(M=17\ \mathrm{ps/(nm\,km)}\):

Bit rate	Allowed \(\Delta t_M\)	\(L_{\max}\)
2.5 Gbit/s	200 ps	34.6 km
10 Gbit/s	50 ps	6.39 km
40 Gbit/s	12.5 ps	0.781 km

The result isolates material dispersion as requested; actual SMF-28 total chromatic dispersion also includes waveguide dispersion.

Problem 9.3: Inferring Samsung fiber geometry

Numerical aperture

The specified relative index difference is \(\Delta=0.0034\). With \(n_1=1.4690\),

\[n_2=n_1(1-\Delta)=1.4640,\]

\[\mathrm{NA}=\sqrt{n_1^2-n_2^2} =\boxed{0.121}.\]

Core-diameter trials

For the mode-field diameter, use the chapter’s Gaussian approximation

\[\frac{\mathrm{MFD}}{d} =0.65+\frac{1.619}{V^{3/2}}+\frac{2.879}{V^6}, \qquad V=\frac{\pi d\,\mathrm{NA}}{\lambda}.\]

Assumed \(d\)	\(V_{1310}\)	MFD at 1310 nm	\(V_{1550}\)	MFD at 1550 nm
8.5 um	2.47	9.18 um	2.09	10.39 um
9.0 um	2.61	9.38 um	2.21	10.52 um
Specified		9.3 um		10.5 um

Both guesses are reasonable, but \(9.0\ \mu\mathrm m\) has the smaller combined error. A least-squares match to both MFD values gives approximately \(d=8.84\ \mu\mathrm m\).

The corresponding ideal step-index cutoff wavelengths are

\[\lambda_c=\frac{\pi d\,\mathrm{NA}}{2.405} =1.34\ \mu\mathrm m\quad(d=8.5\ \mu\mathrm m)\]

and \(1.42\ \mu\mathrm m\) for \(d=9.0\ \mu\mathrm m\). The book asks for the “longest” single-mode wavelength, but the model has no such upper limit: \(V\) falls as wavelength rises. These are the shortest single-mode wavelengths predicted by the model.

Simple fitting routine

This dependency-free Python routine searches for the diameter that best matches both catalog MFD values:

import math

na = 0.1210338
samples = ((1.310, 9.3), (1.550, 10.5))  # wavelength, MFD in um

def predicted_mfd(diameter, wavelength):
    v = math.pi * diameter * na / wavelength
    ratio = 0.65 + 1.619 / v**1.5 + 2.879 / v**6
    return diameter * ratio

best = None
for step in range(50_001):
    d = 6.0 + step * 0.0001
    error = sum(
        (predicted_mfd(d, wavelength) - measured) ** 2
        for wavelength, measured in samples
    )
    if best is None or error < best[0]:
        best = (error, d)

print(best[1])  # approximately 8.84 um

Dispersion distance

Use the same linewidth model as Problem 9.2, now with the specified \(M=18\ \mathrm{ps/(nm\,km)}\) at \(1550\ \mathrm{nm}\):

Bit rate	Allowed broadening	Distance
2.5 Gbit/s	200 ps	32.7 km
10 Gbit/s	50 ps	6.04 km
40 Gbit/s	12.5 ps	0.738 km

Problem 9.4: WDM information bandwidth

The frequency width of the 1530–1560 nm window is more accurate than treating all 30 nm as one constant wavelength conversion:

\[\Delta f_{\mathrm{window}} =c\left(\frac1{1530\ \mathrm{nm}}-\frac1{1560\ \mathrm{nm}}\right) =3.768\ \mathrm{THz}.\]

Using the chapter’s transform-limited convention, a bit rate \(B\) requires optical bandwidth \(2B\). Channel centers are separated by \(1.5(2B)=3B\).

Bit rate/channel	Optical bandwidth	Equivalent width near 1550 nm	Spacing	Whole channels	Aggregate rate
10 Gbit/s	20 GHz	0.160 nm	30 GHz	125	1.25 Tbit/s
40 Gbit/s	80 GHz	0.641 nm	120 GHz	31	1.24 Tbit/s

The nearly equal totals are expected:

\[N_{\mathrm{ch}}B \approx\frac{\Delta f_{\mathrm{window}}}{3B}B =\frac{\Delta f_{\mathrm{window}}}{3}.\]

ITU-T comparison

ITU-T G.694.1 defines a frequency grid, not one mandatory bit rate per channel. Its fixed-grid examples include 12.5, 25, 50, and 100 GHz spacings, anchored at 193.1 THz; its flexible grid uses 6.25 GHz center-frequency granularity and 12.5 GHz slot-width granularity. See the official ITU-T G.694.1 summary.

The ideal 30 GHz spacing is not on the listed fixed examples and the 120 GHz requirement is not a 12.5 GHz slot multiple. On the flexible grid they would round up to 37.5 and 125 GHz. Guard bands, filter shape, laser drift, dispersion, modulation format, and forward-error correction determine the deployable capacity, so the simple result is an upper-level engineering estimate rather than an ITU-compliant network design.