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3 Optical Signal Attenuation and Dispersion
3.9 Assume that a step-index fiber has a V number of 6.0. (a) Using Fig. 3.5,
estimate the fractional power P clad /P traveling in the cladding for the four
lowest-order LP modes. (b) If the fiber in (a) is a glass-core glass-clad fiber
having core and cladding attenuations of 3.0 and 4.0 dB/km, respectively,
find the attenuations for each of the four lowest-order modes.
3.10 Assume a given mode in a graded-index fiber has a power density p(r) = P 0
exp(−Kr
2 ), where the factor K depends on the modal power distribution.
(a) Letting n(r) in Eq. (3.16) be given by Eq. (2.38) with α = 2, show that
the loss in this mode is
α gi = α 1 +
α 2 − α 1
K a 2
Because p(r) is a rapidly decaying function of r and because 1,
for ease of calculation assume that the top relation in Eq. (2.38) holds
for all values of r.
(b) Choose K such that p(a) = 0.1 P 0 ; that is, 10% of the power flows in
the cladding. Find α gi in terms of α 1 and α 2 .
3.11 A 5 km transmission link consists of a step-index multimode fiber that has a
core index n 1 = 1.480 and a core-cladding refractive index difference =
0.01.
(a) Using the approximation on the right-hand side of Eq. (3.15), show that
the delay difference between the fastest and slowest modes is 247 ns.
(b) Using Eq. (3.14a), show that the rms pulse broadening resulting from
intermodal delay is 71.2 ns.
(c) Using Eq. (3.20) and the condition that the maximum bit rate B should
satisfy the condition B < 0.1/T, show that the maximum bit ratedistance product is BL = 4.05 (Mb/s) km.
3.12 For wavelengths less than 1.0 μm the refractive index n satisfies a Sellmeier
relation of the form
n
2
= 1 +
E 0 E d
E
2
0 − E 2
where E = hc/λ is the photon energy and E 0 and E d are, respectively, material oscillator energy and dispersion energy parameters. In SiO 2 glass, E 0 =
13.4 eV and E d = 14.7 eV. Show that, for wavelengths between 0.20 and
1.0 μm, the values of n found from the Sellmeier relation are in good agreement with those shown in Fig. 3.9. To make the comparison, select three
representative points, for example, at 0.2, 0.6, and 1.0 μm.
3.13 (a) An LED operating at 850 nm has a spectral width of 45 nm. If the material
dispersion at that wavelength is 115 ps/(nm km), what is the pulse spreading
in ns/km due to material dispersion? (b) Suppose the material dispersion for
3 Optical Signal Attenuation and Dispersion
3.9 Assume that a step-index fiber has a V number of 6.0. (a) Using Fig. 3.5,
estimate the fractional power P clad /P traveling in the cladding for the four
lowest-order LP modes. (b) If the fiber in (a) is a glass-core glass-clad fiber
having core and cladding attenuations of 3.0 and 4.0 dB/km, respectively,
find the attenuations for each of the four lowest-order modes.
3.10 Assume a given mode in a graded-index fiber has a power density p(r) = P 0
exp(−Kr
2 ), where the factor K depends on the modal power distribution.
(a) Letting n(r) in Eq. (3.16) be given by Eq. (2.38) with α = 2, show that
the loss in this mode is
α gi = α 1 +
α 2 − α 1
K a 2
Because p(r) is a rapidly decaying function of r and because 1,
for ease of calculation assume that the top relation in Eq. (2.38) holds
for all values of r.
(b) Choose K such that p(a) = 0.1 P 0 ; that is, 10% of the power flows in
the cladding. Find α gi in terms of α 1 and α 2 .
3.11 A 5 km transmission link consists of a step-index multimode fiber that has a
core index n 1 = 1.480 and a core-cladding refractive index difference =
0.01.
(a) Using the approximation on the right-hand side of Eq. (3.15), show that
the delay difference between the fastest and slowest modes is 247 ns.
(b) Using Eq. (3.14a), show that the rms pulse broadening resulting from
intermodal delay is 71.2 ns.
(c) Using Eq. (3.20) and the condition that the maximum bit rate B should
satisfy the condition B < 0.1/T, show that the maximum bit ratedistance product is BL = 4.05 (Mb/s) km.
3.12 For wavelengths less than 1.0 μm the refractive index n satisfies a Sellmeier
relation of the form
n
2
= 1 +
E 0 E d
E
2
0 − E 2
where E = hc/λ is the photon energy and E 0 and E d are, respectively, material oscillator energy and dispersion energy parameters. In SiO 2 glass, E 0 =
13.4 eV and E d = 14.7 eV. Show that, for wavelengths between 0.20 and
1.0 μm, the values of n found from the Sellmeier relation are in good agreement with those shown in Fig. 3.9. To make the comparison, select three
representative points, for example, at 0.2, 0.6, and 1.0 μm.
3.13 (a) An LED operating at 850 nm has a spectral width of 45 nm. If the material
dispersion at that wavelength is 115 ps/(nm km), what is the pulse spreading
in ns/km due to material dispersion? (b) Suppose the material dispersion for