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Fiber-Optic Communication Systems

Govind P. Agrawal

Chapter 9

Soliton Systems - all with Video Answers

Educators

Chapter Questions

Problem 1

A $10-\mathrm{Gb} / \mathrm{s}$ soliton system is operating at $1.55 \mu \mathrm{m}$ using fibers with $D=2 \mathrm{ps} /(\mathrm{km}-$ $\mathrm{nm})$. The effective core area of the fiber is $50 \mu \mathrm{m}^2$. Calculate the peak power and the pulse energy required for fundamental solitons of 30 -ps width (FWHM). Use $n_2=2.6 \times 10^{-20} \mathrm{~m}^2 / \mathrm{W}$.

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Problem 2

The soliton system of Problem 9.1 needs to be upgraded to $40 \mathrm{~Gb} / \mathrm{s}$. Calculate the pulse width, peak power, and the energy of solitons using $q_0=4$. What is the average launched power for this system?

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Problem 3

Verify by direct substitution that the soliton solution given in Eq. (9.1.11) satisfies the NLS equation.

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Problem 4

Solve the NLS equation using the split-step Fourier method (see Section 2.4 of Ref. [10] for details on this method). Reproduce Figs. 9.1-9.3 using your program. Any programming language, including software packages such as Mathematica and Matlab, can be used.

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Problem 5

Verify numerically by propagating a fundamental soliton over 100 dispersion lengths that the shape of the soliton does not change on propagation. Repeat the simulation using a Gaussian input pulse shape with the same peak power and explain the results.

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Problem 6

A $10-\mathrm{Gb} / \mathrm{s}$ soliton lightwave system is designed with $q_0=5$ to ensure wellseparated solitons in the RZ bit stream. Calculate pulse width, peak power, pulse energy, and the average power of the RZ signal assuming $\beta_2=-1 \mathrm{ps}^2 / \mathrm{km}^2$ and $\gamma=2 \mathrm{~W}^{-1} / \mathrm{km}$ for the dispersion-shifted fiber.

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Problem 7

A soliton communication system is designed to transmit data over 5000 km at $B=10 \mathrm{~Gb} / \mathrm{s}$. What should be the pulse width (FWHM) to ensure that the neighboring solitons do not interact during transmission? The dispersion parameter $D=1 \mathrm{ps} /(\mathrm{km}-\mathrm{nm})$ at the operating wavelength. Assume that soliton interaction is negligible when $B^2 L_T$ in Eq. (9.2.10) is $10 \%$ of its maximum allowed value.

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01:11

Problem 8

Prove that the energy of standard solitons should be increased by the factor $G \ln G /(G-1)$ when the fiber loss $\alpha$ is compensated using optical amplifiers. Here $G=\exp \left(\alpha L_A\right)$ is the amplifier gain and $L_A$ is the spacing between amplifiers assumed to be much smaller than the dispersion length.

Ajay Singhal
Ajay Singhal
Numerade Educator

Problem 9

$\mathrm{~A} 10-\mathrm{Gb} / \mathrm{s}$ soliton communication system is designed with $50-\mathrm{km}$ amplifier spacing. What should be the peak power of the input pulse to ensure that a fundamental soliton is maintained in an average sense in a fiber with $0.2 \mathrm{~dB} / \mathrm{km}$ loss? Assume that $T_s=20 \mathrm{ps}, \beta_2=-0.5 \mathrm{ps}^2 / \mathrm{km}^2$ and $\gamma=2 \mathrm{~W}^{-1} / \mathrm{km}$. What is the average launched power for such a system?

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Problem 10

Calculate the maximum bit rate for a soliton lightwave system designed with $q_0=5, \beta_2=-1 \mathrm{ps}^2 / \mathrm{km}$, and $L_A=50 \mathrm{~km}$. Assume that the condition (9.3.10) is satisfied when $B^2 L_A$ is at the $20 \%$ level. What is the soliton width at the maximum bit rate?

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Problem 11

Derive Eq. ( 9.3 .15 ) by integrating Eq. (9.3.11) in the case of bidirectional pumping. Plot $p(z)$ for $L_A=20,40,60$, and 80 km using $\alpha=0.2 \mathrm{~dB} / \mathrm{km}$ and $\alpha_p=0.25 \mathrm{~dB} / \mathrm{km}$.

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01:01

Problem 12

Use Eq. (9.3.15) to determine the pump-station spacing $L_A$ for which the soliton energy deviates at most $20 \%$ from its input value.

Raj Bala
Raj Bala
Numerade Educator

Problem 13

Consider soliton evolution in a dispersion-decreasing fiber using the NLS equation and prove that soliton remains unperturbed when the fiber dispersion decreases exponentially as $\beta_2(z)=\beta_2(0) \exp (-\alpha z)$.

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Problem 14

Starting from the NLS equation (9.4.5), derive the variational equations for the pulse width $T$ and the chirp $C$ using the Gaussian ansatz given in Eq. (9.4.6).

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Problem 15

Solve Eqs. (9.4.7) and (9.4.8) numerically by imposing the periodicity condition given in Eq. (9.4.9). Plot $T_0$ and $C_0$ as a function of $E_0$ for a dispersion map made using 70 km of the standard fiber with $D=17 \mathrm{ps} /(\mathrm{km}-\mathrm{nm})$ and 10 km of dispersion-compensating fiber with $D=-115 \mathrm{ps} /(\mathrm{km}-\mathrm{nm})$. Use $\gamma=2 \mathrm{~W}^{-1} / \mathrm{km}$ and $\alpha=0.2 \mathrm{~dB} / \mathrm{km}$ for the standard fiber and $\gamma=6 \mathrm{~W}^{-1} / \mathrm{km}$ and $\alpha=0.5 \mathrm{~dB} / \mathrm{km}$ for the other fiber.

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Problem 16

Calculate the map strength $S$ and the map parameter $T_{\text {map }}$ for the map used in the preceding problem. Estimate the maximum bit rate that this map can support.

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Problem 17

Verify using Eqs. $(9.5 .8)-(9.5 .12)$ that the variances and correlations of amplifierinduced fluctuations are indeed given by Eqs. (9.5.13)-(9.5.15).

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02:31

Problem 18

Prove that the variances of $E, \Omega$, and $q$ are given by Eq. (9.5.17) for the standard solitons using Eq. (9.5.16) in Eqs. (9.5.8)-(9.5.11).

Ryan Hood
Ryan Hood
Numerade Educator
02:31

Problem 19

Derive Eq. (9.5.29) for the timing jitter starting from the recurrence relation in Eq. (9.5.26). Show all the steps clearly.

Ryan Hood
Ryan Hood
Numerade Educator

Problem 20

Find the peak value of the collision-induced frequency and temporal shifts by integrating Eq. (9.7.8) with $b=1$

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Problem 21

Explain how soliton collisions limit the number of channels in a WDM soliton system. Find how the maximum number of channels depends on the channel and amplifier spacings using the condition $L_{\text {coll }}>2 L_A$.

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