Optical Networks: A Practical Perspective - Part 39 pot

350 TRANSMISSION SYSTEM ENGINEERING 5.3 5.4 5.5 (a) You have a transmitter that operates at a wavelength of 1.55/~m, has a spec- tral width of I nm, and an output power of 0.5 mW. The receiver requires -30 dBm of input power in order to achieve the desired bit error rate. What is the length of the longest link that you can build? (b) You have another transmitter that operates at a wavelength of 1.3/zm, has a spectral width of 2 nm, and an output power of i mW. Assume the same receiver as before. What is the length of the longest link that you can build? (c) You have the same 1.3 #m transmitter as before, and you must achieve an SNR of 30 dB using an APD receiver with a responsivity of 8 A/W, a gain of 10, an excess noise factor of 5 dB, negligible dark current, a load resistance of 50 f2, and an amplifier noise figure of 3 dB. Assume that a receiver bandwidth of B/2 Hz is sufficient to support a bit rate of B b/s. What is the length of the longest link you can build? (d) Using the same 1.3/~m transmitter as before, you must achieve an SNR of 20 dB using a pin receiver with a responsivity of 0.8 A/W, a load resistance of 300 f2, and an amplifier noise figure of 5 dB. Assume that a receiver bandwidth of B/2 Hz is sufficient to support a bit rate of B b/s. What is the length of the longest link you can build? Compute the dispersion-limited transmission distance for links with standard single-mode fiber at 1550 nm as a function of the bit rate (100 Mb/s, 1 Gb/s, and 10 Gb/s) for the following transmitters: (a) a Fabry-Perot laser with a spectral width of 10 nm, (b) a directly modulated DFB laser with a spectral width of 0.1 nm, and (c) an externally modulated DFB laser with a spectral width of 0.01 nm. Assume that the modulation bandwidth equals the bit rate and the dispersion penalty is 2 dB. Assume that NRZ modulation is used. Repeat Problem 5.3 for NZ-DSF assuming a dispersion parameter of 5 ps/nm-km. Consider a length L of step-index multimode fiber having a core diameter of 50/zm and a cladding diameter of 200/zm. The refractive indices of the core and cladding are 1.50 and 1.49, respectively. A fixed-wavelength, 1310 nm DFB laser (operating at 0 dBm) is used at one end of the fiber to serve as a 155.52 Mb/s transmitter source. At the far end, a photodetector is used as a receiver. Assume that NRZ modulation is used. (a) Draw and label a diagram that illustrates the above configuration. (b) What would be the corrugation period of the DFB laser at this wavelength? (c) Compute the numerical aperture for this fiber. (d) What would be the maximum acceptable fiber length when operating at this bit rate? (e) Assuming an attenuation of 0.40 dB/km, what would be the output power (in dBm) at the receive end of the fiber? Problems 3 51 5.6 5.7 5.8 (f) Assuming a perfectly efficient photodetector, what would be the resulting photocurrent? (g) If we instead used single-mode fiber for this application, what would be the new requirement on its core diameter? Note that this problem requires you to understand the material in Chapters 2, 3, and 4 as well. Consider a passive WDM link of length L, consisting of single-mode fiber into which five wavelengths are launched through an optical combiner such that the aggregate launch power at its output is 5 mW. These five wavelengths are centered on the 193.1 THz ITU grid, with uniform 100 GHz interchannel spacing. The transmitters all use directly modulated DFB lasers with a spectral width of 0.1 nm. Each channel is transporting a SONET OC-48 (2.5 Gb/s) signal. At the end of this link, the channels are optically demultiplexed and are each received by a direct detection pin receiver. For this problem, neglect all losses and crosstalk associated with the WDM mux and demux. Assume that NRZ modulation is used. (a) Draw and label a diagram illustrating this configuration. (b) Calculate the wavelengths (in nm, to two decimal places) associated with these five channels. (c) Calculate the average launch power per channel at the output of the WDM combiner. (d) Assuming OtdB 0.25 dB/km, D - 17 ps/nm-km, and DpMD 0.5 ps/k~/-k-mm, calculate the worst-case dispersion, PMD, and loss limits for this link. (e) What is the maximum value of L meeting all of these requirements? Consider a point-to-point link connecting two nodes separated by 60 km. This link was constructed with standard single-mode fiber, and a 2.5 Gb/s system is deployed over the link. The transmitter uses a directly modulated 1310 nm DFB laser. The receiver uses perfectly efficient pin photodiodes, and we will assume, for this problem, that they can be modeled as ideal receivers. The bit error rate requirement for this system is 10 -12. Assume lYdB 0.4 dB/km and that NRZ modulation is used. (a) Draw and label a diagram illustrating this configuration. (b) Is this system loss limited or dispersion limited? Briefly explain your rea- soning. (c) What is the required receiver sensitivity (in mW and dBm)? (d) What would be the resulting photocurrent? (e) What would be the required launch power (in dBm)? The link of Problem 5.7 is at full capacity, and we must design a solution that will enable capacity expansion and accommodate further growth. After consid- ering the options, we determine that the most cost-effective solution is to add a 1550 nm point-to-point system over the existing set of fibers, thereby realizing a 352 TRANSMISSION SYSTEM ENGINEERING 5.9 5.10 5.11 5.12 5.13 5.14 two-wavelength (1310 nm/1550 nm) passive WDM configuration. Assume that 3 dB couplers are used to combine the two signals just after the transmitters and separate the two signals just before the receivers. The next step is to determine what bit rate can be supported by this 1550 nm system. Assume that the 1550 nm transmitter uses a directly modulated DFB laser (with spectral width of 0.1 nm). At 1550 nm, assume OldB 0.25 dB/km, D - 17 ps/nm-km, and DpMD 1 ps/k~/-kmm. (a) Draw and label a diagram illustrating this new configuration. (b) What is the launch power now required for the original 2.5 Gb/s 1310 nm system to maintain the same level of receiver performance? (c) If we assume an ideal receiver with the same 10 -12 bit error rate performance for the 1550 nm system, determine the associated receiver sensitivities for both 2.5 Gb/s and 10 Gb/s signals. (d) Calculate bit rate limits based on loss, dispersion, and PMD for the new system. (e) Can 10 Gb/s be suitably transported by this new system? Briefly explain your reasoning. (f) For the 2.5 Gb/s and 10 Gb/s (if it is possible) line rates, calculate the required launch power to successfully transport the signal. Derive equation (5.4). Show that the extinction ratio penalty in amplified systems limited by signal- spontaneous beat noise and spontaneous-spontaneous beat noise is (~-1 ~/r + 1) PP = - 10 log +1,/7+1 Assume that other noise terms are negligible. Consider the amplifier chain discussed in Section 5.5.3. Using equations (5.6) and (5.7), compute the steady-state values of Pout and G in a long chain of amplifiers. Assume Gmax 35 dB, l - 120 km, c~ = 0.25 dB/km, nsp 2, psat __ 10 mW, and Bo - 50 GHz. How do these values compare against the unsaturated gain Gmax and the output saturation power of the amplifier iDsat) Plot the evolution of the signal 9 out" power and optical signal-to-noise ratio as a function of distance along the link. Derive equations (5.11), (5.12), (5.13), and (5.14) when there are N interfering signals rather than just one. Why is equation (5.24) an approximation? Derive a precise form of this equation. Consider the WDM link shown in Figure 5.1. Each multiplexer and demultiplexer in- troduces crosstalk from adjacent channels that is C dB below the desired channel. Problems 353 Figure 5.36 A node in a WDM network for Problems 5.16-5.19. 5.15 5.16 5.17 (a) Compute the crosstalk at the output when N such stages are cascaded. (b) What must C be so that the overall crosstalk penalty after five stages is less than 1 dB? Consider a WDM system with W channels, each with average power P and extinction ratio P1/P0 = r. Derive the interchannel crosstalk power penalty in (5.13) for this system compared to a system with ideal extinction and no crosstalk. What should the crosstalk level be for a maximum 1 dB penalty if the extinction ratio is 10 dB? Consider the WDM network node shown in Figure 5.36. Assume the node has two inputs and two outputs. The multiplexers/demultiplexers are ideal (no crosstalk), but each switch has a crosstalk level C dB below the desired channel. Assume that in the worst case, crosstalk in each stage adds coherently to the signal. (a) Compute the crosstalk level after N nodes. (b) What must C be so that the overall crosstalk penalty after five nodes is less than 1 dB? Consider the WDM network node shown in Figure 5.36. Assume the node has two inputs and two outputs. The mux/demuxes have adjacent channel crosstalk suppressions of-25 dB, and crosstalk from other channels is negligible. The switches have a crosstalk specification of-40 dB. How many nodes can be cascaded in a network without incurring more than a 1 dB penalty due to crosstalk? Consider only intrachannel crosstalk from the switches and the multiplexers/demultiplexers. 354 TRANSMISSION SYSTEM ENGINEERING 5.18 5.19 5.20 5.21 5.22 Consider a WDM system with N nodes, each node being the one shown in Fig- ure 5.36. The center wavelength )~" for each channel in a mux/demux has an ac- curacy of +A~. nm around the nominal center wavelength )~c. Assume a Gaussian passband shape for each channel in a mux; that is, the ratio of output power to input power, called the transmittance, is given by (~_~.~)2 TR()~) - e 2,2 where cr is a measure of the channel bandwidth and U c is the center wavelength. This passband shape is typical for an arrayed waveguide grating. (a) Plot the worst-case and best-case peak transmittance in decibels as a function of the number of nodes N for cr = 0.2 nm, A~. = 0.05 nm. Assume that the laser is centered exactly at )~c. (b) What should A)~ be if we must have a worst-case transmittance of 3 dB after 10 nodes? Consider a system with the same parameters as in Problem 5.18. Suppose the WDM channels are spaced 0.8 nm apart. Consider only crosstalk from the two adjacent channels. Compute the interchannel crosstalk power relative to the signal power in decibels, as a function of N, assuming all channels are at equal power and exactly centered. Compute the crosstalk also for the case where the desired channel is exactly centered at )vi, but the adjacent channels are centered at ~-i-1 q- A~, and ~.i+1 A)v. Consider the simple add/drop element shown in Figure 3.14(b). Suppose we use another circulator instead of the coupler shown in the figure to add the wavelength. This eliminates the loss due to the coupler. Let the input power on the wavelength to be dropped be -30 dBm and the transmitted power on the added wavelength be 0 dBm. Suppose the grating has a reflectivity of 99%. Compute the intrachannel crosstalk power arising from (a) leakage of the added wavelength into the dropped wavelength, and (b) leakage of the dropped wavelength into the added wavelength. Assume that each circulator has a loss of I dB. Will the element work? Show that the optimum choice of the pulse width of an unchirped Gaussian pulse (with narrow spectral width) that minimizes the pulse-broadening effects of chro- matic dispersion over a fiber of length L is To~ pt- v//~2L. If 0 <_ E < 1 is the power-splitting ratio between the two polarization components, the random power penalty in decibels due to PMD is related to the random differential Problems 355 5.23 5.24 5.25 5.26 time delay as Ar 2 PP(dB) - ot-~~(1 -~), where T is the bit period and c~ is a constant depending on the pulse shape and takes values in the range 12-25 f.qr commonly studied pulse shapes [KK97a, Chapter 6]. Note that we have already taken logarithms in the above equation. Thus the random variable PP is a function of the random variables Ar and ~. Assuming a Maxwellian distribution for Ar with mean (At) and a uniform distribution for 6, show that PP has an exponential distribution. What is the mean value of PP? What is the probability that PP >__ 1 dB? Neglecting the depletion of the pump wave, solve (2.14) and (2.15) to obtain the evolution of the SBS pump and Stokes waves. Compute the SBS threshold power for the following systems" (a) a single-channel system using a Fabry-Perot laser with 10 lines, each line having a modulated line width of 1 GHz, (b) a multichannel system with a DFB laser having a modulated line width of 1 GHz, and (c) same as (b) except that the line width is 10 GHz. Consider (5.27) as expressing TL, the pulse width after a distance L, in terms of the initial pulse width To. (a) As in the case of chromatic dispersion, there is an optimum initial pulse width (for a given link length L). Find an expression for this optimum initial pulse width. (b) Assuming a pulse with this optimum width is used, find the maximum link length for a power penalty of 1 dB. Note that this power penalty is due to both SPM and chromatic dispersion. (c) Assume that a pulse of the same initial width is used but that the link has no SPM but only chromatic dispersion. Using (2.13), calculate the pulse width at the end of the link and hence the penalty due to chromatic dispersion. The remainder of the 1 dB penalty is due to SPM. Note that the SPM penalty can be negative for some combinations of link, dispersion, and nonlinear lengths. This occurs when the initial pulse compression due to the chirping caused by SPM results in a narrower pulse at the end of the link, compared to the case when SPM is absent and only chromatic dispersion is present. You are required to design a four-wavelength transmission system operating over dispersion-shifted fiber. The four wavelengths are to be placed in a band from 193.1 THz to 194.1 THz. The possible slots are spaced 100 GHz apart in this band. Pick the four wavelengths carefully so that no four-wave mixing component falls on any of the chosen wavelengths. 356 TRANSMISSION SYSTEM ENGINEERING 5.27 5.28 5.29 5.30 Compute and plot the four-wave mixing limit on the transmit power per channel for a WDM system operating over NZ-DSF. Assume that the channels are equally spaced and transmitted with equal power, and the maximum allowable penalty due to FWM is 1 dB. For the fiber, assume the dispersion parameter D = 3 ps/nm-km in the middle of the transmitted band of channels, and the slope of the dispersion curve is dD/dk = 0.055 ps/nm-km 2. Consider the same three cases as in Figure 5.30: (a) 8 channels spaced 100 GHz apart, (b) 32 channels spaced 100 GHz apart, and (c) 32 channels spaced 50 GHz apart. Why do second-order nonlinearities typically not affect a lightwave system? In discussing the chromatic dispersion penalty, the Telcordia standard for SONET systems [Tel99] specifies the spectral width of a pulse, for single-longitudinal mode (SLM) lasers, as its 20 dB spectral width divided by 6.07. We studed these lasers in Section 3.5.1. Show that for SLM lasers whose spectra have a Gaussian profile, this is equivalent to the rms spectral width. For a narrow but chirped Gaussian pulse with chirp factor K = -6, calculate the chromatic dispersion limit at a bit rate of i Gb/s, in the 1.55/zm band, for a penalty of 2 dB. Compare this with the chromatic dispersion limit for unchirped pulses plotted in Figure 5.19. References [Agr95] G.P. Agrawal. Nonlinear Fiber Optics, 2nd edition. Academic Press, San Diego, CA, 1995. [BA94] E Bruy4re and O. Audouin. Assessment of system penalties induced by polarization mode dispersion in a 5 Gb/s optically amplified transoceanic link. IEEE Photonics Technology Letters, 6(3):443-445, March 1994. [Bak01] B. 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Gain control in erbium-doped fiber amplifiers by an all-optical feedback loop. Electronics Letters, 27:560, 1991. . channel bandwidth and U c is the center wavelength. This passband shape is typical for an arrayed waveguide grating. (a) Plot the worst-case and best-case peak transmittance in decibels as a. ps/nm-km 2. Consider the same three cases as in Figure 5.30: (a) 8 channels spaced 100 GHz apart, (b) 32 channels spaced 100 GHz apart, and (c) 32 channels spaced 50 GHz apart. Why do second-order. values compare against the unsaturated gain Gmax and the output saturation power of the amplifier iDsat) Plot the evolution of the signal 9 out" power and optical signal-to-noise ratio