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This paper presents an all-optical, in-band optical signal-to-noise ratio (OSNR) and chromatic dispersion monitor. We demonstrate monitoring over the 1 nm bandwidth of our signal, which is a 10 GHz pulse train of 8.8 ps pulses. The monitor output power, as measured on a slow detector, has a 1.9 dB dynamic range when the signal OSNR is varied by 20 dB, and a 1.6 dB dynamic range when ±150 ps/nm of chromatic dispersion is applied. Cascaded four-wave mixing occurring in the optical parametric amplifier provides the nonlinear power transfer function responsible for the monitoring. An analysis using the signals’ probability density functions show that the nonlinear power transfer function provides preferential gain to clean undispersed pulses when compared to noisy and/or dispersed pulses. Our analysis includes a consideration of the applicability of the device to high duty cycle systems, and simulations on monitoring of a 40 Gb/s pulse train with a 50% duty cycle.
©2005 Optical Society of America
1. Introduction
As the data rate of optical communications signals rise to accommodate an increasing number of users, the associated signal pulse widths and separations decrease. The reduced tolerance for degradation in these systems require signal processing devices that are highly accurate to maintain signal quality [1]. Signal performance monitors can control compensation devices by providing real-time information on signal impairments [2, 3, 4, 5]. For example, a chromatic dispersion monitor can adjust a tunable dispersion compensator in a feedback loop to minimise residual dispersion [2, 6]. Signal performance monitors can also track system degradation, providing fault warnings. Even more useful, are signal performance monitors which can differentiate between different fault mechanisms such as loss, dispersion or noise. Because these monitors are more specific, they help identify the cause of the degradation and hence assist in isolating and locating faults.
The monitoring of amplified spontaneous emission (ASE) noise and chromatic dispersion, two important degradation mechanisms, would assist in dynamic network optimisation and real time fault analysis. ASE is inherently introduced by optical amplifiers. The optical signal-to noise ratio (OSNR) - a measure of ASE noise - is an important indicator of signal quality. The standard spectral technique to determine OSNR is to measure the ASE power levels adjacent to the signal and interpolate into the signal’s bandwidth, for example see Ref. [7]. This measurement can be inaccurate because signals are filtered in network routers suppressing the ASE adjacent to the signal [3]. Thus a direct measurement which measures OSNR within the signal’s bandwidth would be preferable. Chromatic dispersion arises when frequency components in a pulse separate because of their different group velocities. At high bit rates, not only is the dispersion greater, but there is less space between pulses making signals less tolerant to dispersion induced pulse broadening. Thus dispersion compensators need to be precisely controlled to avoid loss of data [8].
While effective schemes have been developed to measure in-band OSNR or chromatic dispersion for current optical transmission networks [9, 10, 11], they use fast electronic detection of the signal. As data rates increase, signals become increasingly difficult to process and monitor because of the high temporal resolution required, and it becomes increasingly impractical or even impossible to use fast electronic detection [2, 4, 12]. Alternatively all-optical monitors that do not require fast electronics, such as in Ref. [13], are highly attractive for bit rates of 40 Gb/s and beyond.
A class of all-optical techniques uses ultrafast nonlinear optics to probe the signal. These techniques can give ultrahigh resolution because the nonlinear Kerr effect is an intensity dependent process with a femtosecond response time. Methods using the Kerr or other nonlinear effects have been successful in measuring OSNR or dispersion of short pulses. Recently Vorreau et al. presented initial results demonstrating monitoring of dispersion and OSNR using a semiconductor optical amplifier [12]. In this scheme, cross-phase and cross-gain modulation occurs between the data signal and a continuous wave in the semiconductor optical amplifier causing a red frequency shift. The amount of red-shift in the spectrum is reduced for dispersed or noisy signals and so the spectrum can be filtered to provide a monitoring signal. Similar schemes are used in Ref. [14], which also filters a dispersion dependent red-shift, and Ref. [2]in which spectral broadening from self-phase modulation is dispersion dependent. Other all-optical monitoring devices proposed include one using aperiodically poled lithium niobate waveguides to generate a second harmonic generation signal [15], and another using 2-photon absorption to obtain a nonlinear power transfer function [4].
Following our initial report [16], in this paper we present detailed measurements of an all-optical device for monitoring in-band OSNR and chromatic dispersion with the one measurement. The presented monitoring device relies on ultrafast nonlinearities to detect the change in the temporal power distribution of clean and of degraded pulses. It does this by using a nonlinear power transfer function (PTF) which gives preferential gain to distributions which have high instantaneous powers, namely clean pulses. Such a nonlinear PTF maps the power distribution information onto the average power of the output, which is easily measured with a slow detector [16]. Both dispersion and increased ASE levels in an optical signal change the power distribution of the signal and can be monitored by this method. Here we demonstrate in-band OSNR and chromatic dispersion monitoring experiments on a 10 GHz pulse train of 8.8 ps pulses. Furthermore, the device which is based on an optical parametric amplifier (OPA) simultaneously provides gain to the signal wavelength and wavelength conversion to an idler wave [5, 6, 17, 18]. Finally, we consider the monitoring of high duty cycle pulse trains as applicable to standard return-to-zero (RZ) optical communications channels. An analysis of signal power distributions is used to understand the resulting behaviour, while simulations on 40 Gb/s RZ signals with a 50% duty cycle are presented showing they are still able to be monitored by this method.
2. Device concept
In this section we describe how an optical parametric amplifier with a nonlinear PTF can be used to perform in-band OSNR and chromatic dispersion monitoring.
2.1. Role of nonlinear power transfer function in OSNR monitoring
Figure 1 summarizes the principle of using a nonlinear PTF to achieve OSNR monitoring. Figure 1(top) shows two signals - clean pulses and pure noise. The signals are transformed by a quadratic PTF. Figure 1(middle) shows the gain as a function of input power for the nonlinear PTF. That is, for a given instantaneous input power Fig. 1(middle) shows how much gain that part of the input will experience. We see that for this PTF high gains are obtained for high input powers and low gains for low input powers. This is further indicated by the shading which is darker at high gains. Therefore, although the two inputs at the top have the same average power, the pulses can be seen to reach higher gain and receive more average gain than the noise. In Fig. 1(bottom), the output signals are shown. The pulses, which receive more average gain, produce an output with a larger average power than the output produced by the noise. Thus, a simple average power measurement of the transformed signal with a slow detector can then distinguish between pure noise and a clean pulsed input, or the noise fraction in a real signal.
Fig. 1. (top) Pulses reach higher gains than noise of the same average power. The pulses receive more average gain and a larger average output. (middle) Schematic illustration of gain as a function of instantaneous input power for a nonlinear power transfer function. High instantaneous input powers receive high gains. (bottom) Output signals have different average powers because of the different gain received.
This idea can be described mathematically using a probability density function (pdf), which illustrates the power distribution of the signal over all power values. Areas of high density of the pdf are where most of the input signal power lies. That is, the probability, P, of the signal having an instantaneous power p between the values pa and pb is given by
where pdfin is the pdf of the input signal. The average power of the signal transformed by the PTF, can be found by multiplying the device PTF with the pdf of the input and integrating [19]. That is,
Figure 2 is an animation showing the change in the signal power distribution for decreasing OSNR and how it changes the average output power. The input signal is a sech2 pulse train of 8.8 ps pulses (full width at half maximum) with added ASE. These parameters are intended to match to our experiment, to be described in Section 3. In each frame, the right panel shows the pdf of the input pulse train (dashed line), a nonlinear PTF given by pout= (dotted line) and the product of the pdf with the PTF (solid line). The shaded area under the solid curve is the average output power given by Eq. 2. The left panel plots the average output power (that is, the shaded area) against OSNR with a circle around the data point corresponding to the plots in the right panel.
Fig. 2. Frame 1 of the animation. (left) The plot of average output power against OSNR. (right) The pdf of the input (dashed line), a quadratic PTF (dotted line) and the product of the pdf with the PTF (solid line). The shaded area indicates the average output power. [Media 1]
At high OSNRs (clean pulses) the pdf of the signal has two peaks. There is a large peak in the pdf at zero input power, because large spaces between pulses means that the instantaneous power has a high probability of being zero. The smaller peak at 5.6 mW corresponds to the maximum power of the pulses. The rounded top of the pulses gives the instantaneous power a shallow slope and thus a high probability of occurring. The peak at high power is important because it receives the most gain, as can be seen by the matching peak on the solid line. As the OSNR decreases, that is as the pulses get noisier, the peak power of the pulses decrease and the power distribution at the peak power broadens. This combination causes the pdf of the pulse train to move towards the left where the PTF is small. As a result, even though the average power of the input is constant at 0.67 mW, the average output power decreases as the input OSNR decreases.
2.2. Dispersion monitoring
The principle of the dispersion monitoring is the same as for OSNR monitoring [5, 16]. As a pulse disperses, it broadens temporally and decreases in peak power. The probability density function of the pulse train is then compressed from the right to lower powers. Figure 3 are plots showing the affect of dispersion on the pdf of a pulse train and how dispersion effects the average output power a pulse train produces. The probability density shifts from the upper end of the power transfer function to the lower end, results in less gain and hence produces less output power.
Fig. 3. (left) The pdf of the undispersed pulse train (dashed line), a quadratic power transfer function (dotted line) and the product of the pdf with the power transfer function (solid line). The shaded area indicates the average output power. (right) The pulse train is dispersed resulting in pulse broadening and compression of its pdf to lower powers.
2.3. Cascaded four-wave mixing in the optical parametric amplifiers
We obtain a nonlinear PTF using four-wave mixing in OPAs. Four-wave mixing occurs when a strong pump and a weak signal are launched into a nonlinear fibre. When phase matching conditions are met, parametric gain transfers power from the pump wavelength to the signal wavelength, amplifying the signal and efficiently generating an idler wave [5]. At higher signal input powers, this process also generates additional wavelengths through cascaded four-wave mixing. Figure 4 is the optical spectrum at the output of the OPA, showing a strong pump and amplified signal. The generated idler wave and two cascaded four-wave mixing waves C1 and C2 are also present. The wavelengths generated by cascaded four-wave mixing have powers nonlinearly dependent on the signal input power [5]. Thus by filtering out the cascade wave C1 at the output, a device with a nonlinear transfer function is achieved.
Fig. 4. Output of the optical parametric amplifier. Only the signal and pump waves are launched, but many waves are generated.
3. Experiment
The experimental setup is shown in Fig. 5. The ASE noise source is produced by cascading two erbium-doped fibre amplifiers (EDFA). The pulsed source provides a 10 GHz pulse train of 8.8 ps pulses (FWHM) at 1555 nm. The pulse train has a duty cycle of 9%. Both the pulsed and ASE sources are filtered using 1 nm bandpass filters with matched wavelengths before being combined using a 50/50 coupler. Variable attenuators in the pulsed and ASE arms allow the OSNR to be varied. The average combined signal launch power is kept constant at 0.67 mW. The continuous wave OPA pump has a wavelength of 1550.2 nm, launch power of 960mW, and it is dithered with a phase modulator to suppress stimulated Brillouin scattering. The nonlinear medium is 1.5 km of dispersion shifted fibre with a nonlinearity of 2±0.6 W-1km-1, zero-dispersion wavelength of 1549±2 nm and dispersion slope of 0.0816 ps/nm2/km. The output of the nonlinear fibre is filtered with a 1 nm bandpass filter to select only the cascade wave, and its average power is measured. The output is also monitored on an optical spectrum analyser.
Fig. 5. The experimental setup. TDE: Tunable dispersive element, BPF: Bandpass filter, VA: Variable attenuator, C: Coupler, PS: Polarisation scrambler, DSF: Dispersion shifted fibre, ATT: Attenuator.
A polarisation scrambler situated between point ‘A’ and the OPA acts to depolarise the signal, making our method polarisation independent. In our experiment, we used a resonant fibre coil based polarisation scrambler (Advantest Q8163) which produced a large differential group delay of 20 ps. The undesired differential group delay introduced by this element, separated the 8.8 ps input pulses into two sub-pulses of lower peak power. Thus, our launched signal actually had a duty cycle of ≈18%, instead of the 9% duty cycle pulse train emitted from the laser. This pulse splitting could be avoided in future experiments by using a polarisation scrambler with a low differential group delay, for example; a polarisation scrambler based on Lefèvre loops.
The experimental nonlinear PTF of the OPA, relating the input signal power to the output power of C1, is shown in Fig. 6(a). The inset displays the same plot on a log-log scale in which a floor can be seen at powers <0.3 mW. This is the noise floor of the OPA and there is effectively no cascade detectable below this point. Although the monitoring wave C1 does not experience gain when compared to the input signal, gain is not necessary for monitoring. The original signal however, does experience gain as is shown by the gain spectrum in Fig. 6(b). The signal gain spectrum has a maximum at 1558 nm, with a gain of 14 dB. At 1555 nm, the pulsed signal experiences a gain of 13 dB. The instantaneous power transfer function and the signal gain spectrum are obtained by replacing the ASE source with a continuous wave source at 1555 nm, while keeping the pulsed source off.
3.1. Results and Simulations
The monitoring results are shown in Fig. 7. OSNR is defined here as OSNR=10log10(S/N), where S is the average power of the pulsed signal without noise and N is the noise power in a 0.1 nm bandwidth centred at the signal wavelength. Fig. 7(a) shows the dispersion monitoring curves for a clean pulsed input signal and signals with OSNR=15.8 dB and 9.4 dB. Each of the OSNRs were achieved by adjusting the variable attenuators in the pulsed and ASE arms while monitoring their powers at ‘A’. At each setting the dispersion in the pulsed arm was then changed using the tunable dispersive element and the average power of the filtered output measured. For each dispersion setting, the pulse width of the pulses were measured with an autocorrelator at ‘A’ and are shown in Fig. 7(b). A similar method is used to obtain the two OSNR monitoring curves in Fig. 7(c). The solid dots were obtained by keeping the tunable dispersion compensator at zero dispersion and adjusting the variable attenuators to achieve the appropriate OSNR. Likewise the circles were measured by varying OSNR with the tunable dispersion compensator set at -78 ps/nm dispersion. Figure 7(d)is a close-up of the zero dispersion OSNR monitoring curve showing the region of interest for high speed systems, from 15–30 dB [4, 20].
Fig. 6. (a) The experimental nonlinear power transfer function. (b) The gain spectrum of the signal.
The thick lines indicate the results obtained from numerical simulation and the thin lines are their uncertainties. We modelled the pulse source by measuring the noise-free input signal spectrum and used a fast Fourier transform to derive the shape of the pulsed signal. A drift of 0.06 nm in the measured spectrum over the duration of the experiment is the source of the uncertainties in the simulation. The simulated pulse was given an initial chirp so that it had a time-bandwidth product of 0.47, equalling the experimental time-bandwidth product measured at point ‘A’. The simulated pulse and ASE are added together to form a 10 GHz pulse train of noisy 8.8 ps (FWHM) pulses, and 20 ps of differential group delay is applied to it to account for the effect of the polarisation scrambler. The ratio of pulses to ASE was varied to obtain different OSNR values. The combined signal is then transformed by the measured power transfer function to obtain a cascade pulse train, and finally its time average power is found.
The experimental results are seen to agree well with the simulations. Each of the dispersion monitoring curves in Fig. 7(a) have a well-defined peak coinciding with the position of the minimum measured pulse width in Fig. 7(b). This indicates that the output of our pulsed laser was chirped, but that the monitor still peaked at zero dispersion. It can be seen that as the OSNR decreases, the dispersion monitoring curves retain the same qualitative behaviour, but the zero dispersion peak decreases. Similarly, in Fig. 7(c) the OSNR monitoring curves retain the same qualitative behaviour as dispersion increases, but the peak power indicating high OSNR is scaled to a lower value.
The dispersion monitoring curve is shown to have a dynamic range of 1.6 dB in the region ±150 ps/nm, occurring when the pulses are clean. In the OSNR range 6–30 dB, a dynamic range of 1.9 dB is achieved for pulses with no dispersion, with 0.6 dB of this residing in the 16–30 dB region.
Fig. 7. (a) Dispersion monitoring curves for a clean pulse train and two pulse trains of OSNR 15.8 dB and 9.4 dB. The points are experimental data and the thick lines are from numerical simulation with uncertainty indicated by the thin lines. (b) The autocorrelation measured pulse widths at ‘A’. (c) OSNR monitoring curves for an undispersed pulse train and one with -78 ps/nm dispersion. The boxed area is the OSNR region of interest for high speed systems and is shown magnified in (d).
To simultaneously monitor in-band OSNR and chromatic dispersion with this device, the device could be used in conjunction with a dithered tunable dispersion compensator. While dithering the compensator, the average output power measured could be used in a feedback loop to find the location of the zero dispersion peak on the dispersion monitoring curve. Since dithering can keep the signal near the zero dispersion point, the cascade power at that point could be used to determine the OSNR of the pulse train. Thus by using a dithering approach, both dispersion and OSNR of the pulse train would be monitored.
4. Consideration for high duty cycle signals
We have so far demonstrated monitoring for a 10 GHz RZ signal with a 10% duty cycle. However, the duty cycle in standard communications systems is generally higher, for example 33% or 66%. In this section we use the pdf analysis from Section 2 to understand the effect of high duty cycles on the monitoring ability of the device and numerically simulate a 40 Gb/s signal with a 50% duty cycle.
For signals with the same average power, a pulse train with a high duty cycle differs from one with a lower duty cycle primarily in that it has a lower peak power. The signal pulses are shallower with respect to their periods, and thus the power distribution in the signal is based at lower powers than the equivalent low duty cycle signal. Following from the analysis in Section 2, this means that the gain difference between the peaks of clean pulses and ASE is reduced, resulting in reduced monitoring sensitivity. At much higher duty cycles, the peaks of clean pulses are so low that they experience on average less gain than ASE of the same power. Such a case can be seen in Fig. 8(a), in which the pdf of ASE is shown to reach higher powers than a 40 Gb/s RZ pulse train with a 50% duty cycle. Hence the roles of the pulses and ASE are reversed and the average output power produced by the ASE, indicated by the shaded area, is greater than that produced by the pulses. Correspondingly, the OSNR monitoring curve produced when these are passed through a quadratic PTF, shown in Fig. 8(b), is also reversed. Nonetheless, the reversed monitoring curve still allows us to perform monitoring if we reinterpret the monitoring curve to correspond a low OSNR reading for high output powers and vice versa.
Fig. 8. (a) The pdf of a 40 Gb/s signal is compared to the pdf of ASE. The quadratic PTF amplifies the upper part of the ASE giving it more gain than the signal (b) OSNR monitoring curve simulated for a 50% duty cycle signal using a quadratic PTF.
Since the monitoring ability of the device arises from the differing gain experienced at different parts of the nonlinear PTF, it follows that the greater the gain difference, that is the more rapidly the slope of the PTF increases, the greater the dynamic range of the monitor. In effect, the sensitivity of the monitor is not only dependent on the duty cycle of the pulse train, but also on the shape of the nonlinear power transfer function. Thus, the sensitivity of the signal monitor could be improved if a steeper nonlinear PTF, such as an exponential, could be obtained.
Our analysis and simulation shows that OSNR monitoring of high duty cycle systems could potentially be achieved with the proposed device. The resulting curve gives a low power reading for high OSNRs and vice versa. For a 40 Gb/s pulse train with a 50% duty cycle, a 1.8 dB dynamic range in output power is obtained. The sensitivity could be improved with a steeper PTF.
5. Conclusion
We have demonstrated an OPA that directly monitors in-band OSNR and chromatic dispersion. This is achieved using the nonlinear relationship between the optical signal and the new cascaded wave it generates through cascaded four-wave mixing. Despite the pulse splitting experienced, we managed to obtain for the filtered output a 1.9 dB dynamic range in output power over 20 dB of OSNR, and a 1.6 dB dynamic range in output power for a 150 ps/nm dispersion range. In addition, the OPA was able to provide 13 dB of gain to the original signal and a 10 nm wavelength conversion to the idler wave. Our simulations show that a 40 Gb/s signal with a 50% duty cycle will result in a OSNR curve that is reversed but still has a 1.8 dB dynamic range for monitoring.
Acknowledgements
This work was produced with the assistance of the Australian Research Council under the ARC Centres of Excellence and Linkage programs.
References and links
1. S. Wielandy, P. S. Westbrook, M. Fishteyn, P. Reyes, W. Schairer, H. Rohde, and G. Lehmann, “Demonstration of automatic dispersion control for 160 Gbit/s transmission over 275 km of deployed fibre,” Electron. Lett. 40(11), 690–691 (2004). [CrossRef]
2. P. S. Westbrook, B. J. Eggleton, G. Raybon, S. Hunsche, and T. H. Her, “Measurement of residual chromatic dispersion of a 40-Gb/s RZ signal via spectral broadening,” IEEE Photon. Technol. Lett. 14(3), 346–348 (2002). [CrossRef]
3. D. C. Kilper, R. Bach, D. J. Blumenthal, D. Einstein, T. Landosi, L. Ostar, M. Preiss, and A. E. Willner, “Optical performance monitoring,” J. Lightwave Technol. 22(3), 294–304 (2004). [CrossRef]
4. S. Wielandy, M. Fishteyn, and B. Zhu, “Optical performance monitoring using nonlinear detection,” J. Lightwave Technol. 22(3), 784–793 (2004). [CrossRef]
5. T. T. Ng, J. L. Blows, J. T. Mok, R. W. McKerracher, and B. J. Eggleton, “Cascaded four-wave mixing in fibre optical parametric amplifiers: Application to residual dispersion monitoring,” J. Lightwave Technol. 23(2), 818–826 (2005). [CrossRef]
6. T. T. Ng, J. L. Blows, J. T. Mok, J. A. Bolger, P. Hambley, and B. J. Eggleton, “Simultaneous residual chromatic dispersion monitoring and frequency conversion with gain using a parametric amplifier,” Opt. Express 11(23), 3122–3127 (2003). [CrossRef]
7. C. Madsen, J. Wagener, T. Strasser, D. Muehlner, M. Milbrodt, E. Laskowski, and J. Demarco, “Planar waveguide optical spectrum analyzer using a UV-induced grating,” IEEE J. Select. Top. Quantum Electron. 4(6), 925–929 (1998). [CrossRef]
8. B. J. Eggleton, A. Ahuja, P. S. Westbrook, J. A. Rogers, P. Kuo, T. N. Nielsen, and B. Mikkelson, “Integrated tunable fibre gratings for dispersion management in high-bit rate systems.” J. Lightwave Technol. 18(10), 1418–1432 (2000).
9. C. Youn, K. Park, J. Lee, and Y. Chung, “OSNR monitoring technique based on orthogonal delayed-homodyne method,” IEEE Photon. Technol. Lett. 14(10), 1469–1471 (2002). [CrossRef]
10. Q. Yu, Z. Pan, L. Yan, and A. Willner, “Chromatic dispersion monitoring technique using sideband optical filtering and clock phase-shift detection,” J. Lightwave Technol. 20(12), 2267–2271 (2002).
11. M. N. Petersen, Z. Pan, S. Lee, and S. A. Havstad, “Online chromatic dispersion monitoring and compensation using a single inband subcarrier tone,” IEEE Photon. Technol. Lett. 14(4), 570–572 (2002). [CrossRef]
12. P. Vorreau, D. Kilper, and J. Leuthold, “Optical noise and dispersion monitoring with SOA-based optical 2R regenerator,” IEEE Photon. Technol. Lett. 17(1), 244–246 (2005). [CrossRef]
13. J. Lee, D. Jung, and Y. Chung, “OSNR monitoring technique using polarisation-nulling method,” IEEE Photon. Technol. Lett. 13(1), 88–90 (2001). [CrossRef]
14. Y. Shi, M. Chen, N. Ma, and S. Xie, “Chromatic dispersion monitoring technique employing SOA spectral shift in 40 Gbit/s system,” Opt. Comm. 229(1–3), 79–84(2005). [CrossRef]
15. S.-D. Yang, Z. Jiang, and A. M. Weiner, “Extremely low-power intensity autocorrelation and chromatic dispersion monitoring for 10-GHz, 3-ps optical pulses by aperiodically poled lithium niobate (A-PPLN) waveguide,” JWA35 (OFC, 2005).
16. T. T. Ng, J. L. Blows, and B. J. Eggleton, “In-band OSNR monitoring using fibre optical parametric amplifier,” Electron. Lett. 41(6), 352–353 (2005). [CrossRef]
17. J. Hansryd, P. Andrekson, M. Westlund, J. Li, and P. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Select. Top. Quantum Electron. 8(3), 506–520 (2002). [CrossRef]
18. J. Hansryd and P. Andrekson, “Broad-band continuous-wave-pumped fiber optical parametric amplifier with 49- dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. 13(3), 194–196 (2001). [CrossRef]
19. P. Z. Peebles Jr.Probability, random variables and random signal principles (McGraw Hill, New York, 2001).
20. D. C. Kilper and W. Weingartner, “Monitoring optical network performance degredation due to amplifier noise,” J. Lightwave Technol. 21(5), 1171–1178 (2003). [CrossRef]
