Hindawi Advances in OptoElectronics Volume 2019, Article ID 2938652, 6 pages https://doi.org/10.1155/2019/2938652 Research Article Impact of the Four-Sideband and Two-Sideband Theories in Designing of Fiber Optical Parametric Amplifiers 1 2 3 Kumbirayi Nyachionjeka , Hillary Tarus, and Philip Kibet Langat Department of Electrical Engineering, Pan African University Institute of Science Technology and Innovation, Juja, Kenya Directorate of Operations, West Indian Ocean Cable Company (WIOCC), Nairobi, Kenya Department of Telecommunications and Information Technology Engineering, Jomo Kenyatta University of Agriculture and Technology, Juja, Kenya Correspondence should be addressed to Kumbirayi Nyachionjeka; kuuh29@gmail.com Received 31 May 2019; Revised 1 August 2019; Accepted 11 September 2019; Published 7 October 2019 Academic Editor: Jung Y. Huang Copyright © 2019 Kumbirayi Nyachionjeka et al. +is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we seek to compare the two design theories for ﬁber optical parametric ampliﬁer through simulation. +e two- sideband method (standard method) has been the most widely used method in ﬁber optical parametric ampliﬁer design, but it does not predict the gain shrinkage around the pumps. +is technique does not consider the gain shrinking dynamics around the pump(s). +e four-sideband analytical technique is an alternative technique for ﬁber optical parametric ampliﬁer design, and it allows for a simpliﬁed investigation of the gain shrinking dynamics around the pump(s) due to the interaction of the various arising high-order idlers within the vicinity of the pump waves. +e undertaking in this paper is to present a dual-pump ﬁber ampliﬁer based on the highly nonlinear ﬁber and another one based on the photonic crystal ﬁber and ascertain if gain shrinking aﬀects both FOPAs. designed such that an eﬀective transfer of energy occurs 1. Introduction between one or two pump waves, signal, and a generated +e adoption of the ﬁber optical parametric ampliﬁer idler wave [3, 4]. +e dual-pump FOPA can be arranged in (FOPA) in telecommunications will go a long way in en- such a way that the pumps are modulated out of phase. +is abling broadband and quality internet. +e research in this arrangement of using out-of-phase modulated pumps allows area has undergone phenomenal progression in the last few for the reduction of pump depletion due to stimulated years as researchers seek an alternative ampliﬁer to compete Brillouin scattering, and at the same time, it cancels signal with the traditional optical ampliﬁers which are limited in gain distortion or idler spectral broadening [5]. +e FWM their bandwidth and have unstable gain dynamics due to allows for optimized placement of the two pumps so that maximum ﬂat gain can be attained [6]. their material composition [1]. +e imminent launching of 5G around the world has made it even more important to In recent times, FOPAs based on FWM and six-wave have ampliﬁers that can enable broadband communications mixing with bandwidth up to 600 nm and 150 nm, re- and increase the reach of ﬁber networks while reducing costs spectively, have been shown [7]. Using the FWM theory, the [2]. +e many functions that come with FOPA have made it gain bandwidth and the uniformity of parametric ampliﬁers a front contender as a future ampliﬁer in passive optical can be increased signiﬁcantly especially by using dual networks (PONs). +e FOPA can be used in many appli- pumps. It is worth noting that the FWM principle is in- cations such as ultrafast signal processors and optical complete and hence presents an unclear and incomplete wavelength converters; FOPAs utilize the four-wave mixing picture of the gain spectrum estimation, especially when the (FWM) in which the amplifying waveguide dispersion is signal is placed close to one of the pumps. Four-sideband 2 Advances in OptoElectronics waves are birthed by two processes, that is, the Bragg scattering and the modulation instability, and these reso- nantly interact with one another leading to a process called gain shrinking around the pump(s) [3, 8]. When the pump is ω ω ω ω ω ω ω 5 1 3 c 4 2 6 placed very close to one of the pumps, modulation instability Figure 1: Dual-pump ﬁber FOPA process based on the six-wave (MI) becomes the dominant process and is then used to model. deﬁne the resulting interactions. Radic and McKinstrie in- vestigated the concept of four sidebands by using two pumps separated by 25 nm, and the signal was placed 0.9 THz from the ﬁber, i.e., pump 1, pump 2, signal, idler, sideband 1, and the average pump frequency, and in their experiment, they sideband 2. +e waves have angular frequencies denoted by determined the existence of high idlers when a signal is in the ω , ω , ω , ω , ω , and ω , respectively. 1 2 3 4 5 6 vicinity of the pump [9]. In [10], Richter et al. demonstrated When two pump wavelengths are placed symmetrically a two-pump FOPA placed symmetrically around the signal about the zero dispersion wavelength ω and suﬃciently in the ﬁrst case, and in the second scenario, the signal and placed far apart, the gain bandwidth is maximized. +e two- idler were placed symmetrically around one of the pumps. sideband process is phase matched for a signal wave placed +e emphasis was on the inﬂuence of the phase of each between the pump frequencies, and an idler is generated. In individual wave on the gain, and they determined that the such circumstances, the dual-pump FOPA is considered to signal and idler output power depended on the relative be a two-sideband system. Hence, under these conditions, phases of each of the six interacting waves. Bogris et al. the four waves carry much of the power over the whole placed two pumps at 1530 nm and 1590 nm with the signal length of the ﬁber. +e other higher-order waves arising placed at 1554 nm, 6 nm from the average pump wavelength from the FWM are badly phase matched, and they remain of 1560 nm. It was concluded that the analytical and the negligible and thus are not considered. numerical method diﬀered around the pumps’ spectral area, If the signal is placed close to any of the pumps, secondary and a ﬂat gain bandwidth of 40 nm was achieved [11]. A waves are generated. Some of these secondary waves may number of two-pump FOPA designs were theoretically and become phase matched and may attain levels similar to levels experimentally demonstrated by Chavez Boggio et al., as of the signal and idler. In such situations, to get a complete they investigated the gain ripples in the ﬂat gain spectrum. It description of the dynamics, these new higher-order waves was noted that ripples were aﬀected by the length of the ﬁber are considered as they have a bearing on the gain charac- and the fourth-order dispersion coeﬃcient and also that the teristics of the FOPA. +is situation happens when a two- area near the pumps should be avoided when using a two- pump FOPA’s signal frequency is placed close to one of the pump FOPA with a signal placed near the pumps because of pumps. +is gives rise to two other new waves, which are the crosstalk [12]. placed at ω and ω . +e origins of these waves can be 5 6 In this paper, we set out to develop an analytical model explained as follows. When the signal frequency is placed that emphasizes on linear wave-vector mismatches to ex- approximately within 10 nm of pump 1, wave, ω , is generated plain how the sideband and its closest pump interact. due to eﬃcient FWM between the signal frequency and pump Dispersion parameters have a greater role in how the in- 1; this new wave is symmetrically placed relative to the pump, teraction between idlers and pump occurs, and thus mod- and in turn, we obtain ω + ω � 2ω . If idler 1 is near pump 3 5 1 elling around them leads to a more accurate analytical 2, ω which is symmetric to pump 2 is generated due to technique. Subsequently, two dual-pump FOPAs based on eﬃcient FWM between idler 1 and pump 2, and this in- highly nonlinear ﬁbers (HNLFs) and photonic crystal ﬁbers teraction is denoted by ω + ω � 2ω . Due to these very close 4 6 2 (PCFs) are designed using the two-sideband theory and the couplings, when a signal is near the pump, the signal grows at four-sideband theory. +e photonic crystal ﬁber and data the same time with the three idlers, having similar gains. used for FOPA design in this paper are similar to the data Hence, these higher-order idlers lead to reduction in the gain. used by Taghizadeh et al. in [13]. It will be shown that the Firstly, when a signal frequency is located between these four-sideband theory allows the prediction of gain shrinkage symmetrically placed pumps, this results in a phase-matched around the pumps making it appropriate for use in the PC process and an idler of frequency ω � ω + ω + ω . In 4 1 2 3 designing of FOPA [7]. such a case, the ampliﬁer is dominated by the two-sideband theory denoted in Figure 2. +e gain of the PC (two-sideband) as shown in (2) and 2. Theory and Analysis (3) for the idlers ω and ω is obtained from solving 3 4 +e gain of a two-pump FOPA is obtained by solving the equations (1) and (2): nonlinear Schrodinger equation with two high pump ���� zA (1) − j � − Δβ − cP A − 2c P P A , powers at frequencies, ω , and signal frequency at ω . Both 3 1 1 2 4 2 3 zz the pumps are considered to be undepleted corresponding to the linear gain regime. +e four-sideband interaction ���� zA 4 ∗ (2) − j � 2c P P A + Δβ + cP A . shown in Figure 1 can be summed up in terms of three 1 2 4 2 4 zz FWM processes which are modulation interaction (MI), Solving equations (1) and (2) as shown in Section A1.2 of Bragg scattering, and phase conjugation (PC). Six con- [14] gives the gain as tinuous waves can be considered to be propagating along Advances in OptoElectronics 3 two extra-high-order idler waves that arise when the signal is near one of the pumps [3]. +is phenomenon is best de- scribed and modelled by (6)–(9), assuming the pumps are not depleted: ω ω ω ω ω 1 3 c 4 2 zA 5 ∗ Figure 2: Dual-pump ﬁber FOPA process based on the four-wave − j � Δβ + cP A + cP A 5 1 5 1 3 zz model. (6) ���� ���� + 2c P P A + 2c P P A , 1 2 4 1 2 6 2 2 r cosh (gz) 4g (3) G � − , zA Κ ∗ − j � − cP A + − Δβ − cP A 1 5 3 1 3 zz (7) ���� ���� where Κ is the wave number total for all the interacting four ∗ − 2c P P A − 2c P P A , 1 2 4 1 2 waves as shown in (5): 2l 2l 2l ���� ���� zA Κ � c P + P + Δω − Δω , 4 (4) 1 2 s1 p − j � 2c P P A + 2c P P A (2l)! 1 2 5 1 2 3 l zz (8) + Δβ + cP A + cP A , 4 2 4 2 6 where β are the dispersion related parameters that account for all dispersion-associated behaviors deﬁned in equations ���� ���� zA ��������� � ���� − j � − 2c P P A − 2c P P A 2 1 2 5 1 2 2 3 (10)–(13), while g � r − (Κ/2) , r � 2c P P , Δω � zz (9) 1 2 p − cP A + − Δβ − cP A , (ω − ω )/2, Δω � Δω � |ω − ω |, and center frequency 2 1 s1 3 3 c 2 4 6 2 6 ω � (ω + ω )/2. c 1 2 where Δβ are the linear wave mismatches which can be Considering (1) and (2), the following matrix can be gathered into a matrix N as in (5) and i � 3, 4, 5, 6. +e easily obtained: linear wave mismatches give an insight into the in- ∗ ∗ A − Δβ − cP − r A 3 1 teraction of the pump and the idler nearest to it. When 3 3 ⎛ ⎜ ⎞ ⎟ ⎛ ⎜ ⎞ ⎟⎛ ⎜ ⎞ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠ − j � . (5) evaluated near ω using the Taylor series, the linear wave zz A r β − cP A mismatches can be realized as shown in the following 4 4 2 4 equations: Equation (5) is known as the two-sideband model and is of the form zX/zz � iNX, with the solutions shown as l∗3 l l Δβ � (− 1) Δω − Δω , (10) 3 3 p X(z) � exp(iNz)X(0), where P and P are the pump l! 1 2 l�2 powers and c is the ﬁber nonlinearity coeﬃcient. +e linear wave mismatches are shown by Δβ , where l � 3, 4 as in 4 l∗4 l l l equations (10) and (11). Δβ � (− 1) Δω − Δω , (11) 4 4 p l! l�2 +e conditions under which these frequencies in- teract in FWM is depicted as ω + ω � ω + ω , where ω 1 2 4 3 1 and ω are used as pumps and ω and ω are signal and 2 3 4 l∗5 l l Δβ � (− 1) ω − Δω , (12) 5 p idler and can be represented diagrammatically as in l! l�2 Figure 2. Secondly, the waves interact diﬀerently for a six-wave l∗6 l l arrangement (four-sideband) as shown in Figure 1. +e Δβ � (− 1) Δω − Δω , (13) 6 6 p l! interaction is as follows: ω � 2ω − ω and ω � 2ω − ω . 5 1 3 6 2 4 l�2 +e center frequency ω is in a manner that allows the where Δω � Δω � |ω − ω |, Δω � 2Δω − Δω , and s2 4 4 c 5 p s1 following to be possible: 2ω � ω + ω � ω + ω � ω + ω . c 1 2 3 4 5 6 Δω � 2Δω − Δω . Equations (6)–(9) can be written as +e sideband ω is due to pump frequency ω , while 6 p s2 6 2 zX/zz � iNX with the solutions shown as X(z) � exp sideband ω is due to pump frequency ω , and ω and ω are 5 1 5 6 (iNz)X(0). thus the sidebands. Pump 1, pump 2, idler, and signal are represented by A , A , A , and A complex amplitudes, respectively. Lin- 1 2 3 4 3. Methodology and Results ear wave-vector mismatch is given by Δβ for the PCF and HNLF, and z is the distance from the beginning of each In this research, we set out to investigate the merits, de- ﬁber. merits, and impact of two theories by designing FOPA +e arrangement of the system under simulation is such based on each theory. +e main aim was to investigate the that the signal frequency is close to one of the pumps. +e BS behavior of the gain around each pump when a signal wave and MI also become phase matched, and the conversion is placed within 10 nm of the pump(s) wave. +e two- eﬃciencies of the idlers ω and ω are increased. +e four- sideband theory has been extensively researched and has 3 4 sideband theory ensues and describes the interaction of the led to many breakthroughs in the area of FOPA. To achieve 4 Advances in OptoElectronics the aim, two prominent materials used in the design of FOPA 50 were selected for simulation and design of FOPA, i.e., the traditional HNLF with a relatively low nonlinear coeﬃcient and the new material PCF with a very high nonlinear co- eﬃcient. For the HNLF, two ampliﬁers were simulated, one based on the two-sideband theory and the other on the four- sideband theory, and the resulting gain spectrums were an- alyzed; similarly, the same simulation setup was done for the PCF. In all designs, the signal frequency was placed near one of the pumps, i.e., within 10 nm, and this allows for the in- vestigation of the eﬀects of BS and MI on pump performance. +e pump separation was below 100 nm for all the simulations. Equations (6)–(9) were numerically solved and simu- lated in MATLAB using the in-built exponential matrix 1480 1500 1520 1540 1560 1580 1600 1620 1640 1660 operator. Independent solutions to the N matrix were found, Signal wavelength and a FOPA was achieved for a phase insensitive design; the Figure 3: Simulation of a dual-pump HNLF-based FOPA gain initial conditions for the amplitude were set at 0 1 0 0 . using the two-sideband theory. It is adequate to consider only β , β , and β values as these 2 3 4 are the reason for all the behaviors that are dispersion re- lated. +e parametric gain simulation was done for the Pump 1 Pump 2 signals between [ω − Δω ω + Δω ]. 1 p 2 p +e ﬁrst design was based on the standard two-sideband theory. To achieve the design, the following parameters − 1 − 30 2 were used: length � 243 m, β � 6.4 × 10 ps ·km , β � 2 3 − 41 3 − 1 − 55 4 − 1 0.65 × 10 ps ·km , β � − 1.65 × 10 ps ·km , P � − 1 − 1 1.9 W, P �1.3 W, c � 8 W ·km , λ � 1522 nm, and λ � 1 2 1603 nm. Equation (5) was simulated and the following graph was obtained. +e graph is what is typically obtained for the standard FWM model. +e ﬂat gain obtained for this ampliﬁer design is 36 dB over a gain bandwidth of 32 nm as shown in Figure 3. +e same parameters as above were used in simulating − 30 2 a four-sideband design: length � 243 m, β � 6.4 × 10 ps · − 1 − 1 0 − 41 3 − 55 4 km , β � 0.65 × 10 ps ·km , β � − 1.65 × 10 ps · 3 4 1450 1500 1550 1600 1650 1700 − 1 − 1 − 1 km , P � 1.9 W, P �1.3 W, and c � 8 W · km . For the 2 Signal wavelength four-sideband theory, a FOPA was designed as shown in Figure 4: Simulation of a dual-pump HNLF-based FOPA gain Figure 4. +e gain and gain bandwidth achieved with the HNL using the four-sideband theory. ﬁber was 36 dB and 32 nm, respectively, when modelled using the four-sideband theory. Extending the four- and two-sideband theory models to FOPA designs based on PCFs, Figures 5 and 6 depict the gain proﬁles, respectively. Because of the high nonlinearity co- eﬃcient, the gain and bandwidth of the FOPA increase when compared to HNLFs. In [13], a single pulsed-pump FOPA based on the PCF was demonstrated. Adopting the parameters that were re- alized from the PCF design, we were able to simulate a dual- pump FOPA using the two-sideband and the four-sideband theories. +e parameters used were as follows: − 1 − 2 2 length � 35 m, β � − 8.019 × 10 ps ·km , β � 2.881 × 2 3 − 2 3 − 1 − 4 4 − 1 10 ps ·km , β � − 1.852 × 10 ps ·km , P � 1 mW, 4 1 − 1 − 1 P � 1 mW, c � 122.7 W ·km , λ � 1522 nm, and λ � 2 1 2 10 1618 nm. A gain of 50.59 dB and 55 nm gain bandwidth were 1480 1500 1520 1540 1560 1580 1600 1620 1640 1660 1680 realized in the two-sideband model in Figure 5. Signal wavelength While using the four-sideband theory, a gain and gain bandwidth of 50.59 dB and 55 nm were realized, respectively, Figure 5: Simulation of a dual-pump PCF-based FOPA gain using the two-sideband theory. for a dual-pump FOPA based on the PCF. Gain Gain Gain Advances in OptoElectronics 5 70 50 Pump 1 Pump 2 1480 1500 1520 1540 1560 1580 1600 1620 1640 1660 1480 1500 1520 1540 1560 1580 1600 1620 1640 1660 Signal wavelength Signal wavelength Figure 7: Simulation graph of a HNLF-based FOPA with higher Figure 6: Simulation of a dual-pump PCF-based FOPA gain using third-order dispersion coeﬃcient. the four-sideband theory. We also considered the eﬀect of the third-order dis- persion parameter on the ﬂat gain area of the bandwidth. Considering the ampliﬁer of Figure 7, we can show that the ﬂat gain area is inﬂuenced by β . A higher value of β led to 3 3 gain ﬂatness shown in Figure 7, while a lower value led to a gain shape shown in Figure 8. +e value of the third-order dispersion parameter for a ﬁber plays a crucial role in determining the ﬂatness of the gain proﬁle. Hence, it should be considered when designing FOPAs. 4. Discussion A validation of the two-sideband theory and the four- sideband models has been done through the design of a dual- 1450 1500 1550 1600 1650 1700 Signal wavelength pump HNLF-based FOPA and a dual-pump PCF-based FOPA. A gain of 36 dB and gain bandwidth of 32 nm were Figure 8: Simulation graph of a HNLF-based FOPA with lower obtained for the HNLF-based two-pump FOPA, while a gain third-order dispersion coeﬃcient. of 50.59 dB and gain bandwidth of 55 nm were achieved for the PCF-based two-pump FOPA. It is observed that the four- noted that our gain curves were comparable to the practical sideband theory helps in predicting the gain shrinkage gain curves as discussed in Section 6.2 of their article. around the pumps for the ampliﬁer allowing for a complete Similarly, in [11], the theoretical, numerical, and practical design picture, while the two-sideband theory does not FOPAs were designed and the gain curves were compared, predict the gain shrinkage. +e gain shrinkage predicted in and it was observed that the curves diﬀered greatly around four-sideband theory is attributed to two extra FWM the pumps, with the numerical and practical designs pre- sidebands which are due to Bragg scattering and modulation dicting the dips around the pumps. An oﬀhand analysis of instability and occurs just outside the ﬂat gain area on the the curves we obtained using the four-sideband model in- edges of the pumps. +e analytical model put emphasis on dicates that they are far comparable to the practical and the linear wave-vector mismatches to explain how the numerical gain curves obtained in [10–12]. +us, these sideband and its closest pump interact. Dispersion param- eters have a greater role in how the interaction between experimental results in the literature agree with our ana- lytical results and support our claim that the four-sideband idlers and pump occurs, and thus modelling around them model is a possible option for FOPA design especially when leads to a more accurate analytical technique. +e model the signal wavelength is close to the pump wavelength. It is clearly gives a detailed insight into the interactions of the also a model that can be easily used by any new FOPA pump and idler closest to it through the linear wave mis- designer as it is easy to follow and implement. An analysis of matches allowing a designer to manipulate the dispersion the third-order dispersion parameter was also done, and it parameters for a better design. In this model, we were able to was observed that it tends to distort the gain spectrum predict the gain dips around the pumps when a signal is around the ZDW, and this is mainly due to rapid phase placed close to either of the pumps (i.e., 10 nm). In [12], we Gain Gain Gain 6 Advances in OptoElectronics two pumps in opposition of phase,” Journal of Lightwave mismatches that arise because of unavoidable phase mod- Technology, vol. 20, no. 3, pp. 469–476, 2002. ulation between the pumps. It is thus important to consider [6] A. Vedadi, M. E. Marhic, E. Lantz, H. Maillotte, and the third-order dispersion parameter when choosing a ﬁber T. Sylvestre, “Investigation of gain ripple in two-pump ﬁber for use in designing FOPA. Compared to the other theo- optical parametric ampliﬁers,” Optics Letters, vol. 33, no. 19, retical models, we found ours better, as it set to reproduce pp. 2203–2205, 2008. exactly what the practical and numerical techniques achieve, [7] N. Cao, H. Zhu, P. Li et al., “Flat and ultra-broadband two- but it still has room for further improvement by taking into pump ﬁber optical parametric ampliﬁers based on photonic consideration the pump depletion, gain saturation, and ﬁber crystal ﬁbers,” Optical Review, vol. 25, no. 3, pp. 316–322, losses. [8] C. J. McKinstrie and S. Radic, “Parametric ampliﬁers driven by two pump waves with dissimilar frequencies,” Optics 5. Conclusions Letters, vol. 27, no. 13, pp. 1138–1140, 2002. [9] S. Radic and C. J. McKinstrie, “Two-pump ﬁber parametric +eoretical models of the two-sideband and four-sideband ampliﬁers,” Optical Fiber Technology, vol. 9, no. 1, pp. 7–23, models have been developed and validated by designing a dual-pump HNLF-based FOPA and a dual-pump PCF- [10] T. Richter, B. Corcoran, S. L. Olsson et al., “Experimental based FOPA. A gain of 36 dB and gain bandwidth of 32 nm characterization of a phase-sensitive four-mode ﬁber-optic were obtained for the HNLF-based two-pump FOPA, while parametric ampliﬁer,” in Proceedings of the 2012 38th European a gain of 50.59 dB and gain bandwidth of 55 nm were Conference and Exhibition on Optical Communications, pp. 1–3, achieved for the PCF-based two-pump FOPA. It is observed Amsterdam, Netherlands, September 2012. that the four-sideband theory helps in predicting the gain [11] A. Bogris, D. Syvridis, P. Kylemark, and P. A. Andrekson, shrinkage around the pumps for the ampliﬁer allowing for a “Noise characteristics of dual-pump ﬁber-optic parametric complete design picture, while the two-sideband theory does ampliﬁers,” Journal of Lightwave Technology, vol. 23, no. 9, pp. 2788–2795, 2005. not predict the gain shrinkage. A study of the third-order [12] J. M. Chavez Boggio, J. D. Marconi, S. R. Bickham, and dispersion has shown that it has an adverse eﬀect on the gain H. L. Fragnito, “Spectrally ﬂat and broadband double-pumped ﬂatness. ﬁber optical parametric ampliﬁers,” Optics Express, vol. 15, no. 9, pp. 5288–5309, 2007. Data Availability [13] M. Taghizadeh, M. Hatami, H. Pakarzadeh, and M. K. Tavassoly, “Pulsed optical parametric ampliﬁcation +e data used to support the ﬁndings of this study are based on photonic crystal ﬁbres,” Journal of Modern Optics, available from the corresponding author upon request. vol. 64, no. 4, pp. 357–365, 2017. [14] M. E. Marhic, Fiber Optical Parametric Ampliﬁers Oscillators and Related Devices, Cambridge University Press, Cambridge, Conflicts of Interest UK, 2007. +e authors declare that they have no conﬂicts of interest. Acknowledgments +is research was funded by Pan African University. References [1] M. Jazayerifar, S. Warm, R. Elschner et al., “Performance evaluation of DWDM communication systems with ﬁber optical parametric ampliﬁers,” Journal of Lightwave Tech- nology, vol. 31, no. 9, pp. 1454–1461, 2013. [2] I. Sackey, R. Elschner, M. Nolle ¨ et al., “Characterization of a ﬁber-optical parametric ampliﬁer in a 5 × 28-GBd 16-QAM DWDM system,” in Proceedings of the Optical Fiber Com- munication Conference (OFC 2014), pp. 1–3, San Francisco, CA, USA, March 2014. [3] A. Vedadi, E. Lantz, H. Maillotte, and T. Sylvestre, “Gain oscillations in two-pump ﬁber optical parametric ampliﬁers,” in Proceedings of the 2008 IEEE/LEOS Winter Topical Meeting Series, pp. 65-66, Sorrento, Italy, January 2008. [4] M. A. Shoaie, A. Mohajerin-Ariaei, A. Vedadi, and C.-S. Bres, ` “Wideband generation of pulses in dual-pump optical para- metric ampliﬁer: theory and experiment,” Optics Express, vol. 22, no. 4, pp. 4606–4619, 2014. [5] M.-C. Min-Chen Ho, M. E. Marhic, K. Y. K. Wong, and L. G. 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