Abstract

An important challenge for implementing optical signal processing functions such as wavelength conversion or wavelength data exchange (WDE) is to avoid the introduction of linear and nonlinear phase noise in the subsystem. This is particularly important for phase noise sensitive, high-order quadrature-amplitude modulation (QAM) signals. In this paper, we propose and experimentally demonstrate an optical data exchange scheme through cascaded 2nd-order nonlinearities in periodically-poled lithium niobate (PPLN) waveguides using coherent pumping. The proposed coherent pumping scheme enables noise from the coherent pumps to be cancelled out in the swapped data after WDE, even with broad linewidth distributed feedback (DFB) pump lasers. Hence, this scheme allows phase noise tolerant processing functions, enabling the low-cost implementation of WDE for high-order QAM signals. We experimentally demonstrate WDEs between 10-Gbaud 4QAM (4QAM) signal and 12.5-Gbaud 4QAM (16QAM) signal with 3.5-MHz linewidth DFB pump lasers and 50-GHz channel spacing. Error-free operation is observed for the swapped QAM signals with coherent DFB pumping whilst use of free-running DFB pumps leads to visible error floors and unrecoverable phase errors. The phase noise cancellation in the coherent pump scheme is further confirmed by study of the recovered carrier phase of the converted signals. In addition to pump phase noise, the influence of crosstalk caused by the finite extinction ratio in WDE is also experimentally investigated for the swapped QAM signals.

© 2016 Optical Society of America

1. Introduction

In future dynamic optical networks, all optical signal processing functions are expected to be crucial building blocks to fully exploit the capacity of optical fiber beyond point-to-point communication [1]. Among these optical network functionalities, data exchange could realize the bidirectional information swapping between different wavelengths, time slots, or polarizations [2]. In the wavelength domain, this optical wavelength data exchange (WDE) is the swapping of data carried on separate wavelength channels based on parametric depletion through χ (3) or cascaded χ (2): χ (2) nonlinearities in a single device. It consists of simultaneous signal depletion and wavelength conversion processes between two signal channels. Each input signal is power consumed and its corresponding power is shifted to the other channel, resulting in data exchange between two wavelengths in a single device. So far, several WDE implementations have been demonstrated through non-degenerate four-wave mixing (FWM) in highly-nonlinear fiber [3–6] and cascaded second-order nonlinearities in a periodically-poled lithium niobate (PPLN) waveguide [7,8].

Recently, advanced multi-level modulation formats have been widely deployed in optical communication systems to increase the capacity and spectral efficiency. With the increased number of modulation states in such multi-level modulation formats, especially in high-order phase-shift keying (PSK) and quadrature-amplitude modulation (QAM) signals, susceptibility to phase noise becomes a critical impairment. Therefore, it is crucial to avoid the introduction of phase noise when pursuing optical signal processing functionalities for high-order QAM signals. We recently proposed a coherent pumping concept to demonstrate pump-linewidth-tolerant optical wavelength conversion (OWC) [9]. Owing to the phase correlation of the pumps, the phase noise from local pumps can be cancelled out in the resultant converted signals, so it is particularly suitable for OWC of high-order QAM signals. In [10], we have applied this concept to demonstrate pump-phase-noise-tolerant WDE between 4QAM and 16QAM through cascaded sum- and difference-frequency generation (cSFG/DFG) in a single PPLN waveguide with a distributed feedback (DFB) pump laser. Similar concept has been demonstrated through FWM in highly-nonlinear fiber (HNLF) [11]. In this paper, comprehensive experimental demonstration and analysis of pump-phase-noise-free WDE between QAM signals, including WDE between 10-Gbaud and 12.5-Gbaud 4QAM signals, and the WDE between 10-Gbaud 4QAM and 12.5-Gbaud 16QAM signals, are presented with 50-GHz spacing using DFB coherent pumping. The recovered carrier phase of the input and converted QAM signals show similar behavior over time, indicating that no additional phase noise from the pumps is transferred to the converted signal after WDE and error-free operation is achieved after WDE for the swapped QAM signals. With free-running pump lasers, an error floor is observed for the swapped 4QAM signals, and it is impossible to measure the bit-error rate (BER) of the swapped 16QAM signal due to excessive phase noise. Hence, the use of coherent pumps enables the WDE between QAM signals, which is otherwise inhibited due to phase noise originating from the optical pumps. Furthermore, the narrow quasi-phase matching (QPM) band of the PPLN allows 50-GHz channel spacing WDE and provides a larger signal depletion (>30dB), compared to HNLFs with reported signal depletion of ~25dB [12], resulting in a lower inter-channel crosstalk in WDE, and making it suitable for implementation of network switching in dense WDM (DWDM) systems.

In addition to the phase noise, as discussed in [13], with the increase of modulation levels, high-order QAM signals exhibit more sensitivity to in-band crosstalk. In an ideal WDE, the input signals at the original wavelengths would be fully consumed and shifted to the counterpart wavelengths, leading to infinite extinction ratio (ER) and crosstalk-free operation. However, practically it is hard to achieve complete signal depletion in WDE and the residual power left at the original wavelengths remains as a crosstalk-like impairment on the new-generated signals. Hence, the crosstalk in WDE due to finite ER is another issue to be considered when building WDE for high-order QAM signals. In this paper, the optimal operation pump power is determined by experimentally investigating the ER of WDE and BER of converted QAM signals when tuning the pump power. The impact of crosstalk and phase noise on the swapped 4QAM and 16QAM signals is then experimentally investigated, highlighting the different behavior of these impairments in WDE.

2. Operation principle

The principle of operation of the pump-linewidth-tolerant WDE is shown in Fig. 1, with two pumps (P1 at ωp1 and P2 at ωp2) and two input signals (S1 at ωs1 and S2 at ωs2). The WDE between the two signals can be decomposed into two concurring wavelength-shifting processes, each corresponding to the combination of signal depletion and wavelength conversion, promoted by nonlinear interactions. The input data S1(S2) carried at ωs1 (ωs2) is consumed and converted to ωs2 (ωs1), i.e. two wavelength shifting processes, thus achieving WDE functionality in a single device. PPLNs are attractive devices for realizing WDE due to their compactness, negligible frequency chirp and spontaneous noise emission, and immunity to stimulated Brillouin scattering, although we note that the higher FWM efficiency of χ(3) materials may make it attractive in some cases. To satisfy the phase matching condition and optimize the conversion efficiencies, pumps and signals must be arranged symmetrically with respect to the quasi-phase-matching (QPM) wavelength of the PPLN.

 

Fig. 1 Operation principle of the proposed pump-phase-noise-free WDE using coherent pumping.

Download Full Size | PPT Slide | PDF

Assuming negligible pump depletion, lossless propagation, perfect phase-matching, identical pump power, and identical nonlinear coefficients for two wavelength shifting processes, the evolution of the complex amplitude of signals S1 (As1) and S2 (As2) is given by the following simplified equations [14].

A(L)S1=A(0)S1cos(ML)+12+A(0)S2cos(ML)12ei(Δθpump+C)
A(L)S2=A(0)S1cos(ML)12ei(Δθpump+C)+A(0)S2cos(ML)+12
where L is the length of PPLN waveguide, C is a constant term, Δθpump = Δθp1θp2, with Δθp1 and Δθp2 the phase of pump P1 and P2, respectively, andMωSFGκP(0) with к the nonlinear coupling coefficient of the 2nd-order nonlinear interaction, P(0) the input pump power of P1 or P2 (assuming identical power for both pump waves), and ωSFG the angular frequency of SFG wave. According to Eqs. (1) and (2), after WDE, the signals at ω1 and ω2 become mixture combination of swapped and remnant original signals. Here, we define the power ratio between the swapped signal and any remaining signal component at the input wavelength after WDE as extinction ratio (ER). Besides, the phase noise from pumps is transparently transferred together with the original input phase to the swapped output signal. Therefore, in order to realize superior performance of WDE for high-order QAM signals, two important components should be suppressed to preserve the original information, phase noise from pumps and crosstalk due to finite ER.

When the product of ML becomes an odd multiple of π, we can obtain the following equations from Eqs. (1) and (2).

AS1(L)=AS2(0)ei(Δθpump+C)
AS2(L)=AS1(0)ei(Δθpump+C)
This corresponds to the ideal WDE without residual signals after data exchange, i.e. crosstalk-free WDE. To maximize the signal depletion, for a given PPLN waveguide, the pump power should be optimized to improve the ER, thus suppressing the crosstalk. On the other hand, according to the phase term of Eqs. (3) and (4), to reduce the phase noise from the local pumps, a straightforward approach is to deploy narrow-linewidth laser as pump sources, increasing both the implementation complexity and hardware cost. With coherent pumps deployed in WDE, Δθp1 = Δθp2, allowing the phase noise from pumps to be effectively eliminated. The tolerance against the phase noise from pumps with coherent pumping allows the use of lower-cost and large-linewidth lasers as pump sources. The coherent pumps can be synthesized by using Mach-Zehnder modulator (MZM)-based two-tone generator (TTG) driven by a RF clock, or a frequency comb, followed by an optical spectrum shaper.

To demonstrate a pump-phase-noise-free WDE for high-order QAM signals, in the following sections, we first optimize the pump power by measuring the ER and BER of the swapped signals to minimize crosstalk in WDE. With the optimized ER, pump-phase-noise-free WDE between 4QAM signals, and WDE between 4QAM and 16QAM signals are experimentally demonstrated using coherent DFB pumping. The BER performance and the constellations of the swapped QAM signals are experimentally investigated with the proposed coherent pumping, and conventional free-running pumping scheme. The behavior of phase noise and finite-ER-induced crosstalk in the constellations of the swapped signals is also experimentally investigated.

3. Experimental setup

The experimental setup is depicted in Fig. 2. To minimize the phase noise from the input signals, a fiber laser (FL) with linewidth of 10 kHz emitting at 1552.48 nm was deployed as one of the light sources for the input QAM signals. The light from FL was then modulated by an in-phase/quadrature (IQ) modulator, which has a 3-dB bandwidth of around 25 GHz, and a 3.5-V half-wave voltage. In order to generate 12.5-Gbaud 16QAM or 4QAM signals, two de-correlated 4- or 2-level driving electronics originating from 12.5-Gbaud PRBS streams with lengths of 215-1 were generated from a 50-GSa/s arbitrary waveform generator (AWG) to drive the IQ modulator. The other input 4QAM signal was generated from light emitted by a 100-kHz linewidth external cavity laser (ECL) at 1552.9 nm. It was modulated by another IQ modulator, which was fully driven by two de-correlated 10-GBaud binary PRBS streams with length of 215-1 from a pulse pattern generator (PPG).

 

Fig. 2 Experimental setup of WDE between QAM signals with different pumping schemes (coherent pumps or free-running pumps).

Download Full Size | PPT Slide | PDF

For comparison, two different pump configurations were adopted. In the coherent pump configuration, the two pumps were generated from a single DFB laser with 3.5-MHz linewidth at 1548.13 nm using a TTG, which consisted of a high extinction-ratio MZM driven by a 25-GHz clock, resulting in a 50-GHz frequency separation between the two phase-correlated pumps [15]. In this work, the limited bandwidth of the MZM limited the frequency spacing between the correlated pumps, but we note that higher frequency separation may be achieved using optical combs followed by an optical spectrum shaper. In the free-running pumps configuration, two independent free-running DFB lasers emitting at 1547.92 and 1548.32 nm were used as pumps, with 50-GHz spacing. For each configuration, 3.5-MHz linewidth DFBs were deployed as laser sources for the pumps.

After optical amplification and out-of-band noise filtering, the modulated input signal and pumps were combined in a coupler with a splitting ratio of 10% and 90% respectively at the input to the PPLN. Polarization controllers (PCs) were then used to align their polarizations to the optimum axis of the PPLN. The PPLN waveguide was produced on a lithium niobate substrate, doped with 5% of MgO to minimize photorefractive damage. The length, poling period, operation temperature, QPM wavelength and insertion losses of the PPLN device were 6 cm, 19.1 μm, 30.1°C, 1550.4 nm and 3.25 dB, respectively. The cSFG/DFG conversion bandwidth of the PPLN was of about 60 GHz, which allows OWC and WDE without significant signal distortion. After the WDE, the swapped signals were filtered out individually and detected by a digital coherent receiver. The receiver included another FL acting as a local oscillator, an optical 90-degree hybrid and two balanced photo-detectors (PDs). After detection by the PDs, the data was digitized at 50 GSa/s using a digital storage oscilloscope with a 12.5-GHz analog bandwidth, and processed off-line through digital signal processing (DSP). To provide a fair comparison, identical parameters were used in the DSP unit for detecting input QAM signals and the swapped QAM signals in both free-running and coherent pump operations. The Viterbi algorithm [16] is deployed in DSP to estimate and compensate for the frequency offset, and the tap number in smoothing filter is set as 200 for carrier phase recovery. In addition, quadrature phase-shift keying (QPSK) partitioning algorithm [17] is used for frequency offset compensation of 16QAM signals.

4. Pump-phase-noise-free WDE of QAM signals

4.1 Performance optimization of WDE in PPLN

In a typical OWC, the main figure of merit of the process is the conversion efficiency (CE), defined as the ratio of the converted signal power after the PPLN to that of the input signal before the PPLN, regardless of the power at the remaining input signal component. However, the finite ER in WDE process is also crucial since it introduces additional degradation to the output signal at the same wavelength. Fortunately, according to Eqs. (1) and (2), both maximum conversion efficiency and ER are obtained for the same pump power, corresponding to the case when the power of each input channels is totally transferred to the other channel. Another noteworthy conclusion that can be obtained from Eqs. (1) and (2) is that the optimum pump power for WDE is the same as when considering simple OWC, i.e., with AS1(0) or AS2(0) set to 0. Hence, it is possible to determine the best operation condition for WDE by measuring the pump power value at which the ER is maximized in a simple OWC.

Equations (1) and (2) were obtained assuming that ωs2 + ωP2 = ωs1 + ωP1. In real systems, however, the frequency of the light sources of the channels and pump waves may not be exactly correlated and may drift in time, so the channel in ωs2 is converted into a frequency approximately equal, but not necessarily equal to ωs1, and vice-versa. In this case, the dynamics of the system include more nonlinear interactions, but the main source for signal degradation in WDE is still the partial spectral overlap of the converted signals and the remaining power of the input channels. Nonetheless, the optimum conditions for minimized crosstalk should be the same as those devised in Section 2.

In order to experimentally obtain the optimum operation conditions for WDE, the ER and CE for each OWC were measured for different values of the pump power, with the data modulation turned off, i.e., with continuous waves (CW). The experimental results are shown in Fig. 3. As expected, the ER increases with the total pump power due to higher CE, until it reaches a maximum value of more than 20 dB for a total pump power of about 28.5 dBm. At such a pump power value, a CE of −7 dB is obtained, which includes the total insertion losses of the PPLN, of about 3.25 dB.

 

Fig. 3 Variation of the ER (thin red lines) and CE (thick blue lines) with the total pump power for each OWC process, with input signal at 1552.48 nm (solid line) and at 1552.9 nm (dashed line).

Download Full Size | PPT Slide | PDF

For higher pump power values, the power of the converted signal starts to flow back to the original wave and the ER starts to decrease. Contrarily to what was expected, the maximum ER for each OWC processes is not achieved at the same pump power value, but at 28.3 dBm when the input signal is at 1552.48 nm and at 29 dBm for the other case. These results can be explained by a slightly asymmetrical disposition of the frequency of the interacting waves (ωs2 + ωP2ωs1 + ωP1), so that one of the OWC is not perfectly quasi-phase matched, requiring higher pump power values to maximize the ER. According to our numerical simulations, even a small wavelength detuning from the ideal conditions of 0.005 nm for each signal, which was the resolution of the available ECLs, is sufficient to change the optimal pump power by about 1.5 dB. Hence, the optimum operation conditions must be set as a trade-off between the ER of each OWC process, as it will also be shown in the following subsection.

The optimization of the pump power for WDE is critical since even a small deviation of less than 0.5 dB is enough to deteriorate the ER by more than 10 dB. Therefore, stable and precise tuning of the pump power and of the operation temperature of the PPLN waveguide are crucial for WDE.

4.2 Pump power optimization by measuring BER of swapped signals

As mentioned above, due to finite ER, any un-depleted power at input wavelengths becomes deleterious crosstalk impairing the swapped signals, especially for the in-band-crosstalk-sensitive high-order QAM signals. For a given PPLN, ER is mainly dependent on the pump power. As we discussed in Section 4.1, to determine the optimal pump power, the ER has been investigated as a function of the pump power for each wavelength shifting process individually, showing the optimal pump power for wavelength shifting with input at 1552.48 nm of around 28.3 dBm, whereas the optimal one for wavelength shifting at 1552.9 nm is ~29 dBm. Here, to verify the reliability of the observed optimal pump power, we also measure the corresponding BER results of the converted signals in a WDE process, with two input 4QAM signals launched at each signal wavelength. Figure 4 shows the measured BER of the converted 4QAM at optical signal-to-noise-ratio (OSNR) of around 9 dB when tuning the launch power. According to the BER measured as a function of pump power, the optimal pump power for the signal at 1552.48 nm is measured as around 28.5 dBm, while the optimal pump power for input signal at 1552.9 nm is around 28.9 dBm. The obtained results are consistent with those obtained based on the optimal ER. In the experiment, the total pump power was set at 28.5 dBm, in order to obtain acceptable performance for both input signals. It corresponds to ~20-dB ER, indicating that the swapped signals will suffer from around 20-dB in-band crosstalk in WDE. According to the prediction in [13], for 20-dB crosstalk, 0.5-dB and 2-dB OSNR penalties are theoretically predicted for QPSK and 16QAM at BER = 10−3, respectively. This provides a benchmark for the experimental performance investigation in the following sections. With the pump power of 28.5 dBm, the measured input and output optical spectra with different pumping schemes are shown in Fig. 5. As discussed in previous session, similar ER is obtained in these two pumping schemes, which is independent on the coherence of pumps.

 

Fig. 4 Measured BERs of the converted signals after the wavelength exchange process when tuning the pump power (circles: the converted signal at 1552.9 nm; triangles: the converted signal at 1552.48 nm).

Download Full Size | PPT Slide | PDF

 

Fig. 5 Optical spectra after PPLN without pumps (dotted blue line), with coherent pumping (solid black line), and with free-running pumping (red solid line).

Download Full Size | PPT Slide | PDF

We note that in Fig. 4, with the same amount of BER degradation, 2 × 10−4, two input 4QAM signals at different baud rates show different pump power tolerance range (PPTR). For the converted 12.5-Gbaud 4QAM, the measured PPTR is around 0.63 dB, while the PPTR for the converted 10-Gbaud 4QAM is around 1 dB. This should be attributed to the increased susceptibility to crosstalk for higher baud rate signals.

4.3 Pump-phase-noise-free WDE between 4QAM signals

With two input 4QAM signals operating at different baud rates, the WDE between 12.5-Gbaud and 10-Gbaud 4QAMs is first demonstrated. Figure 6 illustrates the reconstructed constellations of the input and the swapped 4QAM signals under different pumping approaches, i.e. coherent and free-running pumping. With coherent pumping, the error-vector magnitude (EVM) of the swapped 4QAM signals is obviously increased compared to that of input signal, which is mainly due to the crosstalk introduced by finite ER in WDE. However, no additional phase noise is observed since phase noise is cancelled out with the coherent pumping. On the other hand, the free-running pumps show both high intensity noise, and high phase noise evident from symbol spreading around the phase angle. BER measurements as function of OSNR (at 0.1nm) at the receiver were also measured to verify this observation. As shown in Fig. 7, with coherent pumping, around 0.8-dB and 0.3-dB power penalty with respect to the input signal is obtained for the swapped 10-Gbaud 4QAM and 12.5-Gbaud 4QAM signals, respectively, after WDE. However, in the case with free-running DFB pumps, 4.8-dB and 4-dB power penalty is observed for the swapped 10-Gbaud and 12.5-Gbaud 4QAM signals, respectively, and a BER floor is observed for both 4QAM signals at BER of around 3 × 10−4, due to the crosstalk and phase noise. We note that, with the same amount of phase noise, the signal at lower baud rate is more sensitive to phase noise, which explains the slightly larger penalty for 10-Gbaud 4QAM in the experiment.

 

Fig. 6 Constellations of input signals: (a) 10-Gbaud (d) 12.5-Gbaud 4QAM; the swapped signals with coherent pumping: (b) 10-Gbaud and (e) 12.5-Gbaud 4QAM, and the swapped signals with free-running pumping: (c) 10-Gbaud and (f) 12.5-Gbaud 4QAM.

Download Full Size | PPT Slide | PDF

 

Fig. 7 BER vs. OSNR of the input signals (squares: 12.5-GBaud 4QAM, circles: 10-Gbaud 4QAM) and the swapped signals with coherent pumping (crosses: 12.5-GBaud 4QAM, stars: 10-GBaud 4QAM), and the ones with free-running pumping (hexagrams: 12.5-GBaud 4QAM, diamonds: 10-GBaud 4QAM).

Download Full Size | PPT Slide | PDF

4.4 Pump-phase-noise-free WDE between 4QAM and 16QAM signals

The WDE between 4QAM and 16QAM was also experimentally investigated. The constellations of the input signals and the swapped signals with different pump configurations are depicted in Figs. 8(a)-8(c) and 8(e)-8(g). With coherent pumps, clear constellations are observed for 4QAM and 16QAM with slightly increased EVM, which is just attributed to the crosstalk due to finite ER, whereas no visible phase noise distortion is observed owing to coherent pumping. However, with DFB free-running pumps, the obtained constellations are distorted due to both the crosstalk and the pump-originating phase noise. The presence of pump phase noise causes clear spreading of the symbols along the concentric circles representing the signal phase, which is more severe for the higher amplitude symbols in 16QAM. This shows that with incoherent DFB pumps, the phase noise from pump severely deteriorates the swapped QAM signals. To investigate the different behavior of crosstalk and phase noise individually in the swapped signals, the performance of converted signals with only one input signal in WDE, i.e. signal wavelength shifting process, was also investigated. Figures 8(d) and 8(h) show the converted constellations for the case of free-running pumps and one input signal for 4QAM and 16QAM respectively. Here, phase noise is visible with symbol spreading along concentric circles, but no degradation caused by crosstalk can be observed, consistent with the QAM signals impaired only by phase noise. Comparing these constellations under different conditions, it is believed that the distortion due to the phase noise from free-running 3.5-MHz-linewidth DFB pumps is the dominant factor to impair the swapped signals in WDE.

 

Fig. 8 Constellations of input signals: (a) 10-Gbaud 4QAM, (e) 12.5-Gbaud 16QAM; the swapped signals with coherent pumping: (b) 4QAM, (f) 16QAM, the swapped signals with free-running pumping: (c) 4QAM, (g) 16QAM, and the converted signals with only one input signals and free-running pumping: (d) 4QAM, (h) 16QAM

Download Full Size | PPT Slide | PDF

The observations in constellations can also be inferred from the measured BER curves as a function of OSNR (measured at 0.1 nm) for the input and the swapped signals with coherent/incoherent pumping schemes. As shown in Fig. 9, with coherent DFB pumping, around 0.6-dB and 3-dB power penalties at BER of 10−3 were obtained for 4QAM and 16QAM, respectively. As discussed above, this is mainly attributed to the crosstalk introduced by finite ER (20 dB) rather than phase noise, since phase noise is eliminated in the coherent pumping. The observed power penalty for both 4QAM and 16QAM are similar to the prediction in [13] with a similar in-band crosstalk level. However, in case of free-running pumping, although it was still possible to obtain a BER curve for the swapped 4QAM signal, ~3.4-dB penalty and visible error-floor at BER of 5 × 10−4 were clearly observed. Due to the high susceptibility of 16QAM to phase noise and crosstalk, it becomes impossible to measure BER of the swapped 16QAM at any achievable noise level. This verifies the effectiveness of the elimination of the pump phase noise in the OWE for high-order QAM with coherent pumping.

 

Fig. 9 BER vs. OSNR of the input signals (squares: 12.5-GBaud 16QAM, circles: 10-Gbaud 4QAM) and the swapped signals with coherent pumping (crosses: 12.5-GBaud 16QAM, stars: 10-GBaud 4QAM), and the ones with free-running pumping (hexagrams: 10-GBaud 4QAM).

Download Full Size | PPT Slide | PDF

4.5 Investigation of the recovered carrier phase in DSP

To detect the received QAM signals, an intradyne coherent receiver was used. The recovery of carrier phase is one of indispensable components in the offline DSP processing. It also allows insight into the impact of phase noise from the pump lasers after WDE. Figure 10 depicts the recovered carrier phase of the original input 4QAM signal where an FL was used as the laser source (dotted red line), and those of the converted 4QAM signal with free-running DFB pumps (solid black line), and with coherent DFB pumps (dashed blue line). It is obvious that within 1.6 μs the phase variation of the converted QAM with coherent DFB pumping is comparable to that of original input QAM signal, i.e. ~0.6 rad. However, using free-running DFB pump lasers, the converted QAM signal exhibits more than 2π rad phase variation. The obtained carrier phase variation is attributed to the beating effect of input light and the LO. Here, a FL laser with 10-kHz linewidth was used as LO for detection which should add little additional phase noise. Therefore, from the observation in the recovered carrier phase further demonstrates that the coherent pump scheme is able to cancel phase noise from pump lasers, even when a DFB laser was used as pump. This also supports the observation that the increase of EVM in the converted QAM signals with coherent pumping (Figs. 8(b) and 8(f)) is not attributed to the phase noise from the pump lasers, but to the crosstalk.

 

Fig. 10 Recovered carrier phase in offline DSP of the original input 4QAM signal with FL as laser source (dotted red line), the converted 4QAM signal with free-running DFB pumps (solid black line), and the converted 4QAM signal with coherent DFB pumping (dashed blue line).

Download Full Size | PPT Slide | PDF

5. Conclusion

We have proposed and experimentally demonstrated a WDE scheme with coherent pumps that allows cancellation of the pump laser phase noise, enabling the use of low-cost pump lasers, even for high-order QAM signals. We have experimentally demonstrated the pump-phase-noise-free WDE between 10-Gbaud 4QAM (4QAM) and 12.5-Gbaud 4QAM (16QAM) with spacing of 50 GHz, suitable for the data exchange and switching of high-order QAM signals in DWDM systems. Even using 3.5-MHz-linewidth DFB lasers as a pump sources, error-free operation was obtained for the swapped 4QAM and 16QAM signals with coherent pumping. The obtained results verify the feasibility of the proposed pump-phase-noise-free WDE scheme.

Acknowledgments

The work was supported in part by Grant-in-Aid for Scientific Research (C) (15K06033) from the Ministry of Education, Science, Sports and Culture (MEXT), Japan. A. Albuquerque also acknowledges Fundação para a Ciência e Tecnologia (FCT) for funding through the PhD grant SFRH/BD/78425/2011.

References and links

1. M. Saruwatari, “All-optical signal processing for terabit/second optical transmission,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1363–1374 (2000). [CrossRef]  

2. J. Wang and A. E. Willner, “Review of robust data exchange using optical nonlinearities,” Int. J. Opt. 2012, 575429 (2012). [CrossRef]  

3. K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8(3), 560–568 (2002). [CrossRef]  

4. C. H. Kwok, B. P. P. Kuo, and K. K. Wong, “Pulsed pump wavelength exchange for high speed signal de-multiplexing,” Opt. Express 16(15), 10894–10899 (2008). [CrossRef]   [PubMed]  

5. M. Shen, X. Xu, T. I. Yuk, and K. K.-Y. Wong, “A 160-Gb/s OTDM demultiplexer based on parametric wavelength exchange,” IEEE J. Quantum Electron. 45(11), 1309–1316 (2009). [CrossRef]  

6. X. Xu, M. Shen, T. I. Yuk, and K. K. Y. Wong, “Optical time-slot swapping based on parametric wavelength exchange,” in Asia Communications and Photonics Conference and Exhibition, Technical Digest (CD) (Optical Society of America, 2009), paper TuC1.

7. J. Wang, Z. Bakhtiari, O. F. Yilmaz, S. Nuccio, X. Wu, and A. E. Willner, “10 Gbit/s tributary channel exchange of 160 Gbit/ssignals using periodically poled lithium niobate,” Opt. Lett. 36(5), 630–632 (2011). [CrossRef]   [PubMed]  

8. J. Wang, Z. Bakhtiari, S. R. Nuccio, O. F. Yilmaz, X. Wu, and A. E. Willner, “Phase-transparent optical data exchange of 40 Gbit/s differential phase-shift keying signals,” Opt. Lett. 35(17), 2979–2981 (2010). [CrossRef]   [PubMed]  

9. G.-W. Lu, A. Albuquerque, B. J. Puttnam, T. Sakamoto, M. Drummond, R. Nogueira, A. Kanno, S. Shinada, N. Wada, and T. Kawanishi, “Pump-linewidth-tolerant optical wavelength conversion for high-order QAM signals using coherent pumps,” Opt. Express 22, 5067–5075 (2014). [CrossRef]   [PubMed]  

10. G.-W. Lu, A. Albuquerque, B. J. Puttnam, T. Sakamoto, M. V. Drummond, R. N. Nogueira, A. Kanno, S. Shinada, N. Wada, and T. Kawanishi, “Pump-linewidth-tolerant optical data exchange between 16QAM and QPSK with 50-GHz channel-spacing using coherent DFB pump, ” in European Conf. on Optical Communications (ECOC, 2014), paper P.2.18. [CrossRef]  

11. A. P. Anthur, R. T. Watts, K. Shi, J. O. Carroll, D. Venkitesh, and L. P. Barry, “Dual correlated pumping scheme for phase noise preservation in all-optical wavelength conversion,” Opt. Express 21(13), 15568–15579 (2013). [CrossRef]   [PubMed]  

12. R. W. L. Fung, H. K. Y. Cheung, B. P. P. Kuo, and K. K. Y. Wong, “Wavelength Exchange with Enhanced Extinction Ratio in Highly Nonlinear Dispersion-Shifted Fiber,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper JTuA39.

13. P. Winzer, A. Gnauck, A. Konczykowska, F. Jorge, and J. Dupuy, “Penalties from in-band crosstalk for advanced optical modulation formats,” in 37th European Conference and Exposition on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Tu.5.B.7. [CrossRef]  

14. J. Wang and Q. Sun, “Theoretical analysis of power swapping in quadratic nonlinear medium,” Appl. Phys. Lett. 96(8), 081108 (2010). [CrossRef]  

15. T. Kawanishi, T. Sakamoto, M. Tsuchiya, and M. Izutsu, “High extinction ratio optical modulator using active intensity trimmers,” in Proc. of European Conference and Exhibition on Optical Communication (ECOC, 2005), paper Th1.6.6. [CrossRef]  

16. A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983). [CrossRef]  

17. I. Fatadin and S. J. Savory, “Compensation of Frequency Offset for 16-QAM Optical Coherent Systems Using QPSK Partitioning,” IEEE Photonics Technol. Lett. 23(17), 1246–1248 (2011).

References

  • View by:
  • |
  • |
  • |

  1. M. Saruwatari, “All-optical signal processing for terabit/second optical transmission,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1363–1374 (2000).
    [Crossref]
  2. J. Wang and A. E. Willner, “Review of robust data exchange using optical nonlinearities,” Int. J. Opt. 2012, 575429 (2012).
    [Crossref]
  3. K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8(3), 560–568 (2002).
    [Crossref]
  4. C. H. Kwok, B. P. P. Kuo, and K. K. Wong, “Pulsed pump wavelength exchange for high speed signal de-multiplexing,” Opt. Express 16(15), 10894–10899 (2008).
    [Crossref] [PubMed]
  5. M. Shen, X. Xu, T. I. Yuk, and K. K.-Y. Wong, “A 160-Gb/s OTDM demultiplexer based on parametric wavelength exchange,” IEEE J. Quantum Electron. 45(11), 1309–1316 (2009).
    [Crossref]
  6. X. Xu, M. Shen, T. I. Yuk, and K. K. Y. Wong, “Optical time-slot swapping based on parametric wavelength exchange,” in Asia Communications and Photonics Conference and Exhibition, Technical Digest (CD) (Optical Society of America, 2009), paper TuC1.
  7. J. Wang, Z. Bakhtiari, O. F. Yilmaz, S. Nuccio, X. Wu, and A. E. Willner, “10 Gbit/s tributary channel exchange of 160 Gbit/ssignals using periodically poled lithium niobate,” Opt. Lett. 36(5), 630–632 (2011).
    [Crossref] [PubMed]
  8. J. Wang, Z. Bakhtiari, S. R. Nuccio, O. F. Yilmaz, X. Wu, and A. E. Willner, “Phase-transparent optical data exchange of 40 Gbit/s differential phase-shift keying signals,” Opt. Lett. 35(17), 2979–2981 (2010).
    [Crossref] [PubMed]
  9. G.-W. Lu, A. Albuquerque, B. J. Puttnam, T. Sakamoto, M. Drummond, R. Nogueira, A. Kanno, S. Shinada, N. Wada, and T. Kawanishi, “Pump-linewidth-tolerant optical wavelength conversion for high-order QAM signals using coherent pumps,” Opt. Express 22, 5067–5075 (2014).
    [Crossref] [PubMed]
  10. G.-W. Lu, A. Albuquerque, B. J. Puttnam, T. Sakamoto, M. V. Drummond, R. N. Nogueira, A. Kanno, S. Shinada, N. Wada, and T. Kawanishi, “Pump-linewidth-tolerant optical data exchange between 16QAM and QPSK with 50-GHz channel-spacing using coherent DFB pump, ” in European Conf. on Optical Communications (ECOC, 2014), paper P.2.18.
    [Crossref]
  11. A. P. Anthur, R. T. Watts, K. Shi, J. O. Carroll, D. Venkitesh, and L. P. Barry, “Dual correlated pumping scheme for phase noise preservation in all-optical wavelength conversion,” Opt. Express 21(13), 15568–15579 (2013).
    [Crossref] [PubMed]
  12. R. W. L. Fung, H. K. Y. Cheung, B. P. P. Kuo, and K. K. Y. Wong, “Wavelength Exchange with Enhanced Extinction Ratio in Highly Nonlinear Dispersion-Shifted Fiber,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper JTuA39.
  13. P. Winzer, A. Gnauck, A. Konczykowska, F. Jorge, and J. Dupuy, “Penalties from in-band crosstalk for advanced optical modulation formats,” in 37th European Conference and Exposition on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Tu.5.B.7.
    [Crossref]
  14. J. Wang and Q. Sun, “Theoretical analysis of power swapping in quadratic nonlinear medium,” Appl. Phys. Lett. 96(8), 081108 (2010).
    [Crossref]
  15. T. Kawanishi, T. Sakamoto, M. Tsuchiya, and M. Izutsu, “High extinction ratio optical modulator using active intensity trimmers,” in Proc. of European Conference and Exhibition on Optical Communication (ECOC, 2005), paper Th1.6.6.
    [Crossref]
  16. A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
    [Crossref]
  17. I. Fatadin and S. J. Savory, “Compensation of Frequency Offset for 16-QAM Optical Coherent Systems Using QPSK Partitioning,” IEEE Photonics Technol. Lett. 23(17), 1246–1248 (2011).

2014 (1)

2013 (1)

2012 (1)

J. Wang and A. E. Willner, “Review of robust data exchange using optical nonlinearities,” Int. J. Opt. 2012, 575429 (2012).
[Crossref]

2011 (2)

J. Wang, Z. Bakhtiari, O. F. Yilmaz, S. Nuccio, X. Wu, and A. E. Willner, “10 Gbit/s tributary channel exchange of 160 Gbit/ssignals using periodically poled lithium niobate,” Opt. Lett. 36(5), 630–632 (2011).
[Crossref] [PubMed]

I. Fatadin and S. J. Savory, “Compensation of Frequency Offset for 16-QAM Optical Coherent Systems Using QPSK Partitioning,” IEEE Photonics Technol. Lett. 23(17), 1246–1248 (2011).

2010 (2)

2009 (1)

M. Shen, X. Xu, T. I. Yuk, and K. K.-Y. Wong, “A 160-Gb/s OTDM demultiplexer based on parametric wavelength exchange,” IEEE J. Quantum Electron. 45(11), 1309–1316 (2009).
[Crossref]

2008 (1)

2002 (1)

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8(3), 560–568 (2002).
[Crossref]

2000 (1)

M. Saruwatari, “All-optical signal processing for terabit/second optical transmission,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1363–1374 (2000).
[Crossref]

1983 (1)

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[Crossref]

Albuquerque, A.

Anthur, A. P.

Bakhtiari, Z.

Barry, L. P.

Carroll, J. O.

Drummond, M.

Fatadin, I.

I. Fatadin and S. J. Savory, “Compensation of Frequency Offset for 16-QAM Optical Coherent Systems Using QPSK Partitioning,” IEEE Photonics Technol. Lett. 23(17), 1246–1248 (2011).

Kanno, A.

Kawanishi, T.

Kazovsky, L. G.

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8(3), 560–568 (2002).
[Crossref]

Kuo, B. P. P.

Kwok, C. H.

Lu, G.-W.

Marhic, M. E.

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8(3), 560–568 (2002).
[Crossref]

Nogueira, R.

Nuccio, S.

Nuccio, S. R.

Puttnam, B. J.

Sakamoto, T.

Saruwatari, M.

M. Saruwatari, “All-optical signal processing for terabit/second optical transmission,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1363–1374 (2000).
[Crossref]

Savory, S. J.

I. Fatadin and S. J. Savory, “Compensation of Frequency Offset for 16-QAM Optical Coherent Systems Using QPSK Partitioning,” IEEE Photonics Technol. Lett. 23(17), 1246–1248 (2011).

Shen, M.

M. Shen, X. Xu, T. I. Yuk, and K. K.-Y. Wong, “A 160-Gb/s OTDM demultiplexer based on parametric wavelength exchange,” IEEE J. Quantum Electron. 45(11), 1309–1316 (2009).
[Crossref]

Shi, K.

Shinada, S.

Sun, Q.

J. Wang and Q. Sun, “Theoretical analysis of power swapping in quadratic nonlinear medium,” Appl. Phys. Lett. 96(8), 081108 (2010).
[Crossref]

Uesaka, K.

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8(3), 560–568 (2002).
[Crossref]

Venkitesh, D.

Viterbi, A.

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[Crossref]

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[Crossref]

Wada, N.

Wang, J.

Watts, R. T.

Willner, A. E.

Wong, K. K.

Wong, K. K.-Y.

M. Shen, X. Xu, T. I. Yuk, and K. K.-Y. Wong, “A 160-Gb/s OTDM demultiplexer based on parametric wavelength exchange,” IEEE J. Quantum Electron. 45(11), 1309–1316 (2009).
[Crossref]

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8(3), 560–568 (2002).
[Crossref]

Wu, X.

Xu, X.

M. Shen, X. Xu, T. I. Yuk, and K. K.-Y. Wong, “A 160-Gb/s OTDM demultiplexer based on parametric wavelength exchange,” IEEE J. Quantum Electron. 45(11), 1309–1316 (2009).
[Crossref]

Yilmaz, O. F.

Yuk, T. I.

M. Shen, X. Xu, T. I. Yuk, and K. K.-Y. Wong, “A 160-Gb/s OTDM demultiplexer based on parametric wavelength exchange,” IEEE J. Quantum Electron. 45(11), 1309–1316 (2009).
[Crossref]

Appl. Phys. Lett. (1)

J. Wang and Q. Sun, “Theoretical analysis of power swapping in quadratic nonlinear medium,” Appl. Phys. Lett. 96(8), 081108 (2010).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Shen, X. Xu, T. I. Yuk, and K. K.-Y. Wong, “A 160-Gb/s OTDM demultiplexer based on parametric wavelength exchange,” IEEE J. Quantum Electron. 45(11), 1309–1316 (2009).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

M. Saruwatari, “All-optical signal processing for terabit/second optical transmission,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1363–1374 (2000).
[Crossref]

K. Uesaka, K. K.-Y. Wong, M. E. Marhic, and L. G. Kazovsky, “Wavelength exchange in a highly nonlinear dispersion-shifted fiber: theory and experiments,” IEEE J. Sel. Top. Quantum Electron. 8(3), 560–568 (2002).
[Crossref]

IEEE Photonics Technol. Lett. (1)

I. Fatadin and S. J. Savory, “Compensation of Frequency Offset for 16-QAM Optical Coherent Systems Using QPSK Partitioning,” IEEE Photonics Technol. Lett. 23(17), 1246–1248 (2011).

IEEE Trans. Inf. Theory (1)

A. Viterbi and A. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory 29(4), 543–551 (1983).
[Crossref]

Int. J. Opt. (1)

J. Wang and A. E. Willner, “Review of robust data exchange using optical nonlinearities,” Int. J. Opt. 2012, 575429 (2012).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Other (5)

X. Xu, M. Shen, T. I. Yuk, and K. K. Y. Wong, “Optical time-slot swapping based on parametric wavelength exchange,” in Asia Communications and Photonics Conference and Exhibition, Technical Digest (CD) (Optical Society of America, 2009), paper TuC1.

R. W. L. Fung, H. K. Y. Cheung, B. P. P. Kuo, and K. K. Y. Wong, “Wavelength Exchange with Enhanced Extinction Ratio in Highly Nonlinear Dispersion-Shifted Fiber,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper JTuA39.

P. Winzer, A. Gnauck, A. Konczykowska, F. Jorge, and J. Dupuy, “Penalties from in-band crosstalk for advanced optical modulation formats,” in 37th European Conference and Exposition on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper Tu.5.B.7.
[Crossref]

G.-W. Lu, A. Albuquerque, B. J. Puttnam, T. Sakamoto, M. V. Drummond, R. N. Nogueira, A. Kanno, S. Shinada, N. Wada, and T. Kawanishi, “Pump-linewidth-tolerant optical data exchange between 16QAM and QPSK with 50-GHz channel-spacing using coherent DFB pump, ” in European Conf. on Optical Communications (ECOC, 2014), paper P.2.18.
[Crossref]

T. Kawanishi, T. Sakamoto, M. Tsuchiya, and M. Izutsu, “High extinction ratio optical modulator using active intensity trimmers,” in Proc. of European Conference and Exhibition on Optical Communication (ECOC, 2005), paper Th1.6.6.
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1 Operation principle of the proposed pump-phase-noise-free WDE using coherent pumping.
Fig. 2
Fig. 2 Experimental setup of WDE between QAM signals with different pumping schemes (coherent pumps or free-running pumps).
Fig. 3
Fig. 3 Variation of the ER (thin red lines) and CE (thick blue lines) with the total pump power for each OWC process, with input signal at 1552.48 nm (solid line) and at 1552.9 nm (dashed line).
Fig. 4
Fig. 4 Measured BERs of the converted signals after the wavelength exchange process when tuning the pump power (circles: the converted signal at 1552.9 nm; triangles: the converted signal at 1552.48 nm).
Fig. 5
Fig. 5 Optical spectra after PPLN without pumps (dotted blue line), with coherent pumping (solid black line), and with free-running pumping (red solid line).
Fig. 6
Fig. 6 Constellations of input signals: (a) 10-Gbaud (d) 12.5-Gbaud 4QAM; the swapped signals with coherent pumping: (b) 10-Gbaud and (e) 12.5-Gbaud 4QAM, and the swapped signals with free-running pumping: (c) 10-Gbaud and (f) 12.5-Gbaud 4QAM.
Fig. 7
Fig. 7 BER vs. OSNR of the input signals (squares: 12.5-GBaud 4QAM, circles: 10-Gbaud 4QAM) and the swapped signals with coherent pumping (crosses: 12.5-GBaud 4QAM, stars: 10-GBaud 4QAM), and the ones with free-running pumping (hexagrams: 12.5-GBaud 4QAM, diamonds: 10-GBaud 4QAM).
Fig. 8
Fig. 8 Constellations of input signals: (a) 10-Gbaud 4QAM, (e) 12.5-Gbaud 16QAM; the swapped signals with coherent pumping: (b) 4QAM, (f) 16QAM, the swapped signals with free-running pumping: (c) 4QAM, (g) 16QAM, and the converted signals with only one input signals and free-running pumping: (d) 4QAM, (h) 16QAM
Fig. 9
Fig. 9 BER vs. OSNR of the input signals (squares: 12.5-GBaud 16QAM, circles: 10-Gbaud 4QAM) and the swapped signals with coherent pumping (crosses: 12.5-GBaud 16QAM, stars: 10-GBaud 4QAM), and the ones with free-running pumping (hexagrams: 10-GBaud 4QAM).
Fig. 10
Fig. 10 Recovered carrier phase in offline DSP of the original input 4QAM signal with FL as laser source (dotted red line), the converted 4QAM signal with free-running DFB pumps (solid black line), and the converted 4QAM signal with coherent DFB pumping (dashed blue line).

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

A (L) S1 =A (0) S1 cos(ML)+1 2 +A (0) S2 cos(ML)1 2 e i(Δ θ pump +C)
A (L) S2 =A (0) S1 cos(ML)1 2 e i(Δ θ pump +C) +A (0) S2 cos(ML)+1 2
A S1 (L)= A S2 (0) e i(Δ θ pump +C)
A S2 (L)= A S1 (0) e i(Δ θ pump +C)

Metrics