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Stable four-mode phase-sensitive (4MPS) process was investigated as a means to enhance two-pump driven parametric multicasting conversion efficiency (CE) and signal to noise ratio (SNR). Instability of multi-beam, phase sensitive (PS) device that inherently behaves as an interferometer, with output subject to ambient induced fluctuations, was addressed theoretically and experimentally. A new stabilization technique that controls phases of three input waves of the 4MPS multicaster and maximizes CE was developed and described. Stabilization relies on digital phase-locked loop (DPLL) specifically was developed to control pump phases to guarantee stable 4MPS operation that is independent of environmental fluctuations. The technique also controls a single (signal) input phase to optimize the PS-induced improvement of the CE and SNR. The new, continuous-operation DPLL has allowed for fully stabilized PS parametric broadband multicasting, demonstrating CE improvement over 20 signal copies in excess of 10 dB.
© 2014 Optical Society of America
1. Introduction
An ideal wavelength multicaster should deliver a large number of signal replicas to spectrally distinct wavelengths, while simultaneously preserving input signal's format and quality. Parametric process in high-confinement waveguide was identified early as an optimal candidate for such wavelength multicasting, possessing both transparency with respect to arbitrary modulation formats [1–3] and high efficiency, a prerequisite for high-count tone generation [4, 5]. Equally important, the phase sensitive (PS) parametric process has the potential to achieve truly noiseless amplification [6], motivating recent research efforts in this direction. Most of these studies have focused on one- and two-mode PS amplification, validating its phase squeezing [7–10] and low noise amplification [11–13] properties. However, in order to obtain high output signal to noise ratio (SNR) and minimize the pump transfer noise, pump recovery (such as injection locking) was employed in most of successful demonstrations, leading, in turn, to basic interferometric PS configuration. Consequently, a practical PS implementation output is sensitive to environmental variations, such as temperature and acoustic fluctuations. To offset these, pump phase tracking system, i.e. phase-locked loop (PLL), was implemented in the recent, one- [7] and two-mode [8, 9, 12] studies.
While one-pump PS scheme possesses inherent simplicity, two-pump four-mode phase-sensitive (4MPS) interaction offers considerably higher performance. It has been applied in amplification [14] and broadband wavelength multicasting [15], demonstrating improved PS gain and conversion efficiency (G/CE), necessary for high-count spectral multicasting. However, these initial demonstrations also faced a basic challenge: phase-stabilization in four-mode interferometric PS architecture. Among the input six waves (i.e. two pumps and four signals) of the 4MPS process, there are three dominating four wave mixing (FWM) processes: modulation instability (MI), phase conjugation (PC) and Bragg scattering (BS). Consequently, the phase matching between the six waves demands that at least three of the input waves be phase-managed for stable and maximized output. In this report, we analyze the phase relation inherent to the 4MPS process, and describe in detail its influence on multicasting G/CE performance. Using this result, a specialized digital phase-locked loop (DPLL) was implemented for the first time to stabilize dual-pump driven 4MPS multicasting (4MPSM). In contrast to earlier (free-running) device [15], this report describes the first, fully-stabilized 4MPSM operation, to the best of our knowledge. Compared to one- and two-mode PS process, two pumps must be independently phase locked to the comb-based four signals to achieve stabilized output. Finally, a selection of four input signal phase states and their use for control of multicasting G/CE is investigated and characterized experimentally.
This paper is structured as follows: Section 2 analyzes the 4MPSM six-wave model and presents the corresponding simulation results. In Section 3, the 4MPSM experimental configuration is demonstrated, and details of DPLL implementation are discussed. The agreement between experimental measurement and theoretical prediction in Section 2 are outlined in this section. The conclusion and limits of the approach are in Section 4.
2. Phase relations in 4MPS process
Dual-pump driven parametric process has been extensively studied [16–18], and includes three distinct FWM processes: MI, BS, and PC. Conventional devices rely on phase insensitive (PI) parametric process, implying that the output G/CE does not depend on the input phases. The MI, BS and PC processes are indicated in Fig. 1(a) when S2 is the only input signal for the dual-pump driven PI parametric process. In contrast, 4MPS process corresponds to a state when all four sidebands (S1/S2/S3/S4) are occupied at the input. Assuming that amplification process does not lead to significant pump depletion and high order mixing tone generation, one input mode (taking S2 as an example) is amplified, while three remaining input modes contribute three idler fields to the amplified tone (S2). Consequently the output mode at S2 is the coherent summation of four fields, expressed as
where S2_out is the output mode, S1-4 are the input modes, and the coefficients μ1-4 are dictated by the characteristics of the parametric device and the input optical waves. Moreover, the parametric G/CEs are governed by the phase matching condition (assuming the dispersion, nonlinear coefficient and interaction length of the nonlinear medium, and optical powers have been optimized), as depicted in Eqs. (2a)-(2c), where θp1,p2 denote the pump phases, θs1-s4 represent the signal phases, and m is an arbitrary integer. Equations (2a)-(2c) indicate that the maximum parametric G/CE mandates the phases of three signal modes must match the phases of the remaining signal modes, as well as that of the two pumps. Provided that the phase conditions in Eqs. (2a)-(2c) are satisfied, and the input signals’ powers and parametric G/CEs on each mode are equalized, a 12-dB coherent gain increase (i.e. four-fold multiple of the single field) is expected when compared to the PI case. We note that this increase directly follows from Eq. (1), and was validated in [14, 15].
Fig. 1 Typical wavelength configuration for PS processes. P1, P2: pumps; S1, S2, S3, S4: signals. (a) Nondegenerate dual-pump driven parametric process. In the phase insensitive case, only S2 (solid line) is the input signal, whereas S1/S3/S4 (dashed lines) are the newly generated idlers from modulation instability (MI), phase conjugation (PC) and Bragg scattering (BS), respectively. In the 4MPS case, S1/S2/S3/S4 are all present with two pumps. (b) Degenerate (left) and nondegenerate (right) one mode parametric amplification, (c) Two mode parametric process, including MI, PC and BS.
In practical terms, four signals are sharing the same transmitting waveguide (e.g. fiber), implying that the relative phases are inherently maintained during the propagation. Since two pumps are regenerated in separate fiber branches, their phases are varying independently due to the thermal and acoustic variations, which lead to time-varying phase relation and fluctuating output power. Consequently, a PLL is required to track and compensate for the phase fluctuations to maintain a relative phase relation between the six waves. Based on Eqs. (2a)-(2c), different solutions for the PLL locking mechanism are admissible. In practice, the four signals are phase locked to each other in one waveguide, and therefore the two pumps' phases can be manipulated independently to achieve a stabilized operation. For example, P1 can be controlled to be phase locked to S1 and S2 (see Eq. (2a)), while P2 is phase locked to P1, S3 and S2 (see Eq. (2b)). In this setting, the relative phase between the six waves are inherently maintained constant. In particular, in order to achieve maximum G/CE state, the phase of a signal wave should be optimized independently in order to satisfy the phase matching (i.e. S4 in Eq. (2c)). In a general case, four signals are not phase locked to each other and all the six waves possess independent phase fluctuations. In the latter setting, locking solutions, different from the one described above, would likely have to be judiciously selected.
The response of the 4MPSM architecture was calculated using an adaptive-step nonlinear Schrödinger equation (NLSE) model, neglecting stimulated Brillouin scattering (SBS) and Raman interaction. In the simulation, two pumps were positioned at 1547.7 and 1550.9 nm, and combined with four signals at 1546.9, 1548.5, 1550.1, and 1551.7 nm, respectively. Subsequently, the six phase-correlated waves were launched into a shock wave parametric mixer [4, 20]. The three-stage multicasting mixer was composed of a 105-m highly nonlinear fiber (HNLF) characterized by a zero dispersion wavelength (ZDW) of 1559 nm, dispersion slope of 0.025 ps/km/nm2 and nonlinear coefficient of 15 km−1W−1, a 6-m single mode fiber (SMF) and a 230-m dispersion flattened HNLF with peak dispersion at −0.05 ps/nm/km. The power of each pump was 27.8 dBm, while the power of four signal modes were equalized; each mode possessing −20 dBm input power (). Quantum noise was modeled as an additive Gaussian white noise at the input of the mixer, with the variance defined by half-photon spectral power density. In the PI case, the 1550.1-nm wave was the only input signal, with power set to. The G/CE of the multicasting is defined as the input/output power ratio for both PS and PI schemes.
The resulting spectra from the simulation are shown in Fig. 2, where the input optical waves were co-polarized. In Fig. 2(a), all the input waves were phase synchronized, i.e. the initial phases were 0°. Whereas in Fig. 2(b), the input phases of P1/P2/S1/S2/S3/S4 were arranged as 15°/60°/10°/20°/55°/65°. Both states satisfy the phase matching condition defined in Eqs. (2a)-(2c), guaranteeing that the multicasting G/CE is maximized, as shown in Fig. 2. We note that no phase fluctuation was induced by environmental variation in the simulation, and thus the optimum phase relation as well as the stable output powers were ideally maintained.
Fig. 2 4MPS multicasting simulated spectra. (a) input phases of P1/P2/S1/S2/S3/S4: 0°, (b) input phase, P1: 15°, P2: 60°, S1: 10°, S2: 20°, S3: 55°, S4: 65°.
The spectral comparison between the PI and PS scheme, and the noise evolution simulation have been studied in [15, 21], respectively; in contrast, this investigation focuses on phase manipulation of the 4MPS process. The phase of each signal in 4MPSM was individually swept from 0° to 360°, while the phases of the remaining input waves were arranged according to the setting in Fig. 2(b). Based on the aforementioned multicasting G/CE definition, the PS G/CEs were calculated with each signal's phase swept. As expected, for the PI case, the G/CE was constant when the phase of the only input wave was swept. Consequently, the PS-to-PI G/CE improvements follow the same trend as the PS G/CE when the input phase is changed: with signal phase swept, corresponding response exhibits sinusoidal characteristics, as shown in Figs. 3(a)-3(d). As an example, consider Fig. 3(a) in which S1’s phase was swept, while the phases of P1/P2/S2/S3/S4 were 15°/60°/20°/55°/65°. The corresponding PS-to-PI gain improvement peaks when the phase of S1 is equal to 10°. In other words, a 12-dB G/CE PS improvement is obtained when the input phases obey the phase matching rule (i.e. Equations (2a)-(2c)). The same conclusion can be attained from sweeping the phase of the other inputs, as shown in Fig. 3: grey curves in Fig. 3 represent newly generated 20 signal copies. As evident, some of the grey curves deviate from optimum phase value, originating from the fact that the higher order mixing tones are dominated by FWM process that is spectrally distant from the original seed.
Fig. 3 The simulated PS-to-PI G/CE improvements response of the central 24 signal copies for individual input (i.e. S1/S2/S3/S4) phase sweep. (a) phase of S1 1546.9 nm was swept from 0° to 360°, (b) phase of S2 1548.5 nm was swept from 0° to 360°, (c) phase of S3 1550.1 nm was swept from 0° to 360° (d) phase of S4 1551.7 nm was swept from 0° to 360°. S1/S2/S3/S4 corresponding outputs are marked as blue, green, magenta and red.
3. Experimental setup and results
We constructed 4MPSM experimental configuration as shown in Fig. 4, consisting of four distinct segments. In the first block, cascaded modulators were used to generate mutually coherent waves, essential for the phase correlation required in the PS process. A narrow linewidth laser centered at 1549.3 nm was launched into concatenated amplitude modulator (AM) and two phase modulators (PMs), driven by a 25-GHz RF signal. An optical comb with 5-nm 10-dB-bandwidth was generated by managing the bias of the AM and the RF phases into the PMs, with output spectrum shown in Fig. 4(a). Erbium doped fiber amplifiers (EDFAs) in Part I were used to compensate for the power degradation induced by the modulator cascade.
Fig. 4 Experimental configuration including four partitions: coherent wave generation, pump recovery, parametric mixer and DPLL. (a) Part I output, an optical comb with 5-nm 10-dB-bandwidth.
Next, the amplified comb was launched into second processing block, used to define the wavelength grid for pumps and signals, to regenerate pumps and finally to combine all polarization-aligned coherent waves before the nonlinear mixer (a multistage fiber mixer with synthesized dispersion in this paper). The two pumps were separated by 400 GHz and were positioned at 1547.7 and 1550.9 nm. Four signals were positioned with 100-GHz offset from the closest pump. Pump seeds and the signals were selected and de-multiplexed into three branches by an optical processor (OP). In each pump branch, injection locking was employed to maintain high SNR and guarantee high degree of phase correlation between the input waves. Subsequently, regenerated pumps were further amplified to 33 dBm, band-pass filtered and re-combined with the four signals at 1546.9, 1548.5, 1550.1 and 1551.7 nm. In the PI case, only one signal 1550.1 nm was extracted by the OP. In the third block, all six coherent waves were launched into the three-stage dispersion-engineered shock wave parametric mixer [4, 20]. The first mixer stage consisted of a 105-m long high gamma HNLF (i.e. HNLF1 in Fig. 4), characterized by a ZDW of 1598 nm, a dispersion slope of 0.018 ps/km/nm2 and a nonlinear coefficient of 22 km−1W−1, which was longitudinally strained to increase the Brillouin threshold [22]. The nonlinear Kerr effects in HNLF1 induced a nonlinear phase shift (a positive chirp) and expanded the initial six tones. In the second mixer stage, the chirped tones were compressed in a 6-m SMF, resulting in optical pulses with high peak power in time domain. This was followed by a nonlinear stage, made of a 230-m long dispersion-flattened HNLF (see HNLF2 in Fig. 4) to further expand high-order tones necessary to achieve broadband multicasting. Provided that all environmental perturbations, such as thermal and acoustic fluctuations were absent, this processor chain was capable of delivering hundreds of copies in PS multicasting role. Unfortunately, the need for pump regeneration also renders this topology to be essentially an interferometer. In practice, thermal and acoustic variations induce length change in all three fiber branches of the second experimental block, resulting in fast (~kHz) fluctuations of the multicast output power.
In contrast to the 4MPS case, driven by three distinct FWM processes (MI, BS, and PC), there is only one FWM process in one-mode and two-mode parametric devices, as shown in Figs. 1(b) and 1(c). As a result, it is sufficient to phase-lock one pump's phase to signal(s) for stabilized output in the latter two cases, as validated in prior studies [7–9, 12]. In particular, the PLL in one- and two-mode applications can be implemented using simple, analog feedback algorithms. In contrast, in the 4MPSM case, the four signals are transmitted in a single waveguide and are, thus, phase locked to each other. On the other hand, the phase relation of the 4MPSM (Eqs. (2a)-(2c)) requires that the two pumps be phase-locked to the four signals, eliminating any simple PLL analog implementation with one dithering frequency tone from the consideration. Digital PLL (DPLL) techniques have been employed in various applications to track and maintain optical coherence [23, 24], since they allow advanced computing algorithm to be utilized in addition to the scaling of the controlled beam count. The first use of DPLL in 4MPSM was demonstrated in [19].
Consequently, we detail specific implementation of DPLL scheme used to increase the G/CE in the 4MPSM architecture. In this paper, the main function of the DPLL is to track the two pumps' phase fluctuations by a single dithering tone, compensate for the phase fluctuations of two pumps and maintain the relative phase relation between the three fiber branches. To obtain the error signal, the multicasting output from the architecture shown in Fig. 4 was split, and 10% of the power was received by the fourth processor section (i.e. DPLL). Instead of a PLL filter and a phase detector in the analog implementation, a microprocessor was utilized to calculate the phase error signals. In order to track and lock the relative phase between the three independent fiber branches, two piezoelectric stretching devices (PZT) were inserted into each pump path, while three fiber lengths were matched to within 1-cm physical difference. Two digital-to-analog-converters (DACs) were used to output the phase error signals with alternating 60-kHz dithering, and subsequently electrically amplified to drive the PZTs. As a result, this phase dithering information was transferred to render 60-kHz power fluctuation of the multicasting output signals. In order to achieve stabilized and equalized multicasting spectrum, three optical channels at 1548.5, 1550.1, and 1551.7 nm were filtered at the output monitor path, detected, electrically amplified and sampled by three analog to digital converters (ADCs). A dedicated microprocessor processed the sampled data in real time, inspected the phase-dithering induced power fluctuation, computed the error signal levels by the gradient descent algorithm [22] and controlled the time sequence of the phase dithering of the two DACs. As indicated in Fig. 4, the phase dithering modulation on two pumps was time division multiplexed. Therefore, the microprocessor was capable of distinguishing phase fluctuations from each pump, a critical functionality in this architecture.
The instantaneous response of the DPLL was recorded by monitoring three channels at 1548.5, 1550.1, and 1551.7 nm, as shown in Fig. 5(a). Prior to the DPLL initiation (0 ms), multicasting output powers fluctuated due to the environmental perturbations. 800μs after DPLL initiation, the digital stabilization algorithm has converged and the system has locked the loop. As seen in Fig. 5(a), the power fluctuations are reduced after this period and the output powers are maximized and maintained at a steady level. Note that the DPLL locking bandwidth is 3 kHz, mainly determined by dithering frequency and error-signal extraction algorithm execution speed in the microprocessor. Figure 5(b) illustrates the DC-coupled drive signals applied to two PZTs, derived from the error-signals computed by the microprocessor. In practical terms, both PZTs were digitally managed in real time to alter the two pumps phases and to compensate for environment-induced phase fluctuations, imposing a strict and stationary phase relation between the input waves.
Fig. 5 DPLL instantaneous performance. (a) Three monitored optical channels’ power fluctuations (b) The DC driving signals on two PZTs.
According to the predictions and simulations in Section 2, one of the input signals' phase also needs to be controlled to obtain the maximum multicasting G/CE, in addition to two-pump phase locking. Consequently, multicasting was optimized by sweeping the phases of the four input signals individually. According to the G/CE definition in Section 2, the PS G/CEs were calculated when the power of each input signal was −18 dBm and each signal's phase was swept over 360° by the OP, while the PI G/CE was obtained for the input signal of 1550.1-nm with power of −18 dBm. Experimental comparison between PS-to-PI G/CE for phase sweep of individual input signal is presented in Figs. 6(a)-6(d), respectively, where a 12-dB G/CE improvement was clearly seen between the PS and PI scheme. Finally, the 12-dB PS induced gain increase over that in the PI case was observed by optimizing the input phase of any one among the four signals.
Fig. 6 Experimental multicasting output (i.e. central 24 copies) power response by changing signal phase. (a) S1, 1546.9 nm phase is swept over 360° (b) S2, 1548.5 nm phase is swept over 360°. (c) S3, 1550.1 nm phase is swept over 360°. (d) S4, 1551.7 nm phase is swept over 360°.
The measurements can be compared with the simulated responses shown in Fig. 2, where phase sweep induced difference of the PS-to-PI coherent gain was more than 6 dB. However, experimental PS-to-PI coherent gain increase was approximately 3 dB, as shown in Fig. 6. This discrepancy is a direct consequence of the developed DPLL algorithm that optimized phases of the two pumps to maximize and stabilize multicasting G/CE when the signal phases were tuned; in contrast, in simulations only one signal phase in the system was tuned and all the other input phases were fixed. However, the measured improvement of PS-induced G/CE in Fig. 6 still illustrates that at least one signal phase should be used for optimization of multicasting G/CE. Note that in the signal phase control, the DPLL served a dual functionality: the DPLL first controlled the relative phase between the pumps and the signals for a maximized multicasting G/CE; additionally the DPLL maintained the relative phase relation between the six waves by compensating for the phase fluctuations in each pump path.
The spectral response of the stabilized PS multicasting was compared with the PI scheme in Fig. 7(a), where one signal was set to the optimized phase value by the OP (e.g. signal 1546.9 nm, phase was manipulated to be 100°), leading to a maximized PS G/CE. The maximum G/CE of the PI multicasting was 5 dB for the input signal at 1550.1 nm having −18 dBm power level. In the PS multicasting case, using the same input power per mode as the PI case, a maximum of 12.5-dB CE improvement compared to the PI multicasting and 24 copies with 3.5-dB power flatness were achieved. A sub-band is shown in Fig. 7(b), indicating that the PS multicasting corresponding to 20-copy count had more than 10-dB gain advantage over the PI case. Equally important, the noise level remained at the same level as the PI condition. Consequently, a 12-dB optical SNR improvement over PI state in 4MPSM was achieved in the case when two schemes had the same input signal power per mode (i.e. −18 dBm).
Fig. 7 Experimental spectra comparison between the PS and the PI multicasting. (a) whole spectrum. (b) a zoom-in view.
Finally, the requirement that the DPLL operates with three simultaneously monitored channels is validated in Fig. 8. As illustrated in Fig. 2, the high order mixing tones require a specific, optimal signal phase, different from the original seedings. In this condition, only localized maximized PS CE can be attained, whereas other multicasting copies will be attenuated. For broadband multicasting, the G/CE must be globally optimized, and thus three wavelength channels (i.e. 1548.5, 1550.1 and 1551.7 nm) were monitored for equalized spectrum, shown as the red curve in Fig. 8, which possessed 1.5-dB flatness over 20 signal copies. When one or two of three channels were selected for phase locking, tilted G/CE spectra were obtained and presented as the blue and black curves in Fig. 8, respectively. Therefore, Fig. 8 confirms that the globally equalized maximum G/CE spectrum mandates at least three monitored channels in the DPLL system. In contrast, only one or two monitored channels were sufficient for stabilization, but the CE will then deviate from copy to copy and will not yield equalized spectral response.
Fig. 8 The signal power spectra with one, two and three monitored channels. Red curve: three monitored channels, blue curves: two monitored channels, black curves: one monitored channel
4. Conclusion
We presented both the theoretical analysis and experimental demonstration of the 4MPSM operating in a continuous mode. We have showed that stabilized multicasting can generate more than 20 signal copies, with more-than-10-dB PS gain improvement over the classical (PI) case. To achieve this, we implemented a new class of multiple-channel, digital phase stabilization algorithm to track relative phase evolution in the parametric mixer. The DPLL was implemented using a real-time processor and the phase dithering technique, allowing for 3 kHz locking bandwidth. The combination of multiple-channel tracking and acoustic bandwidth was critical for phase manipulation of the 4MPS processes. While the presented DPLL required processing of three distinct waves, it is conceivable that more advanced real-time processor and complex algorithms that operate with four- or higher-count inputs would lead to a further increase in the locking bandwidth and would allow the 4MPSM platforms operation in harsh environments exposed to rapid mechanical and temperature shocks.
Acknowledgment
This work is based in part on research sponsored by the Office of Naval Research (ONR). Authors would like to thank Sumitomo Electric Industries for generously providing the HNLFs used in this work.
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