Noise performance of dual-pump, multi-sideband parametric mixer operated in phase-insensitive mode is investigated theoretically and experimentally. It is shown that, in case when a large number of multicasting idlers are generated, the noise performance is strictly dictated by the dispersion characteristics of the mixer. We find that the sideband noise performance is significantly degraded in anomalous dispersion region permitting nonlinear noise amplification. In contrast, in normal dispersion region, the noise performance converges to the level of four-sideband parametric process, rather than deteriorates with increased sideband creation. Low noise generation mandates precise dispersion-induced phase mismatch among pump and sideband waves in order to control the noise coupling. We measure the noise performance improvement for a many-sideband, multi-stage mixer by incorporating new design technique.
©2013 Optical Society of America
Optical frequency multicasting based on dual-pump driven parametric process provides a unique way to spectrally replicate optical fields at strictly-defined frequencies, and in a nearly instantaneously and distortion-less manner [1–3]. The signal and its newly created copies (idlers) can be viewed as sidebands with respect to the pump waves. All sidebands (either phase conjugated or not) are coherent (phase-correlated) since they are derived from only three seed waves. This unique feature is attractive for numerous applications including optical networking , precision metrology  and quantum communications . Moreover, wideband parametric multicaster enables new class of optical processors which rely on spectral parallelization in order to bridge the gap between high-speed optical signals and electronic devices . A travelling-wave parametric multicaster can create high-count signal copies and is a core component in high-speed multiplexing/de-multiplexing , real-time, high-resolution analog-to-digital conversion  and wideband radio-frequency channelization . Recently, this type of multicasting in dispersion-engineered multi-stage parametric fiber mixers was used to demonstrate over 160-nm wavelength range , clearly pointing to the viability of practical parametric devices with a large number of sidebands. In addition to the large idler count, it is also critical that optical multicaster operates with minimal noise, in particular with applications that require high-fidelity signal replication. Consequently, understanding the noise characteristics in multi-sideband parametric mixers is both of fundamental and practical importance.
Noise-figure (NF) is commonly used to assess the noise performance of parametric mixers, and is defined as the ratio between the input and output signal-to-noise ratios (SNR) when the input light is shot-noise limited . We note that this metric is valid in the linear regime, i.e. the sideband power is much lower than that of the pump waves, which is the case in this study. Noise performance of parametric devices with a few power balanced sidebands has been extensively studied [13–15]. In this scenario, it was concluded that the NF of newly generated waves scales with the total sideband number. For example, two-sideband parametric device (generating one idler) has a minimum NF of 2 (3 dB) when both sidebands are with nearly identical powers [13,14], while four-sideband mixer with spectrally equalized output has a NF limit of 4 (6 dB) . The above conclusion has been proved experimentally [16,17], and can be understood by noting the fact that parametric process couples vacuum fluctuations from other interacting sidebands to each output mode [14,15]. More generally, generated idlers can be viewed as internal modes leading to output noise increase . In this regard, one may intuitively conclude that a spectrally equalized N-sideband mixer leads to N-fold output SNR degradation. In practical terms, a many-copy multicasting process should be inherently associated with significantly degraded noise performance. This intuition was supported by both theoretical analysis and experimental measurements, conducted by McKinstrie  and Tong , respectively. However, the former study ignored the waveguide dispersion for simplicity, while in the latter case only eight-sidebands were created and the mixer length was modest (i.e. the dispersion was small). To generate many idlers at the mixer output, a sufficient length of nonlinear waveguide (e.g. optical fiber) as well as wide mixer bandwidth is required, leading to non-negligible dispersion among the sidebands. Therefore, the noise performance of many-sideband distributed parametric process with the presence of specific, finite dispersion remains unclear to date, as only limited research has been conducted in this regard, to the best knowledge of the authors.
Recognizing this problem, we have recently reported an experimental noise characterization of a many-sideband phase-insensitive parametric mixer with multiple fiber stages . We have discovered that, contrary to the previous predictions, when the distributed mixer operates with specific (non-zero) dispersion, the sideband NF does not scale linearly with the copy count . This finding was attributed to the localized parametric interaction induced by a finite normal dispersion of the mixer, which effectively suppresses the noise coupling from farther sidebands. However, in  a general description of noise evolution in multi-sideband parametric mixer was not articulated, and more importantly, the fundamental question remains unanswered: what is the noise limit that such parametric device can reach? In this paper, we try to answer this question and provide a detailed interpretation of the noise mechanism, based on extensive numerical simulations and calibrated experimental measurements. Furthermore, we focus on the conventional phase-insensitive configuration in which only the signal wave is present at the input, resulting in preserved phase information during the multicasting process [13,14]. The results show that 1) when sideband count is limited (e.g. < 10), the mixer NF indeed scales with the copy count, regardless of normal or anomalous mixer dispersion; 2) when sideband count is large, the optimal noise performance can be obtained in the normal dispersion region, with the sideband NF converging to approximately 6 dB, equivalent to that of four-sideband parametric interaction.
We describe the experiments designed to rigorously characterize the noise property of a three-stage, multi-sideband mixer generating more than 25 sidebands. We demonstrate, for the first time, an improved, rather than degraded sideband noise performance as the subsequent mixing stage are introduced. This counter-intuitive result clearly confirms the theoretical analysis described earlier and in the remainder of this paper.
The remaining paper is organized as follows. In Section 2, the sideband noise evolution in dual-pump driven mixer with different dispersion characteristics is simulated. Section 3 provides a detailed description about the noise performance of multi-copy parametric process with normal dispersion, and in Section 4 experimental results of both small- and large-sideband-count conditions are measured and compared. The last section concludes the paper.
2. Sideband noise evolution in parametric mixer
Here we assume that all interacting optical waves are co-polarized in order to generate maximal sideband count. In a dual-pump, travelling-wave parametric device, multiple idlers as well as higher-order pump waves grow with increase in the Nonlinear Figure of Merit (NFoM, defined as a product of interaction length, waveguide nonlinearity and pump power). With constant pump powers, NFoM can be increased with a longer interaction length before the frequency generation becomes limited by dispersion. In this process, the excessive noise coupling among generated sidebands is accumulated along the entire mixer length. Since it is challenging to obtain an analytical description of noise evolution among multiple waves, detailed numerical modeling was carried out to gain an insight into the noise generation in dispersive mixers. An adaptive-step Nonlinear Schrödinger Equation solver was used to simulate the mixer noise performance in a semi-classical way: the pump and sideband fields were treated as classical waves, whereas the quantum noise (vacuum fluctuations) was modeled as additive Gaussian white noise at the mixer input, characterized by a half-photon variance . This semi-classical model was previously shown to be in agreement with the quantum description under the large-photon-number assumption [14,15]. While this work focused on highly-nonlinear fiber (HNLF) as the nonlinear waveguide, the modeling conclusion also applies to any χ(3)-based waveguides.
Representative evolution corresponding to sideband NF and conversion efficiency (CE) in HNLFs possessing different dispersion are shown in Fig. 1. In our simulations, the stimulated Brillouin scattering (SBS) that, in practice, would eventually limit the mixer length, was not considered since the composite (shock-wave) mixer are not subject to conventional threshold consideration . Two pumps centered at 1547.7 nm and 1550.9 nm were launched with 0.6-W (single pump) optical power level, while the signal was at 1548.5 nm with a −10-dBm input power. The HNLFs used in simulations possessed a nonlinear coefficient of 12 W−1km−1. As an initial condition for noise generation, vacuum fluctuations were considered as the dominant noise seed, and the NF of each sideband was calculated through an average of large number of randomly-seeded simulations, as described in . In Fig. 1(a), the HNLF was assumed to be dispersion-free. In this case, the signal NF increases with the HNLF length (and the sideband number), whereas the idler (at 1553.3-nm wavelength) NF initially decreases as the idler wave grows from the vacuum noise. With sufficient power, the idler assumes monotonically increasing trend with the HNLF length, driven by noise contributions from growing count of interacting sidebands. This phenomenon agrees well with the theory in  assuming that all sidebands contribute equally to the noise generation. In Fig. 1(b), the HNLF was characterized by a small anomalous dispersion value ( + 0.05 ps/nm/km) and had zero dispersion-slope in order to achieve efficient sideband generation. As shown in Fig. 1(b), the signal/idler NF curve exhibits similar trend as that in Fig. 1(a), when the sideband number is relatively small (corresponding to a short mixer length). However, as the sideband count grows with the increasing HNLF length, the sideband noise performance is drastically degraded: with longer than 240-m HNLF length, most sidebands lose SNR, leading to coherence collapse. This result is due to the modulation-instability induced noise amplification [14,15,20], which is consistent with previous reports on supercontinuum generation [21,22], indicating that wideband multicasting in anomalous dispersion regime comes with high noise penalty.
In a sharp contrast, for HNLF with low normal dispersion (−0.05 ps/nm/km, with zero dispersion slope), the signal NF increases initially when the sideband number (or the mixer length) is limited, as shown in Fig. 1(c). Similar to the case of anomalous HNLF, the NF trend changes at near 100-m HNLF length, and the NF converges to about 6-dB level. The idler, approaches the same NF level, albeit following a different evolution with HNLF length since it is generated from the quantum noise. Simulated spectra generated from different types of HNLFs (with 200-m length) are shown in Fig. 2, respectively. One can clearly observe the significant coherence degradation (or noise amplification) originating with the anomalous dispersion, and diminished noise level in the normal dispersion regime.
In summary, the simulation results indicate that the noise performance of parametric mixer with a small copy number is insensitive to the dispersion characteristic. However, the noise properties of many-sideband multicaster qualitatively depends on its dispersion, with the best noise performance obtained in the normal dispersion regime. In the following section, we discuss the noise evolution in mixers with finite normal dispersion level.
3. Noise mechanism in many-sideband parametric mixer with normal dispersion
In parametric mixers with finite normal dispersion, modulation-instability induced noise amplification is quenched, and at the same time, nonlinear interaction between two far-apart sidebands is weakened due to dispersion induced phase mismatch. These effects are particularly apparent when mixer is compared to the dispersion-less device. Therefore, the noise coupling among all sidebands will be reduced as well. To better understand this important fact, the output spectrum of a single-pump mixer is simulated in Fig. 3, with zero and small normal dispersion, respectively. It can be seen that, when dispersion is absent (Fig. 3(a)), the spectrum of the amplified quantum noise is flat, implying that the noise coupling is independent of the frequency separation between sidebands: in other words, all sidebands appear to make the same contribution to overall noise accumulation. When the mixer possesses proper normal dispersion, effective noise coupling is restricted to limited bandwidth due to the dispersion induced phase mismatch. As a result, no noise increase is observed at outer frequencies, as shown in Fig. 3(b). This result implies that only spectrally-adjacent sidebands interact efficiently while the distant waves contribute little to the mixer output noise. In addition, longer mixer length (corresponding to higher accumulated dispersion), higher fiber dispersion or lower pump power (i.e. smaller nonlinear interaction) leads to narrower noise coupling bandwidth, implying that fewer sidebands will be involved in the noise coupling process.
By recognizing this critical property, the noise evolution in mixers with normal dispersion can be qualitatively interpreted as follows. When sideband count is small, which requires short mixer lengths or/and low pump launched powers, the dispersion-induced phase-mismatch between pumps and sidebands is small, since the accumulated dispersion as well as the multicasting bandwidth is relatively small. Therefore, the noise contributed by each sideband can be considered approximately uniform, and the noise performance of all sidebands with nearly balanced powers follows the ideal phase-match theory , i.e. signal and idler NFs increase with the increasing sideband number. Conversely, as the mixer length increases, both the accumulated dispersion and pump depletion (originating with the higher-order pump generation) grow, leading to less efficient noise coupling between distant sidebands. In effect, this restricts the global noise coupling (in dispersion-free condition) to spectrally localized interaction among the closest (neighboring) sidebands . Eventually, the localization of parametric procedure in wide-band multicasting mixer can be considered as a sequence of dual-pump, four-sideband parametric processes. In practical terms, this means that the noise contribution to a specific sideband mainly originates with three nearby sidebands centered next to two neighboring pumps. The localization concept and corresponding saturation of sideband noise in large copy-count mixers are depicted in Fig. 4.
The above analysis explains the fact that the noise performance in normal dispersion regime does not follow the achromatic mixer prediction. Another interesting phenomenon is that the NF decreases after a specific mixer length, as shown in Fig. 1(c), which may be explained through the (frequency-wise) multi-mode phase-sensitive interaction among the existing sidebands [24,25]: the CE of sidebands continues to grow via constructive field addition of multiple neighboring sidebands. As a matter of fact, this coherent field summation increases sideband powers more efficiently than the coupled quantum noise, provided that the noise from distinct sidebands is uncorrelated [26,27]. We note that correlated noise terms cannot be obtained until the parametric process becomes substantially spectrally localized, as effective coupling with the newly created copies always introduce uncorrelated noise components. As a result, at a certain point, the phase-sensitive sideband gain will prevail over the accumulated noise and thus leads to NF improvement, until nearly complete nonlinear localization occurs. In addition, the multi-mode phase-sensitive interference also helps to efficiently equalize the sideband powers . Eventually, the mixer NF will converge to a fixed level when noise becomes highly correlated among neighboring sidebands. In other words, sideband and noise then experience the same gain and no further SNR improvement can be derived . As discussed above, the ultimate NF limit is determined by the theoretical limit of the four-sideband process  (6 dB) by assuming that the output sideband powers are identical. This implies that the noise performance of a many-sideband parametric device reduces to the level of four-mode process with sufficient normal dispersion. The apparent 6-dB NF limit may also be understood by noting that dual-pump multicasting originates from the four-mode phase-insensitive procedure, which sets the lower NF bound for subsequent, cascaded interactions. It should be mentioned that quantitative analysis of the phase-sensitive noise mechanism in a multi-sideband process is yet to be addressed.
To further confirm the above conclusions, NF spectra of mixers with different (equalized) sideband numbers are simulated in Fig. 5. In the four- and eight-sideband cases, 100-m and 64-m long conventional HNLFs (with zero-dispersion-wavelengths at 1535 and 1554 nm, respectively) were used. Two pumps were located at 1530, 1570 nm and 1546.5, 1558.9 nm, respectively, possessing 1-W per-pump power. The input signal was centered at 1568 nm and 1555.7 nm for each case. As predicted, NF corresponding to the eight-sideband mixer is higher than that of the four-sideband counterpart by approximately 3 dB in average. In practice, it is difficult to create many sidebands by simply increasing the HNLF length, as it will eventually become limited by SBS. To circumvent this difficulty, multi-stage mixer design comprising a pulse compressor and dispersion-flattened HNLF can be used to realize broadband multicasting [11,20]. Therefore, in the many-sideband simulations, a three-stage (shock-wave) mixer structure was adapted to mimic the setup used in experiments, as will be described in following section. The multi-stage setup includes a 105-m first HNLF stage (HNLF1) characterized by a 1554-nm zero-dispersion-wavelength (ZDW) and a 0.021-ps/nm/km slope, a 4-m standard single-mode-fiber stage (SMF) which cancels the frequency chirp generated by self-phase modulation in HNLF1 and thus compresses the pulse at the output of HNLF1. The third stage is a 350-m long dispersion-flattened HNLF (HNLF2) for efficient higher-order parametric generation. Two pump wavelengths were 1547.7 and 1550.9 nm respectively (400-GHz separation), and the launched signal power was −3 dBm, positioned at 1548.5-nm wavelength. As shown in Fig. 5, HNLF2 stage whose dispersion was within the normal regime (with a maximum dispersion of −0.1 ps/nm/km) leads to an improved NF performance compared to the eight-sideband condition. The central sidebands nearly approach the predicted 6-dB NF limit. In contrast, HNLF2 with a positive peak dispersion (with a 0.02-ps/nm/km maximum dispersion) significantly degrades the output NF. Moreover, it can be observed from Fig. 5 that the sidebands close to the spectrum edges exhibit higher NFs in normal dispersion regime, because they start to exist after a relatively long mixer length, and thus experience insufficient dispersion required for noise localization. Note that there is a clear tradeoff between dispersion induced noise localization and phase-mismatch caused mixer bandwidth reduction. A detailed, quantitative study on mixer optimization (e.g. optimal dispersion profile, mixer length and sideband spectral spacing etc.) is beyond the scope of this paper and will be the subject of future work.
4. Experimental results and discussion
Experimental setup was constructed and measurements were taken for mixers with small (< 10)  and large (> 20) sideband counts , to prove the theoretical predictions. Figure 6 shows the experimental setup for the small-count sideband condition. Two tunable fiber lasers were used as pump seeds. The lasers were amplified individually by two erbium-doped fiber amplifiers (EDFAs) and filtered by two 0.6-nm optical filters. High pump optical SNR (> 60 dB) was maintained with a seeding power of 15.2 dBm. Distributed-feedback laser was used as the signal positioned at 1555.7 nm, and its power was controlled by a variable optical attenuator (VOA). The pumps and signal were combined by two wavelength-division-multiplexing (WDM) couplers and subsequently sent to a 64-m long segment of conventional HNLF with a 1554-nm ZDW. To avoid the pump phase modulation, commonly used for increasing the SBS threshold, the HNLF was longitudinally tensioned with a stepwise increasing strain distribution . In addition, the filter and WDM combination effectively suppressed the EDFA amplified-spontaneous-emission distant from the pump wavelengths, which may significantly degrade the mixer noise performance . After the pump removal block and a tunable optical filter, the selected signal/copy wave was detected and measured by the NF analyzer in electrical domain.
By tuning the wavelengths of pump seed lasers as well as the launched pump powers, varied output sideband number can be obtained. For example, a two-sideband output was generated by turning on only one pump laser at 1558.9 nm, while a four-sideband spectrum was achieved by placing the pump wavelengths at 1535.5 and 1562.5 nm, respectively. An eight-sideband version was generated by using 1546.6- and 1558.9-nm pump wavelengths, through reducing the pump separation to improve the phase-matching. In Fig. 7, the NF performance of the eight-sideband mixer was compared with the two- and four-sideband conditions. It is clear from the figure that the mixer with more sidebands gives a higher NF level, which confirms the theoretical prediction in . The additional noise leading to higher NFs in the experiment than theoretical values is due to the pump transferred noise and Raman phonon induced noise .
A three-stage mixer architecture was used for many-sideband generation, as shown in Fig. 8 [11,20,32]. A narrow-linewidth (< 5 kHz) external-cavity laser was used as the master oscillator centered at 1549.3 nm. It was followed by an amplitude modulator and two concatenated phase modulators generating a flat-top optical frequency comb spanning 5 nm with a 25-GHz pitch. Next, two comb lines with 400-GHz (1547.7- and 1550.9-nm) spacing were selected by a programmable optical processor, and were used to injection lock two distributed-feedback slave lasers characterized by 700-kHz linewidths. This scheme provides phase-correlated pump seeds, mitigating otherwise significant linewidth broadening of higher-order pumps generated in the mixer , and eliminating sideband noise increase from the pump-broadening induced crosstalk. Subsequently, the slave oscillators were amplified to 0.6 W each by high-power EDFAs, filtered, and combined with the injection locked signal laser (1548.5 nm) and then launched into a 3-stage mixer. The first stage of the mixer was constructed using a 105-m long conventional HNLF with a 1554-nm average ZDW. The first HNLF1 section was also longitudinally strained to increase the SBS threshold. The second, compression stage, was composed of a 4-m long SMF matching the frequency chirp induced in the first stage. The third, mixing stage, was made using a 200-m long dispersion-flattened HNLF possessing small normal dispersion. The dispersion of the HNLF2 was precisely controlled by applying spatially constant tension to be well within the normal dispersion region, as shown in inset (a). Due to the low dispersion possessed by the mixer, no dispersion-induced impairments are expected to occur to high-speed signals launched into the multicaster. After a narrow band optical filter, the NF of selected signal/idler was measured by the NF analyzer .
Figure 9 shows the optical spectra after HNLF1 and HNLF2, exhibiting 9 and 26 copies within a 6-dB power uniformity, respectively. Measured NF and CE spectra after HNLF1 and HNLF2 are plotted in Fig. 10(a), respectively. Compared to the NF spectrum after HNLF1, an improvement of more than 0.5 dB (at most 2 dB) in NF was measured after HNLF2, which confirms our theoretical conclusions in the previous section. From Fig. 10(a), eight sidebands have less than 9-dB NFs, and the minimum NF reaches 7.4 dB. The CE spectrum after HNLF2 shows at least 4-dB increase compared to that after HNLF1. As further corroboration of this assessment, additional simulations were carried out, emulating the experimental condition. Results shown in Fig. 10(b) demonstrate an excellent agreement with the experiment. Small discrepancies between theory and experiment are attributed to a longitudinally non-varying ZDW, and absence of polarization mode dispersion and polarization dependent loss. Owing to the relatively-short HNLF2 length used in both experiment and simulation, the 6-dB NF limit was not reached. Finally, the presence of Raman phonon induced noise and pump transferred noise in practice, may increase the theoretical NF level by more than 1 dB , if no inhibition mechanisms are incorporated.
Newly derived noise property of a many-sideband parametric device is critical: firstly, it identifies the optimal noise performance of a phase-insensitive parametric spectral replication process with a large copy count. Indeed, this noise property is considerably better than previously believed and, more importantly, it points to possible direction towards true noiseless spectral replication: multi-mode phase-sensitive broadband multicasting . Since the noise limit of a multi-sideband phase-insensitive process is 6 dB (in the normal dispersion regime), conventional one-mode  or two-mode  seeded phase-sensitive multicasting is not able to approach noiseless operation, even in principle. Consequently the findings we report in this paper appear to be highly attractive for phase-sensitive, low-noise optical frequency multicasting / spectral replication.
Noise performance of multi-sideband parametric mixer possessing small, finite dispersion is rigorously characterized for the first time, both theoretically and experimentally. We show that optimal NF value for many-sideband device can be reached and is not worse than that of a four-sideband mixer. This counter-intuitive result relies on the combined effect of dispersion-induced localization of parametric interactions and the multi-mode phase-sensitive field construction procedure. Experimental results were found to be in excellent agreement with theoretical predictions. The analysis and the results of localized, rather than unrestricted global noise coupling in wide-band parametric mixers are critically important for any applications relying on frequency multicasting, implying that the signal spectral replication may be achieved in a low-noise or noiseless manner by taking advantage of a multi-mode phase-sensitive interaction.
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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