In this paper, we investigate both analytically and numerically four-wave mixing (FWM) in short (80 μm) dispersion engineered slow light photonic crystal waveguides. We demonstrate that both a larger FWM conversion efficiency and an increased FWM bandwidth (~10nm) can be achieved in these waveguides as compared to dispersive PhC waveguides. This improvement is achieved through the net slow light enhancement of the FWM efficiency (almost 30dB as compared to a fast nanowire of similar length), even in the presence of slow light increased linear and nonlinear losses, and the suitable dispersion profile of these waveguides. We show how such improved FWM operation can be advantageously exploited for designing a compact 2R and 3R regenerator with the appropriate nonlinear power transfer function.
©2009 Optical Society of America
Slow light in planar photonic crystal (PhC) waveguides has recently attracted a lot of attention because it holds the promise for realising optical nonlinear functions that are both compact and have low power consumption [1–3]. At the origin of this lies the enhanced interaction occurring between the nonlinear medium and the light that slowly propagates into it. Following the theoretical studies predicting such slow light nonlinear enhancement [4,5], several groups have demonstrated, early 2009, a variety of optical nonlinear phenomena enhanced by slow light, such as third-harmonic generation , self-phase modulation (SPM) [7–10], two-photon absorption [7,10] and free-carrier (FC) related effects , experimentally confirming the potential of slow light PhC waveguides for nonlinear device applications. Although four-wave mixing (FWM) has not been specifically investigated in PhC waveguides yet, the efficiency of this nonlinear process should similarly benefit from slow light.
FWM has been long carried out in waveguiding structures, such as optical fibers, and total-internal reflection ridge waveguides. In that respect, considerable efforts have been pursued for enhancing the underlying nonlinear interaction by exploiting increasingly tightly confining waveguides, such as silicon nanowires, resulting in shorter device lengths . In parallel, there have been several studies devoted to designing optimized waveguides with a suitable dispersion, for ensuring phase matching between the four waves involved in the FWM process . This approach, which was essentially based on optimizing the waveguide dimensions, has led to the demonstration of FWM with both high efficiency and over unprecedentedly large bandwidths [13–15].
In addition to providing slow light propagation, PhC waveguides can be engineered so that slow light modes present a low dispersion. A number of approaches have been investigated and demonstrated successfully to achieve this, for instance by slightly modifying the geometry of the PhC waveguide [16–20] or by using selective liquid infiltration . The possibility of managing the modal dispersion in PhC waveguides combined with the achievement of low group velocities appears highly attractive for optimizing phase matched nonlinear processes like FWM that strongly depend on both nonlinearity and the dispersive characteristics of the waveguide. Furthermore, engineering the dispersion of PhC waveguides appears far more flexible than optimizing the waveguide dimensions of nanowires: the nonlinear interaction (as dictated by the mode area) and the phase matching condition (imposed by the dispersion) can be uniquely tuned almost independently in PhC waveguides.
In this paper, we investigate, both analytically and numerically, how short (80 µm) dispersion engineered slow light PhC waveguides can generate FWM with good performance (high efficiency and over a substantial bandwidth ~10nm). In the section 2, starting from the analytical expression of the FWM efficiency, we study how the specific dispersion of engineered PhC waveguides, calculated by the 3D plane wave method (PWM), allows us to achieve FWM over larger bandwidths than in standard (dispersive) PhC waveguides. We also show that a positive gain can be achieved for both anomalous and normal dispersion regimes, due to the suitable fourth order dispersion that characterizes the flat-band slow light window of these engineered PhC waveguides. In the section 3, we present the model to simulate the propagation of optical pulses in slow light PhC waveguides, in the presence of various additional nonlinear phenomena (TPA, FC effects, Raman), and taking into account the respective slow light enhancement of all these effects. By numerically solving these equations using the split step Fourier method (SSFM), we present, in section 4, the FWM results for various slow light PhC waveguides with different group velocities and dispersion. In particular, we demonstrate that the net FWM efficiency is strongly enhanced by slow light, while the specific dispersion of these waveguides ensures that the process is achieved over a substantial bandwidth. In the section 5, by taking advantage of the improved FWM operation in slow light engineered PhC waveguides, we demonstrate the possibility of designing a compact all-optical regenerator.
2. Improvement of FWM operation through dispersion engineered slow PhC waveguides
The aim of this first section is to demonstrate, from analytical considerations, the potential of dispersion engineered PhC waveguides for FWM. FWM is a phenomenon related to the material third-order non-linear susceptibility, χ(3), where three input waves interact through the medium to give rise to a fourth wave. In the case of partially degenerate FWM, which we focus on here, a single pump (frequency ωpump, wavevector βpump) is mixed with a probe signal (frequency ωprobe, wavevector βprobe). While the probe is amplified through the FWM process, an idler signal (frequency ωidler, wavevector βidler,) is simultaneously generated, as inferred from the expression of the medium nonlinear polarisation PNL at the idler frequency:Eq. (1), ε 0 is the free space permittivity, E pump and E probe are the pump and probe input electric field amplitudes that overlap in time, while the frequency and wavevector differences are respectively given by the following equations:Fig. 1(a) , FWM transfers energy from a strong pump wave to both the probe and idler, upshifted and downshifted in frequency from the pump by an amount .
Assuming that the pump remains undepleted and that the input waves are polarised in the same direction, the conversion efficiency defined as the power Pidler gained by the idler relatively to the initial probe power Pprobe, after a propagation length L is Eq. (4), γ is the nonlinear parameter (see section 3), and is the path average pump power reduced by the associated propagation loss α over a distance L through:Eq. (3) for degenerate FWM. Maximum conversion efficiency is therefore achieved when the linear phase mismatch ΔkL determined by the waveguide properties, is negative and balances the positive non-linear phase mismatch, ΔkNL.
The derivation of the parametric gain forms the starting point for investigating how dispersion engineered slow light PhC waveguides can improve the efficiency of FWM. Note that the conclusions presented here are independent on the specific method used to engineer the dispersion of the PhC waveguides. The dispersion curves that we use below can be commonly achieved through any of the engineering techniques demonstrated recently in PhC waveguides [17–21]. Therefore, we use, without any loss of generality, the technique presented in Ref , and focus on the implication of the shape of such dispersion curves.
The dispersion curves of Fig. 1(b-c) have been calculated by 3D PWM, for TE polarised states sustained by silicon planar PhC waveguides suspended in air. These waveguides are so-called W09, formed within a triangular lattice of air holes (period a of 420 nm, radius r of 126 nm) where one row has been omitted (Fig. 1(a)) and the waveguide width is slightly (0.9 times) smaller than a standard W1 waveguide. For further details, please refer to Ref .
To illustrate the benefit of the engineered PhC waveguide for FWM, we first compare the conversion efficiency associated with a conventional and an engineered PhC waveguide. Figure 1(b) and (c) display the calculated group index (ng) dispersion of the fundamental mode sustained by an engineered PhC waveguide (W09 having its first two rows infiltrated with a fluid of refractive index nf=1.85 ) and a conventional PhC waveguide (W09 non infiltrated). Unlike the conventional PhC waveguide, the engineered waveguide typically exhibits a spectral region (here between 1560nm and 1570nm) where the group velocity is both low (here ng~31) and roughly constant over a bandwidth that can be as large as 10nm. We refer to this window as the “flat-band slow light”.
We plot on Figs. 2(a-b) the group velocity dispersion (GVD) β 2, and β 4 parameters associated with the modes calculated on Fig. 1(b-c). By substituting these values into the Eq. (4) and (6-8), we calculate the normalized FWM conversion efficiency, Gidler, for L=80 µm, P pump=5W, and α=10 dB/cm and plot it as a function of the pump wavelength and the pump-probe detuning on Figs. 2(c-d). For these calculations, we assume that the nonlinear parameter γ, and the propagation loss α are proportional to ng 2 and ng respectively (see section 3), with ng varying as in Fig. 1(b-c). This allows us to account for the effects of slow light nonlinear enhancement and loss increase in the slow light regime  besides dispersion.
For the standard PhC waveguide, although the FWM efficiency increases as the pump frequency is tuned towards slower group velocities, the bandwidth associated with the maximum FWM efficiency monotonously shrinks because the GVD becomes increasingly large. By contrast, both large FWM efficiency and increased bandwidth (about 4 times that of the standard waveguide) can be achieved with the engineered PhC waveguide, when the pump is tuned within the “flat-band slow light” window. Figure 2(d) shows that the variation in the FWM bandwidth follows the dispersion of β2 when tuning the pump wavelength across the “flat-band slow light” window. As expected from Eq. (6), the conversion bandwidth is generally large near the frequency where the amount of β2 is low, reminding us that large dispersion, even negative, reduces the FWM bandwidth. In addition, a positive FWM gain can be achieved both in the anomalous or normal (positive GVD) dispersion regimes. The latter, which is quite unusual for FWM, is made possible by the simultaneously negative β 4 values provided by the engineered PhC waveguide . This highlights an interesting property of dispersion engineering in PhC waveguides, which provides a full control of both the signs and the values of β2 and β4, giving more flexibility to achieve phase matching.
The dispersion of PhC waveguides can be further engineered to achieve even slower (~c/100) regimes for the pump propagation . In particular, we have demonstrated in Ref  dispersionless slow light over a bandwidth between 3nm and 7nm and with a group velocity ranging between c/40 and c/110. Figure 3 shows the group index dispersion and the spectral variation of the FWM conversion efficiency (similar to Fig. 2(c-d)) for three other engineered slow light PhC waveguides having a group index n g equal to 40, 47, and 66 over the flatband window. All of these engineered waveguides exhibit a similar spectral variation of the FWM efficiency to the one discussed above, with a maximum FWM bandwidth obtained when the pump is tuned in the middle of the flat band slow light window. Note that there is however a trade-off between the low group velocity that can be achieved and the bandwidth over which it exhibits a low dispersion. This feature is actually common to all the dispersion engineering techniques that have been reported in PhC waveguides [17–21], and translates into a decrease of the maximum FWM bandwidth achievable for the slowest PhC waveguide.
In the rest of this paper, we use these waveguides to investigate how slow light improves FWM operation, and unless stated otherwise, we set the pump wavelength in the middle of the flatband slow light window (1565.8nm, 1549.5nm, 1537.4nm, and 1519.6nm for the PhC waveguides with n g=31, 40, 47, and 66) to benefit from a larger FWM bandwidth.
3. Modelling of the pulse propagation across PhC waveguides
While the considerations of the section 2 give us a qualitative idea about FWM in PhC waveguides, it is crucial to model all other (linear and nonlinear) processes that will simultaneously affect the pulse propagation, and to take into account the influence of slow light on the various phenomena. For this, we use the following nonlinear Schrödinger equation to simulate the evolution of the slowly varying envelope A(z,t) of the pulse electric field amplitude as it propagates through the waveguide :
The second and third terms on the left hand side of this equation describe the linear propagation loss, and the second, third, and fourth order dispersion, (the odd high order dispersion term being taken into account). The first term on the right hand side of Eq. (9) is related to the combined Kerr (through γ), Two Photon absorption (TPA, through βTPA), and Raman effects. The nonlinear parameter γ is equal to 2π*n2/[λ0*Aeff] with n2 being the nonlinear Kerr coefficient, λ0 the central wavelength of the pulse and Aeff the cross-section area of the PhC mode; ω0 is the average angular frequency, and R(t) = (1-f R)δ(t) + f R h R(t) is the response function, including a Raman contribution hR(t) deduced from the Raman gain spectrum. The parameter f R is found by normalizing .
The last term of Eq. (9) is related to free carrier (FC) effects, as the FC density in silicon (Nc) adds an additional source of absorption (σ parameter) and dispersion (kc). FCs are generated through TPA and recombine within a time τrecomb, as modelled by the equation :
In this work, we investigate low repetition rate signals (typically 10MHz-100MHz), where the time interval between subsequent pulses is much longer than the typical τrecomb (100ps-1ns) that was measured in silicon PhC geometries . In addition, because the FC lifetime is much longer than the pulse duration considered here (12ps), we neglect the second term of Eq. (10) when calculating Nc.
We introduce the modal group velocity into Eqs. (9) and (10) according to Ref , which accounts for the slow light enhancement of the nonlinear (∝ ng 2) and linear (∝ ng) processes (see Table 1 ). The propagation loss dependence on group velocity has been an important question subjected to extensive theoretical studies [26–28] and measurements [27–33], most of which [26–30] highlighted a quadratic dependence, in a regime where losses are dominated by disorder induced incoherent backscattering. However, these studies have been restricted to band-edge slow light in standard PhC waveguides, where the proximity of the mode cut-off seems to be critical . Although the loss dependence on the group velocity for dispersion engineered PhC waveguides, such as the ones we are studying here, is still under investigation, preliminary measurements have recently revealed a different trend where the loss linearly scales with the slow down factor down ng/n [7,34]. Following these relevant studies, we take into account here a linear group velocity dependence of α, with. α = 10dB/cm × (ng/n). The 10dB/cm value is inferred from the measurements of typical propagation loss between 4dB/cm  and 20dB/cm  for fast (c/5) PhC waveguide modes. Note that this is a conservative number for dispersion engineered waveguides, as 27dB/cm was recently reported for c/30 dispersion engineered PhC waveguide modes  Eqs. (9-10) are solved numerically using the split step Fourier method (SSFM) with the various silicon parameters summarized in Table 1.
4. FWM simulation results
We calculate the FWM conversion efficiency using the model above, for 12ps pulsed pump at low repetition rate and continuous-wave (cw) probe input signals, and interpret the results in the light of the analysis carried out in the section 2. The simulations are run for the four 80 μm long engineered slow light PhC waveguides presented in the section 2 using the dispersion calculated on Figs. 2 and 3. We also simulate for comparison a 3mm long ridge waveguide with A eff=6.6×10−13 m 2, β2=−1.7×10−24 s 2/m, and β3=1×10−37 s 3/m, and an 80μm nanowire waveguide with A eff=1.2×10−13 m 2, β2=−2.1×10−23 s 2/m, β3=8.1×10−38 s 3/m, and 1dB/cm propagation loss [11,24]. The characteristics of these waveguides are summarized in Table 2 .
We calculate on Fig. 4(a) the FWM efficiency for the various waveguides as a function of the pump-probe detuning, when setting the pump wavelength in the middle of the flatband slow light window. The pulsed pump and cw probe powers are P pump=5W and Pprobe=1mW. Fig. 4(b) shows, as an example, the corresponding spectra that we obtain in those conditions for the ng=31 engineered PhC waveguide.
Figure 4(a) first demonstrates the net FWM enhancement due to slow light, despite the simultaneous increase of the other nonlinear phenomena, such as TPA or FC. The maximum FWM conversion efficiency achieved from the different PhC waveguides increases along with their group index, and remains much larger than the efficiency obtained with the reference ridge waveguide, even though the latter is about 30 times longer. Figure 4(a) also reveals the trade-off between a large FWM efficiency, as enhanced by slow light, and its achievement over a large bandwidth, as provided by a wide flat-band slow light window. In particular, the maximum conversion efficiency (−13dB) provided by the slowest waveguide (n g=66) is much larger than the efficiency (−25dB) obtained for the moderately slow waveguide (n g=31), but this comes at the expense of the associated bandwidth, which is reduced from 14nm to 7nm. This is a direct consequence of the trade-off between the group index and bandwidth featured by the flat band slow light window of the dispersion engineered PhC waveguides. All slow PhC waveguides also display a larger conversion efficiency (between −25dB and −13dB) than the fast nanowire (−44.6dB) with similar physical length despite the slightly tighter optical confinement and lower loss of the nanowire. This illustrates the specific advantage of slow light PhC waveguides for compact nonlinear devices, at a length where the shortening of the effective length due to the slow light increased (linear and nonlinear) loss relatively to their fast PhC and nanowire [36,37] counterparts is still restricted .
Towards larger pump-probe detunings, an oscillatory behaviour is observed. Indeed, when the net phase mismatch is large, the square of the parametric gain becomes negative, and the resulting idler power (Eq. (4)) varies approximately sinusoidally over the propagation distance of the waveguide with a period z ≈π/g and an amplitude of (γPpump/g)2. This occurs outside of the flat-band slow light window, where the GVD is large. Besides, it can be seen on Fig. 4(a), that as the magnitude of g2 decreases (as for faster PhC waveguides), the sinusoidal variations present both a longer period and a larger amplitude, providing a smoother transition between the two regimes.
Next, we investigate on Fig. 5 the dependence of the conversion efficiency as a function of the pump power for two different pump-probe detunings of 2.5 nm and 4 nm (indicated as vertical dotted lines on Fig. 4(a)).
Figure 5 shows that for values of P pump below 5W, the idler power P idler varies quadratically with P pump as predicted by Eq. (4). However, P idler strongly saturates under higher pump powers, with the clamping arising at lower pump powers as the waveguide is slower. This saturation arises from the slow light enhancement of the nonlinear loss, as accounted for in the simulations, by the nonlinear (∝ ng 2) - and linear (∝ ng) - dependence of βTPA (TPA) and σ (FC absorption) respectively. Moreover, comparing the curves on Fig. 5(a) and (b) for the PhC waveguides with ng ranging between 31 and 47 highlights that such a clamping is “postponed” towards larger pump powers for increasing pump-probe detunings, eventually enabling a larger conversion efficiency to be achieved. The slowest PhC waveguide does not follow this trend though, because the detuning used on Fig. 5(b) is larger than the narrow flat band slow light bandwidth (~4.6nm/2) of this waveguide.
Two main points can be derived from these results. First, they show that flatband slow light regimes can enhance the conversion efficiency of FWM, enabling operation over shorter distances and/ or for lower pump powers. The second point is that dispersion engineered PhC waveguides can generate FWM over a substantial bandwidth, by enabling a full control of both the group index and the high order β2, and β4 dispersion parameters, and most importantly without compromising the mode area, i.e. the γ parameter. Although the associated bandwidth (~10nm) will never compete with the large (>100nm) values that have been reported in long (~cm) dispersion engineered silicon nanowaveguides , it guarantees FWM operation for practical pump and probe pulsed signal bandwidths (~a few nanometers) with an efficiency largely enhanced by slow light. We show next how this can be exploited for realizing compact nonlinear devices for optical signal regeneration.
5. Designing an all-optical regenerator based on FWM in slow light PhC waveguides
All-optical regeneration aims at removing the distortion and noise that degrade an optical signal upon its transmission in real optical systems and through cascaded amplifiers, using nonlinear optical phenomena. Depending on the types of distortion that the device can compensate for, the regenerator is called either 2R (reshaping and re-amplifying), or 3R (2R + retiming) [38–40]. Optical regeneration has been demonstrated in optical fibers and ridge waveguides using various nonlinear phenomena, including SPM [38,40], cross-phase modulation  and FWM [42–44]. While the first process only provides 2R regeneration, the other two phenomena can ensure 3R regeneration. In this section, we investigate numerically the feasibility of low-power, compact all-optical regeneration in 80 μm long dispersion engineered PhC waveguides using FWM.
The input-output power transfer function determines the regenerative capabilities of the optical regenerator. As a prerequisite, the transfer function should be S-shaped to provide noise suppression and reshaping . Creating an appropriate S-shaped transfer function presents a major design challenge in the development of low-power compact all-optical regenerators. Figure 6 shows schematically the principle of 2R regeneration based on FWM using a pulsed pump and cw probe, where the regenerated signal (in this case, the pump) is obtained by spectrally filtering a fraction of the generated idler.
Figure 7(a) shows the simulated output spectra for increasing pump powers when launching a pulsed pump signal (λpump=1565.8 nm) and a cw probe (λprobe=1569.8nm) into the 80μm long dispersion engineered PhC waveguide with n g=31. We numerically filter the idler generated towards shorter wavelengths by integrating the associated power over a gaussian shape window, which simulates the action of an optical band-pass filter (BPF) (denoted as the black rectangle in Fig. 7(a)), characterised by its detuning from the low power idler central wavelength (Δλf=λi-λf) and its full-width-at-half-maximum bandwidth (BP W). The results are displayed on Fig. 7(b) as a function of the pump power.
The different curves plotted on Fig. 7(b) correspond to various filter detunings Δλf ranging between 0nm and 0.4nm, and a filter bandwidth similar to the pump signal one (BP W = 0.3nm). An S-shaped transfer function can be achieved for small filter detuning values. Depending on the requirements, dispersion engineered PhC waveguides can produce a variety of transfer function curves, provided that one appropriately chooses the filter characteristics. In particular, the FC dispersion that is enhanced by slow light in these waveguides manifests as a blue shift of both the pump and the idler signal (noticeable on Fig. 7(a)) for increasing pump powers [7,23]. Using this effect and a suitably detuned filter, we can favour either one of the two functions of reshaping and re-amplifying in the 2R-regeneration. For example, by choosing the central filter wavelength, λf, close to the idler peak wavelength at high pump power, we can obtain a 2R-regenerator with good reshaping performance (Δλf=0.3 nm for P pupm=20W at n g=31). On the other hand, to achieve a 2R-regenerator with both low power operation and high extinction ratio, we should tune the filter wavelength close to the idler central wavelength at low pump power (Δλf=0nm).
To re-emphasize the benefit of slow light for designing a regenerator, we have plotted on Fig. 8(a) the power transfer function for several waveguides investigated in the section 4, with a group index n g equal to 3.5 (long reference ridge), 31, and 66, respectively. For each of them, we have considered a filter bandwidth BP W of 0.3nm and a Δλf detuning equal to 0.1nm, 0.3nm, and 0.9 nm respectively. The slow light enhancement of the FC induced blue shift of the idler indeed led us to choose larger filter detunings for the increasingly slow waveguides in order to obtain an appropriate power transfer function. The pump and probe parameters are similar to that of Fig. 5(a). The resulting power transfer functions displayed on Fig. 8 confirm that noise suppression and signal amplification through FWM should be improved for the slowest PhC waveguide as compared with the faster PhC waveguide (n g=31) and the 3mm ridge waveguide, the last one having an unsuitable linear transfer function. These results therefore clearly illustrate the potential of short slow light PhC waveguides for achieving a S shaped transfer function that is crucial for all-optical regeneration.
To further confirm that the transfer function improvement is due to the slow light effect rather than the different filter parameters considered in Fig. 8(a), we plot on Fig. 8(b), the results corresponding to the same three devices, but with the same filter characteristics (BP W=0.3 nm and Δλf=0.3nm). These results attest that the threshold power characterising the abrupt change in the transfer function strongly decreases (from 66W (not shown) to 6W) when comparing the fast ridge waveguide with the slowest PhC waveguide. The waveguide group index significantly affects the power transfer function shape, and in turn the noise suppression and extinction ratio improvement. Changing the group index, and the filter detuning therefore gives the designer a broader choice of parameters to design a suitable regeneration according the required re-amplifying and reshaping parameters.
The FWM based regeneration scheme can be advantageously extended to provide 3R-regeneration, by replacing the cw probe by a pulsed signal (clock) synchronized with the pump . Indeed the conversion of energy from the pump to the idler through FWM depends on the instantaneous optical intensity of the pump and the probe signals. Therefore, if the clock has a lower timing jitter than the pump signal, the jitter of the converted data will be reduced as compared with the pump signal , providing the missing retiming function.
By performing similar calculations than above but with both (12 ps) pulsed pump and probe signals, we show that we can obtain power transfer functions with equally good performance and a suitable S-shape, as presented on Fig. 9(a) and (b) , for the PhC waveguides with group index of 31 (a-b) and 66 (b).
In conclusion, we have theoretically investigated FWM in short (80 µm) engineered silicon PhC waveguides with low group velocities (between c/31 and c/66) and shown that both the conversion efficiency and the FWM bandwidth can be increased in these waveguides. Our simulations confirm that phase matched nonlinear processes like FWM doubly benefit from the use of engineered PhC waveguides, through both slow light enhancement of the underlying nonlinear process, and dispersion engineering, which ensures phase matching over a larger bandwidth (~10nm) than for a standard dispersive slow light PhC waveguide. The comparison of the respective FWM conversion efficiency through the various slow light PhC waveguides reveals a net increase (of 30dB at 5W peak pump power) for the slowest one, despite the simultaneous reinforcement of the nonlinear losses, as compared with fast nanowires of similar length. These results make these structures attractive for realizing short nonlinear devices like all-optical regenerators, where the exhibited FWM bandwidth is wide enough. In this regard, we have shown the possibility of designing a compact 2R and 3R regenerators with suitable power transfer function by exploiting FWM into these slow light engineered PhC waveguides. Lastly, while the trade-off between the bandwidth and group index of the “flat band slow light” PhC waveguides studied here may be a limitation on some applications, the approach based on PhC waveguides offers a rich playground to the designer. The dispersion of PhC waveguides can be almost freely engineered in a unique way to display an arbitrary shape that would match the requirements for the intended nonlinear device.
The support of the Australian Research Council through its Centre of Excellence and Discovery Grant programs is gratefully acknowledged.
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