We show that optical parametric generation in a nonlinear crystal with a large group velocity mismatch between the pump and nearly-degenerate signal and idler is analogous to laser amplification in the medium with a gain recovery time comparable to the walk-off time. Based on this conclusion we propose to combine an OPO with a nonlinear saturable absorber or Kerr lens to generate directly high peak power sub-picosecond pulses using pump pulses ranging from tens of picoseconds to quasi-CW. Our analytical model predicts better than 80% photon conversion efficiency and pulse lengths that are of the order of a few hundred femtoseconds. Numerical simulations confirm our predictions and show that repetitive passive mode locking is feasible with a quasi-CW pump.
© 2008 Optical Society of America
Optical parametric oscillators (OPO) are efficient sources of tunable radiation due to the exceptionally wide bandwidth of parametric gain that is determined only by the phase mismatch . This gain becomes especially wide near the degeneracy of the idler and signal frequency reaching tens of THz. Spectrally broad radiation can correspond to ultra-short pulses in time domain if all its spectral components are locked in phase – this is exactly what happens in mode-locked lasers where Fourier-limited pulse trains can be generated with a CW or quasi-CW pump. Essentially a mode-locked laser operating with a given duty cycle compresses in time the pump energy by an amount inverse to the duty cycle. It would be common sense then to attempt a similar compressive effect in OPO, but unfortunately the physics of OPO is not conducive to this. While in the laser gain medium the pump energy is stored for a certain gain recovery time, in the OPO the pump energy is constantly on the move with a velocity close to that of the generated signal and idler. Therefore, modulating the cavity loss in an OPO may indeed generate series of short pulses, but only a small fraction of pump energy coming from the pump photons propagating synchronously with the signal and idler will be converted into these short pulses. We checked this fact in a recent work where active mode locking of a CW-pumped OPO has been achieved for the first time . A stable train of short pulses has been obtained, but with a very small efficiency (~15 % of the CW efficiency). In this work we explore the means of increasing the conversion efficiency of mode-locked OPOs pumped by CW radiation or by comparatively long pump pulses.
It had been noticed in  that due to material dispersion the group velocities of the signal (νs) and idler (νi) are larger than the group velocity of the pump νp – hence in the reference frame of the pump the signal photons move and over the length of the crystal cover the interval δtps=L(ν -1 p-ν -1 s). Thus all the pump energy contained in this interval can be converted into the signal and idler. This is the principle behind energy compression in synchronously pumped OPO first suggested in  and then successfully demonstrated by different groups [4–8] where up to a 20-fold pulse compression had been achieved in a BBO crystal pumped by a few picosecond pulses way above threshold. Later on similar results were demonstrated using KTP [9,10], PPLN , and AgGaS2 . Group velocity management can also lead to a better efficiency in parametric amplifiers, especially when the group velocity of the pump lies between those of the signal and idler waves . With the pump compression achieved solely by gain saturation, the shape and the length of the pulses obtained in these experiments depended critically on the detuning between the resonant cavity round-trip time and the pump repetition rate, and the pulse duration was limited by the dispersion. To the best of our knowledge, the only experiment in which passive mode locking of an OPO had been achieved by a Kerr lens effect did not use the temporal walk-off effect thus the compression achieved in that work was insignificant .
In this work we investigate feasibility of introducing a passive mode locker, such as saturable absorber (SESAM)  or Kerr lens, into the OPO characterized by large group velocity mismatch. While most of prior work had relied on numerical analysis, we develop a simple analytical model showing that one can treat the nonlinear crystal with a large temporal walk-off δtps as a saturable gain medium with a gain recovery time comparable to δtps. With that in place, we can apply all the classical mode locking theory developed in the works of Siegman and Haus [16–19], in order to ascertain the efficiency and peak power of mode-locked OPO operating in the steady state regime. We then perform numerical modeling which confirms our analysis and shows that introduction of a passive mode locker into the OPO with a large walk-off leads to generation of stable sub-picosecond pulses, and, furthermore, even with a CW pumping one can achieve repetitive mode locking, similar to harmonic mode locking in lasers.
2. Gain saturation in an OPO with walk-off and in a laser amplifier
We consider the OPO (Fig. 1) to be singly-resonant at the signal frequency. All the amplitudes are normalized in such a way that A 2=p=dn/dt where n is the number of photons. The equations in the nonlinear crystal of length L are then
where ν denotes group velocities. The coupling coefficient is
where nr denotes refractive indices, S is the effective area and η 0=377 Ω. Note that the dimensionality of A is s-1/2 and for κ it is s1/2/cm. In order to operate with real powers rather than photon numbers one can divide κ by the square root of the pump photon energy (ħωp)1/2. For the operation at pump near 1 µm with typical values of χ (2)~20 pm/V which is the case for such state-of-the art materials as GaSe or PPLN and S~10-4 cm2, a typical value of κ≈0.025 cm-1W-1/2 and somewhat higher using quasi-phase matched GaAs  with χ (2)~50 pm/V. We also assume (for now) that the group velocities of signal and idler are close to each other because we operate close to the degeneracy region, thus νs≡νi. Furthermore, at this point we shall consider the case of perfect phase matching – the effect of the imperfect phase matching on the pulse length will be considered in Section. 4. With perfect phase matching one can assume all field amplitudes to be real. If we now introduce a moving system of coordinates with new time t′=t-z/νp we obtain
where we have introduced the rate of walk-off as δν -1=ν -1 p-ν -1 i. In the moving frame as the signal and idler travel a distance Δz they advance in time by Δzδν -1 i.e. by the total walk-off time δtps over the whole length of the crystal. We now consider the signal and idler short pulses defined in an envelope function approximation as
where f(τ), the normalized pulse shape of length ts is assumed not to change as the signal and idler pass through the crystal which is certainly a valid conjecture for the steady-state regime of operation with a stable pulse shape. In the moving frame the signal is delayed by the time δtps=Lδν -1 at the input and emerges from the crystal at t′=0. Besides, in an OPO resonant at the signal frequency and having small cavity losses one can make the so-called mean field approximation that the signal power does not depend on coordinate z. It follows from (3) and (4) that the pump photons associated with time t′ encounter the signal and idler in the spatial window L-t′/δν -1<z<L-t′/δν -1+ts/δν -1; outside of that interval they travel unchanged. We then solve (3) to obtain for the depleted pump power following the signal and idler pulses arrival
by using the substitution τ=t′-(L-z 1)δν -1. Now, if the condition ts≪Lδν -1=δtps is satisfied, i.e. pump photons are being “swept” by the signal and idler pulse over a very short distance δzps=ts/δν -1≪L, the amplitude of the idler stays relatively constant over this “sweep distance”
It is important to note that after being “swept” the pump photons do not get depleted any further and all arrive at the output end of the crystal. Substituting depleted pump (6) into (3) we obtain the propagation equation for the idler
If we multiply both sides of (7) by the Ai(z,t′) and integrate over the whole duration of the pulse, i.e. for t′>(L-z)δν -1+ts, after some calculation we obtain
where we have introduced
by using the substitution τ 2=t′-(L-z)δν -1. For most symmetric pulse shapes γ~0.5 indicating that each signal/idler pulse on average sees only a half-saturated pump level. Solution of (8) can be obtained by using substitution ni=x2 and written as
where we have introduced the saturation photon number as
and the “available pump energy” (in photon number units) as
This energy is exactly the number of pump photons that the signal/idler pulse sweeps by as it propagates distance L in time δtps=Lδν -1. As we have already mentioned above, the pump photons at time t′ emerging from the crystal are depleted at a distance z=L-t′/δν -1 and continue further unchanged after the signal has passed. Hence, substituting (10) into (6) we obtain the result for the pump power at the output face
where again we assumed ts≪δtps.
In Fig. 2(a) we show the variations of the depleted pump strength for different values of ns/n sat under assumption that the pulse length is equal to ts=0.02 δtps. One can see how the pump pulse gets depleted and then recovers after the time δtps. Note that in our approach pump reconversion corresponds to a sign reversal of Ap, as in Fig. 2(a). To define a small signal gain we consider a “probe” pulse following the actual signal pulse and lagging it by some time t – as this signal propagates it encounters the pump power depleted to various degrees. Hence, using (3) and (13) we can find for the small signal gain
where g0 is the unsaturated value at ns→0. The results are plotted in Fig. 2(b) for γ=1/2. Following the signal/idler pulse at t=0 the gain is saturated as
and then it fully recovers over the walk-off time. For small (relative to saturation) values of signal photon flux one can write
This result is no different from the result for the laser gain medium with saturation photon flux γnsat/2≈nsat/2 . Fig. 2(b) shows the curves for the saturation of an “equivalent” laser gain medium for γ=1/2
As one can see the behavior of the OPO gain is almost entirely identical to that of a standard saturable amplifier with a saturation photon energy equal to nsat/2 and a gain recovery time equal to δtps/2. For this reason one should expect the behavior of OPO with a walk-off to be similar to that of the saturable gain medium. For example, in the absence of any mode locking the system still should be capable of generating short gain-switched pulses, characterized by relaxation oscillations. The specifics of OPO though is that the damping time for these oscillations is the walk-off time δtps, which is much shorter than the photon lifetime in the cavity. Therefore only a few or maybe just one strong gain-switched pulse may be observable, as is indeed the case in most of the experiments [4–12], although there has also been clear evidence of more than a single peak in . With passive mode locking by either a Kerr lens or a saturable absorber a single mode-locked pulse should be generated with a pulsed pump, while for a CW pump a harmonically mode-locked solution becomes possible. It should be noticed that when the gain recovery time becomes shorter than the cavity round trip, which is obviously the case for the above-described OPO, one cannot make clear distinction between the gain-switched and mode-locked pulses. Hence, for a pulsed pump, one can expect only a quantitative difference when introducing a mode locker: the mode-locked pulses are single, and have relatively shorter pulse length, while the gain switched pulses are somewhat longer and typically followed by a few much smaller pulses representing relaxation oscillations. However, for CW pumping a much more pronounced difference is expected: since gain switching without saturable absorber or Kerr lens is outright impossible, only mode locking can produce sustainable oscillations.
3. Conversion efficiency
Before testing numerically the results following from our “equivalent saturable gain” model we shall return to (10) in order to ascertain the conversion efficiency of the pump-swept OPO as
where α is the round-trip loss. The plot of this curve is shown in Fig. 3(a). A maximum output efficiency of about 80% actually occurs when the circulating signal pulse energy is, ns,opt≈1.26nsat or the output energy, nopts,sat≈1.26αnsat, and does not depend on γ. Note that we also plot the curve of the depleted small signal gain (15) on the same plot. Clearly the small signal gain gets completely depleted at roughly the same value of circulating signal energy, 1.55 nsat.
Next we can determine the input-output naν-αns characteristics of the pump-swept OPO by transforming (18) into
from which the threshold can be obtained as
Thus the ratio of the threshold available energy to the saturation energy is simply the cavity loss normalized by γ. Substituting (12) and (11) into (20) we obtain the threshold power of the pump
which of course is not different from any other single-resonant OPO . At the same time, the pump energy at which maximum conversion efficiency is achieved can be found as
as is indeed shown in Fig. 3(b) with input available photon number normalized to the threshold value. Thus just as in ordinary singly-resonant OPO maximum conversion efficiency can be achieved relatively soon after the threshold and can realize about 80% of the “compression gain” defined here as G=δtps/ts. This is in marked contrast with the actively mode-locked OPO where only a small fraction of the pump energy was used .
4. Pulse length
Making a precise prediction of the pulse length in the passively mode-locked system is not simple and usually requires numerical estimate. This is especially true for the mode-locked pump-swept OPO in which the bandwidth of the gain depends on the power due to pump saturation. Nevertheless, one can attempt to make an order of magnitude prediction. According to Siegman  one can consider a mode-locked OPO as a sum of two nonlinear filters: low pass gain medium and high pass mode locker, which we consider to have instant response (such a mode locker can be for instance a Kerr lens).
The low pass characteristics of the parametric gain arise from the walk-off between the different spectral components of the signal in the degenerate OPO, and between signal and idler δνsi -1=νsi -1-νsi -1 in the non-degenerate case. Since non-degenerate OPOs are, by their nature, tunable and being singly-resonant are easier to align, we shall consider only non-degenerate OPOs. The phase–matching bandwidth can be approximated as FWHM of the curve sinc2(Δωδνsi -1 L), giving Δω1/2=2.78/(δνsi -1 L). Notice that the phase-matching bandwidth of the parametric process in the spectral domain description is simply inversely proportional to the signal-idler walk-off time in the temporal domain description of the same process. Then the small signal parametric gain can be approximated as a Gaussian curve
where one can show that for the optimum circulating signal energy ns=ns,opt≈1.2nsat, the spectral width of the gain is
In Fig. 4(a) the gain curve of OPO and its Gaussian approximation (23) are shown.
We next characterize the instant mode locker with saturation power pA,sat, introducing saturable losses 
where α0 is the unsaturated loss. When the mode-locked pulse is also Gaussian with FWHM one can show that temporal dependence of the transmission of the mode locker can also be approximated by a Gaussian function
where Δt is the time relative to the peak of the pulse. The maximum transmission T 0 is given by
and the temporal width of the Gaussian transmission is
where we have introduced the saturated loss αsat=α 0/(1+p 0/pA,sat). The passive mode locker temporal characteristics and its Gaussian approximation are shown in Fig. 4(b). It follows from (28) that after each passage through the mode locker the pulse gets compressed in time
and, according to (23), the pulse gets narrowed in spectral domain in the nonlinear material, i.e. it expands in temporal domain in the following fashion
The steady state condition requires the pulse width to stay intact after one round trip, i.e. σt3=σt1 and this finally brings us to
with Δν1/2=Δω1/2/2π. Now let us realistically estimate the value of the expression under the square root sign. The total loss α per round trip is the sum of mirror loss and saturated absorption. In order to get sufficient slope efficiency one should have α=Kαsat where K is of the order of 5–10. That is, the saturated loss of the mode locker should only represent a fraction of the total cavity loss. Typical values are 6 % of cavity loss and 1 % of saturated loss. Furthermore, we assume that absorber gets well saturated i.e. αsat≪α 0<1 and we obtain
The result is quite optimistic as it predicts that one can get a pulse duration not much longer than the Fourier limitation imposed by the phase-matching conditions. We can now also obtain a very instructive approximation of the compression gain
indicating that the maximum peak-to-average power achievable in the pump-swept OPO is of the order of the ratio of two-walk-offs – the one between signal and pump and the one between signal and idler.
5. Material choices
So, what are the typical values of the temporal walk-off that are attainable in the materials of interest? In general the pump photon frequency needs to be fairly close to the band gap where two photon absorption usually presents a problem. The key advantageous point of the mode-locked pump-swept OPO is that one can attain very high instant powers of signal and idler while the pump peak power is kept relatively low. According to (2) and (21) the threshold power density in a typical mode-locked pump-swept OPO with total cavity loss of α~0.1 and crystal length of 1 cm is about
Considering for example QPM GaAs crystal with pump wavelength of 1060 nm – the two photon absorption coefficient at this wavelength is of the order of 0.5×10-8 cm/W indicating that even if we operate at the values that are three times above threshold less than 1% of the pump power will gets absorbed, i.e. only 10 kW/cm2. That should lead to no more than a few degrees rise in temperature. Therefore, as an example we choose bulk QPM GaAs  for which we show the dispersion of group index ng=c/ν g in Fig. 5(a). As one can see for λp=1060 nm group index approaches 4 and the walk-off rate near the degeneracy is δν -1=ν -1 p-ν -1 s≈18 ps/cm. In Fig. 5(b) we show the phase matching bandwidth Δν 1/2 as a function of signal wavelength. Now it appears that for a signal wavelength between about 1800 nm and degeneracy of 2100 nm one can achieve pulses shorter than 1 ps. Therefore we have chosen the following combination of wavelengths and group indices: λp=1060 nm (ng,p=3.947), λs=2000 nm (ng,s=3.421) and λI=2578 nm (ng,i=3.396). For the most divergent, i.e. idler radiation and waist area of 10-4 cm2 the diffraction length is 3 cm thus with 1 cm long crystal one can disregard the change of the cross-section over the length. With 1 cm long QPM GaAs crystal the pump-signal walk-off time is δtps~18 ps and the signal-idler walk-off time is δtsi~0.8 ps with closely related phase-matching bandwidth equal to 0.7 THz. Therefore we can expect conservatively to obtain pulses of a few hundred fs at 2 µm and thus achieve pulse compression by about factor of 50. If one is interested in producing 1 µm light, the same walk-off rate is available in PPLN:MgO using a 532 nm pump source and a longer crystal. A broad bandwidth is also available in some QPM materials away from the degeneracy point within a certain range of pump wavelength, as for PPLN near 800 nm pumping . While this fact would be useful to extend the mode-lockable wavelength range, the corresponding temporal walk-off may be too low for efficient mode-locking in the pumpswept OPO (δtps~5 ps in a 3 cm-long PPLN).
These estimates does not take into account the dispersion of the signal pulse itself but it is expected that the difference between the group velocities of signal and idler is much larger than the range of group velocities contained within the spectrum of the signal pulse itself and thus will be the dominant factor. Indeed for our example one can estimate the dispersion pulse length, i.e. the pulse length over which the pulse broadens by √2 as
indicating that the walk-off between the signal and the idler is expected to limit the shortening of the signal pulse long before the pulse dispersion becomes important.
6. Results of numerical modeling
6.1 Pulsed pumping
Let us consider the results of numerical solution of (1). The equations are solved by the finite difference method – at each time step each field gets advanced by a length proportional to its group velocity (higher order dispersion is not included). First we consider the situation without any mode locker. The effective cross-section of the beam in the crystal is 10-4cm2 and the mirror loss is 3% – thus the threshold power is 75 W peak power. The OPO is synchronously pumped by a series of Gaussian pulses with peak power of 450 W i.e. 6 times above threshold and FWHM t(p) 1/2=20 ps. Thus the pump pulse energy is about 10 nJ. The oscillation starts from random intensity noise (half a photon per mode) and builds up for about 100 round trips, depending on loss and pumping power. In Fig. 6(a) the three curves for the circulating signal, idler (multiplied by 10), and depleted pump (multiplied by 100) are plotted in the steady-state regime. The vertical axis is in units of 1/κ2L2 i.e. in units of photon flux to allow visualizing the photon conversion efficiency. In our example these units correspond to 1.3×1022 s-1. As expected high above threshold the signal and idler are trains of decaying short pulses which look very much like the relaxation oscillations in case of gain switching. The width of the first, strongest and shortest peak is about 1 ps and the maximum output power reaches 990 W – about 2-fold enhancement in peak power or 4-fold in terms of photon flux. The overall photon conversion efficiency is about 71%.
If one increases the cavity outcoupling to 9% as in Fig. 6(b) the threshold will be exceeded only by a factor of two – then only one large gain switched pulse will be excited. The width of the signal pulse is about 0.7 ps, slightly less than that of the idler pulse. Maximum output power reaches 4500 W, a 10-fold enhancement compared to the pump (20-fold in terms of photon flux). These pulses are similar to the ones observed in [5–12]. The overall photon conversion efficiency is about 50%. The pulses are asymmetric as the time-lagging idler pulse continues to contribute to the raise of the trailing end of the signal.
Next we consider the case when an instant mode locker with peak absorption of 3% (equal to mirror loss) and saturation power of 2.5×104 W is placed in the same cavity. The results are shown in Fig. 7(a). Only a single signal pulse is excited with a peak output signal power reaching 11 kW in a pulse with only 250 fs FWHM – this is a 25-fold enhancement in terms of peak power (50-fold in terms of photon flux). Overall conversion efficiency is also increased to 60%. These results are in agreement with our estimates of conversion efficiency. It is lower than the optimum of 80% because the tails of the Gaussian pump do not get converted. In terms of pulse duration, it is actually shorter than expected, i.e shorter than the signal/idler walk-off time. This can be easily explained by the fact that most of the amplification of the signal pulse occurs only towards the end of the nonlinear crystal, hence the effective walk-off is smaller than the nominal value of 0.8 ps. At the same time the width of the idler pulse is comparable with the walk-off time. It is worth noting that even at 250 fs the pulse length is too long for the material dispersion to play an important role.
We have also considered mode locking with a slower mode locker of the SESAM type with a response time of 2 ps and the same peak absorption and saturation power as the fast mode locker in the previous example. The results are shown in Fig. 7(b). A 500 fs-long output pulse is excited with peak power of 2500 W, leading to overall conversion efficiency of 50%. The situation improves when the saturation density of the mode locker is reduced two-fold to 1.25×104 W – the results are shown in Fig. 7(c). Peak output power is now about 10 kW and the pulse length is 300 fs – results comparable to the fast mode locker case. The overall photon conversion efficiency reaches 61%.
6.2 Continuous-wave pumping
Finally we have considered the case of CW pumping of the OPO. First we consider the case with no mode locker (Fig. 8(a)). The signal is initialized with random intensity noise. As expected the solution is CW oscillation with intracavity power of 390 W and overall photon conversion efficiency of 87%. Once the fast mode locker has been inserted, the CW solution becomes unstable and mode-locked pulses appear. In Fig. 8(b) one can see that the pulse trains are periodic with a period equal to cavity round-trip time, as expected. One of the pulses is shown on the expanded time scale in Fig. 8(c). Actually, what we have here is about 20 separate and intermixed mode-locked trains with no phase relations between trains but with phase relation inside each train. Since the pulses originate from random noise the intervals between these pulses during one round-trip are not equal, but, on average they are of the order of being slightly smaller than δtps=18 ps. This can be confirmed by looking at the spectrum of mode-locked pulses shown in Fig. 8(d) that clearly exhibits a peak near 65 GHz. Fig. 8(b) also shows that the peak power and duration of the pulses vary due to different pulse intervals between them, but the highest peak power is comparable to that obtained with a synchronous pulsed pump and their duration is also of the order of 300 fs. The photon conversion efficiency actually achieves 92% because the pump never recovers completely before it gets depleted by the following pulse. Thus the efficiency is higher that the one predicted by our analytical model assuming complete recovery of the pump pulse. Our other numerical simulations have also shown that one attains similar results using a saturable absorber with 2 ps response time. Note that, in order to increase the convergence speed of the simulation we used a large mirror loss of 9 % and a pump power well above threshold. However, the results shown here still qualitatively apply to actual CW-pumped OPOs were the threshold is only a few watts .
In this work we have investigated the feasibility of efficient mode locking of an optical parametrical oscillator, where “efficient” means that the energy of relatively long pump pulses can be nearly completely converted into the signal and idler pulses, whose duration is much smaller than the pump pulse duration. We have considered a singly-resonant OPO with a large temporal walk-off δtps between the pump and the signal as a candidate for mode locking because the short signal pulse can “sweep” most of the pump energy available in the walk-off interval. We have developed analytical expressions for the pump depletion and small signal gain saturation in this mode-locked pump-swept OPO, and have shown that saturation characteristics are equivalent to those of a laser gain medium with a gain recovery time close to δtps/2. Therefore, according to our predictions, the mode-locked pump-swept OPO should be capable of generating short pulses by gain switching, or, in the presence of saturable loss, by mode locking. We have obtained analytical expression for the efficiency of such mode locking and for the duration of the pulses. We then identified QPM GaAs as a good candidate for the development of mode-locked pump-swept OPOs at 2 µm. We performed numerical calculations confirming our analytical results and showing that with both pulsed and CW pumping one can obtain sub-picosecond pulses of mid-infrared radiation. The fact that relatively low (few hundred W) pump power can be used to obtain pulses with peak power in excess of 10 kW mitigates the two-photon absorption effects. While in our model we used QPM GaAs as an example, significant pump-signal walk-off times can be attained in other dispersive materials such as PPLN and GaSe where longer crystals are available. One can consider further enhancement in pulse compression by using mode dispersion in waveguides  or the so-called “slow light” structures, typically with grating, in which the group velocity of the pump can be further slowed down and the walk-off time can approach one nanosecond . That would open a possibility of efficiently “compressing” Q-switched pump pulses into mode-locked signal pulses. Another suggestion would be to use counterpropagating pump and signal to further increase the sweep. Considering that the only other means of obtaining sub-picosecond mid-IR pulses are synchronous pumping with ultrashort pulses, the mode-locked pump-swept OPO offers an interesting alternative. For instance, it could be used as an ultrafast seed source for chirped pulse amplification. An experimental setup is under investigation using a 3-cm-long MgO:PPSLT crystal pumped by a quasi-cw frequency-doubled Nd:YAG laser with 30 W peak power. In this case the Kerr medium could be the crystal itself, or a KTP crystal in the focus of a telescope , since these materials exhibit a rather large nonlinear index coefficient, compared to sapphire for instance . In future works, it would also be interesting to consider the interplay of diffraction and group velocity dispersion with nonlinear amplification, as it can lead to light self-trapping and other unusual phenomena .
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