Modulation-Slippage Tradeoff in Resonant Four-Wave Upconversion
MModulation-Slippage Tradeoff in Resonant Four-Wave Upconversion
A. Griffith, a) K. Qu, and N. J. Fisch Department of Astrophysical Sciences, Princeton University,Princeton, New Jersey 08540, USA (Dated: 8 February 2021)
Following up on a proposal to use four-wave mixing in an underdense plasma at mildly relativistic laser intensitiesto produce vastly more energetic x-ray pulses [V. M. Malkin and N. J. Fisch, Phys. Rev. E, , 023211 (2020)],we perform the first numerical simulations in one dimension to demonstrate amplification of a short high frequencyseed through four-wave mixing. We find that parasitic processes including phase modulation and spatial pulse slippagelimit the amplification efficiency. We numerically explore the previously proposed “dual seed” configuration as acountermeasure against phase modulation. We show how this approach tends to be thwarted by longitudinal slippage.In the examples we considered, the best performance was in fact achieved through optimization of signal and pumpparameters in a “single seed” configuration.
I. INTRODUCTION
Directly producing a megajoule of coherent x rays greatlyexceeds capability of current x-ray technologies such as freeelectron lasers or Compton scattering . An alternativeroute for producing high-power coherent x rays is throughefficient conversion of megajoule ultraviolet pulses, whichare available at existing facilities like the National IgnitionFacility . However, conventional frequency conversionprocesses, such as high harmonic generation in gases ,crystals , or relativistic plasma surfaces , cannot scale tothe appropriate intensity or efficiency in the x-ray regime.The challenges of achieving efficient frequencyupconversion of high-power lasers can be overcomeby working in plasmas. Plasmas can resist the highintensities and high temperatures that disrupt solid or gaseousmediums. Plasmas allow for wave-wave coupling processes,which have been investigated for laser amplification, e.g., Raman scattering , Brillioun Scattering , andmagnetized scattering . Additionally, four-wave mixing inplasmas using atomic levels for frequency conversion andpondermotive gratings for same frequency amplification have been considered.In this paper, we consider through numerical simulationsthe recent proposal to employ four-wave mixing inunderdense plasma to achieve both upconversion andamplification . In this proposal, a cascade of nonlinear,resonant four-wave interactions, based on a relativisticnonlinearity, was suggested to achieve up to a megajoule oflaser energy in the x-ray regime . In each step of the cascade,two pump waves, at frequencies ω and ω , amplify a weakhigher frequency seed wave, frequency ω . An idler wave atfrequency ω is generated to satisfy the resonance conditions, ω j = ω pe + c k j , ω pe = π n e e m e , (1) ω + ω = ω + ω , (2) k + k = k + k . (3) a) Electronic mail: [email protected]
Here, the wave frequency ω j corresponds to wavector k j ( j = , , ,
4) and plasma frequency ω pe , for an unperturbedelectron fluid with particle charge e , mass m e , and density n e . As the idler frequency may be small, the seed frequencyfor each iteration can reach up to the sum of the two pumpfrequencies. Each interaction can thus give a multiplicative,rather than additive, change in frequency. With iteratedinteractions, it might be possible to step up orders ofmagnitude in frequency. It is the aim of this work to simulateone step of this cascade.Ideally, the four-wave mixing can increase frequencywith up to unity efficiency. The maximum efficiency isachieved when the pumps are completely consumed. If thesymmetric pumps are ever depleted simultaneously, the four-wave interaction terminates. For synchronously depletedpumps all the wave energy is in the seed and the idler, andthe pumps cannot regrow. Thus the energy could, in principle,flow from the pumps to the seed and idler and never flowback to the pumps. If the idler frequency is sufficiently low, itcarries away negligible energy, resulting in almost all energybeing consumed by the seed. The unidirectional energytransfer possible in the four-wave mixing process representsa significant advantage compared to three-wave scatteringprocesses, which are susceptible to pump reamplification.However, the elegant resonant four-wave interactionbecomes complicated when taking into account phasemodulation. Phase modulation changes the wave frequenciesasynchronously with amplitude, thereby pushing theinteraction out of resonance. The same nonlinearity in thetransverse direction can also result in filamentation of thepumps or seed. To counteract the problems arising fromphase modulation, the initial proposal suggested using asecond signal and idler pair, namely a dual-seed approach,to balance the self- and cross-beam phase modulation termsagainst each other .In this paper, we confirm, using numerical simulations,that significant seed pulse amplification can occur throughfour-wave mixing, but the efficiency is limited by phasemodulation. The four-wave mixing process is additionallycomplicated by variable group velocities and dynamicenvelope amplitudes. The difference in group velocity tendsto make the dual seed approach ineffective. Instead, betterperformance is achieved, at least for the cases considered a r X i v : . [ phy s i c s . p l a s m - ph ] F e b FIG. 1. The dashed ellipse defines the resonance condition, settingthe possible wave vector pairs, (1,2) and (3,4). Simulations in onedimension are projected onto a single axis (gray) along the midlineof the ellipse. here, with a single seed through selection of the four-waveparameters to minimize phase modulation. We note that thecases that we consider are chosen, in part, for their ease insimulation, and do not represent the full range of possibilities.However, the cases that we consider do expose both the upsidepotential and the key physical issues in realizing four-waveupconversion in underdense plasma.
II. RESONANCE CONDITIONS
The resonance conditions, Eqns. (2)-(3), determine thefrequencies and propagation velocities of the four waves.Four parallel waves are not desirable because they yield onlytwo sets of trivial solutions: either ω = ω correspondingto no frequency upconversion, or ω pe = , but this approach is beyond the scope of ourcurrent paper.The valid seed wavevectors, k = ( k (cid:107) , k ⊥ ) , form anellipse determined by the pump wavectors, k and k . Forconvenience, we rotate the frame such that k ⊥ = − k ⊥ , anddefine quantities 2 k = | k + k | = k (cid:107) + k (cid:107) and 2 ω = ω + ω ,1 − ω pe ω − c k = c ( k (cid:107) − k ) ω + c k ⊥ ω − c k . (4)The ellipse, as illustrated in Fig. 1, represents the complete setof pump-pump and signal-idler wavevector pairs. It has twofoci, located at k ± (cid:115) − ω pe ω − c k . (5)The seed frequency is maximized when k extends beyond theright focus and touches the rightmost point on the ellipse, i.e., max | k | = k + c − ω (cid:115) − ω pe ω − c k . (6) For fixed pump frequencies, the maximum seed frequency isachieved when the pumps are misaligned by an angle θ , ≈ (cid:113) ( c | k || k | ) − ( | k | + | k | ) ω pe . (7)Note that, in depicting the ellipse in Fig. 1, in contrast tothe limiting case portrayed in previous work , all wavevectors are not necessarily chords on the ellipse. The wavevectors only approach chords in the high frequency paraxiallimit when both ω pe / ( ω − c k ) and 1 − ω − c k vanishsimultaneously. In this limit, the ellipse becomes long andthin, and approaches a line segment between the foci at 0 and2 k . Depending on the magnitude of the plasma frequency andthe angle between the pumps, the origin and end point of the k and k or k and k pairs may lie inside or outside theellipse.The required wavevector misalignment results in slippagebetween the four waves. The angles, and consequentlyvelocities, between the rest of the wavevectors are bestinterpreted through considering Fig. 1. As ω increases,it pulls the tip of k towards the rightmost point of theellipse, becoming more parallel with the major axis. Tosatisfy the resonance conditions, k must correspondingly tiltmore inward. There is a resulting ordering of | k ⊥ / k (cid:107) | > | k , ⊥ / k , (cid:107) | > | k ⊥ / k (cid:107) | . In the projection, the misalignmentdrives v (cid:107) > v , (cid:107) > v (cid:107) , causing a slippage between the waves.The slippage can be reduced with smaller pump laserangles, but plasma dispersion must increase to keep the four-wave coupling rate constant. The four-wave coupling rate scales with both the angle between the pump beams and theplasma frequency, i.e.,k , ⊥ ω pe k , √ ω ω (cid:12)(cid:12)(cid:12)(cid:12) eA , m e c (cid:12)(cid:12)(cid:12)(cid:12) . (8)Decreasing misalignment and dispersion both reduce thefour-wave coupling through k , ⊥ / k , and ω pe / √ ω ω respectively. A decrease in either term may be compensatedfor through increasing the magnitude of the pump vectorpotential, A , . But the compensation is capped as pumpstrength may only grow as long as | eA , | (cid:28) m e c to remain inthe mildly relativistic regime. The perpendicular wavevectorcomponent and dispersion contribute similarly to the parallelvelocity, v j (cid:107) / c ≈ − k j ⊥ k − j − ω pe ( c k j ) − . (9)Either misalignment, k j ⊥ k − j , or dispersion, ω pe ( c k j ) − ,may be small, but not both if strong coupling is desired.The misalignment is the dominant effect for largeupconversion. For large upconversion, k , ⊥ ≈ k , θ , / θ , chosen in accordance with Eqn. (7), themissalignment term contributes slippage linear in ω pe / ck , .The missalignment slippage term which is linear in ω pe dominates the dispersion term which is quadratic in ω pe asthe waves remain in the underdense regime. The slippage dueto differences in misalignment can be demonstrated to have asignificant effect on the four-wave upconversion process. III. FOUR-WAVE MODEL
The four-wave interaction is governed by a set of nonlinearwave equations derived through combining Maxwell’sequations, the relativistic equations of motion for a constantdensity mono-energetic electron fluid, and the neutralizingeffect of a static ion background. Each wave has a scaledcomplex envelope b j = √ ω j eA j / ( m e c ) , where A j is thevector potential amplitude for wave j . All the waves arepolarized perpendicular to the plane in which all k j arechosen to lie. The four-wave interaction originates from thelowest order relativistic correction to the electron equations ofmotion, expanding in eA j / ( m e c ) .To pose the problem in 1D (one dimension), thedynamical equations are projected onto the dominant axisof propagation. The axis, denoted x , lies on the center of theellipse governed by Eqn. (4), chosen such that k ⊥ j / k (cid:107) j (cid:28) i ( ∂ t + c k j x ω − j ∂ x ) b j = δ ω j b j + ∂ b ∗ j H , (10) δ ω j = ω pe ω j ∑ l = | b l | ω l × (cid:40) f + , j , l + f − , j , l − , j (cid:54) = lf + , j , l − , j = l (11) H = V , , , b b b ∗ b ∗ + c . c ., (12) V j , l , m , n = ω pe ( f + , j , l + f − , j , n + f − , l , n ) √ ω j ω k ω m ω n , (13) f ± , j , l = c ( k j ± k l ) ( ω j ± ω l ) − ω pe − . (14)The L.H.S. of Eq. (10) describes the wave propgation inthe x direction at group velocity v j = c k j (cid:107) ω − j . TheR.H.S. consists of a modulational term, δ ω j , which resultsin amplitude dependent frequency shifts, and a four-wavecoupling term, captured through the Hamiltonian H .The paraxial equations (10) are evolved numerically tocapture the long time amplification of the seed. We conductthe simulations in a frame moving with the seed to reduce thecomputational domain. The two pump waves are initializedevenly out in front of the signal seed, and the seed runsthrough the waves picking up their energy. The fourth idlerwave, which has the smallest parallel group velocity, quicklyflows out of the left side of the domain. The equations areevolved using Dedalus, a general spectral PDE solver . IV. IDEAL FOUR-WAVE BEHAVIOR
To illustrate the opportunities in four-wave resonantmixing, we first simulate an ideal scenario for the four-wave interaction. In this simulation, phase modulation isassumed to be negligible, which will expose the successes andchallenges intrinsic to four-wave resonant coupling. Considerthen pumps that have the same frequency, but with equal andopposite k ⊥ . The resulting synchronous pumps amplify theseed, which grows monotonically in energy. FIG. 2. Numerical snapshots of b j with upconversion factor ω / ω , = .
4. Snapshots are shown at ω pe t = , , × .Pumps ( b , dotted, and b , dot-dashed) are fed in with relativisticfactor | a , | = .
33, with the seed initialized with relativistic factor | a | = . Figure 2 shows three snapshots of a seed being amplifiedby several orders of magnitude in total energy. The firstsnapshot shows the initial conditions (held constant acrossall simulations). The pumps are just beginning to intersectwith the seed and initiate the linear stage of amplification.When the pumps are still strong, amplification occurs widely,resulting in the long seed tail shown in the second snapshot.As the seed strength grows the pumps become depleted,and amplification occurs closer to the front of the seed.For a sufficiently strong seed, all of the pump energy isconsumed at the seed’s leading edge. The signal then growscontinually steeper in time, taking long duration pump energyand compressing it into a shorter peak. The compressionof pump energy is similar to that in Raman amplification .Like Raman amplification, some energy is lost to a disposablewave, the fourth wave here, or the plasma wave in the caseof Raman amplification. But, unlike Raman amplification, allenergy could ideally be deposited into a single growing peak,without producing the amplified pulse train characteristic ofthe π pulse solution for resonant 3-wave interactions .Thus, considerable upshift and efficiency are easilyachieved in the idealized four-wave resonant interaction.The results shown in Fig. 2 achieve a 40% increase inpump photon energy with ω + ω ∼ ω pe . When pumpdepletion is achieved, the energy conversion efficiency maybecome as high as 70%, with the remaining energy flowingto the idler wave. The frequency-upshifted output wave isadvantageously single-peaked.A larger seed frequency corresponds to a higher limitingefficiency, but, in practice, this higher efficiency is difficultto achieve within a finite plasma length. At higher seedfrequency, upconversion is hampered by the consequentdecrease in the coupling coefficient. Figure 3 showsthe reduction in the realized amplification efficiency with FIG. 3. Increasing ω / ω , decreases coupling, resulting in alonger time to efficiency saturation. Over a finite timescale thisreduces efficiency, even though higher saturation efficiency may bereached. ω = . ω , corresponds with evolution shown in Fig.2. All efficiencies are simulation with Fig. 2 parameters, varying ω . ω = . , . , . ω , correspond to ideal efficiencies of67 . , . , .
5% respectively. increasing seed frequency. Weaker coupling increases thetime to reach the pump depletion regime. Only at pumpdepletion is maximum efficiency and steepening achieved.For a limited interaction region, the maximum efficiency maynever be reached, and upconversion may be strictly worse forhigher frequencies.
V. PHASE MODULATION AND THE FOUR-WAVEINTERACTION
The ideal solution, however, neglected phase modulation.The phase modulation terms must in fact be included tocapture fully the lowest order relativistic behavior. Theseterms can push the four-wave interaction out of resonance.For example, the issue caused by phase modulation can beseen in the case of the exact same wavevectors used in theideal regime, e.g. Fig. 2. The same frequency pumps cancelin the denominator in Eqn. (14), making phase modulationscale ∝ c k ⊥ ω − pe . A large value of ck ⊥ ω − pe is requiredfor significant four-wave coupling, resulting in extreme phasemodulation. For the parameters used in Fig. 2, the strength ofthis term results in a perturbation from resonance that makesamplification untenable, with δ ω , approximately 40 timesthe seed growth rate. The wavevectors must then change forthe interaction to coherently amplify the seed.The resonance drift caused by phase modulation can bereduced through pump detuning. Pump detuning reducesthe strength of the f − , , term to ∝ c k ⊥ ( ω − ω ) − (cid:28) c k ⊥ ω − pe for ω − ω (cid:29) ω pe . But detuning the pumps, whilenecessary to reduce modulation, has a corresponding cost inpump-pump slippage. The frequency detuned pumps havedifferent velocities relative to the seed, so that they no longer FIG. 4. Numerical snapshots of b j at ω pe t = , , × for2 ω / ( ω + ω ) = . δ ω j set to 0, but with pumps detunedsuch that ω − ω = ω pe . Pumps are initialized with | a , | = . | a | = . b j at ω pe t = , , × for2 ω / ( ω + ω ) = . δ ω j governed by Eqn. (11), but withpumps detuned such that ω − ω = ω pe . Pumps are initializedwith | a , | = .
33 and the seed is given amplitude | a | = . move in perfect unison. As a result, the perfectly simultaneouspump depletion of the ideal case is no longer possible. Toisolate the indirect effects, namely slippage, from the directeffects, namely phase modulation, we performed simulationswith the same detuned pumps, both excluding and includingthe δ ω term. The results illustrated by Fig. 4 show thatasynchronous pumps can cause the four-wave interaction towork in reverse, leading to re-amplified pumps and reducedenergy conversion efficiency. In Fig. 5 phase modulationis added into the same simulation. The phase modulationbecomes significant at high seed amplitude, pushing the wavesout of resonance, and further lowering amplification.The combination of detuning and phase modulation FIG. 6. Efficiency evolution corresponding to snapshots presented inFigs. 2, 4, 5, and 8. Symmetric pumps equilibrate at a much longertimescale, but at higher level compared to non-ideal alternatives. Allcases correspond with 2 ω / ( ω + ω ) = . apparently sets a lower achievable maximum efficiency. Theefficiency evolution of the detuned simulations with andwithout phase modulation both perform worse than the earlierideal simulations, shown in Fig. 6. The efficiency rises fasterin the detuned simulations, as detuning moderately increasescoupling. However, it becomes bounded at a lower level fromdetuning, and is driven even lower from phase modulation.Both factors contribute significantly, with slippage on its owndriving a large change in efficiency. The importance ofslippage as an indirect effect will persist even as we attemptto mitigate the modulation through other means. VI. COUNTERBALANCING PHASE MODULATION WITHDUAL SEEDS
Following the original suggestion , to reduce thedetrimental effects of modulation, the four-wave approach canbe qualitatively changed through adding two more waves. Theadditional beams induce cross-beam modulation which could,in principle, counterbalance against the pump/pump phasemodulation. Cross-beam phase modulation between slightlydetuned seeds can oppose the cross-beam phase modulationbetween the slightly detuned pumps.The two additional waves obey the same resonanceconditions (Eqns. (15) and (16)), where wave 5 will be thesecond seeded signal and a wave 6 will be the second idler. ω + ω = ω + ω = ω + ω , (15) k + k = k + k = k + k . (16)The dynamical equations must be adjusted to account forthe two new waves. Evolving two additional waves adds FIG. 7. Resonance detuning, δ ω + δ ω − δ ω − δ ω , may have adifferent sign dependence on seed strength if the dual seed approachis used. A point of perfect resonance is achieved at finite signaland pump amplitude with appropriate application of the dual seedstrategy. additional phase modulation terms, δ ω j = ω pe ω j ∑ l = | b l | ω l × (cid:40) f + , j , l + f − , j , l − l (cid:54) = jf + , j , l − l = j . (17)The phase modulation has been extended to all six waves, withthe novelty primarily contained in the strength of the new f − , , term. The similarly large f − , , doesn’t significantlycontribute as waves 4 and 6 never grow large, slipping behindthe point of interaction much faster than signal and pumpwaves.The Hamiltonian governing the four-wave coupling mustalso be extended to accommodate waves 5 and 6. H = V , , , b b b ∗ b ∗ + V , , , b b b ∗ b ∗ + V , , , b b b ∗ b ∗ + c . c . (18)A second set of four-wave coupling results in symmetricpump-pump to signal-idler transfer for waves five and six, V , , , , as previously only was for waves three and four, V , , , . The resonance conditions imply a novel term whichis the signal-idler to dual signal-dual idler coupling, V , , , .When both the 3 , , ω − ω = . ω pe , and 5 ω pe are compared to the unaltered scheme inFig. 7. Weak detuning results in extreme sensitivity ofthe resonance to the seed to pump ratio and a low relative FIG. 8. Numerical snapshots of b j at ω pe t = , , × for2 ω / ( ω + ω ) = . ω − ω = ω pe and with a secondseed and idler such that ω − ω = ω pe . Pumps are initialized with | a , | = .
33 and the seed is given amplitude | a | = . seed strength at which the terms balance. Larger detuningresults in lower sensitivity, and counterbalancing at larger seedamplitude, where the counterbalancing is needed most. Ofcourse, the perfect resonance may be lost as the waves evolvein time away from the arranged amplitudes.Thus, a fifth and sixth wave are added to the previoussimulations to evaluate the dual signal/idler approach. Thenew initial conditions are shown in the first snapshot of Fig. 8.Now, the second signal seed is given the same initial envelopeas the initial signal, such that initially waves three and fivecompletely overlap. Wave five is detuned from wave threeby 5 ω pe , and slips behind the leading seed as can be seen inthe second snapshot. Finite detuning results in group velocitydifferences, and the new lower frequency second signal wavefalls behind on a faster timescale than the amplification. Theleading signal wave then gains more energy than the secondsignal. The signal-to-signal coupling further amplifies thisissue as it drives an energy transfer between the two signalwaves. The required symmetry between the two signal wavesquickly fades, and the simulation begins to converge towardthe earlier unaltered simulation, where the last snapshots ofFig. 8 and Fig. 5 have similar signal wave envelopes.As with detuning the pumps, there is apparently anunavoidable tradeoff between reducing phase modulationand slippage. The phase modulation dominantly affectsthe resonance through the seed-seed coupling, f − , , . Thesensitivity of the resonance to the seed strength scalesinversely with the detuning, ∂ ( δ ω + δ ω − δ ω − δ ω ) ∂ ( | b , | ) ∼ c k ⊥ ω pe ω ( ω − ω ) . (19)While the sensitivity of the resonance decreases with largerseed-seed detuning, the slippage between the two waves increases linearly, v (cid:107) − v (cid:107) ∼ c ( ω − ω ) k ⊥ ω . (20)Both low sensitivity and low slippage cannot be achievedsimultaneously, but both are needed to make the six-waveapproach effective.The six-wave strategy in our implementation performsworse than the approach without the dual seed. The worseperformance is seen not only in pulse structure, in Fig. 8,but also in the efficiency, as shown in Fig. 6. Only afterthe second seed has fully fallen behind the original seed doesthe efficiency begin to approach that of the unaltered four-wave approach. In our implementation we have not founda parameter regime in which the six-wave solution improvesupon the comparable simpler four-wave approach. VII. CONCLUSIONS
In conclusion, we described, using one-dimensionalsimulations, how idealized four-wave mixing with twobalanced pumps, can amplify, with high efficiency,significantly upshifted pulses. However, this idealizedcase neglected phase modulation terms, which are difficultto cancel out. When phase modulation is considered, thepump wavevectors must be changed to reduce cross-beamphase modulation. With just a simple strategy to mitigatephase modulation, successful upconversion can be achieved,but with efficiency significantly lower than in the idealscenario. At least in the cases that we considered, adding asecond signal-idler pair to mitigate phase modulation has notmanaged to achieve the theoretically available efficiencies.The second seed’s slippage results in asymmetry betweenseeds, which is further exacerbated by additional coupling.The asymmetry between the seeds serves more to reduce theamplification than to remove the limiting effects of phasemodulation. In all the cases that we considered, optimizingfor low slippage was as important as optimizing for favorablefour-wave coupling and phase modulation mitigation.The considerations here are just a first cut at describingnumerically the issues in optimizing the recently proposedfour-wave coupling in underdense plasma to producefrequency upshifted laser power with very high efficiency.While the efficiencies reached in these simulations wereconsiderable, they fell short of the theoretically achievableefficiencies. Nonetheless, the possibilities explored heredo not exhaust what might be attempted to achieve thosetheoretically achievable efficiencies. Other possibilitiesinclude introducing more waves to control the phasemodulation and to vary the waveforms in space. In particular,we have not considered the suggestion of utilizing grazingangle reflections in a channel, which carries perhaps themajor opportunities for realizing the potential of four-waveinteractions . While enlarging the parameter space of themost promising avenues to be explored, these possibilitiesdo come, however, with added complexity in experimentalrealization and computational cost in simulations. Whatwe did explore already showed both the very promisingpotential of the four-wave upconversion effect and the issuesin realizing it. ACKNOWLEDGMENTS
The work is supported by NNSA Grant No. DE-NA0003871.
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