Spectral Control of High Harmonics from Relativistic Plasmas Using

arXiv:1802.00249v2 [physics.plasm-ph] 12 Apr 2018 aie L)fils a enitoue orelativistic to introduced been po- has HHG[ linearly plasma fields, paralleled two (LP) i.e., larized scheme, Recently, field field. two-color two-color a a with field laser the structuring oiincnla ovr ieetfil atrsa well as patterns field different very to super- lead pulses pulses the can as CP position helicity, two and frequency combining situation different when The with drastically envelop. change pulse may varying the slowly following only contains components case this in single force incidence[ one deromotive normal using at by pulse pro- suppressed CP HHG completely ROM almost the is that polarized known cess direc- is circularly opposite it in Interestingly, using rotating tions. pulses on driving based two-color (CP) is which different, den- plasma the modifying ramp[ by achieved sity been the has enhancement Among HHG selected of schemes, limited. control remain spectral spectrum reported pulse[ HHG obtain the to attosecond control control e.g., single to harmonics, out isolated the carried an of structure While been temporal has the duration. investigation attosecond an with many region and ultraviolet spectral provides extreme thus the x-ray in This source range. radiation do- the powerful spectral time a in broad the the comb in to a due pulses of main attosecond har- form and The the domain in light. frequency of emitted that usually mir- to are a close monics as speed acts with which oscillating surface, ror plasma the laser by the reflected of field up-shifting oscillating Doppler relativistic relativistically as interpreted called model[ so (ROM) mirror the is mechanisms h eutn ihhroi yield[ harmonic high resulting the n ae rqec oishrois hc a span can which orders[ harmonics, harmonic its of to thousands frequency extreme driv- fundamental laser an the is ing converts which surfaces process, plasma nonlinear irradiating laser intense ∗ [email protected] eew osdrcnrligtehroi pcrmby spectrum harmonic the controlling consider we Here ihhroi eeain(H)fo relativistically from (HHG) generation harmonic High pcrlcnrlo ihhroisfo eaiitcplasm relativistic from harmonics high of control Spectral 14 rpam ufc morpha[ surface plasma or ] ue.TeHGecec fti ceecnb shg sta dr that as high as be can scheme this polarizati of field. the efficiency HHG and The overall orders, the tuned. harmonic law demonstrate neighboring conservation harmo we allowed related components, of of the field group and driving selected two fields a the laser p only the relativistic that of show from symmetry we (HHG) simulations, generation cell harmonic high control 20 eitouetoclrcutroaigcrual polari circularly counterrotating two-color introduce We ainlKyLbrtr fSokWv n eoainPhysic Detonation and Wave Shock of Laboratory Key National ,wihsosgetptnilt enhance to potential great shows which ], .INTRODUCTION I. 5 – 9 .Tepyia itr a be can picture physical The ]. hn cdm fEgneigPyis inag619,Chi 621999, Mianyang Physics, Engineering of Academy China 22 1 ,bcuetelsrpon- laser the because ], – 4 21 .Oeo h main the of One ]. 10 .Orapoc is approach Our ]. – 13 15 ,mtosto methods ], – 19 ]. Dtd pi 3 2018) 13, April (Dated: iY Chen Zi-Yu Pple n ne omlicdneinteractions. incidence normal under using such although and shown efficiency pulses high also a CP is have har- can As It process CP achieved. HHG an and be laws. can enhancement conservation of generation harmonic related symmetry monic selective the the result, and by their a governed fields and be laser orders to fea- the harmonic found spectral appeared are the harmonic helicity, as The harmonics such high control tures, plasmas. spectral to relativistic also used from bicircu- are be the can HHG Thus fields HHG. gas lar plasma the in relativistic that to discovered find applicable rules and regime selection HHG scheme similar plasma a relativistic such of the effectiveness in the confirm to simulations iesoa 1)PCcd LL(ita ae Plasma Laser (Virtual Lab)[ VLPL code PIC (1D) dimensional quency ple ietyt eaiitcpam H ihu val- be without cannot HHG regime plasma idation. HHG relativistic to gas directly the applied mecha- within results HHG the before two Therefore, obtained the different. since fundamentally not, are relativistic nisms or the regime in work HHG not should is plasma not scheme It has this before. it whether HHG knowledge clear plasmas our surface for to addressed HHG, been gas in extensively ied n t eodhroi 2 harmonic second its and escnb ipae n eetvl nacdi the fields[ in bicircular such enhanced by selectively or- driven and harmonic spectrum displayed of It harmonic sets be ion. specific can parent only ders re- its that and with demonstrated spreading re-collision is lateral one-color of electron’s a probability an by duced to driven due suppressed pulse, HHG the greatly CP whereas also gases, is atomic from process HHG CP in fective frequency fundamental both ω one shown with been configuration already bicircular has It theoretically[ trajectories. charge as 32 n t onerttn eodhroi 2 harmonic second counterrotating its and h iuain eepromduigteone- the using performed were simulations The nti ok euse we work, this In stud- been has configuration field a such Although ]. ∗ 33 e ae ed sanwwyt spectrally to way new a as fields laser zed nsaeo h amncsuc a be can source harmonic the of state on .Tediiglsrfilswt udmna fre- fundamental with fields laser driving The ]. ω .B dutn h nest ai of ratio intensity the adjusting By s. am irr.Truhparticle-in- Through mirrors. lasma 0 H ffiiny h eaieintensity relative the efficiency, HHG i rescnapa wn othe to owing appear can orders nic crepnigt wavelength to (corresponding 23 I IUAINSETUP SIMULATION II. vnb ieryplrzdlaser polarized linearly a by iven ,Isiueo li Physics, Fluid of Institute s, – 26 n experimentally[ and ] suigbcrua fields bicircular using as binitio ab ω na 0 r iclryplrzdin polarized circularly are atcei-el(PIC) particle-in-cell 27 λ – 0 29 ω 0 nm) 800 = htthe that ] a eef- be can 27 – 2

FIG. 2. Electric ﬁeld waveform of the (a) 10th and (b) 11th harmonic shown in Fig. 1(b). In order to see the helicity clearly, only parts of the waveform are shown. The rotation direction is deﬁned as seen by a wave receiver, and the harmonic waves propagate along the -x axis.

III. RESULTS AND DISCUSSION

Figure 1(b) shows the harmonic spectra obtained by fast Fourier transform of the reflected electric field. It is clearly seen that only the harmonic orders with n = 3m 1 (m = 1, 2, 3, ) have been generated, while ev- ery± third one with n···= 3m is missing. As the plasma is FIG. 1. (a) Electric field waveform of the incident laser isotropic and the interaction is at normal incidence, this pulses propagating along the x axis, composed of two-color unique harmonic feature can be explained by the sym- (fundamental frequency and its second harmonic) counterro- metry of the driving field as follows. We follow the argu- tating circularly polarized fields. (b) The corresponding high ments of Kfir et al[29] and Pisanty et al[34]. The laser harmonic spectrum from surface plasmas. field vector, rotated by 120◦ per cycle in the polarization plane, satisfies, L ˆ L the same plane but rotate in opposite directions: E (t + T0/3) = R(2π/3)E (t), (2)

1 ◦ iω0t 2iω0t ˆ E(t)= (E1e eˆ− + E2e eˆ+)+ c.c., (1) where R(2π/3) is the 120 rotation operator. Assuming 2i the emitted harmonic field conforms to the same dynam- ical symmetry and by taking Fourier transform, we can where eˆ± = (eˆ ieˆ )/√2 with eˆ and eˆ the unit vectors y z y z obtain the spectral field of the nth-order harmonic, EHω , along y- and z-axis,± respectively. The three-dimensional n (3D) electric field waveform is shown in Fig. 1(a), where satisfying the following eigenvalue equation − the intensity ratio of the two driving frequencies I2ω/Iω = 2πin/3EHω ˆ EHω 21 2 21 e n = R(2π/3) n . (3) 0.5 with Iω = 2.8 10 W/cm and I2ω = 1.4 10 2 × × W/cm . The Lissajous figure projected in the y z po- −2πin/3EHω − The solutions are (1) n =3m+1, so that e n = larization plane (in gray) displays a characteristic feature e−2πi/3EHω and (2) n =3m 1, so that e−2πin/3EHω = of threefold spatio-temporal symmetry. The electric field n − n e+2πi/3EHω . Apart from these two sets of harmonic or- experienced by the plasma electrons acts roughly as three n ◦ ders, the harmonic orders with n = 3m do not satisfy LP pulses rotated by 120 three times per optical cycle. Eq. (3) and are therefore forbidden by the threefold sym- This largely modified LP field pattern can give rise to metry. harmonic emission as will be shown later. The temporal The preceding selection rules are also consistent with intensity profile has a Gaussian envelope with a full-width the conservation laws for energy, parity and spin angular at half-maximum pulse duration of 30 fs. The laser pulse momentum[28, 34]. Considering the HHG process in a is normally incident onto a plasma target, of which the simple photon-exchange picture, conservation of energy peak density is n = 200n and the thickness is 200 nm. 0 c implies each harmonic frequency can be expressed in the Here n = m ω2/4πe2 is the plasma critical density with c e 0 form of respect to the fundamental laser frequency. An exponen- tial density ne(x) = n0 exp(x/Ls) exits in the front sur- H Ω(n ,n ) = n1 ω0 + n2 2ω0, (4) face of the plasma slab, where the preplasma scale length 1 2 · · Ls = 0.01λ0. The ions are taken to be immobile. The where n1 and n2 are integers. Applying parity operation simulation cell size is λ0/1000, and each cell is filled with on both the incoming and outgoing photons, we have 100 macroparticles. An absorption boundary condition is adopted for both fields and particles. P [h(ΩH )]= 1; (5) (n1,n2) − 3

FIG. 3. Polarization plane projections of the laser electric ﬁelds generated with intensity ratios (a) I2ω/Iω = 0.0, (b) 21 2 I2ω/Iω = 0.1, (c) I2ω/Iω = 0.5, and (d) I2ω/Iω = 2.0. The total intensity is kept constant (Iω + I2ω = 4.2 × 10 W/cm ). (e)-(h) The generated harmonic spectra corresponding to the lasers of panels (a)-(d).

n1+n2 P [h(n1·ω0;n2·2ω0)] = ( 1) , (6) harmonic fields should be close to linear polarization[35]. − This offers an alternative way to generate CP HHG and where P is parity and h represents helicity. Conservation attosecond pulses from relativistic plasmas[36–38]. Such of parity requires that n + n must be odd. In addition, 1 2 radiation sources can find important applications in ul- conservation of spin angular momentum leads to trafast measurement of chiral molecules[39] and magnetic H σ(n1,n2) = n1σ1 + n2σ2. (7) materials[40]. In addition, it was believed previously that CP HHG from relativistic plasmas could be achieved only Since the ω0 and 2ω0 driving pulses are counterrotating, under oblique incidence interactions[36]. The results here we can set σ1 = 1, σ2 = 1. For photon spin, 1 − − ≤ show it is also possible to generate CP surface harmonics σ 1. Then it is easily seen that n2 1 n1 (n1,n2) ≤ − ≤ ≤ at normal incidence. n2 +1, which together with the parity requirement means that n1 and n2 must differ by unity. Therefore, only two Next, we show that the harmonic intensity can be mod- cases are allowed: (1) n1 = m + 1 and n2 = m, leading ulated by adjusting the intensity ratio of the two driving to (m + 1)ω0 + m 2ω0 = (3m + 1)ω0 and (2) n1 = m and pulses. We keep the total intensity of the two fields con- · 21 2 n2 = m + 1, leading to mω0 + (m + 1) 2ω0 = (3m + 2)ω0. stant (Iω + I2ω =4.2 10 W/cm ) as the ratio I2ω/Iω Note 3m + 2 is equivalent to 3m 1. These· two selection varies from 0 to 0.1,× 0.5, and 2. The remaining param- rules also indicate that the n =3−m + 1 harmonic orders eters are the same as in Fig. 1. The Lissajous figures of have the same helicity with the fundamental field, while the superposed electric fields are displayed in Figs. 3(a)- the n =3m+2 orders co-rotate with the 2ω0 field. These (d). The corresponding harmonic spectra are shown in −2πi/3 +2πi/3 are consistent with the eigenvalues e and e of Figs. 3(e)-(h). For the case of I2ω/Iω = 0, i.e., one-color Eqs. (3), which also imply the n =3m+1 and n =3m 1 CP driving laser only, high harmonics with orders n> 3 harmonics have opposite helicity. − are indeed strongly suppressed, almost at the noise level. The aforementioned harmonic helicity is confirmed by As the laser intensity ratio is increased, the harmonic in- the simulation results. Figure 2 shows the 3D electric tensity is significantly enhanced. For I2ω/Iω = 2.0, the field waveform of a representative group of neighboring spectral intensity of all the allowed harmonic orders is harmonics (n = 10 and 11). To see the helicity clearly, close to the I(n) n−8/3 scaling, which is theoretically only parts of the waveform are shown. Each harmonic is derived for the case∝ of LP driving lasers[7] and already circularly polarized. The helicity of the 10th harmonic confirmed by experimental results[9]. This indicates that is the same as the fundamental field, while the 11th har- the scheme of bicircular driving lasers can also be very monic is the same as the second harmonic field. The efficient in high harmonic generation. The enhancement other harmonic groups share the same features. We note of HHG efficiency can be attributed to the different field although the adjacent harmonics have alternating helic- vector pattern induced by a different laser intensity ra- ity, the relative intensity of neighboring harmonics is not tio. As the laser field experienced by the plasma elec- equal (see Fig. 1(b)). Here the n = 3m + 2 harmon- trons gets closer to linearly polarized, stronger harmonic ics are weaker than the n = 3m + 1 harmonics, due to strength can be expected. Also, with increasing I2ω, the the 2ω0 field being less intense than the ω0 field. As a relative intensity of allowed neighboring harmonic orders result, harmonics of all the groups combined can be cir- I3m+2/I3m+1 also increases. This would change the over- cularly or elliptically polarized. Otherwise, the combined all polarization state of the harmonic source. 4

in Fig. 1. The physical origin for this ”abnormal” phe- nomena, however, is not from symmetry-based selection rules, but more likely from plasma effects. On the one hand, the appearance of n = 3m + 1 and n = 3m +2 orders and the absence of n = 3m orders for n < 10 reflect the preservation of the threefold symmetry. On the other hand, as Ls becomes longer, significant plasma waves can be excited and are able to couple to the incident wave. Consequently, various parametric instabilities and self-phase modulation effects[42], together with the chirp effect due to the motion of surface plasmas[43], can result in harmonic spectral modulation, splitting, and frequency shifting. The gradual frequency shifting from the n = 3m + 1 orders to match the n = 3m orders is evident when the harmonics of n > 12 are compared. The absence of the n = 3m +2 orders in this case may be explained as too weak to distinguish. For even longer Ls, however, the whole high harmonic structure would disappear due to strong plasma modulation effects and low efficiency of the ROM mechanism in this case. FIG. 4. High harmonic spectrum for the case of (a) oblique ◦ incidence with incidence angle θ = 45 and (b) plasma scale length of Ls = 0.15λ0. The other simulation parameters are IV. CONCLUSIONS the same as in Fig. 1. In summary, we introduce bicircular counter-rotating Although the Lissajous figures in Figs. 3(b)-(d) have a two-color fields to HHG in the relativistic plasma regime very different appearance, they all keep a threefold sym- for the first time. We numerically demonstrate, by ad- metry. Therefore, the harmonic spectra in Figs. 3(f)-(h) justing the intensity ratio of the two driving fields, that have the same selection rule as n = 3m 1. When the the properties of the high harmonics, including the al- threefold symmetry is lost, the selection± rule described lowed harmonic orders and their overall intensity, the above no longer holds. A variety of harmonic channels relative intensity of allowed neighboring harmonic orders, can open up. The simplest way to break the threefold ro- as well as the polarization state of the harmonics, can be tational symmetry of the interaction is to change the in- controlled. The underlying mechanisms can be under- cidence geometry from normal to oblique. In the oblique stood as a result of the symmetry effect and related con- incidence case, the symmetry breaking can be seen more servation laws. This work thus provides an alternative clearly by considering the interaction in the 1D case. Us- way to spectral control surface harmonics. Also, it offers ing Bourdier’s method to take Lorentz transformations a way to generate CP high harmonics and/or attosec- from the laboratory frame to a moving frame[41], the ond pulses from relativistic plasmas. Moreover, it also laser pulse can be transformed to be at normal incidence shows it is possible to generate CP harmonics from relativistic plasmas at normal incidence with high efficiency. (wave vector k = keˆx). At the same time, the plas- This scheme opens up more possibilities in controlling mas initially stream along the eˆy direction while they − the surface harmonics, as more degrees of freedom can be are stationary along the eˆz direction. Thus the rota- tional symmetry in the (y z) polarization plane is bro- adjusted, such as the frequency, ellipticity, and relative ken. Consequently, the previously− symmetry-forbidden phase of the two drivers in the counterrotating two-color n =3m orders can appear. Figure 4(a) shows the bicircu- fields. lar field-driven harmonic spectrum for the case of oblique incidence with incidence angle θ = 45◦. The other simu- ACKNOWLEDGMENTS lation parameters are the same as in Fig. 1. We see the harmonic orders with n = 3m are indeed shown in this case. This work was supported in part by the National It is interesting to note only the n = 3m harmonic Natural Science Foundation of China (Grants No. orders appear in the spectrum for n > 15 shown in 11505172 and 11705185) and the Presidential Fund Fig. 4(b), where the plasma scale length Ls = 0.15 of China Academy of Engineering Physics (Grant No. while the other simulation parameters are the same as YZJJLX2017002).

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