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Laser cluster interaction in ambient magnetic fields for accelerating electrons in two stages without external injection - Scientific Reports
Results .
New RSM results: electron dynamics with laser and auxiliary \(B_{ext}\) .
Figure 2 Modified RSM results with \((\varvec{E}_{l},\varvec{B}_{l})\) and \(\varvec{B}_{ext}\) along z : showing dynamical variables of the same electron as in Fig. 1 . Panels in the left ( a1,b1,c1 ) and right ( a2,b2,c2 ) columns are with \({B}_{ext} \approx 0.028, 0.0569\) ?a.u. corresponding to non-resonant \(\Omega _{c0}/\omega = 0.5\) and resonant \(\Omega _{c0}/\omega = 1\) (ECR case) respectively. In the ECR case AHR occurs little early around \(t/T \approx 1.85\) , and \(\overline{\mathcal {E}}_A\) reaches up to \(\overline{\mathcal {E}}(\tau )\approx 36\) compared to \(\overline{\mathcal {E}}_A=\overline{\mathcal {E}}(\tau )\approx 2.1\) in Fig. 1 a2; corresponding momenta and excursion also significantly vary after the electron is freed via AHR near \(t/T \approx 1.85\) . Inset plots (in a2,b2,c2 ) show zoomed view of dynamical variables near AHR and inside the cluster due to strong \(\varvec{B}_{ext}\) . Other laser and cluster parameters are as in Fig. 1 .
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Results in Fig. 1 (right column) show that for the chosen \(\varvec{E}_l,\varvec{B}_l\) configuration, a liberated electron from cluster may also gain a mild forward momentum \(\overline{p}_z\) after AHR. The energy-momentum relation \(\overline{p}_z- \overline{p}_{z0}= (\gamma -\gamma _0)c\) for DLA (without space-charge) suggests that to improve energy gain by the electron, its \(\overline{p}_z\) should be increased from the initial \(\overline{p}_{z0} = \gamma _0 c\) . Though magnetic field does not work, an auxiliary \(\varvec{B}_{ext}\) helps bending electron’s trajectory. It may also improve \(\overline{p}_z\) of the freed electron.
Figure? 2 a1,b1,c1 show results with \({B}_{ext}=\vert \varvec{\hat{z}} {B}_{ext}\vert \approx 6.68\) ?kT (corresponding \(\Omega _{c0}= \omega /2\) ) along z for the same \((\varvec{E}_l,\varvec{B}_l)\) as in Fig. 1 a2,b2,c2 which is considered as a reference. Noticeably, variation of dynamical variables are now very different from the corresponding Fig. 1 a2,b2,c2; but the final retained energy of the electron is still \(\overline{\mathcal {E}}_A \approx 2.1\) . The \(\overline{\omega }_{\mathrm {eff}}^2\) starts at \(\approx (\omega _\mathrm {M}^2 + \Omega _{c0}^2)/\omega ^2\) , monotonically drops and passes the AHR line \(\overline{\omega }_{\mathrm {eff}}^2=1\) (horizontal dashed line) at a little early time \(t/T\approx 1.85\) (vertical shaded bar) following Eq. ( 5 ) contrary to its short-time oscillatory nature just before the occurrence of AHR (Fig. 1 a2) near \(t/T\approx 2.1\) . The vanishing of oscillatory nature of \(\overline{\omega }_{\mathrm {eff}}^2\) (Fig. 2 a1) and its smooth passage through the \(\overline{\omega }_{\mathrm {eff}}^2=1\) line is due to additional induced fields \(E_y, E_z\) [though still weak, Fig. 2 c1] due to strong \(\varvec{B}_{ext} = \hat{z} {B}_{ext}\) leading to swirling motion in ( x ,? y ) inside the cluster similar to the driving by a circularly polarized laser field 33 , 73 . Thus an external \(\varvec{B}_{ext} = \hat{z} {B}_{ext}\) may modify electron dynamics inside the cluster and the AHR scenario. The ( \(\overline{x}, \overline{y}, \overline{p}_x, \overline{p}_y\) ) dynamics of the liberated electron tends to follow cyclotron motion; both \(\overline{p}_z\) and \(\overline{\mathcal {E}}\) grow up-to a maximum (note that \(\max {\overline{\mathcal {E}}}\approx 10.5\) now exceeds the conventional \({\overline{\mathcal {E}}}=8\) line without \(\varvec{B}_{ext}\) ) near the pulse peak. But \(\overline{p}_z\) drops later (Fig. 2 b1) leading to lesser (Fig. 2 a1) final energy \(\overline{\mathcal {E}}_A = \overline{\mathcal {E}}(\tau )\approx 2.1\) as in Fig. 1 a2 though electron dynamics drastically differ from Fig. 1 a2,b2,c2.
With a higher \({B}_{ext} = \vert \varvec{\hat{z}} {B}_{ext}\vert \approx 13.37\) ?kT corresponding to \(\Omega _{c0} = \omega\) (ECR), Fig. 2 a2,b2,c2 show a significant jump in the final absorbed energy upto \(\overline{\mathcal {E}}_A\approx 36\) (far exceeding the conventional \(\overline{\mathcal {E}}_A^{max} \approx 3.17\) ) associated with a jump in the corresponding final \(\overline{p}_z\approx 0.3\) . Most of the arguments relevant to Fig. 2 a1,b1,c1 apply here also. Additional inset plots are zoomed view of dynamical variables near AHR and inside the cluster. Due to higher \(\varvec{B}_{ext} = \varvec{\hat{z}} B_{ext}\) , induced fields \(E_y, E_z\) in the cluster (Fig. 2 c2) are also marginally stronger, AHR scenario is marginally modified and note that, even in this case \(\overline{p}_z,\overline{z}\) are almost zero inside the cluster. Distinctly, after the AHR near \(t/T\approx 1.85\) , liberated electron follows almost exact cyclotron motion in the \(x-y\) plane (evident from \(\overline{x}, \overline{y}, \overline{p}_x, \overline{p}_y\) variation) due to the stronger \(\varvec{B}_{ext} = \varvec{\hat{z}} B_{ext}\) , while its \(\overline{p}_z\) and \(\overline{\mathcal {E}}\) continuously increase to \(\overline{p}_z\approx 0.3\) and \(\overline{\mathcal {E}}\approx 36\) during \(t/T\approx 2-4\) followed by saturation, though laser field envelope (Fig. 2 c2) weakens after its peak. Thus an external magnetic field -assisted electron acceleration from a laser-driven cluster is shown to enhance electron’s energy by 10–12 times than the conventional limit of \(\overline{\mathcal {E}}_A^{max} \approx 3.17\) , particularly near the ECR frequency \(\Omega _{c0}= \omega\) . This encouraging new result needs further investigation.
Above results with \({B}_{ext}\) show laser absorption happens mainly in two stages. In the first stage, electron undergoes AHR (may be modified by \({B}_{ext}\) ) and comes out of the cluster with low positive energy and non-zero transverse momentum. Later, in the second stage, it is fully controlled by the remaining \(\varvec{E}_l,\varvec{B}_l\) and \(\varvec{B}_{ext}\) with an increase in \(\overline{\mathcal {E}}\) , i.e., absorption rate (see Fig. 2 a2) for \(t/T\approx 2-4\) . This second stage may be termed as magnetic field assisted DLA. However, to realize the second stage, energy absorption by electron and its liberation from the cluster in the first stage is necessary , otherwise \(\varvec{v}\times \varvec{B}_{ext}\) fails.
Figure 3 Temporal variation of phase angles \(\psi _{v_x}, \psi _{E_x}, \psi _{E_l}\) of respective \(v_x, E_x, E_l\) associated with the same RSM electron (i) in Fig. 1 a2,b2,c2 without \(\varvec{B}_{ext}\) and (ii) in Fig. 2 a2,b2,c2 with \({B}_{ext} = \vert \hat{z} \varvec{B}_{ext}\vert \approx 0.0569\) (ECR case). Phases are numerically calculated by FFT w.r.t. the central frequency \(\omega\) with a sliding time-window duration \(T=2\pi /\omega\) . In (i) relative phase \(\Delta \psi = \vert \psi _{v_x}-\psi _{E_x}\vert\) , on an average, stays close to \(\pi\) (or \(0.75\pi\) ) for a short-while \(\Delta \tau T\) through the pulse peak (before falls to \(\pi /2\) at \(t/T\approx 4\) ) leading to \(\overline{\mathcal {E}}_A \approx 36\) in Fig. 2 a2.
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The temporal phase dynamics. Equation ( 4 ) implies that rate of absorption \({d{\gamma } m_{0} c^2 }/{dt}\) by an electron approaches to zero (or negligible) for phase angle \(\Delta \psi\) between its velocity and the corresponding driving electric field approaching to the odd integral multiple of \(\pi /2\) . From Eq. ( 4 ), one may apparently conclude no role of \(\varvec{B}_{ext}\) for enhanced absorption in Fig. 2 a2,b2,c2. Note that in the second stage of energy absorption, where role of \({\varvec{E}_\mathrm {sc}}\) is nil, Eq. ( 4 ) simplifies to \({d{\gamma } m_{0} c^2 }/{dt} = q {v_x} {E}_x = q v_x E_l\) ; and, though \(\varvec{B}_{ext}\) can not alter \(E_l\) , it may re-orient the phase of \(v_x\) (see the cyclotron orbit) w.r.t. \(E_x\) . To probe this underlying physics , we numerically retrieve phase angles \(\psi _{v_x}, \psi _{E_x}, \psi _{E_l}\) w.r.t. central frequency \(\omega\) of respective \(v_x, E_x, E_l\) (since components along the laser polarization matter the most) for two cases: (i) with \(\varvec{E}_l,\varvec{B}_l\) only for Fig. 1 a2,b2,c2 and (ii) with \(\varvec{E}_l,\varvec{B}_l\) and \({B}_{ext} \approx 13.37\) ?kT for Fig. 2 a2,b2,c2. Those \(\psi _{v_x}, \psi _{E_x}, \psi _{E_l}\) and \(\Delta \psi = \vert \psi _{v_x}-\psi _{E_x}\vert\) vs time are plotted in Fig. 3 (see caption). Vertical shaded bars are the respective AHR regions (see Figs. 1 , 2 ) after which the electron is mostly free from space-charge fields of the cluster and respective \(\psi _{E_x}\) goes hand in hand with \(\psi _{E_l}\) in both cases. Little deviation of \(\psi _{E_l}\) from \(\pi /2\) far away from the pulse center (at \(t/T=2.5\) ) is due to 5-cycle broad-band pulse (ideally it is \(\pi /2\) for a monochromatic pulse \(\sim \sin \omega t\) ). Respective \(\psi _{v_x}, \psi _{E_x}\) in (i) do not differ from (ii) and \(\Delta \psi \approx 0.5\pi\) remains upto \(t/T\approx 1.4\) . After this time, \(\psi _{v_x}, \psi _{E_x}\) in (i) increase slowly for \(t/T\approx 1.4-1.75\) where \(\Delta \psi \approx 0.5\pi \rightarrow \pi\) [ \(\Delta \psi \approx 0.9\pi\) is maintained for a tiny duration \(\Delta \tau\) ] followed by its gradual drop through the AHR region and saturation near \(\Delta \psi \approx 0.5\pi\) afterwards. On the contrary, in (ii) an instantaneous phase swing occurs (near \(t/T\approx 1.4\) ) for \(\psi _{E_x} = -\pi \rightarrow \pi\) after quick dropping to \(-\pi\) . Later, though \(\psi _{E_x} \rightarrow \psi _{E_l} \approx 0.5 \pi\) , the phase \(\psi _{v_x}\) is dynamically tilted in a way that a value of \(\Delta \psi \approx \pi\) is brought about by the auxiliary \({B}_{ext}\) for a long duration \(t/T\approx 1.75-3.0\) (leading to high absorption rate in Fig. 2 a2) from pre-AHR to post-AHR time through the pulse maxima; then \(\Delta \psi\) gradually drops as \(\approx \pi \rightarrow 0.5\pi\) for \(t/T\approx 3.0-4.0\) where absorption slows down and finally saturates at a higher \(\overline{\mathcal {E}}_A\approx 36\) in Fig. 2 a2. Thus an auxiliary \({B}_{ext}\) near the ECR helps maintaining the required \(\Delta \psi \approx \pi\) for enhanced laser absorption in the second stage.
RSM results: absorption with different orientation of \({B}_{ext}\) .
For further understanding of magnetic field-assisted laser-energy coupling we study similar ECR cases with same conditions of Fig. 2 a2,b2,c2 but other orientations of \(\varvec{B}_{ext}\) . For the sake of conciseness, we plot energy vs time in Fig. 4 for: (i) \(\varvec{B}_{ext} = \varvec{0}\) , (ii) \(\varvec{B}_{ext} = \varvec{\hat{z}} {B}_{ext}\) , (iii) \(\varvec{B}_{ext} = \varvec{\hat{y}} {B}_{ext}\) and (iv) \(\varvec{B}_{ext} = \varvec{\hat{x}} {B}_{ext}\) where \({B}_{ext} \approx 13.37\) ?kT. Results show almost same level of enhanced absorption upto \(\overline{\mathcal {E}}_A = \overline{\mathcal {E}}(\tau )\approx 35-36\) only when \(\varvec{B}_{ext}\perp \varvec{E}_l\) [cases (ii) and (iii)], although electron dynamics are different here. When \(\varvec{B}_{ext} \varvec{E}_l\) , there is no enhancement in the final energy [case (iv)] and gives the same level of \(\overline{\mathcal {E}}_A \approx 2.1\) as in the case (i) since \(\varvec{v}\times \varvec{B}_{ext} \approx \varvec{0}\) . Thus RSM quickly identifies possible directions of \(\varvec{B}_{ext}\) for enhanced laser absorption. Now onwards we focus on the results with \(\varvec{B}_{ext} = \varvec{\hat{z}} {B}_{ext}\) only.
Figure 4 RSM results: Time vs absorbed energy in units of \(U_\mathrm {p}\) of the single electron (in Fig. 2 ) for different \(\varvec{B}_{ext}\) : (i) \(\varvec{B}_{ext} = \varvec{0}\) , (ii) \(\varvec{B}_{ext} = \varvec{\hat{z}} {B}_{ext}\) , (iii) \(\varvec{B}_{ext} = \varvec{\hat{y}} {B}_{ext}\) and (iv) \(\varvec{B}_{ext} = \varvec{\hat{x}} {B}_{ext}\) . Laser fields ( \(\varvec{E}_l,\varvec{B}_l\) ) and magnitude of \(\vert \varvec{B}_{ext}\vert \approx 13.37\) ?kT are as in Fig. 2 a2,b2,c2. Cases (ii) and (iii) are only two energetically favorable orientations of \(\varvec{B}_{ext}\) .
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Non-interacting multi-electrons in RSM .
A single-electron dynamics (as studied by RSM above) is important to understand the physics of LCI, but can not answer some other aspects, e.g., fraction of electrons leaving the cluster (outer ionization fraction) and their energy distribution. In the single-electron case, outer-ionization fraction assumes only 0,1 (electron is either inside or outside the cluster). In a real system, however, some electrons may remain bound and outer-ionization fraction may attain any value between (0,1) depending upon laser and cluster parameters. A single-electron case may over-estimate/under-estimate electron energy compared to the realistic multi-electron case where per-electron energy may be averaged out. Moreover, different electrons become free from the cluster at different times, and participate in the magnetic field assisted DLA differently. To answer these aspects we distribute all \(N=2176\) electrons inside the cluster randomly (or uniformly) to mimic a multi-electron system by RSM where electrons are assumed non-interacting among them. For brevity, we compare these multi-electron results of RSM along with detailed PIC simulation in the following section where particle-particle interactions are taken care self-consistently.
Absorption studies with PIC simulation and comparison with RSM .
Figure 5 Comparison of PIC and RSM results: average total absorbed energy \(\overline{\mathcal {E}}(t) = {\mathcal {E}}(t)/NU_\mathrm {p}\) per particle in \(U_\mathrm {p}\) vs t / T with \(B_{ext}=0\) (dashed lines, conventional case of Fig. 1 a2,b2,c2) and with \(B_{ext} = \omega\) (solid lines, ECR case of Fig. 2 a2,b2,c2). RSM results with single-electron (RSM-SP) and non-interacting multi-electrons (RSM-MP) both justify PIC results. Though absorption starts early in PIC, final absorbed energies \(\overline{\mathcal {E}}_A = \overline{\mathcal {E}}(\tau )\) with/without \(B_{ext}\) are comparable with the RSM cases.
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Figure? 5 compares time vs average energy (per electron) between PIC and RSM results for \(B_{ext} = 0\) and \(\vert \varvec{\hat{z}} B_{ext}\vert = \omega\) (ECR case) at \(I_0\approx 7.13\times 10^{16}\text{ W/cm}^{2}\) . RSM results with single-electron (RSM-SP) as in Fig. 4 and non-interacting multi-electrons (RSM-MP) as described in section above are also included. RSM-SP over-estimates the RSM-MP case for final energy \(\overline{\mathcal {E}}_A = \overline{\mathcal {E}}(\tau )\) when \(B_{ext} = 0\) , but PIC result ( \(\overline{\mathcal {E}}_A \approx 0.5\) ) follows RSM-MP more closely. For the ECR case, however, \(\overline{\mathcal {E}}_A \approx 36\) in PIC remains little higher than RSM-MP, which is due to early ejection of electrons with non-zero transverse momentum via AHR from the self-consistently developing potential and electro-static restoring fields (starting from zero) in PIC. Note that \(\overline{\mathcal {E}}(t)\) starts increasing one-period earlier ( \(t/T\approx 1.2\) ) in PIC than the RSM and so as the ECR for those early leaving PIC electrons. Almost \(60-70\) fold increase in \(\overline{\mathcal {E}}_A\approx 0.5 \rightarrow 36\) is obtained in PIC and RSM-MP due to \(\vert \varvec{\hat{z}} B_{ext}\vert\) near ECR.
Figure 6 Comparison of PIC and RSM results: Average absorbed energy \(\overline{\mathcal {E}}_A=\overline{\mathcal {E}}(\tau )\) per particle vs \(\Omega _{c0}/\omega\) for a range of \(\vert \varvec{\hat{z}} B_{ext}\vert \approx (0 - 2\omega )\) with \(n=5\) -cycle pulses of different \(I_0\approx 1.83\times 10^{15}\text{ W/cm}^{2}- 1.83\times 10^{17}\text{ W/cm}^{2}\) . Energy is shown normalized by corresponding \(U_\mathrm {p}\) (left y-axis) and in atomic units (right y-axis). At a low intensity ( a ) absorption peaks almost at the ECR condition \(\Omega _{c0} = \omega\) (vertical dashed line) as clearly exhibited by PIC where electrons undergo AHR at ease and become free with transverse momentum for the ECR in the next stage; whereas RSM-MP shows almost zero absorption since AHR is not met (first stage fails) in RSM. As \(I_0\) increases to moderate values (in b,c ) absorption peaks show-up in RSM-MP due to meeting of AHR followed by ECR. For high intensity RSM-MP justify PIC results quantitatively. Gradual right-shift of the absorption peak from ECR condition \(\Omega _{c0} = \omega\) with increasing \(I_0\) is due to relativistic modification of \(\Omega _{c} = \Omega _{c0}/\gamma\) for \(\gamma >1\) . Absorption peaks \(\approx 65U_\mathrm {p}, 45U_\mathrm {p}\) in ( b,c ) give average energy per electron \({\mathcal {E}}_A\approx 0.27, 0.49\) MeV respectively.
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Scanning through range of values of \(\vert \varvec{\hat{z}} B_{ext}\vert \approx (0 - 2\omega )\) , for different \(I_0\approx 1.83\times 10^{15}\text{ W/cm}^{2}- 1.83\times 10^{17}\text{ W/cm}^{2}\) and same 5-fs pulse duration, results in Fig. 6 are obtained by PIC and RSM-MP in the end of the pulses. At a low intensity (Fig. 6 a) absorption peak (at \(26\,U_\mathrm {p}\) ) occurs almost at the ECR condition \(\Omega _{c0} = \omega\) (vertical dashed line) as clearly exhibited by PIC simulation where electrons can undergo AHR at ease, become free with a transverse momentum for the ECR in the next stage; whereas RSM-MP shows almost zero absorption since AHR is not met (first stage fails) and electrons can’t be freed from RSM potential with a transverse momentum at this low intensity (RSM greatly under-estimates absorption here, since its needs a threshold intensity 31 , 32 ). Therefore, as the peak intensity increases, absorption peaks show-up gradually (Fig. 6 b,c) for RSM-MP due to gradual removal of electrons via AHR (preferably) from surface to the cluster center, but ECR absorption peak occurs always for PIC. Finally, at a higher \(I_0\approx 1.83\times 10^{17}\text{ W/cm}^{2}\) , PIC and RSM (almost overlap) show very good quantitative agreement in Fig. 6 c. Absorption peaks \(\approx 65U_\mathrm {p}, 45U_\mathrm {p}\) in Fig. 6 b,c give average energy \({\mathcal {E}}_A\approx 0.27, 0.49\) MeV. The gradual right-shift of the absorption peak from the ECR condition \(\Omega _{c0} = \omega\) (vertical dashed line) with increasing intensity is due to the relativistic modification of \(\Omega _c = \Omega _{c0}/\gamma\) in dipole-approximation. Since \(\gamma\) is time-varying (during the pulse) and different for different electrons, the time-dependent relativistic-ECR occurs for electrons when \(\Omega _c (t) = \Omega _{c0}/\gamma (t) = \omega\) (call it RECR, see Fig. 7 ). It emphasizes quick slippage of electron from the RECR condition as soon as its \(\gamma (t)>1\) . Therefore, to satisfy the RECR for \(\gamma >1\) , a higher \(\Omega _{c0}\) (or higher \(B_{ext})\) is required—as manifested by gradual right-shift of the absorption peak (Fig. 6 –c) with increasing intensity. Moreover, laser pulse being broadband with frequencies \(\omega , (1\pm 1/n)\omega\) , RECR may happen in a wider frequency range and contribute to broadening of resonance-width about the absorption peak in Fig. 6 .
Figure 7 Time vs frequency analysis for PIC electrons: Normalized \({\Omega }_{\mathrm {eff}}/\omega\) (green, left y-axis) and \(\Omega _{c}/\omega\) (red, right y-axis) of cluster electrons for \(\vert \hat{z} \varvec{B}_{ext}\vert \approx 0.02, 0.0569, 0.07\) at \(I_0 = 7.13\times 10^{16}\,\text{ W/cm}^{2}\) (left column, a1,b1,c1 ) and \(\vert \hat{z} \varvec{B}_{ext}\vert \approx\) 0.02, 0.0569, 0.078 at \(I_0 = 1.83\times 10^{17}\,\text{ W/cm}^{2}\) (right column, a2,b2,c2 ) corresponding to PIC results (at A,B,C ) in Fig. 6 b,c respectively. Vertical shaded region indicates AHR region where \({\Omega }_{\mathrm {eff}}/\omega\) of each electron starts from zero, reaches different maximum, then drops to zero passing through AHR when electron is freed from the cluster potential with excursion \(r/R\gg 1\) (gray) and non-zero transverse momentum. Horizontal dashed lines represent frequencies of the broadband pulse where ECR/RECR are expected. At low \(B_{ext}\) values, ECR is not met ( a1,a2 ), laser absorption is mainly due to AHR occurring for \(t/T\le 2\) (1st stage, vertical shaded region). As \(B_{ext}\) increases, \(\gamma\) of electrons increase, all frequencies of the broadband pulse gradually come under ECR/RECR condition (second stage) with decreasing \(\Omega _{c}/\omega\) as one passes ( b1,b2 ) to ( c1,c2 ). In ( c1,c2 ) ECR/RECR is hit around the peak of the pulse (at \(t/T=2.5\) ) with central frequency \(\omega\) as well as with side-bands \(1.2\omega , 0.8\omega\) leading to higher absorption in ( c1,c2 ) compared to the case ( b1,b2 ). In ( b1,b2 ) ECR is hit in the beginning of the pulse with \(\omega\) when laser field is relatively weak, then RECR with the side-band at \(0.8\omega\) near the pulse peak and in the pulse end (for b2 ). Note that ECR/RECR occurring at very early time ( \(t/T<1.5\) ) or very late time ( \(t/T>4\) ) are less effective due to weak laser field. Other laser and cluster parameters are as in Fig. 1 . See also Fig. 8 for corresponding phase dynamics.
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Figure 8 Time vs phase analysis for PIC electrons: Phase-angles \(\psi _{v_x}, \psi _{E_x}\) of respective dynamical variables \(v_x, E_x\) and phase difference \(\Delta \psi = \vert \psi _{v_x}-\psi _{E_x}\vert\) of all \(N=2176\) electrons for \(\vert \hat{z} \varvec{B}_{ext}\vert \approx 0.02, 0.0569, 0.07\) at \(I_0 = 7.13\times 10^{16}\,\text{ W/cm}^{2}\) (left column, a1,b1,c1 ) and \(\vert \hat{z} \varvec{B}_{ext}\vert \approx 0.02, 0.0569, 0.078\) at \(I_0 = 1.83\times 10^{17}\,\text{ W/cm}^{2}\) (right column, a2,b2,c2 ) corresponding to PIC results (at A,B,C ) in Fig. 6 b,c respectively. Wavy dashed lines indicate respective phase angles for average values \(\sum {v_x}/N, \sum {E_x}/N\) of all N electrons showing average system behavior removing rapid phase fluctuations. At low \(B_{ext}\) values, ECR is not met, \(\Delta \psi\) quickly falls to \(\pi /2\) after initial rise towards \(\pi\) due to AHR (mostly occurring) for \(t/T\le 2\) . As \(B_{ext}\) increases towards ECR, \(\Delta \psi\) is gradually lifted towards \(\pi\) , and it is maintained for a longer duration \(\Delta \tau \approx 60-70\%\) of the pulse through pulse maximum leading to higher absorption in Fig. 6 b,c even after AHR. Phase angles are numerically computed by FFT as in RSM Fig. 3 . See also Fig. 7 for corresponding frequency dynamics. Other laser and cluster parameters are as in Fig. 1 .
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Frequency and phase dynamics of PIC electrons. To elucidate further, we retrieve the relativistic anharmonic eigen-frequency \({\Omega }_{\mathrm {eff}}\) and cyclotron-frequency \(\Omega _{c}\) for each \(k-th\) PIC electron as (see also Eq. ( 5 ))
$$\begin{aligned} {\Omega }_{\mathrm {eff}}^2&= \varvec{\hat{r}} \cdot \varvec{E}_{sc}(r_k)/\gamma _k r_k \end{aligned}$$
(9)
$$\begin{aligned} \Omega _{c}^2&= \varvec{\hat{r}}\cdot (\Omega _{c0}^2 \varvec{\hat{r}_{\perp }})/\gamma _k^2. \end{aligned}$$
(10)
Figure 7 shows temporal variation of \({\Omega }_{\mathrm {eff}}/\omega\) (green, left y-axis) and \(\Omega _{c}/\omega\) (red, right y-axis) of cluster electrons for \(\vert \hat{z} \varvec{B}_{ext}\vert \approx 0.02, 0.0569, 0.07\) ?a.u. at \(I_0 = 7.13\times 10^{16}\,\text{ W/cm}^{2}\) (left column, a1,b1,c1) and \(\vert \hat{z} \varvec{B}_{ext}\vert \approx 0.02, 0.0569, 0.078\) ?a.u. at \(I_0 = 1.83\times 10^{17}\,\text{ W/cm}^{2}\) (right column, a2,b2,c2) corresponding to PIC results in Fig. 6 b,c respectively. Chosen values of \(\vert \hat{z} \varvec{B}_{ext}\vert\) for each intensity represent data points A, B, C (at the tail, at the non-relativistic ECR condition \(\Omega _{c0}=\omega\) , and at the peak) on the PIC absorption curves in Fig. 6 b,c. Normalized position r / R of electrons (gray) show their distances w.r.t. center of the cluster. Corresponding phase-angles \(\psi _{v_x}, \psi _{E_x}\) of respective \(v_x, E_x\) along the laser polarization and the phase difference \(\Delta \psi = \vert \psi _{v_x}-\psi _{E_x}\vert\) for each \(k-th\) PIC electron are computed by FFT (as in RSM Fig. 3 ) and shown in Fig. 8 for both intensities. Wavy dashed lines (Fig. 8 ) indicate respective phase angles for average values \(\sum {v_x}/N, \sum {E_x}/N\) of all N electrons showing average system behavior. Contrary to the RSM, \({\Omega }_{\mathrm {eff}}/\omega\) of each PIC electron starts from zero and reaches different maximum (Fig. 7 ) when its r / R drops towards the potential minimum. Then \({\Omega }_{\mathrm {eff}}/\omega\) each PIC electron drops to zero passing through AHR 33 , 34 , 37 similar to the RSM and electron is freed from the cluster potential with \(r/R\gg 1\) . Shaded vertical bar (in Fig. 7 ) highlights this AHR dominated region (1st stage) during initial time of the laser pulse. Since different electrons undergo AHR at different times and comes out with different non-zero transverse momentum, the exact extent of the 1st stage and the beginning of 2nd stage (ECR stage) with \(B_{ext}\) is difficult to draw (i.e., minor overlap happens and 2nd stage starts early for early leaving electrons via AHR) with all electrons together. However, from the vanishing of \({\Omega }_{\mathrm {eff}}/\omega \rightarrow 0\) and increasing \(r/R\gg 1\) it is clear that AHR domain (1st stage) is mostly limited below \(t/T\approx 2\) for \(I_0 = 7.13\times 10^{16}\,\text{ W/cm}^{2}\) ( \(t/T\approx 1.6\) for \(I_0 = 1.83\times 10^{17}\,\text{ W/cm}^{2}\) ) and shrinks with increasing intensity.
At low \(\vert \hat{z} \varvec{B}_{ext}\vert \approx 0.02\) (or without it) as in Fig. 7 a1,a2, the frequency matching for ECR can not happen, the phase difference \(\Delta \psi\) continues to \(\pi /2\) after initial rise towards \(\pi\) shown in respective Fig. 8 a1,a2 due to short-lived AHR occurring below \(t/T<2\) . Hence absorbed energy remains low ( \(<3U_\mathrm {p}\) ) without initiating the second stage. In these cases not all electrons are freed (Fig. 7 a1,a2), many of them may comeback inside the cluster later time, and may be liberated again through another AHR, e.g., see after \(t/T>4\) .
As \(B_{ext}\) increases (see caption of Fig. 7 ), \(\gamma\) of electrons increase, all frequencies of the broadband pulse (shown by horizontal dashed lines) gradually come under ECR/RECR condition with decreasing \(\Omega _{c}/\omega\) as one passes Fig. 7 b1,b2–c1,c2; accompanied by gradual lifting of \(\Delta \psi\) towards \(\pi\) even after AHR with more time elapsed near \(\pi\) as in respective Fig. 8 b1,b2–c1,c2. Also, as RECR is met with the central frequency \(\omega\) near the pulse peak (Fig. 7 ) at \(t/T=2.5\) and respective \(\Delta \psi\) is maintained near \(\pi\) for a longer duration \(\Delta \tau \approx 60-70\%\) of the pulse through pulse maximum (Fig. 8 ), it leads to higher absorption ( \({\mathcal {E}}_A\) ) in Fig. 6 c, b. Thus, not only frequency matching \(\Omega _{c}/\omega = 1\) for ECR/RECR is satisfied, the required phase matching condition \(\Delta \psi \approx \pi\) is also simultaneously satisfied by PIC electrons for all cases in Fig. 6 (same are checked with electrons in RSM-MP for Fig. 6 , not repeated) for enhanced absorption peak about 30–70 \(U_\mathrm {p}\) . .
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