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Abstract
Electron rescattering using laser-enhancing nanostructures is a topic of particular interest for specialized high-brightness cathodes and attosecond high-harmonic generation. With ever-increasing ponderomotive amplitudes, the structure-induced longitudinal field profile is becoming more impactful on achievable energies. Using a classical approach, we investigate the limiting energies of electron emission and high-harmonic generation from nanostructures in the high-field regime. We perform generalized calculations for these cutoff energies for various structures by taking into account their particular spatial field profiles. We provide the peak electron emission and scattering energies as functions of the spatial adiabaticity parameter for decaying field profiles of the form ${r^{- 1}},{r^{- 2}}$, and ${e^{- r/\delta}}$ for nanoblades, nanotips, and plasmonics, respectively. These results are summarized in double-exponential fitted functions for ease of use. The maximal energies from each field profile differ by up to a factor of the ponderomotive energy. We also perform a simple near-field averaging analysis that produces satisfactorily accurate results. The introduction of a temporal Gaussian envelope decreases energies in most cases and modifies the ponderomotive force factor in a predictable manner. The effects of the carrier–envelope phase are exacerbated for few-cycle and shorter pulses.
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1. INTRODUCTION
The study of strong-laser-field physics on cathodic media is extensive, involving both gaseous [1–3] and solid-state [4–10] systems. The underlying process of elastic electron rescattering results in high-harmonic generation (HHG) [1–3,8] and high-energy electron emission [5–7,9,11,12]. Field enhancement induced by nanostructures, nanoplasmonic effects, or other geometric features permits peak ionizing fields in the V/nm regime with sub-wavelength confinement [4,11–13], resulting in more intense and localized emissions. This field enhancement overcomes the laser beam waist limit for flat or gaseous media; however, solid-state systems invoke the complexities of many-body effects [7,14,15], surface roughness [16], and the potential for target damage [4,7].
Nanotip electron sources provide spatially coherent beams, albeit with low intensities [4] with the yield strictly limited by this damage threshold. One way to increase yield is by utilizing an array of nanotips [17,18]. Extending the emission area from a nanotip to a nanoblade increases the emission area and permits stronger surface fields due to the improved thermo-mechanical properties of the cathode [19–21]. The associated experiments have produced dramatic results, leading to high field emission energies exceeding 1 keV [21]. As such, accurate theoretical predictions for these experimental scenarios are essential. This is the urgent motivation behind the work we present in this paper.
In the strong-field regime, quasi-free electron dynamics dominate much of the phenomena observed. Most notably, the HHG and electron emission energy cutoffs may be extracted by studying the vacuum dynamics. Following a classical model, derived emission cutoff energies for electromagnetic radiation and electron rescattering are commonly quoted as being equal to $3.17{U_p}$ and $10{U_p}$, respectively. More accurately, $3.17{U_p}$ is the maximum scattering energy and is related to the HHG cutoff (or maximum recombination energy) by a factor of the ionization potential [4]. Here the time-averaged ponderomotive energy is ${U_p} = {e^2}E_0^2/4{m_e}{\omega ^2}$, with $\omega$ the laser angular frequency and ${E_0}$ the field amplitude.
These results assume a spatially uniform, temporally sinusoidal, and linearly polarized field. The spatial and temporal uniformity assumptions generally hold true when the field is uniform on the scale of the ponderomotive amplitude ${a_{\!p}} = e\!{E_0}/m{\omega ^2}$ and the pulse length is several wavelengths, $c\tau \gg \lambda$. Nanostructure-based field enhancement induces a field profile that drops in the direction normal to the surface on some length scale, which is predominantly a function of nanostructure geometry. Ignoring the particular design of the nanostructure, this length scale is best approximated as the radius of curvature at the nanostructure apex, $R$. As ${a_{\!p}}$ approaches this length scale, the assumption of a uniform field breaks down. Using an exponential field profile, it has been shown that the quiver motion in the electromagnetic field is altered significantly and the peak energy is reduced for sufficiently short field decay lengths [22]. Additionally, shorter laser pulse lengths induce sensitivity to the carrier–envelope phase (CEP) [23].
Fig. 1. Drawings of the nanoblade (left) and nanotip (right) systems. An incident laser illuminates the structures and induces an enhanced field profile that decays from the apex. The effective radius of curvature $R$ assumes the apices may be modeled by cylinders and spheres. The emitted electrons at the surface may go through a rescattering process, emitting electrons at high energy and collectively producing HHG via scattering.
In this paper, we investigate the classical dynamics of nanostructure-enhanced laser-field-induced electron rescattering. We consider field profiles that closely mimic nanoblade and nanotip geometries (drawings of these systems are shown in Fig. 1), as well as an exponential profile that models plasmonically enhanced fields on flat regions. The exponential profile is also a commonly used theoretical surrogate for arbitrary structures with some near-field drop-off scale. For a configuration of peak field strength, laser wavelength, and nanostructure scale, there is a unitless spatial adiabaticity parameter $\delta$ that effectively characterizes the field profile [22], and we define it to be
with $g(x)$ the field profile with peak field $g(0) = 1$. The parameter $\delta \gg 1$ indicates that the field is effectively uniform, while $\delta \ll 1$ indicates the field gradient is strong enough to significantly alter the dynamics of the system. In Ref. [22], $\delta$ was defined using the $1/e$ decay length of the field, but here we generalize by using the surface gradient, yielding the same value for an exponential field profile, but permitting model-independent use. This preferentially encapsulates the near-field behavior of the profile, which is much more important for moderate ponderomotive amplitudes, $\delta \gtrsim 10$ (e.g., a nanotip, $R = 20\,\,{\rm nm}$, $\lambda = 800\,\,{\rm nm}$, $E \lesssim 31.5\,\,{\rm V}/{\rm nm}$). The field profiles relevant to our current study are listed in Table 1. $\delta$ should not be confused with the Keldysh adiabaticity parameter $\gamma = \sqrt {\frac{W}{{2{U_p}}}}$, which relates the ionization potential to the ponderomotive potential and delineates tunneling ionization from multi-photon ionization.
Table 1. Field Profiles Studied in This Papera
We fit the maximum emission and scattering energies as simple double-exponential functions of ${\delta ^{- 1}} \in [0,5]$ for each nanostructure geometry. These results are corroborated by a 1D time-dependent Schrödinger equation (TDSE) ensemble calculation where the emission energy mitigation due to the field gradient is consistent with our results. We also consider pulsed systems where the ponderomotive force correction is lessened by a temporal termination of the field. Naturally we then consider the effects of the CEP of the incident laser. Finally, we perform a near-field averaging analysis to obtain effective ponderomotive energies for both electron emission and scattering. These closed forms permit quick approximation of the emission and scattering energies and scale more accurately than the double-exponential fits for low $\delta$.
Our electron scattering calculations do not include the electron hole dynamics [22] or any particularities of the band structure [24], which are relevant for solid-based HHG emission. Their inclusion would introduce a separate set of natural units, with field decay length and dispersion relation dependent on the material, and so a more thorough study would need to be done for each unique system.
2. CLASSICAL MODEL
The classical treatment of electron rescattering is as follows. An electron is born into the vacuum at the surface of its source at some time of ionization (TOI) ${t_0}$. As the dominant emission process is typically quantum tunneling at the ponderomotive energy scales involved, we may assume the initial velocity of the electron to be zero.
A. Uniform Field
For a uniform field, the subsequent dynamics are calculated classically:
with $e$ the electron charge, $E$ the electric field magnitude, $m$ the electron mass, and $\omega$ the laser angular frequency. When the electron returns to its source at $x = 0$, it is elastically scattered, reversing its velocity. The velocity after one period (${t_f} = 2\pi /\omega$) will yield the drift velocity as a function of ${t_0}$. The particle continues to be affected by the field after this point. However, as the field is turned off adiabatically, the resultant final velocity is simply this drift velocity.Maximizing the final velocity with respect to ${t_0}$ yields the commonly quoted $10{U_p}$ classical cutoff energy for electron rescattering processes in uniform fields. Maximizing the velocity at the scattering event with respect to ${t_0}$ reproduces the scattering classical cutoff of $3.17{U_p}$, the maximum emitted photon energy.
B. Non-uniform Field
Strong-field photoemission is often achieved without high-power laser systems by utilizing field-enhancing nanostructures, such as nanotips. These nanoscopic structures induce enhanced electric field profiles that may taper off on a scale comparable to the ponderomotive (or quiver) amplitude of the electron:
The final emission energy for a TOI ${t_0} \in \big[{- \frac{\pi}{2},\frac{\pi}{2}}\big]$ necessitates solving Eq. (5) for $t \to \infty$. Numerically we approximate this by evaluating the final velocity at ${t_f} = 2n\pi$ for a sufficiently large integer $n$ (we use $n = 10$). After this point, the field tapering has become gradual enough such that the motion can be split between the fast oscillating motion and the slow drift motion. The asymptotic energy may then be approximated by adding the integral of the remaining ponderomotive force. As ${F_p} = - \nabla {U_p}(x) = - \nabla {g^2}(x)$, this remaining factor is simply ${g^2}({x_f})$. For $\delta \ll 1$, the electron will have traversed most of the field profile by ${t_f}$, and so this factor should be $\approx {g^2}(\infty) = 0$; the ponderomotive force was already accounted for during the calculation interval. For $\delta \gg 1$, this ponderomotive force analysis holds throughout the system, and so the ${F_p}$ correction is $\approx {g^2}(0) = 1$; the electron is effectively in a uniform field, and therefore the ponderomotive force was not entirely included during calculation. This extra factor of $1{U_p}$ results in a classical cutoff for long-pulse or continuous-wave rescattering of approximately $11{U_p}$ when $\delta \gg 1$.
C. Pulsed Systems
Experiments where this system applies often use short (few-cycle) pulses to mitigate target damage. The dynamical equation now follows the form
with $\phi$ the CEP and $\tau$ the full-width half-max power of the laser. In this case, the field felt by the electrons may terminate temporally as opposed to the trivial spatial termination. This results in a reduction of the additional factor of $1{U_p}$.This leads to the introduction of the concept of the temporal adiabaticity parameter. If we have an electron that has gone through the rescattering process and is being emitted at maximum velocity ${v_{\rm{max}}}$, then we may approximate the added energy due to the ponderomotive potential by integrating the ponderomotive force ${F_p}(x,t) = - \nabla {U_p}(x,t)$ with $t \sim x/{v_{\rm{max}}}$ for an exponential profile:
As the maximum energy is around $10{U_p}$ (${v_{\rm{max}}} = \sqrt 5$) for weak field drop-off, the temporal adiabaticity parameter may be estimated using ${\delta _T} \approx \delta _T^* = \frac{1}{2}\sqrt {\frac{5}{{\ln 2}}} \frac{\tau}{\delta} \approx 2\frac{\tau}{\delta}$. With ${v_{\rm{max}}}$ representing different portions of an emission spectrum, this analysis may also be used to investigate above-threshold ionization (ATI) peak shifting. This differential ponderomotive force potential is detectable.
Invoking this same treatment for nanoblade and nanotip fields yields lengthy closed forms. These results maximally deviate from the exponential case by 0.1 at ${\delta _T} = 1.3$ for nanoblades and by 0.05 at ${\delta _T} = 1.2$ for nanotips. We therefore find it sufficient to use the simpler exponential form for the ${E_{{F_p}}}$ correction.
3. NUMERICAL ANALYSES
We further study these systems by numerically solving for the electron emission and scattering velocities. We consider the uniform, exponential, tip, and blade profiles in Table 1, and a Gaussian pulsed system for the exponential case.
A. Numerical Methods
We integrate Eqs. (5) and (6) by reducing them to two first-order equations for position and velocity. The position and velocity are solved for using Heun’s method, modified such that the velocity integration step uses the new position already calculated via Heun’s method instead of Euler’s method. The order of the method remains the same, but the truncation error and the number of operations required are reduced.
We use a default time step of $h{= 10^{- 3}}\;{\rm rad}$. Scattering is included by testing whether $x \le 0$ after the current iteration. If so, the time-step is temporarily set to ${h_s} = ({- |v| + \frac{v}{{|v|}}\sqrt {{v^2} + 2ax}})/a$ for acceleration $\ddot x = a \ne 0$ or ${h_s} = - \frac{x}{v}$ for $a \approx 0$. This ensures the electron is located at the surface, $x = 0$, at the end of the time step. Post-iteration, the velocity is reflected and recorded for scattering studies. Additionally, the time step may deviate in the last iteration of the calculation where it is set to the remaining interval length.
We maximize the energy with respect to the TOI first by a grid search with a density of 1000 samples within ${t_0} \in [{- \frac{\pi}{2},\frac{\pi}{2}}]$. For the Gaussian pulsed calculations, this grid search is done for the central cycle and the two before it. The grid search is then refined by the Nelder–Mead simplex algorithm for emission and the Newton–Raphson method for scattering. For the Gaussian case, the algorithms are performed in and restricted to each of the three intervals of ${t_0}$.
B. Uniform Field Profile
The standard uniform field calculation assumes that the laser–electron interaction terminates once the pulse slowly ends temporally and adiabatically, and the laser field is assumed to have no spatial dependence. In this light, we do not include the extra ponderomotive force contribution to the final velocity as there is none in a uniform field.
The relation between emission and scattering energies and the ionization phase ${t_0}$, which we will now refer to as the emission energy landscape (EEL), is shown in Fig. 2. The direct electron emission portion has an analytical form of $2{U_p} {\sin}^2 {t_0}$ for ${t_0} \in [- \pi /2,0]$. At ${t_0} = 0$, we observe the onset of scattering events, with the first providing the largest emission and scattering energies. With each scattering event, the free electron density diminishes as some transmits through the material instead of backscattering. Thus the multi-scattered peaks are far less detectable in experiment. For ${t_0} \to \pi /2$, the electron does not leave the surface as the extraction impulse is nullified, and so emission at this time is effectively the same as emitting at ${t_0} = - \pi /2$ as far as vacuum dynamics with full reflection probability are concerned.
Fig. 2. Emission and scattering energy dependencies on ${t_0}$ for a uniform field. The surface field as a function of time (dashed cyan, arbitrary units) indicates an extracting force for the first half of the window and a restoring force in the second half, valued zero at ${t_0} = 0$. Direct electron emission (black) has a clear cutoff at $2{U_p}$. At ${t_0} = 0$, scattering begins to occur, and the rescattered electron energy (blue) approaches a peak energy before undergoing multiple scattering events. The subsequent scattering events do not lead to any higher energies. The first (red) and second (green) scattering energies are also shown, the former leading to the HHG classical cutoff.
The aforementioned maximization techniques produce the expected classical cutoff for electron emission of $10.0076{U_p}$ at ${t_0} = 0.2610 \approx \pi /12$. This result is usually applied to spectra of quantum nature with smooth, gentle cutoffs. Therefore, such precision is not typically required, which is why the $10{U_p}$ rounding is common and useful. The scattering cutoff is $3.1731{U_p}$ at ${t_0} = 0.3134 \approx \pi /10$. For the same reason, this result is typically reported as $3.17{U_p}$.
C. Exponential Profile
Contrary to the uniform field profile calculations, with a non-uniform field, it is generally unclear as to whether the laser–electron interaction terminates due to temporal or spatial constraints. A laser pulse of sufficiently short length would end the interaction before the electron traverses the entire field profile, and so the pulse length and the near-to-mid-field behavior would have the largest impact on results. Alternatively, a long pulse would permit the electron to explore most of the field profile, and so the field profile itself would then be the only independent variable. For this reason, we presently continue assuming here an infinitely long pulse, or ${\delta _T} \gg 1$. In this case, even for large $\delta$, we will have an extra factor of $1{U_p}$ added to the final emission energy.
A slowly decaying field profile with $\delta = 5$ yields the EEL shown in Fig. 3(a). The introduction of this field gradient raises the direct electron cutoff from $2{U_p}$ to about $3.5{U_p}$. During the emission process the electron moves away from the surface of the cathode into a region with lesser field strength. Thus, while the extraction force is approximately unchanged, the restoring force is substantially reduced. This results in an overall increase in direct electron energy. This effect arises due to a variant of ponderomotive force, which is always present for spatially varying time-periodic forces.
Fig. 3. Emission energy landscapes for exponential field profiles with (a) $\delta = 5$ and (b) $\delta = 1$. Electron emission energies are the same as Fig. 9 in Ref. [25]. A substantial quenching of the maximum energies is observed even for somewhat large $\delta$.
On the other hand, the rescattering cutoff reduces to about $8.5{U_p}$ and shifts to a later TOI. By the same extraction–restoration imbalance explanation above, to go through a rescattering process, the electron must have an excess restoring force. A larger TOI decreases the extraction impulse, and so the TOI must increase to retain the rescattering event.
The scatter energy decreases to slightly over $2{U_p}$ and also shifts to a later TOI. The reduced restoration force clearly also reduces the energy with which an electron may return to the cathode, and maintaining the rescattering process again requires a shifted TOI.
Increasing the ponderomotive amplitude such that $\delta = 1$ produces the EEL shown in Fig. 3(b). All the effects previously observed are further exacerbated here. The electron quickly traverses into the mid-field for most TOIs. If the electron is emitted towards the end of the ionization half-period, it does not travel very far into the field, and the restoring half-period is enough to induce a low-energy scattering event.
Performing the maximization technique for electron emission and scattering provides the relationship shown in Fig. 4. The cutoff energy monotonically decreases as the ponderomotive amplitude increases and the adiabaticity parameter decreases. The TOI continues to increase as expected from Fig. 3.
Fig. 4. Cutoff energies as a function of the adiabaticity parameter $\delta$ for an exponential field profile. Emission (blue) and scattering (black) cutoff energies (left axis) decrease monotonically with lessening adiabatic fields. Additionally, the time of ionization (right axis) that leads to this peak energy is plotted for the emission (red dashed) and scattering (magenta dashed) cutoffs. The cutoff energies decrease monotonically for decreasing adiabaticity parameter $\delta$.
These cutoffs may be well approximated using a double-exponential fit within the region ${\delta ^{- 1}} \in [0,5]$. The limits of uniform field, $11.0076{U_p}$ and $3.1731{U_p}$, are enforced in these fits. These fits and their root-mean-square errors (RMSEs) are reported in Table 2 for all three profiles. The low-adiabaticity limit is not encapsulated by these functions, and so they should be trusted only within the region ${\delta ^{- 1}} \in [0,5]$. The low-adiabaticity limit may be a topic of future investigation; however, physically achieving $\delta \lt \frac{1}{5}$ is quite difficult without resorting to an effectively static field.
Table 2. Double-Exponential Fitted Cutoff Energies for Exponential, Nanotip, and Nanoblade Profilesa
D. Comparison to Quantum Mechanical Model
In Ref. [20], we calculated the evolution of an ensemble of jellium-bound electrons under the influence of a strong laser field via the TDSE and found the emitted electron spectrum. Here we use an exponential field profile with a linear tangent near the simulation boundary to reach zero field strength at the virtual detector position. No collective effects are included. A strong deviation between the simulated spectra and the $10{U_p}$ uniform field cutoff arises for high wavelengths, even with the change in effective wavelength and peak field from the pulse window taken into account. Figure 5 shows the comparison between the standard $10{U_p}$ approximation and the results found here overlaid on the TDSE spectrum. Our results follow the rescattering cutoff are far closer than the standard $10{U_p}$ result.
Fig. 5. TDSE-derived electron emission spectra (colormap) with the classical $10{U_p}$ cutoff (red dashed) along with the cutoff calculated in this section (black dashed). The TDSE spectra were calculated using an exponential field profile with a peak field strength of 20 V/nm. The cutoffs calculated in this paper closely follow the cutoffs from the TDSE calculations, while the typical $10{U_p}$ result begins notably deviating at $\lambda = 2000\,\,{\rm nm}$.
E. Nanostructure Profiles
A comparison of the emission and scattering energies for the tip and blade fields, represented as a deviation from the exponential field, is shown in Fig. 6. Beginning with electron emission, we see that, for a large adiabaticity parameter, the geometry-specific cutoffs deviate negatively, whereas once $\delta$ decreases to ${\sim}2 - 3$, we see an increase in energy, with diminishing returns quickly setting in. Since $\delta \to 0$ indicates an infinitesimally localized field, we expect the deviation in ponderomotive units to tend towards zero. We note that for the same adiabaticity parameter, the blade-like field spatial profile decays the slowest, then next is the tip profile, and finally the exponential field profile decays the fastest.
Fig. 6. Deviation of emission (solid) and scattering (dashed) energy for tip (red) and blade (blue) fields from the exponential field drop-off. All models are in congruence within a factor of $1{U_p}$.
Considering the electron of maximal emission energy for the exponential field profile with ${\delta ^{- 1}} = 0.3$, the field magnitude at the electron’s position has already diminished to 0.02 of its original strength after two laser cycles ($x \approx 13{a_{\!p}}$). The tip and blade profile magnitudes are reduced to 0.11 and 0.20 at this time, respectively. Because of this, the post-scattering exponential electron experiences a strong extraction force (about the same for all cases), while the following restoring force is greatly reduced (by about a factor of three compared to blade fields) permitting larger emission energies with the same scattering dynamics for the exponential field.
However, once ${\delta ^{- 1}} \gt 0.5$, the scattering dynamics are meaningfully altered. The scattering time is pushed later, and the scattering energy is reduced for the exponential field as compared to the blade and tip fields. Not only is the emission energy for tip and blade fields larger due to the extra scattering energy, but the electrons experience the second extraction field for longer. The earlier scattering time allows for a longer final extraction phase, and the slower decaying fields maintain the extraction further into vacuum.
The increase in scattering energy for blade and tip fields compared to the exponential field has a straightforward explanation: the fields are closer to uniformity because of the slower decay rate (including the nonlinear orders of $\delta$), and so the rescattering dynamics are closer to the ideal uniform system, which attains maximum scattering energy.
These observations together highlight the limitations of the exponential field profile as a general surrogate model of non-uniform rescattering. While useful and interesting, it may lead to erroneous conclusions depending on the precision expected of the calculation.
F. Pulsed Systems
As noted above, a pulsed laser is typically used to avoid damaging the cathode in high-field photoemission. The laser pulse envelope profile may be treated as a Gaussian according to Eq. (6). We will focus primarily on $\phi = \pi /2$ (the field ionizes before and then restores after the peak of the envelope) and $\tau = 6\pi$ (akin to an 800 nm, 8 fs pulse).
The EEL for a laser pulse with a uniform spatial field profile is shown in Fig. 7. The cutoff energies are reduced slightly due to the reduction in field magnitude induced by the envelope. The peak emission and scattering originate from a TOI just before the peak of the envelope for this choice of CEP.
We may also find the peak energies as a function of the adiabaticity parameter with a Gaussian temporal envelope. The cutoff energies for this pulsed laser, using the exponential field profile, are plotted in Fig. 8. One may notice interesting behavior for very small ponderomotive amplitudes (characterized by large $\delta$) where the peak electron energy increases slightly before following the typical decay. This is consistent with correcting the continuous-wave cutoff behavior in Table 2 with the proper ponderomotive force boost ${E_{{F_p}}}$ in Eq. (8).
Fig. 8. Cutoff behavior for pulsed ($\tau = 6\pi$, $\phi = \pi /2$) laser using the exponential field profile. Emission (blue) and scattering (black) cutoff energies (left axis) are shown as a function of the adiabaticity parameter. Additionally, the times of ionization (right axis) that lead to these peak energies are plotted for the emission (red dashed) and scattering (magenta dashed) cutoffs. The double bars indicate a break in the right vertical axis.
At ${\delta ^{- 1}} = 0.18$, there is a ${-}2\pi$ phase shift in peak emission TOI. As the field becomes less adiabatic, the rescattering dynamics are increasingly altered. While the scattering energy in this previous cycle is slightly decreased due to the envelope, the final extraction field is maximized, resulting in the largest emission energy possible. This may also be seen by examining the vector potentials for the uniform field case.
The final momentum in the uniform case is $p = 2A({t_{\rm{scat}}}) - A({t_0})$, with $A$ the vector potential. This factor of two arises due in combination to the restoration impulse returning the electron to the surface and the extraction impulse post-scattering. Thus, to maximize the final momentum, one should preferentially increase the vector potential at the time of scattering. For the Gaussian pulse case, this means maximizing the final extraction field, which occurs at about ${t_0} + 2\pi$.
Then there is another phase shift at ${\delta ^{- 1}} \approx 3.4$. Here the rescattering dynamics are altered to the point that no increase in energy is afforded by rescattering. Thus, the maximum energy comes from the direct emission process. This occurs for nanotips as well, but nanoblades appear to continue to gain energy through rescattering even for low $\delta$.
G. Carrier–Envelope Phase Dependence
To preface this section investigating the effects of the CEP, we found that adjusting results for the envelope-reduced peak field [by dividing $E(x,t)$ by $m(\phi) = \max | {{e^{- 2\ln 2{{({\frac{t}{\tau}})}^2}}}\cos (t + \phi)} |$] does not account for the phenomena we observe. This correction makes the proceeding curves (Fig. 9) roughly piecewise linear, but adds complexity due to the two differential discontinuities of $m(\phi)$ with respect to the CEP. Correcting for either peak extraction or peak restoring field also not only makes the plots roughly linear, but does not aide in interpretation and gives rise to energies much larger than expected (${\sim}15{U_p}$). For these reasons, we continue with the original laser profile in Eq. (6).
Fig. 9. Cutoff energies for a pulsed laser as a function of CEP for systems with either few- or single-cycle pulses ($\tau = 6\pi$ or $3\pi$) and for uniform or exponential field profiles. Decreased pulse length increases sensitivity to the CEP in general. The inclusion of a field gradient decreases the achievable energies and modifies the CEP in which a TOI transition occurs. (a) Electron emission energy cutoffs. In the same order as the legend, the maxima are $10.04U_p$ at $-3.79\;{\rm rad}$, $10.18U_p$ at $-3.91\;{\rm rad}$, $8.48U_p$ at $-4.46\;{\rm rad}$, and $8.40U_p$ at $-4.60\;{\rm rad}$. (b) Electron scattering energy cutoffs. In the same order as the legend, the maxima are $3.17U_p$ at $-3.10\;{\rm rad}$, $3.17U_p$ at $-3.15\;{\rm rad}$, $2.40U_p$ at $-2.96\;{\rm rad}$, and $2.40U_p$ at $-3.00\;{\rm rad}$.
The electron emission cutoff behavior is shown in Fig. 9(a). The uniform field results are equivalent to Fig. 3a in Ref. [23]. As expected, the dependence on CEP is strengthened with a shorter pulse, and including a field gradient decreases the peak energy. For both uniform and exponential fields, the peak energy tends to occur for $\phi \in [- \pi ,0]$, where $\phi = \pi /2$ corresponds to a sine-like pulse where peak extraction follows peak restoring fields, and $\phi = \pm \pi$ corresponds to a peak restoration force at $t = 0$. This may unveil a choice of CEP targeting peak electron energy, which may be generally dependent on the geometry and fields at play. Additionally, the peak emission energy is at about $10.18{U_p}$ for the short pulse with a uniform field.
The sharp vertex in each of these curves indicates that the peak emission time transitioned from one cycle to another. At $\phi \approx \pi /2$, we see a phase transition for the exponential profiles at ${\delta ^{- 1}} = 0.2$, quite close to the transition at ${\delta ^{- 1}} = 0.18$ in Fig. 8.
The CEP’s effect on the scattering cutoff energy is shown in Fig. 9(b). The phase for peak energy is consistently near $\phi = \pm \pi$. This is, as expected, where the peak field at $t = 0$ is restorative. Like the electron emission case, here we see an exaggerated effect for short pulses, and a general drop in scattering energy when including a field gradient.
4. NEAR-FIELD AVERAGING
One possible approximation to arrive at the continuous-wave results is to average the ponderomotive energy of the field throughout one cycle of the electron’s quiver motion near the surface. This spatial averaging results in simple expressions for a modified ponderomotive energy that may then be inserted into the standard uniform field results.
The spatial averaged ponderomotive potential for the scattering energy is
where we have averaged over the first ponderomotive amplitude, the approximate extent of an electron’s trajectory into the laser field. An effective ponderomotive energy for electron emission may be estimated by exponentiating the above result to some power $b$:The formulas for the exponential, nanoblade, and nanotip fields are summarized in Table 3. The cutoffs estimated by these formulas may be calculated by
Table 3. Near-Field-Averaged Effective Ponderomotive Energy for Exponential, Nanotip, and Nanoblade Profilesa
The physical justification for the exponentiation factor $b$ may be as simple as the electron interacting with the field in two instances: one full interaction before scattering and a diminished interaction post-scatter. This is admittedly a slightly speculative argument; however, with reasonably low RMSEs we believe this to be an effective scheme. A similar exponential applied to the scattering ponderomotive energy of 0.95 minimizes the scattering RMSEs. As this power is much closer to one and we have no reason to believe it should not be one, we do not include this exponential factor.
These results together show that this field averaging method may provide a trivial alternative for finding the scattering or electron emission cutoff for a given field profile. Additionally, these forms nearly approach the correct low-adiabaticity limit for direct electron emission from the exponential and tip profiles, $4\delta$ and $8\delta$, respectively (the nanoblade limit has been elusive due to the non-integrability of ${r^{- 1}}$), indicating that they may extend effectively beyond ${\delta ^{- 1}} = 5$. These limits are not embodied by the double-exponential fits. However, in the absence of a rigorous physical interpretation of its underlying mechanisms, this near-field averaging approximation should be used with care.
5. CONCLUSIONS
In this paper, we have utilized a common approach to evaluating the classical dynamics involved with electron rescattering to solve for the maximum emission and scattering energies achievable as functions of the wavelength, enhanced field intensity, and field drop-off scale. We provided easy-to-use fits to these data within the region of inverse adiabaticity parameter ${\delta ^{- 1}} \in [0,5]$, covering well beyond what is currently achieved in a typical experiment. Even for nearly uniform fields, $\delta \approx 10$, the energies are notably reduced by about 10%. We have also performed a straightforward near-field averaging analysis to find effective ponderomotive energies, which are quite accurate within the evaluated region and exhibit nearly correct low-adiabaticity limiting behavior beyond this region. Comparing the exponential profile to the tip and blade profiles revealed its theoretical limitations as a surrogate for the actual field profiles encountered. For strong-field drop-offs, $\delta \sim 1$ or less, the model deviates by up to nearly $0.5{U_p}$ for tips and nearly $0.9{U_p}$ for blades.
We have also considered a pulsed Gaussian-profile incident field. For few-cycle pulse lengths the additive ponderomotive force factor ${E_{{F_p}}}$ behaves like a simple function of the introduced temporal adiabaticity parameter, which itself is a function of the pulse length and the standard adiabaticity parameter. ${E_{{F_p}}}$ tends towards $1{U_p}$ for long pulses and zero for short pulses. We have also found, as expected, that the energy cutoffs become strongly dependent on the CEP for short pulses.
While our results for the electron emission energy should be accurate for most systems, the scattering energy is more dubious when applied to solids, in particular, when considering HHG radiation. Solid HHG involves the dynamics of the ionized electron and its associated electron hole within the material, and the scattering energy is the energy at which these two recombine. The inclusion of the electron hole, which would have a different effective mass, dispersion relation, and/or field profile, obviates the use of the natural units of the vacuum system. Inclusion of these properties is important and has been performed [22,24], although the results are less transferable between systems. Additionally, in cases where the ionization density is large, as is already being explored in experiments, these single-body calculations may not be applicable [15], and a collective approach would be needed.
Funding
National Science Foundation (PHY-1549132).
Acknowledgment
We thank Philip Dienstbier and Professor Peter Hommelhoff for their valuable input on this paper.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
REFERENCES
1. M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,” Nature 414, 509–513 (2001). [CrossRef]
2. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81, 163–234 (2009). [CrossRef]
3. G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L. Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. De Silvestri, and M. Nisoli, “Isolated single-cycle attosecond pulses,” Science 314, 443–446 (2006). [CrossRef]
4. M. Krüger, C. Lemell, G. Wachter, J. Burgdörfer, and P. Hommelhoff, “Attosecond physics phenomena at nanometric tips,” J. Phys. B 51, 172001 (2018). [CrossRef]
5. S. E. Irvine, A. Dechant, and A. Y. Elezzabi, “Generation of 0.4-kev femtosecond electron pulses using impulsively excited surface plasmons,” Phys. Rev. Lett. 93, 184801 (2004). [CrossRef]
6. P. Rácz, S. E. Irvine, M. Lenner, A. Mitrofanov, A. Baltuška, A. Y. Elezzabi, and P. Dombi, “Strong-field plasmonic electron acceleration with few-cycle, phase-stabilized laser pulses,” Appl. Phys. Lett. 98, 111116 (2011). [CrossRef]
7. R. Bormann, M. Gulde, A. Weismann, S. V. Yalunin, and C. Ropers, “Tip-enhanced strong-field photoemission,” Phys. Rev. Lett. 105, 147601 (2010). [CrossRef]
8. S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature 453, 757–760 (2008). [CrossRef]
9. D. Ehberger, J. Hammer, M. Eisele, M. Krüger, J. Noe, A. Högele, and P. Hommelhoff, “Highly coherent electron beam from a laser-triggered tungsten needle tip,” Phys. Rev. Lett. 114, 227601 (2015). [CrossRef]
10. L. Song, C. Ma, and T.-Y. Du, “Optimization of surface high-order harmonic generation by tailoring nanostructures,” J. Opt. Soc. Am. B 39, 2678–2683 (2022). [CrossRef]
11. G. Wachter, C. Lemell, J. Burgdörfer, M. Schenk, M. Krüger, and P. Hommelhoff, “Electron rescattering at metal nanotips induced by ultrashort laser pulses,” Phys. Rev. B 86, 035402 (2012). [CrossRef]
12. M. Krüger, M. Schenk, M. Förster, and P. Hommelhoff, “Attosecond physics in photoemission from a metal nanotip,” J. Phys. B 45, 074006 (2012). [CrossRef]
13. C. Pierce, D. Durham, F. Riminucci, A. Kachwala, S. Karkare, I. Bazarov, J. Maxson, A. Minor, and D. Filippetto, “Towards high brightness from plasmon-enhanced photoemitters,” in Proceedings, North American Particle Accelerator Conference (NAPAC) (2022), paper TUYD4.
14. T. L. Gilton, J. P. Cowin, G. D. Kubiak, and A. V. Hamza, “Intense surface photoemission: Space charge effects and self-acceleration,” J. Appl. Phys. 68, 4802–4810 (1990). [CrossRef]
15. J. Li, S. Fu, H. Wang, X. Zhang, B. Ding, B. Hu, and H. Du, “Limitations of the single-active-electron approximation in quantum simulations of solid high-order harmonic generation,” Phys. Rev. A 98, 043409 (2018). [CrossRef]
16. M. Aeschlimann, C. A. Schmuttenmaer, H. E. Elsayed-Ali, R. J. D. Miller, J. Cao, Y. Gao, and D. A. Mantell, “Observation of surface enhanced multiphoton photoemission from metal surfaces in the short pulse limit,” J. Chem. Phys. 102, 8606–8613 (1995). [CrossRef]
17. C. Spindt, “A thin-film field-emission cathode,” J. Appl. Phys. 39, 3504–3505 (1968). [CrossRef]
18. S. Tsujino, P. Beaud, E. Kirk, T. Vogel, H. Sehr, J. Gobrecht, and A. Wrulich, “Ultrafast electron emission from metallic nanotip arrays induced by near infrared femtosecond laser pulses,” Appl. Phys. Lett. 92, 193501 (2008). [CrossRef]
19. G. Lawler, N. Majernik, J. Mann, N. Montanez, J. Rosenzweig, and V. Yu, “Emittance measurements of nanoblade-enhanced high field cathode,” in Proceedings, 13th International Particle Accelerator Conference (IPAC) (JACoW Publishing, 2022), pp. 709–712.
20. B. Wang, J. Mann, G. Lawler, J. Nangoi, T. Arias, S. Karkare, and J. Rosenzweig, “Simulations of laser field emission from nanostructures with image charge trapping and band structure transitions,” in Proceedings, 13th International Particle Accelerator Conference (IPAC) (JACoW Publishing, 2022), pp. 717–720.
21. T. Paschen, R. Roussel, L. Seiffert, B. Kruse, C. Heide, P. Dienstbier, J. Mann, J. Rosenzweig, T. Fennel, and P. Hommelhoff, “Ultrafast strong-field electron emission and–collective effects at a one-dimensional nanostructure,” ACS Photonics 10, 447–455 (2023). [CrossRef]
22. G. Herink, D. R. Sollie, M. Gulde, and C. Ropers, “Field-driven photoemission from nanostructures quenches the quiver motion,” Nature 483, 190–193 (2012). [CrossRef]
23. M. Krüger, M. Schenk, and P. Hommelhoff, “Attosecond control of electrons emitted from a nanoscale metal tip,” Nature 475, 78–81 (2011). [CrossRef]
24. G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91, 064302 (2015). [CrossRef]
25. K. Echternkamp, G. Herink, S. Yalunin, K. Rademann, S. Schäfer, and C. Ropers, “Strong-field photoemission in nanotip near-fields: from quiver to sub-cycle electron dynamics,” Appl. Phys. B 122, 80 (2016). [CrossRef]
