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Simulation of the Full Spatio-Temporal Evolution of Ultrafast Terahertz Harmonic Generation from Photoionized Holes in Silicon at Cryogenic Temperature
Dieser Artikel untersucht die Simulation ultraschneller Terahertz-Oberschwingungen in Silizium bei kryogenen Temperaturen, wobei der Schwerpunkt auf der Dynamik photoionisierter Löcher und ihren potenziellen Anwendungen liegt. Die Studie verwendet fortgeschrittene Berechnungsmethoden, einschließlich Vollband-Monte-Carlo-Schemata (FBMC) und Finite-Difference-Time-Domain-Schemata (FDTD), um das komplizierte Zusammenspiel von Ionisationsprozessen und Ladungsträgertransport zu modellieren. Zu den Schlüsselthemen gehören die Erzeugung höherer Oberschwingungen, der Einfluss kryogener Temperaturen auf die Trägerdynamik und die Herausforderungen, eine kohärente Rückkühlung für eine höhere harmonische Erzeugung (HHG) zu erreichen. Der Artikel vergleicht außerdem 1 + 1D- und 3 + 1D-Simulationen und betont die Bedeutung der Berücksichtigung lateraler räumlicher Effekte und Beugung in Versuchsanordnungen. Die Ergebnisse zeigen die Machbarkeit vollständiger 3 + 1D-Simulationen und ihre enge Übereinstimmung mit 1 + 1D-Simulationen für On-Axis-Felder und liefern wertvolle Erkenntnisse für zukünftige Experimente und Anwendungen in der Messtechnik und Kommunikation.
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Abstract
We investigate harmonic generation (HG) in p-doped silicon at low temperature with intense terahertz (THz) pulses, extending our previous studies by going to 3+1D simulations combining full-band-Monte-Carlo (FBMC) and finite-difference-time-domain (FDTD) methods to describe the microscopic photoionization and subsequent motion of holes, coupled to the propagation of the THz fields. Notably, the predicted on-axis emitted fields are almost in quantitative agreement with those from 1+1D (plane-wave) FDTD-FBMC simulations, despite the tight focusing of the THz pump beam. However, having access to the transverse profile of the THz fields allows us to inspect the lateral variation of the HG process, and model the subsequent re-imaging of the emitted harmonic fields at the detection plane, as per the geometry used in previous experiments. We show that diffraction leads to modifications in the re-imaged on-axis fields, such that the strong on-axis nonlinear absorption for the fundamental in the sample is not preserved at the detection plane, resolving a previous discrepancy between experiment and theory at high pump fields. This demonstrates that while plane-wave spatio-temporal simulations can provide accurate results for the on-axis fields in nonlinear THz experiments, care must be taken in interpreting the experimentally detected fields when working at the diffraction limit.
1 Introduction
The generation of higher harmonics by intense electromagnetic waves in the THz spectral range not only provides a sensitive probe of high-field carrier dynamics in semiconductors [1‐5], but also can be considered for applications, such as frequency-comb generation for metrology and communications [6, 7]. The underlying nonlinearities in the current response can arise from both the band non-parabolicity and the energy dependence of the scattering rates, whose relative contributions depend on the THz frequency \(\nu _0\) and peak field \(E_0\) [1, 2]. Due to inversion symmetry, one obtains only odd-order harmonics (\(\nu _n=n\nu _0\), \(n=3,5,7,...\)), which we observed experimentally for holes in Si:B up to at least \(n=11\) for \(\nu _0=0.3\text {THz}\) and \(E_0\rightarrow 100\,\,\text {kV cm}^{-1}\) and multi-cycle pulses with duration \({\sim }10\,\,\text {ps}\) [2].
Moreover, at cryogenic temperatures, the carriers are initially bound to their parent dopant ions and are injected into the adjacent band in short bursts at each intracycle extremum of the THz field via tunnel photoionization [2, 8]. In this case, the possibility arises for higher-harmonic generation (HHG), i.e., a broad plateau of odd-order harmonics which can extend to \(n \gg 10\), via subsequent recollision of each carrier with its parent ion, as per gas-phase atoms at higher photon energies [9]. This could provide an attractive alternative to other approaches to solid-state HHG [10, 11]. However, it requires that the carrier motion is sufficiently ballistic during a THz field cycle to allow a coherent return of the carriers to their respective parent dopant sites. In our previous studies [1, 2, 12], the absence of such a HHG plateau could be traced to the loss of coherent motion due to intracycle scattering, primarily due to optical-phonon emission, which in Si:B sets in at the same field strengths required for significant tunnel photoionization. Nevertheless, the quest continues to identify suitable semiconductors/dopants and frequency-field regimes where coherent recollision can be achieved. Here, detailed simulations are important to identify suitable conditions for experiments, which in turn requires clear validation of the predictive value of the theoretical treatment.
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The intricate interaction of the ionization process and charge-carrier transport is difficult to describe by analytical means. In our previous reports [2, 13], we hence applied a full-band-Monte-Carlo (FBMC) treatment [14] to describe the time-dependent carrier dynamics and resultant HG emission. As the propagation/standing-wave effects in the finite-thickness samples strongly affect the HG process, we embedded a large parallel set of FBMC simulations at each spatial point in a 1+1D finite-difference-time-domain (FDTD) wave propagation scheme. In this way, we could achieve near-quantitative agreement with the experimental HG emission [2], although some discrepancies remained, in particular, the transmission of the fundamental as a function of incident pump field \(E_0\). The fundamental should undergo significant self-induced absorption due to the photoionized free-carrier response, although this was not observed experimentally. While this could raise possible doubts as to the rigor of the theoretical method (e.g., the assumed semi-classical ionization rate [8]), we speculated that lateral spatial effects (i.e., field dependence perpendicular to the propagation direction z associated with the pump beam shape) might be responsible, these not being captured by the 1+1D FDTD treatment, which tacitly approximates the propagation of the on-axis fields in terms of plane-waves. Due to the highly nonlinear dependence of the ionization rate on the pump field, the carrier concentration falls rapidly as one goes away from the beam axis, leading to lateral modulation on a sub-wavelength scale. Indeed, treatments of nonlinear THz experiments (e.g., [3]) often invoke this to justify neglecting a consideration of the lateral dependence, as the on-axis field should dominate any newly generated signals. This evidently does not apply to the transmitted fundamental field which rather undergoes nonlinear depletion and dispersion effects in the sample, which affect the beam exiting the sample and hence field re-imaged at the subsequent detector plane. Moreover, while in experiments on ultra-thin samples (such as graphene [3]), one need not consider how the lateral modulation affects the propagation in the sample, here we investigate samples with thickness comparable to the wavelength/Rayleigh range, where they could play a significant role.
To address these issues, here we extend the theoretical treatment to a 3+1D FDTD scheme to describe the full spatio-temporal evolution of the THz fields, where the FBMC treatment of the photoionization and carrier dynamics at each spatial point is efficiently calculated with massively parallel computations. The approach is described in Sect. 2, with results in Sect. 3, including a comparison for the on-axis field with those from 1+1D simulations, which are found to be in close agreement, despite the fact that the fundamental beam is focused to the diffraction limit. Although this might suggest that the 1+1D simulation should already adequately describe the experimentally detected fields, in Sect. 3.3, we apply rigorous diffraction theory to propagate the spatial fields exiting the sample to the detection plane, via two paraboloidal mirrors, which collimate and refocus the THz beam. An analysis of the refocused on-axis THz fields demonstrates that the strong on-axis absorption dip in the fundamental beam exiting the sample does not manifest at the detection plane, due to diffraction effects. However, the re-imaging of the on-axis field for the harmonics \(n\ge 3\) becomes increasingly ideal with n. This resolves a key discrepancy between experiment and 1+1D simulations reported in [2], and also serves as a guiding example for the interpretation of the measured fields in other nonlinear THz experiments.
2 Simulation Approach
The combination of the ensemble MC method with the FDTD approach is a versatile tool to simulate electromagnetic waves in semiconductors with a detailed physics-based model [15‐17]. Since the valence bands are warped, a FBMC approach is required to describe the hole transport [14].
2.1 3+1D FDTD
The THz electromagnetic field is propagated through the computational volume with the FDTD method [18‐20]. We simulate a cuboid of dimensions \(W \times W \times (Z_\text {vac} + Z_\text {si}+ 10 Z_\text {vac})\), where \(Z_\text {vac}\) is the thickness in z direction of the vacuum layer on the left side of the Si layer, \(Z_\text {si}\) the thickness of the Si layer and W the width and height in the x and y directions (Fig. 1a).
Fig. 1
a Schematic representation of the simulation domain in the z, x plane (\(Z_\text {vac} = 120\mu {m}\), \(Z_\text {si} = 272\mu {m}\), \(z_\text {eval} = 400\mu {m}\), \(W = 6000\mu {m}\)). A wave with a Gaussian beam profile is injected at \(z = 40\mu {m}\), taken here as the focal plane, and propagates in the positive z direction. b The x-component of the injected (\(z = 40\mu {m}\)) and transmitted (\(z_\text {eval} = 400\mu {m}\)) electric fields vs. time on the beam axis (\(x=y=0\)) for \(E_0=81\,\,\text {kV cm}^{-1}\). The full simulation period is shown. The injected field is plotted for t and the transmitted one for \(t+t_\text {d}\), where \(t_\text {d}={3.37}{\text {ps}}\) is the time to reach \(z_\text {eval}\) based on the speed of light neglecting the impact of holes
We introduce a grid for the 3D space with a constant spacing of \(\Delta x\) in all three directions [20]
$$\begin{aligned} x_i&= i \cdot \Delta x \text { with } i =-\frac{N_x}{2}, \dots , \frac{N_x}{2} \end{aligned}$$
(1)
$$\begin{aligned} y_j&= j \cdot \Delta x \text { with } j =-\frac{N_y}{2}, \dots , \frac{N_y}{2} \end{aligned}$$
(2)
$$\begin{aligned} z_k&= k \cdot \Delta x \text { with } k =1, \dots , N_z \end{aligned}$$
(3)
and a grid for the time t (\(-t_0 \le t \le t_0\))
$$\begin{aligned} t_l = l \cdot \Delta t \quad \text { with } \quad \Delta t = \frac{\Delta x}{2 c_0} \quad \text {and} \quad l = -\left\lfloor \frac{t_0}{\Delta t} \right\rfloor , \dots , \left\lceil \frac{t_0}{\Delta t} \right\rceil \ . \end{aligned}$$
(4)
As given in Sect. 3.1, we use \(\Delta x=4\mu {m}\) (\(\Delta t = {6.67}{\text {fs}}\), \(N_{x,y} = 1500\), \(N_z = 398\)) to achieve sufficient convergence in the calculated results.
The discrete approximations to the electric field strength E, magnetic field strength H and current density J, are arranged on the grid according to the staggered Yee cell [18] (i.e., with relative spatial and temporal half-cell offsets chosen to achieve a scheme where the discrete approximation to the curl operator is readily evaluated, and where the centered spatial/temporal differences achieve second-order accuracy):
The initial values of all fields are zero for \(l \Delta t \le -t_0\). On all six faces of the cuboid, absorbing boundary conditions are implemented with perfectly matched layers (PML), which are eight grid nodes thick [19‐22].
The parameters of the materials are assumed to be time-independent within each cell. For example, the permittivity is given by
For Maxwell’s equations, the permittivity is required at the center of the edges of the cell. For example, for any edge along the z direction, we obtain
This definition is consistent with the finite-volume method for semiconductor device modeling [23]. In the case of the permeability, we use the vacuum value \(\mu _0\) throughout the whole structure.
The FBMC simulations are performed for the individual cells which belong to the Si region [16]. For the Lorentz force, the electric and magnetic fields are required at the center of the cell, for example,
The current density is calculated at \(t_l\) based on the magnetic field at \(t_{l}\) and electric field at \(t_{l-\frac{1}{2}}\) (forward Euler scheme for MC), because implicit schemes are extremely difficult to implement exactly in the stochastic MC algorithms [24]. On the other hand, the FDTD time step \(\Delta t\) is short for MC simulations of hole densities below \(5\cdot 10^{16}\,\text {cm}^{-3}\) and the first-order error in time of the MC step should be small [25]. The particle ensemble is advanced by \(\Delta t\), i.e., to times \(t_{l+\frac{1}{2}}\) after the MC step. The current density is averaged over the time step and thus evaluated at \(t_l\) yielding, for example, \(J^l_{x,i+\frac{1}{2},j+\frac{1}{2},k+\frac{1}{2}}\). This cell-centered current density is mapped with the same approach as the permittivity (see Eq. (15)) onto the edges of the cells.
In the vacuum next to the LHS PML (Fig. 1), the transverse components of a wave with a Gaussian spatial and temporal envelope, sinusoidal carrier, and plane wavefront (parallel to the x, y plane, corresponding to the focal plane),
are injected, where \(E_x^\text {inj}\) is the x component of the injected electric field, \(H_y^\text {inj}\) the y component of the magnetic field, \(Z_0\) the wave impedance of the vacuum, \(E_0\) the peak field amplitude, \(\omega _0=2\pi \nu _0\) the fundamental angular frequency (\(\nu _0 = {300}{\text {GHz}}\)), \(w_\text {FWHM} = {850}{\mu {\text {m}}}\) the beam’s width at half maximum of intensity and \(\tau _\text {FWHM} = {14.1}{\text {ps}}\) the corresponding quantity in time [2]. The wave is generated at \(z = {40}{\mu {\text {m}}}\) in such a way that it propagates only in the positive z direction and that the source is additive (transparent) to avoid spurious reflections of waves returning to the injection plane [20]. The injected transverse field components form a linearly polarized TEM wave that is inhomogeneous and therefore cannot be a solution of Maxwell’s equations. The missing longitudinal components (\(E_z\), \(H_z\)) of the wave are consistently generated by the FDTD algorithm such that Maxwell’s equations are satisfied, because the FDTD algorithm implicitly solves Maxwell’s two scalar equations under the assumption of charge conservation [19]. Note that while \(E_y^\text {inj}=H_x^\text {inj}=0\), \(E_y\) and \(H_x\) develop to small but non-zero values during spatial propagation.
The total energy of a single pulse of the TEM wave can be calculated by integrating the Poynting vector over the focal plane and time [26]
which holds well for multi-cycle pulses (\(\omega _0 \tau _\text {FWHM} \gg 1\)). In the experiments [2], the repetition rate (50 kHz) is sufficiently low that one can neglect the excitation of the semiconductor from previous pulses and consider the sample at thermal equilibrium before each incident pulse, with all holes bound to their parent ions.
In the vacuum near the exit face of the Si layer at \(z_\text {eval}\) (see Fig. 1a), we evaluate the energy of the transmitted THz pulse in the frequency domain for positive frequencies (one-sided intensity spectrum)
The energy of the emitted pulse is obtained by integration over the positive frequencies.
2.2 1+1D FDTD
In the case of the 1D spatial (plane-wave) treatment, the nonzero field components are: \(E_x(z,t)\), \(H_y(z,t)\), and \(J_x(z,t)\) [20]. Only the grid in z direction is used with: \(E_k^{l+\frac{1}{2}}\), \(H_{k+\frac{1}{2}}^l\), and \(J_k^l\). The material parameters are mapped onto the 1D space grid in the same way as in the 3D case. For the sake of simplicity, the magnetic field is neglected in the Lorentz force for hole propagation [13], which we show below to have a negligible effect on the HG process. The injected wave is a linearly polarized homogeneous plane TEM wave described by Eq. (18), where the dependence on x and y is neglected (\(x=y=0\)). The fluence of the injected wave is given by
where the last equality holds, because the emitted wave is a plane homogeneous TEM wave by construction.
2.3 FBMC
The FBMC simulator is described in detail in Ref. [27]. The valence band structure is calculated by the non-local empirical pseudopotential method [28]. Scattering by ionized impurities and phonons is taken into account [29, 30]. We assume that the holes do not move significantly in space, because the velocity of the holes is orders of magnitude lower than the speed of light [13]. Assuming even a very high drift velocity of \(10^7\,\text {cm}\,\,\text {s}^{-1}\), a hole can move only \({0.1}{{\upmu }{\text {m}}}\) in \({1}\,{\text {ps}}\), which is a very small fraction of the wavelength and cell width. Due to scattering and the oscillating electric field, most holes will move much smaller distances. In practical terms, fixing the position of the holes at the center of the cells improves the stability of the combined FDTD and FBMC algorithms [31]. Since the simulation particles do not migrate between cells, their particle ensembles are independent, allowing the MC simulation to be parallelized on the level of the cells, although the cells are still coupled via the propagating electromagnetic field. These bulk-type MC simulations of non-moving particles correspond to a locally evaluated conductivity that is nonlinear in the electric field and nonlocal in time. On the other hand, neglecting the particle movement does not mean that the simulations could not, in principle, contain plasma oscillations. Since the continuity equation is implicitly contained in Maxwell’s equations, the current density induces a space-charge density, which is not explicitly calculated [19]. This is in contrast to the 1+1D FDTD simulations, which by construction do not contain plasma oscillations. As will be shown below, by comparison of the 3+1D and 1+1D cases, the impact of plasma oscillations is negligible. Furthermore, the impact of the total spurious space charge introduced in the FDTD simulation by the stochastic FBMC current density on the results shown in this paper is negligible, because this spurious space charge is more than six orders of magnitude smaller than the total hole charge. We have checked this by simulations with eight times more simulation particles, which reduces the stochastic noise of the FBMC simulation by about a factor of three and thus the spurious space charge, and we did not find any significant changes in the results.
The electric field leads to tunnel photoionization of the acceptors [8]. This is modeled by the rate given in Ref. [2]
where E is the magnitude of the local electric field and \(w_0\), \(\alpha \) and \(n_{lh}\) are model parameters depending on the dopant ionization potential and effective mass of the adjacent band into which ionization occurs [8], which for Si:B are given by \(w_0=244\,{\text {ps}^{-1}}\), \(\alpha = 253\,\,\text {kV cm}^{-1}\), and \(n_{lh}=0.566\) [2]. Based on this model, if we define a nominal threshold for significant photoionization as \(w=1\,{\text {ps}^{-1}}\), the corresponding threshold field is \(E=42\,\,\text {kV cm}^{-1}\).
At the beginning of each time step, this rate is evaluated, and the corresponding number of holes is generated for the time step, including depletion of the neutral acceptors. The new holes are randomly placed in the light-hole band (as the corresponding photoionization rate for heavy holes is much lower [8]) according to an isotropic Maxwell-Boltzmann velocity distribution for the ambient temperature \(T=4\) K [2], although the energy-width of this distribution is negligible compared to the subsequent displacement in k-space driven by the electric field.
As it is not clear before run-time, how many holes will be generated during a simulation run with a given peak field \(E_0\), the ratio of simulation super-particles to real holes is tuned manually, i.e., adjusted in such a way that the final number of simulation particles is about \(50 \cdot 10^6\) regardless of the electric field \(E_0\). Since the higher harmonics are solely generated by the photoionized holes, a constant number of simulation particles should result in a roughly constant relative stochastic noise level independent of the number of real holes. Thus, even in the case of a small absolute number of holes, the relative stochastic error does not become large. Roughly the same total number of simulation particles is used in the 1D and 3D cases.
3 Results
3.1 3+1D FDTD-FBMC
We inject waves (see Eq. (18)) with a fundamental frequency of \(\nu _0=0.3\text {THz}\), \(\tau _\text {FWHM}={14.1}{\text {ps}}\), \(t_0= 4\tau _\text {FWHM}\) and \(w_\text {FWHM} = {0.85}{\text {mm}}\) (Fig. 1) [2]. The duration of the FDTD simulation is \(2t_0\). The corresponding wavelength of the fundamental harmonic in vacuum is 1mm and in Si 0.292mm. The grid spacing is set to \(\Delta x = {4}{\mu \text {m}}\) to achieve sufficiently fine spatial resolution, as was checked by test simulations using \(\Delta x = {2}{\mu \text {m}}\) and \({8}{\mu \text {m}}\). This results in a time step \(\Delta t = {6.67}{\text {fs}}\). The relative permittivity of the Si layer is 11.7. The acceptor concentration is \(5\cdot 10^{16}\,\text {cm}^{-3}\), as used in the low-temperature experiments of [2]. The temperature is set to 4K, and at equilibrium, the acceptors are not ionized (freezeout).
For \(E_0=81\,\,\text {kV cm}^{-1}\), the wall-clock time of the simulation run is 10 h on a current computer with 64 cores, and about 470 GB of memory are used. About \(54 \cdot 10^{6}\) super-particles are simulated, where each simulation particle represents 25,000 real holes (128 super-acceptors per grid cell for this field). The MC simulation takes about 12% of the total CPU time and less than 2% of the memory.
In Fig. 1b, the x component of the transmitted electric field near the output face of the Si layer in vacuum is shown. The magnitude of the transmitted field is reduced due to reflection losses and self-induced absorption in the Si layer. While the former accounts for the reduced amplitude scale in the leading edge of the pulse, the threshold for tunnel ionization is reached approaching the peak of the pulse, leading to a strong suppression of the remaining pulse due to Drude absorption from generated holes [13]. Note that the y- and z-components of the electric field are negligible at the beam center due to symmetry.
In Fig. 2a, we plot the maximal magnitude of the electric field reached during the simulation in the x-z plane at the vertical center of the beam (\(y=0\)). One clearly sees the effects of wave reflection at both surfaces of the Si layer, which leads to constructive and destructive interference and a pattern similar to a partially standing wave.
Fig. 2
a Maximal magnitude of the electric field over time. b Final hole density in the Si layer on the x-z plane (\(y=0\)) for \(E_0=81\,\,\text {kV cm}^{-1}\)
Since the ionization rate near threshold depends exponentially on the electric field, the holes are essentially generated only where the electric field is strong (Fig. 2a). The final hole density is strongly inhomogeneous and large only close to the beam axis (Fig. 2b). The highest hole density is found near the entry face of the Si sample, where about 50% of the acceptors are ionized. At \(z={0.32}\) mm, the magnitude of the electric field does not exceed \(15\,\,\text {kV cm}^{-1}\) and the number of generated holes in this plane is negligible.
In Fig. 3, the maximal magnitude of the electric field over time is shown for \(E_0 = 81\,\,\text {kV cm}^{-1}\) in the vacuum near the output face of the Si layer, for the cases both with and without holes.
Fig. 3
Maximal value of the x component of the electric field over time at \(z_\text {eval}\) and \(y=0\) with and without holes in the Si layer
a Intensity spectra for \(E_0=81\,\,\text {kV cm}^{-1}\) at \(z_\text {eval}\) with and without holes. In the former case, with and without the magnetic field in the Lorentz force. b The corresponding energy at \(z_\text {eval}\) for the first-, third- and fifth-harmonic, the transmitted energy without holes, and the number of holes generated during the simulation
Maximal value of the magnitude of the x component of the electric field during the simulation at the beam axis in the Si layer by 1+1D and 3+1D FDTD-FBMC simulations and the corresponding hole density
Without generation of holes in the Si layer, the peak value of \(64\,\,\text {kV cm}^{-1}\) results from Fresnel losses, which are compensated to a degree by constructive interference between multiple reflections for waves at \(\nu _0\). As expected in this linear case, the beam retains a Gaussian profile. If we include the ionization of the acceptors in the simulation, holes are only generated near the beam center, and the wave is only absorbed in this region. Farther away from the beam center, the electric field profile is slightly broadened compared to the case without holes, due to the finite effects of diffraction induced by the lateral gradient of the hole concentration.
In Fig. 4a, the intensity spectra are shown both with and without including holes in the simulation. The peaks due to the generation of higher harmonics by the holes at odd multiples of the fundamental frequency are clearly visible up to the 9th order, due to the low noise of the MC simulation, which provides a dynamic range of about seven orders of magnitude. While we restrict the frequency range shown here to \(\nu \le 3\text {THz}\), where sufficient convergence is achieved for the \(\Delta x =4\mu {m}\), test simulations with a finer grid demonstrate that harmonics do arise at least up to the 11th/13th harmonics, following the roll-off shown in Fig. 4a).
Integration of the intensity spectrum over the positive frequencies yields the transmitted energy \(W_\text {tr}={0.37}{\mu \text {J}}\). This is 34% of the initially injected energy \(W_\text {inj}={1.07}{\mu \text {J}}\). Integration of the density of the dissipated electric power \({\textbf {E}}\cdot {\textbf {J}}\) over the Si layer and time yields the total energy absorbed by the holes which is subsequently transferred to the crystal by phonon scattering: \(W_\text {holes}={0.39}{\mu \text {J}}\) [32]. The rest of the injected energy is reflected by the Si layer. Without holes, 69% of the energy is transmitted, which is very close to the analytical result for plane harmonic TEM waves (see [13, 26]). In addition, in Fig. 4a, results are shown with and without the magnetic field in the Lorentz force of the MC simulations. The negligible difference between the two results confirms that the magnetic force can be neglected due to the low velocity of the holes compared to the speed of light.
The energy \(W_n\) for each harmonic is calculated by integrating the spectral intensity \(S_\text {tr}(\nu )\) within a frequency window of \(\pm 0.1\nu _0\) about each center frequency \(\nu _n=n\nu _0\). This energy is shown in Fig. 4b for \(n=1,3,5\) together with the transmitted energy for the case without holes. At low pump fields (\(E_0 \le 40\,\,\text {kV cm}^{-1}\)) the energy of the first harmonic is proportional to the injected energy \(W_1=0.69 W_\text {inj}\), as the density of holes and their absorption is negligible. At higher fields, the higher hole density leads to larger reflection and absorption, and the energy is sub-linear in the injected energy. Turning to the third-/fifth-harmonics (\(W_{3,5}\)), one sees that their energies follow closely a quadratic dependence on the total hole population, whose logarithmic scale in Fig. 4b) is chosen accordingly for comparison. Note that a sharp cut-off in \(W_{3,5}\) down to the noise floor occurs for \(E_0\,{\lesssim }\,30\,\,\text {kV cm}^{-1}\), as the photoionization rate becomes negligible. As discussed in Ref. [2], and exemplified here in Fig. 2, this threshold field is somewhat lower than expected compared to the nominal field transmitted into the Si layer, due to the higher fields reached at z-positions where constructive standing-wave interference occurs.
3.2 Comparison of 3+1D and 1+1D Simulations
In the 1+1D case, one assumes a plane-wave treatment to describe the on-axis behavior of the wave, neglecting the off-axis x- and y-dependence and diffraction effects. The number of simulation particles is similar to the 3+1D case. To assess the differences between both simulation modes, we compare results of the 3+1D and 1+1D simulations on the beam axis. In Fig. 5, the maximal magnitude of the x component of the electric field over the simulation time is shown for both cases in the Si layer and the corresponding hole density.
Notably, both results are very consistent, notwithstanding the larger resultant noise level in the 3+1D case. Although the total number of simulation particles is the same in both simulations, in the 3+1D case, the simulation particles are spread out in the x,y directions, and the number of super-particles in the computational cells on the beam axis is much smaller than in the 1+1D case. Otherwise, the on-axis behavior of the 3+1D simulation is very similar to the 1+1D case, despite the effects of diffraction in the former case. This result also shows that the impact of plasma oscillations, which can not occur in the 1+1D simulations, is negligible.
In Fig. 6a, we plot the intensity spectra where the 3+1D result is evaluated on the beam axis near the right surface of the Si layer. Besides the higher statistical noise in the 3+1D result, again, both results are very similar. These results clearly show that 1+1D simulations accurately describe the on-axis behavior.
Fig. 6
a Intensity spectra for \(E_0=81\,\,\text {kV cm}^{-1}\) at \(z_\text {eval}\) on the beam axis from 1+1D and 3+1D simulations. b Corresponding normalized energy (\(W_n/W_0\)) and fluence (\(F_n/F_0\)) for the harmonics \(n=1,3,5\) at \(z_\text {eval}\) vs pump field \(E_0\)
In Fig. 6b), we compare the pump-field dependence of the energy (3+1D) and fluence (1+1D) of the harmonics, by normalizing relative to the respective injected energy/fluence for \(E_0=81\,\,\text {kV cm}^{-1}\). In the 1+1D case, the saturation of the first-harmonic fluence is much stronger than the 3+1D energy, because the wave is mainly absorbed on the beam axis due to the higher relative number of holes. On the other hand, this is the reason why the relative third- and fifth-harmonic fluence in the 1+1D case is larger than the respective relative energies in the 3+1D case. The close correspondence, however, between the on-axis 3+1D and 1+1D fluence results — despite diffraction effects — raises the following two aspects: (i) It is possible that a reasonable estimate of the 3+1D results could also be obtained by applying 1+1D calculations with a set of input fundamental field strengths between zero and the on-axis value, and performing a weighted sum over the beam profile distribution to estimate the 3+1D pulse energies. This would facilitate nonlinear THz simulations of the full beam profile for studies where large-scale computational resources are not available. However, benchmarking against full 3+1D simulations should best be carried out. (ii) If one experimentally reimages the beam exiting the sample to a subsequent detection plane — ideally forming a replica of the lateral field distribution — and one measures there the on-axis field, then single 1+1D simulations would already provide a quantitative estimate of the detected experimental signal fields. However, as shown in the next section, with such tightly focused beams as considered here, which also develop sub-wavelength (evanescent) lateral structure, as well as using imaging optics with realistic numerical apertures, one must consider how the on-axis fields are modified by such reimaging, in particular for the longer-wavelength (e.g., fundamental) wave components.
3.3 Re-imaging of Emitted Fields to Subsequent Detection Plane
As discussed in the Introduction, while the foregoing simulations allow one to disentangle the microscopic carrier dynamics and propagation effects for the HG process, the confirmation of their validity — in particular, hinging on the semiclassical treatment of the tunnel photoionization and MC description of the dynamics — rests ultimately on comparison with experiments, as was performed in [2] vs. 1+1D simulations. While we found nearly quantitative agreement for the harmonic emission vs. incident pump field \(E_0\), one remaining discrepancy was the temporal form and \(E_0\)-dependence for the transmitted fundamental field component \(E_1\), where the predicted saturation due to Drude absorption from the photoionized holes was not evident in the experiments. At that time, we speculated that the spatial re-imaging of the THz beam at the detection plane might affect the experimental results. In the present study, with the full 3+1D transmitted fields, we are in a position to assess (and ultimately, confirm) this hypothesis, using rigorous diffraction integrals for the subsequent (linear) propagation after the sample.
Fig. 7
Subsequent propagation of spectral components of beam directly after sample (left panels) and traversing detection region (right panels). Filled curves correspond to beam intensity (normalized to peak value at each distance z), with wavefronts plotted as solid red curves. a Overview of full propagation path following the sample, including beam cross-sections for the fundamental wave (\(n=1\)) with low pump peak field \(E_0=30\,\,\text {kV cm}^{-1}\) (sample cryostat omitted for simplicity). b–d Fundamental wave (\(n=1\)) vs. pump peak field \(E_0\) (as indicated). e–h Odd-order harmonics (\(n=3\text{- }9\)) for the highest pump field, \(E_0=81\,\,\text {kV cm}^{-1}\)
A schematic of the experimental optical layout following the Si sample is shown in Fig. 7a, comprising two \(90^\circ \)-off-axis paraboloidal mirrors PM\(_m\), with aperture diameters \(2R_m\) and reflective focal lengths \(f'_m\), chosen according to the experiments in [2], which collimate and refocus the THz radiation, respectively. Here, typical components for THz time-domain-spectroscopy (TDS) setups are used, i.e., \(f'_1=101.6\) mm and \(f'_2=50.8\) mm (\(2R_{1,2}=50.8\) mm), to allow sufficient space about the sample (mounted in a cryostat) and high-numerical aperture focusing into the detector, respectively, while yielding the maximum beam diameter for the collimated beam between PM1 and PM2 for high-pass filters (omitted in Fig. 7 for clarity). Note that the diffraction-limited pump beam incident on the sample corresponds to focusing with a preceding PM with a higher numerical aperture than that of PM1, i.e., with a very large beam which fills the aperture of a PM with \(2R=101.6\) mm and \(f'=152.4\) mm, chosen to achieve a tight focus and high pump field strength. At the detector plane, the THz fields \(E_d(t)\) are detected coherently via co-propagating near-infrared optical pulses in an electro-optical crystal. As the optical beam diameter is significantly smaller than the THz wavelength, one effectively measures the on-axis fields at the focal plane.
In many such THz experiments, one then assumes that the detected field \(E_d\) is a close replica to that directly exiting the sample \(E_s\), i.e., one measures the on-axis field \(E_d=-(f'_1/f'_2)E_s\) (accounting for the relative magnification of the PM pair). Indeed, in the paraxial approximation, with only moderate beam divergence (numerical aperture), and assuming the PMs can be represented as thin lenses, this result can be readily derived [33]. However, in the present case, the paraxial approximation is no longer valid, due to both (i) the tight (diffraction-limited) focusing at the sample and detector, and (ii) the development of lateral spatial modulation on scales comparable with the THz wavelength \(\lambda \). The latter effect is particularly significant here due to the Drude absorption of holes generated by tunnel photoionization [2], which is highly nonlinear. Such beam structure comprises near-fields (i.e., with spatial frequencies \(k_{x,y}>k=2\pi /\lambda \)) which do not propagate to the detection plane. While one could simply apply an angular-spectrum filter to suppress these near-fields in predictions for the subsequently detected fields, non-paraxial effects are significant even for propagating fields with \(k_{x,y} \lesssim k\), and a full non-paraxial treatment of the vector fields should be used for reliable predictions [34].
Moreover, the imaging properties of the PMs must be dealt with rigorously [35, 36], as their reflective field transformations in the non-paraxial regime deviate from the commonly used thin-lens approximation, where the focusing element is reduced to a planar, radially symmetric mask with phase shift \(\varphi =k(x^2+y^2)/2f'\) [33]. Moreover, beam truncation due to the finite PM aperture also plays a role here (see below). To correctly model the vectorial field propagation via the curved PM surfaces, one must employ the more general Stratton-Chu (SC) diffraction integrals [35, 37]. Indeed, our first approaches to these propagation simulations, using the non-paraxial propagator between PMs — i.e., implemented with the vector-field angular spectrum method [34] but retaining the thin-lens treatment — led to qualitatively different results than the rigorous SC results presented here (to be discussed in a subsequent publication).
The SC diffraction integral yields the complex vector field amplitudes \({\textbf {E}}({\textbf {r}})\), \({\textbf {H}}({\textbf {r}})\) at each target point r due to propagation of fields originating from the points R on an arbitrary surface S (inward normal \({\textbf {n}}({\textbf {R}})\)) on which one has the “source” fields \({\textbf {E}}_0({\textbf {R}})\), \({\textbf {H}}_0({\textbf {R}})\), as per [35, 37],
and likewise for \({\textbf {H}}({\textbf {r}})\) by exchanging \({\textbf {E}}\rightarrow Z_{0} {\textbf {H}}\) and \(Z_{0} {\textbf {H}}\rightarrow -{\textbf {E}}\) in Eq. 25. Here \(G(s)=e^{iks}/s\), where \({\textbf {s}}={\textbf {r}}-{\textbf {R}}\), such that
$$\begin{aligned} \nabla G\equiv \nabla _{{\textbf {R}}} G = -\nabla _{{\textbf {r}}} G = (1-iks) \frac{G(s)}{s}\hat{{\textbf {s}}}, \end{aligned}$$
(26)
with \(\hat{{\textbf {s}}}={\textbf {s}}/s\). Here, we neglect the additional contribution from the contour integral around the perimeter of the PM aperture (i.e., we invoke the Kirchhoff approximation [38]), which we found to be negligible for the conditions here. One can show that for a planar source surface S, Eq. 25 reduces to the full vector-field form of the Rayleigh-Sommerfeld integral, which is also mathematically equivalent to the angular-spectrum method [39].
For propagating from the surface of a PM mirror, the input fields \({\textbf {E}}_0\), \({\textbf {H}}_0\) in Eq. 25 are the sum of both incident and reflected fields [35], i.e., \({\textbf {E}}_0={\textbf {E}}_i+{\textbf {E}}_r\), \({\textbf {H}}_0={\textbf {H}}_i+{\textbf {H}}_r\), where we assume ideal metallic reflection, i.e., \({\textbf {E}}_r=-{\textbf {E}}_i + 2({\textbf {n}} \cdot {\textbf {E}}_i ){\textbf {n}}\) and \({\textbf {H}}_r={\textbf {H}}_i - 2({\textbf {n}} \cdot {\textbf {H}}_i ){\textbf {n}}\), and hence \({\textbf {E}}_0=2({\textbf {n}} \cdot {\textbf {E}}_i){\textbf {n}}\), \({\textbf {H}}_0=2{\textbf {H}}_i - 2({\textbf {n}} \cdot {\textbf {H}}_i){\textbf {n}}\) [35]. Note that one then has \({\textbf {n}} \times {\textbf {E}}_0=0\) and \({\textbf {n}} \cdot {\textbf {H}}_0=0\), which reduces the number of terms in Eq. 25. The integrals are carried out numerically over the surface \({\textbf {R}}=[X,Y,Z(X,Y)]\) of each PM (with a square Cartesian grid [X, Y] on the base plane of each PM), using analytic expressions for \({\textbf {n}}({\textbf {R}})\). We also ensure the area elements \(\text {d} S\) preserve the analytic PM surface area \(\iint \text {d} S=A\), and hence maintain total beam energy. Sufficient convergence was found for uniform \(N\times N\) grids with only \(N=160\), i.e., \(\Delta X{\sim }\lambda _1/3\) for the unfocused beam. Hence, while such real-space diffraction integrals are inherently computationally expensive (\(\mathcal {O}(N^4)\)) compared to methods based on 2D fast Fourier transforms (FT, \(\mathcal {O}(N^2 \ln N)\)), in practice, one can use a smaller value of N for the real-space diffraction integrals (Eq. 25). (In any case, FT methods are also essentially limited to propagation from planar surfaces). Moreover, for calculating field cross-sections along a single transverse variable, the cost drops to \(\mathcal {O}(N^3)\).
The calculated field cross-sections for the fundamental beam \({\textbf {E}}_1\) throughout the propagation path (for a pump field \(E_0=30\,\,\text {kV cm}^{-1}\)) are depicted in Fig. 7a, shown in terms of the intensity of the horizontally polarized field component along the horizontal axis at the vertical center, \(|E_{1x} (x,{y=0})|^2\) (normalized to peak value at each value of z), and corresponding wavefronts. One notes that the wings of the fundamental beam are truncated at PM1, which gives rise to a complex modulation in the subsequent collimated beam due to diffraction effects. This situation arises as the numerical aperture for the focusing into the sample (with a larger beam and PM diameter, not shown) is larger than for the re-collimation. While this is generally to be avoided, in such non-linear experiments, one sets a priority to achieve the tightest diffraction-limited focus into the sample, whereas the diameter of the subsequent collimated beam is limited by the use of specialized high-pass spectral filters in the beam between PM1 and PM2, needed to reduce the strength of the fundamental component and improve contrast in the detected signal for the harmonics [2]. While this beam truncation does not cause any severe distortion for the refocused, detected fields, it must be taken into account for quantitative predictions. One also sees that the first PM gives rise to a lateral asymmetry in the collimated beam, due to the asymmetric parabolic profile which does not project constant solid angles of the incident beam onto the reflected beam — a result which can also be confirmed qualitatively considering the density of a diverging ray bundle incident on a PM. To compensate for this for the relative PM geometry [36] used here and in the experiments [2], we position PM2 with a small lateral shift (5 mm, discernible in Fig. 7a) so that the refocusing beam propagates more parallel to the optical axis.
Magnified views of the fundamental beam directly after the sample (\(E_{s1}\), left panels) and traversing the detection focal region (\(E_{d1}\), right panels) are shown in Fig. 7b–d for three values of the pump field, respectively. A comparison shows that the on-axis saturation (c) develops into a dip (d), due to the self-induced Drude absorption of photoionized holes, on a lateral spatial scale smaller than the wavelength, although the evanescent components decay rapidly. (Note that the apparent beam broadening with increasing \(E_0\) is a consequence of plotting the normalized intensities. Also, the absence of the dip in the results in the previous section, Fig. 5, is due to plotting the maximum temporal field there, including the higher harmonics.) In the detection region, however, the beam profiles are almost independent of \(E_0\), as the re-imaging is essentially dominated by the numerical aperture of PM1, which dictates the angular-spectrum components which can subsequently propagate.
Fig. 8
On-axis field strength at detection region vs. distance z for each odd-order harmonic \(n=\)1–9 for a pump field \(E_0=81\,\,\text {kV cm}^{-1}\)
The beam profiles for the generated odd-harmonics \(n=3\text{- }9\) are shown in Fig. 7e–h, respectively. A comparison with the fundamental shows that while the beams directly after the sample are progressively narrower with n (due to the higher nonlinear order vs. n [2]), this narrowing is slower than the decrease in wavelength, such that the divergence — and truncation effects on PM1 — are also smaller. At the detection region, one observes nearly Gaussian refocused beam profiles at the focal plane, although away from this plane one observes more complex beam profiles, which result mostly from the detailed imaging properties of the PMs [36] compared to radially symmetry lenses.
The corresponding on-axis fields vs. distance \(E_{dn}(z)\) are shown in Fig. 8, which approximately exhibit the dependence \(1/\sqrt{1+(z/z_{0n})^2}\) for Gaussian beams with \(z_{0n}\,{\propto }\,1/\nu _n\) in the case of focusing with a given numercial aperture [33]. All harmonics are predicted to focus nearly at a common plane, despite the somewhat complex variation of the beam profiles and wavefronts approaching this plane (Fig. 7b–h). Here, we mention again that other non-paraxial simulations we performed using the thin-lens approximation for PM1 and PM2 predicted a strong dependence of the focal plane position on n, i.e., artifacts that suggested experimentally one would detect a distorted proportionality between the harmonic fields depending on the chosen z position. This emphasizes the need to employ the more rigorous SC treatment here.
Fig. 9
Pump-field dependence of on-axis field for each odd-order harmonic \(n=\)1–9, both directly after the sample (\({E}_{\text {s}}\), dashed lines, left scale) and for the refocused beam at the detection plane (\({E}_{\text {d}}\), right scale, scaled by 2 relative to \(E_s\) to account for paraxial demagnification factor — see text). Note that the vertical scales are compressed in the upper range where \(E_1\) is plotted (i.e., starting at \(E_s=4\,\,\text {kV cm}^{-1}\)/\(E_d=8\,\,\text {kV cm}^{-1}\)). Linear field dependence (\(\eta =1\)) indicated by dotted lines. Inset shows previously reported experimental data for the refocused fundamental (in terms of the on-axis fluence \(F_1{\sim } E_{d1}^2\)) [2], where the predicted on-axis saturation at the sample is absent
Finally, in Fig. 9, we compare the absolute on-axis peak fields directly after the sample \(E_{sn}\) (dashed) and at the detection focal plane \(E_{dn}\) (solid) vs. \(E_0\) for all odd harmonics \(n=1\text{- }9\). As reported previously [2], the signal strengths for the generated odd-harmonics \(n=3\text{- }9\) first rise with an exponential dependence as one crosses the threshold pump field for tunnel photoionization (see Fig. 7 in Ref. [2] for data with a finer spacing of \(E_0\) values), followed by power-law growth and subsequent saturation. This ionization threshold occurs at somewhat lower values than that expected from the nominal pump field, due to the standing-wave effects in the sample (Figs. 2 and 5) which cause field enhancement in certain z-regions [2]. The plot uses separate vertical scales for \(E_s\) and \(E_d\) to account visually for the paraxial (de)magnification factor of \(f'_2/f'_1=0.5\) at the detection plane, which was used here based on the experimental reflective focal lengths for PM1 and PM2 in Ref. [2]. While one has \(E_{dn}/E_{sn}\,{\sim }\,2\) for the highest harmonics (\(n=5\text{- }9\)), \(E_{dn}\) falls progressively below this for \(n=3\) and \(n=1\). This is primarily due to the limited numerical aperture of PM1 and PM2, exacerbated for \(n=1\) by energy losses due to truncation at PM1.
The fundamental field, \(E_{s1}\) (blue dashed curve) shows a sub-linear dependence on the pump field \(E_0\), due to the self-induced Drude absorption from photoionized holes, even leading to a decrease for the highest pump field here \(E_0=81\,\,\text {kV cm}^{-1}\) (corresponding to the on-axis dip in Fig. 7d). The dependence of \(E_{d1}\), however, shows a significantly weaker saturation vs. \(E_0\). This is due to the fact that the on-axis suppression and eventual dip in \(E_{s1}\) directly after the sample have a strong evanescent contribution (i.e., non-propagating near-fields), while the non-paraxial propagation (and truncation effects) allow off-axis contributions in \(E_{s1}\) to contribute to the on-axis contributions to \(E_{d1}\). This provides an explanation for the qualitative discrepancy between experiment and 1+1D simulations previously reported in Ref. [2] (the key results are shown in the inset of Fig. 9), i.e., that the predicted saturation in \(E_{d1}(E_0)\) was not observed experimentally. This discrepancy could have led one to speculate that the photoionized hole densities (and their Drude absorption) were actually lower than in the simulations, and raise the question whether the model for the tunnel ionization rate [8] should be revised. At the time, we found the degree of agreement for the higher harmonics sufficient to support the simulations, which now stands on a strong footing given the results here.
4 Conclusions
We have demonstrated that full 3+1D simulations of the photoionization and harmonic generation driven by intense THz pulses in doped Si with a full-band Monte-Carlo parallel treatment of the local carrier dynamics incorporated in a FDTD propagation scheme are feasible, allowing predictions for the harmonics up to at least ninth-order with low stochastic noise. Remarkably, the results for the on-axis fields/hole dynamics are in close agreement with those from 1+1D (plane wave) simulations, despite the beam divergence effects expected for THz beams focused down to the diffraction limit. This suggests that the key aspects of such nonlinear experiments can be captured with the much lower computation load of 1+1D simulation (allowing also, e.g., larger MC ensembles to achieve lower noise levels). The results here do, however, emphasize the need to consider the lateral profile of the transmitted THz fields, as this can have a strong impact on the fields measured in experiments. Here, the fields emerging from the sample possess sub-wavelength lateral modulation (and hence contain a significant contribution from non-propagating evanescent waves), and also may be distorted by the subsequent propagation/re-imaging to the experimental detection plane.
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