1. INTRODUCTION

The manipulation of electrons and the subsequent dynamics in solids by strong light fields has been a longstanding ambition. The recent advent of strong few-cycle waveform-controlled femtosecond laser pulses [1–5] has renewed the interest in strong-field phenomena in solids [6], namely, the field-induced electron transition. Such examples are absorption [7–10] and transmission [11,12] of solid-state media, control of photoemission in nanostructures [13–16], generation of high-order harmonics [17–22], and the transient semimetallization of insulators on the femtosecond time scale [23–28]. The ultrafast semimetallization of insulators may turn out to be of far-reaching importance to facilitate faster signal processing as the current rate of information processing based on metal oxide semiconductors field-effect transistors (MOSFETs) is limited to 3 GHz due to inherent charging time [29,30]. Heat load from electrical interconnects is another important obstacle to overcome. Owing to such restrictions, contemporary MOSFET-based processors have hit the maximum clock rate at the dawn of the 21st century, and the rate of progress has since slowed down. Signal manipulation by optical means is a potential substitute free from these hindrances. Toward the development of working light-wave electronics devices for ultrafast electric signal manipulation, an abstract idea for a petahertz diode [31] and optical memory devices [32] has been proposed, providing a blueprint for the prospective future.

In view of manipulating a current in insulators through light, some of the important questions to be addressed are as follows: (1) Can a current be induced? (2) How large of a strong-field is needed? (3) Is the induced current controllable or switchable? (4) How does the crystal structure affect the behavior of the induced current? and (5) What role does the atomic constituent play? Recent experimental determinations of the generation and switching of electric currents by strong light fields in quartz, sapphire, calcium fluoride [23,24] and fused silica [27] have answered the first four questions by showing that the induction of currents in insulators by intense laser fields is a general phenomenon [24]. The universality is explained in terms of the Wannier–Stark localization, where at the field strength of a few volts per ångström (V/Å), the Wannier–Stark localization length becomes comparable to or even shorter than the typical lattice constants, and the trapped electrons do not see the long-range ordering. This leads to a similar trend of macroscopic observables among various materials with diverse crystal structures.

 

Fig. 1. (a) Schematic drawing of the experimental setup. A strong ultrashort optical pulse is directed on an ${\text{SiO}_2}$ surface, and the current induced in the gap between two electrodes is measured. (b) Sketch of a notched target with its corresponding crystalline axes.

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Previous works have been done with arbitrarily orientated crystals, implying that the effect of orientation might have been smeared out. Meanwhile, a theoretical investigation predicted the influence of crystal orientation on the phase of the induced current. Wachter et al. have recently discussed, using ab initio time-dependent density functional theory (TDDFT), that the distinguishable crystalline orientations are exhibited in the phases of light-field-driven currents [28]. Motivated by the work, a corresponding experiment with specified crystal axes was performed in order to verify that the orientations lead to a phase shift in the current.

2. EXPERIMENTAL METHOD

In the experimental campaign, single-crystalline quartz substrates with well-defined axes are used. A $\hat y$-cut quartz single crystal is irradiated by a carrier-envelope-phase (CEP, ${\phi _{\text{CEP}}}$)-controlled ${\sim}{4}\;\text{fs}$ visible-near infrared (VIS-NIR) pulse with the wavelength range spanning 450 to 950 nm [Fig. 1(a)]. The laser pulses are focused by an off-axis parabolic mirror, while the pulse energy is controlled by adjusting the opening of an aperture, therefore changing the focal spot size. To determine the focal diameter, the beam is picked off by a flipping wedge and sent to a beam profiler located equidistant to the quartz substrate. To characterize the pulse duration, a second-harmonic interferometric autocorrelator is employed. The measured pulse energy, pulse duration, and focal volume are used to estimate the laser intensity in the air. For direct comparison with simulations, the laser intensity measured in the air is converted into that in the medium by multiplying a factor of ${n}{({\frac{2}{{1 + \sqrt \varepsilon}}})^2}$, where ${n}$ is the refractive index and ${\varepsilon}$ is the dielectric constant of quartz. Note that henceforth the intensity denotes the intensity inside the medium in both experimental and theoretical contexts. The crystal orientation is identified by notches on the substrate [Fig. 1(b)]; the notched edge is parallel to the ${\hat a}$ axis ((1010) in Bravais–Miller index notation) and perpendicular to the ${\hat c}$ axis [(0001) in Bravais–Miller index notation]. The light field ${ \vec E}$ of a linearly polarized laser pulse is oriented parallel to either the ${\hat a}$ or the ${\hat c}$ direction. Two rectangular gold electrodes are put on substrates with a channel (${\sim}{100}\;\unicode{x00B5}\text{m}$) between them being laid perpendicular to the laser polarization. The electrodes are used to capture the charges by maximizing the projection area of charge flux onto the electrodes [23,27].

The irradiation from an intense laser focused onto the gap between the gold electrodes induces polarization inside the dielectric medium. Due to the induced polarization, the surface charges of opposite signs are built up at the boundaries of the dielectric and metallic leads. A portion of the charges is captured by the gold pads, resulting in charge separation between the electrodes. The separated charges yield a current flowing through an external circuit. The current is first directed to a transimpedance amplifier and converted into a voltage with a gain of ${{10}^8}\;\text{V}$ per ampere. Then the amplified voltage signal is sent to a lock-in amplifier, composing an ammeter circuit. As we use a pulse train with a controlled pulse-to-pulse CEP slip of ${\Delta}{\phi _{\text{CEP}}} = {\pi} \text{rad}$ rad, this configuration enables the selective detection of the CEP-dependent component of current in conjunction with a lock-in amplifier referenced at the CEP modulation rate. The magnitude of the output of the lock-in amplifier is on the order of millivolts. Considering the current-to-voltage conversion gain and the repetition rate of the laser, the charge transferred per pulse amounts to a few femtocoulombs (fC).

3. EXPERIMENTAL FINDINGS

At an intensity of ${\sim}{{10}^{13}}\;\text{W}/{\text{cm}^2}$, the CEP is varied by adjusting the transmission length through a glass wedge (Fig. 1(a); Ref. [33]), and a sizeable CEP-dependent current is detected. The physical origin of the measured current is still in debate. The first demonstration of the optical-field-induced current [27] was construed in view of strong-field physics [34–36], where the alteration of the band structure owing to the light field triggers the formation of localized Wannier–Stark states, which prominently enhances the probability of Zener-like tunneling. It generates virtual electron–hole pairs, making a reversible semimetallization [37]. Wachter et al. [28] have done ab initio calculations in TDDFT, covering both multiphoton and tunneling processes and interpreted the observation in view of the field-induced insulator-to-metal transition. On the other hand, a simpler model based on nonlinear optics [38] has been suggested in which virtual carriers (electron–hole pairs) are injected in the conduction and valence bands through odd-order multiphoton excitations and quantum interference among them. All the mechanisms suggest that the generation of the virtual carrier population in the bands eventually accounts for currents in the external circuit. The laser-induced current oscillates at the same period as that of the light field [Fig. 2(a)], imitating the previous works [23,24,27]. The periodicity of ${2}\pi$ in the induced current corraborates the direct control of the instantaneous light field on the current. In addition, the replacement of ${\sim}{4}\;\text{fs}$ (${\lt}{2}$ optical cycles) pulses with ${\sim}{30}\;\text{fs}$ (${\gt}{10}$ cycles) pulses makes the net current nearly zero [black squares in Fig. 2(a)], providing another piece of evidence of the direct impact of the instantaneous light field on the measured current. For multicycle pulses, as the field amplitude of positive and negative half-cycles are less contrasted than few-cycle pulses, the net current almost vanishes.

 

Fig. 2. (a) Five independent measurements of transferred charge ${Q}$ with varying ${\phi _{\text{CEP}}}$ for two crystal orientations at an intensity of $({4.08}\;{\pm}\;{0.14}) \times {{10}^{13}}\;\text{W}/{\text{cm}^2}$. Filled blue symbols are for $\vec E \Vert \hat a$ and open red ones for $\vec E \Vert \hat c$. Different shapes of the symbols represent different runs of repeated measurements; horizontal error bars: $\pm$ standard deviation of the phase slip of an individual curve against the averaged curve over 30 repetitions; vertical error bars: $\pm$ standard deviation of amplitude over 30 repetitions; black squares are for multicycle (${\sim}{30}\;\text{fs}$) pulses. (b) Representative sinusoidal curves derived from the sets in (a); all curves in (a) are fitted to a sine function, and the resultant fitting parameters (phase shift and frequency) are averaged to produce the representative sinusoids. Horizontal error bars, standard deviation of the fitted phase; vertical error bars are not relevant because of normalization. The phase shift ${\Delta}{\phi _{\text{a} - \text{c}}}$ of $\vec E \Vert \hat a$ relative to $\vec E \Vert \hat c$ is $({0.45}\;{\pm}\;{0.09})\pi \;\text{rad}$.

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The CEP-dependent currents for different orientations of the crystal axes are obtained by switching specimens with either an ${\hat a}$ or a ${\hat c}$ axis aligned parallel to laser polarization [Fig. 1(a)]. For a meaningful phase detection, utmost care is paid during the measurement to minimize possible errors such as phase drift with time, the position of the wedge, and focal position relative to the samples. To ensure that the samples are accurately positioned and swappable, ${\vec E \Vert \hat c}$ and ${\vec E \Vert \hat a}$ substrates are attached side by side on a compact printed circuit board (PCB) and mounted on an ${ XYZ}$ stage. The relative position of the focus with respect to the electrodes is monitored by a telezoom microscope, permitting a coarse overlap of the focus onto the electrodes. If a measurable current is detected, transverse ($XY$) and longitudinal ($Z$) micrometers are finely tuned iteratively to converge the sample position to the optimum at which the signal is maximized. With the described procedure, the focal position on the $XY$ plane is reproducible within a fraction of the focal spot diameter (${\sim}{50}\;\unicode{x00B5}\text{m}$). The position inaccuracy in the $z$ direction coupled with substrate swapping motion (made on the $XY$ plane) is negligible compared to the Rayleigh length ${z_R}$, which is ${\sim}\;{2.5}\;\text{mm}$ in the current case. Multiple samples independently fabricated by the same recipe are used, preventing a specific sample-oriented artifact.

For the sake of statistics, experiments have been repeated numerously. At a given intensity (exemplary data taken at $({4.08}\;{\pm}\;{0.14}) \times {{10}^{13}}\;\text{W}/{\text{cm}^2}$ is displayed in Fig. 2) and orientation ($\vec E \Vert \hat a$, for instance), the current as a function of ${\phi _{\text{CEP}}}$ is measured 30 times and then averaged [blue solid symbols in Fig. 2(a)]. It is repeated for the other orientation $\vec E \Vert \hat c$ with the intensity kept constant [red hollow symbols in Fig. 2(a)] to make up a pair of the current signal, which oscillates against ${\phi _{\text{CEP}}}$. The paired measurement is repeated 5 times to acquire five pairs of sine-like curves [distinguished by the shape of symbols in Fig. 2(a)]. By fitting each curve to a sine function, a set of fitting parameters (amplitude, frequency, phase offset, and $y$ offset) is obtained. The amplitude of the signal varies from 0.5 to 2.0 fC, which may be a consequence of intrinsic microscopic differences among samples that occurred during production. The scattering is comparable to that reported in Ref. [24], which was conducted using the same scheme as the current work. In Fig. 3 a of Ref. [24], the data points in the low-field regime ($\delta \; \lt \;{0.7}$) were widely spread from nearly zero up to ${\sim}{2}\;\text{fC}$, elucidating the scattering of the present result. Even in this situation, we note that there is the phase shift between ${\vec E} \Vert \hat a$ and ${\vec E} \Vert \hat c$. Hereafter, to see clearly the change in the phase, the amplitudes are normalized to unity and $y$ offsets are ignored. Representative sinusoids are shown in Fig. 2(b), according to the averaged frequency and phase shift for each orientation. We note the phase shift, $\Delta \phi_{a-c}=\phi (\hat a)-\phi(\hat c)$, the relative phase $\vec E \Vert \hat a$ of with respect to $\vec E \Vert \hat c$ curve.

 

Fig. 3. (a) Graphical presentation of the formation of the tunneling transition channel for different intensities. Blue lines denote Si–O–Si bond. For simplicity, oxygen atoms are hidden. The left and right symmetry indicate the inversion symmetry. Blue ellipse colored by gradation marks the distribution of current density elongated by the field. (b) The phase shift ${\Delta}{\phi _{\text{a} - \text{c}}}$ between different crystal orientations with respect to the laser intensity. Horizontal error bars stand for the fluctuation of laser parameters during the measurement. Vertical error bars result from averaging over repetitions.

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For the given intensity of $({4.08}\;{\pm}\;{0.14}) \times {{10}^{13}}\;\text{W}/{\text{cm}^2}$, ${\Delta}{\phi _{\text{a} - \text{c}}} = ({0.45 \pm 0.09}){\pi}$ rad is pronounced to dominate all the possible phase uncertainties, demonstrating a clear orientation-dependence of light-driven currents in quartz. For an estimated $z$ position error of 0.3 mm, the corresponding Gouy phase ${\psi}({z})$ is only ${\sim}{0.04}\pi \;\text{rad}$, at most, which is far smaller than the observed phase shift of $({0.45}\;{\pm}\;{0.09})\pi \;\text{rad}$. If the case where ${\psi}({z})$ is to be ${0.2}\;\pi$ rad is used as an example, which would considerably change the measured phase shift, the corresponding $z$ is as large as 1.8 mm, much larger than the precision of the alignment procedure. In other words, the Gouy phase effect is almost avoided in the present work by carefully positioning the sample at the focus. It also matches the intensity-dependent CEP shift reported in Ref. [26], the phase fluctuation of $\pm 0.1 \pi \text{rad}$ in the range of ${1} \times {{10}^{13}}$ to ${7} \times {{10}^{13}}\;\text{W}/{\text{cm}^2}$. The value found in Ref. [28] of a phase shift of $\frac{{\pi}}{4}$ rad at ${5} \times {{10}^{13}}\;\text{W}/{\text{cm}^2}$ is qualitatively reproduced in the experiment under a comparable intensity.

In the same way, ${\Delta}{\phi _{\text{a} - \text{c}}}$ are measured for varying intensities $F$ from ${3} \times {{10}^{13}}$ to ${6} \times {{10}^{13}}\;\text{W}/{\text{cm}^2}$ [Fig. 3(b)]. The upper and lower limits of the intensity scope are imposed for practical reasons as the signal became comparable to the noise for $F \lt {3} \times {{10}^{13}}\;\text{W}/{\text{cm}^2}$, and the specimen is at a risk of irreversible dielectric breakdown for $F\; \gt \;{6} \times {{10}^{13}}\;\text{W}/{\text{cm}^2}$. For $F\; \lt \;{4} \times {{10}^{13}}\;\text{W}/{\text{cm}^2}$, the phase difference is not noticeable (${\Delta}{\phi _{\text{a} - \text{c}}} \approx 0$). As the intensity grows to the critical value around $F\;\sim\;{4} \times {{10}^{13}}\;\text{W}/{\text{cm}^2}$, a meaningful phase difference ${\Delta}{\phi _{\text{a} - \text{c}}}\sim 0.5 \pi \,\,\text{rad}$ is observed. For even higher intensities, $F\; \gt \;{4} \times {{10}^{13\:}}\text{W}/{\text{cm}^2}$, the phase shift is reduced to ${\Delta}{\phi _{\text{a} - \text{c}}}\sim 0.1 \pi \,\,\text{rad}$, equivalent to zero within the error bar.

 

Fig. 4. Unit cell of $\alpha$-quartz (${\text{SiO}_2}$) projected onto $\hat {a}-\hat c$ plane. Blue (red) circles denote Si (oxygen) atoms. Some oxygen atoms are hidden to avoid visual confusion. Si atoms in the unit cell are identified by numbers 1 to 6. The lattice constants (a = 4.9 Å, c = 5.4 Å) and O–Si bond length (1.6 Å) are specified.

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4. CORRELATION WITH LATTICE STRUCTURE

The interpretation of the dependence of ${\Delta}{\phi _{\text{a} - \text{c}}}$ on the laser intensity is proposed below. In the absence of an external field, the charge distribution of an atom in a crystalline lattice surrounds the atom. When the lattice is under the influence of an external electric field, the charge distribution gets elongated along the field. If the field is weak, the charge distributions from two neighboring atoms [oxygen–silicon (Si) bond in the case of quartz] do not overlap but behave like localized atoms. In this case, the negligible interplay of adjacent atoms renders the medium isotropic, hence ${\Delta}{\phi _{\text{a} - \text{c}}} \approx 0$ [left column of Fig. 3(a)]. As the field strength approaches the critical value, the charge distribution is stretched further and tilted towards the bond so that they begin to overlap at interstitial space, creating a charge transfer channel [39]. It is referred to as a transition to the tunneling regime, under which the induced polarization density undergoes a phase shift with respect to the incident field [28]. Such formation of the channel occurs more easily for $\vec E \Vert \hat a$ owing to the arrangement of the atomic chain (Fig. 4). For $\vec E \Vert \hat c$, the chain consists of two identical helices side by side, where three Si–O–Si bonds make up a full revolution (1-2-3-4 and 1-5-6-4, according to the numbered Si atoms in Fig. 4). For $\vec E \Vert \hat a$, the chain is composed of two different helices; one with two Si–O–Si bonds (6-4-3) and the other with four Si–O–Si bonds (6-5-1-2-3) per pitch. Since the composite tunneling efficiency of two parallel helical chains is strongly dependent on the segment with the highest individual tunneling probability, the strong field effect is more favorable for $\vec E \Vert \hat a$, which contains the shortest chain (6-4-3). Moreover, by deducing the helix angle from the lattice constants and the number of bonds per pitch, $\vec E \Vert \hat a$ chains are oriented more parallel to the field direction than is $\vec E \Vert \hat c$ chain. It allows the field to better spread the current densities, warranting stronger superposition at interstitial space. Therefore, at the critical intensity, the tunneling transition onsets for $\vec E \Vert \hat a$, while the state along $\vec E \Vert \hat c$ still stays dominantly in the multiphoton regime, inducing nonzero ${\Delta}{\phi _{\text{a} - \text{c}}}$ [central column of Fig. 3(a)]. These geometrical properties of the lattice are reckoned to be most influential above all other contributions, including other possible paths and mutual interaction among them. For intensities above the threshold, as the neighboring current density distributions substantially overlap, the tunneling transition sets in also for $\vec E \Vert \hat c$. Eventually, currents from both orientations are subjected to about the same phase shift relative to the light field, leading to effectively zero ${\Delta}{\phi _{\text{a} - \text{c}}}$ within the precision of our experiment [right column of Fig. 3(a)].

5. CONCLUSION

In conclusion, we have experimentally demonstrated that the phase of light-field-induced currents is sensitive to the anisotropy of quartz crystal, confirming the theoretical study [28]. A phase shift of the induced electric current between the laser polarization along the $\hat c$ and $\hat a$ axes of crystalline quartz is evaluated for various intensities. At the intensity of $({4.08}\;{\pm}\;{0.14}) \times {{10}^{13}}\;\text{W}/{\text{cm}^2}$, where the transition to a strong-field region takes place, the largest phase difference ${\Delta}{\phi _{\text{a} - \text{c}}} = ({0.45 \pm 0.09}){\pi}$ is measured. The intensity at which ${\Delta}{\phi _{\text{a} - \text{c}}}$ is greatest is in accord with simulations [40]; this agreement confirms that the strong-field phenomenon associated with the anisotropy of quartz crystal is responsible for the observed phase shift. This work extends the understanding of the response of solids under a strong field at a level of the atomic lattice.