Attosecond optoelectronic field measurement in solids

The sub-cycle interaction of light and matter is one of the key frontiers of inquiry made accessible by attosecond science. Here, we show that when light excites a pair of charge carriers inside of a solid, the transition probability is strongly localized to instants slightly after the extrema of the electric field. The extreme temporal localization is utilized in a simple electronic circuit to record the waveforms of infrared to ultraviolet light fields. This form of petahertz-bandwidth field metrology gives access to both the modulated transition probability and its temporal offset from the laser field, providing sub-fs temporal precision in reconstructing the sub-cycle electronic response of a solid state structure.


Supplementary Note 1 -Signal processing
The raw waveform data are processed with a bandpass filter of supergaussian form in the frequency domain, The parameters of the filters applied in the relevant figures are listed in Supplementary Table 1.

Supplementary Note 2 -EOS detection and data processing
The detailed measurement of the apparent time delay of the G(t) function of NPS is performed using electro-optic sampling (EOS) of a 1.8-µm field. While in principle, attosecond streaking would also be sensitive to such an offset, it relies on a different pair of pulses at very different photon energies, and would thus not permit a direct comparison of the gating in this case.
The EOS detection system is quite similar to that described in. 1 The EOS crystal used is a type II BBO (such that the probe pulse and detected field have orthogonal polarization), with a phase matching angle (θ) of 50.0 degrees, and thickness of 12.4 µm, as measured with a white light interferometer. The detector response function is calculated using the known optical sampling pulse, using the nonlinear wave equation 1 which describes the evolution of each spectral component of the light E ω (z) as it propagates in the crystal, with wavevector k(ω), refractive index n(ω) and corresponding spectral component (at frequency ω) of the nonlinear polarization P (N L) ω (z) under the slowly evolving wave approximation.
The nonlinear polarization is calculated using the full tensor of the second order nonlinear response for the light on both axes of the crystal, and the third order response using only the self-phase modulation (SPM) and cross-phase modulation terms (XPM), taking its tensor nature into account. 2 Under the experimental conditions, the intensity in the EOS crystal is not sufficient for a significant contribution to the signal from SPM or XPM, which produce a maximal timing shift of 25 attoseconds according to the simulations of nonlinear propagation.
The response function G(t) for the EOS process is obtained by numerically propagating the measured sampling pulse, and a broadband infrared test waveform through the crystal under varying time delays, analyzing the polarization state of the resulting probe light and calculating the resulting signal on the balanced detector based on the known response curve of the photodiodes and transmission of the spectral filter. G(t) is then obtained by inverse Fourier transformation of the ratio of the complex spectrum of the simulated signal to the complex spectrum of the known test waveform, with a bandpass filter applied to include only the near infrared spectral region. This calculated G(t) is then deconvolved from the measured waveform to correct the spectral phases and amplitudes of the EOS signal to produce the electric field at the input facet of the nonlinear medium.

Supplementary Note 3 -Influence of screening and momentum relaxation
As bulk dielectrics have relatively large dielectric constants and the free-carrier densities injected by intense laser fields are significant (on the order of 10 18 cm −3 and above), the form of the electric field incident on the material and the field inside the material will substantially differ from each other.
Each component of the field inside the material may be approximated by where E trans is the field inside the surface of the material transmitted through the surface, E inc is the incident field, and P is the polarization induced by E trans . (This expression simply yields the Fresnel equations for P = χE trans under the boundary condition E trans = E refl + E inc for the reflected field E refl .) We avoid numerical instabilities that appear when P ≈ −4 0 E inc by approximating the RHS of supplementary equation (3) with the first two terms of its Taylor series expansion in P. While the polarization resulting from the linear response of the material can be simply obtained by convolution with a response function χ(t), the polarization due to the laser-induced charge carriers both depends on and influences the transmitted field and the two must be solved self-consistently.
We use the classical equation of motion for the current of free carriers J e (t) driven by the local The field present in the sample in our case is is the free carrier density, m * is the reduced effective mass of an electron-hole pair and γ describes the rate of momentum relaxation. For simplicity, the carrier injection rate w(t) is approximated as w(t) = 10 9 E 8 i (t) a.u. in the following analysis, consistent with the energy transfer dynamics observed in the time dependent density functional theory (TDDFT) calculations discussed in the manuscript. The total current is given by J = J e + J b where J b is the response of the bound electrons, including the linear refractive index and the nonlinear absorption response. The induced polarization follows as The signal from the electrodes will then be proportional to the DC component of P(t).
The presence of a small density of free carriers in a dielectric leads to a reduction in its refractive index, as the polarizations induced by the free and bound carriers oppose one another, reducing the overall polarizability of the material. As this happens in a time-dependent manner during the course of the light-matter interaction, it shifts the relative weight of the portions of the electric field that come before and after the charge injection, which in turn influences the apparent relative timing of the field with respect to the injection of charge carriers. As the screening polarization is sensitive to both To look more specifically at the effect of momentum relaxation on the waveform retrieved through the measurement, the same simulation is performed with fixed electric field strength (0.5 VÅ −1 ) and varying relaxation time in Supplementary Figure 3. The form of the signal waveform is almost completely independent of momentum relaxation. Overall, because the net force on the carriers is always the integral of the electric field after they are injected and the momentum relaxation presents a linear opposing force, the dependence on this parameter is primarily a simple rescaling of the magnitude of the current. This apparent lack of influence over the shape of the waveform is beneficial for field sampling applications, as the true relaxation dynamics are not exactly known. This weak dependence on momentum relaxation time is in contrast to the dependence on the rate of recombination-when it is shorter than the period of the electromagnetic field, the signal is proportional to the electric field, rather than the vector potential. However, this timescale is significantly longer than momentum relaxation, typically longer than picoseconds, and observable via photoemission experiments.
Finally, since the carrier density affects the delay between the vector potential and the sampled waveform, it is important to take its spatial dependence into account when comparing simulations calculated for a single field strength vs the experiment by spatial averaging. This is done by assuming a Gaussian focal spot and performing a weighted average over the simulated intensities. As a result, the overall delay is reduced vs. the single-intensity calculations as shown in Supplementary Figure   4.

Supplementary Note 4 -Time-dependent density functional theory
We have performed ab-initio simulations of energy transfer into the bulk dielectric in order to test and to gauge simplified phenomenological models for the energy that allow to identify different orders where W (t) is the work done on the system, E 0 is the peak field strength, a(t) describes the energy exchange due to the linear response of the system, b(t) describes the energy exchange due to the third-order nonlinear response, and c(t) describes the energy exchange due to the higher-order nonlinear response. We find very good agreement with all data sets for n = 8. The extracted energy transfer after subtracting the transient contributions originating from the first-and third-order responses agrees well with a rate of energy deposition proportional to E 8 (t), i.e. a highly non-linear function of the time-dependent field, but with a small delay of ≈ 75 as.
The identification of the microscopic mechanism for this time delay predicted by TDDFT for strongfield dynamics in condensed matter is still an open question. We note, however, that for strong-field ionization in argon similar time delays have been experimentally observed in the above-threshold ion-ization (ATI) regime. 6 Also, recent ab initio simulations of strong-field ionization of helium indicate that two-photon ionization is considerably delayed relative to one-photon ionization when the same final energy in the continuum is reached. This suggests time delays when high-order photon absorption, as possible in the present case, are involved.

Supplementary Note 5 -Spectral response
The spectral response of the method may be obtained from the results of the TDDFT simulations, simply by Fourier transformation of the energy transfer after the removal of the lower-order terms. As can be seen in Supplementary Figure 7, the cos pulse leads to a relatively flat phase response up to the cut-off frequency of 1200 THz.
The structure of the spectral response can be well understood within the ansatz employed for the The measured signal is the convolution of this function with the timeintegrated electric field. In the spectral domain,G(ω) andÃ(ω) are multiplied. Distortions that arise from a finite duration of G(t) can be understood easily in the frequency domain, and, to a degree, corrected if the injection field is known by dividingS(ω) by the calculatedG(ω) (i.e. deconvolution), with appropriate band-pass filtering to avoid the strong increase of noise in areas whereG(ω) approaches zero.
In the limit of extremely short laser pulses, this approximate spectral response of the measurement approaches a constant value,G(ω) = const. For more readily obtainable laser pulses of few-cycle duration, one cannot generally assume that the injection is always confined to a single half cycle. The simple approximation of G(t) makes it possible to derive the spectral response and pulse duration requirements of the NPS measurement when the injection pulse has a known shape.
For an injection field described by E i (t) = Re[F i (t) exp(−iω L t + iφ CE )] and a gate described by is the Fourier transform of the complex envelope raised to the appropriate power, and C 2n n−k are the binomial coefficients. The spectral response is a series of replicas of the spectrum of the self-gated envelope, repeated with central frequencies ω = 0, ω = 2ω L , ω = 4ω L , and so on, up to a maximum determined by 2nω L , weighted by the binomial coefficients.
Only for sufficiently broadband pulses will the spectra of the replicas at ω = 0 and ω = 2ω L overlap, allowing for gapless sensitivity for all frequencies in the observed range, and only for φ CE = N π (for integer N ) will they have maximal constructive interference. For φ CE = N π + π/2, they will interfere destructively, producing the minimum observed in the measurements (Figure 3) near the carrier frequency ω L . This is also the reason for the π jump in the spectral phase of the response observed in the TDDFT simulations with a sin-like pulse in Supplementary Figure 7.
The response in the spectral range near ω = ω L is strongly dependent on having an extremely short injection field, and will rapidly drop in amplitude relative to the rest of the detected spectrum as the pulse duration is increased. By assuming a Gaussian pulse shape, explicit limits on the duration of the pulse can be imposed. WhenF The loss of spectral amplitude near the carrier frequency when φ CE = 0 can be parameterized simply through the ratio between the amplitudes at ω L and 2ω L , the relative depth of the minimum in the spectrum:G where the approximation has been made that only the terms of the sum for k = 0 and k = 1 contribute (i.e. the replicas of the pulse spectrum centered near ω = 0 and ω = 2ω L ). This amplitude will fall by 1/e for This value can be viewed as a guideline for the maximum applicable pulse duration, assuming that the pulse's complex envelope can be measured accurately via FROG or similar techniques to correct the spectral weights, beyond which the deep hole in the spectrum could be difficult to correct reliably or with adequate signal-to-noise ratio. For example, for n = 4, this corresponds to an intensity full-width-at-half-maximum duration of 3.8 fs at 800 nm carrier wavelength.

Supplementary Note 6 -Harmonic distortions and signal-to-noise ratio
At very low strengths of the driving field, the signal-to-noise ratio (SNR) of the measurement increases as the drive field intensity increases, since the main source of noise is electronic background in the measurement. Above a certain limit, however, the total noise in the measurement will be dominated by the noise contained in the signal itself. At this point, increasing the strength of the driving field does not significantly increase the signal-to-noise ratio of the measurement, at least in the temporal region close to the field maximum. Increasing SNR at this value is dependent on improving the pulse-to-pulse stability of the driving laser system.
At yet higher values of E d , such that the assumption that E d E i is no longer valid, E d will significantly participate in the carrier injection process, resulting in distortions of the measured signal first visible as harmonic distortions. Both of these properties of the measurement are most easily observed through the recorded spectrum over a range of strengths of E d , as presented for two different driving wavelengths (1.8 µm and 750 nm, which provide the largest dynamic range and highest available intensity, respectively) in Supplementary Figure 8 for an injection field strength of 1.7 VÅ −1 .
The onset of these distortions in this set of measurements was when E d approached approximately It can be seen that the measurement enters the range of signal-fluctuation-limited SNR near 0.3 VÅ −1 . In Supplementary Figure 8(b), the "signal" line (208 THz) and "noise" lines (all other frequencies outside the signal bandwidth) increase with approximately the same slope above this value. The SNR is plotted in Supplementary Figure 9, as the ratio between the signal line and average of the noise lines -although it is subject to significant scatter due to the noise term being in the denominator. In the low-driving-field range, the SNR increases with field, and at the higher-field range reaches a plateau.
The combination of a limiting field strength above which harmonic distortions occur, and decreasing benefit in terms of signal-to-noise ratio define an optimal working range in terms of the peak driving field strength. Under our experimental conditions, this was from ≈ 0.2-0.8 VÅ −1 .

Supplementary Note 7 -Comparison of measured spectra
The spectra obtained through Fourier transformation of the waveform data presented in Figure 5 of the main text are shown in Supplementary Figure 10 for those wavelengths for which a grating spectrometer were available. The spectrometers used were an OceanOptics NIR512 for the 1.8-µm pulse and OceanOptics Maya for the three other traces. The spectra are in approximate agreement, although differences may arise from differences in how the measurements take place: the NPS spectra are sampled from a small region of the focused beam, while the spectrometer averages over the spectrum of the whole beam. Thus, spatio-spectral distortions that cause the spectra of different locations in the focus to differ will result in a disagreement between the two measurements.
One may note several features in the spectrum of the driving laser pulse in Supplementary Figure   10(b). The NPS measurement reproduces (within the limits of the spectral resolution) the structure of the spectrum, with the exception of the region above 600 THz (500 nm). This is the limit of the working range of the chirped mirrors used to compress the light. Above this frequency, the reflectivity does not drop to zero, but the phase of the light varies strongly. As a result, the limited range of time delays in the attosecond streaking and NPS measurements does not fully account for the power in this region, as a significant portion of the energy is outside of the measurement window, with the in-window energy being dependent on the higher-order spectral phase.

Supplementary Note 8 -NPS and EOS comparison
The waveforms shown in Figure 4 are zoomed into the maximum of the envelope of the electric field, where the most reliable timing information exists. Supplementary Figure 11 shows a wider view of the pulses measured by the two detection techniques. The agreement between the waveforms is quantitatively better than the comparison with attosecond streaking, but there are still differences visible in the traces at the rising and falling edges of the pulse. This is due to the slightly different effective bandwidths of the measurements, which, in the case of a compressed pulse, lead to differences in the waveform proportional to the gradient of its envelope. For this reason, the timing comparison is done using the cycle of the waveform at the maximum of the envelope, where the time derivative is minimized.

Supplementary Note 9 -Detection limits
It is useful to know how much energy must be contained in a pulse before NPS can detect it. We show in Supplementary Figure 12 that even pulses on the nJ energy scale are detectable with NPS (injection pulse energy 7 µJ), through the observation of the waveform of a near-infrared (1400 nm) pulse generated with difference frequency generation, containing 1.4 nJ, attenuated with a pair of wiregrid polarizers. We compare with a pulse with 1000 times this intensity containing 1.9 µJ.

Supplementary Note 10 -Detection and modulation methods
In the measurement, a lock-in amplifier is used to detect the signal induced by the driving field. This requires the modulation at a fixed frequency. We can employ two methods of modulation for some  f 0 (THz) σ (THz)  2  NPS  600  273  4  EOS and NPS  170  38  5  275-330 nm  960  130  5  375-440 nm  560  170  5  460-1035 nm  460  130  5  1050-2850 nm  195  100  5 3000-5300 nm 80 40 Supplementary Table 1: Bandpass filters used in the processing of raw data. These are the parameters used in supplementary equation 1 for the corresponding figures in the main text.   driving field its intensity is increased. No harmonic distortions appear above the noise floor, but the average noise floor rises, indicating a significant contribution from signal fluctuations. b Lineouts of part a corresponding to the peak frequency of the driving field, its second and third harmonics, and the frequency equidistant from these harmonics. c Spectrum of the signal with 750 nm driving field in the high intensity range. Once the field exceeds ≈ 0.9 VÅ −1 , harmonic distortions appear, primarily at the third harmonic. d Lineouts of part c corresponding to the peak frequency of the driving field, its second and third harmonics, and the frequency equidistant from these harmonics.  Despite the factor of 1000 reduction in intensity, the field remains detectable down to the nJ scale.
The low-intensity pulse is multiplied by the square root of the ratio of the high and low pulse energies (the approximate ratio of electric field strength), for comparison. No bandpass filters have been applied. The measured waveform should be independent of the modulation technique, but differences in amplitude can indicate the persistence of the dipole formed in the material.