Long-term stabilization of carrier envelope phases of mid-infrared pulses for the precise detection of phase-sensitive responses to electromagnetic waves

Recent developments in techniques for generating a strong terahertz (THz) pulse have opened up effective ways using electromagnetic waves to control physical quantities such as polarization and magnetization in solids.^1,2 The control of electronic phases using a THz pulse is also attractive; however, only a few examples of using electromagnetic fields to induce phase transitions have been reported so far.^3,4 It is mainly because the amplitude of a THz pulse is generally difficult to enhance larger than 1 MV/cm, although several techniques have been proposed.^5,6 In a mid-infrared (MIR) region, it is easier to enhance electromagnetic field amplitudes than in a THz region. Strong MIR pulses have indeed been used to induce a metal–superconductor transition in cuprates⁷ and an insulator–metal transition in vanadium dioxide.⁸ A strong MIR pulse can also induce interesting nonlinear phenomena in solids. Studies have reported observations of non-perturbative responses in semiconductors,^9–12 ultrafast polarization reversal in ferroelectrics,¹³ modulation of the on-site Coulomb interaction in Mott insulators,¹⁴ and sum-frequency excitation of Raman-active phonons in diamonds.¹⁵ 

To reveal ultrafast electron and lattice dynamics in phase transitions and other nonlinear phenomena in solids with an MIR pulse, one must detect any changes to optical indices induced by an oscillatory electric field. An MIR pump pulse with a stable carrier envelope phase (CEP) and a probe pulse with duration shorter than the period of the oscillatory fields are indispensable for monitoring these changes. CEP is defined by the phase of the carrier wave in a light pulse with respect to its envelope. Since an MIR pulse with a photon energy of 0.1–0.2 eV (a frequency of ∼24 to 48 THz or a wavelength of ∼12.4 to 6.2 µm) has an oscillation period of 41–21 fs, the electric-field waveform or CEP of the pulse should be stabilized within 1 fs to perform precise phase-sensitive experiments. In general, the CEP of the output of femtosecond laser sources, such as a Ti:sapphire regenerative amplifier (RA), fluctuates unless a feedback device is introduced in the cavity.¹⁶ To this end, several methods have been proposed for the passive stabilization of CEP during frequency down-conversion processes of the output of laser sources to an MIR pulse: (1) inter-pulse difference frequency generation (DFG) between two frequency-detuned near-IR pulses generated separately from a common light source,^17–19 (2) intra-pulse DFG of a spectrally broadened visible/near-IR pulse^20,21 or a two-color pulse generated in a dual wavelength optical parametric amplifier (OPA),²² and (3) four-wave-mixing with two-color filamentation.²³ 

Among those methods, the most general is the use of inter-pulse DFG using outputs from two OPAs,¹⁷ since it uses a relatively simple setup and offers wide tunability of the MIR-pulse frequencies. In this scheme, two signal (or idler) pulses with a constant phase difference Δϕ are generated separately from two OPAs, in which both the excitation pulses and seed white-light pulses come from the same laser source. In an MIR pulse obtained via DFG processing of two signal (or idler) pulses, the original CEP fluctuations are canceled. In an actual experimental setup, however, the optical-path length difference between the two OPAs fluctuates due to jitter that originates from instabilities in temperature and air flow. This gives rise to fluctuations in the phase difference between the two OPA outputs, which result in the instability of the CEP of the MIR pulse. For example, in the case that the photon energies of two OPA outputs are 0.75 eV and 0.6 eV, a change of 300 nm in the path length difference (corresponding to a time difference of 1 fs) induces a change in the CEP of the MIR pulse with 0.15 eV (∼36 THz) of 0.34π rad or 5 fs in time. Note that the temporal difference of the two OPA outputs is magnified during the DFG process. In order to achieve the stability of the CEP within 1 fs, the optical-path length difference should be suppressed within ∼70 nm. The optical-path length of each OPA is typically 2 m so that the stability of the optical-path lengths in the DFG process should be (70 nm)/(2 m) = 3.5 × 10⁻⁸. To achieve such a high stability, a feedback control of the optical-path length difference will be necessary. Manzoni et al. indeed introduced a feedback correction device to an inter-pulse DFG system, in which a CEP (time) stability of 110 mrad (1 fs) over 2 h was achieved at MIR pulses with 17 µm.²⁴ 

In a pump–probe system, in general, the overall path length difference of a pump pulse and a probe pulse may also fluctuate.²⁵ In our case, it leads to a drift in the pump–probe delay time t_d or equivalently an envelope peak position (EPP) of the MIR pulse relative to the ultrashort visible probe pulse, which also gives rise to the fluctuation of the absolute carrier phase of the MIR pulse. In order to suppress the drift of EPP also within 1 fs, the overall path length difference of the MIR pump and visible probe pulses should be stabilized within 300 nm. The overall path length in a pump–probe system is ∼8 m in our setup, as mentioned later, so that the stability of (300 nm)/(8 m) ∼ 3.8 × 10⁻⁸ is necessary, which is comparable to that necessary for the stabilization of CEP (∼3.5 × 10⁻⁸). Therefore, the active stabilization of EPP will also be indispensable as well as CEP, while such a double feedback system has not been reported.

In this paper, we report a high performance feedback control system for MIR-pump visible probe measurements. This system measures the electric-field waveform of the MIR pump pulses with electro-optic sampling (EOS) and detects fluctuations in both the CEP of the MIR pulse and its EPP relative to an ultrashort visible pulse. Feedback devices are included that control the difference between the optical-path lengths of two OPAs and also the difference between the optical-path lengths of the pump and probe pulses. In the constructed system, error in the detection of a phase-sensitive response is reduced to as little as 1.0 fs over a few tens of hours.

Our setup is illustrated in Fig. 1(a). The light source is a Ti:sapphire RA, which generates a pulse with a central wavelength (photon energy) of 800 nm (1.55 eV), duration of 35 fs, repetition rate of 1 kHz, and fluence of 7.5 mJ. The output from the RA is split into two. One is frequency-doubled in a β-BaB₂O₄ crystal and is used as the excitation pulse for a type-I non-collinear OPA (NOPA) with a β-BaB₂O₄ crystal.²⁶ In Fig. 1(b), we show the intensity profile of the NOPA output, obtained with a frequency-resolved optical-gating (FROG) using the retrieval algorithm.²⁷ The full width at half maximum (FWHM) of the pulse is 8.9 fs. Figure 1(c) shows the spectrum of the pulse, which ranges from 1.7 eV to 2.4 eV (from 520 nm to 710 nm). The details of the NOPA were reported in Ref. 26. The other output from the RA is introduced to a dual OPA that includes OPA1 and OPA2, in which a small part of the RA output is used to create a common white-light seed pulse and the residual is used to amplify the seed pulse in each OPA. For the DFG of an MIR pulse, we use GaSe as a second-order nonlinear optical crystal, which enables us to generate a broadband MIR pulse. By introducing two idler pulses from two OPAs to a 250-μm-thick GaSe crystal, an MIR pulse is generated via type-I DFG.²⁸ 

Although the CEP of the original RA output is unstable, the resulting MIR pulse is phase-stabilized.¹⁷ In the experiments reported below, the wavelength and power of the idler pulse from OPA1 were fixed at 0.756 eV (1640 nm) and 390 µJ, respectively. The idler pulse from OPA2 was changed from 0.60 eV (2080 nm) to 0.65 eV (1900 nm), and its power was typically 120 µJ. Under these conditions, MIR pulses with the central frequency (photon energy) of 28–40 THz (116–165 meV) can be obtained.

The diameter of the MIR pulse thus obtained is expanded to 3 cm by two off-axis parabolic mirrors, OAP1 and OAP2 [Fig. 1(a)]. A beam splitter (BS) (a 500-μm-thick Si plate) is introduced to split the MIR pulse into two. The reflected MIR pulse with 60% of the original intensity is used as the pump pulse for pump–probe measurements. Using OAP3, this pulse is focused to a spot 50 µm in diameter (FWHM), the position of which is hereafter referred to as the sample position. The transmitted MIR pulse is used in the EOS to measure its electric-field waveform. Using OAP4, the pulse is focused on a LiGaS₂ crystal, the position of which is hereafter referred to as the control position. The spot diameter at the control position is adjusted to be as large as that at the sample position. The visible probe pulse from the NOPA is also divided into two. Each pulse is focused through a hole drilled in the OAP onto the center of the sample or the control position with a diameter of 20 µm. The former is used as the probe pulse in the pump–probe measurement and the latter as the sampling pulse for EOS.

A schematic of the EOS is shown in the lower-right part of Fig. 1(a). When an electric field is applied to a nonlinear optical crystal, the crystal’s birefringence for the sampling pulse changes in proportion to the electric field through the Pockels effect, which can be measured as the difference in the signal of balanced photodiodes detecting the sampling pulse after passing through a quarter-wave plate and a polarizing BS. Thus, by changing the delay time of the sampling pulse relative to the MIR pulse, one can measure the electric-field waveform. We used a 20-μm-thick LiGaS₂ crystal. LiGaS₂ has a wide optical gap so that absorption can be neglected even for visible probe pulse. The electric-field waveform and Fourier power spectrum of a 33-THz (136 meV) MIR pulse are shown in Figs. 1(d) and 1(e), respectively. At this frequency, the maximum electric-field amplitude of 10.2 MV/cm is obtained. The electric-field amplitude is estimated from the pulse energy, the beam size, and the electric-field waveform [Fig. 1(d)] of the MIR pulse. The detection range is 25–91 THz (100–380 meV) (supplementary material, S1).

The procedures for stabilizing the CEP and EPP are illustrated in Fig. 2(a). In the first step, the electric-field waveform of the MIR pulse at the control position is measured, which is then used as a reference and it is therefore called the reference waveform. In the second step, feedback is used to keep the electric-field waveform of the MIR pulse at the control position identical to the reference waveform in terms of both CEP and EPP. In the loop shown in step 2 in Fig. 2(a), the EOS and pump–probe measurements are performed simultaneously by changing the delay of the sampling pulse and that of the probe pulse to 300 fs, which takes about 30 s. This temporal range of measurements was chosen to strike a compromise between a short feedback interval and coverage of nearly the whole duration of the MIR pulse [see Fig. 1(d)].

To evaluate the drifts in CEP and EPP of an electric-field waveform at the control position relative to those of the reference waveform, we use the cross correlation of the two electric-field waveforms of the MIR pulses and its Hilbert transform. The drifts in CEP and EPP are illustrated in Fig. 2(b). CEP is expressed as the temporal difference between the carrier phase and the EPP multiplied by the carrier’s angular frequency. The cross correlation of the reference waveform and the electric-field waveform in the nth measurement (the nth waveform) forms a fringe pattern. Examples of a reference waveform, an nth waveform, and a cross-correlation profile measured with feedback enabled are shown in Figs. 3(a)–3(c), respectively. τ in the horizontal axis of Fig. 3(c) is the temporal parameter in the calculation of the cross-correlation profile, and the phase at τ = 0 gives the relative change in the carrier phase of the nth waveform with respect to the phase of the reference waveform, i.e., the absolute carrier phase drift. In Fig. 3(c), the absolute carrier phase drift is almost equal to zero.

The EPP drift can be evaluated from the envelope of the cross-correlation profile of the reference and nth waveforms. Assuming that the envelope varies slowly relative to the carrier wave oscillation in the time domain, the envelope of the cross-correlation profile is equivalent to the cross-correlation profile of the envelopes of two waveforms (supplementary material, S2). By extracting the envelopes of cross-correlation profiles using the Hilbert transform, we can determine the EPP of the nth waveform. Calculating the cross-correlation profile before extracting the envelope tends to suppress noise, because uncorrelated noises are averaged out in the calculation. The evaluated EPP values are plotted with open circles in Fig. 3(d). Because of the wide temporal width (FWHM ∼ 100 fs) of MIR pulses, the EPP data vary widely. Therefore, we next use a linear regression of EPP data from 300 consecutive cross-correlation profiles, which include information about waveforms during the last 2.5 h of the experiment. The result is shown with the blue line in Fig. 3(d), in which variations in the raw data are suppressed and appropriate EPP values are obtained. This allows the precise determination of the drift of pump–probe delay. From the difference between the absolute carrier phase drift evaluated using the data of an EOS for 30 s and the EPP drift evaluated using the data accumulated for 2.5 h, we obtain the CEP drift [Fig. 2(b)].

Feedbacks for stabilizing both the CEP of the MIR pulse and its EPP relative to the visible probe pulse are accomplished as follows. The CEP is stabilized by inserting a pair of CaF₂ wedge plates (WP) after OPA2 [Fig. 1(a)]. The WP has a refractive index of 1.43 in the near-IR region and its wedge angle is 4°. One of the WPs is mounted on a motorized stage. By varying its insertion length, we can eliminate the path length differences between the outputs of two OPAs due to fluctuations. The EPP is stabilized by controlling two delay lines DL2 and DL3 [Fig. 1(a)] using EPP data.

To evaluate the performance of our system with the two kinds of feedback applied, we performed continuous EOS measurements at the sample position with and without feedback. The results are highlighted in Figs. 4(a) and 4(b), which show, respectively, the electric-field waveform of MIR pulses at the position shown by the broken line in (b) and those shown in a color contour when feedback is performed. In (b), all waveforms are normalized to the fluence of respective probe pulses. The result demonstrates that the phase is held stable for 20 h. Figure 4(c) plots the absolute carrier phase shifts with and without feedback for about 8 h. The drifts of EPP and CEP without feedback are shown in Fig. S1 in the supplementary material, S3. Our feedback system works well, or the absolute carrier phase would never be stabilized. The stabilized absolute carrier phase has a root-mean-square error of 200 mrad over 20 h, which corresponds to the drift of 1.0 fs in time. Note that a small oscillation of the phase with a period of about 12 min is observed in Fig. 4(b). This oscillatory fluctuation is attributable to the cyclic operation of the air conditioner and the resultant oscillation of the temperature of the laboratory with ±0.05 K. In the absolute carrier phase shift shown in Fig. 4(c), a similar oscillatory fluctuation can be seen as well as fast jitters. In Fig. 4(b), fast jitters are averaged and the oscillatory fluctuation becomes more visible. It should be emphasized again that the drifts in CEP due to the oscillatory variations in temperature are sufficiently suppressed by our feedback system as shown in Fig. 4(c).

In our feedback system, we use two different optical-paths after the MIR pump beam (the visible probe beam) is divided by the beam splitter BS2 (BS3) and therefore the two optical-path lengths from BS2 (BS3) to the sample or control positions may become different by fluctuations. However, under the double feedback controls of CEP and EPP, no drift of absolute carrier phase occurs as can be seen in Fig. 4(c), indicating that such fluctuation is negligible. It is because the optical-path length from BS2 (BS3) to the sample (control) position is about 1 m, which is much shorter than the overall optical-path length (∼8 m) of the pump and probe pulses [see Fig. 1(a)]. Our method can be used to various kinds of sub-cycle pump–probe systems with a pump pulse having any repetition rate and any electric-field cycles, as long as EPP drift occurs as slow as the accumulation time of EOS data necessary to evaluate EPP precisely.

In conclusion, we constructed an MIR-pump visible probe system, in which double feedback correction is achieved. We evaluated the absolute carrier phase drift as well as the drift of the envelope peak positions in MIR pulses by detecting electric-field waveforms. Feedback correction of both the carrier phase and the pump–probe delay allows us to hold the CEP constant within 200 mrad for at least 20 h, which corresponds to a 1.0 fs error in time. Our highly stabilized system is a powerful tool for detecting phase-sensitive responses in solids, such as initial responses of electric-field-induced phase transitions, high order nonlinear phenomena, and the formation of Floquet states by MIR pulses.

See the supplementary material for the details of the electro-optic sampling, the determination of the envelope peak position of MIR pulses using cross-correlation profiles, and the drifts of EPP and CEP without feedback.

T.Y. and N.S. contributed equally to this work.

This work was supported in part by Grants-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (JSPS) (Project Nos. JP18H01166 and JP18K13476) and by CREST (Grant No. JPMJCR1661), Japan Science and Technology Agency. T. Morimoto was supported by JSPS through the Program for Leading Graduate Schools (MERIT) and Research Fellowship for Young Scientists.