1. Introduction

In the last few years, there have been tremendous advances in the generation of strong ultrashort sources with single or even shorter optical cycles by means of an optical waveform synthesizer consisting of multiple optical parametric amplification (OPA) chains [1–6] or a hybrid scheme of waveform synthesis and a cascaded OPA chain [7]. Tailored pulses with such short pulse durations have attracted a lot of interest because their development is highly promising for many areas of pioneering research such as relativistic laser–plasma interactions [8,9], laser wakefield electron acceleration [10,11], electron wavepacket localization on nanostructures [12–14], electric current control in semiconductors and insulators [15], and the manipulation of electronic coherences in molecules and atoms [16]. As the pulse duration becomes ever shorter, not only the pulse profile, corresponding to the intensity profile, but also the carrier wave becomes important in its interaction with matter. It is thus crucial to have a precise and controllable method of manipulating the phase of a carrier wave with respect to the envelope, i.e., the carrier-envelope phase (CEP), which is a key parameter when carrying out applications on the single- and sub-cycle scales.

Several methods are commonly used for CEP control in OPA systems such as implementing a slow feedback as a low-frequency offset to a fast loop [17], inserting a pair of movable glass wedge plates in the beam path [18], changing the translational delay between the pump and seed pulses [5–7], adjusting the separation of gratings in stretchers or compressors [19,20], and manipulating the longitudinal interaction between optical and acoustic beams using an acousto-optic programmable dispersive filter (AOPDF) [21,22]. However, there are some drawbacks for each method. Slow and fast feedback control may lead to the interference between both feedback loops. For other control techniques using mechanized components, such as the movable wedge pairs and gratings, the effect of the mechanical vibration of mechanized devices on the shot-to-shot control should be carefully handled and minimized so as not to significantly affect the CEP measurement and management [18,20]. On the other hand, changing the delay is feasible only if the translational displacement, related to the CEP shift, is operated in a certain range when the temporal overlap between the seed and pump pulses is approximately maintained; otherwise, an obvious discrepancy in the spectral profile is observed [23]. Different from other methods, the AOPDF, widely employed in spectral phase and amplitude modulation for ultrashort pulses over the past decades [24], has been shown to enable CEP control without any coupling of the dispersion, contains no moving elements, and is reasonably insensitive to beam pointing fluctuations. These advantages are beneficial for implementing a precise, quantitative, user-defined function of time to modulate the shot-to-shot CEP shift [21,22]. Nevertheless, for a pulse of single- or sub-cycle duration, usually corresponding to an over-octave-spanning bandwidth, the AOPDF generally cannot directly control the whole range of the over-octave-spanning spectrum due to the restriction of the available bandwidth of the diffracted pulse from the AOPDF.

To solve the bandwidth limitation caused by the AOPDF, one of the approach is to control the CEP with an AOPDF before compressing the pulse towards a few-cycle duration [25]. The CEP of the pre-compressed pulse can be inherited by the compressed few-cycle pulse. Another approach we reported previously is the utilization of a hybrid scheme composed of waveform synthesis and a cascaded OPA chain to manage the dispersion of an over-one-octave spanning short-wave-infrared (SWIR) spectrum by employing two AOPDFs (Dazzler, Fastlite), which control shared spectral components with wavelengths on the short (900–1450 nm) and long (1450–2400 nm) sides [7]. Even though the waveform synthesis technique has become a worldwide trend in the generation or dispersion control of sub-cycle pulses, there has been no discussion on CEP control by operating multiple AOPDFs simultaneously in the waveform synthesis scheme, where each AOPDF manipulates the CEP of the corresponding wavelength range and all the modulated components are finally superposed to form a synthesized field.

In this work, we report the CEP control of synthesized waveforms by employing multiple AOPDFs using the hybrid scheme mentioned above. This paper is organized as follows. In Section 2, we present theoretical simulations of the phase scans for two few-cycle pulses of adjacent spectral ranges along different scanning paths to view the influence of the resultant CEP on their synthesized field. Section 3 presents the main techniques and experimental setup for controlling and characterizing CEP evolution by using two AOPDFs and an $f$-2$f$ interferometer. Section 4 shows the calculated and experimental results of $f$-2$f$ spectrograms for the synthesized field measured by modulating the CEP for each wavelength component with various functions of time through each AOPDF. In Section 5, we perform closed feedback loop control to compensate for the CEP fluctuation by sending the phase shift error estimated from the $f$-2$f$ spectral fringes back to the two AOPDFs simultaneously, which demonstrates the feasibility of the present scheme for the management of CEP fluctuation. The phase control technique presented in this work may provide new insights into the generation and application of tailored pulses on the single- or sub-cycle light–matter interaction by enabling precise and flexible CEP control via multiple AOPDFs in the waveform synthesis scheme.

2. Paradigms of CEP evolution for synthesized waveforms

In the time domain, the positive frequency parts of two few-cycle electric fields with different carrier angular frequencies $\omega _{\mathrm {S}}$ and $\omega _{\mathrm {L}}$ can be written as

The electric field $E(t)$ is dependent on both the CEP average $\overline {\phi _{\mathrm {CE}}}$ and the CEP difference $\Delta \phi _{\mathrm {CE}}$ between the short- and long-wavelength components, but the intensity envelope $I(t)$ is only governed by $\Delta \phi _{\mathrm {CE}}$. In fact, we find from Eq. (4) that the intensity envelope is shortened when $\Delta \phi _{\mathrm {CE}}=0$. This is in contrast to the strong modulation of the intensity envelope with $I(0)=0$ when $\Delta \phi _{\mathrm {CE}}=\pi$.

When we attempt to fix the CEP of the synthesized field to zero by adjusting the two unknown parameters $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$, we can utilize physical phenomena caused by the strong-field interaction, such as above-threshold ionization (ATI) or high-harmonic (HH) generation, because they depend on the instantaneous amplitude of an oscillating electric field. For example, a slight change in the CEP on the $\pi /10$ level has marked effects on the spectral and temporal profiles of the HHG emission [26,27]. This feature is useful for specifying the point where $\phi _{\mathrm {CE}}^{\text {(syn)}}=0$ by observing the evolution of the spectral and temporal profiles of the HHG emission during CEP scans.

Thus, we have calculated the peak value of the amplitude squared ($E^{2}_{\mathrm {pk}}\equiv \max (E^{2}(t)$)) upon scanning $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$, as depicted in Fig. 1(a), to find a contour path in the $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}$–$\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ plane along which the maximum $E^{2}_{\mathrm {pk}}$ can be efficiently searched for. The carrier wavelengths for the short- and long-wavelength components are given as 1.2 $\mathrm{\mu}$m and 1.9 $\mathrm{\mu}$m, respectively, which indicates that the carrier angular frequencies $\omega _\text {S}\,\simeq$1571 THz and $\omega _\text {L}\,\simeq$992 THz. The solid line labeled with I ($\phi ^{\mathrm {(S)}}_{\mathrm {CE}} \simeq 1.56 \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$) expresses a contour of local maxima of $E^{2}_{\mathrm {pk}}$. We can expect to always find a local maximum of $E^{2}_{\mathrm {pk}}$ when we scan $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ along a contour path unparallel to path I, and therefore, the zero CEP might be achieved by repeating the following two procedures: (i) scanning $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ on a contour path unparallel to path I, for example, on the contour path parallel to path (4) obtained by $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}=-\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$, to find the local maximum of $E^{2}_{\mathrm {pk}}$, then (ii) scanning $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ on a path nearly parallel to path I, for example, path (6) obtained by $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}=2\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$, to find the local maximum of $E^{2}_{\mathrm {pk}}$. This scanning route is schematically shown as contour II in Fig. 1(a). In Fig. 1(a), we show some candidate principal contour paths labeled with (1)–(6) to which the contour paths for scanning $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ might be parallel.

 

Fig. 1. (a): Peak value of the square of the synthesized field $E^{2}_{\mathrm {pk}}$ upon changing $\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ and $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}$. Six candidate contour lines to which the paths for scanning $\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ and $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ might be parallel are depicted with labels (1)–(6). Scanning path to the maximum $E^{2}_{\mathrm {pk}}$ is schematically shown as the white dashed two-dot line labeled with II. (b)–(g): Evolutions of $E(t)$ (left column) and $I(t)$ (right column) upon changing $\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ and $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ along the paths numbered (1)–(6) in Fig. 1(a), respectively. (h) Temporal traces of $E(t)$ (solid curve) and $I(t)$ (filled curve) for points A–E marked in (a).

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Paths (1) and (2) represent $\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ = 0 and $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ = 0, respectively. Paths (3) and (4) represent $\phi ^{\mathrm {(S)}}_{\mathrm {CE}} - \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ = 0 and $\phi ^{\mathrm {(S)}}_{\mathrm {CE}} + \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ = 0, respectively. Paths (5) and (6) represent $2\phi ^{\mathrm {(S)}}_{\mathrm {CE}} - \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ = 0 and $2\phi ^{\mathrm {(L)}}_{\mathrm {CE}} - \phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ = 0, respectively. Figures 1(b)–(g) depict the temporal evolutions of $E(t)$ (left column) and $I(t)$ (right column) along scanning paths (1)–(6), respectively. Figure 1(h) demonstrates the temporal traces of $E(t)$ (solid curve) and $I(t)$ (filled curve) for points A–E marked in Fig. 1(a), where A–E are located at $(\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$, $\phi ^{\mathrm {(S)}}_{\mathrm {CE}})$ = $(0,0)$, $(\pi /2, \pi /2)$, $(-\pi /2, \pi /2)$, $(-\pi /2, 0)$, and $( \pi /3, -\pi /3)$, respectively.

These results suggest that the synthesized waveform and pulse profile can be tailored specifically provided we can control and cleverly design the CEP of each composite wavelength component. To realize the scans on the above-mentioned contour paths, it is necessary to prepare an accessible tool for arbitrarily controlling the CEPs of both wavelength components. In the following, we will introduce an experimental scheme using two individual AOPDFs to control the CEPs of two adjacent wavelength components, which form a continuous over-octave-spanning spectrum with partial spectrum overlap. The effectiveness of the phase control is confirmed by $f$-2$f$ interferometry using the concept of spectral interferometry [28].

3. Experimental setup

Figure 2 shows our experimental setup, which consists of two parts: a lab-built BBO-based SWIR OPA [7] and an $f$-2$f$ interferometer. The SWIR OPA, composed of three amplification stages, is employed to prepare an over-octave-bandwidth spectrum, spanning the wavelength range from 900 to 2400 nm, as shown in a graph in Fig. 2, for the input source to the following $f$-2$f$ interferometer. The broad bandwidth of the OPA is obtained by tuning the pump pulse to $\sim$700 nm, which is a distinct wavelength for performing broadband phase-matching in the BBO nonlinear crystals. The pump pulse is delivered from the chirped pulse amplification (CPA) system of a Ti:sapphire laser, whose wavelength is tuned by inserting a dielectric filter in the cavity of a regenerative amplifier located at the first stage of the amplifier chain. This CPA laser system has also been used to generate intense femtosecond optical vortex pulses [29,30]. Another key feature of the system is the division of the spectrum into short- (900–1450 nm) and long- (1450–2400 nm) wavelength components and their synthesis by using a Mach–Zehnder-type interferometer (MZI) involving an AOPDF [AOPDF(1)] in the short-wavelength path in the MZI and another AOPDF [AOPDF(2)] in the long-wavelength path, so that each AOPDF is responsible for controlling the dispersion of each wavelength component in the available spectral range restricted to 900–1700 nm for AOPDF(1) and 1450–3000 nm for AOPDF(2). This MZI is inserted between the second and third OPA stages.

 

Fig. 2. Schematic of experimental setup used for controlling and characterizing the CEP of the over-octave-spanning SWIR spectrum of the output pulse from the lab-built OPA system with two AOPDFs. In Feedback loop 1, the relative CEP for the amplified pulses from OPA2 is determined by an $f$-2$f$ interferometer, and the error signal is fed back to the controller to drive a piezo translation stage for adjusting the delay between the pump and seed pulses in OPA1. The CEP of the amplified pulse from OPA2 is stabilized with this feedback loop because the idler pulse from OPA1 is amplified in OPA2. The CEP for the synthesized field amplified by OPA3 is characterized by another $f$-2$f$ interferometer placed after OPA3, as depicted in the figure. In Feedback loop 2, the phase error estimated from the $f$-2$f$ spectral interference fringes is fed back to the two AOPDFs simultaneously to compensate for the CEP fluctuation of the synthesized waveform. We show the spectrum of the output pulse from OPA3, which was actually measured, to the right of the ‘OPA3’ box.

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The CEP of the output pulse from the second OPA stage (OPA2) is intrinsically stabilized because the idler pulse from the first OPA stage (OPA1), in which the variation of the CEP of the white-light seed pulse should be removed by the same variation of the CEP of the pump pulse in the difference frequency generation process, is amplified in OPA2. Note that the noncollinear scheme for the pump and seed pulses is applied to all the OPA stages with an angle of less than 1 degree. To compensate for the drift of the CEP, we routinely apply a closed feedback loop, in which the error signal of the $f$-2$f$ interference fringes of the output pulse from OPA2 is sent back to a piezo translation stage to adjust the delay between the pump and seed pulses in OPA1, described as Feedback loop 1 in Fig. 2. We previously reported the generation of sub-optical-cycle SWIR pulses by using this lab-built OPA system, which can generate 4.3 fs pulses at 1.8 $\mathrm{\mu}$m with a pulse energy of 32 $\mathrm{\mu}$J and a repetition rate of 200 Hz. More details of the system are given in Ref. [7].

We demonstrate arbitrary control of the CEP for the over-octave-spanning spectrum by implementing various CEP offsets through the controllers to each AOPDF in the MZI contained in this OPA system. Note that the radio-frequency (RF) signal applied to each AOPDF is synchronized with the repetition frequency of the mode-locked oscillator in the pumping Ti:sapphire CPA laser system, and thus, the phase of each RF signal relative to the time when each infrared pulse is injected to each AOPDF is locked, which is the so-called low-jitter mode in the Dazzler. We can control each CEP by adjusting the phase of each RF signal with the phase-locked-loop (PLL) synthesizer contained in each AOPDF controller. A specific CEP value of the short-wavelength component, $\delta \phi ^{\mathrm {(S)}}_{\mathrm {CE}}$, and that of the long-wavelength component, $\delta \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$, are simultaneously implemented in AOPDF(1) and AOPDF(2), respectively, and they are synchronously scanned as a function of time with a period of several tens of seconds, which is sufficiently long to record the $f$-2$f$ interference spectra in a scanning period. We denote the CEPs applied to the AOPDF controllers as $\delta \phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\delta \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ because we could not determine the absolute CEPs of $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\phi ^{\mathrm {(L)}}_{\mathrm {CE}}$, which appear in Eqs. (1) and (2) and Fig. 1(a), only from the CEPs applied to the AOPDF controllers.

To determine the performance of changing the CEP using the two AOPDFs, we set a conventional $f$-2$f$ interferometer after the third stage of the OPA (OPA3). The $f$-2$f$ interferometer is composed of a front aperture, relay lenses (CaF$_2$, Thorlabs. Inc.), a focusing lens, a BBO crystal for second-harmonic generation (SHG), a collimated lens, a polarizer, another focusing lens for the input to the fiber, and a fiber connector, as depicted in Fig. 2. The lenses used in the interferometer all have the same specifications, where the substrate is CaF$_2$ and the center thickness is 4.3 mm. The relay lenses are used for extending the optical path and adjusting the beam size before the beam is focused into the BBO crystal. The length $L_{\mathrm {cry}}$ of the BBO crystal is 3 mm and the crystal is cut at $\theta$ (polar angle) = 21.2$^{\circ }$ to satisfy the type-I phase-matching condition such that the bandwidth of the second-harmonic pulse is $\sim$15 nm. A slight deviation of the cutting angle of the BBO crystal leads to a change in the center wavelength for the SHG. In the experiment, we set the center wavelength at 1045 nm by slightly tilting the BBO crystal. The intensity $S_0$ of the input pulse focused into the BBO crystal is estimated to be $\sim$10 MW/cm$^{2}$. The optical elements mentioned above are all aligned in a commercial cage system (Thorlabs. Inc.), giving the interferometer a rather compact optical footprint. In front of the interferometer, we insert a 7-mm-thick dispersion plate made of s-TIH6 glass (OHARA Inc.) to adjust the period of the interference fringes to resolve the fringes clearly. The incident angle of the input pulse to the s-TIH6 plate is set at $\sim$60.3$^{\circ }$, the Brewster angle at the wavelength of 2090 nm. White-light generation is not required in the interferometer since the bandwidth of the spectrum for the generated SWIR pulse is sufficiently broad for the $f$-2$f$ interferometry.

The output end of the fiber from the $f$-$2f$ interferometer is connected to the entrance of a spectrometer (Maya2000Pro, Ocean Insight) with a high-sensitivity detector (S110, Hamamatsu) to record the spectral fringes. The minimum integration time of the spectrometer is longer than 10 ms (two periods of pulse repetition, 200 Hz). In the experiment, we recorded nine-shot average of the spectral fringes for the CEP measurement.

4. CEP control

We perform the phase scans by using the two AOPDFs to simultaneously change $\delta \phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\delta \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ mentioned in the previous section. Figures 3(a)–(h) show the evolutions of $\delta \phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\delta \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ applied to the AOPDF controllers (first column), the evolutions of the calculated (second column) and measured (third column) $f$-2$f$ spectral interference fringes, and $\delta \varphi _{\mathrm {CE}}\equiv \delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}} - 2\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ (dashed curve in fourth column) and the phases extracted from the spectral interference fringes in the third column (dots in fourth column). Note that $\delta \varphi _{\mathrm {CE}}$ is different from $\Delta \phi _{\mathrm {CE}}$ in Eqs. (3) and (4. The evolutions of the $f$-2$f$ interference fringes in the second column are obtained from Eqs. (7) and (8) in Appendix with the substitution of the following experimental parameters: $L_\mathrm {cry}$ = 3 mm, effective nonlinear coefficient $d_\mathrm {eff}$ = 0.9 pm/V, intensity of the incident fundamental wave $S_0$ = 10 MW/cm$^{2}$, and $\theta$ = $22.55^{\circ }$. We assume the delay of the $2f$-field relative to the $f$-field $\tau _g$ to be $-$566 fs by considering the dispersions of the s-TIH6 glass and the three CaF$_2$ lenses in front of the BBO crystal. The spectral intensities of the $f$- and $2f$-fields are adjusted to be comparable by rotating the polarizer in the $f$-2$f$ interferometer so that the polarizer transmission $a$ for the polarization of the $2f$-field is estimated to be $\sim$40%. To perform the scan of $\delta \phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\delta \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$, we implement a closed feedback loop over OPA1 and OPA2 to stabilize the CEP drift of the incident pulses to the MZI, described as Feedback loop 1 in Fig. 2. In contrast, we do not need to stabilize the arm lengths of the MZI because their drifts are sufficiently small over long-term measurement to demonstrate arbitrary $\delta \varphi _{\mathrm {CE}}$ control. The evolutions of the $f$-2$f$ spectral fringes are measured by nine-shot accumulation of the spectrum for each record; therefore, the number of records, 4500, corresponds to a recording time of 202.5 s since a pulse repetition period is 5 ms.

 

Fig. 3. Evolutions of $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ (green curve) and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ (purple curve) applied to the AOPDFs (first column), the calculated $f$-2$f$ spectrogram (second column), the measured $f$-2$f$ spectrogram (third column), and $\delta \varphi _{\mathrm {CE}}$ (blue dashed curve) calculated from $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ and that extracted from the measured $f$-2$f$ spectrogram (red dots) (fourth column). The scanning conditions are (a) $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}=0$, (b) $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}=0$, (c) and (d) $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}-\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}=0$, (e) and (f) $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}+\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}=0$, (g) $2\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}-\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}=0$, (h) $2\phi _{\mathrm {CE}} ^{\mathrm {(L)}}-\phi _{\mathrm {CE}} ^{\mathrm {(S)}}=0$. The $f$-2$f$ spectrograms in the second column are calculated using Eqs. (7) and (8) in Appendix.

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There are several possible routes for scanning $\delta \phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\delta \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ to search for $\phi _\mathrm {CE}^{\text {(syn)}}$ = 0, because the absolute CEPs in Fig. 1(a) should be expressed as $\phi ^{\mathrm {(S)}}_{\mathrm {CE}}=\delta \phi ^{\mathrm {(S)}}_{\mathrm {CE}}+\phi ^{\mathrm {(S)}}_{\mathrm {CE}_{0}}$ and $\phi ^{\mathrm {(L)}}_{\mathrm {CE}}=\delta \phi ^{\mathrm {(L)}}_{\mathrm {CE}}+\phi ^{\mathrm {(L)}}_{\mathrm {CE}_{0}}$, where $\phi ^{\mathrm {(S)}}_{\mathrm {CE}_{0}}$ and $\phi ^{\mathrm {(L)}}_{\mathrm {CE}_{0}}$ are constant phase offsets. Here, we adopt the contour paths parallel to (1)–(6) shown in Fig. 1(a) to demonstrate the feasibility of this method for finding $\phi _\mathrm {CE}^{\text {(syn)}}$ = 0. First, we scan only $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ as a triangular wave while keeping $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}=0$, as shown in the first column of Fig. 3(a). This scan is equivalent to the contour path parallel to path (2) in Fig. 1(a). The results for scanning only $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ are also shown in the columns in Fig. 3(b). This scan is equivalent to the contour path parallel to path (1).

Figs. 3(c) and (d) show other examples of $f$-2$f$ spectral fringes obtained by implementing identical modulating amplitudes simultaneously so as to maintain the relation $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}-\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}=0$, i.e., scanning $\phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ and $\phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ along the contour path of $\phi _{\mathrm {CE}} ^{\mathrm {(S)}}-\phi _{\mathrm {CE}} ^{\mathrm {(L)}}=\mathrm {const.}$, which is parallel to path (3) in Fig. 1(a). We apply a trianglular wave and a square wave to both $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ for the modulations shown in the first column in Figs. 3(c) and (d), respectively.

In Figs. 3(e) and (f), we apply a trianglular wave and a sinusoidal wave so as to satisfy the condition $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ + $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}=0$. These scans are equivalent to the path parallel to path (4) in Fig. 1(a), expressed as $\phi _{\mathrm {CE}} ^{\mathrm {(S)}}+\phi _{\mathrm {CE}} ^{\mathrm {(L)}}=\mathrm {const.}$.

In the first column of Figs. 3(g) and (h), we show trianglular waves used to modulate $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ so that the conditions $2\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ $-$ $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}=0$ and $2\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ $-$ $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}=0$ are satisfied, respectively. The equivalent scanning paths are parallel to paths (5) and (6) in Fig. 1(a), respectively. Repeating the sequential search along the path $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ $+$ $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}=0$ and the path $2\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ $-$ $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}=0$ may be useful to obtain $\phi _\mathrm {CE}^{\text {(syn)}}$ = 0, as demonstrated with path II in Fig. 1(a). Details of the parameters used to calculate and modulate $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ are described in Appendix.

In Figs. 3(a)–(h), all the calculated and experimental results are in very good agreement, suggesting that the scan of both $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ in the scheme using two AOPDFs is sufficiently accurate to realize the condition $\phi _{\mathrm {CE}}^{\text {(syn)}}$ = 0. The experimental results successfully realize the temporal evolution of the spectral fringes of the $f$-2$f$ spectrograms calculated from Eq. (7) in Appendix. Different scanning waves, such as the square wave in Fig. 3(d) and the sinusoidal wave in Fig. 3(f), are also implemented to demonstrate the capability of the experimental scheme to modulate the CEP with various waveforms other than triangular ones. In the fourth column of all the figures, the phase shifts (red dots) extracted from the measured interference fringes overlap reasonably well with $\delta \varphi _{\mathrm {CE}}$ (dashed curve) calculated from $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {({L})}}$ applied to the AOPDF controllers. These results reveal the feasibility of simultaneous CEP manipulation for the two wavelength components with the two AOPDFs.

A noteworthy example is shown in Fig. 3(h), where the ratio of the modulating amplitudes for the CEPs of the short- and long-wavelength components is 2/1. This ratio leads to $\delta \varphi _{\mathrm {CE}}$ = 0 due to the cancellation of the CEP modulations from the short- and long-wavelength components, followed by the relation $\delta \varphi _{\mathrm {CE}}$ = $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}} - 2 \delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$, resulting in monotonic straight fringes in the $f$-2$f$ spectrogram. The rms deviation of the phase error extracted from the $f$–2$f$ spectrogram for Fig. 3(h) is estimated to be $\sim$642 mrad. This indicates that the simultaneity of the two AOPDFs is sufficiently reliable to ensure the successful cancellation of the CEP shifts.

Other important phase scans are demonstrated in Figs. 3(c) and (d). Under the scanning condition of $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}-\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}=0$, $\Delta \phi _{\mathrm {CE}}=\phi _{\mathrm {CE}} ^{\mathrm {(S)}}-\phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ is fixed to a constant so that the pulse profile $I(t)$ described as Eq. (4) is maintained during the scanning, while only the CEP average $\overline {\phi _{\mathrm {CE}}}$ in Eq. (3) is successfully scanned.

In Figs. 4(a)–(d), we demonstrate more complicated scanning trajectories with respect to $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ by applying triangular waves with the same amplitude but different periods to the AOPDF controllers, as depicted in the first column of Figs. 4(e)–(h). The ratios of the modulating periods for the long- and short-wavelength components are 2:1, 1:2, 4:1, and 1:4 for Figs. 4(a)–(d), respectively, where the trajectories are no longer simple straight lines but 2D Lissajous-like figures. The corresponding evolutions of the $f$-2$f$ spectrograms calculated from Eq. (7) in Appendix and those measured in the experiments are shown in the second and third columns, respectively, in Figs. 4(e)–(h). We find that even though the $f$-2$f$ spectrograms have very intricate spectral fringes, the phase shifts (red dots) extracted from the experimental spectrograms are confirmed to fit reasonably well with $\delta \varphi _{\mathrm {CE}}$ (dashed curve), as depicted in the fourth column of Figs. 4(e)–(h). We also notice that the delicate structures of the calculated spectrograms exhibited in the second column are well reproduced in the experimental spectrograms in the third column, although the fringe period is significantly reduced in these experiments compared within the former experiments. The present results reveal that the quantitative control of $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ with two individual AOPDFs is sufficiently feasible to control the synthesized field $E(t)$.

 

Fig. 4. (a)–(d) Scanning trajectories in the 2D parameter space of $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$. The arrows indicate the scanning sequence. (e)–(h) The four columns in each figure are arranged to exhibit data in a manner similar to those in Figs. 3(a)–(h). The evolutions of $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ (green curve) and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ (purple curve) depicted in the first column of Figs. 4(e)–(h) correspond to the trajectories shown in Figs. 4 (a)–(d), respectively. The ratios of the scanning periods for $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ are (a) and (e) 2:1; (b) and (f) 1:2; (c) and (g) 4:1; (d) and (h) 1:4.

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Moreover, in Figs. 3 and 4, besides of the $f$–2$f$ interference fringes, we also observe another fringe pattern with a periodicity of $\sim$1 nm, which is not modulated with the CEP control. This fringe pattern is caused by the reflection of the light when passing through the Glan-Taylor designed polarizer composed of two air-spaced birefringent crystal prisms in the $f$–2$f$ interfereometer, as shown in Fig. 2. The air gap inside the polarizer is estimated to be $\sim$600 $\mathrm{\mu}$m, corresponding to a spectral fringe with a periodicity of $\sim$1.8 nm at the center wavelength of 1045 nm, which is coincident with our experimental observation.

5. CEP stabilization using two AOPDFs

In the previous sections, we reported the capability of $\delta \varphi _{\mathrm {CE}}$ scanning by simply applying $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ to the AOPDF controllers. At that time, the stabilization of the CEP only relied on the feedback control employed in the first and second OPA stages indicated as Feedback loop 1 in Fig. 2. In the following, we demonstrate the further stabilization of the CEP by introducing a closed feedback loop involving the two AOPDFs, indicated as Feedback loop 2 in Fig. 2.

In this demonstration, the phase error $\delta \varphi _\mathrm {err}$, i.e., the deviation from the phase reference, is extracted from the spectral fringes monitored using the $f$-2$f$ interferometer, then it is fed back to the two AOPDFs to reduce $\delta \varphi _\mathrm {err}$. The AOPDF feedback algorithm consists of the acquisition of nine single-shot $f$-2$f$ spectra, the extraction of the nine corresponding $\delta \varphi _\mathrm {err}$ values, the computation of the mean $\delta \varphi _\mathrm {err}$, and the negative feedback of the mean $\delta \varphi _\mathrm {err}$ to the two AOPDFs. The actual feedback is conducted by adding a simple proportional correction of $-\zeta \delta \varphi _{\mathrm {err}}$ to both $\delta \phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\delta \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ applied to the AOPDF controllers, where we introduce the damping factor $0<\zeta \leq 1$, which is systematically investigated in an experiment to find an optimized value for CEP stabilization. Here we implement the same correction to both the wavelength components by assuming they inherit the same CEP fluctuation, since they originate from the same light source, are amplified in the same OPA stages, and transmit along almost the same beam path except in the MZI, where the drifts of the two arms are found to be sufficiently small to be negligible over long-term measurement. The overall time for the feedback loop is 85 ms, corresponding to a mean stabilization repetition rate of $\sim$12 Hz, where 45 ms is required for data acquisition, 25 ms is required for data transfer and computation, and 15 ms is required for data transfer to the two AOPDFs.

The second row in the left panel of Fig. 5(a) depicts the time evolution of $\delta \varphi _\mathrm {err}$ with only Feedback loop 1. This is assembled from 2000 records, each of which is extracted from nine-shot accumulation of the $f$-2$f$ interference fringes, and thus, the total recording time is 90 s because of the pulse repetition period of 5 ms (a repetition rate of 200 Hz). The histogram of $\delta \varphi _\mathrm {err}$ is also shown in the first row. The width of a bin in the histogram is set as 0.04 rad (25 bins/rad) .

We show the time evolutions of the $\delta \varphi _\mathrm {err}$ values and their histograms when activating Feedback loop 2 in addition to Feedback loop 1 in the right panel of Fig. 5(a). These five records differ in the value of the damping factor $\zeta$ and the recording conditions are all the same as those adopted for the record depicted in the left panel. The rms deviation of $\delta \varphi _\mathrm {err}$ without activating Feedback loop 2 is estimated to be 399 mrad as shown in the text in the left panel of Fig. 5(a), while it is maximally reduced to 237 mrad by activating Feedback loop 2 with $\zeta =0.45$ as shown in the text above the second column in the right panel of Fig. 5(a). It can also be observed in this figure that the stability of the $f$-2$f$ interference fringes deteriorates when $\zeta$ is more than 0.45.

We can intuitively recognize the improvement of the $\delta \varphi _\mathrm {err}$ stability owing to Feedback loop 2 by directly comparing the $f$-2$f$ spectrogram obtained only under Feedback loop 1 [Fig. 5(b)] with that under Feedback loop 1 and Feedback loop 2 with $\zeta$ of 0.45 [Fig. 5(c)]. The present results reveal the feasibility of the closed-loop stabilization of the CEP of the synthesized field by utilizing two AOPDFs to simultaneously control $\delta \phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ and $\delta \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$. For future improvement of the CEP stability, currently limited by the feedback bandwidth of 12 Hz, a higher bandwidth allowing faster feedback is required, which can be realized by applying the hardware feedback for AOPDF or increasing the computation speed.

6. Conclusion

We have employed two individual AOPDFs with different tuning bandwidths (900–1700 nm and 1450–3000 nm) to quantitatively control the CEP of a synthesized waveform by tailoring the CEPs of its two composite wavelength components, which cover the ranges of 900–1450 nm and 1450–2400 nm. The CEPs of the two wavelength components have been systematically scanned along various routes in the 2D parameter space of $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ to confirm the reliability of this scheme for finding the maximum value of $E^{2}_{\mathrm {pk}}$, corresponding to the shortest pulse duration and highest nonlinear effect. The resultant phase shifts of $E(t)$ extracted from the measured $f$-2$f$ interference fringes showed good agreement with $\delta \phi _{\mathrm {CE}}$ calculated from $\delta \phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ and $\delta \phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ applied to the AOPDF controllers. Therefore, we conclude that the present method of simultaneously adopting two AOPDFs to manipulate the CEPs of the composite wavelength ranges for $E(t)$ is sufficiently accurate to perform successful scans of $\phi _{\mathrm {CE}} ^{\mathrm {(S)}}$ and $\phi _{\mathrm {CE}} ^{\mathrm {(L)}}$ for the realization of $\phi _{\mathrm {CE}}^{\text {(syn)}}$ = 0.

To illustrate the feasibility of the present scheme for the stabilization of the CEP, we have also demonstrated the closed-loop feedback control of the two AOPDFs. The rms deviation of the phase error from a phase reference, which was extracted from the $f$-2$f$ spectrogram at a repetition rate of 200 Hz, was reduced from 399 to 237 mrad by adding this feedback control to the existing feedback control. We expect that this new scheme for arbitrary CEP control using multiple AOPDFs will provide new ideas in waveform tailoring for applications in strong-field physics and nonlinear optics.

Appendix

Here, we describe how we calculated the $f$-2$f$ interferograms depicted in the second column in Figs. 3(a)–(h) and Figs. 4(e)–(h). In the frequency domain, the positive frequency part of the electric field of the short-wavelength component $\widetilde {E}^{ (\mathrm {S}) }_{f} (\omega )$, which is identified as the $f$-field, and that of the second-harmonic field of the long-wavelength component $\widetilde {E}^{ (\mathrm {L}) }_{2f} (\omega )$, which is identified as the $2f$-field, can be expressed as follows [31]:

(5)$$\begin{aligned} \widetilde{E}^{ (\mathrm{S}) }_{f} (\omega) &\propto \sqrt{S^{ (\mathrm{S}) }_{f}(\omega) } e^{{-}i \left( \phi^{ (\mathrm{S}) }_{f} (\omega) + \phi^{ (\mathrm{S}) }_{ \mathrm{CE} } \right) }, \end{aligned}$$

(6)$$\begin{aligned}\widetilde{E}_{2f}^{ (\mathrm{L}) }(\omega) &\propto \sqrt{S_{2 f}^{ (\mathrm{L}) } (\omega)} e^{{-}i \left( \phi_{2 f}^{ (\mathrm{L}) } (\omega) + 2\phi_{ \mathrm{CE} }^{ (\mathrm{L}) } + \omega \tau_{\mathrm{g}} \right) }, \end{aligned}$$

where we define the angular frequency as $\omega$, the spectral intensities of the $f$- and $2f$-fields as $S_{f}^{ (\mathrm {S})}(\omega )$ and $S_{2 f}^{ (\mathrm {L})} (\omega )$, and the spectral phases of the $f$- and $2f$-fields as $\phi _{f}^{ (\mathrm {S}) }(\omega )$ and $\phi _{2 f}^{ (\mathrm {L})}(\omega )$, respectively. We assume that the $2f$-field is delayed relative to the $f$-field by $\tau _\mathrm {g}$, which is mainly determined by the dispersive plates employed to adjust the spacing of $f$-2$f$ interference fringes. Because the $2f$-field is the second-harmonic field of the long-wavelength component, the intensity of the $2f$-field can be expressed as [32]

(7)$$S_{2 f}^{ (\mathrm{L}) }(\omega) = \frac{8 \pi^{2} L_{\mathrm{cry}}^{2} d_{\mathrm{eff}}^{2}} { n_{\mathrm{o} }^{2}(\omega/2) n_{\mathrm{e}}(\omega, \theta) \lambda_{\omega/2}^{2} c_0 \epsilon_0 } S_{0}^{2} { \operatorname{sinc}^{2} \left( \frac{\Delta k L_{\mathrm {cry}}}{2} \right) } ,$$

where $L_{\mathrm {cry}}$ is the length of the SHG crystal, $d_{\mathrm {eff}}$ is the effective nonlinear coefficient, $\lambda _{\omega /2}$ is the fundamental wavelength, $c_0$ is the light velocity, and $\epsilon _0$ is the permittivity of vacuum. We assume the type-I phase-matching condition for the SHG, so that the long-wavelength component, i.e., the fundamental field, propagates in the SHG crystal as an o-ray with a refractive index of $n_{\mathrm {o} }(\omega /2)$, and its SH field, or equivalently the $2f$-field, propagates as an e-ray with a refractive index of $n_{\mathrm {e}}(\omega, \theta )$, where $\theta$ is the polar angle of the propagation direction relative to the optical axis of the SHG crystal. The refractive index of the e-ray is given by: $n_{\mathrm {e}}(\omega, \theta )$ = $\sqrt {n_\mathrm {o}^{2}(\omega ) n_\mathrm {e}^{2}(\omega ) / [n_\mathrm {o}^{2}(\omega ) \sin ^{2}(\theta )+ n_\mathrm {e}^{2}(\omega ) \cos ^{2}(\theta ) ] }$ [32], and the principal refractive indices $n_\mathrm {o} (\omega )$ and $n_\mathrm {e} (\omega )$ can be obtained from the Sellmeier equations of the BBO crystal [33] used as the SHG crystal in the experiment. $S_{0}$ is the intensity of the incident fundamental wave. $\Delta k$ is the phase mismatch factor, which is proportional to the difference between the refractive index of the fundamental field and that of the SH field and is expressed as $\Delta k$ = $2k_{ \mathrm {o} }-k_{ \mathrm {e} }$ = $(4\pi /\lambda _{\omega /2})[n_{\mathrm {o} }(\omega /2)-n_{\mathrm {e}}(\omega, \theta )]$. The resultant total intensity of the $f$- and 2$f$-fields is obtained as

(8)$$\begin{aligned} S(\omega) &\propto \vert \widetilde{E}_{f}^{\mathrm{(S)}} (\omega) + \widetilde{E}_{2f}^{\mathrm{(L)}} (\omega) \vert^{2}\\ &= (1-a)\, S_{f}^{\mathrm{(S)}} (\omega) + a\, S_{2f}^{\mathrm{(L)}}(\omega)\\ &\quad+ 2 \sqrt{a\, (1-a) \, S_{f}^{\mathrm{(S)}} (\omega) S_{2f}^{\mathrm{(L)}}(\omega)} \cos \left[ \Delta \phi (\omega) - \omega \tau_{\mathrm{g}} + \delta\varphi_{\mathrm{CE}} + \varphi_{\mathrm{CE}_{0}} \right], \end{aligned}$$

where coefficient $a$ stands for the polarizer transmission for the polarization of the SHG light, $\Delta \phi (\omega )\equiv \phi _{f}^{\mathrm {(S)}} (\omega ) - \phi _{2f}^{\mathrm {(L)}} (\omega )$, and $\varphi _{\mathrm {CE}_{0}}\equiv \phi ^{\mathrm {(S)}}_{\mathrm {CE}_{0}} - 2 \phi ^{\mathrm {(L)}}_{\mathrm {CE}_{0}}$. We approximate $\Delta \phi (\omega )$ as a constant within a narrow bandwidth restricted by the $\operatorname {sinc}^{2}$ function in Eq. (7, and neglect it. This is because the unknown phase offset $\varphi _{\mathrm {CE}_{0}}$ may involve such a constant phase and it is utilized as an adjustable parameter to reproduce the experimental results. To mimic the experimental $f$-2$f$ spectrograms in the third column of Figs. 3(a)–(h), we use values of $\pi /2$, $7\pi /9$, $-2\pi /9$, $-13\pi /18$, $-7\pi /9$, $\pi /2$ $\pi /9$, and $\pi /2$, for $\varphi _{\mathrm {CE}_{0}}$, respectively. We also set $\varphi _{\mathrm {CE}_{0}}$ to $\pi /6$, $\pi$, $8\pi /9$, and $8\pi /9$, to mimic the experimental $f$-2$f$ spectrograms in the third column of Figs. 4(e)–(h), respectively. The good agreement between the calculated and experimental results presented in Figs. 3 and 4 suggests that the approximation of the constant $\Delta \phi (\omega )$ is reasonable.

 

Fig. 5. (a) Left panel, second row: time evolution of $\delta \varphi _\mathrm {err}$ extracted from measured $f$-2$f$ interference fringes when applying only Feedback loop 1. The histogram of $\delta \varphi _\mathrm {err}$ is also shown in the first row. The width of a bin is set to 0.04 rad (25 bins/rad). Right panel, second row: time evolutions of $\delta \varphi _\mathrm {err}$ values extracted from measured $f$-2$f$ interference fringes when applying Feedback loop 1 and Feedback loop 2. The damping factor $\zeta$ in Feedback loop 2 is adjusted to 0.4 (red dots), 0.45 (green dots), 0.50 (blue dots), 0.60 (yellow dots), and 0.70 (indigo dots). The corresponding histograms are depicted in the first row. (b) Temporal evolutions of the $f$-2$f$ spectral interference fringes when applying only Feedback loop 1. (c) Temporal evolutions of the $f$-2$f$ spectral interference fringes when applying Feedback loop 1 and Feedback loop 2 with $\zeta$ = 0.45, which leads to the smallest rms deviation of $\delta \varphi _\mathrm {err}$ compared with those for the other damping factors, as shown in the right panel of Fig. 5(a).

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We summarize the modulation functions for $\delta \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ and $\delta \phi ^{\mathrm {(S)}}_{\mathrm {CE}}$ appearing in the first column in Figs. 3(a)–(h) and Figs. 4(e)–(h) in Table 1. We denote the time as $t'$ and the period as $T\,(=\mathrm {50 \,\,s})$ in this table.

Table 1. Summary of modulation functions for $\delta \phi ^{\mathrm {(L)}}_{\mathrm {CE}}$ and $\delta \phi ^{\mathrm {(S)}}_{\mathrm {CE}}$

View Table

Funding

Core Research for Evolutional Science and Technology (JPMJCR15N1); Ministry of Education, Culture, Sports, Science and Technology (JPMXS0118068681); Japan Society for the Promotion of Science (19H05628, 20K15197, 26220606, 26247068).

Acknowledgments

This work was part of CREST study JPMJCR15N1 commissioned by JST and has contributed to the missions of the Quantum Leap Flagship Program (Q-Leap) commissioned by MEXT of Japan. Y.-C. Lin gratefully acknowledges the financial support from Grant-in-Aid for Early-Career Scientists No. 20K15197 from JSPS, Japan. Y. N. and K. M. gratefully acknowledge the financial support from Grants-in-Aid for Scientific Research Nos. 26247068, 26220606, and 19H05628 from MEXT, Japan.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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