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We demonstrate a method of stabilizing the carrier-envelope phase (CEP) of low-repetition-rate, high-energy femtosecond laser systems such as TW-PW class lasers. A relatively weak high-repetition-rate (~1 kHz) reference pulse copropagates with a low-repetition-rate (10 Hz) high-energy pulse, which are s- and p-polarized, respectively. Using a Brewster angle window, the reference pulse is separated after the power amplifier and used for feedback to stabilize its CEP. The single-shot CEP of the high-energy pulse is indirectly stabilized to 550 mrad RMS, which is the highest CEP stability ever reported for a low-repetition-rate (10-Hz) high-energy laser system. In this novel method, the feedback frequency of the reference pulse from the front-end preamplifier can be almost preserved. Thus, higher CEP stability can be realized than for lower frequencies. Of course, a reference pulse with an even higher repetition rate (e.g., 10 kHz) can be easily employed to sample and feed back CEP jitter over a broader frequency bandwidth.
© 2016 Optical Society of America
1. Introduction
The electric field waveform of a laser pulse can be written as E(t, ω) = f(t)∙cos(ω∙t + φ) where t is time, f(t) is the temporal envelope, ω is the carrier wave angular frequency, and φ is the carrier-envelope phase (CEP). When the duration of a pulse reaches a few cycles, the electric field waveform significantly changes depending on the CEP [1]. Hence, the CEP is a crucial parameter for many interesting ultrafast phenomena that are fast enough to catch up with electric field oscillations, such as high-order harmonic generation (HHG) [2], above-threshold ionization (ATI) [3], electron localization [4], ultrafast dynamics in molecules and atoms [4,5], isolated attosecond pulse (IAP) generation [6–8], and so forth [9–12]. In particular, intense IAPs, which are produced by HHG, are highly attractive for their potential applications in investigating ultrafast dynamics at the attosecond time scale by utilizing an attosecond pump/attosecond probe [13–16], studying nonlinear optics in the XUV region [8,17,18], and more [19,20]. However, the generation efficiency of IAPs by HHG is generally 10−4 to 10−9 [8,21,22]. Thus, an energy scalable scheme for generating intense IAPs is highly desirable [23,24]. Recently, an intense IAP with microjoule-level pulse energy was obtained for the first time using an infrared (IR) two-color gating (TCG) method combined with a loose focusing geometry [8]. Since the IR-TCG method has excellent energy scalability in the generation of IAPs, a higher-energy driving laser can be adopted to increase the energy of IAPs. As discussed in [24], the CEP stability is an important parameter in further improving the contrast of IAPs and their pulse-to-pulse energy stability [25]. Thus, high-energy laser systems with a stable CEP are highly desirable for generating intense IAPs using the IR-TCG method [8,26]. In addition, a CEP-stable high-energy laser is expected to be useful for investigating interesting research topics such as ultrafast phenomena in the interaction of a relativistic intensity laser with plasma [27–29], the waveform synthesis of ultrahigh-intensity lasers [30,31], research on quantum electrodynamics [32–35], and so forth [36,37]. Up to now, the CEP stabilization of chirped pulse amplification (CPA) Ti:sapphire lasers with kHz or higher repetition rates has been well established using an active feedback method [38–43]. Also, feedforward stabilization method has been demonstrated to stabilize the CEP of a seed oscillator [44]. A 10–50 Hz feedback frequency has been used to stabilize the CEP of the amplifiers so far [45–47]. For extremely high power lasers (e.g., 10-TW or 100-TW), the repetition rates are even lower, making it difficult to stabilize their CEPs using active phase stabilization. Recently, Wu et al. attempted the CEP stabilization of a high-energy 10 Hz laser using a bypassed reference pulse [48]. However, the single-shot CEP stability was measured to be 1.3 rad RMS [49], which is not always sufficient for practical applications. To precisely stabilize the CEP of a low-repetition-rate laser, Takahashi et al. demonstrated the CEP stabilization of a 10 Hz/16 TW laser using a collinearly propagating reference pulse train [50]. By extracting a 500-Hz reference pulse train from a 1-kHz pulse train, which was provided by a front-end preamplifier to stabilize its CEP, the measured CEP of the 10-Hz high-energy pulses was stabilized to 670 mrad RMS (single-shot measurement). However, this novel method loses half of the feedback frequency of the reference pulse from the 1 kHz front-end preamplifier. To measure and correct the CEP jitter to achieve higher accuracy, a higher repetition rate (bandwidth) of the reference pulse is preferable for the active feedback method [51–53].
In this paper, we demonstrate another method of precisely stabilizing the CEP of high-energy, low-repetition-rate lasers. Unlike in our previous study [50], the repetition rate of the reference pulse provided by a front-end preamplifier can be almost preserved. Therefore, we can expect to obtain higher CEP stability because the sampling and feedback speeds are almost doubled. Moreover, the vibration and air turbulence induced by an optical chopper are removed. The CEP jitter of the 10 Hz pulses is stabilized to 550 mrad RMS (single-shot measurement), as measured by an out-of-loop f-2f interferometer. The energy-fluctuation-induced CEP measurement error in the f-2f interferometer is also confirmed in this experiment by simultaneously monitoring the pulse energy and its CEP drift. In addition, by changing the repetition rate of the reference pulse from approximately 1 kHz to 50 Hz, we confirm that the higher the repetition rate, the higher the CEP stability. This finding clearly shows our new method is advantageous for precisely stabilizing the CEP in terms of the feedback frequency. Finally, we discuss further improvement of the CEP stability and the possibility of extending the results of this work to stabilizing CEPs of 100-TW-class lasers.
2. Method and experimental setup
Figure 1 shows the method used in this experiment. A high-repetition-rate reference pulse copropagates with a low-repetition-rate high-energy pulse. In this manner, the reference pulse is subjected to almost the same noises as the high-energy pulse. By sampling the CEP jitter and feedback to stabilize the CEP of the reference pulse, the CEP of high-energy pulse can be stabilized indirectly. Clearly, the weak reference pulse must be separated from such a hybrid pulse train before it is used for CEP locking. This strategy is the same as our previously reported method [50] so far. However, in the new method, we employ a polarization characteristic to extract the reference pulse. Here, the polarization of the reference pulse is rotated to s-polarization while the high-energy pulse is p-polarization as shown in Fig. 1(b). By employing a SiC-coated thin window set at the Brewster angle, only the reference pulse is reflected. This is because the reflectivity in the case of s-polarization is several orders higher than that in the case of p-polarization at the Brewster angle incidence of SiC. This technique is normally utilized to efficiently separate an XUV high harmonic (HH) beam and a fundamental pulse (800 nm) [54]. Of course, we can use thin-film polarizers to separate the reference pulse, and the extracted reference pulse can be utilized for CEP characterization and stabilization. In comparison with our previous work [50], this new method of separating the amplified pulse and the reference pulse has two advantages: 1) we can maintain the high frequency of the reference pulse to perform the feedback operation for CEP stabilization, and 2) we can completely eliminate the reference pulse train on the beam path of the high-energy pulse. In our previous work, the amplified 10 Hz pulses had pre- and post-pulses that were unamplified 1 kHz pulses. These pre-pulses are unsuitable for experiments in which a relativistic intensity laser interacts with a solid target. The new method can perfectly separate the amplified and reference pulses using a polarization characteristic.
Figure 2 shows the experimental setup. The configuration of the laser system is the same as that in our previous paper [50]. After a 1 kHz Ti:sapphire regenerative amplifier, a polarization of 10 Hz pulse is rotated by 90° by a Pockels cell, while the polarization of the other pulses (990 shots/s) is not changed. Then the pulses with different polarization are seeding the back-end multipass power amplifier (MPA) pumped by a 10 Hz Nd:YAG laser. In the MPA, only the s-polarized 10 Hz pulses are further amplified to ~400 mJ. The unamplified p-polarized pulses are used as reference pulses. Before entering the pulse compressor, the polarizations of the reference and 10 Hz pulses are changed to s- and p-polarization, respectively. Here we employ a pair of transmission gratings (TGs) to compress the 20 mJ/10 Hz pulses and the 50 μJ reference pulses to ~30 fs (FWHM), because the TGs have a similar efficiency for s- and p-polarized pulses. Note that the efficiency of a diffraction grating in a pulse compressor is normally optimized for a single polarization angle. Moreover, a TG compressor is advantageous for CEP stabilization owing to its compact size compared with a reflection-type grating compressor [41]. The throughput of the TG compressor is ~50% and ~70% for p- and s-polarization, respectively. Both the 990 shots/s reference pulses and the 10 Hz pulses are subjected to similar CEP noises, such as vibrations, air turbulence, and temperature changes and so on, as they pass through the MPA and the pulse compressor. The reference pulses are extracted by a SiC-coated (to a thickness of a few hundred nm) fused silica window set at the Brewster angle, which has reflectivities for s- and p-polarization of Rs>50% and Rp<0.1%, respectively. The CEP value of the reference pulse is measured by an in-loop f-2f interferometer. The error signal is used to feedback to move a grating in the stretcher to stabilize its CEP. We refer to this feedback process as a ‘slow loop’ in this paper. Since all-analog electrical units are used for the detection of CEP fluctuation (Elite CEP, Coherent) and feedback control (CPS Elite, Coherent), the sampling and feedback are sufficiently fast to perform on each laser shot of 1 kHz pulses in the slow loop. Meanwhile, the out-of-loop measurement of the CEP of the reference pulse is recorded using a spectrometer and a computer. Moreover, another out-of-loop f-2f interferometer is used to measure the CEP of the 10-Hz amplified pulses (single-shot measurement). Both f-2f interferometers are synchronized by a 10 Hz external trigger. By stabilizing the CEP of the reference pulses, the CEP of the 10 Hz low-repetition-rate laser pulses is stabilized indirectly. To characterize the effect of energy fluctuations on the CEPs, the energies of all the reference and 10 Hz pulses are also recorded using two synchronized energy meters (PE10-V2 and PE50-V2, OPHIR).
3. Experimental results
3.1 Correlation between CEPs of reference and 10 Hz high-energy pulses
First, we investigated the relationship between the CEPs of the reference and 10 Hz pulses, which were measured by two out-of-loop f-2f interferometers. To avoid the averaging of fast fluctuations and overestimation of the CEP stability [53,55], both measurements were performed for a single-shot rather than multiple shots. The experimental results are shown in Fig. 3. From 0 to 60 s and from 120 to 300 s, we locked the CEP using both fast and slow loops. It can be seen that the CEPs of both the reference and 10 Hz pulses were stabilized to the same level. From 60 to 120 s, the slow loop was shut and only the fast loop was stabilized. The CEPs of the reference and 10 Hz pulses show slow drift. Moreover, their CEPs drift in the same way, which indicates that both pulses were subjected to similar CEP noises after the fast loop was stabilized. To further investigate the correlation between the CEPs of the reference and 10 Hz pulses, we evaluated Spearman’s rank correlation coefficient (ρ). In the time window from 60 to 120 s, the coefficient was calculated to be ρ = 0.942, which proves that the CEPs of the reference and the 10 Hz pulses are strongly correlated. Thus, by stabilizing the CEP of the reference pulse, the CEP of the 10 Hz pulse can be stabilized indirectly. In the time windows of 0–60 s and 120–300 s, the values of ρ are 0.256 and 0.243, respectively, which are smaller than the value for 60–120 s. These smaller values indicate that the CEPs of the reference and 10 Hz pulses were partially correlated after both the fast and slow loops were locked. Thus, other CEP noises independently existed, which might be mainly due to the following reasons.
Fig. 3 Correlation between CEPs of reference (blue) and 10 Hz (red) pulses. Both CEP values were measured for a single-shot.
First, after both loops were locked, the CEP fluctuations decreased to a small value of ~700 mrad RMS for the reference and 10 Hz pulses. The energy-fluctuation-induced CEP measurement error of the 10 Hz pulse through white-light generation in f-2f interferometer cannot be neglected [56,57], which we will discuss in more detail later. For the high-energy 10 Hz pulse, its pulse energy fluctuation is mainly affected by the fluctuation of pump power, which induces independent CEP noise in contrast to the reference pulse. Second, independent measurement noises of two f-2f interferometers exist owing to their different design configurations and, most prominently, different optimization conditions. Third, the detection shot noise of f-2f interferometers also results in measurement errors [55]. The shot noise, which is determined by the number of detected photons, is even obvious for single shot measurement. In this experiment, we could not identify the amount of shot noise in the single-shot measurement since the levels of other types of noise were much higher. Indeed, shot noise becomes a restrictive issue for the CEP stabilization of amplifiers when the CEP stability reaches 100 mrad RMS [55]. Finally, we found that a thermal lens effect in the MPA slightly increased the CEP noise of the reference pulse, although it is much smaller than the other types of noise in the experiment. Details are given below.
In the MPA, a high-energy Nd:YAG laser was employed as a pump laser, which operated at a 10 Hz repetition rate. After each shot of a pump laser pulse passing through a Ti:sapphire crystal, a thermal lens effect remains [58,59]. Thus, the energy fluctuation of the reference pulse is increased, which was predicted in [49]. To characterize the stability of the pulse energy, we measured the energy of the reference pulse for every shot with and without the operation of the pump laser. The results are shown by the red and blue curves in Fig. 4, respectively. The energy of the reference pulse increased after the MPA was pumped by the Nd:YAG laser, which is because the reference pulse was slightly focused by the Ti:sapphire crystal owing to the thermal lens effect. Thus, the fluence of the reference pulse increased after passing through the limited size-mirrors and apertures in the beam path. The inset of Fig. 4 shows an enlargement of the energy of the reference pulse when the Nd:YAG laser was operated. The energy of the reference pulse slowly decreased after each shot of the pump pulse. The difference between the maximum and minimum energies of the reference pulse after each pump laser pulse is approximately 2%. Also, the energy fluctuation of the pulse, which was immediately after each pump pulse and was used to measure the CEP of the reference pulse under 10 Hz sampling, is 0.5% RMS, which induces a relatively small measurement error of the CEP by the f-2f interferometer. Indeed, it was found that thermal lens effect only increased the reference pulse (990shots/) energy fluctuation from 0.5% to 0.7% RMS in one hour. This 0.2% increase in the energy fluctuation resulted in a CEP measurement error, and thus caused a feedback error. However, we could not identify this feedback error by comparing the energy of the reference pulse and the CEP of the 10 Hz high-energy pulse. Thus, the feedback error was small compared with the other types of noise and acceptable for CEP stabilization in this experiment. Contrary to the prediction in [49], it is not necessary for the reference pulse to bypass the power amplifier. The completely collinear propagation of the reference and 10 Hz pulses ensured that they were subjected to almost the same noises after passing through the power amplifier and compressor. Thus, we were able to obtain higher CEP stability than that in [49].
Fig. 4 Energy of the reference pulse for every shot when the MPA was operated with (red) and without (blue) Nd:YAG laser pumping. Inset, enlargement of energy of the reference pulse when the MPA was pumped by a Nd:YAG laser.
3.2 CEP stabilization of 10 Hz high-energy pulse
Figure 5(a) shows the long-term trend of the CEP for the 10 Hz pulse after locking both the fast and slow loops. The single-shot CEP fluctuation was measured to be 643 mrad RMS over 2000 s. In the short term, single-shot CEP was stabilized to 550 mrad RMS as shown in Fig. 5(b). Note that all of these values are measured by an out-of-loop measurement. Figures 5(c) and 5(d) show the CEP distributions over a range of 2π for the long-term and short-term, respectively. Both distributions are similar to a Gaussian distribution. It can be seen that slow drift of the CEP remains in the long-term stability, as shown in Fig. 5(a), which is attributed to the slow energy drift of the 10 Hz pulse [37,42]. To confirm this point, we simultaneously measured the every-shot CEP and energy of the 10 Hz pulse in the time region with unstable energy. The result is shown in Fig. 6(a). The pulse energy is shown by the red curve and clearly exhibits slow drift. Over 3000 s, the pulse energy fluctuation was measured to be 1.6% RMS. At the same time, the CEP drifted in the opposite direction, as shown by the blue curve in Fig. 6(a). The CEP stability was evaluated to be 900 mrad RMS over 3000 s. Moreover, Spearman’s rank correlation coefficient between the pulse energy and the CEP was calculated to be ρ = −0.32 for 0−3000 s, which means that they are partially correlated but have the opposite tendency. The other types of noise of the CEP might have been due to the measurement noise of the f-2f interferometers [55] and the residual noise of the seed laser. In Fig. 6(b), the measured CEP is plotted as a function of the pulse energy in Fig. 6(a). The red solid line is the linear fitting of the data. The slope indicates that a 1% increase in the pulse energy resulted in a CEP measurement error of −152.6 mrad in this experiment. In general, relatively high fluctuation of the pulse energy is a common issue in high-energy femtosecond laser systems such as 10-TW and 100-TW-class lasers [49,50]. To correctly determine the CEP stability, the energy-fluctuation-induced measurement error should be excluded when a white-light-based f-2f interferometer is utilized. Otherwise, another CEP measurement method that is not affected by pulse energy fluctuation must be employed to correctly determine its value.
Fig. 5 Long-term (a) and short-term (b) stabilized CEP of 10 Hz high-energy pulses. (c) and (d) CEP distributions of (a) and (b) over a range of 2π, respectively.
Fig. 6 CEP measurement error induced by fluctuation of (normalized) pulse energy. (a) Pulse energy (red) and CEP shift (blue) of 10 Hz pulse measured in every shot. (b) CEP shift as a function of pulse energy in (a) and its linear fitting (red solid line).
3.3 CEP stabilization under different sampling and feedback frequencies
To improve the CEP stability of a femtosecond CPA laser system, high-speed (e.g., 10 kHz or higher) CEP sampling and feedback are generally preferable [52,53]. This is because fast CEP noise inherited from residual noise of the seed laser exists throughout an entire laser system. Moreover, vibration noise, of whose frequency might reach to 100 Hz level, will disturb a stretcher and a compressor to introduce fast CEP fluctuations. In addition, the shot-to-shot fluctuation of a pump laser used for amplifiers might also induce fast CEP noise [51]. Note that the above noise of the CEP depends on the laboratory environment and the characteristics of the laser system. For these reasons, the repetition rate of the laser pulse should be at least twice as high as frequencies of the various types of noises so as to accurately characterize them. Furthermore, the data collection and feedback units should sufficiently fast to compensate for these noises. To clearly show the advantages of our novel method in terms of the feedback frequency, we measured the CEP stability under different sampling/feedback speed. By using different optical choppers, as shown in Fig. 7, the frequency of the reference pulse was changed to 50 Hz, 100 Hz, 500 Hz, and 990 shots/s (no chopper). Accordingly, CEP sampling and feedback were also performed in the slow loop of the same frequencies. In Figs. 7(a), 7(c), 7(e), and 7(g), single-shot CEP stabilities of 735 mrad RMS, 707 mrad RMS, 662 mrad RMS, and 558 mrad RMS were obtained for the 10 Hz pulse when the repetition rate of the reference pulse were 50 Hz, 100 Hz, 500 Hz, and 990 shots/s, respectively. Figures 7(b), 7(d), 7(f), and 7(h) show the CEP distributions over a range of 2π. The measured CEP values clearly show that a higher sampling/feedback frequency is preferable for achieving higher CEP stability. In fact, our proposed method is capable of employing reference pulse with much higher repetition rates such as 10 kHz and above provided the pulse energy is sufficient (typically microjoule level) for CEP measurement using an f-2f interferometer. By employing such a high-repetition-rate reference pulse for CEP characterization and stabilization over a broad frequency bandwidth, the CEP can be locked with a higher stability. Ultimately the CEP stability would be limited by the CEP stability of the seed laser and the CEP measurement error [51,55].
Fig. 7 CEP stability of 10 Hz pulse under different sampling and feedback frequencies. Left column, optical choppers designed to change the repetition rate of the reference pulse. (a), (c), (e), (g) and (b), (d), (f), (h) show single-shot CEP data and CEP distributions over a range of 2π for the 10 Hz pulses when the repetition rates of the reference pulses are 50 Hz, 100 Hz, 500 Hz and 990 shots/s, respectively.
4. Summary
We demonstrated the CEP stabilization of a high-energy laser with 10 Hz repetition rate. A high-repetition-rate reference pulse, which had a different polarization direction, propagated collinearly with the 10 Hz pulse. By using a SiC-coated fused silica window set at the Brewster angle, only the reference pulse was extracted, which was used to characterize and stabilize its CEP. The CEP of the 10 Hz pulse was indirectly stabilized. The long- and short-term single-shot CEPs were respectively stabilized to 643 mrad RMS and 550 mrad RMS, as measured by an out-of-loop f-2f interferometer. We also characterized the long-term CEP measurement error induced by the fluctuation of the pulse energy in a white-light-based f-2f interferometer. It was found that an energy increase of 1% induced a shift of approximately −152.6 mrad in the CEP measured by the f-2f interferometer in this experiment. This phase shift was the measurement error rather than the actual CEP shift. Considering the energy fluctuation of our 10 Hz laser, the actual long-term CEP stability might be better than the measured values. We also experimentally proved that the sampling and feedback frequencies can affect the CEP stability. After changing the repetition rate of the reference pulse and the feedback speed accordingly, it was confirmed that higher sampling and feedback speeds are preferable for achieving higher CEP stability. Note that the effectiveness depends on the laboratory environment and the characteristics of the laser system.
Here, we briefly discuss how to further improve the CEP stability of a 10 Hz laser system. The residual fast CEP jitter of the seed laser, which is less than 100 mrad integrated from 3 Hz to 1 MHz, is inherited by the reference and 10 Hz high-energy pulses. Thus, by further reducing the noise of the seed laser, the CEP stability of the 10 Hz laser system can be improved accordingly. In fact, the noise level of the seed laser is one of the ultimate limitations on CEP stabilization after the amplifiers [51]. Moreover, since we experimentally proved that a reference pulse with a higher repetition rate is preferable for CEP stabilization, by employing a reference pulse with a repetition rate of 10 kHz or higher to sample noises and feedback over a broader frequency bandwidth, the improved CEP stability of 10 Hz pulses can be expected. In addition, shot noise is another limiting factor for improving the CEP stability of amplifiers, which was thoroughly discussed in [55]. By using a well-designed interferometer and increasing the number of detected photons, shot noise can be reduced. Furthermore, we observed an increase of 0.2% RMS of the reference pulse energy fluctuation, which was induced by the thermal lens effect. Roughly speaking, this increase in the fluctuation might induce a feedback error in CEP stabilization of ~30 mrad RMS, as estimated from Fig. 6(b). This feedback error was negligible in our experiment since it was much smaller than the final stabilized CEP of 550 mrad RMS. However, if the CEP stability approaches the 100 mrad RMS level, such a thermal lens effect can no longer be neglected. In such a case, the energy of the reference pulse must be stabilized to further improve the CEP stability. One way of achieving this is to reduce the thermal lens effect by sufficiently cooling the laser crystal in the MPA. Another method is to build a feedback system to stabilize the energy of the reference pulse, which can be realized by employing well-designed variable neutral density filters and a proportional-integral-derivative program [60].
Finally, we discuss the possibility of using a large TG compressor to extend the results of this work to lasers with an even higher power and a low repetition rate. In a previous experiment, two fused silica TGs with dimensions of 180 × 60 × 1 mm3 were employed to compress reference and 10 Hz pulses. These TGs had similar diffraction efficiencies for both s- and p-polarization [61]. Even though we employed 10 mJ/30 fs 10 Hz pulses after the TG compressor, a pulse energy of more than 60 mJ is attainable using this compressor [61]. If four of the identical TGs are employed for a compressor, a pulse energy of more than 250 mJ can be obtained after compression. The dimensions of the TGs are a limitation on the attainable compressed pulse energy. If larger TGs are utilized [62,63], it will be possible to stabilize the CEP of 100-TW-class femtosecond laser systems or those with even higher power using the method proposed in this paper. For such high-power CEP-stable lasers, pre-and post-pulses are not acceptable for some applications such as investigating the interaction of relativistic intensity lasers with solids [64,65]. However, the copropagating reference pulses, which have orthogonal polarizations, can be removed by using SiC-coated Brewster angle windows or thin-film polarizers. Thus, it will not prevent the applications of such lasers.
Acknowledgments
We are immensely grateful to Dr. Yasuo Nabekawa from RIKEN for providing the transmission gratings and for helpful discussions. We would like to thank Mr. Yuuki Tamaru for technical support and Dr. Chengquan Li from Coherent Inc. for helpful discussions.
This work was supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan through a grant for Extreme Photonics Research, by Japan Society for the Promotion of Science through a Grant-in-Aid for Scientific Research, and by MEXT through a Grant-in-Aid for Scientific Research (B) (25286074) and a Grant-in-Aid for Exploratory Research (16K13704). This work was also part of the Advanced Photon Science Alliance Project commissioned by MEXT.
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