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Abstract
We demonstrate spatially resolved supercontinuum spectral phase interferometry with an isolated attosecond pulse (IAP). The measured spatial-spectral interferogram over the broadband region indicates a high degree of IAP coherence in both spatial and spectral domains. In addition, the spectral-delay interferogram shows periodic temporal oscillations over the full IAP continuous spectrum, which indicates high temporal coherence. The supercontinuum spectral phase interferometry with broadband IAP will contribute to exploring spatiotemporal dispersive electronic dynamics through phase-based spectroscopy in the future.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High-order harmonics in the extreme ultraviolet (XUV) and soft x-ray regions can have high spatial and temporal coherence [1]. Combining spatial and temporal detection by means of self-referencing interferometry has a variety of applications in molecular orbital tomography [2], wavefront reconstruction [3], nanoscale sample imaging [4–6], the characterization of complex electronic responses [7], and electric field reconstruction of attosecond pulses [8].
To date, two phase-locked attosecond pulse trains (APTs) have been used in attosecond phase-based spectroscopy to observe electronic wave packets [2,7,9–11]. Since an APT has discrete odd-order harmonic spectra [12], an isolated attosecond pulse (IAP) with a broadband supercontinuum spectrum will be a more useful tool for characterizing electronic wave packets. In recent years, attosecond streaking spectroscopy with phase-locked IAP and near-infrared (NIR) pulses have been used to measure the static relative group delay dispersion between photoelectrons emitted from ground and shake-up states in helium atoms, and it was observed over full IAP continuous spectral region [13].
Meanwhile, in traditional time-resolved phase-based spectroscopy with the near-infrared (NIR) [14–16] or XUV region [7,17], a reference pulse is commonly used, in addition to pump and probe pulses. This spectroscopy allows direct detection of the transient complex response of samples, both the real and imaginary parts. The advantage of this method is the ability to select an arbitrary temporal gate width between the reference and probe pulses, and perform a continuous delay scan between the pump pulse and the pair of reference-probe pulses. To implement this concept with an IAP with extreme short duration and broadband supercontinuum spectrum, the key issue is the capability of spectral phase interference between two IAPs. However, spectral phase interferometry with an IAP is still technically challenging due to wavefront distortion in the high-order harmonic generation (HHG) process [18–20] and chromatic aberrations with broadband spectrum [21], in addition to the difficulty in constructing a robust interferometer in the XUV region. Here, we demonstrate spatially resolved supercontinuum spectral phase interferometry with an IAP. The measured spatial-spectral and spectral-delay interferograms indicate a high degree of IAP coherence over a broadband spectral region, which will further contribute to the study of dispersive electronic dynamics in atoms, molecules, and solids.
2. Experiment
For the HHG, we used the carrier-envelope phase stabilized a few-cycle NIR driving pulse (1.57-eV center photon energy, 6-fs duration, 500-µJ pulse energy, and 3-kHz repetition rate) from a beam pointing-stabilized Ti:sapphire laser. The IAP was generated using the double optical gating (DOG) technique [22,23]. Here, the DOG field was constructed by using two quartz plates (270- and 480-µm-thick) and a β-BaB2O4 (BBO) crystal (140-µm thick), which has temporal gate width of approximately 1.3 fs (half-cycle of NIR pulse) [24]. The NIR driving pulse (9-mm diameter) was focused by a spherical mirror (radius of curvature of 800 mm) into a gas cell (3-mm interaction length) filled with argon gas. The estimated spot size and Rayleigh length are 22 µm and 1.9 mm, respectively. In the optimized phase-matching condition for higher flux, the backing gas pressure was 9 mbar, and the gas cell location is approximately 2 mm after the geometrical focus position of the NIR pulse, respectively. The estimated target peak intensity is 1.6×1015 W/cm2.
Figure 1(a) shows a schematic of the IAP spectral phase interferometry setup. To construct the interferometer, we used a reflection beam-splitting mirror (BSM). Since transmission beam splitters have strong absorption in the XUV region, a reflection BSMs [25–28] or two-HHG-source scheme [2–5,7,11,17] is commonly used to build an XUV interferometer. The advantage of directly splitting a single IAP into two pulses using the reflection BSM is that the relative phase between IAPs can be simply defined. The BSM used in this experiment was made of fused silica without a coating, and is equipped with a piezoelectric transducer (PZT) with 1-nm distance resolution (SmarAct SLC-1720). It was used in a grazing incidence configuration with the angle of incidence (AOI) of 85 degrees, which reduces the effective distance and improves temporal resolution. The PZT resolution of 1 nm in the normal direction of the mirror is effectively improved by approximately 11 times to 0.09 nm according to the cosine of 85 degrees, which corresponds to 0.6-as delay resolution considering the beam roundtrip path length. The configuration also helps mitigate the mechanical vibration issue in the XUV interferometer.
Fig. 1. Schematic view of the spatially resolved phase interferometry setup with IAP. BSM: beam splitting mirror, which has a delay stage function and is equipped with a piezo-electric transducer with 1-nm distance resolution. Mo/Si: Mo/Si multilayer-coated spherical mirrors with curvature radiuses of 500 and 400 mm (10% reflectivity at 25–70 eV). Grating: XUV diffraction grating with 600 lines/mm. XCCD: X-ray CCD camera with 13.5-µm pixel resolution.
The split IAPs were sent to the first molybdenum silicon (Mo/Si) multilayer-coated spherical focusing mirror (radius of curvature of 500 mm; reflectivity of 10% at 25–70-eV photon energy). The effective focal length was 340 mm considering the beam divergence from the HHG gas cell. The estimated spot size is 7 µm at zero delay between IAPs. For future application, a target sample could be installed at the focus position. After the focus, IAPs were sent to the second Mo/Si focusing mirror (400-mm radius of curvature), which has the function of image transfer optics. The IAPs were refocused and spatially overlapped on the X-ray CCD (XCCD) camera (Newton SO DO940P, Andor Oxford Instruments Inc.) in the XUV spectrometer. Although the spectral phase interferogram can be observed with a diverged beam in principle, a focused beam has higher power density, which allows smaller area detection and thus sufficiently reduces the detector noise of XCCD camera. The object and image distances are 218 and 1850 mm, respectively. The magnification is 8.5 times. The size of a single pixel of the XCCD camera is 13.5 µm. The spectrometer resolution is 150 meV at 45.5-eV photon energy, which was calibrated by the autoionizing state in a neon atom (electronic transition from 2s22p6 to 2sp6(2S1/2)3p) [29]. In this system, the measured delay jitter between IAPs and beam stability were 2.6-as root mean square (rms) and 0.2-µm rms, respectively, which were evaluated from 3,000 laser shot accumulation (for details of the stability evaluation, see Appendix A). In addition, the generated IAP has 386-as coherence time, which was characterized by first-order interferometric autocorrelation using this system (for details of the autocorrelation, see Appendix B).
3. Results and discussion
Figures 2(a) and 2(b) show measured and calculated spatial-spectral interferograms over ±5-fs delay regions, respectively. Here, we refer to the top and bottom beams split by the BSM as IAP1 and IAP2, respectively. The negative delay corresponds to IAP1 later than it does to IAP2, and vice versa for the positive delay. The delay-dependent spatial-spectral interferograms were observed in isolated attosecond supercontinua, as shown in Fig. 2(a), which agrees well with numerically calculated results in Fig. 2(b) based on the diffraction theory [30] (for details of the calculation, see Appendix C). The visibility can be determined from the spatial interference fringe [31], which was approximately 0.6 in Fig. 2(a). The Fourier-transform-limited pulse estimated from the spectrum at zero delay between IAPs has 257-as duration. For the negative delay, the spatial-spectral interferogram presents upward to the right, while the interferogram shows a straight line at the zero delay. For the positive delay, it presents downward to the right. Asymmetrical spatial-spectral interferograms bounded on the zero delay are caused by tilting the IAP spatial phases (wavefronts) in opposite directions [32]. The spatial phase effect is precisely shown in the measured spatial-spectral interferograms. The time-dependent spatial fringe is an important interferometric component, from which the spatial phase information can be extracted through Fourier analysis. This information can relax the constraint of the spectral resolution of the XUV spectrometer [8] and be used for sample imaging [4–6].
Fig. 2. Spatial-spectral interferograms with spatially split IAPs. (a) Measured and (b) calculated spatial-spectral interferograms over ±5-fs delay regions. The negative delay corresponds to IAP1 (top beam split by the BSM) later than IAP2 (bottom beam split by the BSM), and vice versa for the positive delay. The signal in (a) accumulates 30,000 laser shots each delay step, which is averaged for 30 measurements.
Figure 3 shows the normalized intensity I(ω)/I0(ω) of the spectral interferogram over photon energy regions of 32–38 eV, where the I0(ω) is the lineout spectrum with zero delay at center 0-µm space in the spatial-spectral interferogram. The I(ω) corresponds to the lineout spectra with delays from −1 to −6 fs. Note that, in the case of spatial beam splitting, the spectrum at zero delay I0(ω) ideally becomes the same as the original spectrum, which obtains the maximum value of interferograms. The energy gap ΔE of periodic modulation in the spectral interference fringe is expressed as ΔE=2πℏ/τ, where ℏ and τ are Dirac’s constant and the delay between IAPs, respectively. The energy gap ΔE corresponds to 4.1 eV at −1 fs, 2.1 eV at −2 fs, 1.4 eV at −3 fs, 1 eV at −4 fs, 0.8 eV at −5 fs, and 0.7 eV at −6 fs, which are represented well in Fig. 3. The maximum contrast ratio is approximately 70% in our optimized condition. The contrast could be mainly reduced by the shot-by-shot instability with delay and beam pointing jitters. In addition, since the spatial beam-splitting method using the BSM is sensitive to the spatial phase, the effects of wavefront distortion [18–20] and chromatic aberration [21] could also reduce the contrast. Furthermore, the limitation of the spectral and spatial resolutions in the XUV spectrometer would influence the contrast. Nevertheless, while the contrast ratio reaches a certain level of quality in the XUV region, the interferogram can be applied to the Fourier spectroscopy.
Fig. 3. Normalized intensity I(ω)/I0(ω) of spectral interferogram with IAP. Spectral interferograms I0(ω) and I(ω) correspond to zero delay and delays from −1 to −6 fs over photon energy regions of 32–38 eV, respectively. The maximum contrast ratio is approximately 70%.
Figure 4(a) shows measured spectral-delay interferograms with the IAP. The delay step corresponds to 12 as. The spectral-delay interferograms can be obtained from the lineout spectrum for the spatial-spectral interferogram at each delay, as shown in Fig. 2. The spectral-delay interferogram shows periodic temporal oscillations of 103–159 as, which indicates high temporal coherence over the full IAP bandwidth of 26–40 eV. Figure 4(b) shows spectral-spectral interferograms after Fourier transformation for the delay axis in (a). Alternating-current (AC) components appear in the ±26–40-eV region, in addition to the direct-current (DC) component around zero photon energy. Since the AC component has the supercontinuum spectrum corresponding to the IAP, the temporal oscillations in (a) are projected well onto the spectral-spectral interferogram in (b). In this case, a Fourier window can be applied for the AC components. This is an important analytical process because it allows us to remove the DC component, including the detector shot noise, from the interferogram. Consequently, we can enhance the sensitivity of the measurement using attosecond light sources, which suffer from low photon flux [33] compared to other X-ray sources such as a synchrotron radiation source [34] and an x-ray free electron laser [35]. In addition, since the AC component contains information about the relative intensity profile and phase difference between IAPs, corresponding to the real and imaginary parts, the spectral phase interferogram will be a powerful tool for detecting nuclear and electronic dynamics through time-resolved phase-based spectroscopy with IAP in the future.
Fig. 4. Spectral-delay interferogram with IAP. (a) Measured spectral-delay interferogram. The delay step is 12 as. The signal accumulates 6,000 laser shots each delay step, which is averaged for 17 measurements. The periodic temporal oscillations correspond to 103–159 as over the bandwidth of 26–40 eV. (b) Spectral-spectral interferogram after Fourier transformation for delay axis in (a). AC components appear in ±26–40-eV regions, in addition to the DC component around zero photon energy.
4. Conclusion
We demonstrated spatially resolved supercontinuum spectral phase interferometry with IAP. The spatial-spectral interferogram with spatially split IAPs shows asymmetrical patterns with respect to the zero delay. According to space-time phase coupling, the spatial phase (wavefront) effect was observed in the measured spatial-spectral interferograms, which indicate high IAP spatial coherence over the broadband spectral region. The visibility of the spatial interference fringe reaches approximately 0.6. In addition, the measured spectral-delay interferogram shows periodic temporal oscillations over the full IAP bandwidth, which indicates a high degree of temporal coherence. The supercontinuum spectral phase interferometry with broadband IAP will further contribute to exploring spatiotemporal dispersive electronic dynamics through transient phase-based spectroscopy in a variety of atomic, molecular, and solid systems in the future.
Appendix A: Delay and beam position stabilities
This interferometric experiment using IAPs in the XUV region requires very high temporal and spatial stabilities. We evaluated the delay and the beam position stabilities from the spatial-spectral interferogram. Figure 5(a) shows the measured interferogram for delay of −3.2 fs. In this evaluation, we used IAPs with higher photon energy (36–50 eV) than the ones we used previously (Figs. 2, 3, and 4) because the interferogram with a shorter wavelength has higher sensitivity. The upper image in (b) shows lineout spectra from the integrated spatial area of ±13.5 µm in (a). The signal accumulates 3,000 laser shots every spectrum, which corresponds to 1-s exposure time in the XCCD. The value is similar to the exposure time of measurements in Fig. 2, 3, and 4. The bottom graph shows delay and relative phase jitters extracted from the upper image through the Fourier analysis. The rms values of the delay and relative phase jitter measured over 30 min are 2.6 as and 172 mrad, respectively. These values are significantly smaller than the IAP optical cycle of approximately 120-as periodicity in the XUV region.
Fig. 5. Delay and beam position stabilities with IAP. (a) Measured spatial-spectral interferogram for a delay of −3.2 fs. (b) Top: lineout spectra integrated spatial area of ±13.5 µm in (a). The signal accumulates 3,000 laser shots every spectrum. Bottom: delay and relative phase jitters between IAPs extracted from the upper figure through the Fourier analysis. The root mean square (rms) values of the delay and relative phase jitter measured over 30 min. are 2.6-as and 172-mrad, respectively. (c) Top: beam profiles integrated photon energy regions over 36–50 eV in (a). The signal accumulates 3,000 laser shots in every profile, which is synchronized for taking data of (b). Bottom: Gaussian fitted peak value for beam profiles. The beam center position stability measured over 30 min corresponds to 0.2-µm rms.
Furthermore, to evaluate the stability in the spatial domain, the beam position with the interferogram was monitored. The upper part of (c) shows lineout beam profiles, which are integrated photon energy regions over 36–50 eV in (a). The signal accumulates 3,000 laser shots every beam profile, which is synchronized for taking data of (b). The bottom graph shows Gaussian fitted peak value for the beam profile. The beam center position stability measured over 30 min corresponds to 0.2-µm rms. In this experiment, the distance separation of the spatial interference fringe was approximately 60 µm at 42-eV center photon energy. If the stability is larger than the distance separation, the interferogram will be smeared. Consequently, the high spatial and temporal stabilities allow us to resolve the spectral phase interferogram.
Appendix B: First-order interferometric autocorrelation
Figure 6(a) shows the measured IAP spectrum at zero delay between IAPs. The Fourier-transform-limited pulse estimated from the spectrum has 257-as duration. Figure 6(b) shows the measured first-order interferometric autocorrelation trace. The inset shows the trace enlarged over the ±5-fs region, which indicates the pulse isolation. The blue dashed line in (b) corresponds to the reconstructed first-order interferometric autocorrelation trace from (a) through the inverse Fourier transformation, which agrees well with the measured trace (red filled circles and solid line) in (b). The coherence time from the measured trace is 386 as.
Fig. 6. IAP spectrum and first-order interferometric autocorrelation. (a) The blue solid line shows the measured IAP spectrum at zero delay between IAPs. The Fourier-transform-limited pulse estimated from the spectrum has 257-as duration. (b) Measured first-order interferometric autocorrelation trace (red filled circles and solid line), which was observed without a diffraction grating. The delay step is 12 as. The inset shows the trace enlarged over the ±5-fs region. The blue dashed line is the spectrum reconstructed from (a) through inverse Fourier transformation. The estimated coherence time from the measured trace is 386 as. Error bars in (a) and (b) represent the root mean square (rms) over 15 measurements.
Appendix C: Calculation of spatial-spectral interferogram
The spatially split Gaussian beam has tilted spatial phases (wavefronts) at the focal point [32], which affect the spatial-spectral interferogram. Figure 7(a) shows the calculated beam intensity profile with spatially split IAPs, where we refer to the top beam and bottom beam as IAP1 and IAP2, respectively. The (x1,y1) coordinate corresponds to the space on the first Mo/Si mirror after the BSM in the experiment. In this calculation, the input IAP beam radius (before beam splitting) and the focal length were 563 µm and of 340 mm, respectively. These values were obtained from the experimental condition. Figure 7(b) shows images of focused beam intensity profiles (left) and spatial phases (middle), which were numerically calculated based on diffraction theory [30]. The (x2,y2) coordinate corresponds to the space on the XCCD camera in the experiment. Note that these images already consider the image transfer with 8.5 times magnification in this experiment, i.e., they fit the image on the XCCD camera. The upper and lower beams correspond to IAP1 (top beam) and IAP2 (bottom beam), respectively. The right graphs show lineout intensity profiles (red solid line) and phases (blue dashed line) at the space x2=0 µm in the left and middle images. Although IAPs have the same beam intensity profiles at the focal point, the spatial phases tilt in opposite directions. This is an important point for constructing the spatial-spectral interferogram with spatially split IAPs. The synthesized electric field at the focal point is expressed as [32]
Fig. 7. Calculated spatial beam intensity profile and phase with spatially split IAPs. (a) Beam intensity profile of input IAPs (IAP1 for top beam and IAP2 for bottom beam). The (x1,y1) coordinate corresponds to the space on the first Mo/Si mirror after the BSM in the experiment. The input IAP beam radius (before beam splitting) was 563 µm. (b) Images of focused beam intensity profiles (left) and spatial phases (middle), which were numerically calculated based on diffraction theory [30]. The (x2,y2) coordinate corresponds to the space on the XCCD camera in the experiment. The image transfer of 8.5 times magnification is already considered for images of focused beam intensity profiles and spatial phases. The right graphs show lineout beam intensity profiles (red solid line) and spatial phases (blue dashed line) at the space x2=0 µm in left and middle images. The upper and lower images and graphs correspond to IAP1 (top beam) and IAP2 (bottom beam), respectively.
Funding
Japan Society for the Promotion of Science (16H02120, 16H05987, 19H02637, 20H00357, 20H00358, 20H02703); Ministry of Science and Technology, Taiwan (109-2636-M-007-008).
Disclosures
The authors declare no conflicts of interest.
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