1. INTRODUCTION

Synchrotron light sources have been developed for more than 70 years, and currently, ultrashort x-ray pulses are generated with some free electron lasers (FELs) [1–5]. Characterization of the pulses generated from such sources is very challenging due to the short wavelength and short pulse duration.

To characterize ultrashort pulses in the ultraviolet (UV) or extreme UV (XUV) region, it is straightforward to measure the cross-correlation between test and reference pulses. However, it is very difficult to prepare a reference pulse for synchrotron light sources. Several synchronization systems between an ultrashort pulse laser and synchrotron light source have been realized and used for estimation of the duration of pulses from some FELs [6–8]; however, it is always challenging to synchronize such very different light sources within femtosecond timing jitter.

The waveform of spontaneous radiation emitted by a relativistic electron in the undulator is basically defined by the undulator magnetic field. The number of permanent magnets and the gap between magnets define the number of oscillations and the carrier wavelength of the waveform, respectively. In the UVSOR-III synchrotron light source, there is a tandem undulator that can produce two wave packets. The wavelength can be scanned from XUV to the visible region. By changing the gap between the permanent magnets of each undulator, it is possible to change the wavelength of each wave packet individually. A phase shifter, which consists of three pairs of electromagnets and forms a small chicane for the electron beam, between the undulators can control the delay between wave packets in the femtosecond regime with an attosecond accuracy. The system was applied for coherent control of atoms and molecules [9–11]. It is important to characterize the electric field produced in the undulator for such experiments.

We recently reported linear interferometric autocorrelation measurements in the UV region for the spontaneous radiation from the tandem undulator of UVSOR-III [12]. The shapes of the measured autocorrelation traces were well reproduced by the calculations, assuming that the wave packet had the form of a double-pulsed 10-cycle sinusoidal wave. However, the waveform of the wave packet cannot be directly reconstructed only by the autocorrelation trace. Thus, for accurate measurement of the waveform of the electric field emitted by a single electron passing through the undulator, it is essential to introduce a pulse characterization method that allows determining both the spectral phase and amplitude of the light pulse.

In this paper, we report the characterization of the electric-field waveform of the spontaneous radiation from the undulator using an algorithm of spectral phase interferometry for direct electric-field reconstruction (SPIDER) [13,14]. To our knowledge, it is the first time to characterize the UV and XUV electric fields of spontaneous radiation produced by an accelerated electron in the undulator without assuming the shape of the waveform.

2. SPIDER OF SYNCHROTRON RADIATION

SPIDER is a well-established femtosecond pulse characterization method invented in 1998 [13]. The concept of the SPIDER is the retrieval of the spectral phase by analyzing the fringes of the interferogram between the test pulse and a spectrally sheared replica. To obtain such a pair of pulses, a strongly chirped pulse is prepared, and sum frequency between the chirped pulse and test pulse at two different delay times is taken. In this way, two delayed pulses with slightly different center wavelengths, namely, spectrally sheared replicas of the original pulse, are obtained. The fringe deviation of the interferogram from that of two delayed pulses with the same wavelength, namely, zero sheared replicas, corresponds to the derivative of the spectral phase with the frequency, namely, the group delay. By integrating the group delay with the frequency, the spectral phase is obtained, and by calculating the Fourier transform of it together with the power spectrum, it is possible to reconstruct the time-domain picture of the electric field of the test pulse.

SPIDER was also applied for characterization of XUV pulses generated from a seeded FEL by seeding a delayed twin laser pulse [15,16]. However, it has not been applied for characterizing spontaneous radiation emitted by a relativistic electron.

The twin tandem undulator generates a pair of wavelength shifted wave packets with some delay. The interferogram between the pair of the wave packets can be considered as a SPIDER interferogram. Therefore, we can apply the same algorithm as the SPIDER to the interferogram to reconstruct the electric field generated from the undulator.

The important difference from the electric-field reconstruction of ultrashort laser pulses is that synchrotron radiation is longitudinally incoherent. In general, many electrons (${\sim}{10^9}$ in the current case) compose an electron bunch with the duration of a few hundred picoseconds, and circulate together in the storage ring. Each electron in the bunch produces its own wave packet of light, and these wave packets do not interfere with one another. As a result, the generated light pulse consists of randomly superimposed ${\sim}{10^9}$ wave packets within the duration of a few hundred picoseconds. However, each wave packet is supposed to be identical to each other when the effect of electron beam properties, such as energy spread and angular divergence, on waveform shapes [9] is sufficiently weak. This condition holds in the UV and XUV wavelength ranges in the UVSOR-III synchrotron, as demonstrated in optical [12] and quantum [9,10] interference measurements. Thus, by using the SPIDER analysis, we obtain the electric field of the wave packet generated from a single electron. For example, at BL1U of UVSOR-III, a 10-cycle wave packet, corresponding to ${\sim}{1.2}\;{\rm{fs}}$ pulse when operated at 35 eV, is produced. We aim to characterize such a waveform.

3. EXPERIMENTAL

The experiment was carried out at the 750-MeV UVSOR-III storage ring [17]. Figure 1(a) shows a schematic illustration of the tandem undulator installed in the ring. The tandem undulator consisted of twin APPLE-II type variable polarization devices that were operated in the horizontal linear polarization mode. The number of magnetic periods and the period length of the undulators were 10 and 88 mm, respectively. We performed experiments at two different wavelength ranges, UV and XUV. The central photon energy of the fundamental radiation was adjusted to 3.5 and 35 eV, which corresponds to ${{K}}$ values of 5.7 and 1.2, respectively.

 

Fig. 1. Schematic of the experiment. (a) Tandem undulator in the UVSOR-III synchrotron, consisting of two APPLE-II undulators; each relativistic electron in the bunch emits a pair of 10-cycle light wave packets. The undulators were set to horizontal linear polarization mode. The delay time between the wave packets is controlled by the phase shifter magnet between the two undulators. The light wave packets are randomly distributed within the overall pulse length of 300 ps (FWHM), reflecting the length of an electron bunch in the storage ring. (b) Setup for frequency-domain interferometry in the UV region. The UV spectrum of wave packets was measured by using a grating spectrometer. (c) Setup for photoelectron spectroscopy in the XUV region. The XUV spectra were derived from the photoelectron spectrum of helium. A hemispherical electron energy analyzer was used to measure the photoelectron spectrum.

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The spectral width of the radiation pulse was about 10% of the central frequency, which translates to about 10 periods in the time domain. The electron beam current was 30 mA, which corresponds to ${10^9}$ electrons per bunch. Each relativistic electron in the bunch that passes through the tandem undulator emits a pair of wave packets whose waveforms are expected as time-separated 10-cycle oscillations, with longitudinal coherence between them. The spacing between wave packets was tuned with attosecond precision by the phase shifter magnet that controls the electron path length between undulators; this technique is used for a beam energy measurement [18] or the cross undulator system [19].

Figure 1(b) shows the experimental setup used for the UV wavelength range. When we characterized the UV light wave, the undulator radiation was extracted into the air through an ${\rm{A}}{{\rm{l}}_2}{{\rm{O}}_3}$ optical window, which filters out high harmonic radiations. The spectrum of the wave packet was recorded with a grating spectrometer for the wavelength range from 200 to 400 nm with resolution of 0.27 nm. The central part of the undulator radiation was introduced into the entrance slit (${\sim}5\,\, {{\unicode{x00B5}{\rm m}}}$) of the spectrometer located 7 m downstream from the middle point of two undulators.

To characterize the XUV light wave, we measured the photoelectron spectrum of helium by the irradiation of light [see Fig. 1(c)]. The photoelectron spectrum essentially reflects the spectral profile of the ionizing light, as photoionization is proportional to the power spectral density of the double-pulsed ionizing light field [20,21]. Compared to an optical method using a grating spectrometer, photoelectron spectroscopy has the advantage of applicability to XUV or shorter wavelength light. In the measurement, the central part of the undulator radiation was cut out by a 0.4-mm-diameter pinhole located 9 m downstream from the middle point of two undulators. After passing through the pinhole, a toroidal mirror focused the undulator radiation into the gas cell of a hemispherical electron energy analyzer. A 50-nm-thick Al filter was inserted in front of the gas cell, to eliminate the high harmonic components of the undulator radiation. The observation angle of photoelectrons was set to 55 deg with respect to the polarization vector of XUV light. The energy resolution of the analyzer was approximately 30 meV. The spectra of XUV light are derived by correcting the measured photoelectron spectra with the photon energy dependence of the photoionization efficiency obtained from the cross section data [22].

It is important to mention the time-domain view of the photoionization of an atom on interaction with a pair of XUV wave packets. A pair of XUV wave packets produces two photoelectron wave packets in free space. Photoelectron wave packets in pairs spread in free space due to dispersion and overlapping each other, leading to the appearance of interference fringes in spatial distribution. This wave packet interference can be observed as a fringe pattern in the energy or momentum distributions of photoelectrons [20,21]. The fringed profiles in the measured photoelectron spectra are thus obtained.

 

Fig. 2. UV power spectra of synchrotron radiation from the undulator in the downstream. To not generate any light from the upstream, the gap of the upper undulator was open (200.0 mm).

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4. ELECTRIC-FIELD RECONSTRUCTION

The UV spectra of synchrotron radiation from the undulator in the downstream are shown in Fig. 2. The wavelength of the UV wave packet can be tuned by changing the gap of the permanent magnet. The spectra are in the shape of squared sinc-function (${\sin}^2 x/{x^2}$), which indicates that the envelopes of the wave packets are rectangular. In this experiment, the gap was changed from 30.66 to 31.94 mm, which results in the center wavelength of the wave packet being tuned from 365 to 334 nm, respectively. We fixed the gap of the upstream undulator to 31.30 mm, corresponding to the center wavelength of 350 nm and scanned the gap of the downstream undulator to have a spectrally sheared replica.

In the case of the XUV experiment, we fixed the gap of the upstream undulator to 74 mm, corresponding to the center photon energy of 35 eV and scanned the gap of the downstream undulator to have a spectrally sheared replica.

A. Electric-Field Reconstruction of a Rectangular Envelope Pulse

Here we discuss the importance of phase retrieval to characterize a rectangular envelope pulse. As can be seen in Fig. 2, the measured spectrum is similar to a squared sinc-function, and it is easy to predict a rectangular envelope waveform in time domain. However, simple Fourier transformation of the spectrum (assuming a flat phase) never reproduces the right shape in time domain. The phases of the oscillating components in the tail of the spectrum flip by $\pi$ next to each other. Those phase flips are essential to reproduce the rectangular envelope of the pulse. Therefore, the spectral phase measurement is very important to discuss the time-domain waveform of the pulse that has such a spectrum.

In fact, the estimation of the phase difference between spectrally separated components is a challenge for ultrashort pulse characterization because the phase information is lost at the zero intensity region. Even by using SPIDER, it is not straightforward. It is necessary to use a large shear so that the interference between the two separated components can be recorded; however, the resolution of the retrieved spectral phase becomes worse with a larger shear. To avoid the trade-off, the multi-shear algorithm is very useful [23]. The concept of the multi-shear algorithm is that SPIDER spectra with several different shears are recorded and the spectral phase of the target pulse is obtained to be consistent for all shears in the least-squares sense. Here we have adopted the method to reconstruct the electric field of synchrotron radiation.

B. Experimental Results

The UV spectra of the synchrotron light generated from the tandem undulator with several different gaps of the downstream undulator are shown in Fig. 3. The delay between the two wave packets is controlled by using the phase shifter between the two undulators. By recording a reference interferogram with zero shear, the delay between the two wave packets is estimated as 90.25 fs. The minimum shear angular frequency $\Omega$ is $2\pi \times 1.15\,\,{\rm{THz}}$ (4.76 meV), which defines the frequency resolution of the measurement. The recorded shear frequencies are ${\pm}{2^n}\Omega$, where $n$ is from zero to 5. We measured 50 spectra for each shear and analyzed them with the multi-shear algorithm. The averaged spectral phase and power spectrum of the upstream undulator beam is shown in Fig. 4(a). The $\pi$ phase flips at each (nearly) zero-crossing point of the spectra are clearly seen, which corresponds to a step-function-like spectral phase. The phase offset is set to zero at the carrier frequency, 3.6 eV, since it cannot be determined with this scheme. The retrieved electric field is shown in Fig. 4(b). The rectangular envelope of the wave packet is very well retrieved.

 

Fig. 3. UV power spectra of pairs of wave packets generated from the twin undulator. The delay between the two wave packets was set to 90.25 fs. The frequency shifts of the downstream wave packet are (a) $\Omega$, (b) ${-}4\Omega$, (c) $16\Omega$, and (d) ${-}32\Omega$, where $\Omega = 2\pi \times 1.15\,\,{\rm{THz}}$.

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Fig. 4. (a) UV power spectrum of synchrotron light generated from the upstream undulator where the gap is 31.30 mm (shaded curve) and the spectral phase retrieved from the multi-shear algorithm. The numerically simulated power spectrum and spectral phase are also shown as a gray dashed curve and gray dotted curve, respectively. The phase offset is set to zero at 3.6 eV. (b) Waveform of the wave packet retrieved from the power spectrum and spectral phase shown in (a). The numerically simulated waveform is also shown as a gray curve.

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We performed basically the same experiment for XUV. The delay between the two wave packets is estimated as 9.00 fs. The minimum shear angular frequency ${\Omega _X}$ corresponds to a photon energy of 27 meV. The recorded shear angular frequencies are ${\pm}{\Omega _X}$, ${\pm}2{\Omega _X}$, ${\pm}5{\Omega _X}$, ${\pm}11{\Omega _X}$, and ${\pm}23{\Omega _X}$. We measured a photoelectron spectrum for each shear and analyzed them with the multi-shear algorithm. The spectral phase and power spectrum of the upstream undulator beam is shown in Fig. 5(a). The $\pi$ phase flips at each (nearly) zero-crossing point of the spectra are also clearly seen. The phase offset is set to zero at the center photon energy, 36 eV. The retrieved electric field is shown in Fig. 5(b). The rectangular envelope of the wave packet is very well retrieved.

 

Fig. 5. (a) XUV power spectrum of synchrotron light generated from the upstream undulator where the gap is 74 mm (shaded curve) and the spectral phase is retrieved from the multi-shear algorithm. The numerically simulated power spectrum and spectral phase are also shown as a gray dashed curve and gray dotted curve, respectively. The phase offset is set to zero at 35 eV. (b) Waveform of the wave packet retrieved from the power spectrum and spectral phase shown in (a). The numerically simulated waveform is also shown as a gray curve.

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C. Numerical Simulations

We performed numerical simulations for the electric-field generation from a relativistic electron. We calculated the electron trajectories in the undulator magnetic field estimated from the actual arrangement of the permanent magnets. The time-domain picture of the electric field emitted by the electron was computed by means of the Lienard–Wiechert field [24]. Performing Fourier transform on the electric field, we obtained power spectra and spectral phases shown as gray curves in Figs. 4(a) and 5(a), respectively. Filtering out the higher harmonic components and performing inverse Fourier transform, we obtained the waveforms in time domain shown as gray curves in Figs. 4(b) and 5(b), respectively. The $\pi$ phase flips at the zero-crossing points of the power spectra and 10-cycle rectangular envelope waveforms in time domain are consistent with the experimental results.

D. Discussion

Neither of the retrieved waveforms in the experiments and numerical simulations has a perfectly rectangular shaped envelope; there are some temporal wings outside the main part, although it is physically impossible for the electron to radiate light outside the undulator. The reason for the presence of the temporal wings is that only the fundamental radiation was observed in the experiment. In the present experimental conditions, the undulator radiation actually contains high harmonics, and the waveform of the wave packet should be described by a 10-cycle spiky field [25,26]. However, the waveform of the fundamental radiation should be observed as sinusoidal since we eliminate the high harmonic radiations, which contribute to the sharp structures of the wave packet in time domain. At the same time, high harmonic radiations contribute to the sharp rise and fall of the waveform; therefore, the elimination of high harmonic radiation also results in the appearance of some temporal wings in the waveform.

Here we discuss where the measured point of the electric field is. In the case of laser pulse characterization with SPIDER, the recorded spectral phase reflects the spectral phase of the pulse where the sheared replica is produced, namely, where nonlinear wavelength conversion takes place [27]. In our current experimental condition, the sheared replica is produced at the downstream undulator; therefore, we are supposed to measure the spectral phase of the upstream wave packet at the time when the other wave packet is generated in the downstream undulator. Any filtering or dispersive elements after the point do not affect the measured spectral phase but affect the power spectrum. This is the reason that we do not observe any chirp on the recorded electric field even though the UV light passes through several meters of air and several millimeter thick windows, which filter out high harmonic radiations. From the above discussion, the waveform retrieved with the SPIDER method corresponds to the waveform of the fundamental radiation from an electron accelerated with the undulator.

5. CONCLUSION

In conclusion, we have succeeded in the measurement of the electric field of synchrotron radiation by using the multiple-shearing SPIDER algorithm; 10-cycle rectangular envelope UV and XUV electric fields are retrieved. The pulse durations of the UV and XUV wave packets were estimated as 12 and 1.2 fs, respectively. The shape is predicted by a simple theory based on the interaction between a magnetic field and a relativistic electron. The result indicates that each electron in the bunch produces exactly the same electric field. This feature is one of the largest differences from standard incoherent light such as radiation from an incandescent lamp. If each electron in the bunch produces a different wave packet from each other, the SPIDER interferogram never shows up. The SPIDER interferogram is strong experimental evidence of the coherence of synchrotron radiation from an undulator.

Since this method does not need nonlinear wavelength conversion, it would be very useful for shorter wavelength synchrotron radiation such as soft and hard $\rm x$ rays. It is also interesting to apply the method for the characterization of FEL pulses. Although SPIDER itself was applied for the characterization of the XUV pulses generated from a seeded FEL by seeding a delayed twin laser pulse [15,16], our method can in principle be applied for self-amplified spontaneous emission (SASE) FEL, in particular, for two-color FELs [28].

The pulse duration of a hard x-ray pulse from SPring-8 Angstrom Compact Free Electron Laser (SACLA) [2], one of the SASE FELs, has been estimated using an autocorrelation measurement based on two-photon absorption with a tandem undulator [29]. It takes ${\sim}{{37}}\;{\rm{h}}$ to record a single autocorrelation trace because of the weak nonlinear signal in this wavelength range, and it is necessary to assume the shape of the pulse for estimation of pulse duration. We expect that the SPIDER method can dramatically change the situation, since no nonlinear effects are used. The signal can be so large that single shot measurements are possible [30,31]. In addition, we are able to characterize the waveforms of XFEL pulses without any assumption of the pulse shape. Such a dramatic improvement of XFEL pulse characterization has a great impact not only on the development of the XFEL, but also on its applications, such as attosecond science, high field physics, and protein crystallography.