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Abstract
Efficient characterization method for broadband attosecond pulses has become more and more essential, since attosecond pulses with bandwidth spanning few-hundreds electron-volts have been generated. Here we propose a fast phase retrieval algorithm for broadband attosecond pulse characterization with an omega oscillation filtering technique. We introduce a new error function to improve the accuracy of the retrieved phases. More importantly, it can be solved by the steepest descent methods with iterative algorithm, which is much faster than genetic algorithm adopted previously. An experimental spectrogram for isolated attosecond pulses with photon energy covering 52–127 eV and a pulse width of 71 as was successfully retrieved with this method as demonstrated. The proposed technique will help provide real-time feedback on atto-chirp compensation for ultrashort isolated attosecond pulse generation.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Since its first demonstration in 2001 [1], isolated attosecond pulses (IAP) have shown powerful ability in probing ultrafast dynamics in atoms [2,3], molecules [4,5] and solids [6,7] with unprecedented time resolution. Together with the various pump-probe techniques such as attosecond streaking [8,9], attosecond transient absorption spectroscopy [10,11] and attosecond beating [12,13], the pulse duration of the IAP is the key to ensure the high temporal accuracy. Although IAPs with pulse duration of 53 as [14] and 43 as [15] have been demonstrated, ultrashort attosecond pulse generation which requires fine dispersion management is never an easy task. Therefore, quick phase diagnosis of IAP during experiments plays a pivotal role for dispersion compensation with either metal filters [16,17] or gas media [18,19].
Temporal characterization of attosecond pulses is in essence a phase retrieval problem since the spectrum is directly measurable. So far, the temporal characterization of IAP is mainly performed with attosecond streak camera technique [8,20], in which the correlation between the electron replica of the input IAP and a femtosecond streaking laser field yields a spectrogram, known as the streaking trace. With central momentum approximation (CMA), the streaking trace is simplified to have the same form as the trace of frequency resolved optical gating (FROG) [21] in femtosecond pulse measurements. Such that phase retrieval algorithm for FROG, i.e. principle component generalized projection algorithm (PCGPA) [22], can be applied for completely reconstruction of attosecond bursts (CRAB), as dubbed as FROG-CRAB [9]. Many other algorithms based on CMA, such as least-squares generalized projection algorithm (LSGPA) [23] and extended ptychographic iterative engine (ePIE) [24,25], have been developed to accelerate the phase retrieval process. The key point of CMA is to disentangle momentum and time in the spectrogram, so that fast fourier transform (FFT) algorithm can be applied. For ultrashort IAP with bandwidth larger than tens of electron volts, the CMA is no longer valid. To deal with the entanglement of momentum and time, generalized projection scheme based on Volkov transform (VTGPA) [26], direct optimization for parameterized pulses (PROBP) [27,28], neural network method [29,30] and one omega frequency filtering method (PROOF) [31,32] are proposed. The volkov transform in VTGPA and genetic algorithm adopted in PROBP and PROOF are very time consuming, and the phase retrieval process usually takes tens of minutes to converge. The neural network method, on the other hand, requires large number of experimental or numerical datasets which cover as many cases as possible for training. Therefore, a wieldy and fast characterization method for broadband attosecond pulses is needed, especially for quick feedback of atto-chirp compensation diagnosis in ultrashort IAP generation experiments.
In this work, we revisit the phase retrieval procedure in the PROOF technique, and propose a new error function, which avoids extraneous roots and improves accuracy of phase retrieval. Besides, the partial derivatives of the new error function are analytically available, which makes the IAP spectral phase solvable with fast steepest descent methods within less than a second. The proposed algorithm is extensively tested in terms of accuracy, robustness and streaking laser dependence, and an experimental spectrogram for isolated attosecond pulses with photon energy covering 52–127 eV and pulse width (full width at half maximum, FWHM) of 71 as was successfully retrieved with this method.
2. Method
In IAP characterization with streak camera technique, the energy spectrum of photoelectrons $I(\omega,\tau )$, namely the streaking trace, is generated during the ionization of target atoms with ionization potential $I_p$ by extreme ultraviolet (XUV) attosecond pulses $E_X(t)$, in the presence of an infrared (IR) femtosecond laser field $E_L(t)$ with central frequency of $\omega _L$ shifted by a time delay $\tau$. Note that $E_X(t)$ and $E_L(t)$ are expressed as scalars, since in typical experiments the XUV pulses and IR pulses are both linearly polarized and only photoelectrons along the polarization direction are collected. Both of the $E_X(t)$ and $E_L(t)$ can be reconstructed from the two-dimentional trace $I(\omega,\tau )$.
In PROOF characterization, the streaking laser field should be weak enough, so that only one IR photon is involved in the interaction between photoelectron and IR laser field. Thus, the XUV photon with energy $\Omega$ is only coupled by the streaking laser field to its adjacent photons separated by one IR photon, i.e. $\Omega \pm \omega _L$. So we take into account only these discrete XUV photons $\Omega _i=\Omega _0+i\omega _L$ with $\Omega _0$ being the first frequency point in spectrum range, and the corresponding spectral phase is $\phi _i=\phi (\Omega _i)$. Our goal is to find $\phi _i$, or equivalently the phase difference $\delta _i=\phi _{i-1}-\phi _i$. With weak streaking field, three transition pathways, as illustrated in Fig. 1(a), exist for the generation of photoelectrons with energy of $\omega _i=\Omega _i-I_p$ : the direct transition by absorbing one XUV photon with energy of $\Omega _i$, a two-step transition by absorbing one XUV photon $\Omega _{i-1}$ and one IR photon $\omega _L$, and the other two-step transition with absorbing one XUV photon $\Omega _{i+1}$ and emitting one IR photon $\omega _L$. The amplitudes of the three quantum paths $\tilde {P}_0$, $\tilde {P}_1$ and $\tilde {P}_2$ can be simplified with weak streaking field approximation as:
Fig. 1. The principle of PROOF. (a) the XUV photon with energy $\Omega$ is coupled by the streaking laser field to its adjacent photons separated by one IR photon, i.e. $\Omega \pm \omega _L$. Therefore, the modulation of $\Omega$ (one omega oscillation) as a function of the XUV-IR delay contains the information of phase difference between $\Omega \pm \omega _L$ and $\Omega$. (b) the definition of the phase $\alpha$ of one omega oscillation.
The interferences of the three pathways yield the streaking trace $I(\omega,\tau )$ as:
Expanding Eq. (2) will result in 9 terms, in which we focus on only the 4 terms with delay-dependent phase of $e^{j\omega _L\tau }$, i.e. the one omega frequency (OOF) component $I_1(\omega,\tau )=2\textrm{Re}[\tilde {P}_0^{*}\tilde {P}_1 + \tilde {P}_0\tilde {P}_2^{*}]$. To save the tedious mathematical derivation, we denote $\tilde {P}_0^{*}\tilde {P}_1$ and $\tilde {P}_0\tilde {P}_2^{*}$ as vectors $\vec {b}_1$ and $\vec {b}_2$ in the complex plane respectively, as shown in Fig. 1(b), and $\vec {b}$ is their vector sum. The arguments of $\vec {b}_1$ and $\vec {b}_2$ are $\theta _1=\omega \tau +\delta _i-\pi /2$ and $\theta _2=\omega \tau +\delta _{i+1}+\pi /2$ according to Eq. (1). The OOF can thus be written as:
where $A_i$ is the oscillation amplitude, and $\alpha _i$ is the phase of OOF. According to Fig. 1(b), $\alpha _i$ is defined by $\sin \alpha _i=(|\vec {b}_1|\sin \beta _1+|\vec {b}_2|\sin \beta _2)/|\vec {b}|$ and $\cos \alpha _i=(|\vec {b}_1|\cos \beta _1+|\vec {b}_2|\cos \beta _2)/|\vec {b}|$, where $\beta _1=\delta _i-\pi /2$ and $\beta _2=\delta _i+\pi /2$. Then it is intuitive to connect $\delta _i$ with experimental measured OOF phase $\alpha _i$:The phase difference $\delta _i$ of the XUV pulse can then be solved from the simultaneous equations described by Eq. (4). In PROOF, $\tan \alpha _i$ is used to define the OOF phase, and it may yield wrong solutions since tangent function confines $\alpha _i$ in $[-\pi /2,\pi /2]$ instead of $[-\pi,\pi ]$. We will see that later in our simulations. Whereas in iPROOF [32], $\delta _i$ is solved recursively with both $\tan \alpha _i$ and $A_i$. The OOF amplitude $A_i$ is usually noisy and not calibrated during the photoelectron detection with time-of-flight spectrometer. Furthermore, the retrieval error is accumulated during the recursion process. To overcome these drawbacks, we directly solve $\vec {\delta }=(\delta _0,\delta _1,\ldots \delta _N)$ from Eq. (4) by minimizing the following error function:
More importantly, the partial differential of $f(\vec {\delta })$ is analytically available:
3. Results and discussion
3.1 Validity
To test the qPROOF algorithm, we adopted the same simulation example as that in the PROOF demonstration [31]. The streaking trace shown in Fig. 2(a) is produced with ultra-broadband IAP with spectral profile and spectral phase shown in Fig. 2(b) and 20 fs near infrared (800 nm) streaking pulses with laser intensity of 1$\times$10$^{11}$ W/cm$^2$. The OOF phases are extracted from the streaking trace for PROOF and qPROOF algorithms shown as blue triangles and red circles in Fig. 2(c), respectively. Note that the OOF phases for PROOF are wrapped in $[-\pi /2,\pi /2]$, i.e. the shaded area in Fig. 2(c), by the inverse tangent function, while the qPROOF phases occupy the full range of $[-\pi,\pi ]$.
Fig. 2. The numerical demonstration of qPROOF. (a) The streaking trace produced with an ultra-broadband IAP under the streaking of 20 fs near infrared (800 nm) laser pulses with laser intensity of 1$\times$10$^{11}$ W/cm$^2$. (b) The complex spectral profile $U$, spectral phase $\phi _{Act}$ of the IAP used in trace simulation, and the retrieved spectral phase from PROOF (blue triangles) and qPROOF (red circles) algorithm. (c) The OOF phases extracted from the streaking trace for PROOF (blue triangles) and qPROOF (red circles) algorithms, which are reproduced by the PROOF (dotted line) and qPROOF retrieval (solid line). (d) The retrieved temporal IAP profile with either PROOF (dotted line) or qPROOF (dashed line) consists with the actual one (solid line).
GA and Quasi-Newton algorithm are employed to retrieve the IAP spectral phase from the PROOF phases and qPROOF phases, respectively. The retrieved OOF phases from PROOF (dashed line) and qPROOF (solid line) are shown in Fig. 2(c), which consist well with corresponding experimental OOF phases. And the retrieved IAP spectral phases with PROOF (blue triangles) and qPROOF (red circles) are compared with the real spectral phase in Fig. 2(b). The retrieved temporal IAP profile with either PROOF or qPROOF is compared with the actual one in Fig. 2(d). It is seen that both methods work accurately. However, the GA algorithm takes tens of minutes to converge, while qPROOF finishes in less than a second thanks to the analytical partial differentials described in Eq. (6).
The proposed qPROOF is superior to PROOF method not only in retrieval speed, but also in applicability, to demonstrate which we consider a special streaking trace shown in Fig. 3(a). This trace is generated with almost the same simulation parameters as Fig. 2(a) except that an abrupt phase change is introduced at 150 eV, as shown in Fig. 3(b). The spectral phase of IAP is particularly programmed so that the OOF phases have a phase jump of $\pi$ at 150 eV. The $\pi$ phase jump can be only seen by qPROOF but not PROOF, as indicated in Fig. 3(c). The OOF phases extracted by PROOF are the same as that in Fig. 2(c), so the retrieval results in the same spectral phase and temporal profile as the previous example, as shown in Fig. 3(b) and (d), respectively. On contrary, qPROOF finds the true spectral phase and temporal profile accurately and quickly since the $\pi$ phase jump is accounted.
Fig. 3. The numerical demonstration of qPROOF when large abrupt phase jump exists in the IAP. (a) The streaking trace; (b) the IAP spectral intensity (filled line), actual spectral phase (solid line), retrieved spectral phase with PROOF (blue triangle) and qPROOF (red circle); (c) the OOF phases extracted from the trace for PROOF (blue triangle) and qPROOF (red cirle), and the PROOF (dotted line) and qPROOF (solid line) retrieved OOF phases. (d) The temporal profile reconstructed with qPROOF (dashed line) consists with the actual profile (solid line), while PROOF (dotted line) fails.
According to the PROOF principle, both PROOF and qPROOF methods work for ultrabroadband IAP pulses with no limitations in duration or wavelength range. The ability of characterization of sub-50 as pulses of PROOF is already demonstrated previously [31], and the demonstration of PROOF for IAP with high photon energy up to water-window has also been reported [14]. The qPROOF method reported work on the same principle as PROOF and thus can retrieve IAP with ultrashort duration and super high photon energy.
3.2 Robustness
The robustness of the qPROOF algorithm, i.e. the retrieved pulse duration error versus the laser peak intensity, pulse duration, signal noise, time jitter and time step size, is numerically tested in this section. In the above simulations, the spectral shape is too complicate, such that the IAP temporal profile tends to have complex pedestals which makes FWHM pulse duration not a valid description of the pulse profile. Therefore, as shown in Fig. 4(a), a simpler broadband continuum together with a smooth spectral phase were employed for the robustness tests. The temporal FWHM of the IAP is 79 as.
Fig. 4. (a) The spectral intensity (solid line) and spectral phase (dashed line) used for the robustness tests. (b) The error of the qPROOF-retrieved IAP duration for various streaking pulses with duration of 2–10 fs and laser intensities of 1$\times$10$^{10}$–1$\times$10$^{13}$ W/cm$^2$, which suggests that pulse duration has hardly any influence on the retrieval as long as the pulse is longer than 2 fs. And the working laser intensity can go up to 1$\times$10$^{13}$ W/cm$^2$, which is sufficient for most experiments.
The key point of PROOF/qPROOF is to retrieve spectral phase from the OOF phases of the streaking trace, i.e. from the interferences between adjacent frequency components by absorbing/emitting an additional NIR photon. It is better to use narrow band NIR streaking field with well-defined NIR photon energy. Therefore, multi-cycle long pulses are preferred in principle. Furthermore, during the original deduction of PROOF, weak streaking field is assumed so that only one NIR photon absorption/emission is taken into account. Given all that, long and weak laser pulses are needed to perform qPROOF retrieval. However, in the attosecond streaking camera measurements, the streaking pulse, which is a replica of the pulse to generate isolated IAP, is usually few-cycle and has broadband spectrum. And the weak streaking field assumption could hardly be fulfilled neither since intenser streaking field will introduce larger energy modulation in the streaking trace and thus yields better signal-to-noise ratio. Therefore, it is necessary to find the shortest pulse duration and highest laser intensity of streaking laser pulses with which qPROOF is valid.
Streaking traces were simulated with streaking pulse durations of 2–10 fs and laser intensities of 1$\times$10$^{10}$–1$\times$10$^{13}$ W/cm$^2$. The error of the retrieved IAP duration with qPROOF for each trace is shown in Fig. 4(b), which suggests that pulse duration has hardly any influence on the retrieval as long as the pulse is longer than 2 fs. And the working laser intensity can go up to 1$\times$10$^{13}$ W/cm$^2$, which is sufficient for most experiments.
Aside from the streaking laser parameters, the data noise from hardwares, e.g. streak camera noise, pump-probe time jitter and largish step size, are also practical parameters that need to be investigated. In Fig. 5, we show the phase retrieval of IAP from noisy traces under 7 fs, 3$\times$10$^{11}$ W/cm$^2$ streaking field with either 20% electron count noise [Fig. 5(a)], 160 as time jitter [Fig. 5(b)] or 300 as step size [Fig. 5(c)], the retrieved pulse profile is almost identical to the noise-free trace. Further increasing the noise will result in distorted temporal profile. If the three noises coexist, the trace become more realistic, as shown in [Fig. 5(d)]. With the proposed algorithm, the main feature of the temporal profile can still be reconstructed satisfactorily. Therefore, the algorithm can handle very noisy data, and shows good robustness.
Fig. 5. The phase retrieval of IAP from noisy traces under 7 fs, 3$\times$10$^{11}$ W/cm$^2$ streaking field with either 20% electron count noise (a), 160 as time jitter (b) or 300 as step size (c), the retrieved pulse profiles (e–g, solid lines with circle markers) are almost identical to that retrived from noise-free traces (e–g, solid lines with triangle markers). If the three noises coexist (d), the main feature of the temporal profile can still be reconstructed satisfactorily (h, solid line with circle marker).
At last, to ensure the accuracy and correctness of the retrieval results, the carrier envelope phase (CEP) of the few-cycle driving laser pulses, which is critical in IAP generation and characterization according to the three-step model [34] and streak camera principle, needs to be discussed. The spectrum as well as the phase of the IAP is sensitive to the waveform of the laser field, so the IAP temporal profile depends on the CEP of laser. To obtain stable IAP with unchanged pulse width, either CEP stabilized laser system or generalized double optical gating technique [35] should be adopted.
3.3 Experimental data
Finally, qPROOF technique is tested with an experimental streaking trace of IAP generated with double optical gating (DOG). The details of the experimental setup is described in Ref. [36]. The IAP was generated from 50 mbar neon atoms with 5 fs NIR few-cycle laser pulses centered at 750 nm with peak intensity of 1.8$\times$10$^{15}$ W/cm$^2$. The IAPs, after passing through 200 nm zirconium filters, were then converted to photoelectron bursts on a neon gas jet. The energy spectra of photoelectrons were modulated by the streaking laser pulses with intensity of 1$\times$10$^{12}$ W/cm$^2$ and detected by a time-of-flight (TOF) detector. The pulse duration of the streaking pulses is 5 fs as well, which is adequate for successful characterization with qPROOF according to our previous simulations.
The IAP streaking spectrogram measured by TOF is shown in Fig. 6(a), and the IAP spectrum extends from 52 eV to 127 eV, as shown as filled line in Fig. 6(b). Both the qPROOF and PROOF algorithms were employed to retrieve the IAP phases. The qPROOF converged in 0.2 seconds with retrieval error (Eq. (5)) of 0.38, while PROOF took 1041.9 seconds, for the evolution of 100,000 generations with population of 2000, to reach a retrieval error of 1.097. Therefore, in addition to significant accuracy improvement, qPROOF provides more than 5000-fold speed increase over genetic algorithm based PROOF. The retrieved spectal phases ($\phi$) of PROOF (dotted line) and qPROOF (solid line) are shown in Fig. 6(b) as well. The two spectral phase curves are quite consistent with each other except for slight discrepancy in spectrum range above 100 eV. The OOF phases extracted from the experimental trace for PROOF (triangle dots) and qPROOF (circle dots) are compared with the corresponding retrieved ones with PROOF (dotted line) and qPROOF (solid line), as shown in Fig. 6(c), which indicates satisfactory convergence for the two algorithms. With the spectral profile and retrieved IAP phases, the temporal profile of the IAP can be immediately obtained with Fourier transform. As shown in Fig. 6(d), the FWHM of reconstructed IAP with qPROOF (solid line) is 71 as, while PROOF (dashed line) suggests a slightly narrower pulse width. The generation conditions and pulse duration of the demonstrated IAP are very close to the reported shortest IAP generated with 800 nm driving laser [37]. Furthermore, the successful retrieval with streaking laser intensity as high as 1$\times$10$^{12}$ W/cm$^2$, which is much more intense than weak perturbation required by one photon coupling according to PROOF principle, shows great practical utility since prominent energy streaking can be introduced. Especially for ultrabroadband IAP, the streaking is usually not evident with weak streaking pulses. The simulations in Fig. 4 and the experimental demonstration in Fig. 6 show that near non-perturbative laser intensity can be used to achieve good data quality.
Fig. 6. (a) The experimental IAP trace; (b) the photon spectrum (filled line) and retrieved spectral phases with PROOF (dotted line) and qPROOF (solid line); (c) the experimental extracted (Exp.) and retrieved (Retr.) OOF phases with PROOF ($\alpha _{P}$) and qPROOF ($\alpha _{qP}$); (d) the retrieved attosecond pulse temporal profiles with PROOF (dashed line) and qPROOF (solid line), and retrieved temporal phases for PROOF (dashed line with triangle scatter) and qPROOF (solid line with circle marker).
4. Conclusions
In conclusion, we propose a quick PROOF procedure termed as qPROOF for the characterization of ultrabroadband IAPs. Owing to the new error function in accompany with its analytical partial derivatives, the phases can be retrieved in less than a second, which allows for quick diagnosis during the IAP generation experiments. Moreover, qPROOF avoids the wrap-to-$\pi$ operation introduced in PROOF, therefore, it is capable of working with complex phase jumps that PROOF may fail to retrieve. The qPROOF algorithm is numerically tested and proves to be robust against the pulse width and intensity of streaking laser, electron detection noise, pump-probe delay jitter and large scanning step. An experimental streaking trace generated with 71 as IAP was successfully retrieved with qPROOF. The quick IAP reconstruction algorithm is of great importance for ultrashort IAP generation.
Funding
National Natural Science Foundation of China (11974426, 12234020); National Key Research and Development Program of China (2019YFA0307703).
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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