We present a theoretical investigation of the isolated extreme-ultraviolet (XUV) attosecond pulse propagating in the ionizing gas induced by a synchronized strong infrared (IR) laser, with the numerical solution of the nonadiabatic one-dimensional propagation model. Upon scanning the relative delay between the XUV and the IR pulse, it is found that the delay-dependent XUV transmission spectrogram exhibits the unique pattern that is controllable by the chirp of the XUV pulse. Furthermore, we demonstrate that this kind of spectrum modulation can be attributed to the term of the light energy loss involved in the propagation equation. The characteristics of the spectrum modulation dependent on the XUV chirp might provide an all-optical way for the reconstruction of the XUV spectral phase.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
The advent of the attosecond extreme-ultraviolet (XUV) pulse promotes the active research for the real-time observation of the electron dynamics in matters. The implementation is usually achieved by several experimental techniques, for example, the attosecond photoelectron spectroscopy , the attosecond photoion spectroscopy  and the attosecond transient absorption spectroscopy [3–5]. The characterization of such pulses always makes use of the cross-correlation scheme where an IR pulse is synchronized to the XUV pulse with a controllable delay, due to the low photon flux of the XUV pulse. Many algorithms have been proposed to retrieve the spectral phase of isolated attosecond pulses or attosecond pulse trains, such as Reconstruction of Attosecond Beating by Interference of two-photon Transitions (RABBIT) , Frequency Resolved Optical Gating for Complete Reconstruction of Attosecond Bursts (FROG-CRAB) , Spectral Interferometry for Direct Electric Field Reconstruction (SPIDER) , Phase Retrieval by Omega Oscillation Filtering (PROOF)  and recently proposed and successfully performed the Multi-line Volkov-transform Generalized Projection Algorithm (ML-VTGPA) . These methods are mostly combined with the measurement of the kinetic energy distribution of the photoelectron released by the XUV pulses in the presence of an IR laser field. By varying the temporal delay between the two pulses, the photoelectron spectra can be modulated by the IR laser field, and these photoelectron spectra were used to retrieve the temporal intensity envelop and spectral phase of the attosecond pulse and a complete characterization of the IR laser field. However, some approximations are used to retrieve these attosecond pulses, for example, the central momentum approximation used in FROG-CRAB method, which is valid when the bandwidth of the XUV attosecond spectra is much smaller than the central photon energy of the attosecond pulse. This approximation puts a limit on the XUV pulse that can be characterized. Moreover, PROOF and ML-VTGPA algorithm enable to partly remove the limit so that much shorter XUV pulse can be characterized. Recently Gaumnitz et al successfully retrieved a 43-as soft X-ray pulse  based on an ML-VTGPA iterative algorithm, which is a great forward step of attosecond physics.
Various all-optical methods based on the measurement of the spectra have been proposed to achieve the characterization of attosecond pulse trains and isolated attosecond pulse [8, 11–14]. These optical methods are usually referred to as the in situ measurements, where the generation and characterization of the attosecond pulse are entangled. Another scheme reported in , based on the measurement of the spectra of the XUV photons produced by high-order harmonic generation in a gas medium by the combined action of an IAP and an intense IR pulse, can be used to characterize the spectral phase in the target region. These all-optical methods have important advantages of very high detection efficiency and high signal-to-noise ratio in comparison with the techniques based on the measurement of photoelectrons.
In this work, we suggest an all-optical method for characterizing isolated attosecond XUV pulse with high photon energy. A few-cycle, CEP stabilized, intense IR pulse co-propagates with an isolated attosecond pulse through an atomic gas, resulting in the spectra of this attosecond pulse modulated. It is found that the modulated spectra is strongly related to both the temporal delay between the two pulses and the chirp rate of the attosecond pulse. This method has very high detection efficiency and high signal-to-noise ratio. Besides, it is not limited by available optical components and no electron spectrometer is needed to collect photon-ionized-electrons. Similar to other all-optical methods for attosecond pulse characterization, a XUV or soft X-ray spectrometer is needed [8,12–14].
2. Theoretical model and numerical method
In tunneling ionization regime, the electron of an atom will be freed by the strong IR laser during about 1/10 optical cycle around the crest of the oscillating electric field , producing a very steep slope of electron density. When an XUV pulse co-propagates with this kind of IR laser, its spectra will be strongly modulated by participating in the ionization process . The tunneling ionization rate of atom illuminated by an intense laser field is usually described by the frequently used ADK theory .
In our study, the pulse evolution in an ionizing gas is governed by the nonadiabatic one-dimensional (1D) propagation equation, which considers the entire electric field waveform and is valid in the few-cycle regime. Within the coordinate framework moving at the pulse group velocity and by using the slowly evolving wave approximation, the 1D propagation equation for the IR and the XUV field reads :Eq. (1) is the source term, where is the plasma frequency, describing the rapid oscillations of electron density in a gas medium. Here, is the electron density determined from the ionization rate and the atomic density . The Fourier transform is denoted by . In extreme nonlinear optics regime, the nonlinear polarization  is described as , where is electron charge, is the electron density, is the electron displacement caused by the force of the total electric field. Usually the source term is calculated from the time-dependent current density . Similar to the reference , we include two contributions to this current density:. The plasma oscillation term arises from the generation of the free electrons under the total electric field and is responsible for the defocusing and self-phase modulation of the IR pulse. The absorption term describes the energy loss from the XUV pulse to the plasma. The plasma and the absorption terms are determined by:Eq. (3) and Eq. (4) are used to determine the source term in Eq. (2). The absorption term in Eq. (4) is usually very small and is always ignored in the previous work. However, we find that in the range of the parameters used in our simulation, the absorption term of the equation Eq. (4) is responsible for the modulation of the XUV pulse spectra, and in contrast the plasma term can be treated as a small disturbance. This will be discussed later.
After co-propagating through a certain length of medium, the XUV spectra is modulated and the change of the XUV spectra can be obtained by:Eq. (5) represents the difference between the transmitted XUV spectra with and without the IR effect. The final XUV electric field is calculated from equation Eq. (2). Hereafter the XUV spectrum modulation refers to .
Both the IR and XUV pulse are assumed to have a Gaussian temporal profile. The time-dependent vector potentials of a chirp-free IR Gaussian pulse and a chirped XUV Gaussian pulse can be written respectively as :Eq. (7) describes the time profile of an XUV pulse whose duration and peak intensity depend on the chirp rate , but the spectral profile and energy of the XUV pulse do not depend on . In our simulation, the Argon gas cell with a length of 0.1mm and a density of is used. The IR laser central wavelength is , the optical cycle is , the full width at half maximum (FWHM) is , and the intensity of the IR pulse is resulting the Keldysh parameter . Therefore the tunnel ionization is dominating. The XUV pulse intensity is, the FWHM is, and the central photon energy is . Since the absorption coefficients exhibits an almost constant and a small value around 180eV, the gas absorption can be ignored. On the other hand, the high harmonic spectrum from the Argon gas driven by the IR pulse is not in the frequency range we are discussing. Therefore the HHG components are not included in the propagation equation.
3. Results and discussion
We have first analyzed the respective role of the plasma oscillation and the light absorption in the XUV spectrum modulation after considering the propagation effect. Figure 1(a) and 1(b) show the transmitted spectrum of the XUV pulse as a function of the relative delay between the IR and the XUV pulse, calculated for (a) only using the plasma term and (b) only using the absorption term in Eq. (4). We can see a tilted modulation structure along the delay axis in Fig. 1(b), while the modulation does not appear in Fig. 1(a). The comparison shows that the predominant mechanism that contributes to the modulation of the XUV spectra is not the plasma effect but the absorption term in Eq. (4). The result can be understood as follows: For the laser parameters and gas density used in our simulation, the plasma frequency is estimated as , which is far away from the spectral region of the XUV pulse (centered at 187.7 eV with a bandwidth of 18.25 eV). As a result, the XUV pulse is not able to couple with the plasma oscillation and finally cannot be modulated by the electron plasma.
It is worthy further identifying for the absorption term in Eq. (4), which of the electron density in the numerator or the total electric field in the denominator has the direct impact on the delay-dependent XUV modulation. We point out that the time-dependent also depends on the through the ADK ionization rate. Therefore, we can calculate the transmitted XUV spectra by numerically removing or including the XUV electric field in the numerator term or in the denominator term . The result can be classified into three different cases: (i) In Fig. 2 (a), the XUV electric field is included in the calculation of the electron density but not included in the total field . Clearly, the XUV spectra is not modulated with respect to the delay. (ii) Fig. 2(b) shows the modulated XUV spectra under the condition that the XUV electric field is taken into account for the calculations of , but the is calculated from the IR field only. (iii) In Fig. 2(c), the XUV electric field is not included in the calculation of both the electron density and the. Again, the XUV spectra are not modulated. Comparing the Fig. 2(a) and the Fig. 2(c), we can conclude that if the XUV electric field is not contained in the , the transmitted XUV spectrum is not modulated regardless of the situation that whether the XUV electric field is taken into account for the calculation of the or not. Thus the absorption term in Eq. (4) representing the energy loss from the XUV pulse to the plasma induces the modulated XUV spectra. Clearly, Fig. 1(b) and Fig. 2(b) show that the modulation patterns have a relationship with the chirp of the XUV pulse. But how the energy loss term can reflect the chirp information of the XUV pulse?
To answer the above question, we rewrite Eq. (2) in the form of the integral that clearly shows the relationship between the modulated spectra and the chirped XUV pulse:Eq. (8) is obtained by keeping the first two terms of the Taylor series of Eq. (4) under the approximation together with assuming a homogeneous gas density and removing both the linear propagation term of the XUV pulse and the plasma term. It can be confirmed that Eq. (8) is a quite accurate approximation to Eq. (2). The integral in Eq. (8) represents the Fourier transform, revealing the frequency information of the integrand. Thus the half-cycle periodicity of the function and the frequency information of the XUV field can be mapped onto the modulated spectra. In the following, we will discuss the chirp effect of the chirped XUV pulse on these modulation patterns.
Figure 3 shows the modulated spectra of the XUV pulse, calculated with three different XUV chirp rates: (a) , (b) , and (c) , as a function of the temporal delay between the XUV and the IR pulse. Note that the negative (positive) delay represents the XUV pulse arriving after (before) the IR pulse. The fast modulation fringes at large delays are numerical artifact and can be removed when a higher XUV intensity is used for simulation. Additionally, the background occurring at delays O.C. are calculation noise and can completely disappear when a stronger XUV field is used (the results are not shown here). The absence of the XUV spectra at these delays indicates that the XUV spectrum is barely affected by the IR laser pulse since the IR laser field is zero. The increasing of the intensity of the XUV pulse also leads to a higher oscillation amplitude of the transmitted XUV spectra.
The modulation pattern of the transmitted XUV pulse is not washed out by the CEP uncertainties and by the focal volume averaging of the IR laser pulse. We also found that when the transform-limited duration of the XUV pulse is increased from 100as to 300as, the slope of the modulated XUV spectra becomes more flat, and the oscillation strength seems getting a little bit stronger.
Figure 3(a)–3(c) are the main results in our simulation. For the three cases, the transmission spectra of the XUV pulse are all modulated at the overlap region of the two pulses and the largest modulation position occurs around the time delayoptical cycle, corresponding to the peaks of the IR field. We find that the modulation pattern exhibits a different orientation which depends on the attosecond chirp. Along the photon energy axis, the most significant modulation occurs at about 180eV, different from the XUV central photon energy 187.7eV. To obtain more information on this modulation spectra, a mean photon energy at each time delay can be defined as :Eq. (5). We will show that this delay-dependent mean photon energy is directly related to the instantaneous frequency of the XUV pulse later . In order to reveal the essentials, we choose the branch of the modulated spectra around zero temporal delay, where the modulation is the strongest and the feature of the attosecond chirp effect is much clear. The mean photon energy versus the delay is plotted for five different XUV chirp rates: (black-squares), (red-cycles), (purple-lower-triangles), (green-rhombus), (blue-upper-triangles), as shown in Fig. 4. The parameters of the IR and XUV pulses are the same as the ones used in Fig. 1, except for the chirp rate of the XUV pulse. We can see from Fig. 4 that for a certain attosecond chirp rate the mean photon energy in Eq. (9) can be approximated as a straight line.
For the chirp-free XUV pulse, the absorption slope shown by Fig. 4 is zero, which means that different frequency components of the XUV pulse is modulated or absorbed equally. However, when the XUV pulse has negative chirp, the instantaneous frequency under the pulse envelop decreases with time, resulting in the more significant modulation on the lower frequency components than the higher frequency components, which gives the negative absorption slope of the XUV spectra, as shown in Fig. 3 (c) and Fig. 4. Figure 3 and Fig. 4 also indicate that the attosecond chirp rate and the modulation slope is one-to-one correspondence. Thus this modulation method can be used to retrieve the linear attosecond chirp.
The modulation pattern in Fig. 3 and the mean photon energy given by Eq. (9) provide a good approximation to the time-dependent instantaneous frequency of the XUV pulse. To further investigate the validity of this all-optical method, we calculate the modulation pattern for XUV pulse having group delay dispersion (GDD) and the third order dispersion (TOD) in the frequency domain. In this case, the XUV electric field is expressed in the frequency domain as :Eq. (10) has the same spectral profile as the expression given by Eq. (7). Figure 5(a)-(c) show the XUV spectra modulation patterns, calculated for this kind of XUV pulse with different chirp parameters.
Figure 5 (a) shows the spectra modulation for the XUV pulse with zero GDD, and the TOD is fixed at . Figure 5 (b) and (c) show the spectra for the XUV pulse with the same GDD but the opposite TOD: (b) and (c) respectively. Comparing the modulation patterns in Fig. 3(b) and Fig. 5(a), the old-moon-like modulation pattern in Fig. 5(a) confirms that the TOD could be revealed by our all-optical method. Because the GDD effect makes a pulse linearly chirped, thus, when GDD is superimposed to the XUV pulse with TOD, the old-moon-like modulation pattern shall be tilted, resulting in an asymmetric modulation, as shown in Fig. 5(b). The opposite sign of the TOD leads to the opposite orientation of the modulation patterns, as shown by the comparison of Fig. 5(b) and 5(c). Therefore, we can conclude from Fig. 5 that not only the linear chirp of the XUV pulse but also the higher order dispersion could be retrieved by our all-optical method.
The validity of our all-optical method has been demonstrated, but what is the limitation of this model? Could this method be used for retrieving XUV pulse with very large chirp? To answer these questions, a further investigation is performed. It can be seen from Fig. 4 that the two straight lines for chirp rate parameters and show no significant difference (but still can be distinguished), and therefore a saturation effect may exist. In Fig. 4, each curve can be fitted with a straight line, whose slope is directly related to the chirp parameter. When we scan the chirp rate parameter, the slope as a function of chirp rate parameter could be obtained, as shown in Fig. 6. The parameters used in this simulation are the same as Fig. 4 except that the chirp parameter is scanned from −4 to 4. One can see that when the chirp parameter is larger than 3, the saturation effect becomes evident. Therefore, our model does not hold for the chirp of the XUV pulse larger than 3.
Though the IR laser intensity used in our simulation is relatively strong (), it still can be easily obtained by focusing the IR pulse with millijoule energy level to a spot size of hundreds of micrometers. The XUV pulse used in our simulation is very weak, and the XUV photon energy is in the soft X-ray region, which is not very hard to be generated through high-order harmonic generation from gas medium illuminated by a mid-infrared pulse .
Here we point out that when the central XUV photon energy 187.7 eV is increased to a larger one (370eV), the modulation pattern is still preserved. This indicates that our all-optical method has the potential to characterize the attosecond pulse even in the soft X-ray region.
We present a theoretical simulation of the interaction between the Argon gas medium with the combination of the chirped XUV attosecond pulse and an intense IR pulse, and suggest an all-optical method for the attosecond pulse characterization. The ionization process of the electron excited by the XUV and the IR pulse together can induce that the XUV spectra is modulated. By varying the time delay between the two pulses, this modulation process is controlled and exhibits a different pattern strongly dependent on the attosecond chirp, so that the chirp information may be encoded into these patterns. We also demonstrate that the third-order dispersion have a significant effect on the modulation pattern, from which we can retrieve the TOD information. To the best of our knowledge, methods for retrieving the XUV chirp higher than the quadratic phase has never been reported before. The photon energy of the XUV pulse used in our simulation is about 187.7eV, much larger than the XUV photon energy used in the transient absorption in references , extending the photon energy to soft X-ray regime. Thus this method may extend our ability to measure and retrieve the soft X-ray pulse generated from high harmonic generation process in the all-optical way.
National Natural Science Foundation of China (61690223, 11561121002, 61521093, 11127901, 11227902, 11574332, 11774363); Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB16); Youth Innovation Promotion Association CAS.
References and links
1. H. Wang, M. Chini, S. Chen, C. H. Zhang, F. He, Y. Cheng, Y. Wu, U. Thumm, and Z. Chang, “Attosecond time-resolved autoionization of argon,” Phys. Rev. Lett. 105(14), 143002 (2010). [CrossRef] [PubMed]
2. M. Sabbar, H. Timmers, Y.-J. Chen, A. K. Pymer, Z.-H. Loh, G. S. Sayres, S. Pabst, R. Santra, and S. R. Leone, “State-resolved attosecond reversible and irreversible dynamics in strong optical fields,” Nat. Phys. 13(5), 472–478 (2017). [CrossRef]
3. Z.-H. Loh, M. Khalil, R. E. Correa, R. Santra, C. Buth, and S. R. Leone, “Quantum state-resolved probing of strong-field-ionized xenon atoms using femtosecond high-order harmonic transient absorption spectroscopy,” Phys. Rev. Lett. 98(14), 143601 (2007). [CrossRef] [PubMed]
4. Y. Pertot, C. Schmidt, M. Matthews, A. Chauvet, M. Huppert, V. Svoboda, A. von Conta, A. Tehlar, D. Baykusheva, J.-P. Wolf, and H. J. Wörner, “Time-resolved x-ray absorption spectroscopy with a water window high-harmonic source,” Science 355(6322), 264–267 (2017). [CrossRef] [PubMed]
5. M. Schultze, E. M. Bothschafter, A. Sommer, S. Holzner, W. Schweinberger, M. Fiess, M. Hofstetter, R. Kienberger, V. Apalkov, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Controlling dielectrics with the electric field of light,” Nature 493(7430), 75–78 (2013). [CrossRef] [PubMed]
6. Y. Mairesse, A. de Bohan, L. J. Frasinski, H. Merdji, L. C. Dinu, P. Monchicourt, P. Breger, M. Kovačev, R. Taïeb, B. Carré, H. G. Muller, P. Agostini, and P. Salières, “Attosecond synchronization of high-harmonic soft x-rays,” Science 302(5650), 1540–1543 (2003). [CrossRef] [PubMed]
7. Y. Mairesse and F. Quéré, “Frequency-resolved optical gating for complete reconstruction of attosecond bursts,” Phys. Rev. A 71(1), 011401 (2005). [CrossRef]
8. E. Cormier, I. A. Walmsley, E. M. Kosik, A. S. Wyatt, L. Corner, and L. F. Dimauro, “Self-Referencing, Spectrally, or Spatially Encoded Spectral Interferometry for the Complete Characterization of Attosecond Electromagnetic Pulses,” Phys. Rev. Lett. 94(3), 033905 (2005). [CrossRef] [PubMed]
10. T. Gaumnitz, A. Jain, Y. Pertot, M. Huppert, I. Jordan, F. Ardana-Lamas, and H. J. Wörner, “Streaking of 43-attosecond soft-X-ray pulses generated by a passively CEP-stable mid-infrared driver,” Opt. Express 25(22), 27506–27518 (2017). [CrossRef] [PubMed]
11. C. Liu, Z. Zeng, R. Li, Z. Xu, and M. Nisoli, “Mapping the spectral phase of isolated attosecond pulses by extreme-ultraviolet emission spectrum,” Opt. Express 23(8), 9858–9869 (2015). [CrossRef] [PubMed]
12. K. Varjú, Y. Mairesse, P. Agostini, P. Breger, B. Carré, L. J. Frasinski, E. Gustafsson, P. Johnsson, J. Mauritsson, H. Merdji, P. Monchicourt, A. L’Huillier, and P. Salières, “Reconstruction of Attosecond Pulse Trains Using an Adiabatic Phase Expansion,” Phys. Rev. Lett. 95(24), 243901 (2005). [CrossRef] [PubMed]
13. N. Dudovich, J. Levesque, O. Smirnova, D. Zeidler, D. Comtois, M. Y. Ivanov, D. M. Villeneuve, and P. B. Corkum, “Attosecond Temporal Gating with Elliptically Polarized Light,” Phys. Rev. Lett. 97(25), 253903 (2006). [CrossRef] [PubMed]
14. N. Dudovich, O. Smirnova, J. Levesque, Y. Mairesse, M. Y. Ivanov, D. Villeneuve, and P. B. Corkum, “Measuring and controlling the birth of attosecond XUV pulses,” Nat. Phys. 2(11), 781–786 (2006). [CrossRef]
15. P. Corkum and F. Krausz, “Attosecond science,” Nat. Phys. 3(6), 381–387 (2007). [CrossRef]
17. B. M. Penetrante, J. N. Bardsley, W. M. Wood, C. W. Siders, and M. C. Downer, “Ionization-induced frequency shifts in intense femtosecond laser pulses,” J. Opt. Soc. Am. B 9(11), 2032–2040 (1992). [CrossRef]
18. E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. De Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, C. Altucci, R. Bruzzese, and C. de Lisio, “Nonadiabatic three-dimensional model of high-order harmonic generation in the few-optical-cycle regime,” Phys. Rev. A 61(6), 063801 (2000). [CrossRef]
19. M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz, and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83(15), 2930–2933 (1999). [CrossRef]
20. M. B. Gaarde, J. L. Tate, and K. J. Schafer, “Macroscopic aspects of attosecond pulse generation,” J. Phys. At. Mol. Opt. Phys. 41(13), 132001 (2008). [CrossRef]
21. H. Zhao, C. Liu, Y. Zheng, Z. Zeng, and R. Li, “Attosecond chirp effect on the transient absorption spectrum of laser-dressed helium atom,” Opt. Express 25(7), 7707–7718 (2017). [CrossRef] [PubMed]
22. K. W. DeLong, R. Trebino, and D. J. Kane, “Comparison of ultrashort-pulse frequency-resolved-optical-gating traces for three common beam geometries,” J. Opt. Soc. Am. B 11(9), 1595–1608 (1994). [CrossRef]
23. A. Suda and T. Takeda, “Effects of Nonlinear Chirp on the Self-Phase Modulation of Ultrashort Optical Pulses,” Applied Sciences 2(2), 549–557 (2012). [CrossRef]
24. L. Argenti, Á. Jiménez-Galán, C. Marante, C. Ott, T. Pfeifer, and F. Martín, “Dressing effects in the attosecond transient absorption spectra of doubly excited states in helium,” Phys. Rev. A 91(6), 061403 (2015). [CrossRef]