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Abstract
We theoretically investigate high-order harmonic generation (HHG) from solids in two-color fields. It is found that under the premise of maintaining the same amplitude, the intensity of the second plateau can be enhanced by two to three orders in a proper two-color field compared with the result in the monochromatic field with the same frequency as the driving pulse of the two-color field. This can be attributed to the fact that most excited electrons can be driven to the top of the first conduction band due to the larger vector potential of the two-color fields, which leads to the higher electron population of upper conduction bands. Moreover, we also find that isolated attosecond pulses can be generated from solids by choosing a proper two-color field that allows the electrons to reach the top of the first conduction band only once. This work provides a promising method for extending the range of solid HHG spectra in experiments.
© 2017 Optical Society of America
1. Introduction
High-order harmonic generation (HHG) in the gas phase has been extensively studied experimentally and theoretically over the past several decades [1–4]. It provides a significant method to generate attosecond pulses and a source of coherent soft X-ray radiation [3–6]. However, applications of the HHG are restricted by the low conversion efficiency. Due to the high density of solids, it has great potential to break through this bottleneck. Therefore, many researchers turn attention to the HHG from solids, and many remarkable achievements have been obtained [7–18]. In the past few years, solid HHG has been observed in many kinds of crystals and many properties different from gaseous HHG were found such as insensitive ellipticity dependence and linear relation between the cutoff energy and the electric field of the driving laser [7–9]. In the experiments, a main plateau was measured but theoretical simulations [19, 20] predicted the existence of a second plateau. This prediction was confirmed by the later experiments performed on solid rare gases [18] (the rare gases were condensed into a structure similar to the molecular crystal) and single-crystal MgO [21]. The experiments show that the intensity of the second plateau is much lower than that of the first and it becomes distinct only in a very intense laser, which may be the reason why the second plateau is not observed in previous experiments. It also shows the second plateau has broader spectrum and higher cutoff energy [18–22], which means it has brighter prospects for applications. So enhancing the intensity of the second plateau becomes a valuable subject.
Theoretical investigations reveal that the first plateau mainly comes from the transition between the active valence band and the first conduction band, while the second plateau results from transitions from higher conduction bands to the active valence band [19]. So, the intensity of the second plateau can be enhanced by increasing the population of higher conduction bands. On the other hand, solid systems have two special properties. Firstly, electrons have larger probabilities of transition to the higher conduction band as they reach the top of a conduction band [23]. Secondly, a semi-classical theory suggests that the motion of a Bloch electron wave-packet on ε-k plane within a conduction band can be determined by [11, 13, 22, 24, 25]
Here, εn(k), k, k0 and A(t) are the band energy, the crystal momentum, the initial crystal momentum and the vector potential of the external field, respectively. This means that the population of higher conduction bands will be increased, if the vector potential of the external field is sufficiently large so that most excited electrons can reach the top of C1. Therefore, the intensity of second plateau should be enhanced by adopting laser pulses with larger vector potential.In general, vector potential can be effectively enhanced by increasing the intensity of the driving laser. It has been confirmed by previous studies that the intensity of the second plateau can indeed be enhanced by this scheme [21]. Unfortunately, an over-intense laser can create a large number of carriers and when the carrier density reaches a threshold, the stability of the lattice will be broken. Thus solid HHG was restricted by the low damage-threshold intensity. In order to avoid this limit, Du et al. proposed using spatially inhomogeneous fields to enhance the second plateau of solid HHG [26]. However, this scheme has two shortcomings. First, the generation of regular metal-crystal nanostructures (the generator of inhomogeneous fields) is not easy for most ultrafast laboratories. Second, the nanostructure will be damaged after it interacts with lasers many times [27]. In the past twenty years, two-color and multi-color fields have been successfully used to control the electron dynamics processes in atom and molecule systems [28–39]. Such a scheme can effectively enhance the yield and the cutoff energy of HHG or generate attosecond pluses, because it can modify the ionization, acceleration and recombination processes [28–37]. In solid cases, it can also be used to control the interband and intraband processes [15]. Moreover, the large vector potential can be obtained under the same peak electric field by adopting proper two-color fields. Thus the two-color field method has great potential to enhance the second plateau in solid HHG spectra without damaging the structure of solids. However, to the best of our knowledge, there is little research on this aspect.
In this work, we theoretically investigate the enhancement of the second plateau in the solid HHG spectrum using two-color laser fields. Firstly, we manage to find a two-color field that has the same peak electric field as a monochromatic field but has a larger peak vector potential. Then it is confirmed that the second plateau can be effectively enhanced by the two-color field. Secondly, we recover the physical mechanism that the second plateau can be enhanced in such a two-color field. Finally, we propose a method for generating isolated attosecond pulses from solid harmonics using multicycle two-color fields.
2. Theoretical methods
In our simulation, the interaction of lasers with solids is modeled by 1D time-dependent Schrödinger equation (TDSE) in length gauge and dipole approximation. The corresponding time-dependent Hamiltonian can be written as (the atomic units are used throughout unless otherwise stated)
where and V(x) is a periodic lattice potential. In current work, the solid structure is simulated by the Mathieu-type potential: V(x) = −V0[1 + cos(2πx/a0)] with V0 = 0.37 a.u. and lattice constant a0 = 8 a.u. [40]. The two-color laser field is expressed as here Ei and ωi (i = 1, 2) are peak electric fields and frequencies, respectively. The envelop function f (t) is the cos2 envelop and τdelay is the delay time between the two laser pulses. In the following, we will use T1 and T2 to represent the optical cycles of the driving pulse and the controlling pulse, respectively.In order to determine the initial state, we solve the time-independent Schrödinger equation in the space region [−400, 400] a.u. using 8001 points
where εn and ψn(x) are the eigenvalues and the eigenstates of this system, which can also be used to analyze the electron dynamics processes when the solid interacts with the laser pulses.When electrons valence band interact with the laser pulse, they have probability to tunnel to conduction bands. Because only a small portion of electrons near the top of the second valence band can effectively tunnel to conduction bands [26], we choose the highest energy eigenstate on valence bands as the initial state. In our calculations, we employ the Crank-Nicolson method [41] to calculate the time dependent wave function
The time step is set as 0.0889 a.u., which is sufficient to obtain converged results. While the system is evolving, the electron wave-packet may interact with the boundary, which can generate spurious reflections. In order to avoid this, we multiply the wave function by a mask function of the form cos1/8 with |x| > 380 a.u. at every time step. After obtaining the time dependent wave function, the laser induced currents can be calculated bySince the intensity of the second plateau is very low in monochromatic fields, we multiply the time dependent currents by a Hanning window function to increase the signal-noise ratio before the Fourier transform. Finally, the HHG spectrum is obtained by Fourier transform of the time dependent currents.
3. Results and discussions
In order to demonstrate our theoretical framework, we discuss the semi-classical motion of a Bloch electron in a laser field according to Ref. [24]. As shown in Fig. 1, there are three possible ways to transit the electron from the first conduction band to the second conduction band after the electron tunnels to the first conduction band. The first way is that the electron directly transits to C2 through the path one, but this way can be neglected because of the extremely large energy gap. Therefore, the electron driven by the laser field will move along C1. When the vector potential of the laser field is weak, the electron cannot arrive at the top of C1. The electron is mainly transited to C2 through the path two. In this case, the transition probability is still low because the energy gap is still large, though it is smaller than that of the first way. However, when the vector potential of the laser field is large enough, the electron can reach the top of C1. In this case, the third way becomes the main path since the energy gap of this path is smallest. Hence, the second plateau in solid HHG spectra can be increased by adopting a two-color field with larger vector potential.
Fig. 1 The schematic plot of the motion of a Bloch electron. Three different transition paths from C1 to C2 are shown. The band structure is obtained from the Bloch theory.
To confirm the above conjecture, we calculate the harmonic spectra of the 1D solid under two-color fields with different relative phases. For the driving pulse of the two-color fields, the peak electric field, the wavelength and the duration are set as 0.005408 a.u., 3220 nm and 8T1. The parameters of the controlling pulse are E2 = 0.25E1, ω2 = 3ω1, and the duration is 24T2. The relative phases can be changed by changing τdelay in Eq. (3). In Fig. 2, we shown the HHG spectra in the two-color fields with different phases and the dependence of the peak electric field and the peak vector potential on the relative phase. It can be seen that the intensity of the second plateau increases with the increasing peak vector potential before the phase is equal to 0.6π. When the phase is equal to π, the electric field reach the minimum but the intensity of the second plateau is still higher than that at the phase equal to zero. In order to avoid damaging the solid structure, one should choose a two-color field that has highest vector potential and lowest electric field. Therefore, we investigate HHG spectrum in the two-color field with relative phase π in the following. For comparison, we also calculate the HHG spectrum in a monochromatic field, which has the same peak electric field as the two-color field. The specific parameters are set as E = 0.004807 a.u., λ = 3220 nm and duration equal to 8T1, respectively. In Figs. 3(a) and 3(b), we present the electric fields and the corresponding vector potentials of the monochromatic field and the two-color field. One can see that both the monochromatic field and the two-color field have the same peak electric field as shown in Fig. 3(a), but the two-color field has larger vector potential than that of the monochromatic field as shown in Fig. 3(b). Based on above discussions, the second plateau in the HHG spectrum should be enhanced in such a two-color field. In Fig. 3(c), the corresponding HHG spectra are shown in order to confirm it. The black solid line and the red doted line represent the HHG spectrum in the monochromatic field and the two-color field, respectively. One can clearly see that the intensity of the second plateau is about four orders lower than that of the first plateau in the monochromatic field. However, the intensity difference between the two plateaus is only about one order in the two-color field. Although the intensity of first plateau is decreased about one order in the two-color field, the intensity of the second plateau is still two to three orders stronger than that in the monochromatic field. This indicates that we can effectively enhance the second plateau in the solid HHG spectrum without damaging samples by adopting a proper two-color field. In addition, we also find that the cutoffs of the two plateaus are larger in the two-color field compared with them in the monochromatic field.
Fig. 2 The HHG spectra in the two-color fields with different relative phases. The blue line and the red line indicate the dependence of the peak electric field and the peak vector potential on the relative phase, respectively.
Fig. 3 The electric fields, the vector potentials of the laser pulses and the corresponding HHG spectra. The black solid lines and the red doted lines represent the monochromatic field and the two-color field, respectively. (a) The electric fields of the monochromatic and the two-color field. (b) Corresponding vector potentials. (c) The HHG spectra in the two laser fields.
In order to obtain further insights into the physical mechanism of the intensity enhancement, we investigate the electron population of the system. Figures 4(a) and 4(b) present the time-dependent populations from V2 to C4 in the monochromatic field and the two-color field, respectively. The blue areas are the band gaps. Figures 4(c) and 4(d) are the partial enlargements of Figs. 4(a) and 4(b). To compare with the semi-classical motion of an electron, the semi-classical trajectories calculated by Eq. (1) are also shown in the white solid lines in Figs. 4(c) and 4(d). It is clear that the semi-classical results are in good agreement with the most probable trajectories (the trajectory of the maximum of the electron population). This indicates that the semi-classical analysis is valid in our cases. In addition, we also find an unexpected fact that the path three is the main path of transitions from C1 to C2 regardless of laser fields. This is particularly obvious for the two-color field due to the stronger vector potential as shown in Fig. 4(d). Whenever the most probable trajectory reaches the top of C1, a dark red trajectory arises on C2, which means that the population of C2 largely increases. It can also be confirmed by the time-frequency analysis shown in Fig. 4(f) that the four paths in Fig. 4(d) mainly contribute to the second plateau. In the monochromatic field, the path three is also the main path, instead of the path two that is considered to be the main path by most researchers. In Fig. 4(c), the semi-classical trajectory of the electrons transited from C1 to C2 by the first way and the second way is presented in the white dashed line, and the trajectories of the electrons transited to C2 by the third way is presented in white doted lines. It can be seen that the trajectory induced by the first two ways is much thinner than them induced by the third way, and the corresponding time-frequency analysis reveals that the second plateau is mainly emitted in −0.25 o.c. to 0.75 o.c. as shown in Fig. 4(e), which indicates the path three is the main path of transitions from C1 to C2. This is because when the vector potential reaches the maximum, the electron wave-packet is closest to the top of C1, but the electric field is equal to zero so that the electron wave-packet cannot transit to C2 through the path two. Only the electron wave-packet with larger crystal momentum can disperse to the top of C1, then transits to C2 through the path three.
Fig. 4 Electron populations of the solid system in (a) the monochromatic field and (b) the two-color field. Their partial enlargements are shown on (c) and (d), respectively. Blue areas are band gaps. Time-frequency analysis of the time-dependent current in (e) the monochromatic field and (f) the two-color field. The white solid lines represent the semi-classical trajectories of the electrons on C1. The white doted lines and the white dashed line represent the trajectories of the electrons transited from C1 to C2 through the path three and other paths, respectively. The white solid lines, the white doted lines and the white dashed lines are the semi-classical trajectories calculated by εC1[A(t)], εC2[π/a0 + A(t)] and εC2[A(t)], respectively.
According to Fig. 4 and the semi-classical discussions, one can easily conclude the physical picture of the HHG processes in solids. Firstly, the electron wave-packet on the top of V2 tunnels to the bottom of C1. Secondly, a small portion of the electron wave-packet directly transits to higher conduction bands through the path one. But most of the electron wave-packet will evolve on C1. While the electron wave-packet is moving along C1, it can transit to C2 through the path two. At the same time, some electron wave-packet with larger crystal momentum can reach the top of C1, and then transits from C1 to C2 through the path three, which is considered to the main path of transitions from C1 to C2 by us. Finally, the electron wave-packet on C2 will continue to evolve and transit to higher conduction bands or after it goes back to V2 emitting photons.
This picture can help us to interpret the intensity decrease of the first plateau and the intensity enhancement of the second plateau. Firstly, we discuss the intensity decrease of the first plateau. One can qualitatively understand the intensity decrease according to the ADK model [42], where the ionization probability is proportional to exp[−2(2Ip)3/2/|3E(t)|]. Here Ip is the ionization energy. Thus the transition probability is strongly dependent on the electric field and the energy gap. A large electric field and a small energy gap lead to high transition probability. According to Figs. 4(c) and 4(d), when A(t) is equal to zero, the population of the electron wave-packet is mainly concentrated in the top of V2 and the energy gap is minimal. At the same time, |E(t)| is the local maximum in the monochromatic field, but |E(t)| is the local minimum (the trough at t = n0.5 o.c. (n = ±1, 2, 3, 4…) in Fig. 1(a)) in the two-color field. Therefore, the probability of transition from V2 to C1 is lower in the two-color field, which result in the intensity decrease. In order to confirm this, we show the total population on each conduction band. As shown in Figs. 5(a) and (c), the population on C1 indeed decreases in the two-color field. Next, we discuss the intensity enhancement of the second plateau in the two-color field. In the monochromatic field, when the vector potential of the laser field reaches the maximum, only a small portion of the electron wave-packet can disperse to the top of C1, so that the population of higher conduction bands is very low as shown in Figs. 4(c) and 5(b). Thus, the intensity of the second plateau is low. However, in the two-color field, the peak vector potential is strong enough as shown in Fig. 3(b). Then most of the electron wave-packet can reach the top of C1, which leads to much larger population on C2 as shown in Figs. 4(d) and 5(d). Therefore, the second plateau can be obviously enhanced in the two-color field. Moreover, we also find that when the vector potential reaches its maximum, most of the electron wave-packet is closest to the top of C1 but the electric field is around zero at the same time, so that the probability of transition from C1 to C2 is very low. This is one of the reasons why the intensity of the second plateau is lower than that of the first. In addition, the two-color field will not damage the solid. Although it creates larger population on C2, the population is still one order lower than that on C1. The damage of solids is mainly induced by the excessive electrons on conduction band. However, the total population on conduction band is decreased in the two-color field as shown in Fig. 5(c).
Fig. 5 The total electron population on each conduction band. (a) and (c) are the population on C1 in the monochromatic field and the two-color field, respectively. (b) and (d) are the population on high-lying conduction band in the monochromatic field and the two-color field, respectively. We do not show the population on C4 in (b), because it cannot be seen in such a scale.
Since the cutoff energy is determined by the energy difference between the lowest energy that the hole can get and the highest energy that most of the electron wave-packet can obtain, the increased cutoffs can also be explained by this picture. Firstly, we discuss the cutoff of the first plateau. For the monochromatic field, most of the electron wave-packet cannot reach the top of C1 due to the lower vector potential. Thus the cutoff can be calculated by Eq. (1): [εC1(Amax) − εV2(Amax)]/ω0 ≈ 33. However, in the two-color field, the vector potential is large enough so that most of the electron wave-packet can reach the top of C1 and the hole can reach the bottom of V2. So the cutoff is [εC1(π/a0) − εV2(π/a0)]/ω0 ≈ 36. Next, we discuss the cutoff of the second plateau. Although the HHG processes of the second plateau are more complex than them of the first one, the increasing of the cutoff can be still interpreted in the same way. After the electron wave-packet transits from C1 to C2, it will evolve on C2. When the vector potential is decreasing, the electric wave-packet is moving towards the top of C2 because of the band feature of C2 (the negative correlation between εC2(k) and |k| within the first Brillouin zone). When the vector potential is equal to zero, most of the electron wave-packet can reach the top of C2 then transits to C3 regardless of the laser field. But the situation changes on C3 because the band feature of C3 is different from that of C2 (the positive correlation between εC3(k) and |k| within the first Brillouin zone). Due to the weaker vector potential of the monochromatic field, most of the electron wave-packet can just reach the energy range about 40 eV as shown in Fig. 4(a). Thus the cutoff of the second plateau in single color field is [εC3(Amax) − εV2(Amax)]/ω0 ≈ 116. In the two-color field, most of the electron wave-packet can reach the top of C3 and transits to C4 due to the larger vector potential as shown in Fig. 4(b). However, one should pay more attention to the fact that the electron wave-packet on C4 does not move towards the top of C4, but across the band gap, returns to the bottom of C3 while the vector potential is decreasing. The behavior of the electron wave-packet on C4 is very different from that on C2, even though their band features are very similar. Thus the cutoff should be calculated by [εC4(Amax − π/a0) − εV2(Amax)]/ω0 ≈ 127. In comparison with Fig. 3(c), one can find the cutoffs calculated by Eq. (1) are in good agreement with the results obtained by solving TDSE, which reveals that the semi-classical model can effectively estimate the cutoff energy.
Finally, we further discuss the application of such a scheme. In previous part, we find that the motion of the electron wave-packet can be controlled by the vector potential. This means that isolated attosecond pulses can be generated in the frequency range of the second plateau, if the vector potential of the two-color field reaches the sufficient value only once so that the electron wave-packet is only allowed to reach the top of C1 once. To obtain such a two-color field, the parameters are set as: E1 = 0.004807 a.u., λ1 = 3220 nm, E2 = 0.4E1, ω2 = 2.3ω1, τdelay = 0.3T2 and duration is 8T1. In Fig. 6(a), we show the HHG spectrum under the two-color field. It can be seen that a smoothed supercontinuum can be obtained in the second plateau. Then an isolated attosecond pulse with the duration of 552.19 as can be generated by filtering the 64th–84th harmonics as shown in Fig. 6(c). In order to clearly illustrate this scheme, we investigate the electron dynamics processes in the two-color field. In Fig. 6(d), we show the time-dependent electron population. As shown, the main quantum path is largely increased, but other paths are a few orders weaker than the main path. This is because the electron wave-packet reaches the top of C1 at t = −0.23 o.c., and it keeps away from the top of C1 at other time. In this case, the population of C2 is primarily contributed by the main quantum path, so that the second plateau is emitted in −0.16 to 0.34 o.c. as shown in Fig. 6(b).
Fig. 6 (a) The HHG spectrum in the two-color field. (b) Time-frequency analysis of the time-dependent current. (c) The isolated attosecond pulse produced by filtering the 64th–84th harmonics. (d) The time-dependent population of C1 to C3.
4. Conclusion
In summary, we investigated HHG from the 1D solid in two-color fields. The numerical results showed that the intensity of the second plateau can be enhanced by two to three orders in the proper two-color field that has the same peak electric field with the monochromatic field. It is also found that the cutoffs of the two plateaus are extended in the two-color field. To understand their mechanisms, we investigated the electron population and got three conclusions. Firstly, the vector potential of the external field dominates the evolution processes of the solid system. Secondly, the path three (the electron wave-packet reaches the top of C1 then transits from C1 to C2) is the main path of transition from C1 to C2. Thirdly, the stronger vector potential of the two-color field is responsible for the intensity enhancement of the second plateau and the extended cutoffs. Finally, we discussed the application of such a scheme. It was found that isolated attosecond pulses can be generated from solids by adopting a two-color field whose vector potential reaches the sufficient value only once. This work presents a practical method for enhancing the intensity of the second plateau without damaging solid samples, and a promising way to generate isolated attosecond pulses from solids.
Funding
National Natural Science Foundation of China (NSFC) (11404153, 11135002, 11604119, 11405077); Fundamental Research Funds for the Central Universities of China (Grants Nos. lzujbky-2016-29, No. lzujbky-2016-31, No. lzujbky-2016-209, No.lzujbky-2017-14).
References and links
1. T. Brabec and F. Krausz, “Intense few-cycle laser fields: Frontiers of nonlinear optics,” Rev. Mod. Phys. 72(2), 545–591 (2000). [CrossRef]
2. C. D. Lin, L. Anh-Thu, C. Zhangjin, M. Toru, and L. Robert, “Strong-field rescattering physics—self-imaging of a molecule by its own electrons,” J. Phys. B 43(12), 122001 (2010). [CrossRef]
3. F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009). [CrossRef]
4. P. B. Corkum and F. Krausz, “Attosecond science,” Nat. Physics 3(6), 381–387 (2007). [CrossRef]
5. T. Popmintchev, M. C. Chen, P. Arpin, M. M. Murnane, and H. C. Kapteyn, “The attosecond nonlinear optics of bright coherent X-ray generation,” Nat. Photonics 4(12), 822–832 (2010). [CrossRef]
6. M. Chini, K. Zhao, and Z. Chang, “The generation, characterization and applications of broadband isolated attosecond pulses,” Nat. Photonics 8(3), 178–186 (2014). [CrossRef]
7. S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Physics 7(2), 138–141 (2010). [CrossRef]
8. T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, “Extreme ultraviolet high-harmonic spectroscopy of solids,” Nature 521(7553), 498–502 (2015). [CrossRef] [PubMed]
9. Y. Y. S. You, D. A. Reis, and S. Ghimire, “Anisotropic high-harmonic generation in bulk crystals,” Nat. Physics 13(4), 345–349 (2017). [CrossRef]
10. O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8(2), 119–123 (2014). [CrossRef]
11. G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115(19), 193603 (2015). [CrossRef] [PubMed]
12. G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014). [CrossRef] [PubMed]
13. G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91(6), 064302 (2015). [CrossRef]
14. C. Liu, Y. Zheng, Z. Zeng, and R. Li, “Effect of elliptical polarization of driving field on high-order-harmonic generation in semiconductor ZnO,” Phys. Rev. A 93(4), 043806 (2016). [CrossRef]
15. G. Vampa, T. J. Hammond, N. Thire, B. E. Schmidt, F. Legare, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522(7557), 462–464 (2015). [CrossRef] [PubMed]
16. H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, “High-harmonic generation from an atomically thin semiconductor,” Nat. Physics 13(3), 262–265 (2016). [CrossRef]
17. K. F. Lee, X. Ding, T. J. Hammond, M. E. Fermann, G. Vampa, and P. B. Corkum, “Harmonic generation in solids with direct fiber laser pumping,” Opt. Lett. 42(6), 1113–1116 (2017). [CrossRef] [PubMed]
18. G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534(7608), 520–523 (2016). [CrossRef] [PubMed]
19. M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, “High-harmonic generation from Bloch electrons in solids,” Phys. Rev. A 91(4), 043839 (2015). [CrossRef]
20. Z. Guan, X. X. Zhou, and X. B. Bian, “High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses,” Phys. Rev. A 93(3), 033852 (2016). [CrossRef]
21. Y. S. You, M. Wu, Y. Yin, A. Chew, X. Ren, S. Gholam-Mirzaei, D. A. Browne, M. Chini, Z. Chang, K. J. Schafer, M. B. Gaarde, and S. Ghimire, “Laser waveform control of extreme ultraviolet high harmonics from solids,” Opt. Lett. 42(9), 1816–1819 (2017). [CrossRef] [PubMed]
22. M. Wu, D. A. Browne, K. J. Schafer, and M. B. Gaarde, “Multilevel perspective on high-order harmonic generation in solids,” Phys. Rev. A 94(6), 063403 (2016). [CrossRef]
23. P. G. Hawkins, M. Y. Ivanov, and V. S. Yakovlev, “Effect of multiple conduction bands on high-harmonic emission from dielectrics,” Phys. Rev. A 91(1), 013405 (2015). [CrossRef]
24. T. Y. Du and X. B. Bian, “Quasi-classical analysis of the dynamics of the high-order harmonic generation from solids,” Opt. Express 25(1), 151–158 (2017). [CrossRef] [PubMed]
25. T. Y. Du and X. B. Bian, “Mechanism of high-order harmonic generation from periodic potentials,” arXiv:1606.06433 (2016).
26. T. Y. Du, Z. Guan, X. X. Zhou, and X. B. Bian, “Enhanced high-order harmonic generation from periodic potentials in inhomogeneous laser fields,” Phys. Rev. A 94(2), 023419 (2016). [CrossRef]
27. S. Han, H. Kim, Y. W. Kim, Y. J. Kim, S. Kim, I. Y. Park, and S. W. Kim, “High-harmonic generation by field enhanced femtosecond pulses in metal-sapphire nanostructure,” Nat. Commun. 7, 13105 (2016). [CrossRef] [PubMed]
28. I. J. Kim, C. M. Kim, H. T. Kim, G. H. Lee, Y. S. Lee, J. Y. Park, D. J. Cho, and C. H. Nam, “Highly efficient high-harmonic generation in an orthogonally polarized two-color laser field,” Phys. Rev. Lett. 94(24), 243901 (2005). [CrossRef]
29. H. Du, H. Wang, and B. Hu, “Isolated short attosecond pulse generated using a two-color laser and a high-order pulse,” Phys. Rev. A 81(6), 063813 (2010). [CrossRef]
30. H. Du, S. Xue, H. Wang, Y. Wen, and B. Hu, “Generating elliptically polarized isolated attosecond pulses from the 2p0 state of He+ with a linearly polarized two-color field,” Opt. Commun. 338, 422 (2015). [CrossRef]
31. P. Lan, E. J. Takahashi, and K. Midorikawa, “Optimization of infrared two-color multicycle field synthesis for intense-isolated-attosecond-pulse generation,” Phys. Rev. A 82(5), 053413 (2010). [CrossRef]
32. J. G. Chen, S. L. Zeng, and Y. J. Yang, “Generation of isolated sub-50-as pulses by quantum path control in the multicycle regime,” Phys. Rev. A 82(4), 043401 (2010). [CrossRef]
33. Z. Zeng, Y. Cheng, X. Song, R. Li, and Z. Xu, “Generation of an extreme ultraviolet supercontinuum in a two-color laser field,” Phys. Rev. Lett. 98(20), 203901 (2007). [CrossRef] [PubMed]
34. G. Li, Y. Zheng, X. Ge, Z. Zeng, and R. Li, “Frequency modulation of high-order harmonic generation in an orthogonally polarized two-color laser field,” Opt. Express 24(16), 18685–18694 (2016). [CrossRef] [PubMed]
35. C. Jin, G. Wang, H. Wei, A. T. Le, and C. D. Lin, “Waveforms for optimal sub-keV high-order harmonics with synthesized two- or three-colour laser fields,” Nat. Commun. 5, 4003 (2014). [CrossRef] [PubMed]
36. L. E. Chipperfield, J. S. Robinson, J. W. Tisch, and J. P. Marangos, “Ideal waveform to generate the maximum possible electron recollision energy for any given oscillation period,” Phys. Rev. Lett. 102(6), 063003 (2009). [CrossRef] [PubMed]
37. S. Haessler, T. Balciunas, G. Fan, L. E. Chipperfield, and A. Baltuska, “Enhanced multi-colour gating for the generation of high-power isolated attosecond pulses,” Sci. Rep. 5, 10084 (2015). [CrossRef] [PubMed]
38. C. A. Mancuso, K. M. Dorney, D. D. Hickstein, J. L. Chaloupka, J. L. Ellis, F. J. Dollar, R. Knut, P. Grychtol, D. Zusin, C. Gentry, M. Gopalakrishnan, H. C. Kapteyn, and M. M. Murnane, “Controlling nonsequential double ionization in two-color circularly polarized femtosecond laser fields,” Phys. Rev. Lett. 117(13), 133201 (2016). [CrossRef] [PubMed]
39. S. Skruszewicz, J. Tiggesbaumker, K. H. Meiwes-Broer, M. Arbeiter, T. Fennel, and D. Bauer, “Two-color strong-field photoelectron spectroscopy and the phase of the phase,” Phys. Rev. Lett. 115(4), 043001 (2015). [CrossRef] [PubMed]
40. J. C. Slater and J. C. Slater, “A soluble problem in energy bands,” Phys. Rev. 87(5), 807–835 (1952). [CrossRef]
41. M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60(4), 389 (1997). [CrossRef]
42. M. Ammosov, N. Delone, and V. Krainov, “Tunnel ionization of complex atoms and of atoms ions in an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191 (1986).
