Residual current under the combined effect of carrier envelope phase and chirp: phase shift and peak enhancement

Xiaoxue Zhang; Xiaoxue Zhang; Henglei Du; Henglei Du; Wenkang Wang; Wenkang Wang; Huicheng Guo; Huicheng Guo; Chengpu Liu; Chengpu Liu

doi:10.1364/OE.497291

1. Introduction

The interaction of ultrashort laser pulses with solids has been extensively studied from various perspectives, such as high harmonic generation [1–4], terahertz radiation [5,6], and light-field-driven current generation [7–9]. Light-field-driven current with high speed signal processing potential is an important development area in light wave electronics. Many materials such as SiO₂ [10], GaN [11], GaF₂ [12], heterojunctions [13], and graphene [14] have been used in relevant research. Graphene is promising among a group of materials because of its weak screening effect and high damage threshold [15], especially the experimental confirmation that injected currents can propagate significantly in graphene [16]. Understanding and controlling the electronic dynamics progress of the light-field-driving current is fundamental to the implementation of ultrafast optoelectronic integration applications.

Pulse shaping is usually performed by changing the peak field amplitude, phase and polarization [17–19]. The control of carrier envelope phase (CEP) and chirp are both phase shaping. Both CEP and chirp can individually control the electron motion and achieve the reversal of residual currents under strong field regime in graphene [20,21]. Among them, the electron dynamics is the Landau-Zener-Stückelberg (LZS) [22] interference brought by the non-negligible electron wave number variation in strong field, that is, the intraband motion affecting the interband transitions. When the peak field amplitude is fixed, the residual current varies sinusoidally with CEP and is taken to be maximum at ±$\frac{\pi }{2}$. And the residual current with chirp is sinusoidal-like variation within a small range, taken to the maximum at chirp rate ±0.05fs^-2. All of the above are CEP and chirp interacting with graphene alone, then the effect of both together on the residual current is worth exploring. Two characteristics about the combined effect, phase shift and peak residual current enhancement, were obtained by numerical studies. The phase shift can be interpreted as the change brought by the introduction of different chirp rates. The enhancement of the residual current peak is attributed to the change in the physical mechanism of electron motion, where multiphoton interference is involved.

The paper is organized as follows: The second part is a tight-binding model of graphene and a model of the time-dependent Schrödinger equation for calculating the residual current. The third part discusses the effect of the combined interaction of CEP and chirp on the residual current and analyzes the electron dynamics processes. The last section is the conclusion.

2. Theory

We solve the energy band structure of graphene using the tight-binding model of the nearest neighbor atoms A and B. The Hamiltonian matrix can be described as [14,23]

(1)$${H_\textrm{0}}({\mathbf k}) = \left[ {\begin{array}{cc} 0&{ - {\varepsilon_h}f({\mathbf k})}\\ { - {\varepsilon_h}{f^ \ast }({\mathbf k})}&0 \end{array}} \right],$$

with nearest neighbor hopping parameter ${\varepsilon _h} = 3\textrm{ eV}$ and $f({\mathbf k}) = \exp (i\frac{{a{k_x}}}{{\sqrt 3 }}) + 2\exp ( - i\frac{{a{k_x}}}{{2\sqrt 3 }})\cos (\frac{{a{k_y}}}{2}).a = 0.246\textrm{ nm}$ is lattice constant and k_x, k_y are the reciprocal space coordinates. From Eq. (1), the energies of the graphene conduction band and valence band are ${E_{{C / V}}} ={\pm} {\varepsilon _h}|{f({\mathbf k})} |.$ The corresponding eigen wave functions are

(2)$$\phi _{\mathbf k}^V = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{c} {{e^{i{\theta_{\mathbf k}}}}}\\ 1 \end{array}} \right],\textrm{ }\phi _{\mathbf k}^C = \frac{1}{{\sqrt 2 }}\left[ {\begin{array}{c} { - {e^{i{\theta_{\mathbf k}}}}}\\ 1 \end{array}} \right],$$

with phase angle ${\theta _{\mathbf k}} = \arg (f({\mathbf k})).$ Then the analytical expression of the dipole matrix element between the two energy bands is ${\mathbf d}({\mathbf k}) = \left\langle {\phi_{\mathbf k}^V|{e{\mathbf r}} |\phi_{\mathbf k}^C} \right\rangle = \frac{e}{2}{\nabla _{\mathbf k}}{\theta _{\mathbf k}}.$

In the presence of a laser field E(t), the Hamiltonian in the length gauge is $H(t) = {H_0}({\mathbf k}) - e{\mathbf E}(t) \cdot {\mathbf r}$. And the variation of the electron wave number is usually expressed in terms of Bloch acceleration theorem [24], $\dot{{\mathbf k}}(t) = \frac{e}{\hbar }{\mathbf E}(t)$, that is, ${\mathbf k}(t) = {{\mathbf k}_0} - \frac{e}{\hbar }{\mathbf A}(t)$. A(t) is the laser field vector potential. Thus the time-dependent Schrödinger equation in the adiabatic Houston basis is described as

(3)$$i\hbar \frac{\partial }{{\partial t}}{\phi _{{{\mathbf k}_0}}}(t) = H({\mathbf k}(t)){\phi _{{{\mathbf k}_0}}}(t).$$

The wave function of atoms can be expand as ${\phi _{{{\mathbf k}_0}}}(t) = c_{{{\mathbf k}_0}}^A(t)\phi _{{\mathbf k}(t)}^A + c_{{{\mathbf k}_0}}^B(t)\phi _{{\mathbf k}(t)}^B$.$c_{{{\mathbf k}_0}}^A(t)$ and $c_{{{\mathbf k}_0}}^B(t)$ are the coefficients in the atoms representation. After substitution into Eq. (3), we get

(4)$$i\hbar (\dot{c}_{{{\mathbf k}_0}}^A(t)\phi _{{\mathbf k}(t)}^A + \dot{c}_{{{\mathbf k}_0}}^B(t)\phi _{{\mathbf k}(t)}^B) = {H_0}({\mathbf k}(t)){\phi _{{{\mathbf k}_0}}}(t).$$

It can be transformed into matrix form as

(5)$$\frac{\partial }{{\partial t}}\left[ {\begin{array}{c} {c_{{{\mathbf k}_0}}^A(t)}\\ {c_{{{\mathbf k}_0}}^B(t)} \end{array}} \right] = \frac{1}{{i\hbar }}\left[ {\begin{array}{cc} 0&{ - {\varepsilon_h}f({\mathbf k}(t))}\\ { - {\varepsilon_h}{f^ \ast }({\mathbf k}(t))}&0 \end{array}} \right]\left[ {\begin{array}{c} {c_{{{\mathbf k}_0}}^A(t)}\\ {c_{{{\mathbf k}_0}}^B(t)} \end{array}} \right],$$

which is solved by Crank-Nicolson method [25]. To obtain the coefficients $c_{{{\mathbf k}_0}}^V(t)$ and $c_{{{\mathbf k}_0}}^C(t)$ under the valence and conduction band representation, relevant relation is applied:$c_{{{\mathbf k}_0}}^A(t)\phi _{{\mathbf k}(t)}^A + c_{{{\mathbf k}_0}}^B(t)\phi _{{\mathbf k}(t)}^B = c_{{{\mathbf k}_0}}^V(t)\phi _{{\mathbf k}(t)}^V + c_{{{\mathbf k}_0}}^C(t)\phi _{{\mathbf k}(t)}^C$. Therefore, the conduction band electron population is $\rho _{{{\mathbf k}_0}}^C(t) = {|{c_{{{\mathbf k}_0}}^C(t)} |^2}$. Then integrating over the entire Brillouin zone, the residual current density is obtained as

(6)$$J ={-} \frac{{2e}}{{{{({2\pi } )}^2}}}\int_{BZ} {\rho _{{{\mathbf k}_0}}^C(t = \infty )} \frac{1}{\hbar }\frac{{\partial {E_C}}}{{\partial k}}d{\mathbf k}.$$

3. Results and discussion

There are various possibilities to introduce a linear chirp. In this paper, the quadratic term of time is added directly to the chirp-free carrier wave, so that the laser field has the characteristics of a time-dependent frequency, fixed pulse duration and peak field amplitude [26–28]. The expression for the x-directional linearly polarized light interacting with graphene is described as

(7)$${E_x}(t) = {E_0}\exp ( - \frac{{{t^2}}}{{t_p^2}})\cos ({\omega _0}t + \frac{1}{2}\alpha {t^2} + \varphi ),$$

with parameters laser peak amplitude E₀, pulse duration t_p, center frequency ω₀, chirp rate α, and CEP φ. Considering that when chirp and CEP act independently, they all enter the strong field regime in E₀= 1.8 V/nm, so the peak field amplitude is set to E₀= 2.5 V/nm. The incident laser pulse has a wavelength of 800 nm with a pulse duration of 5 fs.

The simulation results of the residual current density obtained by traversing the CEP with a fixed chirp rate are shown in Fig. 1. As the chirp rate increased, there are two obvious characteristics that appear, the phase shift and the enhancement of the residual current density peak. To interpret the reasons for the appearance of the above characteristics, four cases, A, A’, B, C, are selected for analysis.

Fig. 1. Residual current density with chirp rate and CEP combined. A and A’ have the same CEP (φ_{A, A’}= 0.5π) with different chirp rates (α_A= 0, α_A’= 0.01fs^-2). B (α_B= 0.05fs^-2, φ_B= 1.14π) and C (α_C= 0.1fs^-2, φ_C= 1.54π) correspond to the maximum residual current density at the respective chirp rate.

Download Full Size | PDF

As already known before, in the absence of chirp, CEP controls the residual current density taken to the maximum at φ = 0.5π, corresponding to the maximum asymmetric vector potential, that is, case A. Since the electron wavenumber is described by Bloch acceleration theorem, the vector potential is used to break the population symmetry to generate the residual current. To further explain the phase shift, the perturbation perspective is analyzed by fixing the CEP and introducing the chirp rate, i.e., case A’. Figures 2(a) and 2(b) show the conduction band population of A and A’, respectively. It is worth noting that this is only the conduction population around a single K point. The difference on the conduction population is difficult to obtain directly due to the small chirp rate. Therefore, the residual current is integrated only along the k_x direction for different initial k_y, as shown in Fig. 2(c). The numerical value clearly shows a difference between the main peaks of A and A’, with the main peaks of A’ shifted down a little compared to that of A. It means that the total residual current is reduced. This indicates that the introduction of a small chirp rate destroys the maximum asymmetric vector potential in case A. So to go back to the maximum asymmetric vector potential with chirp, it is necessary to change the CEP, which is the reason why the phase shift of the residual current appears.

Fig. 2. Residual conduction band population in (a) case A and (b) case A’. (c) The residual current integrated only along k_x for various initial k_y. The blue line indicates the case A and the red line indicates the case A’. (d) Vector potential of the incident laser in cases A and A’. The inset enlarges the two positive peaks.

Download Full Size | PDF

In [29], the asymmetry parameter Г is introduced by the absolute value of the ratio of the minimum value to the maximum value of vector potential amplitudes in the absence of chirp, $\mathrm{\Gamma} = \left|{\frac{{{A_{xmin}}}}{{{A_{xmax}}}}} \right|$. According to Fig. 2(d), the minimum negative peak of A and A’ overlap, and the main distinction is the maximum positive peak. The enlarged inset shows that in case A, the two positive peaks have the same amplitude, but in A’, the left positive peak is significantly higher than the right positive peak. As mentioned before, in order to return to an asymmetry equivalent with case A in the presence of a chirp, the vector potential waveform is shifted a little to the left overall, which means adding a positive CEP. The existence of the Gaussian envelope suppresses the left positive peak and enhances the right positive peak. Hence the residual current is positively phase shifted at positive chirp as opposed to negative chirp.

In fact, in case A’, CEP plus 0.04π can make Г comparable to that of case A, but the residual current density is still slightly smaller than that of case A. This is because a positive chirp causes the carrier to broaden at low frequencies and compress at high frequencies, which affects the residual current density, that is the part ignored by defining the asymmetric parameters in terms of Г. Due to the fixed chirp, the residual current density brought by the CEP variation is in sinusoidal form, and increasing the CEP on the basis of A’ is at the rising edge, then the ignored part will make the residual current density reach a larger value compared to A at a suitable CEP, which introduces another characteristic, the enhancement of the residual current density peak.

To investigate the light-induced electron dynamics process of residual current density peak enhancement, we focus on three cases A, B, C. Three cases correspond to the maximum residual current density at the respective chirp rate at suitable CEP. Figures 3(a) and 3(b) show the conduction band population of case B and C. Compared with A, population of C is clearly different. For a more intuitive observation, the variation of the residual current with different initial k_y is also obtained by integrating along k_x, as shown in Fig. 3(c). For either case, there are two main positive peaks, looking at the positive peaks because of the sign corresponding to the total residual current. With the increase of chirp rate, the two positive peaks are enhancing, and case C is obviously enhancing more than case A and B. To further discuss the electron motion, the coordinates corresponding to the two positive peaks are selected as the k_y values of the points P₁, P₂ in the reciprocal space, and the specific positions are marked in Fig. 3(b). The solid curve represents the resonance with photon energy of ћω₀. P₁, P₂ are obviously not on the resonance line of the center frequency.

Fig. 3. Residual conduction band population in (a) case B and (b) case C. The solid curve indicates the resonance with photon energy ћω₀. P₁ and P₂ are points on the reciprocal space. (c) Residual current along the k_x integral for different initial k_y in case A (blue), B (orange) and C (green). The positions of the two positive peaks in the three cases are considered, and the appropriate k_y values are chosen as the vertical coordinate of P₁ and P₂.

Download Full Size | PDF

The relative band coupling strength β(t) is introduced to describe the transition probability between energy bands to analyze the electron dynamics process [14]. The expression is $\beta (t) = \frac{{{\mathbf E}(t){\mathbf d}({\mathbf k}(t))}}{{{E_C}({\mathbf k}(t))}}$, where ${\mathbf d}({\mathbf k}(t))$ and ${E_C}({\mathbf k}(t))$ are the dipole matrix element and the conduction band energy, and ${\mathbf E}(t){\mathbf d}({\mathbf k}(t)) = \hbar {\varOmega _R}(t)$. It is worth noting that the Keldysh parameter can be expressed as $\gamma = \frac{{{E_{VC}}}}{{2\hbar {\varOmega _R}}} = \frac{{{E_C}}}{{\hbar {\varOmega _R}}}$ in graphene [30], ${E_{VC}}$ is energy difference between the valence and the conduction band, which means $\beta = \frac{1}{\gamma }$. When γ < 1, the electron motion is into the tunneling regime, corresponding to β > 1. When γ > 1, the electron motion is in the multiphoton absorption regime, corresponding to β < 1.

The temporal evolution of the relative band coupling strength and the conduction band population are calculated for three cases at point P₁, and the results are shown in Fig. 4(a-c). From Fig. 4(a), it can be found that there are positive and negative peak for per optical cycle, corresponding to two transition events per optical cycle under linearly polarized excitation. The dashed line indicates the LZ transition for β > 1, which occurs as LZS interference, corresponding to the conduction band population is constructive interference, thus there is a large population at the end. For Fig. 4(c), although there are four LZ transitions, the result is destructive interference. Then the final population is theoretically zero, but obviously a larger population exists. This illustrates that case C has not only LZS interference, but also another mechanism that makes the population large, which is multiphoton interference. A large linear chirp is introduced in case C, which broadens the spectral width because of $\omega = {\omega _0} + {\alpha _C}t$. Lower order quantum-path interference processes occur, such as single- vs two-photon absorption and two- vs three-photon absorption. The residual current is mainly generated by the interference of single-photon absorption with a photon energy of ћω_k and two-photon absorption with a photon energy of $\frac{1}{2}$ћω_k at different k_x, k_y, as shown in Fig. 4(d). The chirp introduced in case B is in between A and C. The multiphoton interference exists, but the population is mainly the result of the constructive interference from the three LZ transitions shown in Fig. 4(b). It is worth noting that the peak field amplitude remains unchanged from case A to C, where the light-graphene interaction turns from a non-perturbative to a perturbative regime, in contrast to the regime transition via increasing the peak field amplitude [31].

Fig. 4. (a)-(c) Temporal evolution of the relative band coupling strength β(t) and conduction band population ρ(t) at P₁ for the cases of A, B, C. The dashed lines indicate the population magnitude under β > 1. (d) Schematic illustration of single-photon (ћω_k) and two-photon ($\frac{1}{2}$ћω_k) quantum path interference. The gray curve is the energy band slice at a certain k_y.

Download Full Size | PDF

As for P₂, in the three cases β < 1 is calculated, thus multiphoton absorption occurs, as shown in Fig. 5(a-c). Comparing the three cases, the conduction band population is increased. It can be interpreted as the electric field amplitude increases as the chirp increases. At chirp rates of α₁ and α₂, the frequencies ω and $\frac{1}{2}$ω at which multiphoton interference occurs are selected, and their corresponding electric field amplitudes are shown in Fig. 5(d). It can be clearly observed that the electric field amplitude corresponding to α₂ is larger than that of α₁. This is also true for other frequencies. According to the expression $J \propto E_{\frac{1}{2}\omega }^2{E_\omega }$ [32], so the residual current is enhanced.

Fig. 5. (a)-(c) Temporal evolution of the relative band coupling strength β(t) and conduction band population ρ(t) at P₂ for the cases of A, B, C. (d) Laser frequency versus time after the introduction of linear chirp, where α₂ > α₁. The blue curve is the temporal envelope. The asterisks indicate the corresponding laser field amplitudes at different frequencies and chirp rates.

Download Full Size | PDF

The residual current peak can be enhanced by the combination of chirp and CEP, which is the enhancement brought by the change from LZS interference dominance to multiphoton interference dominance around P₁ and the enhancement brought by multiphoton interference around P₂, respectively. In the present variable range, the combined effect is enhanced by 3.7 times compared to CEP alone and by 1.9 times compared to chirp alone.

4. Conclusion

In summary, there are two obvious characteristics of the residual current generated by the interaction of linearly polarized light with graphene under the combined effect of CEP and chirp, namely, phase shift and peak enhancement. The phase shift of the residual current is analyzed from the perturbation perspective as a result of the resistance to the different chirp rates. The residual current peak enhancement is mainly in two regions, namely the enhancement of the transition from LZS interference dominance to multiphoton interference dominance around P₁ and multiphoton interference enhancement around P₂. Further understanding of the excitation dynamics in graphene under different light waveforms can help to better control the properties of two-dimensional materials. The enhanced residual current improves the sensitivity of subsequent detection and contributes to the promotion of optoelectronic device applications, laying the foundation for future high-speed information processing.

Funding

National Natural Science Foundation of China (12074398, 11674342, 11374318).

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.

References

1. 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. Phys. 7(2), 138–141 (2011). [CrossRef]

2. G. Vampa, C. McDonald, G. Orlando, D. Klug, P. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014). [CrossRef]

3. D. Golde, T. Meier, and S. W. Koch, “High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations,” Phys. Rev. B 77(7), 075330 (2008). [CrossRef]

4. J. Schötz, Z. Wang, E. Pisanty, M. Lewenstein, M. F. Kling, and M. Ciappina, “Perspective on petahertz electronics and attosecond nanoscopy,” ACS Photonics 6(12), 3057–3069 (2019). [CrossRef]

5. W. J. Ding, Z. M. Sheng, and W. S. Koh, “High-field half-cycle terahertz radiation from relativistic laser interaction with thin solid targets,” Appl. Phys. Lett. 103(20), 204107 (2013). [CrossRef]

6. H. A. Hafez, X. Chai, A. Ibrahim, S. Mondal, D. Ferachou, X. Ropagnol, and T. Ozaki, “Intense terahertz radiation and their applications,” J. Opt. 18(9), 093004 (2016). [CrossRef]

7. M. S. Wismer, S. Y. Kruchinin, M. Ciappina, M. I. Stockman, and V. S. Yakovlev, “Strong-Field Resonant Dynamics in Semiconductors,” Phys. Rev. Lett. 116(19), 197401 (2016). [CrossRef]

8. F. Langer, Y. P. Liu, Z. Ren, V. Flodgren, C. Guo, J. Vogelsang, S. Mikaelsson, I. Sytcevich, J. Ahrens, A. L’Huillier, C. L. Arnold, and A. Mikkelsen, “Few-cycle lightwave-driven currents in a semiconductor at high repetition rate,” Optica 7(4), 276–279 (2020). [CrossRef]

9. T. Boolakee, C. Heide, A. Garzon-Ramirez, H. B. Weber, I. Franco, and P. Hommelhoff, “Light-field control of real and virtual charge carriers,” Nature 605(7909), 251–255 (2022). [CrossRef]

10. A. Schiffrin, T. Paasch-Colberg, N. Karpowicz, V. Apalkov, D. Gerster, S. Muhlbrandt, M. Korbman, J. Reichert, M. Schultze, S. Holzner, J. V. Barth, R. Kienberger, R. Ernstorfer, V. S. Yakovlev, M. I. Stockman, and F. Krausz, “Optical-field-induced current in dielectrics,” Nature 493(7430), 70–74 (2013). [CrossRef]

11. T. Paasch-Colberg, S. Y. Kruchinin, Ö Sağlam, S. Kapser, S. Cabrini, S. Muehlbrandt, J. Reichert, J. V. Barth, R. Ernstorfer, and R. Kienberger, “Sub-cycle optical control of current in a semiconductor: from the multiphoton to the tunneling regime,” Optica 3(12), 1358–1361 (2016). [CrossRef]

12. O. Kwon and D. Kim, “PHz current switching in calcium fluoride single crystal,” Appl. Phys. Lett. 108(19), 191112 (2016). [CrossRef]

13. J. D. Lee, Y. Kim, and C. M. Kim, “Model for petahertz optical memory based on a manipulation of the optical-field-induced current in dielectrics,” New J. Phys. 20(9), 093029 (2018). [CrossRef]

14. T. Higuchi, C. Heide, K. Ullmann, H. B. Weber, and P. Hommelhoff, “Light-field-driven currents in graphene,” Nature 550(7675), 224–228 (2017). [CrossRef]

15. A. C. Neto, F. Guinea, N. M. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81(1), 109–162 (2009). [CrossRef]

16. T. Boolakee, C. Heide, F. Wagner, C. Ott, M. Schlecht, J. Ristein, H. B. Weber, and P. Hommelhoff, “Length-dependence of light-induced currents in graphene,” J. Phys. B: At., Mol. Opt. Phys. 53(15), 154001 (2020). [CrossRef]

17. A. M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Commun. 284(15), 3669–3692 (2011). [CrossRef]

18. C. Heide, T. Higuchi, H. B. Weber, and P. Hommelhoff, “Coherent electron trajectory control in graphene,” Phys. Rev. Lett. 121(20), 207401 (2018). [CrossRef]

19. F. Navarrete and U. Thumm, “Two-color-driven enhanced high-order harmonic generation in solids,” Phys. Rev. A 102(6), 063123 (2020). [CrossRef]

20. C. Heide, T. Boolakee, T. Higuchi, and P. Hommelhoff, “Sub-cycle temporal evolution of light-induced electron dynamics in hexagonal 2D materials,” JPhys Photonics 2(2), 024004 (2020). [CrossRef]

21. E. H. Wu, C. J. Zhang, Z. S. Wang, and C. P. Liu, “Waveform control of currents in graphene by chirped few-cycle lasers,” New J. Phys. 22(3), 033016 (2020). [CrossRef]

22. O. V. Ivakhnenko, S. N. Shevchenko, and F. Nori, “Nonadiabatic Landau–Zener–Stückelberg–Majorana transitions, dynamics, and interference,” Phys. Rep. 995, 1–89 (2023). [CrossRef]

23. H. K. Kelardeh, V. Apalkov, and M. I. Stockman, “Graphene in ultrafast and superstrong laser fields,” Phys. Rev. B 91(4), 045439 (2015). [CrossRef]

24. V. Grecchi and A. Sacchetti, “Acceleration theorem for Bloch oscillators,” Phys. Rev. B 63(21), 212303 (2001). [CrossRef]

25. J. Crank and P. Nicolson, “A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type,” in Mathematical Proceedings of the Cambridge Philosophical Society (Cambridge University, 1947), pp. 50–67.

26. D. Peng, M. Frolov, L.-W. Pi, and A. F. Starace, “Enhancing high-order harmonic generation by sculpting waveforms with chirp,” Phys. Rev. A 97(5), 053414 (2018). [CrossRef]

27. X. Wang, C. Jin, and C. D. Lin, “Coherent control of high-harmonic generation using waveform-synthesized chirped laser fields,” Phys. Rev. A 90(2), 023416 (2014). [CrossRef]

28. M. Lara-Astiaso, R. Silva, A. Gubaydullin, P. Rivière, C. Meier, and F. Martín, “Enhancing high-order harmonic generation in light molecules by using chirped pulses,” Phys. Rev. Lett. 117(9), 093003 (2016). [CrossRef]

29. C. Heide, T. Boolakee, T. Eckstein, and P. Hommelhoff, “Optical current generation in graphene: CEP control vs. ω+ 2ω control,” Nanophotonics 10(14), 3701–3707 (2021). [CrossRef]

30. C. Heide, T. Boolakee, T. Higuchi, and P. Hommelhoff, “Adiabaticity parameters for the categorization of light-matter interaction: From weak to strong driving,” Phys. Rev. A 104(2), 023103 (2021). [CrossRef]

31. C. Heide, T. Boolakee, T. Higuchi, H. B. Weber, and P. Hommelhoff, “Interaction of carrier envelope phase-stable laser pulses with graphene: the transition from the weak-field to the strong-field regime,” New J. Phys. 21(4), 045003 (2019). [CrossRef]

32. C. Heide, T. Eckstein, T. Boolakee, C. Gerner, H. B. Weber, I. Franco, and P. Hommelhoff, “Electronic coherence and coherent dephasing in the optical control of electrons in graphene,” Nano Lett. 21(22), 9403–9409 (2021). [CrossRef]

Residual current under the combined effect of carrier envelope phase and chirp: phase shift and peak enhancement

Abstract

1. Introduction

2. Theory

3. Results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (5)

Equations (7)

Optics Express