High-order harmonic generation from 2D periodic potentials in circularly and bichromatic circularly polarized laser fields

High-order harmonic generation from 2D periodic potentials in circularly and bichromatic circularly polarized laser fields

Guang-Rui Jia College of Physics and Materials Science, Henan Normal University, Xinxiang 453007, China State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China    Xin-Qiang Wang State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China University of Chinese Academy of Sciences, Beijing 100049, China    Tao-Yuan Du State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China University of Chinese Academy of Sciences, Beijing 100049, China    Xiao-Huan Huang xhuang@hbnu.edu.cn Hubei Key Laboratory of Pollutant Analysis and Reuse Technology, College of Chemistry and Chemical Engineering, Hubei Normal University, Huangshi 435002, China    Xue-Bin Bian xuebin.bian@wipm.ac.cn State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China
Abstract

We studied the high-order harmonic generation (HHG) from 2D solid materials in circularly and bichromatic circularly polarized laser fields numerically by simulating the dynamics of a single active electron in 2D periodic potentials. Circular HHGs with different helicities are generated in the circularly polarized driving lasers. High-order elliptically polarized harmonics are also generated in circular and bicircular mid-infrared lasers. The ellipticity and the intensity of the harmonics can be tuned by the control of the relative phase of the 1 and 2 fields in bicircularly polarized lasers, which can be used to image the structure of the solids.

pacs:
42.65.Ky, 42.65.Re, 72.20.Ht

I Introduction

High-order harmonic generation (HHG) from atomic and molecular gases has been studied extensively Krausz (); Peng (). It has been used to generate attosecond laser pulses. The mechanism is well described by the three-step recollision model Corkum (). Recently, more attention has been attracted to the HHG from solids Ghimire (); Lee (); Huttner (); Yu (); Liucandong (); Liluning () as the development of long-wavelength lasers. Solid HHG demonstrates novel characters different from the HHG from gases. For example, the linear cutoff energy dependence on the amplitude of the laser field Ghimire (), multi-plateau structure in the HHG spectra Ndabashimiye (), and different laser ellipticity dependence Ghimire (). However, the mechanisms of HHG from solids are still under debate. Inter- and intra-band transition models Pronin1 (); Pronin2 (); Guan (); Du2 (); Vampa () are proposed. Three-step models in the coordinate space Vampa () and vector space Du0 (); Du1 () are investigated. The drawback of the solid HHG is the low damage threshold of solid materials. Many efforts have been made to enhance the yield of HHG. For example, two-color laser fields Li (); LiuXi () and plasmon-enhanced inhomogeneous laser fields Du2 (); Vampa2 (); Joel (); Han () are used to manipulate the HHG process, especially for the enhancement of the second plateau of the HHG spectra Du2 ().

In circularly polarized laser fields, HHGs are absent in the atomic systems due to the non-recollision spin motions of electrons in the classical picture Corkum () and forbidden transition in the selection rules in the quantum picture. However, HHG occurs in molecular systems in circularly polarized lasers because of additional recollision centres Yuan1 (). In the solid systems, especially for 2D materials Liu (), HHG is not restricted in circularly polarized laser fields with polarization plane in the same plane of 2D solids because of the multi-centre potential wells.

In bicircular laser fields Milosevic1 (); Milosevic2 (), especially for the counter-rotating case, circular HHG with different helicities can be efficiently generated in atomic and molecular systems Yuan2 (); Mauger (). This has been experimentally demonstrated recently Kfir (). It can also be used to illuminate molecular symmetries Zhu (); Reich (). It is also possible to generate isolated circular attosecond laser pulse Yuan3 (); Xia ().

The HHG from solids in circular and bicircular laser fields is less investigated. In this work, we study the electron dynamics in 2D periodic potentials Hawkins () to simulate the HHG process. Atomic units are used throughout.

Figure 1: (Color online) Illustration of the 2D periodic potentials in the range of a.u.

Ii Numerical Results by Solving Tdse

Figure 2: (Color online) Energy band structure of the 2D potentials in Fig. 1.

In the single-active-electron approximation, the time-dependent Schrödinger equation(TDSE) can be written as

(1)

where and are the components of the laser fields. are two-dimensional periodic Gaussian potentials Pavelich (). The form for one unit cell is

(2)

where represents the maximum depth of the potential well, is the length of the unit cell, (, ) are the coordinates of the center of the potential well. Part of the 2D potentials is illustrated in Fig. 1. In this work, a.u., , .

The wave functions are expanded by B-splines:

(3)

The band structure of the system can be obtained by diagonalization of the field-free Hamiltonian matrix. The results are shown in Fig. 2.

The initial state in TDSE evolution is the state on top of the valence band. The time-dependent wavefunction is obtained by using the Crank-Nicholson method Bianc (); Guan (). The laser-induced currents are:

(4)

The HHG spectra are calculated by Fourier transforms of the above currents. The obtained complex components and can be used to extract the ellipticity , phase , and the orientation angle , respectively. They are defined as Yuan1 (); Born ():

(5)

where

(6)

Iii High-order harmonic generation in circularly polarized laser fields

Figure 3: (Color online) HHG spectra from 2D periodic potentials in circularly polarized laser fields. (a) HHG spectra along the and axes. (b) The ellipticity of each harmonics. (c) Phase difference between the and components. The intensity of the laser is W/cm, the wavelength m.

The circularly polarized laser field is in the following form:

(7)

where the pulse envelope is

(8)

is the pulse duration, which is set to be 10 cycles in this work. a.u. The wavelength is 3.2 m. The HHG spectra are presented in Fig. 3.

Different from the absence of HHG from atoms in circularly polarized laser fields, clear HHG signals are generated from solids. Strong odd harmonics are dominant. The even-order harmonics is very weak. We have increased the pulse duration to 20 cycles, the weak even harmonics remain. It is not from the short laser pulse, but from the weak anisotropy of the potential wells. From Fig. 3, one can observe that circularly polarized odd harmonics with order less than 19 are generated. This energy is close to the minimum band gap between the valance band and the first conduction band: a.u. This is in the perturbertive regime. Without the transition selection rules in the atomic case, circular harmonics with different helicities are allowed to appear. For harmonics with order , they are generated from the inter-band transitions. Elliptically polarized harmonics are presented. The possible reason for non-circular property may come from the anisotropy of the periodic potential wells and band structure in Fig. 2. The cutoff energy is determined by the maximum energy band gap between the first conduction band and the valence band in the range of from our proposed model Du0 (); Du1 () in the momentum space, where is amplitude of the vector potential of the laser fields. From Fig. 2, this value is around 41, agreeing well with the cutoff energy in Fig. 3, suggesting that the HHG model Du0 (); Du1 () is valid even in circularly polarized laser fields.

Iv High-order harmonic generation in corotating circularly polarized laser fields

Figure 4: (Color online) HHG spectra from 2D periodic potentials in corotating two-color circularly polarized laser fields. (a) HHG spectra along the and axes. The inset is the Lissajous curves of the electric field amplitude for one optical cycle of the fundamental field. The phase . (b) The ellipticity of each harmonics. (c) Phase difference between the and components. The intensity of the laser is W/cm, the wavelength m.
Figure 5: (Color online) The same as Fig. 4 except .
Figure 6: (Color online) The same as Fig. 4 except .

The coratating two-color and laser fields are in the following form:

(9)

The intensities of the two lasers are set to be the same. The pulse shape and the durations are the same as that in Eq. (8). The HHG spectra with different phases are illustrated in Figs. 4-6, respectively. Since the laser fields lose the symmetry in the polarization plane, even-order harmonics are obviously generated and their intensity is comparable to their neighbouring odd harmonics. One may find that the harmonics with order are not circularly polarized, but their ellipticities are close to 1. They are less influenced by the phase difference between the two-color laser fields because they are below the minimum band gap. For harmonics with order , their intensity, cutoff energy, ellipticity and phase difference are closely related to the phase of the corotating laser fields. The Lissajous curves of the electric field amplitude for one optical cycle of the fundamental laser field are presented in the above figures. determines the contributions of electrons from different directions to the HHG signals. From Fig.3, the band structure is also anisotropic. As a result, the phase in the corotating driving lasers can be used to control the HHG process, and thus the properties of the HHG signals. Only circularly and elliptically polarized harmonics are generated in the circularly polarized driving laser fields in the above section. Harmonics close to linear polarization are produced in the corotating bicircular laser fields.

V High-order harmonic generation in counter-rotating circularly polarized laser fields

Figure 7: (Color online) HHG spectra from 2D periodic potentials in counter-rotating two-color circularly polarized laser fields. (a) HHG spectra along the and axes. The inset is the Lissajous curves of the electric field amplitude for one optical cycle of the fundamental field. The phase . (b) The ellipticity of each harmonics. (c) Phase difference between the and components. The intensity of the laser is W/cm, the wavelength m.
Figure 8: (Color online) The same as Fig. 7 except .
Figure 9: (Color online) The same as Fig. 7 except .

The counter-rotating bicircular and laser fields with amplitude ratio 1:1 are in the following form:

(10)

The calculated HHG spectra are show in Figs. 7-9.

Circular HHGs with different helicities are generated from atoms in bicircular counter-rotating laser fields Milosevic1 (). However, from Figs. 7-9, no circular HHGs are produced in the case of solids. For the case of HHG from atoms, corotating bicircular lasers are less efficient to generate harmonics compared to the counter-rotating bicircular lasers. However, the efficiencies for generating HHG from 2D solids are comparable in the corotating and counter-rotating bicircular laser fields. Some harmonics close to linear polarization are also generated in counter-rotating bicircular laser fields. The phase can be used to tune the ellipticity, relative intensity, and phase difference of the components of the HHGs.

The amplitude ratio of the bicircular lasers will also affect the HHG signals. The laser frequencies are not restricted to 1 and 2. There are too many ways to combine these parameters to control the HHG processes, their effects will not be discussed in this work.

Vi Summary

In conclusion, we have studied the HHG from 2D solids in circular and bicircular corotating and counter-rotating laser fields. Different from the HHG from atoms, circular HHG from solids can be generated in circular driving lasers. Elliptically and linearly polarized harmonics can be generated in bicircular laser fields. The ellipticity, the relative intensity and phase difference between the harmonic components, and the cutoff can be controlled by the phase of the bicircular driving lasers. This phase dependence reflects the spacial distribution and energy band structure of the solid targets, which can be used as an imaging tool.

Vii Acknowledgements

We thank XuanYang Lai for many very helpful discussions. This work is supported by the National Natural Science Foundation of China(NSFC) (11561121002, 21501055, 11404376, 11674363, 61377109), Youth Science Foundation of Henan Normal University (2015QK03), Start-up Foundation for Doctors of Henan Normal University (QD15217).

References

  • (1) F. Krausz and M. Ivanov, Rev. Mod. Phys. 81, 163 (2009).
  • (2) L.-Y. Peng, W.-C. Jiang, J.-W. Geng, W.-H. Xiong, and Q. Gong, Phys. Rep. 575, 1 (2015).
  • (3) P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).
  • (4) S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, Nature Phys. 7, 138 (2011).
  • (5) K. F. Lee, X. Ding, T. J. Hammond, M. E. Fermann, G. Vampa, and P. B. Corkum, Opt. Lett. 42, 1113 (2017).
  • (6) U. Huttner, M. Kira, and S. W. Koch, Laser Photonics Rev. 11, 1700049 (2017).
  • (7) C. Yu, X. Zhang, S. Jiang, X. Cao, G. Yuan, T. Wu, L. Bai, and R. Lu, Phys. Rev. A 94, 013846 (2016).
  • (8) C. Liu, Y. Zheng, Z. Zeng, and R. Li, Phys. Rev. A 93, 043806 (2016).
  • (9) L.-N. Li and F. He, J. Opt. Soc. Am. B 34, 52 (2017).
  • (10) G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, Nature 534, 520 (2016).
  • (11) K. A. Pronin, A. D. Bandrauk, and A. A. Ovchinnikov, Phys. Rev. B 50, R3473 (1994).
  • (12) K. A. Pronin, and A. D. Bandrauk, Phys. Rev. Lett. 97, 020602 (2006)
  • (13) Z. Guan, X. X. Zhou, and X. B. Bian, Phys. Rev. A. 93, 033852 (2016).
  • (14) T.-Y. Du, Z. Guan, X.-X. Zhou, and X.-B. Bian, Phys. Rev. A 94, 023419 (2016).
  • (15) G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, Phys. Rev. B 91, 064302 (2015).
  • (16) T.-Y. Du and X.-B. Bian, arXiv:1606.06433 (2016).
  • (17) T.-Y. Du and X.-B. Bian, Opt. Express 25, 151 (2017).
  • (18) X. Liu, X. Zhu, P. Lan, X. Zhang, D. Wang, Q. Zhang, and P. Lu, Phys. Rev. A 95, 063419 (2017).
  • (19) J. B. Li, X. Zhang, S. J. Yue, H. M. Wu, B. T. Hu, and H. C. Du, Opt. Express 25, 18603 (2017).
  • (20) G. Vampa, B. G. Ghamsari, S. S. Mousavi, T. J. Hammond, A. Olivieri, E. L. Skrek, A. Y. Naumov, D. M. Villeneuve, A. Staudte, P. Berini and P. B. Corkum, Nat. Phys. (2017). (in press)
  • (21) J. D. Cox, A. Marini, and F. J. G. d. Abajo, Nat. Commun. 8, 14380 (2016).
  • (22) S. Han, H. Kim, Y. W. Kim, Y.-J. Kim, S. Kim, I.-Y. Park, and S.-W. Kim, Nat. Commun. 7, 13105 (2016).
  • (23) K. J. Yuan and A. D. Bandrauk, Phys. Rev. A 81, 063412 (2010).
  • (24) H. Liu, Y. Li, Y. S. You, S. Ghimire, T. F. Heinz, and D. A. Reis, Nat. Phys. 13, 262 (2017).
  • (25) D. B. Milošević, W. Becker, and R. Kopold, Phys. Rev. A 61, 063403 (2000).
  • (26) D. B. Milošević and W. Becker, Phys. Rev. A 62, 011403(R) (2000).
  • (27) K. J. Yuan and A. D. Bandrauk, J. Phys. B 45, 074001 (2012).
  • (28) F. Mauger, A. D. Bandrauk, and T. Uzer, J. Phys. B 49, 10LT01 (2016).
  • (29) O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin, D. Popmintchev, T. Popmintchev, H. Nembach, J. M. Shaw, A. Fleischer, H. Kapteyn, M. Murnane, and O. Cohen, Nature Photon. 9, 99 (2015).
  • (30) X. S. Zhu, X. Liu, Y. Li, M. Y. Qin, Q. B. Zhang, P. F. Lan, and P. X. Lu, Phys. Rev. A 91, 043418 (2015).
  • (31) D. M. Reich and L. B. Madsen, Phys. Rev. Lett. 117, 133902 (2016).
  • (32) K. J. Yuan and A. D. Bandrauk, Phys. Rev. Lett. 110, 023003 (2013).
  • (33) C. L. Xia and X. S. Liu, Phys. Rev. A 87 043406 (2013).
  • (34) P. G. Hawkins, M. Y. Ivanov, and V. S. Yakovlev, Phys. Rev. A. 91, 013405 (2015).
  • (35) R. L. Pavelich and F. Marsiglio, Am. J. Phys. 84, 924 (2016).
  • (36) X.B. Bian, Phys. Rev. A 90, 033403 (2014).
  • (37) M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge, UK, 1999), 7th ed., Chap. I.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
100195
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description