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

###### 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.

## Ii Numerical Results by Solving Tdse

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) |

## Iii High-order harmonic generation in circularly polarized laser fields

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

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

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

(10) |

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).

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