# Valley-Selective High Harmonic Generation and Polarization Induced by an Orthogonal Two-Color Laser Field

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Models

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Krausz, F.; Ivanov, M. Attosecond physics. Rev. Mod. Phys.
**2009**, 81, 163. [Google Scholar] - Corkum, P.B.; Krausz, F. Attosecond science. Nat. Phys.
**2007**, 3, 381–387. [Google Scholar] - Hentschel, M.; Kienberger, R.; Spielmann, C.; Reider, G.A.; Milosevic, N.; Brabec, T.; Corkum, P.; Heinzmann, U.; Drescher, M.; Krausz, F. Attosecond metrology. Nature
**2001**, 414, 509–513. [Google Scholar] [PubMed] - Zhu, X.; Lu, P.; Lein, M. Control of the Geometric Phase and Nonequivalence between Geometric–Phase Definitions in the Adiabatic Limit. Phys. Rev. Lett.
**2022**, 128, 030401. [Google Scholar] [PubMed] - Schafer, K.J.; Yang, B.; DiMauro, L.F.; Kulander, K.C. Above threshold ionization beyond the high harmonic cutoff. Phys. Rev. Lett.
**1993**, 70, 1599. [Google Scholar] - Corkum, P.B. Plasma perspective on strong field multiphoton ionization. Phys. Rev. Lett.
**1993**, 71, 1994. [Google Scholar] [CrossRef] - Lewenstein, M.; Balcou, P.; Ivanov, M.Y.; L’Huillier, A.; Corkum, P.B. Theory of high-harmonic generation by low–frequency laser fields. Phys. Rev. A
**1994**, 49, 2117. [Google Scholar] [PubMed] - He, L.; Lan, P.; Le, A.; Wang, B.; Wang, B.; Zhu, X.; Lu, P.; Lin, C.D. Real–Time Observation of Molecular Spinning with Angular High–Harmonic Spectroscopy. Phys. Rev. Lett.
**2018**, 121, 163201. [Google Scholar] [PubMed] - Ganeev, R.A. High–Order Harmonics Generation in Selenium–Containing Plasmas. Photonics
**2023**, 10, 854. [Google Scholar] - Liu, X.; Zhu, X.; Li, L.; Li, Y.; Zhang, Q.; Lan, P.; Lu, P. Selection rules of high–order–harmonic generation: Symmetries of molecules and laser fields. Phys. Rev. A
**2016**, 94, 033410. [Google Scholar] - Paul, P.M.; Toma, E.S.; Breger, P.; Mullot, G.; Augé, F.; Balcou, P.; Muller, H.G.; Agostini, P. Observation of a Train of Attosecond Pulses from High Harmonic Generation. Science
**2001**, 292, 1689–1692. [Google Scholar] - Chatziathanasiou, S.; Kahaly, S.; Skantzakis, E.; Sansone, G.; Lopez-Martens, R.; Haessler, S.; Varju, K.; Tsakiris, G.D.; Charalambidis, D.; Tzallas, P. Generation of Attosecond Light Pulses from Gas and Solid State Media. Photonics
**2017**, 4, 26. [Google Scholar] - Zhai, C.; Zhu, X.; Long, J.; Shao, R.; Zhang, Y.; He, L.; Tang, Q.; Li, Y.; Lan, P.; Yu, B.; et al. Generation of elliptically polarized attosecond pulses in mixed gases. Phys. Rev. A
**2021**, 103, 033114. [Google Scholar] - Ghimire, S.; DiChiara, A.D.; Sistrunk, E.; Agostini, P.; DiMauro, L.F.; Reis, D.A. Observation of high–order harmonic generation in a bulk crystal. Nat. Phys.
**2011**, 7, 138–141. [Google Scholar] - Kruchinin, S.Y.; Krausz, F.; Yakovlev, V.S. Colloquium: Strong–field phenomena in periodic systems. Rev. Mod. Phys.
**2018**, 90, 021002. [Google Scholar] - Han, S. High–Harmonic Generation Using a Single Dielectric Nanostructure. Photonics
**2022**, 9, 427. [Google Scholar] - Vampa, G.; Hammond, T.J.; Thiré, N.; Schmidt, B.E.; Légaré, F.; McDonald, C.R.; Brabec, T.; Corkum, P.B. Linking high harmonics from gases and solids. Nature
**2015**, 522, 462–464. [Google Scholar] - Wu, M.; Browne, D.A.; Schafer, K.J.; Gaarde, M.B. Multilevel perspective on high-order harmonic generation in solids. Phys. Rev. A
**2016**, 94, 063403. [Google Scholar] - Liu, X.; Zhu, X.; Lan, P.; Zhang, X.; Wang, D.; Zhang, Q.; Lu, P. Time–dependent population imaging for high–order–harmonic generation in solids. Phys. Rev. A
**2017**, 95, 063419. [Google Scholar] - Fu, S.; Feng, Y.; Li, J.; Yue, S.; Zhang, X.; Hu, B.; Du, H. Recollision dynamics analysis of high–order harmonic generation in solids. Phys. Rev. A
**2020**, 101, 023402. [Google Scholar] - Lanin, A.A.; Stepanov, E.A.; Fedotov, A.B.; Zheltikov, A.M. Mapping the electron band structure by intraband high–harmonic generation in solids. Optica
**2017**, 4, 516–519. [Google Scholar] - Li, L.; Lan, P.; He, L.; Cao, W.; Zhang, Q.; Lu, P. Determination of Electron Band Structure using Temporal Interferometry. Phys. Rev. Lett.
**2020**, 124, 157403. [Google Scholar] - Bauer, D.; Hansen, K.K. High–Harmonic Generation in Solids with and without Topological Edge States. Phys. Rev. Lett.
**2018**, 120, 177401. [Google Scholar] [PubMed] - Heide, C.; Kobayashi, Y.; Baykusheva, D.R.; Jain, D.; Sobota, J.A.; Hashimoto, M.; Kirchmann, P.S.; Oh, S.; Heinz, T.F.; Reis, D.A.; et al. Probing topological phase transitions using high-harmonic generation. Nat. Photon.
**2022**, 16, 620–624. [Google Scholar] - Schmid, C.P.; Weigl, L.; Grössing, P.; Junk, V.; Gorini, C.; Schlauderer, S.; Ito, S.; Meierhofer, M.; Hofmann, N.; Afanasiev, D.; et al. Tunable non-integer high-harmonic generation in a topological insulator. Nature
**2021**, 593, 385–390. [Google Scholar] [PubMed] - Yue, L.; Hollinger, R.; Uzundal, C.B.; Nebgen, B.; Gan, Z.; Najafidehaghani, E.; George, A.; Spielmann, C.; Kartashov, D.; Turchanin, A.; et al. Signatures of Multiband Effects in High–Harmonic Generation in Monolayer MoS
_{2}. Phys. Rev. Lett.**2022**, 129, 147401. [Google Scholar] - Tamaya, T.; Akiyama, H.; Kato, T. Shear-strain controlled high-harmonic generation in graphene. Phys. Rev. B
**2023**, 107, L081405. [Google Scholar] - Rana, N.; Mrudul, M.S.; Kartashov, D.; Ivanov, M.; Dixit, G. High-harmonic spectroscopy of coherent lattice dynamics in graphene. Phys. Rev. B
**2022**, 106, 064303. [Google Scholar] - Yoshikawa, N.; Tamaya, T.; Tanaka, K. High-harmonic generation in graphene enhanced by elliptically polarized light excitation. Science
**2017**, 356, 736–738. [Google Scholar] [PubMed] - Heide, C.; Kobayashi, Y.; Johnson, A.C.; Liu, F.; Heinz, T.F.; Reis, D.A.; Ghimire, S. Probing electron–hole coherence in strongly driven 2D materials using high–harmonic generation. Optica
**2022**, 9, 512–516. [Google Scholar] - Cha, S.; Kim, M.; Kim, Y.; Choi, S.; Kang, S.; Kim, H.; Yoon, S.; Moon, G.; Kim, T.; Lee, Y.W.; et al. Gate–tunable quantum pathways of high harmonic generation in graphene. Nat. Commun.
**2022**, 13, 6630. [Google Scholar] [PubMed] - Zheng, W.; Jiang, Y.; Wang, S.; Liu, C.; Bai, Y.; Liu, P.; Li, R. Frequency shift of even–order high harmonic generation in monolayer MoS
_{2}. Opt. Express**2023**, 31, 27029–27040. [Google Scholar] - Castro Neto, A.H.; Guinea, F.; Peres, N.M.R.; Novoselov, K.S.; Geim, A.K. The electronic properties of graphene. Rev. Mod. Phys.
**2009**, 81, 109. [Google Scholar] - Geim, A.K. Graphene: Status and Prospects. Science
**2009**, 324, 1530–1534. [Google Scholar] - Schaibley, J.R.; Yu, H.; Clark, G.; Rivera, P.; Ross, J.S.; Seyler, K.L.; Yao, W.; Xu, X. Valleytronics in 2D materials. Nat. Rev. Mater.
**2016**, 1, 16055. [Google Scholar] - Vitale, S.A.; Nezich, D.; Varghese, J.O.; Kim, P.; Gedik, N.; Jarillo–Herrero, P.; Xiao, D.; Rothschild, M. Valleytronics: Opportunities, Challenges, and Paths Forward. Small
**2018**, 14, 1801483. [Google Scholar] - Langer, F.; Schmid, C.P.; Schlauderer, S.; Gmitra, M.; Fabian, J.; Nagler, P.; Schüller, C.; Korn, T.; Hawkins, P.G.; Steiner, J.T.; et al. Lightwave valleytronics in a monolayer of tungsten diselenide. Nature
**2018**, 557, 76–80. [Google Scholar] - Xiao, D.; Yao, W.; Niu, Q. Valley–Contrasting Physics in Graphene: Magnetic Moment and Topological Transport. Phys. Rev. Lett.
**2007**, 99, 236809. [Google Scholar] - Mak, K.F.; Xiao, D.; Shan, J. Light–valley interactions in 2D semiconductors. Nat. Photon.
**2018**, 12, 451–460. [Google Scholar] - Jiménez–Galán, Á.; Silva, R.E.F.; Smirnova, O.; Ivanov, M. Lightwave control of topological properties in 2D materials for sub–cycle and non-resonant valley manipulation. Nat. Photon.
**2020**, 14, 728–732. [Google Scholar] - Jiménez–Galán, Á.; Silva, R.E.F.; Smirnova, O.; Ivanov, M. Sub-cycle valleytronics: Control of valley polarization using few-cycle linearly polarized pulses. Optica
**2021**, 8, 277–280. [Google Scholar] - Sharma, S.; Elliott, P.; Shallcross, S. Valley control by linearly polarized laser pulses: Example of WSe
_{2}. Optica**2022**, 9, 947–952. [Google Scholar] - Golub, L.; Tarasenko, S. Valley polarization induced second harmonic generation in graphene. Phys. Rev. B
**2014**, 90, 201402. [Google Scholar] [CrossRef] - Rana, N.; Dixit, G. All–Optical Ultrafast Valley Switching in Two-Dimensional Materials. Phys. Rev. Appl.
**2023**, 19, 034056. [Google Scholar] - Mak, K.F.; McGill, K.L.; Park, J.; McEuen, P.L. The valley Hall effect in MoS
_{2}transistors. Science**2014**, 344, 1489–1492. [Google Scholar] [CrossRef] - Mrudul, M.S.; Jiménez–Galán, Á.; Ivanov, M.; Dixit, G. Light–induced valleytronics in pristine graphene. Optica
**2021**, 8, 422–427. [Google Scholar] - He, Y.; Guo, J.; Gao, F.; Liu, X. Dynamical symmetry and valley–selective circularly polarized high–harmonic generation in monolayer molybdenum disulfide. Phys. Rev. B
**2022**, 105, 024305. [Google Scholar] - Chen, J.; Liu, C.; Li, R. Valley-Selective Polarization in Twisted Bilayer Graphene Controlled by a Counter-Rotating Bicircular Laser Field. Photonics
**2023**, 10, 516. [Google Scholar] - Mrudul, M.S.; Dixit, G. Controlling valley–polarisation in graphene via tailored light pulses. J. Phys. B At. Mol. Opt. Phys.
**2021**, 54, 224001. [Google Scholar] - Reich, S.; Maultzsch, J.; Thomsen, C.; Ordejon, P. Tight–binding description of graphene. Phys. Rev. B
**2002**, 66, 035412. [Google Scholar] - Jiang, S.; Wei, H.; Chen, J.; Yu, C.; Lu, R.; Lin, C.D. Effect of transition dipole phase on high–order–harmonic generation in solid materials. Phys. Rev. A
**2017**, 96, 053850. [Google Scholar] - Dimitrovski, D.; Madsen, L.B.; Pedersen, T.G. High–order harmonic generation from gapped graphene: Perturbative response and transition to nonperturbative regime. Phys. Rev. B
**2017**, 95, 035405. [Google Scholar] - Avetissian, H.K.; Avetissian, A.K.; Avchyan, B.R.; Mkrtchian, G.F. Wave mixing and high harmonic generation at two–color multiphoton excitation in two–dimensional hexagonal nanostructures. Phys. Rev. B
**2019**, 100, 035434. [Google Scholar] - Avetissian, H.K.; Mkrtchian, G.F.; Knorr, A. Efficient high–harmonic generation in graphene with two–color laser field at orthogonal polarization. Phys. Rev. B
**2022**, 105, 195405. [Google Scholar] - Vampa, G.; McDonald, C.R.; Orlando, G.; Klug, D.D.; Corkum, P.B.; Brabec, T. Theoretical Analysis of High-Harmonic Generation in Solids. Phys. Rev. Lett.
**2014**, 113, 073901. [Google Scholar] - Liu, X.; Li, L.; Zhu, X.; Huang, T.; Zhang, X.; Wang, D.; Lan, P.; Lu, P. Wavelength dependence of high–order harmonic yields in solids. Phys. Rev. A
**2018**, 98, 063419. [Google Scholar] - Zhang, Y.; Li, J.; Li, L.; Huang, T.; Zhu, X.; Lan, P.; Lu, P. Enhancement of the photocurrents injected in gapped graphene by the orthogonally polarized two–color laser field. Opt. Express
**2021**, 29, 17387–17397. [Google Scholar] [PubMed] - Vampa, G.; McDonald, C.R.; Orlando, G.; Corkum, P.B.; Brabec, T. Semiclassical analysis of high harmonic generation in bulk crystals. Phys. Rev. B
**2015**, 91, 064302. [Google Scholar] [CrossRef] - Guan, Z.; Zhou, X.; Bian, X. High–order–harmonic generation from periodic potentials driven by few-cycle laser pulses. Phys. Rev. A
**2016**, 93, 033852. [Google Scholar] [CrossRef] - Wang, H.; Feng, Y.; Fu, S.; Li, J.; Zhang, X.; Du, H. Complex carrier-envelope-phase effect of solid harmonics under nonadiabatic conditions. Phys. Rev. A
**2019**, 99, 023406. [Google Scholar] - Liu, X.; Li, Y.; Liu, D.; Zhu, X.; Zhang, X.; Lu, P. Effects of quantum interferences among crystal–momentum–resolved electrons in solid high–order harmonic generation. Phys. Rev. A
**2021**, 103, 033104. [Google Scholar] - Zhou, S.Y.; Gweon, G.H.; Fedorov, A.V.; First, P.N.; de Heer, W.A.; Lee, D.H.; Guinea, F.; Castro Neto, A.H.; Lanzara, A. Substrate–induced bandgap opening in epitaxial graphene. Nat. Mater.
**2007**, 6, 770–775. [Google Scholar] - Yankowitz, M.; Xue, J.; Cormode, D.; Sanchez-Yamagishi, J.D.; Watanabe, K.; Taniguchi, T.; Jarillo-Herrero, P.; Jacquod, P.; LeRoy, B.J. Emergence of superlattice Dirac points in graphene on hexagonal boron nitride. Nat. Phys.
**2012**, 8, 382–386. [Google Scholar] - Grujić, M.M.; Tadić, M.Ž.; Peeters, F.M. Spin-Valley Filtering in Strained Graphene Structures with Artificially Induced Carrier Mass and Spin-Orbit Coupling. Phys. Rev. Lett.
**2014**, 113, 046601. [Google Scholar] [PubMed] - Brugnera, L.; Frank, F.; Hoffmann, D.J.; Torres, R.; Siegel, T.; Underwood, J.G.; Springate, E.; Froud, C.; Turcu, E.I.C.; Tisch, J.W.G.; et al. Enhancement of high harmonics generated by field steering of electrons in a two–color orthogonally polarized laser field. Opt. Lett.
**2010**, 35, 3994–3996. [Google Scholar] [PubMed] - Cireasa, R.; Boguslavskiy, A.E.; Pons, B.; Wong, M.C.H.; Descamps, D.; Petit, S.; Ruf, H.; Thiré, N.; Ferré, A.; Suarez, J.; et al. Probing molecular chirality on a sub–femtosecond timescale. Nat. Phys.
**2015**, 11, 654–658. [Google Scholar] - Smirnova, O.; Mairesse, Y.; Patchkovskii, S. Opportunities for chiral discrimination using high harmonic generation in tailored laser fields. J. Phys. B At. Mol. Opt. Phys.
**2015**, 48, 234005. [Google Scholar] - Zhu, X.; Liu, X.; Lan, P.; Wang, D.; Zhang, Q.; Li, W.; Lu, P. Anomalous circular dichroism in high harmonic generation of stereoisomers with two chiral centers. Opt. Express
**2016**, 24, 24824–24835. [Google Scholar]

**Figure 1.**(

**a**) Band structures of the CB and VB for the gapped graphene with a bandgap $\Delta =1\phantom{\rule{4pt}{0ex}}\mathrm{eV}$. (

**b**) K and ${K}^{\prime}$ valleys in the VB.

**Figure 2.**High harmonic spectra from the K and ${K}^{\prime}$ valleys with various relative phases of OTC laser fields: (

**a**) $\phi ={0}^{\circ}$, (

**b**) $\phi ={90}^{\circ}$, (

**c**) $\phi ={180}^{\circ}$, and (

**d**) $\phi ={270}^{\circ}$. The Lissajous figures of the laser fields are plotted in the insets at the upper-right corners of the panels.

**Figure 3.**High harmonic yields from the K and ${K}^{\prime}$ valleys as a function of relative phase $\phi $ of the OTC laser field for (

**a**) ${\mathrm{H}}_{15}$ and (

**b**) ${\mathrm{H}}_{\mathrm{avg}}$. Valley deviation parameter Q as a function of $\phi $ for (

**c**) ${\mathrm{H}}_{15}$ and (

**d**) ${\mathrm{H}}_{\mathrm{avg}}$. ${\mathrm{H}}_{15}$ denotes the 15th-order harmonic. ${\mathrm{H}}_{\mathrm{avg}}$ denotes the average yield of harmonics in plateau, which is calculated as the average yield between 11th-order and 19th-order harmonics.

**Figure 4.**(

**a**) Cutoff orders of harmonics contributed by the K and ${K}^{\prime}$ valleys as a function of $\phi $. (

**b**) The Lissajous figures of adopted OTC laser fields with $\phi ={90}^{\circ}$ and ${270}^{\circ}$. The solid arrow indicates the rotation direction of the laser field. The dashed arrow indicates the helicity of the left/right lobe of the OTC laser field.

**Figure 5.**The time–dependent electron populations ${N}_{\mathrm{c}}$ from K and ${K}^{\prime}$ valleys with various relative phases of OTC laser fields: (

**a**) $\phi ={0}^{\circ}$, (

**b**) $\phi ={45}^{\circ}$, (

**c**) $\phi ={90}^{\circ}$, (

**d**) $\phi ={135}^{\circ}$, (

**e**) $\phi ={180}^{\circ}$, (

**f**) $\phi ={225}^{\circ}$, (

**g**) $\phi ={270}^{\circ}$, and (

**h**) $\phi ={315}^{\circ}$.

**Figure 6.**(

**a**) The electron populations at the end of the laser pulse from K and ${K}^{\prime}$ valleys as a function of $\phi $ for the CB. (

**b**) Valley asymmetry parameter $\eta $ as a function of $\phi $ for the CB.

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**MDPI and ACS Style**

Liu, X.; Liu, D.; Sun, Y.; Li, Y.; Zhang, C.
Valley-Selective High Harmonic Generation and Polarization Induced by an Orthogonal Two-Color Laser Field. *Photonics* **2023**, *10*, 1126.
https://doi.org/10.3390/photonics10101126

**AMA Style**

Liu X, Liu D, Sun Y, Li Y, Zhang C.
Valley-Selective High Harmonic Generation and Polarization Induced by an Orthogonal Two-Color Laser Field. *Photonics*. 2023; 10(10):1126.
https://doi.org/10.3390/photonics10101126

**Chicago/Turabian Style**

Liu, Xi, Dongdong Liu, Yan Sun, Yujie Li, and Cui Zhang.
2023. "Valley-Selective High Harmonic Generation and Polarization Induced by an Orthogonal Two-Color Laser Field" *Photonics* 10, no. 10: 1126.
https://doi.org/10.3390/photonics10101126