# Tunability of Half Cycle Cutoff Harmonics with Inhomogeneously Enhanced Laser Pulse

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Methods

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

HHG | High harmonic generation |

CEP | Carrier envelope phase |

HCO | Half cycle cutoff |

XUV | Extreme-ultraviolet |

## References

- Krausz, F.; Ivanov, M. Attosecond physics. Rev. Mod. Phys.
**2009**, 81, 163. [Google Scholar] [CrossRef] - Baltuška, A.; Udem, T.; Uiberacker, M.; Hentschel, M.; Goulielmakis, E.; Gohle, C.; Holzwarth, R.; Yakovlev, V.; Scrinzi, A.; Hänsch, T.; et al. Attosecond control of electronic processes by intense light fields. Nature
**2003**, 421, 611–615. [Google Scholar] [CrossRef] [PubMed] - Frolov, M.; Manakov, N.; Silaev, A.; Vvedenskii, N.; Starace, A.F. High-order harmonic generation by atoms in a few-cycle laser pulse: Carrier-envelope phase and many-electron effects. Phys. Rev. A
**2011**, 83, 021405. [Google Scholar] [CrossRef] - Frolov, M.; Manakov, N.; Silaev, A.; Vvedenskii, N. Analytic description of high-order harmonic generation by atoms in a two-color laser field. Phys. Rev. A
**2010**, 81, 063407. [Google Scholar] [CrossRef] - Peng, D.; Frolov, M.; Pi, L.W.; Starace, A.F. Enhancing high-order harmonic generation by sculpting waveforms with chirp. Phys. Rev. A
**2018**, 97, 053414. [Google Scholar] [CrossRef] - Taoutioui, A.; Agueny, H. Femtosecond single cycle pulses enhanced the efficiency of high order harmonic generation. Micromachines
**2021**, 12, 610. [Google Scholar] [CrossRef] - Yakovlev, V.S.; Scrinzi, A. High harmonic imaging of few-cycle laser pulses. Phys. Rev. Lett.
**2003**, 91, 153901. [Google Scholar] [CrossRef] - Haworth, C.; Chipperfield, L.; Robinson, J.; Knight, P.; Marangos, J.; Tisch, J. Half-cycle cutoffs in harmonic spectra and robust carrier-envelope phase retrieval. Nat. Phys.
**2007**, 3, 52–57. [Google Scholar] [CrossRef] - Cundiff, S.T. Better by half. Nat. Phys.
**2007**, 3, 16–18. [Google Scholar] [CrossRef] - Xiong, W.H.; Geng, J.W.; Gong, Q.; Peng, L.Y. Half-cycle cutoff in near-threshold harmonic generation. New J. Phys.
**2015**, 17, 123020. [Google Scholar] [CrossRef] - Pfeifer, T.; Jullien, A.; Abel, M.J.; Nagel, P.M.; Gallmann, L.; Neumark, D.M.; Leone, S.R. Generating coherent broadband continuum soft-x-ray radiation by attosecond ionization gating. Opt. Express
**2007**, 15, 17120–17128. [Google Scholar] [CrossRef] [PubMed] - Abel, M.J.; Pfeifer, T.; Nagel, P.M.; Boutu, W.; Bell, M.J.; Steiner, C.P.; Neumark, D.M.; Leone, S.R. Isolated attosecond pulses from ionization gating of high-harmonic emission. Chem. Phys.
**2009**, 366, 9–14. [Google Scholar] [CrossRef] - Cavalieri, A.L.; Goulielmakis, E.; Horvath, B.; Helml, W.; Schultze, M.; Fieß, M.; Pervak, V.; Veisz, L.; Yakovlev, V.; Uiberacker, M.; et al. Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua. New J. Phys.
**2007**, 9, 242. [Google Scholar] [CrossRef] - Guo, Y.H.; Lu, R.F.; Han, K.L.; He, G.Z. Generation of an isolated sub-100 attosecond pulse in a two-color laser field. Int. J. Quantum Chem.
**2009**, 109, 3410–3415. [Google Scholar] [CrossRef] - Zeng, Z.; Cheng, Y.; Song, X.; Li, R.; Xu, Z. Generation of an extreme ultraviolet supercontinuum in a two-color laser field. Phys. Rev. Lett.
**2007**, 98, 203901. [Google Scholar] [CrossRef] - Lan, P.; Lu, P.; Cao, W.; Li, Y.; Wang, X. Carrier-envelope phase-stabilized attosecond pulses from asymmetric molecules. Phys. Rev. A
**2007**, 76, 021801. [Google Scholar] [CrossRef] - Song, X.; Zeng, Z.; Fu, Y.; Cai, B.; Li, R.; Cheng, Y.; Xu, Z. Quantum path control in few-optical-cycle regime. Phys. Rev. A
**2007**, 76, 043830. [Google Scholar] [CrossRef] - Ye, P.; He, X.; Teng, H.; Zhan, M.; Zhang, W.; Wang, L.; Zhong, S.; Wei, Z. Extraction of the in situ temporal information of few-cycle laser pulse from carrier-envelope phase-dependent high order harmonic spectrum. JOSA B
**2014**, 31, 1355–1359. [Google Scholar] [CrossRef] - Geiseler, H.; Ishii, N.; Kaneshima, K.; Kitano, K.; Kanai, T.; Itatani, J. High-energy half-cycle cutoffs in high harmonic and rescattered electron spectra using waveform-controlled few-cycle infrared pulses. J. Phys. At. Mol. Opt. Phys.
**2014**, 47, 204011. [Google Scholar] [CrossRef] - Teichmann, S.M.; Silva, F.; Cousin, S.L.; Biegert, J. Importance of intensity-to-phase coupling for water-window high-order-harmonic generation with few-cycle pulses. Phys. Rev. A
**2015**, 91, 063817. [Google Scholar] [CrossRef] - Kim, S.; Jin, J.; Kim, Y.J.; Park, I.Y.; Kim, Y.; Kim, S.W. High-harmonic generation by resonant plasmon field enhancement. Nature
**2008**, 453, 757–760. [Google Scholar] [CrossRef] [PubMed] - Husakou, A.; Im, S.J.; Herrmann, J. Theory of plasmon-enhanced high-order harmonic generation in the vicinity of metal nanostructures in noble gases. Phys. Rev. A
**2011**, 83, 043839. [Google Scholar] [CrossRef] - Ciappina, M.; Pérez-Hernández, J.; Shaaran, T.; Biegert, J.; Quidant, R.; Lewenstein, M. Above-threshold ionization by few-cycle spatially inhomogeneous fields. Phys. Rev. A
**2012**, 86, 023413. [Google Scholar] [CrossRef] - Ciappina, M.; Biegert, J.; Quidant, R.; Lewenstein, M. High-order-harmonic generation from inhomogeneous fields. Phys. Rev. A
**2012**, 85, 033828. [Google Scholar] [CrossRef] - Du, T.Y.; Guan, Z.; Zhou, X.X.; Bian, X.B. Enhanced high-order harmonic generation from periodic potentials in inhomogeneous laser fields. Phys. Rev. A
**2016**, 94, 023419. [Google Scholar] [CrossRef] - Blanco, M.; Hernández-García, C.; Chacón, A.; Lewenstein, M.; Flores-Arias, M.T.; Plaja, L. Phase matching effects in high harmonic generation at the nanometer scale. Opt. Express
**2017**, 25, 14974–14985. [Google Scholar] [CrossRef] - Ansari, I.N.; Hofmann, C.; Medišauskas, L.; Lewenstein, M.; Ciappina, M.F.; Dixit, G. Controlling polarization of attosecond pulses with plasmonic-enhanced bichromatic counter-rotating circularly polarized fields. Phys. Rev. A
**2021**, 103, 013104. [Google Scholar] [CrossRef] - Piglosiewicz, B.; Schmidt, S.; Park, D.J.; Vogelsang, J.; Groß, P.; Manzoni, C.; Farinello, P.; Cerullo, G.; Lienau, C. Carrier-envelope phase effects on the strong-field photoemission of electrons from metallic nanostructures. Nat. Photonics
**2014**, 8, 37–42. [Google Scholar] [CrossRef] - Adnani, Y.; Taoutioui, A.; Makhoute, A.; Tőkési, K.; Agueny, H. Generation of superintense isolated attosecond pulses from trapped electrons in metal surfaces. Phys. Rev. A
**2022**, 105, 043104. [Google Scholar] [CrossRef] - Mandal, A.; Singh, K.P. High harmonic generation near a bow-tie nanostructure: Sensitivity to carrier envelope phase and plasmonic inhomogeneity. Laser Phys.
**2022**, 33, 015301. [Google Scholar] [CrossRef] - Protopapas, M.; Keitel, C.H.; Knight, P.L. Atomic physics with super-high intensity lasers. Rep. Prog. Phys.
**1997**, 60, 389. [Google Scholar] [CrossRef] - Ciappina, M.F.; Aćimović, S.S.; Shaaran, T.; Biegert, J.; Quidant, R.; Lewenstein, M. Enhancement of high harmonic generation by confining electron motion in plasmonic nanostrutures. Opt. Express
**2012**, 20, 26261–26274. [Google Scholar] [CrossRef] [PubMed] - Zagoya, C.; Bonner, M.; Chomet, H.; Slade, E.; de Morisson Faria, C.F. Different time scales in plasmonically enhanced high-order-harmonic generation. Phys. Rev. A
**2016**, 93, 053419. [Google Scholar] [CrossRef] - Ansari, I.N.; Mrudul, M.; Ciappina, M.F.; Lewenstein, M.; Dixit, G. Simultaneous control of harmonic yield and energy cutoff of high-order harmonic generation using seeded plasmonically enhanced fields. Phys. Rev. A
**2018**, 98, 063406. [Google Scholar] [CrossRef] - de Morisson Faria, C.F.; Dörr, M.; Becker, W.; Sandner, W. Time-frequency analysis of two-color high-harmonic generation. Phys. Rev. A
**1999**, 60, 1377. [Google Scholar] [CrossRef] - Kohler, M.C.; Ott, C.; Raith, P.; Heck, R.; Schlegel, I.; Keitel, C.H.; Pfeifer, T. High harmonic generation via continuum wave-packet interference. Physical Rev. Lett.
**2010**, 105, 203902. [Google Scholar] [CrossRef] - Chirilă, C.; Dreissigacker, I.; van der Zwan, E.V.; Lein, M. Emission times in high-order harmonic generation. Phys. Rev. A
**2010**, 81, 033412. [Google Scholar] [CrossRef] - Sarantseva, T.; Silaev, A.; Romanov, A.; Vvedenskii, N.; Frolov, M. Time-frequency analysis of high harmonic generation using a probe XUV pulse. Opt. Express
**2021**, 29, 1428–1440. [Google Scholar] [CrossRef] - de Morisson Faria, C.F.; Du, M. Enhancement of bichromatic high-order-harmonic generation with a high-frequency field. Phys. Rev. A
**2001**, 64, 023415. [Google Scholar] [CrossRef] - Su, Q.; Eberly, J. Model atom for multiphoton physics. Phys. Rev. A
**1991**, 44, 5997. [Google Scholar] [CrossRef] - Yu, C.; Wang, Y.; Cao, X.; Jiang, S.; Lu, R. Isolated few-attosecond emission in a multi-cycle asymmetrically nonhomogeneous two-color laser field. J. Phys. At. Mol. Opt. Phys.
**2014**, 47, 225602. [Google Scholar] [CrossRef] - Guo, Y.; Liu, A.; Wang, J.; Liu, X. Atomic even-harmonic generation due to symmetry-breaking effects induced by spatially inhomogeneous field. Chin. Phys.
**2019**, 28, 094212. [Google Scholar] [CrossRef] - Manolopoulos, D.E. Derivation and reflection properties of a transmission-free absorbing potential. J. Chem. Phys.
**2002**, 117, 9552–9559. [Google Scholar] [CrossRef]

**Figure 1.**Half cycle cutoff: harmonic spectra (

**a**), time–frequency response of the harmonic generation (

**b**), and classical trajectory calculation of return energy are shown in (

**c**,

**d**). In the false color map in (

**b**) yellow represents high value and blue represents low value of the intensity of the Gabor transformation. S and L represent the short and long trajectories, respectively.

**Figure 2.**Variation of HHG spectra with respect to the strength of inhomogeneity for spatial dependence ${\u03f5}_{q}{x}^{2}$ (

**a**), ${\u03f5}_{m}\left|x\right|$ (

**b**), and $\u03f5x$ (

**c**). Harmonic yield is given in logarithmic scale. The spectra are shifted vertically to improve visualization. Different strength of inhomogeneity are indicated by arrows with corresponding color. An HCO location is indicated with pink arrow.

**Figure 3.**TFR map calculated from the Gabor transformation of dipole acceleration (

**a**–

**d**) and classical trajectory simulation of return energy (

**e**–

**h**) are shown for the three different strengths of inhomogeneity for spatial dependence ${\u03f5}_{m}\left|x\right|$. The color map values are shown on the right of each panel of the TFR map, which represent the logarithm of intensity of the Gabor transformation, i.e., $|{\sigma}_{W}{(\omega ,\tau )|}^{2}$. In CTS plots, open circles represent the starting time of the trajectories and filled circles are the time of recollision. S and L represent the short and long trajectories, respectively.

**Figure 6.**Carrier-envelope-phase-dependent harmonic spectra are shown in color map (the color bars are shown on right side of each panel) for homogeneous (

**a**) and for inhmogeneous driving with spatial dependence ${\u03f5}_{q}{x}^{2}$ (

**b**), ${\u03f5}_{m}\left|x\right|$ (

**c**) and $\u03f5x$ (

**d**).

**Figure 7.**Temporal profiles of intensity of the XUV pulses generated using the harmonics around the half cycle cutoff near harmonic order 100. Different panels are for different types of inhomogeneity. The strength of the inhomogeneity and harmonic order window are mentioned in each XUV profile. Each profile is shifted in time for clarity.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mandal, A.
Tunability of Half Cycle Cutoff Harmonics with Inhomogeneously Enhanced Laser Pulse. *Atoms* **2023**, *11*, 113.
https://doi.org/10.3390/atoms11080113

**AMA Style**

Mandal A.
Tunability of Half Cycle Cutoff Harmonics with Inhomogeneously Enhanced Laser Pulse. *Atoms*. 2023; 11(8):113.
https://doi.org/10.3390/atoms11080113

**Chicago/Turabian Style**

Mandal, Ankur.
2023. "Tunability of Half Cycle Cutoff Harmonics with Inhomogeneously Enhanced Laser Pulse" *Atoms* 11, no. 8: 113.
https://doi.org/10.3390/atoms11080113