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PT symmetry in a fractional Schrödinger equation

Yiqi Zhang Key Laboratory for Physical Electronics and Devices of the Ministry of Education; Shaanxi Key Lab of Information Photonic Technique, Xi'an Jiaotong University, Xi'an, 710049 China Hua Zhong Key Laboratory for Physical Electronics and Devices of the Ministry of Education; Shaanxi Key Lab of Information Photonic Technique, Xi'an Jiaotong University, Xi'an, 710049 China Milivoj R. Belić Science Program, Texas A&M University at Qatar, P.O. Box 23874 Doha, Qatar Yi Zhu Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, 100084 China Weiping Zhong Department of Electronic and Information Engineering, Shunde Polytechnic, Shunde, 528300 China Yanpeng Zhang Key Laboratory for Physical Electronics and Devices of the Ministry of Education; Shaanxi Key Lab of Information Photonic Technique, Xi'an Jiaotong University, Xi'an, 710049 China Demetrios N. Christodoulides CREOL, College of Optics and Photonics, University of Central Florida, Orlando, Florida, 32816-2700 USA Min Xiao Department of Physics, University of Arkansas, Fayetteville, Arkansas, 72701 USA; National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing, 210093 China

Analysis of PDEs mathscidoc:2204.03009

Laser & Photonics Reviews, 10, (3), 2016.5
We investigate the fractional Schrödinger equation with a periodic PT-symmetric potential. In the inverse space, the problem transfers into a first-order nonlocal frequency-delay partial differential equation. We show that at a critical point, the band structure becomes linear and symmetric in the one-dimensional case, which results in a nondiffracting propagation and conical diffraction of input beams. If only one channel in the periodic potential is excited, adjacent channels become uniformly excited along the propagation direction, which can be used to generate laser beams of high power and narrow width. In the two-dimensional case, there appears conical diffraction that depends on the competition between the fractional Laplacian operator and the PT-symmetric potential. This investigation may find applications in novel on-chip optical devices.
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@inproceedings{yiqi2016pt,
  title={PT symmetry in a fractional Schrödinger equation},
  author={Yiqi Zhang, Hua Zhong, Milivoj R. Belić, Yi Zhu, Weiping Zhong, Yanpeng Zhang, Demetrios N. Christodoulides, and Min Xiao},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220428151706325865150},
  booktitle={Laser & Photonics Reviews},
  volume={10},
  number={3},
  year={2016},
}
Yiqi Zhang, Hua Zhong, Milivoj R. Belić, Yi Zhu, Weiping Zhong, Yanpeng Zhang, Demetrios N. Christodoulides, and Min Xiao. PT symmetry in a fractional Schrödinger equation. 2016. Vol. 10. In Laser & Photonics Reviews. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220428151706325865150.
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