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PT symmetry in a fractional Schrödinger equation

We investigate the fractional Schrödinger equation with a periodic ‐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...

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Bibliographic Details
Published in:Laser & photonics reviews 2016-05, Vol.10 (3), p.526-531
Main Authors: Zhang, Yiqi, Zhong, Hua, Belić, Milivoj R., Zhu, Yi, Zhong, Weiping, Zhang, Yanpeng, Christodoulides, Demetrios N., Xiao, Min
Format: Article
Language:English
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Summary:We investigate the fractional Schrödinger equation with a periodic ‐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 ‐symmetric potential. This investigation may find applications in novel on‐chip optical devices. Beam propagation in fractional Schrödinger equation with a periodic PT‐symmetric potential is investigated. At a critical point, the band structure becomes linear in the one‐dimensional case, and cone‐like in the two‐dimensional case, which result in a nondiffracting propagation and conical diffraction of input beams.
ISSN:1863-8880
1863-8899
DOI:10.1002/lpor.201600037