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3The 30th International Conference on Advanced Laser Technologies LM-P-7
ALT'23
The way of the second wave in photonic crystal with PT-symmetry periodic longitudinal and linear transverse modulation
T.A.Khudaiberganov1, I.A.Sidnihin1, S.MArakelian1
1-Department of Physics and Applied Mathematics, Vladimir State University named after A. G. and N. G. Stoletovs,
87 Gorkii st., 600000 Vladimir, Russia e-mail address: thomasheisenberg@mail.ru
In non-Hermitian physics, systems with parity-time (PT) symmetry are of interest, since this type of symmetry, under certain conditions, preserves the integrals of motion, in particular energy. This discovery belongs to Bender [1]. This fields of research is of great interest in photonics, quantum mechanics, and topological protected states [2]. Photonic crystals can used as a convenient platform for studying the properties of PT-symmetric systems. Photonic crystals are inhomogeneous optical materials, which are presence of spatial periodic modulation of the permittivity with a period of the order of the wavelength of light. Topological photonics doing the possibility of realizing stable transport phenomena, which, together with non-Hermitian physics, could cause new effects. It has been observed that dimerization results in improved light retention in the defective waveguide [3].
Consider a photonic crystal with modulation of the real and imaginary parts given by the following expression [4]: £(x,z) = U0(c os(A(z)x) + iysin(A(z)x)), (1)
here U0 and U0y - is modulation amplitudes of gains and losses in a photonic crystal, A(z) - is modulation period of the photonic crystal lattice, which depends linearly on the z coordinate as A(z) = A0(1 + az) (2)
This PT-symmetry photonic crystal are shown in Fig.la. The type of spatial modulation of the permittivity is shown in Fig. 1b.
?j.\ x/A
Figure 1 - (a) Sketch a PT-symmetry photonic crystal with a slow increase in the period of the crystal; (b) the shape of the permittivity of photonic crystal; (c) The shape of the intensity distribution over the photonic crystal; (d) the spectrum of the signal at the output of the photonic crystal, according to the x-coordinate wavenumbers kx; (e) The shape of the intensity distribution over the photonic crystal at the output of the photonic crystal.
When the PT-symmetry is broken, the transmission spectrum of the photonic crystal (in the direction of modulation, i.e. along the x-axis) becomes asymmetric, see Fig.ld (which implies unidirectional light propagation). The asymmetry of the transmission spectrum leads to the fact that for the Gaussian pulse, the wave components directed towards the best signal transmission prevail over the -kx component. This leads to the propagation of the wave energy towards the negative gradient of the imaginary part of the permittivity, i.e. arise x-component Poynting vector. For a transversely incident modulation pulse of a photonic crystal, a secondary wave arises when the PT symmetry is broken. It spreads at an angle to the main signal. The tangent of the angle of this wave is determined by the ratio of the gradients of the propagation constants of the x and z components. For the case of modulation according to the relation (2), the secondary wave propagates along a more difficult trajectory. Moreover, if the secondary wave in the usual case a = 0, see Fig. 1e, is weak compared to the main signal, then when incident at an angle such that the x-component of the wave vector of the secondary wave falls into resonance with the transmission spectrum of the photonic crystal, then it will be amplified many times over and will even dominate the main signal, see Fig.le.
[1] Bender C M, Boettcher S Phys. Rev. Lett. 80 5243 (1998).
[2] Mandal S. et al. Nonreciprocal transport of exciton polaritons in a non-Hermitian chain. PRL. - 2020. v. 125. №. 12. - P. 123902.
[3] T. Eichelkraut, R. et.al., Mobility transition from ballistic to diffusive transport in non-Hermitian latticesNat. Commun., 4, 2533.
[4] Makris K. G. et al. Wave propagation through disordered media without backscattering and intensity variations //Light: Science & Applications. - v. 6. - №. 9. - P. e17035-e17035. 2013.
