The scattering matrix theory has been developed to calculate the third-order nonlinear effect in sphere-grapheneslab structures. By designing structural parameters, we have demonstrated that the incident electromagnetic wave can be well confined in the graphene in these structures due to the formation of a bound state in the continuum (BIC) of radiation modes. Based on such a bound state, third-harmonic (TH) generation and four-wave mixing (FWM) have been studied. It is found that the efficiency of TH generation in monolayer graphene can be enhanced about 7 orders of magnitude. It is interesting that we can design structure parameters to make all beams (the pump beam, probe beam, and generated FWM signal) be BICs at the same time. In such a case, the efficiency of FWM in monolayer graphene can be enhanced about 9 orders of magnitude. Both the TH and FWM signals are sensitive to the wavelength, and possess high Q factors, which exhibit very good monochromaticity. By taking suitable BICs, the selective generation of TH and FWM signals for Sand P-polarized waves can also be realized, which is beneficial for the design of optical devices.
Second harmonic generation from the two-layer structure where a transition-metal dichalcogenide monolayer is put on a one-dimensional grating has been studied. This grating supports bound states in the continuum which have no leakage lying within the continuum of radiation modes, we can enhance the second harmonic generation from the transition-metal dichalcogenide monolayer by more than four orders of magnitude based on the critical field enhancement near the bound states in the continuum. In order to complete this calculation, the scattering matrix theory has been extended to include the nonlinear effect and the scattering matrix of a two-dimensional material including nonlinear terms; furthermore, two methods to observe the bound states in the continuum are considered, where one is tuning the thickness of the grating and the other is changing the incident angle of the electromagnetic wave. We have also discussed various modulation of the second harmonic generation enhancement by adjusting the azimuthal angle of the transition-metal dichalcogenide monolayer.
We reveal that a special exceptional point can act as a Dirac point in a one-dimensional PT-symmetric photonic crystal and prove it in detail using our extended first-principle theory. This theory was developed by applying biorthogonal bases of the non-Hermitian Hamiltonian to the kp method to study the dispersion relations of non-Hermitian systems. By using this theory, we demonstrate that two linear dispersions can cross at a critical touching point, which is a special exceptional point, and the corresponding effective Hamiltonian can be cast into a massless Dirac Hamiltonian under the biorthogonal bases of the non-Hermitian Hamiltonian; therefore, this point can be called the Dirac point, and the linear slope can also be predicted. In contrast to the Dirac point in a Hermitian system, which is induced by the degeneracy of two different eigenstates, the Dirac point here is induced by the degeneracy of parallel eigenstates in a non-Hermitian system. In addition, after increasing the non-hermiticity, the Dirac point evolves into a pair of exceptional points, and their positions in the band structure can be predicted well by our theory. Our findings and theory are important for further understanding the physics of the Dirac point in non-Hermitian wave systems.
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