TY - JOUR
T1 - EXCITATION OF BOUND STATES IN THE CONTINUUM VIA SECOND HARMONIC GENERATIONS
AU - Yuan, Lijun
AU - Lu, Ya Yan
PY - 2020
Y1 - 2020
N2 - A bound state in the continuum (BIC) on a periodic structure sandwiched between two homogeneous media is a guided mode with a frequency and a wavenumber such that propagating plane waves with the same frequency and wavenumber exist in the homogeneous media. Optical BICs are of significant current interest since they have applications in lasing, sensing, filtering, switching, and many light emission processes, but they cannot be excited by incident plane waves when the structure consists of linear materials. In this paper, we study the diffraction of a plane wave by a periodic structure with a second-order nonlinearity, assuming the corresponding linear structure has a BIC and the frequency and wavenumber of the incident wave are one-half of those of the BIC. Based on a scaling analysis and a perturbation theory, we show that the incident wave may induce a very strong second harmonic wave dominated by the BIC and also a fourth harmonic wave that cannot be ignored. The perturbation theory reveals that the amplitude of the BIC is inversely proportional to a small parameter depending on the amplitude of the incident wave and the nonlinear coefficient. In addition, a system of four nonlinearly coupled Helmholtz equations (the four-wave model) is proposed to model the nonlinear process. Numerical solutions of the four-wave model are presented for a periodic array of circular cylinders and used to validate the perturbation results.
AB - A bound state in the continuum (BIC) on a periodic structure sandwiched between two homogeneous media is a guided mode with a frequency and a wavenumber such that propagating plane waves with the same frequency and wavenumber exist in the homogeneous media. Optical BICs are of significant current interest since they have applications in lasing, sensing, filtering, switching, and many light emission processes, but they cannot be excited by incident plane waves when the structure consists of linear materials. In this paper, we study the diffraction of a plane wave by a periodic structure with a second-order nonlinearity, assuming the corresponding linear structure has a BIC and the frequency and wavenumber of the incident wave are one-half of those of the BIC. Based on a scaling analysis and a perturbation theory, we show that the incident wave may induce a very strong second harmonic wave dominated by the BIC and also a fourth harmonic wave that cannot be ignored. The perturbation theory reveals that the amplitude of the BIC is inversely proportional to a small parameter depending on the amplitude of the incident wave and the nonlinear coefficient. In addition, a system of four nonlinearly coupled Helmholtz equations (the four-wave model) is proposed to model the nonlinear process. Numerical solutions of the four-wave model are presented for a periodic array of circular cylinders and used to validate the perturbation results.
KW - bound states in the continuum
KW - harmonic generation
KW - Helmholtz equations
KW - nonlinear optics
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U2 - 10.1137/19M1277539
DO - 10.1137/19M1277539
M3 - RGC 21 - Publication in refereed journal
VL - 80
SP - 864
EP - 880
JO - SIAM Journal on Applied Mathematics
JF - SIAM Journal on Applied Mathematics
SN - 0036-1399
IS - 2
ER -