Excitation of Bound States in the Continuum via Second Harmonic Generations

Abstract

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 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.

Acoustics of a partially partitioned narrow slit connected to a half-plane: case study for exponential quasi-bound states in the continuum and their resonant excitation

“…Furthermore, as in [51,52,56], we shall use conformal mappings [57] to solve certain 'inner' problems that arise as part of the analysis, in this case associated with regions near the partition edge and slit aperture. The present problem differs from those previous singular-perturbation analyses (as well as other asymptotic approaches to slit resonances [50] and quasi-BIC for periodic media [18][19][20][21][22]), however, in that exponentially small terms must be calculated. In particular, k ′′ is exponentially small and beyond all orders of the expansion for k in powers of ǫ.…”

“…Notably, the latter constitutes a singular asymptotic limit, in which, for example, the quality factor diverges. To date, the asymptotic approach of quasi-BIC to BIC has been theoretically studied mainly in the context of embedded guided modes in periodic structures [7][8][9][10][11][12][13][14][15][16][17], in which case perturbation theory [18][19][20][21][22] reveals algebraically singular quality factors scaling like 1/ǫ s , where s is a positive integer (typically s = 2, 4 or 6) and ǫ represents the perturbation of the Bloch wavenumber from its value at which an embedded guided mode exists.…”

Resonant field enhancement in lossy periodic structures supporting complex bound states in the continuum

“…A strong local field is important to many applications in photonics, such as sensing [1][2][3][4] and imaging [5,6], and is essential to the enhnancement of emissive processes and nonlinear optical effects [7][8][9][10][11][12][13]. For structures supporting a high quality factor (Q factor) resonance, if the frequency of the incident wave is close to the resonant frequency, the amplitude of the local field can be much larger than that of the incident wave [14][15][16][17][18][19].…”

Excitation of a bound state in the continuum via spontaneous symmetry breaking

“…Nonlinear polarization induced by the external field can be also used for the excitation of a genuine BIC. Thus, it was shown theoretically that BIC can be excited due to Kerr nonlinearity or second harmonic generation [54][55][56][57][58].…”