Modeling of second-harmonic generation in periodic nanostructures by the Fourier modal method with matched coordinates

Abstract

We present an advanced formulation of the Fourier modal method for analyzing the second-harmonic generation in multilayers of periodic arrays of nanostructures. In our method, we solve Maxwell's equations in a curvilinear coordinate system, in which the interfaces are defined by surfaces of constant coordinates. Thus, we can apply the correct Fourier factorization rules as well as adaptive spatial resolution to nanostructures with complex cross sections. We extend here the factorization rules to the second-harmonic susceptibility tensor expressed in the curvilinear coordinates. The combination of adaptive curvilinear coordinates and factorization rules allows for efficient calculation of the second-harmonic intensity, as demonstrated for one- and two-dimensional periodic nanostructures.

Nonlinear Spectroscopy on the Plasmonic Analog of Electromagnetically Induced Absorption: Revealing Minute Structural Asymmetries

“…In principle, even more sophisticated methods can be used to perform the simulations in which the propagation of the TH signal from the near-to the far-field is taken into account. 61,62 Our findings indicate that the nonlinear response can resolve geometrical asymmetries or structural imperfections, which hardly manifest themselves in the linear regime, but which are unambiguously present in the nonlinear regime. Recently, it has also been shown that symmetry breaking can enhance the third-order nonlinearity in nanotips using high field gradients.…”

“…It should be mentioned that every fabricated nanostructure will of course exhibit a superposition of the introduced asymmetries as a consequence of the imperfections in the fabrication process as well as in the measurement setup. In principle, even more sophisticated methods can be used to perform the simulations in which the propagation of the TH signal from the near- to the far-field is taken into account. , …”

Nonlinear Born-Kuhn Analog for Chiral Plasmonics

“…For the nonlinear simulations, our approach is as follows: First, the electric fields at the fundamental wavelength are calculated by the Fourier modal method under the assumption that any self-phase modulation can be neglected. Then, we use these fields and Miller’s rule to calculate the third-order nonlinear polarization within the undepleted pump approximation. , The nonlinear polarization acts as a source at the third-harmonic wavelength. ,, However, instead of calculating the direct emission problem, we make use of the reciprocity principle , and calculate the near fields for p- and s-polarized plane waves incident from the far fields at the third-harmonic wavelength. Then, the electric field radiated to the far field at the third harmonic can be calculated as the overlap integral of these near fields and the nonlinear source. , …”

“…43,55 The nonlinear polarization acts as a source at the third-harmonic wavelength. 43,55,56 However, instead of calculating the direct emission problem, 57 we make use of the reciprocity principle 58,59 and calculate the near fields for p-and s-polarized plane waves incident from the far fields at the thirdharmonic wavelength. Then, the electric field radiated to the far field at the third harmonic can be calculated as the overlap integral of these near fields and the nonlinear source.…”

Reformulated Fourier Modal Method with improved near field computations

“…This method is wellsuited for analyzing periodic grating metal and dielectric structures, calculating their eigenmodes and high-Q resonances [2][3][4], solar cell simulations [5][6][7][8], in particular, the method is suitable for supercell calculations [9]. In addition, the FMM can be also used to find plasmonic resonances [10] and to solve non-linear optics problems [11,12].…”