## Mode engineering From the [[Mie resonances|last chapter]], we see that the total field pattern is the superposition of different modes. This is also noticeable from the mathematical expression: $\mathbf{E}_s = \sum_{n=1}^{\infty} E_n \left( i a_n \mathbf{N}_{emn} - b_n \mathbf{M}_{omn} \right)$ In this chapter, we will see how to adjust different modes contributed to the total scattering pattern and make it possible for us to tune the total scattering fields as we want. ### Kerker conditions for directional scattering We first define an asymmetry perimeter $g$, who quantifies the degree of directionality of the particle scattering, or more specifically, the direction of the scattered power in a solid angle $\mathrm{d}\Omega = \sin\theta \, \mathrm{d}\theta \, \mathrm{d}\phi$, $g = \int_{4\pi} p \cos\theta \, \mathrm{d}\Omega$ Here $p$ acts as a normalization factor, who is the differential scattering cross-section such that $\int_{4\pi} p \, \mathrm{d}\Omega = 1$. With $g$ defined, we may describe the direction of scattering: - $g=0$, balance between forward and backward scattering; - $g=1$, all forward scattering; - $g=-1$, all backward scattering. ![[Drawing 2024-08-07 21.56.55.excalidraw.svg]] If we adopt the Mie theory results into this $g$, we will have This asymmetry parameter being a function of scattering coefficients $a_{n}$ and $b_{n}$. $g = \frac{4}{x^2} \left[ \sum_n \frac{n(n + 2)}{n + 1} \Re(a_n a_{n+1}^* + b_n b_{n+1}^*) + \sum_n \frac{2n + 1}{n(n + 1)} \Re(a_n b_n^*) \right]$ with $x=ka=\frac{2\pi na}{\lambda}$. If the scattering is dominant by $a_{1}$ and $b_{1}$, we will have $g \propto \Re(a_1 b_1^*)$ or if we write in the Euler form, we have $a_1 = |a_1| e^{i\phi_1}$ $b_1 = |b_1| e^{i\psi_1}$ $g \propto |a_1| |b_1| \cos(\phi_1 - \psi_1)$ In this expression, we can see phase explicitly. This shows that the asymmetry parameter $g$ can be controlled by the interference between the electric and magnetic dipoles. With the interference picture, the things become easier. When magnetic depot and electric diaper were in phase, we will have a maximized forward scattering; when they are in anti-phase, we will have a maximized backward scattering. This corresponds to the first and second Kerker conditions. #### Kerker 1st conditions ![[Drawing 2024-08-08 00.06.02.excalidraw.svg]] Electric fields interfere constructively in the forward direction and destructively in the backward direction. For a dielectric cylinder, the multipole coefficients plot: ![[Drawing 2024-08-08 00.18.13.excalidraw.svg]] We have the same amplitude (real parts) and same phase (imaginary parts). #### Kerker 2nd conditions ![[Drawing 2024-08-08 00.14.50.excalidraw.svg]] Electric fields interfere constructively in the backward direction and destructively in the forward direction. ![[Drawing 2024-08-08 00.29.54.excalidraw.svg]] We have same amplitude and anti-phase. ### Adjusting scattering by varying geometries We want to reach the Kerker conditions at resonance to gain maximized directional scattering. This requires changing the detuning between electric and magnetic dipoles, which may be achieved by changing the geometries (like the diameter of the cylinders) since ED and MD has different dependence on geometry. By engineering 3D array of silicon nano-cylinders, one could achieve ~100% transmission. (although in real world there could be some mismatch between real and imaginary parts intersections) ### Adjusting scattering by BIC modes BIC modes in theory could decrease losses to 0, because it achieves the decoupling of external environment and the optical fields by accurately designing the structure and symmetry of the "medium" based on constructive interference. BIC modes are consequently localized and have very high $Q$ factor. Due to the strong confinement of the field, BIC is an ideal tool for nonlinear optics applications. ### Dielectric resonance properties - As we increase $\Re(\varepsilon)$ we will go first through the plasmonic region ($\Re(\varepsilon) < 0$), with the $a_1$ and $a_2$ plasmonic resonances. As we enter the region of positive values of $\Re(\varepsilon)$, we will have another set of $a_1$ and $a_2$ resonances, along with $b_1$ and $b_2$ higher order resonances. - There are no magnetic resonances ($b_1$ and $b_2$) for the plasmonic regime. This is mostly explained by high losses in metals which prevent circulating current within the metal. - Spectral features for $a_1, a_2$ resonances are nearly the same on both sides. Despite their different physical origin, this is expected considering they are described by the same mathematical expressions. - Dielectric resonances do not exist for very small particles. An intuitive idea is that we have to have big enough particles to act as an FP, due to the resonance origin. However, plasmonic does not have the size limit cause the origin is oscillation of electrons. In actual works, plasmonic, dielectric resonance and BIC are combined to achieve very strong resonance and large Q factor. The combined effects may further facilitate the realizations of optical applications, like heat conversion or catalysis. Besides these, another huge regime is [[Metasurface]].