## Mie resonances Recall in previous chapter we have developed the polarizability of metallic particles, $\alpha = 4 \pi a^3 \frac{\varepsilon_{\text{sph}} - \varepsilon_{\text{d}}}{\varepsilon_{\text{sph}} + 2 \varepsilon_{\text{d}}}$ whose resonance condition is $\varepsilon_{\text{sph}} + 2 \varepsilon_{\text{d}} \rightarrow 0$ and we call it LSP resonance. And the scattering cross-section is $\sigma_{\text{scat}} = \frac{k^4}{6\pi} \left| \alpha \right|^2$ This provides us the possibility to turn the scattering cross-section. In this chapter, the Mie resonance is based on similar an idea. He no longer have a collective oscillation of all three electrons, But we have a dielectric nanoparticle acting as an optical cavity. In this scenario, the size of this dielectric particle becomes important. ![[Drawing 2024-08-07 16.32.30.excalidraw.svg]] ### Mie resonance in dielectric nanoparticles Mie theory describes the scattering of an electromagnetic plant wave by an homogeneous **sphere** of arbitrary radius and constitutive material. The solution is directly solved from the Helmholtz equation in spherical coordinate under given boundary conditions. >[!Note]- > The derivation is relatively complicated, but follow the same principle we applied before: > 1. Write the wave equations in spherical coordinates. > 2. Expand the incident wave as a series of spherical harmonics. > 3. Expand the internal and scattered field as a series of spherical harmonics. > 4. Apply the boundary condition for the $\mathbf{E}$ and $\mathrm{H}$ and interface of this sphere. (continuity) > > In this way, we solve the field everywhere inside and outside the particle. Their revision would not be given here, but the solution has the form: $\mathbf{E}_s = \sum_{n=1}^{\infty} E_n \left( i a_n \mathbf{N}_{emn} - b_n \mathbf{M}_{omn} \right)$ $\mathbf{H}_s = \frac{k_1}{\omega \mu_1} \sum_{n=1}^{\infty} E_n \left( i b_n \mathbf{N}_{omn} + a_n \mathbf{M}_{emn} \right)$ Here $\mathrm{E}_{s}$ and $\mathrm{H}_{s}$ are scattered field, which could be written as an infinite series of vectors spherical harmonics $\mathrm{M}_{n}$ and $\mathrm{N}_{n}$, $e$, $o$ represent the mode, and $n$ represent the order of components. They are the electromagnetic normal moves of the spherical particles and the profiles are given in the figure below. Each of the normal moves is weighted by the scattering coefficients $a_{n}$ and $b_{n}$. >[!Note] > One might be curious about the origin of the mode representation of $e$ and $o$. It seems that this $e$ is from *elektrisch*, while $o$ represents *odd*. There might be some historical reason that *odd* represents magnetic mode. Consider the **electric field** specifically, we have $E_n = i^n E_0 \frac{(2n + 1)}{n(n + 1)}$ Here $E_{0}$ is the amplitude of the incident field. ![[Pasted image 20240807170946.png]] This image is from wikipedia (https://en.wikipedia.org/wiki/Mie_scattering#/media/File:VSHwiki.svg) For each $n$, there are two distinct types of modes: - No radial magnetic field component, sometimes called transverse magnetic (TM), we call it electric type cause the $E$ is field is like an electric dipole, and - No radial electric field component, sometimes called transverse electric (TE), we call it magnetic type, case the $E$ field is circular (and looks like a magnetic dipole) And Because we're focusing on electric field, in the following part, we will refer all $a$ modes as electric and $b$ modes as magnetic, according to the scattering coefficient. The expression for the scattering coefficient $a$ and $b$ is obtained by using the boundary condition for the $\mathrm{E}$ and $\mathrm{H}$ fields at the surface of the sphere. The expression is complicated, and it's only showing in the following notes. >[!Note] >$a_n = \frac{m \psi_n(mx) \psi_n'(x) - \psi_n(x) \psi_n'(mx)}{m \psi_n(mx) \xi_n'(x) - \xi_n(x) \psi_n'(mx)}$ >$b_n = \frac{\psi_n(mx) \psi_n'(x) - m \psi_n(x) \psi_n'(mx)}{\psi_n(mx) \xi_n'(x) - m \xi_n(x) \psi_n'(mx)}$ >Here, $\psi_n(\rho) = \rho j_n(\rho)$, $\xi_n(\rho) = \rho h_n(\rho)$, $\rho=kr$ which is dimensionless. >$m=\frac{n_{s}}{n}$, relative refractive index, and $x=ka=\frac{2\pi na}{\lambda}$, a size parameter. Inside this $x$ we have $\frac{a}{n}$, and it is here the ratio of particle diameter to wavelength comes to play. >$j_{n}$ is the spherical Bessel functions, and $h_{n}$ is the spherical Hankel functions. > >Just a san check, when $m=1$, $a_{n}$ and $b_{n}$ vanish. This is reasonable casual we no longer have contrast. > > One should remember that $a$, $b$ here are both complex. Now let's go back to the cross-section. We want to know how physical quantities like the scattering, absorption and extinction cross-sections vary with the size of constitutive material of the sphere. They are computed already as below: $\sigma_{\text{sca}} = \frac{2\pi}{k^2} \sum_{n=1}^{\infty} (2n + 1) \left( |a_n|^2 + |b_n|^2 \right)$ $\sigma_{\text{ext}} = \frac{2\pi}{k^2} \sum_{n=1}^{\infty} (2n + 1) \Re(a_n + b_n)$ $\sigma_{\text{abs}}=\sigma_{\text{ext}}-\sigma_{\text{sca}}$ With the given analytical expression were able. to compute different cross-sections with respect to the size of particles, resonance modes, particle material and medium refractive index. Here is a website did these calculation: [Mie calculator | Faculty of Physics. ITMO University](https://physics.itmo.ru/en/mie#/spectrum) ### Applications of Mie theory Now we have shown the my theory, and based on the geometry and material property we may develop expression for cross-section. Here we present some examples of how Mie theory could be applied. #### Rayleigh approximation Andrew really approximation, we may consider the sphere as a point source. This is because the really scheduling condition is at the regime where the particle radius is much smaller compared with the wavelength of light ($x\ll \lambda$). Write the Taylor expression of $a_{1}$ and $b_{1}$, $a_1 \approx -i \frac{2}{3} \frac{m^2 - 1}{m^2 + 2} x^3 - i \frac{2}{5} \frac{(m^2 - 2)(m^2 - 1)}{(m^2 + 2)^2} x^5 + \cdots$ $b_1 \approx -i \frac{1}{45} (m^2 - 1) x^5 + \cdots$ Here $m = \frac{n_s}{n} = n_s \rightarrow m^2 = \varepsilon_s$, $x = ka = \frac{2\pi a}{\lambda}$, consider non-magnetic particle in air, we have, $a_1 \approx -i \frac{2}{3} \frac{\varepsilon_s - 1}{\varepsilon_s + 2} x^3 - i \frac{2}{5} \frac{(\varepsilon_s - 2)(\varepsilon_s - 1)}{(\varepsilon_s + 2)^2} x^5 + \cdots$ $b_1 \approx -i \frac{1}{45} (\varepsilon_s - 1) x^5 + \cdots$ For $x\ll 1$, $|b_1| \ll |a_1|$, take only the first term and neglect $b$, we have $a_1 \approx -i \frac{2x^3}{3} \frac{\varepsilon_s - 1}{\varepsilon_s + 2}$ One may notice that the last part is proportional to polarizability. This means in the really approximation, the dominating scheduling coefficient is directly proportional to the electrostatic polarizability of a sphere. But from here, what can also notice that we don't have a geometry dependence, which means the small particle approximation only captures the plasmonic resonance, and fail to describe the dielectric resonance. This would require additional numerical simulation. From the general expressions of the cross-sections and considering $a$ only, we have $\sigma_{\text{abs}} = 4\pi a^2 x \Re \left( \frac{\varepsilon_s - 1}{\varepsilon_s + 2} \right)$ $ \quad \sigma_{\text{sca}} = \frac{8}{3} \pi a^2 x^4 \left| \frac{\varepsilon_s - 1}{\varepsilon_s + 2} \right|^2$ If we assume (which is an okay approximation for dielectric particles) $\left| \frac{\varepsilon_s - 1}{\varepsilon_s + 2} \right|$ only weakly depends on wavelength, then the only $\lambda$ dependent term is $x=\frac{2\pi a}{\lambda}$, therefore, we have the famous dependence: $\sigma_{\text{abs}} \propto \frac{1}{\lambda}$ and $\sigma_{\text{sca}} \propto \frac{1}{\lambda^{4}}$ and this is why the sky is blue. #### Size of particle and regimes of scattering ![[Drawing 2024-08-07 20.18.17.excalidraw.svg]] Here we have the image covering particle and wavelength relaxation for different types of scattering. #### Near field enhancement, mode superposition There are some actual examples of how the modes affect near few distribution associated to different multipoles supported by the nanoparticles. With the near field included, the enhancement near the particle is quite strong. Also with the superposition of different modes., The actual field profile could be much more complicated. The following plot is an illustration. ![[Drawing 2024-08-07 20.41.16.excalidraw.svg]] More about this will be discussed in [[Mode engineering]]. >[!Note] > Why people typically consider high $\varepsilon$ materials like silicon in dielectric nanoparticles? > An intuitive explanation is, we consider nano particles as a tiny FP cavity. And the refractivity at the particle age increases with the refractive index contrast. Therefore, higher $\varepsilon$ could bring larger $r$, which may decrease losses and provide stronger resonance. So we have larger scattering. (Recall $r=\frac{n_{s}-n}{n_{s}+n}$) > Alternatively, one can check the scattering cross-section. The approximation with $a_{1}$ is $\quad \sigma_{\text{sca}} = \frac{8}{3} \pi a^2 x^4 \left| \frac{\varepsilon_s - 1}{\varepsilon_s + 2} \right|^2$. For dielectric material, this value increases with $\varepsilon$.