A Full-Retarded Spectral Technique for the Analysis of Fano Resonances in a Dielectric Nanosphere

Abstract

We introduce a representation of the electromagnetic field scattered by a homogeneous sphere in terms of a set of full-retarded modes independent of its permittivity. Within this framework, we introduce the orthogonality properties of the modes, their resonance conditions, and their classification into narrow and broad modes. We also discuss the role played by the material properties in determining the resonant width of a given mode and in enabling or preventing the multimode interference. We use this theory to unveil the origin of Fano lineshapes in the scattering efficiency of a spherical nanoparticle, by identifying the interfering modes responsible for peaks and dips. Eventually, by using the introduced theoretical approach, we design the permittivity of a homogeneous sphere of size comparable to the incident wavelength to cancel its backscattering through directional multimode interference.

Appendix: Vector Spherical Wave Functions

The explicit expressions of the vector spherical wave functions (VSWF) are [76]:

$$\begin{aligned} \begin{aligned} \mathbf{N}_{\begin{array}{c} e \\ o \end{array}\,m\,n}\left( k \, \mathbf{r} \right)&= n \left( n + 1\right) \left( \begin{array}{cc} \cos m \phi \\ \sin m \phi \end{array} \right) P_n^m \left( \cos \theta \right) \frac{z_n \left( k \,r \right) }{k \,r} \hat{\mathbf{e}}_r \\&+ \left( \begin{array}{cc} \cos m \phi \\ \sin m \phi \end{array} \right) \frac{d P_n^m \left( \cos \theta \right) }{d\theta } \frac{1}{k \,r} \frac{d}{dr} \left[ r z_n \left( k \,r \right) \right] \hat{\mathbf{e}}_ \theta \\&+ m \left( \begin{array}{cc} - \sin m \phi \\ \cos m \phi \end{array} \right) \frac{P_n^m \left( \cos \theta \right) }{\sin \theta } \frac{1}{k \,r} \frac{d}{dr} \left[ r z_n \left( k \,r \right) \right] \hat{\mathbf{e}}_\phi , \\ \mathbf{M}_{\begin{array}{c} e \\ o \end{array}\,m\,n}\left( k \, \mathbf{r} \right)&= m \left( \begin{array}{cc} - \sin m \phi \\ \cos m \phi \end{array} \right) \frac{P_n^m \left( \cos \theta \right) }{\sin \theta } z_n \left( k \,r \right) \hat{\mathbf{e}}_\theta - \left( \begin{array}{cc} \cos m \phi \\ \sin m \phi \end{array} \right) \frac{d P_n^m \left( \cos \theta \right) }{d\theta } z_n \left( k \,r \right) \hat{\mathbf{e}}_\phi . \end{aligned} \end{aligned}$$

(8.55)

where the subscripts e and o denote even and odd, and $P_n^m \left( \cdot \right)$ are the associated Legendre function of the first kind of degree n and order m. Moreover, the superscripts $^{\left( 1\right) }$ and $^{\left( 3\right) }$ are appended to the functions $\mathbf{M}_{\begin{array}{c} e \\ o \end{array}\,m\,n}$ and $\mathbf{N}_{\begin{array}{c} e \\ o \end{array}\,m\,n}$ to denote the function $z_n$, namely Bessel functions of the first kind $j_n$ and Hankel functions of the first kind $h_n$, respectively.

Starting from (8.55) it is possible to derive the expression of (8.45):

$$\begin{aligned} \begin{aligned} \mathbf{N}_{\begin{array}{c} e \\ o \end{array} \, m \, n}^{\left( \infty \right) } \left( \theta ,\phi \right)&= {\left( - i \right) ^n} \left( \begin{array}{cc} \cos m \phi \\ \sin m \phi \end{array} \right) \frac{d P_n^m \left( \cos \theta \right) }{d\theta } \hat{\mathbf{e}}_ \theta + {\left( - i \right) ^n} m \left( \begin{array}{cc} - \sin m \phi \\ \cos m \phi \end{array} \right) \frac{P_n^m \left( \cos \theta \right) }{\sin \theta } \hat{\mathbf{e}}_\phi , \\ \mathbf{M}_{\begin{array}{c} e \\ o \end{array} \, m \, n}^{\left( \infty \right) } \left( \theta ,\phi \right)&= {\left( - i \right) ^{n+1}} m \left( \begin{array}{cc} - \sin m \phi \\ \cos m \phi \end{array} \right) \frac{P_n^m \left( \cos \theta \right) }{\sin \theta } \hat{\mathbf{e}}_\theta - {\left( - i \right) ^{n+1}} \left( \begin{array}{cc} \cos m \phi \\ \sin m \phi \end{array} \right) \frac{d P_n^m \left( \cos \theta \right) }{d\theta } \hat{\mathbf{e}}_\phi . \end{aligned} \end{aligned}$$

(A2)