Dipole resonances of nonabsorbing dielectric nanospheres in the optical range: Approximate explicit conditions for high- and moderate-refractive-index materials

In this work, we discuss the way in which electric and magnetic dipole resonances arising in the optical scattering spectrum of nonabsorbing dielectric nanospheres can be accurately approximated by means of simple explicit expressions that depend on the sphere's radius, incident wavelength, and relative refractive index. We find such expressions to hold not only for high- but also for moderate-refractive-index values, thus complementing the results reported in previous studies.

Figure 1

Calculated scattering efficiency $Q_{\mathrm{sca}}$ (solid line) as a function of size parameter $x$ for a dielectric sphere with $m=3.75$. Dashed and dotted curves show the values of $Q_{\mathrm{sca}}^{a_1}$ and $Q_{\mathrm{sca}}^{b_1}$, respectively.

Figure 2

Calculated $q_1(x,m)$ (a) and $s_1(x,m)$ (b) as a function of $mx/\pi$. Solid, dashed, and dotted curves show the values for $m=2.5,3.75$, and 5, respectively. Vertical dashed-dotted lines mark the values of $x_{\mathrm{res}\left(0\right)}^{a_1,1},x_{\mathrm{res}\left(1\right)}^{a_1,1},x_{\mathrm{res}\left(0\right)}^{b_1,1}$, and $x_{\mathrm{res}\left(1\right)}^{b_1,1}$ in units of $\pi /m$.

Figure 3

Graphical determination of the fundamental electric (a) and magnetic (b) dipole resonances for $m=3.75$. Solid and dashed curves in (a) show the exact values for left- and right-hand sides of Eq. (15), respectively. Open ($\circ$) and solid ($•$) symbols denote their proposed approximations (see text). Same conventions apply for Eq. (5) in (b). Vertical dashed-dotted lines mark the positions of $x_{\mathrm{res}\left(1\right)}^{a_1,1}$ and $x_{\mathrm{res}\left(0\right)}^{b_1,1}$.

Figure 4

Calculated size parameter as a function of $m$ for the fundamental electric (a) and magnetic (b) dipole resonances. Open symbols ($\circ$) in (a) correspond to the numerical solution to Eq. (4), whereas solid, dashed, and dotted curves show the values of the proposed approximations to $x_{\mathrm{res}}^{a_1,1}$ with subscripts $\left(0\right),\left(1\right)$, and (2) (see text). Same conventions apply for Eq. (5) and $x_{\mathrm{res}}^{b_1,1}$ in (b).

Figure 5

(a) Calculated values of $|a_1{|}^2$ (dashed) and $|b_1{|}^2$ (dotted) as a function of size parameter $x$ for a dielectric sphere with $m=3.75$. Vertical solid lines show the approximated values of fundamental dipole resonances with subscript (0), whereas dashed-dotted ones mark the positions of approximated antiresonances with subscript (2). (b) Calculated size parameters as a function of $m$ for the approximations with subscript (2) to the fundamental electric dipole resonance (dotted) and antiresonance (dashed-dotted). Solid curve shows the values of proposed approximations with subscript (0).

Figure 6

(a) Calculated $q_1(x,m)$ (solid) and $q_1^{x\gg 1}(x,m)$ (dashed) as a function of $m$ for increasing values of size parameter. Curves corresponding to $x=2,3,4$ are artificially shifted for the sake of clarity, with their actual zero baselines marked by horizontal dotted lines. (b) Same conventions apply for $s_1(x,m)$ and $s_1^{x\gg 1}(x,m)$.

Figure 7

Calculated size parameter as a function of $m$ for higher-order electric (a) and magnetic (b) dipole resonances of a nonabsorbing dielectric sphere. Solid curves in (a) correspond to the numerical solutions of Eq. (4) for increasing values of $j$, whereas dashed curves show the best fits of our proposed $x_{\mathrm{res}\left(3\right)}^{a_1,j}$ to data points (see text). Dotted curves mark the positions of sequential zeroth-order resonances. Same conventions apply for Eq. (5) and $x_{\mathrm{res}\left(3\right)}^{b_1,j}$ in (b).

Figure 8

Real (solid) and imaginary (dashed) parts of the refractive index as a function of wavelength for Si, ${\mathrm{Cu}}_2\mathrm{O}$, and ${\mathrm{TiO}}_2$, obtained from Refs. [32, 42, 43], respectively. Vertical dashed-dotted line marks the wavelength from which absorption can be neglected.

Figure 9

Electric and magnetic dipole contributions to the scattering efficiency of a dielectric sphere as a function of the incident wavelength and the sphere's radius for three different materials: Si (a), (b), ${\mathrm{Cu}}_2\mathrm{O}$ (c), (d), and ${\mathrm{TiO}}_2$ (e), (f). Open ($\circ$) and solid ($•$) symbols correspond to estimates for resonant radii with subscripts (0) and (2), respectively.

Figure 10

Electric (a) and magnetic (b) dipole contributions to the scattering efficiency as a function of size parameter $x$ for a dielectric sphere with $m=3.75+i{m}^{″}$. Solid curves correspond to increasing values of $m^{″}$. Vertical dashed lines mark the approximated positions of fundamental dipole resonances and antiresonances for $m=3.75$.

Figure 11

Calculated $s_1(\pi ,m)$ (solid) and $s_1^{x\gg 1}(\pi ,m)$ (dashed) as a function of $m$. Vertical dashed-dotted lines mark the values of $m_1^{b_1,j}$ from Eq. (A2).