Eigenstate expansion of the quasistatic electric field of a point charge in a spherical inclusion structure

A point charge in the presence of a metallic nanosphere is a fundamental setup, which has implications for Raman scattering, enhancement of spontaneous emission of a molecule by an antenna, sensing, and modeling a metallic tip in proximity to a nanoparticle. Here we analytically expand the electric field of a point charge in an ${\epsilon }_2$ host medium in the presence of an ${\epsilon }_1$ sphere using the sphere eigenstates, where ${\epsilon }_1$ and ${\epsilon }_2$ can take any complex values. We develop a simple procedure to treat charge distribution, which results in a simple eigenstate expansion for the electric field of charge sources and is able to treat volume sources analytically. The electric field is strongly enhanced when ${\epsilon }_1/{\epsilon }_2$ is close to an ${({\epsilon }_1/{\epsilon }_2)}_l$ eigenvalue of a dominant mode, which is determined by the point charge location and the measurement point. An electric field exists inside the sphere when ${\epsilon }_1/{\epsilon }_2$ is close to a ${({\epsilon }_1/{\epsilon }_2)}_l$ resonance even when ${\epsilon }_1$ is a conductor. Low-order modes generate an electric field far away from the interface, where the $l=1$ mode with a resonance at ${\epsilon }_1=-2{\epsilon }_2$ generates a field at the sphere center. The high-order modes, which are associated with high spatial frequencies, become more dominant when the point charge approaches the sphere surface or when the physical parameters are close the high-order modes resonances. When ${\epsilon }_1/{\epsilon }_2$ is smaller or larger than the eigenvalues of the dominant modes, the modes interfere constructively and generate a strong signal at an angular direction equal to that of the source. The spectral information at the sphere surface may be utilized to calculate the point charge location without knowing its magnitude.

Figure 1

Plot of $|E_{r,l}\left(\mathbf{r}\right)(s-s_l)|$ at $\mathbf{r}=a^{-}\widehat{\mathbf{z}}$ and $\mathbf{r}=a^{+}\widehat{\mathbf{z}}$, i.e., just inside and just outside the sphere, and at $\mathbf{r}=z_0\widehat{\mathbf{z}}$ as a function of $l$ for ${\epsilon }_2=1, z_0=1.5a$, and $a=30\mathrm{nm}$.

Figure 2

Plot of ${\left|\mathbf{E}\right|}^2$ for a point charge at $z_0=1.5a, a=30\mathrm{nm}, s=0.43, {\epsilon }_2=1$, and ${\epsilon }_1=-1.3256$.

Figure 3

Plot of $|E_{r,l}\left(\mathbf{r}\right)(s-s_l)|$ at $\mathbf{r}=a\widehat{\mathbf{z}}$ inside and outside the sphere and at $\mathbf{r}=z_0\widehat{\mathbf{z}}$ as a function of $l$ for ${\epsilon }_2=1$, a point charge located at $z_0=1.15a$, and $a=30\mathrm{nm}$.

Figure 4

Plot of ${\left|\mathbf{E}\right|}^2$ for a point charge at $z_0=1.15a, a=30\mathrm{nm}, s=0.487, {\epsilon }_2=1$, and ${\epsilon }_1=-1.0534$.

Figure 5

Plot of ${\left|\mathbf{E}\right|}^2$ for a point charge at $z_0=2a, a=30\mathrm{nm}, s=1/3, {\epsilon }_2=1.69+0.08i$, and ${\epsilon }_1=-3.38+0.192i$.

Figure 6

Plot of ${\left|\mathbf{E}\right|}^2$ for a point charge at $z_0=3.5a, a=30\mathrm{nm}, s=0.3, {\epsilon }_2=1$, and ${\epsilon }_1=-2.33$.

Figure 7

Plot of ${\left|\mathbf{E}\right|}^2$ for a point charge at $z_0=1.5a, a=30\mathrm{nm}, s=0.492, {\epsilon }_2=1$, and ${\epsilon }_1=-1.0325$.