1. INTRODUCTION

In recent years, optically-induced Fano resonances in nanostructures have been proposed as the basis of biochemical sensing with potentially high sensitivity [1]. A Fano resonance arises from the interference between two resonant modes in a system with two or more interacting structures producing a type of resonant scattering phenomenon exhibiting an asymmetric line shape [2,3]. The sensitivity of the distortion or asymmetry in the spectrum to the local environment from this mode coupling has been suggested as the basis for sensing. Our goal in this report is to study a nanostructure in which a Fano resonance arises, but is sufficiently simple to treat analytically, which then helps clarify the key features of the Fano phenomenon. It is well known that a shell with a nonconcentric core generates multipoles of all orders, and the nonconcentric shell–core system has been analyzed using a multipole expansion of the fields generated by the off-center core [4]. In our study, we also employ a multipole analysis of the nonconcentric shell but show how the matrix elements used to compute the multipole expansion coefficients can be derived analytically to all orders with the aid of the solid-harmonic addition theorem. This analytical solution allows us to show explicitly how the two lowest order multipole terms, the dipole and quadrupole contributions, are coupled and display the classic Fano form. As the core moves further off center, the electric field grows in magnitude just outside the shell boundary near the thinner part of the shell. This field enhancement can be clearly seen in profile calculations of the electric field magnitude through the interior of the shell. As a check, we show that these results are in excellent agreement with finite-element calculations. Our calculations illustrate how the electric field enhancement increases with the core offset. The field enhancement outside the shell surface can be exploited in surface-enhanced spectroscopies, such as surface-enhanced Raman scattering (SERS) since the SERS response scales as the fourth power of the electric field magnitude [5]. A key feature that characterizes the Fano effect is the coupling between a radiant mode (or “bright mode”), which is dipolar and can be stimulated from, and observed in, the far field, and a nonradiant mode (or “dark mode”) that is invisible in the far field but couples to the radiant mode and interferes with the latter. It is this interference that gives rise to a distortion, or asymmetry, in the radiant mode and could, in the absence of damping, theoretically quench the radiant mode entirely at some frequency. In the presence of damping the result is a “Fano dip” in the spectrum rather than complete quenching. A second consequence of this interference is a shift in the peak of the radiant mode. It is primarily the sensitivity to the local environment of the location and size of the nanodip and the bright-mode frequency shift that suggests the possibility of exploiting the Fano phenomenon for sensing applications.

The paper is organized as follows. We first carry out the multipole expansion of the electric field for the nonconcentric nanoshell. The analysis shows that the multipole expansion coefficients are the solution to a linear system, which can be expressed in matrix form. One contribution of this report is to show how the matrix elements in this system can be derived analytically with the aid of the solid-harmonic addition theorem. This gives explicit expressions for the first two coupled multipoles, the dipole and quadrupole modes, which play the role, respectively, of the bright and dark modes in a system exhibiting Fano interference. The resulting expression can be placed in the classic Fano form, which shows how the Fano interference arises or, in particular, how the resonance peak of the system is shifted and how partial quenching (the “Fano dip”) occurs. A number of authors have noted that the classical analog of Fano interference can be demonstrated from an analysis of coupled oscillators [2,6,7]. We also show this by comparing our Fano expression for the off-center shell–core system with that of a system of four masses connected by springs. Finally, we compute electric field profiles through the interior of a nonconcentric shell using the full multipole expansion. This shows how the electric field enhancement on the shell surface increases with increasing core offset. As a check, we confirm the electric field calculations by comparing them to finite-element solutions. We conclude with a brief summary.

2. MULTIPOLE ANALYSIS OF THE NONCONCENTRIC SHELL

Figure 1 shows a spherical shell surrounding a spherical core displaced from the shell center by $L$, where $R_1$ and $R_2$ are the shell and core radii, respectively.

 

Fig. 1. Core is displaced from the shell center by $L$.

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The dielectric constants of the host medium, the shell, and the core are denoted by ${\epsilon }_0$, ${\epsilon }_1$, and ${\epsilon }_2$, respectively. We shall employ the quasi-static approximation, which holds when the shell diameter is small compared to the optical wavelength. In this approximation, the electric field is expressed as the gradient of a potential that obeys Laplace’s equation. We note that when the quasi-static approximation holds, the plasmon frequencies and field enhancements are scale invariant. This follows because the Laplace equation contains no intrinsic length scale. Let ${\psi }_p(r,\theta )$, ${\psi }_s(r,\theta )$, ${\psi }_1(r,\theta )$, and ${\psi }_2(r,\theta )$ denote, respectively, the potential functions associated with the primary (incident) field, the scattered field, and the fields in the shell and core. For the potentials external to and inside the shell, we employ a spherical coordinate system whose origin is at the center of the shell, which we denote by $(r_1,{\theta }_1)$ or, equivalently, by $(r_1,u_1)$, where $u_1\equiv \cos {\theta }_1$. For the potential inside the core, we use a coordinate system centered on the core and denoted by $(r_2,{\theta }_2)$ or $(r_2,u_2)$, where $u_2\equiv \cos {\theta }_2$. Both ${\theta }_1$ and ${\theta }_2$ are measured away from the $z$-axis defined along the direction of the core displacement. As shown later, we shall use the solid-harmonic addition theorem to connect these two systems. We assume an incident plane wave of amplitude $E_0$ polarized along the $z$-axis. The potentials are then given by

3. DIPOLE AND QUADRUPOLE COUPLING AND THE FANO RESONANCE

Here we demonstrate how the Fano coupling occurs between the dipole and quadrupole modes defined by the two lowest order terms in the expansion of Eqs. (25) and (26) ($n=1$ and $n=2$, respectively). Keeping only these terms results in

Now consider a nonconcentric core ($\tilde{L}>0$). Solving Eqs. (35) and (36) for ${\mathbf{v}}_1$ results in

We next employ an analysis similar to that used by Gallinet and Martin [7] in their demonstration of Fano behavior in a system of coupled oscillators. Suppose we expand Eq. (62) about the resonance ${\overline{\omega }}_4$ corresponding to one of the “dark” modes. In the vicinity of ${\overline{\omega }}_4$, the factors ${\omega }^2-{\overline{\omega }}_1^2$, ${\omega }^2-{\overline{\omega }}_2^2$, and ${\omega }^2-{\overline{\omega }}_3^2$ in $D_1$ and $D_2$ are all slowly varying; we then replace them with ${\overline{\omega }}_4^2-{\overline{\omega }}_1^2$, ${\overline{\omega }}_4^2-{\overline{\omega }}_2^2$, and ${\overline{\omega }}_4^2-{\overline{\omega }}_3^2$ and regard these quantities as constants near $\omega ={\overline{\omega }}_4$. We then restore the damping parameter by replacing ${\omega }^2-{\overline{\omega }}_4^2$ in $D_2$ with ${\omega }^2-i{\overline{\omega }}_4\gamma -{\overline{\omega }}_4^2$. To first order in small quantities, we have replaced $\omega \gamma$ with ${\overline{\omega }}_4\gamma$ near the ${\overline{\omega }}_4$ resonance. Defining $C\equiv ({\overline{\omega }}_4^2-{\overline{\omega }}_1^2)({\overline{\omega }}_4^2-{\overline{\omega }}_2^2)({\overline{\omega }}_4^2-{\overline{\omega }}_3^2)$, Eq. (62) becomes

Calculations were carried for $|a_1|$ computed from Eq. (62) for the following five values of the core displacement: $L=\mathrm{0,1},\mathrm{2,3},4\mathrm{nm}$ assuming shell and core diameters of 30 nm and 20 nm, respectively, using the dielectric parameters for gold. The core offsets are illustrated in Fig. 2, and the normalized plots of $|a_1|$ versus frequency are shown in Fig. 3.

 

Fig. 2. Five core offsets.

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Fig. 3. Magnitude of the radiant mode amplitude for five core offsets.

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In Fig. 3 we see the partial quenching of the radiant mode between the two peaks, as well as the frequency shift in the primary resonance peak.

4. MECHANICAL ANALOG

A number of authors have used the analogy of coupled harmonic oscillators to illustrate some of the key features of Fano interference. In a similar fashion, we can show that the equations that describe the response of the nonconcentric shell closely resembles the behavior of four masses connected by three springs, where the masses are confined along a line with small displacements $a_i$, $i=\mathrm{1,2},\mathrm{3,4}$, away from equilibrium with masses $m_i$. Let $K$ denote the spring constant of the springs between masses 1 and 2 and between 3 and 4, while the spring joining masses 2 and 3 has constant $k\ll K$. It is convenient to write $k=\epsilon K$, where $\epsilon \ll 1$. Thus, masses 1 and 2 are weakly coupled to masses 3 and 4. If we drive mass 1 with a force $F=f\exp (-i\omega t)$, we obtain the coupled equations

5. ELECTRIC FIELD PROFILES

Once the potential expansion coefficients are calculated, the electric field is derived from the negative of the gradient of the potentials ($\mathbf{E}=-\nabla \psi$). Using the above multipole solution, we computed profiles of the electric field magnitude along the $z$-axis penetrating the particle. In this calculation, the following simple Drude model for the dielectric function of the shell given by Eq. (34) with ${\omega }_p=8.6\mathrm{eV}$ and $\gamma =0.17\mathrm{eV}$, the approximate values for gold. In our calculations, we assumed a core dielectric constant of silica (${\epsilon }_2=2.43$) and set ${\epsilon }_0=1$. The calculations were carried out assuming $R_1=15\mathrm{nm}$, $R_2=10\mathrm{nm}$ with five values of the core displacement $L=\mathrm{0,1},\mathrm{2,3},4\mathrm{nm}$. The plots in Fig. 4 show the magnitude of the $z$-component of the electric field along a line through the particle. We compared the multipole calculations using $N=10$ terms in the expansion to finite-element calculations obtained with the commercial software package COMSOL Multiphysics [8]. In the figures, the solid and dashed curves are the multipole and COMSOL calculations, respectively, which show close agreement. Each profile calculation was performed at the wavelength corresponding to the peaks in the electric field enhancement at 366, 388, 396, 416, and 468 nm. From the figures, we see that the peak wavelength is redshifted as the core offset increases. In evaluating the multipole expansions for the potentials given by Eqs. (1)–(4), we found that the expansion for the potential ${\psi }_1$ for points in the interior of the shell converged slowly for large core offsets and when $r_1$ was close to the core boundary. This was a more serious problem for the $b_n$ expansion. This problem was solved by using Green’s theorem to compute the interior shell potential by a boundary integration involving the expansion coefficients $a_n$ and $d_n$ only. This works well since the $a_n$ and $d_n$ expansions converge rapidly. The Green’s function analysis is outlined in Appendix B.

 

Fig. 4. Profiles of the magnitude of the $z$-component of the electric field through the particle for different core offsets $L$. Note the change in the vertical scale.

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6. CONCLUSION

The off-center core generates multipoles of all orders and their amplitudes up to order $N$ were obtained by solving an $N$ by $N$ linear system. We showed how the matrix elements in this system can be analytically derived with the aid of the solid-harmonic addition theorem. For $N=10$, the multipole calculations of the electric field profile through the shell were shown to be in excellent agreement with finite-element calculations. Moreover, at the shell’s plasmon resonance, the profile calculations show that the electric field enhancement on the thin side of the shell increases with larger core offset and can be significantly higher than the enhancement for the concentric shell. The expansion coefficients of the two lowest order terms, the dipole and quadrupole resonances, were expressed explicitly in analytic form given by Eq. (35) and (36) resulting after some manipulation in the Fano expression in Eq. (65).