Fanolike resonance due to plasmon excitation in linear chains of metal bumps

Xiao-gang Yin; Cheng-ping Huang; Qian-jin Wang; Wan-xia Huang; Lin Zhou; Chao Zhang; Yong-yuan Zhu

doi:10.1364/OE.19.010485

Since the finding of extraordinary optical transmission (EOT) through an array of subwave-length holes [1], much attention has been paid to the optical properties of nanostructured metal surfaces, due to the researchers’ interest both in the underlying physics and potential applications [2–4]. It was generally acknowledged that the Rayleigh anomaly induces the transmission minimum whereas the surface-plasmon polariton (SPP) mode causes the trans-mission maximum [1,5]. Recently, the Fano lineshape has been employed by several groups to study the light transmission [6–10]. As is well known, the interference between the contributions of two transition channels, i.e., a discrete resonant state and a continuum of nonresonant states, gives rise to a Fano-type resonance characterized by an asymmetrical lineshape [11]. Such a profile has been extended from the atomic physics to various condensed matter systems [12,13]. And, it promises many potential applications in sensors, lasing, switching, and nonlinear and slow-light devices [14]. In the EOT phenomenon, the transmission dip and peak can be attributed, according to the Fano analysis, to the interfering contributions of the nonresonant direct scattering and resonant SPP excitation on the film surface [7,8]. Although the Fano analysis cannot reveal the detailed physics of EOT and fails to describe the dependence of lineshape on the detailed geometrical parameters, the model provides a good fit to the experimental or theoretical results and presents a simple and intuitive description of the effect.

In this paper, we report a new type of Fanolike resonance employing the plasmon wave running along the linear chains of metal bumps instead of the SPP mode supported by a metal surface. The plasmonic structure involved is a modified slit grating, which is dressed, on the slit sidewalls, with the linear chains of metallic bumps. Such a structure owns the characteristics that the subwavelength slits have no cutoff wavelength and support an efficient transmission in a wide spectral region and that the bumps can form the linear metallic chains, thus presenting new collective excitations and “hot spots” near the periodic bumps. The transmission spectrum has been studied numerically and experimentally, showing the Fano profile in a narrow frequency range. Using a semi-analytical model, we attributed the effect to the interference between the direct background transmission of the slits and dipole radiations due to the propagating plasmon wave in the linear chains. We also show that the Fano effect can be tuned dynamically using the dispersion of chain plasmon mode.

Figure 1(a) presents the designed plasmonic structure, which is laid on the substrate of glass. The unit cell along with the corresponding structural parameters is shown in Fig. 1(b). Here, the metal (silver) film thickness is set as t = 100 nm, which is small so that the sharp Fabry-Perot resonance will not be present. Without loss of generality, the grating period and slit width are fixed as d = 600 nm and s = 300 nm, respectively. And, the rectangular bumps with the length l = 80 nm and width w = 150 nm are introduced periodically to the right sides of individual slits (the periodicity is also set as d = 600 nm). To reveal the optical properties of the structure, numerical simulations and experiments have been performed in this study. In the simulation [based on the finite-difference time-domain (FDTD) method], the permittivity of glass is set as 2.25 and the silver is modeled by the Drude dispersion with a plasma frequency of $ω_{p} = 1.37 \times 10^{16}$ rad/s and a collision frequency of $γ = 2 \times 10^{14}$ rad/s. Experimentally, the proposed structure and, for comparison, a pure slit grating were fabricated with the focusedion-beam (FIB) system [the FIB image of a part of sample is shown in Fig. 1(c)] and the zero-order transmission spectra were obtained using an optical spectrum analyzer (the incident light is in the yz plane and transmits from glass to air). The details about the experiments can be referred to Ref [15]. and [16]. In the following, we are interested in the s-polarization of light, i.e., the light electric field is normal to the incident plane [along the x-axis, see Fig. 1(b)]. In this case, the surface charges are accumulated efficiently near the subwavelength slits (the dipole moments are formed in the x-direction), thus giving an efficient transmission.

Fig. 1 (a) Schematic view of the proposed plasmonic structure. (b) The unit cell [marked by dotted lines in (a)] with the corresponding structural parameters (the configuration of incident light is also shown). And (c) The FIB image of a part of the structure.

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At normal incidence, the simulated and measured transmission spectra of the proposed structure (the solid lines) and the referenced pure slits (without metallic bumps, the dotted lines) have been shown respectively in Fig. 2(a) and 2(b) with a reasonable agreement. The deviation between theory and experiment can be attributed to the fabrication errors (the actual bumps are partly rounded, and the metal surface will be contaminative due to FIB milling). The dotted lines for the pure slits exhibit two transmission dips (around 610 nm and 910 nm), which can be attributed to the flat surface SPP resonance modes [17,18]. And as was expected, the spectrum of the pure slits presents a large transmission in a wide wavelength range ( $λ > 950 n m$ ), due to the small film thickness as well as the propagating mode of the subwavelength slits. However, when the periodic bumps are introduced to individual slits, a new and interesting transmission feature can be observed: A narrow transmission maximum-minimum transition around the wavelength 985 nm appears in the slit background spectrum. This is a typical asymmetric lineshape, belonging to characteristics of the Fano resonance. A detailed survey of this effect will be presented in the following. Here three points should be emphasized. First, we did not find any depolarization effect in the simulation, showing that the light polarization is well maintained in the transmission. Second, the transmission dips of pure slits also appear in our structure, with the spectral positions unshifted (see Fig. 2). This means the flat surface SPP modes of pure slits still survive in our structure and play a negative role for the transmission [17,18]. And third, for p-polarization (the magnetic field is normal to the incident yz plane), the transmission spectrum of the structure is almost identical to that of the pure slits and the above effect is not present (not shown here).

Fig. 2 Simulated (a) and measured (b) zero-order transmission spectra for s-polarization. The solid and dotted lines correspond to the modified and pure slit array, respectively. The inset shows the calculated spectra of a single slit with (the solid line) and without (the dotted line) the periodic bumps as well as a Fano fit based on the Eq. (4). Here, the structural parameters are set as p = 600 nm, s = 300 nm, w = 150 nm, and l = 80 nm.

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We have simulated the transmission spectrum of a single slit with the periodic metal bumps (for s-polarization and normal incidence). Remarkably, the result, shown as the solid line in inset of Fig. 2(a), also suggests a similar asymmetric lineshape around the wavelength 985 nm. In contrast, the spectrum of a single slit without the metal bumps (the dotted line) is rather featureless in the studied wavelength range. This indicates that the Fano profile is mainly correlated with the linear chain of single slits (the height of spectrum may be influenced by the periodic slits, but the main feature is unchanged). Moreover, we plotted with the FDTD method the spatial distributions of dominant electric field E_x and current density J for the transmission maximum and minimum (at half the thickness of silver film). The electric-field pattern for the transmission maximum (at 973 nm), as shown in Fig. 3(a) , suggests that the fields are strongly confined on the top ends of metal bumps as well as on the slit sidewalls between the bumps. Correspondingly, a strong current flow exists between the bumps and the nearby slit sidewalls [see Fig. 3(b)] and thus opposite surface charges will accumulate, being consistent with the simulated electric-field pattern. It is interesting to find that the distributions of electric field and current density for the transmission minimum (at 1012 nm) are just similar to those for the maximum (not repeated here). This implies that at the minimum the surface plasmons are also strongly excited and that the transmission maximum and minimum may share a common physical origin.

Fig. 3 Simulated electric-field (a) and current-density (b) patterns for the transmission maximum [on x-y plane (z = −50 nm)]. The results for the minimum are just similar. (c) Schematic view of the linear chain employed in the theoretical model. Here, a_m and b_m represent the quantities of the surface charges, and the thinner red arrows show the surface current flow (the light electric field is denoted by the thicker blue arrow).

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Based on the above results, a theoretical explanation of the effect can be suggested. First, the linear chains of metal bumps can be resonantly excited by the light electric field, generating the electric dipole moment in each unit cell. Second, the electric dipoles in unit cells will emit the radiations, which interfere with the background transmission of the slits, thus leading to the Fanolike resonance. To elucidate this idea, we developed here a semi-analytical model for the single modified slit, where the linear chain was divided into sections of LC circuits [see Fig. 3(c)]. In the mth unit cell of a slit, the surface charges can be driven by the electric field, generating the surface current $d a_{m} / d t$ and $d b_{m} / d t$ on both sides of the bumps. Accordingly, the surface charges $a_{m} + b_{m}$ , $- (b_{m - 1} + a_{m})$ , and $- (b_{m} + a_{m + 1})$ will accumulate, respectively, on the end of bump as well as on the left and right slit sidewalls. Here, a single metal surface (the bump end or slit sidewalls) carrying the charges or electric-field energy can be treated as a capacitor, with the potential proportional to its surface charges ( $u = q / C$ , where C is the capacitance of an “isolated” conductor) [19]. Moreover, a segment of circuit carrying the surface current has an effective inductance L, which includes both a self-inductance L_s associated with the magnetic-field energy and a formal or kinetic inductance L₀ relating to the electronic inertia or kinetic energy [20,21]. Therefore, there are two sections of equivalent LC circuits in a unit cell. Note that the surface current in one unit cell is actually related to the potential difference between the bump end and nearby slit sidewalls and thus to charges of its neighbors. As a result, a coupling between two adjacent bumps is present, enabling a wavelike propagating behavior of plasmon mode in the linear chains. In contrast, the metallic particles inserted into the periodic square holes only support the localized excitations [15,16]. We stress that, unlike the flat surface SPP modes propagating normal to the periodic slits, the chain plasmon mode is running along the slit direction. Nonetheless, the interaction of two types of plasmon modes in our structure is very weak because of the spectral separation between them (this spectral separation can be further enlarged, as can be seen in the following, with the chain mode dispersion).

Applying the circuit equation to each subwavelength LC circuit of the mth unit cell, one can obtain a pair of equations for the plasmonic oscillations:

\begin{array}{l} L \overset{• •}{a_{m}} + R \overset{•}{a_{m}} + \frac{a_{m} + b_{m}}{C_{1}} + \frac{a_{m} + b_{m - 1}}{C_{2}} = E_{m} l, \\ L \overset{• •}{b_{m}} + R \overset{•}{b_{m}} + \frac{a_{m} + b_{m}}{C_{1}} + \frac{a_{m + 1} + b_{m}}{C_{2}} = E_{m} l . \end{array}

Here, L is the total effective inductance, and R is the dc resistance of the equivalent LC circuit; C₁ and C₂ is the capacitance of the bump end and the slit sidewalls, respectively; and

E_{m} = E_{0} \exp i (k_{/ /} m d - ω t)

is the electric field of incident light at the metal surface (k _// is the in-plane wavevector of incident light). The propagating solution of the chain plasmon mode is of the form

(a_{m}, b_{m}) = (a_{0}, b_{0}) \exp i (q m d - ω t)

. With the use of Eq. (1), the phase-matching condition (or momentum conservation) for the chain plasmon-wave excitation can thus be obtained as

q = k_{/ /} = k_{0} n_{d} \sin θ

, where

k_{0}

is the wavevector in the free space,

n_{d}

is the refractive index of the glass substrate, and θ is the incident angle.

The plasmon oscillation of linear bump chains can induce an electric dipole moment $p_{m} \approx (a_{m} + b_{m}) l = p_{0} \exp i (q m d - ω t)$ in each unit cell, where the amplitude $p_{0} = (a_{0} + b_{0}) l$ represents the strength of the chain plasmon mode excitation. In the phase-matching condition, one can deduce from Eq. (1) that

p_{0} = \frac{(2 l^{2} / L) [(1 - \cos q d) ω_{2}^{2} - ω^{2}]}{[ω_{+}^{2} (q) - ω^{2} - i η ω] [ω_{-}^{2} (q) - ω^{2} - i η ω]} E_{0} .

Here,

ω_{1} = 1 / \sqrt{L C_{1}}, ω_{2} = 1 / \sqrt{L C_{2}}

, and

η = R / L = γ / (1 + L_{s} L_{0}^{- 1})

(η is slightly smaller than γ, the collision frequency of free electrons, due to the self-inductance [22]);

ω_{\pm} (q)

denotes the dispersion of the chain plasmon mode, which obeys the following equation:

ω_{\pm}^{2} (q) = ω_{1}^{2} + ω_{2}^{2} \pm \sqrt{ω_{1}^{4} + ω_{2}^{4} + 2 ω_{1}^{2} ω_{2}^{2} \cos q d} .

One can see that the plasmon-mode dispersion has two separated branches and that the resonant excitation of linear chains requires a second condition, i.e., the energy conservation

ω = ω_{\pm} (q)

, to be satisfied. The two dispersion branches can be distinguished by studying the case of zero wavevector (q = 0). In this case, the high-frequency mode corresponds to a nonzero

ω_{+} (0)

and a net dipole moment in the unit cell. Nonetheless, the low-frequency mode gives

ω_{-} (0) = 0

and the dipole moment is canceled completely at the resonance. Hence, the two branches may be termed, respectively, as the “optical” and “acoustic” plasmon modes, according to the classical definition used for the lattice waves [23,24]. Since the “acoustic” branch lies in the long wavelength range, it will not be resonantly excited here.

The electric dipole induced, due to the “optical” mode, in each unit cell can emit the radiation into the far field (when $q < k_{0}$ or $\sin θ < n_{d}^{- 1}$ ), which interferes with the direct transmission of the subwavelength slits $E_{S} = t_{0} E_{0}$ (in the air side, here). To analyze the above effect, we will take the normal incidence as an example. In this case, the dipole moment is simply $p_{0} \propto {(ω_{0}^{2} - ω^{2} - i η ω)}^{- 1} E_{0}$ , which generates a radiation field $E_{R} \propto - ω^{2} p_{0}$ [where $ω_{0} \equiv ω_{+} (0)$ represents the resonance frequency]. Since $p_{0}$ undergoes a π-phase shift when ω passes through $ω_{0}$ , $E_{S}$ and $E_{R}$ will interfere with nearly opposite phase on the two sides of the plasmon resonance. This is the reason for the observed transmission maximum-minimum transition or the asymmetric lineshape (constructive interference for the maximum and destructive interference for the minimum). The light transmission due to the interfering modulation can be expressed in terms of a Fano-like formula

\begin{array}{l} T = {| t_{0} - \frac{α ω^{2}}{ω_{0}^{2} - ω^{2} - i η ω} |}^{2} \\ = t_{0}^{2} \frac{{| (ω^{2} - ω_{0}^{2}) / ω + η (α ω / t_{0} η + i) |}^{2}}{{(ω^{2} - ω_{0}^{2})}^{2} / ω^{2} + η^{2}} \\ ~ t_{0}^{2} \frac{{| ω - ω_{0} + q_{F} η / 2 |}^{2}}{{(ω - ω_{0})}^{2} + {(η / 2)}^{2}} . \end{array}

Here, α is an introduced proportionality factor, and

q_{F} = i + (α ω_{0} / t_{0} η)

is the Fano factor determining the asymmetry of the line profile. In the above approximation,

ω + ω_{0} \approx 2 ω

has been adopted around the resonance wavelength. Due to the absorption loss, the Fano factor obtained here is a complex number, where the real part, associated with the ratio between the resonant and direct transmission contributions, is positive. The complex Fano factor has also been suggested previously in an Aharonov-Bohm interferometer, because of the breaking of time-reversal symmetry in presence of a magnetic field [12]. The fitting of spectrum of a single bump-chain with Eq. (4) is shown in inset of Fig. 2(a) (the open circles), yielding a reasonable agreement (here a resonance wavelength of 985 nm, a damping coefficient of

η = 2 γ / 3

, and a Fano factor of

q_{F} = 0.28 + i

were used). As can be seen, compared with the Fanolike resonances in plasmonic nanoparticles with the narrow localized resonance mode [14], our structure has a relatively small asymmetry factor.

One important result predicted by the theory is that an absorption maximum can be obtained when the linear bump chains are resonantly excited. At the chain plasmon mode resonance [ $ω = ω_{+} (q)$ ], the dipole moment of each unit cell has a maximal imaginary part [see Eq. (2)], which corresponds to an enhanced absorption of incident light. According to the above discussion, the absorption peak or chain plasmon mode resonance will be located between the spectral positions of transmission maximum and minimum (note, once again, the flat surface SPP modes are responsible for the transmission dips, instead [17,18]). To confirm this point, we have calculated by FDTD the transmission (T), reflection (R), and absorption ( $A = 1 - T - R$ ) spectra of the structure (still taking the normal incidence as an example), and the results are shown in Fig. 4 . It can be seen that, at the wavelength 985 nm, which is between the spectral positions of transmission maximum (973 nm) and minimum (1012 nm), a strong absorption of light up to ~60% occurs.

Fig. 4 Simulated transmission, reflection, and absorption spectra of the structure. The chain plasmon mode resonance corresponds to a strong absorption of the light.

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When the light incident angle is enlarged, increases and $ω_{+} (q)$ decreases, according to the dispersion equation. Thus, the chain pla $q = k_{0} n_{d} \sin θ$ smon mode and Fano lineshape will be redshifted to the longer wavelengths. Figure 5(a) presents the simulated transmission spectra for various incident angles, which gives a clear demonstration. When the incident angle is increased from 0 to 20 degrees, a redshift of the spectral transition about 150 nm can be achieved (the dip positions corresponding to the flat surface SPP modes are nearly unchanged). It is worthy of noticing that, in this process, the Fano lineshape is well maintained. The numerical simulations are also supported by the experimental observations presented in Fig. 5(b). In addition, the dependence of chain resonance mode on the incident angle has been mapped in inset of Fig. 5(a), using the numerically determined spectral position of absorption maxima. The above effect enables a dynamic tuning of spectral position of the Fano effect (note that the Fano lineshape is not tuned here).

Fig. 5 Simulated (a) and measured (b) transmission spectra for various incident angles (the Fanolike resonance corresponds, from the left to right, to the incident angle 0, 10, 15, and 20 degrees, respectively). The inset presents the dispersion curve obtained using the spectral positions of simulated absorption maxima.

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The result provides a new method for realizing the Fanolike resonance, employing the interference between the direct background transmission and resonant plasmon excitation in the linear bump chains. The sharp transition around the resonance may allow an efficient tuning of the transmission with a very small wavelength shift. Moreover, the dispersion of chain plasmon mode can be used to tune the spectral position of Fano effect dynamically. The effect is also accompanied by a strong field enhancement as well as an increased absorption, which may be used to amplify the light-matter interactions [25].

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 10874079, 10804051, and 10874082), and by the State Key Program for Basic Research of China (Grant No. 2010CB630703).

References and links

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Fanolike resonance due to plasmon excitation in linear chains of metal bumps

Abstract

Acknowledgments

References and links

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Figures (5)

Equations (4)

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