## Abstract

We investigate the dispersion relation and extinction properties of surface plasmons in an array of gold nanoparticle chains under s-polarized plane wave excitations, through experiment and simulation. Our results reveal that the dispersion and extinction properties of gold nanoparticle chains at an air/glass interface are significantly different from those in a uniform medium. Under total internal reflection, the dispersion is much larger than that above total internal reflection and 100% extinction can be reached. We show that the large dispersion under total internal reflection can be explained by dipole fields and coupling at the air/glass interface.

©2008 Optical Society of America

## 1. Introduction

Surface plasmons in periodic structures composed of noble metal nanoparticles are of great interest because of potential applications in communications and in optical sensing. Periodic chains of metal nanoparticles hold the promise for the realization of miniaturized electromagnetic waveguides with a lateral confinement of the surface plasmon waveguide mode below the diffraction limit [1–3]. The coupling between the surface plasmons of different metal nanoparticles in a periodic chain or array has been demonstrated to enhance and/or tune the localized surface plasmon resonance (LSPR), which is favorable for a number of biological and chemical sensing applications [4–12]. Theoretical and experimental studies have been reported on such coupling interactions and the resulting dispersion and loss properties of the coupled surface plasmon modes in a uniform dielectric medium [2,13–16]. Experiments have also been reported in which the periodically coupled surface plasmon modes at a planar dielectric interface were excited at normal incidence and a few angles [5–10,12,17,18]. We recently reported experimental measurements of the dispersion relations of surface plasmons in an array of gold nanoparticle chains at a air/glass interface using transmission spectroscopy with a range of incident angles [19]. Because transmission spectroscopy was employed, however, the experimental results only covered the portion of the dispersion relations above total internal reflection (TIR). In this paper, we report experimentally obtained transmission and reflection characteristics of the s-polarized surface plasmon mode in an array of gold nanoparticle chains over a full range of incidence angles (0° ~ 90°). Our results reveal significant modifications to the dispersion and extinction properties in the TIR regime. We also report a finite difference time domain (FDTD) numerical approach whose predictions of the dispersion and extinction of the periodically coupled surface plasmon mode are in good agreement with the experimental results. While all the previous theoretical studies on the dispersion and loss properties of metal nanoparticle periodic structures assume a uniform surrounding dielectric medium [2,13–15], in many experiments, the metal nanoparticles were situated at the interface between a substrate and a superstrate with different dielectric constants. In our experiments, we show that an air/glass interface leads to a significantly different dispersion behavior from that with a uniform surrounding dielectric medium. In particular, a sharp increase in dispersion occurs when the incidence angle increases across the TIR critical angle. Under TIR, there exists a range of incidence angles where the reflected radiation loss at resonance is almost zero. In the last part of the paper, we give a theoretical explanation of the dispersion behavior in terms of dipole fields and coupling along an air/glass interface.

## 2. Experiments

The structure under consideration consists of an array of gold nanoparticle chains on a glass substrate. Each chain consisted of gold cylinders that were spaced by 140 nm. Each cylinder had a diameter of ~90 nm and a height of 55 nm. Adjacent chains were spaced by 300 nm. The sample was fabricated by the standard electron beam lithography and lift-off processes. A 5 nm chrome layer was deposited prior to the gold deposition to improve adhesion. The glass substrate was coated with a 15 to 30 nm thick layer of indium tin oxide (ITO) to prevent charging during electron beam writing. Figure 1 is scanning electron micrographs of the sample. Details of the fabrication processes can be found in ref [19].

To characterize the transmission and reflection properties of the array of gold nanoparticle chains over a wide range of incidence angles, the sample was illuminated from the glass side by a collimated beam from a Xenon lamp. The illumination beam was angled along the direction of the chains and was s-polarized in order to excite surface plasmons transversely polarized along the air/glass interface. The incidence angle θ was varied in order to excite surface plasmon modes at different k_{x} values. k_{x} was the wave vector component along the direction of the chains. The inset of Fig. 2(a) is a schematic diagram of the experiment and also gives the definition of x, y and z directions in this paper. Details of the experiment were similar to that of Ref [19].

Power transmission spectra were taken for angles (θ) outside the TIR regime [19], where θ is smaller than the critical angle. Reflection spectra were taken for angles (θ) within the TIR regime, where θ is greater than the critical angle. The 140 nm spacing along the x direction and 300 nm spacing along the y direction did not produce any higher order scattering modes for the range of wavelengths discussed in this paper. Figure 2(a) shows the experimental spectra at several incidence angles (θ). In addition, spectra obtained by a numerical approach are presented in Fig. 2(b), and can be seen to be in good agreement with the experiment. The numerical modeling approach is discussed in greater detail in the following section. The dispersion relation was extracted from these spectra and plotted as Fig. 3. The orange points in Fig. 3 are experimental values of the resonance frequencies at different k_{x} values for the full range of incidence angles, i.e., above the light line of glass. The green points are the numerical results. The resonance frequencies were taken from the positions of the minima of the transmission spectra for incidence angles above the TIR regime and from the positions of the minima of the reflection spectra for incidence angles within the TIR regime. To mitigate against the effects of experimental noise, we fit each spectrum with a fifth order polynomial near the resonance frequency before finding the minimum. It is worth mentioning that that we took the minima of the spectra at each angle instead of the minima of the spectra at each k_{x} value. This was due to the consideration that, at a constant incident illuminating power, the amplitude of the electric field at an air/glass interface changes significantly with the incidence angle near and inside the TIR regime, as shown in Fig. 4. Therefore, the coupling between the illuminating beam and the surface plasmon mode at a constant k_{x} value will change with frequency as the incidence angle changes and consequently shift the minimum position of a fixed k_{x} spectrum away from the actual resonance frequency by a considerable amount. Although by extracting the minima from spectra with fixed incidence angles we were actually comparing surface plasmon modes with different k_{x} values, this method more closely indicates where the resonances are positioned. A distinctive feature of the dispersion relation in Fig. 3 is that the dispersion is much larger within the TIR regime. We discuss our interpretation of this phenomenon with the aid of numerical simulations in the following sections of this paper.

## 3. Numerical simulations

To gain insight into our experimental observation of a distinctive change in the dispersion relation across the TIR line (or the light line of air), we developed a numerical simulation method to obtain the transmission and reflection spectra of metal nanoparticle periodic structures. With this method, the transmission and reflection coefficients are obtained for a range of frequency values in a single simulation. The simulation was done with the FDTD software Fullwave from the RSOFT Design Group, Inc.

In each simulation, a time domain electromagnetic wave pulse covering the spectral regime of interest was launched from the glass side at a single k_{x} value, with a periodic boundary condition applied along the x axis and a symmetric boundary condition applied along the y axis. Therefore, the launch was composed of plane wave components within a range of frequencies, the launching angle of each plane wave component determined by its frequency and k_{x} values. Perfectly Matched Layers (PML) were applied at the top and bottom z-boundaries of the unit cell simulation domain to absorb the transmitted and reflected electromagnetic waves. The transmitted power was found from 1/2 Re(E
_{y}
H
_{x}
^{*}), where E and H monitors were placed on the air side. The reflected power was found from -1/2 Re(E
_{y}
H
_{x}
^{*}), where E and H monitors were placed on the glass side. E and H were Fourier Transforms of the time domain electromagnetic field values. To find the reflected E and H fields, we subtracted the incident fields from the total fields. Because only the fundamental scattering mode exists for the frequency range of interest in the far field, the transmitted and reflected powers can be determined from field monitor points on the air and glass sides, respectively. There is therefore no need to integrate the Poynting vector over a surface in the x–y plane in our case. In each simulation, the transmission and reflection coefficients over a range of frequencies and at one k_{x} value were obtained. Multiple simulations were done to obtain results for all the k_{x} values of interest. The transmission and reflection coefficients at an arbitrary angle were then obtained by linear interpolations of the transmission and reflection coefficients at the same frequency and the nearest k_{x} values. In our simulation, we have simulated k_{x} values with an interval of 0.2π/µm.

Since each simulation contained a range of frequencies, there existed plane wave components with frequencies close to the light lines of air and glass, which propagated approximately parallel to the x axis, either on the air side or on the glass side. These wave components were not efficiently absorbed by PML’s [20], and bounced back and forth inside the simulation domain, causing unphysical oscillatory features in the entire spectral regime of Fourier Transforms for a finite time window if not treated appropriately. To resolve this problem, we calculated the reflected power -1/2 Re(E
_{y}
H
_{x}
^{*}) as -1/2 η|H
_{x}|^{2}/cos(θ), where η was the characteristic impedance of electromagnetic waves in glass. For an s-polarized plane wave at an incidence angle θ, H_{x}=H cos(θ) approaches 0 as θ approaches 90°, while E_{y}=E does not change with θ. Therefore, the Fourier Transform of the time domain H_{x} values suffered much less from the bouncing wave components with large θ values than the Fourier Transform of E_{y}. This method was found to be effective at reducing the effects of the unphysical wave components that bounced between the glass side PML and the air/glass interface due to TIR.

The large angle (θ) transmitted wave components on the air side corresponded to wave components on the glass side with smaller propagation angles (due to Snell’s Law), and therefore underwent reasonable absorption inside the glass side PML. As long as these wave components completely disappeared within the finite simulation time window through absorption by the glass side PML, they do not affect Fourier Transform at other frequencies. Since these wave components would bounce back and forth several times on the air side before they reached the glass side PML, a sufficiently long simulation time was required. We used 1/2 Re(E
_{y}
H
_{x}
^{*}) for the transmitted power instead of 1/2 η|H
_{x}|^{2}/cos(θ) because H_{x} was non-zero for the evanescent waves on the air side. Part of these evanescent waves on the air side was from the large angle bouncing waves on the glass side and did not disappear with time efficiently. This problem did not present itself for the reflected power.

Several other factors needed to be carefully considered in the FDTD simulation in order to obtain accurate transmission and reflection spectra. The transmission and reflection monitors should be located some distance from the metal nanoparticles in order not to detect the near field of the surface plasmons. The pulsed plane wave launch plane should be located some distance from the metal nanoparticles because the pulsed plane wave launch contained an evanescent near field. To ensure adequate absorption by the PML’s of waves at relatively large incidence angles, the PML’s must be sufficiently long. The simulation time must be suitably long in order for most wave components to disappear completely via PML absorption within the simulation time. In our simulation, we placed the transmission E and H monitors 0.9 µm above the air/glass interface in air and the reflection E and H monitors 0.9 µm below the interface in glass. The pulsed plane wave launch plane was 1 µm below the interface. The PML’s were 2 µm long, with a normal incidence reflectivity of 10^{-32}. The simulation time was 100 µm/c, where c is the speed of light in vacuum.

In our simulation, the refractive index of glass was taken as 1.51. The dielectric properties of gold were fit to the Drude-Lorentzian model based on ref [21]. Chrome and ITO were not taken into consideration for simulation results reported in this article. A non-uniform cubic mesh with 5 nm grid size at and near the particles was used. The time step was 2.5nm/c.

The simulation and experimental results agree nicely with each other as exhibited in Fig. 2 and 3. We noticed that by including a 15 to 30 nm thick ITO layer into the simulation, the resonance frequencies red shift, but maintain similar relative changes in dispersion and extinction across the light line of air. The nice agreement between simulation and experiments in Fig. 2 and 3 could be due to the differences in various parameters between simulation and experiments cancelling each other. In Fig. 5, the extinction has been simulated for a wide range of frequencies and k_{x} values. The extinction was found from (1 - transmission - reflection), which is equal to the ohmic absorption. Large differences in dispersion and extinction are clearly present between the above and within TIR regimes. In the TIR regime, the extinction at resonance can reach almost 100%, which means an efficient excitation of surface plasmons. This originates from destructive interference between the surface plasmon radiation and the illuminating plane wave’s reflection, which leads to critical coupling condition between the surface plasmon mode and the TIR plane wave mode. Although there is no radiation loss at a 100% extinction, the linewidth of the resonance spectrum still contains the contribution from radiation, or coupling between the surface plasmon and the TIR plane wave. Efficient excitation is important for some plasmonic devices [22].

## 4. Explanation of dispersion relation

The most distinctive feature of the experimental and simulation results is the large difference in dispersion between the above and within TIR regimes. To gain insight into this phenomenon, FDTD modeling was carried out on various configurations of nanoparticles. A gold nanoparticle chain array with identical parameters to those described earlier, except that it was situated in a uniform glass medium, was considered. A gold nanoparticle chain array with identical parameters, except that the particles were 30 nm in height, was also considered. The simulated dispersion relations are plotted in Fig. 6, along with that of the original structure for comparison. As before, the minima of the transmission spectra, for angles above TIR, and the minima of the reflection spectra, for angles within TIR were used to extract the resonance frequencies. For the uniform glass medium case, only transmission spectra were used. From Fig. 6, it can be seen that the gold nanoparticle chain array in a uniform glass environment does not exhibit the large change in dispersion, which qualitatively agrees with previous theoretical results [2,13,15]. A specious reason for this observed large change in dispersion could be that, as we change the incidence angle under TIR, the electromagnetic wave’s evanescent decay distance on the air side changes, which may consequently change the surface plasmon mode profile and shift the resonance frequency. If this were true, then the change in decay distance would have less effect on the mode profile of shorter nanoparticles. In Fig. 6, however, the 30 nm tall nanoparticle chain array exhibits about the same quantitative change in dispersion as the 55nm tall nanoparticle chain array. Therefore, there must be other important reasons for this dispersion relation feature. In the remainder of this section, we show that this is due to the coupling between surface plasmons of different nanoparticles and the effect of retardation [13,15].

Previous theoretical studies on the coupling and retardation effects in metal nanoparticle chains have frequently made the simplifying assumptions that the surface plasmons in metal nanoparticles can be considered as point dipoles, and that these dipoles interact with each other through a uniform dielectric medium [13,15]. We keep the first assumption. However, the field of a dipole at an air/glass interface is significantly different from that in a uniform medium [23–28]. In the following, we describe numerical simulations of this field pattern, which were also carried out using FDTD. Figure 7 shows the continuous wave field pattern of a line of point dipoles oriented in +y direction on an air/glass interface. The point dipoles were oscillating at the same phase at a frequency of 1.6 c/µm. The point dipoles were spaced by 320 nm in y direction and one of them was sitting at origin. Each point dipole in the line corresponded to one nanoparticle at x=0 nm from one chain in our sample, as illustrated in the inset of Fig. 7. PML boundaries were applied at the x- and z-edges of the computational domain, and symmetric boundary condition was applied on the y-edges. A uniform cubic mesh with a grid size of 20 nm was used. The coupling between this line of dipoles and other lines spaced 140 nm in x direction is determined by E_{y} and changes with frequency and k_{x}. We see from Fig. 7 that the E_{y} pattern along the air/glass interface has both the periodicity of the wavelength in air and the periodicity of the wavelength in glass. Figure 8(a) shows the amplitude and phase of E_{y} along x-axis at the air/glass interface. Figure 8(b) shows the amplitude and phase of E_{y} along x-axis for the same line of dipoles but in a uniform glass environment for comparison with Fig. 8(a). We see that the field along an air/glass interface decays faster than in a uniform dielectric environment.

To conveniently describe the coupling between surface plasmons or point dipoles, we name the lines of nanoparticles or lines of point dipoles at x=(…, -140 nm, 0 nm, 140 nm, 280 nm, …) as (…line -1, line 0, line 1, line 2, …). In Fig. 8, the field from point dipole line 0 has E_{y} values of (…, E_{-1}, E_{0}, E_{1}, E_{2}, …) at x=(…, -140 nm, 0 nm, 140 nm, 280 nm, …) on the x-axis. The induced surface plasmon dipole in nanoparticle line 0 through coupling to surface plasmon dipole fields from the other nanoparticle lines is P_{indu}=α E_{indu}, where α is the polarizability of a single line of nanoparticles, and E_{indu} is a sum of E_{y} field values at the origin from all the other nanoparticle lines. When the in-plane wave vector is k_{x}, we have

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}=\sum _{n=1}^{\infty}{E}_{n}\mathrm{exp}\left(-i{k}_{x}\xb7n\xb7140\mathrm{nm}\right)+{E}_{n}\mathrm{exp}(i{k}_{x}\xb7n\xb7140\mathrm{nm})$$

The phase of the point dipoles in line 0 is set to 0 in our simulation. When E_{indu} has a positive real part, it is in phase with the point dipoles in line 0 and decreases the restoring force on the surface plasmon, which results in a red shift of the surface plasmon resonance; for the same reason, when E_{indu} has a negative real part, there will be a blue shift of the surface plasmon resonance.

Figure 9 shows E_{indu} at a single frequency of 1.6 c/µm for k_{x} values in the first irreducible Brillouin Zone found from Equation (1) and the calculations plotted as Fig. 8. The k_{x} interval between the nearest points in Fig. 9 is a constant. In this calculation we summed up 71 E_{y} values on the second line of equation (1) which corresponds to a 10 µm long simulation domain on one side of line 0, as shown in Fig. 7 and 8. From Fig. 9(b), we see that E_{indu} in a uniform dielectric environment has a sharp and large amplitude peak at the light line. If we assume that the dipole fields vary as exp[i(-ωt+kx)], then the dipole lines interfere constructively at the light line of glass through the first term of the second line of Equation (1). This results in the large amplitude peak. The abrupt change of E_{indu} across the light line of glass is qualitatively consistent with the theoretical calculations of dispersion relations in refs [13,15], which have a sharp red shift of resonances near the light line.

E_{indu} of point dipole lines at an air/glass interface is different. For the uniform glass environment case in Fig. 9(b), E_{indu} undergoes an abrupt change across the lightline; while for the air/glass interface case in Fig. 9(a), the abrupt change in E_{indu} across the light line for the uniform glass environment case is expanded to a fast change across the whole TIR regime. The fast change of the real part of E_{indu} in the TIR regime indicates a corresponding large dispersion, which is consistent with our experimental and simulation results. We have found that in the TIR regime, the value of E_{indu} is also predominantly determined by the first term on the second line of equation (1), which is similar to constructive interference as what occurs at the light line for the uniform glass environment case.

From Fig. 9, it can be seen that there is a fast phase change of E_{indu} in the TIR regime. To understand this phenomenon, it is useful to note that, from Eq. (1), E_{indu} is similar to a spatial Fourier Transform along the x-axis of the field from point dipole line 0. The spatial Fourier Transform of the dipole field is related to a plane wave expansion of the dipole field. The plane wave expansion of the dipole field on the interface is equal to the sum of the plane wave expansion of the dipole field in a uniform glass environment and the corresponding plane waves reflected by the air/glass interface [29]. The plane wave reflection coefficient for the electric field undergoes a fast phase change in the TIR regime, as shown in Fig. 4(b). Therefore, the fast phase change of E_{indu} in the TIR regime can be attributed to the fast phase change of the plane wave reflection coefficient, which results in a large dispersion for our case. It should be clarified that the fast phase change of E_{indu} in the TIR regime may not necessarily be accompanied by a fast change in its real part. For example, for an isolated single chain of particles which is the same as the chains in Fig. 1, its E_{indu} diagram at the frequency of 1.6 c/µm for k_{x} values in the first irreducible Brillouin Zone is shown in Fig. 10. In this case, the fast phase change of E_{indu} in the TIR regime does not involve a significantly fast change in its real part but rather in its imaginary part. Therefore, this single chain structure exhibits a fast drop in loss rate in the TIR regime without a significant higher dispersion in the TIR regime. This topic will be addressed in a future article.

## 5. Summary

We have experimentally obtained the dispersion and extinction properties for the s-polarized surface plasmon mode in an array of gold nanoparticles chains at an air/glass interface by measuring the transmission and reflection spectra. The dispersion is significantly larger within the total internal reflection (TIR) regime. 100% extinction can be reached under TIR. We have outlined an FDTD numerical approach that rejects spurious unphysical waves propagating at large angles. The accuracy of this FDTD approach was verified through comparison with experiment. The large change in dispersion between the above and within TIR regimes has been explained by considering the field pattern of dipoles at an air-glass interface, and the resulting influence upon the coupling between the dipoles.

## Acknowledgments

We thank the Defense Advanced Research Projects Agency (DARPA), the Charles Stark Draper Laboratory and the Harvard Nanoscale Science and Engineering Center (NSEC) for the financial support of this work. The Harvard NSEC is supported by the National Science Foundation (NSF). Fabrication work was carried out at the Harvard Center for Nanoscale Systems (CNS), which is also supported by the NSF.

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