Full analytical model for obtaining surface plasmon resonance modes

Ergun Simsek

doi:10.1364/OE.18.001722

1. Intrnoduction

In the last decade, there has been a great interest in surface plasmons (SPs) due to their potential for developing new types of optical communication devices and sensors. Theoretical and experimental studies have shown that periodically placed nanoparticles (NPs) can lead to giant electromagnetic field enhancement which can be useful for detecting molecules at low concentrations [1, 2, 3, 4, 5, 6]. Applications of SPs include surface [7, 8, 9, 10, 11, 12, 13, 14] or tip [15, 16] enhanced Raman scattering, plasmonic solar cells [17, 18] and nonlinear frequency generation [19]. In these works, researchers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. The specific interest of this work is the design of a full analytical model to calculate SP resonance modes of metal NP structures embedded in a multilayered medium.

Many researchers have studied SP resonance modes of NP chains [6, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] and arrays [41, 3, 42, 43, 44, 45, 46, 47, 48] theoretically, numerically, and/or experimentally. Even though the accuracy of theoretical models based on the coupled dipole approximation (CDA) has been improved over the years [30, 32, 37, 38, 42], a non-negligible discrepancy between experimental and theoretical results still remains [39, 40]. The main reason behind such a difference is due to the fact that the CDA based theoretical models assume a homogeneous background, which is not the case for almost all of the fabricated structures. This is also why numerical techniques (e.g. finite difference-time domain methods) that can handle inhomogeneous backgrounds are preferred to compare with the experimental results [40, 46, 47].

One way of handling inhomogeneity is assigning an effective refractive index for the background [41], and applying homogeneous background techniques, such as CDA. This approach can provide a good estimate of SP resonance modes but can not explain the exact effect of an interface (such as air/glass interface in [49]) on the dispersion and extinction of surface plasmons. In [26, 39], researchers try to implement CDA by the help of image theory (IT) to take air/glass interface into account. Experimental results support the validity of such a theoretical model [39] but it is still unclear how IT can be implemented for structures with more than two layers, especially for the case where the width of the layer (on which NPs are aligned) is less than the half of the height of NPs.

Yang and Crozier propose a semi-analytical model for NPs on top of a glass slide [50] and show good agreement between their model and experimental results for the first transverse mode. Up to our knowledge, this is the first CDA based model, which can describe the effect of the air/glass interface. The main advantage of their model is that it only involves real valued frequencies. The drawback, however, is the requirement of a full wave solver in order to calculate the polarizability of NPs. In this work, we overcome this requirement by appropriately modifying the polarizability factor using layered medium Green’s functions (LMGFs). We also implement CDA with LMGFs and hence obtain a full analytical model that can accurately calculate SP resonance modes of metal NP structures embedded in a multilayered medium. This fully retarded theoretical model includes the effects of retardation, radiative damping, and dynamic depolarization due to the finite size of NPs based on the Modified Long Wavelength Approximation (MLWA) [22]. Unlike [51], the formulation is not limited to three-layer media.

The outline of this paper is as follows. We first briefly explain how to evaluate LMGFs. Then we implement LMGFs (i) to calculate the polarizability of the NPs embedded in a layered medium and (ii) to extend CDA to layered media. Finally, we present numerical results, which show a good agreement with experimental data and support the experimental observations described in [40] about the existence of the second transverse mode. Propagation lengths (PLs) of each mode are also calculated.

Fig. 1. An A-layer medium with source and field points in layer n and layer m,respectively.

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2. Layered medium green’s functions

Consider a general multilayer medium consisting of N layers separated by N-l planar interfaces parallel to the xy plane, as shown in Figure 1. Layer i exists between z_i and z _i-1 and is characterized by permittivity ε_i and permeability μ_i. An arbitrarily directed electric dipole p can be represented in the cartesian coordinates by p = x̂p_x + ŷp_y + ẑp_z. Similarly electric field E created by that dipole can be decomposed as E = x̂E_x + ŷE_y + ẑE_z. The relationship between each component of the field and the dipole is given by the dyadic layered medium Green’s functions, Ḡ(r,r′), as follows

[\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}] = [\begin{matrix} G_{xx} & G_{xy} & G_{xz} \\ G_{yx} & G_{yy} & G_{yz} \\ G_{zx} & G_{zy} & G_{zz} \end{matrix}] . [\begin{matrix} p_{x} \\ p_{y} \\ p_{z} \end{matrix}] .

In this notation, G_ηζ gives η̂ component of the electric field at r due to a ζ̂ directed unit electric dipole located at r′, where η and ζ are either x, y, or z.

As a first step to calculate LMGFs, the problem is transformed from spatial domain to spectral domain. Each layer is represented by a uniform transmission line having the same physical properties; hence electric and magnetic fields can be interpreted as voltage and current, respectively, on a transmission line [52, 53]. By using this transmission line analogy, Green’s functions in spectral domain, called transmission line Green’s functions, can be calculated in compact form. The spectral domain counter parts of the

{\tilde{G}}_{xx} = \frac{k_{x}^{2}}{k_{ρ}^{2}} {\tilde{G}}^{TM, VI} + \frac{k_{y}^{2}}{k_{ρ}^{2}} {\tilde{G}}^{TE, VI}, {\tilde{G}}_{yy} = \frac{k_{x}^{2}}{k_{ρ}^{2}} {\tilde{G}}^{TE, VI} + \frac{k_{y}^{2}}{k_{ρ}^{2}} {\tilde{G}}^{TM, VI}, {\tilde{G}}_{zz} = \frac{k_{ρ}^{2}}{ω^{2} ε_{i} ε_{m}} {\tilde{G}}^{TM, IV},

where subscripts i and m represent source and field layers, respectively; G̃ ^h,PQ is the transmission line Green’s function, which represents voltage or current (P) due to a voltage or current source (Q) for either TE or TM case (h); k_η is the wavenumber along η̂; ω is the radial frequency. The complete expressions for transmission line Green’s functions can be found in [52, 53, 54, 55, 56, 57].

The spatial domain LMGF is the inverse transformation from spectral domain to spatial domain and is known as a Sommerfeld integral [58] given by

G_{ηζ} (ρ, z ∣ z^{'}) = \frac{1}{2 π} \int_{0}^{\infty} {\tilde{G}}_{ηζ} (k_{ρ}, z ∣ z^{'}) J_{v} (k_{ρ} ρ) k_{ρ}^{v + 1} d k_{ρ}

where ν = 0,1, J_ν is ν^th order Bessel function, $ρ = \sqrt{{(x - x')}^{2} + {(y - y')}^{2}}$ and G̃_ηζ(k _ρ,z∣z′) is its spectral domain counterpart.

It should be noted that LMGF’s can be written as a sum of primary field term and reflection terms:

G_{ηζ} = G_{ηζ}^{prim} + G_{ηζ}^{refl} .

The primary field term, the direct interaction between the source and field, can be calculated directly by using Sommerfeld identity [58]. The contribution of the reflections is calculated by taking the integral given in (3). In order to accelerate this process, the singular terms of the integrand are subtracted and their contribution is calculated analytically, as explained in [57]. The remaining integral is computed numerically by using Gaussian quadratures.

3. Layered medium CD A

The dispersion relations are obtained as described in [37]. Assume there is a finite chain of equally spaced metal NPs along the x-axis in a multilayered medium; M is the number of NPs and d is the inter-particle spacing. Again assume NPs can be represented as oscillating dipoles (p_j, j = 1,2, ⋯ , M) and (ω is the oscillating frequency in the absence of an applied field. Note that treatment of nanoparticles as point dipoles means the results are valid only for 3a < d, where a is the particle radius. The induced dipole moment on particle-n because of field generated by dipole-m can be calculated as

p_{n} = \bar{G} (r_{n}, r_{m}) [\begin{matrix} α_{x} (ω) & 0 & 0 \\ 0 & α_{y} (ω) & 0 \\ 0 & 0 & α_{z} (ω) \end{matrix}] p_{m},

where α_η(ω) is frequency dependent polarizability of NP along the η̂-direction.

Surface plasmon resonance occurs when the dipole moment of a single NP becomes equal to the induced dipole moment on that NP due to all other dipoles. Using (5) and the fact that any vector in 3-dimensional space can be written as a combination Fig. 2 shows these three cases: dipoles are parallel to (a) x̂-axis, (b) ŷ-axis, (c) ẑ-axis. Hence, we obtain three dispersion relations:

Fig. 2. Induced moments (red arrows) inside metal NPs (yellow cylinders) on top of a substrate (blue rectangular cubes): for (a) longitudinal, (b) transverse-1, (c) transverse-2 modes. The inter-particle spacing is d along x̂-axis.

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1 - α_{η} (ω) \sum_{n = 1}^{M - 1} G_{ηη} (r_{0}, r_{n}) = 0,

where r_n = nd x̂ assuming r₀ is located at the origin; η = x for the longitudinal mode (L), η = y for the first transverse mode (Tl), η = z for the second transverse mode (T2). For a chain of M metal NPs, Eq. (6) becomes a set of M coupled equations in the M unknown moments of NPs. One can put these equations in a matrix form as explained in [37] and calculate resonance frequencies for which the matrix coupling the dipoles is singular. Then these frequencies are mapped onto the dispersion relations appropriate for the infinite chain. For the complete procedure, reader is kindly referred to [37].

Replacing the free space Green’s functions with layered medium Green’s functions is not enough to extend CDA to layered media; the polarizability of NP should be handled appropriately as well. Assume that we deal with ellipsoidal metal NPs on the xy-plane. The semi-axes (r_η) of the ellipsoids along the (x,y, z) axes are (a,b,c), respectively. The free-space polarizability along the η-axis can be approximated by

α_{η}^{fs} = \frac{ε_{r} - ε_{b}}{ε_{b} + L_{η} (ε_{r} - ε_{b})} \frac{abc}{3},

where

L_{η} = \frac{abc}{2} \int_{s = 0}^{\infty} \frac{1}{(s + r_{η}^{2}) {[(s + a^{2}) (s + b^{2}) (s + c^{2})]}^{0.5}} ds .

In order to include the effects of radiative damping and dynamic depolarization due to the finite size of NPs, the polarizability coefficient, Eq. (7), is modified as,

α_{η}^{mlwa} = {(\frac{1}{α_{η}^{fs}} - i \frac{2}{3} k^{3} - \frac{k^{2}}{r_{η}})}^{- 1},

which is called as Modified Long Wavelength Approximation (MLWA) [22]. However for NPs in a multilayered medium, the polarizability should be even further modified in order to take layered medium into account as explained in [59]

α_{η}^{lm} = {(\frac{1}{α_{η}^{fs}} - i \frac{2}{3} k^{3} - \frac{k^{2}}{r_{η}} - G_{η η}^{r e f l} (r_{0}, r_{0}))}^{- 1},

where r₀ is the mass center of the nanoparticle.

4. Numerical results

In [40, 49, 50], Crozier et al. present a systematic study on the dispersion relations of a metal nanoparticle chain fabricated on top of an indium tin oxide coated (ITO) glass slide. For the experiment, they use cylindrical gold nanoparticles with heights of 55 nm, diameters of 90 nm, and center-to-center distances of 140 nm along the length of the chain. The thickness of ITO-coating is 20 nm. The experiment results indicate that S-polarized illumination couples to a single mode (transverse, T1) and P-polarized illumination couples to a single mode (longitudinal, L) at normal incidence, with an additional mode (transverse, T2) coupled at large angles of incidence. They conclude that this weak mode only occurs for P-polarized light at high incidence angles and it can be associated with nanoparticle dipole moment oriented perpendicular to the substrate. Figure 3 summarizes the results found in [40] for L and T1 modes. In [40], they apply CDA technique to calculate SPR modes: by first assuming point dipoles exist in the air (blue lines in Fig. 3), then in the glass (green lines in Fig. 3). It is observed that experimentally obtained dispersion results (dark grey dots) lay in between these two cases.

We analyze the same structure theoretically using layered medium CDA (abbreviated as LM-CDA). For the optical constants of gold, the experimental values are used [60] rather than Drude model to eliminate any concern regarding the selection of appropriate values for plasmon and relaxation frequencies. We model cylinders as ellipsoids and calculate their polarizability using Eq. (10). The refractive indices of glass, ITO and air are assumed to be 1.51, 1.45 and 1, respectively, and the procedure described in the previous section is followed on the complex ω domain using 20 NPs.

Prior to giving complete picture for the dispersion relations calculated with LM-CDA, let us put them together with classical CDA results to demonstrate the improvement in point dipole modeling from classical CDA to LM-CDA. In Fig. 3, dark red solid line and light red dashed line show the LM-CDA results for the first transverse and the longitudinal modes, respectively, assuming NPs embedded on top of an ITO-coated glass slide. Clearly, (i) there is a very good agreement between the experimental and LM-CDA results, (ii) LM-CDA results are more accurate than the classical CDA and ERI-CDA results (the maximum error is less than 1 % for the LM-CDA, whereas maximum error of CDA implemented with the effective refractive index approximation, abbreviated as ERI-CDA, is about 2.6 % [41]). We can safely say that the use of LMGF and an appropriate polarizability factor is enable us to understand the effect of multilayered background on the dispersion relations for dipolar modes propagating along a chain of metal NPs.

Fig. 3. Experimentally (dark grey circles and squares depict L and T1 modes, respectively) and theoretically calculated dispersion curves. Dark blue solid line and light blue dashed line show the CDA results for the transverse and longitudinal modes, respectively, assuming NPs situated in air; green lines assuming NPs situated in glass, red lines assuming NPs embedded on top of an ITO-coated glass slide. Dashed black lines depict light lines in air and glass.

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Figure 4 provides a detailed picture of the dispersion relations: Fig. 4 (a) compares experimental (red circles) and theoretical (solid lines) results for the longitudinal mode. Blue and green solid lines depict the real and imaginary parts of the resonance frequencies, respectively. Dashed black lines are light lines in air and glass. Similar to the free space case, for the longitudinal mode there is no sharp interaction with the light line, since longitudinal photons cannot propagate along the chain. The lossy nature of the gold causes large imaginary components for the resonance modes.

Figure 4 (b) repeats (a) for the first transverse mode. Because of the phase matching condition between the free photon and the dipolar chain mode, the first transverse mode is strongly perturbed when k_mode ≈ ω/c.

In Figure 4 (c), real and imaginary parts of the resonance frequencies are shown for the second transverse mode. As it is also shown in [41], the second transverse mode also interacts strongly with the light line (when k_mode ≈ ω/c), damping rates diverge near the light line for both transverse modes.

Fig. 4. Dispersion curves for (a) longitudinal, (b) first transverse, (c) second transverse excitation modes, respectively, for the gold nanoparticles with heights of 55 nm, diameters of 90 nm, and center-to-center distances of 140 nm along the length of the chain. Nanoparticles are aligned on top an ITO coated glass. The thickness of ITO-coating is 20 nm. Red dots in (a) and (b) are experimental results [40]. Dashed black lines in (a), (b), and (c) depict light lines in air and glass. (d) Propagation lengths derived from the dispersion curves for each mode.

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Note that both L and T1 modes develop a bandwidth approximately 25 % larger than the case NPs situated in air, resulting in higher group velocities; whereas T2 mode’s bandwidth is much narrower than the bandwidth of the free space transverse mode.

Thus, another important outcome of these results is on the propagation lengths and group velocities supported by this structure. Figure 4 (d) shows the propagation lengths of each mode derived from the dispersion curves. The maximum propagation length that can be obtained with this structure is around 900 nm (and max. group velocity is around 0.38c) for the first transverse mode. Since the longitudinal mode does not interact with the light line, dispersion has a smooth shape, and, hence the propagation lengths are much shorter than the ones belonging to the first transverse mode. As it would be expected from the resonance frequencies with very large imaginary components, the propagation length of the second transverse mode is even shorter than L and T1 modes.

Fig. 5. Dispersion curves for (a) longitudinal, (b) first transverse, (c) second transverse excitation modes, respectively, for the silver nanoparticles with heights of 55 nm, diameters of 90 nm, and center-to-center distances of 140 nm along the length of the chain. Nanoparticles are aligned on top an ITO coated glass. The thickness of ITO-coating is 20 nm. (d) Propagation lengths derived from the dispersion curves for each mode.

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The same numerical study is repeated for the silver NPs. All the parameters are same as the one described above, except the material type of NPs. Fig. 5 shows the dispersion curves and propagation lengths for each mode. Similar to the previous case, radiative losses decrease as we increase the k for the modes below the light line. However, the damping rates are smaller for the chain of silver NPs than the chain of gold NPs, especially below the light line, and hence longer propagation lengths can be achieved using silver NPs (for the first transverse mode, maximum propagation length is about 1.1 μm). Another interesting observation is that maximum propagation lengths of different modes are close to each other for silver NPs, whereas this is not true for gold NPs as shown in Fig. 4 (d).

5. Conclusion

We developed a fully-retarded theoretical model for surface plasmons on metal nanoparticle chains and arrays embedded in a multilayered medium. Polarizability of metal nanoparticles are approximated analytically by taking multilayered medium into account. Numerical results show (i) a very good agreement with the experimental results found in the literature, (ii) the existence of two different transverse modes that strongly couple with light, (iii) propagation lengths around 1 μm are possible in multilayered medium as in the free space case.

Acknowledgements

We acknowledge useful discussions with Prof. Vadim A. Markel from University of Pennsylvania. This research was supported by a Marie Curie International Reintegration Grant within the 7^th European Community Framework Programme. Contract No. PIRG05-GA-2009-247876.

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Full analytical model for obtaining surface plasmon resonance modes

Abstract

1. Intrnoduction

2. Layered medium green’s functions

3. Layered medium CD A

4. Numerical results

5. Conclusion

Acknowledgements

References and links

Cited By

Figures (5)

Equations (10)

Optics Express