Introduction

Extensive studies have been performed on the optical properties and their applications of metallic nanoparticles (NPs), especially the noble metal (such as the gold or the silver) NPs. Metallic NP arrays show interesting collective behaviors (in the absorption/reflection/transmission etc.)[1,2,3,4,5,6] due to the combination effect of near field and far field, for example, Wood anomaly in the reflection/absorption spectra. Metallic NPs arranged in a line can be used as a waveguide for localized surface plasmon (LSP) propagation [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. The dispersion relationships of the modes for LSP propagating in the chain have been analyzed in both theory and experiment [10,11,12,13]. A series of theoretical methods such as the Mie scattering method, and open quantum system approach, were used to describe the characteristics of LSP propagation in metallic NP chains under specific conditions [24,25,26,27,28,29,30,31].

There are a lot of pervious works that point out the exponential decay as the main behavior of the energy transport in metallic nanoparticle chain in most conditions. Some calculation results showed that the non-exponential decay appeared in metallic NP chains [21, 24]. In experiment, the propagation behavior of LSPs with the bright mode showed the non-exponential decay in the chain with close-packed NPs [17]. Though many theoretical and experimental studies have been performed in the past decades, the key factors affecting the LSP propagation in metallic NP arrays are still unclear. It is very important to clarify the propagation behavior of LSP in metallic NP chains, which is very useful in the area of optical communications.

In this paper, taking the silver NP arrays as the sample systems, we apply the coupled-dipole (CD) analysis and the finite-difference time-domain (FDTD) simulation to explore the characteristics of LSP propagating in metallic NP chains. The interplay between the localized plasmon excitation in the individual NP and the lattice collective behavior leads to two phases of different electromagnetic (EM) field transport properties. New propagation mechanisms/properties including power-law/exponential decay of EM fields, universal power-law exponents, and modulation methods have been found.

Two Phases of EM Field Propagation

In our system, a series of Ag NPs with the radius R are arranged in a line with the lattice constant l and the incident field polarization angle with respect to the direction of the chain is \(\theta \)(see the schematic diagram in Fig. 1a). In the coupled-dipole analysis, only the LSP on the first NP is excited by incident light, which could be realized by using a local tip to guide the light onto the first NP in experiment. We first consider the cases of the incident light polarization perpendicular to/parallel to the chain.

Fig. 1
figure 1

(Color online) \(\ln (|I_{D}|^{1/2})\) versus the propagation distance x and \(\ln (x)\) for λ = 320 nm, respectively. The incident light polarization is perpendicular to the chain (𝜃 = 90∘). Here, CD data is the computation results based on coupled-dipole method

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The transport of the EM field in one-dimensional (1D) NP chains is mainly determined by the competition between the loss and the interaction among the NPs, which is the combination effect of the individual NP plasmonic property and the collective behavior from the lattice. The plasmonic property of a single NP can be described by the polarizability \(\alpha _{s}\) as [32]:

$$ \alpha_{s}=\frac{3i}{2k^{3}}\frac{\mu m^{2}j_{1}(m\rho)[\rho j_{1}(\rho)]^{\prime}-\mu_{1}j_{1}(\rho)[m\rho j_{1}(\rho)]^{\prime}}{\mu m^{2}j_{1}(m\rho)[\rho {h_{1}^{1}}(\rho)]^{\prime}-\mu_{1}{h_{1}^{1}}(\rho)[m\rho j_{1}(\rho)]^{\prime}}, $$
(1)

where \(\mu \) and \(\mu _{1}\) are the magnetic permeabilities of the NP and the background, k is the wave-vector in the vacuum, m is the ratio of the index of the metal to that of the background, the dielectric constant of Ag is obtained from the reference [33], \(j_{1}\) and \({h_{1}^{1}}\) are the spherical Bessel functions, and \(\rho =kR\). This expression is suitable to characterize the optical responses of sphere metal nanoparticles with the radius up to tens of nanometers [32], which also agree with the results calculated by the FDTD method. The collective behavior is due to the coupling between the NPs at different lattice positions. In the cases of \(l\gg R\), the interaction between the NP at position nl and the other NP at different lattice position \(n^{\prime }l\) can be described as dipole coupling and can be written as [25]:

$$\begin{array}{@{}rcl@{}} G_{k}(r)=\left( \frac{k^{2}}{|r|}+\frac{ik}{|r|^{2}}-\frac{1}{|r|^{3}}\right)e^{ik|r|} \end{array} $$
(2)

with \(r=nl-n^{\prime }l\) for the incident light polarization perpendicular to the chain and

$$\begin{array}{@{}rcl@{}} G_{k}(r)=\left( -\frac{2ik}{|r|^{2}}+\frac{2}{|r|^{3}}\right)e^{ik|r|} \end{array} $$
(3)

for the incident light polarization along the chain. The coupled-dipole approach works well in the case of \(l\geq 3R\) [25,26,27,28, 32], which we mainly focus on. Here, we apply it to illustrate the propagation characteristic of the silver NP chains. By solving the coupled-dipole equations for the dipole moments \(D_{k}(nl)\) for NPs at position nl,

$$\begin{array}{@{}rcl@{}} D_{k}(nl)=\alpha_{s}[E_{n}+{\Sigma}_{n^{\prime}}G_{k}(nl-n^{\prime}l)D_{k}(n^{\prime}l)] \end{array} $$
(4)

with the incident field \(E_{n}=E\delta _{n,0}\), we obtain the dipole moment of each NP in the chain as [25]:

$$\begin{array}{@{}rcl@{}} D_{k}(nl)=E{\int}_{-\pi/l}^{\pi/l}\frac{exp(iqnl)}{1/\alpha_{s}-S(k,q)}\frac{ldq}{2\pi}, \end{array} $$
(5)

where \(S(k,q)= 2{\Sigma }_{n>0}G_{k}cos(qnl)\). The analytical structure of \(1/\alpha _{s}-S(k,q)\) has important impact on the propagation characteristics. The systems show quite different EM field transport behaviors depending on whether the resonant condition \(Re[1/\alpha _{s}-S(k,q)]= 0\) can be satisfied. The important role of the analytical structure of 1/αs − S(k,q) reflects the interplay between the individual NP plasmonic properties (αs) and the collective behavior (S(k,q)).

We firstly consider the cases with the incident light polarization perpendicular to the chain (𝜃 = 90∘). Here, we use the lattice constant \(l = 180\) nm. The chain with NPs of R = 50 nm is studied to demonstrate the EM field propagating in Ag NP chains for an incident wavelength \(\lambda = 320\) nm as shown in Fig. 1. Here, \(D_{k}\) is normalized as Dk/Eαs. The linear relation of \(\ln (|I_{D}|^{1/2})\) [the local field intensity \(I_{D} \propto |D_{k}|^{2}\)] versus the logarithm of the propagation distance (\(\ln (x)\)) implies an power-law decay of the EM field for \(\lambda = 320\) nm.

At this wavelength, \(Re[1/\alpha _{s}-S(k,q)]\) is always nonzero as shown in Fig. 2, then the field decay is according to a power-law. At the wavelength \(\lambda = 320\) nm, the absolute value of \(1/\alpha _{s}\) is larger than that of \(S(k,q)\). From Eq. 5, we can use perturbation theory and obtain \(D_{k}={\alpha _{s}^{2}}S_{k}E=(\alpha _{s}E)S_{k}\alpha _{s}\), where \(S_{k}(nl)=\int e^{iqnl}S(k,q)ldq/2\pi \). Here, we have the physical picture: \((\alpha _{s}E)\) describes the dipole moment of a NP with polarizability \((\alpha _{s})\), \(S_{k}\) describes the propagation of the dipole field, and the second \(\alpha _{s}\) describes the coupling with the other NP. \(D_{k}\) or \(S_{k}\) shows a universal power-law decay behavior, i.e., \(D_{k}\sim 1/x^{\beta }\), \(\beta = 1\) in the long propagation distance limit. The universal power-law exponent is independent of the material, the size of the nanoparticle. We fit the data from Fig. 1b and obtain the slope \(-\)1.07. The difference between the fitting power-law exponent βf = 1.07 and \(\beta = 1\) is due to the finite fitting propagation distance.

Fig. 2
figure 2

(Color online) Re[1/αs − S(k,q)] versus ql for λ = 320 nm. The incident light polarization is perpendicular to the chain

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The EM field propagation is determined by the analytical structure of \(1/\alpha _{s}-S(k,q)\) (as seen from Eq. 5), which depends on the wavelength. Then, we take another wavelength to show the other phase of the EM field propagation. \(\lambda = 440\) nm is adopted as the working wavelength. \(\ln (|I_{D}|^{1/2})\) versus the propagation distance x and \(\ln (x)\) are shown in Fig. 3, respectively. At \(\lambda = 440\) nm, the linear relation of \(\ln (|I_{D}|^{1/2})\) versus x indicates the exponential decay in the regime of x from the beginning to about 4 μ m as shown in the insert of Fig. 3a, while the linear relation of \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) implies the power-law decay in the regime of \(\ln (x)\) from 1.81 (\(x\sim \) 6.12 μ m) to the end. Thus, the EM field propagates following an exponential law in the short-distance regime and a power-law in the long-distance regime (exponential decay + power-law decay).

Fig. 3
figure 3

(Color online) \(\ln (|I_{D}|^{1/2})\) versus the propagation distance x and \(\ln (x)\) for λ = 440 nm, respectively. The incident light polarization is perpendicular to the chain

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The curve of \(Re[1/\alpha _{s}-S(k,q)]\) versus ql for \(\lambda = 440\) nm is shown in Fig. 4. One can see that there are solutions \(q_{j}\) (j = 1,2...) to the equation Re[1/αs − S(k,q)] = 0. As seen from Fig. 4, the solution \(q_{j}\) is around \(\pm k\). It is related to the singularity around \(q\pm k\), which is caused by the long-range coulomb interaction and has the same origin as the Wood anomaly. For simplicity, we consider a solution \(\bar {q}\). The generalization to the case with multiple solutions is straightforward. Near \(\bar {q}\), \(1/\alpha _{s}-S(k,q)\simeq \frac {\partial Re S(k,q)}{\partial q}|_{q=\bar {q}}(q-\bar {q})+i \eta \equiv Y\), \(\eta =Im (1/\alpha _{s}-S(k,\bar {q}))\). Then, we can rewrite Eq. 5 into two parts as:

$$\begin{array}{@{}rcl@{}} D_{k}(nl)&=&E{\int}_{-\pi/l}^{\pi/l}\frac{e^{iqnl}}{Z}\frac{ldq}{2\pi} =E{\int}_{-\pi/l}^{\pi/l}\frac{e^{iqnl}}{Y}\frac{ldq}{2\pi}\\ &&+ E{\int}_{-\pi/l}^{\pi/l}\frac{e^{iqnl}}{X}\frac{ldq}{2\pi}, \end{array} $$
(6)

where \(Z = 1/\alpha _{s}-S(k,q)\), \(1/X = 1/Z-1/Y\). The first term on the right hand of Eq. 6 is in the form of \(e^{-nl/L}\) (\(L=\frac {1}{\eta }|\frac {\partial S(k,q)}{\partial q}|_{q=\bar {q}}\)), showing an exponential decay behavior. While for the second term, \(Re(X)\simeq \frac {2(\partial S/\partial q)^{2}}{\partial ^{2} S/\partial q^{2}}|_{q=\bar {q}}\) at \(q=\bar {q}\). Considering the very fast change of Z with q near \(\bar {q}\) and the regularization of the singularity of \(1/Z\) (\(1/Z \rightarrow 1/X = 1/Z-1/Y\)), the second term shows power-law decay as that in Fig. 1. We can obtain the power-law exponent as 1.29 from fitting the data in Fig. 3b and the propagation length as 1.54 μ m from fitting the data in Fig. 3a. We note that the propagation length is 500 nm for the lowest mode of Ag nanowire of radius 500 nm. One can see clearly that the inter-particle interaction leads to profound collective effect. It not only affects the propagation length (partially by adding correction to the imaginary part of \(1/\alpha _{s}\), see Eq. 6, but also brings power-law decay in the long-distance regime.

Fig. 4
figure 4

(Color online) Re[1/αs − S(k,q)] versus ql for λ = 440 nm. The incident light polarization is perpendicular to the chain

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Then, the polarization of incident light along the chain (𝜃 = 0∘) is considered. By observing Eqs. 2 and 3, we can see that the Wood anomaly-like singularity disappears and expect some changes in the propagation/transport behavior. The curves of \(\ln (|I_{D}|^{1/2})\) versus the propagation distance x and \(\ln (x)\) for \(R = 50\) nm, \(l = 180\) nm, λ = 320 nm, and 440 nm are shown in Fig. 5, respectively. The clear linear relation of \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) indicates a power-law decay of the EM fields.

Fig. 5
figure 5

(Color online) \(\ln (|I_{D}|^{1/2})\) versus the propagation distance x and \(\ln (x)\) for λ = 320 nm (a–b) and 440 nm (c–d), respectively. The incident light polarization is parallel to the chain

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As shown in Fig. 6, there is no solution to the equation \(Re[1/\alpha _{s}-S(k,q)]= 0\) for these two wavelengths \(\lambda = 320\) and 440 nm. It is quite different from the case with the polarization perpendicular to the chain (Figs. 2 and 4), which is related to the disappearance of Wood anomaly-like singularity. Moreover, the power index should also be different. From Eq. 3, the universal power-law exponent should be 2 in this case. We fit the data from Fig. 5b, d and obtain the slope \(-\)2.01 and \(-\)2.05 in the long propagation distance regime, respectively. There may also exist a small intermediate wavelength range in which there are some \(q_{0}\) to make \(Re[1/\alpha _{s}-S(k,q)]= 0\). And the propagation behavior is the exponential decay + power-law decay.

Fig. 6
figure 6

(Color online) Re[1/αs − S(k,q)] versus ql for λ = 320 nm (a) and 440 nm (b), respectively. The incident light polarization is parallel to the chain

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Here, we would like to note that when there is zero of \(Re[1/\alpha _{s}-S(k,q)]\), the system shows a combination of exponential decay and power-law decay. In the long-distance regime, the power-law decay dominates. In the short-distance regime, it is also possible that the contribution of the power-law decay term is larger than that of the exponential decay term. Then the system shows the effective power-law decay behavior, for example for the case with \(\lambda = 900\) nm.

To further support our study based on coupled-dipole analysis, we perform the FDTD simulation of the EM field propagation. In the simulation, the chain has 200 silver nanoparticles with radius of 50 nm and the lattice constant is 180 nm. A dipole source is located at a distance of \(l = 180\) nm to the first nanoparticle along the chain. The polarization of the dipole source is perpendicular/parallel to the chain. The detecting point is on the side of the nanoparticle with 2.5 nm away from the interface of the nanoparticle along the direction of the dipole. The incident field wavelengths are 320 and 440 nm. The gird is 2.5 nm. The detecting EM field intensity is \(I_{D}\). \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) (x the propagation distance) is shown in Fig. 7.

Fig. 7
figure 7

(Color online) \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) for the polarization of the source dipole perpendicular to (a, b) /parallel to (c, d) the chain with λ = 320 nm (a, c) and 440 nm (b, d), respectively. ID is the EM field intensity, and x is the propagation distance

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From Fig. 7a, b, we find two different field transport characteristics for the source dipole perpendicular to the chain. The linear relation of \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) in Fig. 7a indicates a power-law decay of EM field with the slope of –1.08 for \(\lambda = 320\) nm. Figure 3 shows an exponential decay + power-law decay of the EM field for λ = 440 nm based on CD method. Here, as shown in Fig. 7b, the linear relation of \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) is in the regime from about 1.81(\(x \sim \) 6.12 μ m) to the end, which implies an exponential decay + power-law decay of EM field with the slope of \(-\)1.35 for \(\lambda = 440\) nm. The derivation of the exponent from 1 is due to the finite fitting regime. From Fig. 7c, d, the linear relations of \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) for the the polarization of the source dipole along the chain indicate the power-law decay of EM field with the slopes of \(-\)1.99 and \(-\)2.09, respectively. The power-law exponents based on the coupled-dipole calculations and the FDTD simulations are listed in Table 1. As shown in Table 1, EM field transport characteristics and the power-law exponents from the FDTD simulations agree well with those based on coupled-dipole analysis.

Table 1 The power-law exponents from the coupled-dipole (CD) calculations and FDTD simulations
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Universality and Modulation of EM Field Propagation

We have shown that there are two phases of the EM field propagation in 1D metallic NP chains, which are determined by the analytical structure of \(1/\alpha _{s}-S(k,q)\). The parameters of the excitation field and the structure of the NP chains have important impacts on the EM field transport behavior. As discussed above, the two phases of the EM field propagation can be transformed to each other by tuning the wavelength of the excitation field.

We further explore the EM field propagation with excitation fields of different polarizations. The calculation results in Fig. 8 show that the EM field propagates according to power-law for \(\lambda = 320\) nm. The power-law exponents is 1.08 for incident field polarization angles \(30^{\circ }\), \(45^{\circ }\), \(60^{\circ }\), and \(90^{\circ }\) with respect to the direction of the chain. The power-law exponents is 1.99 for the polarization angle \(0^{\circ }\). In the presence of excitation fields with wavelength \(\lambda = 440\) nm and polarization angles \(30^{\circ }\), \(45^{\circ }\), \(60^{\circ }\), and \(90^{\circ }\), the EM field propagates following exponential law in the short-distance regime and power-law in the long-distance regime. The power-law exponents are 1.43, 1.41, 1.40, and 1.35 for polarization angles \(30^{\circ }\), \(45^{\circ }\), \(60^{\circ }\), and \(90^{\circ }\). These power-law exponents obtained by fitting the \(\ln (|I_{D}|^{1/2})-\ln (x)\) curves in the regimes of finite propagation distance show the combination power-law decay of EM field with exponents 1 and 2. In the very long propagation distance regime, the leading contribution is the power-law decay with exponent 1. For the case of incident field polarization angle \(0^{\circ }\), the EM field propagates following the power-law with exponent 2.09 (by fitting the calculation data in finite propagation distance). Therefore, by tuning the incident field (wavelength and polarization), one can achieve different EM field transport behaviors, i.e., power decay (with different exponents) and exponential decay (with different propagation length) + power-law decay. This modulation of light transport through controlling polarization has practical applications. For example, one may change the polarization state \(\alpha \) of the light before transport into another state β by wave plate. Then after the transportation of light with polarization state \(\beta \), the polarization state \(\beta \) can be changed back into state \(\alpha \). In this way, the EM field transport property of polarization state \(\beta \) has been utilized.

Fig. 8
figure 8

((Color online) \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) for the polarization angles 0∘, 30∘, 45∘, 60∘, 90∘ of the source dipole with respect to the direction of the Ag chain. aλ = 320 nm and b λ = 440 nm

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The geometric parameters of the NP chain also play an important role in the EM field transport. The calculation results of EM field propagation in metallic chains with Ag NPs of radius R = 40 , 45 , and 55 nm are shown in Fig. 9. The general power-law decay of EM field has been found for the cases with the wavelength \(\lambda = 320\) nm (incident field polarization parallel to/ perpendicular to the chain) and the wavelength \(\lambda = 440\) nm (the incident field polarization parallel to the chain) as shown in Fig. 9a, b, c. While for λ = 440 nm (incident field polarization perpendicular to the chain), the EM field transport behavior is exponential decay + power-law decay with the intensity propagation lengths of 43 ,3.88, and 0.56 μ mm, respectively. For NPs with different sizes, the contributions from the exponential decay and the power-law decay are different, which leads to the change of the transition point (from the exponential decay to the power-law decay).

Fig. 9
figure 9

(Color online) \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) for Ag chains with particles of radius R = 40 , 45 , 55 nm, respectively. The polarization of the source dipole is parallel to (a, c) /perpendicular to (b, d) the chain with λ = 320 nm (a, b) and 440 nm (c, d), respectively

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We have focused our studies on the systems with large lattice constant (l > 3R), in which the CD method works well as supported by the FDTD simulations. By using CD method, we are able to provide analytical solutions and clear physical picture/mechanism, also gives the universal power-law exponents. We then consider the cases of lattice constant smaller than \(3R\). We use FDTD method to study the propagation behavior of NP chains with \(l<3R\). In the calculation, we use the NP chain with \(R = 50\) nm, \(l = 120\) nm. Silver and Gold NPs are all considered to demonstrate the universality of the propagation property. The dielectric constants are obtained from the reference [33]. The incident field polarization is perpendicular to the chain and the wavelength \(\lambda \) is 300 nm for Ag NP chain and 514 nm for Au NP chain. \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) are shown in Fig. 10. As shown in Fig. 10, both the two lines show the linear relation with \(\ln (x)\). We fit the two lines and the two slopes are \(-\)1.38 and \(-\)1.16. These results point that the power-law decay may also appear in different metallic NP chains with different lattice constants.

Fig. 10
figure 10

(Color online) \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) for the polarization of the source dipole perpendicular to Ag chain for λ = 300 nm (a) and Au chain for λ = 514 nm (b), respectively. ID is the EM field intensity, and x is the propagation distance

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The non-exponential decay was also pointed out in the experiment with close-packed NP chain [17]. The experiment data can be replotted in the double logarithm form as shown in Fig. 11. The linear relation between \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) (x the propagation distance) indicate a power-law decay of EM field. The pow-law exponent is 1.38. Hence, our analytical and simulation methods provide new ways to clarify the propagation behaviors in 1D metallic NP chains in previous studies [17, 21, 24].

Fig. 11
figure 11

(Color online) a \(\ln (|I_{D}|^{1/2})\) versus the propogation distance x based on the the experiment data at λ = 514 nm [17]; b \(\ln (|I_{D}|^{1/2})\) versus \(\ln (x)\) for λ = 514 nm

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Conclusions

In summary, we have studied the characteristics of the LSP propagation in one-dimensional metallic nanoparticle chain based on the coupled-dipole calculations and FDTD simulations. The LSP propagation shows different behaviors in different phases, depending on the analytical structure of \(1/\alpha _{s}-S(k,q)= 0\), which is determined by both the individual NP plasmonic property and the lattice collective effect. In phase I, a power-law decay of EM field has been identified. While in phase II, EM field propagates in the form of the exponential decay + power-law decay, i.e., exponential decay in the short-distance regime and power-law decay in the long-distance regime. The universal power-law exponents have been found for EM field propagation in the long propagation distance regime. The results of FDTD simulations agree well with our theoretical predictions based on coupled-dipole method. It is found that by tuning the incident field (wavelength and polarization) and the geometric parameters of the NP chains, one can achieve different EM field transport behaviors, i.e., power decay (with different exponents) and exponential decay + power-law decay. Our studies reveal a general picture of the EM field transport in nanoparticle chains, which have applications in the plasmonic waveguide and integrated nanophotonic circuits.