Description and characterization of the complex modes in a linear chain of gold nanospheres

In this paper, complex modes in a linear chain of gold nanospheres are analyzed, accounting for metal losses. Dispersion diagrams are computed for travelling modes with both longitudinal and transverse (with respect to the array axis) polarization states. The procedure outlined in this work allows for the description of single mode evolution varying frequency, thus the modal dispersion diagrams are composed by the superposition of all the different modes in the one dimensional array. Each nanoparticle is modeled as an electric dipole, by adopting the single dipole approximation, and the complex zeroes of the homogeneous equation characterizing the field in the periodic structure are computed. The Ewald method is employed to analytically continue the periodic Green's function into the complex spectral domain and to achieve rapid convergence. Full characterization of the modes is provided in terms of their direction of propagation (forward/backward), their guidance and radiation properties (bound/leaky), the position of their wavenumber on the Riemann sheet (proper/improper), and also in terms of their possible physical excitation in the structure by a source in proximity of the array or a defect (physical/nonphysical modes). Understanding the modes excitable in this kind of structures is essential for possible applications in which the linear chain can be employed, from near-field enhancement to SERS, and innovative sensors.


INTRODUCTION AND STATEMENT OF THE PROBLEM
Periodic arrays of plasmonic nanospheres have been recently adopted in several applications in the optical field (nearfield enhancement and super-resolution [1][2][3][4][5]), in the directive radiation [6], and in the biological and chemical fields (microscopy and spectroscopy using SERS [7][8], biosensors [9][10], DNA detectors [11]). However, despite these applications in which these arrays could be used, it is still quite hard to fully control and determine characteristics with a particular set of excitation wavelengths. Indeed, several difficulties are still encountered in the analysis and interpretation of mode propagation in 1D-and 2D-periodic arrays of metallic nanospheres. The evolution and the interactions between different modes are still unclear, and based on what it has been published so far it is somehow hard to discern if a mode is physical or not. These are the reasons why it is important to perform a thorough mode analysis for this kind of arrays.
In this paper we present and interpret dispersion diagrams for linear chains of gold nanospheres. The reader is referred to [12][13][14] and references therein for detailed theoretical physical explanations and for a list of state of the art work not reported here for the sake of brevity. In the particular case of linear chains of plasmonic nanospheres, in [12][13], the numerical results were relevant to the case of silver nanospheres. In this paper, instead, we consider the case of gold nanospheres shown in Fig. 1. As described in the following sections, gold losses are larger than those of silver, the high frequency parameter of gold almost doubles the silver one and the gold plasma frequency is slightly smaller than the silver one, resulting in different modal dispersion diagrams for the chain.
The structure of the paper is as follows. The mathematical description for the simulation model adopted to perform the analyses described in this paper is briefly summarized in Sec. 2. The dispersion diagrams comprising complex bound and leaky modes in linear chains of gold nanospheres are reported in Sec. 3, considering the presence of losses in the nanoparticles, for both transversal and longitudinal polarization states. The physical interpretation of the obtained results in terms of guidance and radiation, and physical excitation, is provided as well. Conclusions are in Sec. 4. Fig. 1. Linear chain of gold nanospheres, of infinite extent, periodic along the z-direction, embedded in a homogeneous medium with permittivity h ε . r is the radius of each nanoparticle, and d is the center-to-center distance (periodicity). The reference system shown here will be adopted throughout the paper.

FORMULATION
The aim of this section is to briefly describe the formulation adopted to perform mode analysis in a linear chain of metallic nanospheres, and the criteria for determining physical bound and leaky waves. A more detailed formulation and explanation is in [12][13]15]. Notice that the formulation outlined here is general, and is adopted in the following sections to characterize a particular array case.

Definition of the simulation model
The formulation is based on the single dipole approximation (SDA) [15][16]. According to SDA, each plasmonic nanosphere of the linear chain, at optical frequencies, acts as a single-dipole and can be described by a polarization vector where e α is the electric polarizability of the nanosphere, and loc E is the local field produced by all the nanoparticles of the array except the considered nanoparticle plus the external incident field to the array [13,15].
The value of the polarizability e α of a nanoparticle depends on both its geometry and material properties [15,[17][18].
The monochromatic time harmonic convention, , is assumed and is therefore suppressed hereafter. Furthermore, the relative permittivity of the metal of each nanosphere is described by the Drude model where ∞ ε is a high-frequency fitting parameter, p ω is the plasma frequency of the metal (expressed in rad/s) and γ is the damping factor (expressed in 1 s − ).
Consider now a linear chain of nanospheres, for which each nanoparticle is placed at position nd z n = , with 0, 1, 2, n = ± ± K , and d is the period of the chain. Suppose that the array is immersed in a homogeneous background, with relative permittivity h ε , and that the array is then excited by a plane wave or by a quasi-periodic excitation with is computed by using the Ewald's method, which provides two series with Gaussian convergence and only a handful of terms are needed [21][22][23].
The mode dispersion analysis in the linear chain is then obtained by computing the complex z k -zeroes of the homogeneous linear system in (5), after having imposed no excitation source (i.e., ). Therefore, the complex zeros of the determinant of ( ) have to be computed.

Definition of Floquet spatial harmonics
The electric field in the linear chain, due to the periodicity of the array, is quasi-periodic, i.e., it is periodic except for an inter-element phase shift. Thus, it can be described as ( , where d is the period of the periodic structure, is the wavenumber along the array axis, and z y x rẑ y x A mode in the linear chain is expressed in term of a Fourier series expansion, and thus represented as the superposition of Floquet spatial harmonics where , with p the order of the Floquet spatial harmonic, and p e is the weight of the pth harmonic. For more details, the reader is referred to [12][13] and references therein.

Definition of physical modes
The physical modes that can be excited in linear chains of plasmonic nanospheres are summarized in Table 1 in [12], in Table 12.1 in [14] or in [24]. To be physical, a mode has to have a complex wavenumber, i.e., a complex value representing a zero of the determinant of ( ) , that is captured in the integration path outlined in [12][13][14]25]. Looking at the positive z-direction, in order to be captured in the path deformation, a complex wavenumber Moreover, to be physical, not only it needs to have 0 z α ≥ , but also it has to respect the conditions stated in the following. As such, considering the modal harmonic in the fundamental Brillouin zone (BZ), a physical forward wave belongs to the I quadrant of the complex z k plane, whereas a backward one to the II quadrant. Looking at a positive z-direction, z k wavenumbers in the II quadrant representing backward proper (i.e., belonging to the top Riemann sheet) waves (either bound or leaky) are physical; whereas in the leaky region of the I quadrant (i.e., above the light line in the -plane kβ ), wavenumbers representing forward improper (i.e., belonging to the bottom Riemann sheet) leaky waves are physical; last, wavenumbers in the bound region of the I quadrant of the complex z k plane (i.e., below the light line in the -plane kβ ) are physical forward proper bound waves. For more details about physical conditions, path deformation and proper/improper Riemann sheets, the reader is referred to [12][13][14] and references therein.

Definition of bound and leaky modes
A bound mode is a physical mode that has all its Floquet wavenumbers outside the visible region, or Physical guided modes (leaky or bound) travel a long distance when the imaginary part of their wavenumber is small with respect to the host wavenumber, or z k α << .

NUMERICAL SIMULATION RESULTS: MODE DISPERSION DIAGRAMS
The aim of this section is the illustration of dispersion diagrams of linear chains of gold nanospheres, in the particular case of nanoparticles with radius nm 25 = r and center-to-center periodicity nm 75 = d , embedded in free space (i.e., 1 h ε = ).
The Drude parameters to model the permittivity of gold in (2) are taken from [26], adopted also in [27]  , respectively. Although the imaginary part is not perfectly modeled above 550 THz (as can be seen in Fig.  2(b)), the results will provide a trend for the modal dispersion diagrams of linear chains of gold nanospheres.

Modes for transverse polarization
In this section, modal dispersion diagrams are shown for modes travelling along the z-direction polarized along x-or ydirection (transverse polarization, T-pol), in the case of gold nanospheres (accounting losses). The physical bound modes for T-pol considering an observer along the positive z-direction are the following. The modes 'Proper 1' and 'Proper 4' in the II quadrant (Figs. 3(a) and 5(a)) are two bound backward modes; they do not have a small imaginary part with respect to the host wavenumber k, thus they cannot travel long distances from the excitation source. The bound mode 'Proper 1' can be excited for frequency values corresponding to / kd π smaller than the one in 'A' in Fig. 4(a) (for higher frequencies it becomes a proper leaky mode), whereas the bound mode 'Proper 4' can be excited for frequency values corresponding to / kd π higher than the one in 'B' in Fig. 4(a) (at lower frequencies this mode is a proper leaky one). The mode 'Proper 3' in the I quadrant is a bound forward mode; however, it has a quite large imaginary part, thus it decays very fast while travelling. It is physical only in the slow wave region, i.e., it can be excited for frequency values corresponding to / kd π higher than the one in 'E' in Fig. 4(a). The mode 'Proper 2' in the I quadrant is a bound forward mode for frequency values corresponding to / kd π smaller than the one in 'F' in Fig. 4(a), that has the real part of the wavenumber equal to the host wavenumber k (as can be seen in Fig. 3(a), this mode lies on top of the light line); indeed, this mode has a very small positive imaginary part in this frequency range, thus it can travel very long distances from the excitation source.
The physical leaky modes for T-pol considering an observer along the positive z-direction are the following. The modes 'Improper 1' and 'Improper 4', for frequency values corresponding to / kd π between 'D' and 'G', and between 'H' and 'C' in Fig. 4(a), respectively, are radiating forward leaky modes. The 'Improper 1' does not have a small imaginary part with respect to the host wavenumber k, and thus cannot be used as a directive radiator. However, the dashed green ('Improper 4') has a smaller imaginary part than the dashed blue one, near the leaky region 0 z β ≈ , and thus it could be used for radiation. The modes 'Proper 1' and 'Proper 4' in the II quadrant, for frequency values corresponding to / kd π between 'A' and 'D', and between 'C' and 'B' in Fig. 4(a), respectively, are two radiating backward leaky modes. Again, the 'Proper 1' (solid blue) does not have a small imaginary part. However, the solid green ('Proper 4') has a smaller imaginary part than the blue one, near the leaky region 0 z β ≈ , and thus it could be used for moderate directive radiation.
The plots shown in Fig. 5 provide a complete picture of the evolution of the modes for T-pol by varying frequency in the complex z k plane. They also clarify what it has been previously stated about the definition of the physical bound and leaky waves. One can observe the total symmetry with respect to the origin. A zoom is provided in Fig. 6. For the sake of clarity, we show in Fig. 7 only the subset of the excitable modes (bound and leaky) for transverse polarization, assuming an observer along the positive z axis and, for instance, a source near the origin.
(a) (b) Fig. 7. As in Fig. 3, showing only the subset of physical modes (for an observer located on the positive z axis, and a source near the origin, for instance).

Modes for longitudinal polarization
In this section, dispersion diagrams are shown for modes travelling along the z-direction polarized along z (longitudinal polarization, L-pol), in the case of lossy gold nanospheres.
As in Sec. 3.1, the dispersion diagrams for the real and imaginary parts of the wavenumber of each mode are reported in Figs. 8(a) and 8(b), respectively, with a zoom in Fig. 9 for clarity, whereas the evolution of the modes in the complex z k plane varying frequency is presented in Fig. 10. The physical bound modes for L-pol considering an observer along the positive z-direction are the following. The mode 'Proper 1' is a bound forward mode; it has a small imaginary part when it is close to the light line in the -plane kβ , as it can be noticed from the zoom provided in Fig. 9, thus it can travel long distances from the excitation source with small decay. This mode can be excited for frequency values corresponding to / kd π higher than the one in 'A' in Fig. 9(a).
The physical leaky modes for L-pol considering an observer along the positive z-direction and, for instance, a source in proximity of the origin, are the following. The mode 'Improper 2' is a radiating forward mode for frequency values corresponding to / kd π between 'B' and 'C' in Fig. 9(a). It has a small imaginary part with respect to the host wavenumber k, as it can be noticed again from the zoom provided in Fig. 9. However, notice that this mode lies very close to the light line, as can be easily observed in Fig. 10(b); since the dipoles approximating the nanospheres are longitudinally polarized (i.e., along the z-direction), their far-field radiation pattern would behave with classical sin( ) θ shape, with a far field null corresponding to the direction along which the dipoles are aligned. Indeed, in the case of the analyzed linear chain, when z k β = the radiation should take place along the z-direction; this is not possible due to the aforementioned null, present when the structure has finite size; whereas, when z k β < for z k β ≈ , a very small radiation (due to the dipolar sinusoidal far-field behavior) could take place.
(a) (b) Fig. 10. Modes in the complex relative z k plane for L-pol, normalized with respect to (a) the period d, and (b) the host wavenumber k. Note that in (b), crossing "1" in the abscissa means crossing the light line. Arrows indicate direction of increasing frequency. Modes with negative imaginary part have a symmetric pattern with respect to those with positive imaginary part, and they are described with the same color scheme (and thus not shown as dotted lines, as was done in Figs. 8 and 9).
The plots shown in Fig. 10 provide a complete picture of the evolution of the modes for L-pol by varying frequency in the complex z k plane. They also clarify what it has been previously stated about the definition of the physical bound and leaky waves. For the sake of clarity, we show in Fig. 11 only the subset of excitable modes (physical bound and leaky modes) for longitudinal polarization, assuming an observer along the positive z axis and, for instance, a source near the origin.

CONCLUSION
In this work, modal dispersion diagrams, i.e., wavenumber versus frequency, have been computed for modes traveling in a periodic linear array (a chain) of gold nanospheres. The single dipole approximation has been adopted to model the behavior of each nanoparticle at optical frequencies, which is a good approximation when working close to the plasma resonance frequency of the nanospheres; however, more accurate results, for small period comparable with the spheres' radius, would involve multipole field contributions. Physical and nonphysical modes in a chain have been described in this work. This is the first time such a complete physical characterization in a chain of gold nanoparticles has been presented.