Impact of the Excitation Source and Plasmonic Material on Cylindrical Active Coated Nano-Particles

Electromagnetic properties of cylindrical active coated nano-particles comprised of a silica nano-cylinder core layered with a plasmonic concentric nano-shell are investigated for potential nano-sensor applications. Particular attention is devoted to the near-field properties of these particles, as well as to their far-field radiation characteristics, in the presence of an electric or a magnetic line source. A constant frequency canonical gain model is used to account for the gain introduced in the dielectric part of the nano-particle, whereas three different plasmonic materials (silver, gold, and copper) are employed and compared for the nano-shell layers.


Introduction
During the past decade considerable efforts have been devoted to the broad field of metamaterials. The areas of interest encompass potential applications spanning from the microwave [1] to the optical frequencies [2]. As regards the latter, among the vast amount of interesting discoveries, notable OPEN ACCESS attention has been devoted to the design of metamaterials and their applications which incorporate active media with plasmonic materials [3,4]. In particular it has been shown that a properly designed active coated nano-particle (CNP) can lead to novel resonance and transparency effects as the intrinsic losses inherent to the plasmonic materials are overcome by suitable gain impregnation of the CNPs. In the majority of the studies of these effects, the CNPs were taken to be of a spherical shape; this was likewise most often the case in several of the previous studies of small particles with gain [5][6][7].
The properties of the passive coated cylindrical particles in the presence of an arbitrarily located electric line source were thoroughly examined in [8]. The feasibility of a similar configuration to provide resonant active CNPs in the presence of electric as well as magnetic line sources was briefly investigated in a recent work [9]. In there, several super-resonant active CNPs were proposed offering significant enhancements of the radiated power due to a strong excitation of the dipole mode inside the particle. The present work reports an extension of the studies in [9] and includes a more detailed account of the analytical results and, moreover, provides additional numerical results illustrating the super-resonant properties of the investigated CNPs and some of their differences relative to the spherical active CNPs studied in [7]. The CNPs in this work are made of specific dielectric materials, while silver, gold and copper are employed as the plasmonic materials to quantify the amount of gain needed to overcome the losses in the different cases. The gain model employed in the entire analysis is a canonical, single frequency model, and its values are used for different plasmonic materials. The analysis of the cylindrical active CNPs is conducted with a thorough investigation of their near-and far-field properties for altering locations of the ELS. It will clarify the potential application of these CNPs for nano-sensor applications. Specifically, the manuscript is organized as follows: Section 2 introduces the CNP configuration and presents the most important analytical results, while Section 3 discusses the gain and plasmonic material models used. In Section 4 the numerical results are presented and discussed for both electric and magnetic line source illumination of the CNPs. The entire work is summarized and concluded in Section 5. Throughout this work, the time factor exp(jωt), with ω being the angular frequency and t being the time, is assumed and suppressed.

Configuration
The CNP configuration is depicted in Figure 1. It consists of a cylindrical nano-core (region 1) with radius ρ 1 , covered with a concentric cylindrical nano-shell (region 2) with outer radius ρ 2 . The host medium (region 3) of the CNP is that of free-space with the permittivity, ε 0 , permeability, μ 0 , and wave number, k 0 = = 2π/λ, where λ is the free space wavelength. The CNP is excited by an arbitrarily located line source for which two polarizations are considered: the TM (with respect to z) polarization, in which case the source is an electric line source (ELS) with constant electric current I e [A/m], and the TE (with respect to z) polarization, in which case the source is a magnetic line source (MLS) with constant magnetic current I m [V/m]. The two regions 1 and 2 consist of simple, lossy materials with the permittivity, ε i = ε i ' − jε i ", permeability, μ i = μ 0 , and wave number k i = , i = 1 and 2, where the branch of the square root will be discussed below. A cylindrical coordinate system (ρ, φ, z) and an associated rectangular coordinate system (x, y, z) are introduced such that their origins coincide with the center of the CNP. The coordinates of the observation point are (ρ, φ), and those of the line source are (ρ s , φ s ).

Figure 1.
The configuration of a cylindrical CNP excited by either an electric or a magnetic line source.

Theory
The analytical solution to the problem depicted in Figure 1 is rather straightforward to obtain and it has been outlined in detail in [8]. For the purposes of the present work we only present its main points. For both polarizations, the field due to the line source constitutes the known incident field and it is expanded in terms of cylindrical wave functions. The unknown fields due to the CNP in the three regions are likewise expanded in terms of cylindrical wave functions and they involve the unknown expansion coefficients , for TM polarization, and , for TE polarization. For both sets of coefficients, i = 1 for the field in region 1, i = 2 and 3 for the field in region 2, and i = 4 for the field in region 3, while the symbol n is the mode number with n = 0 referring to the monopole mode in the expansion, n = 1 to the dipole mode, etc. for the other modes. The unknown expansion coefficients depend on the location of the line source, and are easily obtained by enforcing the boundary conditions on the two cylindrical interfaces, ρ = ρ 1 and ρ = ρ 2 , for all values of φ.
For the purpose of our investigations of the electromagnetic properties of the CNP excited by an ELS or a MLS, the normalized radiation resistance (NRR) is examined along with the spatial distribution of the electric and magnetic fields as well as the directivity patterns. The NRR is the ratio of the radiation resistance of the line source in the presence of the CNP to its value in the absence of the CNP. In particular, with P CNP representing the total average power radiated by either the ELS or MLS in the presence of the CNP, the corresponding radiation resistance, P CNP , is defined by: for a given constant value of the current I along the line source, where I = I e (I m ) for TM (TE) polarization. In a similar manner, the radiation resistance, R LS , of either the ELS or MLS radiating alone in free space for the same current I along the source is defined by: with P LS representing the total average power radiated by either the ELS or MLS alone in free space. It is noted that the explicit expressions for P CNP and P LS can be found in [8] for the TM polarization case. In mathematical terms, the NRR is thus given by: for TM polarization, and: for TE polarization. In Equations (1) and (2), J n (k 0 ρ s ) is the Bessel function of order n, τ n is the Neumann number, i.e., τ n = 1 for the n = 0 mode and τ n = 2 otherwise, while N max is the truncation limit in the implementation of the exact infinite summation and is chosen to ensure the convergence of the expansion in Equation (1). The directivity, D, defined as the ratio of the radiation intensity to the total average power per unit angle, can be expressed as:

Gain and Material Models
The present work examines three different CNPs. For each of them, region 1, the cylindrical nano-core, is composed of silica-oxide (SiO 2 ) while three different plasmonic materials are considered for region 2, the nano-shell: silver (Ag), gold (Au) and Copper (Cu). The corresponding structures are referred to as the Ag-, Au-, and Cu-based cylindrical CNPs. The radius of the nano-core for all CNPs is set to r 1 = 24 nm, while the outer radius of the nano-shell is set to r 2 = 30 nm, resulting in a 6 nm thick plasmonic nano-shell. This choice matches the spherical active CNP cases considered in [3,4,7].
The permittivity, ε 1 , of the silica nano-cylinder is comprised of a contribution from its refractive index in the frequency region of interest (n = √2.05) and a contribution from the canonical gain model. It is thus expressed as: where κ determines the nature of the nano-cylinder which is lossless and passive for κ = 0, lossy and passive for κ > 0, and active for κ < 0. Thus the amount of gain introduced in the CNP configuration to overcome the plasmonic material losses is tailored by the choice of the parameter κ.
As to the permittivity, ε 2 , of the plasmonic nano-shell we note that its size dependency must be taken into account due to the nano-scale dimension of the CNPs. To this end, empirically determined bulk values of Ag, Au and Cu permittivities have been employed [3] and their real parts, ε 2 ', normalized with the free-space permittivity ε 0 , are shown in Figure 2 for a 6 nm thick Ag, Au, and Cu nano-shell along with the associated values of their loss tangents defined by LT = ε 2 "/|ε 2 '| as functions of the excitation wavelength.
As observed in Figure 2, the real part of the permittivity of the various plasmonic materials under consideration is negative in the depicted wavelength range, and moreover that they all are dispersive and lossy with Ag being the least lossy case.

Results and Discussion
We first present and discuss the results pertaining to the TM polarization cases. Subsequently, those for the TE polarization cases are discussed. Throughout the following numerical investigations, the line currents of the corresponding sources are set to I e = 1 [A/m] for the ELS and I m = 1 [V/m] for the MLS. Figure 3 shows the NRR as a function of the excitation wavelength, λ, for the Ag-based CNP for different values of the parameter κ. The ELS is located in region 1 at (ρ s , φ s ) = (12 nm, 0°). It is rather obvious that for this particular polarization, no resonance, i.e., no large values of the NRR, is in evidence, regardless of the value of κ. The quantity NRR ≈ 0.3 dB for κ = 0 at λ = 324.6 nm, while NRR ≈ 3.13 dB for κ = −0.87 at λ = 320 nm. This is an expected result since the resonance condition [8,10]: In both cases, the field is dominated by that of the ELS alone, i.e., the CNP does not exhibit any influence on the ELS or the local field distribution. Although not shown, similar results were found for the Au-and Cu-based CNPs. Figure 5 shows the NRR as a function of the excitation wavelength, λ, for the three CNPs with (a) κ = 0 and (b) the corresponding super-resonant states which occur with κ = −0.175, κ = −0.262 and κ = −0.310, respectively, for the Ag-, Au-, and Cu-based CNPs. In all cases, the MLS is in region 1 at (ρ s , φ s ) = (12 nm, 0°). For the super-resonant states the NRR values are significantly increased and the intrinsic losses of the plasmonic materials are vastly overcome relative to the passive CNP results reported in Figure 5(a). The NRR values, as well as the values of the parameter κ needed for the super-resonance to occur, are summarized in Table 1 for the three CNPs along with the values of the wavelength λ at which the respective resonances are attained. From Table 1 it follows that the magnitude of the parameter κ needed to attain the super-resonant states is largest for the Cu-based CNP. This is an expected result since copper has the highest losses of the three plasmonic materials utilized in the design of present CNPs, cf., Figure 2      The field is ylindrical su alized to th 56; +0.156] discussed th eir far-field for the supe 2 nm, 0°

Conclusio
Electrom cylindrical s The source plasmonic m While no resonant phenomenon was observed for the case of the electric line source illumination, for the passive as well as active configurations, it was demonstrated that the inclusion of gain in the silica nano-core significantly helps to overcome the intrinsic losses of the plasmonic nano-shells for the case of magnetic line source illumination. In particular, super-resonant states were identified for the silver-, gold-, and copper-based CNPs and their large enhancements of the normalized radiation resistance were quantified. These enhancements were shown to be due to a strong excitation of the dipole mode inside the respective particles, and were largest for the source locations near-by the inner surface of the nano-shells, while being heavily diminished for source locations near to the center of the particles, in agreement with similar findings for the purely passive cases [8]. The far-field radiation patterns further confirmed this dipole mode behavior. The amount of gain required in the three cases was different and was found, as expected, to be largest for the most lossy material. The existence of the super-resonances for the case of the magnetic line source illumination and their absence for the electric line source case was found to be in agreement with the predictions of certain conditions for resonances derived in literature for electrically small metamaterial-based structures. Moreover, the present results provided the basis for an interesting comparison between the currently investigated cylindrical active coated nano-particles and the corresponding spherical ones studied thoroughly in [7]. For the super-resonances to occur a lower amount of gain is needed in the cylindrical than in the spherical cases; the super resonances for the former structures occur at larger wavelengths and result in lower enhancements for the examined range of parameters. In addition, the cylindrical active CNPs were found only to lead to significant enhancements of the radiated power for magnetic source locations outside the CNPs, whereas the corresponding spherical CNPs are known to lead to both enhanced as well as reduced radiation effects as the source is moved outside the CNPs. The illustrated enhancements of the source fields for the TE polarization could have significant impact on their use for nano-sensor applications.