1. Introduction

The ability to guide optical fields through a prescribed route is critically important for realization of compact integrated optical devices and circuits. Strictly speaking, a nano-photonic circuit should have its transverse dimension in the range of sub-100 nanometers. To achieve such length scales, various strategies have been investigated such as array of quantum dots (QDs) [1], plasmonic fibers based on metallic nanowires [2]. There also has been long-standing interest in employing metallic nanosphere (MN) chains to function as optical waveguides below the diffraction limit, in which wave propagation is via near-field coupling [3–6]. The MN chains waveguides should be superior to QD waveguides in terms of crosstalk because of better field confinement generated from MN’s whirlpool-like nanoscale optical vortices [7]. Moreover, compared to the plasmonic fibers, it is more feasible from the nanofabrication viewpoint to realize energy splitter and perform assembly of optical circuit nanoelements where plasmonic and non-plasmonic nanospheres are utilized [8]. However, it is well accepted that radiation and absorption losses in MN waveguide can be serious [5].

A coupled point-dipole approximation is well accepted as a model for the electromagnetic response of such MN chains when$a/d\le 1/3$, where a is the particle radius and d the nearest neighbor center-to-center distance [3–5]. The transmission losses and the dispersion relations of both longitudinal modes (LM, excitation polarized parallel to the chain axis) and transverse modes (TM, excitation polarized normal to the chain axis) have been investigated [3–5]. However, the common feature of previous studies is that the MN waveguides are all based on hosts of either vacuum or silica [3–6], yet in practice the waveguide is generally more likely to be fabricated on substrates or hosts made of other materials. Meanwhile, a host medium, if chosen properly, can lead to optimized plasmonic energy transmission and tunability. Recently, incorporating optical gain into various plasmonic systems, including plasmonic waveguiding on planner metallic structures has been reported [9,10]. It is the objective of this paper to introduce a new design of low-loss MN waveguides based on the gain medium, which may offer real opportunities for practical nanophotonic devices.

2. Device design and theoretical discussion

We consider an infinite gain-assited Ag nanosphere chain waveguide. A schematic of the configuration and a proposed fabrication scheme are shown in Fig. 1 .

 

Fig. 1 Schematic of the Ag nanosphere waveguide. (a) to (d) depict a fabrication based on electron beam lithography. (e) is a 3-dimensional illustration of the device with geometrical parameters included.

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A potential route to realize the proposed waveguide begins with spin-coating R6G-doped polymethylmethacrylate (PMMA) substrate on a silicon/silicon dioxide substrate, then by using electron-beam lithography (EBL) to pattern trenches on the surface of R6G-doped PMMA, one can control the width of the trench and make it can just contain one coated MN in transverse dimension. The coated MN refers to chemically synthesized MN with dye-doped silica layer. One candidate for the doped dye is a composite material containing R6G [11]. This kind of core-shell nanoparticle (NP) may be readily synthesized using current solution-based chemical techniques [11,12]. The selective binding to the bottom of the PMMA trenches can be achieved by DNA-mediated self-assembly and two layer molecular self-assembly procedure as stated in Ref. 1. The precision of spacing and size of the MNs are guaranteed by the EBL and the uniformly chemical synthesized NP. Finally, the linear MN chain is buried by the depositing R6G-doped PMMA using a sol-gel process. With the pump laser (e.g. with $\lambda =532nm,t_{pulse}=10ns$for R6G to achieve the gain of 420$c{m}^{-1}$ [10]), the doped dye can induce a gain to the host environment.

To discuss the proposed waveguide, we begin with the coupled point-dipole approximation with$a=25$nm, $d=75$nm. In ideal modelling, we represent the complex permittivities of the MN and the host medium with${\epsilon }_p={{\epsilon }^{\prime }}_p+{{\epsilon }^{″}}_{p}i$and ${\epsilon }_h={{\epsilon }^{\prime }}_h+{{\epsilon }^{″}}_{h}i$ respectively. Note that compared to the practical parameters, due to the substrate effect, ${{\epsilon }^{\prime }}_h$here should be considered as the effective permittivity with its value bigger than but very close to that of the dye-doped silica, considering the near-field coupling which is the major mechanism of MN chain waveguide occurs mainly in the doped silica and PMMA. The substrate should not affect ${{\epsilon }^{″}}_h$ very much because the high gain region should be in the vicinity of MNs [13]. The induced moment on the nth dipole equals to the polarizability $\alpha (\omega )$ times the sum of electric field generated by other point-like particles [3], that is

Here m is an integer depicting the position of the mth particle along the chain, $\widehat{r}$ is the position vector pointing from the dipole to the field point, c is the speed of light. The quasi-static dipole polarizability has the form$\alpha (\omega )=a^3[\epsilon (\omega )-1)]/[\epsilon (\omega )+2]$, where$\epsilon (\omega )={\epsilon }_p/{\epsilon }_h$. For better accuracy, we use experimental data from Johnson and Christy for the dielectric response of the MN [14]. Meanwhile, we employ the modified long wavelength approximation (MLWA) to account for dynamic polarizability in a more rigorous manner [15]. In the spirit of MLWA, $1/\alpha (\omega )$ is replaced by$1/\alpha (\omega )-2{\omega }^{3}i/3c^3-{\omega }^2/a{c}^2$. For a chain of N MNs, Eq. (1) becomes a set of N coupled equations with its matrix form$MP=0$, where p is an N-rowed column vector of the dipole moments. The matrix M is defined by its elements

Here we use eigenstates of $G(\omega ):|1〉,|2〉\mathrm{...}|n〉$, to get the expansion of P, and $a_n$is an expansion coefficient, $b_n$ is the eigenvalue corresponding to nth eigenstate. Note that the completeness of those eigenvectors, forming a complete set, is different from that of conventional Hermitian operators in quantum mechanics. Here $G(\omega )$ is a complex symmetric operator, whose eigenvectors form a complete, pseudo-orthonormal basis [16]. We use bra vector $〈\overline{n}|$, where the bar denotes complex conjugation of all entries, to pre-multiply Eq. (2). In this work, this operation means a pseudo-inner product defined as the simple Euclidean type inner product [16], then we obtain $a_n=〈\overline{n}|1,\mathrm{0...0}〉/\left[a^3/\alpha (\omega )-b_n\right]$, hence

The physical meaning of Eq. (3) is quite clear. For a certain excitation external field ($\propto e^{-i\omega t}$), if proper ratio $\epsilon (\omega )={\epsilon }_p/{\epsilon }_h$ is fulfilled, the singularity in the denominator of $a_n$ results in an infinite internal field and scattering field, followed by the emergence of a zero-loss (or ultra low-loss) guiding mode. The mode number equals to the MN number as stated in Ref. 3. Hence we obtain the complex dielectric spectra rapidly with the mode-matching condition satisfied. The complex dielectric spectra for Ag NP waveguide (N = 50) at some wavelengths are shown in Fig. 2 . It is not a coincidence that the imaginary parts in Fig. 2 always have negative values. To realize low-loss energy guiding, the MN should be accompanied by a host with negative ${\epsilon }_h^{″}$ representing optical gain. In practical experiments based on the aforesaid scheme, the gain can be contributed from doped R6G. Those modes as revealed in Fig. 2(b) with values of ${\epsilon }_h^{\prime }$ from 2.25 to 3.75 are within the range of effective permittivity offered by the dielectric environment.

 

Fig. 2 Complex dielectric spectra of Ag NP waveguide (N = 50) for mode-matching condition with excitation field of different wavelengths (a): 354nm; (b): 442nm; (c): 633nm; (d): 830nm. The solid dots and the hollow dots represent results for LM and TM respectively.

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It is generally believed that the optimized transmission wavelength is very close to the plasmonic resonance wavelength of single MN [3,4,6] (~375nm for Ag NP). From Fig. 2(a), interestingly, we found that for Ag NP waveguide (N = 50) in vacuum as what has been discussed in Ref. 3, optimized TM transmission will never occur at transmission wavelength ~375nm, because ${\epsilon }_h^{\prime }$ is always less than 1.0. Although a guiding mode with complex permittivity ${\epsilon }_h=0.9892-0.1368i$ (approaches 1.0) does exist for LM, due to the considerable imaginary part, transmission performance is not very good (the decay will be addressed in later sessions).

Analysis of the eigenmodes appears in Fig. 3 . We selected four typical modes as illustration. Here the mode profiles in terms of dipole moments in Fig. 3(a) are normalized to keep the sum of the real-time electromagnetic energy stored inside those dispersive spheres (i.e. the MN chain) constant [17]. Different mode profiles induced by different oscillation phase distributions of MNs are revealed in Fig. 3(b) and 3(d). To the entire MN chain, some of the eigenmodes originate from in-phase oscillation of the MNs, such as LM1; some are from dipole oscillation, such as TM1 (LM1: ${\epsilon }_h=1.9003-0.5583i$, TM1: ${\epsilon }_h=3.3507-0.4384i$). The most common origin is multipole oscillation as demonstrated by LM2 and LM3 (LM2: ${\epsilon }_h=1.9109-0.5481i$, LM3 ${\epsilon }_h=2.9590-0.0935i$). We should point out that besides fundamental excitations that lead to propagation at the dipolar resonances of the individual particles, higher order multipole fields in the interparticle interactions will play more crucial role in MN chain based waveguides. A merit of MN chain waveguide is T or L splitting, in which two waveguides will be jointed perpendicularly for integrated nanooptical circuit [6]. The most favorable host environment for low-loss T or L splitting can also be derived using the aforementioned physical model. The ideal guiding modes, i.e. zero insertion loss, will exist with the appropriate choice of parameters. Since the main mechanism of MN waveguide is near-field coupling, one can consider that T or L splitting as two jointed waveguides. As a reasonable approximation, it can be expected that T or L splitting can be achieved by selecting a host medium that can provide good transmission performance for both TM and LM. This can be satisfied with the value of${\epsilon }_h$ located in the crossover regions of solid dots and hollow dots in Fig. 2. Meanwhile, the best splitting node can be estimated from the mode profile analysis. To the best of our knowledge, this is the first time that this issue has been discussed from this angle and we will continue to explore this point in the future.

 

Fig. 3 Analysis of eigenmodes. The excitation wavelength is 442nm. (a) (b) Normalized dipole moment and phase distribution of LM1, LM2 and TM1. (c) Normalized $P^2$of LM3 and comparison with its ideal guiding case. (d) Phase distributions of LM3.

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To evaluate the decay of the transmission by plotting out the final distribution of $P^2$, we substituted the calculated ${\epsilon }_h$into the matrix$P=M^{-1}|1,\mathrm{0...0}〉$in order to find the solutions. For the “butterfly” mode (i.e. LM3), we demonstrated that when the mode-matching condition is fulfilled exactly (${\epsilon }_h=2.9590-0.0935i$), the loss approaches zero as the comparison shown in Fig. 3(c). Note that for the $P^2$ distribution, the normalization means the first dipole moment should be normalized to 1. Figure 4(a) shows the emergence of the “butterfly” mode when ${\epsilon }_h^{″}$ varies from 0 to −0.0785 with fixed value${\epsilon }_h^{\prime }=2.9590$. It reveals that in practice some deviation from the fully satisfied mode-matching condition is tolerable for achieving low-loss transmission. The gain coefficient ${(2\pi /{\lambda }_0){\epsilon }_h^{″}/({\epsilon }_h^{\prime })}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ for the “butterfly” mode is ~${10}^{3}c{m}^{-1}$, which is within the limits of semiconductor polymers, laser dyes (highly concentrated, or absorbed onto MN) [13]. Actually, for many other modes, the required gain coefficients are all within the limits too, especially when the excitation wavelength has a red shift. In the absence of a gain medium, a material with only optimized ${\epsilon }_h^{\prime }$ can also lead to significantly reduced transmission loss as is shown in Fig. 4(b). The transmission decay of LM and TM at excitation wavelengths 633nm and 830 nm are plotted. As a comparative study, the results in Ref. 3 are also reproduced and shown for LM and TM in plasmonic resonance wavelength of single MN (~375nm for Ag).

 

Fig. 4 (a) Normalized $P^2$with different ${\epsilon }^{″}$ at excitation wavelength 442nm for the “butterfly” mode. (b) Normalized $P^2$for different dielectric host at different wavelengths. The dielectric constant for TM and LM at 633nm are 5.6143 and 7.3651; and at 830nm are 15.9860 and 15.5893 respectively. At 354nm, the medium is vacuum for both TM and LM.

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In the results from Ref. 3 in Fig. 4(b), the initial decay is rapid and non-exponential. This is mainly due to large deviation from the mode-matching condition as revealed in Fig. 2. In the terminal section of TM (354nm), energy transmission is mainly due to superposition of the far field. Even the decay is slow, the value is very small due to absence of sufficient near-field coupling. However, the transmission performance is much better with optimized dielectric host, such as for the case of LM (at 830nm,${\epsilon }_h=15.5893$) in Fig. 4(b), with a global exponential fit$P^2=P_0\exp (-x/\tau )$, where the $1/e$ damping length has a value of approximately 479nm. Note that the damping length here accounts for the global fitting instead of for arbitrarily chosen terminal section in Ref. 3. The proposed optimization can be evaluated at the output terminal of the waveguide. Compared to the results from Ref. 3, at least 3~4 orders higher energy output can be achieved at the 50th MN.

3. Conclusion

In conclusion, we have studied the gain-assisted MN chain waveguide and proposed a fabrication scheme. We demonstrate that the transmission loss can be dramatically reduced, and even completed removed in an ideal case upon choosing an appropriate gain medium. The mode-matching condition in terms of complex dielectric spectra has been identified through the pseudo-orthonormal basis expansion method. We also point out that besides dipolar resonances of individual MNs, majority of the energy guiding in the MN waveguide occurs in multipole collective oscillation mode. The proposed low-loss MN waveguide is also capable of performing efficient power splitting.