## Abstract

In this paper, the dynamic motion of surface plasmon polaritons, spatial Bloch oscillations, in a kind of nanoscale three-dimensional surface plasmon polaritons metal waveguide arrays is presented. The waveguide arrays are composed of 41 three-dimensional plasmonic waveguides with ultra-small cross section, thus the maximum lateral size of the waveguide arrays is only 6.56µm. The gradient of surface plasmon ploartions propagation constants across the waveguide arrays is realized by gradually changing the refractive index of the dielectric layer in the waveguide arrays. Theoretical results from the coupled wave theory show that surface plasmon polaritons propagate in the three-dimensional metal waveguide arrays as breathing and transverse oscillatory mode Bloch oscillations under the conditions of single and multiple waveguide excitations, respectively. All theoretical results are confirmed by finite-difference time-domain numerical simulations. Through the numerical analysis of fabrication tolerance caused by the metal strips uniform shifts, the designed three-dimensional surface plasmon polaritons metal waveguide arrays can resist certain fabrication errors.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Waveguide arrays (WGAs), a kind of discrete optical system, have attracted considerable attention, for their conspicuous performances such as discrete diffraction effect, diffraction control of light, and propagating dark/bright spatial solitons [1–4]. However, conventional dielectric WGAs are difficult to be scaled down to sub-micrometer scale or nanoscale due to the light diffraction limit. Currently, much effort is being devoted to develop surface plasmon polaritons (SPPs) based nanophotonic components and circuits because SPPs, evanescent electromagnetic waves propagating along the interface between a metal and a dielectric, have great potential in carrying optical signals in nanoscale networks [5–9]. It was demonstrated that SPPs not only exhibit a good coherent wave property [10] but also can keep the entanglement of excited photons in the propagation process [11]. So far, SPPs have also been reported in graphene-dielectric system [12], metal-graphene-dielectric system [13], Weyl semimetals [14] and topological insulators [15]. Recently, nanoscale metal-dielectric WGAs as well as graphene-dielectric WGAs, the counterparts of conventional dielectric WGAs in nanoscale, are proposed to support the SPPs propagation effect which mimics the effects of photons in dielectric WGAs in nanoscale [16–20]. In addition, the negative refraction effect [21] and deep-subwavelength focusing effect [22] are also obtained in the nanoscale metal-dielectric WGAs.

As one of the fundamental effects of quantum particles in periodic media under the action of a constant external force, Bloch oscillations (BOs) have been experimentally demonstrated by observing the electrons behavior in semiconductor superlattices [23] seven decades after the first prediction [24]. The effect of other quantum particles, such as atoms, phonons, photons [25–28], etc., has also been observed in recent years, including the photons and SPPs spatial BOs in the WGAs system [29,30]. And a kind of beam splitter and combiner are designed based on the effect of BOs [31]. However, so far, most of the investigations the SPPs spatial BOs focus on the nanoscale one-dimensional WGAs system [30–36]. Recently, a three-dimensional SPPs WGAs (TDSPPWGAs) [37] are proposed and fabricated to realize the SPPs spatial BOs. Through using the dielectric-loaded SPPs waveguides [38,39] to construct the waveguides arrays, the total lateral dimension size of the TDSPPWGAs (with 34 waveguides) is reduced to around 20 µm. In this paper, we introduce a new kind of TDSPPWGAs and reveal the spatial BOs of SPPs in TDSPPWGAs based on the coupled wave theory and further demonstrate the effects by finite-difference time-domain (FDTD) numerical simulations. The proposed TDSPPWGAs are based on a new kind of three dimensional plasmonic waveguide (TDPW) with small cross section [40], and the maximal lateral size of the TDSPPWGAs with 41 waveguides is further reduced to only 6.56 µm.

## 2. The optical properties of TDPW based TDSPPWGAs

Figure 1(a) shows the scheme of TDPW based TDSPPWGAs in the *xy* plane. The cross section of the arrays in the *yz* plane along the dashed line in Fig. 1(a) is shown in Fig. 1(b). The edge-to-edge separation distance is donated as *D*. The width and height of every Ag strips are set as 80 nm and 200 nm, respectively. The refractive index of the dielectric layer is chosen as 1.5, and the thickness of the dielectric layer is 70 nm. The structure of the TDSPPWGAs is easily fabricated, and can be realized by four steps as follows: 1) evaporating a layer of metal (the thickness is more than 200 nm) and a layer of dielectric (the thickness is 70 nm) on a substrate (such as the silicon); 2) coating a layer of photoresist (the thickness is more than 200 nm), and defining the pattern as a grating; 3) based on the grating pattern, evaporating metal (the thickness is 200 nm) again; 4) clearing the solidified photoresist.

Figure 2(a)–2(f) present the FDTD simulated filed distributions of SPPs in the TDSPPWGAs with 41guides as only SPPs modes are coupled into the center waveguide, for *D *= 40 nm, 60 nm, 80 nm, 100 nm, 120 nm, and 140 nm, respectively, at wavelength of 1550 nm (the permittivity of Ag is −129 + 3.23*i* [41]). As the light propagates along the waveguides, the energy spreads into two main lobes with several secondary peaks between them. The field propagating in the *m*th waveguide obeys the function [1]:

*a*

_{0}(0) = 1 and

*a*

_{m}_{≠0}(0) = 0, where

*a*,

_{m}*β*and

*C*are field intensity, the propagation constant, and the coupling coefficient between two adjacent guides, respectively. And

*J*is the Bessel function of order

_{m}*m*(

*m*is an integer).

Then the *C* of the TDSPPWGAs with *D* of 40 nm, 60 nm, 80 nm, 100 nm, 120 nm, and 140 nm can be obtained as 4.0×10^{5} m^{−1}, 4.9×10^{5} m^{−1}, 5.4×10^{5} m^{−1}, 5.0×10^{5} m^{−1}, 4.8× 10^{5} m^{−1}, and 4.5×10^{5} m^{−1}, respectively, by comparing the simulated field distributions and theoretical results. Figure 2(g) displays the curve of the coupling coefficient depending on the edge-to-edge distance. One can find that the coupling coefficient doesn’t decrease linearly with the increase of *D*, however, the coupling coefficient increases with the increase of *D* while it is smaller than 80 nm, then the coupling coefficient decreases with the increase of *D* as it is larger than 80 nm. Such TDPW-based TDSPPWGAs can find interesting applications in such fields as discrete array imaging and controlling the flow of electromagnetic wave in the nanoscale domain. In the next sections, the TDSPPWGAs will be used to realize plasmonic BO effects.

## 3. Theoretic analysis of the BOs effects in TDSPPWGAs

TO observe BOs, it entails an effective refractive index (*n*_{eff}) gradient across the TDSPPWGAs [along the y direction as shown in Fig. 1(a)] which serves as the external electric field on a semiconductor crystal to obtain the electronic BOs effect [23]. The index gradient stems from a variation of propagation constant of SPPs across the waveguide arrays. In terms of the relation of effective refractive index *n*_{eff}=*β*/*k*_{0} (here, *k*_{0} is the wave vector of light in air), a gradient of propagation constant of SPPs across the TDSPPWGAs can be realized by gradually varying the refractive index of the dielectric layer in each guide region with fixed thickness of dielectric layer (70 nm), since the propagation constants (*β*) of SPPs in the TDSPPWGAs depend on the thickness or the refractive index (*n*) of the dielectric layer.

Figure 3(a) shows the cross section of a hetero-TDSPPWGAs with linear increase of refractive index of the dielectric layer from guide(−20) to guide(20) by a step of Δ*n*. The coupled wave theory under the paraxial approximation is applied to investigate the SPPs propagating in the hetero-TDSPPWGAs. Because coupling coefficient of SPPs between the adjacent waveguides of the hetero-TDSPPWGAs is not a constant, the field distributions of SPPs in the hetero-TDSPPWGAs can be read as the following differential equation in the case of ignoring the higher-order coupling of SPPs in the arrays [30]:

*a*(

_{m}*x*) represents the amplitude of electromagnetic field in the

*m*

^{th}-waveguide,

*C*

_{m}_{−1,m}and

*C*

_{m}_{,m+1}are the coupling coefficients between the (

*m*−1)

^{th}- and

*m*

^{th}-guides, and the

*m*

^{th}- and (

*m*+ 1)

^{th}-guides, respectively,

*β*is the propagation constant of SPPs in the

_{m}*m*-th guide, and

*m*is an integer ranging from −20 to 20. Since

*C*

_{m}_{−1,m},

*C*

_{m}_{,m+1}and

*β*are independent of

_{m}*x*in this arrays, so Eq. (3) is typically a constant coefficient linear homogeneous differential equations set. The coefficient matrix of Eq. (3) in this case is simply a matrix:

The general solution of Eq. (3) has the form [30]:

where*χ*is the

_{j}*j*

^{th}eigenvalue of the matrix

*M*,

*Q*is the eigenvector corresponding to

_{j}*χ*, and

_{j}*η*is the coefficient determined by the boundary conditions, and

_{j}*j*is an integer ranging from −20 to 20.

As the refractive index of dielectric in the −20^{th}- guide (*n*_{−20}) of the hetero-TDSPPWGAs is set as 1.5 and the Δ*n* = 0.05, Fig. 3(b) shows the curves of *C _{m}*

_{,m+1}and

*β*depending on the guide numbers at wavelength of 1550 nm with guide to guide distance (

_{m}*D*) of 80 nm. Here,

*C*

_{m}_{,m+1}is approximately obtained as (

*C*+

_{m}*C*

_{m}_{+1})/2, where

*C*or

_{m}*C*

_{m}_{+1}is the coupling coefficient of the

*m*

^{th}- or (

*m*+ 1)

^{th}- homogeneous TDSPPWGAs as shown in Fig. 1 (the total number of the assumed homogeneous TDSPPWGAs is 41) of which the refractive index of the dielectric layer is same as that of in the

*m*

^{th}

*-*or (

*m*+ 1)

^{th}- guide of the hetero-TDSPPWGAs [as shown in Fig. 3(a)]. And

*β*is equal to the propagation constant of the

_{m}*m-*th homogeneous TDSPPWGAs. Furthermore, it can be observed the core region of every guide in the TDSPPWGAs is a metal gap waveguide (MGW) structure, and the cladding region of every guide (namely the coupling region in the TDSPPWGAs) in the TDSPPWGAs is a metal-dielectric-air waveguide (MDAW) structure. The confining SPPs ability in the MGW structure is more powerful than that in MDAW structure, therefore the properties (including propagation constant and the effective width of the propagating wave) of SPPs propagating in the TDPW composed of MGW structure and MDAW structure mainly depends on the core region [40,42], thus the propagation constant of the homogeneous TDSPPWGAs is obtained approximately from the properties of the core region of waveguides. From Fig. 3(b), it can be observed that

*C*

_{m}_{,m+1}increase with the increase of refractive index of dielectric layer, and

*β*is nearly linear increasing with the increase of refractive index of the dielectric layer with a step (Δ

_{m}*β*) of 0.0664

*k*

_{0}. By the same method, the

*C*

_{m}_{,m+1}and

*β*of other wavelength such as 1350 nm, 1450 nm, 1650 nm, and 1750 nm are obtained, and the Δ

_{m}*β*for 1350 nm, 1450 nm, 1650 nm, and 1750 nm are 0.0676

*k*

_{0}, 0.0670

*k*

_{0}, 0.0659

*k*

_{0}and 0.0657

*k*

_{0}, respectively. In the calculations, the permittivity of Ag at wavelength of 1350 nm, 1450 nm, 1650 nm, and 1750 nm are −95.4 + 2.43

*i*, −111 + 2.83

*i*, −147 + 3.80

*i*, −165 + 4.60

*i*, respectively [41].

When the coupling coefficients of a hetero-WGAs with an effective refractive index gradient transversely across the arrays is nearly invariable (such as dielectric WGAs), the period of BOs (*P*_{BO}) can be estimated by [29]:

*P*

_{BO}in the hetero-TDSPPWGAs at wavelength of 1350 nm, 1450 nm, 1550 nm, 1650 nm, and 1750 nm, the

*P*

_{BO}can be gotten as 20.0 µm, 21.6 µm, 23.3 µm, 25.0 µm, and 26.6 µm, respectively. In the case of single waveguide excitation, the incident light is assumed to incident onto the 0-th guide. Then the boundary condition can be written as

*a*

_{m=}_{0}(0) = 1 and

*a*

_{m}_{≠0}(0) = 0. From Eq. (5), the field distributions of SPPs in the hetero-TDSPPWGAs are acquired and shown in Fig. 4(a)–4(e) for the 5 different wavelengths. It can be seen that the excited SPPs couple and spread between waveguides, and the SPP field reassembles at 0

^{th}- guide after propagating a distance, showing a typical periodic breathing mode of the BO effects [29,30]. The field distributions show a little asymmetry around the 0

^{th}- guide, because

*C*

_{m}_{,m+1}in the hetero-TDSPPWGAs is not a constant. The breathing mode

*P*

_{BO}are read as 20.0 µm, 21.8 µm, 22.6 µm, 24.4 µm, and 25.9 µm for the wavelength of 1350 nm, 1450 nm, 1550 nm, 1650 nm, and 1750 nm, respectively, which are very close to the corresponding values gotten from the Eq. (6).

Figure 4(f)–4(j) show the field distributions of SPPs in the hetero-TDSPPWGAs under the condition of multiple waveguides excitations for the wavelength of 1350 nm, 1450 nm, 1550 nm, 1650 nm, and 1750 nm, respectively. In this case, a Gaussian beam with a full width at half maximum of 11 guide widths of the hetero-TDSPPWGAs is assumed to incident onto the arrays from the −16^{th}-guide to the −6^{th}-guide. The boundary conditions are *a _{m}*(0) = exp[-(

*m*+ 11)

^{2}/11

^{2}] (

*m*= −16, −15, … −7, −6) and

*a*(0) = 0 (

_{m}*m*< −16,

*m*> −6). The field distributions in Fig. 4(f)–4(j) exhibit periodic movement with no motion of the beam center, which is characteristic transverse oscillatory mode of BO effects. The

*P*

_{BO}of transverse oscillatory mode are read as 20.0 µm, 22.0 µm, 23.1 µm, 24.9 µm, and 26.5 µm for the wavelength of 1350 nm, 1450 nm, 1550 nm, 1650 nm, and 1750 nm, respectively, and are only a little differences with the corresponding

*P*

_{BO}of breathing mode.

It must be mentioned that the needed total number of waveguides (*N*_{w}) of the hetero-TDSPPWGAs can be estimated by the value of $8\overline C /\Delta \beta $, here, $\overline C $ is the average value of the *C*_{m,m+1} of the hetero-TDSPPWGAs. The *N*_{w} for the wavelength of 1350 nm, 1550 nm, and 1750 nm are estimated as 28, 34, 38, respectively. So the 41 waveguides hetero-TDSPPWGAs are designed. If the Δ*n* < 0.05, then the hetero-TDSPPWGAs with more than 41 waveguides is needed.

## 4. FDTD simulated results of SPPs BOs effect in the hetero-TDSPPWGAs

The three dimensional FDTD numerical method is applied to simulate the behavior of SPPs propagating in the designed hetero-TDSPPWGAs to illustrate the above theoretical results. In the FDTD simulations, a TM-polarized incident light (magnetic field parallel to the *y* axis in the *yz* plane) is used to excite propagating SPPs in the TDSPPWGAs, and the size of cells in the *x*, *y*, and *z* directions are set as Δ*x* = Δ*y* = Δ*z* = 10 nm, and the time step is set as Δ*t* = Δ*x*/(2*c*), where *c* is the velocity of light in the vacuum. The field distributions in the case of a single waveguide excitation for the five wavelengths are shown in Fig. 5. It can be observed that SPPs are excited at the 0^{th}-guide, and propagate along the TDSPPWGAs, at the same time, they experience a discrete diffraction and spread across over several guides. And then, the diffracted SPPs are refocused into the 0^{th}-guide after propagating a certain distance. The simulated SPPs distributions in the TDSPPWGAs also show asymmetric features, which are in agreement with that obtained by the coupled wave theory [Fig. 4(a)–4(e)]. The simulated breathing mode *P*_{BO} are 17.3 µm, 19.2 µm, 21.3 µm, 23.9 µm, and 26.7 µm for the wavelengths of 1350 nm, 1450 nm, 1550 nm, 1650 nm, and 1750 nm, respectively.

Figure 6(a)–6(e) show the simulated field distributions of SPPs in the hetero- TDSPPWGAs with incident light illuminating on multiple waveguides (from the −16^{th} to −6^{th} guides). The field distributions also display the transverse oscillatory mode BOs effect, which are in agreement with the theoretical ones [Fig. 4(f)–4(j)]. The *P*_{BO} is 18.9 µm, 21.0 µm, 23.2 µm, 26.5 µm, and 28.4 µm for the wavelengths of 1350 nm, 1450 nm, 1550 nm, 1650 nm, and 1750 nm, respectively. The SPPs are excited in the guides with relatively low effective refractive index, and propagate along in the TDSPPWGAs, simultaneously, spread to the guides with relatively high effective refractive index, then they are reflected to low index guides by Bragg reflection effect. The trajectory of the SPPs propagating in the TDSPPWGAs is similar to that of photons in the dielectric WGAs [29], and is opposite to that of SPPs in the nano metal-dielectric WGAs [30]. From Fig. 5 and Fig. 6, it can be seen that the simulated results well duplicate the theoretically estimated results by the coupled wave theory as shown in Fig. 4, but the plasmonic wave damps along the *x* axis and gradually disappears as the propagation distance is larger than 25 µm, which is mainly due to the intrinsic loss of metal and scatter loss of wave coupling between waveguides.

Figure 6(f) shows the curves of the dependence of theoretical and FDTD simulated *P*_{BO} on the wavelengths incident light. The two kinds of results are very close, but still have a little differences due to the approximate estimation of the *C _{m}*

_{,m+1}and

*β*in theoretical analysis and the variation of the coupling coefficient across the hetero-TDSPPWGAs. The plasmonic wave in the breathing mode covers transversely nearly $8\overline C /\Delta \beta $ waveguides, and in the transverse oscillatory mode covers transversely nearly $4\overline C /\Delta \beta $ waveguides. Because

_{m}*C*

_{m}_{,m+1}is not a constant in the hetero-TDSPPWGAs, so there is a difference of the

*P*

_{BO}between the breathing mode and transverse oscillatory mode.

## 5. Discussion

Though the similar work has been studied in dielectric and metallic nanoscale WGAs previously, those work mainly focus on the nanoscale one-dimensional WGAs system [30–36]. A kind of TDSPPWGAs with 34 waveguides is also proposed and fabricated to realize the SPPs spatial BOs with the total lateral dimension size of around 20 µm [37]. As the cross section of the designed hetero-TDSPPWGAs with 41 guides is shown in Fig. 3(a), every core region of guide along the *y* axis is 80 nm, and *D *= 80 nm, so the total lateral size along the *y* axis of the designed is only 6.56 µm, which is much smaller that of the previous TDSPPWGAs [37]. Thus, the designed TDSPPWGAs is a structure with very small lateral size to realize SPPs BOs in three-dimensional WGAs. And as a three-dimensional nanoscale photonic structure, it is easily integrated with other photonic devices.

The variation of refractive index of the dielectric layer from the −20^{th}-guide to 20^{th}-guide can be realized by porous silicon with precision etching [43,44]. However, in the fabrication of the hetero-TDSPPWGAs, the metal strips may uniformly shift from the center of the under dielectric layer along the *y* axis. Figure 7(b)–7(d) show the three kinds of possible extreme shifts of the metal strips comparing with Fig. 7(a). Figure 7(b) and 7(c) show that the metal strips shift uniformly 40 nm along the *y* axis to the right or left side of the corresponding under dielectric layer, respectively. Figure 7(d) shows that the metal strips shift uniformly 80 nm along the *y* axis to the center of adjacent two dielectric layers. Other types of uniform metal strips shifts are included in the three kinds of extreme uniform shifts of the metal strips. Figure 7(e)–7(g) display the FDTD simulated results corresponding to the cases of Fig. 7 (b)–7(d), respectively, with incident light of 1550 nm. It can be seen the field distribution also exhibit as transverse oscillatory mode BOs shown in Fig. 6(c), and the *P*_{BO} are also invariable, which means that the hetero-TDSPPWGAs can resist certain fabrication errors caused by the uniform shifts of metal strips. On the other hand, the fabrication of the dielectric layer with a gradient refractive index may also bring out fabrication errors, which are not numerically analyzed in the paper and can be easily theoretically discussed by the coupled wave theory. As we suggest using porous silicon [43,44] as the dielectric layer, the fabrication errors of the dielectric layer should be reduced by precisely controlling the etching time and controlling the moving step distance (160 nm) in practice.

## 6. Conclusions

In summary, a kind of nanoscale TDSPPWGAs to observe spatial plasmonic BOs of SPPs are presented. The gradient of SPP propagation constants across the TDSPPWGAs is realized by gradually changing the refractive index of dielectric layer with fixed thickness of dielectric layer in the arrays. Both results from the coupled wave theory and from FDTD numerical simulations agree well with each other and show that BO effects of SPPs in the TDSPPWGAs manifest themselves as periodic breathing and transverse oscillatory mode motion under the conditions of single and multiple waveguide excitations, respectively. Although the maximum lateral size of the designed TDSPPWGAs composed of 41 ultra-small cross section TDPWs is only 6.56 µm, however, through the numerical analysis of the fabrication tolerance caused by the metal strips uniform shifts, the designed TDSPPWGAs can resist certain fabrication errors.

## Funding

National Natural Science Foundation of China (61205166, 61575145).

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