Design of Power Splitters Based on Hybrid Plasmonic Waveguides

: Plasmonic power splitters based on hybrid plasmonic waveguides (HPWs) are proposed and investigated. The HPW consists of a high-permittivity semiconductor nanowire embedded in a SiO 2 dielectric ﬁlm near a metal surface. The propagation behaviors of Surface Plasmon Polaritons (SPPs) in HPWs are numerically simulated by the 3D ﬁnite-difference time-domain (FDTD) method. The incident ﬁeld is transferred from the middle waveguide to the waveguides on both sides due to the coupling between adjacent waveguides. The intensity distributions can be explained by the multimode interference of SPPs supermodes. According to the ﬁeld intensity distribution of ﬁve HPWs, we design a 1 × 3 power splitter and a 1 × 2 power splitter by reducing the length of some speciﬁc waveguides.


Introduction
Surface Plasmon Polariton (SPP) waveguides can break the diffraction limitation and are regarded as the best candidate to design miniaturized photonic integrated circuits [1,2]. All kinds of plasmonic waveguides have been designed, such as the dielectric loaded metal, metal wedges, slot and gap waveguides, and hybrid plasmonic waveguides [3,4]. Among these plasmonic waveguides, the hybrid plasmonic waveguide (HPW) is one of the most promising alternatives for on-chip nanophotonics due to the ability of subwavelength confinement and long propagation lengths [3].
In photonic integrated circuits, a power splitter is an essential component, which can distribute the energy of incident light to different propagation directions. Various plasmonic power splitters have been proposed, such as 1 × 2 splitters [5][6][7][8], 1 × 3 splitters [9], 1 × N splitters [10][11][12], and variable transmission splitters [13][14][15]. The overwhelming majority of these plasmonic splitters are designed on the basis of metal-insulator-metal (MIM) waveguides, which are composed of two metallic layers and a dielectric layer [5][6][7][8][9][10][11][12][13][14][15]. However, there are few researches on the power splitter based on HPWs. The main reason is that an MIM waveguide could be simplified to a two-dimensional structure and a HPW cannot be simplified to a two-dimensional structure, of which the computational complexity of a 3D structure is much larger than that of a 2D structure. Due to the excellent performance of the HPWs, it is necessary to design a 3D power splitter based on HPWs for the next generation of integrated photonic circuits.
In this paper, we present several new power splitters based on hybrid plasmonic waveguides. The propagation behaviors of Surface Plasmon Polaritons (SPPs) in HPWs are numerically simulated by the 3D finite-difference time-domain (FDTD) method. According to the field distribution of five HPWs, we reduce the length of some specific waveguides and design different types of power splitters. The power splitters based on HPWs are fully compatible with semiconductor fabrication techniques and can be widely used in integrated photonics structures.

Materials and Methods
The hybrid mode in the single HPW was discussed in detail in ref. [3]. Here, we considered a coupled system composed of five HPWs, shown in Figure 1. The HPW consists of a high-permittivity semiconductor nanowire embedded in a low-permittivity dielectric near a metal surface [3]. The semiconductor nanowire was GaAs, of which the relative permittivity is 12.25. The low-permittivity dielectric was SiO 2 , with relative permittivity 2.25. The relative permittivity of Ag was taken to be the measured value of ε = −129 + 3.3j at the telecommunications wavelength λ = 1550 nm [16]. The diameter of the GaAs nanowire was d = 200 nm and the dielectric gap was h = 50 nm. The distance between adjacent waveguides was S = 500 nm. The plasmonic mode supported by a single HPW is shown in Figure 1c, which was consistent with the results of ref. [3].
waveguides and design different types of power splitters. The power splitters b HPWs are fully compatible with semiconductor fabrication techniques and can b used in integrated photonics structures.

Materials and Methods
The hybrid mode in the single HPW was discussed in detail in ref. [3]. Here, sidered a coupled system composed of five HPWs, shown in Figure 1. The HPW of a high-permittivity semiconductor nanowire embedded in a low-permittivity d near a metal surface [3]. The semiconductor nanowire was GaAs, of which the permittivity is 12.25. The low-permittivity dielectric was SiO2, with relative perm 2.25. The relative permittivity of Ag was taken to be the measured value of ε = 3.3j at the telecommunications wavelength λ = 1550 nm [16]. The diameter of th nanowire was d = 200 nm and the dielectric gap was h = 50 nm. The distance adjacent waveguides was S = 500 nm. The plasmonic mode supported by a single shown in Figure 1(c), which was consistent with the results of ref. [3]. We numerically simulated the propagation behavior of SPPs in five HPWs b FDTD method. The grid size was dx = dz = 10 nm and the time increment was Δt A perfectly matching layer was adopted to absorb the outgoing electromagnet around the computational domain.

Results and Discussion
The plasmonic mode was excited by a Gaussian beam, which is normally onto the end of the middle waveguide [17]. The arrangement of the five HPWsi in Figure 2a. The intensity distributions in the XZ plane are shown in Figure 2b, w pattern was very complex due to multimode interference effects [18]. In ref. 18, t ciples and properties of multimode interference devices are presented for conv dielectric waveguides. To the device of five HPWs, supermodes were formed du coupling between adjacent waveguides [19,20]. Customarily, supermodes are th of propagation of a periodic array of waveguides, which consists of N identica guides [20]. We numerically simulated the propagation behavior of SPPs in five HPWs by the 3D FDTD method. The grid size was dx = dz = 10 nm and the time increment was ∆t = ∆s/2c. A perfectly matching layer was adopted to absorb the outgoing electromagnetic wave around the computational domain.

Results and Discussion
The plasmonic mode was excited by a Gaussian beam, which is normally focused onto the end of the middle waveguide [17]. The arrangement of the five HPWsis shown in Figure 2a. The intensity distributions in the XZ plane are shown in Figure 2b, where the pattern was very complex due to multimode interference effects [18]. In ref. [18], the principles and properties of multimode interference devices are presented for conventional dielectric waveguides. To the device of five HPWs, supermodes were formed due to the coupling between adjacent waveguides [19,20]. Customarily, supermodes are the modes of propagation of a periodic array of waveguides, which consists of N identical waveguides [20]. Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 8 The wave functions of the supermodes were written as linear combinations of the unperturbed wave functions of the individual waveguides: The total field in a periodic array of waveguides can be expressed as a sum of all supermodes, which can be written as: s β is the propagation constant of a single supermode, s a is the weight coefficient of excited supermodes [19]. Due to the incident field of SPPs being the input from the middle waveguide, even supermodes can be excited. These excited supermodes produced a multimode interference and formed a self-imaging phenomenon. The corresponding N-fold images were expected to form at distances [18]: where P denotes the periodic nature of the imaging along the waveguides, Lπ = π/(β1 − β2) is the beating length of the two lowest-order supermodes, and β1 and β2 are propagation constants of the two supermodes. In Figure 2b, the arrow A shows the position of a one-fold image (N = 1), where LA = 11.2 μm. According to Equation (3), the positions of two-fold images and three-fold images were at LA/2 and LA/3, which are shown by the arrows B and C.
The coupling transmission of the cross section between adjacent waveguides is shown in Figure 3, which corresponded to different z coordinates. It can be seen from Figure 3 that the energy was transferred from the middle waveguide to the waveguides on both sides due to the coupling between adjacent waveguides.
The total field in a periodic array of waveguides can be expressed as a sum of all supermodes, which can be written as: β s is the propagation constant of a single supermode, a s is the weight coefficient of excited supermodes [19].
Due to the incident field of SPPs being the input from the middle waveguide, even supermodes can be excited. These excited supermodes produced a multimode interference and formed a self-imaging phenomenon. The corresponding N-fold images were expected to form at distances [18]: where P denotes the periodic nature of the imaging along the waveguides, L π = π/(β 1 − β 2 ) is the beating length of the two lowest-order supermodes, and β 1 and β 2 are propagation constants of the two supermodes. In Figure 2b, the arrow A shows the position of a one-fold image (N = 1), where L A = 11.2 µm. According to Equation (3), the positions of two-fold images and three-fold images were at L A /2 and L A /3, which are shown by the arrows B and C. The coupling transmission of the cross section between adjacent waveguides is shown in Figure 3, which corresponded to different z coordinates. It can be seen from Figure 3 that the energy was transferred from the middle waveguide to the waveguides on both sides due to the coupling between adjacent waveguides. Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 8 As the distance increased, the coupling between waveguides became weaker. Figure  4 shows the propagation behavior of SPPs in three HPWs, in which the distance between waveguides was doubled. We removed the 2nd and the 4th waveguides on the basis of Figure 3. As can be seen from Figure 4, the length required for energy coupling from one waveguide to adjacent waveguides became longer. When the length of the HPWs was less than 10 um, the coupling between waveguides could be ignored. This also illustrates that the 5th waveguide had little effect on the 3rd waveguide in Figure 2. Based on the simulation results of Figure 2b, we could design a 1 × 3 power splitter by the self-imaging phenomenon [18]. In Figure 2b, the positions of 3-fold images were at As the distance increased, the coupling between waveguides became weaker. Figure 4 shows the propagation behavior of SPPs in three HPWs, in which the distance between waveguides was doubled. We removed the 2nd and the 4th waveguides on the basis of Figure 3. As can be seen from Figure 4, the length required for energy coupling from one waveguide to adjacent waveguides became longer. When the length of the HPWs was less than 10 um, the coupling between waveguides could be ignored. This also illustrates that the 5th waveguide had little effect on the 3rd waveguide in Figure 2. As the distance increased, the coupling between waveguides became weaker. Figure  4 shows the propagation behavior of SPPs in three HPWs, in which the distance between waveguides was doubled. We removed the 2nd and the 4th waveguides on the basis o Figure 3. As can be seen from Figure 4, the length required for energy coupling from one waveguide to adjacent waveguides became longer. When the length of the HPWs was less than 10 um, the coupling between waveguides could be ignored. This also illustrates tha the 5th waveguide had little effect on the 3rd waveguide in Figure 2. Based on the simulation results of Figure 2b, we could design a 1 × 3 power splitter by the self-imaging phenomenon [18]. In Figure 2b, the positions of 3-fold images were a Based on the simulation results of Figure 2b, we could design a 1 × 3 power splitter by the self-imaging phenomenon [18]. In Figure 2b, the positions of 3-fold images were at L A /3, as shown by the arrows C. When z = 3.2 µm, the field intensity of the 2nd and the 4th waveguides was zero, in which the field energy was coupled to the 3rd and the 5th waveguides. If we reduced the length of the 2nd and the 4th waveguides to 3.2 µm, a 1 × 3 power splitter could be obtained. The scheme is shown in Figure 5a. The intensity distribution is shown in Figure 5b. The field distribution is shown in Figure 5c. It can be seen that the incident field was transmitted into the 1st, 3rd, and 5th waveguides. The ratio of the output power was about 1:1:1. The transmission efficiency, η = P out /P in , was 91%, where P out was the power in the output and P in was the power in the input. The output power P out was the sum of the 1st, 3rd, and 5th waveguides. The energy loss mainly came from two aspects. One was due to the reflection of the waveguide port. When the input field was transmitted in the 2nd and 4th waveguides, part of the energy was reflected by the waveguide ports. The other aspect was the ohmic loss of the waveguide itself [3].
Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 8 LA/3, as shown by the arrows C. When z = 3.2 μm, the field intensity of the 2nd and the 4th waveguides was zero, in which the field energy was coupled to the 3rd and the 5th waveguides. If we reduced the length of the 2nd and the 4th waveguides to 3.2 μm, a 1 × 3 power splitter could be obtained. The scheme is shown in Figure 5a. The intensity distribution is shown in Figure 5b. The field distribution is shown in Figure 5c. It can be seen that the incident field was transmitted into the 1st, 3rd, and 5th waveguides. The ratio of the output power was about 1:1:1. The transmission efficiency, η = Pout/Pin, was 91%, where Pout was the power in the output and Pin was the power in the input. The output power Pout was the sum of the 1st, 3rd, and 5th waveguides. The energy loss mainly came from two aspects. One was due to the reflection of the waveguide port. When the input field was transmitted in the 2nd and 4th waveguides, part of the energy was reflected by the waveguide ports. The other aspect was the ohmic loss of the waveguide itself [3]. We could also design a 1 × 2 power splitter according to the field intensity distribution of Figure 5b. We reduced the length of the 3rd waveguide to 2.6 μm, as shown in Figure 6a. The field intensity distribution is shown in Figure 6b. The incident field was transmitted into the 1st and 5th waveguides. The ratio of the output power was 1:1. The output power was the sum of the 1st and 5th waveguides. The transmission efficiency η was 77%. We could also design a 1 × 2 power splitter according to the field intensity distribution of Figure 5b. We reduced the length of the 3rd waveguide to 2.6 µm, as shown in Figure 6a. The field intensity distribution is shown in Figure 6b. The incident field was transmitted into the 1st and 5th waveguides. The ratio of the output power was 1:1. The output power was the sum of the 1st and 5th waveguides. The transmission efficiency η was 77%.  Another scheme for 1 × 2 power splitter could be designed by three HPWs, as shown in Figure 7a. The length of the middle waveguide was 2.6 μm. The field intensity distri bution is shown in Figure 7b. The incident field was coupled to the adjacent waveguides The ratio of the output power was also 1:1. The transmission efficiency η was 92%, which was more efficient than the scheme of Figure 6. The principal reason was that the numbe of the middle waveguides in Figure 6 was three and the number in Figure 7 was one, in which the input energy was reflected by the output ends of the middle waveguides. We compared the power splitters proposed in this work with other power splitter previously published, which are shown in Table 1. The previous power splitters were mainly based on 2D MIM waveguides. Here, the power splitter was based on 3D HPWs which was more suitable for integrated photonics structures and had a higher transmis sion efficiency.  Another scheme for 1 × 2 power splitter could be designed by three HPWs, as shown in Figure 7a. The length of the middle waveguide was 2.6 µm. The field intensity distribution is shown in Figure 7b. The incident field was coupled to the adjacent waveguides. The ratio of the output power was also 1:1. The transmission efficiency η was 92%, which was more efficient than the scheme of Figure 6. The principal reason was that the number of the middle waveguides in Figure 6 was three and the number in Figure 7 was one, in which the input energy was reflected by the output ends of the middle waveguides. Another scheme for 1 × 2 power splitter could be designed by three HPWs, as shown in Figure 7a. The length of the middle waveguide was 2.6 μm. The field intensity distri bution is shown in Figure 7b. The incident field was coupled to the adjacent waveguides The ratio of the output power was also 1:1. The transmission efficiency η was 92%, which was more efficient than the scheme of Figure 6. The principal reason was that the numbe of the middle waveguides in Figure 6 was three and the number in Figure 7 was one, in which the input energy was reflected by the output ends of the middle waveguides. We compared the power splitters proposed in this work with other power splitters previously published, which are shown in Table 1. The previous power splitters were mainly based on 2D MIM waveguides. Here, the power splitter was based on 3D HPWs which was more suitable for integrated photonics structures and had a higher transmis sion efficiency.  We compared the power splitters proposed in this work with other power splitters previously published, which are shown in Table 1. The previous power splitters were mainly based on 2D MIM waveguides. Here, the power splitter was based on 3D HPWs, which was more suitable for integrated photonics structures and had a higher transmission efficiency.

Conclusions
We presented several new power splitters based on five HPWs. The propagation behaviors of SPPs in five HPWs were numerically simulated by the 3D FDTD method. The incident field was transferred from the middle waveguide to the waveguides on both sides due to the coupling between adjacent waveguides. The intensity distributions could be explained by the multimode interference of SPPs supermodes. According to the field intensity distribution of five HPWs, we reduced the length of some specific waveguides and designed a 1 × 3 power splitter and a 1 × 2 power splitter, which the ratios of the output power being 1:1:1 and 1:1, respectively.