1. Introduction

Nanophotonic wire waveguide structures play a crucial role for compact photonic integrated circuits (PICs) and are deployed for various applications in the design of optoelectronic devices such as power splitters, ring resonators, optical logic gates, wavelength filters, optical switches, and optical-interconnects in H-tree network [1–8]. High-density integrated circuits require the minimization of the dimensions of the photonic devices. In addition, sharp 90° bends are desirable to reduce the overall system size. This strongly militates for the use of dielectric waveguides with high refractive-index contrast. Although standard dielectric waveguides with low-index contrast provide low loss with respect to the transmission efficiency, their deployment requires relatively small bend angles with consequent large bend radii, which is not desirable for high-density PICs [9–11].

Increasing the index contrast may help us improve the optical confinement and increase the density of circuits. For single-mode operation, the dimensions of the waveguide should be reduced. Although Refs. 12 and 13 have reported the propagation losses for straight photonic wires to be ~0.24 dB/mm, losses become excessive as the bends become sharper, especially for 90° bends [5, 14]. At this stage, it may be helpful to emphasize that the bending losses will be of no consequence when bend radius is adjusted to be high enough. On the other hand, while the bend radius is dramatically decreased to have compact devices, several approaches have been described in the literature to reduce severe bending losses. Corner mirrors, waveguide resonators, air trenches, and topology optimization have been proposed [15–19]. In general, the complexity of the proposed solutions and their limited operating bandwidth are issues of concern. Here, we propose a new solution to enhance transmission efficiency of narrow-waveguide T-junctions that can be easily incorporated in H-tree networks. Most of the time, achieving high transmission only within a limited band is not enough and broadband solutions are preferable. We aimed therefore to design bends and branches which provide small bending losses over high bandwidth and occupy small area.

T-junctions are a basic part of a more complicated network configuration such as H-tree circuits that can be used for power distribution and optical-signal clock delivery. An effective design that promotes high power transmission through sharp corners can be systematically duplicated to build larger structures, such as optical interconnects that may contain many branches [20]. Optical clock signals should be distributed to each receiver (photo-detector) with an equal time delay. To achieve that, high-index contrast waveguides with a special layout that resemble an H-tree shape can be used. The potential advantages of optical interconnects over electrical ones include high speed, high bandwidth, and low power consumption. High-speed processors handling large data loads demand the delivery of optical rather than electrical clock signals. Potential drawbacks of electrical interconnects include possible Joule heating that may degrade the attractive features of them such as faster speed and higher density. This argues for high-transmission junctions to minimize requisite optical power levels as well as stray light that may itself contribute to heating. High-efficiency methods that utilize resonant structures, however, may not be robust with respect to wavelength shifts due to any changes occurring in the environment. Such resonant wavelength shifts may occur with the changes in the chip temperature due to the temperature dependence of refractive index of the optical cavities. Hence, our approach is broadband from the start. The waveguide material should be transparent to over the entire range of potential operating wavelengths; consequently, we select Si as a waveguide material.

In this paper, a hybrid structure with a zero bending radius is discussed. The studies reported in Refs. 5, 9, 10, 11 and 14 necessitate small bend angles with consequence of large bend radii. The combination of photonic crystals (PC) and photonic wires is proposed in order to cope with the problem of high bend losses. A photonic dielectric wire is surrounded by air and the PC is composed of a square lattice of dielectric rods that are only present in the vicinity of the junctions where excess bend losses would otherwise appear. By means of the photonic band gap (PBG) mechanism [21–25], the PC behaves as a photonic insulator preventing the light from penetrating into the surrounding region resulting in higher light confinement. After we demonstrate numerically effective signal delivery of T-junction photonic wires both in 2D and 3D, we finally implemented 1 to 4 fanouts in an H-tree configuration in which 1 to 2ⁿ fanouts are possible, where n is an integer. We should emphasize that the proposed solution in Ref. 21 is only for a PC-waveguide structure. However, we here offer a hybrid structure that is composed of a PC and high-index contrast nanowire waveguide. Besides, the PC portion is retained only proximate to the junction itself. The remainder of the structure is Si-waveguide on top of an insulator layer.

The rest of the paper is organized as follows: To address the bend losses in narrow waveguides, we propose optimized waveguide corners designed for H-tree interconnects in Sec. I. Section II describes light transport performance for three cases: standard T-junctions, the same junction surrounded by PC, and the optimized composite version of the structure. Section III outlines 3D investigations of the optimized T-junction. Light transmission in an H-tree network is presented in Sec. IV. We give concluding remarks in Sec. V.

2. Two-dimensional T-junction design and enhanced power transmission

The original T-junction structure is made up of Si ($n_{\text{1}}=\text{3}\text{.45}$) surrounded by air ($n_{\text{air}}=\text{1}\text{.0}$) as shown in Fig. 1 . The width of the waveguide$w$, is represented in terms of a constant, a ($w=0.20a$). The parameter a is defined shortly below. For numerical studies in 2D configurations, a transverse magnetic (TM) guided mode is assumed, in which case the corresponding nonzero electric- and magnetic-field components are E_z, H_x, and H_y respectively. In order to improve the poor power transmission capability of a standard T-junction, some modifications will be made in the area proximate to the junction.

 

Fig. 1 Schematic drawing of the T-junction nanophotonic wire dielectric waveguide is shown. The width and refractive index of the dielectric slab (w) is 0.20a and 3.46, respectively. The equally divided input light pulse is shown in the branches. The arrows indicate the direction of the signal flow.

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For our 2D analysis of T-junction structures, we first carry out simulations for the propagation of electromagnetic waves by using the 2D finite-difference time-domain (FDTD) method [26]. The spectral content of the transmitted power is calculated by taking the ratio of the output power to input power. For the correct normalization of the transmission efficiency, we need to detect both incident and output pulses. The input power is calculated in the absence of sharp corners. Output detector planes are placed outside the engineered area. To eliminate back reflections originating from the ends of the finite computational window, we used perfectly matched layer absorbing boundary conditions [27]. In Fig. 2 the normalized transmission coefficient for the T-shaped structure is presented with a solid line. As can be seen from the figure, this structure has poor transmission efficiency of around 18%. In the lossless case, that value should correspond to 50% efficiency.

 

Fig. 2 The power transmission spectra of three cases are shown. A regular T-junction nanowire dielectric waveguide corresponds to solid-black line. The dashed-green line represents the transmission efficiency of the structure whose schematic drawing is presented in Fig. 4. The T-junction nanowire dielectric waveguide is surrounded by photonic crystal that is composed of dielectric rods with radii, r = 0.20a. The dotted-red line corresponds to the power transmission efficiency spectrum for the case in which $r_{\text{1}}$and $r_{\text{2}}$are adjusted to be 0.18a and 0.26a, respectively. The structure in this case is presented in Fig. 5.

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In Fig. 3 , a snap shot of the pulse propagation for the T-junction structure is shown to illustrate the severe power leakage in the junction area. As can be clearly observed, radiated field leaves the waveguide and consequently the resulting bend loss at the junction of the structure is very high. The 18% transmission efficiency in each branch gives rise to 64% radiation loss in total due to the bend loss.

 

Fig. 3 Steady-state electric field (E_z) distribution of TM mode for the regular T-junction nanowire waveguide when the operating frequency is centered at$\lambda =1.55\mu \text{m}$. The red and blue colors represent the maximum and minimum electric-field values, respectively.

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A hybrid solution composed of a high-index-contrast waveguide and a PC is considered next in order to enhance the transmission efficiency. For that purpose, dielectric rods made of Si with radii r = 0.20a are placed periodically around the junction of the wire as shown in Fig. 4 . The periodicity of the dielectric rods’ locations is indicated by lattice constant a and it is taken to be $a=512$ nm in order to target telecom wavelengths. Such a straightforward placement of the periodic rods suppresses the light scattered at the bend, but only slightly improves the transmission efficiency as compared to standard T-junction. The result is shown with a dashed-green line in Fig. 2.

 

Fig. 4 Schematic of T-junction nanowire dielectric waveguide surrounded by square-lattice photonic crystal that is composed of dielectric rods with radii, r = 0.20a. The box shows the boundary of the region that is engineered to enhance transmission efficiency. It covers an area of (8ax8a = $\text{4}\text{.096}\mu \text{m}\times \text{4}\text{.096}\mu \text{m}$).

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The small increment in the transmission efficiency can be improved upon as we show in the next step. We analyzed the structure carefully and decided to rearrange the locations and number of the PC rods to obtain better results. For this reason, two additional rods marked as “A” in Fig. 5 are located on the diagonal between the second-nearest neighbors as shown and the positions of the corner rods are displaced by (δ_x, δ_y). In a recent study [28], we performed both 2D and 3D studies on sharp 90° bends. For the optimum solution, we applied the same design procedure, mentioned in Ref. 28, for the rods of the sharp-bend structure to obtain efficient T-junction hybrid structures. We optimized the parameters of the crucial unit cells within the junction area. The radii of the designated Si rods ($r_{\text{1}}$,$r_{\text{2}}$) are chosen as (0.18a, 0.26a). The radii of the rest of the rods are kept the same, namely as 0.20a. The power transmission efficiency for the designed structure can be seen in Fig. 2 as the dotted-red line. With the help of that modified hybrid structure, the power transmission efficiency dramatically increases and reaches up to 48.6%. This value gets near to 97%, which implies the system to become highly efficient. We should emphasize that this value is valid under the assumption of absent insertion and scattering losses. As can be seen in Fig. 2, the normalized transmission stays well above 40% (ideal case is 50%) for a bandwidth of nearly $\Delta \omega =206$nm. In Fig. 6 , one can observe a snapshot of pulse propagation in the structure when the normalized frequency of the continuous pulse is chosen to be$a/\lambda =0.33$. It should be noted that the located source size is kept sufficiently small that it can be considered as a point source. In that case, it isotropically emits and some part of the radiated power hits the PC front surfaces. That is why some undesired electric field appears around the source location. The negligible back reflection can be seen both qualitatively (visually) and quantitatively (transmission coefficient). The qualitative explanation is based on the inspection of the time domain snap shot of the electric field presented in Fig. 6. If there were a strong back reflection that may arise due to the presence of an acute corner, the forward and backward propagating waves would interfere. As a result, the spatial periodicity of the field may be disturbed. However, as we can see from Fig. 6, there is no strong back reflection having an effect on the periodicity of the wave oscillation.

 

Fig. 5 Schematic drawing of the modified hybrid T-junction. In the first step, two additional cylinders represented by “A” are embedded along the diagonal direction. In the next attempt, the radii of the cylinder pointed with arrows are arranged in order to enhance the transmission of the light-wave. Finally, (δ_x, δ_y) describes the cylinders movement along the diagonal direction. δ_x and δ_y values are selected with the equal amount to stay on the diagonal direction.

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Fig. 6 Steady-state electric field (E_z) distribution of TM modes for the hybrid structure when the operating frequency is centered at a/λ = 0.33. The red and blue colors represent the positive and negative electric fields, respectively. The arrows indicate the propagation directions.

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The wavelength of the operating signal is taken to be$\lambda =1.55$µm. In that case, the optimal lattice constant is$a=512$ nm as mentioned previously. The efficient division of the optical pulse into the two branches can be observed in Fig. 6.

Even though surrounding the photonic wire waveguide directly by PCs provides slightly better results, the carefully adjusted displacement and incorporation of additional dielectric rods around the junction area dramatically improves the power transmission capability. The efficiency of 40% is the total transmission of the two branches. That means 20% transmission per branch. This can also be inferred from the transmission efficiency vs. frequency graph, shown in Fig. 2 (dashed-green line). However, in the optimized hybrid structure, the total transmission efficiency reaches near to 98% because the numerically reported efficiency at each branch is approximately 49%. The origin of the optimization stage was partially borrowed from the recent work [28]. The starting point of the design is to select the parameters of the periodic dielectric medium that surround the corner region of a standard nano-wire waveguide of width w. The lattice constant, a of the periodic rods that should surround the corner region can be determined according to desired operating wavelength. The radii of dielectric rods, r can be selected in such a way that the band gap of the periodic structure should cover the waveguide mode. After these selections, the critical rods closest to the corner region should be manipulated in terms of their locations and radii to enhance the amount of light transportation. Besides, additional rods can be incorporated in the diagonal directions, viz. the cylinder rods represented by “A” in Fig. 5. The confinement mechanism is replaced by band gap effect instead of total internal reflection due to periodic arrangement of rods around the corner area. The adjustment of locations and radii of rods provide an effective 45° turn while implementing 90° bend. In the same study, we have proved that moving from 2D to 3D FDTD analysis, small deviations occur in the overall performance. The reason for that is the high-index contrast waveguide surrounded by lower refractive indices both above and below the core region. In the next section, 3D analysis of T-junction will be presented.

To check the robustness of the proposed design against some imperfections that may occur during the fabrication procedure we have made some modifications on the upper part of the designed structure as shown in Fig. 5. Firstly, the rod marked “B” as shown in Fig. 5 is shifted back and forth by 0.20a steps. An approximate 1% variation in the power transmission efficiency of the structure was observed. Secondly, the radius of rod B was changed within the interval 0.16a and 0.26a. Again, we obtained only 0.5% variations in the transmission efficiency. Lastly, we rearranged the radii of the rods that are placed next to rod “B”. In this case, the transmission efficiency is only affected by ~0.3%. These observations indicate that the top part of the T-junction is robust against changes in the parameters of the rods and inadvertent displacements from their ideal locations.

3. Three-dimensional analysis of T-junction

While the 2D analysis given above is indicative of performance, a 3D analysis is desirable to obtain more accurate results [29] and to understand the role of out-of-plane and substrate losses occur. A similar 3D data-extraction procedure as in 2D case is conducted in this section. The optimization steps carried out in 2D are repeated in 3D to finely tune and enhance the transmission efficiency of T-junctions. The schematic drawing of the 3D structure is shown in Fig. 7 . The T-junction photonic wire waveguide is surrounded by dielectric Si rods, with $n_{\text{Si}}=\text{3}\text{.45}$ and laid upon a SiO₂ substrate layer. Above the core region, air is used as cladding. The core region is sandwiched (along the z-direction) by SiO₂ from the bottom and air from the top. In addition to that, the corner regions are accompanied by periodic dielectric rods around corner region. The radii of the silicon rods are selected as $r=\text{0}\text{.23}a$ (117.6 nm). This change was made to obtain optimal results in 3D concerning the transmission efficiency. The height of the waveguide is taken to be h = 3a, which is nearly equal to 1.535 µm. The substrate beneath the Si rods and the wire is composed of silica with refractive index$n_{{\text{SiO}}_{\text{2}}}=\text{1}\text{.44}$. The width of the waveguide is 0.20a. The parameters of the structure (waveguide width, radii of rods and slab height) are selected to have large bandwidth for the fundamental guided TM mode. It is known that the slab thickness is usually higher for TM mode than TE mode if one wants to obtain wide bandwidth [30]. Hence, we selected the height of the slab for TM mode according to 3D FDTD analysis that ensures both high transmission for single 90° waveguide bend and single mode operation spanning whole band gap region [25, 28, 30]. The band-gap guided mode appears in the corner region due to surrounding periodic dielectric rods. The other regions are composed of uniform dielectric wire waveguide. As a result, the confinement mechanism is only due to the total-internal reflection. It can be feasible to fabricate such a waveguide with a high aspect ratio, which corresponds to a value of $h/w=(3.00a/0.20a)=15$in our study, thanks to the recent advancements in the semiconductor fabrication technology [31].

 

Fig. 7 The three-dimensional representation of the T-junction is shown.

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The transmission efficiency in 3D configuration was calculated and the result is presented in the solid-blue line in Fig. 8 . As compared to 2D analysis, the transmission efficiency drops 4-5% near the central region of the spectrum, which is actually due to the out-of-plane loss. The finding dictates that the leading loss mechanism in 3D is, again, due to bend loss. For high-index contrast dielectric waveguides, the out-of-plane and substrate to leakage losses are not a major concern as we see from the 3D FDTD results. This is mainly because of the fact that the index contrast between core and cladding/substrate is high enough for the total internal reflection to give strong vertical field confinement. 4-5% efficiency drop corresponds to a power level change that occurs when 3D analysis is performed instead of 2D one. The out-of-plane and leakage to substrate losses are only considered in 3D. As a result, the comparison between 4 and 5% additional loss due to out-of-plane and leakage to substrate losses versus 64% bending loss shows the fact that the main loss contribution in these structures arises because of the sharp corners. It is again important to mention that we excluded insertionand propagation losses in the current work. On the other hand, we should emphasize the importance of the scattering losses that may occur due to sidewall roughness. It will be very crucial to determine the overall performance of fabricated systems; however, we do not address such loss being outside the scope of the present work. The other critical aspect of deployed systems is the input and output coupling. The size mismatch between various waveguides necessitates the implementation of spot-size converters. Novel solutions achieving efficient couplers were proposed in the literature [32, 33].

 

Fig. 8 The power transmission efficiency spectrum for the optimized 3D structure is shown with a solid-blue line. The dashed-red line indicates the efficiency curve obtained by 2D FDTD method.

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The efficient optical pulse propagation in T-junction both in 2D and 3D cases motivates us move to the next stage in which we designed a 1x4 H-tree network to deliver optical signal.

4. 1x4 optical pulse distribution in H-tree network

This section outlines a prototype for optical signal distribution in an H-tree optical waveguide network. It’s well-known that optical clock signal distribution for routing timing signals to all parts of a chip with the same propagation delays plays a key role to reduce skew of clocking in VLSI design [20, 34]. Therefore, the use of H-tree interconnections is inevitable for high-density photonic integrated circuits. Below, we demonstrate numerically transmission of light to 4 fanouts. By duplicating the optimized T-junction in a systematic manner, we obtain an H-tree configuration that is represented in Fig. 9 . This topology ensures the delivery of the input signal to output ports at equal propagation delays. The parameters of the structure are kept the same as before. The source is placed in the input waveguide. The output signals are detected at the end of each branch.

 

Fig. 9 The schematic presentation of the 1x4 optical signal distribution in H-tree network.

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The calculated transmission efficiency of the 1x4 branches is around 23%, as can be seen in Fig. 10 . We observed narrowing in the bandwidth despite the high transmission. Such an observation can be explained as follows: Even though we do not directly incorporate a cavity configuration at the junction, surrounding the waveguide by the dielectric rods gives rise to a very low-Q cavity (Q<10) type of configuration. Each additional T-junction included in the structure contributes to the enhancement of Q associated with the junctions. As a result, the slightly larger Q diminishes the overall bandwidth of the nanowire waveguide.

 

Fig. 10 The transmission efficiency of 1x4 optical signal distribution in H-tree network.

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In order to demonstrate the high transmission capability of the designed branches in the H-tree network configuration, we prepared a snap shot of the pulse propagation down the waveguide and also provided a movie (Media 1). The effective power distribution is apparent in Fig. 11 . Each time the input pulse is divided equally, the color of the electric field in the respective waveguide branch fades. Even though we have achieved 4 fanouts in the H-tree in Fig. 11, the idea is easily generalize to $2^n$ fanouts, with n an integer.

 

Fig. 11 The steady-state e-field distribution of the input signal in H-tree network is shown. T₁ and T₂ correspond to transmitted powers at the junctions (Media 1).

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We summarize the best values of the transmission efficiencies achieved in the present work. It provides us a concise way to judge the performance of the proposed idea. The difference between the values of transmission efficiencies in 2D and 3D configurations corresponds to out-of-plane and leakage to substrate losses. (43x2)% efficiency represents 3D result. The 14% loss includes bending, out-of-plane and leakage to substrate losses for one-junction. On the other hand, (23x4)% efficiency represents the findings of 2D analysis. The 8% loss is due to the 1x4 H-tree network configuration. As a result, loss per junction is 4%. The difference between the 2D and 3D results is 10% that encloses out-of-plane and leakage to substrate losses for one-junction, under the assumption that the bending losses in 2D and 3D structures are equal. When there is no manipulation made to the corner region, the loss per junction is 64%. That means we report a sixteen times improvement in terms of bending loss reduction (64% loss reduces down to 4%).

The flow of the input power at each junction can be monitored as shown in Fig. 11. If we define the power split at first junction as T₁ then at the second junction, the wave splits again. This efficiency is defined as T₂. There exists a relationship between T₁ and T₂ as$T_2=(T_1{)}^2$. When the value of T₁ is taken as$T_1=48.6\%$, as reported in Sec. 2 then T₂ turns out to be${(0.486)}^2=23.61\%$. This number is very close to 23% efficiency value as reported in Sec. 4. We have 0.12 dB bending loss for 1x2 T-junction and 0.377 dB bending loss for 1x4 H-tree network case according to 2D study. On the other hand, 3D analysis gives us a loss value of 0.49 dB that includes other loss mechanisms in addition to bending loss. Ref. 8 experimentally reported 0.15 dB and 2.41 dB excess losses for 1x2 and 1x16 splitter cases, respectively that excluded insertion and propagation losses.

5. Conclusion

The low transmission through T-junctions made of high-index contrast narrow waveguides associated with bend losses is addressed and a novel solution is proposed. Transmission efficiencies extending up to 43% (ideal case corresponds to 50%) are predicted based on a 3D FDTD analysis. Sharp corners appearing along the waveguide branches were optimized for low-loss and uniform light propagation with a balanced propagation delay. Taking advantage of the optimized T-junction, we have shown efficient distribution of an input pulse in a 1x4 H-tree network configuration. The proposed solution in the present study may contribute to the widespread usage of photonic integrated circuits based on high-index Si nanophotonic wires.