Triangular Cross-Section Beam Splitters in Silicon Carbide for Quantum Information Processing

Triangular cross-section color center photonics in silicon carbide is a leading candidate for scalable implementation of quantum hardware. Within this geometry, we model low-loss beam splitters for applications in key quantum optical operations such as entanglement and single-photon interferometry. We consider triangular cross-section single-mode waveguides for the design of a directional coupler. We optimize parameters for a 50:50 beam splitter. Finally, we test the experimental feasibility of the designs by fabricating triangular waveguides in an ion beam etching process and identify suitable designs for short-term implementation.


Introduction
Photonic quantum information processing (QIP) using solid-state single photon emitters has been extensively explored for applications in quantum communications and integrated quantum information circuits.Among the quantum emitters, color centers have been desired solid state qubit candidates due to their spectral homogeneity, long spin coherence times, and availability of spin-spin and spin-photon entangling processes, necessary for QIP applications [1][2][3][4][5][6].Particularly, color centers in silicon carbide (SiC) attracted attention because of emissions in the telecommunication bands which are suitable for sending over Triangular photonics has been the top choice in diamond, silicon carbide and rare earth photonics [16][17][18] that provides pristine color centers and high light and matter interaction.
In this paper, we explore triangular cross-section 50:50 BS in 4H-SiC necessary for performing on-chip quantum interferometry.We construct such a device that splits an input beam into two output beams of equal intensity using two concurrent waveguides.In the front and back regions, the waveguides are far apart (no electromagnetic interaction) on either ends (input, output) of the device, while in the middle overlap region waveguides propagate in close proximity to create an electromagnetic interaction between their fields.When two indistinguishable photons each entangled to a spin of a quantum emitter arrive at the inputs of a 50:50 BS at the same time, they undergo bunching to exit together through the same output, creating remote entanglement between the two spins.Here we specifically study 50:50 BS designs for nitrogen vacancy (NV) color centers in 4H-SiC, with single photon emissions in the telecommunication wavelength range (1176 -1243 nm) [26,27].
Using Finite-Difference Time-Domain (FDTD) method, we first optimize the width of triangular cross-section waveguides to achieve single mode propagation (fundamental TE mode), and high coupling efficiency of the color center (NV in 4H-SiC) emission into that single mode, for different etch angles.Next, we investigate the formation of supermodes when two identical single mode waveguides are placed adjacent to each other.We then study the waveguide mode conversion in bent waveguides and find suitable waveguide bend geometries to maintain the same waveguide mode in the straight and bent regions of the waveguide.We use this understanding to design triangular cross-section 50:50 BS in 4H-SiC.Finally, we explore the fabrication of the simulated triangular cross-section waveguide configurations required for a 50:50 BS.

Single mode triangular cross-section waveguide
Color center emission is dipole-like and such emission can couple to fundamental TE (f-TE), fundamental TM (f-TM) and other higher order modes that are supported in triangular cross-section waveguides [24].It has been shown that for each etch angle there exists an optimal width for single mode (f-TE) propagation in triangular cross-section waveguides [18], which is a necessity for QIP applications [28].So, to achieve high coupling efficiencies for color center emission, the color center should be positioned in a single mode waveguide at the maximum electric field (E-field) intensity point of the f-TE mode [12,18], which is at the centroid of the triangle.
We use FDTD package in Lumerical software to estimate the coupling efficiency of color center emission into the f-TE mode (C fTE ) for different widths (w) and etch angles (α) of a triangular cross-section waveguide (the angle at the apex of the triangle is 2α).We position the dipole emitter at the centroid of the triangle with an emission wavelength of 1230 nm.
We find that for each α there exists a width with highest coupling (> 80%) to the f-TE mode as shown in Figure 1.For our 50:50 BS simulations, we choose the optimal width values (w = 550 nm, 650 nm, 800 nm for α = 30 • , 45 • , 60 • respectively) slightly lower than the widths with the highest C fTE .The reason being that the waveguides with lower width have slightly higher evanescent fields (top panels of inset in Figure 1), necessary to create meaningful coupling between the adjacent waveguides in the BS, while only slightly reducing The mode profile of the waveguide f-TE (f-TM) mode has a E-field intensity maximum closer to the top surface (apex) [24].As the waveguide width decreases, the E-field intensity maximum of the f-TM mode moves closer to the apex and becomes evanascent (not supported in the waveguide).Using MODE package in Lumerical software, we find that at these optimal widths for α = 45 • and 60 • , the waveguide supports only the f-TE mode.However, for α = 30 • , both f-TE and f-TM modes are supported in the waveguide at the optimal and smaller widths.The height of the triangle varies inversely with α and hence the widths needed for the f-TM to become evanescent are much smaller than the optimal width for α = 30 • .In this paper, we design a 50:50 BS using two identical triangular cross-section waveguides with three regions: 1) input and output -where the waveguides are far apart with zero electromagnetic interaction, on either ends of the BS, 2) bent waveguide region, and 3) coupling region -where the waveguides are close enough to electromagnetically interact, as shown in Figure 2a.When light is injected into the f-TE mode of one of the waveguides on the input side (top left), it propagates through the straight and bent regions of that waveguide.
When it reaches the coupling region, some of the light in the top waveguide is coupled into the bottom waveguide as it propagates along.Then the light in both the waveguides pass through the bent and straight regions to reach the output ports (through and drop ports).
The proportion of light coupled into the bottom waveguide is determined by the coupling strength between the individual waveguide modes and the length of the waveguides in the coupling region.
When two identical waveguides are close to each other, the individual modes (f-TE) in each of the waveguides superimpose to form a supermode.The coupling between the two waveguides can be analyzed in terms of a pair of TE supermodes, as shown in the inset of Figure 1.For a BS with power P 0 in one of the waveguide at the beginning of the coupling region, the coupling length (L C ) required for a power P 2 to couple into the other waveguide is given by: where λ 0 is the free space wavelength, ∆n eff is the difference in effective refractive indicies of the two TE supermodes [29].The coupling length is inversely proportional to the coupling strength between the waveguides given by ∆n eff , which in turn depends on the confinement of the individual waveguide modes and gap between the individual waveguides in the coupling region.For a 50:50 BS, where there is 50% light in both the waveguides after the coupling region, the ratio P 2 /P 0 equals 0.5.We use MODE package in Lumerical software to estimate the effective index of the two TE supermodes, for calculating the theoretical values of L C for various BS geometries studied in the following section.

Integrated 4H-SiC beam splitter in triangular geometry
Recent advances in on-chip splitting mostly include rectangular or slab waveguides [15].For performing QIP with integrated color centers in 4H-SiC, we construct the BS structure with symmetric triangular cross-section single mode S-bend waveguides with an overlap region to enable DC. Figure 2a  To test the inter-mode coupling (cross-talk), we inject via the f-TE mode at the input port and collect via the f-TM mode at the through port of the BS.It is observed in Figure 2b that coupling to the f-TM mode (C fTM ) initially increases with the increase in b y and after reaching a maximum value of ∼ 20% for b y = 20 µm, C fTM gradually decreases dropping down to 2.7% with b y = 35 µm.This trend occurs due to the expansion of the curved path with increasing b y , however, after the inversion point, the bending becomes so gradual that the S-bend mimics a straight path.For a better understanding of the f-TE and f-TM mode coupling, we also study the E z fields of the minimum and maximum C fTM as E z field is a good indicator of the existence of the TE and TM modes [25].Figure 2c

Discussion
The modeling results presented in this paper offer an approach to build low-loss triangular crosssection 50:50 BS in 4H-SiC necessary to perform key quantum interferometry operations for QIP applications.Here, triangular cross-section photonics provide a scalable route to integrate color centers into quantum photonic devices and circuits.Moreover, high-performance triangular cross-section photonic structures in 4H-SiC like waveguides, waveguides with integrated SNSPDs, photonic crystal mirrors, and photonic crystal cavities that facilitate efficient generation, collection and detection of single photon emission from color centers, necessary for applications in QIP have been demonstrated [12,18,24,25].Thereby, NV center in 4H-SiC with emission wavelengths near the telecommunications range is most suited for building large-scale quantum communication networks.
Our initial fabrication tests show that triangular cross-section waveguides with gaps greater than 300 nm can be fabricated using the wafer-scale ion beam etch process.From the simulations, we note that for a waveguide gap g ≥ 300 nm in the coupling region, α = 30 • and 45 • , offer 50:50 BS with fabricatable footprints (e.g.coupling region L C ≤ 50 µm), for applications in QIP.Here, supporting structures for waveguide suspension would need to be designed.The ion beam etch process conditions could be further optimized to achieve triangular cross-section waveguides with waveguide gap ≤ 300 nm, by choosing process conditions that result in a predominantly chemical etch.Under such etch conditions, the byproducts of the etch are volatile gases (SiF 4 , CO, CO 2 ), preventing re-deposition, allowing the etch to continue.When there is a reasonable physical etch component, the sputtered material has a higher chance of re-deposition, especially when the gap between the waveguides is small, resulting in a slower etch or sometimes no etch.Another alternative could be choosing a steeper etch angle, because such waveguides would not require a deep etch to release the structure.
Triangular cross-section photonics in 4H-SiC provides an avenue to achieve efficient chip-and wafer-scale integration of color centers, with very little degradation of color center properties compared to bulk, essential for applications in QIP.Our simulation results demonstrate that 50:50 BS necessary for applications in QIP can be implemented using triangular geometry and our initial etch tests show that ion beam etching is suitable for achieving this.

Figure 1 :
Figure 1: The coupling of a TE oriented dipole emission at the centroid of the triangular profile into the f-TE mode ((C fTE )) as a function of the waveguide width for three etch angles (α).Insets show electric field intensity profiles of the f-TE mode, TE supermode 1 and TE supermode 2 (top to bottom), at optimal waveguide widths supporting single mode propagation, for α = 30 • , 45 • , 60 • .The electric field intensity of the TE supermodes were plotted for adjacent identical waveguides with 200 nm gap between them.

Figure 2 :
Figure 2: (a) Top view of the S-bend BS in 4H-SiC and the inset shows the triangular cross-section geometry.(b) The fraction of the f-TE input mode coupling to f-TM (C fTM ) mode along the bend with variations in bend y-span (b y ) for α = 30 • , w = 550 nm, and L C = 30 µm.(c)-(d) Propagation of E z field in the xy plane of the structure described in (b) for b y values of 20 µm and 35 µm, respectively.
shows the top view schematic of the BS with coupling length L C , gap between the waveguides in the coupling region g, S-bend x-span b x , S-bend y-span b y , and the inset shows the triangular waveguide cross-section with w and α.In this paper, we investigate 50:50 BS in triangular cross-section waveguides with α = 30 • , 45 • , 60 • , can be fabricated with the state-of-the-art processes, by varying the gap g between waveguides and consequently L C .In Lumerical FDTD simulations (mesh size = 30 nm), we choose 20 µm for b x and 10 µm for b y , where b x corresponds to the terminal points and b y corresponds to the curvature control points of the S-bend (Bézier curve), for α = 45 • , 60 • .For α = 30 • , we need to account for potential higher-order mode conversion due to the bending curvature as the n eff , one of the most important parameters for understanding mode propagation, of the f-TE and the f-TM modes are virtually the same in the waveguides with α = 30 • .Hence, the f-TE mode can transform into the f-TM due to identical n eff and inter-mode coupling via bends [30].Keeping b x = 20 µm constant, we vary b y for modulating the bend curvature of the {α, w, L C } = {30 • , 550 nm, 30 µm} BS structure.
shows that at the starting point of the bending region with b y = 20 µm, E z has a nodal plane in the triangular waveguide which corresponds to the f-TE mode and at the ending point, E z field starts to develop in the waveguide, confirming the coupling to the f-TM mode.On the other hand, for b y value of 35 µm, there is a constant E z nodal plane indicating the f-TE mode throughout the S-bend, depicted in Figure 2d, which makes this bend structure suitable for maintaining the single mode propagation in the BS.

Figure 4 :
Figure 4: The SEM images of ion beam etched waveguides in 4H-SiC.(a)-(c) 800 nm wide waveguides with gaps of 100 nm, 200 nm and 300 nm respectively.(d) 1000 nm wide waveguides with a gap of 500 nm.Inset shows a FIB-SEM of the fabricated 1000 nm waveguide with a triangular cross-section.The scale bar is 2 µm.