High efficiency focusing double-etched SiN grating coupler for trapped ion qubit manipulation

A one-dimensional focusing grating coupler array based in silicon nitride (SiN) was proposed for trapped ion qubit manipulation. By applying inverse design optimization with a double-etched grating structure, a directionality of 98% was achieved. A small beam diameter of 2.5 μm on the target ion with a low crosstalk of −36 dB was attained. Additionally, the impact of fabrication errors was investigated through a Monte Carlo simulation; within the accuracy of an electron beam lithography-based process, the output efficiency was maintained at 93%.


Introduction
Quantum computers have garnered much interest due to their ability to solve optimization problems considered impractical with conventional von Neumann-type computers. 1) They are particularly promising candidates in the fields of pharmaceuticals, materials science, machine learning, as well as others. [2][3][4][5] One of the most promising ways to construct quantum gate-based computers is with a superconducting ring as a qubit. [6][7][8] Another encouraging approach is through ion trap quantum computing. In such a system, the energy levels of electrostatically trapped ions are used as qubit states. [9][10][11][12] A long coherence time can be achieved by choosing a suitable ion qubit, such as 40Ca+. 13) To suspend the ions, a potential well Paul trap is created with an RF signal and an electrostatic field in a high-vacuum chamber. 14) Laser cooling is used to remove the kinetic energy of the ions, allowing for room temperature operation. 15,16) State manipulation of the qubit is achieved by irradiating laser light with a wavelength corresponding to the ion's energy levels. 17,18) Additionally, motional modes of trapped ion chains can be used for computational purposes, necessitating trapping intervals of several microns. 19) Therefore, ions located every few microns must be individually irradiated for qubit state control.
To realize a scalable quantum computer capable of solving real-world problems, monolithic integration of electrical and optical components is necessary. Nanofabrication shows promise for increasing system integration and scalability. A surface electrode Paul trap, constructed with 2D conducting plates rather than 3D bulk electrodes, has been demonstrated with trapping heights of tens of microns. [20][21][22] Furthermore, integrated photonic control with 2D focusing surface grating couplers (GCs) has been used to diffract guided light from the chip to the ions captured in free space. [23][24][25] Full coherent control of a single trapped ion qubit has since been reported; a chip with four GCs designed for four wavelengths demonstrated fully integrated ion trapping, laser cooling, qubit initialization, and state control. 26,27) However, for multi-qubit control, tightly focused irradiation is needed for ions in a tightly packed chain. This means that a large-footprint GC is required to improve the diffraction limit, making it challenging to place GCs with tight periodicity in a space limited by the ion qubits. In addition, the relatively low diffraction efficiency of GCs causes stray light within the system, resulting in crosstalk with neighboring ions.
In this paper, we report a high efficiency, low-crosstalk SiN GC array for irradiating 729 nm light to a chain of 40Ca + ion qubits. 28) A double-etched GC was used to obtain high selectivity diffraction in the ion direction. An inverse design approach was adopted to design a optimized grating geometry for greater diffraction efficiency. This paper is an extended version of a 26th Microoptics Conference (MOC) paper. 29) We have added a more comprehensive explanation of the theory and the design method, as well as a discussion of the device fabrication errors by way of new data.

Concept and design
For the chip architecture, a silicon photonics platform was chosen as it allows for integration of both surface electrodes and optical waveguides. Because 729 nm light is used to manipulate the state of the 40Ca+ ions, a SiN waveguide, transparent to visible light, was used for this study. A 2D focusing GC is difficult to place on the wafer for the multiion case as a large, diffraction limited footprint is needed. To solve this problem, 1D focusing GCs are placed diagonally as shown in Fig. 1 to form elliptical focusing spots without any overlap. By shifting the GCs by 5 μm in both the X-and Yaxes, it is possible to form focusing spots with a 7.07 μm period in a straight line tilted at 45 degrees.
To design the grating, a double-etched structure was used as shown in Fig. 2. 30) Reflections are cancelled by providing a positional shift between the top and the bottom teeth, giving a half-wavelength phase difference as discussed in the previous report. In addition, by optimizing the thickness of the SiN waveguide and the depth of the grating teeth, it is possible to cancel the diffraction towards the bottom side of the substrate and to achieve high upward selectivity. As surface electrodes are needed on the surface of the wafer, a SiO 2 cladding layer around the SiN waveguide was assumed.
For the design of the high-efficiency diffraction grating, photonic inverse design was used to optimize the geometry of the grating. Lumerical's FDTD solver and LumOpt package were used for this study. 30) The grating design was initially designed through 2D simulation, then extruded to a 3D simulation to confirm the final characteristics. Since the grating parameters vary slowly but continuously, a fifthorder Fourier series expansion was used to define the grating geometry. The following are the details of the design procedure.

Basic parameter selection with a uniform GC
To obtain a high-efficiency GC, major parameters such as the SiN layer thickness T, and the etching depth t were optimized with a uniform grating model. The parameters of the top and bottom gratings were the same but shifted by a constant value.

Theoretical design of a focusing grating
The grating period Λ(n) was set to focus the light onto the target position while keeping D, which is a duty factor of grating, constant as shown in Fig. 3. As in the previous section, the parameters of the top and bottom gratings were made the same but shifted. As the GC is buried in SiO 2 , the diffraction angle at the i th grating teeth position θ 1 , n is obtained by solving the following system of equations, where n 1 is the refractive index of the SiO 2 cladding layer, n 2 is the refractive index of air, θ 1,i is the angle of incidence in the SiO 2 cladding layer, θ 2,i is the angle of incidence in air, x i is the position of the ith grating teeth and x t is the position of the focusing target in the X-axis.
Here, x t is on the negative side. The definitions of these parameters are shown in Fig. 4. When the top and bottom grating teeth do not overlap, and θ 1,i is obtained, the period of the ith grating teeth is calculated In this equation, λ c is the design wavelength, n e2 and n e3 are the effective refractive indices of the cross-section of the etched area, and n e1 is the effective refractive index of the cross-section of the unetched area. When the top and bottom grating teeth do overlap, the grating period can be expressed in a similar manner. 30)

Selection of the ion location
In the first inverse design step, a reference Gaussian light source was placed at the ion location and the geometry was optimized to maximize the coupling efficiency to the waveguide as shown in Fig. 5. To define the reference light source, the parameters such as the beam size, X-axis position, and angle were optimized such that the coupling efficiency was maximized for the focusing GC designed in step (b).

Full optimization of grating geometry
While the previous steps determined the geometry with theory, the inverse design optimizes the parameters with finite difference simulations to maximize the coupling efficiency. The duty factors of the top and bottom gratings D top,i and D bot,i , the grating periods Λ top,i and Λ bot,i , and the

Results and discussion
The procedures discussed in the previous section were used to simulate and design the focusing GC. In this optimization, no Si substrate was assumed, as a radiation boundary was used instead as shown in Fig. 5. This means that light diffracted downwards was lost as radiation. A 227 nm thick SiN layer with a refractive index of 2.0 was found to be optimal. The heights comprising the trapped ion position, H 1 and H 2 , were set to 1.0 μm and 29 μm, respectively (see Fig. 3). The initial value of the duty factor in step (a) was set to a relatively large value of 0.8 in order to reduce the scattering efficiency per grating-this increased the aperture size of the GC allowing for a small focused spot in free space. In step (d), a reference light source with an 18°tilted Gaussian distribution with a beam waist radius of 0.6 μm was used.

2D simulation
After the inverse design, in order to obtain a Gaussian light intensity distribution at the waveguide surface, the duty factor gradually decreases as the grating index increases. The duty factor of the gratings after the optimization process converged in the range from 50% to 90%. Figure 6 shows the light intensity distribution of the optimized GC. High directionality and tight focusing were successfully achieved. The diffraction efficiency, or the ratio of the power detected at the top boundary to the input power, was simulated to be 98.8%. The beam diameter (1/e 2 ) and beam angle at the target position were 1.7 μm and 20.8°, respectively. There was no constraint on the duty factor in the inversed design process.

3D beam shape
The 2D geometry of the GC was extruded to a 3D FDTD model for a more realistic simulation. Assuming a grating shift along the X-and Y-axes of 5 μm, a GC width of 4.0 μm was used to prevent overlap of the GCs as shown in Fig. 1. Figure 7 shows the light intensity distribution at the target position. The beam diameter in the X-axis was measured to be 2.0 μm, in good agreement with the 1.7 μm of the 2D   simulation. Since the grating is unfocused in the Y-axis direction, the diameter was as large as 11.2 μm. The beam shape along a 45°cross section A-A′ is the most important in this case, as the ion positions are located along a straight line on this axis as shown in Fig. 8. The beam diameter in the A-A′ plane was estimated to be 2.5 μm, slightly larger than in the X-axis direction due to the elliptical beam shape. The crosstalk into the neighboring qubits is evaluated by measuring the power intensity 7.07 μm away from the main peak in the A-A′ plane. This estimates the crosstalk into the neighboring qubits to be as low as −36 dB. Tables I and II show the summary of the 3D FDTD simulation result. Beam diameters of 1.7 μm, 11.2 μm and 2.5 μm were measured in the X-axis, the Y-axis and cross section A-A′ respectively. A low crosstalk of −36 dB was obtained because of the tightly focused elliptical beam in the X-axis. The breakdown grating cardinal directionality is shown in Table II. Thanks to the combination of the double-etched grating structure and the inverse design technique, the downward radiation, reflection, and through power were all suppressed to less than 1%-98% of the input light was diffracted upwards.

Fabrication tolerance
Next, to evaluate the impact of fabrication errors on design performance, we performed a Monte Carlo simulation based on the 2D simulation model. The error bounds are summarized in Table III. First, a variation of 10 nm was assumed for the SiN thickness. Since the designed grating contains a fine pattern several tens of nanometers in size, fabrication errors in the electron beam (EB) lithography-based process of 10 nm were assumed for each dimension. The double-etched structure requires multiple exposures for the top and bottom grating teeth, so precise alignment during the lithography process is required. In this case, a relative misalignment of the top and bottom gratings of 25 nm was assumed. The refractive index of the material and the pitch of the gratings were assumed to be constant. Based on these assumptions, 3000 Monte Carlo simulation trials were run. Figure 8 shows the results of the Monte Carlo simulation. The average beam diameter of 1.68 μm agreed with the designed value of 1.7 μm. The beam diameter variation was as small as 0.12 μm, and a maximum value of 1.80 μm was obtained at 2σ variation. The beam position was also affected, laterally shifting the peak position. However, fabrication errors are typically uniform across the die, suggesting that the relative beam position would not change for the whole grating array. Such a simultaneous beam position shift can be compensated by adjusting the bias conditions of the Paul trap, shuttling the ion position along the trap axis. Similarly, the variation of the output efficiency is also shown in Fig. 9. It was found that even when fabrication errors occur, the upward diffraction efficiency was still better than 93.0% in more than 95.4% of the cases. Although the fabrication errors increase the reflection back into the waveguide, the reflection can be managed by inserting an optical isolator into the system. The degradation in performance was mainly due to the misalignment Δd between the top and bottom gratings. This is because the variation of Δd disrupts the selective light diffraction conditions and significantly reduces the efficiency of reflection and diffraction. Other parameters did not have a significant effect on the characteristics. The transmission through the grating could also be suppressed by increasing the number of grating periods. Attention should be paid to the downwards light, as this could cause unwanted manipulation of the neighboring ion qubits. The diffraction efficiency towards the bottom side was less than 3.8% for 2σ variation; this indicates that the stray light in this system is low even when fabrication errors occur.
The Damascene reflow process effectively fabricates double etched GC structure, specifically bottom grating teeth, to obtain accurate and smooth grating structure. 31) In this process, SiO 2 is firstly etched, followed by SiN deposition and planarization. Top grating teeth are then fabricated by another SiN deposition and etching process. We can still adopt EB lithography for the damascene process, meaning that accurate grating geometry is achievable, although attention should be paid to thickness control.
Finally, a 3D FDTD simulation was performed under the condition where the beam diameter was the largest (1.77 μm) among the 3000 Monte Carlo trials to confirm the beam shape. Figure 10 shows the light intensity distribution at the target position. No clear difference was observed between   these two results in Fig. 10(a). Figure 10(b) shows the light intensity along cross-section A-A′. Low crosstalk values of less than 36 dB were attained, regardless of fabrication errors. From these results, it is shown that the proposed 1D focusing GC array can provide well isolated elliptical beams at the target position and suppress the stray light by upwards selective light diffraction.
In this report, we numerically demonstrated the proof of concept of a well-isolated beam focuser. The device specifications in an actual computer depend on the scale and the noise tolerance of the computation, meaning that the grating design and layout of the focusing GCs need to be optimized depending on the system requirement. Such optimizations will be studied in a future report, along with research on improving the performance of ion trap quantum computers.

Conclusion
We proposed a 1D focusing GC array for trapped ion qubit manipulation. A diagonally shifted array prevents the overlap of the beams at the target ion position. By leveraging inverse design optimization and a double-etched grating structure, the output light highly directional, tightly focused, and low in crosstalk. The fabrication tolerance was investigated through a Monte Carlo simulation, and it was confirmed that the diffraction efficiency maintained a high value of 93% with no significant increase in the crosstalk. These results suggest that