High-Performance 2 × 2 Bent Directional Couplers Designed With an Efficient Semi-Inverse Design Method

Directional couplers (DCs) are the basic elements for constructing various silicon photonic devices, including Mach–Zehnder switches (MZSs). Here, we propose a novel 2 × 2 bent directional couplers based on Bézier curves designed with an efficient semi-inverse design method. Two DCs were designed to operate at center wavelengths of 1.55 μm and 2 μm, with splitting ratios maintained between 45 to 55% over broad bandwidths of 130 nm and 160 nm respectively, and excess losses <0.05 dB. The present 1.55/2 μm DCs also have ultra-compact footprints of 16 × 1.8/24 × 2.5 μm2, which are much smaller than conventional DCs. The fabricated DCs show good fabrication tolerance of ±10 nm waveguide width deviation. The present bent DCs are then used for realizing thermo-optic MZSs for the 1.55/2 μm wavelength-bands, which show a bandwidth of >90 nm for achieving low excess losses of <0.3 dB and high extinction ratios of ∼18/20 dB.


I. INTRODUCTION
T HE directional coupler (DC) [1] is one of the most funda- mental functional devices in photonic integrated circuits (PICs).For example, DCs are popularly used as power splitters for Mach-Zehnder interferometers (MZIs) [2] or ring resonators [3], which are usually indispensable for optical switching, modulation, sensing, and optical computing [4], [5].Since a large number of DCs are often used in a PIC, ultralow loss, fabricationfriendly, and compact DCs are highly desirable.A straight DC has almost no excess loss (EL) in theory, but its bandwidth for uniform power splitting is quite limited (e.g., ∼ 30 nm [6]) due to waveguide dispersion.Song et al. reported a strategy utilizing an artificial gauge field to engineer the coupling dispersion, and dispersionless coupling can be realized by the structure with periodically bending modulation [29].Besides, the transmissions are highly sensitive to the waveguide width deviation, resulting in small fabrication tolerances.As an improved alternative, adiabatic asymmetric DCs were proposed to construct 3-dB power splitters with a broad bandwidth.However, their coupling lengths are usually pretty long (>160 μm) [7].In terms of 3-dB couplers, designs such as width-asymmetric DCs [8] or bent DCs [9] can possibly achieve low-loss and high-bandwidth 2 × 2 power splitting.Alternatively, sub-wavelength gratings (SWGs) with dispersion engineering provide a potential solution for realizing compact 2 × 2 DCs with a broad bandwidth [10].However, SWG DCs usually have relatively higher ELs than conventional DCs based on regular waveguides [11].Therefore, it is challenging to achieve low-loss, broadband, fabricationfriendly 2 × 2 DCs with compact footprints.
Recently, inverse design has shown great potential in developing passive silicon photonic devices with ultra-compact footprints.For the case of inverse design, the device usually has a black-box design region defined with many geometry parameters which are to be determined by the evolutionary algorithms (EAs) [11], [12], [13], [14], [15], [16], [17] or artificial neural networks (ANNs) [18].In this way, inverse design devices [12] possibly have a key region much smaller than their conventional counterparts designed according to the classical theories.The most popular structures are based on the topology optimization without shape-constraint [13], [14] or pixel configuration [15], [16], corresponding to search-space dimensions as high as ∼10 3 .In practice, for the design of an application-oriented photonic device, it is often required to solve a multi-objective problem without any appropriate initial individual.In this case, the inverse design becomes very challenging due to the ultra-high search-space dimension.In contrast, low-dimensional design space allows for better optimization [12], which can be achieved through structure definition.
The core strategy of semi-inverse design (SID) method is using multi-stage optimizations with manual interventions.Each stage of optimization is stopped when the FOM comes to a still.The manual interventions are introduced to modify the geometry parameter set or simulation settings.The next stage of optimization uses an initial individual derived from the solution of the last stage of optimization.In this way, the gradually increasing search-space dimension is kept as low as possible under artificial control.Both fast convergence rate and high design degrees of freedom for each dimension can be obtained in the multi-stage optimizations.In each stage of optimization, CMA-ES is used as the optimization algorithm.For the design of 2 × 2 DCs, the geometry framework in this work is based on the freely-shaped waveguides defined by Bézier curves.The design approach is very flexible.For example, in our previous work [28], the semi-inverse design method had been applied to develop several advanced passive photonic devices with sub-wavelength grating structures, including a mode (de)multiplexer, a 90°hybrid, and a wavelength (de)multiplexer.
In this work, 2 × 2 DCs for the wavelength-bands of 1.55/2 μm are designed with a novel geometry definition based on Bézier curve by SID method, which introduces sufficient degrees of freedom for each dimension and low search-space dimension simultaneously.With such a strategy, the computation cost is low while the designed DCs have high performance and compact footprints.Furthermore, the designed DCs are used for the realization of broadband MZSs with low ELs.The fabricated MZSs work well in the wavelength-bands of 1.55 μm and 2 μm, exhibiting high ERs of ∼18 dB and ∼20 dB over bandwidths of >90 nm, respectively.

II. STRUCTURE AND DESIGN
In this work, we focus on the development of 2 × 2 DCs by using a semi-inverse design method based on geometry definition.The key is introducing a special geometry definition, which minimizes the search-space and improves the probability of finding a high performance design.The present DC consists of two bent sections with a flexibly adjustable shape, as shown by the blue part in Fig. 1(a).Meanwhile, there are two transition sections (i.e., the green part) connecting the bent sections and the input/output waveguides.The bent sections and the transition sections are designed with a Bézier-curve center line and identical waveguide widths w.Generally speaking, the nth-order Bézier curve is usually defined with (n+1) control points (see Appendix A), which may not be on the curve.Thus, the locations of the control points cannot intuitively reflect the shape of the whole Bézier curve.
In order to conveniently monitor the minimum gap in the coupling region, the two bent sections are defined by 2n waveguide points {P 1 down …P n down , P 1 up …P n up } located on the two Bézier curves, as shown in Fig. 1(a).As shown in Fig. 1(b), for any Bézier curve, a one-to-one mapping is constructed between the waveguide points and the control points (see details in Appendix A).The waveguide points are used to create individual sets, and the control points are used for drawing the Bézier curve.These control points play an important role for the degree evolution process.As shown in Fig. 1(c), the Bézier curve generated by the optimization at stage #k can be used for the initialization of the optimization at stage #(k+1) by adding control points as well as the corresponding waveguide points (see details in Appendix B).
Such a structure can be equivalent to bent DCs, consisting of multiple sections with independently adjustable radii.As a result, the present bent DC has higher degrees of freedom for design space to possibly performs better than traditional bend DCs based on arc-curves.Thus, it has the potential to obtain better performance and a more compact footprint than traditional adiabatic DCs [7], [25].Meanwhile, the waveguide structures are slowly varied without any abrupt junction, and thus the scattering loss should be negligible.Furthermore, the device is expected to be low-loss if the bending radius is sufficiently large.The fabrication for such a bent DC is friendly because the feature size is chosen to be more than 130 nm (according to the fabrication process).More importantly, the present geometry definition inverse design method enables much lower searchspace dimension than conventional inverse designs [22], [23].From Fig. 1(a), we can see that P n down and P n up are two points defined at the same x value.Therefore, if the waveguide defines n points, the search-space dimension is 3n.For example, when n = 5, there are ten points included and the corresponding search space {x 1 , x 2 …x n , y 1 , y 2 …y n , y n+1 …y 2n } has a low dimension of 15 only.In contrast, those conventional inverse designs usually have a search-space dimension as high as 10 2 -10 3 .The method uses the Covariance Matrix Adaptive Evolution Strategy (CMA-ES) for multi-stage optimization [17].Considering the compact footprint of the present DCs, the 3D-FDTD [15] is used for simulating the light propagation.
Fig. 2(a) shows the optimization process of the 2 × 2 DC to have a splitting ratio of ∼50%:50% at the center wavelength of 1.55 μm.Here, the figure of merit (FOM) and the computation time are shown as a function of iterative generation.To save the computational cost, one-input simulation is performed first.The corresponding FOM for any given wavelength λ i is defined as where S xy is the coupling coefficient from mode #y to mode #x.
Here modes #1-#4 respectively refer to the TE0 modes of ports #1-#4.When it is desired to achieve a broadband DC, multiple operating wavelengths should be considered.As an example, we consider the case involving five wavelengths, and corresponding FOM is given by In particular, we choose the five wavelengths as {1.5, 1.525, 1.55, 1.575, 1.6} μm to make the DC work well at the center wavelength of 1.55 μm, with bandwidth of 100 nm.
For the optimization at stage #1, we set n to 3 (i.e., searchspace dimension = 9) to be simple and efficient.From Fig. 2(a), it can be seen that the FOM comes to a standstill after ∼20 iterations, which indicates that the optimization of stage #1 is suspended accordingly.The final result from stage #1 is then used to initialize the individual with n = 5 (i.e., dimension = 15) for the optimization at stage #2.In this way, the optimization at stage #2 improves the search-space dimensions and the optimization results are improved.Then the optimization is initialized at stage #3 by increasing the lengths of waveguides in the final structure optimized from stage #2, so that the bent waveguides are further smoothened.Notice that the search space dimension is gradually increased in the design, our optimization strategy enables fast convergence and high degrees of freedom for each dimension.As Fig. 2(a) shows, the FOM varies from 2 to 0.1 within 92 generations in the first three stages of optimization, taking 27 hours totally.In order to run the optimization efficiently, only the simulation with light input from one port is run for the optimization process of stage #1-3.Finally, the device performance is verified by running a full simulation with the inputs from ports #1 and #2 at stage #4.In this case, the modified FOM is defined by where FOM 2-in is given by As shown in Fig. 2(a), the FOM for the optimization at stage #4 varies from 0.6 dB to 0.148 dB within 26 generations, taking 32 hours.In a similarly way, a 3 dB 2 × 2 DC at the center wavelength of 2 μm is also designed, and the results are shown in Fig. 2(b).Since the designed 1.55-μm-band DC also shows good performance even operating at the center wavelength of 2 μm, the design parameters for the 1.55-μm-band DC are used to initialize the 2-μm-band DC.In this way, we directly run the simulation with the direct dual-port input at stage #4.The set of wavelength is chosen as {1.9, 1.925, 1.95, 1.975, 2} μm.Finally, the DC obtains a FOM of 0.122 dB within 76 generations, taking ∼80 hours in total.
Fig. 3(a) presents the simulation results of the designed 2 × 2 DC with 440-nm-wide waveguides at the 1.55-μm-band.The device footprint is as compact as 16 × 1.8 μm 2 , and the minimal gap width is ∼110 nm.For the designed 2 × 2 DC, the EL is < 0.026 dB in the wavelength band of 1455-1615 nm in theory.Here the excess loss is defined as EL = max{EL 1 , EL 2 }, where  shows, the performance degradation is negligible when width changed from 620 nm to 650 nm.Fig. 3(d) shows the simulated light propagation fields of the designed DC when operating at the wavelengths of 1870 nm, 1950 nm, and 2030 nm, respectively, indicating that the designed DC can also operate in a broad bandwidth.The bandwidth of the device is ∼160 nm when the width deviation δw is ±10 nm, and decreases to ∼80 and ∼130 nm when δw is −30 nm and 20 nm respectively.
As a summary, it can be seen that high-performance DCs with low ELs, broad bandwidths and compact footprint are achieved with a low computational cost by using the present SID method.This is attributed to that the Bézier-curve-based geometry-defining strategy is introduced for the SID method, which simultaneously enables low search space dimension and excellent initialization for the optimization.When more design points are introduced, it is possible to further improve the performance.

III. FABRICATION AND MEASUREMENT
In order to characterize the DCs, we measured the normalized transmissions |S 31 | 2 and |S 41 | 2 of the fabricated DCs when operating at the center wavelengths of 1.55 μm and 2 μm, as shown in Fig. 4.
Symmetric MZIs were also designed and fabricated by inserting two DCs placed centrosymmetrically (see Fig. 5(a)).The fabrication was carried in Applied Nanotools Inc [19].The designed DCs were fabricated on a silicon-on-insulator (SOI) wafer with a 220 nm-thick top-silicon layer and a 2-µm-thick buried dioxide layer.The silicon photonic waveguides were fabricated by electron-beam lithography and inductively-coupled plasma dry-etching.Then 2.2-µm-thick silicon dioxide cladding was deposited with the chemical vapor deposition.Fig. 5(b) and (c) show the SEM images of the DCs for the 1.55-μm and 2-μm bands, respectively.The fabricated devices were characterized by using a broadband light source and an optical spectrum analyzer (OSA).
The waveguide width deviations of δw = −10, 0, 10 nm were introduced in the experiments.In Fig. 4, the fabricated 1.55/2-μm-band DCs have ELs less than 0.05 dB with bandwidths of 80/100 nm.There is some difference between the   The thermo-optic Mach-Zehnder switches (MZSs) with the designed DCs were also fabricated, as shown in Fig. 6(a).The measured transmissions of the MZS at the 1.55-μm-band are shown in Fig. 6(b).The on-state is achieved when applying a 2.6 V bias and a current of 23.802 mA on the heating electrode by using a source meter (Keithley 2401).It can be seen that one has EL< 0.4 dB and ER>17 dB in the wavelength range of 1520-1610 nm for the ON and OFF states.For the 2-μm-band MZS, one has the ON state when applying a 3.08 V bias and a current of 31.2 mA.Correspondingly, the EL is < 0.6 dB and the ER is >18 dB for both states in the wavelength range of 1900-2000 nm.
Table I gives a comparison of the 2 × 2 DCs as well as the MZIs/MZs reported in recent years.Among them, a bent DC has attracted intensive attention because it has a low EL (∼0 dB) and a large bandwidth (∼ 100 nm) as well as large fabrication tolerance [9], [20].An asymmetric coupler also performs well with a low EL (∼0 dB), a large bandwidth (>100 nm) as well as large fabrication tolerances [21].However, the coupling lengths are as long as ∼45.6 μm.The SWG coupler can work well with a bandwidth as large as hundreds of nanometers [11].However, the EL is usually higher than 0.1 dB, which is unacceptable for the PIC with numerous DCs.For a segmented asymmetric coupler [12], whose widths of different sections were optimized carefully, the EL is low and the bandwidth is large in theory.However, the total length is relatively large.As an alternatively, there have been some inverse design DCs reported [22], [23].The inverse-designed DCs using genetic algorithms were presented to have arbitrary coupling ratios [22].However, often leads to local minimum converge.For a segmented asymmetric coupler [12], whose widths of different sections were optimized carefully, the EL is low and the bandwidth is large in theory.However, the total length is relatively large.Another inverse design DC was demonstrated to have an ultra-compact footprint of 3 × 1.2 μm 2 with an EL of 0.5 dB and a bandwidth of 45 nm [23].The final designs in our work look similar to the bent DCs reported in these works [9], [20], [24].In contrast, our devices are based on Bézier curves with high flexibility for the structure design, and consequently exhibit very decent performance as well as a compact footprint when compared to those previous works.
In the present work, the 1.55-μm-band DC has a very low EL and a very large bandwidth (comparable to the state-of-the-art works), while the footprint is as compact as 16 × 1.8 μm 2 .The fabricated MZS with the 1.55-μm-band DC has a high ER of >15 dB in a large bandwidth of over 90 nm.Similarly, the 2-μm-band DC and MZS also work very well, but the footprint becomes relatively larger due to the longer wavelength.The fabrication tolerances are ±10 nm waveguide width deviations, while the minimum gap width for the 1.55-μm and 2-μm-band DCs are larger than 100 nm to be relatively easy for fabrication.As a result, our DCs can be fabricated conveniently by the regular electron-beam lithography process.The demonstrated MZSs with low ELs and high ERs in a broad bandwidth are attractive for large-scale PICs in the future.

IV. CONCLUSION
In this article, we have designed and demonstrated 2 × 2 3-dB DCs by a semi-inverse design method with a novel geometry definition based on Bézier curves.For the designed DC at center wavelengths of 1.55 µm and 2 µm, the designs region are as short as 8 µm, while the bandwidth is as large as 130/160 nm for achieving splitting ratios between 45% to 55%, as well as the EL<0.05 dB.The MZIs based on these DCs at 1.55/2-µm-band were realized with the bandwidths of 90/100 nm for ELs < 0.3 dB and ERs > 15/20 dB.Furthermore, it has been shown that the present MZSs with bent DCs have excellent reproducibility.In contrast to the previous works, our DCs have comparable performance and compact footprints with fabrication-friendly.The present MZSs with the designed DCs have good reproducibility and the potential for large-scale integration.In the future, the present SID method can be generalized to the design of other photonic devices (such as high-Q ring resonators) with DCs, achieving compact footprints and high performance.

APPENDIX A
A. Details of the Bézier-Curve Geometry Definition Fig. 7 shows the schematic of one waveguide used for the 2 × 2 bent DC.The waveguide consists of three sections, namely, the bent section based on a quadratic Bézier curve, the transition section based on a 3rd-order Bézier curve, and the input/output section.The input/output section (orange parts) is formed by straight single mode optical waveguides.The bent and transition sections have a center line defined by Bézier curves.All the sections have the same width w, and the waveguide outline is determined by extending the points on the Bézier curve with a distance w/2.
Bézier curves have been widely used in two-dimensional graphics programs.A n th -order Bézier curve is determined by a set of control points {C 0 , C 1 , … , C n } in the following way where t is a variable satifying 0≤t≤1, (A2) In details, P (t) satisfies P (t)A 1 /A 1 A 2 = t.by the relationships of A 1 C 1 1 C 2 = t and A 2 C 2 /C 2 C 3 = t, two auxiliary points (A 1 and A 2 ) can be found at the auxiliary lines C 1 C 2 and C 2 C 3 .In this way, the Bézier curve can be drawn by sweeping t from 0 to 1. Similarly, the 3rd order Bézier curve used for the transition sections can be generated, as shown in Fig. 7(c).
With these control points, some waveguide points locating at the corresponding Bézier curve (i.e., the waveguide central line) are then obtained.For example, for a given (2n-3) th curve, one has n waveguide points mapping to the (2n-2) control points.In this way, the gap between the bent waveguides in the coupling region can be controlled conveniently from the waveguide points.Definitely, the control points {C 0 , C 1 , … , C n } can also be generated from any given waveguide points {W 0 , W 1 , … , W n-1 } inversely.In particularly, for an example shown in Fig. 8, the four control points {C 1 , C 2 , C 3 , C 4 } can be generated from the three waveguide points {W 1 , W 2 , W 3 }.The control points C 1 and C 4 are chosen simply as the same as the waveguide points W 1 and W 3 , respectively.On the other hand, the control points C 2 and C 3 are generated with the assistance of points M 1 and M 2 , where M 1 and M 2 are respectively the midpoints of W 1 W 2 and W 2 W 3 , and the following conditions should be satisfied: , where K 0 is the midpoint of M 1 M 2 , and η is a custom factor to control the curve smoothness (e.g., we choose η = 0.8 in this work).

A. The Details of the Bézier-Curve Degree Evolution
As it is well known, the Bernstein function has recursive property, i.e., where k = 0, …n and Here we show how to the n th -order Bézier curve with the control points {C 0 , C 1 …C n } is upgraded to the (n+1) th -order Bézier curve with the control points {D 0 , D 1 …D n , D n+1 }.Here the n th -order and (n+1) th -order Bézier curves are identical and one has D 0 = C 0 and D k+1 = C k .Furthermore, one should have Accordingly, the point D k is determined by where k = 0, 1 …n+1.Apparently, the point D k locates at the line C k-1 C k and divides the line C k-1 C k into two parts with a length ratio of k n+1 /(1 − k n+1 ).In this way, any higher-order Bézier curve can be obtained for the degree evolution.
In order to be clarified, the optimization processes of stages #1 and #2 in Fig. 1(d) and (e) in the main text are described with details.As shown in the Fig. 9, three waveguide points (corresponding to four control points) were used to define the waveguide position in stage #1.After several generations of optimization, the FOM becomes saturated, indicating that the degree of freedom should be upgraded.As mentioned above, the degree evolution of a Bezier curve can be realized by adding control points.As shown in Fig. 9, when one introduces eight new control points, five new waveguide points {W 1 , W 2 , W 3 , W 4 , W 5 } can be mapped accordingly.The midpoints of D 1 D 2 , D 3 D 4 and D 5 D 6 are taken as the waveguide points W 2 , W 3 and W 4 , respectively.In this way, the next-stage optimization can be initialized with an improved individual-dimension of 15 (corresponding to the five waveguide points).

A. Measured Transmissions of Symmetric MZIs With Width Deviation
As shown in the Fig. 10(a), the bandwidth of ER >15 dB is larger than 90 nm with δw = ±10 nm.It should be noted that we cannot accurately measure the bandwidth due to the limitations Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. of the grating couplings.As for band, the bandwidth of ER >16 is about 100 nm These results show that both 1.55μm-band and 2-μm-band DCs have good fabrication tolerances.

Fig. 1 .
Fig. 1.Design of a 2 × 2 DC.(a) Bézier-curve geometry definition for the semi-inverse design method.(b) The waveguide points and the control points.(c) The degree evolution of Bézier curve.

Fig. 2 .
Fig. 2. Figures of merit and consumption time as functions of the iteration generation for the design DCs for the wavelength-bands of 1.55 μm (a) and 2 μm (b).
As shown in Fig.3(a), the DC is designed to operate at a central wavelength of Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

1 .
55 μm, and the splitting ratios |S 3i | 2 and |S 4i | 2 are maintained between 45-55% over a broad wavelength of 1485 to 1615 nm.The results show that the designed DC still works very well even with δw = −20/20 nm.The bandwidth of the DC reduces from 130 nm to 120 nm when the width deviation δw changes from 0 to 20 nm.Fig. 3(c) shows the simulated light propagation fields in the designed DC when the wavelengths are respectively 1485 nm, 1550 nm, and 1615 nm, which further confirm that the designed DC is broadband.Fig. 3(b) and (d) show the theoretical results of the designed DC working at the 2-μm wavelength-band.For this design, the waveguide width is w = 640 nm, and the footprint is 24 × 2.5 μm 2 .The minimal gap width is ∼100 nm.Similarly, the splitting ratio |S 31 | 2 and |S 41 | 2 are maintained between 45%:55% in a broad wavelength of 1870-2030 nm, and EL < 0.03 dB over broad bandwidth of 200 nm.As Fig. 3(b)

Fig. 5 .
Fig. 5. Experimental results of the fabricated symmetric MZIs with DCs.(a) Microscope picture of the PIC for measuring the MZI consisting of two centrosymmetrically-placed DCs and arms with equal lengths.Scanning electron microscope (SEM) images of the fabricated DCs at the (b) 1.55-μm band and at the (c) 2-μm band.Measured transmissions of symmetric MZIs at the (d) 1.55-μm and (e) 2-μm band.ER: Extinction ratio.

Fig. 6 .
Fig. 6.Fabricated MZSs and measured results.(a) Microscope picture of the fabrication thermo-optical MZS.(b) The transmissions of the 1.55-μm-band device in the off-state and the on-state.(c) The transmissions of the 2-μm-band device in the off-state and the on-state.
B k,n (t) = ( n k ) t k (1 − t) n−k is the Bernstein function, and C k represents the k-th control point (k = 0, …, n).Particularly, P(t = 0) and P(t = 1) represents the start and end point of the curve, respectively.As an example, the 2nd order (quadratic) Bézier curve in Fig. 7(b) is determined by

Fig. 9 .
Fig. 9. Degree elevation process of the Bézier curves in the optimization.Here three and five waveguide points are used for the optimization in stage #1 and stage #2.

TABLE I SUMMARY
OF2 × 2 DCS AND RELATED MZIS ON SILICON