Ultra-sharp silicon multimode waveguide bends based on double free-form curves

Shangsen Sun; Zhiqiang Yang; Juanli Wang; Runsen Zhang; Fengchun Zhang; Ning Zhu; Lei Wan; Zhaohui Li

doi:10.1364/PRJ.445529

Abstract

Mode-division multiplexing (MDM) can greatly improve the capacity of information transmission. The multimode waveguide bend (MWB) with small size and high performance is of great significance for the on-chip MDM integrated system. In this paper, an MWB with high performance based on double free-form curves (DFFCs) is proposed and realized. The DFFC is a combination of a series of arcs optimized by the inverse design method. The fabrication of this MWB only needs one-step lithography and plasma etching and has a large fabrication tolerance. MWBs with effective radii of 6 μm and 10 μm are designed to support three modes and four modes, respectively. The proposed method gives the best overall performance considering both the effective bending radius and the transmission efficiency. The fabricated MWB with four mode channels has low excess losses and crosstalks below

- 21 dB

in the wavelength range from 1520 to 1580 nm. It is expected that this design can play an important role in promoting the dense integration of multimode transmission systems.

1. INTRODUCTION

With the development of all kinds of data services, information technology is now facing an extreme challenge in requirements for the expansion of transmission capacity accommodated within a single optical fiber [1 –5]. The newly proposed mode-division multiplexing (MDM) technology utilizes different modes in the same wavelength band, and the amount of carried information can be further increased by this method [6 –9]. In order to compactly route the guided modes in the on-chip multimode communication system, many functional waveguide devices suitable for multimode manipulations have been reported, including tapers [10,11], mode converters [12,13], multimode waveguide crossings [14 –17], and multimode waveguide bends (MWBs) [12,18]. It is well known that when the transmitted light passes through a conventional MWB with compact size, there will be serious inter-couplings between the original guided modes inside the straight waveguide and the distorted modes inside the bent waveguide [19], causing unignorable excess losses and inter-modal crosstalks. Besides, for all kinds of the multimode waveguides, higher order modes have weak optical field confinement and tend to require a larger bending radius to reduce the radiation losses. An optimized waveguide bend requires a larger bending radius to accommodate more modes. Therefore, the design of a bent waveguide with a small radius and supporting as many modes as possible is of great significance to the monolithic MDM system [20].

In recent years, many methods have been proposed to realize the sharp bending of on-chip silicon multimode waveguides. An ultra-sharp silicon MWB based on the meta-surface (grating-like) structures has been proposed. The nonuniform grating mode converter on the straight multimode waveguide converts the “straight modes” to the “bent modes” ahead of the bend by changing the effective refractive index of the straight waveguide [21]. By this method a subwavelength grating (SWG) waveguide supporting four modes is achieved with the effective radius of 45.8 μm. Alternatively, these SWG structures can be also used to compensate for the structural asymmetry in the sharp bending region and convert the “bent modes” back to “straight modes” in the bent waveguide. The proposed MWB with the SWG on the bending part can support four TE modes with a bending radius of only 20 μm [22]. However, the manufacturing processes of these SWG-assisted bends are rather complex, which raises the scattering loss as well as the cost to some extent. Using a specially designed curve as the bending part of an MWB is another way to reduce the effective radius, since a curve with gradually changing radius can greatly reduce the mode field mismatch at the waveguide junction. MWBs based on the Bezier curve supporting three TE modes and on the modified Euler curve supporting four TM modes are designed, with bending radii of 20 μm and 45 μm, respectively [23,24]. Although these kinds of devices usually provide better transmission efficiencies by avoiding the scattering from the subwavelength mode-converting structures, the sizes are limited by the mathematical functions of the curves. While all the designs mentioned above use arc or other curves to bend the light, total internal reflection (TIR) can also be utilized to redirect the light [25]. Based on this principle, multimode waveguide corner bend (MWCB) supporting up to 10 TE modes is reported with the effective radius of 35 μm. Unfortunately, the MWCB requires a long adiabatic taper, which potentially deteriorates the integration density of the device. Other design methods of MWBs such as using mode converters and Etalon lens have also been proposed, but the scalability to higher order modes may be the key problem limiting the applications of these methods [12,18].

Instead of solving the problems forward, some works based at least partially on inverse design methods have also been proposed. An MWB supporting four TE modes with an effective radius of 17 μm is presented with the help of the transformation optics (TO) method [26]. The TO method is to map the traditional waveguide bend in the physical space to the virtual space as a distorted graded index waveguide. The boundary shape of this distorted waveguide in the virtual space is inversely optimized for highly efficient multimode transmission by the quadratic approximation method, and is then mapped back to the physical space to acquire the final sharp bending with high performance. In our previous work, an ultra-compact 90° MWB based on the inverse design has also been proposed [27]. The special curve in this work optimized by the inverse design method is called the free-form curve. The curve is directly optimized for multimode transmission in the physical space. The proposed free-form curve is such a smooth curve that the curvature radius of every point on it can be defined freely without the constraint of any mathematical function, and it is set to be self-symmetric around the angular bisector for convenience. In practice it is discretized by a series of different arcs (the discretization number is $2 D$ considering the symmetry), and each two adjacent arcs have the same tangent at the junction. A combination of the reasonable curvature radii with low propagation loss can be found for these arcs by using the inverse design algorithm [28 –35]. The optimization method used here is called direct range search (DRS) by modifying the famous direct binary search (DBS) method [36], which has been widely used in the field of optical simulations. In this paper, instead of using the MWB with a constant waveguide width, the internal and external curves of the MWB are designed as independent free-form curves. MWBs with double free-form curves (DFFCs) have been designed with even smaller bending radii compared with MWBs with single free-form curves (SFFCs). The reason lies in the effects of the waveguide width on the transmission performance. When the width reduces, the asymmetry between the inner and outer radii of the bent waveguide also reduces. This is helpful to reduce the field asymmetry and consequently the effective bending radius of the MWBs. However, a smaller waveguide width may increase the radiation losses of those high order modes inside the bending region. Thus, there is a trade-off between the loss and crosstalk when changing the width of the bent waveguide. Both the radius and the width everywhere on the bent structure need to be optimized carefully by the inverse design algorithm to reach the optimal bending performance. For the sake of this goal, we demonstrate two designs of the proposed DFFC–MWB with equivalent radii of 6 μm and 10 μm to support three and four TE modes, respectively. The size of the DFFC–MWB with four modes is about 50% smaller than that in our previous work. A comparison of the experimental results is also presented.

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2. DESIGN AND SIMULATIONS

As shown in Fig. 1, the MWB with an equivalent radius of 10 μm is composed of two free-form curves A and B. $W_{S}$ is the width of the MWB, which is not a constant necessarily. The free-form curves are mirror-symmetrical along the angular bisector, and the whole segment number is set as $2 D$ for both curves. The inset of Fig. 1 shows the inner free-form curve A consisting of a series of different curvature radii, with $D = 3$ for convenience of presentation. $R_{i}$ is the curvature radius of the $i$ th arc and should decrease monotonously from $R_{1}$ to $R_{D}$ to shrink the overall bending radius. $O_{i}$ is the center of the $i$ th arc, and $R_{A}$ is the equivalent radius of curve A. The outer curve B has the same structure but is composed of a series of different radii $R_{i}^{'}$ . The figure-of-merit (FOM) function is defined as follows: the device is based on the silicon-on-insulator (SOI) platform for manufacturing compatibility [37,38]. The thickness of the silicon core layer is 220 nm, and the top layer could be air or silica. The 3D finite-difference time-domain (FDTD) method is used to calculate the FOM of the MWB. The FOM function is defined as $FOM = 1 - \frac{1}{n} (1 - α) \sum_{I}^{n} (1 - T_{i}) - \frac{1}{n} α \sum_{I}^{n} X_{i} .$ (1)The excess loss and crosstalk of the $i$ th mode are represented by $T_{i}$ and $X_{i}$ , respectively ( $1 \leq i \leq n$ ). Depending on the emphasis of the simulation results, $α$ is set to different values in the range of (0,1). Usually, $α$ can be increased in the later stages of the simulations to further suppress the inter-mode crosstalks. Ideally, for the proposed device, the FOM is 1 for total transmission without any loss and crosstalk.

Proposed MWB based on the DFFC design. DFFC consists of curve A and curve B, and WS is the width of the MWB. The inset shows a schematic diagram of the change of curvature radius, and RA is the equivalent radius of the free-form curve A.

Figure 1.Proposed MWB based on the DFFC design. DFFC consists of curve A and curve B, and $W_{S}$ is the width of the MWB. The inset shows a schematic diagram of the change of curvature radius, and $R_{A}$ is the equivalent radius of the free-form curve A.

It is well known that setting some reasonable initial parameters is very important and time-saving for the inverse design [39,40]. In our previous work, $R_{i}$ was defined as a random number from 0 to $R_{MAX}$ ( $R_{MAX} = 5 R_{A}$ typically, or more). By randomly selecting $R_{i}$ one by one, it is somehow difficult to find all initial values that can be connected successfully from the input waveguide to the angular bisector, because sometimes the $R_{i}$ decreases too fast to extend the curve properly. Thus, in this paper we use an improved method to obtain the initial values of all $R_{i}$ : $R_{i} = R_{MAX} [1 - {rand}_{i}^{m}] 1 \leq i \leq D .$ (2)Herein the function rand returns a uniformly distributed random number from 0 to 1. The value of $m$ is a real number from 0 to infinity. The higher the value of $m$ , the greater the probability of large $R_{i}$ . Then we can rearrange $R_{i}$ by the sequence of their values to construct the curves. By selecting a proper $m$ value, Eq. (2) tends to find values that can be used as the initial curve pattern efficiently. Moreover, the FOM of the initial curves determined by these $R_{i}$ values also has a greater probability to reach a large value than those curves made of the $R_{i}$ values picked randomly. A large FOM value of the initial waveguide structure can efficiently shorten the optimization time.

After curve A and curve B are obtained from Eq. (2), the width at the narrowest point of MWB is used as a screening criterion. Waveguides failing to support all the required modes with too small width at the narrowest point will be discarded. After comparing the values of the FOM, 100 groups of DFFCs with available performance are found. The segment radii normalized by the effective bending radius are shown in Fig. 2. At the following stage, the curves A and B will be reversely designed using the DRS algorithm described in Ref. [27]. In short, the optimization process can be defined as follows. Each $R_{i}$ of curve A/B is varied in sequence, and the change will be discarded or kept depending on the FOM value. These searching procedures are repeated until the device’s performance meets the criteria. For example, when $R_{1}$ is optimized, values from $R_{2}$ to $R_{MAX}$ at an equal interval of 50 steps can be sampled for $R_{1}$ either in sequence or randomly. Each time the FOM of the MWB is calculated after a new value of $R_{1}$ is updated. If the FOM is greater than the initial value, the updated $R_{1}$ value is reserved, and the search for the next $R_{i}$ ( $R_{2}$ here) is then started. Otherwise, the FOM is recalculated after $R_{1}$ is changed again. If the FOM value is always smaller than the initial value when all the suitable values of $R_{1}$ have been tried, the search for the next $R_{i}$ will be started anyway. The curve A can be obtained after the last radius $R_{D}$ has been optimized. Similarly, the shape optimization of curve B will start in the same way. Repeat the above steps until the FOM of the proposed device is good enough. By this method, the MWB supporting three TE modes with an equivalent bending radius of 6 μm is designed as an example. The input waveguide width of the MWB is $W_{S} = 1.1 μm$ to support three modes, and the segment number $D$ is 20. Figures 3(a)–3(c) show the simulated optical field distributions in the designed MWB based on the DFFC for ${TE}_{0}$ , ${TE}_{1}$ , and ${TE}_{2}$ modes at the wavelength of 1.55 μm, respectively. It is easy to see that the guided waves for all three modes are almost unchanged at the output ports. The calculated transmission spectra of all modes in the device are shown in Figs. 3(d)–3(f). When operating at 1550 nm, the theoretical inter-modal crosstalks for three modes are all less than $- 31 dB$ and the excess losses are less than 0.2 dB. In the wide wavelength range of 1.5–1.6 μm, the inter-modal crosstalk is still very low ( $< 24 dB$ ) for each mode.

Figure 2.Proper combinations of (a) curve A and (b) curve B are given. Here the ordinate represents a multiple of the equivalent radius.

Figure 3.Simulation results of the DFFC–MWB with an input width $W_{S} = 1.1 μm$ . (a)–(c) Simulated optical field distributions in the designed MWB based on the DFFC with ${TE}_{0}$ , ${TE}_{1}$ , and ${TE}_{2}$ modes, respectively. (d)–(f) Calculated transmission spectra for three modes in the wavelength range of 1.5–1.6 μm. Herein the dotted line corresponds to the axis on the left, and the real line corresponds to the axis on the right.

Figure 4.Simulation results of the DFFC–MWB with an input width $W_{S} = 1.48 μm$ and an equivalent radius of 10 μm. (a)–(d) Simulated optical field distributions in the device with four mode channels at 1.55 μm. (e)–(h) Calculated transmission spectra of the four-mode bent waveguide in the wavelength range of 1.5–1.6 μm. Herein the dotted line corresponds to the axis on the left, and the real line corresponds to the axis on the right.

In order to demonstrate the superiority of the DFFC design on the SFFC design in our previous work, an MWB based on the SFFC with the same effective radius and waveguide width is also simulated. The results are compared with each other in Table 1. For simplicity, only the worst cases of all crosstalk values at 1.55 μm are displayed in the table. It should be noted that these crosstalk values do not necessarily come from the same mode. As shown in Table 1, it is obvious that the performances of the DFFC designs are more balanced and better than those of the SFFC ones. The excess losses and crosstalks of the DFFC–MWBs are better than those of the SFFC–MWBs for almost all modes. Only the loss for the ${TE}_{3}$ mode of the SFFC design is slightly lower than that of the DFFC design. This is because the reduction of the waveguide width could increase the radiation loss of the high order mode, which slightly decreases the transmission of the highest order mode in the DFFC bend.

Last but not least, the fabrication tolerance should also be taken into account when selecting different MWB designs. For electron-beam lithography (EBL), the main fabrication error of an MWB device arises from the width deviation caused by the proximity effect in the exposure. In Fig. 5, we give the effects from the width deviation on the excess loss and crosstalk of a four-mode MWB. Since the deviation is much smaller than the waveguide width, we set the deviation $Δ W$ constant everywhere for the whole MWB device. The width error is introduced by increasing or decreasing the magnitude of $Δ W / 2$ to the original curvature radii of each segment on curves A and B. The value of $Δ W$ varies from $- 80$ to 80 nm with a step of 10 nm. Figure 5 shows the transmission efficiencies of all modes at 1.55 μm with varying $Δ W$ up to $\pm 80 nm$ . The worst value of all the crosstalks is around $- 20 dB$ within an error range of $\pm 40 nm$ , which roughly gives the fabrication tolerance. This result shows the good fabrication tolerance of the method reported in this paper.

Figure 5.Transmittance spectra of the MWB based on the DFFC with $Δ W$ varying from $- 80$ to 80 nm. (a)–(d) Transmission efficiencies of the four modes at 1550 nm. Simulation results show that this device has large fabrication tolerance.

3. FABRICATION AND CHARACTERIZATION

The devices fabricated on the SOI platform are tested to verify the above simulation results. The thickness of the silicon core layer and the silica under cladding layer is 0.22 and 2 μm, respectively. The EBL is used to define the patterns, and inductively coupled plasma (ICP) dry etching is applied to transfer the patterns to the silicon core layer. The mode (de)multiplexers with four mode channels are composed of asymmetrical directional couplers (ADCs) to (de)multiplex all the required modes [21,27]. The ${TE}_{0}$ modes in the single mode waveguides are switched by the ADCs to desired high order TE modes. Figures 6(a) and 6(b) show the fabricated multiplexers and demultiplexers, which are connected to the straight multimode waveguide or the eight cascaded 90° MWBs based on the DFFC. A scanning electron microscope (SEM) picture of the cascaded MWBs is shown in Fig. 6(c). For comparison, the eight cascaded 90° MWBs based on the SFFC with the same equivalent radius and waveguide width are also fabricated on the same chip.

Figure 6.Microscopic view of the fabricated silicon multimode bends with four mode channels. The multiplexer and demultiplexer are connected with (a) straight multimode waveguide or (b) eight cascaded 90° MWBs based on the DFFC. (c) SEM image of the cascaded MWBs.

A tunable laser (Keysight 8164B) is used as well as a high-speed photodiode (Agilent 11982A) to measure the transmittance spectra of the fabricated devices. The insertion losses of the mode (de)multiplexers and the vertical grating couplers should be extracted from the transmission curve of each MWB [as shown in Figs. 6(a) and 6(b)]. The measured spectra of a single 90° four-mode bend based on the DFFC are shown in Figs. 7(a)–7(d). For the ${TE}_{0}$ , ${TE}_{1}$ , ${TE}_{2}$ , and ${TE}_{3}$ modes, the waveguide bend has excess losses of 0.01, 0.05, 0.58, and 0.72 dB, respectively. The worst case of all crosstalk values is $- 25 dB$ . In a broadband wavelength range from 1520 to 1580 nm, the crosstalks between all modes are below $- 21 dB$ . Measurement results of the four-mode bend structure based on the SFFC are also shown in Figs. 7(e)–7(h). The crosstalk from ${TE}_{2}$ to ${TE}_{1}$ is about $- 15 dB$ at 1.55 μm. The performance of the four-mode bent waveguide based on the DFFC is obviously better than the one based on the SFFC with the same size, particularly when considering the crosstalk level. Due to the imperfectness in the fabrication, it is reasonable that the measured results are slightly lower than the simulated results.

Figure 7.Measured spectra of the normalized transmittances of a single four-mode waveguide bend. (a)–(h) Measurement results of a single 90° four-mode bend based on (a)–(d) DFFC or (e)–(h) SFFC.

Table 2 shows the comparison between the previously reported MWBs based on different structures and this work. It is obvious that both for the three-mode and four-mode MWBs, this work has better overall performance. To support the same number of TE modes, the MWBs based on the DFFC are almost half the size of those based on the SWGs [22]. The equivalent radius of the proposed device allowing high transmittance of three TE modes is only 6 μm, which is already close to the radius of the silicon single mode waveguide bend, and that of the MWB with four-mode transmission is only 10 μm, which is the smallest four-mode bend to the best of our knowledge.Table 2.

Comparison of the Reported Results of Silicon Multimode Waveguide Bends^a

Reference	Method	$M_{ch}$	$R$ (μm)	EL (dB)		CT (dB)
Reference	Method	$M_{ch}$	$R$ (μm)	Theory	Meas.	Theory	Meas.
[21]	SWG tapers	4	45.8	0.4	1	−20	−20
[22]	SWGs	3	10	0.5	0.7	−30	−20
[22]	SWGs	4	20	0.5	0.8	−26	−15
[23]	Euler curves	4	45	0.1	0.5	−25	−20
[24]	Bezier curves	3	20	0.2	/	−26	−23
[25]	Corner bend	2	7	0.18	0.53	−36	−15
[25]	Corner bend	10	35	0.54	/	−24	/
[26]	TO-optimized	4	17	0.1	0.55	−20	−17
[27]	SFFC	3	9.35	0.04	0.2	−29	−25
This work	DFFC	3	6	0.17	/	−31	/
This work	DFFC	4	10	0.13	0.72	−29	−25

SWG, subwavelength grating; TO, transformation optics; SFFC, single free-form curve; DFFC, double free-form curve; EL, excess loss; CT, crosstalk; $M_{ch}$ , mode-channel number. The units of EL and CT are decibels.

4. CONCLUSION

In this paper, we have proposed and demonstrated an MWB based on the DFFC. The improved DRS algorithm is used to optimize the shape of the DFFC. Unlike the Bezier curve or other mathematical functions, the DFFC utilizes two curves for which the changing of the segment radius is free instead of constraining in a certain mathematical form. The MWBs with effective radii of 6 μm and 10 μm are designed to support three modes and four modes, respectively. Theoretically, the bending losses of the three-mode MWB are calculated to be 0.008, 0.013, and 0.175 dB at 1.55 μm, for the ${TE}_{0}$ , ${TE}_{1}$ , and ${TE}_{2}$ modes, respectively. Each mode of the MWB has the inter-modal crosstalk no more than $- 31 dB$ . Similarly, the MWB with four modes also has quite low excess losses (less than 0.12 dB for all modes) and low crosstalks (less than $- 29 dB$ ). In the experiment, the measured four-mode MWB based on DFFC has excess losses of 0.01–0.72 dB and crosstalks less than $- 25 dB$ for the central wavelength, which is better than the MWB based on the method in our previous work. The method of this paper can be easily extended to more modes and is equally applicable to TM polarization. Besides, the MWB based on free-form curves can be easily applied to waveguides with other materials. The optimization procedures of the proposed device can be combined with other optimization algorithms, such as the artificial neural network and genetic algorithm, to improve the design speed. We believe the present design is of great help to increase the integration density of the on-chip multimode transmission systems.

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