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Abstract
Asymmetric Y-junctions, compared with mode coupling-based devices, possess considerably smaller wavelength dependence and thus are more promising for ultra-broadband mode (de)multiplexing in integrated optics. However, these devices also feature relatively high mode crosstalk and insertion loss. Here, we show that the mode crosstalk and loss of an asymmetric Y-junction can be significantly reduced by optimizing the waveguide shape of the Y-junction using an adjoint-based inverse design. Based on such inverse-designed asymmetric Y-junctions, we realize ultra-compact, broadband, and low crosstalk silicon photonic TE00 & TE1 and TE0 & TE2 mode (de)multiplexers with sizes of only 4.5 × 1.2 µm2 and 6 × 1.4 µm2, respectively. From simulations it is shown that the TE0 & TE1 and TE0 & TE2 mode (de)multiplexers contain wide bandwidths of 160 nm (1460-1620 nm) and 140 nm (1460-1600 nm), respectively, over which the mode crosstalks are below about −20 dB, and the losses are <0.41 dB and <0.88 dB, respectively. The experimental results show that in the corresponding TE0 & TE1 and TE0 & TE2 mode division multiplexing systems, the crosstalks are less than −15.5 dB and −15 dB over the spectral ranges of 1453-1580 nm and 1460-1566 nm, respectively, and the losses are <1.7 dB at 1520 nm and <8.24 dB over the entire measured wavelength range.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the increasing demand for communication capacity, a variety of data multiplexing technologies for photonic integrated circuits have been developed. Among them, the space division multiplexing (SDM) [1] technique shows great potential in further improving the transmission capacity of on-chip optical interconnection. The SDM can be mainly divided into two types: 1) the densely packaged waveguide array-based (DPWA) method [2] and 2) the mode division multiplexing (MDM) method. Compared with the former one, the MDM method can provide a higher space utilization efficiency, and thus is more promising for on-chip data communication. Efficient mode (de)multiplexers are critical components in an on-chip MDM system. These devices have been demonstrated based on asymmetric directional couplers (ADCs) [3–6], multimode interference (MMI) couplers [7–9], optical beam forming [10], pixelated-meta structures [11], subwavelength metamaterials [12,13], waveguide Bragg gratings [14,15], and asymmetric Y-junctions [16–19], Asymmetric Y-junctions, compared with the others, typically have smaller wavelength dependence [20], and thus are promising to achieve ultra-broadband mode (de)multiplexers. However, such devices usually also contain larger mode crosstalk and insertion loss compared with those of ADCs with comparable sizes, due to the mismatch between the modes in the Y-junction and the output waveguides. To decrease the crosstalk and insertion loss of an asymmetric Y-junction, a small branch angle ($< 9^{\circ }$) is generally needed to achieve an adiabatic mode evolution [17]. This, however, can also result in a large device footprint (typically hundreds of microns in length) that cannot meet the needs of highly integrated photonic chips.
In recent years, the inverse design method for the optimization of integrated-optics devices has attracted wide attention [21–25]. Traditional optimization methods for integrated photonic devices typically involve adjusting only a few discrete parameters. The inverse design method, on the other hand, uses a series of efficient algorithms to find all possible structures of devices in a small size range, and then finds the design with the best performance. As a result, when compared to the conventional schemes, this method can provide significantly higher efficiency and much greater design freedom, allowing for the production of integrated photonic devices with both ultra-small sizes and high performance. So far, the inverse design has been employed to optimize a variety of integrated-optics devices, including polarization beam splitters [26], wavelength demultiplexers [27], multimode waveguide bends [28], power splitters [29], etc. Inverse design in silicon photonics can be mainly categorized into several types, including those by optimizing empirical structures [30,31], digital metamaterials [21,32–35], and the topologies and boundary shapes of the integrated photonic structures [22,23,25–28,36–40]. The scheme based on empirical structures, where existing classical structures are used and only a few parameters are tuned, needs the shortest computation time and can also offer high performances, but the device sizes are typically large. In digital metamaterials, regular shapes like circles or rectangles are selectively etched on a periodic 2-D surface. The method can yield much smaller sizes of the designs. Nevertheless, the structures typically contain numerous small nano-features (usually smaller than 120 $\times$ 120 nm) that may be difficult to fabricate uniformly and repeatedly. The topology optimization searches for the full design space of target devices with arbitrary topologies, which can provide the largest number of degrees of freedom and potentially offer the best performance. The produced devices, however, typically contain the most intricate geometric patterns, which places a high requirement on the lithography resolution and precision. On the other hand, for the shape optimization, the boundary shape of the device is arbitrarily varied to optimize the performance. This method can offer several attractive advantages including a sufficiently large number of degrees of freedom, high performances and small sizes of the designs, and being less computationally intensive. Furthermore, as only the waveguide shape is optimized, the structures generally do not contain small feature size/spacing and sharp corners, and thus can be easily fabricated. Owning to these benefits, the scheme has usually been the preferred method for nanophotonic inverse design, and has been exploited to realize various silicon photonic devices, such as 3 dB Y-splitter [25], phase modulators [37], waveguide crossings [38,39], mode converters [40], and multimode maveguide bends [28].
The inverse design was also used to optimize asymmetric Y junctions for mode (de)multiplexing in [33]. The Y-junction was based on digital subwavelength structures, and the performance was improved by optimizing the circular holes’ combination. Through that method, a silicon photonic mode (de)multiplexer for the first two transverse electric (TE) modes (i.e., the $\mathrm {TE_{0}}$ and $\mathrm {TE_{1}}$ modes) with an ultra-small size of $4.8 {\times } 3.6\;\mathrm{\mu}\textrm {m}^2$ and a bandwidth of 60 nm was realized. Nevertheless, although the individual small nano-holes (with radii of 45 nm) are manufacturable by electron-beam lithography, due to the large number of them (several hundreds), it would still be challenging to fabricate all of them uniformly and precisely.
In this paper, we show both theoretically and experimentally that the mode crosstalk and insertion loss of an asymmetric Y-junction can be significantly reduced by optimizing the waveguide shape of the Y-junction. The optimization can be efficiently accomplished by using an adjoint-based inverse design. The obtained Y-junctions have smooth, slowly varying waveguide profiles and thus can be easily fabricated. Further, based on the inverse-designed asymmetric Y-junctions, we realize ultra-compact, broadband, low crosstalk, TE$_0$ & TE$_1$ and TE$_0$ & TE$_2$ mode (de)multiplexers developed on silicon-on-insulator (SOI) platforms, which have sizes of $4.5\times 1.2\;\mathrm{\mu}\textrm {m}^2$ and $6\times 1.4\;\mathrm{\mu}\textrm {m}^2$, respectively. The three-dimensional finite-difference time domain (3D-FDTD) simulation results show that the TE$_0$ & TE$_1$ and TE$_0$ & TE$_2$ mode (de)multiplexers possess broad operation ranges of 160 nm (1460-1620 nm) and 140 nm (1460-1600 nm), respectively, over which the mode crosstalks are less than about −20 dB and the insertion losses are smaller than 0.41 dB and 0.88 dB, respectively. In the experiments, it is shown that for the corresponding TE$_0$ & TE$_1$ and TE$_0$ & TE$_2$ MDM systems with the two cascaded Y-junction based mode (de)multiplexers, the crosstalks are below −15.5 dB and −15 dB over the wavelength ranges of 1453-1580 nm and 1460-1566 nm, respectively, and the insertion losses are <1.7 dB at the wavelength 1520 nm and are <8.24 dB over the entire experimental wavelength range. The proposed inverse-designed asymmetric Y-junctions, owing to their significant advantages in performance and size, provide an attractive solution for mode (de)multiplexing in silicon photonic MDM systems.
2. Design and simulation
2.1 Asymmetric Y-junctions for $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ mode (de)multiplexing
2.1.1 Design and optimization
The devices described in this paper are developed on 220-nm-high strip SOI waveguides with a silicon dioxide cladding layer deposited on the waveguides. We first design and optimize an asymmetric Y-junction that serves as a $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ mode (de)multiplexer. The $\mathrm {TE_{0}}$ and $\mathrm {TE_{1}}$ modes from the input of an asymmetric Y-junction can be finally converted to the $\mathrm {TE_{0}}$ modes of the wider and narrower branches, respectively, when proper widths of the Y-junction and branches are used, due to the adiabatic evolution of the system modes [41]. The original asymmetric Y-junction to be optimized is illustrated in Fig. 1(a), which is simply set to be a uniform straight waveguide. Two S-bends are placed after the Y-junction to serve as the output waveguides. The horizontal and vertical lengths for both S-bends are 3.75 $\mathrm{\mu}$m and 0.15 $\mathrm{\mu}$m, respectively. The width of the Y-junction should be large enough to support TE$_0$-TE$_1$ modes, but should not be too wide as this can increase the device size and could also need long additional tapers to connect the device to routing waveguides. Nevertheless, if the Y-junction is too narrow, this may lead to an overly small width of the narrower output arm (e.g., $<300$ nm), which could increase the bending loss and make the device susceptible to fabrication variations. Based on these considerations, the width of the Y-junction is chosen to be 1 $\mathrm{\mu}$m. The widths of the wider and narrower output ports are 0.54 $\mathrm{\mu}$m and 0.36 $\mathrm{\mu}$m, respectively. These widths have been chosen to ensure sufficient width difference between the two arms, which is essential to achieve a low mode crosstalk, while at same time having a small enough bending loss for the narrower output S-bend. To determine the suitable length of the Y-junction, a series of initial optimizations using various lengths of the Y-junction has been performed, and the results indicate that the final FOM is higher as the Y-junction is longer. Specifically, when the length was set to 3.88 $\mathrm{\mu}$m, 4 $\mathrm{\mu}$m, 4.5 $\mathrm{\mu}$m, 5 $\mathrm{\mu}$m or 5.375 $\mathrm{\mu}$m, the final FOM was 1.912, 1.920, 1.933, 1.940 or 1.945, respectively. Here, the length is chosen to be 4.5 $\mathrm{\mu}$m to obtain a good compromise between the performance and the device size. The width of the design area is set 1.8 $\mathrm{\mu}$m, which is equal to the upper bound of the waveguide width set in the optimization.
Fig. 1. Schematic illustrations of (a) the original and (b) optimized asymmetric Y-junctions designed for $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ mode (de)multiplexing.
The figure of merit (FOM) of the optimization for the $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ mode (de)multiplexer should include the FOM information of the two input modes. To optimize the broadband performance of the device, transmissions at 1000 wavelengths equally spaced between 1500 nm and 1600 nm are calculated, and the average transmission is used to represent the FOM for this mode. The FOM for the $\mathrm {TE_{0}}$ input mode is defined as
The single, total FOM that is the actual optimization objective should include the transmission performances of both modes. Thus, this total FOM can be simply defined to be the sum of the FOMs of the $\mathrm {TE_{0}}$ and $\mathrm {TE_{1}}$ modes:
During the optimization, the FOM should be gradually increased toward the ideal value of 2.The optimized Y-junction is represented by a polygon whose boundary points have fixed, evenly spaced x coordinates, while having variable y coordinates, allowing the geometry to be modified for the optimization. Cubic spline interpolation is then performed between these basic boundary points to produce a smooth polygon geometry. Therefore, the y coordinates of the basic boundary points act as the design parameters. The total number of the basic boundary points, $N_p$, thus determines the number of the design parameters. A smaller $N_p$ could lead to lower degrees of freedom in the design, which may decrease the performance of the final device. A larger $N_p$, however, could result in a longer optimization time and also a smaller minimum curve radius, which could be more likely to cause sharp edges that are difficult to fabricate. Here, $N_p$ is chosen to be 20 to make a good compromise between the optimization performance and the fabrication requirement.
The 3D-FDTD method is used for the electromagnetic computations involved in the optimization. Due to the time-domain nature, this method will produce broadband field and transmission data through a single simulation. The optimization is preformed based on the gradient descent method, where changes in the geometry can be introduced proportional to the gradient of the FOM [42]. The adjoint method instead of the exhaustive searching strategy is used to efficiently calculate the gradient information of the FOM. This method can yield the derivative of the FOM with respect to dielectric permittivity at every point in the design region through only one forward and one reverse adjoint simulation. Specifically, for each input mode, only two simulations are required for the gradient calculation. The first simulation is called “direct simulation”, in which the source is placed at the input of the bend. The second one is called “adjoint simulation”, where the source is put in the position where the FOM monitor was placed, i.e., the output of the Y-junction. Since two FDTD simulations, which are a direct and an adjoint simulations, are required to calculate the gradient information for each input mode, 4 FDTD simulations are needed to be conducted in each iteration. More details of the adjoint method can be found in [25]. During the optimization process, the shape of the waveguide in the design area is allowed to be changed continuously to improve the FOM. The FOM will be improved in the dielectric permittivity where the derivative is positive and vice-versa. Therefore, the geometry boundary will be pushed or pulled when the derivative is positive or negative, respectively. Applied iteratively, this can finally lead to an optimum of the FOM. The optimization will be terminated when the gradient values are less than a specified tolerance value, which is $10^{-5}$ in our optimization. It should be noted that the shape parameters for the current design are only allowed to be varied within a certain range, so that the produced device is practical for fabrication. The upper and lower bounds of the waveguide width are set to 1.8 $\mathrm{\mu}$m and 0.6 $\mathrm{\mu}$m, respectively.
In the FDTD simulations, a symmetric boundary condition is used along the z axis (i.e., the waveguide thickness direction), while perfectly matched layer boundary conditions are used along the other two axes. The time resolution is about 0.021 fs. The mesh step size is 20 nm, which has been chosen to make a good compromise between the simulation time and the accuracy.
After 30 iterations, the optimization converged, and the entire optimization took about 4 hours using a computer with a 64–core central processing unit (with two Intel Xeon Platinum 8375C processors). The geometric evolution of the Y-junction during the optimization process is depicted in Fig. 2. The evolution of the FOM [defined in Eq. (3)] as well as the FOMs of the two modes [expressed in Eqs. (1) and (2)] as a function of iteration number is presented in Fig. 3. The finally produced waveguide shape of the Y-junction is shown in Fig. 2(f).
Fig. 2. Geometry evolution of the asymmetric Y-junction designed for $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ mode (de)multiplexing during the optimization. Iter indicates the iteration number.
2.1.2 Simulation results
The optimized asymmetric Y-junction serving as the $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ mode (de)multiplexer [shown in Fig. 3(b)] is validated using the 3D-FDTD method. The electric field intensity distributions at the wavelength 1550 nm for the original and optimized asymmetric Y-junctions when inputting the different modes are shown in Fig. 4. Significant crosstalk can be seen in the original Y-junction for each input mode, while for the optimized one, no appreciable crosstalk is observed, demonstrating the large improvement in mode (de)multiplexing performance of the asymmetric Y-junction brought by the optimization. More specifically, the transmission efficiency of the $\mathrm {TE_{0}}$ mode into the wider output waveguide is increased from $66{\%}$ to $98{\%}$, while that of the $\mathrm {TE_{1}}$ mode into the narrower output waveguide is improved from $67{\%}$ to $96.6{\%}$ due to the optimization.
Fig. 4. Comparison of electrical field distributions of the original and optimized asymmetric Y-junction designed for $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ mode (de)multiplexing. (a) and (b) are the field distributions of the original Y-junction when the input modes are $\mathrm {TE_{0}}$ and $\mathrm {TE_{1}}$, respectively. (c) and (d) are the field distributions of the optimized Y-junction when the input modes are $\mathrm {TE_{0}}$ and $\mathrm {TE_{1}}$, respectively.
Figures 5(a)–5(b) compare the calculated insertion loss and mode crosstalk, respectively, of the original and optimized Y-junctions over a wavelength range of 1460-1620 nm. A significant decrease in both insertion loss and mode crosstalk over the characterized wavelength range owing to the optimization is observed. Specifically, the insertion losses for the $\mathrm {TE_{0}}$ and $\mathrm {TE_{1}}$ modes of the Y-junction within the simulated spectral range are reduced from <2.04 dB to <0.2 dB and from <2.11 dB to <0.41 dB, respectively, while the mode crosstalks for the $\mathrm {TE_{0}}$ and $\mathrm {TE_{1}}$ modes are decreased from <-6.28 dB to <-20.11 dB and from <-5 dB to <-20.48 dB, respectively. Then, the performances of the corresponding MDM systems are studied, which consist of two identical Y-junction based mode (de)multiplexers cascaded in a symmetrical configuration [inset of Fig. 5(c)]. Figures 5(c)–5(d) compare the insertion loss and mode crosstalk, respectively, of the MDM systems with the original and optimized Y-junctions. The losses for the $\mathrm {TE_{0}}$ and $\mathrm {TE_{1}}$ modes of the systems are reduced from <8.16 dB to <0.49 dB and from <8.76 dB to <0.52 dB, respectively, whereas the crosstalks for the two modes are decreased from <-1.11 dB to <-24.38 dB and from <-1.13 dB to <-24.35 dB within the wavelength range of 1461-1608 nm, respectively. These results confirm that the optimization of individual Y-junctions can bring about significant improvement in the MDM system performance. Note that the reflection of the optimized Y-junction is also found to be <-23 dB over the wavelength range of 1500-1600 nm from the simulation results.
Fig. 5. Performance comparison between the original (dashed) and optimized (solid) asymmetric Y-junctions designed for $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ mode (de)multiplexing. Comparison of (a) insertion loss and (b) mode crosstalk of the original and optimized Y-junctions. Comparison of (c) insertion loss and (d) mode crosstalk of the MDM systems using the original and optimized Y-junctions.
2.2 Asymmetric Y-junctions for $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ mode multiplexing
2.2.1 Design and optimization
To validate the scalability of the optimization methodology, an asymmetric Y-junction for $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ mode (de)multiplexing is designed and optimized using the same strategy. The original asymmetric Y-junction is shown in Fig. 6(a). The initial structural parameters are chosen based on the same criteria as those of the TE$_0$ & TE$_1$ mode (de)multiplexer described above. The Y-junction waveguide has a width of 1.27 $\mathrm{\mu}\textrm {m}$ and thus supports $\mathrm {TE_{0}}$-$\mathrm {TE_{2}}$ modes, and its length is 6 $\mathrm{\mu}\textrm {m}$. The widths of the wider and narrower output ports are 0.85 $\mathrm{\mu}$m and 0.35 $\mathrm{\mu}$m, respectively. The horizontal and vertical lengths of both S-bends are 3.75 $\mathrm{\mu}\textrm {m}$ and 0.15 $\mathrm{\mu}\textrm {m}$, respectively. The upper and lower bounds of the waveguide width are set to 2 $\mathrm{\mu}$m and 0.8 $\mathrm{\mu}$m, respectively. The width of the optimized area is chosen to be 2 $\mathrm{\mu}$m which is equal to the upper bound of the waveguide width.
Fig. 6. Schematic illustrations of (a) the original and (b) optimized asymmetric Y-junctions designed for $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ mode (de)multiplexing.
The optimization aims to demultiplex the input $\mathrm {TE_{0}}$ and $\mathrm {TE_{2}}$ modes in the Y-junction and convert them to the $\mathrm {TE_{0}}$ modes of the wider and narrower branches, respectively, with the minimized insertion losses and mode crosstalk. The geometric evolution of the waveguide during the optimization process is depicted in Fig. 7. The evolution of the FOMs as a function of iteration number is presented in Figs. 8. The upper and lower bounds of the waveguide width are set to 2 $\mathrm{\mu}\textrm {m}$ and 0.8 $\mathrm{\mu}\textrm {m}$, respectively. After 60 iterations, the optimization converged, and the entire optimization took about 6 hours. The finally obtained waveguide shape of the Y-junction is shown in Fig. 7(f).
Fig. 7. Geometry evolution of the asymmetric Y-junction designed for $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ mode (de)multiplexing during the optimization.
2.2.2 Simulation results
The asymmetric Y-junction optimized for $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ mode (de)multiplexing [shown in Fig. 8(b)] is characterized by using the 3D-FDTD method. The electric field intensity distributions at 1550 nm for the original and optimized asymmetric Y-junctions when injecting the $\mathrm {TE_{0}}$ and $\mathrm {TE_{2}}$ modes are shown in Fig. 9. As can be seen, there is considerable crosstalk for the original Y-junction. In contrast, little crosstalk can be observed for the optimized one. More specifically, the transmission efficiency of the $\mathrm {TE_{0}}$ mode into the wider output waveguide is increased from $79{\%}$ to $97.2{\%}$, while that of the $\mathrm {TE_{2}}$ mode into the narrower output waveguide is improved from $69{\%}$ to $94{\%}$ after the optimization.
Fig. 9. Comparison of electrical field distributions of the original and optimized asymmetric Y-junction designed for $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ mode (de)multiplexing. (a) and (b) are the field distributions of the original Y-junction when the input modes are $\mathrm {TE_{0}}$ and $\mathrm {TE_{2}}$, respectively. (c) and (d) are the field distributions of the optimized Y-junction when the input modes are $\mathrm {TE_{0}}$ and $\mathrm {TE_{2}}$, respectively.
Figures 10(a)–10(b) compare the simulated insertion loss and mode crosstalk, respectively, of the $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ mode (de)multiplexers based on the initial and optimized Y-junctions over a wavelength range of 1460-1600 nm. In the wavelength range of 1480-1600 nm, the insertion loss for the $\mathrm {TE_{0}}$ and $\mathrm {TE_{2}}$ modes after the optimization are decreased from <1.01 dB to <0.21 dB and from <2.33 dB to <0.88 dB, respectively, while the mode crosstalks for the $\mathrm {TE_{0}}$ and $\mathrm {TE_{2}}$ modes are reduced from <-10.47 dB to <-19.5 dB and from <-19.2 dB to <-20 dB, respectively. Subsequently, the corresponding $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ MDM system based on the two cascaded Y-junctions are characterized over the wavelength range of 1460-1600 nm. Figures 10(c)–10(d) show the comparisons of the insertion loss and mode crosstalk, respectively, of the MDM systems using the original and optimized designs. In the wavelength range of 1470-1582 nm, the losses for the $\mathrm {TE_{0}}$ and $\mathrm {TE_{2}}$ modes of the systems are reduced from <3.31 dB to <0.65 dB and from <7.77 dB to <0.55 dB, respectively, whereas the crosstalks of the two modes are decreased from <-9.23 dB to <-16.24 dB and from <-9.28 dB to <-16.21 dB, respectively. The reflection of the optimized Y-junction is found to be <-29 dB within the wavelength range of 1500-1600 nm from the calculation results. One may notice that the loss of the TE$_2$ mode for the MDM system here is smaller than that for the single (de)multiplexer. Ideally, however, the loss for the MDM system should be twice that of the single device. This discrepancy from the ideal should be due to the multimode interference happening at the middle of the MDM system caused by the imperfect mode demultiplexing of the (de)multiplexer. Such an interference is very weak due to the low crosstalk of the device, but could slightly vary the power distribution at the two outputs of the MDM system. This could finally result in unexpected small variations in loss from the ideal. Note that fabrication tolerance analysis of the mode (de)multiplexers designed in this work can be found in Supplement 1, Section 1.
Fig. 10. Performance comparison between the original (dashed) and optimized (solid) asymmetric Y-junctions designed for $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ mode (de)multiplexing. Comparison of (a) insertion loss and (b) mode crosstalk of the original and optimized Y-junctions. Comparison of (c) insertion loss and (d) mode crosstalk of the MDM systems using the original and optimized Y-junctions.
It should be noted that the current device is optimized only for the (de)multiplexing performance of the TE$_0$ and TE$_2$ modes. In some application scenarios, a TE$_0$ & TE$_2$ mode (de)multiplexer may be required to allow the intermediate mode (i.e., the TE$_1$ mode) to pass through the device. This can be addressed by including the transmission efficiency of the TE$_1$ mode into the FOM. More details regarding designing such mode (de)multiplexers can be found in Supplement 1, Section 2.
3. Experimental results
3.1 Fabrication and measurement setup
The devices in this work were fabricated in Applied Nanotools, Inc., using electron-beam lithography on a standard commercial SOI wafer. The wafer consisted of a 3 $\mathrm{\mu}\textrm {m}$ thick buried silicon dioxide layer and a 220 nm thick silicon layer. A 2 $\mathrm{\mu}\textrm {m}$ thick silicon dioxide cladding layer was deposited on the etched sample. Vertical grating couplers from the foundry process design kit (PDK), spaced on 127 $\mathrm{\mu}\textrm {m}$ centers, were used to couple light into and out of the chip from a 8-degree polished single-mode optical fiber array with a 127-$\mathrm{\mu}\textrm {m}$ fiber-to-fiber pitch. The TE$_0$ & TE$_1$ and TE$_0$ & TE$_2$ MDM systems were characterized over wavelength ranges of 1440-1580 nm and 1460-1580 nm, respectively. Because of the limited tuning range of 1507-1630 nm of the used tunable laser (Keysight 81960A), the device responses over the wavelength range of 1507-1580 nm were characterized using the tunable laser and a multiport optical power meter (Keysight N7745A), while those over the spectral range of 1440-1510 nm or 1460-1510 nm were measured through a broadband light source (WL-SC400-2-PP) and an optical spectrum analyzer (Yokogawa AQ6370C-20). Due to the limited bandwidth of the grating couplers, we slightly adjusted the incidence angle of the fiber array to tune the center wavelengths of the grating couplers when measuring the different ranges of wavelength.
The multimode transmission responses of our asymmetric Y-junction based mode (de)multiplexers can not be directly tested because only the fundamental modes can be injected into the chip in our measurement setup. Therefore, we measured the corresponding MDM systems to acquire the transmission performances of the Y-junctions. Two different types of MDM circuit illustrated in Fig. 11 were used to characterize the mode (de)multiplexing performance of the asymmetric Y-junctions. The first kind of circuit employs an ADC consisting of two counter-tapered waveguide to serve as the mode multiplexer, while using the asymmetric Y-junction for the mode demultiplexing. For the second type of circuit, two identical but oppositely oriented asymmetric Y-junctions are cascaded to form the MDM system. Figures 12(a) and 12(b) show optical microscope images of fabricated ADC-assisted and cascaded Y-junction based MDM circuits, respectively. Scanning electron microscope (SEM) images of the optimized asymmetric Y-junctions for TE$_0$ & TE$_1$ and TE$_0$ & TE$_2$ mode (de)multiplexing are also presented in Figs. 12(c) and 12(d), respectively. Calibration circuits, which contain only input and output grating couplers and an uniform waveguide connecting the two grating couplers were also fabricated and tested. The results were used to calibrate out the coupling loss of the grating couplers from the measurement results of the devices.
Fig. 11. Schematic illustrations of two different types of MDM integrated-optics circuit to test the performances of the asymmetric Y-junctions in experiments.
Fig. 12. Optical microscope image of fabricated (a) ADC-assisted and (b) cascaded Y-junction based MDM circuits. SEM images of the optimized asymmetric Y-junctions for (c) $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ and (d) $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ mode (de)multiplexing.
3.2 $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ mode (de)multiplexers
The experimental results of the $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ MDM systems using the asymmetric Y-junctions designed in Section 2.1 are presented first. Figures 13(a)–13(d) show the measured normalized transmission responses of the ADC-assisted MDM systems using the original and optimized Y-junctions, respectively. In each of Figs. 13(a)–13(d), the left and right plots show the responses over the different spectral ranges of 1440-1510 nm and 1507-1580 nm, respectively, which are measured through the different experimental setups as mentioned before. It can be seen that the system with the optimized Y-junction exhibits significant improvements in both loss and mode crosstalk within the measured wavelength range compared to those using the original Y-junction. For the original Y-junction based system, the mode crosstalk is as large as <-1.669 dB, and the insertion losses of the $\mathrm {TE_{0}}$ and $\mathrm {TE_{1}}$ modes are 1.39 dB and 3.94 dB at 1520 nm, respectively. In contrast, in the optimized case, the mode crosstalk is below −14.93 dB over the wavelength range of 1460-1580 nm. The losses of the $\mathrm {TE_{0}}$ and $\mathrm {TE_{1}}$ modes are 1.25 dB and 2.16 dB at 1520 nm, respectively, and are below 8.24 dB within the entire measured wavelength range. The experimental insertion losses of the optimized Y-junction are slightly higher than those of the simulation results (Fig. 5), which should be caused by additional loss from the ADC and non-ideal fabrication effects. One may observe that the spectra over the wavelength range of 1440-1510 nm exhibit interference patterns, which is attributed to the light inference occurring in the internal system of the broadband light source. To minimize the impact of such spectral ripples on the estimated crosstalk, the transmission responses have been smoothed using a moving average filter before the crosstalk spectrum calculation. In addition, one can notice that the calibrated transmission responses in Fig. 13 are not flat, which is different from the simulated results (Fig. 5). Such non-flat behaviors of the measured responses are mainly attributed to calibration errors due to the spectral discrepancy between the grating couplers in the device and calibration circuits caused by fabrication non-uniformity. It is also found that such a spectral discrepancy between the grating couplers caused by manufacturing issues is varied under the different incidence angles of the fiber array used in the various setups for measuring the different wavelength ranges. This can result in various impacts on the calibrated responses for the different setups, which could explain the difference between the calibrated transmissions of the MDM system at 1510 nm measured by the different setups observed in some cases [such as Fig. 13(a) and Fig. 14(a)]. Despite of this, the crosstalk, which is a relative parameter, can be much less affected by the fabrication issue-related calibration uncertainty. This ensures the feasibility to evaluate the bandwidths of the devices from the combined crosstalk spectra.
Fig. 13. Experimental transmission responses of the ADC-assisted $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ MDM systems using the (a),(b) original and (c),(d) optimized asymmetric Y-junctions. The left and right figures show the responses over the various spectral ranges of 1440-1510 nm and 1507-1580 nm, respectively, which are measured through the different experimental setups.
Fig. 14. Experimental transmission responses of the cascaded Y-junction based $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ MDM systems using the (a),(b) original and (c),(d) optimized asymmetric Y-junctions. The left and right figures show the responses over the various spectral ranges of 1440-1510 nm and 1507-1580 nm, respectively.
Figures 14(a-d) also plot the measured transmission responses of the cascaded Y-junction based $\mathrm {TE_{0}}$ & $\mathrm {TE_{1}}$ MDM systems with the original and optimized Y-junctions, respectively. Again, the optimized Y-junction based MDM system exhibits large decreases in both insertion loss and mode crosstalk over the whole measurement wavelength range compared with the original Y-junction based one. Specifically, for the system with the original Y-junctions, the mode crosstalk is <10.5 dB, and the insertion losses of the $\mathrm {TE_{0}}$ and $\mathrm {TE_{1}}$ modes are as large as 10.58 dB and 10.14 dB at 1520 nm. By comparison, for the optimized case, the crosstalk is <-15.51 dB over the wavelength range of 1453-1580 nm, and the insertion losses is <5.754 dB. The insertion losses of the $\mathrm {TE_{0}}$ and $\mathrm {TE_{1}}$ modes are 0.25 dB and 1.66 dB at 1520 nm, respectively.
3.3 $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ mode (de)multiplexers
The $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ MDM systems using the asymmetric Y-junctions designed in Section 2.2 were also fabricated and measured. Figures 15(a-b) show the measured transmission responses of the ADC-assisted $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ MDM circuits using the original and optimized asymmetric Y-junctions, respectively, over the measured wavelength range of 1460-1580 nm. A large reduction in insertion loss and mode crosstalk can be observed for the optimized Y-junction based system compared with those with the original design. For the original Y-junction-based circuit, the crosstalk is <-7.03 dB, and the insertion losses of the $\mathrm {TE_{0}}$ and $\mathrm {TE_{2}}$ modes are 1.56 dB and 3.53 dB at 1520 nm, respectively. In contrast, for the optimized case, the mode crosstalk is <-15.92 dB within the measured wavelength range, and the insertion losses is <6.654 dB. The insertion losses of the $\mathrm {TE_{0}}$ and $\mathrm {TE_{2}}$ modes are 0.77 dB and 1.08 dB at 1520 nm, respectively.
Fig. 15. Experimental transmission responses of the ADC-assisted $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ MDM systems using the (a),(b) original and (c),(d) optimized asymmetric Y-junctions. The left and right figures show the responses over the various spectral ranges of 1460-1510 nm and 1507-1580 nm, respectively.
The $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ MDM system based on the two identical cascaded Y-junctions were also characterized, and the results for the original and optimized cases are presented in Figs. 16(a) and 16(d), respectively. Again, we can see that the optimized Y-junction-based MDM system exhibits significantly lower insertion loss and mode crosstalk over the whole measurement wavelength range as compared to those of the original Y-junction-based one. For the original Y-junction-based system, the mode crosstalk is <4.232 dB, and the insertion losses of the $\mathrm {TE_{0}}$ and $\mathrm {TE_{2}}$ modes at 1520 nm are 2.2 dB and 6.75 dB, respectively. By comparison, for the optimized case, the mode crosstalk is <-15 dB within the wavelength range of 1460-1566 nm, and the losses is <5.345 dB. The losses of the $\mathrm {TE_{0}}$ and $\mathrm {TE_{2}}$ modes at 1520 nm are 1.25 dB and 1.63 dB, respectively.
Fig. 16. Experimental transmission responses of the cascaded Y-junction based $\mathrm {TE_{0}}$ & $\mathrm {TE_{2}}$ MDM systems using the (a),(b) original and (c),(d) optimized asymmetric Y-junctions. The left and right figures show the responses over the various spectral ranges of 1460-1510 nm and 1507-1580 nm, respectively.
4. Discussion and conclusion
Table 1 summarizes the measurement results of the MDM systems based on the original and optimized asymmetric Y-junctions. Note that the results for both types of MDM system using 1) an ADC and a Y-junction and 2) two identical cascaded Y-junctions [compared in Fig. 11] are included. The comparison in this table highlights the large performance improvement in both mode crosstalk and insertion loss for the optimized asymmetric Y-junctions compared with the original designs.
Table 1. Comparison of experimental results of the MDM systems based on original (Orig.) and optimized (Opt.) asymmetric Y-junctions; IL: insertion loss; CT: crosstalk; BW: bandwidth.
Table 2 also compares the performance between the current mode (de)multiplexers and other SOI-based ones reported in recent years [33,43–48]. Note that the theoretical performances of the devices in this work shown in the table are obtained from the single mode (de)multiplexers, while the experimental ones are extracted from the corresponding MDM systems based on the two cascaded Y-junctions. As can be seen, our proposed mode (de)multiplexers have the smallest footprints and the broadest experimental bandwidths compared with the previous works. The theoretical and measured mode crosstalks of the current devices are also relatively small, and are only considerably larger than those proposed in [33,48], which, however, were all based on subwavelength structures that may be difficult to fabricate. The experimental insertion losses of our devices are comparable or slightly larger than those of the others. Nevertheless, as the measured losses of our designs are also larger than the theoretical values, the current losses should be further reduced by improving the fabrication process to, e.g., decrease waveguide sidewall roughness, fabrication variations, etc.
Table 2. Comparison of reported mode (de)multiplexers based on SOI platforms; PhC: photonic crystal; SW: subwavelength; SWG: subwavelength gratings.
5. Conclusion
In conclusion, it has been shown that the mode crosstalk and insertion loss for an asymmetric Y-junction can be significantly decreased by optimizing the waveguide shape of the Y-junction using the adjoint-based inverse design. The designed Y-junctions can be easily fabricated because of their smooth, slowly varying waveguide profiles. The large performance improvement due to the shape optimization of asymmetric Y-junctions have been validated through the 3D-FDTD simulations and experiments. Further, based on the inverse-designed asymmetric Y-junctions, TE$_0$ & TE$_1$ and TE$_0$ & TE$_2$ mode (de)multiplexers on SOI have been realized which have ultra-small sizes ($4.5\times 1.2\;\mathrm{\mu}\textrm {m}^2$ and $6\times 1.4\;\mathrm{\mu}\textrm {m}^2$, respectively), large bandwidths, and low crosstalk. From the 3D-FDTD simulations, it has been shown that for the designed TE$_0$ & TE$_1$ and TE$_0$ & TE$_2$ mode multiplexers, the mode crosstalks are less than about −20 dB and the insertion losses are <0.41 dB and <0.88 dB over the bandwidths of 160 nm (1460-1620 nm) and 120 nm (1480-1600 nm), respectively. The experimental results have shown that for the corresponding TE$_0$ & TE$_1$ and TE$_0$ & TE$_2$ MDM systems using the two cascaded Y-junction based mode (de)multiplexers, the mode crosstalks are below −15.5 dB and −15 dB over the wavelength ranges of 1453-1580 nm and 1460-1566 nm, respectively, and the insertion losses are <1.7 dB at the wavelength 1520 nm and <8.24 dB over the entire measurement wavelength span. Finally, the achieved mode (de)multiplexers have been compared with the state-of-the-art works, and the comparison has shown that the present devices have the smallest sizes and broadest experimental bandwidths while possessing relatively low mode crosstalk. These results show great promise of the present inverse-designed asymmetric Y-junctions for mode (de)multiplexing in dense silicon photonics MDM systems.
Funding
National Natural Science Foundation of China (62105089); National Key Research and Development Program of China (2021YFB2801500); the Major Science and Technology Innovation in 2025 projects of Ningbo (2020Z021); the Funds of Natural Science Foundation of Zhejiang (LY21F040004); the Natural Science Foundation of Fujian under Grant (2023J01350).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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