Mode-division multiplexing is a promising and cost-effective way to increase the communication capability of integrated photonic circuits, for both classical and quantum information processing. To construct large-scale on-chip multimode routing systems, the multimode waveguide crossing is one of the key components. However, there have been only a few dual-mode waveguide crossings reported, which can support merely two waveguide modes or two crossing channels. This severely limits the density and capacity of multimode routing systems. Here we demonstrate for the first time, to the best of our knowledge, a universal multimode waveguide star crossing based on transformation optics, which can handle, in principle, any number of waveguide modes and any number of crossing channels as well. The structure is transformed from a Maxwell fisheye and can realize aberration-free imaging for each waveguide mode. A gray-scale electron-beam lithography is adopted to fabricate it on a commercial silicon-on-insulator wafer. The proposed multimode waveguide star crossing has little loss and low crosstalk throughout an ultra-broad wavelength range of . Our study paves the way for realizing highly integrated and large capacity on-chip multimode routing and communication systems.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
CorrectionsShuyi Li, Yangyang Zhou, Jianji Dong, Xinliang Zhang, Eric Cassan, Jin Hou, Chunyong Yang, Shaoping Chen, Dingshan Gao, and Huanyang Chen, "Universal multimode waveguide crossing based on transformation optics: publisher’s note," Optica 6, 125-125 (2019)
5 December 2018: A typographical correction was made to the author listing.
Transformation optics offers great versatility to manipulate light propagation at will via spatial coordinate transformations. Since transformation optics was proposed by Pendry et al.  and Leonhardt , many marvelous optical devices have been designed, such as invisibility cloaks [3–5], remote devices for optical illusion , hyperlenses [7–9], and so on [10–12]. Recently, considerable interest has been drawn to designing on-chip integrated optical devices by conformal mapping (CM) [13–17]. Compared with most traditional methods that rely only on the device shape design, the CM method has the capability of combining the device shape with index profile to achieve amazing functionalities. In this work, we present a universal multimode waveguide star crossing based on a newly developed CM method.
In integrated optics, increasing the communication capability on a photonic chip is an ultimate goal due to the exponential growth of optical interconnect and communication demands for both classical  and quantum [19,20] information applications. Mode-division multiplexing (MDM) is an attractive path to this goal for its advantage of not needing extra expensive multi-wavelength light sources [21,22]. The multimode waveguide crossing is a key component to construct high-density on-chip multimode routing and communication systems [23,24]. Even though various single-mode waveguide crossings, such as multimode interferometers (MMIs) , optimized tapers , and subwavelength grating waveguides [27,28], have been demonstrated, the design of multimode waveguide crossings still remains challenging. To the best of our knowledge, there have been only a few works on dual-mode waveguide crossings reported to date, and they are based on two main design principles. One principle is to let two waveguide modes propagate simultaneously in a traditional  or subwavelength  MMI waveguide crossing by setting the MMI length as the common multiple of self-imaging lengths of the two modes. However, it is difficult for these MMI waveguide crossings to support more waveguide modes due to the difference of self-imaging distances for each mode, poor self-imaging performance for higher order modes, and relatively narrow bandwidth. The second principle makes use of a waveguide Y junction to split the input and modes into two separated modes. Then the two separated TE0 modes are input to the single-mode MMI waveguide crossings . Yet it cannot support other higher order TE modes due to the limited mode splitting ability of a Y junction. Moreover, in the above mentioned two works, only two crossing channels are supported. Recently, another dual-mode crossing based on a Maxwell fisheye (MF) lens was demonstrated , in which the fisheye was partially truncated to reduce the abrupt index contrast between the fisheye boundary and the input/output waveguides. However, this truncation will deteriorate the imaging performance of the fisheye lens. There are obvious index and wave front mismatches between the fisheye lens and the waveguides, which will result in big losses for higher order modes. Therefore, the existing dual-mode waveguide crossings are difficult to expand to support more modes or more channels, which greatly hinders their applications for future high-density multimode routing schemes.
In this paper, we have used transformation optics to design a universal multimode waveguide star crossing to support an arbitrary number of waveguide modes and crossing channels. The structure is transformed from the well-known MF. By taking advantage of the aberration-free imaging property of the MF [31–33], multiple waveguide modes can be imaged from one side to the opposite simultaneously. Furthermore, we developed a new artificial boundary conformal mapping (ABCM) method, together with a semi-analytic form to transform the shape and corresponding index distribution of the MF, by which the distinct index and wavefront mismatches between the MF and the input/output multimode waveguide are eliminated. The proposed multimode waveguide star crossing has high transmission efficiency and low crosstalk across an ultra-wide 1 dB bandwidth, nearly covering the entire O–U optical communication bands, thanks to the adiabatic mode evolution in the fisheye. Based on the gray-scale electron-beam (E-beam) lithography technique, we successfully fabricated the multimode waveguide star crossing on a commercial silicon-on-insulator (SOI) wafer, which accords well on performance with the theoretical predictions. This universal multimode waveguide star crossing has great potential in realizing large-scale on-chip multimode routing and communication systems.
2. DESIGN AND SIMULATIONS
A. Aberration-Free Imaging of the MF
The MF is characterized by a gradient index distribution , where is the index at the fisheye center, is the fisheye radius, and is the distance from the fisheye center. The MF can image an optical field from one point on its circular boundary to the opposite point on the same boundary without aberration . Owing to the aberration-free imaging of MF, many devices, such as optical lenses  and antennas [34,35], have been demonstrated. Since an optical image can be decomposed into a series of waveguide modes, it is reasonable to suppose that MF could image multiple waveguide modes simultaneously without aberration. However, it should be noted that the MF cannot be directly used for waveguide crossing due to the non-negligible reflection and scattering losses  caused by several mismatches. The light wavefront at the boundary of MF is circular , which obviously does not match with the flat wavefronts of waveguide modes. Moreover, the boundary index of the MF is only , which mismatches severely with the effective index (normally equals to ) of the input/output waveguides. The CM method is naturally considered a solution to the above problems. However, such issues are difficult to tackle in the original CM method due to its strict requirement and low flexibility for coordinate transformations. Some quasi-conformal mapping methods have been proposed instead [13–15,37]. However, complex optimization processes are required to reduce the performance degradation caused by the residual material anisotropy.
Here, we propose a strict CM method with much better flexibility. Unlike the previous CM methods that transform only the natural boundaries of the device, we introduce new artificial boundaries at the areas where light seldom passes by cutting the structure open. By this means, we can get more freedom to tune the coordinate transformations in the local cut open area. Such a method is named artificial boundary conformal mapping (ABCM). By ABCM, we reshape the MF to eliminate the index and wavefront mismatches between the MF and input/output waveguides. Meanwhile the whole index range could be well confined between 1.44 and 2.83 to make it feasible for fabrication on a commercial SOI wafer.
B. Multimode Waveguide Crossing Design via Transformation Optics
In our design, an original MF with radius [See Fig. 1(a)] is reshaped to an -channel -mode transformed fisheye multimode waveguide crossing (TF-MWC) by the proposed ABCM method. The result resembles a star [See Figs. 1(b) and 1(c)]. To be specific, we take , and , as examples. We set (, ) and (, ), which are large enough for supporting these channels and modes at the wavelength of 1550 nm. Because the original MF and TF-MWC both have rotational symmetry, the transformation is carried out in polar coordinates for convenience. Since the index and wavefront mismatches exist only at the boundary of the MF, we leave the inner part of the MF with untransformed. For the outer ring region (), we divide it into 2N equivalent sector rings, which are marked in red (, ) and orange (, ) in Fig. 1(a).
Through transforming the outer circular boundary of the sector ring to a flat boundary, the wavefront can match with those of the input/output waveguides. Moreover, owing to the sector ring division beforehand, we can bring a series of artificial boundaries between adjacent sector rings. This greatly increases the deformability of the boundary region of the MF. By properly compressing the outer cut open regions of the MF, the index values of the outer boundary regions can be increased more efficiently. Hence, the index ratio between the center and edge of the MF could be decreased from to , making its boundary index match with the input/output waveguides. However, this ratio needs to be precisely controlled by adjusting the coordinate compressing degree.
The conformal mapping is obtained by solving the Laplace’s equations  in each sector ring [see Supplement 1 (section S1)]. Because of the geometric symmetry, adjacent transformed sector rings could be connected continuously and smoothly. It should be noted that, at the interface between the transformed sector ring and the inner untransformed circle, the index should remain nearly unchanged to guarantee index continuity. On the other hand, at the boundary of TF-MWC, index and wavefront matching with input/output waveguides should be simultaneously obtained. These two objectives could be achieved by optimizing the parameters of the TF-MWC, through coupling the Laplace’s equation module with the parameter optimization module in the COMSOL Multiphysics software (http://www.comsol.com ). In addition, we can express the conformal mapping with a semi-analytic form [see more details in Supplement 1 (section S1)]. The footprints of the TF-MWCs are (, ) and (, ).
It should be noted that the ABCM method is general for designing TF-MWC with nearly arbitrary predefined numbers of waveguide modes and crossing channels. Through appropriately enlarging the radius of the MF and dividing more equivalent sector rings, the TF-MWC could support more waveguide modes and more crossing channels.
Based on the ABCM method, the index distribution of the original MF [Fig. 1(d)] is converted to those of the TF-MWCs [Figs. 1(e) and 1(f)]. As the index of the original MF decreases from the center to the outer boundary, we gradually increase the coordinate compressing degree accordingly to compensate for the index decreasing. The index values of TF-MWC can be almost confined to the range of 1.44–2.83, which corresponds to the effective indices of the 25–220 nm thick silicon slab waveguide. There are only two minor regions whose index values are slightly out of this range. One is at both sides of the flat boundary of the TF-MWC, whose index values are slightly higher than 2.83 due to the inevitable excess coordinate compression there. The other is at the end of the incision, whose index values are slightly lower than 1.44 due to the excess coordinate stretching there. However, the light field propagates mainly in the middle of the sector ring and seldom passes through these two minor regions. Thus, we could truncate the index values out of range to 2.83 or 1.44 directly, which incurs only very little loss and crosstalk levels with respect to the simulation results. Therefore, we have eliminated the index and wavefront mismatches and confined the index values to the possible range on the silicon slab waveguide platform.
In this work, we realize the gradient index of TF-MWC by gradually changing the thickness of the silicon slab waveguide on a SOI wafer . By modulating the silicon core thickness from 25 to 220 nm, the effective index of slab waveguide can be tuned between 1.44 and 2.83, which is the designed index range for the TF-MWC. This can be achieved on-chip by gray-scale E-beam lithography and silicon dry etching processes [See Supplement 1 (section S3)]. Based on the calculated relationship curve between the effective index and silicon thickness [Supplement 1 (section S2)], we can convert the index distribution of TF-MWC to the silicon thickness distribution. Then we have evaluated the performance of the TF-MWC by 3D finite-difference time-domain (FDTD) simulation and fabricated it on a SOI wafer, while successfully characterizing it.
C. 3D FDTD Simulations
For simplicity, we present the simulation results of only a four-channel three-mode TF-MWC here, while the results of a five-channel four-mode TF-MWC are included in Supplement 1 (section S5). All the 3D FDTD simulations in this work were executed by using of Lumerical FDTD software (http://www.lumerical.com ).
To show the advantage of the proposed TF-MWC, we simulated a four-channel three-mode direct multimode waveguide crossing (D-MWC) for comparison. The D-MWC is constructed by simply intersecting the multimode waveguides with each other. The width of the input/output waveguide was chosen as 2 μm, which can support the lowest three order TE modes. Field monitors are placed at the output multimode waveguides, which are 11 μm away from the crossing center for D-MWC and 1.5 μm away from the fisheye boundary for TF-MWC. We use the mode overlapping between the output field and every waveguide eigenmode to calculate the mode losses and crosstalk levels. The simulation results for 1550 nm wavelength are shown in Fig. 2. For the , , and modes, the transmission efficiencies are 74.3% (), 22.6% (), and 1.3% (), respectively. The maximum simulated crosstalk values are , , and 12.01 dB, respectively To show the simulation results clearly, we plot three histograms for three different modes input from the left horizontal port.
In each histogram, the energy efficiency of every mode in every port is listed. The corresponding field distribution is shown below each histogram. We find that the loss and crosstalk deteriorate seriously for the and modes. Large crosstalk exists in the oblique crossing port, which is marked as “C” in Fig. 2. This is caused by the strong diffraction and scattering when light propagates through the crossing. Particularly for the mode, the crosstalk to port C is even higher than the transmission efficiency.
For the four-channel three-mode TF-MWC, the width of the input/output waveguide is also 2 μm, consistent with the D-MWC. The 3D FDTD simulation results for 1550 nm wavelength are shown in Fig. 3. For the , , and modes, the transmission efficiencies are 99.5% (), 97.7% (), and 95.3% (), respectively, and the crosstalk values to other ports or to other modes are all below . The simulation results prove that the proposed four-channel three-mode waveguide star crossing has very high transmission efficiencies and low crosstalk values for all three lowest TE modes. This also confirms that aberration-free imaging could be realized for multiple waveguide modes when the index and wavefront mismatches are removed.
From the field component profiles shown in Figs. 3(d)–3(f), one can see that the wavefront for each mode is smoothly changed and restored without obvious inter-mode coupling or scattering. This means that the propagation of every waveguide mode in the TF-MWC can be regarded as an adiabatic mode evolution, which inherently has very low loss and ultra-wide bandwidth. This discovery may inspire a new roadmap to design ultra-broadband optical devices based on graded index structures.
We have simulated the transmission spectra of the four-channel three-mode TF-MWC and D-MWC by the 3D FDTD method, as shown in Fig. 4. The theoretical spectra of the four-channel three-mode TF-MWC for the , , and modes are shown in Figs. 4(a)–4(c), respectively. Figures 4(d)–4(f) are the spectra of the D-MWC for , , and modes, respectively. From these diagrams, we can find obvious losses and serious crosstalk in the D-MWC, particularly for the and modes. In contrast, for TF-MWC, throughout a broad wavelength range of 1308–1702 nm, the losses of all modes are below and the crosstalk values all below . This shows that our proposed TF-MWC has ultra-broad bandwidth nearly covering the whole O, E, S, C, L, and U communication bands, benefiting from the adiabatic mode evolution.
In Supplement 1, we also give the simulation results of a five-channel four-mode waveguide star crossing. The transmission efficiencies for , , , and modes at 1550 nm wavelength are 99.6% (), 98.6% (), 96.0% (), and 93.1% (), respectively [see Figs. S7 and S8 of Supplement 1 (section S5)]. In addition, the crosstalk values to other ports or to other modes are all below . The 1 dB bandwidth is also ultra-wide from 1241 to 1720 nm. Across this large bandwidth, the losses of all modes are below and the crosstalk values are all below [see Fig. S9 of Supplement 1 (section S5)]. Such performance is comparable with the four-channel three-mode waveguide star crossing, much better than the corresponding D-MWC [see Figs. S5 and S6 of Supplement 1 (section S5)].
In reality, our proposed TF-MWC has the advantage of expandability to the general case of supporting arbitrary numbers of waveguide modes and crossing channels, only needing appropriate enlargement of the original fisheye to accommodate all the input/output channels. This strong expandability is very important in real applications of mode multiplexing systems.
3. FABRICATION AND MEASUREMENT
Because the five-channel four-mode TF-MWC is far more complicated, we experimentally demonstrated only the four-channel three-mode TF-MWC. The TF-MWC was fabricated on a commercial SOI wafer with a 220 nm thick top silicon layer and 3 μm thick buried oxide layer, through gray-scale E-beam lithography [41–44]. This technique is based on modulating the E-beam exposure doses with a gray-scale gradient on resist to form a 3D smoothly varied thickness profile after development. A thermal reflow was followed to reduce the surface roughness of the resist. Then, the 3D profile on resist was transferred to the upper silicon layer of the SOI wafer by inductively coupled plasma (ICP) etching. This fabrication method is widely applied in manufacturing diffraction optical elements [45,46], microelectromechanical systems [47,48], and micro lenses . In our experiment, PMMA 950k is used as the gray-scale lithography resist to fabricate the TF-MWC. By carefully controlling the process parameters, about 10 nm thickness resolution and 3 nm planar resolution were realized, and are adequate to get good device performance. Then, a second aligned E-beam lithography with ZEP520A resist and ICP etching was taken to fabricate the residual optical circuit connected to the TF-MWC. This two-step arrangement is critical to guarantee the fabrication qualities of TF-MWC and other outer optical circuits.
To measure the performance of the TF-MWC, we have designed and fabricated an on-chip MDM system  in each crossing channel. Figure 5(a) shows the microscopic image of the whole circuit. It is composed of the TF-MWC, input/output grating couplers, and mode multiplexers/demultiplexers, which are labeled with the red dashed boxes. A magnified microscopic view of the TF-MWC is exhibited in Fig. 5(b). The TF-MWC region displays smoothly varied colors due to the film interference of the thickness varied silicon layer. The 3D profile of TF-MWC is measured by atomic force microscope (AFM), as shown in Fig. 5(e). Scanning electron microscope (SEM) photos of the grating coupler and mode multiplexer are shown in Figs. 5(c) and 5(d), respectively. The input/output grating couplers comprise a 1D chirped design [Fig. 5(c)], which can increase their mode field overlaps with single-mode fibers and reduce the coupling loss . The coupling angle of fibers was chosen as 10°. The mode multiplexer and demultiplexer  are both formed by an asymmetry directional coupler (ADC) between a narrow single-mode waveguide and a wide multimode bus waveguide [Fig. 5(d)]. The ADC converts mode of the narrow single-mode waveguide to mode of the wide multimode bus waveguide, and vice versa. The ADC for the mode is 46.2 μm long, composed of a 0.5 μm wide single-mode waveguide and a 1.05 μm wide bus waveguide. The ADC for the mode is 57.5 μm long, which has the same 0.5 μm wide single-mode waveguide and a 1.55 μm wide bus waveguide. The gaps between the two waveguides in these two ADCs are both 130 nm. These ADCs are cascaded to realize MDM of the three lowest order TE modes. Different widths of bus waveguides are connected though adiabatic tapers, which are long enough to avoid any inter-mode coupling or radiation loss. Then, the bus waveguide is expanded to the final width of 2 μm to access the TF-MWC. After multimode light passing through the TF-MWC, a mirrored circuit is used to de-multiplex different waveguide modes back to the mode in different output channels. The grating coupler couples the light in each output channel into the single-mode fiber, which is finally linked to the optical powermeter or optical spectrum analyzer (OSA) for measurements [see Fig. S3 of Supplement 1 (section S4)].
For obtaining the transmission efficiencies and crosstalk accurately, we also fabricate the corresponding reference circuits for each channel on the same chip, and they have the same grating couplers and mode multiplexers/demultiplexers, but without a TF-MWC. By subtracting the spectra of these reference circuits, accurate transmission efficiencies and crosstalk values for each mode and each channel could be obtained. And a short straight waveguide with two grating couplers near each circuit was used to calibrate the coupling loss of the grating couplers. Meanwhile, we also fabricated the D-MWC on the same chip to make comparison with the performance of the TF-MWC. The microscope diagrams of all these auxiliary circuits are shown in Fig. S4 of Supplement 1.
4. EXPERIMENTAL RESULTS AND DISCUSSION
For comparison with the simulation results, we also plot the histogram graphs of the experimental data for 1550 nm wavelength. Figures 6(a)–6(c) show the measured efficiencies of the D-MWC. For , and modes, the transmission efficiencies are 71.7% (), 21.4% () and 1.5% () respectively. These measured efficiency values are very close to the simulation results in Figs. 2(a)–2(c), which are 74.3% (), 22.6% () and 1.3% () respectively. The maximum measured crosstalk values for , , and modes at 1550 nm wavelength are , , and 9.08 dB, respectively, which coincide well with the corresponding simulated values of , , and 12.01 dB. As predicted by the simulations, the experimental results verify that the D-MWC has large losses and serious crosstalk, particularly for the and modes.
In contrast, the measured efficiencies of the TF-MWC for the three lowest TE modes are shown in Figs. 6(d)–6(f). For the , , and modes, the measured transmission efficiencies are 87.3% (), 61.4% (), and 53.9% (), respectively. The inter-mode crosstalk values in the throughout port (T) are all below , and the crosstalk values to other ports (C, V, and AC) are all below . The experimental results show that compared with the D-MWC, the TF-MWC has much better performance, not only on transmission efficiencies but also on crosstalk levels. The measured transmission efficiencies are slightly lower than the simulated results in Figs. 3(a)–6(c), which is possibly caused by the etching roughness and fabrication deviation of the 3D thickness profile of the TF-MWC. Further improvement of the transmission could be achieved by reducing the etching roughness via thermal oxidation and hydrofluoric acid removal, or by optimizing the dose distribution of gray-scale lithography.
The measured spectra of the D-MWC are shown in Figs. 7(d)–7(f). The labels T, C, V, and AC correspond to the throughout port, oblique crossing port, vertical crossing port, and anti-oblique crossing port, respectively. The experimental transmission efficiencies of the D-MWC at 1550 nm wavelength are 71.7% (), 21.4% (), and 1.5% (). The maximum measured crosstalk values for the , , and modes at 1550 nm wavelength are , , and 9.08 dB, respectively. As predicted by the theoretical simulations, the experimental results also show that the D-MWC has large losses and serious crosstalk, particularly for the and modes.
In comparison, the experimental spectra of the TF-MWC are shown in Figs. 7(a)–7(c). For the , , and modes, the experimental transmission efficiencies of TF-MWC at 1550 nm wavelength are 87.3% (), 61.4% (), and 53.9% (). So the measured insertion losses of TF-MWC are much lower than the D-MWC, especially for the and modes. The maximum measured crosstalk values for the , , and modes at 1550 nm wavelength are , , and , respectively, while the measured crosstalk values to other ports (C, V, and AC) are all below or close to the noise level of the OSA. Therefore, the TF-MWC demonstrates very low crosstalk between different waveguide modes or crossing channels.
Because the amplified spontaneous emission (ASE) light source used in the measurement has limited bandwidth of 1520–1620 nm and the grating couplers have relatively narrow 3 dB bandwidth of , we could not measure the device spectra in the wide range of O–U bands. Nevertheless, the transmission spectra for all three modes are very flat across the measured wavelength range of 1530–1570 nm, which exhibits a potential wide operation bandwidth of the TF-MWC.
We have demonstrated a universal multimode waveguide crossing based on transformation optics, which has no fundamental restrictions on the number of supported modes and the number of channels. The structure is transformed from a MF, which has the special capability of aberration-free imaging for different waveguide modes simultaneously. A new artificial boundary conformal mapping method together with a semi-analytic form is developed to remove both the index and wave-front mismatches between the fisheye and the input/output waveguide. The proposed TF-MWC exhibits low loss and low crosstalk in an ultra-broad bandwidth, benefiting from the adiabatic mode evolution in the fisheye. Both the theoretical simulations and the experimental results show that the TF-MWC has much better performance than the D-MWC. This universal and ultra-broadband multimode waveguide crossing offers great potential to construct densely integrated MDM routing systems for on-chip ultrahigh capacity communications. It also has important prospects in building large-scale photonic quantum circuits for manipulating mode entanglement. Furthermore, the artificial boundary conformal mapping method will inspire more transformation physics devices in the future, not only for optical applications, but also for other waves, such as acoustic waves and elastic waves.
National Key R&D Program of China (2016YFB0402503); National Natural Science Foundation of China (NSFC) (11374115, 11504435, 11874311, 61261130586); Research Funds for the Central Universities (20720170015).
See Supplement 1 for supporting content.
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