Spatial and polarization division multiplexing harnessing on-chip optical beam forming

On-chip spatial and polarization multiplexing have emerged as a powerful strategy to boost the bandwidth of integrated optical transceivers. State-of-the-art multiplexers require accurate control of the relative phase or the spatial distribution among different guided optical modes, seriously compromising the bandwidth and performance of the devices. To overcome this limitation, we propose a new approach based on the coupling between guided modes in integrated waveguides and optical beams free-propagating on the chip plane. The engineering of the evanescent coupling between the guided modes and free-propagating beams allows spatial and polarization multiplexing with state-of-the-art performance. To demonstrate the potential and versatility of this approach, we have developed a two-polarization multiplexed link and a three-mode multiplexed link using standard 220-nm-thick silicon-on-insulator technology. The two-polarization link shows a measured -35 dB crosstalk bandwidth of 180 nm, while the three-mode link exhibits a -20 dB crosstalk bandwidth of 195 nm. These bandwidths cover the S, C, L, and U communication bands. We used these links to demonstrate error-free transmission (bit-error-rate<10-9) of two and three non-return-to-zero signals at 40 Gbps each, with power penalties below 0.08 dB and 1.5 dB for the two-polarization and three-mode links respectively. The approach demonstrated here for two polarizations and three modes is also applicable to future implementation of more complex multiplexing schemes.


INTRODUCTION
Silicon photonics has been identified as a promising technology to address the communication bottleneck in data centers and longhaul networks [1].On-chip optical transceivers fabricated at large volume using microelectronics facilities could become instrumental in exploiting optical carriers to boost the data bandwidth while reducing the power consumption of communication systems [2,3].Current silicon photonics optical transceivers carry different data channels at distinct wavelengths using wavelength-division multiplexing (WDM) [4].However, as the industry moves forward with the development of next-generation optical networks, other multiplexing schemes will be required to support a higher data capacity per wavelength channel.One promising approach is the use of orthogonal polarizations and spatially-distributed modes to encode more data channels at a specific wavelength [5].The manipulation of higher-order modes and polarization state of light on chip has thus attracted a significant research interest in the past years for the realization of performant mode and polarization (de)multiplexers.A myriad of architectures has been proposed to realize modedivision and polarization-division multiplexing.Mode-division multiplexing (MDM) has been demonstrated based on multimode interference (MMI) couplers, asymmetric Y-junctions and directional couplers (ADC), adiabatic tapers, pixelated-meta structures, and subwavelength metamaterials [6][7][8][9][10][11][12][13].Similarly, multiple devices have been reported for polarization-division multiplexing (PDM), including MMIs, Mach-Zehnder interferometers, diverse types of directional couplers (i.e., symmetric, asymmetric, tapered and bent), photonic crystals, slot waveguides, and subwavelength and tilted nano-gratings [14][15][16][17][18][19][20][21][22][23][24].Despite the great diversity of proposed solutions, crosstalk still remains as one of the main impairments in high-speed optical communications, especially in systems with a high-channel count.An increased crosstalk has a negative impact on the link performance in terms of bit-error-rate (BER) and ultimately results in a power penalty that jeopardizes low power consumption [25].Silicon multiplexers handling more than two modes (e.g., 3 spatial modes with the same polarization) yield poor crosstalk values ranging from -19 dB [11] to -9.7 dB [7].
Here, we propose a simple yet effective strategy to realize highly efficient mode and polarization (de)multiplexing based on engineered on-chip beam forming.Instead of controlling the phase or field distribution matching between two guided modes, we engineer the evanescent coupling between the modes of a photonic waveguide and free-propagating beams on the chip plane.Coupling between guided modes and on-chip free-propagating beams has been achieved using distributed Bragg deflectors [26][27][28][29][30].However, the wavelength-dependent nature of the diffractive coupling seriously limits their use for wideband mode or polarization multiplexers.Conversely, evanescent coupling does not present a strong wavelength dependence, which allowed the demonstration of on-chip beam expanders with a wide bandwidth [31][32][33].In our proposed scheme we engineer the evanescent coupling to make each waveguide mode (or polarization) couple to a different in-plane beam, propagating with a specific angle.As schematically shown in Fig. 1(a), this strategy spatially separates the different waveguide modes, allowing (de)multiplexing.Based on this approach, we experimentally demonstrate a twopolarization link and a three-mode link that allow error-free transmission of multiplexed high-speed data streams.We show transmission of two and three 40 Gbps non-return-to-zero (NRZ) signals, with negligible power penalties at a BER of 10 -9 .To the best of our knowledge, this is the first demonstration of mode and polarization handling enabled by beamforming in an integrated circuit.These devices could be a promising alternative to fixed layout-guided architectures, with excellent potential for the next generation of high-speed and large-capacity on-chip optical interconnects.

MODE DIVISION MULTIPLEXING
A. Operation principle Figure 1(a) shows the top view of a waveguide evanescently coupled to an adjacent slab.The input waveguide has a width of  and is placed at a distance  from the slab.The waveguide supports a discrete set of guided modes with effective indices of    , with  being a natural number indicating the mode order.The slab supports a continuum of vertically-confined (-axis) modes that propagate freely in the -plane with a wavenumber given by: where   is the effective index of the slab,  is the propagation direction angle of the slab mode with respect to the -axis,  0 = 2  0 ⁄ is the vacuum wavenumber, and  ̂ and  ̂ are the unitary vectors in the and -directions, respectively.Phasematching between the guided waveguide modes and the slab modes occurs for Note that Eq. ( 2) is analogous to the grating equation when the zero-order harmonic is considered [34,35].This zero-order operation obviates the strong wavelength dependence of the propagation angle in distributed Bragg deflectors [26][27][28][29][30] For the implementation of the proposed architecture, we consider silicon-on-insulator (SOI) with a thickness of the silicon guiding layer of 220 nm.difference eigenmode (FDE) solver.Figure 1(c) shows the propagation angle of the slab modes satisfying Eq. ( 2), considering a slab index   = 2.85 .Differences in propagation angle exceeding 15° can easily be achieved by properly choosing the waveguide width.

B. Design of the three-mode demultiplexer
The proposed three-mode demultiplexer is presented in Fig. 2(a).The device comprises an input waveguide of width   , a coupling region with linearly varying waveguide width and slab separation, and a final section with fixed waveguide width and slab separation of   and   , respectively.Changing the waveguide width in the coupling region results in a gradual variation of the propagation angle of the slab-propagating beam [see Fig. 1(c)], while the change in the slab gap modifies the coupling strength.These two effects are engineered to focus the slab-propagating beams into a near-Gaussian-shaped profile that is coupled to the fundamental mode of a strip waveguide placed at the focal point.This approach allows coupling each mode of the input waveguide to the fundamental mode of a different output waveguide, thereby performing mode demultiplexing and conversion simultaneously.Note that the approach proposed here can be seamlessly extended to handle a larger number of modes.
The device is designed to maximize the coupling efficiency between each mode of the input waveguide and the corresponding output waveguide while maintaining a low crosstalk.Device dimensions are optimized using three-dimensional finite-difference time-domain (3D FDTD) simulations.The input waveguide has a width of   = 1 µm to support the fundamental, first-order, and second-order modes with transverse-electric (TE) polarization in the wavelength range between 1450 nm and 1650 nm.The dimensions of the coupling region are   = 400 nm ,   = 100 nm ,   = 200 nm and   = 35 µm .These dimensions ensure that all the power in the three input waveguide modes is coupled to slab-propagating beams, simultaneously achieving a near-Gaussian profile for the three slab beams.Further details on the design are provided in Supplement 1, Section 1.
As shown in Fig. 2(a), higher-order modes begin to couple to slab beams at early taper positions owing to their weaker modal confinement.The TE2 (green), TE1 (orange) and TE0 (red) modes are completely radiated along the taper as Gaussian-like beams focused on different output points, namely O3, O2 and O1 with respective angles  2 = 34°,  1 = 37° and  0 = 44°.with a slight increase to 0.65 dB and 0.69 dB for TE1 and TE2 modes, respectively.The degradation of efficiency with wavelength detuning has its origin in chromatic dispersion, which causes a shift of the focal point.Nevertheless, the demultiplexer exhibits a remarkable low crosstalk within a broad bandwidth of 200 nm.Specifically, the attained crosstalk values are better than -41 dB for TE0, -40 dB for TE1 and -28 dB for TE2 demultiplexing over the entire simulated bandwidth.The effect of potential fabrication imperfections on device performance is discussed in Supplement 1, Section 2.

C. Fabrication and experimental characterization of the MDM link
We implemented an MDM link by connecting two mode (de)multiplexers in a back-to-back configuration.The link comprises three input and three output single-mode waveguides and a central multimode waveguide.The input multiplexer couples the fundamental mode of each input waveguide to a different mode of the multimode section.The output demultiplexer couples each mode of the multimode waveguide to the fundamental mode of one output waveguide.Focusing grating couplers optimized for TE polarization are used to inject and extract the light from the chip with a fiber array.We included a reference waveguide on the outermost part of the test structure.The device was fabricated using a 220-nm-thick single crystal Si layer of an SOI wafer, with a 3µm-thick buried oxide (BOX) layer.The patterns were defined by electron-beam lithography (RAITH EBPG 5000 Plus) and transferred via reactive ion etching (ICP-DRIE SPTS).Optical and scanning electron microscope (SEM) images were taken prior the deposition of the upper cladding.The sample was then spin coated with a 1.5-µm-thick PMMA. Figure 3(a) shows optical images of the MDM link, with zoomed-in SEM images of the taper-slab coupling region and the collecting output waveguides.
Experimental characterization of the link transmittance is shown in Figs.3(b), 3(c) and 3(d), when the light is injected through inputs I2, I3, and I4, respectively.The transmittance of each output is obtained by normalizing the measured power at the output ports to the measured power at the reference waveguide output in order to calibrate out the fiber-chip coupling loss.The measured insertion losses are as low as 0.3 dB, 0.9 dB and 1.7 dB at the transmission peak wavelengths with 1-dB bandwidths of 143 nm, 96 nm, and 84 nm for the TE0, TE1, and TE2 mode channels, respectively.Losses of higher-order modes are larger than fundamental mode losses due to slightly lower overlap with the Gaussian-like profile of the collecting output waveguide modes.On the other hand, measured inter-modal crosstalk is below -31.4 dB, -28.3 dB and -25.4 dB at the transmission peak wavelengths for TE0, TE1, and TE2 mode channels, respectively.Additionally, unprecedented crosstalk values reaching -40 dB over a broad bandwidth is observed for TE0 and TE2 channels.Considering 1443 -1638 nm wavelength range (195 nm bandwidth), the crosstalk is better than -20 dB for all the channels.Transmission peaks of the three channels are slightly shifted towards shorter wavelengths.We attribute this small discrepancy with simulations to intrinsic fabrication variability.

POLARIZATION DIVISION MULTIPLEXING A. Design of the two-polarization demultiplexer
We exploit the proposed approach to demultiplex the two orthogonal TE and transverse-magnetic (TM) polarizations in the waveguide.The proposed two-polarization demultiplexer is schematically shown in Fig. 4(a).In this case, we choose a slot waveguide to achieve sufficient angular separation for the slab beams excited by the fundamental TE0 and TM0 waveguide modes, respectively.The propagation angles for the beams phasematched to the TE0 and TM0 modes of a strip waveguide, calculated as a function of the waveguide width (), are shown in Fig. 4(b).The angle difference is lower than 10°, hampering the spatial separation of the two beams.The propagation angle of the slab-propagating beam is governed by the ratio between the effective indices of the waveguide and the slab, as dictated by Eq. ( 2).The effective indices of the TE0 and TM0 modes of a strip waveguide are quite different (e.g., 0.66 for  = 500 nm and 1550 nm wavelength).However, the ratio with the slab indices is very similar (e.g., 0.86 for  = 500 nm and 1550 nm wavelength), resulting in a comparable propagation angle for the slab-propagating beams.This limitation is overcome using a slot waveguide.We fix a slot width of   = 100 nm and calculate the propagation angles for the TE0 (  ) and TM0 (  ) modes of the slot waveguide as a function of the rail width,   [see Fig. 4(c)].For the input waveguide, we choose a rail width of   = 350 nm, yielding initial propagation angles of   = 47° and   = 60°.
The rail width and the separation between the slab and the slot waveguide are linearly reduced along the coupling region to achieve a near-Gaussian profile at the focal points for the two slab beams.The optimized geometrical parameters are   = 400 nm to   = 100 nm,   = 100 nm and   = 20 µm.3D FDTD simulations are carried out to assess the performance of the polarization beam splitter (PBS).The fields radiated into the slab have an MFD of 4 µm for TE0 input and 6.8 µm for TM0 input.The simulated transfer function for each polarization is shown in Figs.4(d) and 4(e).Insertion losses are as low as 0.26 dB for the TE0 mode and 0.15 dB for the TM0 mode at the central wavelength.Crosstalk is -61.9 dB and -47.9 dB at the same wavelength when TE0 and TM0 modes are injected, respectively.Notably, the crosstalk is below -37.4 dB for both modes within the simulated bandwidth.

B. Fabrication and experimental characterization of the PDM link
We fabricated complete PDM links comprising two polarization beam splitters connected in a back-to-back configuration, using the same SOI wafers and fabrication methods described in Section 3.C.Optical and SEM images of the fabricated devices are shown in Fig. 5(a).Two PDM links with nominally identical dimensions for the multiplexers and different grating couplers were fabricated to characterize losses and crosstalk, respectively.The PDM link used for loss characterization has grating couplers optimized for TE polarization for input 4 and output 5 and grating couplers optimized for TM polarization for input 3 and output 6.The PDM link used for crosstalk characterization has TE grating couplers for input 4 and output 6 and TM grating couplers for input 3 and output 5. TE and TM grating couplers have similar radiation angles.Each link includes two reference waveguides on the outermost part to perform the alignment and transmittance normalization for both TE and TM polarizations.
The transmittance of the link is characterized using the experimental setup described in Supplement 1, Section 3. Measured peak insertion losses are as low as 0.5 dB with a 1-dB bandwidth exceeding 100 nm and crosstalk of -40.1 dB for TE0, and as low as 0.7 dB with a 1-dB bandwidth exceeding 108 nm and a crosstalk of -46.7 dB for TM0.Ultra-low inter-modal crosstalk of <-35 dB is attained for both polarizations within the measured 180nm bandwidth.Crosstalk reaching values below -40dB could be observed in the 1542 -1680 nm wavelength range for the TM0 channel.

HIGH-SPEED DATA TRANSMISSION
We have characterized the data transmission performance of the three-mode and two-polarization multiplexed links in terms of biterror-rate.The experimental setup employed is described in Supplement 1, Section 3. The figure of merit used to quantify the dynamic performance is the power penalty, defined as the difference in optical power measured in the receiver with and without signal impairment (i.e., crosstalk), for error-free transmission (BER<10 -9 ) without any correction technique.Figure 6 shows the evolution of the BER as a function of the received power for the reference, single-port transmission, and MDM and PDM operation.For the MDM link, the worst case is when all three signals are introduced simultaneously due to the crosstalk of the aggressor channels.Still, the power penalties are as low as 0.5 dB, 1.5 dB and 0.6 dB for TE0, TE1 and TE2 mode channels, respectively.As expected from static measurements, the highest power penalty is obtained for TE1 mode channel since the crosstalk of the aggressor channels towards this other channel is higher [see Fig.

DISCUSSION AND CONCLUSIONS
In conclusion, we have shown a new approach for spatial and polarization multiplexing, exploiting the excitation of optical beams free-propagating in the chip plane to achieve state-of-the-art performance.To demonstrate the concept, we have developed a three-mode and a two-polarization links allowing error-free propagation of 40 Gbps signals with negligible power penalties.The three-mode link takes advantage of the strong modal dispersion in strip waveguides to realize mode multiplexing and conversion based on free-space-like optical beam forming on chip.The proposed three-mode link comprising a multiplexer and a demultiplexer shows a measured crosstalk lower than -20 dB over a 195 nm bandwidth (1443 -1638 nm) that fully covers the S, C and L telecommunication bands, and partially covers the E and U bands.Furthermore, insertion losses lower than 1.7 dB with a 1-dB bandwidth of 84 nm are attained for all three modes.The twopolarization link harnesses birefringence engineering in slot waveguides to yield an ultra-low crosstalk below -35 dB within the 1500 -1680 nm wavelength range for both polarizations (covering the entire C, L, and U bands, and part of the S band).Measurements also showed low insertion losses (<0.7 dB) for TE0 and TM0, with a 1-dB bandwidth exceeding 100 nm, limited at the upper bound by the wavelength range of the laser available in our setup.
The low crosstalk values (<-40 dB) observed in the MDM link for TE0 and TE2 modes suggest that increasing the separation of the focal points within the slab is a simple but effective way to reduce crosstalk.This could be achieved by increasing the taper length to increase the focal length or by bending the coupling region.On the other hand, insertion losses of higher-order modes could be further reduced by improving the overlap between the modes coupled to the slab free-propagation region and the Gaussian-like profile of the collecting output waveguide modes, for example by implementing a nonlinear taper in the coupling region.Nevertheless, the proposed devices are, to the best of our knowledge, among the mode multiplexers and polarization beam splitters with lowest measured crosstalk within an ultra-broad bandwidth.For the sake of .comparison, Table 1 shows the performance of demonstrated stateof-the-art three-channel MDM links and polarization beam splitters.Note that state-of-the-art polarization beam splitters have been usually measured in standalone configuration, therefore, we have halved the insertion losses measured by our PDM link to ensure a fair comparison.A high-speed optical communications demonstration was also performed to illustrate the applicability of the proposed devices.System-level experiments at 40 Gbps without forward error correction were conducted for both MDM and PDM showing clear and open eye diagrams during the joint transmission of data channels.BER measurements further validated the good transmission capabilities with less than a 1.5 dB power penalty for the MDM link and below 0.08 dB for the PDM link.The ease of scalability of the proposed architecture along with such low penalties can be exploited to implement hybrid WDM-PDM-MDM optical links to significantly enhance the transmission capacity.
The strategy demonstrated here could seamlessly be extended for a larger number of modes and simultaneous modal and polarization multiplexing.The low crosstalk, wide bandwidth and versatility of the proposed multiplexing approach presented in this work will open up unique possibilities in quantum information sciences, optical sensing, on-chip wireless communications and nonlinear photonics.
Funding.French Industry Ministry (Nano2022 project under IPCEI program); Agence Nationale de la Recherche (ANR-MIRSPEC-17-CE09-0041); European Union's Horizon Europe (Marie Sklodowska-Curie grant agreement Nº 101062518) The fabrication of the device was performed at the Plateforme de Micro-Nano-Technologie/C2N, which is partially funded by the Conseil General de l'Essonne.This work was partly supported by the French RENATECH network.
Disclosures.The authors declare no conflict of interest.
Figure 1(b) shows the effective indices of the TE0, TE1, and TE2 modes calculated as a function of the waveguide width .Effective indices are calculated using a commercial finite

Fig. 1 .
Fig. 1.(a) Schematic top view of a waveguide evanescently coupled to an adjacent slab.A silicon strip waveguide of width  is placed at a distance  from a silicon slab.Fundamental (red), first-order (orange), and secondorder (green) TE modes are injected from the left side of the waveguide and propagate along the -axis.Each mode is coupled to a vertically-confined (-axis) beam within the slab with a different propagation angle   .(b) Effective index and (c) propagation angle within the slab of TE0, TE1 and TE2 modes as a function of the waveguide width, calculated for a Si thickness of 220 nm at a wavelength of 1550 nm.
Figures 2(b), 2(c) and 2(d) show the normalized electric field profile at these focusing points (fixed -position) and along the -direction.The radiated fields are fitted to Gaussians with distinct mode field diameters (MFDs), yielding a high overlap integral of 97% for TE0 input, 88% for TE1 input, and 92% for TE2 input.MFDs used for the Gaussians represented by solid blue curves are 2.4 µm, 3.2 µm and 5.4 µm.The calculated transmittance to each output is shown in Figs.2(e), 2(f) and 2(g) when TE0, TE1, and TE2 modes are injected into the strip waveguide, respectively.Simulations show insertion losses as low as 0.14 dB at the central wavelength for TE0,

Fig. 2 .
Fig. 2. (a) Three-dimensional schematic of the proposed three-mode demultiplexer comprising a tapered waveguide and an adjacent slab.The PMMA cladding is not shown for clarity.The simulated electric field distribution in the -plane at mid-height of the Si layer is superimposed on the structure when TE0 (red), TE1 (orange) and TE2 (green) are injected at  0 = 1550 nm.Output waveguide apertures are located at the focal points Om+1 to collect the freepropagating beams.Normalized electric field magnitude |  | for (b) TE0, (c) TE1 and (d) TE2 input.Simulated results are obtained from the leaked-field distribution within the slab at the optimum position in the -direction for each focal point and along the -axis.The simulated transmittance to each of the output as a function of the wavelength when (e) TE0, (f) TE1 and (g) TE2 are launched into the strip nanowire.

Fig. 3 .
Fig. 3. (a) Optical and scanning electron microscope images of the fabricated MDM link.Input and outputs have been numbered from right to left.The topright inset shows details of collecting output waveguides, whereas the bottom-right inset shows the taper-slab coupling region.Measured transmission spectra of the complete multiplexer-demultiplexer link for light input at (b) I2, (c) I3, and (d) I4, which correspond to TE0, TE1, and TE2 channels, respectively.

Fig. 4 .
Fig. 4. (a) Schematic top view of the proposed two-polarization demultiplexer comprising a tapered slot waveguide and an adjacent slab.The PMMA cladding is not shown for clarity.The inset shows the geometry of the slot waveguide.Propagation angle within the slab of TE0 and TM0 modes as a function of the waveguide width for (b) a strip waveguide and (c) a slot waveguide, calculated for a Si thickness of 220 nm at  0 = 1550 nm.The simulated transmittance to each of the output as a function of the wavelength when (d) TE0 and (f) TM0 are launched into the slot nanowire.

Fig. 5 .
Fig. 5. (a) Optical and scanning electron microscope images of one of the fabricated PDM links.Input and outputs have been numbered from right to left.The top-right inset shows details of collecting output waveguides, whereas the bottom-right inset shows the slot-slab coupling region.Measured transmission spectra of the complete multiplexer-demultiplexer link for light input at (b) I4 and (c) I3, which correspond to TE0 and TM0 channels, respectively.

Fig. 6 .
Fig. 6.Crosstalk penalty assessment of MDM and PDM links for a transmission bit rate of 40 Gbps at 1549 nm.Bit-error-rate measurements as a function of the received power for (a) TE0 channel, (b) TE1 channel and (c) TE2 channel of the MDM link, and for (f) TE0 channel and (e) TM0 channel of the PDM link.The insets show the corresponding eye diagrams of the demultiplexed signals (-axis: 5ps/div and -axis: 0.5 mV/div).Ref, Reference; XT, Crosstalk; CH, Channel.
3(b) and 3(d), orange curves].The PDM link, on the other hand exhibits negligible power penalties for both TE0 (0.08 dB) and TM0 (0.03 dB) channels.Power difference between reference (Ref) and single-port (w/o XT) measurements are 0 dB (TE0 channel), 0.04 dB (TE1 channel) and 0.06 dB (TE2 channel) for the MDM link, and 0.25 dB (TE0 channel) and 0.19 dB (TM0 channel) for the PDM link.In all cases, the demultiplexed signals exhibit clear and open eye diagrams indicating a low effective crosstalk, as shown in the insets of Fig. 6.These results indicate that the proposed devices have an excellent potential for error-free data transmission, paving the way for next-generation MDM and PDM communication applications.

Table 1 . Comparison of Demonstrated State-of-the-art Three-channel MDM Links and PBS a MDM/PBS Ref. Architecture IL [dB] BWCT<-20 dB [nm] BWCT<-30 dB [nm] L [µm]
a Values marked with an asterisk correspond to values estimated from figures.IL, Insertion Loss; CT, Crosstalk; BW, Bandwidth; L, Length.