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Abstract
The application of the inverse design method and free-form geometrical optimization in photonic devices endows them with highly tunable functionality and an ultra-compact footprint. In this paper, we implemented this platform to silicon photonic guided-mode manipulation and demonstrated a guided mode-based signal switching architecture. The passive signal switching mechanism is utilized so that no power consumption is needed for routing state maintenance. To solve the explosive increasing design cost in such mechanism when the switching scale is expanded, we illustrate that only a small number of mode switching devices need to be designed as the switching basis. In theory, arbitrary signal routing states can be constructed by cascading some selected basis. The required switching devices can be decreased from factorial N to N - 1 for the N channels switching. For proof of concept, we design and experimentally demonstrate the three-mode cases and the cascade method to combine any three mode-based switching devices. Experiments show that the insertion losses of TE0 - TE1 mode switching unit (U1), TE1 - TE2 mode switching units (U2), and TE0 - TE2 mode switching unit (U3) are less than 2.8 dB, 3.1 dB, and 2.3 dB, respectively. The demonstrated architecture has both arbitrary signal switching capability and ultra-compact footprint, which is promising in the application of mode-division multiplexing communication systems.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The photonics inverse design method converts the dielectric material structural design into a mathematical optimization problem that can be solved by numerical optimization approaches [1,2]. Unlike traditional design methodologies, the inverse design method concentrates solely on the evaluation function and does not rely on existing theoretical solutions. Thus, this approach can search solutions more extensively in the feasible domain without the constraints of the designers’ experience, resulting in many non-intuitive structures, which also allows the inverse-designed devices to have high performance and ultra-compact features [3,4]. The silicon on insulator (SOI) platform has low-cost, high-performance, and CMOS-compatible properties, making it one of the most promising platforms for on-chip optical communications [5–7]. As a powerful design tool, the inverse design facilitates higher density photonic integration on SOI platforms. Many notable results such as wavelength (de-)multiplexers [8–11], polarization rotators [12], power dividers [13–15], waveguide bends [16–18], and mode converters [19–23] have been reported.
In recent years, mode-division multiplexing (MDM) system has been proven to be a promising way to improve communication capacity [24–29], and the guided-mode in the SOI platform is a suitable carrier for implementing on-chip MDM systems [30–34]. Signal switching is one of the important tasks in communication system. In MDM system, passive signal routing based on guided-mode carrier is more suitable due to no power consumption for the routing state maintenance and the reconfigurable mode exchange is hard to realize. However, there are factorial N (N!) routing states for N-channel signals switching, and designing each passive router device to match all of them is expensive, especially as the number of N grows. The inverse-designed mode manipulation devices have small footprints and are more conducive to being cascaded, which yields a viable option of designing a set of mode-based signal switching basis and implementing arbitrary signal routing states through cascading of the basis.
In this paper, based on the adjoint variable method [13,35–37] and the level-set method [10,14,38], we design the mode manipulation devices for the lowest three TE modes in the waveguide: TE0 - TE1 mode switching unit (U1), TE1 - TE2 mode switching unit (U2), and TE0 - TE2 mode switching unit (U3). These mode switching units exchange two modes while keeping the other mode unchanged, which corresponds to a row transformation of the permutation matrix in mode-based signal switching. We show a guided-mode-based passive signal switching architecture implemented by using these mode switching units as a basis. Based on the cascading of signal switching basis, this architecture avoids the explosive demand for passive switching device design when the number of signal channels increases. Only N-1 mode switching devices need to be designed as the basis in our architecture, and achieve arbitrary signal routing states for N channels can be implemented through a finite number of cascading. We fabricated and measured the mode switching devices U1, U2, and U3 and showed that the insertion losses for all switching units are less than 3.1 dB, and the optical signal-noise-ratio are all greater than 9.5 dB for all mode channels.
The remaining sections of this paper are as follows: In Section 2 we describe the SOI on-chip pattern manipulation based on inverse design and the guided-mode-based arbitrary signal switching architecture based on mode switching basis. Section 3 present the simulation and experimental results of the developed U1, U2, and U3 devices and simulation verification of cascading them to implement the arbitrary signal routing state. In Section 4, we conclude.
2. Design principle
2.1 Inverse design of mode manipulation devices on SOI platform
Inverse design transforms a structural design problem into an optimization problem and solves it using numerical optimization algorithms. In passive photonics, design of the structure can be considered as the problem of solving the optimal permittivity distribution ɛr(x,y,z) under the constraints of Maxwell's equations in the frequency domain.
The adjoint variable method is a numerical method of solving the gradient which is widely used in the inverse design of photonics, as follows,
The device design is based on a typical SOI platform with a top Si layer thickness of 220 nm, a 2 µm thick buried oxide layer, and a 2 µm thick cladding, are illustrated in Fig. 1. The manipulation of the modes in the input-output multimode waveguide is achieved by changing the material distribution in the design region. Among the three lowest order TE modes (TE0, TE1, and TE2), there are three mode switching types of two-mode exchanging while keeping the other mode unaltered. Thus, we developed three mode switching components: the TE0 - TE1 mode switching unit (U1), the TE1 - TE2 mode switching unit (U2), and the TE0 - TE2 mode switching unit (U3).
For the mode switching units, the efficiency of the energy coupling between the corresponding input and the target output modes can be used to define the device's FOM, as follows,
Nanostructures are usually designed in two-dimensional (2D) plane, so we use the level-set method to get a higher degree of flexibility in device shape design. The level-set method employs three-dimensional (3D) surfaces to implicitly represent curves in the 2D plane, obviating the requirement for curve clusters or polygons to define the device boundary.
In Fig. 2 we show a schematic of the level-set method using 3D surfaces to represent structures in the 2D plane. The 2D design region in Fig. 2(a) is divided by the curve ∂Ω into two regions and filled with material Si and SiO2, respectively. We define the curve as the intersection line between the plane z = 0 and the surface z = φ(x,y) in Fig. 2(b). When φ(x,y) > 0, the material of the point (x,y) in the plane is Si, and when φ(x,y) < 0, it is SiO2 . For stability in numerical iterations, the surface φ(x,y) we used is the signed distance function. And the following procedure is used to evolve the shapes,
Symmetry is quite useful in designing mode switching devices. As illustrated in Fig. 3(a) and 3(b), linear devices with time-independent and scalar permittivity material have the optical reciprocity: if inputting mode χ from port-1 gets mode ψ at port-2, then inputting mode ψ from port-2 gets mode χ at port-1. When the device possesses axis-symmetric of the y-axis shown in Fig. 3(c) A-A’, meaning the device is equivalent for port-1 and port-2 port inputs, when inputting mode χ from port-1 gets mode ψ at port-2, that inputting mode ψ at port-1 will get mode χ at port-2. This symmetry ensures the device's mode exchanging properties in designing.
Fig. 3. Schematic representations of the passive linear device's reciprocity (a) and (b), and schematic of device geometry shape’s axis-symmetry around the y-axis (c), axis-symmetry about the x-axis (d).
Another symmetry is the axis symmetry around the x-axis illustrated by B-B’ in Fig. 3(d), which is relevant to the symmetry of the mode itself. The guided TE modes can be divided into two categories based on odevity (such as TE0 and TE2 are even-order modes, and TE1 is odd-order mode), and the coupling coefficients between the different modes can be characterized as follows by the coupled mode theory,
2.2 Signal switching architecture through guide-mode carrier
The signal switching relationship within N channels has the property of one input correlating to one output while keeping the number of channels N constant. Thus an N × N order permutation matrix can completely express the signal switching relationship. A permutation matrix is a square binary matrix with one entry of 1 in each row and column and zeros everywhere else. For the signal switching through the guided-mode carrier, the relationship between the signals and the associated guided-modes can be represented by the signal-mode vectors as follows,
In our 3-mode scenario, the TE0, TE1, and TE2 modes are mapped to $|{{\textrm{M}_1}} \rangle $, $|{{\textrm{M}_2}} \rangle$ and $|{{\textrm{M}_3}} \rangle $, respectively. The guided-mode-based signal switching relationships of the U1, U2, and U3 are shown in Fig. 4, which all switching two signals while leaving the other one unaltered, consistent with the row transformations of matrices. And the mode switching units U1, U2 and U3 can be considered as a set of signal switching basis in the three lowest-order TE mode cases, whose corresponding permutation matrices are as follows,
Note that only two mode switching units are actually required in this three mode case, and U3 can be represented as U1 and U2 (U3 = U1·U2·U1 = U1·U2·U1). Based on this set of basis, arbitrary 3 × 3 permutation matrix can be achieved by matrix multiplication (device cascading). Table 1 lists all six 3 × 3 permutation matrices as well as the methods for constructing them using the basis.
Fig. 4. Signal switching states based on guide-mode exchanging corresponding to mode switching unit U1,U2, and U3.
Table 1. Six 3 × 3 permutation matrices and the constructing methods.
3. Result and discussion
3.1 Design result and simulation performance
Figure 5(a) depicts the layout and the simulation result of the TE0 - TE1 mode switch unit U1. The device has a footprint of 9.5 µm × 2.1 µm and swaps TE0 and TE1 modes in multimode waveguide while remain the TE2 mode passes through unaffected. The simulated normalized spectrum as shown, and the insertion loss of the TE0 - TE1 mode exchange is less than 0.75 dB in the wavelength range of 1500 - 1600 nm, while the TE2 mode insertion loss is less than 0.24 dB. Since the TE0 - TE1 and TE1 - TE0 transmission spectrums are theoretically equal due to the device's symmetry, only the TE0 - TE1 spectra is displayed. The simulated shows that the device's crosstalks are all less than −8.8 dB, where there are only four independent spectral lines due to the device's symmetry.
Fig. 5. Structure schematic and simulation results of the mode switching units U1 (a), U2 (b), and U3 (c).
The structure of the TE1 - TE2 mode switch unit U2, which has a footprint of 9.8 µm × 2.2 µm, is shown in Fig. 5(b). The device's electric field Ey distribution shows that it switches between TE1 and TE2 modes while maintaining the TE0 mode's straight-through properties. According to simulations, the TE0 mode's insertion loss is less than 0.37 dB in the 1500 - 1600 nm wavelength region, whereas the TE1 - TE2 mode conversion's insertion loss is less than 0.70 dB. All of the crosstalks are less than −11 dB.
In Fig. 5(c), we display a schematic design of the proposed TE0 - TE2 mode switch unit U3, which has a footprint of 9.8 µm × 2.5 µm. Electric field Ey distribution shows that the TE0 and TE2 modes have been swapped, while the TE1 mode has remained intact. The insertion loss of the TE0 - TE2 mode switching is less than 0.54 dB, while the loss of the TE1 mode is less than 0.17 dB, as illustrated in the transmission spectrum. Crosstalk occurs mostly from the even modes (TE0 - TE0 and TE2 - TE2), which is less than −10.9 dB. As previously stated, the mode coupling between the odd and even modes is not permitted due to the symmetric, resulting in extremely low crosstalk from TE1 mode, which is about −90 dB.
3.2 Experiment result
We fabricate the devices using electron beam lithography and inductively coupled plasma etching. The light is coupled between the chip and the fiber using grating couplers, which have the insertion loss of 4.1 dB/facet at central wavelength 1560 nm. Using an amplified spontaneous emission source and a optical spectrometer, we measured the devices’ static spectrum. Our previous ADC-based mode converters [39,40] generate and characterize the higher-order modes.
Figure 6(a) - (c) show SEM photographs of the U1, U2, and U3 devices showing that the devices are well fabricated. Measured spectrum of the U1 device are shown in Fig. 6(d), the TE0 - TE1 mode exchange has less than 1.9 dB insertion losses and the insertion loss for TE2 mode passing is less than 2.8 dB, in the wavelength range of 1545 - 1590 nm. The crosstalks are all less than −12.6 dB so a greater than 9.5 dB optical signal-to-noise ratio is achieved. In Fig. 6(e) we show the U2 device's transmission spectra, the insertion losses of TE1 - TE2 mode exchange are less than 3.1 dB. The straight-through TE0 mode has insertion loss less than 1.0 dB. All crosstalks are below −13.9 dB, resulting in an optical signal-to-noise ratio of above 10.8 dB. And Fig. 6(f) depicts the U3 device's measured spectrum. The insertion losses of TE0 - TE2 and TE2 - TE0 are less than 1.9 dB, whereas the insertion losses of TE1 - TE1 mode are less than 2.1 dB. And a more than 12.6 dB optical signal-to-noise ratio is achieved due to the crosstalk are all less than −14.7 dB. However, since there exist the mode impurity in the mode generation process and the additional crosstalks introduced by the ADCs, we were unable to observe the ultra-low crosstalks from the TE1 mode input in our experiments.
Fig. 6. (a) - (c) are the SEM photographs of U1, U2 and U3. (d) - (f) are the experimentally measured device transmission spectrums, respectively.
3.3 Demonstration and analysis of the signal switching architecture
Mode-based signal switching basis U1, U2, and U3 and multimode waveguides can already implement the 4 mode-based signal routing states corresponding to the permutation matrices P1, P2, P3, and P6 in Table 1. The remaining two permutation matrices P4 and P5 can be realized by cascading U1 and U2. We show the simulation results of this basis cascading to construct the corresponding mode routing states.
In Fig. 7(a) we show the schematic diagram of the matrix multiplication of U1 and U2 to obtain the permutation matrix P4 in Table 1 and the corresponding signal switching state. The cascaded structure is shown in Fig. 7(b). The distribution of the electric field Ey components corresponding to the device's TE0, TE1, and TE2 mode inputs are shown in Fig. 7(c), verifying the corresponding mode switching functionality. The transmission spectrum shown in Fig. 7(d) and 7(e). At 1550 nm, the simulated mode switching has an insertion loss of 0.65 dB for TE0 - TE1, 0.30 dB for TE1 - TE2, and 0.48 dB for TE2 - TE0, respectively.
Fig. 7. (a) The permutation matrix and its corresponding mode switching functionality. (b) Schematic of the corresponding U1 and U2 device cascades. (c) Electric field responses of the TE0, TE1, and TE2 mode inputs of the cascaded mode-based switching devices. (d) and (e) Simulated transmission spectrum of the cascade structure.
Similarly, the permutation matrix P5 in Table 1 and the functional schematic are shown in Fig. 8(a). The cascaded device structure and its electric field distribution corresponding to the TE0 to TE2 mode inputs are shown in Fig. 8(b) and 8(c). As depicted in Fig. 8(d) and 8(e), the mode conversion insertion losses at 1550 nm wavelength for TE0 - TE2, TE1 - TE2, and TE2 - TE1 mode conversion are 0.64 dB, 0.29 dB and 0.49 dB, respectively according to the simulation.
Fig. 8. (a) Schematic of permutation matrix and the mode-based signal switching functionality. (b) The structure by the U2 and U1 cascading. (c) Devices’ electric field Ey component distributions of TE0, TE1, and TE2 mode inputs. (d) and (e) Simulated transmission spectrum of the cascade structure.
In the verification above, all six mode based signal routing states corresponding to permutation matrices in Table 1 can be implemented by cascading the selected U1, U2, and U3 basis, which validates our signal switching architecture through guided-mode carrier.
4. Conclusion
In conclusion, using the adjoint variable method and level-set method, we implemented three-mode manipulation devices on the SOI platform: the TE0 - TE1 mode switch unit (U1), the TE1 - TE2 mode switch unit (U2), and the TE0 - TE2 mode switch unit (U3). The experimental results show that devices all complete the corresponding mode switching functionality. The U1 device has an optical signal-to-noise ratio of better than 9.5 dB and has less than 2.8 dB insertion loss. The insertion loss of the U2 device is less than 3.1 dB, and the optical signal-to-noise ratio is more than 10.8 dB. The optical signal-to-noise ratio of the U3 device is more than 12.6 dB, with an insertion loss of less than 2.1 dB.
Based on the properties that the mode-based signal routing states can be completely described by the permutation matrices and that the two-mode switching corresponds to the row transformation of the matrix, we illustrate that an arbitrary-mode switching state can be obtained by cascading the two-mode switching devices. Thus, we demonstrate our signal switching architecture through guide-mode carrier in the three-mode scenario with the designed mode switching units U1, U2, and U3 as the selected basis. We verify two cascading structures with the simulation, and the insertion losses are less than 0.7 dB for all modes at the wavelength of 1550 nm. This architecture prevents the passive signal switch's design costs skyrocketing as the number of channels grows and is potential way for building passive signal switching in MDM systems. Moreover, the design approach of selecting specific devices as basis may enlighten the research of other passive devices.
Funding
National Key Research and Development Program of China (2019YFB2203602); National Science Fund for Distinguished Young Scholars (61825504); National Natural Science Foundation of China (61905101, 61975198); Natural Science Foundation of Gansu Province (20JR5RA243).
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.
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