Fig. 1. CM from the arc-bend in the physical space to a rectangle in the virtual space. (a) The arc-bend is composed by two additional rectangles with length La=8  μm and width Ws=6  μm and a 90° arc with radius R=9  μm. (b) The rectangle in the virtual space has length Lt and width Ws=6  μm. Lt is optimized to ensure u and v satisfy the Cauchy–Riemann condition of CM. The optimal Lt is 29.683 μm. All the grids represent coordinates (u,v) in subplots (a) and (b).

Fig. 2. (a) Index distribution of the waveguide arc-bend in physical space. The width of waveguide arc-bend is 2 μm, which can support four TE modes. The waveguide core index is 2.84, which is the TE0 mode effective index of a 220 nm thick silicon slab waveguide with SiO2 upper cladding. (b) Index distribution of the original waveguide in virtual space. It is obtained by Eq. (1) of CM. This original waveguide is not straight and has a gradient index distribution. These two negative factors cause the big loss and intermode cross talk of the traditional multimode waveguide arc-bend.

Fig. 3. (a) Index distribution of the shape optimized original waveguide in virtual space; (b) index distribution of the MWB in physical space. This MWB has a uniform core index, which makes fabrication easier.

Fig. 4. Multimode propagation performance of the shape-optimized original waveguide in virtual space. (a)–(d) The profiles of the Hz field component for (a) TE0, (b) TE1, (c) TE2, and (d) TE3 modes at 1550 nm wavelength, respectively, are shown.

Fig. 5. 3D FDTD simulation results of shape-optimized MWB in physical space. (a)–(d) The profiles of the Hz field component for (a) TE0, (b) TE1, (c) TE2, and (d) TE3 modes at 1550 nm wavelength, respectively, are shown. The insets show the input/output Hz field profiles.

Fig. 6. Simulated transmission spectra of shape-optimized multimode bend waveguide. (a)–(d) The spectra for (a) TE0, (b) TE1, (c) TE2, and (d) TE3 modes, respectively, are shown. At 1550 nm, the excess losses for TE0, TE1, TE2, and TE3 modes are 0.010, 0.010, 0.018, and 0.022 dB, respectively, and intermode cross talks are all below −24  dB. Throughout a broad wavelength range of 1.16–1.66 μm, the losses of all modes are below 0.1 dB and the cross talks are all below −20  dB.

Fig. 7. Simulated transmission efficiencies of shape-optimized MWB at 1550 nm wavelength vary with the width fabrication deviation Δw. (a)–(d) The efficiencies for (a) TE0, (b) TE1, (c) TE2, and (d) TE₃ modes, respectively, are shown. For the intermode cross talk below −20  dB, the theoretical permitted width fabrication deviation of our MWB is ±50  nm.

Fig. 8. Microscopic view of the fabricated PICs and device. (a) The referenced PIC without MWB, including grating couplers, mode multiplexers, and demultiplexers; (b) PIC with MWB; (c) magnified microscopic view of the shape-optimized MWB.

Fig. 9. Measured transmission spectra of the shape-optimized MWB for (a) TE0, (b) TE1, (c) TE2, and (d) TE3 modes, respectively. At 1550 nm wavelength, the excess losses for TE0, TE1, TE2, and TE3 modes are 0.232, 0.327, 0.477, and 0.546 dB, respectively, and intermode cross talks are all below −17  dB.