Ultrashort and broadband silicon polarization splitter-rotator using fast quasiadiabatic dynamics

We propose an ultrashort and broadband silicon mode-conversion polarization splitter-rotator (PSR) consisting of a taper and a Y-junction both designed by the fast quasiadiabatic dynamics (FAQUAD). The FAQUAD is used to homogeneously distribute adiabaticity over the length of the PSR, providing shortcut to adiabaticity at a shorter device length. The total length of the silicon PSR is 39.2 μm. For a wavelength range from 1.5 μm to 1.6 μm, the PSR exhibits a good performance with > 88% transmission and > 11.4 dB extinction ratio (ER). Simulations also show that the designed devices have good fabrication tolerance. © 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement OCIS codes: (130.3120) Integrated optics devices; (130.5440) Polarization-selective devices. References and links 1. T. Tsuchizawa, K. Yamada, H. Fukuda, T. Watanabe, J. Takahashi, M. Takahashi, T. Shoji, E. Tamechika, S. Itabashi, and H. 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Introduction
Silicon waveguides based on the silicon-on-insulator (SOI) structure have attracted a lot of attention due to their good mode confinement, enhanced nonlinearity, and compatibility with commercial complementary metal-oxide-semiconductor (CMOS) process [1].Therefore, high-density integration of optical components and mass production can be realized [2].However, polarization control in SOI structure remains a major issue.Because the SOI structure has large polarization-dependence caused by structural birefringence that induces polarization dependent dispersion and loss, and responds differently to light polarized along different axes [3,4].Therefore, in photonic integrated circuits, polarization control devices like polarization beam-splitter (PBS) which splits the TE0 mode and TM 0 mode into different outputs [5]; polarization rotator (PR) which rotates the polarization by 90 degrees [6]; or polarization splitter-rotator (PSR) which splits the two orthogonal polarizations first, then rotates one of the two polarizations by 90 degrees [7], are desirable.Recently, many types of PSRs have been developed.For instance, the compact PSRs designed using asymmetric directional couplers (ADCs) where one polarization mode can be coupled to the orthogonal mode in the coupled waveguide with equal effective index [8,9].However, ADCs are inherently sensitive to wavelength and fabrication variations because ADCs need precise phase matching.Other PSRs based on tapered directional couplers [10], ADCs cascaded with mode-evolution taper [11], mode-evolution taper cascaded with mode-sorting asymmetric Yjunction [12], and adiabatic couplers [13] have been proposed to enhance the robustness, but the device lengths are relatively long due to their adiabatic designs.
Owing to the analogies between quantum mechanics and wave optics [14], many quantum phenomena have been exploited to implement light manipulation in waveguide structures.A technique called shortcuts to adiabaticity (STA) has been proposed to speed up adiabatic passage in the context of quantum control [15] and has found applications in Bose-Einstein condensate (BEC) [16] and electron spin control [17].In optics, many short and robust coupled waveguide devices have been designed using STA, such as mode converter [18,19], Y-junctions [20], directional couplers [21][22][23][24], mode (de)multiplexers [25,26], and polarization rotators [27].The sensitivity to wavelength and fabrication errors can be minimized favorably in these designs.However, these STA waveguides have complex geometries and could be difficult to fabricate because the waveguide separation and waveguide widths vary simultaneously along the device.Recently, a new STA protocol called the fast quasiadiabatic dynamics (FAQUAD) has been proposed [28].By driving a single control parameter as close to the adiabatic limit as possible during the evolution, slow adiabatic evolution in the device can be significantly sped up.The FAQUAD approach has been applied to the design of Y-junction mode (de)multiplexers using the low index contrast polymer platform [29] as well as the high index contrast SOI platform [30].However, the application of the FAQUAD protocol to polarization manipulation has yet to be investigated.
In this paper, we design a taper polarization mode converter by the FAQUAD first (we call it the FAQUAD taper).An improved FAQUAD protocol which decouples the adiabaticity parameter and the control parameter is introduced.The applicability of the FAQUAD protocol to the taper eigenmodes involving polarization conversions is investigated.Next, we design an asymmetric Y-junction two-mode (de)multiplexer by the FAQUAD (we call it the FAQUAD Y-junction).Finally, we design a PSR by cascading the FAQUAD taper with the FAQUAD Y-junction.The total device length is 32.9 μm, and for a wavelength range from 1.5 μm to 1.6 μm, the PSR exhibits a good performance with > 88% transmission and > 11.4 dB extinction ratio (ER).Performance degradations due to fabrication imperfections are also analyzed, showing that the FAQUAD PSRs have good fabrication tolerance.

Mode conversion in the mode hybridization region
We design the device using the SOI strip waveguide with 220-nm-thick top silicon layer with air cladding and a 2-μm-thick buried oxide layer, as shown in Fig. 1(a).The mode conversion between a fundamental mode and a higher-order mode may occur in a taper structure because of mode hybridization at some specific waveguide widths [11,31].Figure 1(b) shows the effective indices n eff of the guided modes as a function of the width W in the SOI strip waveguide.We can see that there is a region labeled by the circle shown in Fig. 1(b) where the dispersion curves of the TM 0 and TE 1 modes have an anti-crossing due to mode hybridization.In the mode hybridization region, we cannot distinguish the TM 0 and TE 1 modes; and mode conversion between the TM 0 and TE 1 modes will occur if the light propagates along an adiabatic taper structure whose initial waveguide width W 1 and final waveguide width W 2 satisfy the condition: W 1 < W c < W 2 , where W c is the waveguide width at the mode hybridization region.Following [11,31], we first analyze the adiabaticity of a linear adiabatic taper with W 1 = 0.6 μm and W 2 = 0.9 μm that converts the input TM 0 mode to TE 1 mode.

Device adiabaticity parameter
In integrated optics, adiabaticity in a spatially varying device is usually achieved by keeping the coupling between the local eigenmodes (supermodes) below a certain level according to the coupled-mode theory [32].In general, the adiabatic condition to be established must meet the following condition [33]: where , m n are the eigenmodes of an optical waveguide, β m,n are the propagation constants of the eigenmodes, and c(z) is the adiabaticity parameter.The inner product in Eq. ( 1) is the same as the mode orthogonality check of guided modes in optical waveguides; that is * ˆ, where E mt and H nt are the transverse components of the electric field and magnetic field associated with the mth and nth eigenmode of the optical waveguide, and S is the entire waveguide cross section.Therefore, the adiabaticity parameter c(z) in an optical waveguide can be derived by substituting Eq. ( 2) to Eq. ( 1) * ( ) The adiabaticity parameter c(z) for a conventional linear taper from W 1 = 0.6 μm and W 2 = 0.9 μm with a length L is shown in Fig. 2(a) (blue-dashed curve, the top axis labels the corresponding width W of the linear taper), and we can see that the c(z) of the linear taper is small at widths ranging from 0.7 μm to 0.9 μm.However, when the width is between 0.6 and 0.7 μm, c(z) becomes large.Therefore, the linear taper needs to be long enough to compensate for the large c(z) in order to satisfy the adiabatic condition.That is why the linear adiabatic taper has long device lengths in general.

The FAQUAD strategy
In the FAQUAD, we fix the adiabaticity parameter c(z) at a constant ε (to be determined later) and use the chain rule: ( ) , as shown in Fig. 2(a) (green-dotted line).Therefore, the taper length would not need to be so long to compensate for the large c(z) between W = 0.6 μm and W = 0.7 μm in a linear taper, and the adiabatic taper length can thus be shortened by the FAQUAD.The FAQUAD strategy can be understood as a design protocol to homogeneously distribute the adiabaticity parameter to speed up the adiabatic evolution.In order to obtain the functional form of the waveguide width W, we first calculate the adiabaticity parameter of the linear taper c lin by ( ) ( ) , where L is the length of the taper, W i and W f are the initial and final waveguide taper widths, respectively.Substituting Eq. ( 5) into Eq.( 4), we can obtain the following equation ( ) Inverting Eq. ( 6), we obtain The boundary conditions are z(W i ) = 0 and z(W f ) = L.The adiabaticity parameter of the linear taper c lin is strictly related to W and decoupled from z. Integrating Eq. ( 7) and requiring that z(W f ) = L, we obtain ( ) (ε is determined by summing the adiabaticity parameter and redistributing it along W) and finally arrive at the functional form of the FAQUAD taper structure z(W) The key here is to invert Eq. ( 6) and integrate against W instead of z.The result in Eq. ( 8) improves the previous FAQUAD algorithm in [29,30], where the adiabaticity parameter and z are not completely decoupled in the analysis, leading to a less than optimal FAQUAD design.
We use a commercial software (FIMMWAVE, Photon Design) [34] employing a finite difference method (FDM) to calculate the transverse component of the electric field E 2t and magnetic field H 3t associated with the second eigenmode (TM 0 to TE 1 ) and the third eigenmode (TE 1 to TM 0 ) of the taper, and the propagation constants β 2 of the second mode and β 3 of the third mode of the linear taper, and substitute E 2t , H 3t , β 2 and β 3 of the linear taper into Eq.( 3) to get the c lin .We set W i = 0.6 μm and W f = 0.9 μm in the calculation.Despite the eigenmodes going through mode conversion in the process, the adiabaticity parameter can still be obtained.The blue-dashed curves in Figs.2(a) and 2(b) show the c lin and the z(W) of the linear taper, respectively.Then we substitute c lin into Eq.( 8) to obtain the z(W) of the FAQUAD taper, as shown in Fig. 2(b) (red-solid line).Finally, we calculate the c(z) of the FAQUAD taper, and as expected, we find that the adiabatic parameter of the FAQUAD taper is indeed homogenously distributed in Fig. 2(a) (red-solid line).We note that there are small discrepancies between the calculated adiabaticity parameter c(z) and the theoretical value ε.This is because we first obtain z(W) using Eq. ( 8) with uniformly spaced steps in W, and the functional form is resampled with uniformly spaced z steps when we calculate the adiabaticity parameter with the FDM solver.
Previously, the FAQUAD strategy was applied to homogenize the adiabaticity in mode (de)multiplexing Y-junctions where the two modes have the same polarization [29,30].From the analysis above, we have successfully verified that the FAQUAD strategy can be applied to the analysis of eigenmodes which goes through polarization conversion via mode hybridization, showing the versatility of the FAQUAD approach.A new FAQUAD protocol which completely decouples the adiabaticity parameter and the control parameter in the analysis is also introduced.

The FAQUAD taper polarization mode converter
Next, we simulate the FAQUAD taper by using a commercial software (FIMMPROP, Photon Design) employing a full vectorial eigenmode expansion method (EME) [34].The device is simulated at a wavelength of 1.55 μm.The taper mode converter contains three sections as shown in Fig. 3: the first section is a linear taper with L1 = 1 μm and width varying from W 0 = 0.5 μm to W 1 = 0.6 μm; the second part is a FAQUAD taper with L 2 = 22.5 μm and width varying from W 1 = 0.6 μm to W 2 = 0.9μm; the third section is a linear taper with L 3 = L 1 = 1 μm and widths varying from W 2 = 0.9 μm to W 3 = 1 μm.The simulation results of the 24.5 μm long taper polarization mode converter are shown in Fig. 3.We can see that the input TM 0 mode converts to the TE 1 mode along the FAQUAD taper [Fig.3(a)], and the input TE 0 mode goes through the FAQUAD taper unchanged [Fig.3(b)].Figure 4 shows the relation between L 2 and conversion efficiency from TM 0 to TE 1 for the linear taper and the FAQUAD taper.We can find that the FAQUAD taper only needs 22.5 μm or 9.4 μm to achieve 99.71% transmission, and the linear taper needs 78 μm device length to achieve 99.7% transmission.Therefore, we have successfully shortened the linear adiabatic taper by the FAQUAD strategy.The reason we chose L 2 = 22.5 μm is for the better bandwidth at 22.5 μm than at L 2 = 9.4 μm.For more compact devices, L 2 = 9.4 μm can be chosen.

The FAQUAD Y-junction
Asymmetric Y-junctions are often used to as mode sorters in photonic integrated circuits [29,30,35].It has a two-modes stem waveguide and two single-mode output arms with different widths.When the fundamental (second) mode of the stem propagates through the junction, it evolves into the fundamental mode of the wider (narrower) output arm, and vice versa.Previously, we have used the FAQUAD strategy to design an asymmetric Y-junction two-mode (de)multiplexer in silicon buried waveguide [30].In this paper, we use the SOI strip waveguide structure, as shown in Fig. 1(a), to design a FAQUAD Y-junction by the same FAQUAD strategy outlined above.In Fig. 5, the widths of the FAQUAD Y-junction output arms port A and port B are WA = 0.45 μm and W B = 0.55 μm, which satisfy the relationship W A + W B = W 3 of the polarization mode converter in the previous section, and the final waveguide gap is 500 nm. Figure 5 shows the simulated field distributions in the FAQUAD Y-junction with L 4 = 7.4 μm.We can see that the TE 0 mode has evolved to the A port at the output, and the TE 1 mode has evolved to the B port.In Fig. 6, we show the simulated transmission in ports A and B for TE 0 and TE 1 mode inputs as a function of the device L 4 length for both the FAQUAD and the linearly separating Y-junction.We find that the FAQUAD Y-junction only need 7.4 μm long to achieve 99% transmission.On the other hand, the linearly separating Y-junction only achieves 97% transmission at a length of 100 μm.As expected, the FAQUAD Y-junction provides shortcut at a shorter device length.In order to construct a compact device, we choose the length of the FAQUAD Y-junction to be L 4 = 7.4 μm, and the length of stem waveguide L stem = 1 μm.

The combined PSR
Finally, we design the PSR by cascading the FAQUAD taper with the FAQUAD Y-junction. Figure 7 shows the structure of the entire PSR. Figure 8 shows the simulated field distributions in the FAQUAD PSR.The total length L total of the PSR is 32.9 μm, and we can see that when the TM 0 mode is launched into the PSR, we obtain the TE 0 mode output at the A port of the PSR.On the other hand, when the TE 0 mode is launched into the PSR, we obtain the TE 0 mode output at the B port.Therefore, the TE 0 and TM 0 modes of the input light are split, and the TM 0 mode is rotated to the TE 0 mode successfully.We then look at the device bandwidth.Figure 9 shows the wavelength dependence of the transmission of the PSR using the TM 0 and TE 0 inputs, and it can be seen that for a wavelength range from 1.5 μm to 1.6 μm, the PSR exhibits a good performance with > 88% transmission and > 11.4 dB ER.We also simulate the bandwidth of the shorter PSR with L 2 = 9.4 μm in the FAQUAD taper, and the total length of the PSR is 19.8 μm. Figure 10 shows the wavelength dependence of the transmission of the PSR using the TM 0 and TE 0 mode inputs.We can see that for a wavelength range from 1.5 μm to 1.6 μm, the PSR exhibits a performance with > 79% transmission and > 6.8 dB ER.Therefore, the PSR is shortened at the expense of lesser bandwidth.Next, the fabrication tolerance of the designed PSRs are investigated.We simulate the performance variations due to fabrication imperfections by adding Δw to the device width.Figures 11(a

Discussion and conclusion
There is a large body of work on the geometry optimization of photonics devices to improve their performance or functionality.For example, the topology optimization approach has been successfully applied to realize various functionalities such as polarization manipulation [36], wavelength demultiplexing [37], and mode demultiplexing [38].These topology optimized devices in general need long computation times to reach convergence, and the obtained topology has fine features that require electron-beam lithography.Other approaches based on optimization schemes such as the particle swarm [39] or genetic algorithm [40] are also computationally intensive, and the segmentation could leave sharp bends in the designs.The FAQUAD approach to silicon PSR design discussed in this work does not require numerical iterations and results in smooth waveguide parameter variations that are less demanding on fabrication.While the FAQUAD protocol has previously been applied to mode (de)multiplexer Y-junction design, we show, for the first time, that FAQUAD can be used to find shortcuts in polarization manipulation devices in silicon photonics.Combining the FAQUAD polarization mode converter and Y-junction, we obtain an ultrashort and robust PSR, showing the versatility of the FAQUAD approach.
In summary, we propose an ultrashort and broadband silicon mode-conversion PSR which is consisted of a taper and a Y-junction both designed by the FAQUAD.When the TM0 mode is launched into the PSR, we obtain TE 0 mode output from the A port of the PSR.On the other hand, when the TE 0 mode is launched into the PSR, we obtain TE 0 mode output from the B port.Therefore, the TE 0 and TM 0 modes of the input light are split and the TM 0 mode is rotated into the TE 0 mode successfully.An improved FAQUAD protocol which decouples the adiabaticity parameter and the control parameter is introduced.The FAQUAD device provides shortcut to conventional adiabatic devices at a shorter device length, and the total length of the device is only 32.9 μm.For a wavelength range from 1.5 μm to 1.6 μm, the PSR exhibits a good performance with > 88% transmission and > 11.4 dB ER.We simulate the bandwidth of the shorter PSR at 19.8 μm, and the PSR has > 79% transmission and > 6.8 dB ER ranging from 1.5 μm to 1.6 μm.The PSRs also have good fabrication tolerance.

Fig. 1 .
Fig. 1.(a) The cross-sectional view of the SOI strip waveguide.(b) The effective indices n eff of the guided modes as a function of the width of the SOI strip waveguide.

Fig. 2 .
Fig. 2. (a) Device adiabaticity parameter c(z) for the linear taper (blue-dashed curve, the top axis labels the corresponding width W of the linear taper) and FAQUAD taper (red-solid curve) and the ideal FAQUAD taper (green-dotted line).(b) variation of z with W for the linear taper and the FAQUAD taper.

Fig. 5 .
Fig. 5. Simulated field distributions in the FAQUAD Y-junction using the (a) TE 1 and (b) TE 0 modes inputs.The length of the FAQUAD Y-junction L 4 is 7.4μm, and the length of the stem waveguide L stem is 1 μm.

Fig. 6 .
Fig. 6.Simulated transmission as a function of the Y-junction length L 4 at ports A and B of the FAQUAD and linearly separating Y-junctions using (a) TE 1 mode (b) TE 0 mode of the stem waveguide as the input.

Fig. 7 .
Fig. 7. Top view of the FAQUAD PSR consisting of the FAQUAD taper and the FAQUAD Yjunction.

Fig. 8 .
Fig. 8. Simulated field distributions in the FAQUAD PSR using the (a) TM 0 and (b) TE 0 mode inputs.The total length L total of the device is 32.9 μm.

Fig. 9 .
Fig. 9. Simulated transmission of the FAQUAD PSR as a function of the wavelength in the output (a) port A and (b) port B with L 2 = 22.5 μm, L 4 = 7.4 μm.The total length of the device is 32.9 μm.

Fig. 10 .
Fig. 10.Simulated transmission of the FAQUAD PSR as a function of the wavelength in the output (a) port A and (b) port B with L 2 = 9.4 μm, L 4 = 7.4 μm.The total length of the device is 19.8 μm.
) and11(b)  show the fabrication tolerance of the PSR with L 2 = 22.5 μm.We can see that the PSR exhibits a performance with > 80% transmission and > 8 dB ER for Δw from −12.5 nm to 12.5 nm.The performance is largely limited by the FAQUAD Y-junction that has a length of L 4 = 7.4 μm.By increasing the length of the FAQUAD Y-junction to L 4 = 16.5 μm as indicated by the second maxima in Fig.6, the fabrication tolerance can be further enhanced.Figures12(a) and 12(b) shows the fabrication tolerance of the PSR with L 2 = 22.5 μm and L 4 = 16.5 μm (total length = 42 μm), the PSR exhibits a performance with > 86% transmission and > 12 dB ER for Δw from −12.5 nm to 12.5 nm.The simulation results indicate that the designed PSRs are indeed broadband and have good fabrication tolerance.

Fig. 11 .
Fig. 11.Simulated transmission of the FAQUAD PSR as a function of the fabrication error Δw in the output (a) port A and (b) port B with L 2 = 22.5 μm, L 4 = 7.4 μm.The total length of the device is 32.9 μm.

Fig. 12 .
Fig. 12. Simulated transmission of the FAQUAD PSR as a function of the fabrication error Δw in the output (a) port A and (b) port B with L 2 = 22.5 μm, L 4 = 16.5 μm.The total length of the device is 42 μm.