Nonvolatile and ultra-low-loss reconfigurable mode (De)multiplexer/switch using triple-waveguide coupler with Ge2Sb2Se4Te1 phase change material

Mode-division multiplexing (MDM) is a promising approach to dramatically enhance the transmission capacity. A reconfigurable mode (De)multiplexer/switch (RMDS) is a key component for the flexible mode routing in the MDM network. A nonvolatile and ultra-low-loss RMDS is proposed via a triple-silicon-waveguide directional coupler with the Ge2Sb2Se4Te1 (GSST) phase change material (PCM). The nonvolatile property of GSST makes it attractive to reduce the switching power-consumption. Benefiting from the low loss of the GSST-PCM at both amorphous and crystalline states, an RMDS with an ultra-low loss and a high extinction-ratio can be realized. The proposed RMDS is optimally designed by using the full-vectorial finite element method and 3D full-vectorial finite difference time domain method. The numerically simulated results show that a compact RMDS is with the extinction ratios of 18.98 dB and 22.18 dB, ultra-low insertion losses of 0.10 dB and 0.68 dB for the “OFF” and “ON” states, respectively at the operating wavelength of 1550 nm. An ultra-wide bandwidth of 100 nm is achieved for both the “OFF” and “ON” states.


Results
Schematic and Principle. The schematic diagram of the proposed RMDS is shown in Fig. 1(a), consisting of a triple-WG DC and a two-WG DC. The triple-WG DC is comprised of an input SM-WG-1, a central WG-G with GSST, and an external bus WG-2. The input fundamental TM mode can be transferred into the bus WG-2 and outputs at port O 2 , when the state of the GSST is amorphous at "OFF" state of the RMDS. While the c-GSST is triggered at "ON" state, the input quasi-TM 0 mode would propagate along the input SM-WG-1 without mode coupling due to the phase-mismatching. Following this, the input quasi-TM 0 mode will be converted to the quasi-TM 1 mode in the bus WG-3 of the two-WG DC and output at port O 1 . It can be observed from Fig. 1(a) that the coupling lengths are denoted by L OFF and L ON for the triple-and two-WG DCs, respectively. The cross-sections of the triple-and two-WG DCs over the lengths of L OFF and L ON are shown in Fig. 1(b,c), respectively. The widths of the input SM-WG-1, central WG-G, bus WG-2 and -3 are represented by W 1 , W G , W 2 and W 3 , respectively. The height of the silicon layer of these four WGs is denoted by h, which is chosen to be a common height, h = 220 nm of the silicon WG in this case. It can be observed from Fig. 1(b) that for the triple-WG DC, the separation in between each pair of three WGs is denoted by g 1 . The heights of the GSST and intermedium silica layers of the central WG-G are denoted by h G and h s , respectively. It can also be observed that for the two-WG DC, the separation between the input SM-WG-1 and the bus WG-3 is denoted by g 2 . In this case, the refractive indices of the silicon and silicon dioxide are taken as 3.47548 and 1.46, respectively at the operating wavelength of 1550 nm. A silica cladding is used to protect the silicon WGs. The refractive indices of the a-and c-GSST are taken as 3.39 + (1.8 ± 1.2) × 10 −4 i and 5.14 + 0.42i, respectively 17 .
The phase change of the GSST could be supposed to be similar to that of the GST-PCM 18 , which can be triggered via thermo-optic (TO), EO/electro-thermal (ET), and all-optical approaches 19 . In general, the a-GSST can be obtained by heating the c-GSST above the melting point followed by rapid cooling (<1 ns). The c-GSST can be achieved by heating the a-GSST to be crystallized. By introducing a heater above the GSST, the phase state of the GSST can be simply adjusted based on the TO effect. An alternative TO approach could be the use of the pulsed laser to achieve the photo-thermal heating and then change the phase state 20 . For the EO or ET based approach, the silicon sections can be doped as upper and lower contacts and then the applied voltage across the doped silicon can result in the desired thermal phase change via Joule heating current 21 . For the all-optical approach, a pump light can be used to go through the c-GSST section and be partly absorbed by c-GSST due to the relatively high absorption loss of c-GSST, which can also provide the desired thermal phase change for re-amorphization 22 . However, it may be difficult to achieve recrystallization due to the ultra-low extinction coefficient, k = (1.8 ± 1.2 × 10 −4 ) of a-GSST. In this case, the phase transition of the GSST is expected to be achieved by using the pulsed laser induced photo-thermal heating, which can be implemented without any electrode. In future, the approach based on the micro-heater or electro-thermal effect may be preferable for flexible photonic integrated circuits.
Characterization of mode properties. Firstly, the design of the triple-WG DC section with the GSST-PCM is studied by using the FV-FEM. The proposed triple-WG DC operates at two states: (a) for the "OFF" state with a-GSST, the input quasi-TM 0 mode is converted to the quasi-TM 1 mode of the bus WG-2 under the phase-matching condition; (b) for the "ON" state with c-GSST, the input quasi-TM 0 mode propagates along the input SM-WG-1 without any mode coupling. To achieve a maximum mode coupling efficiency at "OFF" state, the phase-matching condition for the input SM-WG-1, central WG-G with a-GSST and the bus WG-2 should be satisfied for the triple-WG DC. Variations of the effective index with the width of the silicon WG are shown in Fig. 2(a), in which the size of the input SM-WG-1 is taken as W 1 × h = 400 nm × 220 nm. The effective index of the quasi-TM 0 mode of the input SM-WG-1 is calculated to be n eff = 1.71, denoted by a horizontally dash-dotted line in Fig. 2(a). The phase-matched width of the bus WG-2 is determined to be W 2 = 1.075 μm. The H x mode fields of the phase-matched quasi-TM 0 and quasi-TM 1 modes are shown in Fig. 2(b,c), respectively.
For the triple-WG DC, more supermodes would be generated compared to the two-WG DC. In this case, the input quasi-TM 0 should be phase-matched with the quasi-TM 1 mode of the bus WG-2. Hence, three supermodes (TM-A, TM-B, and TM-C) are studied and the supermode fields are shown in Fig. 3. In the calculation, the parameters of the triple-WG DC are set as: W 1 = 400 nm, W 2 = 1.075 μm, g 1 = 200 nm, h G = 100 nm and h s = 0 nm. The E y fields of the TM-A, TM-B, and TM-C supermodes are shown in Fig. 3(a-c), respectively. The Poynting vectors, P z (x, y) of these three supermodes are also calculated and shown in Fig. 3(d-f), respectively. To meet the phase-matching condition for this triple-WG DC, the effective indices of the TM-A, TM-B, and TM-C supermodes must satisfy the following condition 23 : where n A , n B , and n C are the effective indices of the TM-A, TM-B, and TM-C supermodes, respectively. Variations of the effective indices of n A , n B , n C , and (n A + n C )/2 with the width of the central WG-G are shown in Fig. 4 and denoted by the dashed blue, dash-dotted black, dotted green and solid pink lines, respectively. The triple-WG DC with the parameters of W 1 = 400 nm, W 2 = 1.075 μm, h = 220 nm, g 1 = 200 nm, h G = 100 nm and h s = 100 nm is taken as an example. The effective indices of the n A , n C and (n A + n C )/2 are increased with the increase of W G , while that of the n B is kept constant due to the mode field of the TM-B supermode only confined in both the WG-1 and WG-2. According to Equation (1), the phase-matched W G is chosen to be W G = 173 nm.
Next, the coupling length of the triple-WG DC can be calculated by using the formula 23 : where β A and β C are the propagation constants of the TM-A and TM-C supermodes, respectively; λ 0 is the operating wavelength. In order to achieve a compact RMDS, the coupling length should be optimised to be as short as possible. Variations of the coupling lengths with the heights of the intermedium silica and a-GSST layers are shown in Fig. 5(a,b), respectively. It can be noted from Fig. 5(a) that the coupling lengths for h G = 50 nm, 100 nm and 150 nm are denoted by the solid black, dash-dotted red, and dotted blue lines, respectively. It can be noted from Fig. 5(a) that the coupling lengths are monotonically increased with the increase of h s . Therefore, a minimum coupling length, L OFF would be obtained in the absence of the intermedium silica layer of the central WG-G. It can also be noted from Fig. 5(a) that a shorter coupling length would be achieved with a larger height  of the a-GSST layer. It can be noted from Fig. 5(b) that the coupling lengths for g 1 = 100 nm, 200 nm and 300 nm are shown by the solid black, dash-dotted red, and dotted blue lines, respectively. A larger separation would induce a longer coupling length due to a weaker mode interaction. For a certain separation, the coupling length is decreased with the increase of the height of the GSST. Particularly, the coupling lengths go to the constants at h G = 50 nm, 70 nm and 90 nm for g 1 = 100 nm, 200 nm and 300 nm, respectively. The index difference, Δn eff between (n A + n C )/2 and n B at c-GSST state means the intensity of the phase-mismatch of the proposed triple-WG DC at "ON" state, which can determine the mode CT. Variations of the index difference with the height of the c-GSST layer are shown in Fig. 6 and denoted by the solid black, dash-dotted red and dotted blue lines for g 1 = 100 nm, 200 nm, and 300 nm, respectively. It can be noted that the index differences are increased with the increase of the height of the c-GSST layer, which can be explained by the fact that a larger index-modulation can be generated via a thicker c-GSST layer. If the index difference goes up to 0.15, the heights of the c-GSST layers need to be larger than 57.9 nm, 54.5 nm and 79.8 nm for g 1 = 100 nm, 200 nm and 300 nm, respectively. The corresponding coupling lengths are calculated to be L OFF = 4.0 μm, 5.89 μm and 8.5 μm. Although a more compact coupling length can be obtained with a narrower separation, a higher  mode CT would be generated due to the stronger mode interaction. In order to balance the coupling length and the mode CT, the propagation characteristics of the proposed triple-WG DC will be studied by using the 3D-FV-FDTD in the following section. For the "ON" state of the proposed RMDS, the input quasi-TM 0 mode would directly propagate along the input SM-WG-1 and then be coupled to the quasi-TM 1 mode of the bus WG-3 via a two-WG DC. Although the phase-matching condition between the isolated SM-WG-1 and bus WG-3 can be obtained from Fig. 2, it is essential to study that of the combined two-WG of the DC, which maybe dramatically different with each other due to the strong mode-interaction. Variations of the effective index with the width of the bus WG-3 are shown in Fig. 7(a). The even-like and odd-like supermodes are denoted by the solid and dash-dotted lines, respectively. The separations, g 2 = 200 nm, 250 nm and 300 nm are shown by the black, blue and red lines, respectively. It can be noted from Fig. 7(a) that the effective indices of the even-like and odd-like supermodes become closer near W 3 = 1.1 μm and get mixed to form two phase-matched supermodes. The coupling length of the two-WG DC can be calculated based on the formula 24 : L c = π/(β even-like − β odd-like ), where β even-like and β odd-like are the propagation constants of the even-like and odd-like supermodes, respectively. Variations of the coupling lengths of the two-WG DC with the width of the bus WG-3 are shown in Fig. 7(b). The coupling length is increased with the increase of the separation. Under the phase-matching conditions (W 3 = 1.15 μm, 1.11 μm and 1.1 μm), the coupling lengths, L ON are calculated to be L ON = 6.9 μm, 8.4 μm and 10.3 μm for g 2 = 200 nm, 250 nm and 300 nm, respectively. The mode fields of two supermodes at g 2 = 200 nm are calculated by using the FV-FEM and shown in Fig. 8. The E-field intensities of the phase-matched even-like and odd-like supermodes are shown in Fig. 8(a,b), respectively. Similar to the triple-WG DC, a narrower separation of the two-WG DC would induce a higher mode CT. The balance between the coupling length and the mode CT will also be studied by using the 3D-FV-FDTD.
Operation and bandwidth. The propagation characteristics of the proposed RMDS are investigated by using the 3D-FV-FDTD. In order to evaluate the performance of the proposed RMDS, the insertion loss (IL) and mode ER are studied and calculated by using the formulas 23 : where P I1 is the input power at port I 1 ; P O1 and P O2 are the output powers at ports O 1 and O 2 , respectively. The mode CT of the proposed RMDS is calculated by CT = −(IL + ER). For the triple-WG DC section, variations of the ER (left y-axis) and IL (right y-axis) with the height of the GSST layer are calculated based on the 3D-FV-FDTD and shown in Fig. 9(a,b) for the "OFF" and "ON" states, respectively. It can be noted from Fig. 9(a) that at "OFF" state, the ER is increased with the increase of h G for both g 1 = 200 and 300 nm, while the IL is decreased. Similarly, it can be noted from Fig. 9(b) that the ER is increased and the IL is decreased with the increase of h G for the "ON" state. Although a higher ER and a lower IL can be obtained with a larger h G , it would induce a critical fabrication process for the central WG due to the large height-to-width ratio (H/W ratio). In this case, the height of the GSST layer of h G = 150 nm is considered. Variations of the ER (left y-axis) and IL (right y-axis) with the gap, g 1 are shown in Fig. 10(a). At both "OFF" and "ON" states, the ERs are monotonously increased as a function of the gap, g 1 . At "ON" state, the IL is dramatically decreased from 3.7 dB at g 1 = 200 nm to only 0.54 dB at g 1 = 500 nm. The IL at "OFF" state is < 0.4 dB in the whole range and goes to only 0.12 dB at g 1 = 500 nm, benefiting from the low-loss of the a-GSST-PCM. However, the absorption loss of the a-GSST [k = (1.8 ± 1.2) × 10 −4 ] should be taken into account, which is calculated by using the formula 25 : L abs = 20log 10 [exp(−β im L OFF )], where β im is the imaginary part of the propagation constant of the central WG-G. Variations of the coupling length (left y-axis) and absorption loss of central WG-G with a-GSST (right y-axis) as a function of the gap, g 1 is shown in Fig. 10(b). It can be noted that with the gap, g 1 changing from 200 nm to 500 nm, the coupling length, L OFF is increased from 5.4 μm to 19.0 μm and then the absorption loss is slightly increased from 0.008 dB (2.1% of IL at "OFF" state) to 0.028 dB (23% of IL at "OFF" state), which is relatively and reasonably small compared to the total IL. Hence, the IL is decreased with the increase of the gap, g 1 , as stated in Fig. 10(a). In this case, an appropriate gap, g 1 = 500 nm is chosen to balance the ER/IL and L OFF . The coupling length of the triple-WG DC is calculated to be L OFF = 19.0 μm. The E y fields along the propagation direction at "OFF" and "ON" states are shown in Fig. 10(c,d), respectively. It can be observed that at "OFF" state, the input optical power of the quasi-TM 0 mode is transferred to that of the quasi-TM 1 mode of the bus WG-2, while the input power propagates along the input SM-WG-1 without mode coupling at "ON" state.
As shown in Fig. 10(c), there is a week coupling at "OFF" state between the SM-WG-1 and WG-2-O 2 even if the central WG-G with GSST is cut off. An s-shape bend waveguide could be added to gradually separate the SM-WG-1 and WG-2-O 2 to reduce unnecessary IL, which is shown in Fig. 11 as an inset. The length, L s and offset of the s-shape bend waveguide are chosen to be 15.0 μm and 1.0 μm, which can provide an enough separation between the SM-WG-1 and WG-2-O 2 . We calculated the normalised residual power in SM-WG-1 by varying the length of the straight section of SM-WG-1, L. The normalised residual power in SM-WG-1 without an s-shape bend waveguide is calculated to be 1.2%, as denoted by a horizontally dashed-black line in Fig. 11. It can be noted that by implementing an s-shape bend waveguide, the normalised residual power in SM-WG-1 is reduced from 1.2% to only 0.62% with an optimal L = 8.0 μm. The IL is slightly decreased from 0.1 dB to 0.078 dB and the mode CT is subsequently decreased from −19.08 dB to −21.9 dB. Next, the propagation characteristics of the two-WG DC section are studied by using the 3D-FV-FDTD. Variations of the mode CT (left y-axis) and L ON (right y-axis) with the gap, g 2 are shown in Fig. 12(a). It can be noted that the mode CT is decreased with the increase of the gap, g 2 , while the coupling length is increased. In this case, the gap, g 2 = 300 nm is selected with an acceptable coupling length of L ON = 10.3 μm and a low mode CT of −19.7 dB. The E y field along the propagation direction is shown in Fig. 12(b), which shows that the input quasi-TM 0 mode is totally converted to the quasi-TM 1 mode of the bus WG-3.
Next, the performance of the optimised RMDS is investigated by utilizing the 3D-FV-FDTD. The optical fields along the propagation direction at "OFF" and "ON" states are shown in Fig. 13(a,b), respectively. At "OFF" state with a-GSST-PCM, the input quasi-TM 0 mode is switched to the first-order TM-mode of the bus WG-2 and outputs at port O 2 . While at "ON" state with c-GSST-PCM, the input quasi-TM 0 mode is switched to the quasi-TM 1 mode of the bus WG-3 and outputs at port O 1 . The ERs (ILs) are calculated to be 18.98 dB (0.10 dB) and 22.28 dB (0.58 dB) for the "OFF" and "ON" states, respectively. The CTs at "OFF" and "ON" states are −19.08 dB and −22.86 dB, respectively. The total coupling length of the optimised RMDS is L OFF + L ON = 29.3 μm, which is an ultra-compact size compared with that of the MZIs based ones (~several hundreds of micrometers) 9 . We need to pay attention to Fig. 13(b) that the output pattern at "ON" state has a certain oscillation, which indicates that the out-coupled field is actually composed of mixed modes. As the bus WG-3 with the optimised width of 1.1 μm, two TM modes (TM 0 and TM 1 ) and three TE modes (TE 0 , TE 1 , and TE 2 ) can be supported in this waveguide. The normalised power of the mixed modes in the bus WG-3 is calculated by using the mode expansion method, which shows that the out-coupled field of the bus WG-3 consists of both the TM 0 and TM 1 modes with 2.155% and 97.845% of total output power, respectively. The IL and mode ER at "ON" state would be slightly deteriorated to be 0.68 dB and 22.18 dB, respectively.  The bandwidth of the RMDS is key to the on-chip MDM systems for constructing hybrid MDM-WDM networks. Variations of the power transmission with the operating wavelength of the optimised RMDS is shown in Fig. 14. It can be noted that the high ER of >17.3 dB can be achieved for both the states over 100 nm spectral bandwidth. The IL ON is lower than 1.0 dB over 80 nm from 1520 nm to 1600 nm and the 1dB-IL OFF bandwidth is over 60 nm from 1520 nm to 1580 nm. Compared to the MZIs based RMDS, the IL is significantly decreased from several dB to less than 1.0 dB 9 . Compared to the micro-rings based RMDS, the bandwidth is dramatically increased from a narrow bandwidth of only < 13 GHz to 100 nm 10 . The proposed RMDS shows a high performance for providing an efficient approach to construct a hybrid MDM-WDM system.
As shown in Fig. 14, the ER at "OFF" state as a function of wavelength has a large fluctuation. The reason for this fluctuation at "OFF" state can be explained as two aspects: variations of the coupling length and the normalised power of mixed modes in bus WG-3 with the operating wavelength. In order to analyze the change of the coupling length with the operating wavelength, we define the ratio of the coupling length as: L c ratio = L c / (L c at 1550 nm). With the optimal parameters of W 1 = 400 nm, W 2 = 1.075 μm, W 3 = 1.1 μm, W p = 134.7 nm, g 1 = 500 nm, and g 2 = 300 nm, variations of the ratio of the coupling length for both L OFF and L ON with the wavelength are shown in Fig. 15. It can be noted that both the L OFF ratio and L ON ratio are monotonously decreased with the increase of the operating wavelength in between 1500 nm and 1600 nm.
For the triple-WG DC over the bandwidth of 1500 nm ~ 1550 nm, the calculated coupling length would be larger than the physical length, L OFF at 1550 nm of the central WG-G, which would induce a residual power remaining in the input SM-WG-1. In addition, a longer coupling length will be achieved for a smaller wavelength. Hence, for the bandwidth in between 1500 nm and 1550 nm, a larger deterioration of the mode ER was induced with a smaller wavelength, as shown in Fig. 14. For the triple-WG DC over the bandwidth of 1550 nm ~ 1600 nm, the calculated coupling length will be shorter than the physical length of the central WG-G. The input  quasi-TM 0 mode will be firstly coupled to the quasi-TM 1 mode in the bus WG-2 and then be coupled back to the central WG-G, and then may be coupled back to the input SM-WG-1 depending on the calculated coupling length, which may also induce a residual power remaining in the input SM-WG-1. However, the mode ER at 1580 nm is larger than that at 1550 nm, as shown in Fig. 14. We checked the propagation field of the triple-WG DC at 1580 nm and found that the back-coupled field in the central WG-G radiates at the end of this waveguide and with only a tiny power (0.62%) coupled back to the input SM-WG-1. But, 1.7% and 0.884% of the total power is coupled back to the input SM-WG-1 for 1560 nm and 1600 nm, respectively. Therefore, a fluctuation is induced in between 1550 nm and 1600 nm, as shown in Fig. 14.
Next, variations of the normalised power of mixed modes in the bus WG-3 with the operating wavelength are calculated and shown in Fig. 16. It can be noted that for the two-WG DC, the power out-coupled to the quasi-TM 0 mode of the bus WG-3 is increased with the increase of the operating wavelength, whereas that out-coupled to the quasi-TM 1 mode is decreased. We should pay attention to the wavelength in between 1580 nm and 1600 nm, where a fluctuation of the normalised power of mixed modes is existed, subsequently further enlarges the variation of the mode ER at "OFF" state shown in Fig. 14.

Conclusion
In conclusion, we have proposed and optimised a nonvolatile and ultra-low-loss RMDS, consisting of a triple-WG DC with the GSST-PCM and a two-WG DC. Due to the self-holding phase-transition between a-and c-states of the GSST-PCM, the nonvolatile capability to sustain the switches' states can be achieved to reduce the power-consumption. The proposed RMDS has been optimally designed by using the FV-FEM and 3D-FV-FDTD. The ultra-low insertion-losses of 0.10 dB and 0.68 dB of the optimised RMDS have been achieved for the "OFF" and "ON" states, respectively, benefiting from the low losses of the GSST-PCM at both a-and c-states. The mode CTs at "OFF" and "ON" states were −19.08 dB and −22.86 dB, respectively. The proposed RMDS was with a compact coupling length of 29.3 μm and high ERs of 18.98 dB and 22.18 dB for the "OFF" and "ON" states,  respectively. A reasonable high ER of >17.3 dB has been achieved for both the states over 100 nm bandwidth, which offers the potential application in the S + C + L band MDM-WDM networks. The proposed RMDS can be applied in the MDM networks to provide a flexible mode routing and switching.