Performance study of silica-on-silicon based multimode interference (MMI) optical coupler

In recent years, the silica-on-silicon based multimode interference (MMI) optical waveguide is an interesting research topic. It is being advanced various researches on the silica based MMI coupler. This paper represents the considerations of the optimal design of the silica-on-silicon based MMI optical coupler for better performance. For that, we have illustrated the simulation results on a particular case of the 4×4 silica-on-silicon based MMI coupler. From the simulation results, it is seen that the performance of the MMI coupler depends on multiple width and length combinations of the MMI waveguide. The results also show that the width of the multimode waveguide could not be too small or too large for optimal performance, and at the widths, 100 μm, 120 μm and 130 μm, the performance could be optimized and be almost 0.62–0.64 in a given length range. Finally, the results have been compared with the optical coupler presently available in the market and show that the silica-on-silicon based MMI coupler is much more efficient in terms of losses and the performance associated with it and the size of the coupler.


Introduction
The ability of optical fiber couplers to transfer light from one fiber to another makes it one of the key components of an optical fiber system. A fiber optic coupler can also be used as a power splitter that allows the light signal to be shared between fibers. Generally, three coupler configurations can be identified such as the bi-directional coupler, star coupler, multimode interference coupler (MMI). Whereas, the bi-directional coupler can only split optical power between two waveguides, and the star coupler and MMI coupler can split the power among more than two waveguides [1]. Now optical couplers are used in photonic integrated circuits [also called planar lightwave circuits (PLCs) or integrated optoelectronic devices] when it is necessary to couple light between different waveguides. These types of couplers have been realized using directional couplers. Unfortunately, these components have been found to be sensitive to the wavelength, fabrication tolerances, and polarization state of light.
[2] However, many of these problems have been solved, leaving PLCs with four major advantages: the enhanced functionality, very low losses, compactness, and potential for the mass production. Generally, PLCs based optical waveguides and devices can be fabricated by using different materials including LiNbO 3 [3], SiO 2 [4, 5], silicon-on-insulator (SOI) [6,7], polymer [8], InP [9,10]. Each material has advantages and disadvantages for a specific required function. Bulk silica (SiO 2 ) and silica-on-silicon (SiO 2 /Si) are by far the most common materials used to manufacture PLCs, due to their refractive index matching with the silica-based optical fiber [11]. Table 1 provides a summary of the material types available for waveguide manufacturing [12]. The basic functions in PLCs are the generation, guiding, splitting, multiplexing, amplification, switching and detection of the light signal. Different types of optical couplers such as 1×4, 1×8, and 1×32 have been made compact on a chip using the Y-branches [13], arrayed waveguide gratings (AWG) [14,15], matrix switches [16], star couplers [17], Mach-Zehnder interferometer [18] or MMI couplers [19][20][21] which are typically based on the PLC technology. The PLC based devices are superior to conventional fused-fiber type splitters in terms of the low cost and mass producibility. PLC type optical couplers have already been put into the practical use in fiber-to-the-home (FTTH) systems that have been considered to be the ultimate solution for future broadband access networks, and their long term reliability for telecommunications has already been established.
Based on the self-imaging principle, MMI couplers offer the advantages of the compact size, low cross-talk, and low power imbalance. Compared to directional couplers and Y splitters, MMI couplers show superiority in scalability, since they do not need to be cascaded in order to achieve large port counts [12]. The multimode interference principle (also known as the self-imaging theory) is developed theoretically based on high-index-contrast optical waveguides using the strong guiding approximation. For that reason, high-index-contrast structures such as InGaAsP/InP and GaAs/AlGaAs are normally used to realize the MMI devices.
But recently, some research works have been done on low-index-contrast optical waveguide MMI devices which are based on silica-on-silicon waveguides, since silica-on-silicon waveguides have several advantages over other optical waveguides such as [29] (1) Silica-on-silicon light wave circuits have a glass composition similar to that of optical fibers.
(2) Silica waveguides can be fabricated using various methods.
(5) Providing perfect field matching to that of optical fibers.
So, silica-on-silicon waveguides are more attractive from both research and commercial perspectives.
Here, we have mentioned current researches that have been studied on silica-on silicon MMI devices in considerable details such as (1) A 3-dB two-mode-interference coupler was proposed based on silica waveguides in 1992 [30].
(2) 1 × 64 MMI splitters were designed and evaluated in 1995 [31]. The loss of the devices was 0.5 dB, and the uniformity was 1.7 dB.
(4) In 2002, genetic algorithms were introduced to do an optimal design of MMI couplers based on weakly guiding waveguides [33].
(5) In 2003, the access waveguide position of weakly guiding MMI couplers was discussed, and the optimal performance devices were obtained [34]. In 2004, a three-dimensional weakly guiding MMI coupler based on ion exchange waveguides was discussed [35].
Hence, it is evident that the MMI couplers based on the silica-on-silicon system is an interesting topic in the integrated optical circuit research. Since the principle of the self-imaging theory in multimode interference established on strong guiding (high-index-contrast waveguides) approximation is not well satisfied in low-index-contrast systems such as silica-on-silicon waveguides, we have studied to realize good performance MMI couplers based on silica-on-silicon structures.

Performance study of Silica-on-Silicon based MMI device
The theory of self-imaging in multimode optical waveguides has been the subject of several studies [36][37][38][39]. The analysis is based on the strong guiding approximation that is satisfied in the high-index-contrast structures. Although silica-on-silicon light wave circuits are low-index-contrast structures, good understanding of the fundamental theoretical analysis is helpful to design silica-on-silicon multimode interference (MMI) devices. The results of the self-imaging theory could be thought as an approximation for the low-index-contrast structures [29]. Figure 1 shows a schematic of an N×N MMI coupler with its related 2D model. As shown in Fig. 1, an MMI coupler can be made of the silica buried waveguide with a doped core layer and undoped cladding layers. The transverse direction of the silica waveguide is designed to be single mode. In the case of 3D waveguides, an equivalent 2D representation can be made by techniques such as the effective index method or the spectral index method [37]. In this section, we have pointed out that the length of the silica-on-silicon based MMI waveguide could be optimized for better performance. As stated in [29], the multimode section can support the maximum M+1 number of modes. For each mode v, the dispersion relation is given by where mmi L is the length of the multimode waveguide and can be estimated as with p = 1 for shortest devices in (3). But L π is related not only to the propagation constant of the two lowest order modes but also to the propagation constant of the fundamental and other higher orders L π is estimated as a series of the minimum values given in terms and max mmi max Now it is noted that there are two important features for MMI based devices such as the insertion loss (LS) and uniformity (UF). With the range of mmi L , the loss and uniformity can be valued to find optimal performance of the MMI device.
The performance of MMI based devices have been studied on these features. As defined in [29], a new parameter to describe the overall performance of the silica based MMI coupler is defined.
This parameter is associated with LS and UF. It is called the performance (PF) that can be expr essed as 10 10 where t = 1, 2,…, N/2 when N is even, and t = 1, 2,…, (N+1)/2 when N is odd.
Since the MMI coupler has multiple input ports, a summation should be done over these ports.
In case of symmetry, summation over all the input ports is not necessary. From (9), it can be seen that the smaller uniformity and loss give the larger performance. The best performance occurs when PF equals 1, and the worst performance occurs when PF is 0. It was reported [29] that the performance of an MMI coupler depended on the width and length of the multimode waveguide section. It is seen that the performance varies within a specified range of the length at a specific width of the waveguide section. Therefore, the width and length of the MMI section can be selected as parameters to design an N×N MMI coupler.

Simulation results and discussions
In this paper, we illustrate the particular case of a 4×4 MMI coupler as an example and analyze the performance for the design of an N×N silica MMI device. In our study, the parameters were selected as the earlier analysis of the non-optimized structure.
That is, the wavelength is 1.55 µm, the refractive indices of the core and cladding are 1.46 and 1.456, respectively, the thickness of the guiding layer and the width of the access waveguide are both 6 µm, and the selected widths of the multimode waveguide are 90 µm, 100 µm, 110 µm, 120 µm, 130 µm and 140 µm. For different widths of the multimode waveguide, mmi L varies within the range found using (7) and (8). Calculations are done for widths of 90 µm, 100 µm, 110 µm, 120 µm, 130 µm, and 140 µm, which corresponded to 10, 11, 12, 13, 14 and 15 modes respectively in the multimode waveguide. On the basis of the above parameters, the simulation results are analyzed.
The PF parameter of a 4×4 device with different lengths and widths of the multimode waveguide is shown in Fig. 2. From the simulation results, it is seen that there are multiple length/width combinations for which there exists similar optimal performance of the coupler. When the width is 120 µm and the length is 15.25 mm, the performance is 0.64 which is similar to the case for which the width is 130 µm and the length is 17.64 mm. This means that optimal performance can be achieved for several choices of the multimode waveguide size. Fig. 2 Performance variation for different widths and lengths for a 4×4 MMI coupler [29].
In case of the width of 110 µm and 140 µm, the performance changes slightly, and the optimal performance can be achieved in the specified range.
Again when the width is 90 µm, the performance of a device drops slightly and then increases slightly in the specified range, and the overall performance is poor. These results show that the width of the multimode waveguide can not be too small or too large for optimal performance.
We have also studied that the PF is associated with the loss and uniformity of a device with the 4×4 MMI coupler, and it is pointed out that the loss and uniformity of a device are different for different lengths and widths as illustrated in Fig. 3. From the simulation results for the widths of 90 μm, 120 μm, 140 μm, it is seen that for the width of 90 µm, when either the port 1 or the port 4 is used as the input, the excess loss and uniformity are around 0.75 dB and 1.5 dB at the length of 8.9 mm, and when either the port 2 or the port 3 is used, the excess loss and uniformity are around 0.49 dB and 1.8 dB at the length of 8.9 mm. Since the uniformity exceeds the loss, the performance can not be improved in this case and is 0.38. For the width of 120 μm, when either the port 1 or the port 4 is used as the input, the excess loss and uniformity are around 0.64 dB and 0.31 dB in a given range at a length of around 15.25 mm, and when either the port 2 or the port 3 is used, the excess loss and uniformity are around 0.66 dB and 0.33 dB in a given range at a length of around 15.25 mm. So the performance is significantly better for this width and is 0.64.
For the width of 140 μm, when either the port 1 or the port 4 is used as the input, the excess loss and uniformity are around 0.57 dB and 0.36 dB in a given range at a length of around 20.35 mm, and when either the port 2 or the port 3 is used, the excess loss and uniformity are around 0.87 dB and 0.98 dB in a given range at a length of around 20.35 mm. Here, the performance is not significantly improved by adjusting the length within the given range and is 0.56. Table 2 provides a summary of the simulation results for the 4×4 MMI coupler.  Form these results, it is seen that the loss can always be improved within the range but the uniformity has a different behavior. Whether uniformity is improved or deteriorated depends on the device width.
Based on the results, it could be concluded that if both the loss and uniformity need to be optimized, not only the length but also the width of the multimode waveguide should be adjusted. Using the results in Fig. 3, an optimal design for both low loss and good uniformity can be found. With the width of the multimode waveguide fixed at 120 µm, the optimal length is found to be approximately 15.25 mm.
The results at the width of 120 μm are compared with the case studied in [31,33] where (3) from the self-imaging theory is used to find the length of the multimode waveguide, mmi L = 14.95 mm, and the access waveguide positions are ±47.30 µm and ±15.80 µm. When the light is input into the port 1 or the port 4, the loss and uniformity are determined to be 1.354 dB and 1.048 dB, respectively at the width of 120 µm, and when light is input into the port 2 or the port 3, the loss and uniformity are 1.037 dB and 1.696 dB, respectively at the width of 120 µm. In this case, the PF of the device is 0.307, where the width and length are not optimized. But using our proposed method, the value of the performance is improved by more than a factor of 2. Now under these considerations, we have compared our proposed coupler with couplers available in the market. A typical standard for the N×N planar coupler/splitter available in the market is presented in Table 3. It is seen that for a typical (4×4) coupler, the loss and uniformity are around 7.0 dB and 0.5 dB, respectively. Tabl e 3 A typical standard for N(1)×N coupler/splitter (source: www.optokon.com) [40]. In this case, the PF is 0.18. So the comparison of our designed coupler with the presently available coupler shows that our designed coupler is much more efficient in terms of the loss and performance associated with it and the size of the coupler.

Conclusions
We have focused on a method that is based on finding the optimal performance by changing the geometrical parameters of the MMI couplers. In this method, the width and length of the multimode waveguide are adjusted to achieve the optimal device performance. It is found that the length of the multimode waveguide could vary in a well-defined range for better performance.
This range is related to the propagation constant spacing of the fundamental and higher modes of the multimode waveguide. Finally, we have proposed a suitable design for a 4×4 silica-on-silicon based MMI coupler. The parameters are that the length and the width are 15.25 mm and 120 µm, the thickness is 6 µm, and the loss and uniformity are around 0.66 dB and 0.33 dB, respectively. The performance is 0.64. While the standard package of the couplers/splitters available in the market is around 40 mm×4 mm× 4 mm, and the loss and uniformity associated with a typical 4(1)×4 coupler/splitter are around 7.0 dB and 0.5 dB, respectively. In this case, the performance is 0.18. So the proposed coupler is much more efficient than the couplers available in the market. But as in our study, the performance of the silica-on-silicon based MMI coupler was analyzed for a single wavelength of 1.55 µm, hence, we need to study its performance for different wavelengths (such as C band) if it is implemented in the optical network like the dense wavelength-division multiplexing (DWDM) system.

Acknowledgement
The author would like to thank Professor Dr. Rezaul Huque Khan and Dr. Sankar Lal Saha for their sincere co-operation.
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