Multimode Waveguides on an SOI Platform for Arbitrary Power Splitting Ratio Couplers

Optical power couplers with arbitrary power splitting ratios are important components for many applications such as Mach-Zehnder interferometer-based structures, filters, switches, dispersion compensations, optical interconnects, and microring resonators. In this chapter, we present a new approach to achieve a very high compact coupler with arbitrary power splitting ratios on silicon on insulator (SOI) waveguides. The proposed device requires only one 4 4 multimode interference (MMI) coupler. We use a passive wide SOI waveguide to achieve the phase shifter. The footprint of the whole device is only about 6 150 μm. A large fabrication tolerance of 50 nm can be achieved. The modified effective index method, beam propagation method, finite difference method, and finite difference-time difference method are used to optimally design the whole device.


Introduction
Integrated optical couplers with arbitrary power coupling ratios are important components in optical communication applications. Such couplers can be used in Mach-Zehnder interferometer (MZI) structures, power taps, and microring resonators, and so on. In principle, any power coupling ratio can be achieved for standard 2Â2 directional couplers [1]. The power coupling ratios can be controlled by adjusting the coupling length and/or the gap between the two waveguides of the directional coupler [2]. In practice, accurate fabrication of the gap requires very tight control of the fabrication process. Moreover, additional loss due to mode conversion loss has been found to be a problem [3].
A multimode interference (MMI) power splitter plays an important role in the development of integrated photonics due to large fabrication tolerance, wide operation bandwidth, and compact size. Therefore, MMI couplers with unequal power splitting ratios have been found in many applications such as optical monitoring, optical microring resonators [4,5], and optical reflectors [6]. In order to achieve a coupler with a variable power splitting ratios, a Mach-Zehnder interferometer (MZI) with a phase shifter at the arm can be used [7][8][9][10][11]. With a conventional MMI coupler with a rectangular shape, it has been shown that such device can only provide limited splitting ratios and therefore it has limited applications in all-optical signal processing and optical networks.
In the literature, there are some methods for obtaining variable power optical couplers based on MMI structures such as butterfly-like MMIs [12], exponential MMIs [13,14], and angled MMIs [15,16]. However, these methods require a very complicated fabrication processes, and they have a low fabrication tolerance. In addition, these methods are not suitable for silicon photonics because of high losses.
In recent years, we have presented some methods to overcome the limitations of the previous approaches such as by cascading three or four MMI couplers [17], using an MZI structure with a multimode waveguide [18], using slot waveguide [19]. Other approaches are to use MMI structure with QR code-like nanostructure [20,21] or double MMI coupler [22]. However, such approaches still require a quite complex fabrication process and they have a large footprint.
In this chapter, we present a new approach to achieve a 2Â2 coupler based on only one 4Â4 MMI coupler with wide waveguide on silicon on insulator waveguides. Our new method has advantages of compact size, ease of fabrication with the current CMOS circuitry.

Theory of MMI matrix
Multimode interference coupler includes three regions: single mode input, output waveguides, and multimode waveguide connecting with two input and output regions. The multimode region has a large width to support some modes. The operation of the MMI coupler is based on Talbot effect [4,23]. Based on locations of input field excitations, there are three main interference mechanisms: (1) general interference (GI) mechanism, which is independent of the modal excitation; (2) restricted interference (RI) mechanism, in which excitation inputs are placed at some special positions so that certain modes are not excited; (3) symmetric interference (SI) mechanism, in which the excitation input is located at the center of the multimode section.
An NxN GI-MMI coupler has length L ¼ L MMI ¼ p N 3L π , where p is a positive integer such that p and N are without a common divisor. In practical designs, the shortest devices are obtained for p = 1. The resulting amplitudes from image input i (i = 1,…,N) to output j (j = 1,…,N) can be given in a compact form [23].
where A ij 2 is the normalized power of the output images. The phases ϕ ij of the equal output signals at the output waveguides can be calculated by [23]: for i + j: even, ϕ ij ¼ where the input ports i (i = 1, 2,…,N) are numbered from bottom to top and the output ports j (j = 1, 2,…, N) are numbered from top to bottom in the MMI coupler. ϕ 0 ¼Àβ 0 L MMI À π 2 is a constant phase that depends upon the MMI geometry and therefore can be implied in the following calculations.
In this chapter, the access waveguides are identical single mode waveguides with width W a . The input and output waveguides are located at [24].
The electrical field inside the MMI coupler can be expressed by [25].
We showed that the characteristics of an MMI device could be described by a transfer matrix [2]. This transfer matrix is a very useful tool for analyzing cascaded MMI structures. The phase ϕ ij associated with imaging an input i to an output j in an MMI coupler. These phases ϕ ij form a matrix Φ, with i representing the row number, and j representing the column number. Then the transfer matrix of the MMI coupler Μ is directly related to Φ, and the output field distribution emerging from the MMI coupler can be written as The superscript T indicates the transpose of a matrix. a i (i = 1,…,N) is the complex field amplitude at input waveguide i and b j (j = 1,…,N) is the complex field amplitude at output waveguide j. Elements of the transfer matrix M are m ij ¼ m ji ¼ A ij e jϕ ij , where A ij is the field amplitude transfer coefficient and ϕ ij is the phase shift when imaging from input i to output j.
Here, we propose a 4Â4 MMI coupler with a width of W MMI , length of L MMI . A single 4Â4 MMI coupler at a length of L 1 ¼ 3L π 4 is described by the following transfer matrix [26][27][28]. If the length is doubled to L MMI ¼ 2L 1 ¼ 3L π =2, a new 4Â4 MMI coupler is formed and its transfer matrix is

Modified effective index and numerical methods
In this chapter, we use our proposed modified effective index method (MEIM) for designing MMI structures based on silicon waveguides as shown in [29]. The principle of this MEIM is to use the beat length L π of the MMI coupler as the invariant. Here, the beat length of an MMI structure can be defined as L π ¼ π= β 0 À β 1 ÀÁ , where β 0 and β 1 are the propagation constants of the fundamental and first order modes, respectively. We shall find a matching value of the cladding index for the effective index method that forces the beat length L π in the equivalent 2D model to be equal to the beat length in an accurate 3D model. By this way, we find an optimal effective cladding index. For our silicon waveguide structure, in this chapter, we found that equivalent effective indices of the core waveguide and the cladding waveguide are to 2.82 and 2.19, respectively.
In order to optimally design the MMI coupler, we use the beam propagation method (BPM). We showed that the width of the MMI is optimized to be W MMI =6μm for compact and high performance device. Figure 1 shows numerical simulations at optimal MMI length of L MMI ¼ 141:7 μm for signal at input port 1 and at input port 2. The simulations show that a very low insertion loss of 0.7 dB for both cases [2]. Emerging Waveguide Technology 232 4. Arbitrary splitting ratio coupler based on a 4Â4 MMI coupler Figure 2 shows the new scheme for achieving arbitrary coupling ratios in only one 4Â4 MMI structure. We use two waveguides with different widths W co1 ¼ 300nm and W co2 ¼ 500nm at the two arms of the structure. The cross-sectional view of the SOI waveguide is shown in Figure 3. Here, the height of the SOI waveguide h co ¼ 220nm, the width of the SOI waveguide varies from 300 nm to 500 nm for single mode operation at wavelength of 1550 nm.
By using the FDM method, the effective refractive index of the SOI waveguide at different waveguide width of 300 nm to 500 nm can be calculated as shown in Figure 4. As an example, Figure 5 shows the mode profile for the waveguide at the width of 300 nm and 500 nm, respectively.
A change in the effective index will induce the change in the phase shift. The phase difference between two waveguide then can be expressed by [30].   where Δn eff is the difference in the effective index, L arm is the length of the waveguide with the width of W co2 . As a result, the length of the waveguide W co2 to achieve the phase shift from 0 to π is presented in Figure 6. We see that the short length of 3.5035 μm is required to achieve a phase shift of π.
Due to the phase shift Δφ, the complex amplitudes at the input ports and output ports of the coupler of Figure 2 can be described in terms of cascaded transfer matrices as  , and κ ¼ cos Δφ 2 . Therefore, if an input signal having power presented at input port 1 of the coupler in Figure 2, the normalized powers at output port 1 and output port 2 can be expressed by: From Eqs. (7) and (9), the normalized output powers can be calculated and plotted in Figures 7  and 8. It is showed that any power splitting ratio can be achieved by changing the phase shift from 0 to π or changing the length of the Wco2 waveguide from 0 to 3:51μm.
Consider the length of the Wco2 waveguide variation, the normalized output powers are shown in Figure 9. The simulation shows that the changes in normalized output powers are very small (nearly 0% in the range of À50 nm to +50 nm). It is feasible for the current CMOS circuitry [31].
Finally, we use FDTD method to simulate our proposed structure and then make a comparison with the analytical theory. In our FDTD simulations, we take into account the wavelength dispersion of the silicon waveguide. We employ the design of the MMI coupler presented in the previous section. A continuous light pulse of 15 fs pulse width is launched from the input to investigate the transmission characteristics of the device. The grid size Δx ¼ Δy ¼ 20nm and Δz ¼ 20nm are chosen in our simulations. The FDTD simulations for the whole device are shown in Figure 10.     Figure 11 shows the FDTD simulations compared with theoretical analysis. The simulations show that the device operation has a good agreement with our prediction by analytical theory.

Conclusions
We presented a compact structure with a footprint of 6Â150 μm 2 based on silicon on insulator waveguides for 2Â2 couplers with free of choice power splitting ratios. The new structure requires only one 4Â4 multimode interference coupler. The wide SOI waveguide is used to achieve the phase shift. By changing the length of the wide waveguide from 0 to 3.51 μm, any power splitting ratios can be achieved. The device operation has been verified by using the FDTD. This coupler can be useful for optical interconnects, microring resonator applications.  Emerging Waveguide Technology 238