#### 2.1. Multimode Interferometers

The fundamental switching element in this work is MMI based phase shifters and thermo-optic based heaters. The design of the 2 × 4 optical switch, considered in this paper is based on single MMI couplers/splitters and their cascaded configuration, the operating principle of which is based on self-imaging theory and total internal reflection [

6,

8,

9,

10,

11,

12,

13,

14,

15,

16,

17]. The device consisted of two identical MMIs: MMI

${}_{1}$ and MMI

${}_{2}$, each with 140

$\mathsf{\mu}$m × 6

$\mathsf{\mu}$m dimension, connected via two smaller phase shifters MMI

${}_{3}$ and MMI

${}_{4}$, each with 16

${\mathsf{\mu}\mathrm{m}}^{2}$, and a metal heating element incorporated in them. The MMI

${}_{1}$, in this case, is designed to act as a power decoupler whereas MMI

${}_{2}$ is designed as a power coupler.

Figure 1 presents an illustrative configuration of the device that contains a 3-dB input MMI decoupler/splitter, two phase shifters, one at each arm of the MMI, and a 3-dB output MMI coupler. The MMIs are designed in such a way that when the input signal is fed through a

${}_{i1}$, the output power is equally split between a

${}_{o1}$ and a

${}_{o4}$. Likewise, when the signal is fed through the input port a

${}_{i2}$, the output power is equally split between a

${}_{o2}$ and a

${}_{o3}$. That means each MMIs can act as two independently operating 3-dB splitters/couplers.

Based on self-imaging properties, a prototype MMI 3-dB power splitter/combiner is first designed. According to [

6], the beat length,

${L}_{\pi}$ between two lowest order modes of an MMI is given by:

where,

${\beta}_{0}$ and

${\beta}_{1}$ denote the propagation constants of the fundamental and first mode, respectively.

As explained by Ulrich [

17] and Soldano [

13], self-imaging is achieved when the input field is reproduced as single or multiple images along the dimension of the MMI slab at a periodic interval. Since the light beam is confined in an MMI slab because of obeying total multiple reflections, the beam undergoes multiple periodically repeating self-interference patterns. For instance, self-generation of two interference patterns with equal output powers can be achieved using the relationship as [

6]:

where

M and

N are any positive integers without a common divisor,

N is the number of self-replicating interferences, and

M defines device length with various

N [

16]. The relationship in (

2) suggests that a compact device is obtained for

M = 1.

For the MMI with the effective width, W

${}_{e}$ and effective refractive index, n

${}_{eff}$, and when excited from either input or output ports, the L

${}_{\pi}$ can also be given by:

That means, ${L}_{\pi}$ is related to the W${}_{e}$, $\lambda $, and ${n}_{eff}$.

For a lateral wave-guided structure, the

${W}_{e}$ is related to the geometric width of the device as [

13]:

where

${n}_{core}$ and

${n}_{cld}$ are the refractive indices of core and cladding layers, respectively. At one beat length and a half, the MMI produces a pair of identical images, making the waveguide a 3-dB coupler.

As shown in

Figure 1, the device is designed to act as two independent optical switches with tapered input/output ports (the tapered section is shown in the inset at the middle top for clarity and the cross-sectional view of the waveguide is shown at the top right). Air, with a refractive index of 1.0 is considered as the upper cladding whereas SiO

${}_{2}$, with a refractive index of 1.45 as the lower cladding.

The individual MMI device parameters were estimated using MATLAB mode solver [

18] and FDTD simulation tool with TE polarized (s-polarized) light at

$\lambda $ = 1550 nm and these are given in

Table 1.

#### 2.3. Phase Shifters

In many circumstances, the phase shifting is an essential characteristic of the optical signal as it traverses through waveguides, including switches and filters. Not all kind of phase shifting is, however, useful for switching application. In most cases, the desired phase shift requires special device design consideration. In the present work, two 1 × 1 MMI devices (MMI

${}_{3}$ and MMI

${}_{4}$) are considered for phase shifting purpose. Symmetric interference theory was used to calculate the L

${}_{MMI}$ to give a single image at the output. In the simulation, the length of the MMIs were varied from 0 to 10

$\mathsf{\mu}$m to obtain corresponding MMI width for single mode operation. The width was varied from 1 to 2

$\mathsf{\mu}$m for a 8

$\mathsf{\mu}$m long MMI to avoid cross-talk between the adjacent waveguides. The chosen waveguide can support up to a maximum of 3 optical modes. These parameters were obtained from the FDTD simulation as well as MATLAB Mode solver [

18]. The grid-size used during FDTD simulation was 5 nm. In analogy to reference [

7], two mesh override regions were used. The calculated parameters of the phase shifter are given in

Table 1.