First we give the transfer matrix expressions of the optical switching unit and waveguide crossing. An optical switching unit has two input ports, *In*_{1} and *In*_{2}, and two output ports; *Out*_{1} and *Out*_{2}. When the optical switching unit is in the “cross” state, two incident lights are guided from its input ports *In*_{1} and *In*_{2} to its output ports *Out*_{2} and *Out*_{1}. When the optical switching unit is in the “bar” state, two incident lights are guided from its input ports *In*_{1} and *In*_{2} to its output ports *Out*_{1} and *Out*_{2}.

The M-Z optical switching unit can be divided into three parts: the front 2×2 multimode interference (MMI) coupler, the two arms and the back 2×2 MMI coupler. The transfer matrix of the front and back 2×2 MMI coupler can be expressed as

$${T}_{\text{coupler}}=\mathrm{(}\begin{array}{cc}{t}_{11}& {t}_{12}\\ {t}_{21}& {t}_{22}\end{array}\mathrm{)},$$(1)

where ${t}_{ik}={a}_{ik}{e}^{j\mathrm{(}{\phi}_{ik}+\delta {\phi}_{ik}\mathrm{)}}$ and $j=\sqrt{-1}$ [31], [32]. *φ*_{ik} is the relative phase shift of the image at output port *k* for input port *I*,

$$\begin{array}{l}{\phi}_{11}={\phi}_{22}=-7\pi /16\\ {\phi}_{12}={\phi}_{21}=\pi /16.\end{array}$$(2)

*δφ*_{ik} is the relative phase deviation of the image at output port *k* for input port *i*. *a*_{ik} is the real field amplitude transfer coefficient from input port *i* to output port *k*. Two arms of the optical switch can be described by the diagonal 2×2 transfer matrix as

$${T}_{\text{ARM}}=\mathrm{(}\begin{array}{cc}{A}_{1}{e}^{j\mathrm{(}{\theta}_{1}+\delta {\theta}_{1}\mathrm{)}}& 0\\ 0& {A}_{2}{e}^{j\mathrm{(}{\theta}_{2}+\delta {\theta}_{2}\mathrm{)}}\end{array}\mathrm{)}.$$(3)

*A*_{1} and *A*_{2} are the loss coefficients of arm 1 and arm 2, respectively, and can be expressed as

$${A}_{i}=\mathrm{exp}\mathrm{(}-\alpha {L}_{i}\mathrm{)},$$(4)

where *α* is the propagation loss of the waveguide. Based on the data from experiment, we assume *α*=2.5 dB/cm.

*L*_{i} is the length of arm *i*, and *θ*_{i} is the phase shift through arm *i* and can be expressed as

$${\theta}_{i}=2\pi {n}_{\text{eff}}^{i}\mathrm{(}\lambda \mathrm{)}{L}_{i}/\lambda ,$$(5)

where *λ* is the wavelength of light in the vacuum and ${n}_{\text{eff}}^{i}$ is the effective refractive index of arm *i*. *δθ*_{i} is the phase shift deviation due to imperfect fabrication. The total transfer matrix of the M-Z optical switching unit is then given by

$${T}_{2\times 2}={T}_{\text{coupler}}^{\text{back}}\cdot {T}_{\text{ARM}}\cdot {T}_{\text{coupler}}^{\text{front}}.$$(6)

The output optical field distributions of the M-Z optical switching unit are expressed by the following matrix equation:

$$\mathrm{(}\begin{array}{c}{E}_{1}^{\text{out}}\\ {E}_{\text{2}}^{\text{out}}\end{array}\mathrm{)}={T}_{2\times 2}\cdot \mathrm{(}\begin{array}{c}{E}_{\text{1}}^{\text{in}}\\ {E}_{\text{2}}^{\text{in}}\end{array}\mathrm{)}.$$(7)

For simplicity, the transfer matrix of an optical switching unit is expressed by

$${T}_{2\times 2}=\mathrm{(}\begin{array}{cc}{T}_{11}& {T}_{12}\\ {T}_{21}& {T}_{22}\end{array}\mathrm{)}.$$(8)

Similarly, the matrix expression of a waveguide crossing (or an ideal optical switching unit in the “cross” state) and waveguide (or an ideal optical switching unit in the “bar” state) can be expressed by

${T}_{\text{crossing}}=\mathrm{(}\begin{array}{cc}0& 1\\ 1& 0\end{array}\mathrm{)}$

$${T}_{\text{bar}}=\mathrm{(}\begin{array}{cc}1& 0\\ 0& 1\end{array}\mathrm{)}$$(9)

As shown in Figure 1B, the architecture of the six-port optical switch can be divided into six columns; each column consists of optical switching units, straight waveguides or waveguide crossings and can be expressed by the transfer matrix *M*_{i} (*i*=1, 2, …, 6). *M*_{i} is a partitioned matrix,

$${M}_{i}=\mathrm{(}\begin{array}{ccccc}{T}^{B/C}& 0& \cdots & 0& 0\\ 0& {T}^{B/C}& 0& \vdots & \vdots \\ \vdots & 0& \ddots & 0& 0\\ 0& \vdots & 0& {T}^{B/C}& 0\\ 0& 0& \cdots & 0& {T}^{B/C}\end{array}\mathrm{)}$$(10)

where

$\{\begin{array}{l}{T}^{B/C}={T}_{\text{cross}},\text{\hspace{0.17em}optical\hspace{0.17em}swiching\hspace{0.17em}unit\hspace{0.17em}is\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}}\u201c\text{cross}\u201d\text{\hspace{0.17em}state}\\ {T}^{B/C}={T}_{\text{bar}},\text{\hspace{0.17em}optical\hspace{0.17em}swiching\hspace{0.17em}unit\hspace{0.17em}is\hspace{0.17em}in\hspace{0.17em}the\hspace{0.17em}}\u201c\text{bar}\u201d\text{\hspace{0.17em}state}\end{array}$

In detail, we show the transfer matrix *M*_{1} as an example:

$${M}_{1}=\mathrm{(}\begin{array}{cccccc}T{\mathrm{(}1\mathrm{)}}_{11}& T{\mathrm{(}1\mathrm{)}}_{12}& 0& 0& 0& 0\\ T{\mathrm{(}1\mathrm{)}}_{21}& T{\mathrm{(}1\mathrm{)}}_{22}& 0& 0& 0& 0\\ 0& 0& T{\mathrm{(}2\mathrm{)}}_{11}& T{\mathrm{(}2\mathrm{)}}_{12}& 0& 0\\ 0& 0& T{\mathrm{(}2\mathrm{)}}_{21}& T{\mathrm{(}2\mathrm{)}}_{22}& 0& 0\\ 0& 0& 0& 0& T{\mathrm{(}3\mathrm{)}}_{11}& T{\mathrm{(}3\mathrm{)}}_{12}\\ 0& 0& 0& 0& T{\mathrm{(}3\mathrm{)}}_{21}& T{\mathrm{(}3\mathrm{)}}_{22}\end{array}\mathrm{)},$$(11)

The input and output optical fields have the following relation:

$${E}^{\text{out}}={M}_{6}\cdot {M}_{5}\cdot {M}_{4}\cdot {M}_{3}\cdot {M}_{2}\cdot {M}_{1}\cdot {E}^{\text{in}}=M\cdot {E}^{\text{in}},$$(12)

where *M* is the transfer matrix of the six-port optical switch.

In the following simulation, the propagation loss of the MMI coupler is 0.25 dB, which is achieved by three-dimensional finite-difference time-domain simulation. The two phase shifters are 200 μm each in length, and the propagation loss of the silicon rib waveguide (structural parameters can be found in Section III) for the two phase shifters is 2.5 dB/cm. The power splitting ratio imbalance of the MMI coupler is caused by fabrication imperfection. In simulation we set the splitting ratio as a random distribution within the range of 49%–51%, which is achieved from the extinction ratio of optical switching unit in experiment.

To study the spectral responses of the six-port optical switch, we focus on its five routing states which cover its 30 optical links (no U turn routing) to constitute an independent analysis scenario. Other reconfigurable routing paths from the same input ports to the same output ports are just the combination of the constituent parts of these specific routing paths. Table shows the states of the 12 optical switching units and the established 30 optical links in the selected five routing states of the six-port optical switch.

Table 1: States of the 12 switching units in selected five routing states of the six-port optical switch which cover its 30 optical links.

In each routing state, the optical signals are guided from six input ports to six output ports. Although the optical signal from a specific input port is aimed to be guided to a specific output port, it is unavoidable that a tiny part of the optical signal is leaked to the other five output ports and becomes the noises to the five output ports. In actual application, the light beams injected from the six input ports are incoherent with each other, so the total noise of one specific optical link is the sum of the noise from the other five input ports individually. In order to maintain consistency with the experiment, the light beams of different input ports are considered as incoherent in simulation. Figure 2 shows the calculated transmission spectra of the optical switch in its five routing states.

Figure 2: Simulated spectral responses in the selected five routing states (OSNR, optical signal-to-noise ratio; RS, routing state).

The propagation loss of the optical signal links fluctuates from 3.4 dB to 5.6 dB, which mainly depends on the number of on-line optical switching units. More on-line optical switching units means larger propagation loss. For each optical link, the signal comes from a specific input port and the noises come from the other five input ports. The five noises have different weights on the optical crosstalk of the optical link depending on the number of on-line optical switching units, power leakage and the length of the routing path. Some periodic intensity fringes can be observed in the transmission spectra of some crosstalk optical links. As the extinction ratio of each optical switching unit is finite, there exists non-ideal light leaking from each switching unit. For a specific optical link, there may exist more than one noise beams due to the leakage. When these beams have comparable power levels, the interferences among them will cause these oscillations. Such a multiple-beam interference characteristic is the inherent property of the optical switch with a reconfigurable non-blocking architecture and may deteriorate the optical crosstalk of the optical switch in specific optical links. Improving the extinction ratio of the optical switching units can degrade its negative effect. The optical switch has optical SNRs of 18.5 dB to 37.2 dB for different optical links in different routing states. For different optical links, the number of on-line optical switching units is different. And the unexpected leakage from the optical switching units are different. These factors cause the differences in optical SNRs.

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