# Compact Photonic-Crystals Based Isolator Using Ni-Zn Gyromagnetic Ferrite Posts
## 3 Photonic-Crystals Isolator  
### Operating Principle of Faraday Rotation Isolator  
In the K waveband, if a ferrite post is magnetized in \(z\)-direction, its magneto-optical effect is usually described in terms of the tensor permeability [25]:  
\[\left[\mu_{r}\right]=\mu_{0}\left[ \begin{array}{ccc}\mu&j\kappa&0\\ -j\kappa&\mu&0\\ 0&0&1\end{array} \right] \tag{1}\]  
where \(\mu=1+\omega_{m}(\omega_{0}+i\alpha\omega)/\left[\omega_{0}+i\alpha\omega)^{2 }-\omega^{2}\right]\) and \(\kappa=\omega_{m}\omega/\left[\omega_{0}+i\alpha\omega)^{2}-\omega^{2}\right]\) with \(\omega_{0}=\mu_{0}\gamma H_{0}\), \(\omega_{m}=\mu_{0}\gamma M_{s}\) and \(\gamma=1.759\times 10^{11}\) C /kg. Here, \(\alpha=3\times 10^{-5}\) and \(M_{s}=2.39\times 10^{5}\) A/ m depend on material parameters. The external DC field is \(H_{0}=3.15\times 10^{5}\) A/ m, which is reduced twofold compared with that in [23]. The symbol of the off-diagonal element \(\kappa\) is decided by the direction of external DC magnetic field and the strength of Faraday rotation effect is determined by the quality factor \(Q=\kappa/\mu\). Isolators based on the Faraday rotation effect have been discussed in infrared, terahertz and optical waveband [26; 27]. The principle of our designed isolator is similar to a T-shaped Faraday rotation circulator with one port shorted. Based on the Faraday rotation effect, the electromagnetic wave splits a pair of rotary degenerate modes, which it transmits into the magnetized ferrite post. The speeds of the two modes are different from each other. Thus, the composite wave of the two modes takes place a Faraday rotation angle when these deviate from each ferrite area. In this simulation, there is a pair of degenerate modes in the ferrite post such as Figure 3 in [28] with \(H_{0}=0\). When the ferrite posts are fully magnetized, the two modes are coupled, which leads to a Faraday rotation of the wave front in the area of ferrite posts. Here, a 45\({}^{\circ}\) rotation of electromagnetic wave can be realized with appropriate operating conditions through each ferrite post. The propagation direction of the electromagnetic waves is deflected in the ferrite area, which mainly attributed to the Faraday rotation effect. In Figure 3, the operating principle of the designed PC isolator is introduced by describing two transmission paths. The electromagnetic wave incoming from W1 takes place a 90\({}^{\circ}\) rotation by the cooperation of the ferrite posts F1 and F4 and is totally reflected by the cavity. The reflecting cavity consists of the ferrite post F4 and the five Al\({}_{2}\)O\({}_{3}\) rods in the orange dotted box as shown in Figure 1c. Then, the electromagnetic wave rotates with an angle of 90\({}^{\circ}\) again when it transmits through the ferrite posts F4 and F3 sequentially. Finally, the signal exactly travels to W2 as shown in Figure 3a. It can be seen that the electromagnetic wave transmission along a straight line is realized by bending twice of 90\({}^{\circ}\) with the three ferrite posts and the aid of the reflecting cavity. Similarly, Figure 3b illustrates the path of electromagnetic waves incoming from W2. The electromagnetic wave is firstly rotated by an angle of 45\({}^{\circ}\) through the ferrite post F3 and then another angle of 45\({}^{\circ}\) through the ferrite post F2. Then a 90\({}^{\circ}\) bend of electromagnetic wave is obtained by the cooperation of the two ferrite posts. Finally, the electromagnetic  
Figure 3: **(a)** The path of electromagnetic wave incoming from W1; **(b)** the path of electromagnetic wave incoming from W2. wave travels into the fan-shaped coupler shown in Figure 1d and is almost all confined in it, thus the W1 is isolating. Here, the built-in fan-shaped coupler works similar to a matched load, whose PBG is different from that of the SLPCs, so the input electromagnetic wave can diffuse in the coupler but is confined in the third arm of the T-shaped PCW. In microwave circuits, the modified port is usually deemed to have'short-circuited'.