## 3 MO Isolator  
### Theoretical Model of MO Isolator  
The transmission characteristics of the MO isolator were based on a Maxwell equation with permeability tensor [\(\mu_{r}\)] like [21]:  
\[\varepsilon^{-1}.\
abla\times\left([\mu_{r}]^{-1}.\
abla\times\overset{\rightarrow }{E}\right)=\omega^{2}/c^{2}.\overset{\rightarrow}{E}. \tag{1}\]  
In millimeter wave band, the gyromagnetic characteristic of ferrite magnetized by external DC magnetic field \(H_{0}\) can be expressed as tensor permeability [\(\mu_{r}\)], like [21]. The elements of tensor permeability using Hilbert type are expressed as follows:  
\[\mu_{r}=1+\omega_{m}(\omega_{0}+i\alpha\omega)/\left[(\omega_{0}+i\alpha \omega)^{2}-\omega^{2}\right], \tag{2}\]  
\[\kappa=\omega_{m}\omega/\left[(\omega_{0}+i\alpha\omega)^{2}-\omega^{2} \right], \tag{3}\]  
\[\omega_{0}=\mu_{0}\gamma H_{0}, \tag{4}\]  
\[\omega_{m}=\mu_{0}\gamma M_{s}, \tag{5}\]  
where the loss coefficient \(\alpha\) of NiZn Ferrite (\(\varepsilon_{r}=13.5\)) is generally chosen to be 0.05. The gyromagnetic ratio \(\gamma=1.759\times 10^{11}\) C/kg, while saturation magnetization \(M_{s}=2.39\times 10^{5}\) A/m. Our MO isolator adopted a similar design to that of H-plane rectangular waveguide resonance isolator, as shown in Figure 4a. A resonance isolator works on the utilization of the fact that at certain regions \(x=x_{1}\) and \(x=b-x_{1}\) in the waveguide's cross section, and the magnetic vector is circularly polarized with one sense of rotation for wave propagation in the forward direction and an opposite sense of rotation for the reverse propagation. In the E-plane resonance isolator, the gyromagnetic phenomena in ferrite is also understood in terms of the circularly polarized components of the magnetic field [22]. In microwave engineering, the effective permeability of right or left circular polarization waves in the magnetized NiZn ferrite sheets is expressed as follows:  
\[\mu_{\pm}=\mu_{r}\mp\kappa=1+\frac{\omega_{m}}{\omega_{0}+i\alpha\omega\mp \omega}. \tag{6}\]  
When the loss of ferrite is considered in Equations (2) and (3), the effective permeability \(\mu_{\pm}\) of right or left circular polarized magnetic field has an imaginary part that represents the loss in Equation (6). The variations of the real part \(\mu^{\prime}\pm\) and the imaginary part \(\mu^{\prime\prime}\pm\) of the effective permeability with \(\omega\omega_{0}\) are, respectively, shown in Figure 4b,c. The effective permeability of the right or left circular polarization waves passing through the magnetized ferrite is introduced in detail in reference [23]. Figure 4: **(a)** The cross section view of the resonance isolator; **(b)** the variation of the real part of the effective permeability with \(\omega\omega_{0}\); and **(c)** the variation of the imaginary part of the effective permeability with \(\omega\omega_{0}\). The resonance isolator can work at the resonance condition \(\omega=\omega_{0}\) under a biasing field \(H_{0}\). This results in the near-resonance absorption of the backward traveling waves due to a fairly large imaginary part of the effective permeability observed by those waves. The forward waves, observing a quite small imaginary part of the effective permeability, experience a relatively low loss propagation, resulting therefore in the one-way transmission function of the isolator. Some typical configurations of the resonance isolator were introduced in detail in [23]. Similarly, the MO isolator in Figure 1 is designed according to the feature of the H-plane rectangular waveguide resonance isolator. We assume that the field vectors of the TE\({}_{10}\) mode satisfy the condition of \(H_{x}\)/\(H_{z}\) = \(\pm\)j when the ferrite sheets are placed at the same position \(x\) = \(x_{1}\) like [23]. The TE\({}_{10}\) mode launched from Port 2 in Figure 2 is a right circular polarization wave for the direction of \(H_{0}\).