# Silicon Waveguide Optical Isolator with Directly Bonded Magneto-Optical Garnet
## 3 Operation Bandwidth  
Although the MZI isolator was designed at a specific center wavelength, phase differences of MZI had a wavelength dependence. Typically, refractive indices of materials and MO effect decreased as wavelength became longer. The phase differences \(\theta_{\text{NPS}}\) and \(\theta_{\text{RPS}}\) decreased as well. Then, an MZI had a resonant spectrum with periods of in- and out-of-phase, the so-called free-spectral range (FSR). When a FSR was wider, the MZI isolator provided a higher isolation for the wider wavelength range. The integer \(m\) in Equation (2) determined the resonant period, and was designed by \(L_{\text{RPS}}\). Figure 3 shows the relation between \(m\) and 20 dB isolation bandwidth. Smaller \(m\) and shorter \(L_{\text{RPS}}\) achieved a wider bandwidth, while it needed a precise fabrication of waveguide dimensions. Figure 4 shows measured transmission spectra of an MZI isolator fabricated with the length \(L_{\text{RPS}}\) of 3.7 \(\upmu\)m, which corresponds to \(m=5\). An optical isolation over 20 dB was obtained at 8-nm wavelength range. The transmittance included \(\sim\)20 dB coupling loss between the fiber and waveguide for two facets by edge coupling. The insertion loss of 13 dB was mainly due to the scattering and reflection at Ce:YIG and air cladding regions' boundaries. Further wideband operation was possible by focusing on the backward operation [20]. In this scheme, \(\theta_{\text{NPS}}\) and \(\theta_{\text{RPS}}\) were designed \(\pi/2\) and \(3\pi/2\), respectively, so that they were subtracted for the backward operation. When the wavelength dependences of \(\theta_{\text{NPS}}(\lambda)\) and \(\theta_{\text{RPS}}(\lambda)\) had similar slopes, the deviations from the designed quantities cancelled each other. Consequently, the wavelength dependence is flat as shown in Figure 5. The slope of phase difference was adjusted by changing the  
Figure 1: Schematic of Mach–Zehnder interferometer (MZI)-based optical isolator with magneto-optical (MO) upper cladding layer. Si waveguides are 450-nm wide and 220-nm high. 3 \(\times\) 2 couplers function as 3-dB power splitter and combiner with directional coupling among three adjacent waveguides. Figure 2: Microscopic image of fabricated MZI isolator on a silicon-on-insulator (SOI) platform. A 1.5-mm square die of cerium-substituted-YIG (Ce:YIG)/substituted-GGG is directly bonded on the Si waveguide. Two MZI arms are separated 500-\(\upmu\)m away to apply external magnetic field in anti-parallel directions with a permanent magnet. temperature dependence of the refractive indices (d\(n\)/d\(T\)) and MO effect (d\(\theta_{F}\)/d\(T\)) of the composed materials. By analyzing the measured temperature dependence of the MZI isolator, the unknown parameters of Ce:YIG film were determined as d\(n\)/d\(T\) = 9.1 \(\times\) 10\({}^{-5}\)/K and d\(\theta_{F}\)/d\(T\) = 44\({}^{\circ}\)/cm/K. These indicated that the refractive index increased and the MO effect became weak at a higher temperature. Temperature-insensitive operation of MZI isolator was proposed by cancelling the temperature dependence of phase differences \(\theta_{\text{NPS}}(T)\) and \(\theta_{\text{RPS}}(T)\). At a higher temperature, \(\theta_{\text{NPS}}(T)\) decreased due to the decrease in the MO effect, whereas \(\theta_{\text{RPS}}(T)\) increased due to the increase in the refractive indices, as described by the following equations. \[\frac{\text{d}\theta_{\text{NPS}}}{\text{d}T}=\frac{\text{d}\Delta\beta}{ \text{d}T}L_{\text{NPS}}, \tag{4}\]  
\[\frac{\text{d}\theta_{\text{RPS}}}{\text{d}T}=\frac{\text{d}n_{\text{eff}}}{ \text{d}T}kL_{\text{RPS}}, \tag{5}\]  
where \(k\) is the wave number. An MZI was designed so that these temperature dependencies cancelled each other in the backward operation. Figure 6 shows the calculated temperature dependence of the total phase difference in an MZI as a function of \(L_{\text{RPS}}\). They were plotted in dots because \(L_{\text{RPS}}\) had to satisfy Equation (2) for integer \(m\).