# High Reliability Evaluation and Lifetime Prediction of 50 GHz Athermal AWG Module
## 2 The Principle of Temperature Independence
### Principle of Temperature Compensation  
This study focused on the linear dispersion of AWG to compensate for this temperature dependence. Figure 2 shows the input slab waveguide of AWG. The symbol S corresponds to the position when \(\mathcal{O}\) = 0 and wavelength \(\lambda_{0}\) in Equation (1), and is expressed as follows under these conditions. \[\mathrm{x}\ n_{s}\frac{D}{L_{f}}+n_{C}\Delta\mathrm{L}=\mathrm{m}\lambda \tag{5}\]  
\[\frac{d\mathrm{x}}{d\lambda}\left(n_{s}\frac{D}{L_{f}}\right)+\frac{dn_{s}}{ d\lambda}\left(\mathrm{x}\frac{D}{L_{f}}\right)+\frac{dn_{c}}{d\lambda} \Delta\mathrm{L}=\mathrm{m} \tag{6}\]  
\[\lambda_{0}=\frac{n_{c}\,\Delta\mathrm{L}}{m} \tag{7}\]  
Figure 2: Input slab waveguide of AWG. If the focus point with diffraction angle \(\partial\)e is defined as E and the distance between S and E as x, then the wavelengths \(\lambda\) and x are expressed as:  
\[\frac{d\mathsf{x}}{d\lambda}=\frac{L_{f}\,\Delta\mathsf{L}}{n_{s}\,D\lambda_{0} }\,\,n_{g} \tag{8}\]  
where \(L_{f}\) is the focal length of the slab waveguide; \(D\) is the pitch of adjacent arrayed waveguide; \(n_{s}\) is the effective refractive index of the slab waveguide; and \(n_{g}\) is the group refractive index of the arrayed waveguide. Equation (8) shows the linear dispersion of the AWG, showing that when the input waveguide is at a distance dx from S, it can input light with a different wavelength from \(\lambda_{0}\) by \(d\lambda\). The input waveguide moves a distance corresponding to the wavelength change \(d\lambda\) to compensate for the temperature change and the temperature dependence of the center wavelength. To implement this principle, this study applied a new concept that separates one of the slab waveguides. Figure 3 shows the structure of this new athermal AWG. The AWG's circuit cuts one slab waveguide and separates it into a large part and a small part. These two parts are attached to the temperature compensation board (Invar), which acts as a compensation plate, and the temperature is compensated by the thermal expansion coefficient of SUS 304 bolts. These bolts are connected to both ends of the temperature compensation board. Figure 4 shows the temperature compensation mechanism. In conventional AWGs, the focus point shifts according to temperature change, and the center wavelength of AWG varies with temperature. However, in an athermal AWG, the focus point also shifts according to temperature change, but the center wavelength remains constant because the output waveguide moves to the shifted focus point by the thermal contraction and expansion of SUS 304 bolts. This temperature compensation mechanism also applies when light is input from the output  
Figure 4: Mechanism of athermal AWG. Figure 3: Structure of athermal AWG. port.thermal AWG uses a technology to control the focus in slab waveguides, so it is not temperature sensitive and has an insignificant effect on optical properties. This implements an athermal AWG with many channels. The compensation is also very stable because the thermal expansion coefficient of the SUS304 bolts is constant and very stable. To accurately compensate for the temperature dependence of the athermal AWG's center wavelength, the distance of the temperature compensation board, determined as follows, should be adjusted by referring to the parameters in Table 1.  
The change in position compensation dx can be expressed as a function of temperature change.