Analytical solutions for stress compensated athermal arrayed waveguide gratings

Analytical solutions are very useful to understand influence factors in the central wavelength temperature sensitivity of AWGs (arrayed waveguide gratings) and convenient to be used in preliminary design. In this paper, the elastic multilayer theory and stress concentration effect are combined to estimate thermal stresses in arrayed waveguides. Effective indices and their temperature coefficients are estimated by use of effective index method. The effects of material properties on temperature sensitivity in AWGs are completely studied. It is theoretically demonstrated that athermal AWGs can be obtained by choosing proper material with negative TEC (thermal expansion coefficient) for the substrate. Subsequently, the theory is extended to study the modified AWGs with a stress plate attached on the bottom, top, or both sides of arrayed waveguides. It is shown that the temperature sensitivity can be controlled effectively by stress plates with different TECs. After attaching a plastic plate on the bottom, the temperature sensitivity of central wavelength can be reduced to -0.003 pm/K for TE mode and 3 pm/K for TM mode, respectively, whereas the thickness is 0.83mm.