Anisotropy in thermo-optic coefficients of polyimide films formed on Si substrates

The temperature dependence of in-plane and out-of-plane refractive indices (i.e., thermo-optic coefficients $\mathrm{dn/dT)}$ was measured for seven kinds of aromatic polyimide (PI) films formed on silicon substrates. The absolute values of ${\mathrm{dn}}_{\mathrm{TE}}\mathrm{/dT}$ (polarization parallel to the film plane) are significantly larger than ${\mathrm{dn}}_{\mathrm{TM}}\mathrm{/dT}$ (perpendicular to the film plane). The $\mathrm{dn/dT}$ for average refractive indices ${\mathrm{(n}}_{\mathrm{av}})$ are $\text{−94}$ to $\text{−58\hspace{0.17em}}\mathrm{ppm/K},$ independent of film thickness. The anisotropies ${\mathrm{(dn}}_{\mathrm{TE}}{\mathrm{/dT-dn}}_{\mathrm{TM}}\mathrm{/dT)}$ are $\text{−9}$ to $\text{−39\hspace{0.17em}}\mathrm{ppm/K}.$ Although the values of $n_{\mathrm{av}}$ are independent of film thickness, the anisotropies slightly increase as the film thickness decreases for flexible PI films. As expected from the temperature derivative of the Lorentz–Lorenz (LL) equation, the amorphous PI films exhibiting high $n_{\mathrm{av}}$ show large ${\mathrm{dn}}_{\mathrm{av}}\mathrm{/dT,}$ however, significant anisotropies are observed even for the PI films exhibiting very small birefringence.