  Proposed near-future upgrades of the current advanced interferometric
gravitational wave detectors include the usage of frequency dependent squeezed
light to reduce the current sensitivity-limiting quantum noise. We quantify and
describe the degradation effects that spatial mode-mismatches between optical
resonators have on the squeezed field. These mode-mismatches can to first order
be described by scattering of light into second-order Gaussian modes. As a
demonstration of principle, we also show that squeezing the second-order
Hermite-Gaussian modes $\mathrm{HG}_{02}$ and $\mathrm{HG}_{20}$, in addition
to the fundamental mode, has the potential to increase the robustness to
spatial mode-mismatches. This scheme, however, requires independently optimized
squeeze angles for each squeezed spatial mode, which would be challenging to
realise in practise.
