I earlier had a doubt based on ray theory that light must reflect and so all wavelengths should propagate through SMF. This Q/A Single-mode fibers and ray-theory of light does specify that we have to consider wave theory for propagation of wave in SMF, but i still don't get why a SMF has a cutoff wavelength.
Any optical fibre - and any optical waveguide in general - has a cutoff wavelength (and therefore a cutoff frequency or a cutoff energy) because wether light is confined in the film region (guided modes) or escapes to the substrate (substrate-radiation modes) depends on the propagation constant, $\beta$, which is related to frequency trough the dispersion relation. $\beta$ is just the quantity introduced to account for the propagation of the wave along the optical axis of the waveguide or fibre, and it appears in the wave equation. Since ray theory doesn't consider this quantity nor the effect of dispersion, it fails to account for the existence of a cutoff wavelength.
This figure shows how $\beta$ and $\omega$ relate in an optical fibre or waveguide (taken from Tamir et al: Integrated Optics - Springer Verlag 1979). The $n_s k$ line corresponds to the cutoff region (the cutoff condition is precisely $N_\nu = n_s$ and therefore $\beta_\nu = k n_s$, where $N_\nu$ is the effective index and $n_s$ the refraction index of the substrate).
I disagree with the other answer. I am pretty sure optical fibers (and other dielectric waveguides) do not have a cutoff wavelength. This is, in theory and within the limits of transparency of the medium, source beam quality, alignment, divergence, etc. All wavelengths can propagate!
They do however have a single-mode cutoff wavelength, which is the wavelength above which only one mode can propagate.
Typically, a fiber has single-mode characteristics only over a limited wavelength range with a width of a few hundred nanometers. The limit towards smaller wavelengths is given by the single-mode cut-off wavelength, beyond which the fiber supports multiple modes. [ ] The long-wavelength limit of the useful single-mode region is usually given by excessive bend losses, by absorption of the material or (for certain fiber designs, e.g. with index-depressed cladding) by leakage into the cladding.