The Cosmic Microwave Background (CMB) radiation is the radiation coming to us from the epoch of recombination, i.e. about 300,000 years after the Big Bang, when the electrons and photons combined to form neutral hydrogen atoms that subsequently decoupled from the ambient radiation, which propagated freely through space. This radiation is lensed by intervening large-scale structure, which are scalar density perturbations, along its path. This lensing effect may be quantified in terms of lensing kernels, which describe the effect of the perturbations on the temperature and polarization power spectra of the CMB. The weak lensing of the CMB by scalar perturbations has been well studied in the literature.

However, a stochastic gravitational wave background, if present, may also contribute to the weak lensing of the CMB photons. In our work, we find that there exist interesting similarities in the structure of the lensing kernels for lensing by large scale structure and gravitational waves, both for the case of temperature as well as for the E- and B- modes of CMB polarization. We perform the analysis both in real space, using a correlation function method, as well as in spherical harmonic space. Our numerical results show that both the above independent methods of analysis lead to the same results for the temperature and polarization power spectra.