Elastic scattering of protons at the TOTEM experiment at the LHC

The TOTEM experiment at the LHC at CERN is optimized to measure elastic and diffractive scattering at the LHC and measures the total proton-proton cross-section with the luminosity-independent method. The TOTEM experiment uses the special technique of movable beam pipe insertions -- called Roman Pots -- to detect very forward protons. The reconstruction of the forward proton kinematics requires the precise understanding of the LHC beam optics. A new method of LHC optics determination is reported, which exploits kinematical distributions of elastically scattered proton-proton data measured by the Roman Pots of the TOTEM experiment. The method has been successfully applied to data samples recorded since 2010. The interpretation of the proton-proton elastic differential cross-section is a challenging task. The geometrical model of proton-proton elastic scattering of Bialas and Bzdak is fitted to ISR data and to data measured by the TOTEM experiment at LHC energy of $\sqrt{s}=7$~TeV. The Bialas-Bzdak model is generalized and improved in order to give a satisfactory and unified description of the ISR and LHC data. The improved model is extrapolated to future LHC energies and beyond.