New calculations of the relative optical air mass function are made over the 0°-87° range of apparent solar zenith angle θ, for various vertical profiles of background aerosol, diamond dust and thin cirrus cloud particle extinction coefficient in the Arctic and Antarctic atmospheres. The calculations were carried out by following the Tomasi and Petkov (2014) procedure, in which the above-mentioned vertical profiles derived from lidar observations were used as weighting functions. Different sets of lidar measurements were examined, recorded using: (i) the Koldewey-Aerosol-Raman Lidar (KARL) system (AWI, Germany) at Ny-Ålesund (Spitsbergen, Svalbard) in January, April, July and October 2013; (ii) the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) satellite-based sensor over Barrow (Alaska), Eureka (Nunavut, Canada) and Sodankylä (northern Finland), and Neumayer III, Mario Zucchelli and Mirny coastal stations in Antarctica in the local summer months of the last two years; (iii) the National Institute of Optics (INO), National Council of Research (CNR) Antarctic lidar at Dome C on the Antarctic Plateau for a typical "diamond dust" case; and (iv) the KARL lidar at Ny-Ålesund and the University of Rome/National Agency for New Technologies, Energy and Sustainable Economic Development (ENEA) lidar at Thule (northwestern Greenland) for some cirrus cloud layers in the middle and upper troposphere. The relative optical air mass calculations are compared with those obtained by Tomasi and Petkov (2014) to define the seasonal changes produced by aerosol particles, diamond dust and cirrus clouds. The results indicate that the corresponding air mass functions generally decrease as angle θ increases with rates that are proportional to the increase in the pure aerosol, diamond dust and cirrus cloud particle optical thickness.