Non-ablative laser heating has recently turned out to be an efficient synthesis method for boron nitride (BN) nanotubes, offering flexibility in the control of experimental parameters. Long tubes are obtained as a hairy growth on the target around laser impact. Tubes are mostly assembled in bundles and thin. Selected area diffraction diagrams reveal that zigzag and armchair helicities are dominant, without preference. A field emission gun enabled to resolve the fine modulations in the diffraction diagrams for multi-walled nanotubes (MWNT) and for bundles of MWNT. The interpretation of the diffraction pattern of nanotubes is generalised for such modulations, considering the analytical expression of the diffracted amplitude at each graphite reflection. The fine modulations in the pattern are linked to roll diameters, bundle lattice, and interlayer distances. Experimentally, several roll diameter modulations are observed. Unusual modulations corresponding to an interlayer distance of 2 layers are also observed. They show a two-roll period, with an alternative shift along tube axis, in the piling of the rolls of MWNT.