With the development of optical communication networks, the need for high-performance broadband mode converters has become increasingly urgent. In this study, we propose and experimentally fabricate strongly-modulated phase-shifted helical long-period gratings (HLPGs) inscribed in few mode fibers, which can realize the function of all fiber broadband orbital angular momentum mode converters. The strongly-modulated HLPG could be fabricated in few mode fibers with short lengths by exposing the fibers to high-power CO₂ laser. The fabricated mode converter has a large bandwidth, which could be further increased considerably by introducing a phase shift in the grating. The 10-dB bandwidth of the high-order mode conversion is greater than 200 nm, and the second- and third-order orbital angular momentum modes could be directly excited within the broadband wavelength range. Broadband wavelength tuning could be realized by applying torsion and strain to the gratings. The proposed HLPGs inscribed in the few mode fibers can be adopted as mode converters to realize high-order orbital angular momentum mode excitation with a large wavelength range and high conversion efficiency.