Optical antennas have received considerable attention in recent years due to their unique ability to convert localized energy to freely propagating radiation and vice versa. Sidelobe level (SLL) is one of the most crucial parameters in antenna design. A low SLL is beneficial to minimize the antenna interference with other optical components. Here a plasmonic optical leaky-wave antenna with low SLL is reported. Shifting spatial frequency by periodically modulating the electric-field amplitude in a plasmonic gap waveguide enables a free-space coupled wave out of the antenna. At the same time, precise control of the aperture fields by the modulation depth allows for reducing SLL. Simulation results indicate that the proposed design can achieve a high directivity of 15.8 dB and a low SLL of -20dB at the wavelength of 1550 nm. A low SLL below -15dB is experimentally demonstrated within the wavelength range from 1527 to 1570 nm. In addition, the low-SLL property is further verified by comparing it with a uniformly modulated antenna. By modulating the guided waves in the plasmonic gap waveguide in different forms, the aperture fields can be flexibly arranged to achieve arbitrary wavefront shaping. It bridges the gap between guided and free-space waves and empowers plasmonic integrated devices to control free-space light, thus enabling various free-space functions.