Orbital angular momentum (OAM)-based mode division multiplexing (MDM) is an emerging physical layer solution for short-distance line-of-sight wireless communications by providing high spectral efficiency with a considerably low detection complexity. Similarly, index modulation (IM) schemes have been extensively studied over the past few years due to their attractive features, such as improved error performance, low complexity, and high energy efficiency. In this paper, we propose the scheme of orbital angular momentum with index modulation (OAM-IM) by using the activated OAM modes themselves to carry additional information through the principle of IM. A general guideline is presented for the construction of optimal OAM-IM schemes along with the formulation of a low-complexity maximum likelihood detector. It is shown via computer simulations as well as theoretical bit error probability calculations that the proposed OAM-IM schemes can achieve considerably better error performance than the reference OAM-MDM schemes, while having the same level of detection complexity.