Traveling-wave modulation is a form of space-time modulation, which has been shown to enable unique electromagnetic phenomena such as nonreciprocity, beam steering, frequency conversion, and amplification. In practice, traveling-wave modulation is achieved by applying staggered time-modulation signals to a spatially discrete array of unit cells. Therefore, the capability to accurately simulate spatially discrete traveling-wave modulated structures is critical to the design. However, simulating these structures is challenging due to the complex space-time dependence of the constituent unit cells. In this article, a field relation (referred to as the interpath relation) is derived for spatially discrete traveling-wave modulated structures. The interpath relation reveals that the field within a single time-modulated unit cell (rather than an entire spatial period) is sufficient to determine the field solution throughout space. It will be shown that the interpath relation can be incorporated into the existing periodic method of moment solvers simply by modifying the source basis functions. As a result, the computational domain is reduced from an entire spatial period to a single time-modulated unit cell, dramatically reducing the number of unknowns. In the context of traveling-wave modulation, this enables researchers to efficiently simulate both complex structures with patterned unit cells in addition to continuous structures with infinitesimal unit cells.
