Cogwheel phase cycling in population-detected optical coherent multidimensional spectroscopy

An integral procedure in every coherent multidimensional spectroscopy experiment is to suppress undesired background signals. For that purpose, one can employ a particular phase-matching geometry or phase cycling, a procedure that was adapted from nuclear magnetic resonance (NMR) spectroscopy. In optical multidimensional spectroscopy, phase cycling has been usually carried out in a "nested" fashion, where pulse phases are incremented sequentially with linearly spaced increments. Another phase-cycling approach that was developed for NMR spectroscopy is "cogwheel phase cycling," where all pulse phases are varied simultaneously in increments defined by so-called "winding numbers." Here we explore the concept of cogwheel phase cycling in the context of population-based coherent multidimensional spectroscopy. We derive selection rules for resolving and extracting fourth-order and higher-order nonlinear signals by cogwheel phase cycling and describe how to perform a numerical search for the winding numbers for various population-detected 2D spectroscopy experiments. We also provide an expression for a numerical search for nested phase-cycling schemes and predict the most economical schemes of both approaches for a wide range of nonlinear signals. The signal selectivity of the technique is demonstrated experimentally by acquiring rephasing and nonrephasing fourth-order signals of a laser dye by both phase-cycling approaches. We find that individual nonlinear signal contributions are, in most cases, captured with fewer steps by cogwheel phase cycling compared to nested phase cycling.