We analyze a recently proposed experimental technique for constructing Poincaré maps in flows exhibiting chaotic advection and develop the theoretical framework that explains the reasons for the success of this approach. The technique is nonintrusive and, thus, simple to implement. Planar laser-induced flourescence is employed to collect a sufficiently long sequence of instantaneous light intensity fields on the plane of section of the Poincaré map (defined by the laser sheet). The invariant sets of the flow are visualized by time-averaging the instantaneous images and plotting iso-contours of the so resulting mean light intensity field. By linking the Eulerian time averages of light intensity at fixed points in space with the Lagrangian time averages along particle paths passing through these points, we show that ergodic theory concepts can be used to show that this procedure will indeed visualize invariant sets of the Poincaré map. As the technique is based on time-averaging, we discuss the rates of convergence and show that inside regular islands the convergence is fast. An example is presented from the application of this technique to visualize the intricate web of regular islands within a steady, three-dimensional vortex breakdown bubble.