While there is a consensus that higher-order effects beyond are present also in high-intensity light-matter interactions, when and how they become apparent needs further study. The central question addressed in this Letter is whether it is possible to design a situation in which they show up before being completely masked by the electrons freed by a high-intensity field. The second question we attempt to answer is how much such observations, if and when feasible, can reveal about the nature of the nonlinear polarization. We answer the first question in the affirmative, but our comparative simulations indicate that distinguishing the higher-order nonlinearity from the third-order polarization can be extremely challenging. We also briefly discuss the implications for the interpretation of the measured values of the nonlinear index.
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