Controllable and dispersive magnetic permeability in negative-index materials (NIMs) offers greatly enhanced design freedom to alter the linear and nonlinear properties. This makes it possible for us to control the propagation of ultrashort pulses at will. In this paper, we will investigate the self-focusing of ultrashort pulses associated with anomalous nonlinear self-steepening (SS) effect and anomalous space–time focus (linear SS) effect in bulk NIMs with a nonlinear electric polarization, trying to disclose some unusual behaviors different from those in naturally positive-index materials. It is found that negative SS acts to push the peak intensity toward the trailing edge of a pulse and the resulting spectrum exhibits a broad redshifted trail; however, positive SS leads to the occurrence of the opposite situation in both temporal and spectral domains, that is, the peak intensity will move toward the leading edge of the pulse by shifting the spectrum toward the blue side. In addition, compared with the case in positive-index materials (PIMs), another most notable property of NIMs is that, whether self-focusing of ultrashort pulses will occur at the leading or the trailing edge can be manipulated freely by engineering the SS effect by choosing the size of split-ring resonator circuit elements. Our analysis is performed by directly numerically solving nonlinear Schrödinger equations as well as by using the moment method, both showing consistent results. These findings demonstrate that NIMs can provide us unique opportunities unattainable in PIMs to manipulate ultrashort pulse propagation.
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