We investigate the usefulness of the wide-angle Fast Fourier Transform-based beam propagation method (FFT-BPM) to investigate important and quite involved problems in the propagation of optical beams. The incident transverse electric (TE) and transverse magnetic (TM) optical beams at the critical angle on a dielectric interface are investigated. Major numerical difficulties associated with the singularity of the transverse derivative of the step-like refractive index at the interface plane (for the TM case) are circumvented via an ‘‘equivalent-index’’ formalism. An arc-tangent function is used to approximate the abrupt change of the refractive index at the interface, such that the singular derivative is eliminated, and the TM problem is transformed to an equivalent TE one with no singular behavior. Further, the propagation of a rectangular incident pulse on an interface at the critical angle is also studied. In this investigation, the large shift of the lateral field and the standing wave pattern resulting from the incident and reflected fields are investigated and justified by simple calculations. A new phenomenon, called ‘‘spatial transient’’ is discussed, concerning the substantial spatial evolution of the pulse over very short propagation distance (fraction of the wavelength). Finally, a parametric study of a plasmonic-type sensor in the Kretschmann configuration is presented using the proposed FFT-BPM to ensure its validity in studying such subwavelength-based phenomenon.