Wave deflection and shifted refocusing in a medium modulated by a superluminal rectangular pulse

We explore the problem of scattering in a medium modulated by a superluminal rectangular pulse, with the pulse modulation realized through transverse excitations. We solve this problem in the moving frame where the modulation appears purely temporal, since the solution to a purely temporal modulation is known. Upon this basis, we use a graphical approach to find the angle and frequency of the wave scattered by the modulation under oblique incidence. This study reveals that superluminal modulations deflect plane waves retrodirectively, and refocus cylindrical waves to a shifted point of space with respect to the original source.

Figure 1

Deflection from a rectangular pulse modulation. (a) Subluminal, (b) superluminal.

Figure 2

Scattering from subluminal and superluminal rectangular pulse modulations for normal incidence in direct and inverse Minkowski diagrams. (a) Subluminal regime, direct space. (b) Superluminal regime, direct space. (c) Subluminal regime, inverse space. (d) Superluminal regime, inverse space. The unprimed and primed axes labels correspond to the stationary and moving frames, respectively.

Figure 3

Scattering of harmonic plane waves from subluminal and superluminal rectangular pulse modulations for oblique incidence. Top: iso-$k_x$ dispersion diagram. Bottom: iso-$\omega$ diagram. This graph is for the particular case of $v_{\text{sub}}/c=c/v_{\text{sup}}$, which conveniently leads to shared Lorentz moving frames. Note that here we have changed the convention angle of Fig. 1 for convenience.

Figure 4

Scattering of a cylindrical wave by subluminal and superluminal pulse modulations. (a) Subluminal case: diffraction. (b) Superluminal case: refocusing.

Figure 5

Scattering of a cylindrical pulse by a superluminal pulse modulation. (a) Before scattering, (b) when the modulation has reached the source point, (c) when the modulation has crossed most of the pulse, (d) after crossing, (e) at the refocus time, and (f) after refocusing [24].