The Complex Charge Paradigm: A New Approach for Designing Electromagnetic Wavepackets

Abstract

Singularities in optics famously describe a broad range of intriguing phenomena, from vortices and caustics to field divergences near point charges. The diverging fields created by point charges are conventionally seen as a mathematical peculiarity that is neither needed nor related to the description of electromagnetic beams and pulses, and other effects in modern optics. This work disrupts this viewpoint by shifting point charges into the complex plane, and showing that their singularities then give rise to propagating, divergence-free wavepackets. Specifically, point charges moving in complex space-time trajectories are shown to map existing wavepackets to corresponding complex trajectories. Tailoring the complex trajectories in this "complex charge paradigm" leads to the discovery and design of new wavepacket families, as well as unprecedented electromagnetic phenomena, such as the combination of both nondiffracting behavior and abruptly-varying behavior in a single wavepacket. As an example, the abruptly focusing X-wave-a propagation-invariant X-wave-like wavepacket with prechosen self-disruptions that enhance its peak intensity by over 200 times-is presented. This work envisions a unified method that captures all existing wavepackets as corresponding complex trajectories, creating a new design tool in modern optics and paving the way to further discoveries of electromagnetic modes and waveshaping applications.

Space-time wave packets

“…We have already seen in Section 10 three such scenarios, including dispersive [286,287], accelerating [45,293], and axially spectrally encoded ST wave packets [292], in addition to the axial dynamics associated with the ST Talbot effect [47]. More recently, a new idea has been developed whereby a propagation-invariant ST wave packet is accompanied by another feature that focuses suddenly at a prescribed axial position without otherwise affecting the propagation-invariant field structure [349]; see Fig. 52(a).…”

Space-time wave packets violate the universal relationship between angular dispersion and pulse-front tilt

“…Taking only positive values for k x and ϕ yields a field with a tilted pulse front whose profile resembles that of a TPF, whereas taking both positive and negative values of k x and ϕ yields a symmetrized TPF structure [33][34][35]; see Fig. 1(f).…”

A universal angular-dispersion synthesizer

“…Producing all the physically realizable field configurations accessible via AD requires: (1) independent tuning of multiple orders of AD, a feature that is not provided by conventional optical components; and (2) access to the newly identified non-differentiable AD, whereby the derivative of the wavelength-dependent propagation angle is undefined at some wavelength [23][24][25][26], a condition that is not produced by any currently available optical device. Non-differentiable AD arises naturally in the study of 'space-time' (ST) wave packets [27][28][29][30][31][32][33], where it undergirds their unique characteristics in free space such as propagation invariance [34][35][36][37][38][39][40], tunable group velocity [30,31,41,42], self-healing [43], Talbot self-imaging in space-time [44,45], accelerating wave packets [46,47], arbitrary dispersion profiles [24,25,48,49], and anomalous refraction [50].…”