**1. Introduction**

Progress in the generation and application of femtosecond and attosecond electromagnetic (EM) pulses have stimulated theoretical research of few-cycle and near-cycle localized exact solutions of the free-space wave equation (see, e.g., [1,2] and references therein). A strong space-time coupling, inherent to such pulses in focusing, makes the classical well-known approximations inapplicable. Moreover, with the decrease in the duration of optical pulses, there is a growing need to go beyond the quasi-monochromatic limit, and spectral representations of fields lose their advantages over direct time-domain ones. There is an ultimate milestone on the road away from the narrow-band limit—where the temporal spectrum of the pulse extends to the frequency scale origin. This causes a merging of positive-frequency and negative-frequency Fourier components, and as a result, the concept of analytic signal with its Hilbert-transform-related real and imaginary parts remains the only sound alternative for determining the envelope of a pulse.

If the spectrum vanishes at the frequency *ω* = 0, the pulse is bipolar. Several decades ago Bessonov [3] introduced the term "strange" for waves whose electric field does not obey the equality

$$\stackrel{\rightarrow}{S}\_{\mathbf{E}}\left(\stackrel{\rightarrow}{r}\right) \equiv \int\_{-\infty}^{\infty} \stackrel{\rightarrow}{E}(\stackrel{\rightarrow}{r},t)dt = 0\tag{1}$$

and called "usual" all waves satisfying Equation (1) at every location →*r* . It is obvious that usual waves are necessarily bipolar.

In a recent paper [4], Popov and Vinogradov studied conditions leading to Equation (1) using a spectral representation. Their main result is that pulses of finite energy in free

**Citation:** Saari, P.; Besieris, I.M. Conditions for Scalar and Electromagnetic Wave Pulses to Be "Strange" or Not. *Foundations* **2022**, *2*, 199–208. https://doi.org/10.3390/ foundations2010012

Academic Editor: Eugene Oks

Received: 26 December 2021 Accepted: 4 February 2022 Published: 7 February 2022

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space are usual and, consequently, bipolar. However, they do not exclude the possibility of the existence of finite-energy strange pulses, although quite exotic, in a vacuum. In other words, they have not proved that the finiteness of pulse energy is a sufficient condition for a pulse to be usual (not strange). Of course, it is not a necessary condition, as a bipolar plane wave pulse can already readily be a usual one.

We will neither question the results of the paper [4] nor prove the sufficiency of the energy condition. Instead, our aims here are to analyze what the relevant necessary and sufficient conditions are and to study the problem with several sample waves.

Specifically, we first consider spherically symmetric pulses converging to and thereafter, at positive times, diverging from a focus at the origin, i.e., pulses described by a wavefunction

$$
\psi\left(\stackrel{\rightarrow}{r},t\right) = \frac{1}{r} [f(t+r/c) - f(t-r/c)],\tag{2}
$$

where *f*(*s*) is an arbitrary nonsingular function depending on the spherical radial coordinate *r* = *x*<sup>2</sup> + *y*2 + *z*2 and time *t*, with *c* being the speed of light in a vacuum.

Second, we consider axially symmetric propagation-invariant pulses [5–10] whose wavefunction and/or energy density depend on the axial coordinate and time solely through the propagation variable *ζ* = *z* − *vt*, where *v* is the group velocity of the pulse. These fields exhibit pronounced space-time coupling. From a theoretical point of view it is interesting that the second type of pulses can be obtained from spherically symmetric ones via relativistic boosts and/or complexifying the axis *z* [5,6,8]. Finally, we study some exotic pulses described by sophisticated solutions of the wave equation.
