**2. Methods**

We primarily analyze scalar waves for the following reasons:

1. Every component of the electric (and magnetic) field is a scalar-valued field that obeys the wave equation. Hence, in order to judge for a chosen wavefunction *<sup>ψ</sup>*<sup>→</sup>*r* , *t* whether the corresponding EM pulse is strange or not, it is sufficient to evaluate the integral

$$S\_{\Psi} \left( \stackrel{\rightarrow}{r} \right) = \int\_{-\infty}^{\infty} \Psi(\stackrel{\rightarrow}{r}, t) dt. \tag{3}$$


It should be pointed out that our approach does not mean resorting to a scalar approximation for EM fields.

In order to determine whether an electric field is strange or not, one can avoid integration according to Equation (1), which in most cases is a computationally difficult task. Instead, one can make use of the expression → *<sup>E</sup>*<sup>→</sup>*r* , *t* = −*∂*<sup>→</sup>*<sup>A</sup>*<sup>→</sup>*<sup>r</sup>* , *t*/*∂t* of the electric field derived from a magnetic vector potential in the Coulomb gauge and consider the vector potential at *t* = ±∞ instead of evaluation the integral of the electric field. It is convenient to derive the vector potential from a scalar wavefunction as → *<sup>A</sup>*<sup>→</sup>*r* , *t* = ∇ × *<sup>ψ</sup>*<sup>→</sup>*r* , *t*→*a z*, where →*a z* is the unit vector along the *z*-axis. In the case of cylindrical symmetry, only the azimuthal component remains, and it takes the simple form *<sup>A</sup>ϕ*(*ρ*, *z*, *t*) = <sup>−</sup>*∂ψ*(*ρ*, *z*, *<sup>t</sup>*)/*∂ρ*.

In the case of spherical symmetry, the unit vector → *a z* needs to first be expressed in a spherical coordinate system, and then for the curl, only the azimuthal component remains, taking a similar simple form *<sup>A</sup>ϕ*(*<sup>r</sup>*, *θ*, *t*) = − sin *θ∂ψ*(*<sup>r</sup>*, *θ*, *<sup>t</sup>*)/*∂<sup>r</sup>*. Hence, if one of the following equalities is fulfilled at every location, in the case of cylindrical or spherical symmetry, respectively,

$$\lim\_{t \to \infty} \frac{\partial}{\partial \rho} [\psi(\rho, z, t) - \psi(\rho, z, -t)] = 0,\\ \lim\_{t \to \infty} \frac{\partial}{\partial r} [\psi(r, \theta, t) - \psi(r, \theta, -t)] \sin \theta = 0,\tag{4}$$

the corresponding electric field is not strange. Equation (4) has been successfully applied for the examples in Section 3.

### *2.1. Evaluation of the Wave Pulse Energy*

The (total) energy of a physically realizable pulse is a time-independent spatial integral (over the whole space) of the energy density, which in the case of an EM wave is given by the well-known expression with squares of strengths of electric and magnetic fields. Energy density of a scalar field is frequently defined as the square of the wavefunction (or modulus squared for complex-valued fields). However, the spatial integral of the latter will not be used below for establishing whether the EM pulse corresponding to a given wavefunction has finite or infinite energy.

For scalar-valued wave fields, another definition of the energy density exists [11], which is consistent with the energy conservation law and the Poynting theorem. It is given by Equation (5) below. We will call the spatial integral of *W* the *Mandel-Wolf total energy* for brevity. Conditions in the spectral domain—analogous to those in [4]—for a scalar field to have finite energy and be usual are discussed in Appendix A.

$$\mathcal{W} = \frac{1}{2} \left[ \frac{1}{c^2} \left( \frac{\partial}{\partial t} \boldsymbol{\Psi} \right)^2 + \nabla \boldsymbol{\Psi} \cdot \nabla \boldsymbol{\Psi} \right]. \tag{5}$$

In order to establish whether a chosen wavefunction gives a pulse of finite energy or not, we used two different packages of scientific calculation for symbolic integration—or, if it turned out to be impossible—numerical integration.

### *2.2. Time-Domain Representation*

As an alternative to using the Fourier expansion of the field as done in [4], for the constituents of the field one may take the singular propagator *<sup>D</sup>*(*<sup>r</sup>*, *t*) (sometimes called the Riemann or Schwinger function)

$$D(r,t) = \frac{1}{4\pi r c^2} [\delta(r/c - t) - \delta(r/c + t)] \equiv \mathbf{G}\_+(r,t) - \mathbf{G}\_-(r,t),\tag{6}$$

where *<sup>δ</sup>*(.) denotes the Dirac delta and *G*± are the causal (retarded) and anticausal (advanced) Green functions, respectively. The function *<sup>D</sup>*(*<sup>r</sup>*, *t*) represents a spherically symmetric delta-shaped pulse wave, first (at negative times *t*) collapsing to the origin (the right term) and then (at positive times *t*) expanding from it.

With this propagator as an elementary constituent, any solution to the three-dimensional homogeneous wave equation can be expressed as the following convolution integral over the whole 3D space:

$$
\Psi\left(\stackrel{\rightarrow}{r},t\right) = \int \left[ D(R,t)h\left(\stackrel{\rightarrow}{r}'\right) + \frac{\partial}{\partial t}D(R,t)g\left(\stackrel{\rightarrow}{r}'\right) \right] d\stackrel{\rightarrow}{r}'.\tag{7}
$$

Here, *R* = → *r* − → *r* and the distributions *g*(.) and *h*(.) are determined by the initial conditions—the field "snapshot" at the time origin moment *g* → *r* = *ψ* → *r* , 0 and

*<sup>h</sup>*<sup>→</sup>*r* = *∂*/*∂t <sup>ψ</sup>*<sup>→</sup>*r* , *<sup>t</sup><sup>t</sup>*=0. However, unlike the solution of a radiation problem, since *D* contains not only the retarded Green function but the advanced one as well, *g* and *h* describe a distribution of fictitious Huygens-type sources, i.e., sources coupled with sinks of the same strength.
