**3. Results**

### *3.1. General Conditions for a Pulse to Be "Usual"* 3.1.1.SufficientConditions

With the help of Equation (7), we can easily find conditions under which the field is not "strange", i.e., conditions of vanishing of the time integral in Equation (3), or—keeping in view the text before it—also conditions for fulfillment of Equation (1). Since *<sup>D</sup>*(*<sup>R</sup>*, *t*) is an odd function with respect to time, the first term of the integrand in Equation (7) in any case does not contribute to *<sup>S</sup>ψ*<sup>→</sup>*<sup>r</sup>* . As the *δ*-function equals zero at infinity, the time integral of the second term vanishes if *R* = →*r* − →*r* remains finite in the spatial integration. Hence, any wave field that is spatially bounded at a certain instant of time (when the pulse is focused) cannot be "strange". Any such pulse, if it has no unphysical singularities, has finite energy irrespective of which of the abovementioned definitions of the pulse energy has been taken. Hence, the sufficient condition formulated here is consistent with the one found by Popov and Vinogradov [4]. The second term also vanishes if *g*<sup>→</sup>*r* = *<sup>ψ</sup>*<sup>→</sup>*r* , 0 = 0 in the whole space. This condition means that any nontrivial *<sup>ψ</sup>*<sup>→</sup>*r* , *t* must be an odd function of time. Therefore, such oddness is another sufficient condition for a pulse to be "usual". The results of the next subsection demonstrate accordance with this condition.

### 3.1.2. Necessary and Sufficient Conditions

For a field to be usual (not strange), the integral in Equation (1) or (3) must vanish everywhere. Therefore, we obtain the necessary condition that it must vanish at the origin →*r* = 0. In this case, we write *R* = →*r* ≡ *r*, omitting the prime for simplicity. After integration from *t* = −*T* to *t* = *T* and subsequently taking the limit *T* → ∞ in accordance with Equation (3), the second term in Equation (7) can be transformed in spherical coordinates as follows:

$$\begin{bmatrix} \int D(R,t)\wp(r,\theta,\varphi)R^2\sin\theta dR d\theta d\varphi \end{bmatrix}\_{-T}^T = $$
 
$$ = \frac{1}{4\pi\epsilon^2} \left\{ \int r^{-1} \begin{bmatrix} \delta(r/c - T) - \delta(r/c + T) - \\ -\delta(r/c + T) + \delta(r/c - T) \end{bmatrix} \mathfrak{g}(r,\theta,\varphi)r^2 \sin\theta dr d\theta d\varphi \right\} = 0 $$
 
$$ = 2\frac{1}{4\pi} \int \mathfrak{g}(cT,\theta,\varphi)T\sin\theta d\theta d\varphi = 2T\langle\mathcal{g}\rangle\_{cT}. $$

Here, *gcT* denotes the average value of the function *g* on the surface of a sphere with radius *cT* and center at the origin. Therefore, a necessary condition for a wave field to be usual is lim *T*→∞*<sup>T</sup> <sup>g</sup>cT* = 0.

Generally, this means that *g*(*<sup>r</sup>*, *θ*, *ϕ*) must asymptotically vanish as *r* → ∞ faster than 1/*<sup>r</sup>*. Of course, the surface average might be zero irrespective to such asymptotic behavior of *g*(*<sup>r</sup>*, *θ*, *ϕ*) if the latter is bipolar on the surface, and due to some symmetry the regions of opposite sign cancel each other. However, if applying the condition for a sphere whose center is shifted from the origin, the symmetry would disappear and the condition would be fulfilled due to only the aforementioned asymptotic behavior of *g*(*<sup>r</sup>*, *θ*, *ϕ*).

The transformation of the integral carried out above remains valid for an arbitrary point →*r* = 0 in Equation (7). The reason for this is that whatever the point in the field is, its radial vector is fixed while → |*r*| runs together with *cT* to infinity. Therefore, →*r* can be neglected in the expression *R* = →*r* − →*r* . Consequently, the necessary and sufficient condition for a wavefield to be "usual" can be stated as follows. The wavefunction *g*(*<sup>r</sup>*, *θ*, *ϕ*)

at *t* = 0 must vanish asymptotically as *r* → ∞ faster than 1/*r*; in other words, the following equality must be fulfilled:

$$\lim\_{r \to \infty} r \psi(r, \theta, \varphi, 0) = 0. \tag{8}$$

Application of the criterion in Equation (8) is especially appropriate when *ψ* → *r* , *t* contains multivalued complex functions and direct temporal integration according to Equation (3) may be hampered due to crossing the branch-cut lines.

### *3.2. Spherically Symmetric Pulses: Some Examples*

Such pulses, the general form of the wavefunction of which is given by Equation (2), are the simplest to analyze. If *ψ* → *r* , *t* in the form of Equation (2) is used to construct a vector magnetic potential, simple relations between *f*(*s*) and EM field vectors and EM pulse energy have been derived in [12] and references therein.

### 3.2.1. Even and Odd Lorentzians

We studied the functions *fe*(*s*) = 1/ 1 + *s*2*<sup>n</sup>* and *fo*(*s*) = *s*/ 1 + *s*2*<sup>n</sup>* (see Equation (2)) where *n* = 1, 2, . . .. Results concerning the strangeness of such fields are as follows:

(a) If *f*(*s*) is an even function, *ψ* → *r* , *t* is odd with respect to time (hence, automatically notstrange)and → *E*(→ *rt*)iseven,butneverthelessnotstrange.Themagneticfield is

 , odd and, hence, not strange. 

(b) If *f*(*s*) is odd, *ψ* → *r* , *t* is even with respect to time (but nevertheless not strange) and → *E*( → *r* , *t*) is odd, i.e., automatically is not strange. The magnetic field is even, but still not strange.

For these pulses the Mandel–Wolf energy and EM energy are both finite.

### 3.2.2. Error Function

The function *f*(*s*) = erf(*s*) is odd, and consequently, *ψ* → *r* , *t* is even. However, it is strange and it is not square-integrable. Nevertheless, its Mandel–Wolf energy and EM energy are both finite. In accordance with point (b) above, → *E*( → *r* , *t*) is odd, i.e., automatically not strange. The magnetic field is an even function of time but still not strange.

### *3.3. Propagation-Invariant Pulses: Some Examples*

### 3.3.1. Superluminal X-Waves

Inspired by the quotation "Therefore, the possibility of the existence of strange and unipolar pulses in a vacuum, although quite exotic, remains" from [4], we turned to the so-called X-waves, which were first intruduced in [5,13] and then studied in numerous papers; see [6–9] and references therein.

The so-called fundamental axisymmetric X-wave is given by

$$\psi(\rho, z, t) = \frac{a}{\sqrt{\left[a + i(z - vt)\right]^2 + \rho^2 \gamma^{-2}}} \,\,\,\tag{9}$$

where *γ* = *v*2/*c*<sup>2</sup> − 1 −1/2 is the superluminal version of the Lorentz factor, including the velocity *v* > *c* of the pulse; *a* is a positive parameter that determines the width of the pulse; and *ρ* = *x*<sup>2</sup> + *y*2. This wavefunction is commonly considered as a white-spectrum superposition of Bessel beams.

Referring to Equation (7), it can be also derived as Liénard–Wiechert potentials for Lorentzian distributions of fictitous "charges" (sources and sinks) flying with the constant velocity *v* along the axis *z*, i.e., the distributions being *a* (*z* − *vt*) 2 + *a*2 −1 *<sup>δ</sup>*(*x*)*δ*(*y*) (for

the imaginary part of *ψ*(.) and (*z* − *vt*) (*z* − *vt*) 2 + *a*2 −1 *<sup>δ</sup>*(*x*)*δ*(*y*) (for the real part) [14]. The real part of Equation (9) respresents a strange scalar field and the imaginary part an ordinary (usual) scalar field. At the origin, *ψ*(0, 0, *t*) = *a*/(*a* − *ivt*), the temporal spectrum of the real part is proportional to exp(−*<sup>a</sup>*|*ω*|/*v*) while that of the imaginary part is proportional to signum(*ω*) exp(−*<sup>a</sup>*|*ω*|/*v*). Hence, although both spectra have their maxima at infinitesimally small frequencies, the spectrum of the imaginary part lacks the constituent at *ω* = 0 exactly, which is in accordance with the established difference in strangeness of the real and imaginary parts.

If the EM vector fields are derived from Equation (9) by standard procedures involving the vector magnetic potential or the Hertz vector, then, due to taking derivatives in the course of these procedures, the EM field turns out to be usual.

A specific procedure to obtain an EM field avoiding the derivatives is to derive the following complex-valued Riemann–Silberstein vector

$$\begin{array}{l} \stackrel{\rightarrow}{F}\left(\stackrel{\rightarrow}{r},t\right) = \frac{i\epsilon^{2}\sqrt{\left(v/c\right)^{2}-1}}{v^{2}Q\left(\stackrel{\rightarrow}{r},t\right)}\frac{P\_{-}\left(\stackrel{\rightarrow}{r},t\right)}{P\_{+}\left(\stackrel{\rightarrow}{r},t\right)}\stackrel{\rightarrow}{a}\_{x} - \frac{\epsilon\sqrt{\left(v/c\right)^{2}-1}}{vQ\left(\stackrel{\rightarrow}{r},t\right)}\frac{P\_{-}\left(\stackrel{\rightarrow}{r},t\right)}{P\_{+}\left(\stackrel{\rightarrow}{r},t\right)}\stackrel{\rightarrow}{a}\_{x} + c^{2}\frac{\left(v/c\right)^{2}-1}{v^{2}Q\left(\stackrel{\rightarrow}{r},t\right)}\stackrel{\rightarrow}{a}\_{z};\\ Q\left(\stackrel{\rightarrow}{r},t\right) = \sqrt{\left((v/c)^{2}-1\right)}\rho^{2} + \left(a + i(z-vt)\right)^{2};\\ P\_{\pm}\left(\stackrel{\rightarrow}{r},t\right) = \sqrt{1 \pm \frac{a+i(z-vt)}{Q\left(\stackrel{\rightarrow}{r},t\right)}}\end{array} \tag{10}$$

which arises from a superposition of vector-valued Bessel beams. It can be expressed as → *F* → *r* , *t* = √*ε*0/2 → *E* + *ic* → *B* , where *ε*0 is the permittivity of free space and → *E* and → *B* are →

real fields obeying the homogeneous Maxwell equations. The *z*-component of *F* →*r* , *t* is essentially the infinite-energy superluminal fundamental X wave, as one can see by inspection of Equations (9) and (10). Hence, the real part of the electric field is a strange field, whereas the magnetic field is a usual field. The EM wave pulse energy is infinite quite analogously to the case of plane waves. This is understandable because the X wave can be thought of as a superposition of plane wave pulses directed along a conical surface.

3.3.2. Subluminal Arctan-Wave

> The expression in Reference [6]

$$\begin{split} \Psi \left( \stackrel{\rightarrow}{r}, t \right) &= \frac{1}{\sqrt{\rho^2 + \gamma^2 \left( z - vt \right)^2}} \tan^{-1} \left( \frac{\sqrt{\rho^2 + \gamma^2 \left( z - vt \right)^2}}{a + i\gamma \left( v/c \right) \left( z - \frac{c^2}{v}t \right)} \right); \\ & \gamma = 1/\sqrt{1 - \left( v/c \right)^2}, v < c, \end{split} \tag{11}$$

where again *ρ* is the polar radial coordinate, is a subluminal localized wave that is relatively undistorted upon propagation depending on the value of the positive free parameter *a*. The real part of *ψ* → *r* , *t* has a finite Mandel–Wolf total energy and is a strange field; however, its imaginary part is normal (usual). Additionally, the finite-energy corresponding electromagnetic fields, constructed within the framework of either a Coulomb gauge or a vector Hertz potential, are usual fields.

### 3.3.3. Luminal Localized Wave

In cylindrical coordinates, the azimuthally symmetric expression

$$\Psi\left(\stackrel{\rightarrow}{r},t\right) = \frac{1}{\sqrt{4b^2\rho^2 + \left[-b^2 + (a\_1 + i\varsigma)(a\_2 - i\eta) + \rho^2\right]^2}},\tag{12}$$

where *ζ* = *z* − *ct*, *η* = *z* + *ct* are the characteristic variables of the one-dimensional scalar wave equation in vacuum and *<sup>a</sup>*1,2 and *b* are positive free parameters, is a spatiotemporally

localized extended splash-mode nonsingular solution to the (3 + 1)-dimensional scalar wave equation under the condition *a*1*a*2 − *b*2 > 0. It can be derived from a superposition of Bessel–Gauss focus wave modes (FWM). For *b* = 0, it reduces to the ordinary first-order splash-mode first derived by Ziolkowski (see, e.g., [5,6]); the latter is not strange. The scalar wave field in Equation (12) is not strange, and the Wolf–Mandel total energy of its real part is finite. The electric and magnetic fields arising from a vector potential → *<sup>A</sup>*<sup>→</sup>*r* , *t* = ∇ × *<sup>ψ</sup>*<sup>→</sup>*r* , *t*→*a z* within the framework of a Coulomb gauge have been examined. The electric and magnetic fields are usual. The total electromagnetic energy is finite.

The reason why the "strangeness" integral of Equation (3) turns out to be zero can be understood from Figure 1.

**Figure 1.** Dependencies of the real part (red) and imaginary part (blue) of the wavefunction in Equation (12) on time (for specificity—in the optical femtosecond domain) along the propagation axis (where *ρ* = 0). Ordinate scales are normalized to *Reψ*(0, 0) (see (**a**)), but notice the change of the scale in (**b**–**d**). Spatial locations: (**a**) *z* = 0 μm; (**b**) *z* = 0.4 μm; (**c**) *z* = 0.8 μm; (**d**) *z* = −0.8 μm. Values of the parameters: *a*1= 0.1 μm, *a*2= 0.2 μm, *b* = 0.1 μm.

At first glance, the real part is unipolar (à la that of the X wave); i.e., the real field seems to be strange. However, closer inspection of Figure 1a indicates that the peak appears on the negative-polarity background, which makes the area under the curve equal to zero, as is the case with the imaginary part. Plots (a) and (b) show that outside the origin, the pulse splits into two counterpropagating ones. Comparison of plots (c) and (d) shows that the real part of the wavefunction is even with respect to simultaneous inversion of the sign of the variables *z*, *t* and the imaginary part is odd with respect to the same inversion. It should be mentioned that neither numerical computation of the "strangeness" integral Equation (3) nor plotting of the wavefunction can be properly accomplished by straightforward application of Equation (12) due to presence of a brach cut in the square root function of a complex-valued argument.

### *3.4. Strange Fields Generated by Sources*

### 3.4.1. Bonnor Fields

In cylindrical coordinates, we identify two regions of space: *ρ* ≥ *a* (outside) and *ρ* ≤ *a* (inside). Motivated by Bonnor's work [15], we specify scalar and vector potentials in the two regions:

$$\begin{array}{l} \Phi\_{0}(\rho,\phi,z,t) = e^{-\left(t-z\right)^{2}} \cos\phi / \rho\_{\prime} \, A\_{0}(\rho,\phi,z,t) = \Phi\_{0}(\rho,\phi,z,t) \stackrel{\rightarrow}{a} z;\\ \Phi\_{i}(\rho,\phi,z,t) = e^{-\left(t-z\right)^{2}} \left(\frac{4\rho^{2}}{a^{3}} - \frac{3\rho^{3}}{d^{4}}\right) \cos\phi \, A\_{i}(\rho,\phi,z,t) = \Phi\_{i}(\rho,\phi,z,t) \stackrel{\rightarrow}{a} z. \end{array} \tag{13}$$

Here, normalization with the speed of light in a vacuum equal to unity has been used. These potential fields satisfy the Lorentz condition. Additionally, the continuity <sup>Φ</sup>0(*<sup>a</sup>*, *φ*, *z*, *t*) = <sup>Φ</sup>*i*(*<sup>a</sup>*, *φ*, *z*, *t*) should be noted.

In the region *ρ* ≤ *a*, the electric volume charge and current densities are determined as follows:

$$\begin{array}{l} \rho\_{vi} = -\left(\nabla^2 - \frac{\partial^2}{\partial t^2}\right) \Phi\_i = -\frac{12e^{-\left(t-z\right)^2} \left(a - 2\rho\right) \cos\phi}{a^4},\\ \overset{\rightarrow}{J}\_{vi} = -\left(\nabla^2 - \frac{\partial^2}{\partial t^2}\right) \overset{\rightarrow}{A}\_i = \rho\_{vi} \overset{\rightarrow}{a}\_z. \end{array} \tag{14}$$

The total charge in this region equals zero. No charges exist for *ρ* ≥ *a*. The electric and magnetic fields in the two regions are given by the expressions

$$\begin{array}{lcl}\stackrel{\rightarrow}{E}\_{0} = -\nabla\Phi\_{0} - \stackrel{\rightarrow}{\partial}\dot{A}\_{0}/\partial t &= \frac{e^{-\left(t-z\right)^{2}}\cos\phi}{\rho^{2}}\stackrel{\rightarrow}{a}\_{\rho} + \frac{e^{-\left(t-z\right)^{2}}\sin\phi}{\rho^{2}}\stackrel{\rightarrow}{a}\_{\rho};\\\stackrel{\rightarrow}{B}\_{0} = \nabla\times\stackrel{\rightarrow}{A}\_{0} = -\frac{e^{-\left(t-z\right)^{2}}\sin\phi}{\rho^{2}}\stackrel{\rightarrow}{a}\_{\rho} + \frac{e^{-\left(t-z\right)^{2}}\cos\phi}{\rho^{2}}\stackrel{\rightarrow}{a}\_{\rho};\\\stackrel{\rightarrow}{E}\_{i} = \frac{e^{-\left(t-z\right)^{2}}\rho\left(-8a+9\rho\right)\cos\phi}{\pi^{4}}\stackrel{\rightarrow}{a}\_{\rho} + \frac{e^{-\left(t-z\right)^{2}}\rho\left(4a-3\rho\right)\sin\phi}{\pi^{4}}\stackrel{\rightarrow}{a}\_{\rho};\\\stackrel{\rightarrow}{B}\_{i} = -\frac{e^{-\left(t-z\right)^{2}}\rho\left(4a-3\rho\right)\sin\phi}{\pi^{4}}\stackrel{\rightarrow}{a}\_{\rho} - \frac{e^{-\left(t-z\right)^{2}}\rho\left(8a-9\rho\right)\cos\phi}{\pi^{4}}\stackrel{\rightarrow}{a}\_{\rho};\end{array} \tag{15}$$

These are transverse electromagnetic (TEM) structures propagating along the *z*-direction with the normalized speed of light in vacuum. Both the electric and magnetic fields are strange. The total energy associated with these fields is finite.

### 3.4.2. Single-Cycle Dipole Electromagnetic Fields

Wang et al. [16] have derived single-cylce electromagnetic fields generated by an oscillating elecric dipole oriented along the *x*-direction. We examined these fields for "strangeness". The electric field is strange but not the magnetic field.
