## 2 Linear Light Bullets
### Ambiguity of the Poynting Vector Maps of Nondiffracting Beams  
Free-space light bullets raise the question about an adequate definition of \"propagation\" and \"beam\" in the Fresnel domain. The use of both terms is not trivial here because in most cases the detected spatial patterns represent time-integrated photon density distributions instead of Poynting vector maps or wavefronts. If the propagation is analyzed on the basis of energy flux, the description has a close relationship to such measuring conditions [27, 28]. A Poynting vector, however, indicates the directional energy flux density per unit area. The local wavefront of a Bessel-like beam results from the interference of two or more partial waves. If one places a Shack-Hartmann wavefront sensor in a Bessel beam, the conical shape of the wavefront [29] and, in combination with a diffractive grating, the spectral phase [30] can be reconstructed. The focal spot of each microlens splits into a ring, as follows from the analysis in [31]. In other words, the sensor which normally delivers a Poynting vector at each position resolves the local ambiguous structure of the wavefront which is more realistic than a formal but physically meaningless vector addition. To illustrate this specific ambiguity in comparison to other types of beams, one can imagine a simplified ray representation in Fresnel domain as schematically drawn in Figure 1. It has to be mentioned that in the extreme near-paraxial case of ultra-small conical angles, the different vectors in WS\({}_{2}\) tend to become nearly identical again so that a wavefront-division based detection technique can be well applied to characterize the beams. This is of relevance to the experiments we will report on. Particular types of beams like Airy beams are often described as accelerating light bullets propagating on \"curved trajectories\". Airy beams were originally introduced by [PERSON] and [PERSON] as a particular nonspreading solution of the 1D-Schroedinger equation [32, 33] and later extended to the 3D-case [34]. They have to be interpreted as caustics [32] rather than as propagating along a bended pathway. Vectorial, nonparaxial types of accelerating beams as recently discussed [35] are sometimes believed to be able to go around a corner (see Figure 2 of ref. [36]) or even to propagate along a circle [37]. To establish bended beams, however, the straight trajectories of all constituting partial beams are necessary (compare [38]) whereas the obstruction by a shading edge distorts the field pattern behind (except in a medium with a gradient index). In medium-free case, the physical significance of this statement can directly be proved by detecting the far field which separates the directional components of the partial waves. The same should apply to Airy-type and other nondiffracting beams. To stay in concordance with the conservation laws of energy and momentum, one has to carefully examine the unambiguity of the Poynting vector maps of accelerating beams (e.g., in [35]) in a similar way as shown for Bessel beams (for a theoretical treatment of the Poynting vector maps of Airy beams, see [39, 40, 41]). We agree with the related comment in ref. [42]: \"As with the Bessel beam's diffraction -free \"propagation\", light doesn't actually propagate along the curved path.\"  
Figure 1: Ambiguity of local wavefronts of nondiffracting beams in comparison to a convergent beam (simplified ray representation): **(a)** Gaussian-type beam with unambiguous Poynting vectors (red arrows) as detected by a Shack-Hartmann wavefront sensor at a certain plane WS\({}_{1}\) (gray); **(b)** Bessel beam with an ambiguity in the superposition zone WS\({}_{2}\) (gray) and a ring-shaped unambiguous angular distribution at a distance WS\({}_{3}\) behind the superposition zone (gray). Linear light bullets in absence of a medium are created by constructive interference in superposition zones like shown in **(b)**.