## 2 Linear Light Bullets
### Pulsed Needle Beams and Highly Localized Wavepackets  
The vast majority of free-space nondiffracting beams including Bessel beams and Airy beams still contains a considerable part of their energy in the outer fringes. In case of Bessel beams, this can be circumvented in two ways: (A) by downstream spatial filtering in a self-apodizing truncation setup, _i.e.,_ matching a diaphragm to the first zero of the intensity distribution so that the central lobe passes without diffraction (but wasting the energy of the fringes) [17,43], or (B) in _aperture-less_ setup by optimizing the parameters (conical angle, waist radius of the input profile) to only generate this central lobe exactly to the first zero (_fulfilling a self-apodizing condition_[43,44]). Because of the distinct needle-shape, this particular type of paraxial single-maximum Bessel-like beams was referred to by us as \"needle beams\" [17,43,44]. To illustrate that, Figures 2b show the intensity distribution functions of a Bessel beam and a needle beam in a transversal plane, respectively. The aperture-less case for self-apodizing geometrical boundary conditions (B) can be found in Figure 3.  
Figure 2: Intensity profiles of (**a**) a Bessel and (**b**) a needle beam schematically represented for an arbitrary plane perpendicular to the propagation axis (color-coded, red = maximum intensity, green = minimum intensity). Needle beams correspond to Bessel beams which are truncated exactly at the first zero (self-apodized truncation). The length of the interference zone \(z_{\text{max}}\) is determined by the finite diameter \(D\) of the initial transversal field under self-apodized conditions [44]. For a wavelength \(\lambda\), it amounts roughly to  
\[z_{\text{max}}=D^{2}/4\lambda \tag{1}\]  
For ultrashort pulses it is an enormous benefit if the spectral and temporal corruption of the outer Bessel fringes (resulting from different delay times of the locally interfering conical waves) is minimized. Special kinds of pulsed Bessel beams were discussed as temporally propagation-invariant localized waves [45,46]. After equation (7) in [45], the field amplitude \(U_{NB}\) of the pulsed needle beam can be represented (after some transformation) by the following Equations (2)-(7):  
\[U_{NB}(r,t)=\text{Re}\Bigg{\{}\sqrt{Z(t)}\cdot\exp\Bigg{[}-\frac{{\tau_{a}}^{2 }+t^{2}}{2{\tau_{0}}^{2}}\Bigg{]}\cdot\exp\Bigg{[}i\,\frac{2\pi\cdot t\cdot c}{ \lambda_{0}}\Bigg{]}\cdot J_{0,r}\left(r,t\right)\Bigg{\}} \tag{2}\]  
with  
\[J_{0,r}=\frac{1}{\pi}\cdot\int\limits_{0}^{\pi}\cos\big{[}\Phi \big{(}r,t\big{)}\cdot\sin\big{(}q\big{)}\big{]}\cdot dq \tag{3}\] \[Z(t)=1+i\,\frac{t\cdot\lambda_{0}}{2\pi\cdot c\cdot{\tau_{0}}^{ 2}}\] (4) \[\Phi(r,t)=Z(t)\cdot 2\pi\cdot r\cdot\Bigg{(}\frac{\sin(\theta)}{ \lambda_{0}}\Bigg{)}\] (5) \[\tau_{a}=\frac{r}{c}\text{sin}(\theta)\] (6) \[\tau_{0}=\frac{\tau_{FWHM}}{2\sqrt{\ln 2}} \tag{7}\]  
Figure 3: Self-apodized truncation condition for the _aperture-less_ Bessel-like needle pulses (schematically, after ref. [44]). Curve on the separated blue plane: radial Bessel intensity profile, red area: central lobe, white area: not generated outer parts (\(\lambda=\) diameter of central lobe, z = propagation axis, \(D=\) axicon diameter, \(\theta_{\text{max}}=\) maximum allowed half conical angle, \(\Delta z_{\text{min}}=\) minimum extension of the needle-shaped nondiffracting zone, \(J_{0}=\) zero order first kind Bessel function). where \(t\), \(r\) and \(k\) are the time, the radial coordinate and the wave number, \(\theta\) and \(\lambda\) are the cone angle and the wavelength, \(q\) is an integration variable, \(c\) is the speed of light in vacuum, \(J_{0}\)(\(r\),\(k_{\sigma}\)) represents a wavelength-dependent Bessel distribution function in radial direction, \(\Phi\)(\(r\),\(t\)) is a complex phase term, \(\tau_{a}\) a radial-dependent time separation between two interfering pulses and \(\tau_{0}\) an input pulse duration (standard deviation).