## Abstract

In this paper we illustrate how the localization of the stationary two-dimensional solution of the propagation equation strongly depends on the features of its spatio-temporal spectral bandwidth. We especially investigate the role of the ultra-broad temporal support and of the spatial bandwidth of the spectrum on the high localization in one spatial dimension of ”Bessel-like” or ”blade-like” beams, quasi-stationarily propagating in normally dispersive materials, and potentially interesting for microfabrication applications.

© 2013 Optical Society of America

## 1. Introduction

In the last few years the localization and stationarity of optical wave packets propagating in linear or in non linear dispersive media have been the object of several studies. Many works have highlighted in the two- or three-dimensional spatial domain the intrinsic conical nature of the localized pulses which are also stationary (non diffractive and non dispersive) during propagation [1–7]. Localized and quasi-stationary (with finite energy) three-dimensional wave-packets (WPs) have been generated also in non linear processes [8–13], with asymptotic features relating to those of non diffracting and non dispersive polychromatic Bessel beams in linear dispersive media. A general description of these linear waves has been given in [14] showing how these can be identified with X-shaped or O-shaped modes of the wave equation in media with respectively normal or anomalous group velocity dispersion (GVD). Also new families of two or three dimensional Bessel X-waves have been shown to be possible in linear bidispersive systems [15].

From the applications point of view often the major issue is the spatial localization of the optical WPs and their stationarity, for an efficient pulse energy to matter transfer. For instance tightly focused high power pulses are widely used in a broad range of technologies in material processing [16] where the challenge is the micromachining precision, and within the same context non diffracting micro-Bessel beams (with very high focal depth) have already been used for the generation of nanochannels or high aspect ratio nanostructures in glass [17, 18]. Analogously, the one-dimensional (1D) spatial localization of stationary optical pulsed WPs should play a crucial role in the writing of sharp trenches [19] or even in the single shot blade cutting (with high aspect ratio) of thick materials (hundred of microns thick). Ideally the (time integrated) beams needed for such applications should have a transverse section with high ellipticity, thus featured by an almost constant intensity along the major section (along the ”blade”) and by an extremely focused peak (down to 1 micron or less) in the perpendicular direction.

Up to date it is not clear how linear WPs can simultaneously be tightly focused in one spatial dimension and stationary in propagation in dispersive materials. Our work proposes to reveal the mechanism for the generation of such non diffracting 1D spatial beams. A complete study highlighting the role of the material dispersion will be later presented in [20] showing the 1D localization in space of stationary non trivial WPs. One example is the spatio-temporal Bessel beam (STBB) already observed and characterized in [21], and analogous to a 2D version of the so-called O-wave stationary in the anomalous GVD regime [14]. Note that the paper by Dallaire et al. [21], being the main significant prior art to the work we present here, clearly explains how to generate non diffracting and non dispersive 2D wavepackets in media with negative GVD. They also show temporal profiles and a complete spatio-temporal characterization of their space-time Bessel pulses.

The work we shall present here can certainly be seen as an extension of [21] to the case of normal dispersion. Nevertheless, in contrast to the results of [21] where the spatio-temporal localization of the STBB is more obvious due to the ring-shaped spatio-temporal spectrum, in our work we put the accent on the fact that in the positive GVD case, stationary X-type 2D WPs could be localized only under some particular conditions, including polychromaticity and the presence of a frequency gap in the supporting spectrum. Also we claim that the simultaneous stationarity *and* localization requires at least 2 dimensions, this being also valid in the anomalous GVD case. More precisely, in this paper we present the explanation for the existence of stationary 1D spatial WPs, which in the normal dispersion regime and with a suitable tailoring could spatially resemble 1D ”Bessel-like” beams featured by a lateral-only flowing energy (in fact truly 2D X-type WPs in the space and time domain). In contrast to the standard conical WPs (Bessel beams, 3D X-waves, etc...), such pulses spatially focused in 1D might become interesting for their potential application to high quality machining or cutting of transparent materials even in single shot.

## 2. Theory

From a theoretical point of view, it turns out that the *simultaneous* features of wave stationarity and localization indeed require infinite energy and at least two ”dimensions”; i.e. if a 2D spatially localized beam (i.e. Bessel Beam) can be monochromatic, a 1D spatially localized stationary beam must have a polychromatic spectral support. The conditions leading to spatial localization of a WP can be understood by analyzing the stationary solutions of the scalar wave equation
$\mathrm{\Delta}E-\frac{1}{{c}^{2}}\frac{{\partial}^{2}D}{\partial {t}^{2}}=0$, that describes the propagation of a linearly polarized light wave in linear transparent media (*E*(*t*, *r*) being the electric field with *r* = (*x*, *y*, *z*) the spatial coordinates, *D*(*t*, *r*) the displacement operator, and
$\mathrm{\Delta}=\frac{{\partial}^{2}}{\partial {x}^{2}}+\frac{{\partial}^{2}}{\partial {y}^{2}}+\frac{{\partial}^{2}}{\partial {z}^{2}}$ the Laplace operator). By using the Fourier transform (FT) it is possible to write down the solution, but we shall focus here on the two-dimensional solution, i.e. a WP modulated in time *t* and featured by one transverse coordinate *x*, propagating along the *z* axis. In the coordinate frame moving with the velocity *u*_{0} and setting
$\eta =t-\frac{z}{{u}_{0}}$, the electric field of such a wave is *E* = *A*(*η*, *x*, *z*)exp{*i*(*ω*_{0}*t* − *k*_{0}*z*)} + *c.c.*, where the complex amplitude *A*(*η*, *x*, *z*) reads as

*ω*−

*ω*

_{0}(

*ω*

_{0}being the carrier frequency).

*S*

_{0}(Ω,

*k*) is the initial spatio-temporal spectrum of the WP, and the effect of dispersion and diffraction is described by

_{x}*k*(

*ω*

_{0}+ Ω) is the dispersion relation function depending on the material properties. In general, the WP stationarity requires the function

*G*(Ω,

*k*) to be linear [1] of the form

_{x}*G*(Ω,

*k*) =

_{x}*γ*

_{1}Ω +

*γ*

_{2}

*k*+

_{x}*γ*where

*γ*

_{1},

*γ*

_{2}and

*γ*are free parameters, so that the field amplitude profile can be written as

*A*(

*η*,

*x*,

*z*) =

*A*

_{0}(

*η*−

*γ*

_{1}

*z*,

*x*−

*γ*

_{2}

*z*)exp(−

*iγz*); thus using Eq.(2) the following relation must be satisfied:

Note that in one spatial dimension a monochromatic wave of frequency *ω*_{0} + Ω (long pulse limit case where the temporal spectrum is described by a Dirac delta function *S*_{0}(Ω, *k _{x}*) =

*S*

_{0}(

*k*)

_{x}*δ*(Ω)), satisfies the quadratic equation (3) for a couple of values

*k*

_{x}_{1},

*k*

_{x}_{2}of the transverse wavevector. Thus, in this case the only stationary solution is a couple of plane waves: no localized solutions are possible. In two spatial dimensions instead, if we replace in Eq. (3)

*k*by ${k}_{\perp}=\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}$, we have for a given frequency

_{x}*ω*

_{0}+ Ω (monochromatic wave) and setting for simplicity

*γ*

_{2}= 0, the spherically symmetric solution given by the monochromatic Bessel beam with

*k*

_{⊥}=

*const*.

In order to have a spatially localized and stationary profile in *one spatial dimension* we must allow for a non monochromatic solution, and thus a temporal bandwidth is needed. The stationarity condition expressed by Eq. (3) shows that the spatial localization and the temporal localization are not independent, since it sets Ω = *f*(*k _{x}*). As a consequence, spatial localization, which implies having a bandwidth of spatial frequencies (from FT properties), necessary implies having a bandwidth of temporal frequencies as well. Thus a 2D linear WP (1D spatial and 1D temporal) can be stationary propagating

*and*localized when its space and time coordinates become entangled via angular dispersion, i.e. when the different temporal frequencies are distributed at different propagating angles. Such a WP can be described by setting its spatio-temporal spectrum for instance as

*S*

_{0}(Ω,

*k*) = 2

_{x}*πS*

_{0}(

*k*)

_{x}*δ*(Ω − Δ(

*k*)) or (as in the case we shall consider from now on)

_{x}*S*

_{0}(Ω,

*k*) = 2

_{x}*πS*

_{0}(Ω)

*δ*(

*k*− Δ(Ω)). The function Δ(Ω) defines the transverse wave-vector

_{x}*k*=

_{x}*k*sin(

*θ*(Ω)) = Δ(Ω) and consequently the angular dispersion

*θ*(Ω) of the stationary WP in the dispersive medium such that

*G*(Ω, Δ(Ω)) =

*α*Ω +

*β*, where

*α*and

*β*are free parameters. Indeed for the longitudinal wave-vector ${k}_{z}=\sqrt{{k}^{2}-{k}_{x}^{2}}$, we have ${k}_{z}(\mathrm{\Omega})=k(\mathrm{\Omega})\text{cos}\left(\theta (\mathrm{\Omega})\right)={k}_{0}+\beta +\mathrm{\Omega}(\alpha +{u}_{0}^{-1})$. The complete propagation invariant wave solution, corresponding to both branches of the spectrum angular dispersion (

*k*= ±Δ(Ω)) travels with group velocity ${u}_{gr}=1/({u}_{0}^{-1}+\alpha )$ and phase velocity

_{x}*v*=

_{ph}*ω*

_{0}/(

*k*

_{0}+

*β*):

*S*(Ω,

*k*) is featured by the characteristic hyperbolic curves described in the paraxial approximation by $\mathrm{\Delta}(\mathrm{\Omega})=\sqrt{{k}_{0}(g{\mathrm{\Omega}}^{2}-2\alpha \mathrm{\Omega}-2\beta )}$, and corresponding to stationary modes with normal GVD. It also turns out that the extension of the temporal frequency bandwidth along the angularly dispersed spectral branches, the angular gap between the latter, and also the thickness of these, have a strong influence on the WP localization.

_{x}Figure 1 illustrates the localization features of the space-time intensity profile of a stationary (1D spatial, 1D temporal) WP in a normally dispersive material, obtained after computing the integration in Eq. (4). The case without spatial gap in the spectral distribution (*β* = 0, see inset on right corner: solid line is *k _{x}* = Δ(Ω), while

*S*

_{0}(Ω) is shown by dash) is shown in Fig. 1(a). The WP has a sharp peak, but the tails are constant and non decaying. In contrast to the 2D spatial case, the introduction of a gap in the spatio-temporal spectrum (

*β*< 0) is here a necessary condition for having a decay in the tails of the intensity profile (Fig. 1(b)). This can be shown analytically by analyzing the asymptotic behavior of Eq. (4). In order to demonstrate the gap influence on the tails asymptotic features we considered the solution given by Eq. (4) at a given distance

*z*(since it is invariant with propagation we can take

*z*= 0), and an angular dispersion described by $\mathrm{\Delta}(\mathrm{\Omega})=\sqrt{{k}_{0}(g{\mathrm{\Omega}}^{2}-2\beta )}$ (

*α*= 0,

*β*< 0). We can introduce the ”characteristic” coordinates along the X-shaped tails: $\xi =t-x\sqrt{{k}_{0}g}$ and $\zeta =t+x\sqrt{{k}_{0}g}$. If

*β*= 0, we obtain from Eq. (4)

*A*(

*ξ*,

*ζ*) =

*A*

_{0}(

*ξ*) +

*A*

_{0}(

*ζ*), with ${A}_{0}(t)=\frac{1}{2\pi}{\int}_{-\infty}^{\infty}{S}_{0}(\mathrm{\Omega})\text{exp}(i\mathrm{\Omega}t)\text{d}\mathrm{\Omega}$ being the temporal WP profile corresponding to an initial spectrum

*S*

_{0}(Ω). The solution

*A*(

*ξ*,

*ζ*) represents nothing more than the superposition of two tilted wavepackets in the space-time plane. If we consider one of the two tails (for instance

*ζ*= 0,

*ξ*≠ 0) we have

*A*(

*ξ*, 0) =

*A*

_{0}(

*ξ*)+

*A*

_{0}(0). For a spectrum

*S*

_{0}(Ω) related to any ”bell shaped” pulse profile featured by the asymptotic behaviour

*A*

_{0}(

*t*)|

_{|}

_{t}_{|→∞}= 0, we obtain a WP characterized by a central spike of amplitude

*A*(0, 0) = 2

*A*

_{0}(0), however

*A*(

*ξ*, 0)|

_{|}

_{ξ}_{|→∞}=

*A*

_{0}(0) =

*const.*i.e. the tail is not decaying. This result is in accordance with the amplitude profile features obtained by integration of Eq. (4), as also shown in the corresponding intensity profile presented in Fig. 1(a). In contrast, a spatio-temporal spectrum with the presence of a gap between the two hyperbolic branches (

*β*< 0) leads to an X-shaped WP with decaying tails. This can be shown analytically by taking a rectangular spectrum

*S*

_{0}(Ω)|

_{|}

_{Ω}

_{|<}

*=*

_{a}*S*

_{00},

*S*

_{0}(Ω)|

_{|}

_{Ω}

_{|>}

*= 0. In this case from Eq. (4) we obtain*

_{a}*σ*

^{2}= −2

*β/g*.

From Eq. (5) we can see how the tail of the 2D X-type wave packet is in this case decaying as
$A\left(\xi ,0\right)~\frac{1}{\xi}$. The numerical integration of Eq. (4) also revealed that in the case of a Gaussian spectrum *S*_{0}(Ω) (with the same width as the rectangular spectrum considered above), the tails decay occurs even faster. Note however, that in both cases illustrated in Fig. 1(a) and Fig. 1(b), the infinite energy of the theoretical solution supports the non-decaying tails in the *time-integrated* intensity profiles, shown just below the space-time profiles.

The important role of polychromaticity in the 1D spatial localization is confirmed by the result of Fig. 1(c), obtained for a WP angular dispersion identical to that of Fig. 1(b), but with a temporal bandwidth 10 time smaller. In that case an oscillating spatial profile (due to the cosine function in Eq.(4)) appears, and the spatial localization vanishes. Figure 1(d) shows the effect of the angular dispersion line (*k _{x}* = Δ(Ω)) ”thickness” which acts as a beam apodization effect, thus leading to a time-integrated profile with decaying tails (in this case the solution is energy limited and thus quasi-stationary). A suitable tailoring of the spatio-temporal spectral properties of a 2D WP may therefore lead to a quasi-stationary beam highly localized in one spatial dimension, potentially interesting for many microfabrication applications. Indeed in this context a good high contrast spatial shape of the time-integrated profile is the essential feature needed, for an effective radiation absorption close to the beam core where the pulse duration is shorter, and thus where the intensity, the multi-photon absorption and the energy transfer to matter are larger.

## 3. Experiment

The generation and study of physical, energy limited, linear 1D spatially localized WPs which could propagate quasi-stationarily in media with normal dispersion have been carried out by using an experimental set-up in a folded geometry analogous to that used in [21]. The strategy to produce the wanted beam consists in generating its far field by positioning a reflective mask in a pulse shaper designed to split the temporal and spatial frequencies of a short laser pulse. Here our mask was created by applying on a flat metallic mirror a black paper where the spectral shape of a typical normal dispersion curve [14,20] (two hyperbolic branches in the *S*(Ω, *k _{x}*)- or equivalently

*S*(

*λ*,

*θ*)- space), was engraved (this mask is different from the ring-shaped mask of [21] necessary for the generation of the STBB). The laser source was a Ti:Sapphire laser (Amplitude Technologies Ltd.) emitting 40 fs pulses at 20 Hz repetition rate, with a spectrum of 30 nm width centered around 800 nm. A portion of the input beam (gaussian beam, 4mm full width at half maximum) was selected by means of a vertical rectangular slit with 2

_{x}*mm*width along the horizontal direction. As shown in Fig. 2, the temporal inverse FT of the signal first dispersed by a grating and then reflected back by the mask was performed along the horizontal axis by means of a first cylindrical lens (in our case of focal length 100 mm). A second cylindrical lens of focal length 300 mm, was then used to perform the inverse FT in the spatial domain (along the vertical axis). The dispersion of our system was minimized by adjusting the distance between the grating and the first cylindrical lens, and by slightly pre-chirping the laser input pulse. Our principal aim here was the spatial characterization of the resulting beam observed in the Fourier plane of the second cylindrical lens. This has been done by recording the temporally integrated spatial profile of the beams, featured by different spectral supports, by means of a 14 bit CCD camera (Sony ICX205AL, WincamD).

In Figs. 3(a), 3(b) and 3(c), we report the spatial beam images (and in Figs. 3(d), 3(e) and 3(f), the corresponding transverse profiles along *x* taken at the beam center) of the generated WPs recorded in air for three different values of the gap between the two branches of the spectrum. We observe, in accordance with theoretical predictions, that when the gap increases, the interference process of the different spatio-temporal bandwidth portions, leads to an increase of the number of lateral decaying fringes (similar to Fig. 1c). The angular gap has an analogy with the cone angle of the standard 2D spatial Bessel beam. Indeed the bigger the angle, and the smaller the core size of the beam. In our case this is reflected in the transverse size of the fringes characterizing the 1D ”Bessel-like” beam generated, that decreases when the spectral angular gap increases. On the other hand by increasing the thickness of the spectral branches, we have observed a better 1D localization in space (along the x axis) with the corresponding quenching of the lateral tails. Indeed a broader spatial spectral support in the transverse wave vector domain naturally corresponds to a stronger apodization of the resulting WP in the real space domain (see Fig. 4, Fig. 1(d)).

Figure 4 shows the recorded spatial images of a WP generated by means of a spectral mask having an angular gap of about 0,008 rad and characterized by very thick branches. The sub-diffractive nature of the 1D spatial beam, which has a central peak dimension of 50 *μm* (FWHM) has been verified. In Fig. 4(a) we have reported the spatial evolution of the beam in air along the propagation direction *z*, showing a quasi stationary behaviour of the central blade (constant size) over a length of at least 20 mm (5 times the Rayleigh range of a Gaussian beam of similar dimensions). Moreover the ratio between the central peak intensity and the intensity of the lateral lobes of the transverse intensity profile shown in Fig. 4(b) remains constant along 15 mm of propagation (data not shown). Note that while the limitations in the dimension and maximum intensity of the blade-like beam are given by the input beam apodization and the energy for what concerns the major section, the limitations in the minimum dimension (perpendicular direction) and in the duration are dictated by the spatio-temporal spectral bandwidth support. At z=50mm the far field structure of the 1D beam appears. The generated WP has also been launched in rectangular slabs of glass with different lengths, allowing us to check its spatial localization for distances up to 30–40mm. Note that cross-correlation measurements performed by mixing in a second order non linear crystal the 1D-like spatial beam with a portion of the laser beam have shown that the pulses generated with our set up are featured by a total duration of 350 fs full width at half maximum.

The results reported until here are related to the use of a limited central portion of the typical spatio-temporal spectral support of stationary wave modes in normal GVD material, thus actually excluding the extended angular dispersion part. The goal of a second experiment was to show the role of the polychromaticity of the spectral bandwidth extended along the angularly dispersed branches, on the spatial localization of the generated WP. Because of our limited bandwidth laser source we implemented this test by using a ”rescaled” mask that would reflect our 30nm bandwidth pulse, through an arbitrary spectral support featured by an ”exaggerated” angular dispersion. In that case we focused our attention on the localization properties of the resulting beam, forgetting for a while about the real stationarity conditions. Clearly the process is rescalable, and this proof of principle experiment can be applied to fs pulses (100nm bandwidth of more), with the correct spatiotemporal spectral curve describing the far-field of a stationary wave in a given normally dispersive material. Note that for this experiment, the mask placed in the Fourier plane of the cylindrical lens after the grating, was featured by spectral branches slightly less than 1mm thick and by a gap angle of 0,01 rad. Although the spatial quality of the resulting beam was poorer than in the previous cases (also due to irregularities presented in the hand made mask), the mask used has allowed us to highlight the effect of the bandwidth enlargement of the WP spectral support.

The results presented in Fig. 5 illustrate the effect of the temporal spectral selection on the 1D localization of the Bessel-like WP. In the left column are reported the spatio-temporal spectral masks used in the pulse shaper, with the red lines highlighting the effectively used internal portions of the mask (the rest being covered with black paper). As expected, the spatial localization and the quenching of the lateral decaying tails get enhanced when a broader temporal spectrum is considered.

## 4. Conclusion

To conclude, we have shown that a 2D (1D spatial and 1D temporal) stationary WP in a linear dispersive medium can also be sharply localized (in both dimensions x and t) provided that its spectrum is featured by spatial and temporal coordinates entangled via angular dispersion. The results presented support the concept that at least two dimensions (where space and time are equivalent) are needed in order to have simultaneous stationarity and localization of a WP (in analogy for instance with the 2D spatial monochromatic Bessel beam). We have presented the 2D spatio-temporal X-like stationary WP solution of the wave equation in normally dispersive media, showing the effect of its spectral features on its spatio-temporal pattern, generated thanks to the constructive interference of the various angularly dispersed spectral wave components. Experimentally we have focused on the 1D spatial localization features of the time-integrated and apodized beams, and we have shown that 1D Bessel-like spatial beams, which may be stationary in suitable transparent normally dispersive media, can be linearly generated by using the appropriate hyperbolic spatio-temporal spectrum. Moreover we have highlighted the effect of the polychromaticity and of the angular dispersion on the transverse spatial localization of the high ellipticity time integrated beam, by using in the beam shaping experiment a spectral WP support with a strong angular dispersion applied to our limited bandwidth. Also note that our results are in agreement with those of interference fringe measurements, and show the expected fringes for the time integrated profiles of the 2D X-wave WPs.

The results of the proof of principle experiments presented here indicate that by opportunely tailoring the spatio-temporal spectrum of the stationary wave mode in a given material, it should be possible to generate suitable 1D-spatially localized quasi-stationary pulsed beams that (after suitable beam demagnification and imaging inside the material to avoid instabilities at the air/material interface [11]) may also find applications in microfabrication experiments. Note that in order to demonstrate stationarity, full space-time measurements in propagation through a dispersive medium would be required (and are not shown in this work). Finally, a complete study of the localization properties of 2D stationary wave packets in different dispersion regimes is in progress [20].

## Acknowledgments

This work was supported by Cariplo foundation and Regione Lombardia. The authors thank D. Faccio and A. Parola for helpful discussions.

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