We describe a family of dispersion-free and diffraction-free optical beams consisting in two-dimensional wave packets with a spatiotemporal Bessel (STB) profile propagating in media with anomalous dispersion. We also describe quasi-invariant optical beams with a spatiotemporal Bessel-Gauss (STBG) profile; these wave packets have finite dimensions and energy, conditions to be representative of physical beams. The paper provides a detailed account of the properties of STB and STBG beams, including their spatially resolved frequency spectrum, their far-field behaviour and a comparison of the propagation of STBG beams with that of Gaussian wave packets. An experimental setup based on a folded pulse shaper has allowed to generate STBG beams using the ultrashort pulses from a Ti:sapphire laser. The analysis of the spatially resolved frequency spectrum and of the spatial and temporal profiles obtained experimentally shows good agreement with theory.
©2009 Optical Society of America
Diffraction-free and dispersion-free optical beams or wave packets are special cases of optical wave propagation that have caught general attention in the last decades. The generation of beams with spatial or temporal invariance requires a thorough knowledge of the physical laws governing their propagation in a given medium. Temporal solitons were the first class of optical signals to be recognized as being invariant under propagation  and to be experimentally demonstrated . Solitons require a nonlinearity to preserve their shape: in the case of temporal solitons, the effects of nonlinearity and dispersion cancel each other while, for spatial solitons, nonlinearity compensates for diffraction. It was pointed out by Durnin that nonlinearity was not a requirement for invariant propagation; indeed Bessel beams can propagate in a purely linear material without deformation .
The diffraction-free character of ideal Bessel beams has triggered immense interest. It was soon realized that, due to practical considerations, only quasi-Bessel beams could be produced [3,4]. Gori et al. derived the complete solution describing the evolution of transverse Bessel-Gauss beams during propagation . These monochromatic solutions to the wave equation have been generalized to polychromatic fields; for instance, pulsed Bessel and Bessel-Gauss beams [6–9] and Bessel-X waves [10–12] have caught a lot of attention since they lead to interesting spatiotemporal coupling phenomena. Other diffraction-free beams have also been investigated, such as diffraction-free optical bottles , periodically focused propagation-invariant beams  and helicon waves , to name a few.
In recent papers, Longhi  and Porras and Trapani  have introduced a family of invariant three-dimensional wave packets propagating in media with anomalous dispersion that they have called “envelope O waves” or “O-waves”, in reference to their spatiotemporal concentric structure. The amplitude of these “O-waves” along their spatiotemporal radius is modulated by a sinc function that induces a structure with multiple concentric shells.
In this paper, we explore in detail the properties of a family of two-dimensional (x and t) optical wave packets that we call “spatiotemporal Bessel beams” (STB beams) and that we have introduced in Ref . These wave packets are characterized by a spatiotemporal profile that corresponds to a Bessel function of first kind of order m. This set of solutions to the wave equation is neither subject to dispersion nor diffraction and it is restricted to optical materials with anomalous dispersion. However, unlike solitons, the propagation of STB beams takes place in a linear optical medium.
The fundamental solution (m = 0) of STB beams is similar to the localized wave packets described by Malaguti et al.  for the case of anomalous dispersion. In practice, it would be impossible to generate a pure STB beam because of its infinite dimensions and energy. The introduction of a Gaussian envelope in the Bessel-type solution allows to circumvent this difficulty. The resulting beam will be called a “spatiotemporal Bessel-Gauss” beam, or STBG beam. The Gaussian envelope induces a z-dependent evolution of beam parameters which can be described analytically. The spread of an STBG beam during propagation is found to be much lower than that of a spatiotemporal Gaussian wave packet matching the central lobe of the STBG beam. We have experimentally demonstrated how to synthesize STBG beams through spatiotemporal reshaping of the ultrashort pulses emitted by a Ti:sapphire laser. The folded pulse shaper developed for that purpose can be viewed as a spatiotemporal axicon. Experimental measurements of the spatial and temporal profiles, along with the analysis of the spatially resolved spectrum, show good agreement with theory.
This paper is organized as follow. In Section 2, we present the STB beam as a solution to the two-dimensional wave equation in media with anomalous dispersion. In Section 3, we introduce STBG beams which are physical solutions with finite energy due to their Gaussian envelope. The propagation of STBG beams is described in Section 4. In Section 5 we examine the spatially resolved spectrum of STB beams; propagation invariance is shown to result from a proper selection of the plane wave spectrum constituting this type of optical wave packet. The pulse shaper that we have designed to produce STBG beams is described in Section 6. In Section 7, we present our experimental results showing that optical wave packets with a spatiotemporal Bessel profile have been generated using the pulse shaper.
2. Spatiotemporal Bessel beams
We consider an optical field defined by the following expression20]:
The operator is the transverse Laplacian, defined as . The wavenumber is with being the refractive index and the speed of light in vacuo. For a relatively narrow spectrum, can be expanded as a Taylor series around with its frequency dependent component defined as:Eq. (2) since << , where and is the index of refraction at . The wave equation in a reference frame moving at the pulse group velocity is obtained by defining a new time coordinate . By limiting the expansion of to second order in and assuming that the field has no dependency in y, an inverse Fourier transform of Eq. (2) leads to the following wave equation in the reference frame travelling with the pulse at the group velocity
A class of solutions exhibiting propagation invariance can be obtained from Eq. (4) by introducing the Laplacian operator in polar coordinates:
This operator is expressed in terms of the spatiotemporal radial and angular coordinates defined as and , respectively. In order that ρ be real for all combinations of x and T, parameter has to be negative; in other words, this treatment is valid for materials with anomalous dispersion. In terms of coordinates ρ and θ, Eq. (4) becomes:
The last equation can be solved by separation of variables with , as outlined at page 1259 of ref . To obtain a general solution to that equation, we assume the angular dependency to have the form , m being the integer associated to a particular azimuthal order and the corresponding amplitude. One can also obtain with , where a is a constant introduced for the separation of variables. The radial function is , where is the well-known Bessel function of first kind of order m. A solution to Eq. (4) is then given by:Equation (7) is the general expression of a spatiotemporal Bessel (STB) beam of order m. Except for the factor representing the carrier, this expression is clearly independent of z in the reference frame moving at group velocity . Therefore, during propagation, the spatiotemporal profile of an STB beam remains unchanged.
Figure 1 shows the intensity pattern of an STB beam of order m = 0 along normalized axes and . This beam consists in a series of concentric annuli with a high intensity central lobe that propagates without deformation along the z-axis, as shown in the video of Fig. 2 . The total width of the central lobe along the x-axis and its duration along the T-axis, both taken at the first zero of Bessel function (as shown in Fig. 3 ), are respectively given by
In the case of a medium with positive GVD (i.e. normal dispersion), ρ would become imaginary for certain values of x and T, leading to a solution involving a modified Bessel function of first kind with an asymptotic divergence.
The spatiotemporal frequency content of the STB beam in the polar coordinate system is given by the Hankel transform of . Using , one finds , where is the Dirac delta function. This result means that the STB beam is formed by a set of plane waves whose spectrum forms a ring of infinitesimal thickness in the polar coordinate system. From the definition of the spatiotemporal coordinate ρ one finds that , where is the spatiotemporal frequency in the polar coordinate system and is the projection of along the x-axis.
3. Spatiotemporal Bessel-Gauss beams
The spatiotemporal Bessel beam defined by Eq. (7) has infinite extent along x and t (or z) and, as such, it would carry infinite power . To circumvent this problem, one can limit the extent of the beam by introducing a spatiotemporal Gaussian envelope in the expression that defines the field of an STB beam . The resulting wave packet, named spatiotemporal Bessel-Gauss beam (or STBG beam), can be expressed as follows in the plane of beam waist:Fig. 3.
The solution describing an STBG beam of arbitrary order m at any position z can be obtained by the following Huygens-Fresnel integral in cylindrical coordinates:
The temporal size of the Gaussian envelope can be obtained from as:
4. Propagation of STBG beams
Equation (12) describes the evolution of STBG beams as they propagate in media with anomalous dispersion. A typical example of STBG beam propagation is shown in Fig. 4 , where the movie reveals two different beam structures: 1) co-propagating concentric annuli near the beam waist (near field) and 2) a single diverging ring far from the waist (far field).
4.1 Near-field distribution
Figure 5(a) shows the evolution of the relative intensity distribution of an STBG beam (order m = 0) near its waist during propagation in fused silica. The envelope of the signal is centered at 1550 nm and its spectral width is 40 nm. For the purpose of comparison, the evolution of a spatiotemporal Gaussian wave packet with a beam size matched to the central lobe of the STBG beam is shown in Fig. 5(b). The size is obtained by taking the width of the central lobe at 1/e 2 of its normalized intensity as it is defined for a Gaussian beam. The relation between the size of the central lobe of the STBG beam and beam parameter a is . The Rayleigh distance zc associated to the beam size is defined as . For the example shown in Fig. 5(a), the size wo of the Gaussian envelope has been set to 10.
Near the beam waist, the Bessel profile is the dominant contribution of the STBG beam distribution. Figure 5(a) shows that the basic profile of the STBG beam is maintained over a long distance (15 zc), even if the intensity of the central lobe is in fact reduced by 98% (see Fig. 5 (c)). For the same distance, the residual intensity at center of the Gaussian wave packet is even lower (by a factor of 6). Energy redistribution from the inner structure to an outer ring gradually takes place during propagation, as seen in Fig. 5(a). The Gaussian envelope has a direct impact on the profile of the propagated beam; a broader envelope will produce quasi-invariant propagation over a longer distance than a narrower envelope. Figure 5(c) shows the intensity at center as a function of propagation distance; the intensity at center of the STBG beam decreases by 50% at a distance exceeding 6 zc, e.g. 6 times longer than for the Gaussian wave packet. The ratio can be expressed as
4.2 Far-field distribution
Far away from its waist, the STBG beam transforms into a spatiotemporal ring whose radius grows linearly as a function of distance z. This annulus can be interpreted as being due to the spatial and temporal separation of multiple pulsed beams travelling at different angles with respect to the z-axis. As these pulsed beams propagate toward the waist, they spatially and temporally converge to a central point moving at group velocity ; when they overlap, interference occurs, creating a circular pattern with the shape of a Bessel function. Once the circular structure appears near the waist, the dimension of the concentric annuli remains invariant. The visibility of the concentric annuli forming the Bessel profile increases closer to the waist until a maximum is reached at z = 0. Beyond the waist, the combined effect of divergence and dispersion has the consequence that the original pulsed beams separate spatially and temporally, resulting in the gradual fading of the circular beam structure which evolves into a ring distribution in the far field. In the far field the beam pattern reduces to a spreading and fading spatiotemporal donut, whose intensity distribution is asymptotically given by:
Equations (15-18) are valid only for z >> zc. Equation (15) indicates that the Bessel structure has vanished in the far field where the only remaining feature is an annular Gaussian profile having an increasing radial offset and a decreasing intensity maximum as it propagates along z. A typical far-field profile is shown in Fig. 6 . For a Gaussian envelope of very large size, the far-field envelope becomes extremely thin; this is consistent with the fact that the Hankel transform of an STB beam is an annular Dirac delta function, as discussed in Section 2.
The donut beam profile in the far field produces a sequence of two pulses along the z-axis. The two pulses are separated by a maximum delay at transverse position x = 0 ; that delay decreases for increasing values of |x| to produce a single pulse at the edges of the donut. Equation (17) gives the spatiotemporal radial position of maximum intensity in the far field; one then finds directly the delay between the front and rear short pulses:
It should be noted that the offset is independent of the size of the Gaussian envelope at the beam waist. We can see from Eq. (17) that the transverse offset increases as , leading to the following relation defining the divergence angle δ along the propagation axis:5]. It points out that √a can be viewed as the spatiotemporal frequency of the STB beam in the coordinate system (), as discussed in Section 2.
5. Spatially resolved spectrum of STB beams
In order to generate propagation invariant two-dimensional wave packets, it is important to know their spectrum of optical frequencies as a function of lateral position x. For the STB beam this information is obtained by a temporal Fourier transform of expression (7). Putting m = 0 and using Eq. (6).677.3) of ref , one finds the following analytical expressions defining the spatially resolved spectrum of plane waves constituting the STB beam:Fig. 7 , the spectrum of optical frequencies needed to generate an STB beam is limited by two cutoffs. This is due to the condition that must be real for propagating wave packets. From Eq. (22) one finds that the boundaries of the spectral distribution are given byFig. 8 ). Equation (27) can be used to rewrite Eqs. (8-9) in terms of physical parameters only, showing that the scale of the temporal variation of the Bessel function at the beam waist depends only upon the spectral half-width according toEq. (8), a spatial size 1500 μm for a spectrum of 10-nm half-width centered at 1550 nm.Eq. (22) establishes that the range of values of is limited by
Wave vector components outside these boundaries will induce distortions to the ideal Bessel profile. It is easily observed from (22) that if ; this means that, in order to verify the dispersion relation of the STB beam, its frequency components at center of the spectrum are propagating at the largest angle with respect to the z-axis. Conversely, its frequency components at the boundaries of the spectrum are propagating parallel to the z-axis. In order to generate an STB beam, one has to use a set of plane waves such that their frequency ω and the component of their wave vector along the x-axis verify Eq. (22).
Finally, it should be observed that Eq. (23) predicts that the group velocity at all frequencies ω is constant () along the z-axis. Propagation invariance can be viewed as the consequence of the fact that all waves interfering to produce the STB beam have the same group velocity along the z-axis.
6. Experimental setup
The annular intensity distribution presented in Fig. 6 is the spatiotemporal Fourier transform of an STBG beam. Diffraction has the effect of separating the spatial frequencies of the beam, and dispersion does the same with its optical frequencies. The resulting far-field distribution can be viewed as a mapping of optical frequencies as a function of transverse position x and of spatial frequencies as a function of time T. A strategy to produce an STBG beam would consist of generating its far field by positioning a reflective mask in a pulse shaper designed to split the optical and spatial frequencies of a short laser pulse; if the mask has an annular shape, it is then possible to obtain an STBG beam at its waist by doing the inverse Fourier transform in the space and time domains of the signal reflected by the mask.
We have designed a pulse shaper that performs the required Fourier transforms. The experimental setup is composed of a Ti:sapphire laser emitting femtosecond pulses with a spectrum of 10- to 30-nm width centered at 800 nm, a spatiotemporal pulse shaper (shown in Fig. 9 ), and beam/pulse diagnostics (autocorrelator, CCD camera, etc.). The pulse shaper is folded by the use of a reflective annular mask; temporal and spatial Fourier transforms are made sequentially. The primary goal of our experiments is to demonstrate a pulse shaping technique allowing to generate the proper field distribution at the waist of an STBG beam.
Figure 9 shows the path followed by an short laser pulse in the pulse shaper. First the optical frequencies of the laser pulse are spatially dispersed by the grating. After being collimated by cylindrical lens #1 of focal length f 1 = 100 mm, the spectrally dispersed beam is reflected by the circular mask that selects the proper optical frequencies along the horizontal axis. The beam is then focused by lens #1 on the grating; this set of operations generates the inverse Fourier transform in the time domain of the signal reflected by the mask. The beam propagates up to cylindrical lens #2 (f2 = 500 mm) which does the inverse Fourier transform in the spatial domain (i.e. along the vertical axis) of the signal reflected by the mask. The resulting STBG beam is obtained at the Fourier plane of lens #2. The focal length of the lens that generates the spatial Fourier transform must be matched to the half-width of the spectrum using Eq. (27):
The pulse shaper, based on a folded geometry, allows to control independently the dispersion and the diffraction acting on the propagated wave packet. Its global dispersion is set to zero by properly adjusting the distance between lens # 1 and the grating. Parameters have been chosen such that diffraction from the grating up to the observation plane has little effect on the structure of the packet along the T-axis.
7. Experimental results
We have characterized the STBG beam profile by monitoring it independently along the space and time axes. A homemade autocorrelator has been used to acquire its temporal profile, integrated along the transverse axis; its temporally integrated spatial profile has been acquired with a CCD camera placed at the beam waist. Furthermore, the spatiotemporal intensity distribution of the STBG beam has been analyzed in terms of its spatially resolved spectrum of optical frequencies, in view of the fact that its temporal (spectral) profile is different at every position x. By extracting this information, it is possible to compute its spatiotemporal intensity distribution by means of an inverse Fourier transform.
Figure 10 shows an autocorrelation trace obtained with an STBG beam produced with pulses having an 18-nm bandwidth. One important element to mention is that the two-photon photodiode employed in the autocorrelator integrates over the entire lateral x-axis, meaning that all the annuli of the spatiotemporal Bessel-Gauss profile are spatially integrated. The main consequence of this operation is that the autocorrelation signal has no zero. A very good agreement is found between the theoretical trace and the experimental measurements, as shown in Fig. 10. The theoretical trace was calculated from the exact expression of the spatially and temporally dependent autocorrelation of an STBG beam, followed by an integration along the spatial axis for all values of the temporal coordinate.
We have also recorded the spatial profile of the STBG beam by placing a CCD camera at its waist. This measurement produces a time-integrated spatial profile along the x and y axes (see Fig. 11(a) ). The two-dimensional picture shows that the beam profile has no structure along the y-axis (except for a Gaussian envelope), verifying our assumption of a planar geometry along the x-axis. Here again, the measurement does not lead to a Bessel-Gauss intensity distribution due to the temporal integration over a sequence of short pulses whose structure varies with position x; furthermore the theoretical intensity distribution does not fall to zero at any point along the x-axis, as shown in Fig. 11b. The experimental spatial profile shown in Fig. 11a is in very good agreement with the theoretical prediction. The intensity distribution at the center of the beam, shown in Fig. 11c, reveals that the minima and the peaks of the intensity distribution are located very closely to the corresponding positions expected from theory.
We have also investigated the spatially resolved spectrum of the STBG beam by placing a diffraction grating in the plane of its waist. The spatially resolved spectrum was recorded on a CCD camera. A typical result is shown in Fig. 12 . According to theory, the spectrum of optical frequencies of the STBG beam evolves along the x-axis; this is a consequence of the nature of the STBG beam which, at each position x, is made of a specific sequence of pulses in the time domain (see Figs. 3 and 4). The presence of the Gaussian envelope modifies the spatially resolved spectrum found for STB beams; the comparison of Figs. 7 and 12(a) shows that the Gaussian envelope softens the sharp discontinuities at the spectral boundaries and limits the distribution along the x-axis. The experimental spatially resolved spectrum, presented in Fig. 12(b), is in satisfactory agreement with the theoretical prediction shown in Fig. 12(a).
Figure 13 shows the spatiotemporal profile of the STBG beam retrieved from the measured spatially resolved spectrum and the assumption that it is unchirped. This assumption is justified by the fact that we used transform-limited pulses and a zero-dispersion pulse shaper. The obtained spatiotemporal distribution exhibits the characteristic circular profile expected from an STBG beam. Some distortion along the concentric annuli can be observed, caused by excess intensity along the space and time axes, and possibly some spatiotemporal astigmatism. Based on the spectral bandwidth found in Fig. 12(b), the central lobe of the STBG beam would last approximately 70 fs (at full width at half maximum).
We have described the properties of quasi-invariant, two-dimensional (x and t) optical wave packets propagating along the z-axis in materials with anomalous dispersion. In contrast to solitons, these wave packets propagate in linear optical media. The functional expression of these wave packets involves a Bessel function of first kind of order m and a Gaussian envelope that limits their size. In the limit case of a Gaussian envelope with infinite size, full invariance during propagation would take place; however such beams would have infinite extent and, in practice, it would be impossible to generate them. Quasi-invariant propagation results from the mutual compensation of diffraction and anomalous dispersion when beam parameters are properly selected. We have analyzed the spatially resolved frequency spectrum of spatiotemporal Bessel beams. This analysis has revealed that these beams are produced by a superposition of plane waves of different optical frequencies and directions of propagation traveling at the same group velocity along the z-axis; the plane waves produce an interference pattern that takes the shape of a space-time Bessel function. The Gaussian envelope used to limit the size of a spatiotemporal Bessel beam induces a variation of the beam profile during propagation; however this variation is much slower than that of a Gaussian wave packet matched to the central lobe of the STBG beam. The intensity of the central lobe of an STBG beam fades gradually to zero during propagation; the beam energy is redistributed from the Gaussian envelope to an outer ring whose radius grows linearly with propagation distance. In the far field, the beam profile takes the form of a diverging spatiotemporal donut.
The equations describing the propagation of STB and STBG beams involve only the x- and z-axes, thus assuming a constant field along the y-axis. In the case of propagation in a bulk dielectric with anomalous dispersion, these wave packets would then have a constant field along the y-axis; this would produce concentric cylinders of light propagating along the z-axis. STB and STBG beams could also be produced in guided-wave optics; they represent field distributions suitable for quasi-invariant propagation in planar waveguides with anomalous dispersion. In both cases, it would be interesting to investigate the effect on optical nonlinearities on their propagation and, in particular, their resistance to self-focusing and filamentation.
STBG beams have been synthesized experimentally with a femtosecond laser and a pulse shaper designed to provide their spatiotemporal frequency spectrum. The pulse shaper is effectively emulating a spatiotemporal axicon. It allows to match the spatial and temporal sizes of an STBG beam to the bandwidth of the pulse spectrum. The experimental measurement of the spatially-resolved spectrum of the so-produced wave packets has led to the spatiotemporal reconstruction of their intensity distribution. Future avenues being considered experimentally include the generation of STBG beams with an orbital angular momentum (i.e. ) and the shaping of three-dimensional O-waves. To verify the propagation invariance of STBG beams, we are considering two strategies: propagation at 800 nm in a dispersion line made of diffraction gratings providing anomalous dispersion, or direct propagation in a dielectric with anomalous dispersion, such as silica glass at 1550 nm.
This work was supported by grants from Natural Sciences and Engineering Research Council of Canada (NSERC), Fonds québécois de la recherche sur la nature et les technologies (FQRNT) and Institut canadien pour les innovations en photonique (ICIP/CIPI).
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