Abstract
Controlling the propagation of intense optical wavepackets in transparent media is not a trivial task. During propagation, low and highorder nonlinear effects, including the Kerr effect, multiphoton absorption and ionization, lead to an uncontrolled complex reshaping of the optical wavepacket that involves pulse splitting, refocusing cycles in space and significant variations of the focus. Here we demonstrate both numerically and experimentally that intense, abruptly autofocusing beams in the form of accelerating ringAiry beams are able to reshape into nonlinear intense lightbullet wavepackets propagating over extended distances, while their positioning in space is extremely well defined. These unique wavepackets can offer significant advantages in numerous fields such as the generation of high harmonics and attosecond physics or the precise microengineering of materials.
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Introduction
Finiteenergy Airy beams were first introduced in the field of optics in 2007 (refs 1, 2) as new nondiffracting wavepackets in one and two spatial dimensions. The Airy function, describing the spatial electric field profile of these beams, is the only nonspreading onedimensional (1D) solution of the Schrödinger equation discovered in the field of quantum mechanics in 1976 (ref. 3). The most notable feature of the linear Airy beam is the transverse acceleration, which causes the beam to propagate in a parabolic trajectory^{4}, while keeping a constant central lobe diameter. This exotic behaviour is forced by a strong spatial cubic phase which is imprinted on the beam profile. This spatial chirp causes a constant energy flux from the oscillating tail of the beam towards the main lobe, effectively forcing the beam to accelerate in the transverse direction. In addition, the Airy beam proves very resilient to perturbations, as it is able to linearly reconstruct (selfheal) itself^{5}. Further research activity on Airy beams in the nonlinear regime has revealed stationary nonlinear 1D solutions^{6}, (3+1)D nonlinear light bullets^{7}, solitons in selfdefocusing media^{8} and other newly observed spatial mechanisms^{9,10}. Recently, the Airy wavepacket concept has also been demonstrated for electron packets^{11}.
The cylindrically symmetric Airy beam (or ringAiry beam), which is able to abruptly autofocus in the linear regime, was introduced theoretically^{12} and was demonstrated experimentally^{13} to be able to deliver highenergy pulses inside thick transparent samples without damaging the material before the focus. The abrupt intensity increase at the focus of the ringAiry makes it easy to combine a long working distance with a small focal volume. The remarkable abilities of the ringAiry beam make it an ideal candidate for laser ablation applications in previously hardtoreach environments.
Here we demonstrate that the nonlinear dynamics of highpower ringAiry wavepackets are even more impressive. Using numerical simulations, we compare the attributes of highpower ringAiry wavepackets to those of Gaussian beams carrying the same power and identify a number of exciting features, which are very useful for practical applications^{14,15,16,17}. These include a minor nonlinear focal shift and the reshaping of the wavepacket into an intense nonlinear dynamic light bullet structure. In addition, we elucidate the physical origin of these interesting features by correlating them to the typical ringAiry spatiotemporal reshaping mechanism in the focus. Finally, our theoretical findings are confirmed by experiments in bulk fused silica.
Results
Nonlinear ringAiry beams
To explore the nonlinear propagation dynamics of ringAiry beams, we compare them with Gaussian beams. We consider two types of Gaussian beams that can be deemed equivalent to a given ringAiry beam^{13}, the equivalent envelope Gaussian beam (EEGB) defined as the Gaussian beam with fullwidth at half maximum (FWHM) equal to the diameter of the ringAiry beam and the equivalent peak contrast Gaussian beam (ECGB), which, in the linear regime, reaches the same peak intensity contrast value at the focus (more details can be found in the beam description of the Methods section). The term peak contrast refers^{12} to the ratio of the peak intensity in the beam focus to the peak intensity at the beam origin. In our simulations, both beams are focused at the focus of the ringAiry using a lens. Both of them carry the same power; thus, the EEGB and the ECGB only differ by their beam width and initial peak intensity. Regardless of the spatial profile of the beams used, the temporal profile in our simulations was in all cases Gaussian with a FWHM of 200 fs at a central wavelength of 800 nm. Additional information about the description of the simulated wavepackets and the numerical model used in the simulations can be found in the Methods section.
Focus shift in the nonlinear regime
We performed numerical experiments by increasing the input power for all three beams. As expected, the focus is shifted towards the laser source as the input power is increased. Figure 1 shows a comparison of the position of the nonlinear focus of the ringAiry beam with those obtained for the two Gaussian beams. Interestingly, in the nonlinear regime, the ringAiry beam focus shift is significantly smaller compared with the two Gaussian beams as the beam power increases. For the ringAiry, the focus position moves from 24 cm to roughly 22 cm, whereas the input power is gradually increased up to 10 P_{cr} (P_{cr} being the critical power for selffocusing, see Methods for exact definition). Further increase in power, up to 24 P_{cr}, seems to have a negligible effect on the focus position. For both Gaussian beams, the focus shift is much larger compared with the ringAiry case. Especially for the ECGB, the focus position is shifted down to 11 cm for 24 P_{cr}, which is more than half the initial distance. For the EEGB, the focus shift is not as strong, but still reaches 6 cm, that is, about three times more than for the ringAiry case. The difference between the two Gaussian beams is explained by the wider beam width for the EEGB and the fact that the nonlinear focus position is proportional to the Rayleigh length, that is, scales as (w_{0}^{2})^{18,19}. In the Discussion section below, we provide an analytical approach for calculating precisely the small focus shift of the ringAiry beams.
Spatial dynamics and filament formation
The peak intensity of the ringAiry beam is shown in Fig. 2a as a function of the propagation distance for five different input powers. As the input power is increased, the distribution of the peak intensity along the beam propagation changes significantly, showing an escalating abrupt increase near the focus. For example, at 5 P_{cr}, the first peak appears much more abruptly and the consecutive secondary peaks, although spatially separated in the linear regime, merge into an almost uniform filament sustaining high peak intensities over a longer propagation distance, compared with the linear case (black continuous line). As the power is gradually increased and nonlinearities come into play, the secondary peaks gradually merge and rise in intensity until the formation of a continuous intensity plateau at P_{in}>10 P_{cr}. At these powers, the peak intensity is almost constant ≅2.5 × 10^{13} W cm^{−2} along the plateau, without significantly increasing as the input power is further increased. This can be understood in terms of intensity clamping^{20} due to the presence of nonlinear losses, which are strong enough to limit the increase of the peak intensity. The resulting structure is a long filament similar to filaments obtained by propagation of highpower Gaussian beams^{18}, except that it results from the collapse of a ringAiry beam as detailed below. This is clearly demonstrated in Fig. 2b, where the peak intensity of a ringAiry beam at the plateau region (z=40 cm) is shown in comparison with two Gaussian beams as a function of the input power. The peak intensity for all beams is practically identical for input powers above 5 P_{cr}. A graphic representation of the transformation of a ringAiry beam, where the discrete sharp intensity peaks in the linear regime transform into a uniform filament in the nonlinear regime, is shown in Fig. 2c.
Spatiotemporal effects and light bullet formation
Besides the minor focal shift and the formation of a filament, interesting spatiotemporal dynamics take place beyond the focal region. Figure 3 depicts the intensity isosurfaces for an EEGB (first row), an ECGB (second row) and a ringAiry wavepacket (third row) for various propagation distances beyond the nonlinear focus.
Each of the three wavepackets carries ten critical powers. Because of the different nonlinear dynamics (Fig. 1), the actual focus position (denoted as 0 cm in Fig. 3) along the propagation axis is different for each beam.
The two Gaussian beams exhibit a typical severe spatiotemporal reshaping process extensively studied in the literature^{18,21}. As shown in Fig. 3, the initial Gaussian wavepacket is reshaped into a ring structure with an intense spot that is split in the temporal domain. We can clearly see that for both cases the space and time distributions of the wavepacket are constantly evolving in a complex way, as expected in filamentation of Gaussian beams. The dynamic reshaping action of nonlinear effects diminishes after ∼15 cm. After this point, both Gaussian wavepackets spread owing to linear diffraction and dispersion.
On the other hand, the spatiotemporal dynamics of the ringAiry beam are very different. The wavepacket has the form of a ring with an intense core in its centre, whereas no pulse splitting is obtained in the temporal domain. This nonlinear lightbullet structure propagates without significant changes over 15 cm. Only after 18 cm of propagation, the nonlinear dynamics start to weaken and the ringAiry wavepacket starts to spread due to the action of diffraction and dispersion. The annular structure that surrounds the ringAiry intense peak at 15 cm after the focus, although resembles a similar annular feature formed in the Gaussian wavepacket, is generated by different mechanisms. In the case of the nonlinear ringAiry beam, the ring is sustained by the combination of the preorganized energy flux from the tail to the main lobe of the Airy profile^{12,13} and light coming from the nonlinear focus, diffracted by the generated plasma. In the case of the Gaussian beam, the ring is generated by a spontaneous transformation of the Gaussian beam into a Bessellike beam induced by nonlinear effects^{22} and featured by conical emission^{23}. This type of nonlinear ring formation depends on the multiphoton order of the nonlinear absorption and is discussed in details in the case of twophoton absorption in ref. 24.
Experimental demonstration
To confirm the validity of the simulation findings, we performed nonlinear propagation experiments of ringAiry beams in fused silica. The use of fused silica is beneficial both in terms of the energy needed (P_{cr} is much lower than air) and also in the visualization of the nonlinear propagation by exploiting the accompanying twophoton red fluorescence emission^{25}. Detailed information about the experimental procedure and the analysis used to estimate the intensity profile of the ringAiry beam can be found in the Methods section. Typical nonlinear fluorescence emission images are shown in Fig. 4a. It is clearly visible that the nonlinear focus shifts as the input energy is increased. Furthermore, lowintensity secondary peaks, characteristic to the ringAiry, are emerging after the primary focus. The experimentally measured intensity distribution along the propagation axis I(0,z) for various ringAiry input energies is depicted in Fig. 4b.
As predicted by the theoretical simulations in the nonlinear propagation regime, the consecutive peak contrast is reduced as the beam energy is increased. At highenough energies, the intensity maxima are merged to a single intense filament along the propagation axis. This trend is verified in Fig. 4c by numerical simulations we performed in fused silica for the same input parameters as in the experiments. One may also note in both experiments and simulations the gradual decrease of the focus shift with increasing input energy as with the simulations in air above.
Discussion
An important question to answer for the purpose of predicting the position of the focus in a given experimental situation (as the one shown in Fig. 2a) is what is the mechanism leading to the small nonlinear focus shift for the ringAiry, and, even more importantly, can a scaling law be derived?
As the spatial profile of the ringAiry beam is significantly different from that of a typical Gaussian beam, the wellknown Marburger’s formula^{19} cannot be used to predict the collapse distance. The collapse of a ringAiry beam can be described in two stages: (i) a quasilinear stage, during which the ring structure shrinks over a propagation distance f_{min} and follows a parabolic trajectory (see Methods section, equation (4)); (ii) a second stage from f_{min} to the collapse distance f_{rAi}, during which the intensity of the ringAiry beam increases more abruptly and the assumption of a quasi1D trajectory for the peak intensity no longer holds. This happens when the primary ring reaches a sufficiently small diameter to induce significant mutual attraction between opposite parts of the ring, in analogy with the mutual attraction and merging of two individual filaments induced by cross Kerr effects when they are separated by less than approximately three typical filament diameters^{26}. Pursuing this analogy, we assume that in the case of ringAiry beams as well, nonlinear effects will take over when the diameter of the primary ring is ∼3 times that of the diameter of a typical filament in the medium.
Using equation (4) from the Methods section, we can estimate the propagation distance f_{min} from the condition R(f_{min})=3w_{α}/2:
where w_{α}≅90 μm is the typical filament diameter in air and R_{0}∼980 μm.
In our case, the ringAiry focus in the linear regime is f=23.95 cm. For the second stage, a simple empirical formula that fits nicely the simulation results for the nonlinear focus shift as a function of the ringAiry input power P_{in} is given by:
where P_{cr} refers to the critical power^{19} of a Gaussian beam with the same central wavelength and γ≡P_{Ring}/P_{in} is a scaling parameter referring to the fraction of power contained in the primary ring of the ringAiry distribution. The quantity f−f_{min} corresponds to the maximum shift that can be reached at very large powers, where the optical Kerr effect abruptly focuses on the primary ring. A comparison of the focus shift estimated by our numerical simulations and the prediction of the empirical formula of equation (2) is shown in Fig. 5a.
To confirm that equation (2) adequately predicts the nonlinear focus position for different ringAiry beams, we have performed simulations for a range of ringAiry beam parameters (different radii and widths). The predicted and simulated focus position for various ringAiry beams is shown in the inset of Fig. 5a. In all cases, equation (2) clearly predicts the focus position of the ringAiry beam. To further validate equation (2) we compare its prediction in Fig. 5b, against the measured focus shift as a function of the input energy for the experimentally generated ringAiry beam in fused silica (see Fig. 4a). Again, although the propagation medium is now different, the theoretical prediction using equation (2) clearly fits the experimental findings.
In conclusion, ringAiry wavepackets present unique features in the nonlinear regime. The nonlinear focus shift, as the beam power is increased, is much smaller for a ringAiry beam compared with equivalent Gaussian beams and can be precisely predicted with a simple semiempirical formula we derived considering the Kerrinduced ring collapse. Although, the most exciting feature is that beyond the focal region, the otherwise localized, in the linear regime, ringAiry wavepacket is reshaped into an intense nonlinear light bullet. Finally, we have also offered the first experimental demonstration of these predictions performing experiments in fused silica. The level of control achieved with these wavepackets offers unique advantages for applications that require regulated paths of high intensities, such as harmonic generation, laser micromachining and remote sensing.
Methods
Beam description
Three different laser beam profiles were used in the simulations, a ringAiry beam and two Gaussian beams. All simulations were done in air at atmospheric pressure. The ringAiry beam electric field^{13} is described by the following equation:
where Ai() corresponds to the Airy function^{3}, r is the radial coordinate, r_{0} represents the radius of the primary ring, α is a truncation constant and w is a scaling factor. More precisely, the primary ring of the beam has its peak intensity at a radius of R_{0}≡r_{0}–w·g(α), where g(α) denotes the first zero of the function Ai′(x)+α·Ai(x), whereas its FWHM is ∼2.28 w (ref. 13). The power content carried by the beam is numerically calculated by integration of the starting spatial intensity profile over the whole computational box (5 mm radius). The linear focus of the ringAiry beam can be analytically described by the beam parameters , by considering large ringAiry beams as quasi1D structures, that is, the trajectory of the intensity maximum of the ringAiry beam in the linear regime is given by the quadratic acceleration for the 1D Airy beams^{1,13}:
In our simulations, we used r_{0}=921 μm, w=61.4 μm and α=0.05, leading to g(α)∼−0.97. Using these parameters, the ringAiry beam is autofocusing at a position z=23.95 cm. The widths at 1/e^{2} radius of the EEGB and ECGB are 1,738 and 855 μm, respectively. Figure 6 shows the radial intensity distribution of the ringAiry and both Gaussian beams.
Numerical model
The numerical model^{18} that is used in the simulations resolves a nonlinear envelope equation governing the propagation along the z direction of the pulse envelope E(r,z,t) in cylindrical coordinates. It takes the form of a canonical unidirectional propagation equation in the generalized fewcycle envelope approximation^{21,27} and reads in the Fourier domain:
the first (linear) term on the righthand side includes the effects of diffraction, dispersion and space–time coupling: , where and the dispersive properties of the medium are described by a Sellmeier relation k(ω) for air^{28}, and denote the carrier wavenumber at the central frequency ω_{0}, the linear refractive index in air at ω_{0} and .
The nonlinear dispersion function describes the selfsteepening operator where the nonlinear polarization represents the optical Kerr effect, c is the speed of light in vacuum and ɛ_{0} is the vacuum permittivity. The derivation is explained in details in ref. 21. The nonlinear terms in equation (1) are all included in the nonlinear polarization envelope and current envelope , which are calculated in the space–time (τ,r,z) domain, where denotes time in the local pulse frame. The nonlinear polarization describes the optical Kerr effect:
where the coefficient for the Kerr nonlinearity in air n_{2}=3.2 × 10^{−19} cm^{2} W^{−1} leads to a critical power for selffocusing ∼3.2 GW.
The current describes plasma absorption and plasma defocusing in the framework of the Drude model, with collision time τ_{c}:
where ρ denotes the density of the plasma generated by multiphoton ionization and avalanche. The various parameters in equations (6) and (7) are the cross section for inverse Bremsstrahlung =5.5 × 10^{−20} cm^{2} calculated from the Drude model with the collision time τ_{c}=350 fs in air, the cross section for multiphoton absorption β_{K}=Kħω_{0}ρ_{nt}σ_{K}=4 × 10^{−95} cm^{13} W^{−7}, and σ_{K}=3.4 × 10^{−96} cm^{16} W^{−8} s^{−1} denotes the coefficient for multiphoton ionization with K=8 photons at 800 nm for ionizing oxygen molecules with an initial density ρ_{nt}=0.5 × 10^{18} cm^{−3}.
Equation (5) is coupled with an evolution of equation (8) for the electron density, which describes plasma generation by multiphoton ionization with rate W_{MPI}=σ_{K}I^{K} and avalanche:
U_{i}=12.03 eV denotes the ionization potential of oxygen molecules.
Experimental implementation
As shown in Fig. 7, we used 160 fs, 800 nm laser pulses, with a Gaussian spatiotemporal distribution, produced by a 50Hz Ti:Sapphire laser system. To generate the ringAiry beam, we followed a Fourier transform approach described in detail in ref. 13. In brief, the method is based on modulating the phase of a Gaussian beam (using a Hamamatsu LCOSX104682 phase reflecting only spatial light modulator), which is then Fourier transformed by a lens. The ringAiry distribution is then generated in the Fourier transform plane of the lens after blocking the zeroorder diffraction. To further reduce the effective focal length of the generated ringAiry beam, we reduced its transverse size by ∼17 times, using a 4f optical system composed by a 300mm lens and a × 8 (0.2 numerical aperture) microscope objective.
The red fluorescence emission, induced by the high intensity of the ringAiry beam in the focus region, was imaged using a compact microscope system (0.2 numerical aperture) and a linear chargecoupled device camera (12 bit, 1,224 × 968 pixels). Using the fluorescence emission images, we were able to estimate the intensity distribution of the propagating beam. More specifically, the observed fluorescence is actually an Abel transform of a radially symmetric fluorescence emission in the focal volume. By applying inverse Abel transform on the images, we estimate the radial distribution of the fluorescence emission F(r,z). The beam intensity I(r,z) distribution can be then extracted from F(r,z)I^{2}(r,z). Although this method does not provide an absolute estimation of the beam intensity, it can be used to accurately estimate its spatial distribution.
Additional information
How to cite this article: Panagiotopoulos, P. et al. Sharply autofocused ringAiry beams transforming into nonlinear intense light bullets. Nat. Commun. 4:2622 doi: 10.1038/ncomms3622 (2013).
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Acknowledgements
This work was supported by the THALES project ‘ANEMOS’ and Aristeia project ‘FTERA’ (grant number 2570), cofinanced by the European Union and Greek National Funds.
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All authors have contributed to the development and/or implementation of the concept. P.P. and A.C. performed the simulations, D.G.P. and S.T. designed and performed the experiments, and S.T. supervised the research. All authors contributed to the discussion of the results and to the writing of the manuscript.
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Panagiotopoulos, P., Papazoglou, D., Couairon, A. et al. Sharply autofocused ringAiry beams transforming into nonlinear intense light bullets. Nat Commun 4, 2622 (2013). https://doi.org/10.1038/ncomms3622
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DOI: https://doi.org/10.1038/ncomms3622
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