Abstract

We experimentally generate third-harmonic (TH) vortex beams in air by the filamentation of femtosecond pulses produced in a lab-built Ti:sapphire chirped pulse amplifier. The generated TH beam profile is shown to evolve with increasing pump energy. At a sufficiently high pump energy, we observe a conical TH emission of the fundamental vortex and confirm that the conical radiation follows the conservation law for orbital angular momentum. We further investigate the far-field angularly resolved spectra of the TH wave to analyze the conical emission angle. We theoretically verify that the formation of the conical TH vortex results from the phase-matching between the fundamental and TH waves during the filamentation process.

© 2016 Optical Society of America

1. Introduction

Over the past decade, third-harmonic (TH) generation during ultrashort laser filamentation has been intensively investigated owing to its promising applications in the remote detection of atmospheric pollutants [1–3], microwave channeling [4], and supercontinuum generation [1,5]. The most common scheme of TH generation is regarded as being collinear, indicating that all contributing light beams are oriented in the same direction to ensure the enhancement of the interaction length. Nevertheless, in a high-intensity ultrashort laser system, conical TH emission with a non-collinear geometry can also be realized [6–9]. Aközbek et al. presented the first observation and a rigorous theoretical analysis of conical TH emission during infrared (IR) pulse filamentation in air [6]. Numerous explanations have also been given for the mechanism of conical emission during the filamentation process [10–13]. So far, the presented works have predominantly been carried out using TEM00 Gaussian beams as the fundamental pulses since they can be produced by most ultrashort laser systems. In spite of this, scientists have also used light beams with more complex spatial and phase structures; a prominent example is an optical vortex [14].

Optical vortices, characterized by their inherent feature of a spiral phase dislocation, have been employed progressively in research on singular optics [15], light and matter interaction [16], and quantum optics [17]. The light beams exhibiting well-defined orbital angular momentum (OAM) [18] have gained considerable attention with the advent of various applications [19–21]. In the past few years, scientists have devoted much effort to embedding a phase dislocation into an ultrashort pulse [22–27] seeing the high nonlinearities such a pulse can access in materials. The development of ultrashort vortex pulses thus opens new possibilities for the generation of filamentation and supercontinuum in light with complex structures. Several works have successfully demonstrated the nonlinear phenomena in bulk Kerr media [28] and atmospheric air [29]. More recently, Sukhinin et al. performed an original theoretical work on the co-propagation of two-color filaments in air by utilizing the IR wavelength and its TH counterpart [30]. It is proposed that the existence of the TH vortex filament benefits the stabilization of the IR filament and provides a more suitable mode for propagation. This implies that two-color vortex filaments merit thorough investigation not only to understand their physics but also to discover novel applications employing their propagation characteristics. However, there have been few studies focusing on the generation of multicolor vortex filaments owing to their lack of accessibility.

In this work, we successfully observe TH vortex beams in air by exploiting a lab-built Ti:sapphire chirped pulse amplifier (CPA) to induce the filamentation of intense near-IR femtosecond pulses. At a low pump energy, we initially produce a TH vortex and manifest that its formation originates from the collinear scheme for TH generation. With increasing pump energy, a conical TH emission is generated that surrounds the existing vortex beam. We experimentally confirm that the conical TH emission is also an optical vortex that follows the conservation law for OAM. To obtain a deeper insight into the mechanism of TH radiation, we also demonstrate angularly resolved spectra at different pump energies. Analyzing the divergence angle using the angular spectra, we theoretically verify that the conical TH vortex is realized as a result of the phase-matching between the fundamental and TH waves via the filamentation process. It is expected that the presented work will provide some new insight into the generation of multicolor vortex filaments [30] and high-harmonic vortex beams [31].

2. Experimental setup and results

Figure 1 shows the experimental configuration, which consisted of two parts including a lab-built Ti:sapphire CPA laser system and the arrangement for the filamentation experiment. The Ti:sapphire CPA laser system was composed of two Ti:sapphire-based amplifiers and a 4-f vortex converter placed between them. The pulses emitted from a Ti:sapphire laser resonator were first stretched by a grating-based pulse stretcher and preamplified by a lab-built Ti:sapphire regenerative amplifier. After the first stage of amplification, the output obtained from the regenerative amplifier was 200 μJ with a repetition rate of 200 Hz. We further enlarged the beam radius to ~3 mm at the 1/e2 level to avoid damaging the spatial light modulator (SLM, Hamamatsu Photonics X10468-01) in the 4-f OV converter utilized to transform the pulses into vortex beams. We applied the brazed phase grating structure with a period of 200 μm to the SLM in order to maximize the diffraction efficiency. The position of the singular point in the grating was adjusted to the center of the input beam. We set an aperture near the Fourier plane of the concave mirror used to configure the 4-f OV converter to select the beam of the first order diffraction. The throughput of the 4-f OV converter was approximately 42 %. The output beam from the 4-f OV converter was collimated with a 4/5-telescope consisting of two convex lenses for the mode matching to the following 5-pass amplifier. We must carefully adjust the incident angle to the convex lens to compensate for the astigmatism in the output beam from the telescope. The pulse energy of the resultant vortex beam with a single-ring Laguerre-Gaussian (LG) mode [32] at this stage was ~75 μJ. The pulse energy of the vortex beam was amplified to 3.7 mJ in the 5-pass amplifier, and then the chirp was compensated by a pair of gratings. As a result of the two-stage amplification, we were able to generate an ultrashort vortex pulse with a topological charge equal to one having a central wavelength at 720 nm, a repetition rate of 200 Hz, a transform-limited pulse duration of τ = 60 fs (FWHM), and an energy of 1.5 mJ. The throughput of the compressor is relatively low due to the degradation of gold coating on the gratings with a long time use.

 figure: Fig. 1

Fig. 1 Upper: Schematic experimental setup. L1: focusing lens with focal length f=600 mm. L2: focusing lens with f=40 mm. M1: dielectric mirrors with high-reflectivity coating for wavelengths of 220–270 nm. Lower: Schematic setup for generation of intense vortex pulses.

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The generated near-IR vortex pulses were then focused in air to realize filamentation using a lens with 60 cm focal length (L1). Beyond the filament, the generated TH and fundamental waves were separated by utilizing dielectric mirrors (M1) with high reflectivity within the spectral range from 220 to 270 nm at an incident angle of 45 and high transmission for visible and near-IR light. Furthermore, we employed an aperture mounted on a translation stage to selectively transmit different transverse elements of the TH beam. The emission was then collimated into a calibrated spectrometer (Ocean Optics, HR2000) using a lens (L2) with a focal length of 4 cm to obtain the angularly resolved spectra.

Figure 2 shows the far-field patterns of the fundamental and TH beams measured at three different pump powers beyond the filament. The pump powers for Figs. 2(a)–2(c) are 3.13 GW, 7.05 GW, and 10.97 GW, respectively. We employed input powers below and above the critical power Pcrit() = {22 +1Γ( + 1)Γ( + 2)]/[2Γ(2 + 1)]}Pcrit(0) = 4.13 GW [33] required for self-focusing and filamentation, where = 1, Pcrit() is the critical power for a vortex beam of topological charge , and Pcrit(0) is the critical power for a TEM00 Gaussian beam in the same medium. The critical power of the TEM00 Gaussian beam is given by Pcrit(0)λ02/[4πn(ω0)n2], where λ0 = 720 nm, ω0 is the central angular frequency of the fundamental field, n(ω0) ≈ 1 in air, and n2 ≈ 4 × 10−19 cm2/W is the nonlinear refractive index in air [34]. The width of the vortex rings in Figs. 2(a)–2(c) decreases with increasing pump power. This results from the optical Kerr effect, which leads to the self-focusing of the fundamental beam. Furthermore, we evaluated the peak intensity, Ipk, of the fundamental wave associated with Fig. 2(a) at the geometrical focus and obtained Ipk =3.112 × 1013 W/cm2, which is sufficient to generate a TH wave in air [5,6]. Thus, in Figs. 2(d)–2(f) we present the TH counterparts corresponding to Figs. 2(a)–2(c). At a lower pump power below Pcrit(1), we observe only a single TH ring with half-angle divergence of 3 mrad, identical to that of the fundamental pattern in Fig. 2(a). By increasing the pump power to 7.05 GW, above Pcrit(1), another TH ring (conical emission) with a larger cone angle of ~8 mrad is generated that surrounds the previous ring. When the pump power is further increased to 10.97 GW, the inner ring can hardly be seen, whereas the width of the conical emission ring increases.

 figure: Fig. 2

Fig. 2 Far-field patterns. (a)–(c) Fundamental patterns of = 1 at pump powers of 3.13 GW, 7.05 GW, and 10.97 GW, respectively. (d)–(f) TH counterparts corresponding to (a)–(c).

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Previous research [31] has demonstrated that the high-harmonic generation of a vortex beam follows the conservation law for OAM, i.e., the topological charge of the TH wave here should be three times larger than that of the fundamental wave. To verify this interpretation, the topological charges of both the inner and outer rings should be carefully determined. For this purpose, we employed a cylindrical lens commonly used for the determination of topological charges for polychromatic singular beams [35]. Figures 3(a)–3(d) illustrate the results of theoretical simulations on the conversion of LG beams in Figs. 3(a) and 3(b) with two different beam divergences into their corresponding Fourier-transformed patterns in Figs. 3(c) and 3(d) via a cylindrical lens. The beam divergences in Figs. 3(a) and 3(b) were decided on the basis of the experimental results in Figs. 2(d) and 2(f). The focal pattern can be calculated by solving the paraxial equation for a collimated input LG beam [32] multiplied by the lens transmission function, T = exp[−ikvac(3ω0)y2/2fcyl] [35], where kvac(3ω0) = 3ω0/c is the free-space wave number and fcyl = 30 cm is the focal length of the cylindrical lens. From a previous study [35], it is known that the number of dark stripes is equal to the modulus of the vortex topological charge, namely, the theoretical results presented in Figs. 3(c) and 3(d) correspond to = 3 for the three dark stripes. The tilted direction of the stripes also implies that the OAM of the LG beams rotates in the clockwise direction [35]. On the other hand, the experimental focal patterns converted from the TH beams in Figs. 2(d) and 2(f) are displayed in Figs. 3(e) and 3(f), respectively. It is clearly demonstrated that the experimental and simulation results are in good agreement, which confirms that the generated TH rings are vortex beams with = 3. This suggests that both the inner emission and the conical emission are optical vortices that follow the conservation law for OAM. Nevertheless, further experimental investigation is desirable to determine the mode purity of the conical TH emission.

 figure: Fig. 3

Fig. 3 (a)–(d): Results of theoretical simulations for vortex beams converted via a cylindrical lens with a focal length of 30 cm. (a) and (b) LG beams of = 3 with different divergence angles. (c) and (d) Focal patterns for (a) and (b), respectively. (e) and (f) Experimental focal patterns transformed from Figs. 2(d) and 2(f), respectively.

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Using a calibrated photon detector, we further measured the conversion efficiencies of the total TH beam, the inner TH vortex ring, and the outer TH vortex ring (conical emission) for different pump energies, as depicted in Fig. 4(a). For comparison, in Fig. 4(b) we also demonstrate the energy ratios of the inner and outer vortex components to the total TH beam. For pump energies between 10 μJ and 200 μJ, over 60% of the total energy is attributed to the inner vortex ring, as shown in Fig. 4(b). With increasing pump energy, the ratios of the inner and outer rings to the total TH remain roughly constant between 230 μJ and 550 μJ, suggesting that filamentation occurs at a pump energy of approximately 230 μJ [6]. This closely corresponds to the estimated critical power of Pcrit(1) = 4.13 GW, which is equivalent to a pump energy of ~ 263 μJ for a Gaussian-shaped pulse with a duration of τ = 60 fs. When the pump energy is increased to above 600 μJ, the ratio of the outer ring to the total TH begins to rise again. For a pump energy of up to 1 mJ, the outer vortex ring contributes to approximately 80% of the total energy, while the contribution of the inner ring is ~20%.

 figure: Fig. 4

Fig. 4 (a) Measured energy for total, inner, and outer TH vortex beams as a function of the pump energy. (b) Ratios of the inner and outer TH vortex beams to the total TH energy as a function of the pump energy.

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To obtain a more profound physical understanding, we further measured the angularly resolved spectra of the TH waves generated at three different pump energies using the experimental setup shown in Fig. 1. Figures 5(a)–5(c) illustrate the angularly resolved spectra at pump energies of 455 μJ, 550 μJ, and 715 μJ, respectively. The evolution of the spectra closely corresponds to the transformation of the TH beam profiles in Fig. 2. The spectrum distribution of the inner ring is at a cone angle between 3 and 4 mrad. The conical emission has a divergence angle of ~8 mrad. In addition, the inner vortex ring displays clear spectral interference, which might be due to pulse splitting [36] or pulse steepening [7].

 figure: Fig. 5

Fig. 5 Angularly resolved spectra of TH waves measured at 410 mm beyond the central point of the filament at pump energies of (a) 455 μJ, (b) 550 μJ, and (c) 715 μJ (in a logarithmic scale and normalized by the peak intensity of the TH waves). Dashed lines were obtained by simulation using Eq. (10) at the clamped intensity of Ic = 8.34×1013 W/cm2.

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3. Theoretical analysis and discussion

To determine the reason for the occurrence of the TH radiation, here we discuss the phase-matching condition between the fundamental and TH fields by adopting the propagation equation for the slowly varying envelope (SVE) of the TH field, A3ω0(rT,z) [37],

{iz+T22k(3ω0)}A3ω0(rT,z)=3ω02n(3ω0)cχ3{Aω0(rT,z)}3ei{k(3ω0)3k(ω0)}z,
where the subscript, T, signifies the transverse component of vectors, T2=2/x2+2/y2 is the transverse Laplacian, rT is the transverse position, ω0 and 3ω0 are the central angular frequencies of the fundamental and TH fields, respectively, and χ3 is the third-order susceptibility of air. The wave numbers of the fundamental and TH fields, k(ω0) and k(3ω0), respectively, are given by k(ω0)= ω0 n(ω0)/c and k(3ω0)= 3ω0 n(3ω0)/c, where the refractive indices n(ω0) and n(3ω0) are composed of linear and plasma parts so that n(ω0)= nL(ω0) + nplasma(ω0) and n(3ω0)= nL(3ω0)+nplasma(3ω0). The linear part of the refractive index can be calculated from the dispersion formula in air of nL(ω) = 1 +η/[ξ − (ω/2πc)2] +η1/[ξ1 − (ω/2πc)2], where η = 0.05792105, ξ = 238.018, η1 = 0.00167917, ξ2 = 57.362, and the unit of ω is megahertz [38]. The refractive index of the plasma, nplasma(ω), is equal to (1ωplasma2/ω2)1/21ωplasma2/(2ω2)<1, in which the plasma frequency, ωplasma, is defined as ωplasma2qe2Ne/(ε0me) [37] (qe: electron charge; me: electron mass; Ne: electron density). Note that we have neglected the group delay difference between the fundamental and TH fields to obtain Eq. (1). The spatial field amplitudes of the fundamental and TH waves, Eω0(rT,z) and E3ω0(rT,z), are written as Aω0(rT,z)exp[ik(ω0)z] and A3ω0(rT,z)exp[ik(3ω0)z], respectively.

Multiplying ∫d2rT exp(−ikT · rT) to both sides of Eq. (1) and using the Helmholtz equation, (T2+kT2)A3ω0(rT,z)=0, we have the equation

izA˜3ω0(kT,z)=3ω02n(3ω0)cχ3d2rTeikTrT{Aω0(rT,z)}3ei{k(3ω0)3k(ω0)kT22k(3ω0)}z,
where A˜3ω0(kT,z) is defined as
A˜3ω0(kT,z)d2rTeikTrTA3ω0(rT,z)eikT22k(3ω0)z.
The Fourier amplitude, A˜3ω0(kT,z), is obtained by integrating the right-hand side of Eq. (2) with respect to z, and thus, the phase of this term should not rapidly change with z to significantly contribute to the z-integration. This is the phase-matching (wave-number matching) condition. Note that the wave-number difference in the brackets of the exponential term in the right-hand side of Eq. (2) is approximated as k(3ω0)3k(ω0)kT2/[2k(3ω0)](k2(3ω0)kT2)1/23k(ω0)=kz(3ω0)3k(ω0).

For an optical vortex beam, the SVE of the fundamental field, Aω0(rT,z), can be expressed as Aω0(rT,z)=|Aω0(rT,z)|exp(iϕ)exp{i[ϕr(ω0)+ϕGouy(ω0)+ϕKerr(ω0)]}. The amplitude |Aω0(rT,z)| of a single-ring LG mode is described by |Aω0(rT,z)|[w0/w(z)][r/w(z)]||exp[r2/w2(z)], and the phase terms are

ϕr(ω0)k(ω0)zr22(z2+zR2),
ϕGouy(ω0)(||+1)arctan(zzR),
ϕKerr(ω0)ω0cn2Ipkz,
where w0 is the beam waist of the fundamental field, w(z) = w0[1 + (z/zR)2]1/2 is the spot size at z, and the Rayleigh length, zR, is equal to k(ω0)w02/2. The radial coordinate r coincides with |rT|. Thus, the wave-number mismatch, ΔK(kT;3ω0,ω0), at z = 0 and r = w0 (||/2)1/2, where the intensity is maximized, is evaluated as
ΔK(kT;3ω0,ω0)=k(3ω0)3{k(ω0)+z[ϕr(ω0)+ϕGouy(ω0)+ϕKerr(ω0)]z=0}kT22k(3ω0)=3ω0cΔnkT22k(3ω0).
Here we define the index difference Δn as
Δn=[nL(3ω0)nL(ω0)]+n(ω0)k2(ω0)w02(||+2)n2Ipk+VNe,
where the effective volume of plasma is given by V(4/9)qe2/(ε0meω02).

The first term in Eq. (8)nL(3ω0) − nL(ω0), is the usual index mismatch originating from dispersion in the air. The second term originates from ϕr(ω0) +ϕGouy(ω0). The third term is due to the nonlinear phase shift of the fundamental beam with the Kerr effect, and the fourth term results from the change in the index caused by the plasma. The free electron density, Ne, nonlinearly increases with increasing peak intensity, Ipk.

Letting ΔK(kT;3ω0,ω0) = 0 for the phase-matching condition, we obtain

|kT|2Δnkvac(3ω0).
Here k(3ω0) has been approximated to kvac(3ω0). Therefore, the divergence angle of the TH radiation originating from the phase-matching condition, θ, is expressed as
θ=arctan(|kT|kvac(3ω0))arctan(2Δn).

To further elucidate the effect of the electron plasma on the index difference, in Fig. 6 we plot Ne and Δn as a function of the peak intensity, Ipk. The free electron density, Ne, in air is given by NO2PO2+NN2PN2, where NO2=0.21N0 and NN2=0.78N0 are the number densities of oxygen and nitrogen molecules, respectively, and N0 = 2.5 × 1019 atoms/cm3 is the number density of air at room temperature and atmospheric pressure. The ionization probability of the molecules, Px, is written as Px=1exp[Rx()dt], where x stands for O2 or N2 molecules. The ionization rate, Rx(), is a function of the field amplitude, , at z = 0 and r = w0(|l|/2)1/2. The field amplitude is given by (rT,z,t)=E(rT,z)D(t), where D(t) is the temporal profile of the pulse. To obtain Rx(), here we adopt the Perelomov, Popov, and Terent’ev (PPT) theory [39] (see Appendix A), which is valid for describing ionization processes ranging from multiphoton ionization to tunneling ionization. The contribution of the TH field to Ne can be neglected here since the conversion efficiency of the TH radiation is low, as shown in Fig. 4. Moreover, we have approximated the temporal profile, D(t), as a square pulse of full width τ = 60 fs. The parameters used here are = 1, w0 = 50μm, and n2 = 4 × 1019 cm2/W. The beam waist, w0, is obtained by measuring the beam profiles of the fundamental field at the geometrical focus.

 figure: Fig. 6

Fig. 6 Electron density, Ne, (solid line) and difference in the refractive index, Δn, (dashed line) as a function of the peak intensity, Ipk.

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In Fig. 6 it is shown that the index difference Δn initially decreases owing to the nonlinear Kerr effect corresponding to the term n2Ipk in Eq. (8), while it starts to increase as the plasma index, VNe, takes on a dominant role. At higher peak intensities of above 3 × 1014 W/cm2, the index difference eventually saturates owing to the saturation of Ne since almost all the target molecules are ionized in this regime. Here we define the clamped intensity, Ic, at which the dynamic between the self-focusing due to the Kerr effect and the defocusing of the plasma is balanced, namely, −n2 Ic +VNe = 0 [40]. As a result of the balance, it has been previously shown that the peak intensity can be clamped inside the filament with increasing pump power and that only the length of the filament is increased [6]. In Fig. 4 we have shown that the TH efficiency at pump energies between 230 μJ and 550 μJ does not increase with increasing input energy; this is expected since the intensity inside the filament is clamped to Ic. In our case, we obtain Ic = 8.34 × 1013 W/cm2, in good agreement with the value of 8 × 1013 W/cm2 reported by Kasparian et al. [40] and Gaarde et al. [41]. However, clamped intensity inside the filament is strongly dependent on the external focusing geometry [37,42,43] and the complicated temporal transformation such as pulse splitting [36] and pulse shortening [7]. Numerous experiments and simulations have thus shown values of clamped intensity ranging from few 1013 W/cm2 to 1015 W/cm2 [40–47]. More recently, a value of 1.45×1014 W/cm2, 2-3 times larger than the widely quoted intensity, has also been reported by direct measurement of the fluence and intensity inside the air filament generated with a loosely focused beam [48]. So far, the precise value of the clamped intensity in air filament is still a debated issue that deserves profound investigation. We expect our evaluations and observation could provide some insight into the determination of the clamped intensity in air filament.

Figure 7 depicts the divergence angle, θ, given by Eq. (10). θ first slightly decreases and then increases to θ ~ 8.3 mrad at the clamped intensity, Ic. For Ipk above Ic, θ exhibits rapid growth with increasing Ipk. Apparently, once the phase-matching condition is satisfied, we can generate a TH beam with the divergence angle exhibiting the same behavior as depicted in Fig. 7. In Fig. 5 we illustrate θ as a function of the central wavelength of the TH waves (dashed lines) at Ic, where filamentation occurs and the balance is attained. The dashed lines are shown to fit very well with the spectrum distribution of the conical TH emission in Fig. 5. This provides strong evidence that the occurrence of the conical TH vortex is due to the nonlinear phase-matching between the fundamental and TH fields. The conical TH vortex is verified to appear at Ic, at which the dynamic between the self-focusing and defocusing is balanced during the filamentation process. Most importantly, we inform from Eq. (1) that A3ω0(rT,z)exp(i3ϕ) by substituting the analytical expression for Aω0(rT,z) into the right-hand side. This confirms our experimental observation that the conical TH emission of the vortex beam exactly follows the conservation law for the topological charges.

 figure: Fig. 7

Fig. 7 Divergence angle, θ, as a function of the peak intensity, Ipk.

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However, the formation of the inner TH vortex does not result from the nonlinear phasematching mechanism. From Figs. 2(a) and 2(d), we find that the far-field profiles of the fundamental and inner TH waves have an identical divergence angle, which implies that k0) ║ k(3ω0). According to this relation, since for an LG beam, k0) has no magnitude of the transverse component at the beam waist, the transverse wave number, |kT|, of the inner TH ring should vanish at z = 0. For |kT| = 0, we find that ΔK(kT;3ω00) ≠ 0 from Eq. (7), which shows that the inner TH ring is not phase-matched. The inner TH ring can be understood by considering the “on-axis” TH component described in [13], in which the on-axis TH field appears to be locked with the fundamental that drives it, and thus the two components propagate together. This suggests that the spatial field amplitude of the inner TH ring is simply proportional to the cube of the fundamental field, namely, [Eω0(rT,z)]3.

In previous works, there are theoretical models that can well explain the conical emission from the filament generated by a Gaussian beam, such as the nonlinear X-Wave model [11, 12] and the Cerenkov emission model [10]. We note that these models are always employed to predict the conical emission near the fundamental wavelength region, which should be equivalent to the outer ring emission of the fundamental wave in our experimental condition using a vortex beam. Nevertheless, the far-field profiles of the fundamental wave do not reveal such an outer ring even when the peak power increases to 10.97 GW as shown in Fig. 2(c). Therefore, we do not adopt these theoretical models to analyze our experimental data.

In Fig. 8, we further demonstrate the divergence angle of various topological charges as a function of the peak intensity. The parameters used here are the same as previous simulations with w0 = 50μm and n2 = 4×10−19 cm2/W. From the relation of −n2 Ic +VNe = 0, we obtain the clamped intensity to be the value of 8.34 × 1013 W/cm2, independent of the topological charge. At Ic, in Fig. 8 it is evident that θ increases with increasing . This originates from the second term in Eq. (8), which is associated with the Gouy phase shift. For higher-order transverse modes, the Gouy phase shift is stronger. The presented results may provide some insight into the generation of conical emission for higher-order transverse modes.

 figure: Fig. 8

Fig. 8 Divergence angle, θ, of various topological charges, , as a function of the peak intensity, Ipk.

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4. Conclusion

We have experimentally generated TH vortex beams in air by performing the filamentation of the intense near-IR femtosecond vortex pulses with a topological charge of = 1. The TH beam profiles were shown to be composed of two concentric vortex rings. The inner TH vortex, associated with a lower pump threshold, was demonstrated to be locked to the fundamental field. By increasing the pump energy, we successfully generated a conical TH beam of the fundamental field. We used a cylindrical lens to demonstrate that both the inner vortex and the conical radiation conserve the OAM. The topological charge for the TH waves was verified to be ℓ= 3. Furthermore, we systematically measured the far-field angularly resolved spectra of the TH radiation at various pump energies to investigate the divergence angles of the TH beams. We adopted the propagation equation for the SVE of A3ω0(rT,z) to obtain the divergence angle, θ, originating from the phase-matching condition. The simulated result for θ showed good agreement with the measured cone angle of the conical TH vortex, which confirmed that the formation of the conical emission resulted from the nonlinear phase-matching mechanism. For a particular intensity Ipk, we found that increasing leads to larger divergence angles, θ, which stems from the stronger Gouy phase shift of the higher-order transverse modes.

Appendix A

The PPT model employed here is from Ref. [39]. In this model the ionization rate of a molecule in a linearly polarized laser beam of electric field amplitude can be expressed in atomic units as

R()=6π|Cn*,l*|2fl,mAm(γ)(201+γ2)2n*|m|32e203g(γ),
where
|Cn*,l*|2=22n*n*Γ(n*+l*+1)Γ(n*l*),
fl,m=(2l+1)!(l+|m|)!2|m|(|m|)!(l|m|)!,
Am(γ)=43π1|m|!γ21+γ2κ>v+e(κv)α(γ)Φm[(κv)β(γ)],
Φm(x)=ex20x(x2y2)|m|ey2dy,
α(γ)=2[sinh1(γ)γ1+γ2],
β(γ)=2γ1+γ2,
g(γ)=32γ[(1+12γ2)sinh1γ1+γ22γ].
We use the conventional notation for the gamma function, Γ, in Eq. (12). Here, 0(2Ui)3/2, γ=ω2Ui/ is the Keldysh parameter [39], Ui is the ionization potential of the molecule, and ω is the laser angular frequency. For γ ≫ 1 multiphoton ionization dominates the ionization process, while for γ ≪ 1 tunneling ionization plays a critical role. In addition, ν = (Ui)[1 + (2γ2)] and κ = 〈(Ui) + 1〉 + S, in which S = 0,1,… and the symbol 〈〉 indicates the integer part of the value inside. On the other hand, the effective principal quantum number is given by n*=Zeff/2Ui, with Zeff signifying the effective residual charge of the ion. Accordingly, the effective orbital quantum number is written as l* = n* −1, l is the orbital quantum number, and m is the magnetic quantum number.

The parameters used for the calculation of the ionization rate in the presented results were determined on the basis of Ref. [49], in which the quantum numbers are l = m = 0 for molecules O2 and N2 and the ionization potential is 12.55 eV for O2 and 15.58 eV for N2. The effective residual charge, Zeff, has been measured experimentally [49] to be 0.53 for O2 and 0.90 for N2.

Acknowledgments

We thank Dr. K. Isobe for providing the spatial light modulator. This work was financially supported by the Advanced Photon Science Alliance commissioned by MEXT and partly contributed to the objectives of CREST studies commissioned by JST. Y. N. and K. M. gratefully acknowledge the financial support from Grants-in-Aid for Scientific Research (A) No. 26247068 and Grants-in-Aid for Scientific Research (S) 26220606 from MEXT, Japan.

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2. P. Rairoux, H. Schillinger, S. Niedermeier, M. Rodriguez, F. Ronneberger, R. Sauerbrey, B. Stein, D. Waite, C. Wedekind, H. Wille, L. Wöste, and C. Ziener, “Remote sensing of the atmosphere using ultrashort laser pulses,” Appl. Phys. B 71, 573–580 (2000). [CrossRef]  

3. A. Ting, I. Alexeev, D. Gordon, E. Briscoe, J. Peñano, R. Hubbard, P. Sprangle, and G. Rubel, “Remote atmospheric breakdown for standoff detection by using an intense short laser pulse,” Appl. Opt. 44, 5315–5320 (2005). [CrossRef]   [PubMed]  

4. M. Chateauneuf, J. Dubois, S. Payeur, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92, 091104 (2008). [CrossRef]  

5. F. Théberge, W. Liu, Q. Luo, and S. L. Chin, “Ultrabroadband continuum generated in air (down to 230 nm) using ultrashort and intense laser pulses,” Appl. Phys. B 80, 221–225 (2005). [CrossRef]  

6. N. Aközbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. 89, 143901 (2002). [CrossRef]   [PubMed]  

7. H. Xiong, H. Xiong, H. Xu, Y. Fu, Y. Cheng, Z. Xu, and S. L. Chin, “Spectral evolution of angularly resolved third-order harmonic generation by infrared femtosecond laser-pulse filamentation in air,” Phys. Rev. A 77, 043802 (2008). [CrossRef]  

8. M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. 92, 253901 (2004). [CrossRef]   [PubMed]  

9. V. Vaičaitis, V. Jarutis, and D. Pentaris, “Conical third-harmonic generation in normally dispersive media,” Phys. Rev. Lett. 103, 103901 (2009). [CrossRef]  

10. E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysyrowicz, “Conical emission from self-guided femtosecond pulses in air,” Opt. Lett. 21, 62–64 (1996). [CrossRef]   [PubMed]  

11. D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation from spontaneous X–Wave formation in air,” Opt. Express 16, 1565–1570 (2008). [CrossRef]   [PubMed]  

12. D. Faccio, P. Di Trapani, S. Minardi, A. Bramati, F. Bragheri, C. Liberale, V. Degiorgio, A. Dubietis, and A. Matijosius, “Far-field spectral characterization of conical emission and filamentation in Kerr media,” J. Opt. Soc. Am. B 22, 862 (2005). [CrossRef]  

13. M. Kolesik, E. M. Wright, A. Becker, and J. V. Moloney, “Simulation of third-harmonic and supercontinuum generation for femtosecond pulses in air,” Appl. Phys. B 85, 531 (2006). [CrossRef]  

14. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974). [CrossRef]  

15. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]  

16. K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110, 143603 (2013). [CrossRef]   [PubMed]  

17. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001). [CrossRef]   [PubMed]  

18. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992). [CrossRef]   [PubMed]  

19. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012). [CrossRef]  

20. K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440, 935 (2006). [CrossRef]   [PubMed]  

21. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826 (1995). [CrossRef]   [PubMed]  

22. K. Yamane, Y. Toda, and R. Morita, “Ultrashort optical-vortex pulse generation in few-cycle regime,” Opt. Express 20, 18986 (2012). [CrossRef]   [PubMed]  

23. K. Bezuhanov, A. Dreishuh, G. G. Paulus, M. G. Schätzel, and H. Walther, “Vortices in femtosecond laser fields,” Opt. Lett. 29, 1942–1944 (2004). [CrossRef]   [PubMed]  

24. I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13, 7599–7607 (2005). [CrossRef]   [PubMed]  

25. I. Zeylikovich, H. I. Sztul, V. Kartazaev, T. Le, and R. R. Alfano, “Ultrashort Laguerre-Gaussian pulses with angular and group velocity dispersion compensation,” Opt. Lett. 32, 2025–2027 (2007). [CrossRef]   [PubMed]  

26. V. G. Shvedov, C. Hnatovsky, W. Krolikowski, and A. V. Rode, “Efficient beam converter for the generation of high-power femtosecond vortices,” Opt. Lett. 35, 2660–2662 (2010). [CrossRef]   [PubMed]  

27. M. Bock, S. K. Das, C. Fischer, M. Diehl, P. Börner, and R. Grunwald, “Reconfigurable wavefront sensor for ultrashort pulses,” Opt. Lett. 37, 1154–1156 (2012). [CrossRef]   [PubMed]  

28. D. N. Neshev, A. Dreischuh, G. Maleshkov, M. Samoc, and Y. S. Kivshar, “Supercontinuum generation with optical vortices,” Opt. Express 18, 18368–18373 (2010). [CrossRef]   [PubMed]  

29. P. Polynkin, C. Ament, and J. V. Moloney, “Self-focusing of ultraintense femtosecond optical vortices in air,” Phys. Rev. Lett. 111, 023901 (2013). [CrossRef]   [PubMed]  

30. A. Sukhinin, A. Aceves, J.-C. Diels, and L. Arissian, “On the co-existence of IR and UV optical filaments,” J. Phys. B: At. Mol. Opt. Phys. 48, 094021 (2015). [CrossRef]  

31. G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113, 153901 (2014). [CrossRef]   [PubMed]  

32. A. E. Siegman, Lasers (University Science, 1986).

33. V. I. Kruglova, Yu. A. Logvina, and V. M. Volkov, “The theory of spiral laser beams in nonlinear media,” J. Mod. Opt. 39, 2277 (1992). [CrossRef]  

34. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650 (1997). [CrossRef]  

35. V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17, 23374–23379 (2009). [CrossRef]  

36. A. Couairon, E. Gaižauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E 73, 016608 (2006). [CrossRef]  

37. S. L. Chin, Femtosecond Laser Filamentation (Springer, 2010). [CrossRef]  

38. P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Phys. Lett. 35, 1566 (1996).

39. A. M. Perelemov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924 (1966).

40. J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71, 877–879 (2000). [CrossRef]  

41. M. B. Gaarde and A. Couairon, “Intensity spikes in laser filamentation: diagnostics and application,” Phys. Rev. Lett. 103, 043901 (2009). [CrossRef]   [PubMed]  

42. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007). [CrossRef]  

43. P. P. Kiran, S. Bagchi, C. Arnold, S. R. Krishnan, G. R. Kumar, and A. Couairon, “Filamentation without intensity clamping,” Opt. Express 18, 21504 (2010). [CrossRef]   [PubMed]  

44. H. R. Lange, A. Chiron, J. F. Ripoche, A. Mysyrowicz, P. Breger, and P. Agostini, “High-order harmonic generation and quasi-phase matching in xenon using selfguided femtosecond pulses,” Phys. Rev. Lett. 81, 1611–1613 (1998). [CrossRef]  

45. D. S. Steingrube, E. Schulz, T. Binhammer, M. Gaarde, A. Couairon, U. Morgner, and M. Kovacev, “High-order harmonic generation directly from a filament,” New J. Phys. 13, 043022 (2011). [CrossRef]  

46. S. Xu, X. Sun, B. Zeng, W. Chu, J. Zhao, W. Lei, Y. Cheng, Z. Xu, and S. L. Chin, “Simple method of measuring laser peak intensity inside femtosecond laser filament in air,” Opt. Express 20, 299–307 (2012). [CrossRef]   [PubMed]  

47. X. Sun, S. Xu, J. Zhao, W. Liu, Y. Cheng, Z. Xu, S. L. Chin, and G. Mu, “Impressive laser intensity increase at the trailing stage of femtosecond laser filamentation in air,” Opt. Express 20, 4790–4795 (2012). [CrossRef]   [PubMed]  

48. S. I. Mitryukovskiy, Y. Liu, A. Houard, and A. Mysyrowicz, “Re-evaluation of the peak intensity inside a femtosecond laser filament in air,” J. Phys. B: At. Mol. Opt. Phys. 48, 094003 (2015). [CrossRef]  

49. A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163, 29 (1999). [CrossRef]  

References

  • View by:

  1. N. Aközbek, A. Becker, M. Scolora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
    [Crossref]
  2. P. Rairoux, H. Schillinger, S. Niedermeier, M. Rodriguez, F. Ronneberger, R. Sauerbrey, B. Stein, D. Waite, C. Wedekind, H. Wille, L. Wöste, and C. Ziener, “Remote sensing of the atmosphere using ultrashort laser pulses,” Appl. Phys. B 71, 573–580 (2000).
    [Crossref]
  3. A. Ting, I. Alexeev, D. Gordon, E. Briscoe, J. Peñano, R. Hubbard, P. Sprangle, and G. Rubel, “Remote atmospheric breakdown for standoff detection by using an intense short laser pulse,” Appl. Opt. 44, 5315–5320 (2005).
    [Crossref] [PubMed]
  4. M. Chateauneuf, J. Dubois, S. Payeur, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92, 091104 (2008).
    [Crossref]
  5. F. Théberge, W. Liu, Q. Luo, and S. L. Chin, “Ultrabroadband continuum generated in air (down to 230 nm) using ultrashort and intense laser pulses,” Appl. Phys. B 80, 221–225 (2005).
    [Crossref]
  6. N. Aközbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. 89, 143901 (2002).
    [Crossref] [PubMed]
  7. H. Xiong, H. Xiong, H. Xu, Y. Fu, Y. Cheng, Z. Xu, and S. L. Chin, “Spectral evolution of angularly resolved third-order harmonic generation by infrared femtosecond laser-pulse filamentation in air,” Phys. Rev. A 77, 043802 (2008).
    [Crossref]
  8. M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. 92, 253901 (2004).
    [Crossref] [PubMed]
  9. V. Vaičaitis, V. Jarutis, and D. Pentaris, “Conical third-harmonic generation in normally dispersive media,” Phys. Rev. Lett. 103, 103901 (2009).
    [Crossref]
  10. E. T. J. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysyrowicz, “Conical emission from self-guided femtosecond pulses in air,” Opt. Lett. 21, 62–64 (1996).
    [Crossref] [PubMed]
  11. D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation from spontaneous X–Wave formation in air,” Opt. Express 16, 1565–1570 (2008).
    [Crossref] [PubMed]
  12. D. Faccio, P. Di Trapani, S. Minardi, A. Bramati, F. Bragheri, C. Liberale, V. Degiorgio, A. Dubietis, and A. Matijosius, “Far-field spectral characterization of conical emission and filamentation in Kerr media,” J. Opt. Soc. Am. B 22, 862 (2005).
    [Crossref]
  13. M. Kolesik, E. M. Wright, A. Becker, and J. V. Moloney, “Simulation of third-harmonic and supercontinuum generation for femtosecond pulses in air,” Appl. Phys. B 85, 531 (2006).
    [Crossref]
  14. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [Crossref]
  15. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
    [Crossref]
  16. K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110, 143603 (2013).
    [Crossref] [PubMed]
  17. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [Crossref] [PubMed]
  18. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
    [Crossref] [PubMed]
  19. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
    [Crossref]
  20. K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440, 935 (2006).
    [Crossref] [PubMed]
  21. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826 (1995).
    [Crossref] [PubMed]
  22. K. Yamane, Y. Toda, and R. Morita, “Ultrashort optical-vortex pulse generation in few-cycle regime,” Opt. Express 20, 18986 (2012).
    [Crossref] [PubMed]
  23. K. Bezuhanov, A. Dreishuh, G. G. Paulus, M. G. Schätzel, and H. Walther, “Vortices in femtosecond laser fields,” Opt. Lett. 29, 1942–1944 (2004).
    [Crossref] [PubMed]
  24. I. G. Mariyenko, J. Strohaber, and C. J. G. J. Uiterwaal, “Creation of optical vortices in femtosecond pulses,” Opt. Express 13, 7599–7607 (2005).
    [Crossref] [PubMed]
  25. I. Zeylikovich, H. I. Sztul, V. Kartazaev, T. Le, and R. R. Alfano, “Ultrashort Laguerre-Gaussian pulses with angular and group velocity dispersion compensation,” Opt. Lett. 32, 2025–2027 (2007).
    [Crossref] [PubMed]
  26. V. G. Shvedov, C. Hnatovsky, W. Krolikowski, and A. V. Rode, “Efficient beam converter for the generation of high-power femtosecond vortices,” Opt. Lett. 35, 2660–2662 (2010).
    [Crossref] [PubMed]
  27. M. Bock, S. K. Das, C. Fischer, M. Diehl, P. Börner, and R. Grunwald, “Reconfigurable wavefront sensor for ultrashort pulses,” Opt. Lett. 37, 1154–1156 (2012).
    [Crossref] [PubMed]
  28. D. N. Neshev, A. Dreischuh, G. Maleshkov, M. Samoc, and Y. S. Kivshar, “Supercontinuum generation with optical vortices,” Opt. Express 18, 18368–18373 (2010).
    [Crossref] [PubMed]
  29. P. Polynkin, C. Ament, and J. V. Moloney, “Self-focusing of ultraintense femtosecond optical vortices in air,” Phys. Rev. Lett. 111, 023901 (2013).
    [Crossref] [PubMed]
  30. A. Sukhinin, A. Aceves, J.-C. Diels, and L. Arissian, “On the co-existence of IR and UV optical filaments,” J. Phys. B: At. Mol. Opt. Phys. 48, 094021 (2015).
    [Crossref]
  31. G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113, 153901 (2014).
    [Crossref] [PubMed]
  32. A. E. Siegman, Lasers (University Science, 1986).
  33. V. I. Kruglova, Yu. A. Logvina, and V. M. Volkov, “The theory of spiral laser beams in nonlinear media,” J. Mod. Opt. 39, 2277 (1992).
    [Crossref]
  34. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650 (1997).
    [Crossref]
  35. V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17, 23374–23379 (2009).
    [Crossref]
  36. A. Couairon, E. Gaižauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E 73, 016608 (2006).
    [Crossref]
  37. S. L. Chin, Femtosecond Laser Filamentation (Springer, 2010).
    [Crossref]
  38. P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Phys. Lett. 35, 1566 (1996).
  39. A. M. Perelemov, V. S. Popov, and M. V. Terent’ev, “Ionization of atoms in an alternating electric field,” Sov. Phys. JETP 23, 924 (1966).
  40. J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71, 877–879 (2000).
    [Crossref]
  41. M. B. Gaarde and A. Couairon, “Intensity spikes in laser filamentation: diagnostics and application,” Phys. Rev. Lett. 103, 043901 (2009).
    [Crossref] [PubMed]
  42. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
    [Crossref]
  43. P. P. Kiran, S. Bagchi, C. Arnold, S. R. Krishnan, G. R. Kumar, and A. Couairon, “Filamentation without intensity clamping,” Opt. Express 18, 21504 (2010).
    [Crossref] [PubMed]
  44. H. R. Lange, A. Chiron, J. F. Ripoche, A. Mysyrowicz, P. Breger, and P. Agostini, “High-order harmonic generation and quasi-phase matching in xenon using selfguided femtosecond pulses,” Phys. Rev. Lett. 81, 1611–1613 (1998).
    [Crossref]
  45. D. S. Steingrube, E. Schulz, T. Binhammer, M. Gaarde, A. Couairon, U. Morgner, and M. Kovacev, “High-order harmonic generation directly from a filament,” New J. Phys. 13, 043022 (2011).
    [Crossref]
  46. S. Xu, X. Sun, B. Zeng, W. Chu, J. Zhao, W. Lei, Y. Cheng, Z. Xu, and S. L. Chin, “Simple method of measuring laser peak intensity inside femtosecond laser filament in air,” Opt. Express 20, 299–307 (2012).
    [Crossref] [PubMed]
  47. X. Sun, S. Xu, J. Zhao, W. Liu, Y. Cheng, Z. Xu, S. L. Chin, and G. Mu, “Impressive laser intensity increase at the trailing stage of femtosecond laser filamentation in air,” Opt. Express 20, 4790–4795 (2012).
    [Crossref] [PubMed]
  48. S. I. Mitryukovskiy, Y. Liu, A. Houard, and A. Mysyrowicz, “Re-evaluation of the peak intensity inside a femtosecond laser filament in air,” J. Phys. B: At. Mol. Opt. Phys. 48, 094003 (2015).
    [Crossref]
  49. A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163, 29 (1999).
    [Crossref]

2015 (2)

A. Sukhinin, A. Aceves, J.-C. Diels, and L. Arissian, “On the co-existence of IR and UV optical filaments,” J. Phys. B: At. Mol. Opt. Phys. 48, 094021 (2015).
[Crossref]

S. I. Mitryukovskiy, Y. Liu, A. Houard, and A. Mysyrowicz, “Re-evaluation of the peak intensity inside a femtosecond laser filament in air,” J. Phys. B: At. Mol. Opt. Phys. 48, 094003 (2015).
[Crossref]

2014 (1)

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113, 153901 (2014).
[Crossref] [PubMed]

2013 (2)

P. Polynkin, C. Ament, and J. V. Moloney, “Self-focusing of ultraintense femtosecond optical vortices in air,” Phys. Rev. Lett. 111, 023901 (2013).
[Crossref] [PubMed]

K. Toyoda, F. Takahashi, S. Takizawa, Y. Tokizane, K. Miyamoto, R. Morita, and T. Omatsu, “Transfer of light helicity to nanostructures,” Phys. Rev. Lett. 110, 143603 (2013).
[Crossref] [PubMed]

2012 (5)

2011 (1)

D. S. Steingrube, E. Schulz, T. Binhammer, M. Gaarde, A. Couairon, U. Morgner, and M. Kovacev, “High-order harmonic generation directly from a filament,” New J. Phys. 13, 043022 (2011).
[Crossref]

2010 (3)

2009 (3)

V. Denisenko, V. Shvedov, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, A. Volyar, M. Soskin, and Y. S. Kivshar, “Determination of topological charges of polychromatic optical vortices,” Opt. Express 17, 23374–23379 (2009).
[Crossref]

V. Vaičaitis, V. Jarutis, and D. Pentaris, “Conical third-harmonic generation in normally dispersive media,” Phys. Rev. Lett. 103, 103901 (2009).
[Crossref]

M. B. Gaarde and A. Couairon, “Intensity spikes in laser filamentation: diagnostics and application,” Phys. Rev. Lett. 103, 043901 (2009).
[Crossref] [PubMed]

2008 (3)

H. Xiong, H. Xiong, H. Xu, Y. Fu, Y. Cheng, Z. Xu, and S. L. Chin, “Spectral evolution of angularly resolved third-order harmonic generation by infrared femtosecond laser-pulse filamentation in air,” Phys. Rev. A 77, 043802 (2008).
[Crossref]

M. Chateauneuf, J. Dubois, S. Payeur, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92, 091104 (2008).
[Crossref]

D. Faccio, A. Averchi, A. Lotti, P. Di Trapani, A. Couairon, D. Papazoglou, and S. Tzortzakis, “Ultrashort laser pulse filamentation from spontaneous X–Wave formation in air,” Opt. Express 16, 1565–1570 (2008).
[Crossref] [PubMed]

2007 (2)

2006 (3)

K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440, 935 (2006).
[Crossref] [PubMed]

A. Couairon, E. Gaižauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E 73, 016608 (2006).
[Crossref]

M. Kolesik, E. M. Wright, A. Becker, and J. V. Moloney, “Simulation of third-harmonic and supercontinuum generation for femtosecond pulses in air,” Appl. Phys. B 85, 531 (2006).
[Crossref]

2005 (4)

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M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic nonlinear X waves for femtosecond pulse propagation in water,” Phys. Rev. Lett. 92, 253901 (2004).
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2003 (1)

N. Aközbek, A. Becker, M. Scolora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
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2002 (1)

N. Aközbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. 89, 143901 (2002).
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2001 (2)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
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A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
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2000 (2)

P. Rairoux, H. Schillinger, S. Niedermeier, M. Rodriguez, F. Ronneberger, R. Sauerbrey, B. Stein, D. Waite, C. Wedekind, H. Wille, L. Wöste, and C. Ziener, “Remote sensing of the atmosphere using ultrashort laser pulses,” Appl. Phys. B 71, 573–580 (2000).
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J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71, 877–879 (2000).
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1999 (1)

A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163, 29 (1999).
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1998 (1)

H. R. Lange, A. Chiron, J. F. Ripoche, A. Mysyrowicz, P. Breger, and P. Agostini, “High-order harmonic generation and quasi-phase matching in xenon using selfguided femtosecond pulses,” Phys. Rev. Lett. 81, 1611–1613 (1998).
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1997 (1)

1996 (2)

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826 (1995).
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1992 (2)

V. I. Kruglova, Yu. A. Logvina, and V. M. Volkov, “The theory of spiral laser beams in nonlinear media,” J. Mod. Opt. 39, 2277 (1992).
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1974 (1)

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H. R. Lange, A. Chiron, J. F. Ripoche, A. Mysyrowicz, P. Breger, and P. Agostini, “High-order harmonic generation and quasi-phase matching in xenon using selfguided femtosecond pulses,” Phys. Rev. Lett. 81, 1611–1613 (1998).
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N. Aközbek, A. Becker, M. Scolora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
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N. Aközbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. 89, 143901 (2002).
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P. Polynkin, C. Ament, and J. V. Moloney, “Self-focusing of ultraintense femtosecond optical vortices in air,” Phys. Rev. Lett. 111, 023901 (2013).
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M. Kolesik, E. M. Wright, A. Becker, and J. V. Moloney, “Simulation of third-harmonic and supercontinuum generation for femtosecond pulses in air,” Appl. Phys. B 85, 531 (2006).
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N. Aközbek, A. Becker, M. Scolora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
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N. Aközbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. 89, 143901 (2002).
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L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
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J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
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D. S. Steingrube, E. Schulz, T. Binhammer, M. Gaarde, A. Couairon, U. Morgner, and M. Kovacev, “High-order harmonic generation directly from a filament,” New J. Phys. 13, 043022 (2011).
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N. Aközbek, A. Becker, M. Scolora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
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N. Aközbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. 89, 143901 (2002).
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X. Sun, S. Xu, J. Zhao, W. Liu, Y. Cheng, Z. Xu, S. L. Chin, and G. Mu, “Impressive laser intensity increase at the trailing stage of femtosecond laser filamentation in air,” Opt. Express 20, 4790–4795 (2012).
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S. Xu, X. Sun, B. Zeng, W. Chu, J. Zhao, W. Lei, Y. Cheng, Z. Xu, and S. L. Chin, “Simple method of measuring laser peak intensity inside femtosecond laser filament in air,” Opt. Express 20, 299–307 (2012).
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F. Théberge, W. Liu, Q. Luo, and S. L. Chin, “Ultrabroadband continuum generated in air (down to 230 nm) using ultrashort and intense laser pulses,” Appl. Phys. B 80, 221–225 (2005).
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N. Aközbek, A. Becker, M. Scolora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
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N. Aközbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. 89, 143901 (2002).
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H. R. Lange, A. Chiron, J. F. Ripoche, A. Mysyrowicz, P. Breger, and P. Agostini, “High-order harmonic generation and quasi-phase matching in xenon using selfguided femtosecond pulses,” Phys. Rev. Lett. 81, 1611–1613 (1998).
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G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113, 153901 (2014).
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D. S. Steingrube, E. Schulz, T. Binhammer, M. Gaarde, A. Couairon, U. Morgner, and M. Kovacev, “High-order harmonic generation directly from a filament,” New J. Phys. 13, 043022 (2011).
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A. Sukhinin, A. Aceves, J.-C. Diels, and L. Arissian, “On the co-existence of IR and UV optical filaments,” J. Phys. B: At. Mol. Opt. Phys. 48, 094021 (2015).
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J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
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A. Couairon, E. Gaižauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E 73, 016608 (2006).
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G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113, 153901 (2014).
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H. Xiong, H. Xiong, H. Xu, Y. Fu, Y. Cheng, Z. Xu, and S. L. Chin, “Spectral evolution of angularly resolved third-order harmonic generation by infrared femtosecond laser-pulse filamentation in air,” Phys. Rev. A 77, 043802 (2008).
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D. S. Steingrube, E. Schulz, T. Binhammer, M. Gaarde, A. Couairon, U. Morgner, and M. Kovacev, “High-order harmonic generation directly from a filament,” New J. Phys. 13, 043022 (2011).
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M. B. Gaarde and A. Couairon, “Intensity spikes in laser filamentation: diagnostics and application,” Phys. Rev. Lett. 103, 043901 (2009).
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G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113, 153901 (2014).
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J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
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N. Aközbek, A. Iwasaki, A. Becker, M. Scalora, S. L. Chin, and C. M. Bowden, “Third-harmonic generation and self-channeling in air using high-power femtosecond laser pulses,” Phys. Rev. Lett. 89, 143901 (2002).
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J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71, 877–879 (2000).
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M. Chateauneuf, J. Dubois, S. Payeur, and J.-C. Kieffer, “Microwave guiding in air by a cylindrical filament array waveguide,” Appl. Phys. Lett. 92, 091104 (2008).
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G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113, 153901 (2014).
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M. Kolesik, E. M. Wright, A. Becker, and J. V. Moloney, “Simulation of third-harmonic and supercontinuum generation for femtosecond pulses in air,” Appl. Phys. B 85, 531 (2006).
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D. S. Steingrube, E. Schulz, T. Binhammer, M. Gaarde, A. Couairon, U. Morgner, and M. Kovacev, “High-order harmonic generation directly from a filament,” New J. Phys. 13, 043022 (2011).
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S. I. Mitryukovskiy, Y. Liu, A. Houard, and A. Mysyrowicz, “Re-evaluation of the peak intensity inside a femtosecond laser filament in air,” J. Phys. B: At. Mol. Opt. Phys. 48, 094003 (2015).
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D. S. Steingrube, E. Schulz, T. Binhammer, M. Gaarde, A. Couairon, U. Morgner, and M. Kovacev, “High-order harmonic generation directly from a filament,” New J. Phys. 13, 043022 (2011).
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N. Aközbek, A. Becker, M. Scolora, S. L. Chin, and C. M. Bowden, “Continuum generation of the third-harmonic pulse generated by an intense femtosecond IR laser pulse in air,” Appl. Phys. B 77, 177–183 (2003).
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A. Sukhinin, A. Aceves, J.-C. Diels, and L. Arissian, “On the co-existence of IR and UV optical filaments,” J. Phys. B: At. Mol. Opt. Phys. 48, 094021 (2015).
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J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6, 488–496 (2012).
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K. I. Willig, S. O. Rizzoli, V. Westphal, R. Jahn, and S. W. Hell, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis,” Nature 440, 935 (2006).
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Opt. Lett. (5)

Phys. Rep. (1)

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–189 (2007).
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Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
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Phys. Rev. E (1)

A. Couairon, E. Gaižauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E 73, 016608 (2006).
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P. Polynkin, C. Ament, and J. V. Moloney, “Self-focusing of ultraintense femtosecond optical vortices in air,” Phys. Rev. Lett. 111, 023901 (2013).
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Figures (8)

Fig. 1
Fig. 1 Upper: Schematic experimental setup. L1: focusing lens with focal length f=600 mm. L2: focusing lens with f=40 mm. M1: dielectric mirrors with high-reflectivity coating for wavelengths of 220–270 nm. Lower: Schematic setup for generation of intense vortex pulses.
Fig. 2
Fig. 2 Far-field patterns. (a)–(c) Fundamental patterns of = 1 at pump powers of 3.13 GW, 7.05 GW, and 10.97 GW, respectively. (d)–(f) TH counterparts corresponding to (a)–(c).
Fig. 3
Fig. 3 (a)–(d): Results of theoretical simulations for vortex beams converted via a cylindrical lens with a focal length of 30 cm. (a) and (b) LG beams of = 3 with different divergence angles. (c) and (d) Focal patterns for (a) and (b), respectively. (e) and (f) Experimental focal patterns transformed from Figs. 2(d) and 2(f), respectively.
Fig. 4
Fig. 4 (a) Measured energy for total, inner, and outer TH vortex beams as a function of the pump energy. (b) Ratios of the inner and outer TH vortex beams to the total TH energy as a function of the pump energy.
Fig. 5
Fig. 5 Angularly resolved spectra of TH waves measured at 410 mm beyond the central point of the filament at pump energies of (a) 455 μJ, (b) 550 μJ, and (c) 715 μJ (in a logarithmic scale and normalized by the peak intensity of the TH waves). Dashed lines were obtained by simulation using Eq. (10) at the clamped intensity of Ic = 8.34×1013 W/cm2.
Fig. 6
Fig. 6 Electron density, Ne, (solid line) and difference in the refractive index, Δn, (dashed line) as a function of the peak intensity, Ipk.
Fig. 7
Fig. 7 Divergence angle, θ, as a function of the peak intensity, Ipk.
Fig. 8
Fig. 8 Divergence angle, θ, of various topological charges, , as a function of the peak intensity, Ipk.

Equations (18)

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{ i z + T 2 2 k ( 3 ω 0 ) } A 3 ω 0 ( r T , z ) = 3 ω 0 2 n ( 3 ω 0 ) c χ 3 { A ω 0 ( r T , z ) } 3 e i { k ( 3 ω 0 ) 3 k ( ω 0 ) } z ,
i z A ˜ 3 ω 0 ( k T , z ) = 3 ω 0 2 n ( 3 ω 0 ) c χ 3 d 2 r T e i k T r T { A ω 0 ( r T , z ) } 3 e i { k ( 3 ω 0 ) 3 k ( ω 0 ) k T 2 2 k ( 3 ω 0 ) } z ,
A ˜ 3 ω 0 ( k T , z ) d 2 r T e i k T r T A 3 ω 0 ( r T , z ) e i k T 2 2 k ( 3 ω 0 ) z .
ϕ r ( ω 0 ) k ( ω 0 ) z r 2 2 ( z 2 + z R 2 ) ,
ϕ Gouy ( ω 0 ) ( | | + 1 ) arctan ( z z R ) ,
ϕ Kerr ( ω 0 ) ω 0 c n 2 I pk z ,
Δ K ( k T ; 3 ω 0 , ω 0 ) = k ( 3 ω 0 ) 3 { k ( ω 0 ) + z [ ϕ r ( ω 0 ) + ϕ Gouy ( ω 0 ) + ϕ Kerr ( ω 0 ) ] z = 0 } k T 2 2 k ( 3 ω 0 ) = 3 ω 0 c Δ n k T 2 2 k ( 3 ω 0 ) .
Δ n = [ n L ( 3 ω 0 ) n L ( ω 0 ) ] + n ( ω 0 ) k 2 ( ω 0 ) w 0 2 ( | | + 2 ) n 2 I pk + V N e ,
| k T | 2 Δ n k vac ( 3 ω 0 ) .
θ = arctan ( | k T | k vac ( 3 ω 0 ) ) arctan ( 2 Δ n ) .
R ( ) = 6 π | C n * , l * | 2 f l , m A m ( γ ) ( 2 0 1 + γ 2 ) 2 n * | m | 3 2 e 2 0 3 g ( γ ) ,
| C n * , l * | 2 = 2 2 n * n * Γ ( n * + l * + 1 ) Γ ( n * l * ) ,
f l , m = ( 2 l + 1 ) ! ( l + | m | ) ! 2 | m | ( | m | ) ! ( l | m | ) ! ,
A m ( γ ) = 4 3 π 1 | m | ! γ 2 1 + γ 2 κ > v + e ( κ v ) α ( γ ) Φ m [ ( κ v ) β ( γ ) ] ,
Φ m ( x ) = e x 2 0 x ( x 2 y 2 ) | m | e y 2 d y ,
α ( γ ) = 2 [ sinh 1 ( γ ) γ 1 + γ 2 ] ,
β ( γ ) = 2 γ 1 + γ 2 ,
g ( γ ) = 3 2 γ [ ( 1 + 1 2 γ 2 ) sinh 1 γ 1 + γ 2 2 γ ] .

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