## 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 TEM_{00} 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 *μ*J with a repetition rate of 200 Hz. We further enlarged the beam radius to ~3 mm at the 1/*e*^{2} 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 *μ*m 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 *μ*J. 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.

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 *P*_{crit}(*ℓ*) = {2^{2}^{ℓ}^{+1}Γ(*ℓ* + 1)Γ(*ℓ* + 2)]/[2Γ(2*ℓ* + 1)]}*P*_{crit}(0) = 4.13 GW [33] required for self-focusing and filamentation, where *ℓ* = 1, *P*_{crit}(*ℓ*) is the critical power for a vortex beam of topological charge *ℓ*, and *P*_{crit}(0) is the critical power for a TEM_{00} Gaussian beam in the same medium. The critical power of the TEM_{00} Gaussian beam is given by
${P}_{\text{crit}}(0)\approx {\lambda}_{0}^{2}/[4\pi n({\omega}_{0}){n}_{2}]$, where *λ*_{0} = 720 nm, *ω*_{0} is the central angular frequency of the fundamental field, *n*(*ω*_{0}) ≈ 1 in air, and *n*_{2} ≈ 4 × 10^{−19} cm^{2}/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, *I*_{pk}, of the fundamental wave associated with Fig. 2(a) at the geometrical focus and obtained *I*_{pk} =3.112 × 10^{13} W/cm^{2}, 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 *P*_{crit}(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 *P*_{crit}(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.

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[−*ik*_{vac}(3*ω*_{0})*y*^{2}/2*f*_{cyl}] [35], where *k*_{vac}(3*ω*_{0}) = 3*ω*_{0}*/c* is the free-space wave number and *f*_{cyl} = 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 *ℓ* = 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 *ℓ* = 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.

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 *μ*J and 200 *μ*J, 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 *μ*J and 550 *μ*J, suggesting that filamentation occurs at a pump energy of approximately 230 *μ*J [6]. This closely corresponds to the estimated critical power of *P*_{crit}(1) = 4.13 GW, which is equivalent to a pump energy of ~ 263 *μ*J for a Gaussian-shaped pulse with a duration of *τ* = 60 fs. When the pump energy is increased to above 600 *μ*J, 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%.

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 *μ*J, 550 *μ*J, and 715 *μ*J, 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].

## 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, ${A}_{3{\omega}_{0}}({\mathit{r}}_{T},z)$ [37],

*T*, signifies the transverse component of vectors, ${\nabla}_{T}^{2}={\partial}^{2}/\partial {x}^{2}+{\partial}^{2}/\partial {y}^{2}$ is the transverse Laplacian,

*r**is the transverse position,*

_{T}*ω*

_{0}and 3

*ω*

_{0}are the central angular frequencies of the fundamental and TH fields, respectively, and

*χ*

_{3}is the third-order susceptibility of air. The wave numbers of the fundamental and TH fields,

*k*(

*ω*

_{0}) and

*k*(3

*ω*

_{0}), respectively, are given by

*k*(

*ω*

_{0})=

*ω*

_{0}

*n*(

*ω*

_{0})

*/c*and

*k*(3

*ω*

_{0})= 3

*ω*

_{0}

*n*(3

*ω*

_{0})

*/c*, where the refractive indices

*n*(

*ω*

_{0}) and

*n*(3

*ω*

_{0}) are composed of linear and plasma parts so that

*n*(

*ω*

_{0})=

*n*

_{L}(

*ω*

_{0}) +

*n*

_{plasma}(

*ω*

_{0}) and

*n*(3

*ω*

_{0})=

*n*

_{L}(3

*ω*

_{0})+

*n*

_{plasma}(3

*ω*

_{0}). The linear part of the refractive index can be calculated from the dispersion formula in air of

*n*

_{L}(

*ω*) = 1 +

*η*/[

*ξ*− (

*ω*/2

*πc*)

^{2}] +

*η*

_{1}/[

*ξ*

_{1}− (

*ω*/2

*πc*)

^{2}], where

*η*= 0.05792105,

*ξ*= 238.018,

*η*

_{1}= 0.00167917,

*ξ*

_{2}= 57.362, and the unit of

*ω*is megahertz [38]. The refractive index of the plasma,

*n*

_{plasma}(

*ω*), is equal to ${\left(1-{\omega}_{\text{plasma}}^{2}/{\omega}^{2}\right)}^{1/2}\simeq 1-{\omega}_{\text{plasma}}^{2}/(2{\omega}^{2})<1$, in which the plasma frequency,

*ω*

_{plasma}, is defined as ${\omega}_{\text{plasma}}^{2}\equiv {q}_{\mathrm{e}}^{2}{N}_{\mathrm{e}}/({\epsilon}_{0}{m}_{\mathrm{e}})$ [37] (

*q*

_{e}: electron charge;

*m*

_{e}: electron mass;

*N*

_{e}: 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, ${E}_{{\omega}_{0}}({\mathit{r}}_{T},z)$ and ${E}_{3{\omega}_{0}}({\mathit{r}}_{T},z)$, are written as ${A}_{{\omega}_{0}}({\mathit{r}}_{T},z)\mathrm{exp}[ik({\omega}_{0})z]$ and ${A}_{3{\omega}_{0}}({\mathit{r}}_{T},z)\mathrm{exp}[ik(3{\omega}_{0})z]$, respectively.

Multiplying ∫*d*^{2}*r** _{T}* exp(−

*i*

*k**·*

_{T}

*r**) to both sides of Eq. (1) and using the Helmholtz equation, $\left({\nabla}_{\mathit{T}}^{2}+{\mathit{k}}_{\mathit{T}}^{2}\right){A}_{3{\omega}_{0}}({\mathit{r}}_{\mathit{T}},z)=0$, we have the equation*

_{T}*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 $k(3{\omega}_{0})-3k({\omega}_{0})-{\mathit{k}}_{T}^{2}/[2k(3{\omega}_{0})]\simeq {({k}^{2}(3{\omega}_{0})-{\mathit{k}}_{T}^{2})}^{1/2}-3k({\omega}_{0})={k}_{z}(3{\omega}_{0})-3k({\omega}_{0})$.

For an optical vortex beam, the SVE of the fundamental field, ${A}_{{\omega}_{0}}({\mathit{r}}_{T},z)$, can be expressed as ${A}_{{\omega}_{0}}({\mathit{r}}_{T},z)=|{A}_{{\omega}_{0}}({\mathit{r}}_{T},z)|\mathrm{exp}(i\ell \varphi )\mathrm{exp}\{i[{\varphi}_{r}({\omega}_{0})+{\varphi}_{\text{Gouy}}({\omega}_{0})+{\varphi}_{\text{Kerr}}({\omega}_{0})]\}$. The amplitude $|{A}_{{\omega}_{0}}({\mathit{r}}_{T},z)|$ of a single-ring LG mode is described by $|{A}_{{\omega}_{0}}({\mathit{r}}_{T},z)|\propto [{w}_{0}/w(z)]{[r/w(z)]}^{\left|\ell \right|}\mathrm{exp}[-{r}^{2}/{w}^{2}(z)]$, and the phase terms are

*w*

_{0}is the beam waist of the fundamental field,

*w*(

*z*) =

*w*

_{0}[1 + (

*z/z*)

_{R}^{2}]

^{1}

^{/}^{2}is the spot size at

*z*, and the Rayleigh length,

*z*, is equal to $k({\omega}_{0}){w}_{0}^{2}/2$. The radial coordinate

_{R}*r*coincides with |

*r**|. Thus, the wave-number mismatch, Δ*

_{T}*K*(

*k**;3*

_{T}*ω*

_{0},

*ω*

_{0}), at

*z*= 0 and

*r*=

*w*

_{0}(|

*ℓ*|/2)

^{1}

^{/}^{2}, where the intensity is maximized, is evaluated as

*n*as

The first term in Eq. (8)*n*_{L}(3*ω*_{0}) − *n*_{L}(*ω*_{0}), is the usual index mismatch originating from dispersion in the air. The second term originates from *ϕ _{r}*(

*ω*

_{0}) +

*ϕ*

_{Gouy}(

*ω*

_{0}). 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,

*N*

_{e}, nonlinearly increases with increasing peak intensity,

*I*

_{pk}.

Letting Δ*K*(*k** _{T}*;3

*ω*

_{0},

*ω*

_{0}) = 0 for the phase-matching condition, we obtain

*k*(3

*ω*

_{0}) has been approximated to

*k*

_{vac}(3

*ω*

_{0}). Therefore, the divergence angle of the TH radiation originating from the phase-matching condition,

*θ*, is expressed as

To further elucidate the effect of the electron plasma on the index difference, in Fig. 6 we plot *N*_{e} and Δ*n* as a function of the peak intensity, *I*_{pk}. The free electron density, *N*_{e}, in air is given by
${N}_{{\mathrm{O}}_{2}}{P}_{{\mathrm{O}}_{2}}+{N}_{{\mathrm{N}}_{2}}{P}_{{\mathrm{N}}_{2}}$, where
${N}_{{\mathrm{O}}_{2}}=0.21{N}_{0}$ and
${N}_{{\mathrm{N}}_{2}}=0.78{N}_{0}$ are the number densities of oxygen and nitrogen molecules, respectively, and *N*_{0} = 2.5 × 10^{19} atoms/cm^{3} is the number density of air at room temperature and atmospheric pressure. The ionization probability of the molecules, *P _{x}*, is written as
${P}_{x}=1-\mathrm{exp}[{\displaystyle {\int}_{-\infty}^{\infty}{R}_{x}(\mathcal{E})dt}]$, where

*x*stands for O

_{2}or N

_{2}molecules. The ionization rate, ${R}_{x}(\mathcal{E})$, is a function of the field amplitude, $\mathcal{E}$, at

*z*= 0 and

*r*=

*w*

_{0}(|

*l*|/2)

^{1}

^{/}^{2}. The field amplitude is given by $\mathcal{E}({\mathit{r}}_{\mathit{T}},z,t)=E({\mathit{r}}_{\mathit{T}},z)D(t)$, where

*D*(

*t*) is the temporal profile of the pulse. To obtain ${R}_{x}(\mathcal{E})$, 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

*N*

_{e}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

*ℓ*= 1,

*w*

_{0}= 50

*μ*m, and

*n*

_{2}= 4 × 10

^{−}^{19}cm

^{2}/W. The beam waist,

*w*

_{0}, is obtained by measuring the beam profiles of the fundamental field at the geometrical focus.

In Fig. 6 it is shown that the index difference Δ*n* initially decreases owing to the nonlinear Kerr effect corresponding to the term *n*_{2}*I*_{pk} in Eq. (8), while it starts to increase as the plasma index, *VN*_{e}, takes on a dominant role. At higher peak intensities of above 3 × 10^{14} W/cm^{2}, the index difference eventually saturates owing to the saturation of *N*_{e} since almost all the target molecules are ionized in this regime. Here we define the clamped intensity, *I*_{c}, at which the dynamic between the self-focusing due to the Kerr effect and the defocusing of the plasma is balanced, namely, −*n*_{2} *I*_{c} +*VN*_{e} = 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 *μ*J and 550 *μ*J does not increase with increasing input energy; this is expected since the intensity inside the filament is clamped to *I*_{c}. In our case, we obtain *I*_{c} = 8.34 × 10^{13} W/cm^{2}, in good agreement with the value of 8 × 10^{13} W/cm^{2} 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 10^{13} W/cm^{2} to 10^{15} W/cm^{2} [40–47]. More recently, a value of 1.45×10^{14} W/cm^{2}, 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, *I*_{c}. For *I*_{pk} above *I*_{c}, *θ* exhibits rapid growth with increasing *I*_{pk}. 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 *I*_{c}, 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 *I*_{c}, at which the dynamic between the self-focusing and defocusing is balanced during the filamentation process. Most importantly, we inform from Eq. (1) that
${A}_{3{\omega}_{0}}({\mathit{r}}_{\mathit{T}},z)\propto \mathrm{exp}(i3\ell \varphi )$ by substituting the analytical expression for *A _{ω}*

_{0}(

*r*,

_{T}*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.

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 ** k**(ω

_{0}) ║

**(3ω**

*k*_{0}). According to this relation, since for an LG beam,

**(ω**

*k*_{0}) has no magnitude of the transverse component at the beam waist, the transverse wave number, |

*k**|, of the inner TH ring should vanish at*

_{T}*z*= 0. For |

*k**| = 0, we find that Δ*

_{T}*K*(

*k**;3ω*

_{T}_{0},ω

_{0}) ≠ 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, ${\left[{E}_{{\omega}_{0}}({\mathit{r}}_{\mathit{T}},z)\right]}^{3}$.

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 *ℓ* as a function of the peak intensity. The parameters used here are the same as previous simulations with *w*_{0} = 50*μ*m and *n*_{2} = 4×10^{−19} cm^{2}/W. From the relation of −*n*_{2} *I*_{c} +*VN*_{e} = 0, we obtain the clamped intensity to be the value of 8.34 × 10^{13} W/cm^{2}, independent of the topological charge. At *I*_{c}, in Fig. 8 it is evident that *θ* increases with increasing *ℓ*. 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.

## 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 *ℓ* = 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
${A}_{3{\omega}_{0}}({\mathit{r}}_{\mathit{T}},z)$ 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 *I*_{pk}, we found that increasing *ℓ* 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 $\mathcal{E}$ can be expressed in atomic units as

*U*is the ionization potential of the molecule, and

_{i}*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*= (

*U*)[1 + (2

_{i}/ω*γ*

^{2})] and

*κ*= 〈(

*U*) + 1〉 +

_{i}/ω*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 $n*={Z}_{\text{eff}}/\sqrt{2{U}_{i}}$, with

*Z*

_{eff}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 O_{2} and N_{2} and the ionization potential is 12.55 eV for O_{2} and 15.58 eV for N_{2}. The effective residual charge, *Z*_{eff}, has been measured experimentally [49] to be 0.53 for O_{2} and 0.90 for N_{2}.

## 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.

## References and links

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