Propagation-invariant vortex Airy beam whose singular point follows its main lobe

In order to introduce a new-type vortex Airy beam, we start by examining the paraxial equation of diffraction,

$$\begin{equation}\left(2\mathrm{i}{\partial }_{\tilde{z}}+{\partial }_{\tilde{x}}^{2}+{\partial }_{\tilde{y}}^{2}\right)\varphi =0,\end{equation} \tag{ 1 }$$

where $\tilde{s} = s/{s}_{0}(s = x,y,z)$ is a normalized axis, x₀(=y₀) is a scaling factor of the transverse plane, ${z}_{0}(= k{x}_{0}^{2})$ is a scaling factor of the propagation axis, k = 2πn/λ is the wavenumber with the wavelength λ and the refractive index n. The electric field envelope of a 2D-Airy beam φ_Airy is a solution to equation (1) [9, 28, 29]:

$$\begin{equation}{\varphi }_{\text{Airy}}(\tilde{x},\tilde{y},\tilde{z};{\tilde{x}}_{\mathrm{d}},{\tilde{y}}_{\mathrm{d}})=\prod\limits _{\tilde{s}=\tilde{x}+{\tilde{x}}_{\mathrm{d}},\tilde{y}+{\tilde{y}}_{\mathrm{d}}}\mathrm{A}\mathrm{i}[\tilde{s}-{\tilde{z}}^{2}/4+\mathrm{i}{a}_{0}\tilde{z}]\mathrm{exp}\left[{a}_{0}\left(\tilde{s}-{\tilde{z}}^{2}/2\right)-\mathrm{i}\frac{\tilde{z}}{2}\left(\frac{{\tilde{z}}^{2}}{6}-{a}_{0}^{2}-\tilde{s}\right)\right],\end{equation} \tag{ 2 }$$

where $({\tilde{x}}_{\mathrm{d}},{\tilde{y}}_{\mathrm{d}})$ gives the lateral constant shift of the Airy beam, Ai(⋅) represents the Airy function, and a₀ is an exponential truncation factor.

The proposed Airy vortex beam is composed of two conventional Airy beams which are superimposed with one another. As shown in figure 1(a), the Laguerre–Gaussian mode with the radial index p = 0 and the azimuthal index ℓ = 1 (LG₀₁ mode) can be expressed as the superposition of Hermite–Gaussian modes of order (i, j) = (1, 0) and (0, 1) (HG₁₀ and ${\mathrm{H}\mathrm{G}}_{01}\left.\right)$ with the phase shift of π/2 [30]. Now, we can regard parts of an Airy beam as a higher order Hermite–Gaussian mode. For example, the part of an Airy beam which comprises the main lobe and its left neighbor lobe can be approximated as the Hermite–Gaussian mode of order (i, j) = (1, 0). Thus, the superposition of two Airy beams which have a relative phase of π/2 and a lateral shift, is expected to be a vortex Airy beam carrying ℓ = +1 OAM. In fact, there will be an azimuthal phase shift of 2π around the main singular point (the singular point of the main vortex lobe). In this manuscript, we call this beam a new-type vortex Airy beam with ℓ = +1 OAM (figure 1(b)). Similarly, the new-type vortex Airy beam with ℓ = −1 OAM can be generated via the superposition of two Airy beams with both a lateral shift and a −π/2-phase shift.

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We here express a new-type vortex Airy beam with ℓ = ±1 OAM as

$$\begin{equation}{\varphi }_{\text{new}-\text{type}}^{\pm }(\tilde{x},\tilde{y},\tilde{z})={\varphi }_{\text{Airy}}(\tilde{x},\tilde{y},\tilde{z};{b}_{1},{b}_{1}^{\prime })\pm \mathrm{i}{\varphi }_{\text{Airy}}(\tilde{x},\tilde{y},\tilde{z};{b}_{1}^{\prime },{b}_{1}),\end{equation} \tag{ 3 }$$

where b_k and ${b}_{k}^{\prime }$, respectively, represent the kth real zeros of Ai(⋅) and ${\mathrm{A}\mathrm{i}}^{\prime }(\cdot )\left(\right.{b}_{1}\simeq -2.34$ and ${b}_{1}^{\prime }\simeq -1.02\left.\right)$ [31]. The total power of the new-type vortex Airy beam in the beam cross section is derived from Parseval's theorem as

$$\begin{equation}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\vert {\varphi }_{\text{new}-\text{type}}^{\pm }(\tilde{x},\tilde{y},\tilde{z}){\vert }^{2}\mathrm{d}\tilde{x}\enspace \mathrm{d}\tilde{y}=\frac{1}{4\pi {a}_{0}}\enspace \mathrm{exp}\left(\frac{4}{3}{a}_{0}^{3}\right).\end{equation} \tag{ 4 }$$

An a₀ = 0 new-type vortex Airy beam, while being a non-real solution to the paraxial equation owing to its infinite power, is truly propagation-invariant. We consider this beam to be a perfect vortex Airy beam. As expected, when a₀ > 0, the new-type vortex Airy beam has a finite energy distribution.

New-type vortex Airy beams with higher-order OAM can be obtained through the superposition of more than two Airy beams. For example, a new-type vortex Airy beam with ℓ = ±2 OAM is expressed as follows:

$$\begin{equation}{\varphi }_{\text{new}-\text{type}}^{\pm 2}(\tilde{x},\tilde{y},\tilde{z})={\varphi }_{\text{Airy}}(\tilde{x},\tilde{y},\tilde{z};{b}_{2}^{\prime },{b}_{1}^{\prime })\pm \mathrm{i}{\varphi }_{\text{Airy}}(\tilde{x},\tilde{y},\tilde{z};{b}_{1},{b}_{1})-{\varphi }_{\text{Airy}}(\tilde{x},\tilde{y},\tilde{z};{b}_{1}^{\prime },{b}_{2}^{\prime }).\end{equation} \tag{ 5 }$$

We show that the new-type vortex Airy beam has a vortex lobe which remains stationary (i.e. the position of the singular point does not change with respect to the parabolic trajectory) as the distance from the focus $(\tilde{z} = 0)$ changes. This is in contrast to a conventional vortex Airy beam; the characteristic of which is shown in figure 2(a). The deformation of the intensity distribution of the conventional vortex Airy beam is attributed to the main singular point leaving the parabolic trajectory $(\tilde{x},\tilde{y})=({\tilde{z}}^{2}/4,{\tilde{z}}^{2}/4)$ with respect to propagation distance (further discussion is described in appendix A). In the case of the perfect new-type vortex Airy beam (with ${a}_{0} = 0\left.\right)$ the main singular point does follow a parabolic trajectory; this can be seen through solution of equation (3) as follows:

$$\begin{equation}{\varphi }_{\text{new}-\text{type}}^{\pm }(\tilde{x},\tilde{y},\tilde{z};{a}_{0}=0)\propto {\varphi }_{\text{new}-\text{type}}^{\pm }\left(\tilde{x}-\frac{{\tilde{z}}^{2}}{4},\tilde{y}-\frac{{\tilde{z}}^{2}}{4},\tilde{z}=0;{a}_{0}=0\right).\end{equation} \tag{ 6 }$$

Moreover, the perfect new-type vortex Airy beam maintains the same intensity distribution for any propagation distance; this is shown in figure 2(b).

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While the intensity distribution of the perfect new-type vortex Airy beam can be theoretically modeled, it is a non-real solution to the paraxial equation. Real solutions with a₀ > 0 have also been investigated in this work. In such cases, we observe that there is some deviation of the main singular point from the parabolic trajectory, as the beam propagates away from focus. The amount of deviation $\tilde{d}$ (as described in appendix B) is however smaller than that observed in a conventional vortex Airy beam in cases where a₀ < 0.27. Plots of deviation as a function of axial position $(\tilde{z})$ for different values of a₀ are shown in figure 3.

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When a₀ = 0.27, the amplitude envelope of the new-type vortex Airy beam decays to 1/e times in the main vortex lobe since ${b}_{1} > -{a}_{0}^{-1} > {b}_{2}( \simeq -4.09)$. Usually, the exponential truncation factor is experimentally made small (a₀ ≪ 1). The smaller a₀ is, the smaller the amount of deviation $\tilde{d}$ is at the same propagation distance for the new-type vortex Airy beams, in comparison with conventional vortex Airy beams. If we consider the case where a₀ = 0.04, the 1/e decay of the new-type vortex Airy beam is in its 26th side lobe since $-{a}_{0}^{-1}\simeq {b}_{27}(\simeq -25.1)$. Here, the new-type vortex Airy beam preserves the ring shape of the main vortex lobe (as shown in figure 2(d)) as it propagates, whereas the main vortex lobe of the conventional vortex Airy beam separates as it propagates (as shown in figure 2(c)). We find that for ${a}_{0}\tilde{z}$ values ≲0.1, the ring shape of the main vortex lobe is well-preserved upon propagation and the singular point follows the parabolic trajectory. This is detailed in appendix B.

As vortex Airy beams are not symmetric around the main singular point, they carry not only ℓ = +1 OAM, but also the other OAM across their profile. Here we examine the OAM spectrum of these vortex Airy beams. The OAM spectrum of the electric field envelope φ at $\tilde{z} = 0$ is defined by

$$\begin{equation}{D}_{m}(\tilde{r})=\frac{1}{2\pi }{\int }_{0}^{2\pi }\varphi \left(\tilde{x},\tilde{y},\tilde{z}=0\right){\text{e}}^{-\mathrm{i}m\phi }\enspace \mathrm{d}\phi ,\end{equation} \tag{ 7 }$$

where m represents the topological charge (or OAM in a reduced Planck constant $\hslash \left.\right)$, $\tilde{r} = \sqrt{{\tilde{x}}^{2}+{\tilde{y}}^{2}}$ and $\phi = \mathrm{arctan}(\tilde{y}/\tilde{x})$ are the normalized radius and the azimuthal angle of the polar coordinates in the transverse plane respectively [32]. Figure 4 shows the absolute amplitude distributions of the OAM spectra of the perfect conventional vortex Airy beam with ℓ = +1 OAM and the perfect new-type Airy vortex beam with ℓ = +1 OAM. In the inner part of the main vortex lobe $(\tilde{r} \leqslant {b}_{1}-{b}_{2} \simeq 1.75)$, both of the beams mainly contain m = 1 optical vortex modes. Figure 5 shows a plot of the OAM spectrum of the inner part of the main vortex lobe of both conventional and new-type vortex Airy beams, where the intensity (I_m ) is derived as ${I}_{m} = {\int }_{0}^{{b}_{1}-{b}_{2}}\vert {D}_{m}{\vert }^{2}\tilde{r}\enspace \mathrm{d}\tilde{r}/{\sum }_{n}{\int }_{0}^{{b}_{1}-{b}_{2}}\vert {D}_{n}{\vert }^{2}\tilde{r}\enspace \mathrm{d}\tilde{r}$. The mode purity of the perfect new-type vortex Airy beam is 91%, while that of the perfect conventional Airy vortex beam is 81%. Thus, the new-type vortex Airy beam is superior in terms of OAM mode purity as well as beam propagation characteristics.

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We experimentally investigated the propagation dynamics of new-type vortex Airy beams, generated from an in-house-built Ti:sapphire regenerative amplifier pulsed laser system. The output from this laser was horizontally polarized and had a Gaussian spatial profile with a beam radius of 3 mm. Spectral purity was maintained by passing the beam through a bandpass filter (central wavelength, 800 nm; bandwidth, 5 nm). The laser beam was then incident on a spatial light modulator (SLM) which acted as a phase mask. The details of the phase mask displayed on the SLM can be found in appendix C. The spatially phase-modulated laser beam was then transformed into a new-type vortex Airy beam by a converging lens $\left(\right.f = 300$ mm). Using a CMOS imaging sensor in conjunction with a mechanical stage, we recorded the profiles of the generated vortex Airy beams at different propagation distances, in the vicinity of the focus. For comparison purposes, we generated a conventional vortex Airy beam by implementing a phase mask with a sum of a cubic and a spiral phase distribution on the SLM [33].

Figure 6 shows the experimental propagation dynamics of a finite-energy new-type vortex Airy beam with ℓ = +1 OAM and a finite-energy conventional vortex Airy beam with ℓ = +1 OAM. The factors of the transverse plane, the propagation axis and the exponential truncation were evaluated to be x₀ = y₀ = 0.06 mm, z₀ = 30 mm and a₀ = 0.04, respectively. The propagation dynamics of the conventional vortex Airy beam (figure 6(a)) and the new-type vortex Airy beam (figure 6(b)) clearly agree well with the numerical simulations shown in figures 2(c) and (d), respectively. The main vortex lobe of the finite-energy conventional vortex Airy beam deformed at $\tilde{z}=1$ and finally divided into two spots at $\tilde{z}=2$, which as mentioned is attributed to the main singular point deviating from the parabolic trajectory. In contrast, the main vortex lobe of the new-type vortex Airy beam maintained its ring shape even at $\tilde{z}=2$. This is consistent the main singular point of this new-type vortex Airy beam following the parabolic trajectory (as expected, given ${a}_{0}\tilde{z}$ was small).

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In order to show that a new-type vortex Airy beam has an ℓ = 1 singular point in the main vortex lobe, we implemented an interference measurement at $\tilde{z}=0$ with a reference beam (figure 7). This was done using the random mask encoding method [21] wherein both object and reference beams were simultaneously generated from the same phase mask. The interference image had a two-pronged fork pattern in the main vortex lobe (figure 7(a)) and this was consistent with that predicted via numerical simulation (figure 7(b)). Thus, the dominant topological charge of the main vortex lobe was +1 [34–36], which was consistent with the OAM spectrum shown in figure 5. These results indicate that these new-type vortex Airy beams comprise a new family of vortex Airy beams. We note that there are other families of vortex Airy beams such as circular [37], elliptical [38, 39], and symmetric [40–42] vortex Airy beams. Since having an abruptly autofocusing property [43, 44], these beams are totally different from the new-type vortex Airy beams.

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We discuss whether the new-type vortex Airy beams can be generated by the astigmatic transformation. Astigmatic mode converters can output an LG mode from a certain HG mode [45, 46]. From the analogy shown in figure 1, one may think that the new-type vortex Airy beam can be generated by the astigmatic transformation of an Airy beam rotated by a 45 degree angle. The astigmatic transformation is based on the synthesize of eigenmodes with phase shifts among them; in addition, the eigenmodes share a common beam axis. Since being a superposition of Airy beams whose beam axes are parallel shifted from each other, a new-type vortex Airy beam does not match the beam generated through the astigmatic transformation of an Airy beam rotated by a 45 degree angle. In contrast to that, the specular Airy beams, having mirror symmetry in its intensity profile, can be transformed into the vortex mode through an astigmatic mode converter [47]. The astigmatic transformation of an Airy beam and its family is an interesting topic of future work.

We comment on future applications of the new-type vortex Airy beams. It has been pointed out that the Airy beam carries relatively more optical power in the main lobe compared with a Bessel beam [16]. The power fractions in the main vortex lobe of the new-type vortex Airy beam are calculated to be 28% (for $a = 0.04\left.\right)$ and 71% (for $a = 0.08\left.\right)$. Thus, we expected the new-type vortex Airy beam can carry high power in the main lobe, which benefits microscopy in terms of improving the signal noise ratio and applications need high power laser beam. Moreover, the dark spot of the singular point is well-preserved in the region of ${a}_{0}\vert \tilde{z}\vert \lesssim 0.1$, although the shape of the main vortex lobe is not perfectly symmetric especially on propagation. The new-type vortex Airy beams are expected to possess the self-healing properties since they are indeed composed of Airy beams, while it is needed to examine of their self-healing properties in future work. Thus, they can be useful for the STED beam in STED-SPIM. Light-sheet imaging [11–14], material processing [48–50] and nonlinear optics [51] may be another fruitful direction.