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Abstract
Airy beams have provided exciting inspiration in the field of optical communication, particle manipulation, and imaging. We investigate the propagation properties of the exponential truncation Airy beams (ETABs) on constant Gaussian curvature surfaces (CGCSs) in this paper. The analytical expression of the electric field of ETABs propagating on the CGCSs is derived. It shows that the equivalent periodical accelerations of the trajectories of ETABs on the curved surface are always larger than the constant one on the flat surface because the CGCSs have a strong focusing ability. For the same reason, the non-diffraction propagation of ETABs is found when the focusing ability of the CGCSs is strong enough. Moreover, we investigate the self-healing length of ETABs on CGCSs and explore that the ability of self-healing is related to the geometry of CGCSs besides the width of the block and the size of the beam. The self-healing length gets larger with the increase of radius of CGCSs and finally consists with that on the flat surface. These propagation characteristics are different from those in the flat space and are useful for the future applications of ETABs in particle manipulation on waveguides, light-sheet fluorescence microscopy, curved nanophotonics, and so on.
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1. Introduction
Airy wave packets were first proposed in 1979 by Berry and Balazs as non-spreading wave packets with infinite energy [1]. It was not until 2007, Siviloglou $et$ $al.$ found exponential truncated Airy beams theoretically and experimentally [2,3], that Airy beams were applied to optics. Last few years, several new types of Airy beams have been investigated, like circular Airy Gaussian vortex beams [4,5], symmetric Airy beams [6,7], partially coherent Airy beams [8,9], radially polarized Airy beams [10,11] and so on. Airy beams have asymmetric wave-front phase and energy distribution, which leads to several unique properties, such as non-diffraction, self-focusing [12,13], self-acceleration, and self-healing. Each property has been researched deeply in recent years. Many investigations have been carried out on self-acceleration, including the trajectories controlling [14–18] and trajectories in different mediums [19–21]. Concerning self-healing, many pieces of research have been done, such as the self-healing properties of various types of Airy beams [22–25], Poynting vector explaining this unique property [26], and quantitative descriptions to describe this property [27,28]. Based on these superior properties, a lot of potential applications of Airy beams have been discovered in the last decade, like the areas of light bullets [29], optical micromanipulation [30], tomographic microscopy [31], and generating supercontinuum [32].
Recently, the study of laser transmission in curved space has aroused wide interest. The field equations of the Gaussian beams in curved space are derived from Maxwell equations [33]. The Gaussian beams show the properties of focusing and refocusing periodically, propagating without diffraction on the positive CGCSs [34]. The CGCSs can be regarded as a kind of fractional Fourier transform lens, and different shapes of curved surfaces affect the degree of refocus and non-diffraction properties of Gaussian beams [35]. Airy beams show some excellent properties in flat space, and we further wonder how the curved surfaces work on Airy beams. Specifically, the propagation of Airy beams along non-geodesic trajectory and the periodical change of the wave structure are calculated by wave equations on the CGCSs [36]. The non-geodesic trajectory of Airy beams can be explained by the interaction between the interference and the curvatures of the curved surfaces. And the periodical change of the wave structure can be explained by the fact that the curved surface regards as a lens for beams propagation [37]. Transmission properties of several kinds of beams in curved space have been performed on the waveguide experimentally [37–39]. Recently, a hollow parabolic waveguide has been realized by multiphoton polymerization in nanophotonic [40]. This polymerization method can be applied to produce more types of curved surfaces. Then some theoretical works can be turned into reality based on these surfaces in the future. But until now, the self-acceleration and self-healing properties of Airy beams on the curved surfaces are still under inadequate research due to the complicated structure of the electric field.
In this paper, we focus on the self-acceleration and self-healing properties of ETABs propagating on the CGCSs. We use the ABCD matrices of the CGCSs and Collins formula to describe the propagation of Airy beams on curved surfaces under the paraxial approximation. It can be deduced that the analytical expression of the electric field of Airy beams propagating on the CGCSs is very similar to the expression in flat space. According to this expression, we further investigate the propagation trajectories of ETABs on CGCSs and discuss the dependence of the trajectories of ETABs on the parameters of the CGCSs. We compare the propagation properties on curved surfaces with those on flat surfaces. The equivalent self-acceleration of Airy beams on curved surfaces is always larger than that on flat surfaces, and the self-healing properties of Airy beams on curved surfaces are also stronger.
2. Theoretical model of ETAB propagation on the CGCSs
The CGCSs are kinds of rotating surfaces. The radius between the rotation axis and the surface can be written as $\rho =r_0 cos(h/r)$, where $r_0$ and $r$ represent radius parameters of the CGCSs. $h$ is the arc length from the alternative point towards the maximum rotational circuit. $\theta$ is the angle between the input plane and the output plane revolving around the rotation axis. Based on these definitions, the ABCD matrices of CGCSs can be described as follows, which is deduced in our previous research [41]:
3. Transmission characteristics of ETABs on the CGCSs
Now we can analyse the expression of the trajectories of ETABs based on Eq. (3), firstly. The effect of the radius $r$ of the CGCS on the propagation trajectory is shown in Fig. 1, when $\lambda =632.8nm$, $h_0=0.1mm$ and $\alpha =0.3$. Figure 1(a1), (b1) and (c1) show the geometries of the CGCSs with $r=0.5r_0$, $r=r_0$ and $r=2r_0$ respectively. Meanwhile, Fig. 1(a2), (b2) and (c2) show the propagation trajectories of ETABs on the CGSSs. From Fig. 1, it is found that the degree of the diffraction of the propagating beam enlarges with the increase of the ratio $r/r_0$. This phenomenon can be attributed to the stronger focus ability of the curved surface when the ratio $r/r_0$ is smaller. Also, the intensity of the ETABs along the propagation axis is a periodic function with period $\theta =2\pi r/r_0$. The period is twice as the other beams propagating on the CGCSs [41], because of the special self-bending property of ETABs.
Fig. 1. The elements of left column are the geometries of the CGCSs. The elements of the right column are the propagation trajectories of ETABs. (a1) - (a2) $r=0.5r_0$. (b1) - (b2) $r=r_0$. (c1) - (c2) $r=2r_0$.
Secondly, we investigate some details of the self-acceleration property of Airy beams on curved surfaces based on Eq. (3). The expression of non-geodesic propagation trajectories of the main lobe can be written as below in $h'-\theta$ plane:
Thirdly, as spherical surfaces are the simplest curved structure and spherical waveguides have been commonly realized [34,37], the non-diffraction property of beams on spherical surfaces ($r=r_0$) is discussed in the following. The ability of non-diffraction can be described as transmission invariance ($TI$), which is a squareness ratio [42] defined as:
From Fig. 2, when $\alpha =0$, the infinite Airy beams still keep non-diffraction during their propagation on the curved surfaces, and the focusing effect of CGCSs does not destroy it. That is because for the infinite Airy beams, the different geometries of CGCSs do not affect the maxmum transverse intensity of the main lobe during its propagation. Once $\alpha \ne 0$, diffraction property of Airy beams appears in flat space, but the Airy beams emerge transmission invariance on curved surfaces with small radius $r$, as Fig. 2 shows. This is because the curved surfaces with smaller radii have a stronger focusing effect. Figure 2 shows if the radius $r$ is small enough, the degree of non-diffraction remains invariant even $\alpha$ ranges from 0 to 1. The transmission invariance can be explained by that the focusing effect of the curved surfaces compensates for the diffraction property of ETABs. When they reach a balance, the non-diffraction propagation of beams appears on the CGCSs.
Finally, we investigate the self-healing property of ETABs to explore more novel characteristics of ETABs caused by curved surfaces. Gaussian aperture is chosen to block the main lobes of Airy beams. The Gaussian aperture can be described as:
where $R$ presents the width of the block. In such a condition, the electric field of ETABs in the input plane can be rewritten as:Airy beams can be regarded as a bundle of oblique rays, and the main lobe lacks some rays after blocking. These lacked rays will appear again after propagating a distance which is called self-healing distance [35]. In this paper, the self-healing distance is defined by the parameter $\theta _h$. The relationship between the width of the block and the self-healing distance is given by $R = \frac {{r_0^{2}\theta _h^{2}}}{{4{k^{2}}h_0^{3}}} + \frac {{r_0^{4}\theta _h^{4}}}{{8{k^{2}}{r^{2}}h_0^{3}}} + O\left ( {\theta _h^{6}} \right )$. The radius of the curved surface affects the self-healing length. If $r\to \infty$, the curved surface unfolds to a flat one, the self-healing length converts to a constant $z = {r_0}\theta _h = 2k\sqrt {h_0^{3}R}$. The constant matches with the consequence on the flat surface obtained before [30]. When ETABs transmit on the CGCSs, the corresponding self-healing length approximately is ${\left ( {{r_0}{\theta _h}} \right )^{2}} = {r^{2}}\left [ {\sqrt {1 + 8{k^{2}}h_0^{3}R/r^{2}} - 1} \right ]$. It shows that the self-healing length gets larger with the increase of $r$ until the self-healing length becomes a constant $2k\sqrt {h_0^{3}R}$. This phenomenon can be interpreted by the fact that the CGCS with smaller $r$ has the stronger focusing effect. And this effect leads the energy of beams in the side lobes to flow to the main lobe faster.
The self-healing ability is usually described as similarity $F$ [27,28]:
As shown in Fig. 4(a), the smaller width of the block indicates a shorter self-healing distance and the stronger ability of self-healing. Moreover, from Fig. 4(b), the larger truncation factor means that the energy of the main lobe makes up a larger proportion of the whole energy, so a larger proportion of energy of ETABs is blocked at the beginning of the propagation. Airy beams with larger truncation factors are also accompanied with a larger degree of diffraction during propagation. Based on these two reasons, the self-healing abilities $F$ of the beams with different $\alpha$ almost reach the same maximum after propagating the same distance as shown in 4(b), which is unlike the phenomena in flat space [23].
4. Conclusion
In conclusion, by using the ABCD matrix and Collins formula, an analytical expression of Airy beams on the CGCSs is deduced. The expression in curved space is very similar to the expression on the flat plane. Based on this, we demonstrate that Airy beams propagate along non-geodesic trajectories on the CGCSs, and the trajectories can be controlled. There still are self-acceleration and self-healing properties when ETABs propagate on CGCSs. However, several novel differences compared with that on the flat surfaces appear in curved space. The periodical accelerations of the trajectories of ETABs on the curved surfaces are always larger than that on the flat surface. And the self-healing length gets larger by the increase of radius of CGCSs until the self-healing length becomes a constant equalling to that on the flat surface. More deeply, it shows that the non-diffraction propagation of ETABs on the CGCSs occurs because of the focusing effect of the CGCSs.
In the end, we propose some future use of Airy beams. At the micro level, Airy beams can capture and manipulate particles on the waveguide, especially in biology for the Airy beams can bypass obstacles to locate cells and molecules. Light-sheet microscopy can be optimized for controllable trajectories based on our research. Also, Airy beams can bring curved nanophotonic more attractive phenomena for their special properties. Concerning the macro level, lots of the black holes and planets have a large mass and curve the space-time, so maybe Airy beams can be used to explore them by surveying and mapping the trajectories of Airy beams.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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