## Abstract

We study theoretically and experimentally the varying polarization states and intensity patterns of self-accelerating vector beams. It is shown that as these beams propagate, the main intensity lobe and the polarization singularity gradually drift apart. Furthermore, the propagation dynamics can be manipulated by controlling the beams’ acceleration coefficients. We also demonstrate the self-healing dynamics of these accelerating vector beams for which sections of the vector beam are being blocked by an opaque or polarizing obstacle. Our results indicate that the self-healing process is almost insensitive for the obstacles’ polarization direction. Moreover, the spatial polarization structure also shows self- healing properties, and it is reconstructed as the beam propagates further beyond the perturbation plane. These results open various possibilities for generating, shaping and manipulating the intensity patterns and space variant polarization states of accelerating vector beams.

## Introduction

Airy beams, as part of the group of wave-functions coined sometimes “non-diffracting beams” or “self-accelerating beams” have drawn significant scientific interest, due to their shape-preserving and accelerating propagation dynamics in space or in time. Ever-since their first introduction^{1}, Airy beams have been investigated in a wide range of physical systems^{2,3,4,5}, in various symmetries^{6,7}, in linear and nonlinear mediums^{4,8}, and suggested for many applications such as micro-particles manipulations^{9}, optical routing^{10} and super resolution imaging^{11}. The research of Airy beams has also triggered the developing of other forms of accelerating beams^{12,13,14,15}. One of the most intriguing properties of self-accelerating beams is their ability to self-heal from blocking obstacles^{16}, a property now researched in the context of other beam types and polarizations^{17,18,19}. The common way to characterize accelerating beams is by tracing the trajectory of the highest intensity lobe as it curves while propagating through space. To date, in the vast majority of reported experiments, a linearly polarized Gaussian beam has been used as the excitation source. While such beams are probably the most common form of coherent light source, we are recently witnessing a growing interest in light with a spatially varying polarization, generally known as vector beams^{20}.

One of the many types of vector beams is the radially polarized mode, which exhibits a spatially varying polarization, with the transverse electric field directed outward from the optical axis, and which is an axial-symmetric solution to Maxwell’s equations^{21,22,23}. Now implemented in different systems, radially polarized light was proved to be efficient in nano-focusing^{24,25,26,27,28,29}, microscopy and particle manipulations^{30}.

The combination of Airy patterns and vector beams (shown in Fig. 1), which can be defined as Airy vector beams, has been somewhat overlooked by researchers, with very little reports so far^{7,31}. In such beams, the interplay between the Airy pattern and polarization states, symmetry differences and different propagation dynamics, rises many interesting outcomes.

In this report we present theoretical predictions along with numerical and experimental results showing the propagation dynamics, self-healing mechanisms and polarization distribution of radially-polarized vector Airy beams (RAVB). While RAVBs are the basic form of accelerating vector beams, this research can be further extended to any accelerating wave packet and any type of polarization structure.

This paper will show that when a radially polarized Gaussian beam illuminates a cubic phase mask and focused, RAVBs are generated at the focal plane. As illustrated in Fig. 1c, the main lobe of RAVB’s is divided into two lobes by a radially shaped polarization singularity (zero-order topological charge^{32}). The checkerboard pattern of the RAVB tail at the focal plane is also formed by the radial polarization as every two adjacent lobes interfere destructively with anti-phase relations.

Furthermore, as the beam propagates, it will be shown that the usual form of accelerating Airy beams can be observed, with a linearly polarized main lobe appearing and accelerating in space, while the polarization singularity is broadened and drifted into the beam’s tail.

## Theoretical Background

Radially polarized light can be described as a solution of the wave-equation in cylindrical coordinate system^{20,21,22,23}. Its transverse electric field component is given in the form.

Where *J*_{1}(*x*) is the first order radial Bessel function, *r* and are the radial coordinate and the radial unit vector respectively, and k is the propagation wavenumber in the propagation (z) direction. The radially polarized beam has a distinct signature in its Stokes parameters, namely the four-lobed structure of S1 and S2 parameters as depicted in Fig. 2 (top row). This unique signature will be used further in this report to establish the polarization properties of radially polarized Airy beams.

The diffraction of a radially polarized beam by a grating will result in the creation of multiple diffraction orders, each of them still holds radial polarization distribution, as demonstrated previously^{29}.

An Airy beam is created by diffracting light through a binary cubic phase-grating with transmission function of the following shape: ^{33,34}. This transmission function creates the diffraction pattern presented in Fig. 1c, which is noticeably different from the linearly polarized Airy pattern presented in Fig. 1b, by the splitting of the main lobe, and the “checkerboard” pattern of the beam’s tail. These differences are direct outcomes of the polarization structure, and not of the intensity distribution alone. Under the paraxial approximation, this diffraction pattern may be analyzed by the convolution theorem^{35}.

Where F(z) representsthe optical propagation by Fourier transform to plane z = f, where f is the focal length of the lens, is the incident radially polarized beam and T(x, y) is the transmission function of the diffractive element generating the Airy beam. It is understood that represents the diffraction of the radially polarized beam and is the Airy function. While the diffraction of most beams is characterized by broadening, the Airy beam will preserve its shape, and shift laterally^{1}. The convolution in Eq. 2 means that the propagation of the combined RAVB is the propagation of the two functions (Radially polarized incident beam, and Airy pattern), superimposed. Therefore, the Airy’s curved trajectory will result in a shift of the Airy pattern with respect to the origin of the radial polarization distribution as shown in Fig. 1(a). Similar separation has been observed with Airy-Vortex beams where the phase singularity is separated from the Airy’s main lobe^{31,36}. In this paper we observed and measured for the first time, this separation dynamics for the polarization singularity and highest intensity lobe. It must be mentioned that the singularity in Vortex-Airy beams is originated from a singular phase distribution, while in our case, it evolves directly from the polarization structure, which will yield different propagation characteristics in non-isotropic media.

## Experiments Description

Our experimental setup is fairly simple (see Fig. 1a): a He-Ne laser beam (*λ* ≅ 633 nm) is passed through a radial polarization converter (Altechna) followed by a binary phase grating and a lens. The binary phase grating is fabricated on a double-polished glass substrate by standard photolithography techniques, with a transmission function defined by:

We have fabricated several gratings with different values of the parameters P_{1} to P_{4} to create Airy beams with different characteristics. Parameter P_{1} inversely controls the diffraction angle in the horizontal direction, while the combination of P_{2} along with P_{3} or P_{4} controls the Airy acceleration in the horizontal or vertical directions, respectively. The values of parameters P_{1} to P_{4} were chosen to fulfill the following requirements: (a) easy fabrication by standard photolithography procedures, which dictates that feature size is kept above 2 μm, and is also beneficial as Airy beams of high spatial frequency tend to smear over short propagation distances; (b) clear detection of all Airy lobes by our camera (i.e. separation between Airy lobes at the focal plane must be significantly larger than pixel size); and (c) the full diffraction pattern must fit into the CCD area. Typical numbers for these parameters (considering the use of standard lenses with focal lengths of 5–20 cm) are 1 < P_{1} < 10, 1000 < P_{2} < 10^{5}, 1 < P_{3},P_{4} < 10, where the lateral coordinates x and y are measured in meters. The camera used to detect the Airy pattern was a sCMOS camera (Hamamatsu Orca Flash 4 V2), which is mounted on a rail to allow capturing the Airy beam at the focal plane and beyond, without losing alignment.

Stokes measurements at various distances were made by a rail-mounted set of a linear polarizer and a quarter wave plate.

## Experimental Results

Our experimental results are presented in Figs 2, 3, 4 and 5. Figure 2 presents the intensity distribution and Stokes parameters S_{1} and S_{2} of the RAVB at the focal plane and 10 cm beyond the focal plane. We observe experimentally the splitting of the main lobe and the checkerboard pattern of beams’ tail. From observing the Stokes parameters, we notice that the radial polarization structure is indeed imprinted in all the Airy lobes, and the checkers-board structure is the result of destructive interference of anti-phase fields of adjacent lobes. Next, we measured the Stokes parameters of a beam propagating beyond the focal plane, presented in Fig. 2. From these measurements we learn that the change in the beam’s shape is the result of the gradual separation between the radial polarization structure and the Airy pattern. This understanding allows us to study the propagation dynamics of the intensity and polarization of these beams, presented in Figs 3, 4 and 5. As shown in Fig. 3, the radial polarization center is diffracted at the angle expected according to the grating periodicity, whereas the Airy lobes are accelerated and we notice the predicted separation between these structures, i.e. the polarization singularity of the radial beam drifts away from the main lobe towards the RAVB’s tail. This effect is much more pronounced when the acceleration is large, as depicted in Fig. 4, which compares numerically and experimentally between similar beams with a factor of 40 in their acceleration constant. It is seen that for high-acceleration beam (beam A in Fig. 4’s top panels) the separation of the radial polarization and Airy pattern occurs over much shorter distances. Finally, self-healing experiments were made to fully understand the propagation characteristics of RAVBs. These results, alongside with numerical calculations are shown in Fig. 5.

The self-healing experiments were done by introducing a perturbation to the beam at the focal plane and examining the propagation of the beam as described previously.

We have considered two types of perturbations: A - polarizing the main lobe, B - completely blocking the main lobe. Both cases are presented in Fig. 5. The first case was studied numerically, while the latter was studied both numerically and experimentally.

The first row of Fig. 5 shows the propagation of the non-perturbed RAVB. This will be used as a reference for the following cases. The second row of Fig. 5 shows the RAVB after applying a uniform linear polarizer in the direction of the white arrow. Here, the radial pattern is essentially lost and the beam resembles the more conventional linearly polarized Airy beam.

When polarizing the main lobe alone, as described in the middle row of Fig. 5, we only introduce a small perturbation to the beam, and the reconstructed pattern is nearly indistinguishable comparing with the unperturbed case (top row).

The last two rows (numerical calculation and experimental measurements, respectively), describe the case in which the main lobe of the beam is completely blocked. As can be seen, in spite of the significant perturbation, the RAVB shows fantastic polarization self- healing characteristics, as evident by the stokes parameters in which the four-lobed pattern can be observed after propagating 15 centimeters away from the focal plane. This result can be explained by the Fourier-optics approach: as the main lobe is blocked in the Fourier plane, the beam will maintain its spatial structure, but will suffer a decrease in contrast. This is because each point along the reconstruction plane is being generated by coherent contributions from all points along the beam at the obstacle plane. The polarization distribution is thus only weakly affected by the blocking, as the other lobs maintain their polarization structure and contribute to the vector field reconstruction as the beam propagates away from the obstacle. We can thus conclude that the information of the local radial polarization singularity is carried in a nonlocal manner by the entire wave-front, i.e., the vectorial propagation dynamics is not described by a local vectorial variation dynamics near the main lobe, but instead by the vectorial states of the light which emerges from the additional sidelobes^{37}.

## Conclusions

We have reported on the propagation and self-healing dynamics of Airy vector beams. These radially polarized Airy beams belong to the broader family of accelerating vector beams. While such beams are rarely considered, they potentially hold many intriguing phenomena. Our results show that for a freely-propagating beam, the acceleration of the Airy beam has a major role in the evolution of the beam’s polarization state.

For RAVB passing an obstacle, the self-healing process is polarization insensitive - a single reconstructed main lobe appears in all cases examined, with polarization properties similar to those of a freely-propagating beam.

The polarization reconstruction of the self-healed RAVB is an important result. The RAVB feature which enables this reconstruction is the singularity data imprinted in the Airy beam’s tail. The reconstruction therefore shows that the airy beam’s tail carries the full information (amplitude, phase and polarization) needed to reconstruct the beam, and not the information of intensity alone^{37}.

The polarization of light is a degree of freedom often overlooked, but can be of great significance in many fields such as beam-shaping and micro-particle manipulations. It is our hope that this work will be followed by additional demonstrations to provide further advancements in the field. For example, one may consider generating arbitrary accelerating wave packets with even more complex polarizations structures. Furthermore, the propagation dynamics of these beams can be investigated inside birefringent media, for which each polarization component may have a different phase velocity. One may also consider the case where the beam trajectory goes beyond the scalar approximation. Finally, it might be of interest to test the self-focusing properties of these beams inside Kerr type nonlinear medium, which can also affect the polarization and intensity patterns of the RAVBs.

## Additional Information

**How to cite this article**: Bar-David, J. *et al.* Unveiling the propagation dynamics of self-accelerating vector beams. *Sci. Rep.* **6**, 34272; doi: 10.1038/srep34272 (2016).

## References

- 1.
Siviloglou, G. A., Broky, J., Dogariu, A. & Christodoulides, D. N. Observation of Accelerating Airy Beams.

*Phys. Rev. Lett.***99**, 213901 (2007). - 2.
Voloch-Bloch, N., Lereah, Y., Lilach, Y., Gover, A. & Arie, A. Generation of electron Airy beams.

*Nature***494**, 331–335 (2013). - 3.
Polynkin, P., Kolesik, M., Moloney, J. V., Siviloglou, G. A. & Christodoulides, D. N. Curved Plasma Channel Generation Using Ultraintense Airy Beams.

*Science***324**, 229–232 (2009). - 4.
Bekenstein, R., Schley, R., Mutzafi, M., Rotschild, C. & Segev, M. Optical simulations of gravitational effects in the Newton-Schrodinger system.

*Nat Phys***11**, 872–878 (2015). - 5.
Fu, S., Tsur, Y., Zhou, J., Shemer, L. & Arie, A. Propagation Dynamics of Airy Water-Wave Pulses.

*Phys. Rev. Lett.***115**, 34501 (2015). - 6.
Vaveliuk, P., Lencina, A., Rodrigo, J. A. & Matos, O. M. Symmetric Airy beams.

*Opt. Lett. OL***39**, 2370–2373 (2014). - 7.
Zhang, P.

*et al.*Trapping and guiding microparticles with morphing autofocusing Airy beams.*Opt. Lett. OL***36**, 2883–2885 (2011). - 8.
Ellenbogen, T., Voloch-Bloch, N., Ganany-Padowicz, A. & Arie, A. Nonlinear generation and manipulation of Airy beams.

*Nat Photon***3**, 395–398 (2009). - 9.
Baumgartl, J., Mazilu, M. & Dholakia, K. Optically mediated particle clearing using Airy wavepackets.

*Nat Photon***2**, 675–678 (2008). - 10.
Rose, P., Diebel, F., Boguslawski, M. & Denz, C. Airy beam induced optical routing.

*Applied Physics Letters***102**, 101101 (2013). - 11.
Jia, S., Vaughan, J. C. & Zhuang, X. Isotropic three-dimensional super-resolution imaging with a self-bending point spread function.

*Nat Photon***8**, 302–306 (2014). - 12.
Kaminer, I., Bekenstein, R., Nemirovsky, J. & Segev, M. Nondiffracting Accelerating Wave Packets of Maxwell’s Equations.

*Phys. Rev. Lett.***108**, 163901 (2012). - 13.
Greenfield, E., Segev, M., Walasik, W. & Raz, O. Accelerating Light Beams along Arbitrary Convex Trajectories.

*Phys. Rev. Lett.***106**, 213902 (2011). - 14.
Alpmann, C., Bowman, R., Woerdemann, M., Padgett, M. & Denz, C. Mathieu beams as versatile light moulds for 3D micro particle assemblies.

*Optics Express***18**, 26084 (2010). - 15.
Bandres, M. A. & Rodríguez-Lara, B. M. Nondiffracting accelerating waves: Weber waves and parabolic momentum.

*New J. Phys.***15**, 13054 (2013). - 16.
Broky, J., Siviloglou, G. A., Dogariu, A. & Christodoulides, D. N. Self-healing properties of optical Airy beams.

*Opt. Express, OE***16**, 12880–12891 (2008). - 17.
Milione, G.

*et al.*Measuring the self-healing of the spatially inhomogeneous states of polarization of vector Bessel beams.*J. Opt.***17**, 35617 (2015). - 18.
Vyas, S., Kozawa, Y. & Sato, S. Self-healing of tightly focused scalar and vector Bessel–Gauss beams at the focal plane.

*J. Opt. Soc. Am. A JOSAA***28**, 837–843 (2011). - 19.
McLaren, M., Mhlanga, T., Padgett, M. J., Roux, F. S. & Forbes, A. Self-healing of quantum entanglement after an obstruction.

*Nature Communications***5**, 3248 (2014). - 20.
Zhan, Q. Cylindrical vector beams: from mathematical concepts to applications.

*Adv. Opt. Photon. AOP***1**, 1–57 (2009). - 21.
Wolf, E. Electromagnetic Diffraction in Optical Systems. I. An Integral Representation of the Image Field.

*Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences***253**, 349–357 (1959). - 22.
Richards, B. & Wolf, E. Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System.

*Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences***253**, 358–379 (1959). - 23.
Schimpf, D. N., Putnam, W. P., Grogan, M. D. W., Ramachandran, S. & Kärtner, F. X. Radially polarized Bessel-Gauss beams: decentered Gaussian beam analysis and experimental verification.

*Opt. Express OE***21**, 18469–18483 (2013). - 24.
Quabis, S., Dorn, R., Eberler, M., Glöckl, O. & Leuchs, G. Focusing light to a tighter spot1.

*Optics Communications***179**, 1–7 (2000). - 25.
Dorn, R., Quabis, S. & Leuchs, G. Sharper Focus for a Radially Polarized Light Beam.

*Phys. Rev. Lett.***91**, 233901 (2003). - 26.
Lerman, G. M., Yanai, A. & Levy, U. Demonstration of Nanofocusing by the use of Plasmonic Lens Illuminated with Radially Polarized Light.

*Nano Lett*.**9**, 2139–2143 (2009). - 27.
Sheppard, C. J. R. & Choudhury, A. Annular pupils, radial polarization, and superresolution.

*Appl. Opt. AO***43**, 4322–4327 (2004). - 28.
Yanai, A. & Levy, U. Plasmonic focusing with a coaxial structure illuminated by radially polarized light.

*Opt. Express OE***17**, 924–932 (2009). - 29.
Bar-David, J., Lerman, G. M., Stern, L., Mazurski, N. & Levy, U. Generation of a periodic array of radially polarized Plasmonic focal spots.

*Opt. Express***21**, 3746–3755 (2013). - 30.
Zhan, Q. Trapping metallic Rayleigh particles with radial polarization.

*Opt. Express OE***12**, 3377–3382 (2004). - 31.
Zhou, J., Liu, Y., Ke, Y., Luo, H. & Wen, S. Generation of Airy vortex and Airy vector beams based on the modulation of dynamic and geometric phases.

*Opt. Lett. OL***40**, 3193–3196 (2015). - 32.
Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. & Woerdman, J. P. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes.

*Phys. Rev. A***45**, 8185–8189 (1992). - 33.
Dai, H. T., Sun, X. W., Luo, D. & Liu, Y. J. Airy beams generated by a binary phase element made of polymer-dispersed liquid crystals.

*Opt. Express OE***17**, 19365–19370 (2009). - 34.
Lee, W.-H. Binary computer-generated holograms.

*Appl. Opt., AO***18**, 3661–3669 (1979). - 35.
Goodman, J. W.

*Introduction to Fourier Optics*. (Roberts and Company Publishers, 2005). - 36.
Dai, H. T., Liu, Y. J., Luo, D. & Sun, X. W. Propagation dynamics of an optical vortex imposed on an Airy beam.

*Opt. Lett. OL*.**35**, 4075–4077 (2010). - 37.
Kaganovsky, Y. & Heyman, E. Wave analysis of Airy beams.

*Optics Express***18**, 8440 (2010).

## Author information

## Author notes

- Jonathan Bar-David
- & Noa Voloch-Bloch

These authors contributed equally to this work.

## Affiliations

### Department of Applied Physics, The Benin School of Engineering and Computer Science, The Center for Nanoscience and Nanotechnology, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel

- Jonathan Bar-David
- , Noa Voloch-Bloch
- , Noa Mazurski
- & Uriel Levy

## Authors

### Search for Jonathan Bar-David in:

### Search for Noa Voloch-Bloch in:

### Search for Noa Mazurski in:

### Search for Uriel Levy in:

## Contributions

J.B.-D and N.V.-B conducted the experiments, analyzed the results and wrote the manuscript. Diffraction gratings were designed and fabricated by J.B.-D., N.M. and U.L. oversaw the experiments and data analysis, and reviewed the manuscript.

## Competing interests

The authors declare no competing financial interests.

## Corresponding author

Correspondence to Uriel Levy.

## Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/