## Abstract

Nondiffracting light beams have been attracting considerable attention for their various applications in both classical and quantum optics. Whereas substantial investigations on generation of the nondiffracting beams were made, their lateral dimension is much larger than optical wavelength. Here we present both theoretically and experimentally a study of the generation of nondiffracting light beams at deep-subwavelength scale. The highly localized light field is a result of in-phase interference of high-spatial-frequency waves generated by optical sharp-edge diffraction with a circular thin film. It is shown that the generated beam can maintain its spot size below the optical diffraction limit for a distance of up to considerable Rayleigh range. Moreover, the topological structure of both the phase and polarization of a light beam is found to be preserved when it passes through the diffractive configuration, which enables generating nondiffracting vortex beams as well as transversely polarized vector beams at deep-subwavelength scale. This work opens a new avenue to manipulate higher-order vector vortex beams at subwavelength scale and may find intriguing applications in subwavelength optics, e.g., in superresolution imaging and nanoparticle manipulation.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

The spreading of a light beam is a universal property of a wave owing to its spatial diffraction and temporal dispersion in a dispersive medium. Overcoming optical diffraction/dispersion so as to obtain linearly nonspreading light beams is intriguing and has long been an active research since the first diffraction-free Bessel beam was generated by Durnin [1,2]. The Bessel beam, an exact solution to the paraxial Helmholtz wave equation, was found to preserve its original identity during propagation. The Bessel beam was followed by the discovery of other interesting nondiffracting solutions [3–7] including the recently observed optical Airy beams [4], which not only maintain their shapes but also self-accelerate along a parabolic trajectory during propagation. A spatiotemporal wave packet that preserves its shape in both space and time (called a linear light bullet) was also realized by combining a Bessel beam with an Airy pulse [8]. In addition, nondiffracting light beams with structured polarization states [9–13] were studied. Due to their intriguing properties, the research on nonspreading beams has drawn much attention, and various applications can be found in the areas of particle manipulation [14,15], curved plasma channel generation [16], quantum entanglement [17], superresolution imaging [18], and numerous other fields.

As counterparts, the concept of diffraction-free beams has been studied in other physical systems such as surface plasmonics [19–23], electronics [24–26], acoustics [27,28], and hydrodynamics [29–31]. We note that the nondiffracting Airy beams have also been extended to nonlinear optics, which enables generation and manipulation of such beams at different wavelengths [32,33].

To date, substantial studies on nonspreading light beams have been made. In most situations, their lateral dimensions are far beyond the optical wavelength. It is clear that the next challenge is to achieve these intriguing light beams at subwavelength scale [34,35]. Since the nondiffracting beam is resulted from superposition of many plane wave components [1,2], to generate such beams at extremely small scale, the components should contain high spatial frequencies and be in phase, i.e., their corresponding wave vectors exhibit cylindrical symmetry in reciprocal space. We note that a phenomenon of light waves interacting with surface plasmon in a metallic structure can lead to nondiffracting beams at subwavelength scale [36,37]. The mechanism relies on surface plasmon resonance (SPR), which requires matching the plasmonic structure to particular spatial polarization of illumination [38–40]. Generally, a cylindrical vector beam with radial polarization is required [36–40], enabling excitation of these high-spatial-frequency surface waves from all azimuthal directions. This SPR phenomenon provides a way to achieve nondiffracting waves at subwavelength scale. However, the beam is evanescent, and limited to only a certain regime of magnitude of wavelength where the evanescent Bessel beam dominates [36–40].

In this paper, we demonstrate a new technique for generating non-evanescent Bessel light beams at deep-subwavelength scale. Different from the SPR excitation, the high-spatial-frequency waves having cylindrical symmetry are generated by optical diffraction of a circular sharp-edge obstacle (SEO). Contrary to the evanescent nondiffracting beam, the resultant Bessel beam can maintain its subwavelength property up to considerable propagation distance. In addition, we reveal both theoretically and experimentally an unnoticed feature of the diffracted field in that it can retain its initial phase and polarization properties when it passes through the circular thin film, permitting subwavelength generation of nonspreading vector vortex beams, which are inaccessible with SPR-based techniques. These intriguing phenomena allow us to study generation of nondiffracting structured light beams at deep-subwavelength scale, in both scalar and vector frameworks.

## 2. RESULTS AND DISCUSSION

We start by considering the principle of the SEO configuration for realizing the subwavelength nondiffracting light beams in the scalar regime. As mentioned, it requires coherently interfering optical waves having high spatial frequencies. On one hand, the sharp edge would generate significant higher-spatial-frequency wavevectors than the original one ${k_0}$ [41,42], where ${k_0} = 2\pi /\lambda$, with $\lambda$ being the incident wavelength. The reason is that an incident plane wave is diffracted seriously when it crashes against the edge of the obstacle described by an ideal step function $f(x) = {\rm step}(x)$. In the reciprocal space, the step function exhibits a reverse function of ${f_x}$ in the Fourier space, i.e., $F({f_x}) = 1/2\delta ({f_x}) + 1/(i2\pi\! {f_x})$, where $\delta$ denotes the Dirac delta function, ${f_x}$ is spatial frequency with respect to $x$ coordinate, and $i$ is the imaginary unit. It means that in addition to the zero-order diffraction component, the generated diffractive waves have a continuously non-zero distribution of the higher-order wavevectors. On the other hand, the circular obstacle described by $f(\rho) = 1\; -$ circ $(\rho /{\rho _0})$, where ${\rm circ}(\cdot)$ denotes the circle function, i.e., inside the circle, $\rho \lt {\rho _0}$, ${\rm circ}(\rho)=1$, whereas outside the circle, $\rm circ(\rho)=0$ [here $\rho = ({x^2} + {y^2}{)^{1/2}}$ and ${\rho _0}$ is the radius], generates amplitude sprungstelle along a circle in the $x-y$ plane, which gives rise to the high-spatial-frequency diffraction components having cylindrical distributions in the reciprocal space, as illustrated in Fig. 1. For a specific frequency, the corresponding wavevectors ${k_j}$($j = 1,2,3,\ldots$) are in phase, leading to constructive interference. To be expected, it would generate a subwavelength localized field having Bessel form, which shows a manifestation of the famous Poisson–Arago spot forming in the shadow of the obstacle. The higher-frequency wavevector results in a larger diffractive angle; however, the interference strength becomes relatively weaker.

To verify the assertions, we theoretically investigate the diffraction behavior with the SEO described by $f(\rho)$. Using the Rayleigh–Sommerfeld (RS) diffraction integral [43], the light beam $E(x,y,z)$ beyond the SEO that is placed at $z = 0$ can be obtained as follows:

In experiment, a key issue lies in achieving perfect binary modulation of the incident amplitude near the edge of the SEO. We achieved this goal by means of fabricating a metallic disc (${\rho _0} = 7.5\,\,\unicode{x00B5}{\rm m}$) with thickness of only 60 nm (50 nm gold film plus a 10 nm chromium film as an adhesion layer) on a glass substrate. An expanded and collimated He–Ne laser working at $\lambda = 632.8\;{\rm nm} $ was utilized to normally illuminate the sample from the substrate side (see the experimental setup in Fig. 3). A high-magnification objective lens (Nikon, CFI EPI $150 \times$, numerical aperture NA = 0.9) was used to detect the diffracted fields beyond the sample. Together with a tube lens, their transverse intensity patterns were imaged onto a charge-coupled device (CCD) with pixel size of $1.4\;\unicode{x00B5}{\rm m}$. To observe the nondiffracting property, we recorded each cross section of the light fields along $x$ coordinate and aligned them in $z$ axis, with the result shown in Fig. 2(b). We observed that the resultant subwavelength light wave resists the diffraction-induced broadening, preserving its Bessel form during propagation. We note that, owing to the limited objective NA, the recorded light fields start after a short distance. Even so, the experimental results match well with the theory, as can be further confirmed by the simulated and measured intensity distributions shown in Figs. 2(c) and 2(d), as well as their corresponding cross sections displayed in Fig. 2(e). The measured on-axis intensity ratio in Fig. 2(d) to the incident intensity is 45.2%, approximately in accord with that in Fig. 2(c). Furthermore, the lateral size along the distance was measured, with the result illustrated in Fig. 2(f). The generated Bessel beam maintains its width below the diffraction limit up to a distance of $z \approx 8.0\;\unicode{x00B5}{\rm m}$, confirming its subdiffraction property. We attribute the slowly broadening effect to the relatively smaller spatial frequency of the wavevector when increasing the distance, as illustrated in Fig. 1.

Structuring the light beam with a vortex phase provides an additional degree of freedom for optics [44] and its related applications [45,46]. Using the above configuration, we present generation of diffraction-free vortex light beams at the subwavelength dimension. To this end, we structure the initial condition by adding a vortex phase, and modify the RS diffraction integral as

Intriguingly, Eq. (7) shows that the light wave with an initially structured vortex phase preserves its topological property when it propagates through the SEO. Moreover, it provides a technique to convert a diffracting Laguerre–Gauss (LG) vortex beam into a nondiffracting Bessel vortex beam with arbitrary topology at the subwavelength scale. It means that the generated Poisson–Arago spot becomes a ring with a null at the geometric center, when the incident light has orbital angular momentum. This can be attributed to the fact that the diffracted waves from opposite edges of the obstacle have opposite phases and therefore interfere destructively at the center. We point out that if $l = 0$, Eq. (7) admits a solution to the RS diffraction integral shown by Eq. (1).

Figure 4 presents theoretical and experimental results confirming the conservation of topological charge, as well as the prediction of a subwavelength nondiffracting vortex beam. Figures 4(a) and 4(b) show the initially structured phases with topological charges of $l = 1$ and $l = - 1$, respectively. In experiment, the vortex phases were realized by inserting a vortex plate before the SEO (see the red box in the setup in Fig. 3). The topological charge of the vortex plate can be tuned, enabling generation of a subwavelength diffractionless vortex beam with arbitrary phase charge. To detect the vortex wave front, we utilized an outer doughnut area of the incident vortex field, which can be considered as a plane wave approximately [48], to interfere with the diffracted field. The resultant interference patterns are presented in Figs. 4(c) and 4(d), corresponding to the cases of $l = 1$ and $l = - 1$, respectively. We indeed observed a dislocation in the interference pattern for both cases, confirming the vortex phase structure. To examine the nonbroadening effect of the vortex phase, we scanned the light field along the distance. Figure 4(f) depicts the measured intensity of the generated vortex field as a function of distance with $l = - 1$ (owing to the identical intensity pattern, the case of $l = 1$ is not shown). It is seen that a nearly nondiffracting vortex light beam at subwavelegnth scale was generated. The measurements agree well with the theoretical result based on Eq. (7) [see Fig. 4(e)].

Finally, we show the ability of this configuration to generate the subwavelength nondiffracting light beams with transversely structured polarizations. In this scenario, we explore the vectorial diffraction behaviors by considering the vector wave equation $\nabla \times \nabla \times \textbf{E} - k_0^2\textbf{E} = 0$. The vector beams with typical polarization states mapped on the equator of the first-order ($l = 1$) Poincar $e^\prime $ sphere are considered [49,50]. Therefore, the initially structured light beam is replaced by

It is worth noting that the resultant subwavelength vortex and vector beams having identical topology ($|l| = 1$) share similar nondiffracting behavior, as can be judged from the solutions shown in Eqs. (7) and (9). To elaborate more on the diffracted field structure for the higher-order vortex and vector beams, we examine their beam dimensions as a function of distance, with results presented in Fig. 6. Figure 6(a) depicts a lateral distribution of the vortex field with $l = - 1$ at a distance of $z = 6.3\,\,\unicode{x00B5}{\rm m}$, while Fig. 6(b) shows its corresponding cross section (red curve) along $x$, matching the first-order Bessel function (blue curve) according to Eq. (9). From the results shown in Figs. 6(c) and 6(d), both the spot size and the ring size exhibit a slowly diffracting, compared to a seriously diffracting, LG vortex beam having the same topological charge and initial beam width. It is evident that the generated main lobes of the vector and vortex beams can maintain their spot sizes below the diffraction limit up to considerable distance. Increasing the azimuthal index, we find that both the spot size and the ring size are also increased accordingly, while these higher-order light beams still exhibit a nondiffracting property during propagation, as seen in Fig. 6(d).

## 3. CONCLUSION

To summarize, we presented the first demonstration of nondiffracting structured light beams at deep-subwavelength scale (below the diffraction limit) in both the scalar and vector regimes. We realized these intriguing structured light beams by taking into account an almost forgotten phenomenon: the Poisson–Arago spot. Contrary to the previously demonstrated evanescent Bessel beams, the developed technique enables generating a non-evanescent Bessel light beam with beam size below the diffraction limit. Furthermore, we demonstrated that the topological structures of the phase and polarization can be maintained while the incident beam propagates through the element. This phenomenon provides us with an opportunity to fully structure the subwavelength nondiffracting light beams with complex phases and polarizations, in addition to the amplitude. We believe these results are new and may find interesting applications in subwavelength optics. Owing to the similarity to wave nature, our findings may also inspire subwavelength generations of nonspreading waves in other physical systems such as plasmonics [21,22], acoustics [27], and electronics [24,26].

## Funding

Pearl River Talent Project (2017GC010280); Natural Science Foundation of Guangdong Province (2017B030306009); National Natural Science Foundation of China (11704155, 11974146, 61935010); Key-Area Research and Development Program of Guangdong Province (2020B090922006); Science and Technology Planning Project of Guangdong Province (2018B010114002).

## Disclosures

The authors declare no competing interests.

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