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 [37] 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 [913] 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 [1923], electronics [2426], acoustics [27,28], and hydrodynamics [2931]. 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 [3840]. Generally, a cylindrical vector beam with radial polarization is required [3640], 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 [3640].

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.

 

Fig. 1. Schematic illustration for achieving subwavelength diffraction-free light beams based on a circular sharp-edge obstacle. The sharp edge enables binary amplitude modulation of the incident waves, which generates significant high-spatial-frequency wavevectors, e.g., marked as ${k_1}$, ${k_2}$, and ${k_3}$, in addition to the original one ${k_0}$. The circular geometry allows generating high-spatial-frequency components having cylindrical distribution, leading to in-phase superposition. Thereby, a subwavelength beam having Bessel form will be expected.

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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:

$$E(x,y,z) = \frac{{{E_0}}}{{i\lambda}}\int \int f(x^\prime ,y^\prime)\frac{z}{{{r^2}}}\exp (i{k_0}r){\rm d}x^\prime {\rm d}y^\prime ,$$
where ${E_0}$ represents the incident plane wave amplitude, $z$ denotes the propagation distance, and $r = [(x - x^\prime {)^2} + {(y - y^\prime)^2} + {z^2}{]^{1/2}}$. A typical radius of the SEO is set to ${\rho _0} = 7.5\,\,\unicode{x00B5}{\rm m}$, and a plane wave condition with $\lambda = 632.8\;{\rm nm} $ is adopted. Figure 2(a) shows the simulated intensity distribution of the light fields in the $x - z$ plane. As expected, it demonstrates a weakly diverging feature of the generated light beam during propagation. The nearly diffraction-free property is resulted from the generation of a Bessel profile, e.g., see the transverse intensity distribution at $z = 6.5\,\,\unicode{x00B5}{\rm m}$ in Figs. 2(c) and 2(e). The beam dimension at full width at half maximum (FWHM) is calculated as 280 nm, smaller than half of the wavelength. Note that the peak intensity of the subwavelength main lobe in Fig. 2(c) can reach 42% of the incident intensity. However, the power efficiency, i.e., the power carried in the subwavelength light field compared to the power of the incident beam, is extremely low (0.19%), since the integral is just over the subwavelength area. These simulated outcomes suggest that a subwavelength Bessel beam can be possible using the SEO configuration.
 

Fig. 2. Generation of subwavelength diffraction-free light beam with circular SEO (${\rho _0} = 7.5\,\,\unicode{x00B5}{\rm m}$) in the scalar regime. (a), (b) Theoretical (a) and measured (b) intensity distributions in the $x - z$ plane. (c)–(e) Lateral intensity distribution of the light field with FWHM value measured as 280 nm at $z = 6.5\,\,\unicode{x00B5}{\rm m}$: (c) theory, (d) experiment, and (e) normalized intensity profiles along $x$ at $y = 0$. (f) FWHM values of the generated beam as a function of distance.

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Fig. 3. Experimental setup. QWP, quarter-wave plate; VWP, vortex wave plate; QP, $q$-plate; OB, objective; TL, tube lens; CCD, charge-coupled device.

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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

$$\begin{split}{E(x,y,z)} &= {\frac{{{E_0}}}{{i\lambda}}\int\! \int f(x^\prime ,y^\prime)\!\exp (il\varphi ^\prime)}{\frac{z}{{{r^2}}}\!\exp\! \left({i{k_0}r} \right)\!{\rm d}x^\prime {\rm d}y^\prime ,}\end{split}$$
where $l$ is topological charge of the optical vortex, and $\varphi ^\prime = \arctan (y^\prime /x^\prime)$. To reveal the nondiffracting Bessel beam bearing vortex phase structure, it is convenient to rewrite the diffracted light field in a cylindrical form:
$$\!\!\!E(\rho ,\varphi ,z) = \frac{{{E_0}}}{{i\lambda}}\int_0^{2\pi}\!\! \exp (il\varphi ^\prime){\rm d}\varphi ^\prime \int_{{\rho _0}}^\infty \frac{{z\rho ^\prime\! \exp (i{k_0}r)}}{{{r^2}}}{\rm d}\rho^\prime ,\!$$
where $x^\prime = \rho ^\prime \cos (\varphi ^\prime)$, and $y^\prime = \rho ^\prime \sin (\varphi ^\prime)$. We note that $\rho ^\prime d\rho ^\prime \approx rdr$ for a transverse point whose vertical distance from the $z$ axis is small enough by comparison to the radius of the circular element. Following the formula [47] $\int_a^b g(r)\! \exp(i{k_0}r)dr = g(r)\!\exp (i{k_0}r)/(i{k_0})|_a^b$, Eq. (3) is simplified as
$$\!\!\!E(\rho ,\varphi ,z) = \frac{{{E_0}}}{{2\pi}}\int_0^{2\pi} \exp (il\varphi ^\prime)\!\exp [{i{k_0}R(\varphi ^\prime)} ]\frac{z}{{R(\varphi ^\prime)}}{\rm d}\varphi ^\prime ,\!$$
where
$$R(\varphi ^\prime) = {\left[{{{(x - {\rho _0}\cos \varphi ^\prime)}^2} + {{(y - {\rho _0}\sin \varphi ^\prime)}^2} + {z^2}} \right]^{\frac{1}{2}}}.$$
To analytically solve the integral form of Eq. (4), we perform series expansion of $R(\varphi ^\prime)$, and neglect the higher-order terms, resulting in $R(\varphi ^\prime) \approx {r_0} + {\rho ^2}/{r_0} - {\rho _0}\rho \cos (\varphi ^\prime - \varphi)/{r_0}$, where $r_0^2 =$${z^2} + \rho _0^2$. This is valid for $\rho ^\prime \ge {\rho _0} \gg \rho$. With these conditions, the integral of Eq. (4) can be reduced to
$$\begin{split}{E(\rho ,\varphi ,z)}& = {\frac{{{E_0}z}}{{2\pi {r_0}}}\exp \left[{i{k_0}\left({{r_0} + \frac{{{\rho ^2}}}{{{r_0}}}} \right)} \right]\exp (il\varphi) }\\ &\quad\times{\int_{- \pi}^\pi\! {\exp} [{il(\varphi ^\prime - \varphi)} ]\exp [{- i\xi \cos (\varphi ^\prime - \varphi)} ]{\rm d}\varphi ^\prime ,}\end{split}$$
where $\xi = {k_0}{\rho _0}\rho /{r_0}$. By virtue of the formula ${J_l}(\xi) = \frac{1}{{2\pi}}\int_{- \pi}^\pi \exp [i(l\tau - \xi \cos \tau)]{\rm d}\tau$, where ${J_l}$($l = 0,1,2,\ldots$) is the Bessel function of order $l$, we obtain the diffracted light field taking the form of
$$E(\rho ,\varphi ,z) = \frac{{{E_0}z}}{{{r_0}}}\exp \left[{i{k_0}\left({{r_0} + \frac{{{\rho ^2}}}{{{r_0}}}} \right)} \right]\exp (il\varphi){J_l}(\xi).$$

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).

 

Fig. 4. Generation of subwavelength nondiffracting light beams with structured vortex phase. (a), (b) Initial phase maps with topological charges of $l = 1$ (a) and $l = - 1$ (b). (c), (d) Experimentally recorded plane wave interference patterns of the light fields at distance of $z = 8.2\,\,\unicode{x00B5}{\rm m}$, in the case of $l = 1$ (c) and $l = - 1$ (d). (e), (f) Theoretical (e) and experimental (f) intensity distributions of the diffracted fields in the $x - z$ plane, under the case of $l = - 1$. Note that the sample used here is the same as that in Fig. 2.

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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)].

 

Fig. 5. Generation of subwavelength nondiffracting light beams with structured polarizations. (a), (b) Initial polarization states: (a) radial polarization ($\alpha = 0$) and (b) azimuthal polarization ($\alpha = \pi /2$). (c), (d) Illustrate theoretically the $x$ component of the vector polarized beam before the sample, while the insets show experimentally the $x$ component after the sample ($z = 6.5\,\,\unicode{x00B5}{\rm m}$), in the cases of (c) $\alpha = 0$ and (d) $\alpha = \pi /2$. (e), (f) Theoretical (e) and experimental (f) field distributions in the $x - z$ plane, under the case of $\alpha = 0$.

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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

$$\textbf{E}(\rho ,\varphi ,z = 0) = f(\rho)\left[{\cos (\varphi + \alpha)\vec x + \sin (\varphi + \alpha)\vec y} \right],$$
where $\vec x$ and $\vec y$ are the unit vectors associated with $x$ and $y$ axes, respectively, and $\alpha$ denotes the polarization angle. Equation (8) indicates that the vector beam consists of two orthogonal phase-modulated components. Since the diffracted wave of each component retains the same phase as the original one, as indicated by Eq. (7), the initial polarization state is supposed to be maintained when passing through the SEO. To confirm this assertion, we solve the RS diffraction integral with the initial condition displayed in Eq. (8) in the vector regime. By derivations similar to Eqs. (3)–(7), the vectorial light field beyond the SEO can be achieved, expressed as
$$\begin{split}{\textbf{E}(\rho ,\varphi ,z) }&={\frac{{{E_0}z}}{{{r_0}}}\exp \left[{i{k_0}\left({{r_0} + \frac{{{\rho ^2}}}{{{r_0}}}} \right)} \right]{J_1}(\xi) }\\ &\quad\times{\left[{\cos (\varphi + \alpha)\vec x + \sin (\varphi + \alpha)\vec y} \right].}\end{split}$$
It is clearly seen that the topological structure of the polarization state is also maintained, indicating that a nondiffracting solution to a light beam having transversely polarized structure is found. To experimentally generate these vector beams, the vortex plate was replaced by a $q$-plate with charge of $q = 1/2$, while keeping the setup unchanged (see the blue box in the setup in Fig. 3). Note that different polarization states can be obtained via rotating the $q$-plate, i.e., changing the polarization angle $\alpha$. Figures 5(a) and 5(b) present typical polarization states with $\alpha = 0$ and $\alpha = \pi /2$, respectively. The preservation of the polarization states can be further verified from their similar patterns of the $x$ or $y$ components before and after the sample. Figures 5(c) and 5(d) depict theoretically the initial $x$ components for both cases of $\alpha = 0$ and $\alpha = \pi /2$, while the insets present the experimental results after the sample. To reveal the nonspreading property of the generated vector beam, Fig. 5(f) shows experimentally its propagation along the distance in the case of $\alpha = 0$, which matches well with the theoretical outcome [see Fig. 5(e)] based on Eq. (9). Equation (9) further reveals that the generated nondiffracting vector beam could have an arbitrary polarization state, not limited to the cases discussed here.
 

Fig. 6. (a), (b) Typical intensity distribution of nondiffracting vortex beam with $l = - 1$ at $z = 6.3\,\,\unicode{x00B5}{\rm m}$: (a) measured transverse distribution and (b) normalized profiles along $x$ at $y = 0$. (c), (d) Beam dimensions as function of distance: FWHM values of the main lobe (c) and the ring size (d) [see the arrows at (b)] for the generated vortex and vector beams. Note that the experimentally generated vortex and vector beams have identical topology ($|l| = 1$). In comparison, the diffraction of a LG vortex beam with $l = 1$ is considered, presented in (d).

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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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25. R. Shiloh, Y. Tsur, R. Remez, Y. Lereah, B. A. Malomed, V. Shvedov, C. Hnatovsky, W. Krolikowski, and A. Arie, “Unveiling the orbital angular momentum and acceleration of electron beams,” Phys. Rev. Lett. 114, 096102 (2015). [CrossRef]  

26. V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of nondiffracting electron Bessel beams,” Phys. Rev. X 4, 011013 (2014). [CrossRef]  

27. P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014). [CrossRef]  

28. Z. Lin, X. Guo, J. Tu, Q. Ma, J. Wu, and D. Zhang, “Acoustic non-diffracting Airy beams,” J. Appl. Phys. 117, 104503 (2015). [CrossRef]  

29. S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, “Propagation dynamics of Airy water-wave pulses,” Phys. Rev. Lett. 115, 034501 (2015). [CrossRef]  

30. S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, “Propagation dynamics of nonspreading cosine-Gauss water-wave pulses,” Phys. Rev. Lett. 115, 254501 (2015). [CrossRef]  

31. U. Bar-Ziv, A. Postan, and M. Segev, “Observation of shaped-preserving accelerating underwater acoustic beams,” Phys. Rev. B 92, 100301 (2015). [CrossRef]  

32. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009). [CrossRef]  

33. I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108, 113903 (2012). [CrossRef]  

34. K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett. 36, 4335–4337 (2011). [CrossRef]  

35. E. Greenfield, R. Schley, I. Hurwitz, J. Nemirovsky, K. G. Makris, and M. Segev, “Experimental generation of arbitrarily shaped diffractionless superoscillatory optical beams,” Opt. Express 21, 13425–13435 (2013). [CrossRef]  

36. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006). [CrossRef]  

37. W. Chen and Q. Zhan, “Realization of an evanescent Bessel beam via surface plasmon interference excited by a radially polarized beam,” Opt. Lett. 34, 722–724 (2009). [CrossRef]  

38. W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9, 4320–4325 (2009). [CrossRef]  

39. A. Yanai and U. Levy, “Plasmonic focusing with a coaxial structure illuminated by radially polarized light,” Opt. Express 17, 924–932 (2009). [CrossRef]  

40. Q. Jin, G. Liang, G. Chen, F. Zhao, S. Yan, K. Zhang, M. Yang, Q. Zhang, Z. Wen, and Z. Zhang, “Enlarging focal depth using epsilon-near-zero metamaterial for plasmonic lithography,” Opt. Lett. 45, 3159–3162 (2020). [CrossRef]  

41. D. Weisman, S. Fu, M. Goncalves, L. Shemer, J. Zhou, W. P. Schleich, and A. Arie, “Diffractive focusing of waves in time and in space,” Phys. Rev. Lett. 118, 154301 (2017). [CrossRef]  

42. W. B. Case, E. Sadurni, and W. P. Schleich, “A diffractive mechanism of focusing,” Opt. Express 20, 27253–27262 (2012). [CrossRef]  

43. G. D. Gillen and S. Guha, “Modeling and propagation of near-field diffraction patterns: a more complete approach,” Am. J. Phys. 72, 1195–1201 (2004). [CrossRef]  

44. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [CrossRef]  

45. K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996). [CrossRef]  

46. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum,” Nat. Photonics 6, 488–496 (2012). [CrossRef]  

47. R. L. Lucke, “Rayleigh-Sommerfeld diffraction and Poisson’s spot,” Eur. J. Phys. 27, 193–204 (2006). [CrossRef]  

48. P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016). [CrossRef]  

49. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011). [CrossRef]  

50. S. Fu, C. Guo, G. Liu, Y. Li, H. Yin, Z. Li, and Z. Chen, “Spin-orbit optical Hall effect,” Phys. Rev. Lett. 123, 243904 (2019). [CrossRef]  

References

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    [Crossref]
  3. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams Georgios,” Opt. Lett. 32, 979–981 (2007).
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  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
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  5. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
    [Crossref]
  6. P. Aleahmad, M. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
    [Crossref]
  7. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108, 163901 (2012).
    [Crossref]
  8. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
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  9. Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
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  10. C. Pfeiffer and A. Grbic, “Controlling vector Bessel beams with metasurfaces,” Phys. Rev. Appl. 2, 044012 (2014).
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  11. I. Moreno, J. A. Davis, M. M. Sánchez-López, K. Radham, and D. M. Cottrell, “Nondiffracting Bessel beams with polarization state that varies with propagation distance,” Opt. Lett. 40, 5451–5454 (2015).
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  15. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
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  16. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
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  21. L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011).
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  22. A. Libster-Hershko, I. Epstein, and A. Arie, “Rapidly accelerating Mathieu and Weber surface plasmon beams,” Phys. Rev. Lett. 113, 123902 (2014).
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  23. J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
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  25. R. Shiloh, Y. Tsur, R. Remez, Y. Lereah, B. A. Malomed, V. Shvedov, C. Hnatovsky, W. Krolikowski, and A. Arie, “Unveiling the orbital angular momentum and acceleration of electron beams,” Phys. Rev. Lett. 114, 096102 (2015).
    [Crossref]
  26. V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of nondiffracting electron Bessel beams,” Phys. Rev. X 4, 011013 (2014).
    [Crossref]
  27. P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
    [Crossref]
  28. Z. Lin, X. Guo, J. Tu, Q. Ma, J. Wu, and D. Zhang, “Acoustic non-diffracting Airy beams,” J. Appl. Phys. 117, 104503 (2015).
    [Crossref]
  29. S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, “Propagation dynamics of Airy water-wave pulses,” Phys. Rev. Lett. 115, 034501 (2015).
    [Crossref]
  30. S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, “Propagation dynamics of nonspreading cosine-Gauss water-wave pulses,” Phys. Rev. Lett. 115, 254501 (2015).
    [Crossref]
  31. U. Bar-Ziv, A. Postan, and M. Segev, “Observation of shaped-preserving accelerating underwater acoustic beams,” Phys. Rev. B 92, 100301 (2015).
    [Crossref]
  32. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
    [Crossref]
  33. I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108, 113903 (2012).
    [Crossref]
  34. K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett. 36, 4335–4337 (2011).
    [Crossref]
  35. E. Greenfield, R. Schley, I. Hurwitz, J. Nemirovsky, K. G. Makris, and M. Segev, “Experimental generation of arbitrarily shaped diffractionless superoscillatory optical beams,” Opt. Express 21, 13425–13435 (2013).
    [Crossref]
  36. Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. 31, 1726–1728 (2006).
    [Crossref]
  37. W. Chen and Q. Zhan, “Realization of an evanescent Bessel beam via surface plasmon interference excited by a radially polarized beam,” Opt. Lett. 34, 722–724 (2009).
    [Crossref]
  38. W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9, 4320–4325 (2009).
    [Crossref]
  39. A. Yanai and U. Levy, “Plasmonic focusing with a coaxial structure illuminated by radially polarized light,” Opt. Express 17, 924–932 (2009).
    [Crossref]
  40. Q. Jin, G. Liang, G. Chen, F. Zhao, S. Yan, K. Zhang, M. Yang, Q. Zhang, Z. Wen, and Z. Zhang, “Enlarging focal depth using epsilon-near-zero metamaterial for plasmonic lithography,” Opt. Lett. 45, 3159–3162 (2020).
    [Crossref]
  41. D. Weisman, S. Fu, M. Goncalves, L. Shemer, J. Zhou, W. P. Schleich, and A. Arie, “Diffractive focusing of waves in time and in space,” Phys. Rev. Lett. 118, 154301 (2017).
    [Crossref]
  42. W. B. Case, E. Sadurni, and W. P. Schleich, “A diffractive mechanism of focusing,” Opt. Express 20, 27253–27262 (2012).
    [Crossref]
  43. G. D. Gillen and S. Guha, “Modeling and propagation of near-field diffraction patterns: a more complete approach,” Am. J. Phys. 72, 1195–1201 (2004).
    [Crossref]
  44. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [Crossref]
  45. K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
    [Crossref]
  46. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum,” Nat. Photonics 6, 488–496 (2012).
    [Crossref]
  47. R. L. Lucke, “Rayleigh-Sommerfeld diffraction and Poisson’s spot,” Eur. J. Phys. 27, 193–204 (2006).
    [Crossref]
  48. P. Miao, Z. Zhang, J. Sun, W. Walasik, S. Longhi, N. M. Litchinitser, and L. Feng, “Orbital angular momentum microlaser,” Science 353, 464–467 (2016).
    [Crossref]
  49. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
    [Crossref]
  50. S. Fu, C. Guo, G. Liu, Y. Li, H. Yin, Z. Li, and Z. Chen, “Spin-orbit optical Hall effect,” Phys. Rev. Lett. 123, 243904 (2019).
    [Crossref]

2020 (1)

2019 (1)

S. Fu, C. Guo, G. Liu, Y. Li, H. Yin, Z. Li, and Z. Chen, “Spin-orbit optical Hall effect,” Phys. Rev. Lett. 123, 243904 (2019).
[Crossref]

2017 (1)

D. Weisman, S. Fu, M. Goncalves, L. Shemer, J. Zhou, W. P. Schleich, and A. Arie, “Diffractive focusing of waves in time and in space,” Phys. Rev. Lett. 118, 154301 (2017).
[Crossref]

2016 (2)

2015 (6)

I. Moreno, J. A. Davis, M. M. Sánchez-López, K. Radham, and D. M. Cottrell, “Nondiffracting Bessel beams with polarization state that varies with propagation distance,” Opt. Lett. 40, 5451–5454 (2015).
[Crossref]

Z. Lin, X. Guo, J. Tu, Q. Ma, J. Wu, and D. Zhang, “Acoustic non-diffracting Airy beams,” J. Appl. Phys. 117, 104503 (2015).
[Crossref]

S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, “Propagation dynamics of Airy water-wave pulses,” Phys. Rev. Lett. 115, 034501 (2015).
[Crossref]

S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, “Propagation dynamics of nonspreading cosine-Gauss water-wave pulses,” Phys. Rev. Lett. 115, 254501 (2015).
[Crossref]

U. Bar-Ziv, A. Postan, and M. Segev, “Observation of shaped-preserving accelerating underwater acoustic beams,” Phys. Rev. B 92, 100301 (2015).
[Crossref]

R. Shiloh, Y. Tsur, R. Remez, Y. Lereah, B. A. Malomed, V. Shvedov, C. Hnatovsky, W. Krolikowski, and A. Arie, “Unveiling the orbital angular momentum and acceleration of electron beams,” Phys. Rev. Lett. 114, 096102 (2015).
[Crossref]

2014 (7)

V. Grillo, E. Karimi, G. C. Gazzadi, S. Frabboni, M. R. Dennis, and R. W. Boyd, “Generation of nondiffracting electron Bessel beams,” Phys. Rev. X 4, 011013 (2014).
[Crossref]

P. Zhang, T. Li, J. Zhu, X. Zhu, S. Yang, Y. Wang, X. Yin, and X. Zhang, “Generation of acoustic self-bending and bottle beams by phase engineering,” Nat. Commun. 5, 4316 (2014).
[Crossref]

A. Libster-Hershko, I. Epstein, and A. Arie, “Rapidly accelerating Mathieu and Weber surface plasmon beams,” Phys. Rev. Lett. 113, 123902 (2014).
[Crossref]

C. Pfeiffer and A. Grbic, “Controlling vector Bessel beams with metasurfaces,” Phys. Rev. Appl. 2, 044012 (2014).
[Crossref]

G. Wu, F. Wang, and Y. Cai, “Generation and self-healing of a radially polarized Bessel-Gauss beam,” Phys. Rev. A 89, 043807 (2014).
[Crossref]

M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014).
[Crossref]

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8, 302–306 (2014).
[Crossref]

2013 (2)

2012 (7)

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

W. B. Case, E. Sadurni, and W. P. Schleich, “A diffractive mechanism of focusing,” Opt. Express 20, 27253–27262 (2012).
[Crossref]

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108, 113903 (2012).
[Crossref]

J. Lin, J. Dellinger, P. Genevet, B. Cluzel, F. de Fornel, and F. Capasso, “Cosine-Gauss plasmon beam: a localized long-range nondiffracting surface wave,” Phys. Rev. Lett. 109, 093904 (2012).
[Crossref]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[Crossref]

P. Aleahmad, M. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[Crossref]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108, 163901 (2012).
[Crossref]

2011 (5)

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett. 107, 116802 (2011).
[Crossref]

P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett. 36, 3191–3193 (2011).
[Crossref]

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011).
[Crossref]

K. G. Makris and D. Psaltis, “Superoscillatory diffraction-free beams,” Opt. Lett. 36, 4335–4337 (2011).
[Crossref]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

2010 (1)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

2009 (5)

W. Chen and Q. Zhan, “Realization of an evanescent Bessel beam via surface plasmon interference excited by a radially polarized beam,” Opt. Lett. 34, 722–724 (2009).
[Crossref]

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9, 4320–4325 (2009).
[Crossref]

A. Yanai and U. Levy, “Plasmonic focusing with a coaxial structure illuminated by radially polarized light,” Opt. Express 17, 924–932 (2009).
[Crossref]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[Crossref]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[Crossref]

2008 (1)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams Georgios,” Opt. Lett. 32, 979–981 (2007).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

2006 (2)

2004 (1)

G. D. Gillen and S. Guha, “Modeling and propagation of near-field diffraction patterns: a more complete approach,” Am. J. Phys. 72, 1195–1201 (2004).
[Crossref]

2002 (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[Crossref]

1996 (1)

1995 (1)

Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Abeysinghe, D. C.

W. Chen, D. C. Abeysinghe, R. L. Nelson, and Q. Zhan, “Plasmonic lens made of multiple concentric metallic rings under radially polarized illumination,” Nano Lett. 9, 4320–4325 (2009).
[Crossref]

Ahmed, N.

J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum,” Nat. Photonics 6, 488–496 (2012).
[Crossref]

Aleahmad, P.

P. Aleahmad, M. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[Crossref]

Alfano, R. R.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref]

Arie, A.

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[Crossref]

Phys. Rev. X (1)

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[Crossref]

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[Crossref]

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[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. Schematic illustration for achieving subwavelength diffraction-free light beams based on a circular sharp-edge obstacle. The sharp edge enables binary amplitude modulation of the incident waves, which generates significant high-spatial-frequency wavevectors, e.g., marked as ${k_1}$, ${k_2}$, and ${k_3}$, in addition to the original one ${k_0}$. The circular geometry allows generating high-spatial-frequency components having cylindrical distribution, leading to in-phase superposition. Thereby, a subwavelength beam having Bessel form will be expected.
Fig. 2.
Fig. 2. Generation of subwavelength diffraction-free light beam with circular SEO (${\rho _0} = 7.5\,\,\unicode{x00B5}{\rm m}$) in the scalar regime. (a), (b) Theoretical (a) and measured (b) intensity distributions in the $x - z$ plane. (c)–(e) Lateral intensity distribution of the light field with FWHM value measured as 280 nm at $z = 6.5\,\,\unicode{x00B5}{\rm m}$: (c) theory, (d) experiment, and (e) normalized intensity profiles along $x$ at $y = 0$. (f) FWHM values of the generated beam as a function of distance.
Fig. 3.
Fig. 3. Experimental setup. QWP, quarter-wave plate; VWP, vortex wave plate; QP, $q$-plate; OB, objective; TL, tube lens; CCD, charge-coupled device.
Fig. 4.
Fig. 4. Generation of subwavelength nondiffracting light beams with structured vortex phase. (a), (b) Initial phase maps with topological charges of $l = 1$ (a) and $l = - 1$ (b). (c), (d) Experimentally recorded plane wave interference patterns of the light fields at distance of $z = 8.2\,\,\unicode{x00B5}{\rm m}$, in the case of $l = 1$ (c) and $l = - 1$ (d). (e), (f) Theoretical (e) and experimental (f) intensity distributions of the diffracted fields in the $x - z$ plane, under the case of $l = - 1$. Note that the sample used here is the same as that in Fig. 2.
Fig. 5.
Fig. 5. Generation of subwavelength nondiffracting light beams with structured polarizations. (a), (b) Initial polarization states: (a) radial polarization ($\alpha = 0$) and (b) azimuthal polarization ($\alpha = \pi /2$). (c), (d) Illustrate theoretically the $x$ component of the vector polarized beam before the sample, while the insets show experimentally the $x$ component after the sample ($z = 6.5\,\,\unicode{x00B5}{\rm m}$), in the cases of (c) $\alpha = 0$ and (d) $\alpha = \pi /2$. (e), (f) Theoretical (e) and experimental (f) field distributions in the $x - z$ plane, under the case of $\alpha = 0$.
Fig. 6.
Fig. 6. (a), (b) Typical intensity distribution of nondiffracting vortex beam with $l = - 1$ at $z = 6.3\,\,\unicode{x00B5}{\rm m}$: (a) measured transverse distribution and (b) normalized profiles along $x$ at $y = 0$. (c), (d) Beam dimensions as function of distance: FWHM values of the main lobe (c) and the ring size (d) [see the arrows at (b)] for the generated vortex and vector beams. Note that the experimentally generated vortex and vector beams have identical topology ($|l| = 1$). In comparison, the diffraction of a LG vortex beam with $l = 1$ is considered, presented in (d).

Equations (9)

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E ( x , y , z ) = E 0 i λ f ( x , y ) z r 2 exp ( i k 0 r ) d x d y ,
E ( x , y , z ) = E 0 i λ f ( x , y ) exp ( i l φ ) z r 2 exp ( i k 0 r ) d x d y ,
E ( ρ , φ , z ) = E 0 i λ 0 2 π exp ( i l φ ) d φ ρ 0 z ρ exp ( i k 0 r ) r 2 d ρ ,
E ( ρ , φ , z ) = E 0 2 π 0 2 π exp ( i l φ ) exp [ i k 0 R ( φ ) ] z R ( φ ) d φ ,
R ( φ ) = [ ( x ρ 0 cos φ ) 2 + ( y ρ 0 sin φ ) 2 + z 2 ] 1 2 .
E ( ρ , φ , z ) = E 0 z 2 π r 0 exp [ i k 0 ( r 0 + ρ 2 r 0 ) ] exp ( i l φ ) × π π exp [ i l ( φ φ ) ] exp [ i ξ cos ( φ φ ) ] d φ ,
E ( ρ , φ , z ) = E 0 z r 0 exp [ i k 0 ( r 0 + ρ 2 r 0 ) ] exp ( i l φ ) J l ( ξ ) .
E ( ρ , φ , z = 0 ) = f ( ρ ) [ cos ( φ + α ) x + sin ( φ + α ) y ] ,
E ( ρ , φ , z ) = E 0 z r 0 exp [ i k 0 ( r 0 + ρ 2 r 0 ) ] J 1 ( ξ ) × [ cos ( φ + α ) x + sin ( φ + α ) y ] .

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