Generation of a sub-diffracted Bessel beam via diffraction interference in a combined amplitude structure

We have theoretically investigated the use of a simple combined amplitude structure to produce a sub-diffracted Bessel beam via diffraction interference. This powerful structure is composed of a spiral slit and radial grating. When a vortex beam illuminates this combined amplitude structure, a subwavelength Bessel beam with a size of 0.39λ and a long working distance of approximately 100 μm is numerically realized. By tailoring the parameters of the spiral slit, we can obtain a longer sub-diffracted Bessel beam. Moreover, the observed Bessel beam has low-energy side-lobes. The peculiar features of our theoretically generated Bessel beam have numerous potential applications, such as in nanoparticles manipulation, super-resolution imaging, and lithography. © 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

simple combined structure excited by a vortex beam, we can realize a long working distance sub-diffracted Bessel beam, which also possesses the low-energy side band. Our results not only enrich the conventional Bessel beams, but also open doors for a variety of potential applications.

Theory
The idea is motivated by the diffraction of vortex beams from the amplitude radial grating [24], in which a hotspot with maximum intensity appears on the optical axis when the topological charge of the vortex beam is equal to the spoke number of the radial grating. To obtain a sub-diffracted Bessel beam, we employ a combined amplitude structure, which consists of a radial grating and spiral slit. With the aid of a vortex beam, the radial grating produces a zeroth Bessel beam in the center along the optical axis, where the side lobes are very close to the central focal spot. By contrast, the spiral slit generates a discontinuous beam with low-energy side lobes in the center along the optical axis. The diffraction interference between the zeroth Bessel beam and the discontinuous beam in the free space shrinks the point spread functions below the diffraction limit and leads to a subwavelength Bessel beam. In the current scheme, the main function of the spiral slit is to shape the zeroth Bessel beam to subwavelength size and push the side band away from the central beam. As indicated in Fig. 1, we devise a combined amplitude structure, which contains a central cosine radial grating and a peripheral spiral slit, where black and white represent 0 and 1, respectively. Under this configuration, the prerequisite to realize sub-diffracted Bessel beam with low side lobes is to satisfy both m = |L| and l = −L conditions, where m is the spoke number of the cosine radial grating, L is the topological charge of the vortex beam, and l is the topological charge of the spiral slit. In addition, we can tune the working distance of the sub-diffracted Bessel beam by changing the parameters of spiral slit. Owing to the sub-diffracted focusing effect, the combined amplitude structure can also be considered as an optical super-oscillatory lens. In comparison with the binary-amplitude metal super-oscillatory lens [21], the current structure does not involve a specific algorithm or complicated nanostructures. To confirm that our design indeed produces a sub-diffracted Bessel beam, we perform numerical simulations for the simple combined structure. For the sake of simplicity, we consider the cylindrical coordinate system. Assume that a vortex beam propagates along the z-axis and illuminates the amplitude structure at the z = 0 plane. If the combined structure is sufficiently thin, the distribution of the electric field behind the structure is approximately equal to the product of the illuminating electric field and transmission function of the composite structure. The transverse electric field behind the structure is given by: where − → e x and − → e y denote the unit vectors along the x-and y-axis, respectively. t(r, θ) is the transmission function of the combined structure, and E 0 (r) is the input light field function. In our simulation, they can be written in the form of where θ is the angular coordinate, d is the width of the slit, r θ is the radial coordinate, and r 0 is the initial radius of the spiral slit. As the input light illuminates the combined amplitude structure, Z l is the largest propagation distance, where one can still obtain the sub-diffracted beam. This condition can be satisfied by adjusting the parameter l, whose value is opposite to the topological charge of the input vortex beam. λ represents the input light wavelength, and w 0 denotes the beam waist of the input light. The matter-light interaction in our current system is described by the vectorial angular spectrum theory [23,25]. Applying algebra, the corresponding components of the electric field at a propagation distance of z are obtained to govern the dynamics of the light transmission: where HT 0 and HT 1 are the zeroth-order and the first-order Hankel transformations, respectively.
Here, f x , f y , and f z are frequency components along the x-, y-, and z-axis, respectively.

Simulation
In the simulation, we assume that the central radial grating has a minimum spoke number of m = 2, which is almost symmetrical along the y-axis, as shown in Fig. 1. The illumination vortex beam has a topological charge L = −2 and its wavelength is 633 nm. The initial radius of the spiral is r 0 = 200 µm, and the width of the slit is d = 1 µm. The largest observation distance was set at Z l = 98 µm with l = 2. The beam waist of the incident vortex beam is 150 µm. Under these considerations, Fig. 2 shows the simulated intensity distributions of the beam in the xz and yz planes in terms of Eqs. (7) -(9). Figure 2(a) and Fig. 2(b) show the intensity evolution of the generated beam within z = 200 µm, which clearly presents the excellent diffraction-free performance, that is, the transverse intensity distribution is independent of the propagation distance z. We can observe an excellent diffraction-free Bessel beam, whose side lobe is far away from the central main beam in Fig. 2(a) and 2(b). While propagating in the z direction, the intensity of the Bessel beam decreases slowly from z = 0 µm to z = 100 µm in Fig. 2(c) and Fig. 2(d), which correspond to the marked areas of Fig. 2(a) and Fig. 2(b), respectively. Moreover, the beam size almost does not change within 100 µm along the propagation direction. It is worth to emphasize that the Hankel transform, which is a paraxial approached method, is not applicable in the region near (typically a few or a dozen wavelengths) the amplitude structure. We need to strictly solve the Helmholtz equation to achieve the diffraction field distribution of input light near the structure, where the transmitted light still propagates in its original way and behaves like a particle. Therefore, the particle of light dominates in this region due to the wave has not yet developed. The light will gradually behave like a wave when it near the Fresnel region. After confirming the Bessel beam, we are ready to look for the sub-diffracted characteristics at some propagation distance in both the xz and yz planes. In our simulation, we calculated a series of subwavelength-focused spots at different propagation distances. Figure 3 shows some typical images with sub-diffracted focused spots. According to the cross-section profile of the spot shown in Fig. 3, (a) Gaussian line shape is selected for fitting in order to better estimate the beam size. As expected, without any data post-processing, the profile is well fitted with the Gaussian curve. For λ = 633 nm, at a distance of z = 50 µm, a sub-diffracted spot is identified with a full width at half maximum (FWHM) of 248 nm (i.e., 0.39λ in Fig. 3(b)) along the x-axis and 207 nm (i.e., 0.33λ in Fig. 3(c)) along the y-axis, respectively. At z = 98 µm (Fig. 3(d)), the obtained sub-diffracted spot has a FWHM of 306 nm (in Fig. 3(e), close to half of the wavelength 316.5 nm) along the x-axis and 254 nm along the y-axis (in Fig. 3(f)). It should be noted that because of the spiral slit, the beam cross-section is not rotation-symmetric in the x-y plane. We can hence find that the side band in the x-axis is further away from the center main spot than that in the y-axis as shown in Fig. 3(a) and 3(d). In addition, most of the energy is distributed in the center spot while the side fringes contain less power. The observed astigmatic diffraction patterns can be used to identify the direction of the Bessel beam's phase rotation and the order of the Bessel mode. Further, we investigated the dependence of the FWHMs of the hotspots on the positions of the observation planes in the x-and y-axis within z = 100 µm. As illustrated in Fig. 4, the FWHMs of the Bessel beam in the y-axis (182 nm at minimum, i.e., 0.29λ, shown by the blue triangle line) are always smaller than those in the x-axis (230 nm at minimum, i.e., 0.36λ, shown by the red circle line). Interestingly, as shown in Fig. 4, the FWHMs of the Bessel beam are always less than the diffraction limit (λ/2, i.e., 316.5 nm, the black dotted line) and gradually increase as the propagation distance increases, and then jump to a much higher value after a characteristic distance (98 µm). This distance is equivalent to the largest one that we previously set when designing the spiral slit. At z = 94 and 96 µm, the FWHMs of the hotspots in the x-axis are almost same as the diffraction limit. As expected, when the propagation distance is increased to z = 100 µm, the size of the Bessel beam is very close to the λ/2 along the y-axis, while the FWHM is larger than λ/2 along the x-axis. This implies that the sub-diffracted beam becomes hard at observation distances greater than 98 µm, which is determined by the parameter Z l of the spiral slit. Therefore, we can obtain a much longer sub-diffracted Bessel beam by tuning the parameter Z l . Due to constructive/destructive interference, we can find the FWHM exhibiting an oscillatory behavior below 98 µm. It should also be noted that the combined structure for realizing the sub-diffracted hotspots at different positons along the x-and y-axis basically remains the same, with an FWHM difference of approximately 50 nm, within 84 µm.
Through the destructive/constructive interference, a sub-diffracted Bessel beam can be observed up to 100 µm away from the combined amplitude structure excited by a vortex beam. The achievable sub-diffracted Bessel beam depends on parameters, such as the initial radius of the spiral slit, the spoke number of the radial grating, and the topological charge of the input light. By judiciously tuning these parameters, it is possible to obtain a longer working distance sub-diffracted Bessel beam. Moreover, the combined amplitude structure does not involve any complicated algorithms and has structural parameters that are dimensionally larger than the wavelength of light, which may be fabricated easily in practice. Owing to the high optical focusing ability of the combined amplitude structure, one may reduce the focused spot-size down to tens of nanometers. Furthermore, our investigation may offer new opportunities for performing nanoparticle manipulation, subwavelength imaging, and developing focusing devices.

Conclusion
In conclusion, we theoretically propose a new method to generate a sub-diffracted Bessel beam with a vortex beam in a combined amplitude structure, which consists of a cosine radial grating and spiral slit. The effect is capable of long working distance imaging and may reduce the associated defocusing of the beam. Thus, it could be useful for rapid and detailed in vivo observation of biological cells. In practice, the image quality may be affected by the finite working distance and size of the input probe. We also noticed that our scheme might be able to transport and separate nanoparticles in optical micromanipulation. In addition, our present investigation further indicates that such a configuration is also useful for application in focusing devices. Disclosures. The authors declare no conflicts of interest.