Sub-diffraction quasi-non-diffracting beams with sub-wavelength transverse size are attractive for applications such as optical nano-manipulation, optical nano-fabrication, optical high-density storage, and optical super-resolution microscopy. In this paper, we proposed an optimization-free design approach and demonstrated the possibility of generating sub-diffraction quasi-non-diffracting beams with sub-wavelength size for different polarizations by a binary-phase Fresnel planar lens. More importantly, the optimization-free method significantly simplifies the design procedure and the generation of sub-diffracting quasi-non-diffracting beams. Utilizing the concept of normalized angular spectrum compression, for wavelength λ0 = 632.8 nm, a binary-phase Fresnel planar lens was designed and fabricated. The experimental results show that the sub-diffraction transverse size and the non-diffracting propagation distances are 0.40λ0–0.54λ0 and 90λ0, 0.43λ0–0.54λ0 and 73λ0, and 0.34λ0–0.41λ0 and 80λ0 for the generated quasi-non-diffracting beams with circular, longitudinal, and azimuthal polarizations, respectively.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
In recent years, due to their unique properties of diffractionless propagation, high local intensity, and self-healing, non-diffracting beams have attracted significant interest for a variety of applications, including optical manipulation , optical fabrication , optical imaging [3,4], Raman spectroscopy , and control of pulse propagation . Because of the required infinite energy, an ideal non-diffracting beam is not physically available. Therefore, the generation of quasi-non-diffracting (QND) beams has become a major area of focus. Extensive investigations have been made on the methods of generating QND beams, using conventional axicons , diffraction gratings , aberrating lenses , annular-type photonic crystals , liquid crystals , computer-generated holograms , sub-wavelength annular apertures , Fresnel zone plates (FZP) , Fresnel axicons , spatial light-modulation devices , and meta-surfaces . However, the conventional optics utilized to generate QND beams are diffraction-limited in nature, and it remains a great challenge to generate QND beams with sub-diffraction features. Although, great endeavors have been made to generate such beams with varieties of theoretical proposals [18–23], it was not until very recently that the experimental generation of QND beams with sub-diffraction features larger than one wavelength have been demonstrated with a spatial light modulator [24–26] based on the concept of super-oscillation . At the sub-wavelength level, sub-diffraction optical focus with a depth-of-focus (DOF) of approximately 5λ–20λ have been experimentally demonstrated with circular polarization [28, 29], azimuthal polarization [30, 31], and longitudinal polarization . Such sub-diffraction optical needles have shown promising potential in label-free microscopy with resolutions beyond λ/6 [33–35], and in high-density data storage with feature sizes smaller than 50 nm . However, the length of the needles is limited to only a few wavelengths, owing to the difficulties in theoretical design of long-DOF lenses for sub-wavelength and sub-diffraction focusing. To overcome this restriction, we recently proposed a method based on the idea of normalized angular spectrum compression (NASC)  and the concept of a local grating for generating ultra-long sub-diffraction QND beams with sub-wavelength transverse size. However, this method is based on a particle swarm optimization algorithm, and it is time-consuming, especially for large size lenses. To address these problems, in the present work, based on an optimization-free approach, we demonstrate the possibility of generating sub-diffraction QND beams with different polarizations by a single lens. Utilizing the concept of NASC, a binary-phase planar Fresnel lens is designed and fabricated. Ultra-long sub-diffraction QND beams with sub-wavelength transverse size are then generated with circular, longitudinal, and azimuthal polarizations, respectively. This approach greatly simplifies the design and generation of sub-diffraction QND beams. Furthermore, the design method is totally optimization-free, providing a fast design tool for sub-diffraction QND beams.
2. Theoretical considerations
As mentioned above, QND beams with azimuthal polarization have been demonstrated in our previous work , in which a pair of parameters, i.e., f and λ, were first determined for the given target focal length f0 and propagation distance Zmax at working wavelength λ0. A sub-diffraction focusing lens (SDFL) was then designed with a focal length of f at a wavelength of λ, which is usually smaller than λ0, using particle swarm algorithms and the vectorial angular spectrum method. The vectorial angular spectrum method is used to calculate the diffraction pattern on the desired focal plane at z = f, and the particle swarm algorithm  is employed to optimize the target diffraction field for single-spot sub-diffraction focusing at z = f. This design procedure is extremely time-consuming because of the following two reasons. First, the calculation of vectorial wave diffraction is realized by a fast Hankel transform with oversampling in a geometric progression scheme , whose number of sampling points can be on the order of 220 for a lens with a radius of several hundreds of wavelengths. And the computational complexity will increase exponentially with the lens radius. Second, the particle swarm algorithm might require tens of thousands of iterations for optimization, particularly when the lens radius is on the scale of several hundred wavelengths. Therefore, an increase in lens radius will significantly increase the computational complexity, especially in the case of designing a sub-diffraction focal spot on a sub-wavelength scale, due to the deep sub-wavelength oversampling utilized in the diffraction calculation, and the increased number of genes used in the particle swarm algorithm. This computational complexity makes it impossible to further increase the lens radius for longer sub-diffraction quasi-non-diffraction beams.
According to the idea of NASC, the key to creating sub-diffraction quasi-non-diffraction beams is to design a sub-diffraction single point focusing lens at wavelength λ, and then sub-diffraction quasi-non-diffraction beams can be generated with the designed lens at a shorter wavelength of λ0. To significantly reduce the computational complexity in the present work, an optimization-free approach was proposed. More importantly, the proposed method has no requirement for diffraction-pattern calculation, and therefore greatly simplifies the design of sub-diffraction quasi-non-diffraction beams with long propagation distance. As will be shown later, this method not only just simplifies the design procedure, but also gives better results compared with that reported in the previous work . More importantly, the present method has no restriction on the length of sub-diffraction quasi-non-diffraction beams.
In the following, to make a comparison between the two methods, the lens proposed in present work is given the same values for target parameters Zmax and f0 as those of the previously reported one at wavelength of λ0 = 632.8 nm.
3. Reexamination of the far-field focusing properties of classical FZP
Near-field and quasi-near-field sub-wavelength focusing have been demonstrated by modified FZP  and classical Fresnel zone plates (CFZPs) [41–44] with a focal length of approximately one wavelength and a large numerical aperture (NA) with a value close to unity. It was found that the well-known CFZP can focus waves into sub-diffraction or even super-oscillation spots in the far field (more than hundreds of wavelengths distance away from the CFZP) for different polarizations, such as circular, radial and azimuthal polarizations. To demonstrate this sub-diffraction or super-oscillation focusing performance, a binary-phase CFZP was designed with a radius of 600λ and focal length of 200λ at wavelength λ = 672.8 nm. According to the well-established design scheme, the CFZP consists of a serial of concentric ring belts, and the radii at which the ring belts switch from phase 0 to π is given by Eq. (1) , where n is an integer, λ the wavelength, and f the focal length. The maximum value of n is 865. This phase configuration ensures constructive interference of the waves from all ring belts at the designed focal point.
Figures 1(a)–1(f) are the numerical simulation results of the CFZP for incident waves with circular, azimuthal, and radial polarizations, respectively. The results for circular polarization are given by the vectorial angular spectrum method [31, 32, 46], while the data for radially and azimuthally polarized incident waves are obtained by COMSOL Multiphysics. Figures 1(a) –1(c) are the two-dimensional (2D) optical intensity distributions on the xz propagation plane, showing a clear single focal spot, either solid or hollow, in the vicinity of the designed focal point, as expected. Figures 1(d)–1(f) give the corresponding distributions of optical peak intensity (PI, red-solid line), transverse full width at half-maximum (FWHM, blue-solid line), and sidelobe ratio (SR, the ratio of maximum sidelobe intensity to the peak intensity, green-solid line) along the optical axis, where the diffraction-limit (0.5λ/NA, DL) and the super-oscillation criteria (0.38λ/NA, SOC)  are plotted as a magenta-dashed line and black-dashed line with a NA of approximately 0.95, respectively. In all three cases, the maximum intensity appears around the designed focal point at z = 200λ, and the DOF is approximately 3λ, 8λ, and 8λ for each case, respectively. The FWHM has the largest value of 0.46λ in the case of circularly polarized incident wave, where only transverse components were taken into account. The spot formed by the radially polarized waves has a FWHM of 0.48λ, smaller than the DL. The inner FWHM of the hollow spot generated by the azimuthally polarized wave is only 0.34λ, smaller than the SOC. The sidelobe ratios are 10%, 14%, and 20% at the peak position for incident waves with circular, radial, and azimuthal polarizations, respectively.
This is quite impressive, since CFZP and the corresponding Eq. (1) have been known for a long time. It seems that the far-field sub-diffraction performance of the CFZP was not fully recognized, because numerous endeavors have been conducted in sub-diffraction focusing of waves using a high-NA objective lens in combination with mask filters [21,48] and phase-shifters [49,50]. This excellent property is what we have been looking for to generate sub-diffraction quasi-non-diffraction beams without complex numerical calculation of diffraction patterns. More importantly, the lens structure is simply determined by Eq. (1), which saves a great deal of time compared to time-consuming optimization algorithms.
4. Lens design
4.1 Design method
According to the ideas of NASC and a local grating , the focused optical field at z(r) within the propagation range from z = f0 to Zmax corresponds to the local grating located at the radial coordinate of r on the lens. This relation can be expressed by Eq. (2).Eqs. (3) gives the values of f and λ, which are later used to calculate rn given in Eq. (1), and then the structure of the CFZP is completely determined.
The proposed lens design method is summarized in the flowchart of Fig. 2(a), and the previously reported method  is presented in Fig. 2(b) for comparison. It is clearly seen that the proposed scheme based on binary-phase CFZP is extremely simple, and only requires solving two nonlinear equations, i.e., (3), which can be simply carried out with a few lines of MatLab code, and only takes a few seconds to obtain the final results. However, the previously reported method might need tens of days to finish the design because of the significant amount of numerical computation of the diffraction pattern, and the huge number of iterations for optimization. Here, it should be emphasized that the designed binary-phase lens is not a CFZP in the classical sense, because its structure is designed at wavelength λ, while its working wavelength is actually λ0 (<λ).
To make a comparison with the previously reported approach and hence show the advantages of the present design scheme in terms of both computation complexity and lens performance, a binary-phase lens based on a CFZP was designed with the same target parameters, λ0 = 632.8 nm, f0 = 226λ0, Zmax = 96λ0, as those of the pervious reported SDFL , and therefore the values of λ and f are 672.8 nm and 200λ, respectively, which are then substituted into Eq. (1) to obtain the radius rn for each ring belt on the binary-phase lens.
4.2 Numerical simulation
To understand the performance of the designed binary-phase lens in sub-diffraction QND beam generation at the working wavelength of λ0 = 632.8 nm, numerical simulations were carried out with different incident polarizations. Figures 3(a)–3(d) depict the intensity distribution of ultra-long QND beams on the xz propagation plane for incident waves with different polarizations, i.e., circular, radial, and azimuthal polarizations. Figures 3(a) and 3(b) are the intensities of total optical field and transverse optical field in the case of a circularly polarized incident wave. It is clear that the transverse electrical component has a narrower transverse size compared to the total optical field. Figures 3(c) and 3(d) give the intensity distributions for the incident waves with radial and azimuthal polarizations, respectively. The QND beam generated with a radially polarized wave is mainly composed of a longitudinal electrical field, while the one generated from an azimuthally polarized wave consists of only a transverse component. In all the three cases, the propagation distance of the QND beams is approximately 110λ0 between z = 220λ0 and 330λ0.
Figures 3(e)–3(h) plot the peak intensity (PI, red-solid line), FWHM (blue-solid line), and sidelobe ratio (SR, green-solid line) distributions against the optical axis for the corresponding intensity distributions given in Figs. 3(a)–3(d), respectively. The magenta-dashed line and black-dashed line denote DL (0.5λ/NA) and SOC (0.38λ/NA), respectively. A clear oscillation is observed in the optical intensity as the beam propagates forward in all four cases. This intensity variation is attributed to the sharp intensity change at the lens edge in the simulation, which acts as a hard aperture . According to Figs. 3(e) and 3(f), the central peak intensity is the same for the total optical field and the transverse component, implying that the central peak intensity only consists of a transverse component, i.e., circular polarization. The total optical field has a FWHM greater than 0.6λ0 along the entire QND beam, while its transverse component has a much smaller FWHM of 0.39λ0–0.53λ0 between z = 238λ0 and 334λ0, resulting in a 96λ0-long sub-diffraction QND beam with sub-wavelength transverse size. The beam becomes even super-oscillatory with a FWHM varying between 0.39λ0 and 0.42λ0 within the range from z = 285λ0 to 340λ0. It should be emphasized that such an ultra-long sub-diffraction transverse polarized QND beam is of extreme importance in realizing 3D, label-free, far-field super-resolution microscopy , in which only the point-spread function of the transverse optical field plays a key role in determining the optical resolution. The broadening of the total optical field is caused by the contribution from the longitudinal field. The sidelobe ratios for the total optical field and the transverse polarization only have a slight difference in the entire range of the QND beam. For the radial polarized incident wave, Fig. 3(g) shows a sub-diffraction propagation distance of 87λ0 with a transverse FWHM varying from 0.545λ0 to 0.45λ0 between z = 260λ0 and 334λ0. The beam overall sidelobe ratio is less than 18%, except for the position before z = 240λ0. In the azimuthally polarized hollow QND beam, as depicted in Fig. 3(h), the sub-diffraction propagation distance is approximately 108λ0 with a transverse inner FWHM of 0.33λ0–0.53λ0 between z = 233λ0 and 334λ0, in which the transverse FWHM varies from 0.41λ0 to 0.33λ0, being smaller than the SOC, in the range from z = 243λ0 to 340λ0. The overall sidelobe ratio is less than 0.36 in most parts of the beam.
4.3 Comparison between the binary-phase lens based on CFZP and SDFL
To make comparison, axial distributions of peak intensity, transverse FWHM (or inner FWHM for azimuthally polarized wave), and sidelobe ratio are depicted in Fig. 4 for numerical simulation results of the binary-phased lens based on a CFZP designed using the present (solid) method and the previously reported SDFL (dashed). Figures 4(a)–4(c) are results obtained under the illumination of light with circular, radial, and azimuthal polarization, respectively. It is also interesting to find that the SDFL works for all polarizations, although its performance was only optimized for azimuthal polarization. It can be clearly seen that both lens can generate sub-diffraction quasi-non-diffraction beams with a propagation distance of approximately 100λ0, while, compared with the binary-phased lens based on the CFZP (red), that based on the SDFL (blue) shows little fluctuations in the distributions of peak intensity, transverse size, and sidelobe ratio along the optical axis, particularly in the intensity and sidelobe distributions. The reason for these smooth distributions is that, unlike the randomly generated phase distribution in SDFL design, the binary-phased lens based on a CFZP has a quasi-continuous change in the zone radius and width. The performance indicates that the binary-phased lens based on a CFZP has advantages over those based on the SDFL in the generation of sub-diffraction quasi-non-diffraction beams, although its design method is extremely simple compared to the one used for SDFL. However, it should be pointed out that the sub-diffraction property of the binary-phased lens based on a CFZP is determined once the values of R, f, and λ are obtained, while, for given values of R, f, and λ, the SDFL might have certain control on the transverse size of a single focal spot at the wavelength of λ during the optimization process.
For further comparison, the phase distribution is plotted along in the radial direction for CFZP (red) and SDFL (blue) respectively in Fig. 5. It is clearly seen that the phase distribution of CFZP is quite different from that of the SDFL. Therefore, the SDFL could not be simply treated as an approximation of CFZP.
5. Experimental methods
5.1 Lens fabrication
According to the designed lens phase distribution, utilizing electron-beam lithography and dry etching, Si3N4 dielectric ring belts are generated on a glass substrate. Figures 6(a) and 6(b) show the scanning electron microscopy (SEM) images of the binary-phase planar lens taken with a NOVA Nano SEM 430 + EDS.
5.2 Experimental setup
The QND beams generated with azimuthally polarized waves have only a transverse component, while those created with circularly polarized waves or radially polarized waves have both transverse and longitudinal polarizations. The transverse optical field can be simply recorded with a high-NA microscope . However, to measure both polarizations, a tilted nano-fiber probe has to be adopted . Compared with the 3D step-by-step scan when using a nano-fiber probe, the advantage of using a microscope is that it only requires a 1D scan along the z axis, because the in-plane optical intensity distribution can be recorded by a high-resolution digital camera with a single shot, and is therefore time-saving.
The experimental setup is illustrated in Fig. 7. The light source is a He-Ne laser (HNL 210 L, Thorlabs, Inc., USA) emitting at a wavelength of λ0 = 632.8 nm. A zero-order quarter-wave plate (Edmund Optics 49-220) is used to convert the linear polarization into circular polarization. In addition, an S-wave plate (SWP; RPC-632.8-06-188, Workshop of Photonics) is employed to generate azimuthally polarized and radially polarized incident waves. The incident waves normally illuminate the binary-phase lens based on CFZP. After the lens, the diffraction pattern is collected with either a microscope system, or a scanning system based on a nano-fiber probe. The microscope system consists of a 100 × objective lens (CF Plan 100 × /0.95, Nikon, USA) with NA = 0.95, an infinity-corrected tube lens (ITL200, Thorlabs, Inc., USA), and a complementary metal-oxide-semiconductor (CMOS) camera with a pixel size of 1.67 × 1.67 µm2 (acA3800-14 µm, Basler AG, Germany). The objective lens was mounted on a 1D nano-positioner (PZT; EO-S1047, Edmund Optics, USA) with a linear scanning range of 100 μm. The nano-fiber-probe scanning system includes a tilted nano-fiber probe with a diameter of 100 nm (Nanonics Imaging Ltd., CFN-100), a 3D piezo nano-positioner (Physik Instrumente, P-561.3CD), and a single-photon-counter (Thorlabs, SPCM50A/M). The optical intensity is collected by the nano-fiber probe and detected by the single-photon counter. The 3D optical intensity distribution is collected by step-by-step scanning with the 3D nano-positioner.
6. Experimental results
Experimental investigations are conducted for multiple types of incident polarizations to demonstrate the possibility of generating sub-diffraction QND beams with different polarizations.
6.1 Generation of sub-diffraction QND beam with circular polarization
For the circularly polarized incident wave at a wavelength of λ0 = 632.8 nm, a clear QND beam is observed in the lens diffraction pattern using a microscope and nano-fiber-probe system, respectively. The experimentally recorded optical intensity distributions are depicted in Figs. 8(a) and 8(b) within the range from z = 220λ0 to 330λ0 after the lens for the transverse optical field and total optical field, respectively. It is noted that the transverse component has a narrower transverse distribution compared to the total optical field. For further comparison, the normalized peak intensity, transverse FWHM, and sidelobe ratio are plotted against the propagation distance for experimental (red) and theoretical (blue) data, as shown in Figs. 8(c)–8(e) for the transverse polarization, and in Figs. 8(f)–8(h) for the total optical field, respectively. The magenta-dashed line and black-dashed line represent the DL and SOC, respectively. Good agreement is seen between the experimental and theoretical results. According to Figs. 8(c) and 8(f), both data show a propagation distance of over 100λ0 between 220λ0 and 330λ0. The intensity variation has a similar profile, except for the slightly faster attenuation at the far end of the experimentally obtained results. As shown in Figs. 8(d) and 8(g), the experimental and theoretical FWHMs also share a quite similar trend of variation along the optical axis for both transverse polarization and the total optical field. For the transverse polarization, excellent agreement is found in the transverse FWHM between the theoretical data and the experimental data, showing a sub-diffraction QND beam as long as 90λ0 with a FWHM varying from 0.54λ0 to 0.40λ0 between z = 240λ0 and 330λ0. As mentioned above, such an ultra-long transversely polarized QND beam is critical to a 3D super-resolution microscope. As for the total optical field, the beam transverse FWHM is greater than 0.47λ0, and most of its portion is greater than the DL. According to Figs. 8(e) and 8(h), the sidelobe ratio is smaller than 0.5 between z = 237λ0 and 320λ0 for the transverse optical field and between z = 224λ0 and 272λ0 for the total optical field, respectively. However, the overall value is comparatively greater than the theoretical prediction, especially in the far end area of the beam, where the peak intensity gradually decreases to zero.
6.2 Generation of sub-diffraction QND beam with longitudinal polarization
In generation of the QND beam with longitudinal polarization, the linearly polarized laser beam is converted into a radially polarized beam by the S-wave plate first, which then normally impinges on the planar lens. The diffraction pattern after the lens is collected by the nano-fiber-probe system. The experimentally obtained optical intensity is displayed in Fig. 9(a), which shows a bright needle-shaped structure surrounded by several weaker sidelobes. The beam propagates from z = 230λ0 to 330λ0, resulting in a 100λ0-long longitudinally polarized QND beam. Figures 9(b)–9(d) plot the normalized peak intensity, FWHM, and sidelobe ratio with respect to the optical axis for both experimental and theoretical data. Both the intensity and FWHM show good agreement between the experimental and theoretical results, while the measured sidelobe ratio is larger than the theoretical one. Excluding several points, the experimental FWHM shows an oscillatory change between the DL (0.54λ0) and SOC (0.43λ0) in the range from z = 257λ0 to 330λ0, leading to a longitudinally polarized sub-diffraction QND beam with sub-wavelength transverse size for a propagation distance greater than 73λ0. To our knowledge, such an ultra-long longitudinally polarized sub-diffraction QND beam has never been experimentally reported.
6.3 Generation of sub-diffraction QND beam with azimuthal polarization
To generate an azimuthally polarized sub-diffraction QND beam, the incident wave with azimuthal polarization is first obtained by converting the linearly polarized laser beam with the S-wave plate. Since it consists of only a transverse electrical field, the diffracted optical pattern is simply recorded by the microscope system. The experimentally obtained intensity in the xz propagation plane is depicted in Fig. 10(a), displaying a clear hollow QND beam with gradually attenuated sidelobes. Figures 10(b)–10(d) present distributions of the normalized (blue-solid line) distributions of peak intensity, inner FWHM, and sidelobe ratio obtained from the experimental data along the optical axis, while their theoretical counterparts (red-solid line) are also plotted for comparison. It is found that the experimental data are in good consistency with theoretical results. The measured peak intensity shows a propagation distance greater than 104λ0 between z = 226λ0 and 330λ0. Compared with the theoretical data, the experimentally obtained inner FWHM has a comparatively larger value at the near end of the beam before z = 274λ0, while the two data are quite close in the area between z = 272λ0 and 330λ0. It is also noted that the beam is of sub-diffraction type in the range from z = 250λ0 to 330λ0, and becomes super-oscillatory with an inner FWHM varying from 0.41λ0 to 0.34λ0 between z = 258λ0 and 330λ0. Therefore, the experimentally generated QND beam has a sub-diffraction length of 80λ0 and a super-oscillatory length of 72λ0, respectively. As shown in Fig. 10(d), the measured sidelobe ratio is slightly smaller than 0.35 between z = 250λ0 and 300 λ0, which is close to the theoretical value. However, there is a fast increase in the sidelobe ratio after z = 250λ0.
For all three different incident polarizations, i.e., circular, radial, and azimuthal polarizations, although there are still differences between the experimental and theoretical results, the experimentally generated sub-diffraction QND beams show good consistency with theoretical prediction. The discrepancies between the two groups of data is mainly caused by several issues, including fabrication errors, the difference in incident field distribution between the theoretical design and the experiment, and the difficulty of alignment, especially in the case of radially and azimuthally polarized incident waves.
7. Discussions and conclusions
In summary, sub-diffraction QND beams have attracted interest for multiple important applications. Although QND beams with sub-diffraction features have been experimentally demonstrated with a spatial-light-modulator, such sub-diffraction features are surrounded by 10-times-stronger huge sidelobes. In our recent work , a planar binary-phase lens designed based on the idea of NASC and particle swarm algorithm was reported for the generation of sub-diffraction azimuthally polarized QND beams. However, the design process is based on optimization algorithm and the complex diffraction pattern calculation. Restricted by those issues, the previously reported approach is not only time-consuming, but also not applicable for design of lens with large radius. To overcome these problems, in our present work, we provide a fast, efficient and practicable design approach for generating sub-diffraction QND beams. The lens design utilizes the formula of CFZP, which is optimization-free and significantly simplifies the design. A binary-phase lens based on FZP was designed and fabricated for a working wavelength of λ0 = 632.8 nm. Experimental results demonstrate the generation of circularly, longitudinally, and azimuthally polarized QND beams with sub-diffraction propagation distances as long as 90λ0, 73λ0, and 80λ0, respectively. Although the FWHM of the QND beam generated with a circularly polarized wave is larger than that of the DL, it has no restriction to the application in 3D super-resolution, where the size of the transverse component is critical for achieving super-resolution beyond λ0/6 [33–35]. To our knowledge, no any experimental report on generating sub-wavelength QND beams with circular or longitudinal polarizations with a propagating distance greater than 70λ0 exists. Our approach provides an optimization-free tool for realizing sub-diffraction QND beams with sub-wavelength transverse size for multiple types of polarizations, which greatly simplifies the design procedure. Such lenses would find applications in optical nano-manipulation, optical nano-fabrication, super-resolution optical imaging, etc.
National Natural Science Foundation of China (61575031); National Key Basic Research and Development Program of China (Program 973) (2013CBA01700); Open Fund of State Key Laboratory of Information Photonics and Optical Communications (University of Electronic Science & Technology of China), P. R. China; Fundamental Research Funds for the Central Universities (106112016CDJXZ238826, 106112016CDJZR125503); National Key Research and Development Program of China (2016YFED0125200, 2016YFC0101100).
Authors also thank LetPub (www.letpub.com) for their linguistic assistance during the preparation of this manuscript.
References and links
1. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008). [CrossRef]
2. M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photonics Rev. 6(5), 607–621 (2012). [CrossRef]
3. N. Weber, D. Spether, A. Seifert, and H. Zappe, “Highly compact imaging using Bessel beams generated by ultraminiaturized multi-micro-axicon systems,” J. Opt. Soc. Am. A 29(5), 808–816 (2012). [CrossRef] [PubMed]
6. A. Dudley, M. Lavery, M. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photonics News 24(6), 22–29 (2013). [CrossRef]
7. J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44(8), 592–597 (1954). [CrossRef]
8. P. García-Martínez, M. M. Sánchez-López, J. A. Davis, D. M. Cottrell, D. Sand, and I. Moreno, “Generation of Bessel beam arrays through Dammann gratings,” Appl. Opt. 51(9), 1375–1381 (2012). [CrossRef] [PubMed]
9. R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991). [CrossRef]
10. H. Kurt and M. Turduev, “Generation of a two-dimensional limited-diffraction beam with self-healing ability by annular-type photonic crystals,” J. Opt. Soc. B 29(6), 1245–1256 (2012). [CrossRef]
11. A. Hakola, A. Shevchenko, S. C. Buchter, M. Kaivola, and N. V. Tabiryan, “Creation of a narrow Bessel-like laser beam using a nematic liquid crystal,” J. Opt. Soc. B 23(4), 637–641 (2006). [CrossRef]
13. Y. Y. Yu, D. Z. Lin, L. S. Huang, and C. K. Lee, “Effect of subwavelength annular aperture diameter on the nondiffracting region of generated Bessel beams,” Opt. Express 17(4), 2707–2713 (2009). [CrossRef] [PubMed]
16. L. Gong, Y. X. Ren, G. S. Xue, Q. C. Wang, J. H. Zhou, M. C. Zhong, Z. Q. Wang, and Y. M. Li, “Generation of nondiffracting Bessel beam using digital micromirror device,” Appl. Opt. 52(19), 4566–4575 (2013). [CrossRef] [PubMed]
17. H. Gao, M. Pu, X. Li, X. Ma, Z. Zhao, Y. Guo, and X. Luo, “Super-resolution imaging with a Bessel lens realized by a geometric metasurface,” Opt. Express 25(12), 13933–13943 (2017). [CrossRef] [PubMed]
18. H. Dehez, A. April, and M. Piché, “Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent,” Opt. Express 20(14), 14891–14905 (2012). [CrossRef] [PubMed]
21. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]
22. Z. Man, C. Min, L. Du, Y. Zhang, S. Zhu, and X. Yuan, “Sub-wavelength sized transversely polarized optical needle with exceptionally suppressed side-lobes,” Opt. Express 24(2), 874–882 (2016). [CrossRef] [PubMed]
23. J. Guan, J. Lin, C. Chen, Y. Ma, J. Tan, and P. Jin, “Transversely polarized sub-diffraction optical needle with ultra-long depth of focus,” Opt. Commun. 404, 118–123 (2017). [CrossRef]
25. 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(11), 13425–13435 (2013). [CrossRef] [PubMed]
26. J. Wu, Z. Wu, Y. He, A. Yu, Z. Zhang, Z. Wen, and G. Chen, “Creating a nondiffracting beam with sub-diffraction size by a phase spatial light modulator,” Opt. Express 25(6), 6274–6282 (2017). [CrossRef] [PubMed]
27. F. Huang, Y. Chen, F. J. G. Abajo, and N. I. Zheludev, “Optical super-resolution through super-oscillations,” J. Opt. A, Pure Appl. Opt. 9(9), S285–S288 (2007). [CrossRef]
28. E. T. F. Rogers, S. Savo, J. Lindberg, T. Roy, M. R. Dennis, and N. I. Zheludev, “Super-oscillatory optical needle,” Appl. Phys. Lett. 102(3), 031108 (2013). [CrossRef]
29. G. Yuan, E. T. F. Rogers, T. Roy, G. Adamo, Z. Shen, and N. I. Zheludev, “Planar super-oscillatory lens for sub-diffraction optical needles at violet wavelengths,” Sci. Rep. 4(1), 6333 (2014). [CrossRef] [PubMed]
30. F. Qin, K. Huang, J. Wu, J. Jiao, X. Luo, C. Qiu, and M. Hong, “Shaping a sub-wavelength needle with ultra-long focal length by focusing azimuthally polarized light,” Sci. Rep. 5(1), 9977 (2015). [CrossRef] [PubMed]
31. G. Chen, Z. Wu, A. Yu, K. Zhang, J. Wu, L. Dai, Z. Wen, Y. He, Z. Zhang, S. Jiang, C. Wang, and X. Luo, “Planar binary-phase lens for super-oscillatory optical hollow needles,” Sci. Rep. 7(1), 4697 (2017). [CrossRef] [PubMed]
32. A. P. Yu, G. Chen, Z. H. Zhang, Z. Q. Wen, L. R. Dai, K. Zhang, S. L. Jiang, Z. X. Wu, Y. Y. Li, C. T. Wang, and X. G. Luo, “Creation of sub-diffraction longitudinally polarized spot by focusing radially polarized light with binary phase lens,” Sci. Rep. 6(1), 38859 (2016). [CrossRef] [PubMed]
33. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M. R. Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nat. Mater. 11(5), 432–435 (2012). [CrossRef] [PubMed]
34. T. Roy, E. T. F. Rogers, G. Yuan, and N. I. Zheludev, “Point spread function of the optical needle super-oscillatory lens,” Appl. Phys. Lett. 104(23), 231109 (2014). [CrossRef]
35. F. Qin, K. Huang, J. Wu, J. Teng, C. W. Qiu, and M. Hong, “A supercritical Lens optical label-free microscopy: sub-diffraction resolution and ultra-long working distance,” Adv. Mater. 29(8), 1602721 (2017). [CrossRef] [PubMed]
36. G. Yuan, E. T. F. Rogers, T. Roy, Z. Shen, and N. I. Zheludev, “Flat super-oscillatory lens for heat-assisted magnetic recording with sub-50 nm resolution,” Opt. Express 22(6), 6428–6437 (2014). [CrossRef] [PubMed]
37. S. Zhang, H. Chen, Z. Wu, K. Zhang, Y. Li, G. Chen, Z. Zhang, Z. Wen, L. Dai, and A. L. Wang, “Synthesis of sub-diffraction quasi-non-diffracting beams by angular spectrum compression,” Opt. Express 25(22), 27104–27118 (2017). [CrossRef] [PubMed]
38. N. Jin and Y. Rahmat-Samii, “Advances in particle swarm optimization for antenna designs: real-number, binary, single-objective and multiobjective implementations,” IEEE Trans. Antenn. Propag. 55(3), 556–567 (2007). [CrossRef]
39. V. Magni, G. Cerullo, and S. D. Silvestri, “High-accuracy fast Hankel transform for optical beam propagation,” J. Opt. Soc. Am. A 9(11), 2031–2033 (1992). [CrossRef]
40. R. G. Mote, S. F. Yu, W. Zhou, and X. F. Li, “Subwavelength focusing behavior of high numerical-aperture phase Fresnel zone plates under various polarization states,” Appl. Phys. Lett. 95(19), 191113 (2009). [CrossRef]
41. Y. Fu, W. Zhou, L. E. N. Lim, C. Du, and X. Luo, “Plasmonic microzone plate: superfocusing at visible regime,” Appl. Phys. Lett. 91(6), 061124 (2007). [CrossRef]
42. V. V. Kotlyar, S. S. Stafeev, Y. Liu, L. O’Faolain, and A. A. Kovalev, “Analysis of the shape of a subwavelength focal spot for the linearly polarized light,” Appl. Opt. 52(3), 330–339 (2013). [CrossRef] [PubMed]
43. S. S. Stafeev, V. V. Kotlyar, and L. O’Faolain, “Subwavelength focusing of laser light by microoptics,” J. Mod. Opt. 60(13), 1050–1059 (2013). [CrossRef]
44. V. V. Kotlyar, S. S. Stafeev, A. G. Nalimov, M. V. Kotlyar, L. O’Faolain, and E. S. Kozlova, “Tight focusing of laser light using a chromium Fresnel zone plate,” Opt. Express 25(17), 19662–19671 (2017). [CrossRef] [PubMed]
45. D. Vaughan, X-Ray Data Booklet (Inorganic Organic Physical & Analytical Chemistry, 1985).
46. G. Chen, Y. Li, A. Yu, Z. Wen, L. Dai, L. Chen, Z. Zhang, S. Jiang, K. Zhang, X. Wang, and F. Lin, “Super-oscillatory focusing of circularly polarized light by ultra-long focal length planar lens based on binary amplitude-phase modulation,” Sci. Rep. 6(1), 29068 (2016). [CrossRef] [PubMed]
47. K. Huang, H. Ye, J. Teng, S. P. Yeo, B. Lukyanchuk, and W. Cheng, “Optimization-free superoscillatory lens using phase and amplitude masks,” Laser Photonics Rev. 8(1), 152–157 (2014). [CrossRef]
48. H. Guo, X. Weng, M. Jiang, Y. Zhao, G. Sui, Q. Hu, Y. Wang, and S. Zhuang, “Tight focusing of a higher-order radially polarized beam transmitting through multi-zone binary phase pupil filters,” Opt. Express 21(5), 5363–5372 (2013). [CrossRef] [PubMed]
50. G. Y. Chen, F. Song, and H. T. Wang, “Sharper focal spot generated by 4π tight focusing of higher-order Laguerre-Gaussian radially polarized beam,” Opt. Lett. 38(19), 3937–3940 (2013). [CrossRef] [PubMed]
51. G. Chen, Z. X. Wu, A. P. Yu, Z. H. Zhang, Z. Q. Wen, K. Zhang, L. R. Dai, S. L. Jiang, Y. Y. Li, L. Chen, C. T. Wang, and X. G. Luo, “Generation of a sub-diffraction hollow ring by shaping an azimuthally polarized wave,” Sci. Rep. 6(1), 37776 (2016). [CrossRef] [PubMed]