## Abstract

Quasi-non-diffracting beams are attractive for various applications, including optical manipulation, super-resolution microscopes, and materials processing. However, it is a great challenge to design and generate super-long quasi-non-diffracting beams with sub-diffraction and sub-wavelength size. In this paper, a method based on the idea of compressing a normalized angular spectrum is developed, which makes it possible and provides a practical tool for the design of a quasi-non-diffracting beam with super-oscillatory sub-wavelength transverse size. It also presents a clear physical picture of the formation of super-oscillatory quasi-non-diffracting beams. Based on concepts of a local grating and super-oscillation, a lens was designed and fabricated for a working wavelength of λ = 632.8 nm. The validity of the idea of normalized angular spectrum compression was confirmed by both numerical investigations and experimental studies. An optical hollow needle with a length of more than 100λ was experimentally demonstrated, in which an optical hollow needle was observed with a sub-diffraction and sub-wavelength transverse size within a non-diffracting propagation distance of 94λ. Longer non-diffracting propagation distance is expected for a lens with larger radius and shorter effective wavelength.

© 2017 Optical Society of America

## 1. Introduction

In 1987, Drunin and associates proposed the concept of non-diffracting optical beams based on the scalar-wave equation [1]. Such beams can travel an infinite distance without altering their transverse intensity pattern. In addition, non-diffracting beams also have properties of high local intensity and self-healing. Because of these unique properties, non-diffracting beams are attractive for a variety of applications, including optical trapping [2,3] and tweezing [4], materials processing [5], imaging [6,7], stimulated Raman scattering [8], and controlling ultrashort pulse propagation in dispersive media [9]. However, the realization of ideal non-diffracting beams requires infinite energy. Alternatively, extensive studies have been conducted to realize quasi-non-diffracting beams to approach the impressive properties of the ideal ones. It was first suggested by Drunin *et al.* that, by illuminating a circular slit placed on the focal plane of a lens with a collimated light, a zeroth-order Bessel beam can be obtained directly behind the lens within a finite distance *Z*_{max} [10]. Although there are various methods of generating quasi-non-diffracting beams, including conventional axicons [11], diffraction gratings [12,13], aberrating lenses [14], annular-type photonic crystals [15], liquid crystals [16], computer-generated holograms [17,18], sub-wavelength annular apertures [19], Fresnel zone plates [20], Fresnel axicons [21,22], spatial light-modulation devices [23,24], and meta-surfaces [25], the basic principle lies in the fact that the quasi-non-diffracting beams can be treated as the superposition of plane waves, which accumulate almost the same phase change in the propagation direction. This ensures that the beam profile remains unchanged within a limited propagation distance [10].

Sub-wavelength and nearly non-diffracting beams can be obtained with longitudinally polarized waves [26]. A concave conical mirror [27] has been proposed to generate an ultra-long sub-diffraction longitudinal optical needle by focusing radially polarized beams. Utilizing the spherical aberration in a concave mirror, the generation of an ultra-long sub-diffraction longitudinal optical needle was theoretically studied [28]. However, in both cases, the proposed ultra-long optical needle lies in the area surrounded by the reflection surface, which limits its real application. Moreover, such bulky optical elements make experimental implementation extremely difficult due to system alignment issues. There has not been, to our knowledge, any experimental report on the generation of such optical needles based on these methods. Using a high-numerical-aperture objective lens and a binary phase optical filter, the generation of sub-wavelength optical needles has been theoretically investigated by focusing a radially polarized beam [29], a hybridly polarized beam [30], and an azimuthally polarized beam with orbital angular momentum [31]. Quasi-non-diffracting optical beams with sub-diffraction features have been demonstrated with a spatial light modulator using superposition of different order Bessel beams [32, 33] and a specially designed phase mask [34].

In recent years, there has been growing interest in developing a new class of optical components based on sub-wavelength structures for generation of quasi-non-diffracting beams with sub-diffraction and sub-wavelength size for super-resolution purposes. The sub-diffraction beam transverse size can be achieved using the concept of optical super-oscillation [35], which allows the generation of sub-wavelength and sub-diffraction size beams without the presence of evanescent waves [36]. Various super-oscillation focusing lenses [37–52] have been experimentally demonstrated, among which several types of lenses were proposed to generate sub-diffraction optical needles with a depth of focus (DOF) of less than 20λ. Similar to the case of the spherically aberrating lens method [14], central stops were adopted to block the light through the central part in the point-focusing super-oscillatory lenses to achieve a DOF as long as 5λ–20λ; however, the realization of such an extended DOF was at the expense of degradation in the focus transverse size [45–49]. In another way, a super-Gaussian function was used to describe the extended longitudinal profile of an optical needle, and the binary particle swarm optimization algorithm and Rayleigh-Sommerfeld integrals were used to optimize the super-oscillatory lens design and achieve minimal variance between the actual field distribution and the merit function, leading to experimental demonstration of a 15λ-long DOF [50]. Using a binary particle swam optimization algorithm and the vectorial angular spectrum method, a longitudinally polarized optical needle [51] and an azimuthally polarized optical hollow needle [52] with sub-diffraction transverse size were demonstrated with DOFs of approximately 5λ and 10λ, respectively. Using the above two methods, the transverse patterns must be calculated at different points with sub-wavelength intervals along the propagation axis, which gives little physical information on the formation of such an optical needle, and also tremendously increases the computational difficulties and make them impossible to overcome in the design of ultra-long DOFs or of quasi-non-diffracting beams.

To solve these problems, a method of designing super-oscillatory quasi-non-diffracting beams was proposed based on the idea of normalized angular spectrum compression, which presents a clear physical picture and makes it possible to design a planar lens for generation of an ultra-long optical needle with sub-diffraction transverse size and a propagation distance of more than hundreds of wavelengths. To verify this idea, a super-oscillatory binary phase planar lens was designed and fabricated. Experimental results show the generation of an optical hollow needle of more than 100λ, in which a sub-diffraction optical hollow needle were observed with a length of 94λ at a wavelength of λ = 632.8 nm.

## 2. Compression of Normalized angular spectrum

#### 2.1 Concept of Compressed Normalized Angular Spectrum

As pointed out above, quasi-non-diffracting beams can be treated as the super-position of plane waves, which accumulate almost the same phase change in the propagation direction [10]. This implies that the value of *k _{z}*/

*k*should distribute in a narrow range; equivalently, this requires that the value of

*k*/

_{r}*k*and the angle θ distribute in a narrow range, where

*k*and

_{z}*k*are the axial and radial components of wave vector

_{r}*k*, and θ is the angle between the wave vector and optical axis. The normalized radial spatial frequency

*k*/

_{r}*k*is related to the angle θ through the equation

*k*/

_{r}*k*= sin

*θ*. For an ideal non-diffracting beam, the values of

*k*,

_{z}*k*, and θ are the same for all plane waves consisting of the beam. Figure 1(a) illustrates the formation of a quasi-non-diffracting beam with a conical lens, where the refractive angle is a constant θ within the effective lens area, which implies a single spatial frequency of

_{r}*n/λ*sin

*θ*in the entire angular spectrum, or a single refraction angle. For such a quasi-non-diffracting beam, the narrower the range of the angle θ, or the smaller the normalized angular spectrum bandwidth, the longer the non-diffracting distance.

In our previous work [52], it was found that, when a 10λ-long super-oscillatory optical hollow needle with azimuthal polarization was immersed in water, its non-diffracting propagation distance was almost doubled without obvious degradation of its super-oscillatory transverse size. Since the transverse electrical field is continuous at the air-water interface, the electrical field in the water side is the same as that in the air side. In the two mediums, the only difference is that the effective wavelength is reduced by a factor of 1/*n* in water, where *n* is the refractive index of water. For a plane wave with an incident angle of θ* _{i}* in the air side, Fresnel’s law gives

*n*sinθ

*= sinθ*

_{r}*, where θ*

_{i}*is the refraction angle in water. Because water has a larger refractive index than air, the refractive angle has a smaller value than the incident angle. As shown in Fig. 1(b), when passing through the air-water interface, the value of*

_{r}*k*/

_{r}*k*

_{0}

*= nk*/

_{r}*k’*, where

*k*

_{0}and

*k’*are the wavenumbers in air and water, respectively. The angular spectrum remains unchanged before and after the air-water interface, while the cutoff spatial frequency is increased from 1/λ to

*n*/λ. This reveals the fact that the normalized angular spectrum bandwidth (NASB, the ratio of the angular spatial spectrum bandwidth to the cutoff frequency

*n*/λ) is compressed by a factor of

*n*in water. Such compression in the NASB makes the beam less diffracting, and results in an extended optical needle in water. This phenomenon suggests that a lens might achieve a super-long DOF focus, or a quasi-non-diffracting beam, by compressing the NASB.

#### 2.2 Compression of NASB with Binary Phase Grating

Compared with other meta-surface-based planar lenses, a binary phase lens is much easier to fabricate. As shown in Fig. 1(c), the binary phase planar lens consists of concentric dielectric ring belts, which can be treated as a quasi-periodic grating with different local grating constants at different radii. For single-point focusing, the normal incident wave is diffracted towards the designed focal point by these local gratings. The diffraction angle has different values at different radii to ensure the formation of a focal spot. The range of the diffraction angle {θ_{m}} corresponds to the normalized angular spectrum bandwidth. As discussed above, to realize a less diffractive beam, it is required to compress the NASB. There are two ways to realize NASB compression. One is to magnify the size of the lens structure; the other is to reduce the effective wavelength. In the first case, the spatial angular spectrum is directly compressed, and the incident beam profile must be changed by the same magnitude. However, this will also result in the change in the size of the focus. The advantage of the latter case is that, except for the effective wavelength, all of the lens parameters and incident-beam parameters remain unchanged, as is discussed later in this paper.

For a diffractive point-focusing lens, compressing the normalized angular spectrum can be well understood by the concept of a local grating. The transmission function of a binary phase (0,π) planar lens can be expressed by *T*(*r*) = *t*(*r*) + [1−*t*(*r*)]exp(*j*π), where *t*(*r*) and [1−*t*(*r*)]exp(*j*π) describe the transmission functions of those ring belts with phase 0 and π, respectively. For a lens with a focal length of *f*_{0}, it is required that the light at radius *r* is diffracted with an angle of θ*’*(*r*) by the local grating at radius *r*, which satisfies the relation of tanθ*’*(*r*) = *r*/*f*_{0}. Therefore, the local grating constant *d(r)* at radius *r* can be obtained by the grating equation, *nd*(*r*)sinθ*’*(*r*) = λ*’*, where *n* is the refractive index of the medium behind the lens and is treated as a constant for the small wavelength range in our case, and λ*’* is the incident wavelength in vacuum. The grating equation can be also rewritten as sinθ*’*(*r*) = λ*’*_{eff}*/d*(*r*), where the effective working wavelength is λ*’*_{eff} = λ*’/n*, or the wavelength in the dielectric medium. As shown in Fig. 1(d), when the effective wavelength λ*’*_{eff} is replaced by λ_{eff}, the refraction angle at radius *r* will satisfies the equation sinθ(*r*) = (λ_{eff}*/*λ*’*_{eff})sinθ*’*(*r*). Clearly, according to this equation, a reduced effective wavelength λ_{eff} will make the wave being diffracted have a smaller angle θ(*r*), and the incident wave at different radii will be diffracted into different angles and will intercept the optical axis at different positions, as given in Eqs. (1)–(3), to form a quasi-non-diffracting beam:

This also can be described by *k _{r}*/

*k =*(λ’

_{eff}

*/*λ

_{eff})

*k’*/

_{r}*k’*, which shows a clear evidence of compression of the NASB for the case of (λ’

_{eff}/λ

_{eff})<1. Such NASB compression can be achieved with a shorter effective wavelength λ

_{eff}by either increasing the refractive index or by shortening the incident wavelength.

## 3. Theoretical design and numerical analysis

To investigate how the compression of the NASB works for the generation of a quasi-non-diffracting beam, a point-focusing binary phase planar lens was first designed with a radius of R = 638λ*’* and a full width at half-maximum (FWHM) of 0.35λ’ at a wavelength of λ’ *=* 672.8 nm for an azimuthally polarized Gaussian-Lagrange wave, as given in [52]. For a working wavelength λ = 632.8 nm, the focal length *f* ranges from 226λ to 322λ.

#### 3.1 Lens design

According to the above requirement, the wavelength λ*’* and the corresponding focal length *f*_{0} were first determined by scanning *f*_{0} and λ*’* utilizing Eqs. (1)–(3). One pair of values, 200λ’ and 672.8 nm, were found for *f*_{0} and λ’, respectively. Figure 2 is the relation between the points on the optical needle and positions of their corresponding local gratings. As expected, the focal depth ranges from 226λ to 322λ along the *z* axis. It is clearly seen that the contribution to a given point on the optical needle comes primarily from a certain belt area on the lens. However, this contribution is not uniform: the central part with a radius of less than 316λ only contributes the first 27λ of the length of the total 96λ-long optical needle, as indicated by area I in Fig. 2. In the remaining parts of the lens, the local gratings contribute almost uniformly to the rest of the 69λ length of the optical needle, which is indicated by the nearly linear relation between the local grating position and the position of the corresponding focal spot on the optical needle between *z* = 253λ and *z* = 322λ along the *z* axis, as indicated by areas II, III, and IV in Fig. 2.

Based on the idea of NASB compression, using the vectorial angular spectrum method [52] and a particle swarm optimization algorithm [53], a point-focusing binary phase planar lens was first designed at the wavelength 672.8 nm with a focal length of *f*_{0} = 200λ*’* and a transverse size of 0.35λ*’* (corresponding to 0.37λ at λ *=* 632.8 nm) under the illumination of an azimuthally polarized Gaussian-Lagrange wave with a beam width of *w*_{0} = 331 μm. The lens consists of 443 ring belts with phase change of π. The widths of ring belts and the gaps between ring belts have values of integer times of 300 nm, which are determined in the optimization. And then the lens is applied to a working wavelength of λ = 632.8 nm to generate a quasi-non-diffraction hollow beam, or a super-long optical hollow needle.

#### 3.2 Numerical Investigation

To investigate the lens performance at the working wavelength of 632.8 nm, COMSOL Multiphysics^{®} software (COMSOL, Inc., USA) was adopted to conduct the numerical simulation, in which the Si_{3}N_{4} layer was used to construct the lens ring belt structures. The thickness of the Si_{3}N_{4} concentric ring belts is 348 nm for a relative phase change of π at the working wavelength of 632.8 nm, where the experimentally obtained Si_{3}N_{4} refractive index of 1.91 was used in the simulation.

Figure 3(a) gives the numerical result of the intensity distribution on the propagation plane. An optical hollow needle was found between *z* = 225λ and *z* = 328λ along the optical axis, which is in excellent agreement with results obtained by numerical simulation and predicted by the local grating approach. It is clearly seen that the transmitted light was diffracted into the zeroth-order and first-order as illustrated by the bright light traces. The zeroth-order diffraction beam propagates in the direction parallel to the optical axis, while the first-order diffraction beams intercept the optical axis and form the optical needle, which is similar to the formation of a quasi-non-diffracting beam with conical lenses. In order to better understand the formation mechanism and the physical properties of such a super-long hollow optical needle, the Poynting vector distributions of the optical field around and within the optical hollow needle are plotted in Figs. 3(b) and 3(c), respectively. The arrows indicate the Poynting vectors, which correspond to the light traces in the geometrical optics. In the lower half of Fig. 3(b), the Poynting vectors almost have the same conical angle, and are direct to the optical needle. Within the needle area, the Poynting vector directions have complicated distributions. In the upper part of Fig. 3(b), the Poynting vectors keep their original direction after crossing the optical axis. This again shows good agreement with the local grating approach. A magnified illustration of the Poynting vectors is presented in Fig. 3(c), which shows the cross-section of the optical needle and the corresponding Poynting vector distributions. The main lobe of the hollow needle is surrounded by a series of weak ring lobes with gradually decayed intensity. It was found that inside the lobes the Poynting vectors are parallel to the beam propagation direction, or that the energy propagates parallel to the optical axis. Energy exchanges were also found between the neighboring lobes, as indicated by the arrows pointing from one lobe to another one.

Based on the numerical simulation, major parameters of the optical hollow needle are obtained, including the transverse FWHM of the central dark area, the sidelobe ratio (the ratio of the maximum side lobe intensity to the central lobe intensity), and the central lobe intensity, which are plotted along the propagation direction in Fig. 4. The optical needle has a length of approximately 103λ*’*, within the area between *z* = 225λ and *z* = 328λ. There is a clear fluctuation in the central lobe intensity, similar to the quasi-non-diffracting beam generated by the conical lens with a hard aperture [14]. In most of its area, the hollow needle has a sub-diffraction transverse size that is less than the diffraction limit of 0.53λ (0.5λ/NA) (where NA denotes numerical aperture). It was found that the transverse size gradually decreases from 0.53λ at *z* = 233*λ* to 0.31λ at *z* = 328λ, as the beam propagates forward. The hollow needle becomes super-oscillatory with a transverse size of less than 0.40λ (0.38λ/NA), when *z* is larger than 242λ, resulting in a super-long, super-oscillatory optical hollow needle with a length of 86λ. A clear fluctuation in the sidelobe ratio was also observed along the propagation direction, while in most of the area the sidelobe ratio is approximately 27– 40 %.

To understand the formation of such a super-long hollow needle, the contribution from different belt area on the lens was investigated by numerical simulation. In Figs. 5(a)–5(d), the central lobe intensity distribution (solid curves) is plotted in the propagation direction, when the areas of 0<*r*<316λ, 316λ<*r*<423λ, 423λ<*r*<527λ, and 527λ<*r*<638λ were blocked, respectively, with zero transmittance corresponding to the areas marked I, II, III, and IV in Fig. 2. For comparison, the intensity obtained without blocking area was also plotted in dashed curves in the plots of Fig. 5. As predicted based on the local grating approach in Fig. 2, the central half-part of the entire lens only contributes to a 22λ-long needle, a small fraction of the entire optical needle. It was found that, similar to the case for a conical lens, the different parts of the optical needle are formed by the wave diffracted from different ring belt areas, as shown in Fig. 2, which correspond to the formation of an optical needle in the areas of 225λ<*z*<247λ, 247λ<*z*<276λ, 276λ<*z*<299λ, and 299λ<*z*<328λ along the propagation direction, showing very good agreement with the numerical simulation and local grating approach. It is also interesting to note that the largest oscillation in the optical intensity at *z* = 269λ is caused primarily by the outermost belt area, as seen in Fig. 5(d), where this oscillation is suppressed to a certain extent when the outermost ring belt area was blocked. As pointed out above, such an oscillation is attributed to the abrupt change in the illumination intensity at the outer boundary area, due to the hard aperture effect [54].

## 4. Experimental methods

#### 4.1 Lens Fabrication

To fabricate the binary phase lens, a 348-nm-thickness Si_{3}N_{4} dielectric layer was first deposited on top of the sapphire substrate using plasma-enhanced chemical vapor deposition. The thickness corresponds to the relative phase change of π. According to the optimized ring belt distribution, electron-beam lithography and dry etching were used to form the concentric dielectric ring belts on the lens. Figure 6(a) presents the scanning-electron-microscopy (SEM) images of the lens sample, and Fig. 6(b) is a zoom-in picture of the lens, which gives a close view of its central part.

#### 4.2 Experimental Setup

The optical needle is azimuthally polarized with only a transverse polarization component, and the super-oscillatory field can propagate into the far field; therefore, its sub-diffraction futures can be captured with a high-numerical-aperture optical microscope [52]. 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 λ = 632.8 nm. The linearly polarized laser beam is converted into an azimuthally polarized Laguerre-Gaussian wave with an *S*-waveplate (SWP; RPC-632.8-06-188, Workshop of Photonics). The beam is normally incident on the binary phase lens. Then, the pattern of the diffracted field is obtained by a microscope system, consisting 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 CMOS camera with a resolution of 3856 × 2764 and pixel size of 1.67 µm × 1.67 µm (acA3800-14 µm, Basler AG, Germany). The objective lens was mounted on an open-loop objective lens nano-positioner (PZT; EO-S1047, Edmund Optics, USA) with a linear scanning range of 100 μm.

## 5. Experimental Results

The experimentally acquired transverse diffraction pattern within the optical needle is depicted for propagation distances of *z* = 250λ, 260λ, 270λ, 280λ, 290λ, and 300λ in Fig. 8, while the numerical simulation result is also presented for comparison. Good agreement was found between them. At these positions, the measured FWHMs are 0.34λ, 0.34λ, 0.38λ, 0.36λ, 0.34λ, and 0.34λ in x-direction, respectively, which are close to their theoretical values of 0.38λ, 0.36λ, 0.35λ, 0.34λ, 0.34λ, and 0.33λ, respectively. The theoretical in-plane phase distribution is also plotted in Fig. 8, which shows clear evidences of optical super-oscillation, as indicated by the abrupt phase changes of π at radial positions with minimum optical intensity.

To have a comprehensive understanding of the characteristics of the optical needle, a long scan was taken along the propagation direction. The optical intensity between *z* = 220λ and *z* = 325λ in the propagation plane is illustrated in Fig. 9(a), and its theoretical result is presented in Fig. 9(b) for comparison. Although there is a difference in the intensity distribution, excellent agreement was found between them: the propagation distance of the optical needle is approximately 102λ, and the central lobe is surrounded by several gradually decaying sidelobes. The optical intensity of the central lobe, the hollow needle FWHM, and the sidelobe ratio were also plotted as functions of propagation distance in Figs. 9(c)–9(e). The experimentally obtained peak intensity shows less fluctuation compared to that predicted by numerical simulation. This suppression in intensity distribution is attributed to the fact that, in the experiment, the area outside of the designed lens area is transparent to the incident beam, which results in a smooth change in the transmitted optical intensity at the lens boundary. However, the outside area is ignored in the numerical simulation for the sake of reducing the size of the model, which leads to an abrupt change in the optical intensity at the lens edge, and therefore results in a Sevier fluctuation in the optical intensity along the propagation direction, as already pointed out in previous research [54]. This is also consistent with the theoretical result presented in Fig. 5(e), as was already discussed above. It is clearly seen that the optical needle ranges from *z* = 220λ to 322λ, resulting in a 102λ-long hollow needle. Also seen is good agreement in the transverse size between the experimental and theoretical results. Because the transverse intensity distribution is not symmetrical, an average FWHM is used by taking the mean of FWHMs measured in different directions. As shown in Fig. 9(d), in most of the part of the needle between 240λ and 334λ the transverse size of the hollow needle is less than the diffraction limit of 0.53λ (0.5λ/NA), leading to a 94λ-long sub-diffraction sub-wavelength optical hollow needle. It is also noticed that between z = 258λ and z = 334λ the transverse size is less than or very close to the super-oscillatory criteria of 0.4λ (0.38λ/NA). The discrepancy between the theoretical and experimental results is attributed to the difficulties in the optical alignment, especially for such ultra-long optical needle. As illustrated in Fig. 9(e), the sidelobe ratio oscillates around the value of 0.37, which is slightly higher than its theoretical value of 0.35. Generally, the experimental results are in good agreement with the numerical simulation.

## 6. Discussion and Conclusions

To study the behavior of the optical needle at the air-water interface, a practical issue in the microscopy of bio-chemical applications, numerical simulation was carried out to obtain the diffraction pattern when the entire optical needle is immersed in the water.

Figure 10(a) depicts the optical intensity distribution in the propagation plane between *z* = 220λ and 450λ, which shows a clear super-long optical hollow needle with a length of approximately 220λ, or 139.2 μm. The central lobe intensity distribution, transverse size, and sidelobe ratio are plotted in Fig. 10(b), which also shows a clear fluctuation along the propagation direction. In the area between *z* = 260λ and 440λ, the transverse size of the hollow needle is less than the super-oscillatory criteria, showing a super-oscillatory hollow needle as long as 180λ with a transverse size of approximately 0.35λ–0.4λ. The fluctuation was found in the sidelobe ratio along the propagation direction with a value in the range between 0.4 and 0.8, where the largest variation was observed near the far end of the needle. To further investigate the formation of the extended optical needle, the Poynting vector distributions in the longitudinal cross-section were plotted within the needle area and the air-water interface. As shown in Fig. 10(c), the Poynting vectors are refracted into smaller angles at the interface, which can be clearly seen at z = 139.26 μm (220λ). The reduction in the propagation angle results in the extended non-diffracting propagation distance.

In conclusion, the idea of normalized angular spectrum compression was proposed to create quasi-non-diffracting beams. It is found that the quasi-non-diffracting beam can be realized by increasing the cutoff spatial frequency by reducing the effective working wavelength, while keeping the angular spectrum unchanged. A method based on the concept of the local grating and super-oscillation was developed to design a binary phase lens consisting of concentric dielectrical ring gratings for generation of super-oscillatory quasi-non-diffracting beams. Although other method has been proposed for optical needle design [55], it requires calculating diffraction patterns at multiple points within the desired optical needle, which tremendously increases the computational complexity and makes it difficult in designing long DOF optical needle. Significantly reducing the design difficulty, our method makes it possible, and provides a practical tool, to design a quasi-non-diffracting beam of super-oscillatory sub-wavelength transverse size. It also provides a clear physical picture of the formation of such super-oscillatory quasi-non-diffracting beams. In the experiment, a super-long optical hollow needle was obtained with a propagation distance of approximately 102λ. A sub-diffraction sub-wavelength hollow needle was also observed within the 94λ-long non-diffracting propagation distance. Excellent air-water interface penetration is also observed by numerical investigation, which shows a nearly doubled propagation distance of 180λ, while keeping the super-oscillatory transverse size at approximately 0.31λ–0.4λ. The validity of the idea of synthesizing super-long, sub-diffraction optical hollow needles by normalized angular spectrum compression was well verified by the numerical and experimental results. Longer non-diffracting propagation distance is expected for a lens with larger radius and shorter effective wavelength. This idea and the corresponding design approach can be extended to waves with different wavelength ranges.

## Funding

National Key Basic Research and Development Program of China (Program 973) (2013CBA01700); China National Natural Science Foundation (61575031, 61177093); Fundamental Research Funds for the Central Universities (106112016CDJZR125503); Open Fund of State Key Laboratory of Information Photonics and Optical Communications (University of Electronic Science & Technology of China), P. R. China; Program for New Century Excellent Talent in University (NCET-13-0629).

## Acknowledgments

Authors also thank LetPub (www.letpub.com) for their linguistic assistance during the preparation of this manuscript.

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