Abstract
Bessel beams are of great interest due to their unique nondiffractive properties. Using a conical prism or an objective paired with an annular aperture are two typical approaches for generating zerothorder Bessel beams. However, the former approach has a limited numerical aperture (NA), and the latter suffers from low efficiency, as most of the incident light is blocked by the aperture. Furthermore, an additional phasemodulating element is needed to generate higherorder Bessel beams, which in turn adds complexity and bulkiness to the system. We overcome these problems using dielectric metasurfaces to realize metaaxicons with additional functionalities not achievable with conventional means. We demonstrate metaaxicons with high NA up to 0.9 capable of generating Bessel beams with full width at half maximum about as small as ~λ/3 (λ=405 nm). Importantly, these Bessel beams have transverse intensity profiles independent of wavelength across the visible spectrum. These metaaxicons can enable advanced research and applications related to Bessel beams, such as laser fabrication, imaging and optical manipulation.
Introduction
Nondiffracting Bessel beams are a set of solutions of the free space Helmholtz equation and have transverse intensity profiles that can be described by the Bessel functions of the first kind. Since their discovery in 1987^{1}, they have exhibited many interesting properties such as nondiffraction^{1}, selfreconstruction^{2} and even providing optical pulling forces^{3, 4}. The scalar form of Bessel beams propagating along the z axis can be described in cylindrical coordinates (r,φ,z) by:
where A is the amplitude, k_{z} and k_{r} are the corresponding longitudinal and transverse wavevectors that satisfy the equation (where λ is the wavelength). Equation (1) shows that the transverse intensity profiles of Bessel beams are independent of the z coordinate, which gives rise to their nondiffracting characteristic. It also indicates that any higherordered Bessel beam (n≠0) must carry orbital momentum and have zero intensity along the z axis at r=0 because of the phase singularity resulting from the term.
Ideal Bessel beams are not spatially limited and carry infinite energy; therefore, they can only be approximated within a finite region by the superposition of multiple plane waves. This can be achieved by symmetrically refracting incident plane waves toward the optical axis of a conical prism, an axicon, to generate a J_{0} Bessel beam. Figure 1a shows the schematic diagram of a conventional axicon. The numerical aperture (NA) of an axicon is related to the angle α (Figure 1a) by:
where n is the refractive index of the constituent material, often glass. This equation and Figure 1a show that for a given refractive index, achieving high NA axicons requires an increase in α. Considering a refractive index of 1.5, typical of most silica glasses, total internal reflection occurs when α>42°. Thus, the NA of a conventional axicon cannot exceed 0.75 (Supplementary Fig. S1). This, in turn, also limits the minimum achievable full width at half maximum (FWHM) of the Bessel beam. In addition, the tip of a refractive axicon is rounded instead of being perfectly sharp because of limitations in glass polishing, which again affects the FWHM of the Bessel beam^{5}. Herein, the FWHM of the zerothorder Bessel beam J_{0} is defined as the distance between two points at half of the maxima intensity of the center bright spot, and can be derived from Equation (1) as:
where . Similarly, the FWHM of a J_{1} Bessel beam is defined as twice the distance from the dark spot center to the point at its closest ring with the half maxima intensity, and given by:
In the case of conventional axicons, the NA is almost constant within the visible region due to the weak dispersion of glass. Thus, the FWHM of the J_{0} beam is proportional to wavelength and varies accordingly. For example, changing the wavelength from 400 to 700 nm can result in a difference of 175% in the FWHM. Alternatively, a high NA objective lens paired with an annular aperture is usually used to generate Bessel beams with subwavelength FWHMs^{6}, as shown in Figure 1b. However, this configuration is not efficient as most of the incident light is blocked by the aperture. Both methods require the addition of phasemodulating elements, such as spatial light modulators or spiral phase plates, to generate higherordered Bessel beams.
In recent years, metasurfaces, consisting of subwavelengthspaced phase shifters, have been demonstrated to fully control the optical wavefront^{7, 8, 9, 10, 11, 12}. Various compact optical components have been reported including lenses^{13, 14, 15, 16, 17}, holograms^{18, 19, 20, 21}, modulators^{22, 23, 24} and polarizationselective devices^{25, 26, 27, 28}. Metallic and dielectric metasurface axicons were reported in Ref. 29 and Ref. 27, respectively; both had low NAs. Unlike conventional phasemodulating devices (for example, spatial light modulators), metasurfacebased devices can provide subwavelength spatial resolution, which is essential in order to deflect light by very large angles. This is mandatory to realize high NA optical components, including axicons and lenses capable of generating beams with even smaller FWHM. Various applications, including (but not limited to) scanning microscopy^{6, 30, 31}, optical manipulation^{32, 33, 34} and lithography^{35, 36}, all require subwavelength FWHM to achieve high spatial resolution, strong trapping force and subwavelength feature sizes, respectively. Here we report metaaxicons with high NA up to 0.9 in the visible region that are capable of generating, not only the zerothorder, but also higherordered Bessel beams with FWHM about onethird of the wavelength without the use of additional phase elements. In addition, by appropriately designing the metasurfaces’ phase shifters, the transverse field intensity profiles are maintained independent of the wavelength.
Materials and methods
Figure 1c shows a schematic diagram of a metaaxicon. The basic elements of the metaaxicon are identical rotated titanium dioxide (TiO_{2}) nanofins with height h, length L and width w, arranged in a square lattice. To maximize the performance of the metaaxicon, each nanofin should act as a halfwaveplate at the design wavelength, converting incident circularly polarized light to its orthogonal polarization state. To tailor the required phase profiles, we use geometric phase associated with the rotation angle of the nanofin, known as the Pancharatnam–Berry phase^{37, 38}. The h, L and w parameters are determined using the threedimensional finite difference time domain (FDTD from Lumerical Inc.) method to maximize the circular polarization conversion efficiency^{39, 40}. At the design wavelength λ_{d}=405 nm, simulated polarization conversion efficiencies >90% were obtained. The efficiencies decrease as NA increases due to the sampling criterion (Supplementary Fig. S2). To determine the polarization conversion efficiency, we arranged an array of TiO_{2} nanofins in such a way to deflect light to a particular angle and then the efficiency is calculated by dividing the total deflected optical power by the input optical power. Perfectly matched layer boundary conditions were used normal to the propagation of the incident circularly polarized light and periodic conditions were used for the remaining boundaries.
For the generation of a zerothorder Bessel beam, a metaaxicon requires a radial phase profile φ(r) with a phase gradient
This can be understood from the generalized Snell’s law^{9} as the condition for all light rays to be refracted by the same angle θ at the design wavelength λ_{d}, where sin(θ) is the NA. Integrating Equation (5) gives:
where . The generation of the highorder Bessel beams requires the addition of a term nφ, where is the azimuthal angle, which represents the phase of an optical vortex imparted to the deflected light. Equation (6) then becomes
This phase profile is imparted by the rotation of each nanofin at a position (x, y) by an angle for the case of lefthanded polarized incidence. TiO_{2}based metaaxicons are fabricated using the approach described in Ref.41, which can prevent tapered sidewall^{42}. Figure 1d shows a scanning electron microscope image of the center part of a fabricated metaaxicon. We used a custombuilt microscope to characterize the metaaxicons. A schematic diagram of the setup and the optical components used can be found in Supplementary Fig. S3.
Results and discussion
Figure 2a and 2d shows the measured transverse intensity profile of the J_{0} and J_{1} Bessel beams at λ=405 nm, whereas Figure 2b and 2e shows the intensity along a horizontal cut across the centers of Figure 2a and 2d, respectively. The measured FWHM of the J_{0} Bessel beam is observed to be ~163 nm with 3.5 nm standard deviation, which is very close to its theoretical limit of 160 nm (Equation (3)). The measured FWHM of the J_{1} Bessel beam is 130 nm with 1.75 nm standard deviation, which agrees well with its theoretical value of 131 nm (Equation (4)). Figure 2c and 2f shows the intensity profile along the beam propagation direction of the J_{0} and J_{1} Bessel beams. Their FWHMs at different planes normal to the propagation z axis are provided in Supplementary Fig. S4. Both the J_{0} and J_{1} Bessel beam have a depth of focus of 75 μm (150λ). This value is close to the theoretical value using geometric optics, that is, , where D=300 μm is the diameter of the metaaxicons.
Polarization properties of a Bessel beam generated by a high NA metaaxicon can be very different from that of a corresponding axicon with low NA^{43, 44}. In order to understand the polarization properties of the J_{0} and J_{1} Bessel beams generated by the metaaxicons with NA=0.9, we show in Figure 3 their theoretical (first row) and simulated (second row) normalized electric field intensities Ex^{2}, Ey^{2} and Ez^{2}. Only a portion of the metaaxicon (30 μm in diameter), but with the same NA, was simulated due to limited computational resources. A slight deviation of the simulations from theory results from the uncoupled light of nanofins and the effects of the boundary in the simulations. We note that for either Ex^{2} (Figure 3a and 3d) or Ey^{2} (Figure 3b and 3e), the shape of the J_{0} and J_{1} Bessel beams at their respective centers is elliptical rather than circular. For example, as shown in Figure 3d and 3e, for J_{1}, the center regions are accompanied by two brighter spots at the end of the long axis of the ellipse. Moreover, the intensity of Ex^{2} and Ey^{2} show variation on their side lobes. These are also observed in their corresponding simulation results (Figure 3j and 3k). In addition, Ez^{2} is described by a Bessel function one order larger than its transverse electric field. These properties can be explained by considering the vector form of Bessel beams. The theoretical vector solutions of the electric field of a Bessel beam propagating along the z direction in cylindrical coordinates are:
where C_{TM} and C_{TE} are complex numbers associated with the constituent transverse magnetic (TM) and transverse electric (TE) waves of Bessel beams, and η is the phase difference between C_{TM} and C_{TE}^{3, 45}). For our case (circularly polarized Bessel beams), C_{TE} is equal to C_{TM}, , NA=0.9, and n=0 and n=1 for J_{0} and J_{1} Bessel beam, respectively. For high NA Bessel beams, the electric field of the z component results from TM waves: the higher the NA, the higher the contribution the TM waves make to . The and components mainly result from TE wave contribution, as the term is relatively small for the high NA case. The term contributes to localized intensity near the center spot or the most inner ring, as for large r, such that J_{n}(k_{r}r) and J_{n+2}(k_{r}r) cancel each other due to a π phase difference. The intensity distribution of side lobes away from the center of a Bessel beam is due to another term . When we transform cylindrical (r,φ) to Cartesian coordinates (x,y) using , we will have cos (φ) modulation resulting in the elliptical center, and a corresponding modulation of sin (φ) for the side lobe, which is shown in Figure 3a and 3d, respectively. This is a signature feature of high NA Bessel beams (see a comparison with a lower NA in Supplementary Fig. S5). Due to the spatially varying intensity of E_{x} and E_{y}, the Bessel beams for high NA are not homogeneously polarized, but rather show spacevariant polarization states (see Supplementary Fig. S6 for plots of ellipticity and polarization orientation angle). Therefore, only the center part of the J_{0} Bessel beam can be circularly polarized in the case of high NA case.
By the judicious design of our metasurface, we compensate the wavelength dependency of the FWHM of Bessel beams (Equations (3) and (4)). As mentioned previously, the transverse intensity profile is determined by the factor . In our case, the , where φ follows Equation (7). Therefore, k_{r} only depends on the phase gradient ∇φ(x,y,λ), which can be designed to be wavelengthindependent using the Pancharatnam–Berry phase concept. In this case, the phase gradient is a constant, and the NA is only proportional to the wavelength λ. This manifests experimentally, in the form of the increasing ring diameters in the Fourier plane of the metaaxicons for increasing wavelength (Supplementary Fig. S7). To demonstrate this unique characteristic across a broad wavelength region, we use the same method to design two metaaxicons for J_{0} and J_{1} with the NA=0.7 at the wavelength λ=532 nm. Each nanofin (L=210 nm, W=65 nm and h=600 nm) for this case is arranged in a square lattice, with a lattice constant of 250 nm. We show in Figure 4a–4d and 4f–4i the corresponding J_{0} and J_{1} Bessel beams in false color for different wavelengths (at λ=480, 530, 590 and 660 nm) with a bandwidth of 5 nm at the z plane about 60 μm away from the surface of metaaxicons. Note that the efficiency of the metaaxicon is dependent on wavelength. The efficiency was measured and is shown in Supplementary Fig. S8. The FWHM_{J}_{0} and FWHM_{J}_{1} for each wavelength spanning 470–680 nm are shown in Supplementary Fig. S9. Figure 4 explicitly indicates that the intensity profile for different wavelengths vary weakly, confirming wavelengthindependent behavior. It is notable that for these measurements the distance between the metaaxicon and objective lens was kept unchanged. We also repeated the measurements using a supercontinuum laser of bandwidth 200 nm centered at 575 nm (see Supplementary Fig. S10 for its spectrum). The intensity profiles (Figure 4e and 4j) for both J_{0} and J_{1} remarkably changed weakly. It is important to note that in order to generate high NA and wavelengthindependent Bessel beams, the Nyquist sampling theorem and wavelengthindependent phase gradient conditions both need to be satisfied. According to the Nyquist sampling theorem, one needs to sample the phase profile given by Equation (7) in the spatial domain with a rate that is at least twice the highest transverse spatial frequency. This requires the size of the unit cell to be equal to or smaller than , which cannot be satisfied by conventional diffractive elements. For example, spatial light modulators usually have ~6 μm pixel sizes^{46} and photoaligned liquid crystal devices are usually limited to a phase gradient of ~π/μm^{47}, corresponding to a maximum achievable NA of about 0.03 and 0.26 in the visible region, respectively. It is also important to note that the phase profile of metasurfaces can also be designed by varying the geometrical sizes (length, width or diameter and so on) of the nanostructures, pixel by pixel^{7, 9}. However, these metasurfaces, not designed by the Pancharatnam–Berry phase, are accompanied by strong amplitude differences between each pixel at wavelengths away from the design wavelength. This becomes more significant within the absorption region of the constituent materials used. In addition, the unwanted amplitude difference between each pixel can result in the deflection of light to multiple angles^{48}, changing the profile of the Bessel beams. Utilizing the Pancharatnam–Berry phase approach minimizes the relative amplitude difference between each nanofin for all wavelengths in the case of circularly polarized illumination, as each nanofin is identical, and consequently, has the same size. We experimentally demonstrate this concept by measuring metaaxicons consisting of silicon nanofins from the nearinfrared to the visible spectrum, where silicon becomes intrinsically lossy (Supplementary Fig. S11). It is clearly observed that the sizes of the Bessel beam remain constant over the wavelength range of 532–800 nm.
Conclusions
In summary, as a superior alternative to using conventional prism axicons or an objective paired with an annular aperture, we demonstrate high NA metaaxicons capable of generating Bessel beams of different orders in a single device in a much more efficient and compact way. The FWHM of J_{0} and J_{1} Bessel beams are shown to be as small as ~160 and 130 nm at the design wavelength λ=405 nm. This size is maintained for an exceptionally large distance of 150λ (depth of focus). Their polarization is spacevariant due to high NA. By tailoring the phase profile of the metaaxicons, the FWHMs of generated Bessel beams are made independent of the wavelength of incident light. These metaaxicons can be massproduced with large diameter using today’s industrial manufacturing (deep ultraviolet steppers). These properties show great promise in potential applications ranging from laser lithography and manipulation to imaging.
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Acknowledgements
This work was supported in part by the Air Force Office of Scientific Research (MURI, grant# FA95501410389), Charles Stark Draper Laboratory, Inc. (SC0010000000959) and Thorlabs Inc. WTC acknowledges postdoctoral fellowship support from the Ministry of Science and Technology, Taiwan (1042917I564058). RCD is supported by a Charles Stark Draper Fellowship. AYZ thanks Harvard SEAS and A*STAR Singapore under the National Science Scholarship scheme. This work was performed in part at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure (NNCI), which is supported by the National Science Foundation under NSF award no. 1541959. CNS is a part of Harvard University. We thank E. Hu for the supercontinuum laser (NKT ‘SuperK’).
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Keywords
 Bessel beam
 high numerical aperture
 light angular momentum
 metasurface
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