Generation of diffraction-free beam with winding trajectory based on metasurface holography

The diffraction-free beams with curved trajectories and shaped wavefronts have wide application prospects in many fields. This paper proposes the generation of diffraction-free beam with winding trajectory and spiral wavefront based on holographic metasurface. The holographic metasurface consists of rotated rectangular nanoholes and the winding trajectory for the generated diffraction-free beam may be in two or three dimensional space under the control of the rotated nanoholes. The multiple diffraction-free beams are exemplified and the performance of holographic metasurfaces are testified by the simulation and experiment results. The utilization of compact metasurface enables the flexible generation of the diffraction-free beams with complex trajectories and tailored wavefronts. It may bring more new applications of diffraction-free beams with on-demand trajectories and customized wavefronts.


Introduction
Diffraction-free beams point to the beams without obvious spread during the propagation, and the common diffraction-free beams include the Bessel beam [1], Airy beam [2,3], Mathieu and Weber beam [4].Diffraction-free property makes the applications of this kind of beams appear in optical imaging [5,6], particle manipulating [7][8][9] and biochemical sensing.Diffraction-free beams with various trajectories may bring more convenience for practical applications like micro-machining [10] and light bullets [11,12].
The generation of diffraction-free beam with on-demand trajectory is a prerequisite for its applications.Till now, a variety of approaches have been developed to generate diffraction-free beams, where the spatial light modulator (SLM) is the common technique [13][14][15][16][17].As we know, the pixel of SLM is relatively large and it needs the electric drive.Liquid crystal plate is also used to generate beams with arbitrary trajectories, but it also requires electric field drive [18].Recently, metasurface consisting of nanounits has attracted much attention because it can effectively control the optical field in the nanometer scale and behaves the remarkable advantage in the design of planar devices [16].Metasurfaces have been also used to generate diffraction-free beams with uniform wavefronts [19] and bending trajectories [20,21].
With comparison to the diffraction-free beams with monotonic trajectories, the ones with winding trajectories are more desired for potential applications.As we know, most of non-monotonic curves are unintegrable and the traditional stationary phase method cannot be utilized.Thus, the empirical expression for the wavefront phase was proposed to generate the direction-reversible accelerating light beam [22].Relatively, the superimposition of segmented trajectories may be utilized [23], and yet it brings inevitably the complex operation.Therefore, it is still a challenge to conveniently generate the diffraction-free beams with complex trajectories.Vortex beam carrying one spiral phase can provide the orbital angular momentum for the interaction of light and matter.The combination of winding trajectory and spiral wavefront may bring more possibilities for the applications of diffraction-free beams in microscopy imaging, optical sensing, quantum communication and multi-dimensional optical manipulation [24][25][26].Due to the interference of the secondary fringes of diffraction-free beam, it is difficult to generate an ideal diffraction-free vortex beam especially with complex propagation trajectory.The holographic imaging is supposed to shape the trajectory of vortex light [27].
In this work, we propose the generation of diffraction-free beams with in-plane sinusoidal and three-dimensional helix trajectories based on holographic metasurfaces.The trajectory curve of diffraction-free beam can be taken as the combination of lots of points.The multi-point pattern is formed through Fresnel holography and the metasurface hologram is performed through rotating the nanometer rectangular nanoholes.The design principle of metasurface hologram is shown in section 2. The numerical simulations verify the generation of diffraction-free beams with controllable trajectories and wavefronts, which are detailed in section 3. The experiment measurement for the generated diffraction-free beam with sinusoidal trajectory is conducted in section 4 and the results testify the availability of the proposed method.Section 5 gives the conclusions of this work.The proposed method is benefit to the flexible generation of diffraction-free beams with any trajectories and wavefronts and it may be helpful for expanding their applications of diffraction-free beams in related fields.

Design principle
As we know, the Fresnel diffraction field of three-dimensional object E(x f , y f , 0) can be expressed, where f (x, y, z) denotes the function of object, (x f , y f , 0) denotes the coordinates of Fresnel diffraction plane, , k is the wave number with respect to the wavelength of λ.One can construct Fresnel hologram through extracting the phase distribution of E(x f , y f , 0).As the plane light illuminates the Fresnel hologram, the image of object can be reproduced.Figure 1(a) shows the imaging principle of Fresnel holography [28][29][30][31], and the top picture denotes the recording process and the down picture denotes the reproducing process.Where h(x 1 -x 2 ;y 1 -y 2 ;± z) represents the point spread function for the Fresnel diffraction with the propagation distance of ±z.The integration of equation ( 1) means that Fresnel holographic field can be taken as the superposition of Fresnel diffraction fields of lots of point sources at object plane.The diffraction-free beam with the default winding trajectory contains light points distributing continuously at the propagation trajectory.Thus, the above equation may be rewritten as the summation of exp(ikR j ) with R j = [(x f −x j ) 2 +(y f −y j ) 2 + z j 2 ] 0.5 .In fact, it can be equivalent to the transmission superposition of series of metalenses [32].For the diffraction-free vortex beam, one spiral phase needs to be added and the diffraction field can be simply written by [33], where the integer of l denotes the order of vortex and θ is the azimuthal angle.The Fresnel hologram is obtained in terms of equation ( 2) and the metasurface hologram is realized with the help of rotated rectangular nanoholes etched in silver film deposited on the glass substrate, where the rotation angles of nanoholes take half of the phases at the corresponding positions of Fresnel hologram.This is because the introduced phase by the rotated nanohole is twice of its rotation angle [33].The sketch map in figure 1(b) shows the generation of diffraction-free beam based on holographic metasurface.
In practice, the parameters of metasurface including the length and width of nanoholes, the separation of two adjacent nanoholes and the thickness of silver film, need to be optimized so as to get the maximum transmission under the limit of sampling theorem.The optimization process is achieved using finite-difference time-domain method [34].During the simulation, the incident wavelength takes 633 nm, the minimum grid length is set at 2 nm, the boundary along the z direction takes the perfectly matched layer and the boundaries along the x and y directions take the periodic boundary condition or the perfectly matched layer, which depends on the trajectory of diffraction-free beam.The dielectric constants of glass and silver are taken the values given by Palik ED [35].
The optimized parameters of nanoholes are taken as the length of l = 240 nm, the width of w = 140 nm, the thickness of silver film of h = 180 nm and the separation of adjacent nanoholes of d x = d y = 300 nm, which satisfies the sampling principle and ensures the maximum transmission.With these above parameters, the polarization conversion of a nanohole can reach 48%.

Numerical simulations
Firstly, we generate the diffraction-free beam with in-plane sinusoidal trajectory, and the trajectory curve in three-dimensional space is shown in figure 2(a).The position coordinates of any point on the sinusoidal trajectory can be written as: here a represents the amplitude of sinusoidal trajectory, z 1 denotes the initial position of sinusoidal trajectory and ∆z denotes the step along z direction.Insert equation (3) into equation ( 2) and let l = 0, and then extract the phase of the diffraction field, one can construct the metasurface hologram through encoding this phase distribution with the help of the geometric phase delay introduced by rotating nanounits.Then, we design the metasurfaces to generate the diffraction-free beam and vortex beam with three-dimensional helix trajectory, and their trajectory curves in three-dimensional space are shown in figures 3(a) and (c).The position coordinates at any point on the helix trajectory can be written as, Figure 3(b) shows the transverse intensity distributions of the generated diffraction-free beam with the parameters taking l = 0, a = 1 µm, z 1 = 6 µm, and ∆z = 24 µm.Where the propagation distances from the first pattern to other patterns take 15 µm, 18 µm, 20 µm, 21 µm, 25 µm, 29 µm 33 µm and 34 µm.For convenience observation, the dashed circles are inserted in the patterns.From the given results, one can see clearly bright spot rotates along anti-clockwise direction with increase of the propagation distance.These results mean that the diffraction beam proceeds spirally and they confirm the generation of the diffraction-free beam with the helix trajectory.
Figure 3(d) shows the transverse intensity distributions of the generated diffraction-free vortex beam.The parameters of the trajectory take l = 3, a = 1.8 µm, z 1 = 20 µm, and ∆z = 80 µm.Where the propagation distances for the patterns from x to take 18 µm, 38 µm, 62 µm, 90 µm, 108 µm, 120 µm, 130 µm and 160 µm.The given results show the annular intensity distributions and they also rotate along anti-clockwise direction with increase of the propagation distance.These results indicate that the diffraction light proceed spirally and they confirm the generation of diffraction-free vortex beam with helix trajectory.It needs to be pointed out that the obtained rings are not perfectly symmetric due to the interference of the secondary fringes of diffraction-free beam.

Experiment measurement
In order to testify the performance of metasurface, we manufacture one metasurface sample to generate the diffraction-free beam with helix trajectory and perform practical experimental measurement of longitudinal and transverse intensity distributions of diffraction-free beam.The manufacture of sample can be divided into two steps.The first step is the deposition of the silver film on a glass substrate using magnetron Then the sample is placed into the optical path shown in figure 4(a).The light beam emitted from laser illuminates the metasurface sample from the glass substrate and the size of light spot is much larger than the size of metasurface.The quarter wave plate QWP 1 is used to obtain the circular polarization state.A two-dimensional charge-coupled device (CCD) receives the transmitted intensity distribution amplified by the 40× microscope objective (MO).The combination of the polarizer P and the quarter wave plate QWP 2 is used to remove the initial circular polarization.A density filter D is used to control the incident intensity.
Figure 4(b) shows the projections of the measured longitudinal intensity distributions of holographic metasurface in xoz plane, which is composed of 32 one-dimensional data extracted from two-dimensional transverse intensity distributions recorded by CCD.From the given results, we can clearly see that the trajectory of longitudinal intensity distributions closes to the sinusoidal curve and almost two periods take on.This trajectory tallies with the theoretical model.Figure 4(c) shows 11 transverse intensity distributions with the equal longitudinal separation, which correspond to 11 propagation distances of z 1 -z 11 denoted in figure 4(b).It is easy to see that the bright spot rotates along the inserted circle.As the propagation distance increases, the beam rotates nearly two turns.These results are consistent with the simulation results.The helix trajectory of diffraction-free beam can be intuitively seen from the video (Supplemental document).

Conclusions
This paper proposes the generation the diffraction-free beams with winding trajectories based on holographic metasurfaces.The propagation trajectory of diffraction-free beam may wind in two-or three-dimensional space and the wavefront may be the uniform and spiral.The metasurface hologram is performed in terms of principle of metalenses, and the propagation trajectory and wavefront of diffraction-free beam are changed only through rotating the nanoholes.The concise theoretic model, simple mathematical operation and compact metasurface hologram are benefit to manipulating the formation of diffraction-free beam with winding trajectory and tailored wavefront.The numerical simulations and experimental measurement verify the generation of diffraction-free beams with controllable trajectories.This work provides one more effective way to generate the diffraction-free beams with winding trajectories with respect to the caustic method, stationary phase method, trajectory segmentation method and angular spectrum-based method.We anticipate this work will bring convenient applications of diffraction-free beams in wider fields including object navigation, optical micro-manipulation, micro-machining and plasmonic guiding.

Figure 1 .
Figure 1.(a) Schematic diagram of Fresnel holographic imaging and (b) sketch map for the generation of diffraction-free beam based on holographic metasurface.

Figure 2 .
Figure 2. Schematic diagrams of the trajectory curves of diffraction-free beam (a) and vortex beam (d) in three-dimensional space, (b), (e) the theoretical results and (c), (f) simulated results for the reproduced image of diffraction-free beam (b and c) and vortex beam (e and f) with sinusoidal trajectory.
Figure 2(b)  shows the reproduced image of diffraction-free beam with sinusoidal trajectory in xoz plane and the image in yoz plane is also inserted in the lower right corner.Figure2(c) gives the intensity distributions of holographic metasurface.Where the parameters of metasurface take a = 1 µm, z 1 = 1 µm, and ∆z = 30 µm.Theoretically, two periods of the sinusoidal trajectory appear.From figures 2(b) and (c), one can see the simulated results are almost the same as the theoretical results.The diffraction-free beam with two periods of the sinusoidal trajectory takes on in xoz plane and three bright spots appear in the yoz plane.

Figure 3 .
Figure 3. Schematic diagrams of diffraction-free beam (a) and vortex beam (c) with helix trajectory, and the transverse intensity distributions of the generated diffraction-free beam (b) and vortex beam (d).

Figure 2 (
Figure 2(d) shows the sinusoidal trajectory of the diffraction-free vortex beam in three-dimensional space, and the corresponding parameters take l = 3, a = 1.0 µm, z 1 = 20 µm and ∆z = 40 µm.Figures 2(e) and (f) give the theoretical and simulated results of the generated diffraction-free vortex beam.Obviously, the simulated results are almost the same as the theoretical results.The diffraction-free vortex beam with near one period of the sinusoidal trajectory takes on in xoz plane and the inserted transverse intensity distributions tally with the annular shape rule of vortex field.Then, we design the metasurfaces to generate the diffraction-free beam and vortex beam with three-dimensional helix trajectory, and their trajectory curves in three-dimensional space are shown in figures 3(a) and (c).The position coordinates at any point on the helix trajectory can be written as,

Figure 4 .
Figure 4. (a) Experimental setup and SEM image of the metasurface sample, (b) the measured longitudinal intensity distributions in xoz plane and (c) the transverse intensity distributions at different propagation distances, where the dashed circles are inserted.