2D Exotic Optical Lattice via a Digital‐Coding Circular Airy Beam

Optical lattices have been widely used from classical to quantum physics. The tunable and scalable fabrication of lattices would be of great significance in lattice‐based multipartite applications. This work demonstrates first that a circular Airy beam (CAB), which has the peculiar properties of self‐healing and abrupt autofocusing, can be used to generate two‐dimensional (2D) optical lattices in propagation when encoded by a programmable spatial mask, resulting in the formation of large‐scale and tunable optical lattices with both axis and axial symmetry, and even high‐orbital kaleidoscope shapes. The efficient diffraction of CAB during the spatial crosstalk with the mask enables the realization of tunable lattices with rich periodicity and complexity. The study shows a flexible method to manipulate lattices with large‐scale and versatile structures for potential applications in integrated and scalable optical and photonic devices.


Introduction
Optical lattices have emerged as valuable tools in numerous areas of optics, [1] atoms, [2][3][4][5] condensed matter, [6] and photonics. [7,8]igh-dimensional and tunable optical lattices, such as triangular, hexagonal, and movable optical lattices have been implemented to explore rich phenomena in quantum physics, [9][10][11][12] where the optical lattices were prepared by the interference of Gaussian laser beams.Furthermore, optical lattices with tailorable structures have been experimentally created in various optical systems including photorefractive crystal, [13] atomic medium, [14][15][16][17][18][19] photonic crystal fibers, [8,20] and artificial structures, [21][22][23] and have been adopted to control both the classical and quantum behaviors of light.26] In recent years, the structured light of the vortex beam has been introduced to prepare optical vortex lattices, which carry the orbital angular momentum of photons within each array and have an additional degree of tunability to the lattice structure.35][36][37][38] Here, we propose an alternative scheme to engineer 2D optical lattices utilizing a structured circular Airy beam (CAB) and a digital spatial mask.Compared to the traditional ways of generating optical lattice by interference between several laser beams, our scheme utilized only a single laser beam of CAB, which is encoded with a spatial mask for programmable modulation of its amplitude and phase distributions that induce structural crosstalk between the CAB and the digital mask, resulting in the formation of large-scale and tunable optical lattices with both axis and axial symmetry, and even high-orbital kaleidoscope shapes.In this proposed scheme, the abrupt autofocusing (AAF) and self-healing properties of CAB, which allow for autotunable focal depth and sharpness in free space, also play important roles in the manipulation of the complex lattices near the far-field plane, paving the way for the creation of exotic lattices through a multi-interference process.This simple yet flexible scheme is the first demonstration of steered beam arrays by a structured CAB beam via programmable modulation of its propagation, offering multifunctional controllability. [39,40]he CAB possesses a radially symmetric Airy intensity distribution and is known for its unique self-accelerating property toward the center of symmetry. [41]As a result of its distinctive properties, CAB has also been termed as an AAF beam, which refers to the sudden increase of intensity at the focal point in free space. [42,43]Owing to its fascinating features, CAB has found numerous applications in various fields, such as optical trapping, [30,[44][45][46] light bullets creation, [47] photopolymerization, [48] and advanced manipulations. [49,50]One of the crucial aspects of CAB is its focusing behavior, which is characterized by spot size, depth-of-focus, and intensity contrast between Airy rings.58][59][60] By leveraging AAF, self-healing, and the stable superposition behavior of the CAB, we propose and experimentally demonstrate the generation of 2D optical lattices in the propagation of CAB through a spatial digital mask.The process is analogous to the diffraction of CAB during the crosstalk with a spatial structure, [28] giving rise to the superposition and interference of numerous edge beams of a CAB and the formation of rich 2D optical lattice structures with steerable shapes and sizes.In our scheme, we first show the dual images of lattices relating to a real and a "virtual" focus, highlighting the peculiar AAF effect of CAB propagation described by the Janus wave (JW). [61]Moreover, complex amplitude and phase modulation are first introduced to replace the conventional amplitude diffraction optics, enabling the realization of tunable lattices with rich periodicity and complexity.

Generation of 2D Optical Lattice via a Modulated CAB
The observation of the generated lattice is illustrated schematically in Figure 1.A programmatically modulated CAB is prepared by a continuous wave laser source (λ ¼ 532 nm) and a spatial light modulator (SLM) loaded with superimposed information of the original CAB and a designed spatial mask.Subsequently, a 4Àf system consisting of two lenses with focal length f 0 and an aperture (AP), is used to convey the initial light field information.The rear focal plane of the 4Àf system also serves as the initial plane (IP) of CAB (z = 0).A lens L with focal length f is placed at the IP.Finally, a charge-coupled device (CCD) camera situated at the observation plane (OP), which is located at a distance z away from the lens L, records the resulting tunable optical lattice.
Correspondingly, the field distribution E ρ, z ð Þ at OP is given by the Collins formula [62] E ρ, in which ρ and r represent the radial coordinates at OP and IP, respectively, k is the wave number, T M ⋅ ð Þ denotes the modulation coefficient of the mask, Ai ⋅ ð Þ represents the Airy function, r 0 denotes the radius of CAB primary ring, w denotes the arbitrary scaling factor, and α is the exponential decay factor.For a free-space optical system, we have A = 1, B = z, C = 0, D = 1, while for a lens-optical system with focal length f, we have In the following statement, the mask is chosen to be inscribed with the circle of radius r 0 to achieve optimal results for the lattice.
As shown in Figure 1, the CAB is generated by the SLM while it is modulated by loading the mask information.The special AAF effect of the CAB is described by the JWs with mirror-twin focus at f AAF (real) and Àf AAF (virtual) when propagating in free space. [61]Thus, the modulated CAB follows the same autofocusing process (see Figure 1b).f AAF denotes the focal length of the AAF of CAB in free space and is written as [43] f Þ (without lens) can be further expressed as a superposition of two conjugated waves with equal amplitude, ψ ρ, z ð Þand ψ Ã ρ, À z ð Þ, which, after the lens, are imaged as , respectively [63] ρ denotes the transverse magnification of the lens L, and κ represents the size ratio of the dual images.Figure 1b shows the case of z ¼ f AAF , resulting in When a nonmodulated CAB is focused by a thin lens, the dual focuses appear in z FÀ and z Fþ , respectively.Interestingly, as proof of the AAF effect, the transverse wave distribution of CAB between z FÀ and z Fþ shows spatial interference between two conjugate waves. [61]Unlike the propagation of the nonmodulated CAB, in our scheme, the propagation of CAB is modulated by a digital spatial mask, which acts as a complex diffraction optics.Through amplitude and phase modulation, CAB with superimposed conjugate waves is diffracted to form a pair of mirror-twin diffraction patterns (DPs), which are composed of a real and a virtual one, shown in Figure 1b.Due to the AAF effect, the multiple diffracted beams further focused by the thin lens, lead to a pair of similar and inverted dual images of mirror-twin DPs at z FÀ and z Fþ .Furthermore, we demonstrate that a series of such dual images can be observed in the space near or between two foci z FÀ , z Fþ ð Þ , as shown in Figure 2a,i, or c,g.It is worth mentioning that certain peculiar optical lattices can be observed in this interval (as seen in Figure 2d-f ) under a selection of the specific parameters of the scheme, which we will discuss analytically in the following sections.Notably, the DP in Figure 2e at the focal plane, as a far-field diffraction distribution, manifests the most pronounced and stable optical lattice structure.

Experimental Realization and Simulation Results of Lattice
Using the stability criterion for far-field diffraction, it is possible to achieve and steer a set of stable 2D lattice arrays in the focal plane (or far-field plane) by manipulating the focal parameters of the CAB (w and r 0 in Equation ( 3)) and the structural shape of the digital mask.When the CAB is modulated by a digital mask, the lattice is induced by the spatial crosstalk between the CAB and the mask, which is equivalent to a diffraction barrier that the CAB is bent or diffracted around the mask edges.The DP is therefore formed by the interference of bending waves.Due to the AAF of the CAB, the bending waves are further focused along the center of the mask and are interfered to create optical lattices in the transverse plane shown in Figure 3. Specifically, for the odd-angle mask, the odd number of bending waves along the center of the mask are interfered to form the arrays with construction points that are twice as numerous as the original number of angles.Conversely, for the even-angle mask, the even number of bending waves along the center of the mask are interfered to form the arrays of construction points with an equal number of original angles.Figure 3a(1-4) and b(1-4) provides large-scale lattice arrays in the case of triangular, square, and star shape masks.Figure 3a(5-6) presents the theoretical results of pentagonal and hexagonal masks, the lattice arrays induced by these polygonal masks could hardly be constructed, since these shapes are more inclined toward annular apertures, which have been proven to increase the focal depth in the circular beam, [54] and (d,f ).e) The far-field diffraction case.The insets in the middle row show the details of the lattices.Here, a typical triangular mask is used, and other parameters are chosen to be f ¼ 72 mm, r 0 ¼ 1 mm, w 0 ¼ 0.05 mm, and α ¼ 0.05.and the induced diffraction is likely to be weaker than that with polygonal star masks in Figure 3a(3-4) and b(3-4).Utilizing the polygonal star masks, the CABs are appropriately encoded (blocked), which has been shown to enhance the intensity contrast of the CABs due to their self-healing and AAF properties, [64,65] and further interfered to achieve large-scale lattice structures with higher intensity contrast.
In Figure 3, the 2D optical lattices with regular shapes are achieved via employing typical binary (0 and 1) masks, which are analogous to amplitude apertures in the diffraction system.Most strikingly, the steering of 2D exotic lattices can be realized by utilizing atypical digital masks, as shown in Figure 4. Here, the atypical digital mask means that the modulation values can be set to multiple and arbitrary values, enabling the simultaneous modulation of both amplitude and phase information on the CAB. Figure 4a shows four different atypical masks, and Figure 4b,c shows the corresponding simulated and experimental lattices, respectively.Compared to Figure 3, the optical lattices in Figure 4b,c are no longer densely distributed in the center but present a kaleidoscopic distribution with a hollow center surrounded by the arrays, which is similar to the singular lattice generated by the interference of multiple vortex beams. [33]The characteristics of these lattices can be explained through the interference of a circular Airy wave packet diffracted by the edges of the mask.The modulation with a value of À1 in the mask corresponds to a Àπ change in phase, giving rise to complex interference processes at the edge circle than that in the center.The design and digital modulation of a mask  are relatively convenient, and the value range can even be extended to the complex domain to explore non-Hermitian situations.
To further advance our understanding of the steering of optical lattices by CAB, we undertook a comprehensive investigation of the CAB parameters w and r 0 , while selecting an atypical mask, as demonstrated in Figure 5. Remarkably, we found that when r 0 remains constant and w is decreased, the optical lattice becomes more pronounced and abundant in arrays.A similar trend is observed when w is fixed and r 0 is increased, the size of the array is reduced while its structure appears to be more abundant, but at the cost of reducing the contrast of the central lattice arrays.The principle of the fast Fourier transform algorithm suggests that r 0 , as a crucial parameter controlling the radius of the CAB main ring, undergoes an inverse change of the lattice size at OP when varied at IP, as clearly demonstrated in Figure 5a-c.In addition, the ring width factor w is linked to the self-healing effect and corresponds more closely to the variation of the AAF focal length f AAF (see the form of Equation ( 3)).The smaller w, the narrower the ring width, gives rise to the smaller f AAF (i.e., stronger focusing) and higher self-healing behavior.This variation of w has a higher impact on the generation of large-scale lattices, as vividly illustrated in Figure 5(1-3).In summary, the parameters r 0 and w play crucial roles in determining the structural size and contrast of the lattice distribution, respectively.Consequently, by suitably adjusting r 0 and w, optical lattices of diverse sizes and contrasts can be effortlessly obtained.

Conclusion
To conclude, we have successfully demonstrated the generation and steering of a 2D optical lattice via the propagation of CAB expertly encoded by a digital mask.By utilizing the autofocusing and self-healing properties of CAB, we have investigated the underlying mechanism of lattice formation.Our findings reveal the presence of inverted and similar dual lattices, in the vicinity of the far-field plane of the lens system, confirming the existence of the AAF effect during the propagation of encoded CAB.Furthermore, we have implemented both typical and atypical exotic optical lattices through programmable modulation, which can be extended to complex domains to explore intriguing non-Hermitian cases.By manipulating the multiparameters of the system, we have succeeded in exploring the manipulation of the size and shape of lattice arrays.Crucially, our work harbors potential for numerous fields of physics, including but not limited to atomic, optical, condensed, and quantum physics, as well as the tailoring of structured light beams for multiplexed optical tweezer and imaging.

Figure 1 .
Figure 1.a) Schematic for observation of optical lattice.By loading the corresponding phase and amplitude information in SLM, the modulated CAB is to be generated at z = 0. SLM: spatial light modulator; AP: aperture; CCD: charge-coupled device; IP: initial plane; OP: observation plane.b) The quasi-imaging AAF phenomenon of a modulated CAB.

Figure 2 .
Figure 2. Normalized intensity distributions of dual images at different z. (a,i) are a pair of dual images, ditto (b,h), (c,g),and (d,f).e) The far-field diffraction case.The insets in the middle row show the details of the lattices.Here, a typical triangular mask is used, and other parameters are chosen to be f ¼ 72 mm, r 0 ¼ 1 mm, w 0 ¼ 0.05 mm, and α ¼ 0.05.

Figure 3 .
Figure 3. Normalized intensity distributions of optics lattices at the plane of z = f.a) Simulation (a1-a6) and b) experimental (b1-b4) results.The numeric labels 1-6 represent the different masks, which are shown as insets in (a1-a6).Other parameters are the same as in Figure 2 except for f = 200 mm.

Figure 4 .
Figure 4. a) Atypical modulation masks and b,c) normalized intensity distributions of simulated and experimental lattices when w ¼ 0.03 mm and r 0 ¼ 2 mm.The numbers 1-4 denote cases of different masks.Remaining parameters are α ¼ 0.05 and f ¼ 200 mm.

Figure 5 .
Figure 5. Normalized intensity distributions of simulated optical lattices at the plane of z = f for the different w (1-3) and r 0 (a-c).Remaining parameters α ¼ 0.05 and f ¼ 200 mm.