Controlling self-healing of optical field based on moiré dual-microlens arrays

Optical self-healing is a repairing phenomenon of a beam in the propagation, as it is perturbed by an opaque object. In this work, we demonstrate experimentally and theoretically that the moiré distributed dual-microlens array enables to generate optical fields with better healing ability to withstand defects than their counterparts of a single microlens array. By utilizing the double parameter scanning method, the self-healing degree of the optical field is significantly affected by both the interval distance and the relative angle of the dual-microlens arrays. The self-healing level is decreased significantly by lengthening the interval between the two microlens array with a small twist angle, while increasing the angle enhances the self-healing degree. Further study manifests the self-healing process with respect to the size and central location of the obstacle. The research results provide a simple and effective method to generate self-healing optical wave fields, which have potential applications including optical communication, assisted imaging technology, and even intense laser physics.


Introduction
Self-healing effect refers to the ability of a system to automatically detect and repair defects or faults that occur within itself. This can be seen in various areas such as self-healing materials and chemical self-healing. The defect in the materials can self-healing themself by applying physical or chemical modulation such as controlling the temperature, electric field, using chemical reagents, and so on [1][2][3]. Recently, self-healing of the optical fields has attracted much attention in many research fields, and the properties of self-healing have already been applied in several physical areas, such as quantum communication [4], optical manipulation [5], optical pumps [6], optical imaging [7,8], and creation of three-dimensional optically trapped microstructures [9]. One significant advantage of optical self-healing is that the beam can propagate in highly scattered and turbulent media. Moreover, the defective optical fields with self-healing properties can also be reconstructed in the propagation when an opaque obstacle obstructs parts of the beam. In the last several decades, many experiments and simulations have been carried out to concern the self-healing properties of the non-diffracting beam. Beyond the classical Bessel beam, the self-healing of Airy and its family beams has also been studied [10][11][12][13][14][15][16][17]. Furthermore, the other ingeniously designed structured beams based on fundamental non-diffracting beams [18], such as caustic beam [19], Mathieu beam [20], Weber beam [21], Pearcey beams [22], pillar beam [23], optical ring lattice [24], helico-conical beams [25], can recover the intensity profile of the optical field. Recently, the self-healing of self-rotating beams has been proposed by Niu et al, which can overcome obstacles and regenerate after a characteristic distance [26]. However, these complex optical fields are limited within many applications due to their unique intensity patterns and the restorable optical field areas. Recently, the partially coherent beam with a special design has also been shown to be self-healing in intensity correlation [27]. The polarization, degree of coherence function, and phase information can also obtain from the defective optical field [28][29][30].
Moiré technique refers to a process of overlapping two identical or similar periodic structures (called sublattices) to form a composite structure. Due to the difference in the angle between the two sublattices, the new period lattice can be generated, which is called moiré fringes. The moiré technique is first proposed in the physical area of van der Waals heterojunction [31,32]. This technology has been widely used in multiple physical areas, such as two-dimensional materials [33], acoustics [34], mechanics [35], thermodynamics [36], and even in cold atomic systems [37]. Moiré optical lattice with periodic or aperiodic pattern facilitates the progress of optical localization and spatial soliton [38,39]. Recently, the self-healing property of holographically generated moiré lattice wavefields was investigated [40]. However, the moiré lattice wavefields are also non-diffracting beams, and the self-healing distance is independent of the twist angle. In addition, the generation of complicated design of optical fields highly depends on the expensive phase modulation devices such as spatial light modulators (SLMs) or digital micromirror devices (DMDs) in experiments [41,42]. These devices always limit the application of self-healing beams in intense or multiple-color laser fields. Recently, Zhao et al combined Talbot effect with a non-diffracting beam, giving the beam the self-heal ability in laser transmission [41]. Unfortunately, the optical field passing through a single Talbot optical element with diffraction effects fails to reconstruct itself well according to the previous investigation [43].
In this work, inspired by the moiré technique, we experimentally and theoretically demonstrate the generation of self-healing optical fields by using moiré distributed dual-microlens arrays. The microlens array is a typical and crucial diffractive optical element that is widely used in various optical fields, such as fiber coupling [44], sensing [45], beam shaping [46], artificial compound eyes [47], solar cells [48], imaging technology [49,50] and even nonlinear filamentation control [51]. As a comparison, the propagation of optical field modulated by a single microlens array is also presented. By gradually altering the interval distance and moiré angle between the two microlens arrays, the self-healing degree of the optical field is systematically studied via the similarity degree equation. In addition, the optical self-healing property affected by the size and central position in the transverse direction is also investigated, which can be enhanced by the moiré angle and interval distance of the dual-microlens arrays. This work shows the moiré modulation application in self-healing optical fields.
As compared with previous approaches for self-healing beam, the method of moiré microlens array is anticipated to involve applications or advantages as follows. (i) The self-healing degree of the optical field at an arbitrary position is adjustable by simply changing the relative angle or the interval distance of the two microlens arrays. (ii) In our experiment, the self-healing optical field with uniform distributed multiple independent optical spots are generated. Although there are multiple spots in other self-healing beams such as Airy beam, Mathieu beam, Weber beam, and Pearcey beams consisting of multiple spots, not all spots can be used in practice applications due to asymmetric and nonuniform pattern [18]. Our scheme works will be helpful in the application of multipoint light manipulation and laser machining. (iii) The generated whole optical field can overcome the size of the light field defects at the millimeter level. (iv) The method of moiré microlens arrays does not need to use an expensive system such as SLM and DMD, regardless of their damage threshold and wavelength limitations, thus the method also can be used in intense or multi-color laser fields.

Experiment
The experimental setup is schematically illustrated in figure 1. A continuous-wave laser Gaussian beam with a wavelength of 800 nm and maximum input laser power of 200 mW passes through the telescope system to be expanded as a broad Gaussian beam with a full width at half maximum of ∼5 mm. The laser beam is reflected and collimated by a pair of highly reflective mirrors with a center wavelength of 800 nm. The output power of the laser is adjusted by a half-wave plate and ultra-broadband wire grid polarizers (WP25L-UB, Thorlabs). Firstly, the laser beam is focused by a 10 mm × 10 mm × 1.2 mm microlens array (MLA1) (pitch = 1.015 mm, f = 213 mm, Edmund Optics). The MLA1 is placed horizontally with a 0 • angle of rotation. Then the laser beam is passed through another same parameter microlens array MLA2. A multi-dimensional adjustable optical platform controls the interval distance and coaxial of the MLA1 and MLA2. A precise optical adjustment element adjusts the relative angles θ of MLAs. The obstacle of a thin magnetic circle steel disk plate induces the light field defect. A digital camera (Canon, ESOPR) records the diffusion pattern along the transmission direction to identify the evolution of the self-healing optical field. Several suitable ND filters are placed before the camera to avoid saturation.

Numerical simulation
According to the experimental setup, we perform a simulation on the self-healing of optical wavefields U (x, y, z). The optical propagates along z-axis, and the second microlens (MLA2) is placed in the plane z = 0. The transmittance of the first microlens (MLA1) composed by 100 (10 × 10) lenses is where x n and y n are the central coordinates of lenses; l represents the side length of lenses with the value of 1.015 mm; λ and f are the wavelength and focal length of lenses. The first term of equation (1) establishes a square hole, and the second term is the transmittance function of thin lens. The MLA1 is deployed in front of MLA2 with an interval of d and a moiré twist angle of θ. To illustrate the twisted angle between the two microlens arrays, a new transverse coordinate (x r , y r ) is constructed as the expression of To illustrate the twisted microlens (MLA2), we constructed a new coordinate (x r , y r ) that rotates clockwisely by an angle of θ with respect to the original coordinate (x, y). The new transverse coordinate where is is the two-dimensional rotation matrix [33,52]. Then the transmittance of MLA2 is Considering the Gaussian source g (x, y) passing the two microlens arrays, the wavefield in the plane z = 0 can be expressed as where the convolution operation represents the Fresnel approximation of optical propagation in free space. The Gaussian beam is expressed by , where a controls the waist width.
According to the previous study, the self-healing distance is affected by the diameter of the obstruction [13].
Here, we utilized a disk which is characterized by a circle function. After the obstacle, the amplitude of the wavefield is where D is the diameter of the obstruction. The optical field on the observation plane is ] .
The above equation is valid due to the experimental observation is within the Fresnel approximation range. By calculating equation (7), all the optical fields at the arbitrary plane perpendicular to z axis in this experiment can be expressed. Figures 2(a) and (b) present the ray tracing simulation with a single and dual microlens array. The ray tracing suggests a physical picture to understand the self-healing processes for single and dual microlens arrays. We can see that the parallel rays transmit for the single microlens array, then are focused by the microlens array. The rays are dispersed after the focal point. An obstacle with a size of 1.5 mm is set at z = 0 mm, where the rays fail to pass. We notice that the rays begin to overlap at a propagation distance of 663 mm after passing through the obstacle. The optical field will interact at the overlap region after passing the obstacle. Thus, the optical field will self-reconstruct at the overlap region. The physics process is similar to that in the Bessel nondiffracting beam observed by Durnin et al [11]. For the case of dual microlens arrays, the rays overlap significantly earlier than the former case, which occurs at the distance of 464 mm after the obstacle, indicating that the optical field possesses a faster self-healing. From the analysis of the above geometric optics results, the self-healing of the optical field can be improved by using dual microlens array. In general, geometric optics cannot present the interference dynamics of the self-healing optical fields. Therefore, we further simulate the self-healing processes by the diffractive optics simulation. For comparison, the expanded Gaussian beam is launched into a single microlens array with an obstruction. The optical propagation trajectory is simulated and displayed in figure 2(c). It shows that the light from edges of defect fills the central defect after 170 mm propagation, then interfere with each other, and forms a distribution similar with defect-free optical field at z = 370 mm. However, it is clearly seen that the filled structure differs from its surroundings slightly. Therefore, the optical field induced by a single microlens array cannot get self-healing well [43]. As shown in figure 2(d), once the dual-microlens arrays are utilized, the optical field is healed at z = 210 mm. We can also notice that the refilled region is consistent with other area, showing a better periodic distribution. The above results manifest that the beam from dual-microlens arrays is enhanced in self-healing.

Simulation
The optical propagation behaviors are further simulated by changing the twist angle and interval between the two microlens arrays to elucidate the optical self-healing ability of dual-microlens arrays. According to equation (8), a simulation of the optical wavefields propagating along z axis is performed. For quantificationally depicting the self-healing distance under different conditions [7,27,53,54], we introduce the similarity degree with the expression of where I wt and I ob represent the wavefield intensity at an arbitrary point (x, y) in the transverse plane, without and with obstruction, respectively. The larger D p indicates the higher self-healing level of an optical field. By scanning the twist angle and interval at steps of 1 • and 10 mm, respectively, the average values of D p in the last 50 mm are recorded and shown in figure 3(a). The figure shows that all the optical fields enable self-healing, and the self-healing level is affected by both twist angle and interval. When the interval is limited in d < 220 mm, the self-healing level is high at twist angle θ < 33 • . The self-healing effect becomes weak if  the twist angle is larger (<90%). For the interval d > 220 mm, the situation is reversed: the self-healing ability is weak at a small twist angle and becomes stronger at θ > 22 • . The weakest self-healing effect occurs around (θ, d) = (10 • , 400 mm). Figure 3(b) depicts the variation of D p when the single microlens is performed, it manifests that the optical field cannot self-heal gradually in a diffraction distance of 2000 mm, although light energy fulfills the defective region ( figure 2(a)). The interval between peaks of D p corresponds to Talbot recurrence distance, indicating the Talbot effect renders the optical self-healing. Figure 3(b) depicts the variation of D p when the single microlens is performed. It manifests that the optical field cannot self-heal gradually in a diffraction distance of 2000 mm, although light energy fulfills the defective region ( figure 2(a)). Instead, there are some specific distances where the optical field is close to self-healing state, corresponding to the peaks of D p . This is because the Talbot effect originating from the diffraction of periodic object [55]. The interval between peaks of D p corresponds to Talbot recurrence distance, indicating the Talbot effect renders the optical self-healing at some specific distances, indicating the Talbot effect renders the optical self-healing at some specific distances. In figures 3(c)-(e), the variations of D p are displayed under different twist angles when the interval between the two microlens arrays is fixed as 100 mm, 220 mm, and 300 mm, respectively. For d = 100 mm, the value of D p increases within half focal distance, then decrease between half focal distance and one focal distance and finally gets close to 1 gradually. The self-healing distance becomes longer with a larger twist angle. When the interval d is set as 220 mm, the values of D p almost synchronized with different twist angles, indicating the self-healing processes are similar. Once the interval d is set as 300 mm (>215 mm), the light cannot reach focal after the MLA2, thereby the values of D p increases monotonically. In this situation, the larger the twist angle, the shorter the self-healing distance and the higher healing level.

Experiment
The transverse distribution of optical intensities at different propagation distances are simulated and experimentally achieved to study the self-healing process of the optical fields after the dual-microlens arrays. First, we keep the twist angle of θ = 0 • , and the influence of the interval distance between the two microlens arrays on the self-healing degree is studied, as displayed in figure 4. One can notice that the edges of optical fields are different between experimental results and simulation, which originates from the gripper for MLA2 blocking the lenslets located at the edge, thereby forming a circular type of spots at the edges in the experiment. From top to bottom, the evolution of the defective optical fields at different intervals of 100 mm, 200 mm, and 300 mm is depicted in figures 4(a)-(c), respectively. The optical intensity of each case is individually normalized in the simulation for enhancing the contrast ratio. The artificial light defects are introduced in the middle of the light fields with a size of D = 1.5 mm. For an interval distance of d = 100 mm (up two rows), the weak diffraction at the edges of the obstruction can be observed after the laser translation of 100 mm. As the translation distance increases, the defect of the optical field is gradually filled by light energy. The center of the defect is firstly filled as the laser transmission distance increases beyond 300 mm. Then, the remaining area is subsequently compensated after 400 mm propagation and fully self-reconstructed at z = 600 mm. We can observe that the intensity pattern is uniform and periodic, although the beam's intensity decreases along the propagation. The phenomenon manifests that the optical field is completely self-healing. Furthermore, similar self-reconstruction processes of the optical fields can be observed when the interval distance is increased to d = 200 mm and 300 mm. However, the self-healing capacity is weakened. For example, in the case of the interval distance of d = 300 mm, the center region of the optical field is not fully self-healing, although the self-healed areas show perfect periodicity and consistency. The perfect consistence between simulation and experiment demonstrates the validity of our results, and the decayed self-healing degree with increasing the interval between the two microlens arrays is in accordance with the result shown in figure 3(a).
Our above results show that the optical field with defects induced by dual microlens arrays with twist angle θ = 0 • has perfect self-reconstruction capacity. Here, the twist angle is manipulated from 0 • to 45 • when the interval distance d = 300 mm between the two microlens arrays is fixed. The influence of the moiré twist angle on the light field restoration is investigated. Figure 5 shows the evolution of the optical field with defects along the propagation with different moiré angles (a) θ = 15 • , (b) θ = 30 • , (c) θ = 45 • , respectively. In these situations, the optical lobes with different periodic are observed, originating the optical field moiré superimposition. The generated optical field is similar to the spatial optical lattice. For the moiré angle θ = 15 • , the self-healing ability is improved as compared to that of moiré twist angle θ = 0 • , but the repair is still not complete. If the moiré angle is increased to θ = 30 • , the optical field can self-heal after transmission beyond 600 mm. The effect of optical field interference superimposition generates the seemingly confusing optical field. As the moiré angle is further increased to 45 • , the completely self-healing of the periodic optical field occurs at z = 400 mm. From these results, the self-healing distance decreases, and the self-healing capability enhances with the increase in twist angle for d = 300 mm. The experimental results are well agreed with the theoretical prediction in figure 3(a).
Finally, the self-healing property is studied when different sizes of defects are considered in the optical fields. The simulation on the variations of D p is calculated for different sizes of defects, which are displayed in figure 6(a). The optical field heals itself quickly for a 1 mm optical field defect. As the size of the optical defect increases, the self-healing ability becomes weak gradually, and the self-healing distance is elongated for the moiré angle of θ = 10 • and interval distance d = 300 mm. However, the self-healing ability is re-enhanced when the moiré angle is increased to 30 • . As shown in figures 6(c) and (d), experimental results are fully consistent with the theoretical results. In addition, the central position of the optical field defect is altered, the results are shown in figure 6(b). A similar self-healing process is observed for the three central positions. The transverse distribution of the beam is depicted in figure 6(e), in which the defect is still healed. The results show that the self-healing degree for different defect sizes and positions can be arbitrarily manipulated according to the simulation results in figure 3(a).

Conclusions
We theoretically and experimentally study the self-healing optical field generation from the moiré distributed microlens arrays. This self-healing of optical fields is better than those generated by a single microlens array. Through double parameters scanning of the twist angle and interval between the two microlens arrays, the self-healing degree is reduced with increasing the twist angle when the interval is less than 220 mm, while the situation is reversed for the interval beyond 220 mm. The simulated and experimental results for transverse optical fields show high coincidence. Our work shows that the self-healing degree can be enhanced by modulating the interval distance and moiré twist angle for arbitrary defect position on the optical field and a significant defect. This scheme provides a simple and effective method to obtain a self-healing optical field with strong self-rebuilding capabilities, which may be used for multiple research fields such as imaging, optical manipulation, optical tweezers, and filamentation.

Data availability statements
All data that support the findings of this study are included within the article (and any supplementary files).

Conflict of interest
We declare that we have no conflict of interest.