Since the experimental demonstration of Airy beams in 2007 [1,2], self-accelerating beams have attracted great attention due to wide applications in the generation of curved plasma channels in air [3], accelerating electrons [4], optical manipulation of particles [5], laser microfabrication [6,7], optical signal transmission [8], and microscopy [9,10]. By extending the Airy function to the cylindrical coordinate, circular Airy beams (CABs) were obtained [11,12]. The CAB is a circularly symmetric beam with an abruptly autofocusing property, which has triggered a great deal of research on optical tweezers [13], light bullets [14,15], and femtosecond laser polymerization [16]. Inspired by the CAB, the symmetric Airy beam (SAB) arose [17–20], which first auto-focalizes with a single central lobe and then accelerates into four Airy lobes in quadrate symmetry. The SAB is one of the caustics, which are the loci where rays are locally focused [21,22], so it can be inferred that the object can be imaged along the four caustic curves. The optical imaging is widely employed in both scientific research and industrial communities. Up to now, great efforts have been devoted to improving imaging quality, such as breaking the diffraction limit [23,24] and correcting geometrical and chromatic aberrations [25,26]. Nevertheless, research on dynamic imaging is rare, and it has potential in optical encryption. Furthermore, additional transformation of SABs to other Airy-type beams may encourage more promising applications.

For the generation of SABs, the most common method is based on symmetric cubic phase modulation by spatial light modulators (SLMs). However, the pixels of commercial SLMs are in micrometer size, which limits the accuracy of the phase gradient. The efficiency of SLMs is still a challenge (usually<50%) [27]. Furthermore, the bulky footprint of SLMs imposes restrictions on the application in integrated optics. Attempts like photorefractive crystals [28], liquid crystals [29,30], digital micromirror devices (DMD) [20], nanoantennas [31], and metasurfaces [32,33] can be adopted to avoid some of these problems. However, the above choices cannot simultaneously satisfy the conditions of continuous phase variation, high efficiency, and compact construction. Recently, femtosecond laser micro/nanofabrication [34–38] has been demonstrated to be a promising technique to process three-dimensional (3D) optical microdevices like photonic crystals [39], waveguides [40], and microlens groups integrated on the fiber facet [36,41] in high precision and flexibility, which can be a great choice for precise fabrication of micro-nano optical elements with high performance and flexible integration.

In this paper, we experimentally realize miniature symmetric and asymmetric cubic phase plates (SCPPs and ACPPs) by femtosecond laser nanofabrication based on spectral coordinate transformation of cubic phase, and we generate symmetric and asymmetric Airy beams (SABs and AABs) for dynamic imaging. The fabricated SCPP and ACPP demonstrate the benefits of compact construction, continuous variation of phase, high efficiency (∼81% at 532 nm), and broadband light modulation (405–708 nm), which can be helpful in integrated optics. All these advantages enable the generation and application of SABs and AABs to be more efficient, ensuring the achievement of dynamic imaging, which means a certain input signal but various imaging results during propagation. On the other hand, broadband operation makes the imaging enigmatic. Only by extracting the element of exact wavelength at the certain propagation distance can we get the precise signal, which may provide a potential method for message extraction. Moreover, flexible fabrication of femtosecond laser capacitates the SCPP on a soft substrate, enabling the SCPP to perform smartly in multifarious conditions. The SCPP fabricated on a polydimethylsiloxane (PDMS) substrate exhibits high mechanical flexibility and robustness. The SCPP can maintain its shape and size, even optical properties unchanged during severe stretching.

Solving the normalized paraxial equation of diffraction under the condition of introducing an exponential aperture function results in the envelope of a finite Airy beam [1,2]: Here Ai[s−(ξ/2)2+iaξ] represents the Airy function, a is the decaying factor to truncate the tail of the infinite Airy beam, and a≪1. s=x/x0 represents a dimensionless transverse coordinate, where x0 is an arbitrary transverse scale. ξ=z/kx02 is a normalized propagation distance, k=2πn/λ is the wave number, n is the refractive index of the medium, and λ is the free-space wavelength. The Fourier spectrum of the Airy beam is given by This implies that the finite Airy beam can be generated through the Fourier transform of a Gaussian beam with a cubic phase. The cubic phase spectrum [Fig. 1(a1)] for a two-dimensional (2D) Airy beam is in the form of φAB(kx,ky)=kx3+ky3, where kx and ky are Fourier spectrum coordinates, and the simulated intensity distribution and propagation dynamic of the corresponding Airy beam are shown in Figs. 1(a2) and 1(a3), respectively.

By converting the spectral coordinates with the absolute value in the spectral cubic phase term expressed as φSAB(kx,ky)=|kx|3+|ky|3 [Fig. 1(b1), kx, ky: −12π to 12π], the SAB can be generated, and the complex amplitude of the SAB can be expressed as [19] where Gi(s)=1π∫0∞sin(13t3+st)dt. To carry out the simulation with the phase mask, the diffraction theory is adopted: Ein=Aexp(−k2/ω2) is the incident Gaussian light, t(kx,ky)=exp[−ik(kx2+ky2)/2f] is the phase transfer function of the lens, f is the focal length, and the dimensionless coordinate is normalized by sx=x/x0, sy=y/y0, ξξf=z/ξf, and ξf=f/kx02. As shown in Figs. 1(b2) and 1(b3), the SAB first auto-focalizes with a single central lobe, and then it splits and accelerates into four Airy lobes in quadrate symmetry. Since all four Airy lobes present equal acceleration, the propagation dynamic of the SAB versus (sx,ξ) is consistent with the one versus (sy,ξ) as illustrated in Fig. 1(b2). The main lobe of SAB demonstrates absolute symmetry and shows the same shape in both lateral and vertical directions at arbitrary propagation planes [Fig. 1(b3)].

Further coordinate transformation can occur by introducing a ratio factor in the kx or ky coordinate that enables the spectral phase turn into the form φAAB(kx,ky)=|kx|3+t|ky|3, where t is a transformation factor, and t>0,t≠1, resulting in the SAB converting into another Airy-type beam. It can also be realized by giving different values of transverse scales x0 and y0 [42], but the introduction of the transformation factor normalizes the relationship between two transverse scales. As shown in Fig. 1(c1), with the transformation factor t=1.5, the phase mask exhibits a denser phase range in ky coordinate (kx: −12π to 12π, ky: −18π to 18π) compared to the symmetric cubic phase mask [Fig. 1(b1)], and thus we call it an asymmetric cubic phase mask. Thanks to the transformation factor in the ky coordinate of the spectral phase, the corresponding beam shows different autofocusing and accelerating profiles [Fig. 1(c2)]. The beam auto-focalizes longer and accelerates slower in the sy−ξ plane than that in the sx−ξ plane, and the main lobe is first elongated in the vertical direction during autofocusing and then elongated in the lateral direction when splitting and accelerating [Fig. 1(c3)]. Since the beam is not consistent in both directions, we call it an asymmetric Airy beam (AAB). Although SABs and AABs both display similar propagation features like autofocusing and accelerating, they show different accelerating frames (SAB: quadrate, AAB: rectangular) and disparate main lobe shapes (SAB: circle, AAB: ellipse).

These Airy-type caustics (SAB and AAB) are a series of rays forming local foci, and each ray is tangent to the main lobe at one point. Thanks to the prominent propagation features of SABs/AABs, they have potential in dynamic imaging. Corresponding to the propagation dynamic of the SAB, we infer that a letter “T” imaged by the SAB first auto-focalizes and becomes smaller, and then it splits into four of the same “T”s and accelerates in quadrate symmetry, forming diverse imaging results at different planes. The schematic illustration of dynamic imaging of a letter “T” by an SAB is shown in Fig. 1(d). To verify this inference, the simulation of “T” imaged by an SAB/AAB was carried out. As displayed in the inset of Fig. 1(d), the result by an SAB presents a letter “T” during the autofocusing term, and then it turns into a Chinese word “开” at ξ/ξf=1.97, which means “open” in English. Therefore, dynamic imaging by an SAB can convey different information during propagation. As for dynamic imaging carried by an AAB, the initial result of a letter “T” is the same as the one by an SAB, except for a little elongation in the vertical direction, which corresponds to the main lobe of AAB. However, dynamic imaging during propagation is quite disparate owing to different accelerating frames, such as the Greek symbol “π” formed at ξ/ξf=1.37.

The femtosecond laser source (Coherent Chameleon) has an 80 MHz repetition rate, 800 nm center wavelength, and 75 fs pulse width. The photoresist [SZ2080 mixed with 1 wt.% 4,4-bis(diethylamino)benzophenone (BIS) as photoinitiator, provided by IESL-FORTH, Greece] [43] dropped on a glass was prebaked at 100°C for half an hour to evaporate the solvent in the SZ2080. The SCPP/ACPP was fabricated by femtosecond laser layer-by-layer writing, and the fabrication process was controlled by an XY scanning unit while the step between two layers was realized by a nano-positioning stage [Fig. 2(a)]. The optimal fabrication parameter was 7.8 mW laser power at 1 ms exposure time. After polymerization, the sample was developed in 1-propanol for 30 min until all unpolymerized photoresist was washed off. The fabrication process on a PDMS substrate is the same as above, but we should pretreat the PDMS substrate before fabrication. First, PDMS was mixed with the crosslinker (Sylgard 184, Dow Corning) at a mass ratio of 10:1. After that, the mixture was spin-coated on glass at 300 r/min for 200 s, followed by curing at 80°C for 6 h, which resulted in 150 μm thickness of PDMS. Then the PDMS was peeled off from the glass, cut into slices, and covered on a glass slide. Afterwards, it was treated by oxygen plasma at 75 W for 50 s (Mingheng PDC-MG). Later, the treated PDMS was soaked in a solution mixed with poly(vinyl alcohol) (PVA), oxalic acid, and water in a 1:10:500 mass ratio to increase the adhesion between the photoresist and PDMS for 12 h. Finally, it was taken out from the solution and blow-dried. After photoresist drop, prebake, fabrication, development, and peeling off the PDMS from the glass slide, the SCPP/ACPP was fabricated on a PDMS substrate.

The scanning electron microscope (SEM) images were taken with a secondary electron SEM (ZEISS EVO18) operated at an accelerating voltage of 10 keV with a working distance of 8 mm after depositing ∼10nm gold. The topography of the SCPP/ACPP was scanned at a rate of 0.5 Hz in air in tapping mode with a commercial atomic force microscope (AFM, MFP3D-origin OXFORD) using a Tap300Al-G tapping mode tip. The optical images were taken by a commercial optical microscope (Leica DMi8-M, Germany).

A Gaussian beam from a 532 nm laser propagated through a 40× objective lens and the SCPP/ACPP, and after magnification by another 40× objective lens, the generated SAB/AAB was captured by a CCD camera at and after the focal plane of the first objective lens [Fig. 3(a)]. A series of images was taken at various propagation planes. The position and full width at half-maximum (FWHM) of the main lobe in each image at a certain propagation distance were measured and marked in Figs. 3(b)–3(d). To generate the chromatic SAB/AAB, we replaced the laser source with a halogen lamp, and when signal masks were placed after the light source, dynamic imaging of signals was realized.

Four guide screws coupled with clamping pieces controlled by linear motors constitute our homemade stretch device. The PDMSs were fixed on two clamping pieces in the x direction for a single-direction stretch test. The stretched length was precisely given by linear motors driven by their controllers. The homemade stretch device was installed on an inverted microscope (Leica DMi8-M, Germany) to observe the SCPP during the stretch process, and it was introduced into the SAB generation setup to verify the consistency of intensity distribution before, during, and after stretch.

Compared to phase modulation by an SLM, generation of SABs and AABs through the specially designed phase devices with continuous phase variation shows space saving as well as higher optical efficiency. Therefore, we decided to transform the phase profile to the geometric height, which can be fabricated by femtosecond laser direct writing precisely [Fig. 2(a)]. As shown in Fig. 2(b), a symmetric cubic phase mask is first converted through the thickness-accumulated phase equation h(x,y)=λφ(x,y)/2π(nm−1) and then discretized. λ is the free space wavelength of incident light and is set to 550 nm, and nm represents the refractive index of the photoresist (∼1.5). Therefore, the corresponding height range is about 0–1100 nm. The beam quality can be evaluated by the mean square error (MSE) between the continuous phase and discrete phase with different step N and normalized by using the value for N=1 written as MSE(N)=∑i,j[Icontinuous(i,j)−IN(i,j)]2∑i,j[Icontinuous(i,j)−I1(i,j)]2 [44]. For the case N=11, the MSE tends to be 0.002, demonstrating a good quality. Furthermore, the step tends to disappear after fabrication in the course of unpolymerized resin rinsing (self-smoothing effect) [45], forming a continuous phase. However, the MSE increases with the decrease of N [MSE(5)=0.0216]; meanwhile, a greater step can cause invalidation of the self-smoothing effect. Therefore, the step of discretization is set to 100 nm, and the number of layers is N=11 [Fig. 2(b)].

The fabricated SCPP with a size of 60μm×60μm×1.1μm presents high quality, which is verified by the SEM micrograph and AFM image [Fig. 2(c)]. Topography of the fabricated SCPP extracted from the AFM result along the dashed line in Fig. 2(c) reveals the great phase continuity and agrees well with the desired profile [Fig. 2(d)]. Some tiny fluctuations caused by the fabrication and measurement errors are unavoidable. On the other hand, the high aspect ratio of local areas makes it difficult to dip the AFM probe into the bottom, which results in a mismatch between the extracted height and the desired one near the corner of the SCPP. An ACPP can be designed and fabricated similarly to the SCPP, and the top-view SEM and three-dimensional AFM images are depicted in Figs. 2(e) and 2(f), respectively, demonstrating great fabrication quality.

The SAB/AAB can be realized through the phase modulation of SCPP/ACPP [Fig. 3(a)]. To characterize the propagation feature of the generated SAB/AAB, we measured a series of intensity patterns [Fig. 3(b1)] in the 0–200 μm range, where z=0 coincides with the focal plane position of the first objective lens. The experimentally generated SAB first auto-focalizes during 60 μm propagation and then splits to four main lobes that accelerate in quadrate symmetry. As for the AAB [Fig. 3(b2)], it also has an autofocusing property during the first 60 μm propagation but is elongated in the vertical direction. After splitting, it exhibits inconsistent acceleration that leads to a discrepancy in the lateral and vertical directions. These experimental results [Figs. 3(b1) and 3(b2)] are in good agreement with the numerical simulations [Figs. 1(b3) and 1(c3)].

We also quantitatively analyzed the distinction of the propagation features between the SAB and AAB. Figure 3(c) depicts the transverse acceleration of the SAB/AAB as a function of beam propagation distance. It is obvious that the SAB shows identical propagation dynamics in both the x−z and y−z planes, which means the same autofocusing distance and equal acceleration. However, the measured result reveals that the AAB owns a longer autofocusing distance but accelerates much slower in the y−z plane than in the x−z plane. The denser phase range in the ky coordinate caused by the transformation factor t=1.5 can account for these results. Moreover, our experimental measurements [pink (x−z) and purple (y−z) points toward the right (SAB) and left (AAB)] are consistent with the theoretical trajectories [solid (x−z) and dashed (y−z) lines with black color towards the right (SAB) and red color towards the left (AAB)] extracted from the simulated results [Figs. 1(b2) and 1(c2)].

Although the SAB and AAB display similar propagation features, the shapes of their main lobes are clearly distinct. After obtaining the normalized intensity along the lateral and vertical directions of SAB/AAB at z=0,60,120, and 180 μm, we measured the FWHMs of their main lobes, and the variation tendency is recorded in Fig. 3(d). The FWHM of the main lobe of the SAB drops rapidly during the first 60 μm (autofocusing duration); after splitting, it still decreases but much less (60–120 μm), and finally it increases little. The results in both the lateral and vertical directions are consistent. For AAB, the FWHM of the main lobe is quite different in the two directions. In the lateral direction, the FWHM first decreases slowly during the autofocusing phase (0–60 μm), then it increases during the next 60 μm acceleration, and finally it becomes almost constant. However, the vertical result reveals that the FWHM drops fast during 120 μm propagation, and then the decrease slows down quickly and tends to fade away. These can be explained by the different autofocusing distances and shapes of the main lobes in the lateral and vertical directions, which have been discussed above. All these experimental results are in accordance with the simulated distributions [Figs. 1(b3) and 1(c3)].

Defined as the rate of laser power of the transformed SAB/AAB divided by the laser power before the sample, the optical efficiency of the SCPP was measured. Figure 4(a) depicts the relationship between the optical efficiency of SCPP and the wavelength of incident light. Notice that the efficiency differs slightly at the wavelengths of 405 nm (∼59%), 590 nm (∼66%), 633 nm (∼65%), and 780 nm (∼60%) but exhibits a large gap between 488 nm (∼29%) and 532 nm (∼81%). As we described in the former section, the height profile corresponding to the phase variation is related to the wavelength of incident light. Therefore, the incident light with a wavelength away from 550 nm suffers from incomplete phase modulation and low efficiency. Nevertheless, an optical efficiency around 60% is still considerable. It is satisfactory that high and stable optical efficiency (81%±4%) can be realized at the wavelength of 532 nm due to the integrity and continuity of the phase. The transmission loss from the glass substrate and some tiny flaws on the SCPP result in the loss of laser power after transmission, which is unavoidable. Another special point is that the SCPP shows a poor 29% efficiency at 488 nm caused by the absorption of the photoresist in the spectral range of 450–500 nm [43].

Intensity distributions of SAB/AAB over a visible light band (405–780 nm) are recorded as well. As displayed in Fig. 4(a), the intensity distributions of the SAB/AAB at the same propagation distance (z=180μm) exhibit different frame sizes with similar features. We surmised that the acceleration is relevant to the wavelength of incident light, which has been confirmed in Airy beams. To verify this assumption, we measured the length between two main lobes of the SAB and AAB in the lateral and vertical directions at different propagation planes at the wavelengths of 488, 532, and 633 nm. It can be seen in Fig. 4(b) that the generated SAB at 488 nm shows the shortest autofocusing distance (d=0 means autofocus) and accelerates the fastest, while the SAB at 633 nm auto-focalizes the longest distance and accelerates much slower. As for AAB, though the distances in the lateral and vertical directions are unequal (dx>dy), the results in both directions are consistent, which is coincident with the conclusion of SAB. The autofocusing distance shortens and the acceleration increases with the decrease of wavelength. All the points in Fig. 4(b) represent the measured results, and solid and dashed lines are fitted curves.

After replacing the monochromatic light with a polychromatic light (halogen lamp), the generated SAB and AAB exhibit chromatic characteristics. Here we name them chromatic SAB and chromatic AAB. Figure 4(c1) depicts the simulated (top row) and experimental (bottom row) results at z=0,60,120, and 180 μm. Chromatic SAB is the same as the SAB but color. Though the multicolor presentation during propagation has to be avoided in most imaging applications, it may be useful in multicolor imaging display. Chromatic SAB is a beam comprising a series of SABs with various wavelengths. Thanks to the disparate autofocusing distances and acceleration of SABs at different wavelengths, contemporaneous SABs from a polychromatic light shift differently during propagation, resulting in the formation of chromatic SAB. As shown in Fig. 4(c1), with the increase of propagation distance, the multicolor becomes more and more obvious, which can be obtained from the side-view result as well [Fig. 4(c2)]. All the experimental distributions agree well with the simulated results. Similar to chromatic SAB, the chromatic AAB also displays multicolor during propagation with distinct acceleration in the lateral and vertical directions as shown in Figs. 4(d1) and 4(d2).

Because the main lobe of the SAB/AAB is formed by a series of rays locally focused during propagation, every point of the main lobe of the SAB/AAB is a focus, which makes it suitable for imaging. Dynamic imaging was experimentally realized by the SAB/AAB when we imposed a signal mask after the light source [Fig. 5(a)]. As shown in Fig. 5(b1), an English letter “T” imaged by the SAB at different propagation distances displays diverse results. It becomes smaller during the first 60 μm propagation, which is caused by the autofocusing effect, and then it splits into four of the same “T”s that accelerate in quadrate symmetry. These four “T”s partly overlap with each other [Fig. 5(b1), z=118μm] in the initial period of acceleration but finally completely separate [Fig. 5(b1), z=180μm]. After elaborately observing the process of overlap and separation, we found that four overlapping “T”s form a Chinese character “开” which means “open” in English at the distance of 118 μm. It is significant that a special signal can be transformed from one language to another language, expressing entirely different meaning by the SAB at a certain propagation distance. Similarly, the imaging of an English letter “T” by the AAB also auto-focalizes from 0 to 60 μm, zooming out the size of letter, then bursts into four “T”s, and accelerates in the rectangular frame [Fig. 5(b2)]. Therefore, the overlapping result is fairly different from the one by the SAB. Due to the slower acceleration in the vertical direction, the “T”s in the vertical direction completely overlap together, while a part of them separates in the lateral direction, which constitutes a Greek symbol “π” at the unique distance of 82 μm [Fig. 5(b2), z=82μm]. All the experimental results [Figs. 5(b1) and 5(b2)] show a good agreement with the simulations [Fig. 1(d), inset]. Note that if we want to pass a “π” signal to the receiver, we can send a “T” message to the receiver, who can obtain the true information via an AAB at a certain propagation distance, and the choice of “开” signal can be realized by an SAB. This provides us with an opportunity for information transformation during propagation.

As demonstrated in the former section, the SCPP and ACPP show a broadband operation from 405 to 780 nm. We also show the dynamic imaging at the wavelengths of 633 nm and 488 nm with different shapes of signals. As displayed in Fig. 5(c1), a “square” shows various imaging results at different propagation planes via the SAB at the wavelength of 633 nm. At the initial autofocusing term, the picture shows a Chinese character “�?�,” which is the same shape as the square signal [Fig. 5(c1), z=0], meaning “mouth.” Further acceleration enables the formation of the Chinese character “田” [Fig. 5(c1), z=146μm], meaning “cropland,” and the ancient Chinese character “㗊” [Fig. 5(c1), z=177μm], meaning “noisy.” On the other hand, a “circle” can realize various imaging results via the AAB during propagation at the wavelength of 488 nm [Fig. 5(c2)]. Due to the elliptical main lobe of the AAB, the imaging result of a “circle” shows an ellipse like a number “0” at z=0. After propagating 67 μm, it turns into a horizontal number “8” and then transforms into a double-digit number “88” at z=122μm. Here we demonstrate some special symbols whose imaging results are common symbols or characters. Actually, any other symbols can be used for the application, because all symbols can be defined artificially as valuable information at some propagation planes, although they look meaningless.

The dynamic variation process of the signal imaging under white light illumination is similar to that under a monochromatic laser illumination. The difference is that the images do not show multicolor during the autofocus process but exhibit chromatic multiple letters during acceleration [the left column in Fig. 5(d)] because of different accelerations according to various wavelength elements. It provides two choices that we can utilize the autofocusing term without chromatism in image observation and take advantage of accelerating phase presenting rich color in a multicolor display. Furthermore, a mixture of ingredients with various colors blurs the picture and makes it hard to distinguish useful information at some propagation planes, which should be avoided in multicolor display but may be valuable in confidential information transmission and reception. The second image of the left column in Fig. 5(d) displays the blurry image of a letter “T” at z=90μm via the AAB, in which we are unable to read some useful information. However, after extracting blue (488 nm), green (532 nm), and red (633 nm) elements in the picture, different information emerges with a Chinese word “开” in the blue ingredient and a Greek symbol “π” in the red ingredient. Hence, with the aid of polychromatic light, the valid result can only be obtained in a certain ingredient of wavelength at a unique propagation distance with an exact SCPP/ACPP.

Since the SCPP/ACPP has demonstrated so much superior performance as mentioned above, given its strong mechanical flexibility and robustness, SCPP/ACPP has promising potential in reforming integrated optical systems by introducing miniaturized and fully flexible optical elements. To realize this aim, the SCPP is fabricated on a PDMS substrate, and then mechanical stretch is imposed on the PDMS substrate [Fig. 6(a) inset]. The stretch coefficient of the PDMS substrate with an SCPP defined as ε=Δd/d0, where Δd is the increment during the stretch and d0 represents the original length between two clamps, can be controlled flexibly. As shown in Fig. 6(a), given the stretch coefficient in the x direction varying in a wide range from 0% to 85%, the SCPP can still maintain its shape. Meanwhile, the efficiency remains still as well, showing a minor fluctuation from 67% to 74% at 532 nm. Moreover, the efficiency presents a good consistency during a 100 cyclic stretch with a stretch coefficient of 50% [Fig. 6(b)].

We also verified the consistency of the generated SAB during the stretch process. As illustrated in Figs. 6(c1)–6(c3), the SCPP (first row) almost remains the same, and the corresponding SAB shows no difference both at the side view (second row) and at a certain propagation distance (third row) before [Fig. 6(c1)], during [Fig. 6(c2)], and after stretch [Fig. 6(c3)], as does the imaging result of the letter “T” (last row) at z=118μm. Further stretch with a higher stretch coefficient (>90%) makes the structure break along the vein. Surprisingly, the cracked parts join together perfectly, and the SCPP is restored after releasing the stretch. The crack is too tiny to be distinguished under an optical microscope, which can be verified by the SEM image [Fig. 6(d)]. The unique architectural feature with dense rings, which enhances the tensile strength, may account for the robustness of the SCPP. Since the SCPP displays such a nice restoration, the corresponding intensity profiles and imaging of the letter “T” are not influenced [Fig. 6(d)]. The robustness of the SCPP integrated on the soft substrate makes it flexible to fit different scenarios.

Symmetrization of the spectral cubic phase of an Airy beam enriches the accelerating Airy-type beams, such as autofocusing CABs, which have circular symmetry with an Airy radial profile, autofocusing SAB with a quadrate accelerating frame. Our introduction of a ratio factor in a symmetric cubic phase results in the autofocusing AAB with a rectangular accelerating frame and elliptical main lobes. The similar but different properties between SAB and AAB provide a choice for information extraction on demand [“开” in Fig. 5(b1) and “π” in Fig. 5(b2)]. In addition, our effort on the coordinate transformation of spectral cubic phase provides a potential for manipulating the structured light with diverse features to meet various requirements in application. The design and fabrication of the SCPP/ACPP overcomes the shortcomings of the SLM, such as the limitations of phase gradient accuracy, low optical efficiency (<50%), and bulky footprint in optical setup. Further fabrication on the soft substrate endows the SCPP/ACPP with the capacities of flexibility and robustness, which may play a significant role in integrated optical systems. The dynamic imaging via the SAB/AAB, which presents the distinct information from the input signal, broadens the application of Airy-type beams, though the CAB and SAB have been widely used in optical tweezers, light bullet, biomedical treatment, and femtosecond laser polymerization. Broadband operation across the visible region enhances the difficulty of identification by introducing the variable of wavelength.

In summary, we demonstrate the design and fabrication of an SCPP/ACPP with superior performance: compact construction (60μm×60μm×1.1μm), continuous variation of phase, high optical efficiency (∼81% at 532 nm), and broadband operation from 405 to 780 nm, through which the experimental generation of an SAB/AAB with an autofocusing feature and quadrate/rectangular self-accelerating frame is realized under both monochromatic light and polychromatic light. Dynamic imaging of different signals is realized via the SAB/AAB, showing different results during propagation with a certain initial signal, which provides a potential for the encryption and decryption of information during propagation. Furthermore, the SCPP fabricated on a soft substrate demonstrates mechanical robustness and flexibility, which can maintain its size, optical efficiency, generated intensity distribution, and even imaging results unchanged before, during, and after stretching. Our research on high-performance SCPPs/ACPPs constructs a promising platform for the creation of compact optical elements with high efficiency and broadband operation for integrated optical systems, and the findings on dynamic imaging provide potential applications in imaging science.