Cryptographic Strain-Dependent Light Pattern Generators

Refractive freeform components are becoming increasingly relevant for generating controlled patterns of light, because of their capability to spatially-modulate optical signals with high efficiency and low background. However, the use of these devices is still limited by difficulties in manufacturing macroscopic elements with complex, 3-dimensional (3D) surface reliefs. Here, 3D-printed and stretchable magic windows generating light patterns by refraction are introduced. The shape and, consequently, the light texture achieved can be changed through controlled device strain. Cryptographic magic windows are demonstrated through exemplary light patterns, including micro-QR-codes, that are correctly projected and recognized upon strain gating while remaining cryptic for as-produced devices. The light pattern of micro-QR-codes can also be projected by two coupled magic windows, with one of them acting as the decryption key. Such novel, freeform elements with 3D shape and tailored functionalities is relevant for applications in illumination design, smart labels, anti-counterfeiting systems, and cryptographic communication.


Introduction
Controlling the spatial profile of light beams is critically important in various scientific and technological fields, including high resolution microscopy, [1] endoscopy, [2] lithography and additive manufacturing, [3] optical manipulation of micro-objects, [4] wireless communication [5] and computation. [6] Various methods have been reported to this aim, mostly based on diffractive elements and digital holography, which exploit arrays of micromirrors, [7] liquid crystal-based modulators [8] or metasurfaces. [9] While such techniques allow high spatial resolution in modulated beams as well as both static and dynamic light patterns to be generated, they typically need highly complex optical elements. This has recently pushed the attention toward refractive freeform optics, that can redistribute the intensity profile from a light source into an arbitrary and pre-determined pattern through simple and robust devices, in which at least one surface has no translational or rotational symmetry with respect to the axis normal to the component main plane. [10] The surfaces of freeform optical elements can be precisely designed in order to produce a desired intensity pattern, [11] defining the involved geometries as sum of spherical or aspherical lenses, or through Q-polynomials description and non-linear partial differential equations. [10,12] The advantages of this method comprise relevant system miniaturization, wider field of view and higher imaging resolution, [2,13,14] . Manifold manufacturing technologies are generally required, involving grinding, polishing, and ultra-precision turning, [15,16] which is highly time-consuming, costly, and poorly versatile, thus preventing freeform optical systems to be realized rapidly and with features variable by external gates. Alternative manufacturing methods are available through 3-dimensional (3D) printing technologies, that can generate objects with unprecedented complex geometries. [17][18][19] 3D printing comprises a variety of processes to fabricate unconventional architectures by different materials. [20][21][22] In the field of optics and optoelectronics, additive manufacturing has been already employed to produce aspheric lenses, micro-optics, waveguides, photonic crystals, light-emitting diodes (LEDs), detectors and sensors. [19,23,24] Though 3D printing of macroscopic objects with optical quality and submicrometric resolution is still challenging, [25] a number of approaches have been proposed for improving the achievable accuracy, as well as the rate of printing and the size of the printed objects [26][27][28] Importantly, some applications might exploit light patterns generated from surfaces with lower quality, taking advantage of the design flexibility and customization offered by 3D printing technologies. A relevant example is given by cryptographic labels, [29,30] where the capability to recognize the generated light patterns by naked eye or by a low-cost scanner, without the need of bulky optics and complex optical systems, is highly desirable. [31,32] Furthermore, additive manufacturing combined with flexible materials, such as polymeric elastomers, might lead to optical components with high compliance to nonplanar surfaces and large strain. [33] In this respect compression, bending, or stretching, exploited so far for tuning deformable photonic devices, [34][35][36] could be rethought as effective gating fields to provide 3D optical systems with new functionalities, including controllable properties and cryptographic capability.
Here we introduce 3D printed stretchable magic windows (MWs). [37] MWs are transparent refractive components whose surface reliefs are designed by the inverse Laplacian of a target light pattern, [37] and are capable to reshape an incoming light beam into the target image. Our MWs are manufactured by digital light processing (DLP), a fast and cheap 3D printing technology (see Experimental Section for details). The 3D printed MWs, assessed through the achieved spatial distribution of the light intensity, evidence the possibility to design the desired shape of a transmitted optical beam in the whole visible range. Cryptographic systems are demonstrated, in which the information (micro-QR-codes) carried by a light pattern is encrypted in 3D shaped MWs and unveiled by projection of the light pattern through coupled MWs. Moreover, stretchable MWs are made by replica molding of the 3D printed optics, finding that the resulting light pattern can be varied across pre-configured geometries by the Published in Advanced Materials Technologies, DOI: 10.1002/admt.202101129 (2022). 4 application of uniaxial strain to MW elastomers. More specifically, MWs can be designed in order to project indecipherable patterns, whose encoded information (micro-QR-codes) controllably becomes readable upon applying a calibrated uniaxial strain to the devices.

Results and Discussion
The working principle of MWs is illustrated in Figure 1a: it relies on the refraction of light passing through a textured interface between two media with different refractive indices.
As schematized in the inset of Figure 1a, the incident light rays are deflected by a certain angle, which depends on the surface texture and the refractive index mismatch. The refracted rays, collected at a defined "focal" distance, fMW, produce a specific spatial distribution of the light intensity. Given the intensity pattern to be realized, the surface profile of the MW can be calculated by solving the inverse problem, for which various approaches have been reported. [16] Here the method proposed in Ref. [37] is followed: given the desired distance between the plane of projection of the image and the MW (fMW), the MW surface profile h(x,y) (where x,y are the in-plane coordinates and h is the thickness of the MW, see Figure 1b Figure S1 of the Supporting Information): [37] where R={x,y}, and n is the refractive index (n=1.5 in our calculations). From Equation (1), the surface profile h(x,y) is calculated by finite difference approach (see Experimental Section for details) and used for the design of the 3D MW by computer assisted design (CAD). shown in the inset. The corresponding 3D printed MW is shown in Figure 1c. The MW is almost transparent in the visible range (optical attenuation data are shown in Figure S2 of the Supporting Information) and has no pattern on its surface which is discernible to the naked eye.
Importantly, at variance with diffractive architectures and metasurfaces, the overall technology is entirely based on geometrical optics, thus leading to wavelength-independent components made of transparent materials which might exhibit unequalled broadband operation.
Several MWs are designed and printed, with different target patterns: a flower, a Yin Yang symbol, and a square perimeter (Figure 2). They are illuminated with various LED sources ( Figure S3, and the Experimental Section for details), and the corresponding light patterns projected on a screen are shown in Figure 2, evidencing that the 3D printed MWs can reproduce the desired intensity patterns with high accuracy at several different wavelengths.
All images in Figure 2 are obtained at the same focal distance, although this aspect can be in principle affected by the wavelength dispersion of the refractive index of the photo-polymerized MW material (1.48-1.58 for visible light). [38]  7 The properties of the light pattern cast by the MWs are evaluated by three figures of merits: the full width at half maximum (FWHM) of a selected feature (the flower stem in Figure   3), and two contrast parameters, i.e. the contrast-to-noise ratio (CNR) and the Weber contrast (WC), defined as: where < >, < >, and are the average intensity and standard deviation of light intensity of the white and black regions, respectively (red squares in Figure 3c). WC is a measure of the image contrast normalized to the background luminosity, whereas CNR takes into account also the noise of the image. [39,40] In Figure 3e, the intensity profile of a detail of the projection (red line in Figure 3c) is shown as a function of the distance, d. From such data the FWHM of the investigated feature as a function of d is obtained (Figure 3f). The measured size of the flower stem image is found to match the size of the corresponding feature of the target image at d=3 mm, in accordance with the value of fMW used for the MW design. Figure   3g shows the d-dependence of the contrast parameters, both featuring a maximum at d=3 mm, in agreement with the analysis of the feature width. Similar results are obtained for the other studied patterns ( Figure S5).
To investigate the minimum feature size that can be projected by the MWs printed by DLP, a set of samples capable of projecting patterns with various geometries (squares and parallel lines) and with features size ranging from 5 mm down to 100 µm are realized ( Figure   the obtained values with the spatial resolution of various 3D printing technologies. Indeed, current 3D printing methods provide for a wide range of spatial resolutions, [28] and the most suitable technique can be selected depending on the designed MW features. As a general trend, MWs with longer focal distance and smaller lateral size require 3D printing with higher spatial resolution, such as multiphoton stereolitography. [41] Less stringent requirements for spatial resolution are needed for MWs having larger lateral size and shorter focal distances ( Figure   S12).
To   The above encryption mechanism can be further strengthened by utilizing two coupled MWs, whose encrypted light pattern can be unveiled only if the two optical components are placed in series, while remaining undecipherable if only one of them is used (Figure 6). In such approach, one of the two MWs acts as unique key for decryption. starting from a pattern of the micro-QR code that is intentionally encrypted in a way that it cannot be read by a QR scanner unless a pre-defined uniaxial strain, ε=ΔL/L0, (where L0 is the lateral size of the unstretched MW) is applied (Figure 7a,b). We realize two MWs textured with different surface profiles, namely designed to work upon calibrated elongation of 15-20% ( Figure 5c) and of 20-30% (Figure 7d), respectively. Slight reductions of the set elongation points might be related to the algorithms exploited for error correction by QR scanners, and to possible damage or deformation compensation of the 2-dimensional code pattern. [42,43] An exemplary behavior of the MW under stretching is also shown in the Supporting Movie.

Conclusion
In summary, we introduced cryptographic, 3D optical components capable of projecting a pre-defined light pattern, upon illumination with light in the visible range, including white sources. Enhanced by inverse Laplacian design and replication methods as done in this work, DLP can enable the fabrication of freeform elements also with stretchable materials. [3,44] We anticipate that 3D printed MWs might be also used for large-scale, broadband beam shaping and control, which is especially useful in those spectral intervals where beam shaping is more demanding, such as for the THz or Mid-IR wavelengths. For such spectral ranges, the larger wavelengths would also lead to less stringent requirements of printing in order to achieve highquality optical surfaces, and various stereolithographic methods are currently available to this aim. [28] Printable resins with low attenuation losses in these spectral ranges have been recently demonstrated, [45] and could be used for the additive manufacturing of high-quality, freeform optical systems. Stretchable MWs with tailored functionality might find application in fields as various as illumination design, smart labels, anti-counterfeiting, and cryptographic systems.

Experimental Section
Calculation of the surface relief of the MW. Given a generic freeform interface with sufficiently gentle height variation, the intensity profile, I(R), that is generated by the refraction of incident light rays satisfies the relation: [37] ( ) = with L the side length of the MW (here assumed to be squared). If this step is not performed the height profile is superimposed on a concave or convex parabolic profile. [37] Indeed, the possibility of modifying the surface relief of the MW, by considering a normalization constant in Eq. (6) A>L 2 (A<L 2 ) for a convex (concave) shape gives an additional flexibility for the design of the MW, which can be better tailored to the 3D printing process. In fact, this normalization method is exploited to further engineer the MW and create a convex profile, which is important because the convex shape prevents the air from getting stuck under the previously printed layers, creating bubbles and ripples in the final 3D printed MW ( Figure S14). The variation of the normalization constant, A, has two consequences: (i) the focal distance will slightly decrease (increase) due to the superimposed convex (concave) profile which changes the angle of incidence on the surface features and (ii) the contrast between black and white areas of the projected intensity profile will slightly decrease as the black regions in the digital image will now become gray.
In order to design the stretchable MW, one has to consider that the replica molding results in a PDMS MW with an inverted surface profile with respect to the 3D printed template.
Therefore, the PDMS MW will project the same image but with black and white areas inverted.
To realize a PDMS MW that projects the original image, the template MW was designed and 3D printed starting from a negative target image. The flexible MW can be also designed to project a distorted image, for example shrunken in one direction, so that the undistorted, desired image can be obtained by uniaxial stretching of the MW. To this aim, the thinning of the material when stretched has to be accounted for. The relation between the stretched length and the height variation can be derived through the Poisson's ratio, ν, of the deformable material: where ℎ 0 and 0 are the thickness and length at rest, h and L are corresponding stretched values, respectively (ν=0.49 for PDMS). [46] The surface profile of the MW is firstly calculated for the undistorted image, and then shrunk along one axis, e.g. the x axis, by a certain percentage. The thickness of the deformed MW is then derived through Equation (7).        (2) and (3) of the main text, respectively) utilizing the images projected by the 3D printed (circles) and elastomeric (squares) MWs ( Figure S6d).      Figure S15. Spectrum of the MW-generated light intensity, transmitted by a polyethylene screen used for imaging.