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Abstract
The design of an image with dynamic effect usually adopts an interlacing strategy, but it suffers from the crosstalk between the neighboring images. Here, by introduction of the weak domain transformation to the micro pattern array (MPA), striking motion effects are realized with no crosstalk. A complete theoretical model based on both the moiré and fix point theorem is developed for interpretation of the kinetics. A microlens array-based moiré magnifier is fabricated to demonstrate the visually striking motion effect. The moiré magnifier is of primary interest for a wide variety of applications in appealing decoration, information encryption, and displacement measurement, etc.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Static image that displays motion effect when the viewing angle changes has drawn considerable interests in security and decorative devices. Usually, interlaced pattern is printed and then bonded to a microlens array (MLA) to produce such device [1–2]. However, the image quality is greatly deteriorated by the crosstalk between the neighboring interlaced images beneath the MLA, thus it is uncomfortable for observers to view the dynamic effect.
Moiré phenomenon can be widely found at the intersection of repetitive structures such as superposed combs, bridge side railings, fabric, etc. Moiré-like schlieren, such as CCD moiré and moiré metamaterials, can be used to measure deformation field or to form plasmonic superlattices [3–4]. Introduction of micro-focusing elements to one of the superposed layer, so-called moiré magnifier, has also attracted great interest. The MLA as revealing layer can enlarge repetitive icon, e.g. text or color motifs in the MPA layer (as base layer) to form enlarged synthetic image. Hutley et al. dealt with MLA combined with MPA whose period is nearly the same size with the characteristic of the orthoparallactic movement [5–6]. Cadarso et al. proposed one-dimensional band moiré image incorporating periodic lenticular array as an alternative that overcomes the design limitations of two-dimensional MLA [7]. By using a superposition of randomly distributed MLA and a corresponding MPA, we have demonstrated dynamic Glass pattern, characterized by only one single magnified three-dimensional moiré image concentrated about the fixed point in the superposition and fading out if one going away from this point, and further reported a reflective moiré magnifier having the advantages of the relatively short focal length, immunity to external stain and independence of illumination condition owing to its flat working surface [8–9].
In this paper, instead of using the interlacing strategy, we demonstrate the adoption of the weak domain transformation to the MPA layer to create magnified synthetic image with fascinating motion effects. A pictorial, intuitive approach based on the fixed point theorem over a rigorous mathematical treatment is present to offer a deep insight into the motion effect. By carefully choosing the weak domain transformation, it can provide not only the parallactic/orthoparallactic but also curvilinear motion effect, such as the rotating or radial movement effect. With the combination of nanoimprinting and embedded nano-printing techniques, the imaging film is fabricated on 70 µm-thick transparent biaxially oriented polypropylene (BOPP) film, which shows no crosstalk in the observed image. The micro-optic integrated moiré magnifier with dynamic effect will be of primary interest for a wide variety of applications: to create new designs and to synthesize visually appealing decoration, to increase the sensitivity of displacement measurement, to improve the performance of information encryption and authentication device.
2. Theoretical model
A theoretical model for interpretation of the kinetics is investigated firstly. Figure 1 depicts the diagram of the moiré magnifier with weak domain transformation. A plurality of MLA with square arrangement is disposed on top surface of the transparent BOPP film. The periodic lattice is chosen as 50 µm. The MPA layer (in $x - y$ plane) is situated at the focal plane of MLA, whose distance h (70 µm) is exactly the same as the thickness of the BOPP film plus the residual layer of the ultra-violet resin.
Fig. 1. Illustration of the moiré magnifier. The top-left inset is a diagram at tilt viewing angle $\varphi$ or $\phi$.
There are several methods that have been proposed to analyze the geometric shape of moiré fringes, such as the Fourier series formulism [10], the indicial equation method, [11] and the spectral trajectory method [12], but they can only work in the case of superposition of two or more periodic (or quasi-periodic) structures, which not suitable for the MLA-based moiré magnifier with the transformed MPA. There are two prerequisites in the model: one is that moiré images appear at the location of the fixed points according to the fixed point theorem [13]; the other is that as the moiré magnifier oscillates around the horizontal or vertical axis, the sampling point shifts from the unit Pi to the neighboring unit Pi±1, as shown in top-left inset in Fig. 1. The shift distance is decided by the rotating angle ($\varphi$ and $\phi$) and the focal length h. Then the location of the fix point under the weak domain transformation ${\textbf g}({x,y} )$ can be calculated as
3. Fabrication process
Fabrication process is shown in Fig. 2(a). Firstly, positive photoresist islands (RJZ-390, RUIHONG Electronics Chemicals) are fabricated using laser direct-writing technique, then the microlens array is achieved through thermal reflow method. The thickness of the photoresist is 6.0 µm and the diameter of the island is 42 µm. Squared-arrangement pattern with a period lattice of 50 µm is adopted here. The heating temperature is 150°C for 30 min. The MLA nickel mold with a concave profile is obtained by electroforming. Similar process is applied to the fabrication of MPA nickel mold except the baking process. Secondly, ultra-violet nanoimprting lithography is used to replicate the microstructures on the double surfaces of the transparent BOPP film respectively. The refractive index of BOPP (∼1.5) matches to that of the UV resin, thus it exhibits better optical performance than polyester (PET) film whose refractive index is 1.65 in visible range. The imprinting pressure is typically less than 0.3 MPa due to the low viscosity of the UV resin. It takes about 20 s to cure the resist under a 365 nm light-emitting diode with an intensity of 1000 mW/cm2 at a distance of 1 cm. The alignment is dealt manually in our experiment. Finally, the pigment paste (concentration of 70%, viscosity of 25 cps, and particle diameter in the range of 100 to 200 nm, Suzhou Betely Polymer Materials Co., Ltd) is filled into the micro-grooves through the scrape technique. After 30 min sintering at 60 °C, the paste is solidified in the groove. A wiping process with nontoxic organic solvents can be employed to clean the surface. The sag height and curvature radius of the MLA is 8.0 µm and 30 µm respectively. Figures 2(b) and 2(c) show the profiles of the fabricated MLA and MPA of the character “E”. The linewidth of the icon is about 2.6 µm and its aspect ratio is 1:1, which is helpful for filling ink into the micro-groove through a gravure-like printing technique [8]. The printing resolution is 9700 dpi much higher than traditional screen printing technique.
Fig. 2. (a) Fabrication progress of the moiré magnifier. (i) pattern generation, (ii) development, (iii) thermal reflow to form MLA, (iv) MLA nickel master mold by electroplating, (v) MPA nickel master mold by the similar process, (vi) MLA replication by UV nanoimprinting, (vii) MPA replication and pigment paste filling, (viii) actual light path in the device. The microscopic pictures of the fabricated MLA (b) and MPA (c) at 45° from top view.
4. Results and discussion
4.1 Parallactic movement
Firstly, a scaling transformation ${\textbf g}({x,y} )$ is applied to the MPA layer, and the coordinate of the fix point is calculated as
A quick inspection of Eq. (3) reveals that the parallactic movement can be achieved under the scaling transformation. The moiré images move along the x-direction with a magnified factor of $\frac{1}{{1 - s}}$ as the sample oscillates around the vertical axis at an angle of $\phi$. The solution of the second function in Eq. (3) denotes there is no displacement in the y-direction. Similar result can be achieved when $\phi$ is set to zero and the moiré images move along the y-direction as the sample oscillates around the horizontal axis at an angle of $\varphi$. It is well-known that the scaling transformation leads to the stereoscopic effect on the moiré images appearing to lie beneath $({s > 1} )$ or above $({s < 1} )$ the $x - y$ plane [9]. The fix points of the floating and deep images move in the opposite direction in terms of Eq. (3), which can provide enhanced depth cue to three-dimensional effect, as shown in Visualization 1 and the single-frame excerpt shown in Fig. 3(a). The scaling factor s and the corresponding magnification factor is 0.993, 142.9 and 1.006, -166.7, respectively. Since the magnification factors have the opposite signs, the two MPA sets are central symmetric to each other, as shown in Fig. 3(b).
Fig. 3. (a) A single-frame excerpt from Visualization 1. (b) The microscopic picture of the MPA. The scale bar in (a) and (b) is 16 mm and 50 µm, respectively.
4.2 Orthoparallactic movement
Secondly, a rotation transformation ${\textbf g}({x,y} )$ is applied to the MPA layer, the coordinate of the fix point is calculated as
Equation (5) shows the orthoparallactic movement effect can be achieved under rotation transformation. The displacement in the y-direction is $\left|{\cot \frac{\alpha }{2}} \right|$ times greater than that in the x-direction as the moiré magnifier oscillates around the vertical axis at rotating angle of $\phi$, as shown in Visualization 2 and a single-frame excerpt shown in Fig. 4(a). The skew angle $\alpha$ is 0.28°and the magnification factor is about 200. According to the moiré vector theory, the moiré image rotates at an angle of $\textrm{9}{\textrm{0}^ \circ }\textrm{ - }\frac{\alpha }{2}$ to the MPA, as shown in Fig. 4(b) [14]. Such orthoparallactic movement is counter-intuitive to the normally expected parallactic movement. It has been used in several current banknotes to create novel security features [15].
Fig. 4. (a) A single-frame excerpt from Visualization 2. (b) The microscopic picture of the MPA. The scale bar in (a) and (b) is 16 mm and 50 µm, respectively.
4.3 Rotating movement
What is the most interesting part of the story is that curvilinear movement with no crosstalk can be achieved by using nonlinear weak domain transformation. Here the transformation function can be chosen as
Figures 5(a)–5(b) show the square-arranged icons “E” with 50-micron lattice constant before and after the geometrical transformation in terms of Eq. (6). In the central area, the local pattern is violently changed to extremely deformed tiny icons in ring arrangement. Meanwhile, much slight distortion occurs as farther away from the center. The microscopic pictures of the fabricated character in the central and neighboring area in the destination plane are shown in insets in Fig. 5(b).
Fig. 5. The square-arranged icons “E” with 50-micron lattice constant before (a) and after (b) the geometrical transformation in terms of Eq. (6). The insets show the microscopic pictures of the fabricated character in the central and neighboring area. The scale bar is 50 µm. (c) and (d) are illustration of the rotating movement effect clockwise and counter-clockwise as the moiré magnifier oscillates around the horizontal axis. The icons in light red are the original moiré image and those in deep red are the observed moiré image at a rotation angle of ${\pm} {15^o}$, respectively. The arrows show the direction of the movement. (e) and (f) are single-frame excerpts from Visualization 3.
Since there is no analytic solution to Eq. (6), we use weak transformation approximation to predict the position of the fixed point as
When the moiré magnifier oscillates around the horizontal axis ($\phi = 0$), the trajectory of the fixed point $\left( {x_{F} = \cos \frac{{h\tan \varphi }}{\varepsilon },y_{F} = \sin \frac{{h\tan \varphi }}{\varepsilon }} \right)$ is a circle as changing $\varphi$ according to Eq. (8). Simulation results are shown in Figs. 5(c)–5(d), where $\varepsilon = 0.0954$. The character “E” in light and deep red represents the moiré image before and after oscillation respectively. As $\varphi$ changes to ${\pm} {15^\textrm{o}}$, the resulting moiré image rotates around the alignment center at an angle of ${\pm} {11.2^\textrm{o}}$, as shown in Visualization 3 and the single-frame excerpts in Figs. 5(e)–5(f). The experimental results are consistent with the theoretical expectation.
4.4 Radial movement
Finally, by employing the same geometrical transformation, as the magnifier oscillates around the vertical axis ($\varphi \textrm{ = 0}$), the moiré image displaces along the radial direction with continuously changing size. Simulation and corresponding experimental results are shown in Figs. 6(a)–6(d), in which Figs. 6(c)–6(d) are single-frame excerpts from Visualization 4. The locally spatial frequency $({u,v} )$ in moiré space is given by the gradient of the transformation function Eq. (7) [16]
Fig. 6. The schematic of the radially inward (a) and outward (b) movement effect. The icons in light red is the moiré image without oscillation and those in deep red is the moiré images at a rotation angle of ${\pm} {15^o}$ round the vertical axis, respectively. The arrows show the direction of the movement. (c) and (d) are single-frame excerpts from Visualization 4.
The solution curves of $u$ consist of a family of radial straight lines $y = cx$ passing through the origin of coordinate, where c is an arbitrary constant. It suggests that the local vector in the x-axis direction is projected to its radial direction (along the straight line $y = cx$) in image space. The solution curves of v consist of a family of straight lines perpendicular to those solution curves to u. It suggests that the local vector in the y-axis direction is projected to the direction perpendicular to its radial direction in image space. By combination of Eqs. (8) and (9), it can be obtained that as the moiré magnifier oscillates around the vertical axis, the shape of the character “E” is perfectly preserved, as shown in Visualization 4. The local magnification factor M equals to the reciprocal of the square root of the local frequency
When $\varepsilon$ is set to 0.0954, the moiré image at a distance of 2 mm from the alignment center is enlarged to 755 µm $\times$ 775 µm which can be discerned by the naked eye. The moiré image moves outward with an increasing magnification factor. Inversely, a decreasing magnification factor is achieved.5. Conclusion
In conclusion, this paper describes the adoption of domain transformation in moiré magnifier in order to realize diverse kinetic motion effect while swaying with no crosstalk. The main reason without any crosstalk lies in the fundamental theory that the synthetically magnified images only appear at the locations of the fixed points in the moiré imaging phenomenon, which are well-defined by the transformation function. The underlying theoretical foundation is quantitatively explored by carefully choosing the transformation function, which can completely determine the magnified factor, orientation, distribution and movement effect of the moiré images. With the combination of nanoimprinting and gravure-like printing techniques, 70 µm-thick moiré magnifier on transparent BOPP film is fabricated to demonstrate rectilinear/curvilinear movement effect. The methodology can be applied to create new design of visually appealing decoration, to increase the sensitivity of displacement measurement, and to improve the performance of information encryption and authentication device.
Funding
National Natural Science Foundation of China (61405133, 61505134, 61575133, 61775076); Natural Science Foundation of Jiangsu Province (BK20140357); Major Basic Research Project of the Natural Science Foundation of the Jiangsu Higher Education Institutions (14KJB140014); Science and Technology Project of Suzhou (ZXG201427); Priority Academic Program Development of Jiangsu Higher Education Institutions.
Acknowledgments
We thank the support from Shine Optoelectronic Co. Ltd. for experiment.
Disclosures
The authors declare no conflicts of interest.
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