## Abstract

Moiré effect is a well-known interference phenomenon occurred between repetitive structures. Conventional moiré fringes are produced by superposing gratings or dots arrays. However, when the micro-image array as base layer and the corresponding micro-focusing elements as revealing layer overlap each other, a special kind of moiré effect, so-called moiré magnifier, can be observed. Micro-image units in the base layer are enlarged and projected to moiré space. To our knowledge, there has no complete design methodology for the realization of the moiré magnifier. With the combination of the Fourier transform and spectral approach, a new algorithm based on transfer matrix is investigated, which is capable of predicting the location of any arbitrary point in the base layer mapping to the moiré space, thus it provides a simple way to explore the physical insight into the field of moiré imaging. The magnification factor and the orientation of the synthetically enlarged image are determined not only by the scaling ratio but also by the interrelation between the primitive vectors in the base and revealing layers. Experimental results are in good agreement with theoretical predictions. By using the proposed method, the moiré magnifier can extend appealing applications in esthetic security devices, highly accurate measurements and precise color printing.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Moiré phenomenon is an interesting optical effect, which has found wide applications in optical alignment, displacement measurement, document anti-counterfeiting, and even just for fun. Classical moiré pattern is produced by superimposing two sets of gratings or dots arrays whose frequencies and structures satisfy certain conditions [1]. Compared to its counterpart, a moiré image, so-called moiré magnifier, appears when superposing a base layer made of micro pattern array (MPA) and a revealing layer made of micro-focusing elements with similarly topological structure. Since the repetitive MPA contains the same icon, e.g. text or color motifs, the moiré magnifier comprises of the enlarged repetitive images and has several striking visual effects, such as three-dimensional (3D) scene and parallactic or orthoparallactic motion. H. C. Hutley et al. calculated how many the original arrays get magnified and rotated as a function of the periods and orientations of the two layers [2]. H. Kamal et al. discussed the conditions for erect and inverted moiré magnifications [3]. V. J. Cadarso et al. studied band moiré image incorporating lenticular array as an alternative that overcomes the design limitations of two-dimensional (2D) microlens array (MLA) [4]. We have demonstrated the dynamic Glass pattern, characterized by only one single magnified 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 [5,6]. Moiré magnifier has potential for design, fabrication and characterization for utility as esthetic security features for currency, document, and product authentication by the general public.

Several methods have been developed to explore the geometry of moiré fringes, such as indicial equations method, local frequency method, and spectral trajectories method [7–9]. The indicial equation method is simple and straightforward but only suitable for analyzing superposition of the families of lines. The local frequency method is basically equivalent to the indicial equations method, which is simply different facet of the same mathematical realization. I. Amidror has proposed a Fourier-based approach describing the behavior of the superposition layer as the convolution of their respective spectra [10,11]. However, when the individual bands in the base layer do not comprise the line-gratings but MPA, by carefully placing a revealing layer of micro focusing elements over it, the enlarged moiré image is not the same as the original MPA in shape and arrangement. The above-mentioned moiré analysis methods have great difficulty in explaining the linear transformation of the moiré image.

In this paper, we explore the basic design methodology for moiré magnifier. A mathematical model in transfer matrix form is investigated that can enable deriving the geometric projection between the original MPA and the synthetically enlarged moiré image. The model specifies the mapping of any arbitrary point in the base layer into the moiré image space, thus it can provide physical insight in the field of moiré phenomenon. The orientation of the MPA, the direction of the moiré repetition vector, and the shape of the enlarged moiré image can be calculated as a function of the geometric layouts of the base and revealing layers. Thus, it provides complete design methodology for the realization of moiré magnifier. Reflective 1D and transmissive 2D moiré magnifiers based on micro-focusing elements are fabricated. Experimental results are in good agreement with theoretical calculations. The linear transformation of the enlarged moiré image can be clearly predicted due to the angle misalignment between the base and revealing layers. Because of the capability of providing quantitative description on the behavior of the moiré magnifier, the proposed method will help to span diverse array of applications for moiré magnifier, such as optical anti-counterfeiting by the naked eyes, precise measurement for in-plane linear transformation and strain analysis in experimental mechanics [12–16].

## 2. Methodology

#### 2.1 Derivation of the transfer matrix based on the Fourier transform and spectral approach

Assume the base layer, revealing layer and moiré lattice in their periodicity by primitive vectors $\left[{b}_{1},{b}_{2}\right]$, $\left[{r}_{1},{r}_{2}\right]$ and $\left[{m}_{1},{m}_{2}\right]$ in cartesian coordinate, as shown in Fig. 1. It also represents the image domain and covers all possible 2D Bravais lattices. The geometric transformation between the base vectors ${b}_{i}$ and the revealing vectors ${r}_{i}(i=1,2)$ can be given by a $2\times 2$ matrix ${T}_{1}$ as

_{${T}_{1}$} represents an anisotropically scaled and rotated transformation. We denote the Fourier transform of the above vectors by $\left[{B}_{1},{B}_{2}\right]$ and $\left[{R}_{1},{R}_{2}\right]$, which are connected by

*I*is unit matrix.

According to the theory of Fourier transform and spectral approach, the moiré vectors in the spatial frequency domain $\left[{M}_{1},{M}_{2}\right]$ are the convolution of the base spectral vectors $\left[{B}_{1},{B}_{2}\right]$ and revealing spectral vectors $\left[{R}_{1},{R}_{2}\right]$. It can be represented in matrix form as

The relationship between the moiré vectors and the base vectors in the image domain can be calculated by

The transformation matrix T can be obtained by the manipulation of Eqs. (2)-(7)

withThe advantage of the derived algorithm lies in the fact that it cannot only calculate conventional moiré parameters such as the magnification factor and rotation angle, but also provide a simple way to predict the location of any arbitrary point $({x}_{b},{y}_{b})$ in the base layer mapping to the moiré space $({x}_{m},{y}_{m})$ through Eq. (11). What is worthy to note is that human visual system is equivalent to a low-pass filter which can merely capture low spatial frequency signal, thus a strong enough impulse in the spectrum of the image superposition falling inside the visibility circle can be discerned by human eyes [18]. The original MPA can be anisotropically scaled and rotated along the directions of the moiré vectors, therefore the enlarged moiré image is a linear transformation of the original pattern in the base layer. It is almost impossible to be interpreted with the indicial equation method or spectral approach.

#### 2.2. Case 1: 1D moiré magnifier

We demonstrate that the proposed algorithm can solve 1D moiré imaging problem when the lenticular array is used as the revealing layer. In 1D moiré magnifier, the angle $\beta $ is ${90}^{\xb0}$ and ${r}_{1}$ is infinite. The module of the vector ${r}_{2}$ equals to the periodic of the line grating ${T}_{r}$. Define ${t}_{x}={b}_{2}\mathrm{cos}\alpha ,{t}_{y}={b}_{2}\mathrm{sin}\alpha $, and assume ${b}_{1}$ in the x-axis direction. According to Eqs. (8)-(10), the transformation matrix *T* and the moiré vector matrix $\overline{\overline{m}}$ can be calculated as

#### 2.3. Case 2: 2D moiré magnifier

For 2D case, the moiré magnifier is much complicate but more interesting. Consider the base vector matrix $\overline{\overline{b}}$ and the release vector matrix $\overline{\overline{r}}$ containing a rotation matrix of the angle $\theta $, then combine Eq. (11)

We can get

whereThe moiré magnification factors ${\kappa}_{1},{\kappa}_{2}$ denote the length ratio of the moiré lattice vectors ${m}_{1},{m}_{2}$ and their counterparts ${b}_{1},{b}_{2}$ in the base layer. According to Eqs. (6) and (16)-(18), we can get

By using of Eqs. (16)-(22), we can calculate the mapping relationship from any arbitrary vector in the MPA into the moiré image in 2D space. It provides an effective solution to the achievement of the enlarged moiré image from the original MPA, which is a comprehensive effect of the base vector $b$, the revealing vector $r$ and the arrangement of the MPA itself.

## 3. Experimental results and discussions

In order to show the capability of the proposed method, we design and fabricate two types of moiré magnifiers. Micro-focusing elements such as lenticulars and microlens array (MLA), are used as revealing layer. The purpose of the introduction of the micro-focusing elements is to contribute to sampling the multiple images and to rendering the process more efficient than pinholes. The micro-focusing elements and MPA are fabricated by using optical lithography and ultra-violet (UV) nanoimprinting replication technique. Fabrication details can be found in our previous papers [5, 6].

#### 3.1 1D reflective moiré magnifier

The schematic of the 1D reflective moiré magnifier is shown in Fig. 2(a). A thin layer of silver mirror is thermally deposited on the surface of the lenticular array with a period of ${T}_{r}=$ $110\mu m$, a sag height of 10.4 *μm*, and a curvature radius of 150 *μm*. The focal length of the reflective focusing elements is relatively short, which can lead to ultrathin device architecture. The thickness of the film is 75 *μm*. Microscopic picture and contour shape of the lenticular array is shown in Fig. 2(b). The reflective focal point is located on the other side of the substrate, as schematically shown in Fig. 2(c). The height of the microtext (‘SOE’) 66.5 *μm* is elongated to 13.3 *mm* in the moiré image, thus the absolute magnification factor $\kappa $ is 200 in the y-axis direction. According to Eqs. (12)-(13), there are two solutions to achieve an upright moiré text. One solution is to use upright MPA under the condition of ${t}_{x}=0$, which represents a conventional solution. The other solution is to use oblique base bands under the condition of ${t}_{x}\ne 0$. In order to demonstrate it, we design two moirés in which ${t}_{x}=10\mu m$, ${t}_{y}=$ $110.55\mu m$ (with a magnification factor $\kappa $ of −200) and ${t}_{x}=10\mu m$, ${t}_{y}=109.45\mu m$ (with a magnification factor $\kappa $ of + 200), representing two cases of ${t}_{y}>{T}_{r}$ and ${t}_{y}<{T}_{y}$ respectively. The angle between the stroke in the microtext and the baseline is calculated as 3.15* ^{o}* and 176.85

*by using Eqs. (12)-(13). The deformed microtext in the base layer is shown in Figs. 6-8 in Appendix. Figure 2(d) and 2(e) show the local detail of the letter ‘E’ in the MPA and the replication vectors $({t}_{x},{t}_{y})$ are shown in the insets. There are two differences between the two cases. One is about the visual 3D effect. When ${t}_{y}>{T}_{r}$, the synthetically magnified image stereoscopically appears to be lie above the plane of the substrate. When ${t}_{y}<{T}_{y}$, the synthetically magnified image stereoscopically appears to lie beneath the plane of the substrate. The other lies in the effect of the dynamic motion, which means that when moving the sampleup and down, the moiré texts move along the opposite direction. The video in Visualization 1 shows the fabricated sample and the motion effect. Single-frame excerpt from the video is illustrated in Fig. 3. It can be clearly seen that an upright moiré image can be obtained from the oblique microtext array according to the transformation relationship discussed.*

^{o}#### 3.2 2D transmissive moiré magnifier

Figure 4(a) shows the schematic of the 2D transmissive moiré magnifier. Assume $\theta ={0}^{\circ}$ and the MLA in the hexagonal arrangement is chosen, which means the angle $\beta $ equals to ${60}^{\circ}$. The distance between the microlens is $\left|{r}_{1}\right|=\left|{r}_{2}\right|=35\mu m$. The sag height of the microlens is 6.05 *μm*, as shown in Fig. 4(b). Contrary to its reflective counterpart, the MLA is on the same side of the eye when observing, as shown in Fig. 4(c). By solving Eqs. (12)-(22), one can find two sets of solutions for obtaining a synthetically magnified upright letter “F” perpendicular to the base line. The magnification factors of ${\kappa}_{1},{\kappa}_{2}$ are fixed at 200. One set of the solutions is regular and intuitive, in which $\alpha =\beta ={60}^{\circ}$. It suggests that the moiré image be just elongated of the original MPA in equal proportion along the directions of the moiré repetitive vector. The magnified upright letter “F” is shown in Fig. 4(d) and its dimension size is 4.2 *mm* x 2.0 *mm*. It has the same arrangement with that in the microtext array as shown in Fig. 4(e). The other potential solution is non-intuitive but can be predicted by the equations, in which the angle $\alpha $ between the vertical stroke in letter F and the bass line is ${59.88}^{\circ}$. We set $\varphi ={90}^{\circ}$, which means the directions of the two moiré vectors are perpendicular to each other and the magnified moiré image is in the orthogonal arrangement, as shown in Fig. 4(f). However, the arrangement of MPA is different from the moiré image, as shown in Fig. 4(g). The same result can also be obtained in 2D reflective moiré magnifier.

By using of the proposed method, the relationship between the moiré image, the original MPA, and the alignment can be clearly revealed. The magnification factors of ${\kappa}_{1},{\kappa}_{2}$, the orientation angle ${\gamma}_{1}$, and the spanning angle $\varphi $ between the moiré vectors are plotted as a function of the rotation angle $\theta $ shown in Fig. 5(a). Here we set ${b}_{1}=35.17\mu m$,${b}_{2}=$ $34.83\mu m$, $\alpha ={60}^{\circ}$ and ${\beta}_{1}={60}^{\circ}$. The curve of ${\kappa}_{1}$ is almost supererimposed to that of ${\kappa}_{2}$ by using Eq. (19)-(20). The microlens film is placed over the MPA substrate. The alignment of the rotation angle $\theta $ can be adjusted manually in the experiment. Figure 5(b) shows the confocal microscopic picture of the MPA in the base layer. The red and black arrows in the circle shows the variation of the rotation angle $\theta $. The moiré image is being deformed as $\theta $ varies in the range from + 1* ^{o}* decreased to −1

*, as shown in Visualization 2. Figure 5(c)-5(g) show the single-frame excerpts from the video in Visualization 2. As*

^{o}*θ*decreases towards 0.49°, $\Delta $ approaches 0 according to Eq. (17). Therefore, the magnification factor $({\kappa}_{1},{\kappa}_{2})$ increases drastically and can even reach infinite under the ideal condition. At the same time, the spanning angle

*Φ*between the two moiré vectors approaches zero. However, in reality, the imperfect scaling matching of the base layer and revealing layer results in limited enlarged effect, as shown in Fig. 5(d). The orientation angle $\text{\gamma}$ has an abrupt $\text{\pi}$ phase change and become positive at the sampling point ii, which means the opening direction of the moiré image is reversal to its initial state. The magnification factor has a local minimum about 170 at the sampling point iii, as shown in Fig. 5(e). When

*θ*approaches −0.16°, the magnification factor can reach infinite again, as illustrated in Fig. 5(f). The spanning angle

*Φ*is 180

*in this situation, which means the two strokes of the icon is along a straight line. The orientation angle $\text{\gamma}$ undergoes another abrupt $\text{\pi}$ phase change. If the MLA film rotates continuously, the magnification factor becomes smaller and the moiré image cannot be distinguished by human eyes. The linear transformation of the moiré magnifier is vividly recorded changing with the rotation angle $\text{\theta}$ and the results are in good agreement with the theoretical calculation.*

^{o}## 4. Conclusion

Moiré magnifier is more complicate than that in the basic interference pattern produced by superimposing two sets of gratings or dots arrays. The resulting moiré image can be seen as the geometrically enlarged linear transformation of the underlying MPA. Combined with Fourier transform and spectral approach, a mathematical model that enables deriving the layout of the base layer capable of creating a 2D moiré image is depicted. The algorithm is able to calculate the location of any arbitrary point in the base layer mapping to the moiré space, thus it provides a simple way to explore the physical insight into the field of the moiré phenomenon. Non-intuitive solution to the moiré magnifier can be discovered. Moiré magnifier film is fabricated by using the unique microfabrication technique. Experimental results are in good agreement with theoretical prediction. Since the moiré technique is effective and potentially full-field optical, the proposed method is useful for the design of novel moiré magnifier which can find appealing applications in esthetic security devices, highly accurate measurements and precise color printing.

## Appendix Design of the 1D reflective moiré magnifier

The 1D reflective moiré magnifier is formed by the superposition of a rectilinear base layer and a reflective lenticular as revealing layer. The period of the lenticular is 110 *μm*. The thickness of the device is 75 *μm* and. The focal length of the reflective lenticular can be roughly estimated as R/2, in which R is the curvature radius of the focusing elements. Here the curvature radius of the lenticular is 150 *μm*.

The base layer comprises the tilted and vertically flattened text ‘SOE’. The height of the microtext is 66.5 μm, and the width of a single letter is 10.6 mm.

According to Eqs. (11) and (12), one can get

Here we set ${t}_{x}=10\mu m$, ${t}_{y}=110.55\mu m$ (for magnification factor $\kappa $ of −200) and ${t}_{x}=$ $10\mu m$, ${t}_{y}=109.45\mu m$(for magnification factor $\kappa $ of +200), representing the two cases of ${t}_{y}>{T}_{r}$ and ${t}_{y}<{T}_{r}$, respectively. The angle between the stroke in the microtext and the baseline is calculated as 3.15° and 176.85°, respectively.

The orientation angle direction of the moiré repetition vectors $\varphi $ can be calculated as

then $\varphi ={84.83}^{\circ}$ for magnification factor $\kappa $ of −200 and $\varphi ={84.78}^{\circ}$ for magnification factor $\kappa $ of + 200.According to the above calculation, we can design the microtext in the MPA, as shown in Figs. 6 and 7. Schematic of the array arrangement is illustrated in Fig. 8.

The design is realized in bmp format and then is input to the computer for laser direct writing.

## Funding

National Science Foundation (NSF) (61575133, 61405133); National High Technology Research and Development Program 863 (No. 2015AA042401); Natural Science Foundation of Jiangsu Province (No. BK20140348); Project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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