Optical Engineering 37 (11) 3007-3014 (November 1998) 1 Properties of moiré magnifiers Hala Kamal* Reinhard Völkel** Javier Alda* * Departamento de Optica Universidad Complutense de Madrid. Facultd de Físicas. Ciudad Universitaria s/n-28040 Madrid, Spain Phone: +34.1.394.4555; Fax: +34.1.394.4674 E-mail: j.alda@fis.ucm.es ** Institute of Microtechnology Université de Neuchâtel, Rue A.-L. Breguet 2 CH-2000 Neuchâtel, Switzerland Phone: +41.32.718.3279; Fax: +41.32.718.3201 E-mail: reinhard.voelkel@imt.unine.ch 1 Abstract: Moiré magnification can be observed visually if an array of identical objects is viewed through an array of identical microlenses with a dif- ferent period. Theoretical analysis and experimental results of the moiré image obtained by moiré magnifier are presented. Conditions for erect and inverted moiré magnifications are derived and interpreted. Virtual erect images are observed only when the period of the lens ar- ray is larger than that of the object array, while inverted images are ob- tainable in both cases. For equal periods, uniform field of view results. The relation between the relative size of the periods and the distance between object and lens array are derived. Expressions for image mag- nification, orientation and size are deduced. The condition to obtain a demagnified moiré pattern is deduced. Rotation of the lens array with respect to the object array results in rotation of the erect and inverted moiré pattern in similar and opposite directions, respectively. Subject terms: Moiré effect, microlens arrays 2 Introduction The term moiré is a French word meaning ‘watered mohair’, a glossy cloth with wavy alternating patterns that change forms as the wearer moves. It is also re- ferred to the ‘watery and wavy’ appearance when lay- ers of silk are pressed together by special tech- niques[1]. Physically it is an optical phenomenon re- sulting from the superposition of two or more periodic grid structures. The resulting moiré pattern is influ- enced by changing any of the three geometrical pa- rameters characterizing the individual grid structures, namely, period, orientation and shape.[1, 2, 3] Although there is a long history in the investigation of superposition moiré effects, only few publications ex- ist which describe the moiré effects appearing if a pe- riodic array of identical objects is observed through a periodic microlens array.[4, 5] However, this moiré magnification effect, as the authors of this article call the phenomenon, is well known from integral images and integral photography.[6, 7] The frequent appear- ance of similar phenomenon in the usage of microlens array in conjunction with arrays of light sources, photo-detectors, liquid crystals displays, CCD-chips, etc. motivates to further investigations about this phe- nomena.[8] The term ‘moiré magnifier‘ is suggested by the con- junction of a periodic object seen through a periodic array of optical elements or systems. The image is formed as the composition, side by side, of the indi- vidual images generated by the optical array. The im- age formation is equal to superposition compound eyes as nocturnal insects and deep-water crustaceans have.[9] The foundations of optical array design also demonstrate the existence of cooperative effects.[10] In optical array design, the composed image is named as the synthetic image. By using this last approach, the ‘moiré magnifier’ system should be named as ‘syn- thetic magnifier’. However, we prefer to keep the term ‘moiré magnifier’ because of the good acceptance of this term by the optics community after the presenta- tion of the device by Hutley et. al.. The article from Hutley et. al. [4] dealt with an object array combined with a lens array whose period is nearly the same size. Experimental results for the magnification and orientation of moiré images and their fundamental dependence on the orientation of the lens array have been given. In our investigation, we explain the formation of erect and inverted moiré im- ages for moiré magnifier consisting of microlens and object arrays with different periods. We derive the re- lation between the relative period difference of the ar- rays and the object array position in x, y, z and θ. We describe the rotation properties of moiré magnifiers. 3 Basic properties of a moiré magnifier The basic configuration of a moiré magnifier is shown in Figure 1. An array of identical objects (Figure 1-b) is imaged by an array of identical lenses (Figure 1-c). For certain combinations of object and lens arrays, one or more magnified moiré images of the object motif are observed (Figure 1-d). We will now derive the general properties of moiré magnification. For sim- plicity, we restrict our investigation on symmetrical square arrays. However, a variety of different array types (hexagonal, rhombic, etc.) might be used for moiré magnification in a similar way. The object array shown in Fig. 1 consists of O × O identical objects or motifs at a period Λ O . The lens ar- ray consists of L × L identical lenses of Ø L aperture at a period Λ L . The focal length of the lenses is ƒ > 0 (positive lens). Each lens images a part of the object array. The actual type or shape of the lenses (circular or square aperture, refractive or diffractive lens, etc.) is not relevant for our investigation. We assume that the lens size is equal to the lens period Ø L = Λ L , and ig- nore all rays not hitting the lenses. The plane of the object array is parallel to the plane of the lens array.