Static moiré patterns in moving grids

Abstract

We describe an optical phenomenon of unmovable moiré patterns in sliding (moving) grids and gratings. the phenomenon was observed visually in the planar straight movement of the black-andwhite gratings with a period of several mm. This is a velocity-independent effect confirmed analytically and in a computer simulation based on the spatial averaging. We found the static directions of the moiré patterns in the regular grids, but our technique can be also applied to other objects. the orientation and period of the static moiré patterns are not obvious, especially in the presence of the distance effect. The phenomenon can be practically used in security applications. The moiré effect is an optical interaction (interference) between superposed layers with periodically modulated transmittance and close geometrical characteristics (period and orientation). The moiré patterns appear when two or more "repetitive structures (such as screens, grids or gratings) are superposed or viewed against each other" 1. The mathematical representation of such interference is the point-by-point multiplication of the reflectance/ transmittance functions of the superposed (overlapped) gratings. In discussing the moiré effect, one cannot fail to mention fundamental papers and books by famous authors, such as I. Amidror, O. Kafri, K. Patorski, D. Post, O. Bryngdal, C.A. Sciammarella, P.S. Theocaris and other authors who laid a foundation of the moiré theory and applications. Their groundbreaking works 1-10 made a great impact on the theoretical and experimental investigation of the moiré effect. The moiré effect is well investigated in optics 1-3,6,11,12 including the optical measurements 8,13. Note that the involved gratings are not diffraction gratings, rather regular structures with a period of several mm or cm, which is much longer than the wavelength of the visible light. Therefore, the moiré effect can be described in terms of linear optics, i.e., by rays. In visual displays (dynamic screens, still photographs, and printouts), the moiré effect may create an unwanted image overlaid with a useful image, because the moiré patterns may appear in some unintended areas of the screen; as a result, the image quality drops down. Therefore, in displays, the moiré effect is an undesirable visual effect 14,15. This is especially important for autostereoscopic three-dimensional (3-D) displays 16 because of their typical structure consisting of layers with close periods. This is a moiré effect in the macro world. The moiré phenomenon can be observed not only under the visible light, but also in other rays, for instance, X-rays 17-19 , electron beams 20-23 , and infrared light 24,25. In the nano-world, the moiré effect has been, particularly, observed in nanoparticles under an electron microscope (EM). In nano-layers, the moiré patterns are often referred to as the moiré superstructure or superlattice 26,27. The moiré effect in cylindrical nanoparticles, for instance, in single-and double-walled nanotubes (SWNT, DWNT) is also not unknown 28-30. The...

Anomalous periodicity in superpositions of localized periodic patterns

“…This gives rise to a phase gradient which changes K S as a function of Δφ, for any κ, as demonstrated in figure 3(b) (see a quantitative analysis in appendix D). The dependence of K S on T 2 in figure 2, is understood as an interplay between the fundamental parameters κ and Δφ of equation (1). The specific conditions formed in our experiment and explaining figure 2, are depicted by the red trajectory in the κ-Δφ plane in figure 3(c).…”

“…One example are Moiré patterns, which are a result of alternating constructive and destructive interference between periodic patterns with a slightly shifted periodicity, or relative rotation. Such patterns appear in many interesting contexts (e.g., [1]). While interference in general is the basis of classical and quantum wave mechanics, it still remains a rich ground for new experimental findings and theoretical interpretations.…”

Nanometric Moiré Stripes on the Surface of Bi2Se3 Topological Insulator

“…Figure 4 b shows the mathematical modeling of the interaction between the two periodic gratings obtained by the expression [1 + cos( k 1 x )][1 + e -z/λ ) cos( k 2 x )]. 24 This results from the convolution of two periodic functions with wavevectors k 1 = 2 π/0.478 nm –1 and k 2 = 2π/0.418 nm –1 corresponding to the strained outermost layer ( a = 0.478 nm) and the underlying layer ( a = 0.418 nm) of Bi 2 Se 3 , respectively. The factor e -z/λ considers the contribution to the STM signal from a depth z , 25 with λ being the tunnel electron wavelength.…”

“…Any different deformation of one of the lattices with respect to the other, such as a twist around the same center point or a uniform strain along both a and b , would give rise to different moiré patterns (some examples are given in Figure S6). Figure b shows the mathematical modeling of the interaction between the two periodic gratings obtained by the expression [1 + cos­( k 1 x )]­[1 + e -z/λ ) cos­( k 2 x )] . This results from the convolution of two periodic functions with wavevectors k 1 = 2 π/0.478 nm –1 and k 2 = 2π/0.418 nm –1 corresponding to the strained outermost layer ( a = 0.478 nm) and the underlying layer ( a = 0.418 nm) of Bi 2 Se 3 , respectively.…”

Observation of Anomalous Moiré Patterns

“…Our work also uncovers a broader universal 1D moiré phenomena in Fourier space, even when the two envelopes do not overlap in real space. As moiré patterns are used in a wide range of applications [1], such features may prove both inspiring and useful for a variety of fundamental studies of one-dimensional periodic phenomena, as well as for technological applications.…”