1. Introduction

“The moiré effect is a well known phenomenon which occurs when repetitive structures (such as screens, grids or gratings) are superposed or viewed against each other” [1]. Essential is that “a new pattern of alternating dark and bright areas which is clearly observed at the superposition, although it does not appear in any of the original structures” [1]. Generally speaking, the moiré effect is always present, once gratings are superposed. Its visibility depends on particular geometric characteristics of the repetitive structures (layers), for example, the ratio of periods of the layers. More formally, the spatial frequency of moiré patterns is smaller than the spatial frequencies of superimposed component structures [2–4]; and the strongest visual effect appears when certain relationships between the layers are satisfied.

The moiré effect can be successfully used, for example, to provide highly accurate measurements in moiré metrology [5], to ensure the security of documents [6], as well as in many other fields. In visual displays, however, its high sensitivity to the change of parameters makes the moiré phenomenon an adverse negative effect. The control (or minimization) of the moiré effect is therefore an important issue in printing [7], in three-dimensional displays [8] especially at finite distances [9], in scanned images [10], and in other research areas.

A particular practical problem is the minimization of the moiré effect in autostereoscopic three-dimensional displays. Their typical design makes the moiré effect an often guest in displays of that type, where the moiré patterns become especially unstable because of a possible movement of observer(s) [9]. This makes the moiré minimization very important in high quality three-dimensional displays; a stable moiré-free state should be found. Therefore, various aspects of the problem should be investigated in order to find and ensure the stable minimization conditions for the autostereoscopic displays including the multi-view displays and the integral imaging displays especially.

In this paper we further develop the spectral approach for an arbitrary number of gratings [11]. We have found equations of spectral peaks in the complex plane and trajectories of four kinds, the number of geometric elements, as well as the regular geometric structures in spectra. In examples, we describe in details the orthogonal case which can be often met in applications. Every example is confirmed in simulation, many confirmed in experiment. In the most of experiments, the gratings and grids with the rectangular profile are used, although the theory is most advanced for the sinusoidal case.

Assumptions. A grating is a one-dimensional periodic structure consisting of repeated continuous parallel lines. The gratings can be arranged in grids or layers, so as in transformations, a layer behaves as a solid unit; a typical example is a square grid as a product of two orthogonal gratings. We rely upon the multiplicative model [1], in which the visual effect of superposition is expressed as the product of transparency functions of gratings. Also, in the paper [3], it is stated that the additive model is more suitable for grids projected onto a screen, while the multiplicative for transparencies in contact. Correspondingly, we talk about transparencies here. The affine transformations with a fixed point (rotation and scaling) are applied to the gratings. Basically we consider two-dimensional power spectra. We do not deal with amplitudes; only the presence of spectral peaks and their locations in the spectral domain are considered. The functions with sparse and limited spectrum are under consideration. The former term means that the most area of the spectral surface is “empty”, i.e., lies below a certain level and therefore is negligible; there are relatively narrow and high peaks above a relatively uniform low-level background. For example, a sparse matrix is a matrix populated primarily with zeros, while the image of moiré fringes produced by one-dimensional gratings is sparse in the spectral domain. The latter term means that instead of unlimited (generally, infinite) number of spectral peaks, we only take into account several of them. In this paper, we consider the black-and-white gratings only.

Trajectories appear when a parameter is gradually changing. In the spectrum analysis, a small change of a parameter causes the corresponding incremental change of the spectrum. The spectral trajectories can be met in various research areas, in particular, in remote sensing [12, 13], in speech recognition [14, 15], and in medicine [16].

Principally, each peak of a sparse spectrum is recognizable and can be tracked individually; the path it follows can be referred to as a trajectory. However, the tracking could be a complicated mathematical procedure itself [17]. Nevertheless, if the increment of each step is small, then some simpler procedures can be applied instead.

Previously we suggested summation of successive sparse spectra with one parameter varying [11], which procedure gives virtually the result similar to tracking, clearly recognizable trajectories. Also, we have proposed using complex numbers to describe locations in the spectral domain for the two-dimensional Fourier transformation [18]. The locations in the spectral domain can be described by ordered pairs of coordinates (u, v). The ordered pairs of numbers can be treated as complex numbers (u, v) = u + iv and graphically drawn in the complex plane. This way one may employ the perfectly developed powerful and convenient mathematical tools to describe the moiré effect. Note that using the complex numbers and vector summation to describe moiré patterns in the Fourier space were primarily suggested earlier in [2]; these features are presented there in the form of a general equation. The corresponding equation in our paper looks naturally similar. We develop it further, proposing the spectral trajectories for N gratings with some specific parameters. The topics include the number of elements (particularly in the compact matrix form as powers), the derivatives of the spectral trajectories of moiré waves. To illustrate the technique, we provide a number of analytical consequences from the basic equation together with graphical examples obtained in simulation and in experiment. In particular, we describe the regular structures in the spectrum, their visibility and appearance in the spatial domain, as well as relationships between them. The experiments are performed with non-sinusoidal gratings (square wave profile) for all four possible combinations of geometric characteristics of layers and are presented for two varying parameters.

The paper is organized as follows. The general case of N one-dimensional gratings is considered in Section 2, including the locations of the spectral peaks, the spectral trajectories with one parameter varying, and the number of elements. Section 3 covers the special case of 4 gratings arranged in 2 layers by orthogonal pairs which is important for practical applications; the theory and simulation for sinusoidal gratings is considered, including the trajectories and the number of elements; this includes the case of N = 2. The regular geometric structures in the spectrum are described in Section 4. The experimental data are presented in Section 5. The terminative Section 6 finalizes the paper by discussion and conclusion.

2. General case of N gratings

2.1 Spectral peaks

Locations of spectral peaks of a one-dimensional periodic grating are distributed uniformly along a straight line comprising an impulse train or comb in the direction of the wavevector, since the spectrum consists of equidistant spectral peaks. The complex plane is a convenient tool to describe locations of the two-dimensional spectrum. The power spectrum is a real function R(z) of the coordinates (u, v) or of the complex number z = u + iv. Generally, such a function can be drawn as a surface, located “above” the complex plane, so as the height at each point z is the value of the function R(z). However, this paper is focused on locations of the spectral peaks, especially in the neighborhood of the origin. This allows a simpler description of the two-dimensional moiré effect, which is capable to predict many essential features. This approximation gives the answer in terms of the wavevector, i.e., wavelength and direction of the moiré waves.

In the complex plane, the location of a p-th peak of a comb can be written directly as the product of two numbers, integer and complex,

In Eq. (1), the complex number k represents the fundamental spatial frequency; its polar coordinates are the modulus |k| and the argument Arg k, i.e., the fundamental wavenumber and the direction (orientation) angle, resp.

Generally, the value of Q can be assigned to a finite number, if necessary, basing on the decay of Fourier coefficients and depending on the required accuracy.

Among the affine transformations with a fixed point (the origin), the uniform scaling is a proportional change of the modulus of the basic wavenumber k and the rotation of the grating about the origin by angle α can be expressed as the product of the basic wavenumber Eq. (1) and the complex number e^iα basing on the Euler identity. Therefore, the locations of Q peaks of the transformed grating with a limited spectrum are given by

The multiplicative model of superposition is implied. For a sparse spectrum, the coordinates can be found by the vector summation, which represents the convolution theorem in this case. Then, the following expression can be obtained from Eq. (2) for the spectral peaks of N superposed gratings,

 

Fig. 1 Vector sums in a spectrum. The direction of the fundamental wavevectors of two superposed gratings are the real axis and the dotted line.

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The Eq. (4) describes all possible combination terms which may technically be needed. For the visual effect, however, the only terms located near the origin are important. A certain neighborhood of the origin is called the visibility circle [1]; it represents a model of the human visual system. By this concept, the only visible are spectral components located within a certain radius from the origin. For instance, among four peaks near the imaginary axis shown in Fig. 1, only two peaks can be perceived visually.

As an example, consider two sinusoidal gratings $T_1=p_1{k}_1{e}^{i{\alpha }_1}$ and $T_2=p_2{k}_2{e}^{i{\alpha }_2}$ arranged in a rectangular grid (in which case, max |p₁| = max |p₂| = 1, k₁ = kσ₁, k₂ = k, α₁ = 0, α₂ = π/2), so as

In Eq. (5) it is assumed that one of the gratings has a fixed wavevector.

Similarly, for a double layer structure, with each layer consisting of two orthogonal gratings and the following set of parameters,

In obtaining Eq. (7), the identity e^iπ/2 = i (derived from the Euler’s formula) was implicitly applied. The set of parameters in Eq. (6) is convenient for the rectangular grids, see Fig. 2. In this set, the parameters σ₁ and σ₃ are aspect ratios (the ratio of the vertical and horizontal periods of the rectangular cell of a grid), and ρ is the relative size of grids (the ratio of the vertical periods of two grids).

 

Fig. 2 The layout of two orthogonal pairs of gratings.

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2.2 Spectral trajectories

The general equation for spectral trajectories looks like Eq. (3) or (4) with one of parameters k or α being a function of another variable t; the variable t is not necessarily the time, though.

For the spectral peaks, there is only one type of geometric element in the complex plane, the point, while for trajectories the elements of two distinctive types can be recognized, the line (curve) and the point (center) around which the lines are arranged. These elements can be characterized as the elements of non-zero length and of the zero length elements, resp. The former are the segments of generalized circles, i.e., the circular arcs or the segments of straight lines.

The centers are the terms of Eq. (3) which do not depend on the current varying parameter. For example, for the varying variable k_m, the equation of trajectories looks like the follows,

As inherited from the spectral peaks, the trajectories are symmetric with respect to the origin. Particularly, the arrangement of centers is symmetric about the origin (2-fold rotational symmetry). There also exists an additional local symmetry for trajectories. The depending terms describe the lines which are symmetrically arranged in pairs around the centers. The m-th depending term includes the integer number between -q_m and + q_m as a coefficient. In other words, the pairs of trajectories are locally symmetric with respect to their centers.

A derivative depicts how the spectral peaks are moved when a parameter is changed. The derivatives of the spectral trajectories for the parameter k(t) can be found as follows,

Assuming a linear dependence like k(t) = A_kt + B_k, and recalling Eq. (2),

In the case of the running angle, α(t) should be used instead of k(t), and therefore

With the linear change of the angle (say, α(t) = A_αt + B_α),

Both Eqs. (10) and (12) involve the multiplication of the m-th term by a complex constant. Therefore, in both cases of the linear dependence k(t) or α(t), the derivative is proportional to the m-th component, which depends on the varying parameter.

2.3 Number of elements

The peaks of a grating are added to combinations of other gratings. Therefore, in counting the total number of the geometric elements in the spectrum, the amount of spectral components (peaks) of each grating should be multiplied by the amount of peaks of other gratings. There are N terms in Eq. (3); the j-th term contains Q_j peaks. Then, the total number of the geometric elements in the spectrum can be expressed as follows,

For two superposed one-dimensional gratings (Q₁ and Q₂ spectral peaks each), the spectral peaks are arranged in the parallelogram (see Fig. 2) with Q₁ and Q₂, peaks along each side. Therefore, there are Q₁ ∙ Q₂ peaks within this parallelogram.

For gratings with identical Q_j = Q₀, instead of Eq. (13), a simpler expression can be obtained,

Equation (13) does not depend on the current varying parameter. Correspondingly, given a set of gratings, the total number does not depend on varying parameters,

For Eq. (8), the number of lines equals Q_m, where m is the index of the parameter-dependent term. Particular cases can be analyzed in details as the orthogonal gratings in the next section.

3. Special case of 4 gratings in 2 layers by orthogonal pairs

3.1 Trajectories

To continue the previous example with the orthogonal gratings arranged in 2 pairs (layers), refer to Fig. 2. There are two pairs of orthogonal gratings in this layout; one of pairs can be rotated. For sake of convenience, each pair can be considered as a solid unit, a layer (a grid). Figure 2 represents a structure of a three-dimensional autostereoscopic display, in which one layer is a monitor while the second one is an attached optical plate. This is a typical layout of a multiviev display of an integral imaging display. Both layers consist of small units (e.g., the pixels of an LCD monitor and microlenses of an optical plate) regularly distributed across the screen. The structure of layers of this example can be conveniently described by quantities d₁ and d₂ (the geometric characteristics of the structure of the layers), the values of which are assigned as follows, 1 for a grating (one-dimensional structure) and 2 for a grid (two-dimensional). When these quantities are equal to each other, the common value d = d₁ = d₂ can be used. The convenient set of parameters is described in the previous section, see Eq. (6). With using it, the trajectories of four kinds can be obtained from Eq. (7),

Four distinctive sorts of trajectories in Eqs. (16)–(19) are the arcs, the horizontal and slanted parallel line segments, and the non-parallel line segments. All four kinds of trajectories were observed in the experiments, see Sec. 5.

In the case of varying angle, Eq. (16) can be refined for four combinations of d₁ and d₂ from (1, 1) to (2, 2) as follows,

As can be seen from Eqs. (20)-(23), the following relations between d_j and p_j are satisfied,

The derivatives show the change of trajectories for each center. The derivatives of Eqs. (16)-(19) look like follows,

In the case of the linear change of parameters (similar to At + B), one can obtain

The graphical examples of derivatives by Eqs. (30)-(33) are given in Fig. 3. The lines with arrowheads in Fig. 3 are shown for illustration purpose only, i.e., to indicate the radial and the tangential directions of change.

 

Fig. 3 Derivatives of the spectral trajectories in the orthogonal case. Varying α in (a) and (b), ρ in (c) and (d). The combinations of d₁ and d₂ are as follows, (1, 1) in (a) and (c); (1, 2) in (b) and (d). The squares in (b) and (d) are only shown for the graphical clearness of the illustration.

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3.2 Number of elements

The number of centers for the orthogonal case can be calculated basing on Eqs. (16)-(19). The modified Eqs. (18) and (19) with the isolated dependent terms explicitly describe the centers and the lines as follows,

Equation (34) describes the inclined line segments, while Eq. (35) describes the horizontal line segments only. In the sinusoidal case, each term consists of 3 integers p₁, p₂ and p₃ between −1 and + 1; this means the triple amount of centers for varying σ₁ and σ₃, as compared to varying α and ρ.

Once the stationary terms are isolated, the locations of centers can be found. For varying ρ and α, the centers are always located along a horizontal line (for d₁ = 1) or the sides of a square (for d₁ = 2) by Eqs. (16) and (17). For varying σ, the locations of centers are different: the horizontal/slanted line (d = 1), the rotated square or parallelogram (d₁ ≠ d₂), and 3 displaced squares/parallelograms (d = 2). In the cases d₁ = 2, the locations of centers include vertically displaced locations for d₁ = 1 and corresponding d₂. Illustrations of that can be found in [18].

Similarly to Eq. (14), the number of centers in the orthogonal sinusoidal case can be expressed as follows:

The value of d_c in (36) varies, depending on the combinations of the current varying parameter and numbers d_j. To show the number of elements for all these combinations, the matrix form can be used as follows

The Eq. (39) represents Eq. (38) in powers. All matrix elements of matrices M_tot and M_c are powers of 3; this exactly corresponds to Eq. (36). The matrix elements of M_l are the products of powers of 2 and 3. One can also find the ratio of N_tot and N_c,

The Eqs. (40) and (41) describe the number of all elements per center and the number of lines per center; for illustration refer to Fig. 3. The particular values of the ratio Eq. (41) are powers of 2 for g < 3 (as it is in this paper). Alternatively, the ratios (40), (41) can be written in the matrix form with the division (/) treated as an entry-by-entry matrix operation,

The matrices Eqs. (42) and (43) represent the number elements of any length and of non-zero length per center, resp. All the matrix elements of the above matrices are powers of either 2 or 3. The numerical values in Eq. (38) and consequently in Eqs. (42) and (43) were confirmed by direct counting the geometric elements in simulation.

4. Regular geometric structures in spectrum

In this section, the proposed technique is applied to find some regular structures in the spectrum. Both examples relate to the superposed gratings for the case of two pairs of two orthogonal gratings as in the previous section for d = 2. In this section, the only varying angle is implied, so as the functional notation “(t)” in expressions like “α(t)” will be omitted here.

A regular polygon is a polygon with equal sides and equal angles. Since the layout of this example is orthogonal, then among the regular polygons, the squares and octagons are searched in the neighborhood of the origin. For that purpose, geometric conditions like the colinearity of some points should be proven; the distances and angles between them should be tested as well.

The condition of collinearity of three points can be expressed through the zero area of the triangle determined by these points:

The condition of orthogonality of complex vectors z₁ and z₂ means that their dot product equals zero,

4.1. Square

Among all the spectral peaks depicted by Eq. (16), there are the peaks located along the sides of a square: at the edges and midpoints of sides. Such squares in the spectral domain can be observed in many cases.

Consider the trajectories within the visibility circle. A plausible suggestion is that a square in the neighborhood of the origin of the spectrum means a similar figure in the spatial domain. This indicates a certain similarity between the spatial and spectral domains, see Fig. 4.

 

Fig. 4 A square moiré patterns: (a) in the spatial domain and (b) in the spectral domain. The spectral peaks arranged into a square are located inside the visibility circle in (b).

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Among trajectories Eq. (16), the following trajectories may comprise a square, see Fig. 5.

 

Fig. 5 Eight spectral trajectories arranged along the sides of a square.

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This suggestion needs to be proven by finding the exact position of the endpoints of these trajectories. After that, the size and the orientation of a square can be found.

Among Eqs. (48)-(55), the four trajectories Eqs. (49), (51), (53), and (55) form the corners of the square; the four others the midpoints. Equations (48), (50), (52), and (54) are the particular cases of Eq. (16) with two integers equal to zero as p₁ = p₃ = 0, p₂ = -p₄ ≠ 0 in Eqs. (48), (52), and p₂ = p₄ = 0, p₁ = -p₃ ≠ 0 in Eqs. (50), (53). The remaining formulas Eqs. (48)-(55) are particular cases with non-zero integers as p₁ = p₂ = 1, p₃ = p₄ = −1 in Eq. (49), p₁ = p₄ = 1, p₂ = p₃ = −1 in Eq. (51), p₃ = p₄ = 1, p₁ = p₂ = −1 in Eq. (53), and p₂ = p₃ = 1, p₁ = p₄ = −1 in Eq. (55).

Consider, e.g., the right edge of the square. Substituting Eqs. (48), (49) and (55) into Eq. (45), the identically zero determinant can be obtained,

Thus, the above three of eight trajectories are always collinear for any parameters σ₁, σ₃, and ρ. The similar statements about zero determinants can be proven for three other triplets. Therefore, the trajectories Eqs. (48)–(55) lie at the corners and at the midpoints of a quadrilateral with parallel sides.

Then, the angle 90° at the corners should be proven. Consider, for example, the trajectories Eqs. (48) and (50). In this case, for Eq. (46) we have

From Eq. (47),

When σ₁ = σ₃, the dot product Eq. (59) equals zero, as required. It means a rectangle, not a parallelogram. The side of the rectangle is equal to the distance between two corners, say, Eqs. (49) and (51), as follows,

On the other hand, from Eqs. (49) and (55), the other side is

The sides a₁ and a₂ in Eqs. (60) and (61) are identical when σ₁ = σ₃ = 1. This condition means a square.

The tangent of the slant angle of the upper side of the square is (v₁ – v₂)/(u₁ - u₂), where u_j and v_j are defined after Eq. (45). From Eqs. (54) and (55) one can find that

In the case of ρ = 1, the side of the square from Eq. (60) is

In the case of ρ = 1, Eq. (62) is simplified

The squares in the spectral domain are shown in Fig. 6 for θ between 10° and 90°.

 

Fig. 6 Squares in the spectral domain and the trajectory of the corner of the square.

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The square always exists, except for the origin exactly; however generally, it alone may not produce a visible moiré. Vertices of the square follow the circles with the radius √2, one of them is drawn in Fig. 6; it is the trajectory Eq. (55).

4.2. Octagon

The next expected polygon is octagon. For sake of simplicity, in this subsection the case of σ₁ = σ₃ = ρ = 1 is only considered.

The candidate trajectories are shown in Fig. 7. Their equations are the following,

 

Fig. 7 Spectral trajectories forming the octagon.

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The sides can be found as the Euclidean distances between trajectories. In particular, for Eqs. (66), (70), and (71), the distances are

When α = π/4, d_eb = d_bg = $\sqrt{10-7\sqrt{2}}$. The interior angle of a regular polygon is equal to π - θ, where θ is the central angle. The central angle can be determined analytically through the difference of two arctangents Eqs. (66) and (69) as follows

Substituting α = π/4 obtained above into Eq. (76), one can find the central angle π/4. Therefore, the interior angle of the polygon is equal to π - π/4 = 3π/4, as required for the regular octagon. Similar conditions can be proved for other pairs and triplets of vertices in Eq. (65)-(72). Thus, all angles are equal, as well as all sides are. Therefore, the trajectories Eqs. (65)-(72) at π/4 describe a regular octagon.

It might be expected that similarly to the square, the octagon in spectrum means that something like a set of octagons covering the plane in the spatial domain. However, it is known that the plane can be filled with regular polygons of three kinds only, the triangles, the squares, or the hexagons; the octagon is not among them. However, the plane can be filled non-periodically with using slightly distorted octagons. See the illustration in Fig. 8 and the related notes in [1], where the non-periodical moiré patterns are described.

 

Fig. 8 (a) The repeated but non-periodic structure in the spatial domain. (b) Its spectrum with the octagon inside the visibility circle.

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5. Experiments

The physical experiments were made for four combinations of d₁ and d₂ with σ = 1, ρ between 0.71 and 1.4. The profile of gratings was rectangular (non-sinusoidal). The varying angle α changed from 0° to approx. 70°.

The gratings were printed on the paper and the transparent film, superposed over each other and photographed. This was repeated for the incremented angle up to an upper limit. Then, either ρ or one of d was changed and the procedure repeated.

The photographs of the superposed gratings were subjected to the Fourier transformation. The trajectories were obtained by summation of successive two-dimensional power spectra of photographs of superposed gratings.

5.1. Varying angle α

Examples of the experimental trajectories for varying α are shown in Figs. 9, 10, 11, and 12, where, the superposed spectra are only presented without intermediate stages of processing. The resulting examples of trajectories simulated by Eqs. (16)-(19) are presented in [18] and might be considered in parallel to Figs. 9–12.

 

Fig. 9 Experiment. Two one-dimensional gratings (d₁, d₂) = (1, 1). Varying angle 0 – 55°; ρ = 0.71, 1.0, 1.4 in (a), (b), (c), resp.

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Fig. 10 Experiment. One-dimensional grating and two-dimensional grid (d₁, d₂) = (1, 2). Varying angle 0 – 50°; ρ = 0.71, 1.0, 1.4 in (a), (b), (c), resp.

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Fig. 11 Experiment. Two-dimensional grid and one-dimensional grating (d₁, d₂) = (2, 1). Varying angle 0 – 70°; ρ = 0.71, 1.0, 1.4 in (a), (b), (c), resp.

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Fig. 12 Experiment. Two two-dimensional gratings (d₁, d₂) = (2, 2). Varying angle 0 – 60°; ρ = 0.71, 1.0, 1.4 in (a), (b), (c), resp.

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An apparent similarity can be found between the experimental Figs. 9–12 and the abovementioned simulation, although the experiments show the presence of higher spectral components. The difference between sinusoidal simulation and non-sinusoidal experiments will be explained in the subsection 5.3.

5.2. Varying size ρ

The full set of the experimental data is multidimensional and contains many various combinations of parameters. Initially, the data are ordered by the angle as in Figs. 9-12. However it is not impossible to rearrange the existing experimental data. For instance, the data sequences for the same angle but different ρ can be found. Such chunks of data can be assembled and treated as regular experiments with longer time interval between the successive measurements. As far as the time issues are not considered in this paper, this difference matters nothing, and all trajectories obtained by rearranging the existing data are equally valuable. The experimental trajectories obtained this way are shown in Fig. 13, each paired with corresponding sinusoidal simulation.

 

Fig. 13 Experimental ((a), (c), (e), (g) at the left side) and corresponding simulated trajectories ((b), (d), (f), (h) at the right side) for the varying ratio ρ from 0.7 to 1.4. The number of gratings by layers (d₁, d₂) and the angle are as follows, (a), (b): (1, 1) and 15°, (c), (d): (1, 2) and 20°, (e), (f): (2, 1) and 38°, (g), (h): (2, 2) and 42°.

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The trajectories for varying ρ, experimental at the left and simulated at the right side of Fig. 13, are evidentially similar.

5.3. Analysis of experimental data

The experimental data can be analyzed in details. For instance, in this subsection, Fig. 11(b) is compared with the simulation for all recognized spectral peaks. The comparison shows that the experimental trajectories of rectangular gratings can be treated as a linear combination of trajectories for ρ = 1 and ρ = 2. This means the experimental observation of the moiré effect in non-sinusoidal gratings with q ≤ 2. The simulated sinusoidal trajectories (Q ≤ 5, up to 25 centers) for the identical periods (ρ = 1) and for the period of grating twice shorter than grid (ρ = 2) are shown in Fig. 14. This illustration represents the sketches of separate spectral components of the non-sinusoidal gratings.

 

Fig. 14 Simulated sketches for grid and grating of the square profile with (d₁, d₂) = (2, 1), σ₁ = σ₃ = 1 with different ρ: (a) ρ = 1, (b) ρ = 2. The trajectories involved in experimental data are shown by bold lines. The corresponding active centers are shown by bold dots. The non-sinusoidal trajectories are drawn in gray color.

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At the same time, not all trajectories from 25 centers were experimentally observed. It seems that the centers involved in the experimentally observed trajectories (might be called as active centers), are located within the circle of the radius, approximately equal to the side of the square into which these Q x Q centers are inscribed. There are 13 such centers for ρ = 1 and 5 for ρ = 2, as shown in Fig. 14(a) and Fig. 14(b), respectively.

In Fig. 15, the sketches ρ = 1, 2 are drawn over the experimental trajectories Fig. 11(b). This provides the direct comparison of several spectral components with experiment.

 

Fig. 15 Experimental and simulated trajectories together.

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The comparison shows the clear match of sketches and the experimental data within the abovementioned circles (Fig. 14), except for a few points. Namely, a mismatch is observed in 3 points vs. 16 matched trajectories (as average 15 points each), i.e. about in 1.3% only. Such a good agreement of the details of the experimental spectra with the simulation indicates that the above values of ρ and q provide a suitable explanation of the experimentally observed spectral trajectories in non-sinusoidal gratings.

6. Discussion and conclusion

The equations of spectral peaks and spectral trajectories for case of N overlapped one-dimensional gratings are found in the closed form. The practically important case of two layers with orthogonal gratings, typical for the autostereoscopic 3D display, is considered in details. All formulas of this case are confirmed by computer simulation. The simulation program is based on two-dimensional Fourier transformation and the visibility circle concept [1].

Equation (1) and following are enough for describing the combinations of one-dimensional periodical structures. We put aside the amplitude itself, because important here is the presence of peaks and their locations.

In sub-section 4.1 it was proved that in the case σ₁ = σ₃ = 1, there always exists a square in the spectrum for any angle α. In sub-section 4.2 it was proved that in the case of the case of σ₁ = σ₃ = ρ = 1, the angle α = π/4 provides the regular octagon in the spectrum. The corresponding structure in the spatial domain is a non-periodic tiling of the plane by almost regular octagons.

At 45°, the inradius of the square in the spectral domain is $\sqrt{2-2\sqrt{2}}=0.7654$, while the circumradius of the octagon is $\sqrt{2}-1=0.4142$. Therefore, the square lies outside of the circle with radius 0.7, while the octagon inside the circle with the radius 0.5. This indicates that when the octagon appears, the square is outside the visibility circle and does not produce a visual effect. The relative size and position of the two figures are shown in Fig. 16; the visibility circle lies between them. Also, none of trajectories comprising the square is involved in the octagon; this can also be seen from Eqs. (48)-(55) and (65)-(72). From this point of view, the square and the octagon are independent structures, although not completely because their trajectories are subsets of the same generic Eq. (16).

 

Fig. 16 The square and octagon in the spectrum at α = 45°. The background trajectories are drawn for α from 0 to 50°.

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Figure 13 is based on the rearrangement of the experimental database; therefore, the number of experimental points per line segment is different in these series and varies from approx. 5 to approx. 10, depending on the particular combination of parameters. As a result, some experimental trajectories for varying ρ may look less continuous than trajectories in Figs. 9–12. However, with an increased number of points per segment, the trajectories for varying ρ should become similarly continuous, even if obtained by rearrangement. Trajectories for varying σ can be built in a similar way, basing on the data for varying angle with different parameter σ.

The reduction of the number of centers involved in generation of the observed trajectories (subsection 5.3), probably, reflects relationships on their amplitudes. The decay of Fourier coefficients about 1/n along each coordinate virtually makes similar decay along any direction; practically it looks like a circle. Formally speaking, the number of the active centers is equal to the number of points of norm less than or equal to n² in square lattice. The explanation of the experiment Fig. 11(b) additionally shows that practically, it is unnecessary to consider the infinite spectra. For practical purpose (like an estimation of the visual moiré effect in printed gratings), it is sufficient to consider the limited spectrum consisting of approx. 5 peaks per center (as in this paper).

In this paper the overall picture of spectral trajectories is presented. While the case of two layers of orthogonal gratings is considered in details, the general formulas allow considering any non-orthogonal layouts. It should be noted here, that the cases of 3 and 4 gratings are primarily considered in the spatial domain [3]; in our examples we tried to discover the instances in the spectral domain and to show their influence to the visual effect. The examples illustrate various aspects of trajectories, their transformations, and regular geometric structures in the spectrum. Using complex numbers makes analytical investigation convenient.

The paper proposes a technique, in which the moiré-free areas can be found graphically, and the presence of spectral peaks near the origin can be estimated with using the complex numbers. This was confirmed in many particular cases and after that the simulation program draws sketches of trajectories directly by Eqs. (16)-(19). This feature does not require the Fourier transformation at all, and therefore the estimation of the moiré effect can be obtained very fast. This technique seems to allow a quick and efficient look through various ranges of parameters. There is a good agreement between the theory, simulation and experiment. The authors hope that the proposed technique could be conveniently used in practical applications, especially in the minimization of the moiré effect in three-dimensional autostereoscopic displays.