Maximum and minimum amplitudes of the moir&#x00E9; patterns in one- and two-dimensional binary gratings in relation to the opening ratio

Vladimir Saveljev; Sung-Kyu Kim; Hyoung Lee; Hyun-Woo Kim; Byoungho Lee

doi:10.1364/OE.24.002905

1. Introduction

The moiré effect is a visual optical effect observed in regular superposed layers. To a considerable extent, the moiré effect is caused by the periodicity of layers, such as the highly regular pixel grid of a digital screen, a lenticular or barrier plate of a typical autostereoscopic three-dimensional (3D) display, etc. The moiré fringes form a patterned background which is effectively added to a displayed image, and the image quality degrades. To provide the high quality, the visibility of the moiré patterns should be estimated and the moiré effect should be minimized [1].

On the other hand, it is not impossible to make precise measurements [2,3] or to display images [4] using the moiré effect as the main physical principle. In similar cases, the moiré effect should be maximized.

The amplitude plays a key role for the visibility. The higher amplitude is consistent with the stronger visual effect, whereas the lower amplitude reduces it. The appearance of the moiré patterns depends on such geometric factors as the angle and the gap between gratings, the shape, layout and size of gratings, etc. Based on the previous research studies [5–7], we may consider the period and the opening ratio as the two key characteristics of gratings which mostly affect the moiré patterns in 3D displays. The opening ratio r is the ratio of the blank (transparent) interval g between the dark (opaque) lines of the grating to its period p; a time-domain equivalent of the opening ratio is the duty cycle of the time-dependent signals. In this paper we consider the connection between the opening ratio and the amplitude of the moiré patterns, especially the extrema.

Several approaches to the visibility of moiré patterns are known. For instance, I. Amidror in his excellent book [8] defines and effectively uses the concept of the visibility circle among many other very useful concepts. The visibility circle models the response of the human visual system in the spectral domain. The period of the moiré patterns is known since 1887, see [8,9]. Recently, some novel moiré topics were discovered, for instance, the wavelet transform very efficiently applied to the visualization and separation of fringes [10].

There are papers considering the contrast of the moiré patterns. The contrast is classically defined by A. Michelson [11]. The intensity of the moiré patterns in the superposed gratings of the rectangular profile (“binary” gratings) shows a trapezoidal shape [12]; therefore, the moiré period can be divided in four phases which are indicated by the letters A through D in Fig. 1.

Fig. 1 Intensity profile of the moiré patterns in binary gratings. The letters A through D show the phases.

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According to the paper [13], the maximum contrast in one-dimensional line gratings (1D gratings) is reached when the opening ratio is 50%. The research [12] estimates the moiré contrast from the Fourier coefficients, and the maximum is also reached at 1/2. The simulation [7] indicates the same value for 1D binary gratings. The mentioned research papers point to the same value 0.5. At the same time, a direct experimental confirmation of that value was needed, including the visual observation.

Another related topic is sharpening. The sharpening of the moiré fringes [14,15] means a particular case of the moiré profile with a reduced or eliminated flat top (phase C in Fig. 1) which corresponds to narrow patterns. This feature is useful for the improvement of the accuracy of the linear measurements. Although the amplitude of the narrow patterns with a triangular top is generally higher than in the general case, but strictly speaking, this is not sufficient to maximize the amplitude.

At the same time, the display devices involve some more complicated layouts than a simple combination of two line gratings (i.e., 1D case). For instance, one layer could be a two-dimensional (2D) pixel grid. Therefore, an investigation of the moiré amplitude in 2D gratings was necessary, because of the lack of the analytical research on the amplitude of the moiré patterns, especially in the 2D case. In this paper we developed a theory including the 2D case and present the corresponding experiments.

To estimate the amplitude, the minimum and maximum intensities I_min and I_max of the moiré profile should be known. These can be found by averaging across the period of gratings like in [16]. In the current paper, the peak-to-peak amplitude A is defined as the difference

A = I_{\max} - I_{\min}

The width of various bell-shaped curves can be described in terms of the full width at half maximum (FWHM). For example, the FWHM of the triangular pulse of the unit height is 1, for the Gaussian it is 2.36σ, while for the parabola vertically displaced by c, it is 2(c/2)^1/2.

For 2D functions (surfaces), however, the curve-based FWHM should be generalized, for example, as the radius of an “equivalent” circle crossing the surface at one half of the maximum. To find it, one may use the principle of equal areas. Implying two FWHMs for orthogonal directions to be axes of an ellipse, the equivalent radius of the circle can be obtained as a geometric mean of these FWHMs.

The paper is organized as follows. In Sec. 2, we obtain analytical expressions in a closed form for the maximum and minimum amplitude of the moiré patterns in 1D and 2D binary gratings depending on the opening ratio. Also, we derive the width of the distinctive curves in terms of the FWHM. In the experiments presented in Sec. 3, we measure the minimum and maximum intensities across the samples of various opening ratios arranged in a matrix, estimate the amplitude of the moiré patterns, and find its extrema. In Sec. 4 we discuss the visual observation and the equivalence of the periods and angles.

2. Theory

A grating modulates the intensity of the transmitted or reflected (scattered) light. Generally, the either reflectance or transmittance function can be used depending on the technology and the substrate material of the gratings. In this research, we use the gratings printed on the paper and therefore, the reflectance functions are applied.

Consider two parallel binary gratings of slightly different periods p₁ and p₂; besides the second grating is obtained by a uniform rescaling of the first grating, and

p_{2} = (1 - ε) p_{1}, ​ ​ ​ ε < < 1

The gratings can be one- or two-dimensional. The 1D layout involves two line gratings, see Fig. 2(a), while the 2D layout consists of two square grids, see Fig. 2(b).

Fig. 2 Moiré patterns in superposed 1D/2D gratings ε = 0.065; opening ratio is 0.5 in (a), and 0.625 in (b). Minimum and maximum are labeled as A and C.

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In the printed binary gratings, the intensity of the reflected light of the black ink in the opaque dark lines is I₀, while that of the white paper in the blank areas is I₁. We use the multiplicative model, in which the visual effect of superposition is proportional to the product of the reflectance functions of gratings.

The variation of the intensity arises due to different relative position of lines of superposed gratings in different fields of the gratings. The gratings at the extrema are schematically shown in Fig. 3.

Fig. 3 Reflectance functions of gratings and moiré intensity in phases A (minimum) and C (maximum).

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In the binary gratings, the intensity of the scattered light is proportional to the area with the I₁. The area can be estimated as the average value of the intensity across the period; the maximum and minimum can be found as the weighted sums of I₀ and I₁ with the corresponding relative (per period) areas as the weights. In the experiments, the intensity is averaged by the camera across more than one period of the gratings; this is an equivalent of the spatial averaging [16]. Consider four phases of the period.

Phase A. The minimum intensity, see Fig. 3(a). The blocked area (printed lines) is maximal, i.e., the lines of gratings do not overlap each other or overlap in the minimal extent. In this case, the blocked area per period is the sum of line widths. The corresponding minimum per period is presented as follows,

I_{\min} = I_{0} \min (w_{1} + w_{2}, p) + I_{1} (p - \min (w_{1} + w_{2}, p))

where w₁ and w₂ are the widths of the lines, while p = min(p₁, p₂) = p₂.

Equation (3) involves the minimum function to ensure the non-negative value I_min ≥ 0, which condition might not be satisfied when p < w₁ + w₂.

Phase B. The transition between the minimum and the maximum. The lines touch each other and begin to overlap; the blocked area is decreased while the intensity rises.

Phase C. The maximum intensity, see Fig. 3(b). The lines of gratings completely overlap each other. In this case, the open area is maximal and

I_{\max} = I_{0} \max (w_{1}, w_{2}) + I_{1} (p - \max (w_{1}, w_{2}))

Phase D. The transition between the maximum and the minimum. The area of the overlapped gratings increases; the intensity falls.

Consider the “normalized” gratings printed on an ideal medium with

\begin{array}{l} I_{0} = 0 \\ I_{1} = 1 \end{array}

Then, in 1D case shown in Fig. 2(a), we have from Eqs. (3) and (4),

I_{\min 1 D} = p - \min (w_{1} + w_{2}, p)

I_{\max 1 D} = p - \max (w_{1}, w_{2})

Normalizing Eqs. (6) and (7) by the period, and using the definitions p = w + g and r = g/p, one can re-express these equations in terms of the opening ratio, i.e. the ratio of the open area to the period. The normalized Eq. (6) is as follows,

I'_{\min 1 D} = {\begin{cases} 0, if p_{2} \leq w_{1} + w_{2} \\ r_{1} + r_{2} - 1 + ε F_{1} (r_{1}, r_{2}), ​ if p_{2} > w_{1} + w_{2} \end{cases}

where r₁ and r₂ are the opening ratios of the gratings, the parameter ε is defined by Eq. (2), F₁(r₁, r₂) is a linear function of r₁ and r₂ which also depends on the type of normalization: F₁(r₁, r₂) = - r₂ for the normalization by p₁ and F₁(r₁, r₂) = 1- r₁ by p₂. The variables r₁ and r₂ vary between 0 and 1; we will refer to this interval as a domain.

Equation (7) can be rewritten similarly,

I'_{\max 1 D} = {\begin{cases} r_{2} + ε F_{2} (r_{1}, r_{2}), if w_{2} \geq w_{1} \\ r_{1} + ε F_{3} (r_{1}, r_{2}), if w_{1} > w_{2} \end{cases}

where the functions F₂(r₁, r₂) and F₃(r₁, r₂) are the linear function of r₁ and r₂ similar to F₁; F₂ equals -r₂ and 0, while F₃(r₁, r₂) equals - 1 and −1-r₁ for two mentioned types of normalization.

In Eqs. (8) and (9), the terms of the second order by ε are neglected, while the terms of the first order are presented by the linear functions. The conditions in Eqs. (8) and (9) can be also normalized in a similar manner; then, instead of p₂ > w₁ + w₂ we should write r₁ + r₂ > 1 + εr₂, while instead of w₁ > w₂ we write r₂ > r₁ - ε + εr₂.

Taking the above into account, we can rewrite Eqs. (8) and (9) as follows

I'_{\min 1 D} = {\begin{cases} 0, if r_{1} + r_{2} \leq 1 + ε r_{2} \\ r_{1} + r_{2} - 1 + ε F_{1} (r_{1}, r_{2}), ​ if r_{1} + r_{2} > 1 + ε r_{2} \end{cases}

I'_{\max 1 D} = {\begin{cases} r_{2} + ε F_{2} (r_{1}, r_{2}), if r_{2} \leq r_{1} - ε + ε r_{2} \\ r_{1} + ε F_{3} (r_{1}, r_{2}), if r_{2} > r_{1} - ε + ε r_{2} \end{cases}

The meaning of Eq. (11) is to select the minimum of r₁ and r₂. The amplitude can be found from Eqs. (10) and (11) based on the definition Eq. (1),

A' = {\begin{cases} \min (r_{1}, r_{2}) + ε F_{4} (r_{1}, r_{2}), if r_{2} + r_{1} \leq 1 + ε r_{2} \\ \min (r_{1}, r_{2}) - (r_{1} + r_{2} - 1) + ε F_{5} (r_{1}, r_{2}), ​ if r_{2} + r_{1} > 1 + ε r_{2} \end{cases}

where F₄ and F₅ are the linear combinations of the functions F₁, F₂, and F₃. The meaning of Eq. (12) is to select minimum between r₁, r₂, 1- r₁, and 1- r₂ .

Besides, we do not have continuous functions in an experiment. Instead, we typically deal with a discrete set of parameters (opening ratios) with the increment δ. Then, if the increment is larger than ε, the impact of a linear function of r₁ and r₂ is generally larger than of another linear function multiplied by a small parameter. Practically it means that in Eqs. (10)-(12) we may omit the terms of the first order too, because the condition δ > ε is satisfied in our experiments. Then the formulas for the minimum, maximum and amplitude can be presented as follows,

I'_{\min 1 D} = {\begin{cases} 0, if r_{1} + r_{2} \leq 1 \\ r_{1} + r_{2} - 1 ​ if r_{1} + r_{2} > 1 \end{cases}

I'_{\max 1 D} = \min (r_{1}, r_{2})

A'_{1 D} = \min (r_{1}, r_{2}, 1 - r_{1}, 1 - r_{2})

Equations (13)-(15) describe the surfaces shown in sketches Fig. 4. Among them, the minimum and maximum surfaces consist of two intersecting planes, the amplitude surface of four. According to Eq. (15), the amplitude is proportional to the distance in the parameter space (r₁, r₂) from the nearest edge of the domain, and there is the single maximum in the middle of the domain.

Fig. 4 Sketches of theoretical surfaces of minimum, maximum and amplitude in 1D case drawn by Eqs. (13)-(15).

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The intensity levels I₀ and I₁ do not depend on the structure of gratings including their dimension; these levels are equal in both 1D and 2D cases, see Fig. 2. However in the 2D analogs of Eqs. (6) and (7) which are built basing on the multiplicative model, the weights of the summation are changed from p – min(w₁ + w₂,p) and p – max(w₁,w₂) to p ²– (min(w₁ + w₂,p)) ² and p ²– (max(w₁,w₂)) ², i.e.,

I'_{\min 2 D} = {\begin{cases} 0, r_{1} + r_{2} \leq 1 \\ {(r_{1} + r_{2} - 1 - ε r_{2})}^{2}, r_{1} + r_{2} > 1 \end{cases}

I'_{\max 2 D} = {(\min (r_{1}, r_{2}) - ε)}^{2}

As in 1D case, the terms with ε can be neglected. It yields

I'_{\min 2 D} = {\begin{cases} 0, r_{1} + r_{2} \leq 1 \\ {(r_{1} + r_{2} - 1)}^{2}, r_{1} + r_{2} > 1 \end{cases}

I'_{\max 2 D} = {(\min (r_{1}, r_{2}))}^{2}

Then the 2D equation for the amplitude is presented as follows,

A'_{2 D} = {\begin{cases} {(\min (r_{1}, r_{2}))}^{2}, r_{1} + r_{2} \leq 1 \\ - {(\max (r_{1}, r_{2}))}^{2} - 2 r_{1} r_{2} + 2 (r_{1} + r_{2}) - 1, r_{1} + r_{2} > 1 \end{cases}

The sketches of the surfaces defined by Eqs. (18)-(20) are schematically shown in Fig. 5.

Fig. 5 Sketches of theoretical surfaces of minimum, maximum and amplitude in 2D case drawn by Eqs. (18)-(20).

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While each amplitude surface defined by Eqs. (15) and (20) has the maximum in the middle of the domain, the minima (zero amplitude) are reached at the edges of the domain. Particularly, the minimum of the both cases is reached at the perimeter r_i = 0 or r_i = 1.

The surfaces can be characterized by cross-sections. Based on the symmetry of the problem, consider the vertical cross-sections along the diagonal r₁ = r₂. The expressions for the minimum and maximum along the diagonal obtained from Eqs. (13), (14), (18) and (19) look as follows,

I_{\min 1 D d} = {\begin{cases} 0, r \leq 1 / 2 \\ 2 r - 1, r > 1 / 2 \end{cases}

I_{\max 1 D d} = r

I_{\min 2 D d} = {\begin{cases} 0, r \leq 1 / 2 \\ {(2 r - 1)}^{2}, r > 1 / 2 \end{cases}

I_{\max 2 D d} = r^{2}

Consequently, the amplitudes along the diagonal are

A_{1 D d} = {\begin{cases} r, r \leq 1 / 2 \\ 1 - r, r > 1 / 2 \end{cases}

A_{2 D d} = {\begin{cases} r^{2}, r \leq 1 / 2 \\ - 3 r^{2} + 4 r - 1, r > 1 / 2 \end{cases}

Equations (25) and (26) represent the piece-wise polynomials of the first and the second order. The theoretical graphs of the minimum, maximum, and amplitude along the diagonal are shown in Fig. 6 according to Eqs. (21)-(26).

Fig. 6 The cross-sections of 1D and 2D theoretical surfaces along diagonal; (a) minimum and maximum, (b) amplitude.

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The maximum value of the piece-wise linear (triangular) function Eq. (25) is 0.5; it is reached at r = 0.5, see also Fig. 6(b),

r_{\max 1 D} = \frac{1}{2}

To find the location of the 2D maximum, consider the zeroes of parabolas. The first component of the piece-wise quadratic function Eq. (26) has zero at 0, while the second component has zeroes at 1/3 and 1. The first parabola has the minimum 0 at r = 0, while the maximum of the second parabola is 1/3; it lies between the roots, i.e.

r_{\max 2 D} = \frac{2}{3}

The minima of the diagonal cross-section are located at 0 and 1 in both 1D and 2D cases.

The FWHM of the theoretical amplitude curves shown in Fig. 6(b) is 0.5 in 1D case and 0.51 in 2D case. The FWHMs of the corresponding 1D and 2D surfaces are 0.5 and 0.36.

3. Experiment

In the experiments, the 1D and 2D samples (like in Fig. 2) were arranged in square matrices, the central parts of which are shown in Fig. 7.

Fig. 7 Central part of 1D and 2D matrices of samples (computer-generated image); r₁ and r₂ between 0.4 and 0.8.

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The opening ratio of one grating is changed along the abscissa of this matrix, another -along the ordinate with the increment 0.1 (the values 0 and 1 are not included). Such matrix includes all combinations of opening ratios between 0.1 and 0.9.

The matrices were printed on A0 size paper (118.9 cm x 84.1 cm), installed in a metal frame, and photographed using a digital camera. The size of the sample is 6.85 cm x 6.85 cm, the size of the matrix is 65 cm x 65 cm. The period of the first grating is 0.25 cm; the parameter ε of the second grating is 0.05. The distance 8 m ensures that the individual lines of the samples are not resolved by the camera; an example of the experimental photograph is shown in Fig. 8.

Fig. 8 Digital photographs of the 1D and 2D matrices. One region of measurement is indicated by dashed square in (a).

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As the digital photograph is acquired, its content can be processed, and the numerical characteristics of the moiré patterns can be obtained. A processed scan line in the middle of the matrix is shown in Fig. 9.

Fig. 9 Example of the scan line.

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In printed binary gratings, the value I₀ means the reflectance of the printed black lines, while I₁ means the intensity of the reflected light of the paper between the lines. Ideally, the intensities I₀ and I₁ do not vary across the image; the lightest area of any sample is I₀ while the darkest area is I₁. Practically, the values I₀ and I₁ are affected by the external illumination conditions. The pictures were taken under direct sunlight; the uniformity of the background was verified in the corners of each photograph and confirmed within 1-2%.

At the same time, the values I_min and I_max are different in the samples across the matrix, see Fig. 9. To find the moiré amplitude, I_min and I_max of each sample are needed. The processing is arranged in accordance to the structure of the matrix as follows. The identical regions smaller than samples are located in all samples of the experimental photograph; one of regions is shown by a small dashed square in the central sample Fig. 8(a). The minimum and the maximum intensities are detected within these regions. As a result, two matrices are obtained, the matrix of minimum values and the matrix of maximum values.

Based on the minimum and maximum matrices, the amplitude matrix can be calculated term by term by Eq. (1). The amplitudes of 1D and 2D cases are graphically shown in Fig. 10. The experimental amplitude matrices have the single maximum near the center of the matrix.

Fig. 10 Examples of experimental moiré amplitude maps of surfaces shown in Fig. 10 (1D and 2D cases).

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In order to compare the photographs to each other, as well as to compare the experiments to the theory, we normalize the measured intensities. For that purpose the lowest and highest intensities of the photograph are used. The lowest value was subtracted from all matrix elements of minimum and maximum matrices of the same photograph. After that, the matrix elements were normalized according to the highest value. As a result, the normalized intensities vary between 0 and 1 as in Eq. (5).

To characterize the experimental data numerically, the shape of the normalized surfaces was estimated by the surface fitting. In particular, the polynomial surfaces of the 2nd order were applied to approximate the entire surface. Based on the fit data, the location of the maximum of the surface is numerically calculated from the coefficients of the polynomial. The numerically estimated coordinates of the maxima in 1D case is 0.52, while in 2D cases 0.67. The RMS difference between the theory and normalized experiment is 5-7%. This shows a good match between the theory and experiment. The experimental and theoretical data along the diagonal are shown in Fig. 11.

Fig. 11 1D and 2D experimental points and piece-wise theoretical curves along diagonal.

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To find the FWHM of the curves, the abscissas are needed, where the function is equal to one half of the maximum. However, the required exact values could be absent in the experimental data. Therefore, the experimental FWHM was found basing on the linear approximation of the experimental data between the neighboring points where the value is lower than one half of the maximum and another is higher. The experimental FWHM along the diagonal is 0.49 and 0.47 in 1D and 2D cases; for the surfaces, the FWHMs based on the principle of equal areas are 0.53 and 0.46, resp.

4. Discussion

At different stages of this research, different layouts of gratings were used: the parallel gratings of different periods, as well as the identical gratings installed at a small angle. Nevertheless in both layouts, the visual effect appears to be similar. Moreover, if a certain relationship between the geometric parameters is satisfied, the period of the moiré patterns is exactly the same in both layouts. Such corresponding parameters can be referred to as equivalent to each other.

Correspondingly this section describes the visual observation of the moiré patterns of various opening ratios, and discusses the equivalence of periods and angles by their influence on the period of the moiré patterns.

4.1 Visual observation

The direct visual observation of the gratings was made additionally [17], where we observed 1D and 2D binary gratings under the room light. The opening ratios of gratings were 1/8, 3/8, 5/8, and 7/8. The size of samples is 1.7 cm x 8.7 cm. The samples are shown in Fig. 9.

In contrast to the main experiment (Sec. 3), the samples for the visual observation were made of gratings of the same periods installed at the angle 4°, see Fig. 12. The samples were arranged in a square matrix 4x4 as shown in Fig. 13.

Fig. 12 (a) 1D and (b) 2D samples for visual observation (opening ratio is 5/8).

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Fig. 13 Matrix for visual experiment (computer-generated image). The location of the origin is different from Figs. 7 and 8.

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The participants were asked to rate the moiré patterns visually using the 10-grade scale. Involved were 9 persons (male and female) aged between 27 and 59. Since the responses of participants were to a certain degree dissimilar, their responses were averaged. The averaged response configures a surface in (r₁, r₂) space. Similarly to individual responses, the averaged surface has a single maximum near the center. The maxima are located along the main diagonal. These features correspond to the theory and the main experiment (Sections 2 and 3).

The estimated visual surface has the maximum at the opening ratio 0.36 in 1D case and at 0.52 in 2D case, see Fig. 14. Although the locations of maxima of the visual observation are different from the theory, the trend is the same, i.e., the difference in the coordinates of maxima between 1D and 2D cases is 0.16, similarly to the main experiment and to the theory.

Fig. 14 Estimated average visual amplitude of the moiré patterns for (a) 1D and (b) 2D cases.

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4.2 Equivalence

For any given opening ratio of gratings, the same visual effect with a low spatial frequency can be observed either in two parallel gratings of close periods, or in the identical gratings installed at a small angle. The only difference is in the initial orientation of gratings (vertical or near horizontal). It means that for its influence on the period, the ratio of periods is an equivalent of a small angle. The analytical representation of such equivalence is found below.

The moiré patterns in two different layouts of gratings are shown in Fig. 15; despite this difference, the period of the moiré patterns is the same in both layouts.

Fig. 15 Moiré patterns in superposed gratings: (a) parallel gratings of different periods, (b) non-parallel gratings of identical periods.

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The equivalence can be verified analytically basing on the spectral theory [18], according to which, the general equation of spectral peaks in the complex plane is

T = \sum_{n = 1}^{N} p_{n} k_{n} e^{i α_{n}}

where N is the number of gratings, while other variables describe the n-th grating (n = 1, …, N) as follows, k_n is the basic complex wavenumber, α_n is the rotation angle, while p_n is an integer number between the limits -q_n and + q_n. Equation (29) represents the summation by gratings; it is not a Fourier series.

The period of the moiré patterns λ_m is the inverse wavenumber which is the modulus of the wavevector. The analytical expression for the wavenumber can be obtained from the general Eq. (29). Particularly, for non-identical gratings installed at an angle with a gap, the moiré period is

λ_{m} = \frac{λ}{\sqrt{ρ^{2} + s^{2} - 2 s ρ \cos α}}

where λ is the period of one grating, ρ is the ratio of periods, α is the angle between them, s = 1 + d/z is a geometric parameter of the layout, z is the distance to an observer, while d is the gap between the gratings. The equation for the simpler coplanar case looks as follows,

λ_{m} = \frac{λ}{\sqrt{1 + ρ^{2} - 2 ρ \cos α}}

This well-known formula (see [8] and references therein) was firstly developed in 1887 by Righi, see also the references in [9]. Similar equations in slightly modified forms can be found in [19] and [20]. For small angles, Eq. (31) becomes

λ_{m} = \frac{λ}{\sqrt{{(1 - ρ)}^{2} + ρ α^{2}}}

The influence of the expressions at the left and right sides of the summation sign under the square root function in Eq. (32) on the period of the moiré patterns is identical. Therefore, this formula associates the angle with the rational function of the size ratio; by their influence on the period, these variables are equivalent and can mutually replace each other as follows,

{(1 - ρ)}^{2} / ρ \leftrightarrow α^{2}

Specifically, in Fig. 15(a), the parameter ε is 0.065; according to Eq. (33), the equivalent angle is 4°, see Fig. 15(b) where this angle is actually used.

Since the results of the visual observation of moiré patterns by several participants (Sec. 4.1) are close to the main experiment, the visual observation confirms the equivalence. In accordance to Eq. (33), the observed moiré patterns of both layouts have the same period and consequently, the same number of the moiré patterns per sample; compare Figs. 12(a) and 13.

5. Conclusion

In the current paper, the amplitude of the moiré patterns in binary gratings is analyzed theoretically and measured experimentally in gratings with slightly different periods depending on the opening ratio.

The developed theory states that in printed gratings, the maximum amplitude of the moiré patterns of 1D case is 0.5 of the brightest level (white paper); and it is 0.33 in 2D case. The maximum amplitude of the moiré patterns is obtained at the opening ratio 0.52 in 1D case (line gratings) and 0.67 in 2D case (square gratings). The amplitude is low (i.e., near zero) in gratings with narrow openings or narrow lines. The minimum amplitude is obtained at the perimeter of the domain (r is 0 or 1) in both cases.

The experimental measurements confirm the maximum close to the above theoretical values. This also corresponds to the 1D case described previously by independent authors in [7], and [12–14].

The similar numerical values of all the above were also obtained in the visual observation of the moiré patterns by several participants, although it was made in the different layout of gratings, the identical gratings installed at an angle vs. the parallel gratings of different periods. Figure 12 demonstrates that the moiré period is the same in these two different layouts. The visual observation provides an additional confirmation of the equivalence of the angles and the size ratios (Sec. 4).

The theoretical FWHM along diagonal is 0.5 and 0.49 in 1D and 2D cases; the corresponding experimental values are 0.49 and 0.47. The FWHM obtained basing on the principle of equal areas in 1D and 2D cases are 0.5 and 0.36 from the theory, and 0.53 and 0.46 from the experiment. It means that the FWHM along diagonal is almost the same in both 1D and 2D cases; while the average FWHM of surfaces in 1D case is 20-30% wider than that in 2D case.

In the further research, we intend improving the accuracy of measurements, with using the smaller increment of the discrete opening ratios. This should refine the shape of the experimental surfaces, and particularly, the FWHM. Also, improved visual experiments with more participants should be performed.

The results may be efficiently applied firstly, to the minimization of the moiré patterns, for instance, in autostereoscopic 3D displays with 1D and 2D parallax (the horizontal parallax only and the full parallax), i.e., to multiview and integral imaging displays; and secondly, to other applications where the maximal amplitude of the moiré patterns is needed including the autostereoscopic moiré displays and the measurements.

Funding Information

This work was supported by 'The Cross-Ministry Giga KOREA Project' grant from the Ministry of Science, ICT and Future Planning, Korea.

References and links

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Maximum and minimum amplitudes of the moiré patterns in one- and two-dimensional binary gratings in relation to the opening ratio

Abstract

1. Introduction

2. Theory

3. Experiment

4. Discussion

4.1 Visual observation

4.2 Equivalence

5. Conclusion

Funding Information

References and links

Cited By

Figures (15)

Equations (33)

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