Diffraction Structures Formed by Two Crossed Superimposed Diffraction Gratings

The diffraction patterns generated by surface relief diffraction structures that consisted of two identical crossed, superimposed gratings formed in the As2S3 thin films by electron beam recording and subsequent chemical etching were studied. The angle between the gratings with grating periods equal to 2 μm was varied between 2° and 90°. Additional diffraction gratings formed by a set of the intersection nodes of the basic grating lines were considered to simulate the diffraction patterns. The circular symmetric diffraction structure composed of four crossed superimposed gratings with equal periods has been considered as a combination of pairs of crossed, superimposed gratings. A good agreement between the calculated and experimental diffraction patterns is observed.


INTRODUCTION
The diffraction structures formed by superimposing several diffraction elements allow a wide application of the phenomenon of light diffraction in optoelectronic devices. The simplest diffraction structure consists of two superimposed rectilinear diffraction gratings with uniform periods. A pair of the crossed diffraction gratings was described for the first time in 1874 by J.W. Rayleigh in an extensive paper dedicated to the production and theory of diffraction gratings [1]. It is shown in this work that two superimposed single rectilinear diffraction gratings with equal periods directed at a small angle to each other create a moiré pattern, which presents a regular alternation of parallel bright and dark bands. The data presented in this paper became the basis for using a moiré pattern for the optic tests of objects.
In the initial investigations, two diffraction gratings, closely spaced above one another, could move with respect to each other. According to [2], these crossed gratings were last considered in the mid-1950s in monograph [3] dedicated to the analysis of the diffraction grating quality with the help of a moiré pattern. Later on, the crossed gratings that present an integrated diffraction structure consisting of some superimposed diffraction gratings were studied. The influence of the displacement of two separate reliefphase diffraction gratings with a rectangular profile with respect to each other on the diffraction properties of this structure was studied in [2]. The grating structures of two crossed gratings with equal periods Λ with relatively high values of approximately 10 and 20 μm were composed with this aim. A combination of a long grating period and a small angle between the gratings (from 1 to 10 mrad) makes it possible to obtain long sections of structures (up to 10 mm) along which the distance between the lines of different gratings can be smoothly varied. It is shown that, at a relative displacement of the lines of different gratings equal to Λ/2, the generalized diffraction grating with a period of Λ/2 produces the diffracted beams with a dominant intensity.
The paper [4] presents a theoretical investigation of the diffraction structures composed of two superimposed rectilinear diffraction gratings for three possible variants: (1) different periods and identical orientations, (2) identical periods and different orientations, and (3) different periods and orientations. From the general moiré period expression (variant 3), the period of the moiré pattern for the first two variants has been derived. In the paper [5], the analysis of structures consisting of two crossed diffraction gratings with equal grating periods was performed with the help of the Fourier analysis. Furthermore, each grating was considered as a plane amplitude grating with a rectangular profile of the spatial transmission distribution. The corresponding moiré patterns (terminology of the authors) reflect the distribution of the diffraction spots in the diffraction patterns produced by a pair of crossed gratings. A theoretical method developed by the authors in [6] to study gratings was applied to the relief structure consisting of two orthogonally directed diffraction gratings. The intensities of the first-order diffracted beams were calculated. The theoretical evaluation showed the distinctive features of the optical properties of the relief surface with a two-dimensional periodicity. Thus, according to the received spectral curves, in the spectral range 0.5 to 0.7 μm, the sample of gold with this surface (the period of each grating is 0.5 μm) absorbs much more light energy than the sample with a smooth surface. It was also shown theoretically that this relief surface can perform the function of an antireflective coating.
The double exposure of the photoresist at a consecutive holographic recording of two superimposed crossed diffraction gratings results in the formation of a two-dimensional diffraction structure. The summation of the photoresist exposure in the line intersection nodes of the crossed gratings leads to a significant variation in the photoresist dissolution rate in these regions. This fact is used in one of the methods to form two-dimensional photonic crystals [7][8][9][10][11]. If the sample is rotated 90° or 60° before recording the second grating, then a structure of a square or hexagonal twodimensional photonic crystal, respectively, can be formed using the subsequent etching of the sample. In the paper [10], the distribution of the total dose of the light illumination in a light-sensitive material at the double and ternary exposure of it was simulated. In the latter case, the relief structures of the hexagonal photonic crystal formed in the positive photoresist were also simulated. The experimentally produced relief structures demonstrated a good agreement with the simulated ones.
Experience in the application of amorphous chalcogenide films to form diffraction structures with the help of both the direct optic and electron-beam recording and photo-and electron-beam lithography has been accumulated at present. In this case, the formation of the relief diffraction elements in the chalcogenide films can be performed with the help of both positive and negative chemical etching [12,13]. In particular, in the As-containing films, interference lithography was used to form the relief structures consisting of two orthogonally directed crossed diffraction gratings with a period of 370 nm [13]. The hexagonal two-dimensional photonic crystals with a grating constant of 550 nm were formed in the As 2 S 3 films using the method of a double exposure and a negative chemical etching [11]. The ordered deep dimples in the unirradiated film areas were formed in this case.
It is shown in [14] that nanodimensional structures can be formed with the help of electron-beam lithography in the chalcogenide films. For example, the diffraction gratings with a period of 30 nm containing deep grooves with width about 7 nm were formed in the arsenic sulfide films. The ability of chalcogenide films to accumulate an electron irradiation dose allowed the formation of one-dimensional relief diffraction structures with a complex profile in the As 38 S 18 Se 44 films with the help of a unified etching procedure [15].
We used the arsenic sulfide films for the electronbeam recording of the superimposed diffraction gratings. Earlier [16], diffraction structures consisting of two or three unidirectional superimposed diffraction gratings with the different values of the grating period were studied. This work is dedicated to the diffraction structures consisting of two crossed, superimposed gratings formed in the arsenic sulfide films with the help of the electron-beam recording and chemical etching of the samples. The diffraction patterns produced by two identical crossed diffraction gratings with different relative orientation angles were studied.

EXPERIMENTAL
Arsenic sulfide films were prepared by thermal evaporation onto glass substrates in vacuum. Film thickness was about 1.4 μm. For the leakage of electric charge, the films were coated with a semitransparent layer of aluminum. Electron-beam recording was performed with the help of a BS 300 (Tesla) scanning electron microscope at an accelerating voltage of 27 kV using its own raster scan system. The linear dose of electron irradiation was about 0.7 mC/cm 2 . The successive exposure method was used to record the structures consisting of two (N = 2) superimposed diffraction gratings DG1 and DG2 with an equal grating period Λ of 2 μm and different orientation. The angle between the directions of the grating vectors (β) was varied in the range 2°-90°, and it was set before the second grating recording by the rotating the sample.
The relief surface of the recorded structures was formed with the help of the chemical etching of the samples in the KOH aqueous solution. The upper layer of aluminum was previously removed in the diluted hydrochloric acid.
The first-order diffraction efficiency of gratings was determined in the transmission at the perpendicular incidence of the laser beam (λ =0.633 μm). Its value was calculated as an intensity ratio between the diffracted beam and the beam passing near the diffraction structure.
The surface of the relief diffraction structures was studied using an atomic-force microscope (AFM). The diffraction patterns of the diffraction structures were produced at the perpendicular incidence of the laser beam (λ = 0.633 μm). Figure 1 presents the AFM surface images of the diffraction structures consisting of two superimposed gratings (with a period of 2 μm) with relative orientation angles β of (a) 90°, (b) 7°, and (c) 2°. The dark lines are for the grooves formed in the irradiated areas of the As 2 S 3 film due to the chemical etching. In the case of orthogonally directed gratings (Fig. 1a), an ordered structure typical for two-dimensional crystals is formed. In this case, the line intersection nodes of the superimposed gratings are at the vertices of squares with a side equal to the diffraction grating period. The grooves in the line intersection nodes are widened due to the electron reirradiation of these regions when recording the second grating.

RESULTS AND DISCUSSION
The surface of the relief structures formed at small angles between two superimposed gratings (Figs. 1b, 1c) greatly differs from that shown in Fig. 1a. In this case, it is hard to discern two gratings in the grating structure or to localize the line intersection nodes. These diffraction structures consist of the alternate areas with various overlapping of lines of different gratings. The periodic variation in the line overlapping level causes the formation of a moiré pattern on the relief structure surface. When the periods of the intersecting gratings are equal (Λ 1 = Λ 2 = Λ) the moiré bands are perpendicular to the intersection angle bisecting line and the moiré period Λ moire , according to the Rayleigh formula [1], is: (1) Figure 2 presents the images of the structure area where the lines of different gratings are the most separated. When angle β decrease, this area widens in the direction of its bisecting line. In the fragments presented in Fig. 2 this area, for (a) β = 7° and (b) β = 4°h as the form of a comb breakdown of a single grating. For (c) β = 3° and, especially, for β = 2° (a lower section in Fig. 2d), this area has the form of a single grating with the twice reduced grating period (Λ/2), which corresponds to the data in [2].
The structure consisting of two crossed, superimposed gratings with equal periods is drawn schematically in Fig. 3. The grating structure under consider-moire cos(β 2) . sin β ation has a two-dimensional periodicity with a rhombic unit cell.
The line intersection nodes of the grating structure form a matrix of the spatially ordered dot elements. Differently directed lines can be drawn through the intersection nodes. The equidistant parallel nodal lines form a diffraction grating. The diffraction gratings with different periods and different orientations are formed in the matrix of the intersection nodes as is shown in Fig. 3 for some variants.
The nodal lines that form additional gratings pass along the long and short diagonals of the parallelograms composed of the rhombic cells. The distance between the neighboring line intersection nodes along the line of every grating is equal to the rhombus side length a = Λ/sinβ. The number of these intercepts (indices m and n) determines the side lengths of the composite parallelograms. It should be noted that m and n are combined in such a way that the diagonals of the composite parallelograms are not composite, i.e., they do not contain any line intersection nodes. For this purpose, even one index must be odd. Besides, m ≠ n except for the rhombic unit cell for which m = n = 1.
The periods of gratings formed by the nodal lines are determined by the following relationships: The superscript S corresponds to the direction of the nodal grating line along the short diagonal of the parallelogram, and the superscript L corresponds to the direction along the long diagonal. From these relationships, it follows that the grating periods When the lines of the diffraction gratings are formed by the rows of separated dots, the intensity of the diffracted light beam decreases with the growth in the distance between the dots. Thus, it is to be expected that the additional diffracted beams from the gratings formed by the nodal lines with small internode distances are the most intensive. Precisely these lines of diffraction gratings are shown schematically in First of all, the closeness of the intersection nodes occurs when they are placed at the vertices of the rhombic unit cells (m = n= 1). In this case, the DG3 and DG4 gratings are formed whose nodal lines are directed along the diagonals of the rhombic cell. These gratings are oriented mutually perpendicular at any angle β, and thus determine the symmetry axis of the diffraction pattern. The periods of these additional diffraction gratings can be also determined by the following relationships: Relationship (5) is identical to relationship (1), i.e., the period is just a period of the observed moiré pattern.
Next in order of significance is the variant of the position of the intersection nodes at the vertexes of the parallelograms composed of two rhombic unit cells. First, these are gratings with the same periods of DG5 and DG6 composed of the lines that pass along the short diagonals of the mentioned parallelograms. Then, there are similar gratings DG7 and DG8 composed of the lines that pass along the long diagonals. In both cases, the gratings have equal periods that, from general relationships (2) and (3), can be presented as: The presented relationships show that, at small angles β for which cosβ ≈ 1, the values of , and weakly depend on β, and they are approximately equal Λ to Λ, Λ/2, and Λ/3, respectively. In this case, the firstorder diffraction angles corresponding to the additional diffraction gratings almost coincide with the angles of the first, second, and third order diffraction, which correspond to the basic diffraction gratings.
The lines that pass along the diagonals of the rhombic cells can be convenient for use as basic ones to determine the orientation of the additional gratings formed in the matrix of the intersection nodes. Considering the structures with a small angle β, the lines that pass along the long diagonals of the rhombic cells appear to be preferable.
The angles between the nodal line of the additional grating and the long diagonal of the rhombus are determined from relationships (8) (9) in the case of the lines that pass along the long and short diagonals of the composed parallelograms, respectively. Here, as earlier, m and n are the number of the internode distances a, which determines the composed parallelogram lengths. It is interesting to note that the orientation of each additional grating does not depend on the period of the basic grating. That is, a set of the orientations of the propagation planes of the additional diffracted beams depends only on the angle of the relative orientation of the basic gratings. At the perpendicular incidence of the laser beam, the diffraction angles of the additional diffracted beams are determined from the relationships (10) (11) where i is the diffraction order. The combination of the diffraction angles and the corresponding angles of the propagation plane orientation determines the totality of the directions of the diffracted beams. It allows the simulation of the diffraction pattern formed on the screen by the multibeam diffraction of light, including the light diffraction from the additional diffraction gratings.
The diffraction pattern calculated for the case of the interaction between the perpendicularly incident beam (0.633 μm) and the orthogonally directed (β = 90°) superimposed diffraction gratings with periods of 4 μm is shown as an example in Fig. 4.
From now on, not a complete diffraction pattern but its fragment is meant by a diffraction pattern. The fragment presented in Fig. 4 is restricted to fourth order the diffraction maxima from the basic gratings. The corresponding diffraction spots are situated along the horizontal and vertical axes of the diffraction pattern. To calculate the parameters of the additional gratings, the indices m and n took the values 1, 2, and 3. The calculated diffraction pattern includes the diffraction spots that correspond to the diffraction max- The calculated and observed diffraction patterns of two superimposed gratings with periods of 2 μm and relative orientation angle of 25° are compared in Fig. 5.
In the calculated diffraction pattern near the firstorder diffraction spots, the periods of the basic and additional gratings generating the corresponding diffracted beams are shown. There is a good agreement between the calculated and observed diffraction patterns. It should be noted that the diffraction spots of the gratings formed by the lines directed along the diagonals of the composed parallelograms, which correspond to the combinations of the index values 1-3 (3-1) and 2-3 (3-2), were hardly visible, and they are not reflected in the photo of the diffraction pattern. Figure 6 shows the diffraction patterns of the pairs of gratings with small angles between gratings.
The diffraction spots similar to those mentioned at β = 25° were observed at β values decreased down to 3°. At β = 2° (Fig. 6d), the diffraction spots of the first-order diffraction from different gratings, due to a high overlapping, coalescence into one diffraction spot. However, at the second order diffraction, one can recognize two diffraction spots corresponding to each basic diffraction grating. The existence of distinct additional diffraction spots is indicative of a twodimensional periodicity of the position of the line intersection nodes in the formed structures, including the cases at the small angles β.
The results showed that, as expected, the additional diffracted beams with a high intensity are generated by the additional gratings formed by the nodal lines with short internode spacings. In these cases, the indices m and n take on the values 1 and 2. Six additional gratings are formed that can be considered in pairs. The nodal lines and the corresponding periods of these gratings are shown schematically in Fig. 3. It should be noted that the decrease in the basic grating period is accompanied by the reduction in the internode spacings. As a result, contrary to the structure under consideration with Λ = 2 μm, a notable contribution to the diffraction pattern can be also observed from the additional gratings for which the values of the indices m and n are greater than 2.
The values of the periods of the additional gratings DG3-DR8, as well as the corresponding values of the first order diffraction angles calculated for the angles β in the relative orientation of the gratings β in the range from 2° to 90°, are presented in Table 1. For the pairs of gratings DG5-DG6 and DG7-DG8, the calculated values of the relative orientation angles of and , respectively, are also presented. It should be added that the basic diffraction gratings with a period of 2 μm produce the diffracted beams with the first and second order diffraction angles of 18.45° and 30.26°, respectively. According to  occurs at β ≤ 10°. At the same time, the period and the corresponding diffraction angle alter appreciably with a decrease in the angle β. In particular, at β = 3°, the diffraction angle is approximately only 1°. It may be noted that, at β ≤ 10°, the relative orientation angle of the gratings DG7 and DG8 becomes less than 3°a nd the first-order diffracted beams from these gratings can hardly be distinguished even at β = 4°. The gratings DG5 and DG6 are characterized by a strong dependence of the relative orientation angle on β. It should be noted that, at a basic grating period of 2 μm within the whole range of the angle β variation, the additional grating period ( ) is of a submicron size, and its value became close to the incident laser beam wavelength of 0.633 μm when β < 30°. This results in relatively large diffraction angle of approximately 71°. Therefore, the corresponding diffraction spots are not Λ reflected in the calculated diffraction pattern at β = 25° (Fig. 5), though they are present in the calculated diffraction pattern at β = 90° (Fig. 4). Relationships (2) and (3) show that the periods of the additional gratings linearly decrease with the decrease of the basic grating period. Thus, as Table 1 shows, at Λ < 2 μm, the periods , are of a submicron size except for some cases. For instance, at Λ = 1 μm, the first-order diffraction maxima will be observed only from the grating with the period and only at large angles β. Figure 7 shows the central regions (the first-order diffraction spots) of the diffraction patterns from two superimposed gratings with (a) β = 90° and (b) β = 60°a t the perpendicular incidence of the laser beam as well as (c) from the symmetric structure consisting of three (N = 3) superimposed gratings.
At β = 60°, the value of the additional grating period coincides with the value of the basic grating period Λ =2 μm. Furthermore, the directions of the corresponding additional first-order diffracted beams L mn Λ L 11 Λ 2γ Fig. 7. Diffraction patterns of two crossed, superimposed gratings with relative orientation angles of (a) 90° and (b) 60° as well as of (c) three crossed, superimposed gratings.
coincide with the directions of the diffracted beams from one of three (N = 3) superimposed gratings. In consequence, the central regions of the diffraction patterns of two and three superimposed gratings are identical, as it can be seen in Figs. 7b and 7c. The diffraction efficiency of the superimposed gratings considered in this paper is about half as high as the diffraction efficiency of a single grating of approximately 20%. Within the whole range of β variation (2°-90°), the first-order diffraction efficiency of the superimposed gratings (each of the two) varied from 8 to 9%. A weak dependence of the diffraction efficiency on the angle β can be presumably explained by the fact that with the decrease in β the elongation in the line intersection regions is accompanied with the increasing spacing between them.
With the formation of diffraction structures consisting of more than two superimposed diffraction gratings, the nodes of intersection of more than two lines, for instance, three or four, can appear. Regardless of the probability that more than two lines can intersect in one node, the nodes of the intersection of two lines are obviously formed, and the corresponding matrix of nodes can be considered as a basic one. (The structure that consists of three crossed superimposed gratings where all the intersection nodes can be the nodes of the intersection of three lines is a special case). From this point of view, the structure consisting of N superimposed diffraction gratings can be considered in the form of a complex of the pairs of the crossed superimposed diffraction gratings. The number of the pair combinations of the basic gratings is defined by the following expression: (12) Under this approach, a set of additional diffraction gratings formed in the total matrix of the intersection nodes of two lines can be mainly presented as a complex of the additional diffraction gratings formed by the nodal lines in each pair of the superimposed gratings. On this basis, one can determine the propagation directions of the additional diffracted beams from the group of the N superimposed gratings with a different orientation.
On the basis of these assumptions, we calculated the parameters (periods and orientation) of the additional diffraction gratings formed in the matrix of the intersection nodes of two lines for the diffraction structure with a circular symmetry consisting of four crossed superimposed gratings with equal periods of 2 μm. According to (12), six pairs of the basic gratings are formed at N = 4. For four pairs angle β = 45° and for two pairs β = 90°.
In Fig. 8, the fragments of the experimental (upper part) and calculated (lower part) diffraction patterns generated by this diffraction structure are brought together for comparison. It should be noted that the orientation of the additional grating in the diffraction structure under consideration can coincide with the orientation of the basic grating. An additional diffraction grating with a period of spatial beating is formed by the pair of the unidirectional diffraction gratings with different periods. The corresponding diffraction spots were also taken into account in the calculated diffraction pattern.
The brightest diffraction spots in Fig. 8 correspond to the first-order diffraction beams from four basic gratings. As is obvious, the calculated diffraction pattern is in good agreement with the observed one. This supports the assumption that the additional gratings are mainly form by the intersection nodes of two lines and that the structure composed of N > 2 superimposed gratings can be considered as a combination of the pairs of the superimposed gratings. It should be noted that some weak diffraction spots which are not produced by any pair of diffraction gratings can be also seen in the observed diffraction pattern. Apparently, the origin of the corresponding diffracted beams is due to the additional gratings formed by the nodal lines typical for only the combined matrix of the intersection nodes. It should be noted that the contribution of these diffracted beams to the diffraction pattern is negligible.

CONCLUSIONS
The relief structures consisting of two crossed superimposed diffraction gratings with periods of 2 μm and the relative orientation angles varying from 2 to 90° were formed in the arsenic sulfide films with the help of electron-beam recording and subsequent chemical etching. The comparison of the calculated and observed diffraction patterns showed that, in the case of the interaction of the incident laser beam with the diffraction structures considered in the article, the additional diffracted beams are caused by the light diffraction from diffraction gratings formed by the nodal lines in the matrix of the intersection nodes of two lines. It was found that a significant contribution to the additional light diffraction is made by the gratings formed by the nodal lines with short internode spacings.
It was shown that the diffraction structure consisting of more than two differently directed superimposed gratings can be considered as a combination of the pairs of crossed gratings. A good agreement between the calculated and observed diffraction patterns is noted in this case. Thus, it is confirmed that, in the diffraction structure consisting of a group of differently directed superimposed diffraction gratings, the main contribution to the formation of additional gratings is made by the intersection nodes of two lines.