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Abstract
A simple method for generating 2D binary amplitude structure with additive superimposition of mutually orthogonal 1D amplitude gratings is proposed. Its implementation requires software generated three binary amplitude gratings, i.e., the crossed Ronchi, checker board and 1D Ronchi gratings with aspect ratio equal to 0.5. Their computer processing involves only two steps. First the checker grating is multiplied by a high frequency 1D grating. Next the product is added to the crossed grating. In result 3-level transmittance (0, 0.5, 1) hybrid diffraction structure is obtained. The intermediate level results from the use of a dense 1D grating. The zero diffraction order, well separated from the rest of the spectrum, consists of crossed spectra of additively superimposed 1D Ronchi gratings. Detailed heuristic explanation of the process aided by spectrum domain analyses is presented. Additionally, simulations and experiments conducted in the Fresnel diffraction field exemplify the invented structure properties in comparison with the multiplicative superimposition crossed Ronchi grating. Up to authors’ best knowledge the Fresnel field (self-imaging phenomenon or Talbot effect) properties of 2D periodic structure with additive superimposition of component 1D gratings have not been published in the literature.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Among diffraction gratings binary amplitude structures are probably the oldest ones. In spite of spectacular progress in the technology of phase gratings providing very high diffraction efficiency both in transmission and reflection, binary amplitude structures are still of interest because of their utmost simplicity. For example, considering scientific and engineering instrumentation several low cost grating interferometers have been proposed [1–9]. Even more pronounced importance relates to a possibility of conducting fast and simple experiments to corroborate theoretical studies in various fields of optics. They include diffraction [10,11] and the moiré fringe technique [12] implementations in such applications as strain and stress analysis, displacement measurements, optical alignment techniques, moiré topography, interferometry, color printing [13] and 3D displays [14].
Simple crossed Ronchi grating is the most commonly used two-dimensional binary amplitude structure. It is formed by a multiplicative superposition of two orthogonal 1D binary gratings. The product of two functions leads to a convolution operation in the spectrum domain. The crossed Ronchi grating has a rich spectrum in the form of a 2D array of diffraction spots. In many applications this abundance of diffraction orders necessitates its reduction either by filtration in a coherent optical system [12,15] or by digital processing of the output interference pattern [16–18]. In this way the measurand carrying fringes can be separated. In a natural way the following question arises: is it possible to construct a 2D diffraction structure with reduced spectrum (reduced number of spectral components) by employing simple and binary amplitude elements? One way is the additive type superimposition of component diffracting structures instead of widely used multiplicative one. The Fourier transform of the sum of two signals is equal to the sum of their transforms. Such a spectrum is much simpler when compared with the spectrum of the product of two functions. However, for example, direct addition of two Ronchi gratings (of binary transmittances 0 and 1) results in three transmittance levels 0, 1 and 2 which should be precisely mutually controlled. One simple way to solve this problem is to employ the technique which generates the intermediate transmittance level (e.g., 0.5 in the case of two remaining ones assumed to be equal to 0 and 1) using a structure with equal areas of 0 and 1 transmittance within the required regions. The randomly encoding method with tiny encoded pixels on the mask can be used for that purpose [19,20]. Generating additive type two-dimensional sinusoidal wave form gratings was implemented with the use of LCD technology [21].
In this paper we propose an approach without resorting to the LCD technology or the three-step design of randomly encoded binary amplitude grating. Its result is similar to the above mentioned one with regions containing fine transparent and opaque elements of equal area. It deploys binary amplitude structures only which are readily generated in the integrated circuit mask technology. The idea is to add two 2D gratings, i.e., the crossed Ronchi grating [16–18] and checker (chessboard) grating [22,23] multiplied by a dense 1D Ronchi grating. The use of three Ronchi type structures (with the opening/aspect ratio equal to 0.5 and appropriate spatial frequencies) is essential to provide the required 0.5 transmittance areas in the final grating. Firstly, the checker grating is multiplied by a dense 1D Ronchi line grating. The convolution operation in spectrum domain of the product results in repeating the chessboard grating spectrum along the direction of diffraction spots of line grating. Because of high frequency of the 1D grating the repeated checker grating spectra are well separated. Next, the crossed Ronchi grating is digitally added to the above mentioned product. The transmissive square elements of the crossed grating coincide with opaque ones in the checker grating structure. The crossed and checker gratings are built of square (transparent and opaque) elements of the same dimensions. In result the three-level transmittance hybrid diffraction structure is obtained. It consists of transparent, opaque and containing dense line Ronchi grating square elements. Their transmittances are equal to 1, 0 and 0.5, respectively. It can be shown that the spectrum of the resulting binary structure, in its central part, is very similar to the one obtained for the crossed binary structure formed by adding two orthogonal line gratings (with spectrum limited to two orthogonal directions only). Our method although conducted with incoherently superimposed structures finds fine explanation in the spatial frequency domain prevailingly exploited in the coherent light system investigations. This approach allows for a smooth transition to investigate, in the second part of the paper, the self-imaging phenomenon of the three-level transmittance 2D grating.
We provide a detailed heuristic explanation of the formation steps of the proposed hybrid structure accompanied by spectra calculations of component gratings. Numerical and experimental studies of the self-imaging (Talbot effect) phenomenon of the proposed hybrid diffraction structure corroborate its unique and novel properties.
2. Three-level transmission 2D grating formation
Figures 1 and 2 show simulated component 2D structures for our method, i.e., the crossed Ronchi and checker gratings together with calculated modulus of their Fourier transforms. Simulated patterns had spatial periods equal to 100 µm.
Fig. 1 Magnified central part of: (a) the crossed Ronchi grating and (b) the modulus of its Fourier transform. Four lowest, spatial beating formed side orders are encircled in (b), see text.
Fig. 2 Magnified central part of: (a) the checker grating and (b) the modulus of its Fourier transform. Four lowest fundamental harmonics are encircled in (b).
Structures in Figs. 1 and 2 are composed of transparent and opaque square elements of identical dimensions. The crossed Ronchi grating is oriented to be periodic along diagonals of Fig. 1 (directions perpendicular to the sides of square elements), whereas the checker periodicity is along horizontal and vertical directions. This orientation was chosen for clear presentation of the spectra of subsequently generated structures. The checker grating has smaller period than the crossed grating and, correspondingly, its diffraction orders are more sparsely distributed than orders of the latter one. We will exploit this feature in one of the following steps to generate a hybrid, three transmission level structure.
Figure 3 shows the result of the multiplicative superposition of the checker grating and the 1D Ronchi grating with vertical lines. The latter one has much higher spatial frequency in comparison with the crossed and checker grating frequencies.
Fig. 3 Magnified central part of: (a) the product of the checker grating, Fig. 2(a), and 1D Ronchi grating (dense vertical lines); (b) the modulus of the Fourier transform of (a).
Figure 4(b) shows the central part of the spectrum of the final hybrid structure of Fig. 4(a). In the central part it contains only two mutually orthogonal directions populated with diffraction orders. Actually, it resembles very much the spectrum of two additively superposed 1D Ronchi gratings. Note the absence of even diffraction orders characteristic to binary amplitude Ronchi grating. On the left and right hand sides of Fig. 4(b) we see the parts of repeated spectra of the checker grating resulting from the convolution of the checker and 1D line grating spectra, Fig. 3(b).
Fig. 4 Magnified central part of: (a) the result of adding the structures shown in Figs. 1(a) and 3(a); (b) the modulus of the Fourier transform of (a).
We will present now a heuristic interpretation of the spectrum distribution in the central region of Fig. 4(b). We believe that this heuristic explanation, aided by simple component grating spectra calculations, provides deeper physical interpretation of the proposed grating formation rather than resorting to the rigorous mathematical description.
The next step is to digitally sum the structures shown in Fig. 1(a) and Fig. 3(a). The result and the modulus of its Fourier transform are shown in Fig. 4.
The explanation bases on the following main observations:
- 1. The comparison of the spectra of the crossed Ronchi grating, Fig. 1(b), and the checker grating, Fig. 2(b), indicates an overlap in space of the encircled diffraction orders in both spectra. Let us limit our considerations, for the discussion simplicity only, to the lowest side diffraction orders, i.e.,
- a) orders generated by the spatial beating phenomenon [12] between two orthogonal, multiplicatively superposed line gratings of the crossed grating (or in mathematical terms, spatial frequencies formed by the convolution operation in the spectral domain). These are orders (+ 1,-1) and (−1, + 1) which lie on the horizontal central axis (running through the spectrum central zero order) in Fig. 1(b), and orders (+ 1, + 1) and (−1,-1) along the central vertical axis in Fig. 1(b). The numbers in parenthesis indicate diffraction order numbers along directions perpendicular to the lines of the overlapped component line gratings;
- b) fundamental frequency orders of the checker grating encountered along directions of the x (horizontal) and y (vertical) axes in Fig. 2(b).
- 2. Our design concept of the hybrid diffracting structure with reduced spectrum is based on the assumption of possible mutual subtraction of the above mentioned encircled overlapping orders, Fig. 4(b). This spectrum central region can be treated as the zero diffraction order of the whole hybrid structure of Fig. 4(a).
For our assumption to occur two conditions have to be met: a) overlapping orders from the two added gratings, Fig. 1 and Fig. 3, have to be of equal amplitude, and b) they have to be in an anti-phase relationship (mutually phase shifted by π). Amplitude considerations require simple spectra calculations and their comparison, see Appendix 1. The conclusions from Appendix 1 are directly exploited for spectral amplitude considerations given in point 3 below. The anti-phase relation between the two additively overlapped structures, Fig. 1(a) and Fig. 3(a), is explained below in points 4 and 5;
- 3. The most important conclusion following from Eqs. (6) and (7) and Fig. 14 of Appendix, and the point 2 above is that four fundamental harmonics of the checker grating have twice as high amplitude (equal to 2(a12)checker) than the amplitudes of four spatial frequencies (diffraction orders) of the crossed Ronchi grating generated by the spatial beating phenomenon (and equal to (a12)Ronchi). These two sets of four diffraction orders overlap in space in the central region of spectrum shown in Fig. 4(b). Note that four diffraction orders from the checker grating result from its product with 1D line Ronchi grating, Fig. 3. Therefore, in the spectrum domain the amplitude of these orders is given by the convolution of the amplitudes of orders from the checker grating itself with the zero order of the 1D Ronchi grating with the amplitude equal to a0 = 0,5. In result the equality of amplitudes of the overlapping two groups of diffraction orders from the crossed Ronchi and checker gratings is encountered.
- 4. It can be seen that along diagonal directions of Fig. 1(a) and Fig. 3(a) the structures of crossed Ronchi and modified checker grating (multiplied with vertical 1D Ronchi grating) are in an anti-phase. To be specific, transparent square elements of the crossed grating overlap with opaque elements of the checker and vice versa, square elements with vertical lines of Ronchi grating (with their transmittance equal to 0,5, Fig. 3(a)) overlap with opaque elements of the crossed Ronchi grating. At the same time it is helpful to note that the checker grating can be obtained by simultaneously translating the crossed Ronchi grating along orthogonal component line directions by half its period [22]. Correspondingly, the checker grating can be considered as the anti-phase structure with respect to the crossed grating.
- 5. The anti-phase relationship between central side-orders of the two summed structures can be proven for higher order spatial beat frequencies as well. In result the reduced spectrum is only obtained with diffractions orders localized along two mutually perpendicular directions, Fig. 4(b). Each direction contains, besides the DC term, odd diffraction orders only, similarly to the case of binary 1D Ronchi grating. Five central orders dominate the spectrum because of their higher amplitude.
- 6. The most important features of the spectrum (see the central region of Fig. 4(b)) follow from the addition of complex amplitude transmittances of the crossed and checker gratings. Their zero order amplitudes add and besides the diffraction orders along two mutually orthogonal directions, all other spatial beat frequencies (higher diffraction orders) mutually subtract. In result the spectrum of two additively superimposed 1D Ronchi gratings is obtained. We will prove it by means of numerical simulations of the self-imaging (Talbot effect) phenomenon of two sinusoidal and binary amplitude transmittance gratings being multiplicatively and additively superimposed. Finally, heuristic interpretation and numerical studies will be corroborated by Fresnel diffraction field laboratory experiments.
At the end of this Section describing the formation of the 3-level transmission 2D structure and its properties we would like to emphasize that our 2D binary amplitude structure with reduced spectrum has been obtained by adding the product of the checker grating and 1D (line) grating to the crossed grating. Another approach using the product of the crossed grating and 1D grating added to the checker grating does not yield the same result any more. In this case the amplitudes (see point 3 above) of overlapping crossed grating spatial beat orders (+ 1, + 1), (+ 1,-1), (−1, + 1) and (−1,-1) and amplitudes of the fundamental harmonics of the checker grating are not equal. Specifically, the former ones are four times smaller than the latter ones. In consequence, another 2D binary amplitude structure with reach spectrum is obtained in this case, see Fig. 5.
Fig. 5 Magnified central part of (a) the sum of the product of the crossed grating and the 1D (line) grating added to the checker grating; (b) the modulus of its Fourier transform.
Finally, for the sake of curiosity and completeness, Fig. 6 shows the result of direct addition (1D carrier grating is not involved) of the checker and crossed gratings and its spectrum. The result can be treated as the negative of the crossed Ronchi grating, Fig. 1.
Fig. 6 Magnified central part of (a) direct addition of the crossed and checker gratings; (b) the modulus of its spectrum.
3. Fresnel diffraction field of 3-level transmittance 2D structure and its properties
Previous Section presents a heuristic explanation of the proposed method to obtain 2D binary amplitude grating with reduced spectrum, i.e., the spectrum specific to a 2D binary structure generated by additive superimposition of mutually orthogonal 1D binary gratings. Here we would like to corroborate the result by studying and comparing Fresnel diffraction fields of multiplicatively and additively superimposed orthogonal gratings. The motivation to choose this approach is twofold:
- 1. Different intensity patterns at the characteristic planes of the Fresnel field of 2D gratings with multiplicative and additive superimpositions of component 1D gratings allow easy distinction of the type of superimposition;
- 2. Up to the authors’ best knowledge, the Fresnel diffraction field properties of 2D periodic structures with additive superimposition of component 1D gratings have not been presented in the literature to date.
Our investigations will be conducted for component gratings with cosinusoidal and binary amplitude transmittances.
As is well known, the Fresnel diffraction field of periodic objects is characterized by the so-called self-imaging phenomenon (Talbot effect) [10,11,24]. For example, in the case of a 1D periodic object with spatial period d illuminated by spatially and temporarily coherent plane wave front beam, the object plane complex amplitude distribution repeats itself at the distances z = Md2/λ, where M stands for integer and λ is the radiation wavelength. For M = even integer the complex amplitude distribution is in-registry with the object amplitude transmittance, whereas for M = odd integer lateral shift by d/2 in the direction perpendicular to object lines is encountered. Additional planes of interest lie in the middle between the above mentioned planes, i.e., (M + 0.5)(d2/λ). In the case of 1D cosinusoidal amplitude transmittance objects, three interfering diffraction orders form a double frequency cosinusoidal intensity pattern at the intermediate planes [11,25]. For the square wave binary amplitude grating, a uniform intensity distribution occurs at those planes [11] (for the uniform field to be generated a sufficient number of grating diffraction orders must contribute to the diffraction image formation). Similar features for cosinusoidal and binary amplitude gratings are found for 2D structures with multiplicative, orthogonal superimposition of component gratings [11,26,27].
We will show below that intensity distributions at the intermediate distances z = (M + 0.5)d2/λ most clearly distinguish the Fresnel diffraction field of 2D amplitude periodic structures with multiplicative and additive orthogonal superimpositions of 1D amplitude periodicities.
3.1 Fresnel diffraction field properties of crossed cosinusoidal amplitude gratings
3.1.1 The case of a multiplicative superimposition of orthogonal 1D gratings
Detailed theoretical and numerical studies of this case can be found in [26,27]. The Fresnel diffraction field complex amplitude Ecm(x,y,z) at a distance z from multiplicatively crossed cosinusoidal grating can be written as
where a0 and a+1 = a-1 = a1 denote the amplitudes of the zero and side order beams interfering behind the grating. Other symbols were explained before. The intensity distribution is calculated as
Exemplifying intensity distributions at distances z = d2/λ and z = (1.5)d2/λ calculated for the amplitude values a0 = 0.5 and a1 = 0.318 (corresponding to the amplitudes of the three lowest orders of the binary amplitude Ronchi grating) are presented in Fig. 7. In the second case (b), a doubled spatial frequency pattern in comparison with case (a) is observed [10,24]. The spatial period of simulated patterns in Section 3 was equal to 100 µm (sinusoidal and binary ones).
Fig. 7 Central regions of the Fresnel field intensity patterns at the distances z = d2/λ (a) and z = (1.5)d2/λ (b) from multiplicatively superimposed two orthogonal cosinusoidal amplitude gratings.
3.1.2 The case of an additive superimposition of the orthogonal 1D cosinusoidal gratings
The Fresnel field complex amplitude distribution Eca(x,y,z) at a distance z from additively crossed 1D cosinusoidal gratings can be simply written as
The corresponding intensity distribution is calculated according to Eq. (2). Exemplifying intensity distributions at the distances z = d2/λ and z = (1.5)d2/λ calculated for the same values of amplitudes a0 and a1 as above are presented in Fig. 8.
Fig. 8 Central regions of the Fresnel field intensity patterns at the distances z = d2/λ (a) and z = (1.5)d2/λ (b) from additively superimposed two orthogonal cosinusoidal amplitude gratings.
The comparison of Fig. 7(b) and Fig. 8(b) enables easy distinction of the types of superimposition of two component 1D cosinusoidal amplitude gratings.
3.2 Fresnel diffraction field properties of crossed 1D amplitude binary gratings
3.2.1 The case of a multiplicative superimposition of orthogonal 1D binary amplitude gratings
Complex amplitude distribution Ebm(x,y,z) of the Fresnel diffraction field at a distance z from the grating of this type is expressed as
Corresponding intensity distributions are calculated by the Eq. (2), i.e., from the product of Eq. (4) and its complex conjugate. Figure 9 shows the results for z = d2/λ and z = (1.5)d2/λ.
Fig. 9 Central regions of the Fresnel field intensity patterns calculated for the distances z = d2/λ (a) and z = (1.5)d2/λ (b) from multiplicatively superimposed two orthogonal binary amplitude Ronchi gratings. Eleven positive and eleven negative side diffraction orders together with the zero order for both orthogonal directions x and y were taken into calculations.
Note the quasi-uniform intensity Fresnel pattern for z = (1.5)d2/λ, Fig. 9(b), which is well known for the crossed binary Ronchi grating. Thin dark crossed lines are encountered over the uniform intensity distribution because of a limited number of grating harmonics taken into calculations. At the same time, paraxial approximation for higher diffraction orders must be satisfied.
It is worthwhile to add that at the distances z = (d2/4λ) from the self-image planes (z = Md2/λ) and from the uniform intensity distribution planes (z = (M + 0.5)d2/λ), the Fresnel images very similar to the ones found in the self-image planes, but with slightly reduced contrast, are encountered. Similar reduced contrast patterns (which can be easily misjudged as the grating “self-images”) occur at the same distances z for 1D binary Ronchi grating [28,29]. This is due to an apparent (subjective) exaggeration in the sensation of the contrast between adjacent parts of the object; this visual response phenomenon is known as the subjective (simultaneous) contrast effect [30].
3.2.2 The case of additive superimposition of orthogonal 1D binary amplitude Ronchi gratings
Complex amplitude distribution of the Fresnel diffraction field at a distance z from the grating of this type is expressed as
Figure 10 presents calculated intensity patterns for the distances z = d2/λ and z = (1.5)d2/λ.
Fig. 10 Central regions of the Fresnel field intensity patterns at the distances z = d2/λ (a) and z = (1.5)d2/λ (b) generated by additively superimposed two orthogonal 1D binary amplitude gratings.
The most prominent difference in comparison with multiplicative superimposition occurs at the planes z = (M + 0.5)d2/λ. The checker board intensity patterns in place of uniform intensity distributions are encountered (Fig. 10(b)).
We have to add that for the additive superimposition of component 1D binary Ronchi gratings, both at the self-image localization distances z = Md2/λ as well as at the above mentioned distances d2/4λ from the self-image planes (before and behind them) the crossed type patterns exhibit more diversified intensity differences in square elements. This is an obvious consequence of the fact that at the plane of additively crossed binary grating itself we face three intensity levels. They are equal to 0 (in the places the zero transmittance bands/lines of two gratings cross), 4 (in the bright square elements) and 1 in the remaining ones (in the multiplicative superimposition case intensity value in those elements is equal to 0). Figure 11 shows the intensity distribution in the additively superimposed simulated binary grating (displayed nonlinearly for enhanced visualization; upper intensity value limited to 3 for better visualization of the range of lower intensities).
Fig. 11 Central region of the intensity distribution at the plane of orthogonally, additively superimposed binary grating (nonlinear display for enhanced visualization of diversified intensity levels).
Detailed analysis of intensity patterns at distances different than z = (M + 0.5)d2/λ is out of scope of this paper. We treat the most prominent difference between intensity patterns at distances z = (M + 0.5)d2/λ as the sufficient proof of the grating superimposition type.
4. Experiments
Laboratory experiments aimed at corroborating the additive-like construction of the resulting 2D binary diffraction structure generating the energy spectrum shown in the central region of Fig. 4(b). The spectrum of interest is composed of diffraction orders, distributed along two mutually orthogonal directions in a manner characteristic to the spectrum of 1D Ronchi binary amplitude grating, i.e., without even orders. For this purpose, the 3-level transmittance binary amplitude structure shown in Fig. 4(a) was designed and produced by conventional etching technology of a thin chromium layer on a glass substrate. Next, both sides of the plate were antireflection coated. The dense vertical lines of 1D Ronchi grating had a spatial period equal to 6.25 μm, and the period of the crossed Ronchi grating was equal to 108 μm. The grating was placed at the input plane of a conventional coherent optical processor; at its spatial frequency (Fourier transform) plane the square opening filter was used to pass the spectrum of interest. The filter allowed the diffraction orders from the −9th to + 9th for each of the two spectrum directions to contribute to the studied Fresnel diffraction field at the output of the optical system. He-Ne laser served as the light source.
Figure 12 shows two Fresnel field diffraction images of the low-pass filtered (passed central region of the spectrum in Fig. 4(b)) 3-level transmission 2D binary amplitude structure. The first image was recorded in one of the self-image planes, z = Md2/λ, and the second one was encountered at the distance z = (M + 0.5)d2/λ. Characteristic intensity patterns perfectly agree with the ones presented in Fig. 10. Certainly, we have to take into account different angular orientations of the simulated structure (Fig. 4, with crossed grating lines drawn at 45 and 135 degrees to facilitate simpler further analysis of the spectra at successive steps of the final 3-level hybrid grating formation, see explanation in Section 2) and simulated ones in Section 3 (crossed grating lines along direction of 0 and 90 degrees).
Fig. 12 Recorded experimental Fresnel diffraction images formed by the diffraction orders from the spectrum central region (composed of two mutually orthogonal directions of diffraction orders) of the developed 3-level binary amplitude grating (numerically calculated spectrum is shown in Fig. 4(b)). Figure 12(a) shows the intensity pattern of one of the self-images whereas image Fig. 12(b) was recorded in one of the self-image intermediate planes.
5. Conclusions
The method for generating 2D binary amplitude grating with reduced spectrum corresponding to the one of additively superimposed two orthogonal 1D gratings and its self-imaging have been described. The method simplicity follows from the use of computer to prepare simple auxiliary binary amplitude gratings (the crossed, checker and 1D line Ronchi type gratings) and their digital superimpositions to obtain the required structure. The final binary mask represents the sum of the crossed Ronchi amplitude grating with the product of the checker and 1D Ronchi grating. The crossed and checker gratings are built of square elements of the same dimensions; the spatial frequency of the line grating is much higher than the frequency of 2D gratings. In result the three-level transmittance structure is generated with transmittances of component square elements equal to 1, 0.5 and 0. The intermediate 0.5 transmittance level regions (with equal areas of 0 and 1 transmittance) were obtained by deploying fine 1D Ronchi grating with opening/aspect ratio equal to 0.5. In this way equal areas of 0 and 1 transmittance within the required regions were generated. The binary character of the resulting mask is to be emphasized since it can be produced by conventional etching technology of a thin chromium layer on a glass substrate. The overall zero diffraction order of our hybrid structure (which is well separated from the rest of the spectrum) consists of two mutually orthogonal directions only. Each one contains the spectrum of the 1D Ronchi grating. Therefore, the overall zero order spectrum is equivalent to the one of the crossed binary structure formed by adding orthogonal Ronchi line gratings. A detailed heuristic explanation of the method and obtained result is given. The interpretation is conducted in the spectrum domain of superimposed structures (for each of the above mentioned processing stages of auxiliary gratings) and supported by simple calculations of their Fourier transforms.
To prove the additive character of the overall zero diffraction order of our hybrid structure we decided to exploit the self-imaging phenomenon which takes place in the Fresnel diffraction region of a periodic object. This approach is innovative itself since up to the authors’ best knowledge the self-imaging of a 2D structure formed by addition of mutually orthogonal 1D periodic objects has not been studied in the literature up to now. Published works deal with multiplicative superimposition 2D objects only. It has been shown numerically and experimentally that characteristic intensity patterns encountered at the distances lying in the middle between the self-image planes, i.e., z = (M + 0.5)d2/λ, clearly distinguish the cases of additive and multiplicative superimposition of orthogonal 1D amplitude gratings.
Finally, we would like to add that the proposed approach to find specific 2D structures providing, upon addition of their complex amplitude spectra, the characteristic and expected spectrum distribution can be exploited for the design and interpretation of other 2D structures.
The intensity distribution shown in Fig. 12(b), characteristic to diffraction planes at distances z = (M + 0.5)d2/λ, well corresponds to the pattern of Fig. 10(b) taking into account the above explained difference in angular orientation (rotation of structure lines by 45 deg). This fact corroborates the additive-like character of the spectrum in the central region of Fig. 4(b).
Appendix
A comparison of diffraction spectra of the binary amplitude crossed and checker gratings is given here to explain the result obtained in Fig. 4.
Figure 13 shows both gratings and their Fourier spectra (modulus of the Fourier transform of grating amplitude transmittances). The first straightforward observation is that the checker grating spectrum is simpler as compared with the crossed Ronchi grating one.
Fig. 13 Binary amplitude crossed Ronchi (a) and checker grating (c) together with their Fourier transform modulus (b), and (d), respectively. Gratings can be considered as built of transparent and opaque square elements of the same dimensions. Figures 13(b) and 13(d) show enlarged central parts of spatial frequency spectra (displayed with the same nonlinear scale for their enhanced visualization).
The well known mathematical expressions for the diffraction efficiency of diffraction orders of the squared binary Ronchi and checker gratings can be found in [22]. For the binary cross-type structure, the diffraction order amplitudes are given by
where m and n are diffraction order numbers along x and y directions (perpendicular to multiplicatively superposed lines of component gratings), respectively. The ratio of the width of the space between lines to the period of the grating is called the grating opening number (or aspect ratio) and denoted by (h/d). For the Ronchi grating the opening number is equal to 0.5. The amplitude of diffraction orders of the amplitude checker grating is given by [22]
where symbols mean the same as in Eq. (6). Note that for all four combinations of m = ±1 and n = ±1 corresponding to first diffraction orders encountered along 45 and 135 deg diagonals, the value of the square parentheses term on the right hand side of Eq. (7) is equal to 2. Computation results are shown in the next figure.
Figure 14 shows the spectra of the conventional squared Ronchi grating, Fig. 13(a) and the amplitude checker grating, Fig. 13(b). The following conclusions can be drawn from Fig. 14:
Fig. 14 Cross-sections through the calculated modulus of Fourier transforms presented in Fig. 13(b) and 13(d) shown in linear scale. Horizontal cross-sections through the zero order of the crossed Ronchi and checker gratings are shown in Figs. 14(a) and 14(b), respectively, and 45 deg diagonal cross sections through the zero order of the crossed Ronchi and checker gratings are shown in Figs. 14(c) and 14(d), respectively.
- 1. first order diffraction efficiency along 45/135 deg diagonals in the Fourier spectrum of the amplitude checker grating is the same as the diffraction efficiency of the crossed Ronchi grating, Fig. 13(a), along horizontal and vertical directions;
- 2. first order diffraction efficiencies along diagonals (frequencies formed by spatial beating between fundamental harmonics along the x and y directions) in the Fourier spectrum of the squared Ronchi gratings, Fig. 13(a), is twice as small as the one of the diagonal direction first diffraction orders of the checker grating;
- 3. zero order diffraction efficiency of the amplitude checker grating is approx. 1.5 times higher than the one of the crossed grating.
From the point of view of the task of this paper, i.e., the interpretation of the effect of additive superposition of structures shown in Fig. 1 and Fig. 3, the most important is the property mentioned in point 2 above. It is exploited in the main text of the paper.
Funding
National Science Center Poland (NCN) (Grant 2017/25/B/ST7/02049); Statutory Funds Faculty of Mechatronics Warsaw University of Technology.
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