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Abstract
For positioning Talbot encoder and Talbot lithography, etc., properties manipulation of Talbot imaging is highly expected. In this work, an investigation on the distance and depth modulation of Talbot imaging, which employs a specially designed grating structure, is presented. Compared with the current grating structure, the proposed grating structure is characterized by having the phase layers with uneven thicknesses. Such a specific structural design can cause the offset of Talbot image from its nominal position, which in turn generates the spatial distance modulation of self-imaging and imaging depth expansion. Theoretical analysis is performed to explain its operating principle, and simulations and experiments are carried out to demonstrate its effectiveness.
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1. Introduction
Talbot imaging, also known as grating self-imaging, refers to the replication of the intensity pattern at a certain distance from the grating when it is illuminated by a collimated coherence beam. This phenomenon was first discovered by Talbot in 1836 [1], and observed by Rayleigh in 1881 [2]. Nowadays, Talbot imaging is widely used in industrial applications, including spectrometry instruments [3–7], lithographic equipment [8–13], positioning sensors [14–17], and refractive-index measurement apparatus [18,19], etc. Among them, the excellent quality and precise spatial location of the Talbot imaging patterns [20–23] play a crucial role in ensuring the functionality and accuracy of the above devices. The harmonic distortion of Talbot imaging pattern is a typical example that will limit the fabrication accuracy of micro-electronics as well as the measurement accuracy of Talbot positioning encoders [24–26]. As a result, property manipulation and position control of Talbot imaging are greatly anticipated in practice.
Harmonic suppression of Talbot imaging patterns was extensively investigated by Ye et al. and Ieki et al., and approximately harmonic-free Talbot imaging patterns with sinusoidal intensity distribution were obtained [27–30]. Rasouli et al. proved that by appropriately engineering the transmission profile and opening number of the grating, it is possible to enhance the contrast of the sub-images generated at quarter-Talbot distance [31].
A close look into the above studies reveals that they all focus on the intrinsic performance of Talbot imaging, while research on the extrinsic spatial position modulation of Talbot imaging is relatively rare. Hebri et al. found that for 2D orthogonal non-separable periodic structures with Fourier coefficients of only with odd indices, the Talbot distance is greatly influenced by the number theoretic properties of the structure [32]. Rasouli et al. further demonstrated that the depth of focus for 2D Talbot images was increased compared to 1D Talbot images, due to the contrast enhancement that occurred in the orthogonal superimposition of the two 1D gratings [33]. An in-depth investigation of the spatial modulation of Talbot imaging will be especially helpful for improving the installation-tolerance for positioning Talbot encoders [25], and broadening the application of Talbot lithography on non-planar substrates [34].
In this work, we present a distance and depth modulation of Talbot imaging via the specified design of a 1D grating structure. The grating structure here contains the grating itself and the phase layer behind it. Different from the conventional grating structure, the proposed grating structure features non-uniform thicknesses phase layers. Talbot distance variations in different grating regions induced by these non-uniform thickness phase layers will in turn generate the distance modulation and the depth extension of the Talbot image. The modulation mechanism of the designed grating structure is explained in detail, and simulations and experiments are conducted to confirm its effectiveness.
2. Theoretical analysis
2.1 Definition of the self-imaging distance and self-imaging depth
Figure 1 shows the phenomenon of Talbot imaging. When a parallel coherent light beam is irradiated perpendicularly on a diffraction grating, the light beam distribution that meets the self-repetition phenomenon appears at a certain distance behind the grating, as illustrated in Fig. 1(a). This certain distance is defined as the self-imaging distance, and its values can be given by [35]:
where m is an integer, zm represents the distance of the m-order Talbot image, d is the period of the grating, λ is the wavelength of incident light in a vacuum, zT is the Talbot length [36] and n0 is the refractive index of the background medium behind the grating, such as air, water, etc. The planes located at Talbot distance from the scale grating are usually called Talbot planes.
Fig. 1. Schematic diagram for Talbot imaging of a diffraction grating. (a) Illustration of the self-imaging distance. (b) Illustration of the self-imaging depth, which reflects the allowable imaging interval around the Talbot planes.
In practical applications, such as optical encoders, a certain gap tolerance between the two gratings should be permitted, and such a gap tolerance is strongly related to the depth of self-imaging. The contrast value of the Talbot image reaches its maximum at the Talbot planes, and gradually decreases as it moves away from the Talbot planes, as shown in Fig. 1(b). As a result, the range of focus of the Talbot images can be used to determine the depth of self-imaging [37]. For getting a good signal with a maximum decay of less than 10%, the minimum contrast must be greater than 0.9. Taking the contrast attenuation of 10% as the threshold, self-imaging depth is defined as:
where ε is self-imaging depth and z+(c = 0.9) and z-(c = 0.9) respectively refer to z-position corresponding to c = 0.9 on the both sides of each Talbot plane. In Eq. (2), the image contrast of it is defined as: where Imax and Imin refer to the maximum light intensity and minimum light intensity, respectively, on a certain image plane.2.2 Distance and depth modulation of Talbot imaging via specified design of the grating structure
Equation (1) gives the self-imaging distance when the light is diffracted by a diffraction grating and travels freely. The self-imaging distance control work is analyzed from the perspective of phase relationship, and the schematic diagram of the entire optical path is shown in Fig. 2. Point P is a point on the plane where the grating is located, and point P’ is the imaging point corresponding to point P along the z-direction. Assume that the diffraction beam with a diffraction angle θ is indicated as W+θ, and its wavefront phases at P point and P’ point are marked as φθ and φθ’, respectively. Similarly, the zero-order diffracted beam is denoted as W0, and its wavefront phases at P point and P’ point are φ0 and φ0’, respectively. The refractive index of the phase layer with a thickness h is n, and the refraction angle created after the plane wave W+θ enters it is θ’. Then the phase relations of each train of plane waves at P and P’ are as follows:
Fig. 2. Schematic diagram for Talbot imaging of the diffraction grating containing a phase layer. the red line represents the plane wave W+θ with diffraction angle θ, the green line represents the plane wave W0 with zero order diffraction, and the blue line represents the plane wave W-θ with diffraction angle -θ.
In Eqs. (4)–(5), za is the distance between the grating and the phase layer, zb is the distance between the phase layer and the Talbot image, n0 is the refractive index of the background medium, and k is the wave vector in free space, which is expressed as 2π/λ. Based on Eqs. (4)–(5), the phase difference relationship between two positions can be obtained, which is given by:
In order to ensure that the phases at P and P’ can satisfy the self-imaging phenomenon, the phase difference between the two positions should satisfy Eq. (7).
Considering the paraxial small-angle approximation, the formula (1-sin2θ)1/2 = 1-(sin2θ)/2 can be approximately expressed. According to this approximate condition, Eq. (8) is obtained:
Based on the grating diffraction formula dn0sinθ = Nλ, N = 1,2,3…, the imaging distance zm,N, which is determined by N-order diffraction wave interference, is calculated as.
The largest spatial periodicity inside the Talbot carpet is the Talbot length. This distance corresponds to the first phase matching between the zeroth and projected first diffraction order along the mask’s perpendicular axis [38]. Thus, the modulated self-imaging distance is defined as:
In Eq. (10), the value of h exists in a restricted interval, and an excessively large h may cause imaging inside the phase layer, or exceed the near-field imaging range. Under normal circumstances, the background medium is air, so the refractive index n0 = 1. Comparing Eq. (1) with Eq. (10), it can be found that the self-imaging distance is shifted backward by (1-1/n)h, which is defined as the modulation factor of self-imaging distance.
All the above calculations indicate that the quantitative movement of the Talbot image can be realized by adding a certain thickness of the phase layer. Figure 3 further shows the feasibility of achieving spatial self-imaging depth combinations by adjusting the self-imaging distance. In Fig. 3, only two kinds of phase layers are shown. In practical applications, combinations of more kinds of phase layer can be used to achieve a reasonable extension of the self-imaging depth.
Fig. 3. Schematic diagram for the Talbot imaging of a diffraction grating with a stepped phase layer. (a) Illustration of the self-imaging distance, which shows the variation in imaging distance. (b) Illustration of the self-imaging depth, which shows the combination of allowable imaging interval.
3. Simulation results
To illustrate the correctness of the above control work for self-imaging distance, numerical simulations for various grating structures with the different phase layers are performed in this section. According to the composition of the modulation factor of self-imaging distance, the simulation verification will be conducted from two aspects: the thickness h and the refractive index n of the phase layer. For comparison, the electric field information of the traditional grating structure (that is, the grating without any phase layer) is also recorded.
3.1 Thickness of the phase layer
A 633 nm parallel light is used to irradiate the diffraction grating vertically. The grating is set to an amplitude grating with a period of 80 µm and a fill factor of 0.5. Based on Eq. (10), when the background medium is air, the theoretical calculations show that the Talbot planes will move back 1 mm and 2 mm respectively, under the action of phase layers of 3 mm and 6 mm thickness with a refractive index of 1.5.
The simulation results of the grating models corresponding to phase layers of different thicknesses are shown in Fig. 4. In Fig. 4(a), under the case of no phase layer, the position of the first Talbot plane is at z = 20.18 mm. In Fig. 4(b), when the thickness of the phase layer is 3 mm, the position of the first Talbot plane is found at z = 21.19 mm. In Fig. 4(c), when the thickness of the phase layer is 6 mm, the position of the first Talbot plane appears at z = 22.15 mm. Compared with the first Talbot plane position shown in Fig. 4(a), the self-imaging distances in Fig. 4(b) and Fig. 4(c) are increased by 1.01 mm and 1.97 mm respectively, which are almost consistent with the theoretical move values of 1 mm and 2 mm.
Fig. 4. The simulated electric field distribution of the diffraction grating with the different thick phase layers. (a) The electric field distribution under the condition of traditional grating structure (b) The electric field distribution for the grating with a 3 mm thick phase layer. (c) The electric field distribution for the grating with a 6 mm thick phase layer.
3.2 Refractive index of the phase layer
Figure 5 shows the Talbot image position distribution under the action of the phase layers with the same thickness but different refractive index. The thickness of the phase layer is set as a constant value, that is, h = 6 mm. In the theoretical calculations, the Talbot plane will be moved back by 1.2 mm and 2 mm under the action of the phase layers with refractive indexes of 1.25 and 1.5, respectively, according to the modulation factor of self-imaging distance (1-1/n)h.
Fig. 5. The simulated electric field distribution of the diffraction grating with the different refractive index of phase layers. (a) The electric field distribution under the condition of the traditional grating. (b) The electric field distribution of the grating with a refractive index of 1.2 phase layer. (c) The electric field distribution of the grating with a refractive index of 1.5 phase layer.
In the three cases of no phase layer, phase layer refractive index set to 1.25 and phase layer refractive index set to 1.5, the position of the first Talbot plane appears at z = 20.18 mm, z = 21.41 mm and z = 22.15 mm, respectively, as shown in Fig. 5. That is, compared to Fig. 5(a), an increase in the self-imaging distance of 1.23 mm and 1.97 mm was obtained in Fig. 5(b) and Fig. 5(c), respectively. This again agrees well with the theoretical calculated move values, i.e., 1.2 mm and 2 mm. Therefore, it is feasible to control the self-imaging distance by adjusting the refractive index of the phase layer structure behind the grating.
4. Experiments and results
4.1 Experimental setup
Based on the schematic diagram shown in Figs. 1–3, the experimental setup is presented in Fig. 6(a). A He-Ne laser with a wavelength of 633 nm was used as the light source, and its spot diameter was about 5 mm. The grating was a chromium plated amplitude grating with a period of 80 um and a fill factor of 0.5. The grating without phase layers is shown in Fig. 6(c). The single phase layer used in the experiments was the borosilicate glass substrate with a refractive index of 1.4714 and its thickness was 1 mm, 2 mm and 3 mm, respectively, as shown in Fig. 6(d). In Fig. 6(e), the stepped phase layer was formed by the superimposition of two phase layers and its specific parameter description is shown in Fig. 6(b). The CMOS Camera (MQ013CG-E2, XIMEA Inc.) was used to detect light intensity distribution, with a sensor active area of 6.9 × 5.5 mm2 and a pixel size of 5.3 µm.
Fig. 6. Illustration of the experimental setup. (a) The overall construction of the experimental device. (b) Illustration of thickness parameters corresponding to (e) model. (c) No phase layer model. (d) Single phase layer model. (e) Stepped phase layer model.
The distance between the photosensitive surface of the CMOS Camera and the diffraction grating should be as narrow as feasible to get clear imaging patterns. The coherent light beam emitted by the He-Ne laser passed through the amplitude grating and led to a diffraction phenomenon. Under the action of the phase layer, the phase relationship at the same location was changed. Light intensity information was finally displayed on the computer terminal. By moving the movable object to adjust the distance between the CMOS camera and the amplitude grating, the distribution of light intensity in the x-y plane at different z-positions was obtained. During the 25 mm stroke along the z-axis, we recorded 251 data samples.
4.2 Experimental results and discussion of self-imaging distance modulation
According to the light intensity information obtained in the above experiments, the contrast at different z-positions can be calculated. Figure 7 shows the contrast variation along z-axis, where the 0 mm position indicates the closest position of the CMOS Camera to the diffraction points, and the z-value corresponding to each maximum contrast value represents an imaging position. Taking the case of no phase layer in Fig. 7(a) as a comparison, it can be concluded that as the thickness of the phase layer increases, each imaging position in Figs.7 (b-d) produces a certain amount of movement towards z-direction.
Fig. 7. Imaging contrast curves corresponding to different thick phase layers. (a) Imaging contrast curve without phase layer. (b-d) Imaging contrast curves with the phase layer corresponding to thickness of 1 mm, 2 mm, and 3 mm, respectively.
Table 1 shows the specific movement value of each image under the action of phase layers, and 1st-Image, 2nd-Image and 3rd-Image respectively refer to the images corresponding to the positions of three maximum contrast values along z-axis in turn. In order to reflect the overall situation of the imaging movement in the experiments, the movement values of three images corresponding to each thickness phase layer are averaged. In Table 2, the theoretical movement values can be calculated according to the modulation factor of self-imaging distance (1-1/n)h, and movement deviation can be obtained by using the formula E = |Δze - Δzt|, where Δze represents the experimental movement value of the self-imaging and Δzt is the theoretical movement value. The magnitude of E indicates the deviation of the experiment from the theory. Based on the calculation data shown in Table 2, it can be seen that the movement deviation values corresponding to three thickness models of 1 mm,2 mm and 3 mm are 0.02 mm, 0.09 mm and 0.10 mm respectively, which is basically consistent with theoretical calculations.
Table 1. Movement value of each imaging position in the experiment
Table 2. Movement deviation under the condition of the different thickness phase layer
In addition, according to the experimental results obtained in Tables 1 and 2, the error bars of the experimental movement values and the line chart of theoretical movement values of the Talbot images for different phase layer thicknesses are plotted in Fig. 8, from which the consistency between theoretical and experimental polylines of self-imaging can be observed. Therefore, it is further confirmed that the self-imaging distance can be controlled by adding a certain thickness of the phase layer behind the grating. This type of deep modulation for period structure imaging has the potential to create a more constructive spatial periodic structure light field, which will provide assistance for optical measurement and lithography process, etc.
Fig. 8. Comparison of the self-imaging movement value for theory results (red solid line) and experiment results (black solid line) under different thickness phase layers.
4.3 Experimental results and discussion of self-imaging depth modulation
To meet the self-imaging depth application of the modulation factor of self-imaging distance, phase layers of varying thicknesses were arranged along the gate line direction (that is x-direction) to form a stepped phase layer, as shown in Fig. 6(e). A total of 81 light intensity samples were performed within the 4 mm travel range, resulting in a 0.05 mm movement step. By slowly increasing the distance between the CMOS Camera and the diffraction grating, the first clear grating image under the condition of no phase layer was obtained at z = 0.85 mm, as shown in Fig. 9(a). Due to the different positions corresponding to the different thickness phase layers along the x direction, the next imaging height would be changed. Therefore, the CMOS Camera sampling height with the change of imaging position along the x-direction was adjusted. Then, as the distance between the CMOS Camera and the grating grew, the second clear grating image corresponding to the 4 mm thick phase layer was obtained at z = 2.1 mm, as shown in Fig. 9(b). In the same way, the third clear Talbot image under the condition of the 7 mm thick phase layer was obtained at z = 3.1 mm, as shown in Fig. 9(c).
Fig. 9. Light intensity distribution at different positions under the action of the stepped phase layer.
According to the light intensity information collected by CMOS Camera during the whole movement, the contrast-position curve is drawn in Fig. 10. Since the maximum contrast value represents one imaging, there are three times of imaging within 4 mm range along z-direction in Fig. 10, but only one imaging appears in the 10 mm range in Fig. 7(a), which means that the widening of special self-imaging depth can be achieved by increasing the number of imaging within a certain distance range.
Fig. 10. The contrast-position curve corresponding to the stepped phase layer. The red zone is the contrast analysis without the phase layer. The green zone and the blue zone are the contrast characterizations in the cases of 4 mm phase layer thickness and 7 mm phase layer thickness, respectively. The red dotted line is the threshold line where the contrast is equal to 0.9.
According to Eq. (2), z+(c = 0.9) and z-(c = 0.9) on the both sides of each Talbot plane need to be obtained to calculate the self-imaging depth ε. It can be concluded from the red zone in Fig. 10 that the self-imaging depth is between z-(c = 0.9) = 0.48 mm and z+(c = 0.9) = 1.19 mm under the condition of no phase layer, and its size is 0.71 mm. When a stepped phase layers are placed behind the amplitude grating, the self-imaging depth is also formed at z-(c = 0.9) = 1.78 mm to z+(c = 0.9) = 2.58 mm and z-(c = 0.9) = 2.85 mm to z+(c = 0.9) = 3.60 mm, and their sizes are 0.80 mm and 0.75 mm, respectively, as shown in the green zone and blue zone in Fig. 10. Therefore, the total depth interval changes from 0.71 mm to 2.26 mm, and the self-imaging depth is almost three times the original self-imaging depth. If the self-imaging distance is further adjusted to change the distance between imaging positions, a continuous imaging interval can be formed.
5. Conclusion
In this paper, a method to modulate the self-imaging distance and self-imaging depth has been presented. The method employs a specified designed grating structure with non-uniform phase layers, that could obtain different Talbot distances corresponding to different grating regions. Thereby it is achievable to obtain the distance and depth modulation of Talbot imaging. The validity and effectiveness of the proposed method have been confirmed by both simulations and experimental results. The results obtained in this work are especially helpful for designing high-installation-tolerance optical encoders and the application of research results in optical encoders will be carried out in subsequent work. In addition, we hold a strong belief that it will offer a new opportunity for expanding the application of Talbot lithography on non-planar substrates.
Funding
Key Research and Development Projects of Shaanxi Province (2020ZDLGY14-03); National Natural Science Foundation of China (52011530186, 52075430).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
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