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Abstract
A grating-based interferometric cavity produces coherent diffraction light field in a compact size, serving as a promising candidate for displacement measurement by taking advantage of both high integration and high accuracy. Phase-modulated diffraction gratings (PMDGs) make use of a combination of diffractive optical elements, allowing for the diminishment of zeroth-order reflected beams and thus improving the energy utilization coefficient and sensitivity of grating-based displacement measurements. However, conventional PMDGs with submicron-scale features usually require demanding micromachining processes, posing a significant challenge to manufacturability. Involving a four-region PMDG, this paper establishes a hybrid error model including etching error and coating error, thus providing a quantitative analysis of the relation between the errors and optical responses. The hybrid error model and the designated process-tolerant grating are experimentally verified by micromachining and grating-based displacement measurements using an 850 nm laser, confirming the validity and effectiveness. It is found the PMDG achieves an energy utilization coefficient (the ratio of the peak-to-peak value of the ±1st order beams to the 0th-order beam) improvement of nearly 500% and a four-fold reduction in 0th-order beam intensity compared with the traditional amplitude grating. More importantly, this PMDG maintains very tolerant process requirements, and the etching error and coating error can be up to 0.5 µm and 0.6 µm, respectively. This offers attractive alternatives to the fabrication of PMDGs and grating-based devices with wide process compatibility. This work first systematically investigates the influence of fabrication errors and identifies the interplay between the errors and the optical response for PMDGs. The hybrid error model allows further avenues for the fabrication of diffraction elements with practical limitations of micromachining fabrication.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Displacement detection using optical interferometers allows for anti-electromagnetic and noncontact measurements of small displacements [1–3], outperforming other methods in terms of its high precision and resolution. In addition, optical interferometers are not subject to scaling laws [4,5], which means they can achieve a miniaturized size without compromising the displacement measurement performance. Thus, optical displacement detection schemes with compact sizes have been used extensively for applications ranging from consumer electronics to cutting-edge instruments [6–8]. Among the various types of optical interferometers, the grating-based interferometric cavity is a typical demonstration, which brings together the advantages of both high integration and sub-nano- or even picometer accuracy [9–11]. As shown in Fig. 1, a grating-based interferometric cavity comprises a diffraction grating and a reflection mirror. The interference beam is formed by coherently superimposing the direct reflection diffraction beam and the beam after being reflected by the mirror. The phase difference of the cavity length is carried by the light intensity of the interferometric diffraction beam, which can be used to detect the displacement of movable components with high sensitivity [11]. Grating-based interferometric cavities have found widespread applications, such as atomic force microscopy applications [12,13], accelerometers [14–16], microphones [17–19], and light switches [20–23].
While the scheme incorporates several advantages, the positioning and integration of optoelectronic components, as well as the rapidly tightening requirements of compactness and sensitivity, pose multiple challenges. One of the most serious concerns is the suppression of undetectable orders (usually the detectable orders are the ±1st orders). Because the reflection of the 0th-order beam causes instability of the laser source, the system’s main energy is occupied by undetectable orders (especially the 0th-order), which decreases the signal-to-noise ratio (SNR). Many studies have reported the progress of grating-based interferometric cavities. For example, one study attempted to introduce an angle to the incident light to make it not completely perpendicular to the grating, preventing the 0th-order beam from returning to the laser and negatively impacting the system's performance [24,25]. However, this scheme is unable to fundamentally improve the energy without suppressing the 0th-order beam. The oblique incidence also brings difficulties in setting up. Other researchers, such as Degertekin and Hall et al. [26–28], have altered the structure of the light grid to address this issue. Degertekin achieved the inversion of the ±1st order beams and then regulated the light intensity of the interferometric diffraction beam by designing two adjacent opposite phase elements. Hall et al. interleaved the out-of-phase regions into a single periodic grating based on Degertekin's scheme, building a phase-modulated diffraction grating (PMDG) with four out-of-phase regions in one period. Therefore, the 0th-order reflected beam was eliminated in principle. Conventional gratings are usually fabricated by lithography and etching, holography lithography, electron beam lithography, and focused ion beam writing technique, etc. [29–38]. However, the PMDG existed useless higher-order diffraction beams and required demanding micromachining processes. In addition, the selection of the grating substrate and metal was not investigated.
In this paper, we systematically investigate the influence of fabrication errors and identify the interplay between the errors and the optical response for PMDGs, providing practical guidance for the fabrication of diffractive devices. On the basis of the hybrid error model, a process-tolerant grating with a period of 16 µm, a line width of 4 µm, and relative thicknesses of 100 ± 5 nm, 0 ± 5 nm, 240.75 ± 5 nm, and 416.83 ± 5 nm, benchmarked against the minimum thickness, is proposed. The designated refined PMDG exhibits wide process compatibility and is experimentally verified by micromachining with 0.5 µm machining precision and grating-based displacement measurements. The grating and the hybrid error model not only provide a specific design for grating-based displacement measurement devices but also allow systematic analysis of diffraction elements with practical fabrication limitations.
2. Theory and error model
2.1 Theoretical basis
As illustrated in Fig. 2(a), the grating can be separated into four sections in a single period, and the 0th-order beam can be obtained by adding the complex phasors returned from the four sections. Ui = exp(Φi) represents the complex vector of the beam reflected from the ith section, where Φi is the phase of each section. By adjusting the region thicknesses, the phase difference between U1 and U3 is 180°, and similarly, the phase difference between U2 and U4 is 180° at any distance between the PMDG and the reflecting mirror [26]. Figure 2(b) depicts the geometric representation of the vectors on a complex unit circle where the 0th-order beam can be eliminated. It requires ha and hb to be approximately ha = λ/4nsub and hb = λ/4 (nsub - 1), where λ is the wavelength of the incident laser in air, and nsub ≈ 1.45 is the refractive index of the grating substrate. The intensities of the interference of the magnitudes of the 0th- and ±1st order beams can be obtained in the Fraunhofer approximation for scalar diffraction:
Fig. 2. (a) Complex vectors of regions within a single period of a PMDG. (b) Geometric representation of vectors on a complex plane, Φ2-Φ1 = 2[(dgap-hb) + (hb-ha)×nsub]/λ.
The aforementioned calculation is based on scalar diffraction theory, which may contain a certain error. Therefore, we apply FDTD (Finite-Difference Time-Domain) and RCWA (Rigorous Coupled Wave Analysis) simulation to analyze the light-displacement responses precisely. Figure 3(a) depicts the results of the Fraunhofer approximation, FDTD and RCWA simulation. It shows that the +1 and -1 beams are complementary, allowing differential measurement that reduces the effects of common mode error. For FDTD simulation setting, we use the plane wave at the wavelength of 850 nm as input, with periodic boundary conditions in the horizontal direction and PML (Perfectly matched layer) in the other directions. The length, breadth, and height of the FDTD simulation region are 16, 20, and 2 µm, respectively. The mesh accuracy of the simulation is 7, and the smallest mesh size is 0.00025 µm, which is significantly lower than the wavelength. The FDTD results are consistent with the Fraunhofer approximation and RCWA. The FDTD simulation is a completely numerical method, and its accuracy depends on the mesh accuracy, boundary condition settings, and other factor settings, which explains the magnitude disparity. However, the high degree of concordance between the three outcomes suggests that the results are accurate and valid.
Fig. 3. (a) The comparison of the Fraunhofer approximation, FDTD simulation, and RCWA simulation results of normalized light intensity versus the displacement of the moveable mirror of different orders, where the red curves represent the Fraunhofer approximation results, and the blue curve represents the FDTD simulation results. (b) Electromagnetic field distribution of the grating in TM mode. (c) Electromagnetic field distribution of the grating in TE mode.
Figure 3(b) and (c) describe the electromagnetic field distribution of TM and TE modes, respectively. It can be seen that interference beams are formed through the grating and the reflector, and some of the light is absorbed by the boundary, so the total intensity in Fig. 3 (a) is not 1. Table 1 shows the maximum intensity of three central diffracted beams and energy efficiency of the grating in the two-polarization mode. The energy utilization coefficient is defined as the ratio of the total light intensity of the detectable order to the input light intensity, which can be expressed as η = Iout/Iin. In order to facilitate the analysis, we use the parameter of $\eta^{\prime}=I_{\pm 1}^{\max } / I_0^{\max }$ to refer to the previous coefficient of η because usually the 0th-order beam is the undetectable order, and the detectable orders are the ±1st orders. Herein, $I_{\pm 1}^{\max }$ and $I_0^{\max }$ represent the maximum light intensity of the detectable orders (±1st) and the maximum light intensity of the 0th-order. The TM-polarized light source has a better diffraction effect and is the investigation object in our simulation.
Table 1. The performance of the grating in the two polarization modes
2.2 Fabrication
Figure 4 (one period illustration) depicts the micromachining process of a PMDG, wherein the thickness of the fused silica substrate (SiO2) is 300-500 µm and the process flow is as follows: (a) the bare fused silica substrate is cleaned; (b) the positive photoresist is spin coated on the fused silica substrate, which is covered with mask plates on regions 1 and 2 and exposed to ultraviolet light, and the photoresist is dissolved in the developer solution after exposure, forming a pattern on regions 1 and 2; (c) reactive ion etching (RIE) is used to form the pattern shown in (b); (d) based on the same principle, the pattern is formed in region 3; (e) the pattern formed in (d), i.e. region 3, is etched to form a groove of a specified depth in order to obtain a stepped fused silica substrate, known as the second etching; (f) the stepped fused silica substrate is spin coated with the negative photoresist, which is covered with a mask plate on regions 1 and 3 and exposed to ultraviolet light, and the remaining regions are dissolved in the developer solution after exposure, resulting in a pattern on regions 2 and 4; (g) the electron beam evaporation coating technique is used to grow a Cr layer of a certain thickness on the fused silica substrate exposed in (f), i.e., regions 1 and 3, while the remaining ultraviolet adhesive is peeled off. Figure 4(h) is a schematic representation of the grating, where hb represents the first etch depth and hb-ha represents the second etch depth. All steps are based on existing, mature micromachining techniques with an accuracy of 1/8 linewidth to meet the requirements of grating interference cavities.
Fig. 4. Process flow of a PMDG. (a)-(g) The micromachining process. (h) Schematic diagram of the grating profile, where ha and hb are the heights of regions 3 and 4, respectively.
2.3 Error model
The PMDG process consists of two steps of etching and one step of coating, and etching and coating errors cannot be avoided in reality. The initial etching is performed on a single block of fused silica substrate, which is straightforward. The second etching is performed on the etched structure and is more challenging. Notably, the second etching and coating steps introduce errors relative to the ideal design, which can be mixed in the practical micromachining process. Herein, we construct a hybrid error model considering etching and coating errors. This model contains eight specific models, as shown in Fig. 5, which combine the alignment error of the coating step, as shown in Figs. 5(a)-(h), respectively.
Fig. 5. Error model for a PMDG with a period of 16 µm. (a) Model one, Δ = -8 µm. (b) Model two, Δ = -6 µm. (c) Model three, Δ = -4 µm. (d) Model four, Δ = -2 µm. (e) Model five, Δ = 0 µm. (f) Model six, Δ = 2 µm. (g) Model seven, Δ = 4 µm. (h) Model eight, Δ = 6 µm.
The grating period (P) is 16 µm, and the width of the four regions in a single period is 4 µm, which divides the single period from left to right into regions 1, 2, 3, and 4. The initial etching region consists of regions 1 and 2, the second etching region is region 3, and the coating region includes regions 1 and 3. The etching error Δ refers to the distance between the actual position of the second etching and region 3 when maintaining a line width of 4 µm. Following the corresponding model, the coating shifts from regions 1 and 3 to regions 2 and 4, holding a 4 µm line width, and the change in the coating position is referred to as the coating error d. Figure 5 depicts the ideal model and the eight specific models as follows: (a) Model one, Δ = -8 µm. The real position of the second etching is region 1, which is 8 µm from where it should be. (b) Model two, Δ = -6 µm. (c) Model three, Δ = -4 µm. (d) Model four, Δ = -2 µm. (e) Model five, Δ = 0 µm. (f) Model six, Δ = 2 µm. (g) Model seven, Δ = 4 µm. (h) Model eight, Δ = 6 µm. The second etching position is Δ off from where it should be. And the coating position is ɛ µm off from where it should be. Here, ɛ is a variable with values ranging from 0 ∼ 4 µm. This model is not limited to a period of 16 µm and can be easily extended to other scales by multiplying a factor. The etching and coating errors are continuously distributed. We assume that the error is a two-dimensional distribution centered at Δ = 0 and ɛ = 0 on -P/2 ∼ P/2, -P/8 ∼ P/8, as shown in Fig. 6(a). Most of the error models fall near ɛ = 0, nearly 90%. In extreme cases, error models are distributed in ɛ = 0, as shown in Fig. 6(b). Figure 6(b) shows the probability distribution of the error margins, with Δ = 0 as the center. The coating error ɛ in the range of 0 ∼ P/4 is equivalent to -P/8 ∼ P/8. Therefore, the subsequent discussion of the coating error interval is replaced by the 0 ∼ P/4.
Fig. 6. (a) The probability distribution of the error margins as a function of (Δ, ɛ) at P = 16 µm. (b) The normalized probability distribution of the error margins as a function of Δ at P = 16 µm and ɛ = 0 µm.
The primary target of the phase-modulated grating is to suppress the intensity of the 0th-order beam, so the energy utilization coefficient is an important parameter to evaluate the grating. All error models are analyzed with the accuracy of P/80 in the range of -P/2 ∼ P/2 for Δ and 0 ∼ P/4 for ɛ, obtaining enough response results. Figure 7 shows the normalized peak-to-peak value as a function of (Δ, ɛ) at P = 16 um. Figure 7(a), (b), and (c) represent the results of the +1st, -1st, and 0th-order beams for P = 16 µm, respectively. Since the major part of the error models are centered around ɛ = 0, we take the profile of ɛ = 0 from the 3D surface graph. Figure 7(d), (e), and (f) represent the results of the +1st, -1st, and 0th-order beams for P = 16 µm and ɛ = 0. It can be seen that the margins we choose are either the extreme point in the interval or the mutation point, especially at -6 ∼ -4 and 0 ∼ 8 µm. In order to highlight the intensity changes of the working orders (±1st), we take a machining precision of 2 µm as the unit of etching error models, because it is resource-saving and more effective. Regarding the coating error models, 0.2 µm machining precision is small enough and can be regarded as semi-continuous.
Fig. 7. The normalized peak-to-peak value as a function of (Δ, ɛ) at P = 16 µm. (a), (b) and (c) The normalized peak-to-peak value of +1st, -1st, and 0th-order beams as a function of (Δ, ɛ) at P = 16 um. (d), (e) and (f) The normalized peak-to-peak value of +1st, -1st, and 0th-order beams as a function of Δ at P = 16 µm m and ɛ = 0 µm.
3. Tolerance analysis
The performance indicators of a PMDG include absolute light intensity, optical contrast, and symmetry, whereas the influencing factors include the period, etching error Δ, and coating error ɛ. Herein, we analyze the hybrid error models with two period settings of 8 µm and 16 µm in detail to investigate the impact of process compatibility for a PMDG. Eight etching error models were established with an increment of Δ equal to P/8. Then, the cavity length and the coating error were swept simultaneously based on the etching error models to obtain the light-displacement responses. The swept cavity length was 1 µm, and the coating error ɛ varied from 0 to P/4.
Figure 8 depicts the normalized peak-to-peak light intensity of the ±1st and 0th-order beams as a function of Δ for the two-period settings and different ɛ. The larger the ɛ is, the weaker the curve color is. Regarding the case of P = 16 µm, it is obvious that a larger ɛ leads to a more deteriorated light intensity for the ±1st order beams. The primary function of the phase-modulated grating is to suppress the intensity of the 0th-order beam, so we should make the intensity of the 0th-order beam as small as possible and the working order intensity as strong. By setting the standard as follows: the maximum normalized light intensity of the ±1st order beams is larger than 0.4, while the maximum normalized light intensity of the 0th-order beam is smaller than 0.1, we obtained that the etching error model of Δ = 0 µm had the largest coating error tolerance range, which was up to 600 nm. In contrast, the etching error model of Δ = 4 µm exhibited the poorest coating error tolerance. It is easy to understand that the smaller Δ is, the larger the tolerance range of ɛ, which is also valid for the case of P = 8 µm. This indicates that doubling the period hardly affects the normalized peak-to-peak light intensity; in other words, a larger period brings up better process compatibility without compromising the absolute light intensity. Of course, P is not as larger as better. When P is large, it will cause the diffraction angle to be very small, which is not conducive to measurement. In a more extreme case, light may not be diffracted when it hits the grating, so P has an upper bound. In addition, the number of diffraction orders increases with the increase in P, and it inevitably decreases the energy utilization of the detectable orders (usually ±1st order beams).
Fig. 8. The normalized peak-to-peak value as a function of Δ. (a), (b) and (c) P = 16 µm. The normalized peak-to-peak value of the +1st, -1st and 0th-order beams as a function of Δ with six different ɛ. (d), (e) and (f) P = 8 µm. The normalized peak-to-peak value of the +1st, -1st, 0th-order beams as a function of Δ with six different ɛ.
The contrast is another major concern related to the sensitivity and energy utilization of the displacement measurement, which has the form C = (Imax - Imin)/(Imax + Imin). Imax is the maximum light intensity of the displacement-light intensity curve, Imin is the minimum light intensity. The displacement sensitivity is the slope of the curve. Figure 9 depicts the normalized light intensity of three central diffracted beams as a function of the displacement of the reflector for different Δ at ɛ = 0. The smaller the Δ is, the weaker the curve color is. Figure 9(a), (b), and (c) represent the results of the +1st order, -1st order, and 0th-order beams for P = 16 µm, respectively. The results show that each model exhibits a perfect periodicity of 420 ± 10 nm (λ/2). Table 2 shows the variation of contrast of the ±1st order beams with etching error Δ. All models have high optical contrast. The corresponding light-displacement sensitivity of the ±1st order beams at Δ = 0 µm is approximately 0.25%/nm. Figure 9(d), (e), and (f) represent the results of the ±1st and 0th-order beams for P = 8 µm. The high degree of consistency demonstrates that doubling the period barely affects the performance of the contrast and periodicity for a PMDG.
Fig. 9. Intensity for three central diffracted beams as a function of the grating-to-reflector distance at different Δ. (a), (b) and (c) P = 16 µm. The intensity of +1st, -1st and 0th-order beams as a function of the grating-to-reflector distance. (d), (e) and (f) P = 8 µm. The intensity of +1st, -1st and 0th-order beams as a function of the grating-to-reflector distance.
Table 2. Contrast of the +1st and -1st order with different etching errors Δ at P = 16 µm and ɛ = 0
Figure 10 shows the normalized peak-to-peak light intensity of the orders as a function of ɛ for different Δ. The tolerance range of the coating error can be obtained by using the aforementioned standard. Figure 9 (a), (b), and (c) represent the results of the +1st- order, -1st-order, and 0th-order for P = 16 µm, respectively. Herein, we obtained the coating error tolerance for P = 16 µm, as shown in Table 3. Table 3 shows the coating tolerance ranges for various models. Figures 7(d), (e), and (f) represent the results of the ±1st and 0th-order beams for P = 8 µm. The results show that for P = 8 µm, the tolerance range of the coating error is almost reduced to 1/2 of that for P = 16 µm.
Fig. 10. The normalized peak-to-peak value of three central diffracted beams of the eight models as a function of the coating error ɛ. (a), (b) and (c) P = 16 µm. The normalized peak-to-peak value of the +1st, -1st, and 0th-order beams as a function of ɛ. (d), (e) and (f) P = 8 µm. The normalized peak-to-peak value of the +1st, -1st, and 0th-order beams as a function of ɛ.
Table 3. The coating error tolerance as a function of etching errors Δ at P = 16 µm
The impact of the thickness variation of the coating films is also investigated by sweeping the thickness. Figure 11 illustrates the comparison of cases where the coating thickness h is smaller than, comparable to, and larger than the penetration depth of Cr. The penetration depth of Cr is 0.08 nm. It is noted that the light-displacement responses with different thicknesses are almost identical, indicating that the thickness of the coating films is not the key factor. This is well understood in principle because the coating films serve as an ideal reflector; thus, the thickness variation can be realistically tolerated as long as it is larger than the penetration depth of Cr.
Fig. 11. The intensity as a function of the grating-to-reflector distance for three different h. (a), (b) and (c) The intensity of the +1st, -1st, and 0th-order beams as a function of the grating-to-reflector distance for seven different h.
4. Experimental verification
To prove the validity of the model, a self-built setup, as illustrated in Fig. 12(a), was constructed with two designated gratings corresponding to error model five, wherein ɛ equals 0 and 4 µm. The red beam represents the invisible measurement light path, and the blue light (He-Ne laser) represents the visible indication light path which is used to indicate the invisible measurement optical path. A laser beam generated from a single longitudinal mode source was first collimated by a beam shaping device and then passed through an isolator. This beam normally struck the cavity, consisting of a fixed grating and a movable Ag reflector stuck to a piezoelectric ceramic. The intensity of the output light was obtained by using a photodetector in the perpendicular direction of the +1st-order beam.
Fig. 12. (a) Experimental configuration of the grating-based displacement measurement. (b) Output voltage of the laser as a function of time.
The input light stability was first tested by placing the photodetector between the beam prism and the grating, as shown in Fig. 12(b). The average intensity was 1.0623 V, and the root means square error (RMSE) was 0.2332 mV, on the order of 10−4. The light source diameter is 1.5 mm, covering 93 grating periods, and the divergence angle is 0.286°. The laser output is relatively stable and can be used in the experiment.
This experiment employed two grating samples, A and B, with distinct engraving flaws, not exactly consistent with the ideal error models. Figure 13(a) is a scanning electron microscopy (SEM) photograph of grating sample A, where V1, V2, V3, V4, and V5 are the lengths of the depicted parts. The heights of the regions shown in Fig. 13(b) are H1, H2, H3, and H4. Consequently, the corresponding model of grating sample A is depicted in Fig. 13(c), in which layer 2 and layer 5 represent Cr, and the rest is fused silica. The total period is 16.1127 µm, and a = 3.393 µm, b = 4.196 µm, I = 0.5357 µm, d = 3.482 µm, and f = 4.506 µm. Figure 13(d) is an SEM photograph of grating sample B, where V1 and V2 represent the line widths of the first and second etching regions, and V3 represents the line width of the coating region. The corresponding model of grating sample B is shown in Fig. 13(f), where layer 2, layer 4, and layer 5 represent Cr, and the remaining layers are fused silica. The period is 15.7133 µm, and a = 0.2963 µm, b = 3.9371 µm, d = 3.4997 µm, e = 0.4444 µm, f = 7.733 µm, I = 0.2593 µm, and g = 3.925 µm.
Fig. 13. SEM photographs of grating sample A with (a) in-plane dimensional parameters and (b) out-of-plane dimensional parameters. (c) Schematic profile of grating sample A. (d) SEM photographs of grating sample B with in-plane dimensional parameters and (e) out-of-plane dimensional parameters. (f) Schematic profile of grating sample B.
Figure 14 shows the simulation results and experimental results of the two interferometric cavities. Regarding sample A, the light-displacement responses of the simulation and experiment are shown in Fig. 14(a) and (b), whose responses are normalized to the average value of the incident light intensity. Although a small deviation of the actual values is observed, the comparison demonstrates similar trends of all orders with an identical periodicity of λ/2, which confirms the validity of the error model to some extent. The deviation can be considered to have been caused by the tiny difference between the actual structure and the measured dimensional parameters. In addition, the system errors of the setup, such as non-parallelism, may result in deviation. The non-parallelism of the cavity would inevitably lead to incomplete interference between two coherent beams, resulting in the decline of the contrast. Figure 14(c) and (d) show the light-displacement responses of the simulation and experiment for sample B, respectively. The curve trend of the experimental result is nearly consistent with that of the theoretical result, which effectively proves the suppression effect of the 0th-order. There is a certain difference between the actual values of the experimental and simulation results. In addition to the aforementioned reasons for sample A, this could be due to the undercutting phenomenon of the interference measurement, as shown in the bottom region of blue and red curves in Fig. 14(d), which is attributed to the inappropriate choice of the initial bias of the piezoelectric ceramic.
Fig. 14. Simulations and experimental results of normalized light intensity versus displacement for two grating samples. (a) Simulation results of grating sample A. (b) Experimental results of grating sample A. (c) Simulation results of grating sample B. (d) Experimental results of grating sample B.
In conclusion, although the experimental and simulation results show slight differences, the trends and features demonstrate high consistency. This not only proves the effectiveness of the designated process-tolerant phase-modulated grating but also further confirms the effectiveness of the error model, which helps to lay the foundation for the analysis and application of the phase-modulated grating.
5. Conclusion
In this work, a hybrid error model of a PMDG combining etching error and coating error is established, which further contains eight sub-models of etching error and twenty-one sub-models of coating error. A quantitative analysis of the relation between the errors and the optical response is provided for process-tolerant PMDGs with different periods. This PMDG maintains very tolerant process requirements, and the etching error and coating error can be up to 0.5 µm and 0.6 µm, respectively. Additionally, the impact of the thickness variation of the coating film is found to be almost negligible. The effectiveness of the hybrid error model is experimentally verified by micromachining two types of gratings and the corresponding grating-based displacement measurements. The accordance of the theoretical results with the experimental results confirms the validity of the model and analysis. The designated PMDG with great process compatibility serves as a promising candidate for diffractive optical elements for displacement measurements. Moreover, the hybrid error model can provide theoretical guidance for the fabrication of diffraction elements with limitations of micromachining fabrication.
Funding
National Natural Science Foundation of China (62004166); Natural Science Foundation of Zhejiang Province (LY23F040002); Natural Science Foundation of Ningbo (202003N4062); Equipment Development Department Rapid Support Project (80917020109-1); National Postdoctoral Program for Innovative Talents (BX20200279); Natural Science Basic Research Program of Shaanxi Province (2020JQ-199).
Disclosures
All of the authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
References
1. T. Kouh, D. Karabacak, D. H. Kim, and K. L. Ekinci, “Diffraction effects in optical interferometric displacement detection in nanoelectromechanical systems,” Appl. Phys. Lett. 86(1), 013106 (2005). [CrossRef]
2. D. Mason, J. Chen, M. Rossi, Y. Tsaturyan, and A. Schliesser, “Continuous force and displacement measurement below the standard quantum limit,” Nat. Phys. 15(8), 745–749 (2019). [CrossRef]
3. Y.-S. Jang, H. Liu, J. Yang, M. Yu, D.-L. Kwong, and C. W. Wong, “Nanometric precision distance metrology via hybrid spectrally resolved and homodyne interferometry in a single soliton frequency microcomb,” Phys. Rev. Lett. 126(2), 023903 (2021). [CrossRef]
4. M. L. Kuntzman, C. T. Garcia, A. G. Onaran, B. Avenson, K. D. Kirk, and N. A. Hall, “Performance and modeling of a fully packaged micromachined optical microphone,” J. Microelectromech. Syst. 20(4), 828–833 (2011). [CrossRef]
5. J. Dong, M. Liao, Q. Xu, L. Zhang, M. Tang, and J. Gong, “Detection and displacement characterization of landslides using multi-temporal satellite SAR interferometry: A case study of Danba County in the Dadu River Basin,” Eng. Geol. 240, 95–109 (2018). [CrossRef]
6. L. Yuan, J. Yang, and Z. Liu, “A compact fiber-optic flow velocity sensor based on a twin-core fiber Michelson interferometer,” IEEE Sensors J. 8(7), 1114–1117 (2008). [CrossRef]
7. S. Ghosh, S. keyvaninia, Y. Shoji, W. Van Roy, T. Mizumoto, G. Roelkens, and R. G. Baets, “Compact Mach–Zehnder interferometer Ce: YIG/SOI Optical Isolators,” IEEE Photonics Technol. Lett. 24(18), 1653–1656 (2012). [CrossRef]
8. Y. Han, B. Liu, Y. Wu, Y. Mao, J. Wu, L. Zhao, and Y. Liu, “Ultra-compact silicon-microcap based improved Michelson interferometer high-temperature sensor,” Opt. Express29(5), 6703 (2021). [CrossRef]
9. S. Gao, Z. Zhou, X. Bi, and L. Feng, “A low cross-axis sensitivity micro-grating accelerometer with double-layer cantilever beams,” IEEE Sensors J. 21(15), 16503–16509 (2021). [CrossRef]
10. J.-Y. Lee and G.-A. Jiang, “Displacement measurement using a wavelength-phase-shifting grating interferometer,” Opt. Express21(21), 25553 (2013). [CrossRef]
11. Q. Lu, C. Wang, J. Bai, K. Wang, W. Lian, S. Lou, and G. Yang, “Subnanometer resolution displacement sensor based on a grating interferometric cavity with intensity compensation and phase modulation,” Appl. Opt. 54(13), 4188 (2015). [CrossRef]
12. D. Rugar, H. J. Mamin, and P. Guethner, “Improved fiber-optic interferometer for atomic force microscopy,” Appl. Phys. Lett. 55(25), 2588–2590 (1989). [CrossRef]
13. N. M. Kronenberg, A. Liehm, J. A. Steude, J. G. Knipper, G. Borger, M. C. Scarcelli, and Gather, “Long-term imaging of cellular forces with high precision by elastic resonator interference stress microscopy,” Nat. Cell Biol. 19(7), 864–872 (2017). [CrossRef]
14. Q. Lu, Y. Wang, X. Wang, Y. Yao, X. Wang, and W. Huang, “Review of micromachined optical accelerometers: from mg to sub-µg,” Opto-Electron. Adv. 4(3), 200045 (2021). [CrossRef]
15. T. Li, R. Wang, and X. Qiao, “Demodulation of Fiber Bragg Grating accelerometer using in-line Sagnac interferometers,” IEEE Sensors J. 22(5), 4077–4082 (2022). [CrossRef]
16. F. Guzmán Cervantes, L. Kumanchik, J. Pratt, and J. M. Taylor, “High sensitivity optomechanical reference accelerometer over 10 kHz,” Appl. Phys. Lett. 104(22), 221111 (2014). [CrossRef]
17. N. A. Hall, Wook Lee, and F. L. Degertekin, “Capacitive micromachined ultrasonic transducers with diffraction-based integrated optical displacement detection,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr. 50(11), 1570–1580 (2003). [CrossRef]
18. N. A. Hall and F. L. Degertekin, “Integrated optical interferometric detection method for micromachined capacitive acoustic transducers,” Appl. Phys. Lett. 80(20), 3859–3861 (2002). [CrossRef]
19. N. Bilaniuk, “Optical microphone transduction techniques,” Appl. Acoust. 50(1), 35–63 (1997). [CrossRef]
20. Y. L. Li, N. N. Li, D. Wang, F. Chu, S. D. Lee, Y. W. Zheng, and Q. H. Wang, “Tunable liquid crystal grating based holographic 3D display system with wide viewing angle and large size,” Light: Sci. Appl. 11(1), 188 (2022). [CrossRef]
21. A. Ravindran, D. Nirmal, D. G. N. Prajoon, and Rani, “Optical grating techniques for MEMS-based spectrometer—a review,” IEEE Sensors J. 21(5), 5645–5655 (2021). [CrossRef]
22. U. Bortolozzo, S. Residori, and J.-P. Huignard, “kHz-speed optically induced phase gratings with liquid crystal light valves in transient dynamic mode,” Opt. Lett. 46(19), 4730 (2021). [CrossRef]
23. M. Mühleis and J. Hohl-Ebinger, “Programmable spectral shaping demonstrated at the solar spectral irradiance distribution,” Opt. Express29(6), 8223 (2021). [CrossRef]
24. C. Li, B. Yang, X. Zheng, X. Guo, Z. Sun, L. Zhou, and X. Huang, “2 ng/√Hz-resolution optomechanical accelerometer employing a three-dimensional MEMS interferometer,” Opt. Lett. 47(7), 1883 (2022). [CrossRef]
25. X. Wang, L. Feng, B. Yao, and X. Ren, “Sensitivity improvement of micro-grating accelerometer based on differential detection method,” Appl. Opt. 52(18), 4091 (2013). [CrossRef]
26. B. Van Gorp, A. G. Onaran, and F. L. Degertekin, “Integrated dual grating method for extended range interferometric displacement detection in probe microscopy,” Appl. Phys. Lett. 91(8), 083101 (2007). [CrossRef]
27. R. P. Williams, S. K. Hord, and N. A. Hall, “Optically read displacement detection using phase-modulated diffraction gratings with reduced zeroth-order reflections,” Appl. Phys. Lett. 110(15), 151104 (2017). [CrossRef]
28. D. Shin and B. Kim, “A laser interferometer encoder with two micromachined gratings generating phase-shifted quadrature,” J. Micromech. Microeng. 21(8), 085036 (2011). [CrossRef]
29. E. H. Anderson, C. M. Horwitz, and H. I. Smith, “Holographic lithography with thick photoresist,” Appl. Phys. Lett. 43(9), 874–875 (1983). [CrossRef]
30. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature 404(6773), 53–56 (2000). [CrossRef]
31. B. Liu, C. Zhan, X. Yao, S. Yan, and B. Ren, “Nanobowtie arrays with tunable materials and geometries fabricated by holographic lithography,” Nanoscale 12(41), 21401–21408 (2020). [CrossRef]
32. S. Kishimoto, M. Egashira, and N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32(3), 522–526 (1993). [CrossRef]
33. R. Winkler, J. D. Fowlkes, D. Rack, and H. Plank, “3D nanoprinting via focused electron beams,” J. Appl. Phys. 125(21), 210901 (2019). [CrossRef]
34. D. Yan, J. Cheng, and A. Apsel, “Fabrication of SOI-based nano-gratings for Moiré measurement using focused ion beam,” Sens. Actuators A Phys. 115(1), 60–66 (2004). [CrossRef]
35. P. Li, S. Chen, H. Dai, Z. Yang, Z. Chen, Y. Wang, and H. Duan, “Recent advances in focused ion beam nanofabrication for nanostructures and devices: Fundamentals and applications,” Nanoscale 13(3), 1529–1565 (2021). [CrossRef]
36. D. Post, B. Han, and P. Ifju, High sensitivity Moiré: experimental analysis for mechanics and materials, Springer-Verlag, New York, Chap.4 (1994). [CrossRef]
37. J. Shao, X. Chen, X. Li, H. Tian, C. Wang, and B. Lu, “Nanoimprint lithography for the manufacturing of flexible electronics,” Sci. China: Technol. Sci. 62(2), 175–198 (2019). [CrossRef]
38. J. McKelvie, D. Pritty, and C. A. Walker, “An automatic fringe analysis interferometer for rapid Moiré stress analysis,” in 4th European Electro-Optics Conference (SPIE, Bellingham), pp. 175–188 (1979).
