Study on the influence of surface roughness on the diffraction efficiency of two-dimensional gratings

Yaowen Ban; Guobo Zhao; Zhenghui Zhang; Bangdao Chen; Bingheng Lu; Hongzhong Liu

doi:10.1364/OE.494470

1. Introduction

Two-dimensional gratings, as important optical components, possess characteristics of high precision, good reliability, and easy integration, making them widely used in the field of nanoscale two-dimensional displacement interferometry [1,2]. In the measurement of two-dimensional grating interferometry, the excellent diffraction characteristics of ±1st order diffraction of the two-dimensional gratings are crucial for obtaining optical signals and measuring displacements accurately [3–5]. The macroscopic parameters and microscopic surface features of the two-dimensional grating directly affect its diffraction efficiency, thus influencing the performance of the measurement system. Previous studies have mainly focused on the impact of macroscopic parameters of gratings on diffraction efficiency [6–8], but in practical applications, surface roughness is inevitably introduced during the manufacturing process of gratings, and the surface roughness of gratings also has a significant impact on diffraction efficiency, especially when measuring accuracy further extends to the nanoscale.

It is generally believed that surface roughness has a negative impact on diffraction efficiency [9–11]. Previous research methods have analyzed the impact of surface roughness using simplified mathematical models for specific types of gratings which cannot meet the research needs of studying the coupling effects of various roughness in more complex two-dimensional gratings [12–14]. Therefore, simulation software is often used to obtain computational solutions. In this study, the authors established a roughness model of two-dimensional gratings using commercial FDTD optical simulation software and analyzed for the first time the relationship between the roughness of two-dimensional gratings and diffraction efficiency by rigorously solving the Maxwell's equations. The roughness of the two-dimensional grating includes ridge surface roughness (RSR), groove surface roughness (GSR), and sidewall surface roughness (SSR). First, the significance of the impact of the three roughnesses on diffraction efficiency was studied using orthogonal experiments. Then, the impact of RSR, GSR, and SSR on diffraction efficiency was studied separately. These research results are of great significance for further understanding the diffraction characteristics of two-dimensional gratings and optimizing grating design and manufacturing processes, especially for high-precision optical applications such as nanoscale displacement interferometry.

2. FDTD simulation analysis

2.1 Two-dimensional grating roughness model established using FDTD

The research presented in this study focuses on a two-dimensional grating with a period of 4µm. When the distance between the grating and the observation screen is sufficiently large, Fraunhofer diffraction occurs, and the conditions for its occurrence can be expressed as [15]:

(1)$$\frac{{{p^2}}}{{z\lambda }} \ll 1, $$

where p is the period of the grating, z is the distance between the grating and the observation screen, and λ is the wavelength of the incident light.

A two-dimensional diffraction grating is an optical element with a two-dimensional periodic structure, and its Fraunhofer diffraction field has the spatial distribution as shown in Fig. 1 The diffraction orders include (0,0), (−1,0), (1,0), (0,−1), (0,1), (1,1), (−1,0), (−1,−1), and (1,−1). Unlike one-dimensional gratings, a two-dimensional diffraction grating has four additional diffraction spots along the diagonal directions. The intensities of the ±1st order diffraction fields in the x and y directions are higher than those in the diagonal directions. The ±1st order diffraction fields in the x and y directions are defined as (−1,0), (1,0), (0,1), and (0,−1). In most cases of interferometric measurements using two-dimensional gratings, the ±1st order diffraction fields in the x and y directions are used for two-dimensional displacement measurements. Therefore, we mainly investigate the diffraction efficiency of the ±1st order diffraction fields in the x and y directions of the two-dimensional grating. The general diffraction intensity distribution of a two-dimensional diffraction grating in the Fraunhofer diffraction field is given by:

(2)$$I({\sin {\theta_x},\sin {\theta_y}} )= {i_0} \cdot {\left( {\frac{{\sin {\alpha_x}}}{{{\alpha_x}}}} \right)^2} \cdot {\left( {\frac{{\sin {\alpha_y}}}{{{\alpha_y}}}} \right)^2} \cdot {\left( {\frac{{\sin {N_x}{\beta_x}}}{{\sin {\beta_x}}}} \right)^2} \cdot {\left( {\frac{{\sin {N_y}{\beta_y}}}{{\sin {\beta_y}}}} \right)^2}, $$

where

(3)$${\alpha _x} = \frac{{\pi a\sin {\theta _x}}}{\lambda }, {\alpha _y} = \frac{{\pi b\sin {\theta _y}}}{\lambda },{\beta _x} = \frac{{\pi {p_x}\sin {\theta _x}}}{\lambda },{\beta _y} = \frac{{\pi {p_y}\sin {\theta _y}}}{\lambda },$$

where a is the width of the grating ridge, b is the width of the grating groove, p is the grating spatial period, θ is the diffraction angle, α is the intensity unit factor, β is the intensity structure factor, i₀ represents the diffracted light intensity at the zero-order center of a single slit, which corresponds to the geometrical image point, λ is the wavelength of the incident light, and N is the number of unit cells within the periodic structure. The grating equation is as follows:

(4)$$p\sin \theta = k\lambda (k = 0, \pm 1, \pm 2, \ldots ). $$

Fig. 1. Schematic diagram of the spatial distribution of the Fraunhofer diffraction field for a two-dimensional diffraction grating and simulated spatial distribution of ±1st order diffracted field.

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The conditions for ±1st order diffraction light are k=±1. The intensity of ±1st order diffraction spots can be expressed as follows:

(5)$${I_{({ - 1,0} )}} = I\left( { - \frac{\lambda }{{{p_x}}},0} \right), {I_{({1,0} )}} = I\left( {\frac{\lambda }{{{p_x}}},0} \right),{I_{({0,1} )}} = I\left( {0,\frac{\lambda }{{{p_y}}}} \right),{I_{({0, - 1} )}} = I\left( {0, - \frac{\lambda }{{{p_y}}}} \right).$$

The diffraction efficiency of the ±1st order diffraction fields of a two-dimensional grating can be expressed as the ratio of the diffraction intensity I _± 1 to the incident intensity I₀:

(6)$${\eta _{ {\pm} 1}} = \frac{{{I_{({ - 1,0} )+ }}{I_{({1,0} )+ }}{I_{({0,1} )+ }}{I_{({0, - 1} )}}}}{{{I_0}}}. $$

The actual diffraction efficiency of a grating is influenced by the roughness of its surface. Typically, the grating surface is modeled as a Gaussian random micro-rough surface. As shown in Fig. 2, the surface morphology is usually determined by the mean height (h_r), root mean square roughness (σ), and correlation lengths (τ_x, τ_y, or τ_z). For RSR and GSR, the height function on the Gaussian random rough surface is given by:

(7)$$\left\langle {H({{{\vec{r}}_1}} )- {h_r}} \right\rangle \left\langle {H({{{\vec{r}}_2}} )- {h_r}} \right\rangle = {\sigma ^2}\textrm{exp} \left[ { - \frac{{|{{x_1} - {x_2}^2} |}}{{\tau_x^2}} + \frac{{|{{y_1} - y_2^2} |}}{{\tau_y^2}}} \right], $$

where $H\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_1}} \right)$ and $H\left( {{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} }_2}} \right)$ are the surface elevation heights at positions ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _1}$ and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _2}$, respectively. The angle brackets 〈〉 is the averaging operator, h_r is the mean height, σ is the root mean square roughness, and τ_x and τ_y are the correlation lengths in the x and y directions, respectively. For RSR and GSR, (x₁, y₁) and (x₂, y₂) are the coordinates of ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _1}$ and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _2}$, respectively.

Fig. 2. Schematic representation of two-dimensional grating surface roughness parameters.

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For SSR, the height function on the Gaussian random rough surface is given by:

(8)$$\left\langle {H({{{\vec{r}}_1}} )- {h_r}} \right\rangle \left\langle {H({{{\vec{r}}_2}} )- {h_r}} \right\rangle = {\sigma ^2}\textrm{exp} \left[ { - \frac{{|{{y_1} - {y_2}^2} |}}{{\tau_y^2}} + \frac{{|{{z_1} - z_2^2} |}}{{\tau_z^2}}} \right], $$

where τ_z and τ_y are the correlation lengths in the z and y directions, respectively. For SSR, (y₁, z₁) and (y₂, z₂) are the coordinates of ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _1}$ and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over r} _2}$, respectively.

The diffraction process of a two-dimensional grating with a rough surface is highly complex, and currently lacks a well-defined theoretical derivation, making numerical computations the primary approach. Therefore, this paper investigates the impact of surface roughness on the diffraction efficiency of the two-dimensional grating using a rigorous electromagnetic field solver - the Finite-Difference Time-Domain (FDTD) method. The FDTD method was first introduced by K.S. Yee in 1966 [16]. This method directly solves the Maxwell's equations to describe the spatiotemporal evolution of electromagnetic fields. The fundamental idea involves discretizing the electric field component (E) and magnetic field component (H) of the electromagnetic wave in both time and space using Yee cells, which are the discrete units in the computational grid. By this discretization, the Maxwell's equations, formulated in terms of time, can be transformed into difference equations, enabling a step-by-step advancement in time to solve for the electromagnetic field distributions within the spatial domain at each time instant. The analysis method for establishing roughness models in the FDTD has been widely applied in the field of micro and nano-optics. In this study, the rigorous electromagnetic field simulation of the two-dimensional grating roughness is performed using the commercially available FDTD Solution (Lumerical Solutions, Canada) numerical simulation software, which has been widely recognized for its reliability [17–20]. Wangpei Yu et al. used this roughness model in FDTD to investigate the impact of surface roughness on ellipsometric parameters [17]. Yu Wang et al. also employed this roughness model in FDTD to study the effects of surface roughness on device performance [18]. Their research demonstrated the capability of FDTD in analyzing the mechanisms of surface roughness on light propagation performance. Building upon their relevant work, this paper utilizes the FDTD method with a roughness model to analyze the influence of surface roughness on ±1 order diffracted light in two-dimensional gratings. The Advanced Surface Roughness module in the Rough Surface category of the Components tool is used to model the surface roughness [21,22]. A schematic diagram of the modeling process is shown in Fig. 3.

Fig. 3. Schematic diagram of roughness modeling for a two-dimensional grating in FDTD.

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In FDTD, the wavelength of the light source is set to λ=632.8 nm. The material of the grating is transparent glass with a refractive index of n = 1.4. Calculations based on Eq. (1) yield p²/λ=25.28µm. In the FDTD simulation, the distance between the 2D grating and the screen is set to z = 230µm, which approximately satisfies the condition p²/zλ<<1. The screen size is chosen to fully capture the ±1st order diffracted light, with dimensions of −69µm ≤ x ≤ 69µm and −69µm ≤ y ≤ 69µm. Perfectly matched layer (PML) boundary conditions are employed in the FDTD simulation. Additionally, due to the symmetric structure of the 2D grating, symmetric/anti-symmetric boundary conditions are applied to reduce memory requirements and improve computational efficiency.

2.2 Significance analysis of RSR, GSR, and SSR on diffraction efficiency

The macro parameters of the two-dimensional grating include groove type, groove depth (h), and duty cycle (a/p), where p is the period of the grating, and a is the length of the grating ridge. Previous researchers have conducted extensive studies on the influence of these parameters on diffraction efficiency. It has been found that a square groove type with a groove depth of π phase (0.791µm) and a duty cycle of 1/2 achieves the maximum diffraction efficiency for ±1st order diffraction. Building on previous research, we further investigate the impact of surface roughness on the diffraction efficiency of ±1st order diffraction. During the fabrication process of the two-dimensional diffraction grating using etching techniques, surface roughness is inevitably introduced, including RSR, GSR, and SSR. According to manufacturing experience, the roughness generated by etching is generally limited to no more than 0.5µm. For this study, the surface roughness is also specifically restricted to be less than 0.5µm.

First, we analyze the comprehensive influence of the three surface roughness factors on the diffraction efficiency of ±1st order diffraction. In practical situations, roughness does not occur separately on the three surfaces. Usually, each surface has roughness, and the roughness values on each surface are different. Therefore, in simulation experiments, it is necessary to consider the coupling analysis of the three surface factors simultaneously. Under the three surface factors, there are 9 root mean square (σ) variables for each roughness, resulting in a total of 729 experimental groups if all possible combinations of parameters are analyzed for their impact on diffraction efficiency. The scale of the simulation experiment is very large and cumbersome. Therefore, we chose to use orthogonal experimental design method for scientific analysis to reduce the complexity of experiments. The orthogonal experimental design includes three factors: RSR, GSR, and SSR, each with 9 levels of roughness root mean square (σ). Statistical analysis software SPSSAU was used to generate an orthogonal experimental table L81.9.3, which contains a total of 81 experimental groups.

A three-factor analysis of variance (ANOVA) was conducted to compare the proportional impact of three factors, namely different levels of surface roughness, on diffraction efficiency in simulated data. Degrees of freedom (DF) indicate the number of independent observations that can vary in the model or error. A higher degree of freedom implies a larger number of observations that can independently vary, indicating potential influence on experimental results. DF_Model= a_F−1, where a_F represents the number of levels for each factor. In the experimental design, there are three factor models, each having 9 degrees of freedom. Therefore, in this configuration, DF_RSR= 8, DF_GSR= 8, DF_SSR= 8. DF_Total= N−1, where N represents the total number of observations in the experimental data. In this study, there are 81 experimental groups, therefore, the total degrees of freedom for this configuration is DF_Total= 80. DF_Error = DF_Total- DF_Model(RSR + GSR + SSR) = 56. Sum of Squares refers to the sum of the squared values of a set of data. A higher sum of squares suggests larger effects of factors or errors in the model, which may have a significant impact on experimental results. The Sum of Squares can generally be obtained through relevant mathematical formulas. In practical applications, the values of the Sum of Squares can be directly calculated using SPASSU or Origin software. Mean Square is the result obtained by dividing the sum of squares by the corresponding degrees of freedom. A higher mean square implies larger variance in the effects of factors or errors in the model, which may have a greater explanatory power on experimental results. Mean Square = Sum of Squares/DF. P-value is the probability of observing values as extreme as the ones observed, given the data, in hypothesis testing. The P-value is obtained by referring to the F-distribution table based on the F-value and the degrees of freedom (DF_Model and DF_Error). Using ANOVA for significance testing, a larger F-value indicates more significant effects of factors in the model, potentially influencing experimental results to a greater extent. F-Value = Model Mean Square / Error Mean Square. From Table 1, it can be observed that the three factors have varying degrees of impact on the diffraction efficiency of ±1 order diffraction, with RSR having the largest impact, followed by GSR, and then SSR. Therefore, particular attention should be paid to the impact of RSR and GSR on the diffraction efficiency of ±1 order diffraction.

Table 1. Three-Factor ANOVA table

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2.3 Impact analysis of RSR, GSR, and SSR on diffraction efficiency

Next, we need to separately analyze how three types of surface roughness affect the ±1st order diffraction efficiency under nine experimental groups with different root mean square (σ) of surface roughness. First, we analyze the SSR which has the smallest proportion of impact.

Figure 4 shows the impact of different SSR σ on the ±1st order diffraction efficiency. The dashed line in the figure represents the diffraction efficiency without roughness (σ=0), the gray region represents the region of decreased diffraction efficiency, and the blue region represents the region of increased diffraction efficiency. Simulation data indicate that with the increase of SSR σ, the diffraction efficiency shows an upward trend, and the diffraction efficiency is always in the blue region. This phenomenon is different from previous understanding, which usually assumes that roughness would decrease the diffraction efficiency. However, SSR σ enhances the diffraction efficiency. Therefore, when manufacturing two-dimensional gratings, there is no need to worry about the introduction of SSR σ reducing the diffraction efficiency. At the same time, the impact of SSR σ on the diffraction efficiency is very small in proportion, so the influence of SSR σ can be neglected when considering strategies to optimize diffraction efficiency.

Fig. 4. Curves showing the influence of SSR σ on the diffraction efficiency of ±1st order diffracted light.

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Figure 5 shows the impact of different RSR σ on the ±1st order diffraction efficiency. Simulation data indicate that with the increase of RSR σ, the diffraction efficiency shows a decreasing trend, and the diffraction efficiency is always in the gray region. This phenomenon is consistent with the view that roughness would decrease the diffraction efficiency, which was held previously. Table 1 shows that the impact of RSR σ on the diffraction efficiency is the most significant, and should be the focus of consideration when manufacturing gratings. The increase of RSR σ is detrimental to the diffraction efficiency, so efforts should be made to achieve smaller RSR σ during manufacturing. Since the grating is manufactured using an etching process, where the grating groove is the etched area and the grating ridge is the non-etched area, the RSR σ is mainly determined by the surface roughness of the glass substrate itself, so a glass substrate with a smoother surface should be selected. Additionally, during the photolithography process, the photoresist adheres to the grating ridge surface, so the process parameters should be controlled during the resist stripping process to ensure thorough removal of the photoresist attached to the grating ridge surface, and minimize the RSR σ. Efforts should be made to overcome the inhibitory effect of RSR σ on the diffraction efficiency.

Fig. 5. Curves showing the influence of RSR σ on the diffraction efficiency of ±1st order diffracted light.

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Figure 6 illustrates the impact of different GSR σ on the diffraction efficiency of ±1 order diffracted light. Simulation data suggests that as the GSR σ increases, the diffraction efficiency initially increases and then decreases. In contrast to the case where the RSR σ suppresses the diffraction efficiency, the diffraction efficiency with the presence of GSR σ is higher than that without roughness, and the diffraction efficiency values are in the blue region. This indicates that GSR σ has a beneficial impact on the diffraction efficiency, and thus the introduction of GSR σ during the fabrication of 2D gratings is not expected to reduce the diffraction efficiency. As shown in Table 1, the impact of GSR σ on the diffraction efficiency is relatively significant, second only to the RSR σ, and should be given due consideration.

Fig. 6. Curves showing the influence of GSR σ on the diffraction efficiency of ±1st order diffracted light.

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In practical situations, the roughness of both the ridge surface and the groove surface, σ, may coexist, and coupling effects may occur when both roughnesses interact. Figure 7 displays the impact of the simultaneous presence of RSR σ and GSR σ on the diffraction efficiency of ±1 order diffracted light, where the simulation assumes that both roughnesses have the same value. Simulation data reveals that with an increase in surface roughness σ, the diffraction efficiency exhibits an initial increase followed by a decrease. Based on Table 1, as well as Fig. 5 and 6, it can be inferred that the weight of the RSR σ is the highest, and overall, the diffraction efficiency should decrease with an increase in roughness σ. However, under conditions of low roughness, the impact of GSR σ on the diffraction efficiency exceeds that of RSR σ, meaning that low roughness may actually enhance the diffraction efficiency.

Fig. 7. Curves showing the influence of simultaneous existence of RSR and GSR on the diffraction efficiency of ±1st order diffracted light.

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In summary, the fabrication process should aim to minimize the RSR σ as much as possible, overcoming the inhibitory effect of RSR σ on the diffraction efficiency, taking into consideration the overall impact of both RSR and GSR σ in a comprehensive manner.

3. Diffraction efficiency experiment

3.1 Manufacturing

The processing flow for 2D photolithography on glass substrates is depicted in Fig. 8. Initially, the glass substrate is cleaned using the SFQ-402LDKS cleaning equipment, following the standard Acid Clean and Etch (ACE) process for 10 minutes, followed by Quick Dump Rinse (QDR) for 10 minutes, Standard Piranha Mixture (SPM) for 10 minutes, Quick Dump Rinse (QDR) for 10 minutes, and Spin Dry, as shown in Fig. 8(a). Next, the photoresist is applied using the BEST TOOLS-PE coating equipment, with a rotation speed of 3000 rpm/min and a coating time of 40 seconds, as depicted in Fig. 8(b). Subsequently, the coated glass substrate is baked in the BEST TOOLS oven at 110℃ for 10 minutes. The baked substrate is then exposed using the Nikon G6 exposure equipment for 5.5 seconds with a light intensity of 8.8KJ/cm², as shown in Fig. 8(c). The exposed substrate is developed in the AC200-SE developer using a 2.38% tetramethylammonium hydroxide solution for 45 seconds. The developed substrate is hard baked at 115°C for 5 minutes in the BEST TOOLS hard bake equipment, as illustrated in Fig. 8(c). The substrate is then etched for 7.1 minutes using the Oxford Plasma Pro 100 Cobra300 etching equipment with CHF₃, O₂, and SF₆ gases, as shown in Fig. 8(d). Finally, the resist is removed using the IPC3000 stripping equipment with O₂ gas for 10 minutes, as depicted in Fig. 8(e). The processed glass substrate is then cleaned using the SFQ-402LDKS cleaning equipment, following the standard cleaning process of SPM for 10 minutes, QDR for 10 minutes, and Spin Dry.

Fig. 8. Process flow diagram for manufacturing 2D gratings.

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3.2 Characterization

In the experiment, we first evaluated the quality of the fabricated two-dimensional grating. A model OLS4000 3D measuring laser microscope (LSCM) produced by Olympus was used to test the grating samples, obtaining the period distribution and groove profile of the two-dimensional grating. In the LSCM, the equation for calculating surface roughness is the root mean square roughness (σ). The root mean square roughness (σ) is a statistical measure used to quantify the average roughness of the surface by analyzing deviations in surface height. Assuming a set of height data from N measurement points is represented as {z₁, z₂, …, z_N}, the equation for the root mean square roughness (σ) is given by:

(9)$$\sigma = \sqrt {\frac{1}{N}\mathop \sum \limits_{i = 1}^N z_i^2}.$$

As shown in Fig. 9, an approximately square groove profile was obtained, with a grating period (p) of 4µm, groove depth (h) of 0.790µm in the x-direction, groove depth (h) of 0.786µm in the y-direction, and a duty cycle of 1/2.

Furthermore, the roughness of the samples was detected using the roughness measurement module in the LSCM. Figure 10 shows the root mean square roughness (σ) data for 10 sets of measurements taken from the ridge surface and groove surface in the x-direction. The surface roughness measurement results for the sample are shown in Table 2, with average values of 0.0058µm and 0.0609µm for the ridge surface and groove surface, respectively. Similarly, using the same method, 10 sets of root mean square roughness (σ) data were measured for the ridge surface and groove surface in the y-direction, with average values of 0.0041µm and 0.1098µm, respectively.

Fig. 9. Photographs of the two-dimensional grating manufactured using LSCM for characterization of grating parameters.

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Fig. 10. Photographs of the roughness parameters of the two-dimensional grating manufactured using LSCM for characterization.

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Table 2. Results of surface roughness measurements using LSCM on the samples.

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3.3 Experiment

Further experiments were conducted to measure the diffraction efficiency. First, a two-dimensional grating diffraction efficiency measurement setup was constructed, as shown in Fig. 11, with a He-Ne laser (model 25-LHP-151-230) produced by Melles Griot as the light source, which emits linearly polarized light, and a digital optical power and energy meter (model PM100D) produced by Thorlabs was used to measure the diffraction light of ±1st order. In order to eliminate the influence of other diffraction orders on the measurements, we blocked those orders and conducted the experiment. This procedure was repeated for 10 sets, and the experimental results are summarized in Table 3. Figure 12 shows the results of the diffraction efficiency experiment for the fabricated grating sample, showing a measured diffraction efficiency of 0.4353, which represents one of the measurement sets. The average value of the diffraction efficiency measurements is 0.4352, and the standard deviation (STD) is 0.0007.

Fig. 11. Experimental setup diagram for measuring diffraction efficiency.

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Fig. 12. Experimental results of diffraction efficiency for the manufactured two-dimensional grating.

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Table 3. Experimental results of diffraction efficiency from 10 measurements.

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3.4 Discussion

The contribution of manufacturing errors to slight overcompensation of diffraction efficiency should be considered, with the main macroscopic parameters of the manufacturing errors being groove shape, groove depth (h), and duty cycle (a/p). The diffraction efficiency is maximized when the duty cycle 1/2 with ±1 order diffraction, and the diffraction intensity decreases when the duty cycle deviates from 1/2. The actual average groove depths in the x and y directions are 0.790µm and 0.786µm, respectively, and the maximum diffraction efficiency occurs at h = 0.791µm (π phase). When the phase deviates from π, the diffraction intensity decreases. Therefore, manufacturing errors in duty cycle and groove depth do not contribute to slight overcompensation of diffraction efficiency [8]. Table 4 summarizes the diffraction efficiency values of ±1st order for different conditions. The actual manufactured grating groove shape is not a standard square, and simulations show that the diffraction efficiency of a square groove shape with rounded edges is lower compared to that of a standard square groove shape. Based on the experimental parameters, we conducted simulations, and the simulated diffraction efficiency of the grating corresponding to the experimental parameters is 0.4335, which is basically match with the experimental results. Considering duty cycle, groove depth, and groove shape, we believe that small roughness actually enhances the diffraction efficiency.

Table 4. Values of ±1st order diffraction efficiency corresponding to different conditions.

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In addition, we will discuss the possible reasons why GSR and SSR can enhance the diffraction efficiency. The authors propose two potential reasons for this enhancement. One is macroscopic Effects. The macro parameters of the two-dimensional grating include groove type, groove depth (h), and duty cycle (a/p), where p is the period of the grating, and a is the length of the grating ridge, as shown in Fig. 13. Variations in these macroscopic parameters directly impact the diffraction efficiency. For GSR, the fluctuations in roughness on the groove surface can be equivalent to variations in groove depth, affecting the phase changes in light propagation, possibly leading to increased diffraction efficiency. For SSR, the fluctuations in roughness on the sidewall surface can be equivalent to variations in duty cycle, which may also enhance the diffraction efficiency. The other is microscopic Effects. Within the grating groove, there exists a structure resembling a microcavity, and its influence is only affected by GSR and SSR. In this microcavity-like structure, GSR and SSR can increase light scattering and multiple reflections, creating additional scattering centers. This leads to more phase changes in light on the rough surface, allowing more light to participate in the diffraction process, potentially leading to an increase in diffraction efficiency. Moreover, the microcavity-like structure formed by the roughness in the groove can efficiently couple various light waves with local surface plasmon resonance modes, creating a unique surface structure that enhances the diffraction effect and, consequently, the diffraction efficiency.

Fig. 13. Schematic diagram of the impact of surface roughness on light propagation.

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4. Conclusion

In this paper, the influence of surface roughness on the diffraction efficiency of two-dimensional gratings was studied using the FDTD method. The roughness of the RSR has an inhibitory effect on the diffraction efficiency, while the roughness of the GSR and SSR has an enhancing effect on the diffraction efficiency. Overall, the presence of roughness reduces the diffraction efficiency, but small roughness actually enhances the diffraction efficiency. Experimental investigations were carried out to study the diffraction efficiency of two-dimensional gratings, and the obtained experimental results are in substantial agreement with the conclusions. The findings of this research have important implications for the application and development of two-dimensional gratings in manufacturing and precision measurement techniques, providing valuable insights.

Funding

Fundamental Research Funds for the Central Universities (xzy022019002).

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 may be obtained from the authors upon reasonable request.

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Source of Variation	DF	Sum of Squares	Mean Square	F-Value
RSR	8	0.06128	0.00766	66.74961
GSR	8	0.04544	0.00568	49.5006
SSR	8	0.00357	4.47e-04	3.89203
Error	56	0.00643	1.14753e-04	0

		σ₁ (µm)	σ₂ (µm)	σ₃ (µm)	σ₄ (µm)	σ₅ (µm)	σ₆ (µm)	σ₇ (µm)	σ₈ (µm)	σ₉ (µm)	σ₁₀ (µm)	Mean (µm)
x	RSR	0.008	0.006	0.003	0.005	0.006	0.005	0.004	0.007	0.006	0.009	0.0058
x	GSR	0.076	0.053	0.057	0.058	0.055	0.056	0.061	0.072	0.063	0.054	0.0609
y	RSR	0.004	0.003	0.004	0.005	0.005	0.008	0.004	0.003	0.004	0.006	0.0041
y	GSR	0.115	0.103	0.144	0.128	0.112	0.095	0.074	0.111	0.125	0.081	0.1098

Source of Variation	DF	Sum of Squares	Mean Square	F-Value
RSR	8	0.06128	0.00766	66.74961
GSR	8	0.04544	0.00568	49.5006
SSR	8	0.00357	4.47e-04	3.89203
Error	56	0.00643	1.14753e-04	0

		σ₁ (µm)	σ₂ (µm)	σ₃ (µm)	σ₄ (µm)	σ₅ (µm)	σ₆ (µm)	σ₇ (µm)	σ₈ (µm)	σ₉ (µm)	σ₁₀ (µm)	Mean (µm)
x	RSR	0.008	0.006	0.003	0.005	0.006	0.005	0.004	0.007	0.006	0.009	0.0058
x	GSR	0.076	0.053	0.057	0.058	0.055	0.056	0.061	0.072	0.063	0.054	0.0609
y	RSR	0.004	0.003	0.004	0.005	0.005	0.008	0.004	0.003	0.004	0.006	0.0041
y	GSR	0.115	0.103	0.144	0.128	0.112	0.095	0.074	0.111	0.125	0.081	0.1098

Study on the influence of surface roughness on the diffraction efficiency of two-dimensional gratings

Abstract

1. Introduction

2. FDTD simulation analysis

2.1 Two-dimensional grating roughness model established using FDTD

2.2 Significance analysis of RSR, GSR, and SSR on diffraction efficiency

2.3 Impact analysis of RSR, GSR, and SSR on diffraction efficiency

3. Diffraction efficiency experiment

3.1 Manufacturing

3.2 Characterization

3.3 Experiment

3.4 Discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (13)

Tables (4)

Equations (9)

Optics Express