1. Introduction

With the rapid development of wireless communication technology and associated electronic devices, the issue of unwanted radio frequency (RF) emission has become a serious cause for concern [1,2] in terms of how it affects human health and the operation of electronic equipment [3]. Thus, electromagnetic interference (EMI) shielding has assumed vital importance in this context. EMI shielding is particularly difficult in applications wherein visual observation is needed, for example, optical windows utilized in aerospace equipment, electronic displays, mobile communication devices, etc [4,5]. For such applications, researchers have examined the use of transparent conductive films [6–8], thin metal films [9], transparent EMI shielding materials comprising graphene [10], carbon nanotubes (CNTs) [11] and carbon-based nanocomposites [12,13], band-pass frequency selective surfaces (FSSs) [14,15], and metallic mesh filters [16] to achieve EMI shielding. In particular, metallic meshes with sub-millimeter periods are widely used as high-pass filters to achieve high transmittance from the visible to infrared bands while providing wide-band low-frequency microwave shielding [17]. However, the inherent tradeoff between achieving high optical transmittance and satisfactory EMI shielding performance for the widely used square metallic meshes greatly limits their application. In such meshes, the high-order diffraction energy is mainly distributed along the two axes parallel to the mesh line directions, which contributes to the concentration of stray light and thus degrades the imaging quality [18]. Moreover, the hexagon mesh pattern, with the six sides of the hexagon mesh unit arranged in the three mesh line directions [19], has not proved to be a significant improvement over the square mesh pattern. In the context of the mesh pattern design, Murray et al. have reported on the randomized overlapping circle mesh pattern, which was shown to possess a favorable diffraction performance [20]; however, the shielding performance of this kind of mesh has not been the focus of many studies to the best of our knowledge. Therefore, it becomes essential to design new mesh structures to achieve high optical transmittance, strong EMI shielding, and uniform diffraction patterns while simultaneously guaranteeing reduced imaging-quality degradation.

In a recent work [21], we proposed a novel mesh design, in which “rotated” sub-ring arrays are introduced to decrease the high-order diffraction energy without affecting the transmittance. However, due to the structural complexity of the ring and sub-ring based meshes, it is a significant but not easy work to establish an optical transmission model for this kind of mesh. Therefore, in this study, we focus on the establishment of the optical transmission model; this approach can also be used to analyze and optimize mesh structures with other types of rings and sub-ring arrays. Moreover, we study the shielding performance both from theoretical and experimental aspects for the TRM with sub-rings. More importantly, we present a fast shielding effectiveness evaluation method for the TRM with sub-rings; it is the first time the model is established for the ring and sub-ring-based meshes, which can also further be expanded to other ring and sub-ring-based metallic meshes. The above works for EMI shielding analyzing along with the optical transmission model establishment provide complete theoretical basis for the design and optimize for the ring and sub- ring-based metallic meshes Both our theoretical analysis and experimental results show that the establishment of the optical transmission model and SE evaluation method are valid. And our proposed mesh design significantly mitigates the high-order diffraction energy and improves the SE when compared with the corresponding performances of the previously reported single-layer square mesh [17,18]. Importantly, the tradeoff between the transmittance and the SE is significantly reduced. Our proposed type of mesh pattern could find use in transparent EMI shielding applications, touchscreen devices, and displays where high optical transmittance, strong shielding effectiveness, an ultra-uniform diffraction pattern of stray light, and minimal image degradation are required.

2. Structural description and fabrication of proposed mesh

The ring unit has an advantage on homogenizing diffraction distribution over the other basic shapes because it has no fixed side arrangement direction [16,19]. In order to further overcome the concentration of the high-order diffraction energy of metallic meshes by increasing rings with different diameters and spatial positions, we present a ring mesh design with rotated sub-rings. Generally, the main physical mechanism that determines the uniform diffraction pattern is increasing the rings with different diameters at different spatial positions, which will generate more diffractive spots at different positions and lower single diffraction peak values. In this section, by adopting a previous reported triangular ring mesh with rotated sub-rings as an example [21], we briefly introduce the structure of this kind of mesh.

Figure 1(a) schematically shows microwaves and visible light passing through the proposed metallic mesh. In comparison with visible light, there is a strong reflection and rapid attenuation of microwaves passing through the metallic mesh. Figure 1(b) depicts the detailed structures of the mesh unit cells in the metallic mesh. A mesh unit cell is composed of a main ring (blue solid lines) with four inscribed sub-rings (yellow dashed lines) that are mutually tangential to each other. The diameter of the main ring is ($\sqrt{2}$ + 1) times that of the sub-ring in order to guarantee tangency between the main ring and sub-rings in the unit cells. In the figure, Unit Cells 2, 3, and 4 can be obtained by rotating Unit Cell 1 clockwise around the center of the main ring by angles of 22.5°, 45°, and 67.5°, respectively.

 

Fig. 1 (a) Schematic of metallic mesh. (b) Mesh unit cells with different angles of rotation.

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Triangular ring mesh (TRM) samples with “non-rotated” sub-rings and rotated sub-rings were fabricated on quartz-glass substrates with the use of the standard ultraviolet-photolithography process, which is a mature lithographic technique that offers high resolution.

In the study, the mesh metal material on the quartz-glass substrate was made of aluminum, which was selected for lower costs and favorable conductivity; further, aluminum film strongly adheres to the quartz-glass substrate while being highly corrosion-resistant. Chrome mesh masks were fabricated via electron beam direct writing. The overall size of the mesh sample was 4 inches, as shown in Fig. 2(a). Figure 2(b) shows the mesh unit cells arranged in a triangular distribution with the cells being electrically interconnected each other to ensure conductivity. Both the mesh samples had a unit cell length of 518 μm and thus a diameter of 214.6 μm for all the sub-rings, but for a fabrication error, the measured line width was a = 2.83 μm for the sample TRM with the rotated sub-rings and a = 4.35 μm for the TRM with non-rotated sub-rings.

 

Fig. 2 (a) Mesh sample photographs of transparent triangular ring mesh (TRM) with rotated sub-rings. (b) Micrograph of mesh sample acquired with a Nikon SMZ1500 stereomicroscope. (c) Atomic force microscopy (AFM) image of the mesh sample obtained with a Bruker Dimension Icon AFM.

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3. Transmission characteristics analysis and experiments

3.1 Model used for analysis of transmission characteristics

In order to illustrate the transmission characteristics of the TRM with rotated sub-rings, we constructed an optical intensity distribution model of Fraunhofer diffraction based on the Huygens–Fresnel diffraction theory. Figure 3(a) shows the schematic metallic mesh with a round aperture and Fig. 3(b) illustrates basic array of the TRM with rotated sub-rings. The entire mesh pattern can be obtained by periodically arranging the basic arrays in the horizontal and vertical directions. The basic array is composed of mesh unit cells numbered 1,2,3,4 as shown in Fig. 1(b). Obviously, the arraying period for the main ring arrays is (D, $\sqrt{3}$D) in two directions respectively as a triangular distribution. Compared to the main ring arrays, rotating makes a larger arraying period for sub-rings, which is (4D, $\sqrt{3}$D) as shown in Fig. 3(b). As the mesh pattern is a combination of main rings and sub-rings, the mesh transmission pupil function can be divided into two parts. Firstly, the pupil function of the main ring, arranged as a triangular distribution in a round aperture with diameter Dr = ND can be expressed as:

 

Fig. 3 (a) Schematic of the mesh with round aperture. (b) Basic array of the triangular ring mesh with rotated sub-rings. (c) Spatial coordinates of the mesh unit cell.

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Here, circ(r) represents a circle function, which is defined as circ(r) = 1, when r≤1;otherwise, circ(r) = 0. A the basic array of the main ring which consists of two rings, ** the 2D convolution symbol, ∑∑δ(x-nD)δ(y-√3mD) a 2D comb function, N the number of mesh unit cells arranged in the horizontal direction along the diametric length of the round aperture (Fig. 3(a)), D the diameter of the main ring, and a the line width of the ring.

It can be observed from Fig. 3(c) that the other sub-rings (indicated by dashed lines) in one unit cell can be obtained by rotating one sub-ring (solid lines), which can simplify the arithmetic expressions with the use of rotational coordinates. Similarly, from Fig. 2(b), we note that the rotating angular difference between the adjacent sub-rings in neighboring unit cells is the same, which is 22.5° as per design (see Fig. 1(b)). Thus, we can conclude that all the sub-rings can be obtained by rotating and translating one sub-ring in the main ring, which is described as expressions in the Eqs. (5) and (6). Based on this point, we adopt the rotation transformation formula to derive expressions for the sub-rings:

Here, D and d represent the diameters of the main ring (unit cell length) and sub-ring, respectively. A_k and B_k are the basic arrays for the sub-rings. In the formulas, when translating unit cells, 22.5° is the angular difference between neighboring unit cells, while 90° is the rotation angle when rotating one sub-ring to get other sub-rings in one unit cell. Based on the above expressions, we can express the transmission pupil function of the TRM with rotated sub-rings as follows:

According to the Fourier optics principle, the intensity point spread function (PSF) of an optical system is the modulus squared of the Fourier transform of its transmission pupil function [16], and thus, the normalized intensity of the mesh sample can be calculated as:

Here, $(\xi ,\eta )$ represents the spatial frequency corresponding to (x, y) on the simulation area, F(t) the Fourier transform and F*(t) its conjugate pair.

3.2 Transmission analysis and experiments

For optical/IR imaging applications, the uniform diffraction pattern is vital important when the light beam passes through the metallic mesh film. In fact, only the zeroth-order diffraction beam is beneficial to the imaging and observation, while the other high-order diffraction spots contributes to stray light, which causes the false target and degrades imaging quality.

The total transmittance for meshes here are designed to be 95.39% (zero-order transmittance, 91%), with the basic geometrical parameter D = 518 μm at the line width a = 2.5μm. By using the Eqs. (1) to (8), we can get the simulated diffraction distribution and normalized optical intensity results. It can be seen from Fig. 4(a) that TRM with non-rotated sub-rings shows a uniform high-order diffraction distribution, and TRM with rotated sub-rings exhibits a better result in Fig. 4b. Figures 4c-d show the normalized optical intensity of the two mesh patterns, and they are the projection profiles of the diffraction spots along the x-axis direction between y = −40 mm and y = 40 mm in Figs. 4a-b respectively.

 

Fig. 4 Simulation results of the diffraction distribution and the normalized optical intensity for different mesh patterns. Panes (a) and (c) correspond to the triangular ring mesh (TRM) with non-rotated sub-rings; (b) and (d) correspond to the TRM with rotated sub-rings.

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The two meshes show a zeroth-order transmittance of 91% (−0.04dB), which agrees with the value calculated by the obscuration ratio and demonstrates that rotating the sub-rings has no influence on the transmittance. Though the maximal normalized optical intensity (MNOI) is low for the TRM with non-rotated sub-rings, considering the effect of rotation, it continues to decrease from 0.0167% (−3.77dB) to 0.0064% (−4.19dB), and a 61.68% drop is achieved. The increasing trend in the number of diffraction spots and the decreasing trend in the MNOI illustrate that the new mesh designs own excellent diffraction characters which is favored in the applications where the uniform diffraction pattern are required.

Figure 5(a) depicts the schematic of the experimental setup used for the optical diffraction measurements and Figures. 5b-c shows the diffraction distribution patterns of the two mesh samples on the observation screen, as captured by a Nikon D810 camera. The laser used in the experiment was a single-frequency He–Ne laser with circular polarization, providing a Gaussian beam with a wavelength of 632.8 nm. The diameter of the laser beam emitted from the laser was 0.7 mm, which was expanded to 5 mm by means of a beam expander. Subsequently, the laser beam was focused onto the mesh sample, forming a diffraction pattern on an observation screen 3 m away.

 

Fig. 5 (a) Schematic of the optical diffraction measurement setup. (b) and (c) Experimental results of the diffraction distribution for the two mesh samples. (d) and (e) Measured gray values of optical intensity for mesh samples corresponding to (b) and (c).

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The measured visible transmittance and diffraction distribution of the two mesh samples (with rotated and non-rotated sub-ring arrays) have been reported in a recent work [21]. The normalized visible transmittances for the TRM with rotated and non-rotated sub-ring array samples are nearly 0.95 and 0.93, respectively. These ultra-uniform diffraction patterns in Figs. 5b-c verify the excellent stray light homogenizing effect that is achieved via introducing and rotating the sub-rings. And the corresponding MNOIs were 0.0999% and 0.0167%, obtained by comparing gray values in Figs. 5d-e to that of a standard square mesh (80, 0.1332%), respectively [21].

It is worth to point out that the rotation angle of the sub-rings can be selected and optimized based on the length of basic array and the number of sub-rings in one mesh unit cell, which provides huge flexibility in the design of this kind of mesh. Basically, the longer of the basic array, the better diffraction performance would be obtained. Generally, the most important principle of engineering the metallic meshes is increasing the rings with different diameters at different spatial positions. As shown in Eqs. (1) to (8), we have established the optical transmission model for TRM with four rotated sub-rings. If the mesh pattern is designed to possess different number of sub-rings or to be other ring-based patterns, the corresponding optical transmission model can be established by the similar process.

4. Electromagnetic shielding performance analysis and experiments

The shielding performance of material is evaluated by the shie lding effectiveness (SE), which is defined as the logarithmic ratio of incident power (Pi) to transmitted power (Po); that is, SE (dB) = −10log(Po/Pi). Obviously, the higher the dB level is, the less electromagnetic power transmitted through the materials. The equivalent circuit method, first proposed by Ulrich [22], is effective for analyzing the electromagnetic shielding effectiveness of square meshes. Kohin et al. [18] further expanded the scope of application of the equivalent circuit model to the case of oblique incidence of electromagnetic waves by considering the mesh as an equivalent film, and they proposed a real equivalent refractive index model. Moreover, Ciddor et al. proposed a complex equivalent refractive index model [23] to evaluate the SE of periodic metallic grids/meshes. Following this model, the real and imaginary parts of the complex refractive index can be expressed as below:

Here, n_e = n – ik represents the complex equivalent refractive index of the mesh-equivalent film, the incident electromagnetic wavelength, d_e the equivalent film thickness (which is always considered to be a small value), X the equivalent reactance, and R the loss resistance for the mesh.

In order to evaluate the SE of the ring mesh pattern with sub-rings with the equivalent circuit method, it is necessary to obtain the exact models of the equivalent reactance and loss resistance. Here, we establish the modified equivalent reactance and resistance model (MERRM) to obtain the equivalent reactance and loss resistance for the TRM with sub-rings as follows:

Here, h represents the number of sub-rings in one main ring, g the equivalent period of the TRM with sub-rings,$\omega =g/\lambda$ the normalized incident frequency, ${\omega }_0=1-0.41a/g$the normalized resonant frequency, ${\epsilon }_0$the dielectric constant of free space, c the velocity of light in vacuum, and σ = 3.45 × 10⁷ S/m is the bulk DC conductivity of the alumium.

As the incident wavelength in the RF shielding band is considerably larger than the unit cell length of the mesh sample, the rotating of the unit cells has little influence on the shielding performance. Furthermore, it has been previously reported [16] that the maximal mesh aperture is an important structural parameter to determine the microwave shielding performance of metallic meshes. We can observe from Figs. 2(b) that the maximal aperture of the mesh sample is determined by the diameter of the sub-rings, which lies on the diameter of the main ring and the number of sub-rings. Consequently, we proposed the equivalent mesh period g expressed in Eq. (11) as a key parameter in the SE evaluation, which allows simultaneous consideration of the diameter of the main ring (i.e., unit cell length D) and the number of sub-rings h in one unit cell and the equivalent mesh period g is also the diameter of the sub-ring. Equations (12) and (13) are semi-empirical formulas to obtain equivalent reactance and loss resistance of the mesh sample, which contain the key parameters of the mesh structure period g and line width a. Here, we obtained the equivalent period for the proposed mesh structures

Based on the MERRM, we obtained the complex equivalent refractive index n_e of the TRM with sub-rings using Eqs. (9) and (10). The metallic mesh and quartz-glass substrate can be treated as a thin film at low frequencies as per the equivalent film method (EFM) [23]. Subsequently, the reflectivity of the thin-film system calculated using the Fresnel formula method [17] can be used to characterize the SE since the metallic mesh film has a negligible effect on the absorption of electromagnetic energy in the low-frequency bands. In the study, in order to confirm the validity of the proposed model, we utilized the CST Microwave Studio software package, which is a numerical simulation software affording high accuracy, to analyze the SE for TRMs with various numbers of sub-rings h. The theoretical simulations obtained with the frequency domain solver (FDS) in CST and the EFM based on the MERRM using Eqs. (9) to (13) for TRMs with various numbers of sub-rings are plotted in Fig. 6(a). The agreement in the SE amplitudes and trends (Fig. 6(a)) between the two methods illustrates the validity and accuracy of the established model. At the same time, in comparison with the CST simulation, it is noteworthy that the EFM based on MERRM using Eqs. (9) to (13) can yield SE results thousands of times faster than the CST methods, which factor is a valuable consideration in designing and optimizing the structural parameters of this type of mesh.

 

Fig. 6 (a) Simulation results of the shielding effectiveness for the triangular ring mesh (TRM) with different numbers of sub-rings in one unit cell as obtained via EFM-MERRM and CST methods. (b) Measured curves and CST simulation results for the two fabricated mesh samples. #1 represents TRM with rotated sub-rings sample and #2 represents TRM with non-rotated sub-rings sample.

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The electromagnetic SE was obtained by means of a measurement system composed of an Agilent E8363B PNA series network analyzer, transmitter antenna, and receiver antenna. The mesh samples were placed at the central position between the transmitter and receiver antennas. This system was utilized to measure the transmission coefficient S₂₁ in the Ku band (12–18 GHz). From Fig. 6(b), it can be observed that the rotating of the unit cells has little influence on the shielding performance; there is only a small enhancement in the shielding performance with respect to that of the metallic mesh with non-rotated sub-rings, both in the measurement and CST simulation results. The simulated SE values are higher than the measured ones because of the inhomogeneity of the metal thickness, inhomogeneity in the line width of the fabricated mesh samples, the electrical conductivity decrease in the thin sputtered aluminum used in the mesh lines (when compared with bulk aluminum), and the surface oxidation of the aluminum mesh lines; these factors degrade the samples’ SE. We also need to consider the measurement error of less than ± 1 dB for the system used to measure the SE.

As the shielding mechanism is also an important aspect for an EMI shielding material [28,29], according to the sample size of our fabricated mesh, we measured the reflection coefficient S₁₁ in the Ka band (26.5-40 GHz) using a Radar-Cross-Section (RCS) measurement system in a microwave chamber. As shown in the Fig. 6(b), the measured reflectance of > – 0.6 dB (or reflectance > 87%) is observed in the range of 26.5–40 GHz, which is also verified by CST simulation. It can be seen that reflection is the major shielding mechanism which occupies most of the total energy in this band. In the microwave band, metal film behaves similarly and it can be concluded that shielding mechanism for the metallic mesh is reflection-dominant and absorption is little and secondary.

As shown in Fig. 7(a), based on the MERRM, we obtained simulated SE values for single-layer TRM with rotated sub-rings with different substrate thickness on a wider range of incident electromagnetic waves. As the double-layer metallic structures are used to achieve stronger shielding performance, the simulation results of the double-layer structures are calculated and shown in Fig. 7(b). It is easy to observe SE peaks and valleys in both figures, which are caused by the interference effect of electromagnetic waves corresponding to the wavelength/frequency and substrate thickness. It can be concluded from the simulation results that we can adopt appropriate substrate’s thickness and multi-layer structures to optimize shielding performances according to the different applications. For example, for a substrate thickness of 1 mm and incident electromagnetic wave frequency lower than 40 GHz, the SE of a single-layer TRM with rotated sub-rings is larger than 11dB, and that for a double-layer structure exceeds 31 dB, both of which are all favorable results in comparison to the traditional square mesh.

 

Fig. 7 Simulated SE values for single-layer (a) and double-layer (b) TRM with rotated sub-rings with different substrate thickness on an incident electromagnetic wave range of 1 – 200 GHz.

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In the study, we obtained a Ku-band SE >17 dB for the TRM with the rotated sub-rings, whose value indicates a favorable shielding performance with respect to those of the traditional square mesh and the reported typical EMI shielding materials listed in Table 1. It can be seen from the Fig. 6(b) that in the band of 12-18GHz, SE of the TRM with rotated sub-rings sample ranges from 21dB to 17dB, which reaches 21dB at the frequency of 12GHz. It is to be noted that the SE increases with decrease in electromagnetic wave frequency, as indicated both by simulation results in Fig. 6(a) and the trend of the SE curve in Fig. 6(b). We can then extrapolate that in the X band (8–12 GHz), the mesh sample can yield an SE >21 dB, which value is comparable or larger than the SE values of the EMI shielding materials listed in Table 1.

Table 1. Electromagnetic Shielding, Normalized Visible Transmittance and MNOI for Different Electromagnetic Interference (EMI) Shielding Materials

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5. Conclusion

In conclusion, we developed an optical transmission model and a fast SE evaluation method for the metallic mesh with rings and sub-ring arrays, whose approach can be applicable to other ring-based designs. The experimental results indicate that the TRM with rotated sub-rings affords a normalized visible transmittance that is greater than 95% and a Ku-band shielding effectiveness over 17 dB (98% energy attenuation) along with an ultra-uniform diffraction pattern. In comparison with the traditional single-layer square mesh, the new mesh design homogenizes the stray light caused by diffraction and reduces the MNOI while yielding almost the identical transmittance along with a >3-dB improvement in the shielding performance. This approach allows for the design of high-transparency EMI-shielding optical elements with high transmittance, strong shielding effectiveness, and reduced imaging-quality degradation, which cannot be achieved by the widely used square mesh.