Spatially multiplexed broadband and narrowband wavefront control metasurface devices and their design methods

By designing a superstructured surface device using a stacked dielectric structure layer, using a combination of a dual-nano column structure and a diagonal nano-column array, broadband and narrowband wavefront regulation is achieved, solving the problems of single functions and low space utilization in the prior art, and improving the versatility and applicability of the device.

CN118938360BActive Publication Date: 2025-05-13INNER MONGOLIA NEIGONG JIANGCHENG LOW ALTITUDE TECHNOLOGY CO LTD
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Patent Information

Application Number
CN202411050990.X
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2024-08-01
Publication Date
2025-05-13
Estimated Expiration
2044-08-01

AI Technical Summary

Technical Problem

Existing superstructure surface devices are difficult to achieve wideband and narrowband wavefront regulation at the same time, with low space utilization and relatively single functions.

Method used

A spatially multiplexed broadband and narrowband wavefront regulation superstructure device is designed, and a bottom-layer dielectric structure layer, an intermediate base dielectric layer and a top-layer dielectric structure layer are stacked in sequence from bottom to top. The bottom-layer dielectric structure layer is a double-nano column structure, and the top-layer dielectric structure layer includes a pair of nano-columns arranged on one diagonal line or two pairs of nano-columns arranged on two diagonal lines. Wavefront regulation is achieved by regulating the transmission phase and geometric phase.

Benefits of technology

The wideband and narrowband wavefront regulation is independently implemented in the entire space, improving the spatial utilization and functional diversity of superstructure surfaces, and is suitable for more complex application environments.

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Abstract

The present invention discloses a spatially multiplexed broadband and narrowband wavefront control metasurface device, comprising: a bottom dielectric structure layer, an intermediate base dielectric layer and a top dielectric structure layer stacked in sequence from bottom to top, wherein the bottom dielectric structure layer is a double nanocolumn structure, and the top dielectric structure layer comprises a pair of nanocolumns arranged on a diagonal line or two pairs of nanocolumns arranged on two diagonals, and the pair of nanocolumns on the diagonals have the same size; the present invention also discloses a design method for a spatially multiplexed broadband and narrowband wavefront control metasurface device; the present invention can independently realize broadband and narrowband wavefront control in the entire space.
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Description

Technical Field

[0001] The present invention relates to the technical field of electromagnetic wave phase and amplitude control, and in particular to a spatially multiplexed broadband and narrowband wavefront control metasurface device and a design method thereof. Background Art

[0002] Traditional optical systems achieve spatially multiplexed broadband and narrowband wavefront control by combining optical elements such as lenses, filters, and quarter-wave plates. The combination of these devices makes the entire system bulky and complex in design, which does not conform to the development trend of modern optical systems towards integration, planarization, miniaturization, and multifunctionality. In recent years, with the continuous development of optical technology, metasurfaces have emerged. Metasurfaces are functional film devices composed of artificially designed and arranged subwavelength structures, also known as two-dimensional metamaterials, which can flexibly control the phase, amplitude, polarization, etc. of electromagnetic waves, providing a new way for the miniaturization and planarization of optical systems.

[0003] At present, the designed metasurfaces mainly work in transmission or reflection space, with low space utilization and relatively simple functions. In order to increase the functions of metasurface devices, polarization multiplexing, wavelength multiplexing, orbital angular momentum multiplexing, and space multiplexing metasurfaces have emerged. Among them, spatial multiplexing is often achieved by multi-layer dielectric metal structures, regional arrangement structures, etc., but the design is relatively complex and there is a large crosstalk. In order to further simplify the design, a composite phase-based spatial multiplexing metasurface is generated. The transmission phase β is obtained by adjusting the size of the subwavelength structure, and the direction angle θ of the structure is adjusted to obtain the geometric phase = ±2θ. The two are combined to achieve spatial multiplexing. This method can improve the spatial utilization of the metasurface and increase the function while simplifying the design. However, limited by the broadband wavefront control characteristics of the geometric phase, it is difficult for the composite phase spatial multiplexing metasurface to achieve broadband and narrowband wavefront control at the same time.

[0004] Therefore, how to design spatially multiplexed broadband and narrowband wavefront control metasurfaces has become an urgent problem to be solved. Summary of the invention

[0005] In order to solve the problems existing in the prior art, the purpose of the present invention is to provide a spatially multiplexed broadband and narrowband wavefront control metasurface device and a design method thereof. The present invention can independently realize broadband and narrowband wavefront control in the entire space.

[0006] To achieve the above object, the technical solution adopted by the present invention is: a broadband and narrowband wavefront modulation metasurface device with spatial multiplexing, comprising: a bottom dielectric structure layer, a middle substrate dielectric layer, and a top dielectric structure layer stacked in sequence from bottom to top. The bottom dielectric structure layer is a double-nanopillar structure, and the top dielectric structure layer includes a pair of nanopillars disposed on one diagonal or two pairs of nanopillars disposed on two diagonals, and the pair of nanopillars on the diagonal have the same size.

[0007] As a further improvement of the present invention, the bottom dielectric structure layer, the middle substrate dielectric layer, and the top dielectric structure layer are selected from any one or two of silicon, silicon dioxide, germanium, magnesium fluoride, barium fluoride, titanium dioxide, and aluminum oxide.

[0008] As a further improvement of the present invention, the thickness of the bottom dielectric structure layer is H2, and λ0 / 20 < H2 < λ0; the thickness of the middle substrate dielectric layer is t, and λ0 / 20 < t < λ0; the thickness of the top dielectric structure layer is H1, and λ0 / 20 < H1 < λ0; the structural period of the metasurface device is P, and λ0 / 3 < P < λ0; where λ0 is the central wavelength of the incident light.

[0009] As a further improvement of the present invention, the shape of the double-nanopillar structure of the bottom dielectric structure layer is elliptical, rectangular, triangular, or circular, and when the double-nanopillar structure is elliptical, rectangular, or triangular, there is a relative rotation angle between the two nanopillars of the double-nanopillar structure, and the rotation angle is and

[0010] As a further improvement of the present invention, the shape of the nanopillars of the top dielectric structure layer is rectangular or elliptical, the relative rotation angle between adjacent two nanopillars is 45°, and when the shape of the nanopillars is rectangular, each rectangular nanopillar has 4 chamfers C, and 0 < C < λ0 / 2.

[0011] The present invention also provides a design method for the broadband and narrowband wavefront modulation metasurface device with spatial multiplexing as described above, comprising the following steps:

[0012] Step 1: Use the middle substrate dielectric layer as the substrate to support the bottom dielectric structure layer and the top dielectric structure layer;

[0013] Step 2: Form a double-nanopillar structure on the bottom dielectric structure layer, form a pair of nanopillars on one diagonal of the top dielectric structure layer, or form two pairs of nanopillars on two diagonals, and the pair of nanopillars on the diagonal have the same size.

[0014] As a further improvement of the present invention, forming a double-nanopillar structure on the bottom dielectric structure layer is specifically as follows:

[0015] Select any one of an ellipse, a rectangle, a triangle or a circle as one of the nano-columns in the double nano-column structure;

[0016] Align the center of the nano-column with the position (P / 4, P / 2), and then copy and move it by a distance D; λ0 / 10 < D < λ0;

[0017] The distance between the centers of the two ellipse structures is D, and λ0 / 10 < D < λ0;

[0018] When the double nano-column is an ellipse, a rectangle or a triangle, rotate the first nano-column by an angle, and the second nano-column is not rotated to form a double nano-column structure.

[0019] As a further improvement of the present invention, when the double nano-column is an ellipse, the major axis of the ellipse structure is d1, and λ0 / 20 < d1 < λ0; the minor axis of the ellipse structure is d2, and λ0 / 20 < d2 < λ0.

[0020] As a further improvement of the present invention, form a pair of nano-columns on one diagonal of the top dielectric structure layer, or form two pairs of nano-columns on two diagonals as follows:

[0021] Select any one of an ellipse and a rectangle as one of the nano-columns;

[0022] Align the center of the nano-column with the position (P / 4, 3P / 4), rotate it by an angle θ, copy it and move it to the diagonal position (3P / 4, P / 4) to form a pair of nano-columns. Align the center of the other nano-column with the position (P / 4, P / 4), rotate it by an angle θ + 45°, copy it and move it to the diagonal position (3P / 4, 3P / 4) to form another pair of nano-columns;

[0023] Place the two pairs of nano-columns in sequence to form the top dielectric structure layer.

[0024] As a further improvement of the present invention, when the nano-column is a rectangle, the length of the first pair of nano-columns is L2 and the width is W2, the length of the second pair of nano-columns is L1 and the width is W1, and λ0 / 10 < L1, L2 < λ0 / 2; λ0 / 10 < W1, W2 < λ0 / 2..

[0025] When left-handed circularly polarized light is incident, it is converted into right-handed circularly polarized light and reflected in a broadband range, and the phase is regulated by the combined action of the transmission phase and the geometric phase; while when right-handed circularly polarized light is incident, it is directly transmitted in a narrowband range, and the phase is regulated only by the geometric phase. The metasurface of the present invention can replace some optical components in a full-space point cloud generator and a virtual reality display system, meeting the requirements of optical system integration and device miniaturization with diversified functions.

[0026] The metasurface device designed by the present invention can realize broadband and narrowband wavefront control with spatial multiplexing, and independently realize broadband and narrowband wavefront control in the whole space. When left-handed circularly polarized light is incident, it is converted into right-handed circularly polarized light and reflected in a broadband range, obtaining a phase delay of Φ=2θ+β (θ is the direction angle of the subwavelength structure, β is the transmission phase); when right-handed circularly polarized light is incident, it is directly transmitted in a narrowband range, obtaining a phase delay of Φ=2θ.

[0027] The beneficial effects of the present invention are:

[0028] The present invention adopts a new double-layer structure, which can independently realize broadband and narrowband wavefront control in the entire spatial range; the metasurface has the advantages of multifunctional integration and planarization, and is suitable for more complex application environments. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 It is a structural schematic diagram of an embodiment of the present invention;

[0030] Figure 2 Schematic diagram of the simulation results of the amplitude and phase of the transmitted and reflected light in an embodiment of the present invention;

[0031] Figure 3 Schematic diagram of bandwidth simulation results according to an embodiment of the present invention.

[0032] Reference numerals:

[0033] 1-bottom dielectric structure layer, 2-middle base dielectric layer, 3-top dielectric structure layer. DETAILED DESCRIPTION

[0034] The embodiments of the present invention are described in detail below with reference to the accompanying drawings.

[0035] Example

[0036] like Figure 1 As shown in FIG. 1 , a schematic diagram of a unit structure of a metasurface device embodiment of spatial multiplexing broadband and narrowband wavefront control by composite design of geometric phase and transmission phase is shown; the metasurface device includes a bottom dielectric structure layer 1, an intermediate dielectric base layer 2, and a top dielectric structure layer 3 stacked from bottom to top. The bottom dielectric structure layer 1 is a plurality of unit structures, each unit structure includes two elliptical nanocolumns, and the rotation angle between two adjacent elliptical structures is The top dielectric structure layer 3 includes a plurality of super unit structures, and the super unit is composed of two pairs of rectangular nano columns.

[0037] The bottom dielectric structure layer, the middle dielectric base layer, and the top dielectric structure layer are selected from any one or two of silicon, silicon dioxide, germanium, magnesium fluoride, barium fluoride, titanium dioxide, and aluminum oxide; the thickness of the bottom dielectric structure layer is H2 and λ0 / 20 < H2 < λ0; the thickness of the middle dielectric base layer is t and λ0 / 20 < t < λ0; the thickness of the top dielectric structure layer is H1 and λ0 / 20 < H1 < λ0; the structural period is P and λ0 / 3 < P < λ0; where λ0 is the central wavelength of the incident light.

[0038] The double-elliptical nanocolumns can be modified into double-rectangular, double-triangular, and circular nanocolumns, and there is a relative rotation angle between the two elliptical nanocolumns, and the rotation angle is and The supercell is 2 pairs or 1 pair of rectangular or elliptical nanocolumns.

[0039] The design method of the metasurface device unit structure is as follows:

[0040] Taking the middle dielectric layer as the substrate to support the bottom dielectric structure layer and the top dielectric structure layer;

[0041] Form a plurality of double-elliptical nanocolumns on the bottom dielectric structure layer, and form a plurality of supercells composed of 2 pairs of rectangular nanocolumns with different sizes on the top dielectric structure layer.

[0042] The formation of the unit structure of the bottom dielectric structure layer 1 includes the following steps:

[0043] S1. Select any one of an ellipse, a rectangle, and a triangle;

[0044] S2. Align the center of the ellipse with the position (P / 4, P / 2), and then copy and move it by a distance D; the major axis of the elliptical structure is d1, and λ0 / 20 < d1 < λ0; the minor axis of the elliptical structure is d2, and λ0 / 20 < d2 < λ0; the distance between the centers of the two elliptical structures is D, and λ0 / 10 < D < λ0;

[0045] S3. Rotate the first elliptical cylinder by angle, and the second elliptical cylinder does not rotate to form a double-elliptical cylinder structure;

[0046] The formation of the supercell of the top dielectric structure layer includes the following steps:

[0047] S3. Select any one of an ellipse, a rectangle, a triangle, and a rhombus;

[0048] S4. Rotate a rectangular nanocolumn with length L2 and width W2 by an angle θ with its center corresponding to the position (P / 4, 3P / 4), copy it and move it to the diagonal position (3P / 4, P / 4) to form a pair of nanocolumns. Rotate another rectangular nanocolumn with length L1 and width W1 by an angle θ + 45° with its center corresponding to the position (P / 4, P / 4), copy it and move it to the diagonal position (3P / 4, 3P / 4) to form another pair of nanocolumns. The lengths of the two rectangular structures are L1 and L2, and λ0 / 10 < L1, L2 < λ0 / 2. The widths of the two rectangular structures are W1 and W2, and λ0 / 10 < W1, W2 < λ0 / 2.

[0049] S5. Place the two pairs of nanocolumns in sequence to form a supercell.

[0050] To design the above-mentioned spatially multiplexed broadband and narrowband wavefront modulation metasurface, the technical solution adopted by the present invention is: a design method for a spatially multiplexed broadband and narrowband wavefront modulation metasurface, which includes the following steps:

[0051] 1) Design the unit structure of the bottom dielectric layer;

[0052] 2) Continuously copy the unit structure to form the bottom dielectric structure layer (1);

[0053] 3) Design multiple unit structures of the top dielectric layer supercell;

[0054] 4) Rotate the designed top unit structure by an angle θ where 0 ≤ θ ≤ 360°, and arrange multiple supercell structures with different sizes to form the top dielectric structure layer;

[0055] 5) Place the bottom dielectric structure layer below the middle dielectric layer and place the top dielectric structure layer above the middle dielectric layer.

[0056] Among them, step 1) includes the following steps:

[0057] S1. Select any one of an ellipse, a rectangle, and a triangle;

[0058] S2. Align the center of the ellipse with the position (P / 4, P / 2), then copy it and move it by a distance D;

[0059] The major axis of the ellipse structure is d1, and λ0 / 20 < d1 < λ0. The minor axis of the ellipse structure is d2, and λ0 / 20 < d2 < λ0. The distance between the centers of the two ellipse structures is D, and λ0 / 10 < D < λ0. Rotate the first elliptical column by angle, and do not rotate the second elliptical column to form a double-elliptical column structure. The formation of the top dielectric structure layer supercell includes the following steps:

[0060] S4. Select any one of an ellipse, a rectangle, a triangle, and a rhombus;

[0061] S5. Rotate a rectangular nanocolumn with length L2 and width W2 by an angle θ with its center corresponding to the position (P / 4, 3P / 4), copy it and move it to the diagonal position (3P / 4, P / 4) to form a pair of nanocolumns. Rotate another rectangular nanocolumn with length L1 and width W1 by an angle θ + 45° with its center corresponding to the position (P / 4, P / 4), copy it and move it to the diagonal position (3P / 4, 3P / 4) to form another pair of nanocolumns. The lengths of the two rectangular structures are L1 and L2, and λ0 / 10 < L1, L2 < λ0 / 2. The widths of the two rectangular structures are W1 and W2, and λ0 / 10 < W1, W2 < λ0 / 2.

[0062] S6. Place the two pairs of nanocolumns in sequence to form a supercell.

[0063] The thickness of the bottom dielectric structure layer is H1 and λ0 / 20 < H1 < λ0. The thickness of the middle dielectric substrate layer is t and λ0 / 20 < t < λ0. The thickness of the top dielectric structure layer is H2 and λ0 / 20 < H2 < λ0. The unit cell structure period is P and λ0 / 3 < P < λ0. Here, λ0 is the central wavelength of the incident light.

[0064] As Figure 1 shown, in the broadband range of 10 - 11.5 μm, when the central wavelength λ0 is 10.6 μm, the unit cell structure period P = 8.1 μm. A designed metasurface device consists of a bottom dielectric structure layer (Si layer in this embodiment) with a thickness H2 = 4.1 μm, a middle dielectric substrate layer 2 (MgF2 layer in this embodiment) with a thickness t = 10 μm, and a top dielectric structure layer (Si layer in this embodiment) with a thickness H1 = 6 μm. In this embodiment, the bottom dielectric structure layer is composed of multiple double-ellipse nanocolumn structures with the same size. Figure 1 In the right figure, the major axis d1 of the elliptical nanocolumn is 5.25 μm, the minor axis d2 is 1.77 μm. Horizontally copy it and move it by a distance D = 4.05 μm to form a second elliptical nanocolumn. Rotate the first elliptical nanocolumn Composition unit structure double elliptical nanocolumns. The top dielectric structure layer is composed of multiple super unit structures, and the super unit is composed of 2 pairs of chamfered rectangular nanocolumns. The rectangular nanocolumns with a length of L2, a width of W2, a 45° chamfer and C = 0.32μm, the center of which corresponds to the position (P / 4, 3P / 4), are rotated at an angle of θ, copied and moved to the diagonal position of (3P / 4, P / 4) to form a pair of nanocolumns, and another rectangular nanocolumn with a length of L1 and a width of W1 corresponds to the position (P / 4, P / 4) at the center, rotated at an angle of θ+45°, copied and moved to the diagonal position of (3P / 4, 3P / 4) to form another pair of nanocolumns; 2 pairs of rectangular nanocolumns constitute a super unit. In this embodiment, a total of 8 super units are designed, and the structural dimensions are shown in Table 1:

[0065] Table 1

[0066]

[0067] The phase and amplitude of the 8 designed structures are as follows: Figure 2 As shown, when left-handed circularly polarized light (LCP) is incident, it is converted into right-handed circularly polarized light (RCP) and reflected, and the transmission phase β covers the range of 0-2π, and the reflection amplitude exceeds ~0.85. When RCP light is incident, it is directly transmitted, and the transmission transmission phase varies in the range of 3π / 4 to 5π / 4, which is approximately unchanged, and the transmission amplitude exceeds ~0.8. According to the principle of geometric phase, when the structural rotation angle θ varies in the range of 0 to 180°, the geometric phase can also cover the range of 0-2π (not listed in this embodiment). Therefore, the reflection space can obtain a phase of Φ=2θ+β, and the transmission space can obtain a phase of Φ=-2θ, which can realize independent regulation of the phase in the entire space.

[0068] To further demonstrate its bandwidth control capability, taking the 8th structure as an example, when LCP is incident, it is converted into RCP light and reflected. In the wavelength range of 10-11.5μm, the average amplitude is about 85%, achieving broadband control. When RCP light is incident, it is directly transmitted. At 10.6μm, the transmission amplitude is about 0.9, achieving narrowband transmission.

[0069] The above design process, embodiments and Figure 3 The simulation results shown verify the present invention well.

[0070] The above-mentioned embodiments only express the specific implementation of the present invention, and the description thereof is relatively specific and detailed, but it cannot be understood as limiting the scope of the present invention. It should be pointed out that, for ordinary technicians in this field, several variations and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. A spatially multiplexed broadband and narrowband wavefront control metasurface device, characterized in that: Comprising: A bottom dielectric structure layer, an intermediate base dielectric layer and a top dielectric structure layer are stacked sequentially from bottom to top, wherein the bottom dielectric structure layer is a double nanocolumn structure, and the double nanocolumn structure of the bottom dielectric structure layer is in the shape of an ellipse, a rectangle, a triangle or a circle, and when the double nanocolumn structure is in the shape of an ellipse, a rectangle or a triangle, there is a relative rotation angle between the two nanocolumns of the double nanocolumn structure, and the rotation angle is and The top dielectric structure layer includes a pair of nano-pillars arranged on a diagonal line or two pairs of nano-pillars arranged on two diagonal lines, and the pair of nano-pillars on the diagonal lines have the same size.

2. The spatially multiplexed broadband and narrowband wavefront control metasurface device according to claim 1, characterized in that: The bottom dielectric structure layer, the middle substrate dielectric layer, and the top dielectric structure layer are selected from any one or two of silicon, silicon dioxide, germanium, magnesium fluoride, barium fluoride, titanium dioxide, and aluminum oxide.

3. The spatially multiplexed broadband and narrowband wavefront control metasurface device according to claim 1, characterized in that: The thickness of the bottom dielectric structure layer is H2, and λ0 / 20 < H2 < λ0; the thickness of the middle substrate dielectric layer is t, and λ0 / 20 < t < λ0; the thickness of the top dielectric structure layer is H1, and λ0 / 20 < H1 < λ0; the structural period of the metasurface device is P, and λ0 / 3 < P < λ0; where λ0 is the central wavelength of the incident light.

4. The spatially multiplexed broadband and narrowband wavefront control metasurface device according to claim 1 or 3, characterized in that: The shape of the nanocolumns in the top dielectric structure layer is rectangular or elliptical, the relative rotation angle between two adjacent nanocolumns is 45°, and when the shape of the nanocolumn is rectangular, each rectangular nanocolumn has 4 chamfers C, and 0 < C < λ0 / 2.

5. A method for designing a spatially multiplexed broadband and narrowband wavefront control metasurface device as claimed in any one of claims 1 to 4, characterized in that: Including the following steps: Step 1: Using the middle substrate dielectric layer as the substrate to support the bottom dielectric structure layer and the top dielectric structure layer; Step 2: Forming a double-nanocolumn structure on the bottom dielectric structure layer, forming a pair of nanocolumns on one diagonal of the top dielectric structure layer, or forming two pairs of nanocolumns on two diagonals, and the sizes of the pair of nanocolumns on the diagonal are the same.

6. The method for designing a spatially multiplexed broadband and narrowband wavefront control metasurface device according to claim 5, characterized in that: Forming a double-nanocolumn structure on the bottom dielectric structure layer is specifically as follows: Selecting any one of elliptical, rectangular, triangular, or circular as one of the nanocolumns in the double-nanocolumn structure; Aligning the center of the nanocolumn with the position (P / 4, P / 2), and then copying and moving a distance D; λ0 / 10 < D < λ0; The distance between the centers of the two elliptical structures is D, and λ0 / 10 < D < λ0; When the double nanopillars are elliptical, rectangular or triangular, rotate the first nanopillar Angle, the second nanopillar does not rotate, forming a double nanopillar structure.

7. The method for designing a spatially multiplexed broadband and narrowband wavefront control metasurface device according to claim 5, characterized in that: When the double-nanocolumn is elliptical, the major axis of the elliptical structure is d1, and λ0 / 20 < d1 < λ0; the minor axis of the elliptical structure is d2, and λ0 / 20 < d2 < λ0.

8. The method for designing a spatially multiplexed broadband and narrowband wavefront control metasurface device according to claim 5, characterized in that: Forming a pair of nanocolumns on one diagonal of the top dielectric structure layer, or forming two pairs of nanocolumns on two diagonals is specifically as follows: Selecting any one of elliptical or rectangular as one of the nanocolumns; Aligning the center of the nanocolumn with the position (P / 4, 3P / 4), rotating by an angle θ, copying it and moving it to the diagonal position (3P / 4, P / 4) to form 1 pair of nanocolumns, aligning the center of the other nanocolumn with the position (P / 4, P / 4), rotating by an angle θ + 45°, copying it and moving it to the diagonal position (3P / 4, 3P / 4) to form another 1 pair of nanocolumns; Placing the 2 pairs of nanocolumns in sequence to form the top dielectric structure layer.

9. According to the space-division multiplexing broadband and narrowband wavefront modulation metasurface device described in claim 5 The design method is characterized in that When the nanocolumn is rectangular, the length of the first pair of nanocolumns is L2, and the width is W2, The length of the second pair of nanocolumns is L1, and the width is W1, and λ0 / 10 < L1, L2 < λ0 / 2; λ0 / 10 < W1, W2 < λ0 / 2.

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