Method for 3D printing a multi-directional permittivity continuously graded luneburg lens

Abstract

The application discloses a method for manufacturing a multi-directional dielectric constant continuously gradient dragon-burn lens through 3D printing, and relates to the technical field of dragon-burn lens manufacturing, and comprises the following steps: determining a dielectric constant distribution model of the dragon-burn lens; determining a working frequency and a wavelength; introducing a super material unit concept and determining the size thereof; determining the size of the lens and establishing a dielectric constant space distribution formula; expounding a dielectric constant regulation principle of the super material unit; defining the external shape, internal structure and size of the super material unit; establishing a functional relationship between the structure wall thickness of the super material unit and the dielectric constant; dividing a sphere into super material units and performing a differential operation; assigning the structure wall thickness to the differential super material units corresponding to each point in the space of the dragon-burn lens, and modeling the dragon-burn lens model; and manufacturing the dragon-burn lens through 3D printing technology; the dielectric constant of different regions of the dragon-burn lens is matched through the design process, so that the gain and efficiency of the dragon-burn lens antenna are improved.

Description

Technical Field

[0001] This invention relates to the field of Luneburg lens manufacturing technology, specifically a method for 3D printing Luneburg lenses with continuously varying dielectric constants in multiple directions. Background Technology

[0002] Luneburg lenses were first proposed in 1944 by American mathematician Rudolf Karl Lüneburg. Traditionally manufactured Luneburg lenses are composed of nested layered structures with different dielectric constants, similar to an onion structure. This results in problems such as the inability to achieve natural gradient changes, high interlayer interface losses, and unstable dielectric constants in each foamed layer. Conventional 3D-printed Luneburg lenses can only achieve discontinuous uniform gradient dielectrics because the size of each metamaterial unit constituting the Luneburg lens remains constant and the wall thickness of the internal structure is fixed. This allows for only intermittent changes in dielectric constant, rather than continuous and uniform variations. In existing technologies, Luneburg lenses are mainly manufactured layer by layer using fused deposition modeling (FDM) or solidification laser ablation (SLA) techniques. Gradual changes in dielectric constant are achieved by adjusting the material ratio. For example, some companies use thermoplastic materials mixed with ceramic powder to simulate gradient structures by adjusting the interlayer composition. Furthermore, some solutions use 3D printing to prefabricate molds and then fill them with artificially designed multilayered metamaterial structures (such as the spherical layer-by-layer filling technology of Guangdong Haixin Communication). However, this requires the separate fabrication of metamaterial units, making the process chain complex. Traditional 3D printing relies on adjusting the mixing ratio of materials, making it difficult to achieve continuous gradients in multiple directions. This results in a lens transmittance of only about 50%, which cannot meet the requirements of complex electromagnetic wave control. It also leads to insufficient precision in controlling the change of dielectric constant. In addition, multiple printing-filling-curing cycles are required, such as in metamaterial solutions, which can take up to several weeks. High-precision printing equipment and special materials, such as gallium nitride, drive up manufacturing costs. Furthermore, due to material and process limitations, the interlayer bonding strength is low, and delamination is prone to occur under high temperature environments. This makes it difficult to meet the requirements for long-term aging resistance and cannot achieve uniform gradient changes, limiting the performance of millimeter-wave bands and resulting in limited structural performance. Summary of the Invention

[0003] This invention provides a method for 3D printing Luneburg lenses with continuously gradient dielectric constants in multiple directions. This method effectively solves the problems mentioned in the background section, where traditional 3D printing relies on adjusting material mixing ratios, making it difficult to achieve continuous gradients in multiple directions. This results in a lens transmittance of only about 50%, failing to meet the requirements of complex electromagnetic wave control and causing insufficient precision in controlling dielectric constant changes. Furthermore, it requires multiple printing-filling-curing cycles, such as metamaterial solutions, which can take weeks. High-precision printing equipment and special materials, such as gallium nitride, drive up manufacturing costs. Additionally, due to material and process limitations, interlayer bonding strength is low, delamination easily occurs under high-temperature environments, making it difficult to meet long-term aging resistance requirements and preventing the achievement of uniform gradient changes, thus limiting millimeter-wave band performance and resulting in limited structural performance.

[0004] To achieve the above objectives, the present invention provides the following technical solution: a method for 3D printing to fabricate a Luneburg lens with a continuously gradient dielectric constant in multiple directions. Through a unique metamaterial unit structure design constituting the Luneburg lens, the method enables the fabrication of a Luneburg lens with a continuously gradient dielectric constant in multiple directions using 3D printing, thus more closely resembling an ideal Luneburg lens and improving gain and efficiency. The method includes the following steps: Step S1: Determine the dielectric constant distribution model of the Luneburg lens; Step S2: Determine the operating frequency and wavelength; Step S3: Introduce the concept of metamaterial units and determine their dimensions; Step S4: Determine the lens size and establish the spatial distribution formula for the dielectric constant; Step S5 explains the principle of dielectric constant modulation of metamaterial units; Step S6: Define the external shape, internal structure, and dimensions of the metamaterial unit; Step S7: Establish the functional relationship between the wall thickness and dielectric constant of the metamaterial unit structure; Step S8: Divide the sphere into metamaterial units and perform differentiation operations; Step S9: Assign the structural wall thickness to the metamaterial units corresponding to the differentials of each point in the Luneburg lens space to model the Luneburg lens model. Step S10: Manufacture Luneburg lenses using 3D printing technology.

[0005] According to the above technical solution, in step S1, for an ideal Luneburg lens with radius R, the relevant dielectric constant at a distance r from the center of the sphere can be calculated according to the following formula, where the dielectric constant at the center of the sphere is 2, and the dielectric constant at the surface of the sphere is 1: ; An ideal Luneburg lens needs to satisfy the following conditions: the dielectric constant at the center of the sphere is 2, the dielectric constant at the surface of the sphere is 1, and the change in dielectric constant from the center to the surface of the sphere is a uniform gradient. Due to the limitations of materials and manufacturing processes, it is difficult to achieve a surface dielectric constant of 1. Therefore, the dielectric constant of the sphere surface is generally controlled between 1 and 1.3. For a Luneburg lens of radius R under non-ideal conditions, the relevant dielectric constant at a distance r from the center of the sphere can be calculated using the following formula, where It is the dielectric constant at the center of the sphere. It is the dielectric constant of the surface of the sphere: .

[0006] According to the above technical solution, in step S2, the applicable electromagnetic wave operating frequency range of the Luneburg lens should first be determined, and its wavelength should be calculated.

[0007] According to the above technical solution, in step S3, the Luneburg lens made by 3D printing is composed of several metamaterial units with different structural sizes to achieve a gradient effect. According to the equivalent medium theory in metamaterial research, when the size of a unit structure composed of different materials is much smaller than the working wavelength, electromagnetic waves cannot distinguish its internal structure, and this metamaterial unit can be regarded as a uniform medium. Choose the size of a metamaterial unit, which is generally less than 1 / 4 of the wavelength, usually 1 / 5 or 1 / 10.

[0008] According to the above technical solution, in step S4, the diameter of the Luneburg lens is determined, and the radius of the Luneburg lens is set to R. The dielectric constant of the Luneburg lens is designed to be such that the center of the sphere is... The dielectric constant of the sphere surface is ; Establish a three-dimensional coordinate system, with the center point (x, y, z) of the sphere as (0, 0, 0). The dielectric constant of each point on the sphere should satisfy the following formula: ; This allows us to obtain a data table of points in coordinate space and their dielectric constant values.

[0009] According to the above technical solution, in step S5, the metamaterial unit is composed of dry air and one or more other materials, and the dielectric constant of the metamaterial unit is ε. m The dielectric constant of dry air is ε0, and the volume percentage of dry air in the metamaterial unit is ν0. The dielectric constant of material 1 is ε1, and the volume percentage of material 1 in the metamaterial unit is ν1. The dielectric constant of material n is ε. n The volume percentage of material n in the metamaterial unit is ν. n ; According to the Brown linear model in the Lichtenecker-Rother (LR) equations, the dielectric constant of the metamaterial unit is: ; The volume percentage ν of material n in the metamaterial element can be changed by altering the structural properties of material n within the corresponding element. n ; The proportion of dry air is ν0 = 1 - ν1 - ν2 - ... - ν n Therefore, when the material composition of the metamaterial unit is fixed, the properties of the unit structure corresponding to material n have a functional relationship with the dielectric constant of the metamaterial unit. The dielectric constant of the metamaterial unit can be changed by changing the properties of the unit structure, thereby achieving the customization of the dielectric constant. In addition, by changing the material n with different dielectric constants, the maximum dielectric constant value of the metamaterial unit can be controlled. The following explanation is based on the metamaterial unit being composed of dry air and another material.

[0010] According to the above technical solution, in step S6, the metamaterial unit can adopt a variety of external shapes, including cuboids, cubes, and polyhedra. The dimensions of the metamaterial unit in the X, Y, and Z directions can be equal or unequal, as long as they satisfy the equivalent medium theory that the dimensions are much smaller than the wavelength. The unit can also adopt a variety of internal structures. This paper proposes a metamaterial unit that can achieve uniform gradient deformation. This unit can perform one or more deformation operations without changing the structural volume ratio and wall thickness characteristics, including the following methods: unidirectional or multidirectional movement; uniaxial or multiaxial rotation; unidirectional or multidirectional scaling; uniaxial or multiaxial torsion; unidirectional or multidirectional bending; and merging of multiple sub-metamaterial units. The dimensions of the individual metamaterial unit in the X, Y, and Z directions before merging must be much smaller than the wavelength, thus satisfying the equivalent medium theory.

[0011] According to the above technical solution, in step S7, multiple samples composed of metamaterial units with different structural wall thicknesses are printed using 3D printing technology. The dielectric constant of the samples is tested at the electromagnetic wave operating frequency, and the metamaterial unit structural wall thickness L is obtained by fitting using computer technology. n With dielectric constant ε m From the functional relationship, we can further obtain the relationship between the corresponding spatial coordinate points and the wall thickness of the corresponding metamaterial element. This is explained here using the material's relational function, as follows: ; The calculated data table of each point in space and the corresponding metamaterial element wall thickness is as follows: ; Further mathematical calculations can be used to obtain Right now The relationship between the x, y, and z coordinates.

[0012] According to the above technical solution, in step S8, the sphere with radius R is divided into multiple metamaterial units according to the size of a*b*c metamaterial units. The external shape of the metamaterial unit is a cube or cuboid, wherein a, b, and c are less than or equal to the size of the metamaterial unit selected in step S3. The large metamaterial unit is divided into the smallest sub-metamaterial units. The coordinates of the point are taken from the structural centerline of each sub-metamaterial unit. The structural unit can be differentiated along the direction of the structural centerline. The structural wall thickness L at this point is calculated according to the formula in step S7. L is assigned to the wall thickness of the differential structure to which the centerline belongs. It can be seen that the smaller metamaterial unit of the differential structure conforms to the relationship between structural wall thickness and dielectric constant in step S7. At the same time, the size is much smaller than the electromagnetic wave wavelength, which satisfies the equivalent medium theory and meets the application requirements. Thus, the dielectric constant can be continuously and uniformly varied in the X and Y axis directions, and the requirements of an ideal Luneburg lens can be met in the X and Y directions. Similarly, the Z-direction can be differentiated. Taking the coordinates of a point on the Z-direction structural centerline of each metamaterial unit, the structural unit can be differentiated along the structural centerline to calculate the structural wall thickness L at that point. This L is then assigned to the wall thickness of the Z-direction differential structure to which the centerline belongs. It can be seen that the smaller metamaterial units of the differential structure conform to the relationship between structural wall thickness and dielectric constant in step S7, and their size is much smaller than the electromagnetic wave wavelength, satisfying the equivalent medium theory and meeting application requirements. This allows for continuous and uniform variation of the dielectric constant in the X, Y, and Z axes, satisfying the requirements of an ideal Luneburg lens in all three directions. For applications requiring only gradient changes in specific directions, only one or more of the X, Y, and Z directions need to be differentiated.

[0013] According to the above technical solution, in step S9, the structural wall thickness is assigned to the metamaterial unit corresponding to the differential at each point in the Luneburg lens space to model the Luneburg lens model. When the Z-axis is fixed, when viewed in the X and Y directions, X=0→X=R, the dielectric constant changes uniformly. Therefore, L is emitted from the center point of the sphere to the surface of the sphere and decreases uniformly. This can be regarded as a gradient change and there is no layered structure. When the X or Y axis is fixed, when observed along the Z direction, Z=0→Z=R, the dielectric constant decreases uniformly. Therefore, L is emitted from the center point of the sphere to the surface of the sphere and decreases uniformly. This can be regarded as a gradient change and there is no layered structure. In step S10, the designed Luneburg lens model is manufactured using 3D printing technology. 3D printing technology includes, but is not limited to, fused fiber deposition modeling (FFF), stereolithography (SLA), digital light processing (DLP), mask stereolithography (MSLA), 3D printing (3DP), selective laser sintering (SLS), multi-jet melting (MJF), binder jetting (BJ), material jetting (MJ), and layered solid modeling (LOM).

[0014] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention utilizes a unique metamaterial unit structure design to construct a Luneburg lens, enabling the fabrication of a Luneburg lens with continuously gradient dielectric constants in multiple directions using 3D printing. By employing unique structural features of the metamaterial units in the X, Y, and Z directions, the invention addresses the issue of abrupt dielectric constant changes between metamaterial units in existing 3D-printed Luneburg lens designs. Through a design process, it achieves dielectric constant matching with different regions of an ideal Luneburg lens, thereby improving the gain and efficiency of the Luneburg lens antenna. This invention is original in the field of Luneburg lens design and manufacturing. Furthermore, Luneburg lenses designed using 3D printing include, but are not limited to, technologies such as FDM, SLA, DLP, MSLA, SLS, 3DP, MJF, BJ, MJ, and LOM. 3D printing can achieve very complex structures and precision that are impossible with traditional processes. In addition, the development of high-precision 3D printing can help achieve metamaterial unit structures with finer edges and thicker walls, thereby achieving a lower dielectric constant on the spherical surface, making it closer to the ideal Luneburg lens, and improving gain and efficiency. Attached Figure Description

[0015] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used together with the embodiments of the invention to explain the invention and do not constitute a limitation thereof.

[0016] In the attached diagram: Figure 1 This is a flowchart of the manufacturing method of the present invention; Figure 2 This is a schematic diagram of metamaterial unit 1 of the present invention; Figure 3 This is a schematic diagram of the metamaterial unit 1 of the present invention moving and deforming along the X-axis direction; Figure 4 This is a schematic diagram of the metamaterial unit 1 of the present invention being rotated 90° along the Z-axis direction and deformed. Figure 5 This is a schematic diagram of the scaling and deformation of the metamaterial unit 1 of the present invention in the Z-axis direction; Figure 6 This is a schematic diagram of the metamaterial unit 1 of the present invention undergoing torsional deformation along both the Y-axis and the Z-axis simultaneously; Figure 7 This is a schematic diagram of the metamaterial unit 1 of the present invention undergoing bending deformation along the X-axis direction; Figure 8 This is a schematic diagram of the two deformed sub-metamaterial units 1 of the present invention merging into one metamaterial unit in the Z direction; Figure 9 This is a schematic diagram of the metamaterial unit 2 of the present invention (composed of two metamaterial units 1 combined and rotated 90° between each). Figure 10 This is a schematic diagram of the division of the ø240mm Lumborghäuser lens of the present invention into multiple metamaterial units (metamaterial unit size: 1*1*0.4mm). Figure 11 This is a schematic diagram of the structural centerline of the metamaterial unit of the present invention along the X and Y axes; Figure 12 This is a schematic diagram of the differential X / Y metamaterial unit of the present invention; Figure 13 This is a schematic diagram of the differential X / Y metamaterial unit that imparts structural wall thickness according to the present invention; Figure 14 This is a schematic diagram of the metamaterial unit of the present invention taking the structural center line in the Z-axis direction and simultaneously differentiating the metamaterial unit in the X & Z / Y & Z directions; Figure 15 This is a schematic diagram of the differential metamaterial unit that imparts wall thickness to the X&Y&Z structure according to the present invention; Figure 16 This is a schematic diagram in the X / Y direction of dividing the Luneburg lens sphere into metamaterial units (Z=-0.2 to Z=0.2mm) according to the present invention; Figure 17 This is a schematic diagram of the X / Y direction model design of the metamaterial unit (Z=-0.2 to Z=0.2mm) of the Luneburg lens sphere cut off in this invention; Figure 18 This is an enlarged detail of the X / Y direction model design of the metamaterial unit (Z=-0.2 to Z=0.2mm) of the Luneburg lens sphere used in this invention; Figure 19 This is a schematic diagram of the Z-direction of the metamaterial units (X=0 to X=1mm) used to cut the entire sphere of the Luneburg lens in this invention; Figure 20 This is a schematic diagram of the Z-direction model design of the metamaterial unit (X=0 to X=1mm) of the Luneburg lens sphere cut off in this invention; Figure 21 This is a magnified detail of the Z-direction model of the metamaterial unit (X=0 to X=1mm) of the Luneburg lens sphere cut out in this invention; Figure 22 This is a schematic diagram of the Luneburg lens sphere design of the present invention. Detailed Implementation

[0017] The preferred embodiments of the present invention will be described below with reference to the accompanying drawings. It should be understood that the preferred embodiments described herein are for illustration and explanation only and are not intended to limit the present invention.

[0018] Example: Figure 1 As shown, this invention provides a technical solution: a method for 3D printing to fabricate Luneburg lenses with continuously gradient dielectric constants in multiple directions. Through a unique metamaterial unit structure design constituting the Luneburg lens, it achieves the fabrication of Luneburg lenses with continuously gradient dielectric constants in multiple directions using 3D printing, thus more closely resembling the ideal Luneburg lens and improving gain and efficiency. The method is characterized by the following steps: Step S1: Determine the dielectric constant distribution model of the Luneburg lens; Step S2: Determine the operating frequency and wavelength; Step S3: Introduce the concept of metamaterial units and determine their dimensions; Step S4: Determine the lens size and establish the spatial distribution formula for the dielectric constant; Step S5 explains the principle of dielectric constant modulation of metamaterial units; Step S6: Define the external shape, internal structure, and dimensions of the metamaterial unit; Step S7: Establish the functional relationship between the wall thickness and dielectric constant of the metamaterial unit structure; Step S8: Divide the sphere into metamaterial units and perform differentiation operations; Step S9: Assign the structural wall thickness to the metamaterial units corresponding to the differentials of each point in the Luneburg lens space to model the Luneburg lens model. Step S10: Manufacture Luneburg lenses using 3D printing technology.

[0019] Based on the above technical solution, in step S1, for an ideal Luneburg lens with radius R, the relevant dielectric constant at a distance r from the center of the sphere can be calculated according to the following formula, where the dielectric constant at the center of the sphere is 2, and the dielectric constant at the surface of the sphere is 1: ; An ideal Luneburg lens needs to have a dielectric constant of 2 at the center and a dielectric constant of 1 at the surface, with a uniform gradient of dielectric change from the center to the surface. Traditional processes and materials make it difficult to achieve this. At the same time, due to limitations in materials and manufacturing processes, it is difficult to produce low dielectric materials with a dielectric constant of 1. Therefore, the dielectric constant of the surface is generally controlled between 1 and 1.3, while still maintaining good overall electromagnetic performance.

[0020] For a Luneburg lens of radius R under non-ideal conditions, the relevant dielectric constant at a distance r from the center of the sphere can be calculated using the following formula, where It is the dielectric constant at the center of the sphere. It is the dielectric constant of the surface of the sphere:

[0021] Based on the above technical solution, in step S2, the applicable electromagnetic wave operating frequency range of the Luneburg lens should first be determined, and its wavelength should be calculated. Specifically, 30GHz corresponds to a wavelength of 10mm. The following explanation is based on 30GHz.

[0022] Based on the above technical solution, in step S3, the Luneburg lens made by 3D printing is composed of several metamaterial units to achieve a gradient effect. According to the equivalent medium theory in metamaterial research, when the size of a unit structure composed of different materials is much smaller than the working wavelength, electromagnetic waves cannot distinguish its internal structure, and this metamaterial unit can be regarded as a uniform medium. The size of the metamaterial unit is selected to be less than 1 / 4 of the wavelength, usually 1 / 5 or 1 / 10. 1 / 10 of the 30GHz wavelength is selected, which is 1mm.

[0023] Based on the above technical solution, in step S4, the diameter of the Luneburg lens is determined. The diameter of the Luneburg lens is 240mm, and the dielectric constant of the Luneburg lens is designed to be 2 at the center of the sphere and 1.2 at the surface of the sphere. Establish a three-dimensional coordinate system, with the center point (x, y, z) of the sphere as (0, 0, 0). The dielectric constant of each point on the sphere should satisfy the following formula: ; This allows us to obtain a data table of points in coordinate space and their dielectric constant values.

[0024] Based on the above technical solution, in step S5, the metamaterial unit is composed of dry air and one or more other materials, and the dielectric constant of the metamaterial unit is ε. m The dielectric constant of dry air is ε0, and the volume percentage of dry air in the metamaterial unit is ν0. The dielectric constant of material 1 is ε1, and the volume percentage of material 1 in the metamaterial unit is ν1. The dielectric constant of material n is ε. n The volume percentage of material n in the metamaterial unit is ν. n ; According to the Brown linear model in the Lichtenecker-Rother (LR) equations, the dielectric constant of the metamaterial unit is: ; The volume percentage ν of material n in the metamaterial element can be changed by altering the structural properties of material n within the corresponding element. n The specific attribute is wall thickness; The proportion of dry air is ν0 = 1 - ν1 - ν2 - ... - ν n Therefore, when the material composition of the metamaterial unit is fixed, the properties of the unit structure corresponding to material n have a functional relationship with the dielectric constant of the metamaterial unit. The dielectric constant of the metamaterial unit can be changed by changing the properties of the unit structure, thereby achieving the customization of the dielectric constant. In addition, by changing the material n with different dielectric constants, the maximum dielectric constant value of the metamaterial unit can be controlled. The following explanation is based on the metamaterial unit being composed of dry air and another material.

[0025] Based on the above technical solution, in step S6, the metamaterial unit can adopt a variety of external shapes, including cuboids, cubes, and polyhedra. The dimensions of the metamaterial unit in the X, Y, and Z directions can be equal or unequal, as long as they satisfy the equivalent medium theory that the size is much smaller than the wavelength. The metamaterial unit can also adopt a variety of internal structures. This section proposes metamaterial units that can achieve uniform gradients, such as... Figure 2 As shown, this unit can undergo single or multiple deformation operations without changing the structural volume ratio and wall thickness characteristics, including the following methods: unidirectional or multidirectional movement, such as... Figure 3 As shown; rotation along a single axis or multiple axes, such as Figure 4 As shown; scaling in one or more directions, such as Figure 5 As shown; single-axis or multi-axis torsion, such as Figure 6 As shown; bending in one or more directions, such as Figure 7 As shown; the merging of multiple sub-metamaterial units, where the Z-direction dimension of each individual metamaterial unit before merging is still much smaller than the wavelength, also satisfies the equivalent medium theory, such as... Figure 8 , 9 As shown; The following explanation is based on metamaterial unit 2, such as... Figure 9 As shown, the unit is composed of two sub-metamaterial units 1 merged together. Each sub-metamaterial unit is rotated 90 degrees along the Z-axis. The X-direction dimension of the large metamaterial unit is set to 1 mm, the Y-direction dimension is set to 1 mm, and the Z-direction dimension is set to 0.4 mm. The Z-direction dimension can also be taken to a smaller value. The smaller the Z-direction dimension, the more refined the dielectric constant in the Z-direction will be, matching the gradient requirements of the Luneburg lens.

[0026] Based on the above technical solution, in step S7, multiple samples composed of metamaterial units with different structural wall thicknesses are printed using 3D printing technology. The dielectric constant of the samples is tested under 30GHz electromagnetic waves, and the metamaterial unit structural wall thickness L is obtained by fitting using computer technology. n With dielectric constant ε mFrom the functional relationship, we can further obtain the relationship between the corresponding spatial coordinate points and the wall thickness of the corresponding metamaterial element. Here, we illustrate this with an example of the relationship function for a material, as follows: ; The calculated data table of each point in space and the corresponding metamaterial element wall thickness is as follows: ; .

[0027] Based on the above technical solution, in step S8, the sphere with a size of Φ240mm is divided into multiple metamaterial units of 1*1*0.4mm size, such as... Figure 10 As shown; The large metamaterial unit is then divided into two sub-metamaterial units. Coordinates are taken from the structural centerline of each sub-metamaterial unit. (See structural centerline diagram). Figure 11 The blue line segment can be used to differentiate the structural unit along the centerline of the structure, such as... Figure 12 As shown, the structural wall thickness L at this point is calculated, and L is assigned the wall thickness of the differential structure to which the centerline belongs, as follows. Figure 13 As shown, the smaller metamaterial unit of the differential structure conforms to the relationship between the structural wall thickness and dielectric constant in step S7. At the same time, the size is much smaller than the electromagnetic wave wavelength, which satisfies the equivalent medium theory and meets the application requirements. Thus, the dielectric constant can be continuously and uniformly varied in the X and Y axis directions, while meeting the requirements of an ideal Luneburg lens in the X and Y directions. Similarly, the Z-direction can be differentiated, and the coordinates of points can be taken from the Z-direction structural centerline of each metamaterial unit. (See Z-direction structural centerline...) Figure 14 The red line segment represents the structural unit, which can be differentiated along the centerline to calculate the wall thickness L at that point. L is then assigned to the wall thickness of the differentiated structure in the Z direction, corresponding to the centerline. Figure 15 As shown, the smaller metamaterial units of the differential structure conform to the relationship between the structural wall thickness and dielectric constant in step S7. Simultaneously, their size is much smaller than the electromagnetic wave wavelength, satisfying the equivalent medium theory and meeting application requirements. This allows for continuous and uniform variation of the dielectric constant in the X, Y, and Z axes, while also meeting the requirements of an ideal Luneburg lens in all three directions. For applications requiring only gradient changes in specific directions, differentiation can be performed in only one or more of the X, Y, and Z directions.

[0028] Based on the above technical solution, in step S9, the structural wall thickness is assigned to the differential metamaterial units corresponding to each point in the Luneburg lens space, thus modeling the Luneburg lens model. For example... Figure 22As shown. Common parametric modeling software such as CADRhino3D & Grasshopper, VoxelDance Design, nTop, and Altair Inspire can be used for modeling.

[0029] With the Z-axis fixed, when viewed in the X and Y directions, the dielectric constant changes uniformly from X=0 to X=120mm. Therefore, L, emitted from the center of the sphere to the surface, decreases uniformly, which can be considered a gradient change without a layered structure. A metamaterial unit from Z=-0.2mm to Z=0.2mm is used for illustration, as detailed below. Figure 16-18 As shown; With the X or Y axis fixed, observing along the Z direction, from Z=0 to Z=120mm, the dielectric constant decreases uniformly. Therefore, L emitted from the center of the sphere to the surface decreases uniformly, which can be considered a gradient change without a layered structure. We will use a metamaterial unit from X=0mm to X=1mm for illustration, specifically as follows... Figure 19-21 As shown.

[0030] Finally, in step S10, the designed Luneburg lens model is manufactured using 3D printing technology. 3D printing technology includes, but is not limited to, fused fiber deposition modeling (FFF), stereolithography (SLA), digital light processing (DLP), mask stereolithography (MSLA), 3D printing (3DP), selective laser sintering (SLS), multi-jet melting (MJF), binder jetting (BJ), material jetting (MJ), and layered solid modeling (LOM).

[0031] Finally, it should be noted that the above descriptions are merely preferred embodiments of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for 3D printing a Luneburg lens with continuously graded dielectric constant in multiple directions, characterized in that: Includes the following steps: Step S1: Determine the dielectric constant distribution model of the non-ideal Luneburg lens; Step S2: Determine the operating frequency and wavelength; Step S3: Determine the external dimensions of the metamaterial unit, which are less than 1 / 4 of the working wavelength. The Luneburg lens is composed of several metamaterial units with different structural dimensions. Step S4: Determine the lens size and establish the spatial distribution formula for the dielectric constant; Step S5: Determine the material of the metamaterial unit, which is composed of dry air and one or more other materials; Step S6: Define the external shape, internal structure, and dimensions of the metamaterial unit; Step S7: Establish the functional relationship between the metamaterial unit structure wall thickness and the dielectric constant, including: printing multiple samples composed of metamaterial units with different structural wall thicknesses, testing the dielectric constant of the samples, and fitting the metamaterial unit structure wall thickness L. n With dielectric constant ε m The functional relationship is: ； Furthermore, the relationship between the corresponding spatial coordinates x, y, z and the corresponding metamaterial element wall thickness was obtained, and a data table of each point in space and the corresponding metamaterial element wall thickness was calculated: ； Get L n即 The relationship between the x, y, and z coordinate values, where It is the dielectric constant at the center of the sphere. It is the dielectric constant of the surface of the sphere; Step S8: Divide the sphere into metamaterial units and perform differentiation operations; Divide the Luneburg lens sphere with radius R into multiple metamaterial units; Divide the metamaterial unit into sub-metamaterial units; Take the coordinates of a point from the structural centerline of each sub-metamaterial unit; Differentiate the sub-metamaterial unit along the direction of the structural centerline; The differentiation direction is one or more of x, y, and z; Calculate the structural wall thickness L at that point according to the formula in step S7; Assign L to the wall thickness of the differential structure to which the centerline belongs. Step S9: Assign the structural wall thickness to the metamaterial units corresponding to the differentials of each point in the Luneburg lens space to model the Luneburg lens model. Step S10: Manufacture Luneburg lenses using 3D printing technology.

2. The method for 3D printing a Luneburg lens with continuously varying dielectric constant in multiple directions according to claim 1, characterized in that: In step S1, for a Luneburg lens with radius R under non-ideal conditions, the relevant dielectric constant at a distance r from the center of the sphere is calculated according to the following formula. .

3. The method for 3D printing a Luneburg lens with continuously varying dielectric constant in multiple directions according to claim 1, characterized in that: In step S2, the applicable electromagnetic wave operating frequency range of the Luneburg lens is determined, and its wavelength is calculated.

4. The method for 3D printing a Luneburg lens with continuously varying dielectric constant in multiple directions according to claim 1, characterized in that: In step S3, the Luneburg lens, which is 3D printed, is composed of several metamaterial units with different structural sizes.

5. The method for 3D printing a Luneburg lens with continuously varying dielectric constant in multiple directions according to claim 1, characterized in that: In step S4, a three-dimensional coordinate system is established, with the center point (x,y,z) of the sphere as (0,0,0). The dielectric constant of each point on the sphere should satisfy the following formula: ； This yields a data table of points in coordinate space and their dielectric constant values.

6. The method for 3D printing a Luneburg lens with continuously varying dielectric constant in multiple directions according to claim 1, characterized in that: In step S5, the dielectric constant of the dry air is ε0, and the volume percentage of the dry air in the metamaterial unit is ν0, while the dielectric constant of material 1 is ε1, the volume percentage of material 1 in the metamaterial unit is ν1, and the dielectric constant of material n is ε n The volume percentage of material n in the metamaterial unit is ν. n ; According to the Brown linear model in the Lichtenecker-Rother (LR) equations, the dielectric constant of the metamaterial unit is: ； The volume percentage ν of material n in the metamaterial unit can be changed by altering the structural properties of material n within the corresponding unit. n ; The proportion of dry air is ν0 = 1 - ν1 - ν2 - ... - ν n The properties of the unit structure corresponding to material n have a functional relationship with the dielectric constant of the metamaterial unit. By changing the properties of the unit structure, the dielectric constant of the metamaterial unit can be changed, thereby achieving customization of the dielectric constant. By changing the material n with different dielectric constants, the maximum dielectric constant value of the metamaterial unit can be controlled.

7. The method for 3D printing a Luneburg lens with continuously varying dielectric constant in multiple directions according to claim 1, characterized in that: In step S6, the metamaterial unit adopts various external shapes, including cuboids, cubes, and polyhedra. The dimensions of the metamaterial unit in the x, y, and z directions must satisfy the equivalent medium theory. The unit adopts various internal structures. The metamaterial unit achieves uniform gradual change by performing one or more deformation operations without changing the structural volume ratio and wall thickness characteristics, specifically including: unidirectional or multidirectional movement, and uniaxial or multiaxial rotation. Scaling in one or more directions; torsion in one or more axes; bending in one or more directions; merging of multiple sub-metamaterial units, where the individual sub-metamaterial units before merging satisfy the equivalent medium theory.

8. The method for 3D printing a Luneburg lens with continuously varying dielectric constant in multiple directions according to claim 1, characterized in that: In step S7, multiple samples composed of metamaterial units with different structural wall thicknesses are printed using 3D printing technology. The dielectric constant of the samples is tested at the electromagnetic wave operating frequency, and the metamaterial unit structural wall thickness L is obtained by fitting using computer technology. n With dielectric constant ε m The functional relationship.

9. The method for 3D printing a Luneburg lens with continuously varying dielectric constant in multiple directions according to claim 8, characterized in that: In step S8, the external shape of the metamaterial unit is a cube or a cuboid.

10. The method for 3D printing a Luneburg lens with continuously varying dielectric constant in multiple directions according to claim 9, characterized in that: In step S10, a Luneburg lens model is manufactured using 3D printing technology. The 3D printing technology includes fused fiber deposition modeling (FFF), stereolithography (SLA), digital light processing (DLP), mask stereolithography (MSLA), 3D printing (3DP), selective laser sintering (SLS), multi-jet melting (MJF), binder jetting (BJ), material jetting (MJ), and layered solid modeling (LOM).

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