Method for measuring the propagation constant of electromagnetic waves in two-dimensional periodic structures

Abstract

The application discloses a method for measuring electromagnetic wave propagation constant of two-dimensional periodic structure. The method is based on field values of five points in the structure, and uses an analytical expression containing the field values of the five points to respectively express phase constants and attenuation constants of two dimensions of the two-dimensional periodic structure. The method is simple, fast and accurate. The method can be used for two-dimensional uniform structure, two-dimensional periodic structure, closed transmission line structure, open transmission line, two-dimensional waveguide structure, antenna radiation structure, non-loss structure and loss structure.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic field microwave technology, specifically to a method for measuring the propagation constant of electromagnetic waves in a two-dimensional periodic structure. Background Technology

[0002] In the field of electromagnetic fields and microwaves, the design of microwave components and antennas often requires knowledge of the propagation constants of two-dimensional periodic structures. This includes, for example, the phase and attenuation constants of electromagnetic waves in a two-dimensional periodic frequency-selective surface, as well as the phase and attenuation constants of a two-dimensional leaky-wave antenna or a two-dimensional periodic metasurface. These electromagnetic wave propagation constants are closely related to the transmission and reflection characteristics of these structures and the spatial radiation characteristics of antennas. We typically use the propagation constant to characterize the propagation characteristics of electromagnetic structures; it includes both the phase and attenuation constants. For two-dimensional periodic structures, the propagation constant has two directional components, usually referred to as the transverse and vertical axes. For convenience, the x-axis is typically set along the transverse direction, and the y-axis along the vertical direction.

[0003] Currently, there is no mature method for measuring the propagation constant of electromagnetic waves in two-dimensional periodic structures. For one-dimensional periodic structures, the methods are basically based on scattering parameters. This involves first measuring the scattering parameters of one or more microwave network elements within an electromagnetic structure, then transforming these parameters into network transfer parameters, and finally obtaining eigenvalue equations with the propagation constant as the unknown. Solving these eigenvalue equations yields the propagation constant of the electromagnetic wave in the structure. For two-dimensional periodic structures, it is difficult to maintain a fixed phase difference at the periodic boundary in one direction while simultaneously measuring the scattering parameters at the port in another direction. Therefore, this method based on microwave equivalent networks is difficult to implement. More importantly, this method relies on the assumption that the electromagnetic properties of the electromagnetic structure elements can be represented by a microwave network. However, the electromagnetic properties of some electromagnetic structure elements cannot be represented by microwave networks. For example, if an element is a perfectly serrated periodic waveguide, the equivalent voltage and current cannot be defined due to the lack of uniform waveguide segments, making it impossible to represent it using a microwave equivalent network. Another method to obtain the electromagnetic wave propagation constant of a two-dimensional periodic structure is to use the eigenmode solver in a full-wave electromagnetic field analysis tool. However, this method only yields the phase constant, not the attenuation constant. Furthermore, since eigenmodes can only handle lossless and closed structures, they cannot be used for lossy or open structures. Many two-dimensional periodic structures are typically lossy, with even greater losses due to radiation in leaky antennas. On the other hand, most two-dimensional periodic structures are open structures, such as frequency-selective surfaces; and two-dimensional leaky antennas are always open structures, and eigenmode solvers are inherently incompatible with open structures. Summary of the Invention

[0004] The purpose of this invention is to address the shortcomings of existing scattering parameter measurement methods by providing a method for measuring the electromagnetic wave propagation constant of a two-dimensional periodic structure. This method is applicable to various types of structures, including uniform or periodic structures, and lossy or open radiating structures.

[0005] To achieve the above functions, this invention designs a method for measuring the propagation constant of electromagnetic waves in a two-dimensional periodic structure. The two-dimensional periodic structure includes multiple unit structures with the same shape, size, and material. Each unit structure is arranged in the horizontal and vertical directions with a preset period. For the two-dimensional periodic structure, the following steps S1-S5 are performed to obtain its electromagnetic wave propagation constants in the horizontal and vertical directions:

[0006] Step S1: Within a two-dimensional periodic structure, take this point as the midpoint 5, and take four points at preset vertical and horizontal distances, respectively as the upper point 4, lower point 3, left point 1, and right point 2 of the midpoint.

[0007] The coordinates of left point 1, right point 2, bottom point 3, top point 4, and middle point 5 are represented as (ap) x ,a), (a+p x ,a), (a,ap) y (a, a+p) y (a, a), where the distances on the horizontal coordinate axis between the left point 1, the right point 2, and the middle point 5 are all p. x The distances between the top point 4, the bottom point 3, and the middle point 5 on the vertical coordinate axis are all p. y ;

[0008] Step S2: Measure the field value of any electromagnetic field component in the same direction at the five points: midpoint 5, top point 4, bottom point 3, left point 1, and right point 2.

[0009] Step S3: Calculate the complex value b based on the field values ​​at the five points. x ;

[0010] Step S4: Calculate the complex value b based on the field values ​​at the five points. y ;

[0011] Step S5: Based on complex value b x Complex value b y The distance p between the top point 4, the bottom point 3, and the middle point 5 on the vertical coordinate axis y The distance p between left point 1, right point 2, and middle point 5 on the horizontal coordinate axis x Calculate the electromagnetic wave propagation constants of a two-dimensional periodic structure, including the transverse phase constant, transverse attenuation constant, longitudinal phase constant, and longitudinal attenuation constant.

[0012] As a preferred embodiment of the present invention: when the two-dimensional periodic structure is a two-dimensional uniform structure, p x p is any length not exceeding half the transverse length of the entire structure. y It is any length not exceeding half the vertical length of the entire structure, but it must be ensured that the five points, namely the middle point 5, the top point 4, the bottom point 3, the left point 1, and the right point 2, are all within the structure.

[0013] As a preferred technical solution of the present invention: in a two-dimensional periodic structure, p x It is an integer multiple of the transverse period length of the structure, and not greater than half of the transverse length of the entire structure, p y It is an integer multiple of the vertical period length of the structure, and not greater than half of the vertical length of the entire structure, but it must be ensured that the five points, namely the middle point 5, the top point 4, the bottom point 3, the left point 1, and the right point 2, are all within the structure.

[0014] As a preferred technical solution of the present invention: the complex value b mentioned in step S3 x The calculation is as follows:

[0015] ;

[0016] ;

[0017] In the formula, F -1,0 F 1,0 F 0,0 The values ​​of any identical component of the electromagnetic field at points 1 on the left, 2 on the right, and 5 in the middle are, respectively. x It is an intermediate variable.

[0018] As a preferred technical solution of the present invention: the complex value b mentioned in step S4 y The calculation is as follows:

[0019] ;

[0020] ;

[0021] In the formula, F 0,-1 F 0,1 The values ​​of any identical component of the electromagnetic field at points 3 (bottom) and 4 (top) are given in order. y It is an intermediate variable.

[0022] As a preferred embodiment of the present invention, the transverse phase constant mentioned in step S5 is as follows:

[0023] ;

[0024] In the formula, β xIndicates the transverse phase constant. Representing complex values The phase angle.

[0025] As a preferred embodiment of the present invention, the transverse attenuation constant in step S5 is as follows:

[0026] ;

[0027] In the formula, α x This represents the transverse attenuation constant.

[0028] As a preferred embodiment of the present invention, the longitudinal phase constant in step S5 is as follows:

[0029] ;

[0030] In the formula, β y Represents the longitudinal phase constant. Represents the complex value b y The phase angle.

[0031] As a preferred embodiment of the present invention, the longitudinal attenuation constant in step S5 is as follows:

[0032] ;

[0033] In the formula, α y This represents the longitudinal attenuation constant.

[0034] To measure the electric or magnetic field values ​​of any component in the same direction at five points—midpoint 5, top point 4, bottom point 3, left point 1, and right point 2—you can use direct measurement methods, such as near-field probes, indirect measurement methods, or calculation methods using electromagnetic field simulation software.

[0035] Beneficial effects: Compared with the prior art, the advantages of the present invention include:

[0036] This invention presents a method for measuring the electromagnetic wave propagation constant of a two-dimensional periodic structure. This method uses complex field values ​​of the electric or magnetic fields measured at five points within the structure to obtain the propagation constants in two directions of the two-dimensional periodic structure, including the phase constant and attenuation constant, using simple analytical expressions. This method is simple, fast, and accurate. It can be applied to both two-dimensional homogeneous and two-dimensional periodic structures; both closed and open transmission line structures; both waveguide structures and antenna radiating structures; and both lossless and lossy structures. Attached Figure Description

[0037] Figure 1This is a schematic diagram of a method for measuring the propagation constant of electromagnetic waves in a two-dimensional periodic structure according to an embodiment of the present invention;

[0038] Figure 1 Zhong: 1. Left point; 2. Right point; 3. Bottom point; 4. Top point; 5. Middle point. Detailed Implementation

[0039] The present invention will be further described below with reference to the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solution of the present invention, and should not be used to limit the scope of protection of the present invention.

[0040] The method for measuring the electromagnetic wave propagation constant of a two-dimensional periodic structure provided in this embodiment of the invention refers to... Figure 1 The two-dimensional periodic structure comprises multiple unit structures of the same shape, size, and material. Each unit structure is arranged in the horizontal and vertical directions with a preset period. For the two-dimensional periodic structure, the following steps S1-S5 are performed to calculate its electromagnetic wave propagation constant:

[0041] Step S1: Within a two-dimensional periodic structure, take this point as the midpoint 5, and select four points at predetermined vertical and horizontal distances, respectively, as the upper point 4, lower point 3, left point 1, and right point 2 of the midpoint 5.

[0042] The coordinates of left point 1, right point 2, bottom point 3, top point 4, and middle point 5 are represented as (ap) x ,a), (a+p x ,a), (a,ap) y (a, a+p) y (a, a), where the distances on the horizontal coordinate axis between the left point 1, the right point 2, and the middle point 5 are all p. x The distances between the top point 4, the bottom point 3, and the middle point 5 on the vertical coordinate axis are all p. y ;

[0043] When the two-dimensional periodic structure is a two-dimensional homogeneous structure, p x p is any length not exceeding half the transverse length of the entire structure. y It is any length not exceeding half the vertical length of the entire structure, but it must be ensured that the five points, namely the middle point 5, the top point 4, the bottom point 3, the left point 1, and the right point 2, are all within the structure.

[0044] In a two-dimensional periodic structure, p x It is an integer multiple of the transverse period length of the structure, and not greater than half of the transverse length of the entire structure, p yIt is an integer multiple of the vertical period length of the structure, and not greater than half of the vertical length of the entire structure, but it must be ensured that the five points, namely the middle point 5, the top point 4, the bottom point 3, the left point 1, and the right point 2, are all within the structure.

[0045] Step S2: Measure the field value of any one of the same components of the electromagnetic field at the five points: midpoint 5, top point 4, bottom point 3, left point 1, and right point 2.

[0046] Step S3: Calculate the complex value based on the field values ​​at the five points. ;

[0047] The aforementioned complex values The calculation is as follows:

[0048] ;

[0049] ;

[0050] In the formula, F -1,0 F 1,0 F 0,0 The values ​​of any identical component of the electromagnetic field at points 1 on the left, 2 on the right, and 5 in the middle are, respectively. x As an intermediate variable; to reduce errors caused by numerical calculations, the measured field value F is used. -1,0 F 1,0 F 0,-1 F 0,1 F 0,0 When measuring the electric or magnetic field components with larger field values, it is advisable to select the components with larger field values. When measuring the field value of any component of the electric or magnetic field in the same direction at the five points (midpoint 5, top point 4, bottom point 3, left point 1, and right point 2), you can use direct measurement methods, such as the near-field probe method, indirect measurement methods, or calculation methods using electromagnetic field simulation software.

[0051] Step S4: Calculate the complex value b based on the field values ​​at the five points. y ;

[0052] The complex value b y The calculation is as follows:

[0053] ;

[0054] ;

[0055] In the formula, F 0,-1 F 0,1 The values ​​of any identical component of the electromagnetic field at points 3 (bottom) and 4 (top) are given in order. y It is an intermediate variable.

[0056] Step S5: Based on complex values Complex value b y The distance p between the top point 4, the bottom point 3, and the middle point 5 on the vertical coordinate axis y The distance p between left point 1, right point 2, and middle point 5 on the horizontal coordinate axis x Calculate the electromagnetic wave propagation constants of a two-dimensional periodic structure, including the transverse phase constant, transverse attenuation constant, longitudinal phase constant, and longitudinal attenuation constant.

[0057] The transverse phase constant is as follows:

[0058] ;

[0059] In the formula, β x Indicates the transverse phase constant. Representing complex values The phase angle.

[0060] The lateral attenuation constant is as follows:

[0061] ;

[0062] In the formula, α x This represents the transverse attenuation constant.

[0063] The longitudinal phase constant is as follows:

[0064] ;

[0065] In the formula, β y Represents the longitudinal phase constant. Represents the complex value b y The phase angle.

[0066] The longitudinal attenuation constant is as follows:

[0067] ;

[0068] In the formula, α y This represents the longitudinal attenuation constant.

[0069] A uniform structure can be viewed as a periodic structure with an arbitrary period.

[0070] When this method is applied to a uniform structure, p x and p y It should not be too small, so as to avoid the field values ​​of the five points being too close, resulting in a large error in measurement and calculation.

[0071] When used in periodic structures, p x and p yIt should not be too small, so as to avoid the field values ​​of the five points being too close, which would cause a large error in the measurement and calculation. Therefore, if the period length is relatively small, the interval can be the length of multiple periods.

[0072] This method still applies when the horizontal and vertical dimensions of a two-dimensional periodic structure are not perpendicular to each other, but rather oblique.

[0073] Periodic structures are a prerequisite for the validity of this invention. Therefore, this invention can also be applied to periodic structures in other technical fields. The embodiments of this invention in the electromagnetic field are not intended to limit the application of this invention in other fields, and such applications should all be included within the scope of protection of this invention. For example, this invention can also be used to calculate the sound wave propagation constant of a two-dimensional acoustic structure, and it can also be used to calculate the characteristic parameters of electronic waves in a two-dimensional crystal structure.

[0074] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those skilled in the art, various changes can be made without departing from the spirit of the present invention.

Claims

1. A method for measuring the propagation constant of electromagnetic waves in a two-dimensional periodic structure, characterized in that, The two-dimensional periodic structure comprises multiple unit structures of the same shape, size, and material. Each unit structure is arranged in the horizontal and vertical directions with a preset period. For the two-dimensional periodic structure, the following steps S1-S5 are performed to calculate its electromagnetic wave propagation constant: Step S1: At a point in the two-dimensional periodic structure, take the point as the middle point (5), and take four points at preset distances in the up, down, left and right directions respectively as the upper point (4), lower point (3), left point (1), and right point (2) of the middle point (5). The coordinates of the left point (1), right point (2), bottom point (3), top point (4), and middle point (5) are respectively represented as (ap x ,a), (a+p x ,a), (a,ap) y (a, a+p) y ), (a, a), where the distances between the left point (1), the right point (2), and the middle point (5) on the horizontal coordinate axis are all p. x The distances between the top point (4), the bottom point (3), and the middle point (5) on the vertical coordinate axis are all p. y ; When the two-dimensional periodic structure is a two-dimensional homogeneous structure, p x p is any length not exceeding half the transverse length of the entire structure. y It is any length not greater than half the vertical length of the entire structure, but it must be ensured that the five points, namely the middle point (5), the top point (4), the bottom point (3), the left point (1), and the right point (2), are all within the structure; p in a two-dimensional periodic structure x It is an integer multiple of the transverse period length of the structure, and not greater than half of the transverse length of the entire structure, p y It is an integer multiple of the vertical period length of the structure and not greater than half of the vertical length of the entire structure, but it must be ensured that the five points, namely the middle point (5), the top point (4), the bottom point (3), the left point (1), and the right point (2), are all within the structure; Step S2: Measure the field value of any one of the same components of the electromagnetic field at the five points: the middle point (5), the top point (4), the bottom point (3), the left point (1), and the right point (2); Step S3: Calculate the complex value b based on the field values ​​at the five points. x ; The complex value b x The calculation is as follows: ； ； In the formula, F -1,0 F 1,0 F 0,0 The values ​​of any equal component of the electromagnetic field at the left point (1), right point (2), and middle point (5) are given in sequence. x As an intermediate variable; Step S4: Calculate the complex value b based on the field values ​​at the five points. y ; The complex value b y The calculation is as follows: ； ； In the formula, F 0,-1 F 0,1 The values ​​of any identical component of the electromagnetic field at the lower point (3) and the upper point (4) are given in sequence, a. y As an intermediate variable; Step S5: Based on complex value b x Complex value b y The distance p between the top point (4) and the bottom point (3) and the middle point (5) on the vertical coordinate axis y The distance p between the left point (1), the right point (2), and the middle point (5) on the horizontal coordinate axis x Calculate the electromagnetic wave propagation constants of a two-dimensional periodic structure, including the transverse phase constant, transverse attenuation constant, longitudinal phase constant, and longitudinal attenuation constant.

2. The method for measuring the propagation constant of electromagnetic waves in a two-dimensional periodic structure according to claim 1, characterized in that, The transverse phase constant mentioned in step S5 is as follows: ； In the formula, β x Indicates the transverse phase constant. Represents the complex value b x The phase angle.

3. The method for measuring the propagation constant of electromagnetic waves in a two-dimensional periodic structure according to claim 1, characterized in that, The transverse attenuation constant mentioned in step S5 is as follows: ； In the formula, α x This represents the transverse attenuation constant.

4. The method for measuring the propagation constant of electromagnetic waves in a two-dimensional periodic structure according to claim 1, characterized in that, The longitudinal phase constant mentioned in step S5 is as follows: ； In the formula, β y Represents the longitudinal phase constant. Represents the complex value b y The phase angle.

5. The method for measuring the propagation constant of electromagnetic waves in a two-dimensional periodic structure according to claim 4, characterized in that, The longitudinal attenuation constant mentioned in step S5 is as follows: ； In the formula, α y This represents the longitudinal attenuation constant.

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