A complex frequency synthesis method and device for realizing the second type of Kerker scattering

Abstract

The application discloses a kind of complex frequency synthesis method and device for realizing second type Kerker scattering, belong to electromagnetic scattering directional control technical field.Method includes: selecting working frequency range, determining scatterer parameter;According to the forward scattering intensity distribution of scatterer in complex frequency plane, determine the forward scattering complex frequency zero point, construct artificial pole;Establish spatial coordinate system, set antenna center and scatterer center at the same height, form angle between incident wave vector and scattering wave vector;Plane wave is radiated to scatterer using horn antenna, steady-state electromagnetic field distribution under real frequency is measured, and wideband real frequency scattering response data is obtained;Based on complex plane contour integral and residue theorem, weighted linear superposition is carried out to real frequency data, and second type Kerker scattering is realized.The application is equivalent to the space scattering field of target under complex frequency excitation by weighted integration wideband real number frequency scattering measurement data, using complex plane contour integral and residue theorem, realizes zero forward scattering.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic scattering directional control technology, specifically to a method and apparatus for complex frequency synthesis to achieve second-type Kerker scattering. Background Technology

[0002] Directional manipulation of electromagnetic scattering has important applications in radar imaging, beamforming, and other fields. Kerker scattering theory provides the foundation for achieving directional scattering: Type I Kerker scattering (zero backscattering) has been realized in passive systems and is often used to design high-performance nanoantennas or anti-reflective metasurfaces; in passive systems, forward scattering intensity can be suppressed as much as possible by designing the size of the scatterer and selecting the dielectric constant. However, Type II Kerker scattering (zero forward scattering) is theoretically limited to situations with optical gain (active systems). But practical gain media inevitably suffer from inherent fabrication defects, large-area inhomogeneities, and temporal instabilities. These factors severely restrict the practical engineering application of Type II Kerker scattering technology.

[0003] To overcome the limitations of active gain media, researchers have proposed a theoretical scheme using complex frequency waves as incident radiation. Complex frequency waves are electromagnetic waves whose amplitude decays exponentially in the time domain. Injecting such waves can excite the forward scattering null point of the scattering system located in the complex frequency plane, thus theoretically realizing Kerker scattering of the second kind in passive systems. However, this scheme faces significant technical challenges in practical implementation, especially in high-frequency bands such as microwaves and millimeter waves. The precise generation and high-precision measurement techniques for complex frequency electromagnetic waves with exponentially decaying time characteristics are not yet mature, making it difficult to meet the needs of practical applications.

[0004] Complex frequency synthesis, as an emerging signal processing technique, can equivalently synthesize complex frequency responses by performing weighted integration on broadband real frequency spectrum data, thus avoiding the technical difficulties of directly generating physical complex frequency waves. Currently, this technology has been preliminarily applied in fields such as super-resolution imaging and ultra-sensitive biosensing, demonstrating promising application prospects. However, the feasibility of applying complex frequency synthesis technology to complex resonant scattering systems has not been systematically studied, particularly regarding whether this technology can be used to stably and accurately realize second-type Kerker scattering in passive systems; no effective solution exists at present. Summary of the Invention

[0005] To address the problems existing in the prior art, the purpose of this invention is to provide a method and apparatus for complex frequency synthesis of second-type Kerker scattering.

[0006] To achieve the above objectives, the present invention provides the following technical solution: In a first aspect, the present invention provides a method for complex frequency synthesis to realize second-type Kerker scattering, comprising the following steps: Step 1: Select the operating frequency range and determine the size and relative permittivity of the scatterer; Step 2: Based on the forward scattering intensity distribution of the scatterer in the complex frequency plane, determine the zero point of the forward scattering complex frequency and construct an artificial pole; Step 3: Establish a spatial coordinate system, set the center of the antenna and the center of the scatterer at the same height, and form an angle between the incident wave vector and the scattered wave vector; Step 4: Use a horn antenna to emit a plane wave to illuminate the scatterer, and use a probe to measure the steady-state electromagnetic field distribution at the real frequency to obtain broadband real-frequency scattering response data of the scatterer within the operating frequency range. Step 5: Post-process the real frequency scattering response data, and synthesize the complex frequency response by weighting the complex plane enclosed channel integral and residue theorem to realize the second type of Kerker scattering.

[0007] Furthermore, the scatterer is a spherical scatterer with a diameter of 31.75 mm and a relative permittivity of 10.2.

[0008] Further, in step 2, the zero point of the forward scattering complex frequency is determined by the forward scattering intensity formula of the H-plane in spherical Mie scattering. Setting the forward scattering intensity to 0, the zero point of the forward scattering complex frequency on the complex frequency plane is obtained by solving the formula: in, The angle between the incident wave vector and the scattered wave vector is... , denoted as the Mie scattering coefficient, and k is the incident wave number.

[0009] Furthermore, in step 2, the artificial poles are constructed by introducing poles on the complex frequency plane, separating them from the intrinsic poles of the scattering system and making their imaginary parts small, so as to ensure that the contribution of artificial poles is dominant in the synthesis result.

[0010] Further, in step 5, the post-processing step specifically includes: weighting and superimposing the measured discrete real frequency scattering field data according to the following synthesis formula: in, For an ideal target complex frequency wave, It is a complex frequency scattering response synthesized from a measured real frequency scattering field. It is a set of measured discrete real frequencies. It is the frequency step size. That is, real frequency scattering field The measured real frequency total field With the incident field Get something by doing something bad. For time variables, This is the zero point of forward scattering.

[0011] In a second aspect, the present invention provides a complex frequency synthesis apparatus for realizing second-type Kerker scattering, comprising: A test and data acquisition system for measuring broadband real-frequency steady-state scattering response data of a scatterer within a selected operating frequency range; The complex frequency synthesis module is communicatively connected to the test and data acquisition system. It is used to perform weighted linear superposition of the broadband real frequency scattering response data based on the complex plane coaxial integral and residue theorem to synthesize the scattering field at the target complex frequency, thereby realizing the second type of Kerker scattering.

[0012] Furthermore, the testing and data acquisition system includes a vector network analyzer, a horn antenna, a three-dimensional scanning platform, a receiving probe, a scanning controller, and a computer. The first port of the vector network analyzer is electrically connected to the horn antenna as a plane wave excitation source, and the second port of the vector network analyzer is electrically connected to the receiving probe to receive the amplitude and phase information of the scattered field. The receiving probe is fixedly mounted on the three-dimensional scanning platform. The scanning controller is electrically connected to both the computer and the three-dimensional scanning platform, and is used to drive the three-dimensional scanning platform to move within the target plane according to the control signals from the computer. The computer is communicatively connected to the vector network analyzer, and outputs control signals to the scanning controller, thereby controlling the movement of the three-dimensional scanning platform in the target plane, causing the receiving probe to measure the electric field amplitude and phase on the plane, transmitting the measured information to the second port of the vector network analyzer. After the vector network analyzer analyzes the data, it is transmitted to the computer. Finally, the complex frequency synthesis module performs complex frequency synthesis on the acquired data to obtain a complex frequency scattered field, realizing the second type of Kerker scattering.

[0013] Furthermore, the vector network analyzer is a model with an operating frequency covering 8-12.5GHz, and the receiving probe is an electric field probe that simultaneously measures the amplitude and phase of the electric field.

[0014] Based on the above technical solution, the embodiments of the present invention can produce at least the following technical effects: (1) This invention achieves a new mechanism for second-type Kerker scattering (zero forward scattering) by weighting the broadband real frequency spectrum data based on the complex plane coaxial integral and the residue theorem and successfully exciting the forward scattering complex frequency zero of the target scatterer in a passive medium system.

[0015] (2) By employing post-processing technology for complex frequency synthesis data, this invention cleverly avoids the huge technical bottleneck of directly generating and measuring physical complex frequency waves with time exponential decay characteristics in high-frequency bands such as microwaves, and achieves extremely high accuracy and temporal stability of scattering field reconstruction. It has broad application prospects in the fields of directional scattering control such as radar imaging and beamforming.

[0016] (3) This invention provides a deeper understanding of the complex frequency electromagnetic control mechanism in complex resonant scattering systems and provides a new approach for the design of super-resolution electromagnetic imaging, ultra-sensitive biosensing and novel high-directional antenna devices. Attached Figure Description

[0017] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0018] Figure 1 This is a schematic diagram of the structure of the microwave real-frequency scattering measurement and post-processing system provided in an embodiment of the present invention; Figure 2 This is a schematic diagram of the complex plane enclosed-channel integral and residue theorem for the principle of synthesizing complex frequency scattering in an embodiment of the present invention; Figure 3 The electromagnetic field distribution and far-field scattering cross section under ideal complex frequency excitation; Figure 4 The electromagnetic field distribution and far-field scattering cross section are synthesized by superimposing real frequency data; Figure 5 The residual field distribution at the extracted scattering poles is shown in the diagram. Figure 6 The results show the comparison between the target scattering field distribution and radiation pattern at different real frequencies; Figure 7 The comparison results are shown between the target scattering field distribution and the radiation pattern at the synthesized complex frequencies. Figure 8 This is a schematic diagram of the waveform of a complex frequency incident electric field as a function of time. Figure 9 A graph showing the evolution of the target's forward scattering cross section over time; Figure 10 The electromagnetic field distribution and radiation pattern under steady-state excitation at complex frequencies. Detailed Implementation

[0019] The technical solutions in the embodiments of the present invention will be clearly and completely described below. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. In addition, the technical solutions of the various embodiments can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0020] This invention discloses a method and apparatus for synthesizing complex frequencies to achieve second-type Kerker scattering. The core of this method lies in measuring the broadband real-frequency scattering response of the scatterer and then performing a weighted linear superposition based on the complex plane enclosed-channel integral and the residue theorem to equivalently synthesize the scattering field at the target complex frequency, thereby realizing second-type Kerker scattering (zero forward scattering) in a passive system.

[0021] The apparatus for implementing this invention includes a test and data acquisition system and a complex frequency synthesis module. The test and data acquisition system is used to measure the real-frequency scattering response of the scatterer; the complex frequency synthesis module is communicatively connected to the test and data acquisition system and is used to linearly superimpose the acquired broadband real-frequency scattering data according to weights to synthesize a complex frequency response to achieve Kerker scattering of the second kind.

[0022] like Figure 1 The diagram shows a schematic of the microwave real-frequency scattering measurement and post-processing system provided in an embodiment of the present invention. The system specifically includes a vector network analyzer, a horn antenna, a three-dimensional scanning platform, a receiving probe, a scanning controller, and a computer. The first port of the vector network analyzer is electrically connected to the horn antenna, serving as a plane wave excitation source; the second port of the vector network analyzer is electrically connected to the receiving probe, used to receive the amplitude and phase information of the scattered field; the receiving probe is fixedly mounted on the three-dimensional scanning platform; the scanning controller is electrically connected to both the computer and the three-dimensional scanning platform, used to drive the three-dimensional scanning platform to move within the target plane according to the control signals from the computer; the computer is communicatively connected to the vector network analyzer, and outputs control signals to the scanning controller, thereby controlling the movement of the three-dimensional scanning platform within the target plane, causing the receiving probe to measure the electric field amplitude and phase on the plane. The measured information is transmitted to the second port of the vector network analyzer, the data is analyzed by the vector network analyzer and then transmitted to the computer, and finally, the complex frequency synthesis module performs complex frequency synthesis on the collected data to obtain a complex frequency scattered field, realizing the second type of Kerker scattering. In this embodiment, the vector network analyzer is a model with a working frequency covering 8-12.5GHz, the positioning accuracy of the three-dimensional scanning platform is better than 0.1mm, and the receiving probe is an electric field probe, which can simultaneously measure the amplitude and phase of the electric field.

[0023] The implementation method of this invention specifically includes the following steps: Step 1: Select the operating frequency range and determine the geometric and electromagnetic parameters of the scatterer. In this embodiment, the operating frequency range is selected as 8-12.5 GHz. Taking a spherical scatterer as the research object, its diameter d and relative permittivity ε are determined. r .

[0024] Step 2: Based on the forward scattering intensity distribution in the complex frequency plane of the scatterer, determine a suitable forward scattering complex frequency zero point and construct an artificial pole.

[0025] This invention calculates the forward scattering intensity distribution of a scatterer based on the spherical Mie scattering theory and finds the forward scattering zero point on the complex frequency plane.

[0026] For a plane wave polarized in the x-axis and incident in the z-direction, the expression for its incident electric field is: in, The x-axis is the unit vector in a rectangular coordinate system. The imaginary unit (i.e.) ), , Let be the incident complex frequency. The real part of the complex frequency, For the imaginary part of the complex frequency, The time variable is the incident wave number. , It is the speed of light in a vacuum. Let be the amplitude of the incident field. For a coordinate system... The radius of the origin is A spherical scatterer, its scattering field It can be represented as: in, It is a radial unit vector. It is the azimuth angle. The angle between the incident wave vector and the scattered wave vector is denoted as . Let be the order of the multipole expansion. as well as , It is a first-order connected Legendre polynomial. and It is the third kind of Riccati-Bessel function and its first derivative. Mie scattering coefficients. and It can be obtained from the boundary conditions: in, Let the radius of the sphere be . It is the wavenumber inside the scatterer. and Here are the relative permittivity and relative permeability of the scattering body. and It is the first kind of Riccati-Bessel function and its first derivative.

[0027] For the far-field radiation mode, take ,at this time Will due to Approaching 0, we obtain the horizontal component: in, Define H-plane ( Planar differential scattering cross section The differential scattering cross section formula and forward scattering intensity are obtained: Let forward scattering intensity The zero point of the forward scattering complex frequency can then be obtained by solving on the complex frequency plane. This is used as an artificial pole for subsequent complex frequency synthesis.

[0028] Step 3: Establish a spatial coordinate system, place the center of the antenna and the center of the spherical scatterer at the same height, set the horizontal plane as the yz plane, and the angle between the incident wave vector and the scattered wave vector.

[0029] Step 4: Illuminate the scattering object by emitting a plane wave using a horn antenna, and measure the steady-state electromagnetic field distribution at the real frequency in the yz plane using a probe to obtain broadband real-frequency scattering response data of the scattering object within the operating frequency range. In this embodiment, the frequency step size for real-frequency measurement is [missing information]. GHz.

[0030] Step 5: Post-process the real frequency scattering response data, and synthesize the complex frequency response by weighting the complex plane enclosed channel integral and residue theorem to realize the second type of Kerker scattering, even if the forward scattering intensity is zero.

[0031] The principle of complex frequency synthesis in this invention is based on the complex plane enclosed integral and the residue theorem, such as... Figure 2 As shown. Consider the auxiliary function. ,in This represents the scattered electric field. As can be seen from the figure, there are two types of poles in the complex frequency plane: one is... It is the inherent forward scattering pole of the scattering system, by Given, another type of pole is the artificial pole, which is... Provided.

[0032] According to the residue theorem, the loop integral ① of the artificial pole is: Will When the forward scattering zero point is selected, the scattered field of the ideal complex frequency zero forward scattering corresponding to that frequency can be obtained.

[0033] The real frequency axis integral ② is the composite complex frequency scattered field. : The residues of the intrinsic poles and ③ represent the residual field. : The field integral of an infinitely large semicircular arc ④ tends to 0 according to Jordan's lemma: in , The radius modulus of the integral arc is, i.e. , For complex numbers Polar angle in the complex plane.

[0034] Since ② + ④ = ① + ③, we get: When the selected artificial pole location satisfies the conditions of being far from other intrinsic poles and having a relatively small imaginary part, the following will occur: In the case where artificial poles dominate, the synthesized complex frequency scattered field can be approximated as: in, It is a set of measured discrete real frequencies. It is the frequency step size (the step size is used in this embodiment). GHz) real frequency scattering field The measured real frequency total field With the incident field You get what you deserve by doing bad things.

[0035] In a specific implementation case, a spherical scatterer is used as an example for theoretical and simulation verification. The diameter of the spherical scatterer is selected as d=2a=31.75mm, and the relative permittivity ε is set as follows: r =10.2. The forward scattering zero point is determined by formula (11). GHz. For example, Figure 3 As shown, this is the electromagnetic field distribution and far-field scattering cross-section of the target under ideal complex frequency excitation. Substituting this specific complex frequency into the scattering field expression, the scattering field and radiation pattern of the yz plane (H plane) are plotted. The results show that its forward scattering is significantly suppressed, satisfying the characteristics of the second type of Kerker scattering. Furthermore, by applying the complex frequency synthesis formula, a set of discrete real frequency scattering field values ​​are weighted and superimposed to obtain the following... Figure 4 The synthesis results are shown. A comparison reveals that a complex frequency field with near-zero forward scattering characteristics was successfully synthesized using a real frequency field. Figure 5 The residual field distribution corresponding to the system's intrinsic pole residues is shown (the radiation pattern is magnified tenfold). Since the selected artificial poles dominate, the residual error is extremely small, confirming that the real frequency synthesis results meet the requirements of type II Kerker scattering.

[0036] Based on the above theoretical and simulation verification, this embodiment further conducted physical experimental measurements and synthesis. Using methods such as... Figure 1 The microwave real-frequency measurement device shown acquires the target's scattered field distribution at real frequencies such as 8.5 GHz, 10.5 GHz, and 12.5 GHz (e.g., Figure 6 (As shown). Comparing the field distribution and radiation pattern obtained from numerical calculations and experimental measurements, the two are highly consistent, verifying the accuracy of the basic real-frequency data acquisition. Subsequently, the acquired broadband real-frequency data was post-processed to synthesize the scattered field at the corresponding complex frequency. (See figure). Figure 7 As shown, the combined results of simulation and experiment both exhibit near-zero intensity in the 0° direction (forward), successfully realizing the second type of Kerker scattering using real frequency data in the experiment.

[0037] Furthermore, the synthesized complex frequency field belongs to a spatiotemporal dynamic evolution process. For example... Figure 8 The figure shows the waveform variation of a complex frequency incident wave in the interval from 0 to 25T0. This waveform is forced to zero before t=0 to ensure energy convergence. Figure 9 This is the normalized evolution curve of the forward scattering intensity of the corresponding synthetic field. In the initial stage ( , (The period corresponds to the real part of the synthesized complex frequency). The incident complex frequency wave packet does not fully interact with the scatterer, and the forward scattering intensity is still relatively large; after evolving to a steady state over time ( The forward scattering intensity rapidly decreases to near zero, marking the formation of type II Kerker scattering. For example... Figure 10 As shown, at different times in steady state, the synthesized scattering field and its pattern consistently maintain zero forward scattering characteristics, verifying that the ideal contribution represented by the selected artificial pole dominates the synthesized complex frequency response under the rule of causality.

[0038] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed. The scope of protection of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for synthesizing complex frequencies to realize second-type Kerker scattering, characterized in that, Includes the following steps: Step 1: Select the operating frequency range and determine the size and relative permittivity of the scatterer; Step 2: Based on the forward scattering intensity distribution of the scatterer in the complex frequency plane, determine the zero point of the forward scattering complex frequency and construct an artificial pole; Step 3: Establish a spatial coordinate system, set the center of the antenna and the center of the scatterer at the same height, and form an angle between the incident wave vector and the scattered wave vector; Step 4: Use a horn antenna to emit a plane wave to illuminate the scatterer, and use a probe to measure the steady-state electromagnetic field distribution at the real frequency to obtain broadband real-frequency scattering response data of the scatterer within the operating frequency range. Step 5: Post-process the real frequency scattering response data, and synthesize the complex frequency response by weighting the complex plane enclosed channel integral and residue theorem to realize the second type of Kerker scattering.

2. The method for complex frequency synthesis of second-type Kerker scattering according to claim 1, characterized in that, The scatterer is a spherical scatterer with a diameter of 31.75 mm and a relative permittivity of 10.

2.

3. The method for complex frequency synthesis of second-type Kerker scattering according to claim 1, characterized in that, In step 2, the zero point of the forward scattering complex frequency is determined by the forward scattering intensity formula of the H-plane in spherical Mie scattering. Setting the forward scattering intensity to 0, the zero point of the forward scattering complex frequency on the complex frequency plane is obtained by solving the formula: ； in, The angle between the incident wave vector and the scattered wave vector is... , Here, k is the Mie scattering coefficient, and k is the incident wave number. Let be the incident complex frequency.

4. The method for complex frequency synthesis of second-type Kerker scattering according to claim 1, characterized in that, In step 2, the artificial poles are constructed by introducing poles on the complex frequency plane, separating them from the intrinsic poles of the scattering system and making their imaginary parts small, so as to ensure that the contribution of artificial poles is dominant in the synthesis result.

5. The method for complex frequency synthesis of second-type Kerker scattering according to claim 1, characterized in that, In step 5, the post-processing step specifically includes: weighting and superimposing the measured discrete real frequency scattering field data according to the following synthesis formula: ； in, For an ideal target complex frequency wave, It is a complex frequency scattering response synthesized from a measured real frequency scattering field. It is a set of measured discrete real frequencies. It is the frequency step size. That is, real frequency scattering field The measured real frequency total field With the incident field Get something by doing something bad. For time variables, This is the zero point of forward scattering.

6. A complex frequency synthesis apparatus for realizing Kerker scattering of the second kind, used to implement the method according to any one of claims 1-5, characterized in that, include: A test and data acquisition system for measuring broadband real-frequency steady-state scattering response data of a scatterer within a selected operating frequency range; The complex frequency synthesis module is communicatively connected to the test and data acquisition system. It is used to perform weighted linear superposition of the broadband real frequency scattering response data based on the complex plane coaxial integral and residue theorem to synthesize the scattering field at the target complex frequency, thereby realizing the second type of Kerker scattering.

7. The complex frequency synthesis apparatus for realizing second-type Kerker scattering according to claim 6, characterized in that, The testing and data acquisition system includes a vector network analyzer, a horn antenna, a 3D scanning platform, a receiving probe, a scanning controller, and a computer. The first port of the vector network analyzer is electrically connected to the horn antenna, serving as a plane wave excitation source. The second port of the vector network analyzer is electrically connected to the receiving probe, used to receive the amplitude and phase information of the scattered field. The receiving probe is fixedly mounted on the 3D scanning platform. The scanning controller is electrically connected to both the computer and the 3D scanning platform, used to drive the 3D scanning platform to move within the target plane according to the computer's control signals. The computer is communicatively connected to the vector network analyzer, outputting control signals to the scanning controller, thereby controlling the movement of the 3D scanning platform within the target plane, causing the receiving probe to measure the electric field amplitude and phase on the plane. The measured information is transmitted to the second port of the vector network analyzer. After analysis, the data is transmitted to the computer, and finally, the complex frequency synthesis module performs complex frequency synthesis on the acquired data to obtain a complex frequency scattered field, realizing Kerker scattering of the second type.

8. The complex frequency synthesis apparatus for realizing second-type Kerker scattering according to claim 7, characterized in that, The vector network analyzer is a model with an operating frequency coverage of 8-12.5GHz, and the receiving probe is an electric field probe that simultaneously measures the amplitude and phase of the electric field.

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