Optical Device and Implementation Method for Detecting Chirped Signals Based on Transformation Materials

Abstract

The present invention relates to an optical device and an implementation method for detecting a chirped signal based on transformation materials, belonging to the technical fields of transformation materials and optical signal processing. Based on the traditional Luneburg lens, both the input surface and the output surface are expanded into planes, and through conformal transformation, the transformation relationship with the traditional Luneburg lens is obtained, and then the medium parameters are obtained by using the method of transformation materials; the implementation includes the following steps: performing conformal transformation W = tan(Z) on the traditional Luneburg lens to obtain a flat lens, and after transformation, the arcs on both the left and right sides become straight lines; intercepting the central part and adjusting the refractive index to obtain the optical device of the present invention; when the wavelength λ of the input chirped signal and the chirp rate m satisfy the relationship m = 8π / λ, the focal point is located on the output surface, and the chirp rate detection is realized by scanning the wavelength. The present invention does not require spatial scanning on the optical axis; it can accurately measure high-chirp-rate chirped signals; it ensures isotropy, which is beneficial to engineering preparation.

Description

Technical Field

[0001] The present invention relates to an optical device and an implementation method for detecting a chirp signal based on transformation materials, belonging to the technical fields of transformation materials and optical signal processing. Background Art

[0002] The chirp signal, also known as the chirp signal, is one of the most representative non-stationary signals in the field of signal processing and is widely used in fields such as radar, sonar, communication, biology, and geological exploration. The chirp signal in the spatial domain is widely used in the field of optical signal processing, such as optical measurement, optical estimation, and image processing. An important feature of the chirp signal is the chirp rate, that is, the rate of change of frequency, which is usually the target of chirp signal detection.

[0003] Most of the detection methods for the chirp rate of chirp signals are based on time-frequency, such as the Wigner-Ville distribution, fractional Fourier transform, high-order ambiguity function, high-order phase function, non-linear least squares method, etc. These methods usually need to traverse and search each row or column of the signal or image to be measured, which is very time-consuming, has a relatively large computational complexity, and the detection accuracy is limited by the scanning step size.

[0004] On the other hand, based on the fractional Fourier transform property of the quadratic refractive index lens, the spatial chirp signal detection lens is a gradient refractive index lens that locks the pulse position corresponding to the chirp signal on the internal optical axis through spatial search, and then determines the chirp rate of the chirp signal; further, the exposed lens exposes the optical axis inside the traditional gradient refractive index lens on the surface of the lens, which is convenient for direct measurement in three dimensions; and a new type of exposed lens that improves the detection range of the chirp rate of the exposed lens;

[0005] For the spatial chirp signal detection based on the gradient refractive index lens and its variants, the chirp rate detection function depends on obtaining the pulse position corresponding to the chirp rate of the input chirp signal. When the working area is narrow, it is not convenient to use the gradient refractive index lens for spatial search; at the same time, the method based on the gradient refractive index lens is restricted by the paraxial approximation. When the chirp rate is high and the focal point is close to the input surface, the error is large.

[0006] The two-dimensional Luneburg lens is a circular gradient refractive index lens with a special refractive index distribution. The refractive index increases radially from the outer surface to the center and has the ability to focus and collimate waves. Parallel light incident in any direction can converge to a point on the other side of the lens surface.

[0007] Transformation optics is a method for solving the inverse problem of electromagnetic field distribution - electromagnetic parameters. According to the form invariance of Maxwell's equations, the problem of solving the electromagnetic parameters corresponding to the wave field is transformed into the problem of geometric shape design and the calculation of coordinate transformation parameters. The material obtained by transformation optics is called transformation material.

[0008] How to combine the above methods to obtain a device and implementation method for measuring linear frequency modulation signals to solve the limitations of the prior art. Summary of the Invention

[0009] The purpose is to provide an optical device and implementation method for detecting linear frequency modulation signals based on transformation materials. When the processing effect of the linear frequency modulation signal in the optical device is known in advance, that is, the light field distribution of the optical device is clear, the trajectory of the light wave is controlled according to the properties of the transformation medium, and then the medium parameters of the optical device are determined by the method of transformation materials, guiding the light wave to propagate in the optical device along a preset path, so as to achieve the expected light field distribution.

[0010] The purpose of the present invention is achieved through the following technical solutions: Through conformal transformation, a one-to-one correspondence with the traditional Luneburg lens can be maintained, and a two-dimensional planar Luneburg lens based on transformation materials is realized, specifically including: an input surface, a two-dimensional planar Luneburg lens for detecting spatial linear frequency modulation signals based on transformation materials, and an output surface;

[0011] The input surface is the incident surface where the input signal enters the two-dimensional planar Luneburg lens for detecting spatial linear frequency modulation signals based on transformation materials, and the input surface is a plane:

[0012] The input surface is obtained by conformal transformation of the input surface of a traditional Luneburg lens with a radius R = 1 and a center at (0, 0) with a radially varying refractive index, that is, the right semi-circular arc on the surface of the traditional Luneburg lens becomes a straight line of x = π / 4, and the part in the middle of -1 ≤ y ≤ 1 in the y direction is intercepted;

[0013] Further, on the basis of the input surface of the traditional Luneburg lens, a conformal transformation is performed, and the shape changes, but in terms of mathematical and physical meanings, the points on the input surface and the points on the input surface of the Luneburg lens before conformal transformation are in a one-to-one correspondence relationship;

[0014] Compared with the traditional Luneburg lens, the two-dimensional planar Luneburg lens for detecting spatial linear frequency modulation signals based on transformation materials has a shape changed from circular to rectangular. By means of transformation optics, the radially varying refractive index distribution of the traditional Luneburg lens is changed. Specifically, after conformal transformation, the refractive index of the lens is axially symmetric about x = 0 and y = 0 respectively. The refractive index distributions of the input surface and the output surface are the same, being 2 at the center point and gradually decreasing towards both sides. The part where -1 ≤ y ≤ 1 in the y direction is intercepted, and the refractive index less than 1 is filled with 1.

[0015] The refractive index distribution determines the propagation path of light in the lens. The two-dimensional planar Luneburg lens for detecting spatial linear frequency modulation signals based on transformation materials, under the condition of satisfying the refractive index distribution, is not restricted by the shape of the lens.

[0016] The output surface is a straight line of x = -π / 4 formed by the left semi-circular arc on the surface of the traditional Luneburg lens. When the wavelength λ of the input linear frequency modulation signal and the frequency modulation rate m satisfy m = 8π / λ, after passing through the two-dimensional planar Luneburg lens for detecting spatial linear frequency modulation signals based on transformation materials, the linear frequency modulation signal is focused on the output surface, and the frequency modulation rate of the linear frequency modulation signal is detected directly on the output surface by scanning the wavelength.

[0017] An optical device for detecting linear frequency modulation signals based on transformation materials disclosed by the present invention has an implementation method including the following steps:

[0018] Step 1: Perform conformal transformation W = tan(Z) on the traditional two-dimensional circular Luneburg lens to obtain an infinite two-dimensional planar Luneburg lens. By means of transformation materials, determine the permittivity and permeability of the new lens after transformation, obtain the refractive index after transformation, and determine the waveform change on the input boundary.

[0019] The traditional two-dimensional circular Luneburg lens corresponds to the original complex plane W = u + iv, which is a virtual space, where u is the real part of the original complex plane W and v is the imaginary part of the original complex plane W.

[0020] The two-dimensional planar Luneburg lens corresponds to the transformed complex plane Z = x + iy, which is a physical space, where x is the real part of the transformed complex plane Z and y is the imaginary part of the transformed complex plane Z.

[0021] Furthermore, when the original complex plane W is conformally transformed into the transformed complex plane Z, according to the coordinate invariance of Maxwell's equations, the relationship between the physical space (x, y, Z) and the virtual space (u, v, W) is shown in Equation (1):

[0022]

[0023] Among them, ε and μ are the permittivity and permeability of the virtual space W respectively, ε' and μ' are the permittivity and permeability of the physical space Z respectively, detA is the determinant of the matrix A, and A is the Jacobian matrix, as shown in Equation (2):

[0024]

[0025] The space transformation relationship between the virtual space W and the physical space Z is shown in Equation (3):

[0026]

[0027] Among them, represents partial differential;

[0028] Furthermore, according to Fermat's theorem, the relationship between the transformed optical path of the physical space Z and the optical path of the virtual space W is shown in Equation (4):

[0029]

[0030] Furthermore,

[0031] Among them, d· represents differential, n w is the refractive index of the virtual space W, n z is the refractive index of the physical space Z,

[0032] After the virtual space W is conformally transformed into the physical space Z, the right semi-circular arc boundary of the traditional circular Luneburg lens before transformation is the input surface, and the left semi-circular arc boundary is the output surface. After transformation, the curved surface becomes a straight surface, the midpoint is compressed, and it is compressed first and then stretched towards both sides;

[0033] Determine the waveform transformation on the input boundary: On the plane of the virtual space W, on the input boundary of the traditional Luneburg lens with a radius R = 1 and the center at (0,0), the arc length S is shown in Equation (5):

[0034]

[0035] Among them, R is the radius of the traditional Luneburg lens, θ is the angle between the line connecting a point on the input surface of the traditional Luneburg lens and the center of the circle and the positive semi-axis of the u-axis, and u and v are the horizontal coordinate and the vertical coordinate respectively;

[0036] According to the conformal transformation W = tan(Z), the relationship between u, v and x, y is shown in Equation (6):

[0037]

[0038] Furthermore, set the radius of the circle \(R = 1\). The right semi-circular arc on the \(W\)-plane after transformation becomes the straight line \(x=\frac{\pi}{4}\) on the \(Z\)-plane. The relationship between the arc length \(S\) on the virtual space \(W\)-plane and the ordinate \(y\) on the physical space \(Z\)-plane is shown in Equation (7):

[0039]

[0040] On the virtual space \(W\)-plane, the horizontal distance \(\Delta u\) from a point on the right semi-circular arc to the point \((R,0)\) is shown in Equation (8):

[0041]

[0042] When a plane wave with an initial phase of 0 propagates horizontally to the input surface and is incident on the right semi-circular arc, the signal is shown in Equation (9):

[0043]

[0044] where \(A\) is the amplitude, \(k = \frac{2\pi}{\lambda}\) is the wave number, and \(\lambda\) is the wavelength;

[0045] Furthermore, according to Equation (7) and Equation (9), the signal incident on the right semi-circular arc is shown in Equation (10):

[0046]

[0047] Even further, the signal incident on the right semi-circular arc is expressed in the form of a Taylor expansion, as shown in Equation (11):

[0048] \(f(S)=A\cdot\exp[i(-k(a_1y 2 -a_2y 4 +a_3y 6 +\cdots+a n y 2n +\cdots))]\ (11)

[0049] where \(a n =[2 2n (-1) n+1 E n / [(2n)!]\) is a constant, \(E n is the Euler number. When \(n = 1\), then \(a_1 = 2\);

[0050] When \(y\) is very small, ignoring the high-order terms and only retaining the quadratic term, the signal incident on the right semi-circular arc is approximately a linear frequency modulation signal, as shown in Equation (12):

[0051]

[0052] The traditional Luneburg lens focuses parallel light on the output plane. Then, the conformally transformed planar Luneburg lens can focus a linear frequency-modulated signal with a specific input chirp rate on the output plane. Since the linear frequency-modulated signal is usually expressed as f(y) = exp[-i(1 / 2)my 2 , the chirp rate of the linear frequency-modulated signal is determined according to Equation (12) as shown in Equation (13):

[0053]

[0054] The inherent characteristic of the conformally transformed planar Luneburg lens is to focus a linear frequency-modulated signal with a specific chirp rate on the output plane. The specific chirp rate at which the linear frequency-modulated signal can be focused is called the eigen-chirp rate, and the eigen-chirp rate is only related to the wavelength;

[0055] Step 2: Adjust the shape and refractive index of the lens:

[0056] Intercept the middle part -1 ≤ y ≤ 1 of the infinite strip lens obtained in Step 1, and fill the part with a refractive index less than 1 with 1 to obtain a rectangular lens with a width of π / 2, a height of 2, and a refractive index greater than or equal to 1;

[0057] So far, through Steps 1 to 2, the generation of an optical device for detecting linear frequency-modulated signals based on transformation materials of the present invention is completed;

[0058] Further, it further includes Step 3: Measuring the chirp rate of the linear frequency-modulated signal by using an optical device for detecting spatial linear frequency-modulated signals based on transformation materials of the present invention:

[0059] When the light source is a spatial linear frequency-modulated signal, the illumination wavelength determines the chirp rate actually transmitted to an optical device for detecting linear frequency-modulated signals based on transformation materials of the present invention. When the chirp rate satisfies m = 8π / λ, that is, the modulation input chirp rate is equal to the eigen-chirp rate, the signal produces the best focusing effect on the output plane of the planar lens. By scanning the illumination wavelength within a certain range and through the focusing effect, the wavelength at which the best focusing effect is obtained is determined, and the chirp rate of the spatial linear frequency-modulated signal is determined according to the wavelength and Equation (13).

[0060] Beneficial effects:

[0061] 1. An optical device for detecting linear frequency-modulated signals based on transformation materials of the present invention measures the chirp rate of the linear frequency-modulated signal directly on the output plane by scanning the wavelength, without the need for spatial scanning on the optical axis, and has unique flexibility;

[0062] 2. An optical device for detecting linear frequency modulation signals based on transformation materials according to the present invention. When the frequency modulation rate of the linear frequency modulation signal is extremely large, the focal point still lies on the output surface, enabling precise measurement of high-frequency linear frequency modulation signals.

[0063] 3. An implementation method of an optical device for detecting linear frequency modulation signals based on transformation materials according to the present invention. By using conformal transformation to ensure isotropy, it is beneficial for engineering preparation. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 is a schematic structural diagram of an optical device for detecting linear frequency modulation signals based on transformation materials according to the present invention;

[0065] Figure 2 is a schematic diagram of the virtual space W and the physical space Z of the conformal transformation in the implementation method of an optical device for detecting linear frequency modulation signals based on transformation materials according to the present invention;

[0066] Figure 3 is a schematic diagram of a plane wave horizontally incident on the surface of a traditional Luneburg lens in the implementation method of an optical device for detecting linear frequency modulation signals based on transformation materials according to the present invention;

[0067] Figure 4 is a simulation result diagram of the frequency modulation rate detection of a linear frequency modulation signal by an optical device for detecting linear frequency modulation signals based on transformation materials according to the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0068] To better illustrate the purpose and advantages of the present invention, the following further describes the content of the invention in conjunction with the drawings and examples.

[0069] Example 1:

[0070] An optical device for detecting linear frequency modulation signals based on transformation materials is a two-dimensional planar Luneburg lens based on transformation materials. Through conformal transformation, it can maintain a one-to-one correspondence with the traditional Luneburg lens, including: an input surface, a two-dimensional planar Luneburg lens for detecting spatial linear frequency modulation signals based on transformation materials, and an output surface;

[0071] As Figure 1 shown, ABCD is the refractive index distribution of the planar Luneburg lens obtained by conformal transformation of the traditional Luneburg lens. The right boundary AB is the input surface, and the left boundary CD is the output surface.

[0072] The input surface is the incident surface for the input signal to enter the two-dimensional planar Luneburg lens for detecting spatial linear frequency modulation signals based on transformation materials, and the input surface is a plane:

[0073] The input surface is obtained by conformal transformation of the input surface of a traditional Luneburg lens with a radius R = 1 and a center at (0,0) having a radially varying refractive index. That is, the right semi-circular arc of the surface of the traditional Luneburg lens becomes a straight line at x = π / 4, and the part in the y-direction where -1 ≤ y ≤ 1 in the middle is intercepted.

[0074] Furthermore, the input surface undergoes a conformal transformation based on the input surface of the traditional Luneburg lens, and its shape changes. However, in terms of mathematical and physical meanings, the points on the input surface have a one-to-one correspondence with the points on the input surface of the Luneburg lens before conformal transformation.

[0075] Compared with the traditional Luneburg lens, the two-dimensional planar Luneburg lens for detecting spatially chirped signals based on transformation materials changes from a circular shape to a rectangular shape. By means of transformation optics, the radially varying refractive index distribution of the traditional Luneburg lens is changed. Specifically, the refractive index of the lens after conformal transformation shows axial symmetry about x = 0 and y = 0. The refractive index distributions of the input surface and the output surface are the same, with a value of 2 at the center point and gradually decreasing towards both sides. The part in the y-direction where -1 ≤ y ≤ 1 in the middle is intercepted, and the refractive index less than 1 is filled with 1.

[0076] The refractive index distribution determines the propagation path of light in the lens. The two-dimensional planar Luneburg lens for detecting spatially chirped signals based on transformation materials is not restricted by the shape of the lens under the condition of satisfying the refractive index distribution.

[0077] The output surface is a straight line at x = -π / 4 formed by the left semi-circular arc of the surface of the traditional Luneburg lens. When the wavelength λ of the input chirped signal and the chirp rate m satisfy m = 8π / λ, after passing through the two-dimensional planar Luneburg lens for detecting spatially chirped signals based on transformation materials, the chirped signal is focused on the output surface, and the chirp rate of the chirped signal is directly detected on the output surface by scanning the wavelength.

[0078] As Figure 2 shown, an optical device for detecting chirped signals based on transformation materials, and the implementation method includes the following steps:

[0079] Step 1: Perform a conformal transformation W = tan(Z) on a traditional two-dimensional circular Luneburg lens to obtain an infinite two-dimensional planar Luneburg lens. By means of transformation materials, determine the permittivity and permeability of the new lens after transformation, obtain the refractive index after transformation, and determine the waveform change on the input boundary.

[0080] The traditional two-dimensional circular Luneburg lens corresponds to the original complex plane \(W = u+iv\), which is a virtual space. Here, \(u\) is the real part of the original complex plane \(W\), and \(v\) is the imaginary part of the original complex plane \(W\).

[0081] The two-dimensional planar Luneburg lens corresponds to the transformed complex plane \(Z = x + iy\), which is a physical space. Here, \(x\) is the real part of the transformed complex plane \(Z\), and \(y\) is the imaginary part of the transformed complex plane \(Z\).

[0082] As Figure 2 shown in (a), in the two-dimensional coordinate system under the virtual space \(W\), for the traditional circular Luneburg lens with a gradually changing refractive index along the radial direction, the radius \(R = 1\), the center of the circle is \((0,0)\), the input surface is the right semi-circular arc, and the output surface is the left semi-circular arc. Transform the circular lens shown in Figure 2 (a) conformally to the two-dimensional coordinate system under the physical space \(Z\) shown in Figure 2 (b). Before and after the transformation, Figure 2 the right semi-circular arc of (a) becomes Figure 2 the straight line \(x=\pi / 4\) in (b), Figure 2 and the left semi-circular arc of (a) becomes Figure 2 the straight line \(x =-\pi / 4\) in (b).

[0083] Furthermore, when the original complex plane \(W\) is conformally transformed into the transformed complex plane \(Z\), according to the coordinate invariance of Maxwell's equations, the relationship between the physical space \((x,y,Z)\) and the virtual space \((u,v,W)\) is shown in Equation (1):

[0084]

[0085] where \(\varepsilon\) and \(\mu\) are the permittivity and permeability of the virtual space \(W\) respectively, \(\varepsilon'\) and \(\mu'\) are the permittivity and permeability of the physical space \(Z\) respectively, \(\det\cdot\) is the value of the determinant, and \(A\) is the Jacobian matrix, as shown in Equation (2):

[0086]

[0087] The space transformation relationship between the virtual space \(W\) and the physical space \(Z\) is shown in Equation (3):

[0088]

[0089] where represents the partial derivative;

[0090] Furthermore, according to Fermat's theorem, the relationship between the transformation optical path in the physical space \(Z\) and the optical path in the virtual space \(W\) is shown in Equation (4):

[0091]

[0092] Furthermore,

[0093] where d· represents differentiation, and n w is the refractive index of the virtual space W, and n z is the refractive index of the physical space Z.

[0094] After the conformal transformation of the virtual space W into the physical space Z, the right semi-circular arc boundary of the traditional circular Luneburg lens before the transformation is the input surface, and the left semi-circular arc boundary is the output surface. After the transformation, the curved surface becomes a straight surface, the midpoint is compressed, and it is compressed first and then stretched towards both sides.

[0095] Determine the waveform transformation on the input boundary: As Figure 3 shown, on the plane of the virtual space W, on the input boundary of the traditional Luneburg lens with a radius R = 1 and the center at (0, 0), the arc length S is as shown in Equation (5):

[0096]

[0097] where R is the radius of the traditional Luneburg lens, θ is the angle between the line connecting a point on the input surface of the traditional Luneburg lens and the center and the positive semi-axis of the u-axis, and u and v are the horizontal coordinate and the vertical coordinate respectively.

[0098] According to the conformal transformation W = tan(Z), the relationship between u, v and x, y is as shown in Equation (6):

[0099]

[0100] Furthermore, setting the radius of the circle R = 1, the right semi-circular arc on the W plane after the transformation becomes the straight line x = π / 4 on the Z plane, and the relationship between the arc length S on the plane of the virtual space W and the ordinate y on the plane of the physical space Z is as shown in Equation (7):

[0101]

[0102] On the plane of the virtual space W, the horizontal distance Δu from a point on the right semi-circular arc to the point (R, 0) is as shown in Equation (8):

[0103]

[0104] When a plane wave with an initial phase of 0 propagates horizontally to the input surface and is incident on the right semi-circular arc, the signal is as shown in Equation (9):

[0105]

[0106] where A is the amplitude, k = 2π / λ is the wave number, and λ is the wavelength.

[0107] Furthermore, according to Equation (7) and Equation (9), the signal incident on the right semi-circular arc is as shown in Equation (10):

[0108]

[0109] Even further, the signal incident on the right semi-circular arc is expressed in the form of a Taylor expansion, as shown in Equation (11):

[0110] f(S) = A·exp[i(-k(a1y 2 -a2y 4 +a3y 6 +…+a n y 2n +…))] (11)

[0111] where a n = [2 2n (-1) n+1 E n / [(2n)!] is a constant, E n is the Euler number. When n = 1, then a1 = 2;

[0112] When y is very small, ignoring the high-order terms and only retaining the quadratic terms, the signal incident on the right semi-circular arc is approximately a chirp signal as shown in Equation (12):

[0113]

[0114] The traditional Luneburg lens focuses parallel light on the output surface. Then, the conformally transformed planar Luneburg lens can focus a chirp signal with a specific input chirp rate on the output surface. Since the chirp signal is usually expressed as f(y) = exp[-i(1 / 2)my 2 (44)], the chirp rate of the chirp signal is determined according to Equation (12) as shown in Equation (13):

[0115]

[0116] The inherent characteristic of the conformally transformed planar Luneburg lens is to focus a chirp signal with a specific chirp rate on the output surface. The specific chirp rate at which the chirp signal can be focused is called the eigen-chirp rate, and the eigen-chirp rate is only related to the wavelength;

[0117] Step 2: Adjust the shape and refractive index of the lens:

[0118] Cut out the middle part -1 ≤ y ≤ 1 of the infinite strip lens obtained in Step 1, and fill the part with a refractive index less than 1 with 1 to obtain a rectangular lens with a width of π / 2, a height of 2, and a refractive index greater than or equal to 1, as Figure 1as shown;

[0119] So far, through Step 1 to Step 2, the generation of an optical device for detecting a linear frequency modulation signal based on transformation materials according to the present invention is completed;

[0120] Furthermore, it further includes Step 3: measuring the frequency modulation rate of the linear frequency modulation signal by using an optical device for detecting a spatial linear frequency modulation signal based on transformation materials according to the present invention:

[0121] When the light source is a spatial linear frequency modulation signal, the illumination wavelength determines the frequency modulation rate actually transmitted into the optical device for detecting a linear frequency modulation signal based on transformation materials according to the present invention. When the frequency modulation rate satisfies m = 8π / λ, that is, the modulation input frequency modulation rate is equal to the intrinsic frequency modulation rate, the signal produces the best focusing effect on the output surface of the flat lens. By scanning the illumination wavelength within a certain range and through the focusing effect, the wavelength at which the best focusing effect is obtained is determined, and the frequency modulation rate of the spatial linear frequency modulation signal is determined according to the wavelength and formula (13);

[0122] In the embodiment, the frequency modulation rate of the input linear frequency modulation signal is m = 359, that is, the intrinsic wavelength λ = 8π / m ≈ 0.07, and COMSOL Multiphysics is used for frequency modulation rate detection simulation;

[0123] In the embodiment, the wavelength parametric scanning range is (0.055 - 0.085), the step size is 0.005, the normalized window size is 0.4, and the refractive index of the traditional Luneburg lens on the W plane The refractive index of the optical device for detecting a linear frequency modulation signal based on transformation materials according to the present invention is n z = n w |dW / dZ| = n w / γ, and the main elongation γ = |dZ / dW| = |1 / (sec(Z)) 2 | = |1 / (1 + W 2 )| of the input surface and the output surface compared with the traditional Luneburg lens. The input surface is the right boundary of the flat Luneburg lens;

[0124] Table 1 The first zero amplitude at each scanned wavelength when the intrinsic wavelength is 0.07

[0125]

[0126] As Figure 4 shown, Figure 4 (a), Figure 4 (b), Figure 4 (c), Figure 4 (d), Figure 4 (e) and Figure 4(f) are the wave field distribution diagrams with scanning wavelengths of 0.055, 0.06, 0.065, 0.07, 0.075, and 0.08 respectively. Figure 4 (g), Figure 4 (h), Figure 4 (i), Figure 4 (j), Figure 4 (k) and Figure 4 (l) are the amplitude distribution diagrams of the output surface with scanning wavelengths of 0.055, 0.06, 0.065, 0.07, 0.075, and 0.08 respectively.

[0127] From Figure 4 (g), Figure 4 (h), Figure 4 (i), Figure 4 (j), Figure 4 (k) and Figure 4 (l) the amplitude distribution of the output surface, it can be obtained that when the scanning wavelength is 0.065, the electric field value of the first zero point of the output line graph is closest to 0, the waveform is closest to the Fourier transform of a square wave, and the focusing effect is the best. From m = 8π / λ, m = 8π / 0.065 ≈ 386.6576 can be obtained. Compared with the actual tuning frequency of 359, the relative error is about 7.7%. An optical device based on transformation materials for detecting linear frequency modulation signals of the present invention can achieve the focusing of a linear frequency modulation signal with a specific tuning frequency on the output surface and determine its tuning frequency.

[0128] The above specific description further details the purpose, technical solution, and beneficial effects of the invention. It should be understood that the above is only a specific embodiment of the present invention and is not used to limit the protection scope of the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. An optical device for detecting a chirp signal based on transformation materials, characterized in that: Through conformal transformation, a one-to-one correspondence with the traditional Luneburg lens can be maintained to realize a two-dimensional planar Luneburg lens based on transformation materials, specifically including: an input surface, a two-dimensional planar Luneburg lens for detecting spatially linearly chirped signals based on transformation materials, and an output surface; The input surface is the incident surface for the input signal to enter the two-dimensional planar Luneburg lens for detecting spatially linearly chirped signals based on transformation materials, and the input surface is a plane; The input surface is obtained by conformal transformation of the input surface of a traditional Luneburg lens with a radially varying refractive index, where the radius R = 1 and the center is (0, 0). That is, the right semi-circular arc on the surface of the traditional Luneburg lens becomes a straight line at x = π / 4, and the part in the y-direction with -1 ≤ y ≤ 1 in the middle is intercepted; Furthermore, based on the input surface of the traditional Luneburg lens, a conformal transformation is performed, and the shape changes. However, in terms of mathematical and physical meanings, the points on the input surface have a one-to-one correspondence with the points on the input surface of the Luneburg lens before conformal transformation; Compared with the traditional Luneburg lens, the shape of the two-dimensional planar Luneburg lens for detecting spatially linearly chirped signals based on transformation materials changes from circular to rectangular. By means of transformation materials, the permittivity and permeability of the new lens after transformation are determined, the refractive index after transformation is obtained, and the waveform change on the input boundary is determined, changing the radially varying refractive index distribution of the traditional Luneburg lens. Specifically, the refractive index distribution of the lens after conformal transformation is axisymmetric about x = 0 and y = 0 respectively. The refractive index distributions of the input surface and the output surface are the same, being 2 at the center point and gradually decreasing towards both sides. The part in the y-direction with -1 ≤ y ≤ 1 in the middle is intercepted, and the refractive index less than 1 is filled with 1; The refractive index distribution determines the propagation path of light in the lens. When the two-dimensional planar Luneburg lens for detecting spatially linearly chirped signals based on transformation materials satisfies the refractive index distribution, its shape is not restricted by the shape of the lens; The output surface is a straight line at x = -π / 4 formed by the left semi-circular arc on the surface of the traditional Luneburg lens. When the wavelength λ of the input linearly chirped signal and the chirp rate m satisfy m = 8π / λ, after passing through the two-dimensional planar Luneburg lens for detecting spatially linearly chirped signals based on transformation materials, the linearly chirped signal is focused on the output surface, and the chirp rate of the linearly chirped signal is detected directly on the output surface by scanning the wavelength.

2. An optical device for detecting a chirped signal based on transformation materials according to claim 1, characterized in that: The implementation method includes the following steps: Step 1: Perform a conformal transformation W = tan(Z) on the traditional two-dimensional circular Luneburg lens to obtain an infinitely large two-dimensional planar Luneburg lens. By means of transformation materials, the permittivity and permeability of the new lens after transformation are determined, the refractive index after transformation is obtained, and the waveform change on the input boundary is determined; The traditional two-dimensional circular Luneburg lens corresponds to the original complex plane \(W = u+iv\), which is a virtual space. Here, \(u\) is the real part of the original complex plane \(W\), and \(v\) is the imaginary part of the original complex plane \(W\). The two-dimensional planar Luneburg lens corresponds to the transformed complex plane \(Z = x + iy\), which is a physical space. Here, \(x\) is the real part of the transformed complex plane \(Z\), and \(y\) is the imaginary part of the transformed complex plane \(Z\). Furthermore, when the original complex plane \(W\) is conformally transformed into the transformed complex plane \(Z\), according to the coordinate invariance of Maxwell's equations, the relationship between the physical space \((x,y,Z)\) and the virtual space \((u,v,W)\) is shown in Equation (1): where \(\varepsilon\) and \(\mu\) are the permittivity and permeability of the virtual space \(W\) respectively, \(\varepsilon'\) and \(\mu'\) are the permittivity and permeability of the physical space \(Z\) respectively, \(\det\cdot\) is the value of the determinant, and \(A\) is the Jacobian matrix, as shown in Equation (2): The space transformation relationship between the virtual space \(W\) and the physical space \(Z\) is shown in Equation (3): Among them, represents partial differentiation; Furthermore, according to Fermat's theorem, the relationship between the transformed optical path in the physical space \(Z\) and the optical path in the virtual space \(W\) is shown in Equation (4): Furthermore, where d· denotes differentiation, n w is the refractive index of the virtual space W, n z is the refractive index of the physical space Z After the virtual space \(W\) is conformally transformed into the physical space \(Z\), the right semi-circular arc boundary of the traditional circular Luneburg lens before transformation is the input surface, and the left semi-circular arc boundary is the output surface. After transformation, it changes from a curved surface to a straight surface, and is compressed at the midpoint, and compressed first and then stretched towards both sides. Determine the waveform transformation on the input boundary: On the virtual space \(W\) plane, on the input boundary of the traditional Luneburg lens with a radius \(R = 1\) and the center at \((0,0)\), the arc length \(S\) is shown in Equation (5): where \(R\) is the radius of the traditional Luneburg lens, \(\theta\) is the angle between the line connecting a point on the input surface of the traditional Luneburg lens and the center and the positive semi-axis of the \(u\)-axis, and \(u\) and \(v\) are the horizontal coordinate and the vertical coordinate respectively. According to the conformal transformation \(W=\tan(Z)\), the relationship between \(u\), \(v\) and \(x\), \(y\) is shown in Equation (6): Furthermore, setting the radius of the circle \(R = 1\), the right semi-circular arc on the \(W\) plane after transformation becomes the straight line \(x=\frac{\pi}{4}\) on the \(Z\) plane. The relationship between the arc length \(S\) on the virtual space \(W\) plane and the ordinate \(y\) on the physical space \(Z\) plane is shown in Equation (7): On the virtual space \(W\) plane, the horizontal distance \(\Delta u\) from a point on the right semi-circular arc to the point \((R,0)\) is shown in Equation (8): When a plane wave with an initial phase of 0 propagates horizontally to the input surface and is incident on the right semi-circular arc, the signal is shown in Equation (9): where \(A\) is the amplitude, \(k = \frac{2\pi}{\lambda}\) is the wave number, and \(\lambda\) is the wavelength. Furthermore, according to Equation (7) and Equation (9), the signal incident on the right semi-circular arc is shown in Equation (10): Even further, the signal incident on the right semi-circular arc is expressed in the form of a Taylor expansion, as shown in Equation (11): f(S) = A·exp[i(-k(a1y 2 -a2y 4 +a3y 6 +…+a n y 2n +…))] (11) where a n = [2 2n (-1) n+1 E n / [(2n)!] is a constant, E n is the Euler number. When n = 1, then a1 = 2; When \(y\) is very small, ignoring the high-order terms and only keeping the quadratic terms, the signal incident on the right semi-circular arc is approximately a linear frequency modulation signal, as shown in Equation (12): The traditional Luneburg lens focuses parallel light on the output surface. Then, the conformal transformed planar Luneburg lens can focus a linear frequency modulated signal with a specific input chirp rate on the output surface. Since the linear frequency modulated signal is usually expressed as f(y) = exp[-i(1 / 2)my 2 , the chirp rate of the linear frequency modulated signal is determined according to Equation (12) as shown in Equation (13): The inherent characteristic of the flat Luneburg lens after conformal transformation is to focus the chirp signal with a specific chirp rate on the output plane. The specific chirp rate at which the chirp signal can be focused is the eigen-chirp rate, and the eigen-chirp rate is only related to the wavelength. Step 2: Adjust the shape and refractive index of the lens: Intercept the middle part -1 ≤ y ≤ 1 of the infinite strip lens obtained in Step 1, and fill the part with a refractive index less than 1 to 1, to obtain a rectangular lens with a width of π / 2, a height of 2, and a refractive index greater than or equal to 1. So far, through Steps 1 to 2, the generation of an optical device for detecting chirp signals based on transformation materials is completed. Furthermore, it further includes Step 3: Measuring the chirp rate of the chirp signal by applying an optical device for detecting spatial chirp signals based on transformation materials. When the light source is a spatial chirp signal, the illumination wavelength determines the chirp rate actually transmitted into an optical device for detecting chirp signals based on transformation materials. When the chirp rate satisfies m = 8π / λ, that is, the modulation input chirp rate is equal to the eigen-chirp rate, the signal produces the best focusing effect on the output plane of the flat lens. By scanning the illumination wavelength within a certain range and through the focusing effect, determine the wavelength when the focusing effect is the best, and determine the chirp rate of the spatial chirp signal according to the wavelength and formula (13).

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