Transform medium-based high-resolution Fourier device and design method

Abstract

The invention provides a high-resolution Fourier device based on a transformation medium and a design method, and belongs to the technical field of optical signal processing and transformation materials. For the incident boundary of the lens, the incident boundary can be simply filled with homogeneous materials and directly expanded into a straight edge; for the emergent boundary, a sector near the output surface of the Lunburg lens needs to be taken for transformation. The characteristics of the transformation medium are utilized to control the trajectory of the light wave, so that the light wave is guided to be propagated in the optical device according to a preset path, and when the focusing effect of the light wave in the high-resolution Fourier device is known in advance, the medium parameter of the device is determined by utilizing the medium transformation method, so that the expected light field distribution is achieved. The device solves the problems that a Luneburg lens is difficult to manufacture, an output signal is difficult to detect and a system is cascaded. According to the method, smaller aberration can be generated, a higher resolution effect is achieved, and engineering preparation is facilitated.

Description

Technical Field

[0001] The invention relates to a high-resolution Fourier device based on a transformation medium and a design method, belonging to the technical field of optical signal processing and transformation materials. Background Art

[0002] In 2006, JB Pindry et al. proposed a coordinate transformation method, which follows the invariance of the Maxwell equations under the coordinate transformation form, and deduces the values ​​of the dielectric constant and permeability of the transformed medium. The materials of the optical devices designed based on this method are all anisotropic. In the same year, U. Leonhardt proposed a conformal transformation, which follows the invariance of the Helmholtz equation in two-dimensional conformal transformation. The refractive index distribution of the medium before and after the transformation can be directly derived, and the obtained dielectric constant and permeability also meet the isotropic conditions. Transformation optics is a powerful tool for controlling wave fields. It is a method for solving the inverse problem of electromagnetic field distribution-electromagnetic parameters. It uses the formal invariance of Maxwell equations, based on the equivalence of "geometric space" and "material space", and is obtained from known parameters through "mapping (transformation)". The medium that controls the wave field properties obtained by this method is usually called "transformation medium".

[0003] There are many ways to achieve Fourier transform through optics. The simplest method is that when a diffraction hole is illuminated by coherent light, the resulting diffraction pattern observed at a distance from the hole is the Fourier transform of the incident light. However, the light output of this lensless Fourier transform is very small, a difficulty that can be overcome by adding a lens after the diffraction aperture. Another common method is to use a diffraction grating to split and diffract the light wave into several beams, whose diffraction pattern corresponds to the Fourier transform, but the production and adjustment of the diffraction grating is relatively complicated. An optical Fourier transformer is a device that uses lens optical elements to perform Fourier transforms. The spatial distribution of the incident light field is first converted into its Fourier spectrum. An optical element is placed on the focal plane of the optical system to record this frequency information, but its disadvantages are high cost and complex manufacturing and debugging processes. In some holographic techniques, Fourier transform information of the object is generated and encoded into the recording medium. The most common method to achieve pure optical Fourier transform is thin lenses, which can form discrete pure optical Fourier transform systems such as 2f and 4f. However, their application is limited by the influence of geometric size and paraxial approximation in large wavelength (such as millimeter wave) and high-precision Fourier transform scenarios; and discrete thin lens systems are challenging in complex system design and integration. Another way to achieve optical Fourier transform is to use GRIN (Gradient Index) lenses, which have the advantages of flat end faces, small size, and easy installation and debugging. Common GRIN lenses include HS (Hyperbolic-secant) lenses and birefringent lenses. Chen et al. found that HS lenses can be obtained by mapping the maxwell fisheye lens, so it is easy to find that its Fourier transform also works under paraxial approximation conditions from the way the latter guides light. The birefringent lens is an approximation of the HS lens, and its aberration on the focusing plane is larger, and the accuracy of the Fourier transform is further affected; but because it simplifies the radial refractive index distribution, it is still a common Fourier transform device.

[0004] With the continuous expansion of the application of Fourier optics in various fields, especially the development of emerging wavelength fields such as millimeter waves, lenses that can achieve high-precision optical Fourier transform in a large wavelength range are becoming increasingly important. In theory, the Luneburg lens is a perfect Fourier transform system because it can focus light from various specific directions, has no geometric aberrations on the focusing plane, and is not constrained by wavelength and paraxial approximation. However, since its output surface, i.e. the focusing plane, is curved, signal detection, system cascading, etc. become difficult, and the manufacturing difficulty is also increased, which greatly limits its application, so it is usually not used for optical Fourier transform.

[0005] The present invention proposes a high-resolution Fourier device and design method based on a transformation medium, which adopts transformation optics of deformed viewing angle to transform the Luneburg lens and transform the output surface into a straight surface, thereby overcoming its fatal defects, retaining the characteristics of a large range and high precision with relaxed requirements for paraxial approximation, and improving the performance of the device. Summary of the invention

[0006] The purpose of the present invention is to transform the output surface of the Luneburg lens into a straight surface in order to solve the problems that the circular surface of the Luneburg lens is difficult to manufacture in practical applications, the signal of the output surface is difficult to detect, and the system is difficult to cascade. A high-resolution Fourier device and design method based on a transformation medium are proposed. The input and output surface boundaries of the Luneburg lens are both arcs, and are homogeneous along the boundaries. For the incident boundary, it can be simply filled with homogeneous materials to directly expand it into a straight edge; for the exit boundary, it is necessary to take the fan-shaped near the output surface of the Luneburg lens for transformation: the inner boundary of the fan-shaped is fixed to meet the continuity condition, and the outer boundary of the fan-shaped needs to be straightened during design. According to the application requirements, the thickness and center angle of the fan-shaped can be selected; the deformation of the upper and lower boundaries of the fan-shaped is not important, and the shape of the high-resolution Fourier lens after the transformation becomes a rectangle.

[0007] The core idea of ​​the present invention is to control the trajectory of light waves by using the properties of transformation media, thereby guiding the light waves to propagate in optical devices along a preset path. When the focusing effect of light waves in a high-resolution Fourier device is known in advance, that is, the light field distribution of the optical device is clear, the medium parameters of the device are determined by using the method of transformation media, thereby achieving the expected light field distribution.

[0008] The high-resolution Fourier device based on transformation medium and the design method thereof include a high-resolution Fourier device based on transformation medium and a design method thereof; wherein the high-resolution Fourier device based on transformation medium is referred to as the present device, and the design method thereof is referred to as the present method;

[0009] Wherein, the device is an input surface, a high-resolution Fourier lens designed based on a transformation medium, and an output surface;

[0010] The lens is a high-resolution Fourier lens designed based on a transformation medium;

[0011] The left half plane of the Lunburg lens is filled with a homogeneous material, and a part of the sector of the right half plane is conformally transformed. The shape of the obtained high-resolution Fourier lens is a rectangle. Therefore, the input surface is the incident surface of the input signal entering the high-resolution Fourier lens, that is, the left side of the rectangle, and the output surface is the right side of the rectangle. The device after the conformal transformation maintains a one-to-one correspondence with the original device in both mathematical and physical senses;

[0012] Compared with the Luneburg lens, the shape of the lens in this device has changed. The circular Luneburg lens has been changed to a rectangular lens. Through the technical means of transformation optics, the uniform refractive index distribution of the right half plane of the Luneburg lens is changed, so that the refractive index of the lens presents a higher value in the middle than on both sides; because the distribution of the refractive index of the lens determines the propagation path of light in the lens, when the light wave is input to the input surface of the device, it passes through the lens and is focused on the output surface. Therefore, in the case of the refractive index distribution of the device, when the light wave is input to the input surface of the lens, after entering the lens, it is a Fourier transform on the output surface, and a focusing point can be seen;

[0013] Output plane: that is, the spectrum plane. When the input light wave passes through the high-resolution Fourier lens, the output wave is a Fourier transform signal of the input wave.

[0014] The design method of the device is as follows: for a radius of 1 and a refractive index of The outer boundary of the left half plane of the Luneburg lens is filled with a medium with a refractive index of 1, where r is the distance from each point in the Lunburg lens to its center, and r0 is the radius of the Lunburg lens. The partial fan-shaped conformal transformation of the right half plane is divided into two steps and is achieved by the following steps:

[0015] Step 1: Transform each arc of the sector. The curvature of the arc gradually increases from left to right. The left boundary of the sector remains unchanged while the right boundary becomes a straight line. The length of the arc remains unchanged during the transformation, and the midpoint of the arc remains unchanged. The points before and after the transformation have a one-to-one mapping relationship. Let P(r1,θ) be the point before the transformation, and P′(r′,θ′) be the point after the transformation. is the distance between point P and its pole before transformation, i.e., the polar diameter; θ = β1 = arctan(Y g / X g ), θ is the angle between the line connecting point P and its pole and the horizontal coordinate before the transformation, that is, the polar angle; β1 is the angle between the line connecting point P and its center and the horizontal coordinate before the transformation, r′ is the distance between point P′ and its pole after the transformation, θ′ is the angle between the line connecting point P′ and its pole and the horizontal coordinate after the transformation, X g , Y g is the coordinate of point P before transformation in the Cartesian coordinate system.

[0016] The relationship between the radius corresponding to the arc length after transformation and the radius before transformation is:

[0017]

[0018] Where R1 is the radius corresponding to the arc length of the inner boundary of the transformed sector, R2 is the radius corresponding to the arc length of the outer boundary of the transformed sector, and r2 is the radius corresponding to the arc length after transformation. Note that the center of the circle corresponding to r2 is changed after transformation. The arc length before and after transformation remains unchanged, S1 is the arc length before transformation, S2 is the arc length after transformation, and β2 is the angle between the line connecting point P′ and its center after transformation and the horizontal coordinate, that is,

[0019]

[0020] The Cartesian coordinates of point P′ after transformation are

[0021] x p =r2cos(β2)-(r2-r1),y p =r2sin(β2), (3)

[0022] where x p is the horizontal coordinate of P′, y p is the ordinate of P′. In this transformation, since the arc length remains unchanged, the angular principal stretch λ θ =1, and the radial main stretch λ r ≠1, the resulting mapping is not conformal, which will lead to anisotropic materials. Based on the deformation perspective of transformation optics, it is necessary to adjust the local principal stretching to ensure that the principal elongation in the plane tends to be equal to approximate isotropy. This is the purpose of step 2.

[0023] Step 2: After calculating the coordinates of P' in step 1, calculate the principal elongation of the in-plane spatial microelement formed by the deformation in step 1. Deformation gradient tensor A = ▽x', A is the Jacobi transformation matrix, A ij Define for the elements in A, To find the sign of the partial derivative, the left Cauchy-Green deformation tensor is B = AA T , B ij (i,j=1,2,3) is the element definition in B, in the Cartesian coordinate system

[0024]

[0025] The principal in-plane stretch is

[0026]

[0027] Since the arc length is constant, the minimum principal stretch is λ1=1 along the tangent direction of the arc, and the maximum principal stretch λ2 is orthogonal to it and along the normal direction of the arc. Therefore, if a local stretch of λ2 is applied in the tangent direction of each point of the arc, the two principal stretches of the local deformation are equal. Let dS2 be the local arc length before adjustment in step 2, and dS2′ be the local arc length after adjustment, then we have

[0028] dS2′=λ2dS2, taking the integral along the arc length, the arc length starting from the midline becomes

[0029] is the adjusted arc length,

[0030] Therefore, the center angle β2′ corresponding to the arc length satisfies

[0031] S2′=β2′r2,

[0032] Since r2 is constant, we have

[0033]

[0034] That is, after adjustment in step 2, the coordinate β2 of each point becomes β2′ after transformation.

[0035] Then the local deformation is adjusted to conformal, and the Cartesian coordinates of point P′ after transformation become

[0036] x p ′=r2cos(β2′)-(r2-r1),y p ′=r2sin(β2′), where x p ′,y p ' is the Cartesian coordinate of point P' after the fan transformation. When λ2=1, β2′=β2, then x p ′=x p ,y p ′=y p , that is, the left boundary remains unchanged. Here, the high-order changes of the original λ2 caused by the adjustment in step 2 are ignored. In order to reduce the influence of errors, the final microelement stretching rate takes the geometric average of the main stretching. Where λ1 and λ2 represent the final minimum and maximum principal stretches, respectively. Therefore, the refractive index after the fan-shaped transformation is n = n0 / λ, where n0 is the refractive index of the lens before the transformation.

[0037] After steps one and two, the design method of the device is completed.

[0038] Beneficial Effects

[0039] The present invention proposes a high-resolution Fourier device and design method based on a transformation medium, which has the following beneficial effects compared with the existing optical devices:

[0040] 1. The device solves the problems of difficult manufacturing of Luneburg lenses, difficult detection of output signals, and system cascading.

[0041] 2. The lens of the method can produce smaller aberration when a light wave with an oblique incident angle is input.

[0042] 3. The lens of the method has a higher resolution effect on two light waves with a smaller difference in oblique incident angles.

[0043] 4. The design method of the device adopts conformal transformation, which ensures the isotropy of the lens material and is conducive to engineering preparation. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 It is a schematic diagram of the composition of the device of the "high-resolution Fourier device and design method based on transformation medium" in Example 1 of the present invention;

[0045] Figure 2 This is a diagram of the design method of the device in Example 2 of the "high-resolution Fourier device and design method based on transformation medium" of the present invention;

[0046] Figure 3 This is a comparison diagram of geometric simulation results of the aberrations of the device in Example 3 of the present invention "High-resolution Fourier device and design method based on transformation medium" when the device is applied to two light waves with a small difference in oblique incident angles, compared with those of the traditional secondary GRIN lens and HS lens;

[0047] Figure 4 This is a comparison chart of the wave simulation results of the two focusing points separated by the traditional secondary GRIN lens and HS lens when the device in Example 4 of the present invention "High-resolution Fourier device and design method based on transformation medium" is applied to the input of two light waves with a small difference in oblique incident angles. DETAILED DESCRIPTION

[0048] The present invention will be described in detail below with reference to the accompanying drawings and embodiments, but the specific implementation of the present invention is not limited thereto.

[0049] Example 1

[0050] The device designed according to this method has an operating wavelength that can be applied to the millimeter or sub-millimeter level. The advantages of the lens, such as lightness, smallness, stability, and versatility, are more obvious in this band. It can be applied to optical measurement, and the spectral information of the input signal can be outputted conveniently and quickly through the device. The lens can be realized under the current material manufacturing technology (such as metamaterials, 3D printing, etc.), and a complex refractive index distribution can be printed to further manufacture the lens.

[0051] This embodiment describes the specific implementation of the composition and design method of the device in the "high-resolution Fourier device and design method based on transformation medium" of the present invention.

[0052] from Figure 1 As shown in Figure 1 This is a diagram of the composition of this device.

[0053] The device transforms the Luneburg lens by simply filling the incident boundary with homogeneous material to directly expand it into a straight edge. For the exit boundary, it is necessary to transform the fan-shaped area near the output surface of the Luneburg lens, straighten the outer boundary of the fan-shaped area, and keep the inner boundary of the fan-shaped area unchanged to meet the continuity condition. Figure 1 It can be seen that the left side of the rectangle is the input surface of the device, and the right side of the rectangle is the output surface. Specifically in this embodiment, the light wave is incident from the left side of the rectangle, and after passing through the lens, it is focused on the output surface, that is, the right side of the rectangle. Therefore, in the case of the refractive index distribution of the device, when the light wave is input to the input surface of the lens, after entering the lens, it is a Fourier transform on the output surface, and a focusing point can be seen. The design method of this device adopts conformal transformation, which ensures the isotropy of the lens material and is conducive to engineering preparation. According to the application requirements, the thickness and central angle of the fan can be selected; the deformation of the upper and lower boundaries of the fan is not important.

[0054] Example 2

[0055] This embodiment describes the process and specific implementation of the method in the present invention "High-resolution Fourier device and design method based on transformation medium".

[0056] Figure 2 It is the process and specific implementation of the method in the present invention "High-resolution Fourier device and design method based on transformation medium".

[0057] For a radius of 1, the refractive index is The outer boundary of the left half plane of the Luneburg lens is filled with a medium with a refractive index of 1, where r is the distance from each point in the Lunburg lens to its center, and r0 is the radius of the Lunburg lens. The partial sector conformal transformation of the right half plane is divided into two steps, see Figure 2 , this method comprises the following steps:

[0058] Step 1: Transform each arc of the sector. The curvature of the arc gradually increases from left to right. The left boundary of the sector remains unchanged while the right boundary becomes a straight line. The length of the arc remains unchanged during the transformation, and the midpoint of the arc remains unchanged. The points before and after the transformation have a one-to-one mapping relationship. Let P(r1,θ) be the point before the transformation, and P′(r′,θ′) be the point after the transformation. is the distance between point P and its pole before transformation, i.e., the polar diameter; θ = β1 = arctan(Y g / X g ), θ is the angle between the line connecting point P and its pole and the horizontal coordinate before the transformation, that is, the polar angle; β1 is the angle between the line connecting point P and its center and the horizontal coordinate before the transformation, r′ is the distance between point P′ and its pole after the transformation, θ′ is the angle between the line connecting point P′ and its pole and the horizontal coordinate after the transformation, X g , Yg is the coordinate of point P before transformation in the Cartesian coordinate system.

[0059] The relationship between the radius corresponding to the arc length after transformation and the radius before transformation is:

[0060]

[0061] Where R1 is the radius corresponding to the arc length of the inner boundary of the transformed sector, R2 is the radius corresponding to the arc length of the outer boundary of the transformed sector, and r2 is the radius corresponding to the arc length after transformation. Note that the center of the circle corresponding to r2 is changed after transformation. The arc length before and after transformation remains unchanged, S1 is the arc length before transformation, S2 is the arc length after transformation, and β2 is the angle between the line connecting point P′ and its center after transformation and the horizontal coordinate, that is,

[0062]

[0063] The Cartesian coordinates of point P′ after transformation are

[0064] x p =r2cos(β2)-(r2-r1),y p =r2sin(β2), (3)

[0065] where x p is the horizontal coordinate of P′, y p is the vertical coordinate of P′.

[0066] Specifically in this embodiment, Figure 2 (a) is a schematic diagram of the fan-shaped transformation before and after, where the dotted line represents the uncertain shape of the line after the transformation, the red line indicates that the left boundary remains unchanged before and after the transformation, the green and blue solid lines and the green and blue dotted lines correspond to the boundaries before and after the transformation respectively, and the black line indicates that the right boundary changes from an arc to a straight line before and after the transformation. Figure 2 (b) is a schematic diagram of the transformation in step 1. The arc length remains unchanged before and after the transformation, but the central angle changes.

[0067] In this transformation, since the arc length remains unchanged, the angular principal stretch λ θ =1, and the radial main stretch λ r ≠1, the resulting mapping is not conformal, which will lead to anisotropic materials. Based on the deformation perspective of transformation optics, it is necessary to adjust the local principal stretching to ensure that the principal elongation in the plane tends to be equal to approximate isotropy. This is the purpose of step 1.

[0068] Step 2: After calculating the coordinates of P' in step 1, calculate the principal elongation of the in-plane spatial microelement formed by the deformation in step 1. Deformation gradient tensor A = ▽x', A is the Jacobi transformation matrix, A ij Define for the elements in A, To find the sign of the partial derivative, the left Cauchy-Green deformation tensor is B = AA T , B ij (i,j=1,2,3) is the element definition in B, in the Cartesian coordinate system

[0069]

[0070] The principal in-plane stretch is

[0071]

[0072] Since the arc length is constant, the minimum principal stretch is λ1=1 along the tangent direction of the arc, and the maximum principal stretch λ2 is orthogonal to it and along the normal direction of the arc. Therefore, if a local stretch of λ2 is applied in the tangent direction of each point of the arc, the two principal stretches of the local deformation are equal. Let dS2 be the local arc length before adjustment in step 2, and dS2′ be the local arc length after adjustment, then we have

[0073] dS2′=λ2dS2, taking the integral along the arc length, the arc length starting from the midline becomes

[0074] is the adjusted arc length,

[0075] Therefore, the center angle β2′ corresponding to the arc length satisfies

[0076] S2′=β2′r2,

[0077] Since r2 is constant, we have

[0078]

[0079] That is, after adjustment in step 2, the coordinate β2 of each point becomes β2′ after transformation.

[0080] Then the local deformation is adjusted to conformal, and the Cartesian coordinates of point P′ after transformation become

[0081] x p ′=r2cos(β2′)-(r2-r1),y p ′=r2sin(β2′), where x p ′,y p ′ is the Cartesian coordinate of point P′ after the fan transformation.

[0082] Specifically in this embodiment, Figure 2 (c) is a schematic diagram of the transformation in step 2, stretching along the tangential direction of the arc to adjust the principal elongation of the microelement in the radial and tangential directions to be equal.

[0083] When λ2=1, β2′=β2, then x p ′=x p ,yp ′=y p , that is, the left boundary remains unchanged. Here, the high-order changes of the original λ2 caused by the adjustment in step 2 are ignored. In order to reduce the influence of errors, the final microelement stretching rate takes the geometric average of the main stretching. Where λ1 and λ2 represent the final minimum and maximum principal stretches, respectively. Therefore, the refractive index after the fan-shaped transformation is n = n0 / λ, where n0 is the refractive index of the lens before the transformation.

[0084] After steps one and two, the device is completed.

[0085] Example 3

[0086] This example describes the ray trajectory simulation results of comparing the device in the present invention "High-resolution Fourier device and design method based on transformation medium" with the traditional secondary GRIN lens and HS lens. This example uses COMSOL Multiphysics for computer simulation.

[0087] The parameters in this embodiment are set as follows: The following related parameters ignore the length unit and can be set as needed. The wavelength is 0.03, the lens length is 2, the width is 2, the transformation sector thickness is 0.2, the center angle is 90°, and the maximum refractive index is set to The input window of the light wave is 0-0.6.

[0088] In the geometric optics simulation, two oblique incident angles (incident direction (x:5.5, y:-1, 10.3 degrees), incident direction (x:5.75, y:-1, 9.87 degrees)) are input to the input surface of the traditional quadratic GRIN lens, HS lens and high-resolution Fourier lens at the same time. The focusing results of the Fourier transform of the three lenses are shown as follows Figure 3 (ac), Figure 3 (a) is the ray trajectory of the traditional secondary GRIN lens when the oblique incident light wave is 10.30° and 9.87°. Figure 3 (b) is the ray trajectory of the HS lens when the oblique incident light wave is 10.30° and 9.87°. Figure 3 (c) is the ray trajectory of the high-resolution Fourier lens when the oblique incident light waves are 10.30° and 9.87°. The two oblique incident light waves have large aberrations in the traditional quadratic GRIN lens. The two focal points on the output surface of the traditional quadratic GRIN lens overlap and cannot be distinguished. The aberrations of the output surfaces of the HS lens and the high-resolution Fourier lens are small. The two focal points on the output surface of the HS lens also have overlapping areas. The two focal points of the high-resolution Fourier lens can be clearly separated.

[0089] Therefore, through this device, when inputting light waves, although the high-resolution Fourier lens is not a perfect Fourier transform device, its resolution is significantly improved.

[0090] Example 4

[0091] This example describes the wave optics simulation results of comparing the device in the "High-resolution Fourier device and design method based on transformation medium" of the present invention with the traditional secondary GRIN lens and HS lens. This example uses COMSOL Multiphysics to perform computer simulation according to the parameter settings described in Example 3.

[0092] The parameters in this embodiment are set as follows: The following related parameters ignore the length unit and can be set as needed. The wavelength is 0.03, the lens length is 2, the width is 2, the transformation sector thickness is 0.2, the center angle is 90°, and the maximum refractive index is set to The input window of the light wave is 0-0.8.

[0093] In the wave optics simulation, the same light wave is input into the traditional quadratic GRIN lens, HS lens, and high-resolution Fourier lens respectively. The reference incident angles of the light waves of these three lenses are set to 5°, 10°, 15°, and 20°. The reference incident angle is increased in steps of 0.1 degrees until the Fourier transform results of the two beams are distinguishable under the Rayleigh criterion. Figure 4 The field intensity distribution of the wave with a reference angle of 20° is given, Figure 4 (a) is the electric field intensity diagram of the traditional secondary GRIN lens at 20 degrees. Figure 4 (b) is the electric field intensity diagram of the traditional secondary GRIN lens at 27 degrees. Figure 4 (c) is the electric field intensity distribution diagram of the traditional secondary GRIN lens at 20 degrees and 27 degrees. Figure 4 (d) is the electric field intensity diagram of HS lens at 20 degrees. Figure 4 (e) is the electric field intensity diagram of HS lens at 26.7 degrees. Figure 4 (f) is the electric field intensity distribution diagram of the traditional secondary GRIN lens at 20 degrees and 26.7 degrees. Figure 4 (g) is the electric field intensity diagram of the high-resolution Fourier lens at 20 degrees. Figure 4 (h) is the electric field intensity diagram of the high-resolution Fourier lens at 22.2 degrees. Figure 4(i) is the electric field intensity distribution diagram of the high-resolution Fourier lens at 20 degrees and 22.2 degrees. According to the Rayleigh criterion, the traditional secondary GRIN lens can just separate the light waves of 20° and 27° at the output surface, the HS lens can just separate the light waves of 20° and 26.7° at the output surface, and the high-resolution Fourier lens can just separate the light waves of 20° and 22.2°. In addition, the focusing effect of the traditional secondary GRIN lens and HS lens at the output surface is not as good as that of the high-resolution Fourier lens. The complete data with the reference incident angle set to 5°, 10°, 15°, and 20° is shown in Table 1. The results show that compared with the traditional secondary GRIN lens and HS lens, the high-resolution Fourier lens has a stable minimum separation angle and high resolution.

[0094] Table 1 compares the minimum separation angles that can be distinguished by the output surfaces of traditional secondary GRIN lens, HS lens, and high-resolution Fourier lens under the Rayleigh criterion

[0095]

[0096] The above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A design method for a high-resolution Fourier device based on a transformation medium, characterized in that: The Fourier device relied on includes: input surface, high-resolution Fourier lens based on transformation medium design and output surface; The design method of the Fourier device is as follows: for a radius of 1 and a refractive index of The outer boundary of the left half plane of the Luneburg lens is filled with a medium with a refractive index of 1, where r is the distance from each point in the Lunburg lens to its center, r0 is the radius of the Lunburg lens, and the partial fan-shaped conformal transformation of the right half plane is divided into two steps, including the following steps: Step 1: Transform each arc of the sector. The curvature of the arc gradually increases from left to right. The left boundary of the sector remains unchanged while the right boundary becomes a straight line. The length of the arc remains unchanged during the transformation, and the midpoint of the arc does not move. There is a one-to-one mapping relationship between the points before and after the transformation. Let P(r1,θ) be the point before the transformation, and P′(r′,θ′) be the point after the transformation. is the distance between point P and its pole before transformation, i.e., the polar diameter; θ = β1 = arctan(Y g / X g ), θ is the angle between the line connecting point P and its pole and the horizontal coordinate before the transformation, that is, the polar angle; β1 is the angle between the line connecting point P and its center and the horizontal coordinate before the transformation, r′ is the distance between point P′ and its pole after the transformation, θ′ is the angle between the line connecting point P′ and its pole and the horizontal coordinate after the transformation, X g , Y g is the coordinate of point P before transformation in the Cartesian coordinate system; The relationship between the radius corresponding to the arc length after transformation and the radius before transformation is: Where R1 is the radius corresponding to the arc length of the inner boundary of the transformed sector, R2 is the radius corresponding to the arc length of the outer boundary of the transformed sector, r2 is the radius corresponding to the arc length after transformation, note that the center of the circle corresponding to r2 is changed after transformation, and the arc length before and after transformation remains unchanged, S1 is the arc length before transformation, S2 is the arc length after transformation, β2 is the angle between the line connecting point P′ and its center after transformation and the horizontal coordinate, that is, The Cartesian coordinates of point P′ after transformation are x p =r2cos(β2)-(r2-r1),y p =r2sin(β2), (3) where x p is the horizontal coordinate of P′, y p is the ordinate of P′. In this transformation, since the arc length remains unchanged, the angular principal stretch λ θ =1, and the radial main stretch λ r ≠1, the resulting mapping is not conformal, which will lead to anisotropic materials. Based on the deformation perspective of transformation optics, it is necessary to adjust the local principal stretching to ensure that the principal elongation in the plane tends to be equal to approximate isotropy; Step 2: After calculating the coordinates of P' in step 1, calculate the principal extension of the in-plane spatial microelement formed by the deformation in step 1 and the deformation gradient tensor A is the Jacobi transformation matrix, A ij Define for the elements in A, To find the sign of the partial derivative, the left Cauchy-Green deformation tensor is B = AA T , B ij (i,j=1,2,3) is the element definition in B, in the Cartesian coordinate system The principal in-plane stretch is Since the arc length is unchanged, the minimum principal stretch is λ1=1 along the tangent direction of the arc, and the maximum principal stretch λ2 is orthogonal to it along the normal direction of the arc. Therefore, if a local stretch of λ2 is applied in the tangent direction of each point of the arc, the two principal stretches of the local deformation are equal. Let dS2 be the local arc length before adjustment in step 2, and dS2′ be the local arc length after adjustment, then we have dS2′=λ2dS2, taking the integral along the arc length, the arc length starting from the midline becomes S2′ is the adjusted arc length, Therefore, the center angle β2′ corresponding to the arc length satisfies S2′＝β2′r2, Since r2 is constant, we have That is, after the adjustment in step 2, the coordinate β2 of each point becomes β2′ after transformation; Then the local deformation is adjusted to conformal, and the Cartesian coordinates of point P′ after transformation become x p ′=r2cos(β2′)-(r2-r1),y p ′=r2sin(β2′), where x p ′,y p ′ is the Cartesian coordinate of point P′ after the fan transformation. When λ2=1, β2′=β2, then x p ′=x p ,y p ′=y p , that is, the left boundary remains unchanged. Here, the high-order changes of the original λ2 caused by the adjustment in step 2 are ignored. In order to reduce the influence of errors, the final microelement stretching rate takes the geometric average of the main stretching. Where λ1 and λ2 represent the final minimum and maximum principal stretches, respectively. Therefore, the refractive index after the fan-shaped transformation is n = n0 / λ, where n0 is the refractive index of the lens before the transformation; After steps one and two, the design method of the device is completed.

2. The design method of a high-resolution Fourier device based on a transformation medium as claimed in claim 1, characterized in that: The lens is a high-resolution Fourier lens designed based on a transformation medium, wherein the left half plane of the Lunburg lens is filled with a homogeneous material, and a partial sector of the right half plane is conformally transformed, so that the shape of the obtained high-resolution Fourier lens is a rectangle. Therefore, the input surface is the incident surface of the input signal entering the high-resolution Fourier lens, that is, the left side of the rectangle, and the output surface is the right side of the rectangle. The device after the conformal transformation maintains a one-to-one correspondence with the original device in both mathematical and physical senses.

3. The design method of a high-resolution Fourier device based on a transformation medium as claimed in claim 2, characterized in that: Compared with the Luneburg lens, the shape of the lens in the Fourier device has changed. The circular Luneburg lens has been changed into a rectangular lens. Through the technical means of transformation optics, the uniform refractive index distribution of the right half plane of the Luneburg lens is changed, so that the refractive index of the lens presents a higher value in the middle than on both sides.

4. The design method of a high-resolution Fourier device based on a transformation medium as claimed in claim 3, characterized in that: Since the distribution of the refractive index of the lens determines the propagation path of light in the lens, when the light wave is input to the input surface of the device, it passes through the lens and is focused on the output surface.

5. The design method of a high-resolution Fourier device based on a transformation medium as claimed in claim 4, characterized in that: When the Fourier device satisfies the refractive index distribution, the light wave is input to the input surface of the lens. After entering the lens, it undergoes a Fourier transformation on the output surface, and a focusing point is seen.

6. A method for designing a high-resolution Fourier device based on a transformation medium as claimed in any one of claims 1 to 5, characterized in that: The output surface is a spectrum surface. When the input light wave passes through a high-resolution Fourier lens, the output wave is a Fourier transform signal of the input wave.

Images