Four-focus long-focal-depth broadband achromatic superlens design based on improved weighted GS algorithm

Abstract

The invention discloses a four-focus long-focal-depth broadband achromatic superlens design based on an improved weighted GS algorithm, and belongs to the field of micro-nano optics. The super lens comprises a silicon dioxide substrate and a titanium dioxide cylindrical nano column array. In the initial iteration stage of the GS algorithm, focusing and axicon superposition phases are adopted to construct a complex amplitude field, and a dynamic amplitude weighted feedback mechanism is introduced. According to the design, four off-axis focuses with highly uniform energy are realized on a target focal plane, and focal length drift caused by broadband dispersion is covered by using an expanded super-long focal depth. Simulation shows that within the range of 640-740 nm, the average focal depth of the device reaches 6.33 microns, the focus intensity deviation is smaller than 5%, the average focusing efficiency is 62.47%, and broadband achromatic focusing is effectively achieved. The method has a wide application prospect in the fields of parallel laser processing, high-throughput imaging and the like.

Description

Technical Field

[0001] This invention belongs to the field of micro-nano optics and diffractive optics, specifically relating to a superlens device and its design method that utilizes metasurface structures to achieve multifocal beam shaping, depth of focus extension, and broadband achromaticity. Background Technology

[0002] Metasurfaces, as two-dimensional metamaterials, can flexibly control the amplitude, phase, and polarization of light waves through subwavelength structural units. Multifocal superlenses have significant application value in parallel laser processing, optical communication beam splitting, particle trapping, and high-throughput microscopy. Existing multifocal superlens designs often employ the Gerchberg-Saxton (GS) iterative algorithm to solve for the phase hologram. However, when generating multifocal phase distributions, the traditional GS algorithm, limited by its phase retrieval mechanism, often suffers from poor convergence and uneven energy distribution, easily getting trapped in local optima. This results in significant differences in focal intensity at different locations on the target plane (poor uniformity), making it difficult to meet the application requirements of high-precision parallel processing or imaging.

[0003] Furthermore, traditional superlens devices generally suffer from severe chromatic aberration. The focal point generated using conventional algorithms is typically a diffraction-limited Gaussian spot with a short Rayleigh length (i.e., depth of focus). When the operating wavelength changes, the focal length axial drift caused by material and structural dispersion rapidly exceeds the coverage of this short depth of focus, leading to defocusing and imaging failure, severely limiting the device's ability to operate under broadband light sources. Although some design schemes based on achromatic unit structures (such as complex nanopillars utilizing group delay modulation) exist, these often involve extremely high computational costs, high aspect ratio structures, and high fabrication costs, making large-scale application difficult.

[0004] While some explorations have been made in current technologies regarding depth-of-focus extension and chromatic aberration control for superlenses, limitations remain. For example, some studies have used superoscillatory phase modulation optimization methods to design single-focus long-depth-of-focus superlenses, achieving extended depth of focus, but these are mainly limited to single-point focusing and do not address the complex field manipulation of multi-focus areas. Other researchers have used phase masks such as Cubic or ShiftedAxicon to generate longitudinally extended focal points to compensate for dispersion drift and achieve full-color imaging; however, these methods are mostly applied to imaging lenses, with relatively few studies on their application in multi-beam parallel beam splitting. Furthermore, while high numerical aperture multifocal designs based on geometric phase and GS algorithms can achieve clear multi-point focusing, the characteristics of the generated focal points are still limited by traditional diffraction rules, resulting in a standard Gaussian spot with a short depth of focus. Due to the lack of a synergistic design for "multifocal uniformity" and "broadband achromatic aberration," current technologies struggle to maintain stable multifocal focusing performance under non-designed wavelengths or broadband light sources; once the wavelength shifts, the focus deviates from the target plane. Therefore, there is an urgent need for a superlens design scheme that is simple in structure and can simultaneously achieve highly uniform multifocal beam splitting and broadband achromaticity. Summary of the Invention

[0005] The purpose of this invention is to provide a four-focal-length, long-depth-of-field, broadband achromatic superlens based on an improved weighted GS algorithm and its design method, aiming to solve the problems of poor energy uniformity, narrow bandwidth, and chromatic aberration sensitivity of existing multifocal superlenses.

[0006] The first aspect of the present invention provides a four-focal-length-depth-of-focus broadband achromatic superlens based on an improved weighted Gerchberg-Saxton (GS) algorithm. The superlens includes a transparent substrate and a plurality of subwavelength unit structures arranged in an array on one side surface of the transparent substrate. The subwavelength unit structures are cylindrical nanopillars, and their phase distribution profiles are calculated and generated by the improved weighted Gerchberg-Saxton (GS) algorithm.

[0007] Furthermore, the transparent substrate is made of silicon dioxide (SiO2), and the subwavelength unit structure is made of titanium dioxide (TiO2); the cylindrical nanopillars have a lattice period of 350 nm, a height of 1 μm, and a radius that varies from 50 nm to 150 nm.

[0008] Furthermore, the superlens is composed of 60×60 unit structures, with a designed wavelength of 680 nm and a numerical aperture of approximately 0.62.

[0009] A second aspect of the present invention provides a design method for the above-mentioned superlens, comprising the following steps:

[0010] Step 1: Establish a unit structure database. Use the finite-difference time-domain method (FDTD) to scan the correspondence between the radius of the cylindrical nanopillar and the transmission phase and amplitude, and construct a phase response database.

[0011] Step 2: Set the target light field. Set the target light field of the superlens to four off-axis Gaussian spots distributed in a rectangle on the focal plane.

[0012] Step 3: Initialize the phase. In the initial iterative stage of the improved weighted GS algorithm, a complex amplitude field superimposed on the focused phase and the axial pyramid phase is loaded as the initial optical field distribution, rather than a random phase; the introduction of the axial pyramid phase factor gives the superlens the physical characteristic of long focal depth.

[0013] Step 4: Iterative optimization. Perform iterative calculations. During the forward propagation of each iteration, calculate the peak intensity of the four focal points on the focal plane and introduce a dynamic amplitude weighted feedback mechanism. Dynamically adjust the weighting factor of the target light field according to the non-uniformity of the intensity of each focal point, forcing the energy to be evenly distributed among the four focal points until phase convergence.

[0014] Step 5: Structure Mapping. Extract the converged phase distribution and map the continuous phases to the corresponding nanopillar radius arrays according to the database described in Step 1 to obtain the final superlens structure.

[0015] The beneficial effects of this invention are as follows:

[0016] High uniformity: By introducing a dynamic weight feedback mechanism into the GS algorithm, the problem of traditional algorithms getting trapped in local optima is effectively overcome, making the energy intensity deviation of the four foci less than 5%, which is significantly better than traditional methods.

[0017] Broadband achromatic: Utilizing the ultra-long depth of focus (average depth of focus up to 6.33 μm) generated by axial prism phase initialization, a relatively long common focusing channel is formed along the axis. This channel effectively covers the focal length drift caused by material and structural dispersion in broadband light (640nm-740nm), thus achieving "quasi-achromatic" focusing in the broadband range with a simple single-layer structure without the need for complex achromatic unit structures. Furthermore, the average focusing efficiency reaches as high as 62.47% in the 640nm-740nm range, while the focal length (FWHM) remains at the sub-micron level (approximately 870nm).

[0018] Simple structure: It adopts conventional cylindrical TiO2 nanopillars, which have mature technology and are easy to process and integrate. Attached Figure Description

[0019] Figure 1This is a schematic diagram of the structure of the four-focal long focal depth superlens provided in the embodiment of the present invention; wherein, (a) is a schematic diagram of the overall working principle of the superlens under illumination, (b) is a three-dimensional perspective schematic diagram of the subwavelength unit structure, and (c) is a top view schematic diagram of the subwavelength unit structure.

[0020] Figure 2 This is a flowchart of the superlens design method provided in the embodiments of the present invention.

[0021] Figure 3 This is a schematic diagram of the initial phase construction principle in the design method of this invention; wherein, (a) is the spherical wave phase distribution of the standard focusing lens, (b) is the conical wave phase distribution of the axial pyramid, and (c) is the composite phase distribution used for algorithm initialization after the two are superimposed.

[0022] Figure 4 The above are electromagnetic simulation scanning results of the cylindrical nanopillar unit structure in the embodiments of the present invention; wherein, (a) is the characteristic curve of the transmission phase as a function of the nanopillar radius, and (b) is the characteristic curve of the transmittance as a function of the nanopillar radius.

[0023] Figure 5 The above are simulation results of the light field of the superlens designed in the embodiment of the present invention; wherein, (a) is the normalized light intensity distribution of the focal plane (XY plane), and (b) is the normalized light intensity distribution of the axial propagation plane (XZ plane).

[0024] Figure 6 These are comparison diagrams of focal intensity distribution generated by the improved weighted GS algorithm and the traditional standard GS algorithm in this embodiment of the invention; wherein, (a) and (b) are the transverse cross-sectional light intensity distribution curves of the four focal points (f1, f4 and f2, f3) obtained by the improved weighted GS algorithm of this invention; (c) and (d) are the transverse cross-sectional light intensity distribution curves of the four focal points (f1, f4 and f2, f3) obtained by the traditional standard GS algorithm.

[0025] Figure 7 This is a graph showing the axial light intensity distribution characteristics of the four focal points generated by the superlens designed in this embodiment of the invention along the optical axis (Z-axis).

[0026] Figure 8 These are simulation results of the focusing performance of the superlens designed in this embodiment of the invention at different wavelengths in a broadband range of 640 nm to 740 nm; the top row of images shows the light intensity distribution on the axial propagation plane (XZ plane), and the bottom row of images shows the light intensity distribution on the focal plane (XY plane).

[0027] Figure 9 This is a graph showing the focusing efficiency variation of the superlens designed in the embodiment of the present invention in the broadband range of 640 nm to 740 nm. Detailed Implementation

[0028] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0029] Example 1: Structural design of a superlens.

[0030] like Figure 1 As shown, this embodiment provides a four-focal-length, long-depth, broadband achromatic superlens, which, from bottom to top, includes a substrate layer and a subwavelength structure layer. Figure 1 As shown in (b), the substrate material is transparent silicon dioxide (SiO2), and the subwavelength structure layer consists of periodically arrayed titanium dioxide (TiO2) cylindrical nanopillars. Specific structural parameters are as follows: lattice period... =350 nm, this subwavelength period can effectively suppress higher-order diffraction effects; nanopillar height =1 μm, sufficient height to ensure adequate modulation and accumulation of the light wave phase; nanopillar radius It varies between 50 nm and 150 nm. The designed center wavelength is... =680 nm. The entire superlens device consists of 60×60 unit structures, with a device radius of approximately 10.5 μm and a designed focal length of... =13.125μm, corresponding numerical aperture ≈0.625.

[0031] Figure 1 (a) illustrates the working principle of the superlens under illumination. When the light beam travels along... When incident on the axis, the particles are modulated by the metasurface and converge at the focal plane to form four off-axis focal points. It should be noted that, although... Figure 1 (a) This is illustrated using linearly polarized light as an example. However, because the cylindrical nanopillar used in this invention has rotational symmetry, the superlens is insensitive to the polarization state of the incident light and is also applicable to circularly polarized or unpolarized light. Furthermore, as... Figure 1 As shown in (c), the distribution of nanopillars on the surface of the superlens exhibits specific symmetry characteristics. It is important to emphasize that this invention employs a complex amplitude superposition design method, rather than spatial partitioning. The geometric dimensions of each nanopillar unit are determined by the composite phase calculated from the linear superposition of the complex amplitude functions of the four target focal points, thus enabling the entire aperture to participate in the manipulation of the optical field at the four focal points.

[0032] Example 2: Design method and algorithm flow.

[0033] like Figure 2 As shown, the superlens design method in this embodiment includes the following steps:

[0034] Step S1: Establish a phase response database.

[0035] Electromagnetic simulations of a single nanopillar unit were performed using the finite-difference time-domain (FDTD) method. Periodic boundary conditions were set, and a plane wave was incident to scan the nanopillar radius from 50 nm to 150 nm. The simulation results are as follows: Figure 3 As shown. Figure 3 (a) shows the curve of transmission phase as a function of radius, which shows that within the design range, the phase achieves a change from 0 to 2. It provides full coverage and meets the requirements of arbitrary wavefront modulation. Figure 3 (b) shows the change in transmittance with radius, with transmittance remaining above 90% for most dimensions, ensuring the high efficiency of the device.

[0036] Step S2: Construct an improved weighted GS algorithm.

[0037] Traditional GS algorithms, when processing multifocal holograms, often start iterating from a random phase, which can easily lead to uneven energy distribution across focal points. This invention makes two core improvements to the traditional algorithm:

[0038] Axial pyramid phase initialization: such as Figure 4 As shown, the initial light field on the algorithm input plane Instead of using random noise, a "focused phase" is employed. Figure 4 a) and “axial pyramid phase” Figure 4 b) Superimposed complex amplitude field ( Figure 4 c). Its physical logic is based on the focus of each target. Apply an action to make it focus on ( , , The lens phase and an axial pyramidal phase that produces its long depth of focus characteristics The specific formula is: in The modulation factor is a pyramidal axis; in this embodiment, it is taken as... = 0.15. The complex amplitudes of the four focal points are linearly superimposed to form the total initial field. This step is crucial for achieving a long depth of focus.

[0039] Dynamic amplitude-weighted feedback: In the iterative loop, the peak intensities at the four focal points on the focal plane are monitored in real time. Assume the... In this iteration, the intensities of the four foci are respectively , , , Its average value is When generating new target plane amplitude constraints, weights are applied to each focal region. The weight update rule is as follows: If the intensity of a focal point is below the average, its weight increases in the next iteration; otherwise, it decreases.

[0040] Step S3: Iterative optimization and structure mapping.

[0041] The algorithm constructed in step S2 was run in MATLAB. After approximately 100 iterations, the phase distribution tended to converge. The calculated phase hologram exhibited a smooth gradient. Finally, based on the database from step S1, the phase value of each pixel in the hologram was matched to the nearest nanopillar radius to generate the final superlens array layout.

[0042] Example 3: Device Performance Verification

[0043] To verify the effectiveness of the design, vector diffraction simulation was performed on the generated structure.

[0044] Verification of focal plane light intensity distribution and uniformity: such as Figure 5 As shown in (a), at a design wavelength of 680 nm, the focal plane (XY plane, (13.125 μm) Four off-axis focal points with a rectangular distribution are clearly visible. To quantify their uniformity, the transverse cross-sectional light intensity distribution curves of the four focal points were extracted, and the results are as follows. Figure 6 As shown. Figure 6 (a) and (b) show the results using the improved weighted GS algorithm of this invention. It can be seen that the peak light intensity heights at the four focal points f1-f4 are highly consistent, and the calculated intensity deviation is less than 5%. For comparison, Figure 6 (c) and (d) show the results of the traditional standard GS algorithm, demonstrating significant intensity differences between the focal points. This fully demonstrates the significant advantages of the dynamic amplitude weighted feedback mechanism of this invention in improving the uniformity of multifocal points.

[0045] Verification of depth of field (DOF) characteristics: such as Figure 5 As shown in (b), the axial optical field of the XZ plane was scanned. The results show that the beam exhibits a significant "needle-like" stretching pattern along the optical axis (Z-axis), rather than the "point-like" convergence of a typical lens. Further quantitative analysis is as follows... Figure 7 The figure shows the light intensity distribution characteristics along the Z-axis at the four focal points. By measuring the full width at half maximum (FWHM), the average depth of focus (DOF) of the four focal points was calculated to be approximately 6.33 μm. This extended depth of focus creates a longer axial tolerance range, providing a physical basis for achieving broadband achromatic light.

[0046] Broadband achromatic performance verification: To evaluate the device's broadband operating capability, full-wavelength simulations were performed in the 640 nm to 740 nm band. The results are as follows: Figure 8 As shown. Figure 8 The XZ-plane light intensity distribution diagram at the top shows that as the wavelength increases from 640 nm to 740 nm, the actual physical focal length gradually shortens from approximately 14.87 μm to approximately 11.97 μm due to dispersion (focal length drift of approximately 2.9 μm). However, thanks to the ultra-long depth of focus of 6.33 μm designed in this invention, this focal length drift always remains within the coverage of the depth of focus (the two white dashed lines in the figure indicate the common overlapping area). This means that on any fixed plane within this overlapping area, the superlens can effectively focus on all test wavelengths. Figure 8 The XY plane intensity distribution diagram at the bottom further confirms that the spot shape remains clear and distortion-free throughout the entire broadband range. Quantitative analysis of the lateral full width at half maximum (FWHM) of the off-axis focus reveals that within the 640-740 nm range, the FWHM value fluctuates slightly between 872 nm and 890 nm (with a fluctuation range of only about 18 nm), and remains consistently at the sub-micron level. This indicates that the design of this invention not only eliminates defocus caused by chromatic aberration but also effectively suppresses spot distortion caused by broadband off-axis aberrations, achieving high-quality achromatic focusing.

[0047] Focus on efficiency analysis: such as Figure 9 The diagram illustrates the focusing efficiency variation of the superlens over a broadband range of 640 nm to 740 nm. The focusing efficiency peaks near the design center wavelength (680 nm) (approximately 66%); as the wavelength shifts towards both ends, the efficiency decreases slightly, but the lowest value remains above 54%. Calculations show that the average focusing efficiency of this superlens reaches 62.47% across the entire 100 nm bandwidth. This demonstrates that the design proposed in this invention maintains extremely high energy efficiency while achieving multifocal beam splitting and long focal depth achromatic characteristics.

[0048] It should be noted that although the embodiments described above are illustrative, they are not intended to limit the invention. Therefore, the invention is not limited to the specific embodiments described above. Any other embodiments obtained by those skilled in the art under the guidance of this invention without departing from its principles are considered to be within the protection scope of this invention.

Claims

1. A four-focal-point long focal depth broadband achromatic superlens based on an improved weighted GS algorithm, characterized in that, The superlens comprises a transparent substrate and a plurality of subwavelength unit structures arranged in an array on one side surface of the transparent substrate; the subwavelength unit structures form a specific phase distribution profile Φ on the surface of the superlens , ); the phase distribution profile is generated by iterative calculation of an improved weighted Gerchberg-Saxton (GS) algorithm; in the initial stage of iteration, the improved weighted GS algorithm adopts a complex amplitude field obtained by linear superposition of a focusing phase and an axial prism phase as an initial light field distribution; through modulation of the phase distribution profile, the superlens forms four off-axis focal points in a rectangular distribution at a target focal plane under irradiation of incident light; the introduction of the axial prism phase enables each focal point to form an extended depth of focus (DOF) along the optical axis direction, and the extended depth of focus covers the axial shift of focal length caused by different wavelengths of incident light, thereby realizing achromatic focusing in a wideband range.

2. The four-focal-point long-depth-of-focus broadband achromatic superlens based on the improved weighted GS algorithm according to claim 1, characterized in that, The material of the transparent substrate is silicon dioxide (SiO2), and the material of the subwavelength unit structure is titanium dioxide (TiO2); the subwavelength unit structure is a cylindrical nanopillar, the lattice period P is 350 nm, and the height is 1 μm; the radius of the nanopillar varies in the range of 50 nm to 150 nm to achieve a phase full coverage of 0 to 2π at a design center wavelength of 680 nm.

3. The four-focal-point long-axial-depth broadband achromatic superlens based on the improved weighted GS algorithm according to claim 2, characterized in that, The design parameters of the superlens satisfy: a radius R = 10.5 μm, a design focal length f = 13.125 μm, and a numerical aperture NA ≈ 0.625; and the superlens is composed of an array of 60x60 subwavelength unit structures.

4. The four-focal-point long-depth-of-focus broadband achromatic superlens based on the improved weighted GS algorithm according to claim 1, characterized in that, The working bandwidth of the superlens covers at least a wavelength range of 640 nm to 740 nm; in this wavelength range, the average focal depth length of the four focal points is greater than 6 μm, and the full width at half maximum (FWHM) of each focal point on the focal plane is maintained at a diffraction limit level.

5. The four-focal-point long-depth-of-focus broadband achromatic superlens based on the improved weighted GS algorithm according to claim 1, characterized in that, The improved weighted GS algorithm includes an amplitude feedback mechanism, so that the energy intensity deviation between the four focal points is less than 5% at any wavelength within the working bandwidth.

6. A method of designing a four-focal-point long focal depth broadband achromatic superlens according to claim 1, characterized in that, The method comprises the following steps: Step S1: establishing a unit structure database, scanning the geometric parameters of the subwavelength unit structure by the finite difference time domain (FDTD) method, and obtaining the mapping relationship between the transmission phase, transmittance and geometric parameters; Step S2: setting the target light field of the superlens as four off-axis Gaussian light spots in a rectangular distribution on the focal plane; Step S3: Constructing the improved weighted GS algorithm, calculating the initial light field distribution loading the initial light field distribution to the input plane of the algorithm as the iteration starting point; Step S4: performing iterative optimization, in the forward propagation process of each iteration, calculating the peak intensity of the four focal points on the target plane, and dynamically adjusting the amplitude weight factor of the target light field according to the intensity difference of each focal point until the phase distribution converges; Step S5: extracting the converged phase distribution, combining smoothing processing, discretizing the continuous phase according to the mapping relationship in step S1, and constructing the corresponding subwavelength unit structure array on the substrate.

7. The method of designing according to claim 6, wherein, In step S3, the initial light field distribution is obtained by calculating the superposition of the "focusing phase" and the "axicon phase" for each of the four target foci, followed by complex amplitude summation; the phase expression of the initial light field distribution is given by: where arg represents a taking argument operation, is an imaginary unit, represents the first focus point, ( , ) is the center coordinate of the first focus point, is a design center wavelength, is a focal length, is an axial prism modulation factor, ( , ) is a superlens plane coordinate.

8. The design method of claim 7, wherein, The value of the axial prism modulation factor α ranges from 0.1 to 0.2, and the preferred value is 0.15, which is used to control the balance between the length of the focal depth and the width of the central light spot.

9. The method of claim 6, wherein, In step S4, the specific way of dynamically adjusting the amplitude weight factor of the target light field is: calculating the average peak intensity of the four foci in the current iteration step ; Regarding the first Each focus point updates its weight. If its peak intensity Below If so, increase the weight; if Higher than If so, then reduce the weight; The weight update formula is: wherein is the number of iterations.

10. The superlens according to any one of claims 1 to 5 in parallel laser processing, optical communication wavelength division multiplexing, optical tweezer capture or high-throughput holographic microscopic imaging systems.

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