Design method of spatial multiplexing off-axis bifocal multi-order diffractive lens based on blue noise optimization

Abstract

This invention relates to a design method for a spatially multiplexed off-axis bifocal multi-order diffractive lens based on blue noise optimization. The method involves determining the off-axis angle and off-axis focal length to generate an ideal off-axis hyperbolic phase distribution φ. L and φ R Calculate the local density of the initial random mask, iteratively optimize and generate a blue noise mask, and then apply φ... L and φ R Mapping to a blue noise mask yields an off-axis bifocal multiplexed phase. A phase library of multi-order diffraction structures is constructed. Phase discrete encoding is performed based on minimizing the phase residual, mapping the discrete phase to a height distribution of the multi-order diffraction structure, thus completing the lens construction. This invention optimizes traditional spatial multiplexing masks based on blue noise, weakening local clustering features in the mask distribution, thereby reducing the mask's spectral characteristics and achieving low background noise and high energy consistency for ideal bifocal phases. Multi-order diffraction structures are used for phase encoding to realize the generation of bifocal phase devices.

Description

[0001] This invention relates to a design method for a spatially reusable off-axis bifocal multi-order diffractive lens based on blue noise optimization. Background Technology

[0002] Off-axis bifocal (multifocal) optical field manipulation has significant application value in optical systems such as laser processing, optical control, and bioimaging. Compared to monofocal lenses, off-axis bifocal lenses can simultaneously perform multi-position energy distribution and wavefront manipulation on the same optical plane, offering advantages in improving system functional density and integration.

[0003] Spatial multiplexing, as a commonly used design scheme for off-axis bifocal planar optical elements, has the advantages of intuitive implementation and flexible design. Its basic idea is to map the hyperbolic phase distribution corresponding to different target focal points onto different regions of the spatial multiplexing mask, and achieve bifocal focusing through full-aperture compounding. However, off-axis bifocal planar optical elements based on spatial multiplexing still have the following problems:

[0004] In their paper "Design of Intensity-Ratio Controllable Bifocal Superlenses Based on Spatial Multiplexing," Mu Yonghao et al. explored the focusing behavior of off-axis bifocal plane lenses using two types of periodic spatial multiplexing masks (stitching method and ring multiplexing method). The stitching method-based off-axis bifocal plane lens exhibits significant multifocality in the propagation direction, poor bifocal intensity consistency at the focal plane, and noticeable background noise. Subsequently, the authors proposed a ring-distributed spatial multiplexing method, which further improved the bifocal intensity consistency; however, the multifocality problem in the propagation direction remained unresolved, and significant focal length and offset distances persisted.

[0005] Jo et al. from Sungkyunkwan University in South Korea confirmed that random spatial multiplexing masks have significant background noise and that there is a significant deviation in the focus intensity ratio between focal spots.

[0006] To address these issues, the team introduced complex amplitude modulation on top of the randomized spatial multiplexing mask, effectively improving background noise and enhancing the consistency of focal spot intensity. However, the introduction of light field optimization further increased the design complexity.

[0007] Therefore, the distribution of spatial multiplexing masks directly affects the spectral characteristics of the mixed phase wavefront. This can be seen from the different types of spatial multiplexing masks:

[0008] The implementation of periodic spatial multiplexing masks is relatively simple and does not involve optimization processes. However, during the diffraction propagation process, it introduces obvious diffraction components into the focal plane, resulting in multifocal phenomena caused by interference in the light field in the propagation direction, and the consistency of the focal points is poor.

[0009] Although randomized spatial multiplexing masks can avoid fixed diffraction orders, the masks themselves still have local clustering characteristics, resulting in the retention of strong low-frequency components in the mask spectrum. This leads to significant background noise between focal points, which affects the consistency of energy distribution between the two focal points.

[0010] For example, Publication No. 51-1346 / O4 proposes a design method for a geometric phase-based off-axis multifocal plane lens based on holographic optimization, using a phase retrieval algorithm to achieve lens design; however, it suffers from poor intensity consistency among multiple focal points. Publication No. 31-1252 / O4 proposes a design method combining holographic optimization with complex amplitude superposition, designing an achromatic off-axis bifocal plane lens through inverse optimization, but it fails to effectively reduce focal background noise or improve focal intensity consistency. Patent application No. 202511584293.7 discloses an off-axis multifocal plane lens for the visible light band, achieving multi-position focusing through annular spatial mask phase multiplexing, but it still suffers from the problem of failing to effectively reduce focal background noise or improve focal intensity consistency. Patent application No. 202511584293.7 discloses an off-axis multifocal plane lens for the visible light band, achieving multi-position focusing through annular spatial mask phase multiplexing, but it still suffers from the problem of failing to effectively reduce focal background noise or improve focal intensity consistency.

[0011] Planar optical elements offer a new research path for the miniaturization and integration of bifocal optical systems, demonstrating good application potential in scenarios such as multi-channel control and functional multiplexing. Most known off-axis bifocal planar optical elements employ superlens schemes, characterized by diverse phase modulation methods and high design freedom. However, a drawback is that many designs are difficult to apply practically. This is because the feature size of the superlens is often required to be no larger than half the operating wavelength, leading to excessively high manufacturing difficulty and cost, making it impractical. For example, patent number 202411445570.1 describes a design method for a bifocal self-correcting focus scanning superlens, achieving off-axis bifocal focusing by increasing the design freedom of planar lenses, including a double-layer cascaded planar lens scheme. However, this method suffers from high manufacturing difficulty and assembly challenges.

[0012] Therefore, there is an urgent need for a method that can weaken the local clustering characteristics in the mask distribution, thereby reducing the spectral characteristics of the mask and achieving low background noise and high energy uniformity of the ideal dual-focal phase. Summary of the Invention

[0013] To address the problems of existing periodic masks easily generating regular diffraction components and random masks easily generating local clusters and low-frequency background noise, this invention provides a design method for a spatially multiplexed off-axis bifocal multi-order diffraction lens based on blue noise optimization.

[0014] To overcome the problems existing in the prior art, the technical solution adopted in this invention is: a design method for a spatially reusable off-axis bifocal multi-order diffraction lens based on blue noise optimization, the steps of which are as follows:

[0015] Step 1: Obtain the ideal off-axis hyperbolic phase distribution;

[0016] Define the off-axis angles of the left and right foci as θ respectively. L and θ R Substituting these values ​​into equation (1), we obtain the ideal off-axis hyperbolic phase distribution φ at the left and right foci. L and φ R ;

[0017] (1)

[0018] Where θ is the off-axis angle, k = 2*π / λ is the wave number; the coordinates x and y are the coordinate distributions of the lens plane, determined by R and w;

[0019] Step 2: Perform blue noise optimization on the initial mask M0 with a given duty cycle and effective aperture constraint;

[0020] Treating the blue noise optimization of the mask as a spatial distribution optimization problem, and using a Gaussian kernel G... σ For M0 convolution, the local density function D(x,y) is defined as:

[0021] (2)

[0022] Where σ is the local statistical scale; M0∈ {0,1} represents the distribution of the two wavefront phases in the mask space;

[0023] If the duty cycle of one of the wavefronts is ρ, then ∑M = ρN must be satisfied.

[0024] Where N is the total number of samples within the effective aperture, and D(x,y) is the degree of spatial clustering of the mask in the neighborhood. A larger D(x,y) corresponds to dense local pixels, and vice versa.

[0025] Step 3: By iteratively swapping pixel positions in high-density and low-density regions, local clustering is gradually reduced while maintaining the duty cycle, so that the spatial clustering degree D(x,y) of the mask in the neighborhood is close to the target duty cycle ρ; thus obtaining a spatially multiplexed mask M that satisfies the statistical characteristics of blue noise. Blue The objective function to be optimized is:

[0026] (3)

[0027] Where Ω represents the set of spatial locations;

[0028] Step 4, based on the spatial multiplexing mask M Blue The ideal off-axis hyperbolic phase φ of the left and right focal points obtained in step (1) is used. L and φ R The off-axis dual-focus reset phase is obtained, and the discrete phase mapping formula is:

[0029] (4)

[0030] Step 5: Calculate the phase shift distribution applied at different heights of the multi-order diffraction structure, and construct a phase library that is height- and phase-dependent.

[0031] (5)

[0032] Where n(λ) is the refractive index of the material at the working wavelength; i is the height index of the multi-order diffraction structure;

[0033] Step 6: Complete phase discrete encoding by minimizing the phase residual.

[0034] (6)

[0035] Step 7, discrete phase φ hybrid Mapped to the height distribution of a multi-order diffraction structure, a multi-order diffraction lens is constructed through the mapping of phase to physical structure:

[0036] (7)

[0037] Where δ is the Kronecker function.

[0038] Furthermore, the phase library in step 5 covers the phase from 0 to 2π.

[0039] Furthermore, in step 7, the multi-order diffraction structure is a block unit with a constant bottom width.

[0040] Compared with the prior art, the present invention has the following advantages:

[0041] 1) Blue noise sampling is commonly used in digital halftone, image dithering, computer graphics rendering, and texture generation. Its main function is to suppress low-frequency noise and make the sampling error present a high-frequency distribution. This invention applies blue noise, which is often used in dithering technology in digital audio, to an off-axis dual-focal spatial multiplexing lens. Blue noise has high energy in the high-frequency band and gradually decreases in the low-frequency band. In the spectrum, the power density of blue noise is inversely proportional to the frequency. In the optimization process of mask spatial distribution, the blue noise distribution weakens the local clustering features in the mask distribution, thereby reducing the spectral characteristics of the mask. It can effectively reduce the correlation and aggregation phenomena in low-frequency components, so that the sample is more uniformly distributed in space. The phase distribution with blue noise characteristics diffracts and propagates to the dual-focal light field, and the spectral characteristics are consistent with the intensity of the dual focal point at low background noise.

[0042] 2) This invention employs a multi-order diffraction structure to encode the ideal off-axis phase, realizing a device with a dual-focal phase. The multi-order diffraction structure is a block unit with a constant bottom width, featuring a low aspect ratio and a large feature size. It can be fabricated using common semiconductor technologies. The phase shift is obtained by adjusting the structure height, thereby establishing a phase library from 0 to 2π. This structure has the following characteristics:

[0043] Firstly, it has the characteristic of being insensitive to polarization, making it widely applicable in practical lighting conditions;

[0044] Secondly, it can achieve feature sizes comparable to the operating wavelength, and has good machinability and structural stability.

[0045] 3) This invention optimizes traditional spatial multiplexing masks based on blue noise, retaining the ease of use of traditional spatial multiplexing methods while avoiding the increased design difficulty caused by additional optical field optimization. Through a multi-order diffraction structure phase encoding strategy, the integration of off-axis bifocal related systems is improved. While miniaturizing the material, the manufacturing difficulty and cost are significantly reduced, achieving manufacturability of off-axis bifocal phases and providing a solution for the design of off-axis bifocal plane lenses.

[0046] 4) This invention does not simply combine the blue noise optimization method with the multi-order diffraction structure phase encoding method. The blue noise optimized spatial multiplexing mask improves the light field quality of the off-axis dual-focal phase distribution, while the multi-order diffraction structure phase encoding ensures that the optimized phase can be realized in a form that is easy to process, low in cost, and structurally stable. The two work together to enable this invention to simultaneously achieve low background noise, high dual-focal intensity consistency, and good processing feasibility, overcoming the problems of insufficient light field quality in traditional spatial multiplexing schemes and high processing difficulty in superlens schemes.

[0047] 5) This invention is based on a blue noise-optimized spatial multiplexing mask, which retains the ease of use of traditional spatial multiplexing methods while avoiding the increased design difficulty caused by additional optical field optimization. Subsequently, to achieve the practical manufacturability of off-axis dual-focus phase, a multi-order diffraction structure phase encoding strategy is adopted, which significantly reduces the processing difficulty and cost while improving the integration and miniaturization of the off-axis dual-focus related system. Attached Figure Description

[0048] Figure 1 A flowchart of the present invention.

[0049] Figure 2 Comparison of two spatial multiplexing masks; (a) random mask; (b) blue noise mask.

[0050] Figure 3 A comparison of the radial average power of two types of masks under the same design parameters.

[0051] Figure 4 Comparison of normalized light intensity distribution of two types of masks under the same design parameters.

[0052] Figure 5 Multi-order diffraction structure phase library.

[0053] Figure 6 Schematic diagram of an off-axis bifocal plane lens.

[0054] Figure 7 Light field distribution diagram of a spatially reusable off-axis bifocal multi-order diffraction lens optimized for blue noise; where (a) and (b) are the XZ and XY plane light field distributions calculated based on theory, respectively; and (c) and (d) are the XZ and XY plane light field distributions simulated based on simulation, respectively. Detailed Implementation

[0055] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0056] The binary distribution of the spatial multiplexing mask in this invention exhibits a blue noise distribution.

[0057] This embodiment provides a design method for a spatially reusable off-axis bifocal multi-order diffractive lens based on blue noise optimization. The design parameters of the spatially reusable off-axis bifocal multi-order diffractive lens based on blue noise optimization involved in this embodiment are: lens radius R = 20μm, size w = 0.4μm, working wavelength λ = 1.064μm, numerical aperture of 0.65, and off-axis angles (θ) of the left and right focal points. L and θ R All are 31°.

[0058] In this embodiment, the binary distribution of the spatial multiplexing mask exhibits a blue noise distribution.

[0059] Specific methods are as follows Figure 1 As shown,

[0060] Step 1: Determine the off-axis angle and off-axis focal length to obtain the ideal off-axis hyperbolic phase distribution:

[0061] Substituting the above design parameters into equation (1), we obtain the ideal off-axis hyperbolic phase distributions φ at the left and right foci, respectively. L and φ R :

[0062] (1)

[0063] Where θ is the off-axis angle, k = 2*π / λ is the wave number; the coordinates x and y are the coordinate distributions of the lens plane, determined by R and w;

[0064] Step 2: Subsequently, for the initial mask M0 with a given duty cycle and effective aperture constraint, as follows... Figure 2 As shown in (a), blue noise optimization is performed;

[0065] Specifically, the blue noise of the mask is optimized as a spatial distribution optimization problem under a given duty cycle constraint. To characterize the local spatial distribution features of the mask, a Gaussian kernel G is introduced. σ Perform M0 convolution on the x and y, and define the local density function D(x,y) as follows:

[0066] (2)

[0067] Where σ is the local statistical scale; M0∈ {0,1} represents the distribution of the two wavefront phases in the mask space;

[0068] If the duty cycle of one of the wavefronts is ρ, then ∑M = ρN must be satisfied.

[0069] Where N is the total number of samples within the effective aperture; D(x,y) reflects the degree of spatial clustering of the mask in the neighborhood: a larger D(x,y) corresponds to dense local pixels, and vice versa;

[0070] Step 3: Further, by iteratively swapping pixel positions in high-density and low-density regions, local clustering is gradually reduced while maintaining the duty cycle. This makes the spatial clustering degree D(x,y) of the mask in the neighborhood close to the target duty cycle ρ. From an optimization perspective, this process is equivalent to minimizing the spatial fluctuation of local density relative to ρ, thus obtaining a spatially reused mask M that satisfies the statistical characteristics of blue noise. Blue ,like Figure 2 As shown in (b), the optimization objective function is described by equation (3). The obtained blue noise mask exhibits characteristics of reduced low-frequency energy and increased mid-to-high-frequency energy in the spectral space. Figure 3 As shown,

[0071] (3)

[0072] Where Ω represents the set of spatial locations;

[0073] Step 4: Based on the spatial multiplexing mask M Blue Reuse the ideal off-axis hyperbolic phase φ of the left and right focal points obtained in step 1 L and φ R The specific phase mapping form is described by equation (4). At the focal plane light field, the off-axis phase based on the blue noise mask exhibits lower background noise after diffraction propagation than that of the random mask, significantly improving the intensity consistency of the dual-focal light. Figure 4 As shown,

[0074] (4)

[0075] Step 5: Calculate the phase shift distribution applied at different heights of the multi-order diffraction structure, and construct a phase library that is height- and phase-dependent.

[0076] (5)

[0077] Where n(λ) is the refractive index of the material at the working wavelength; i is the height index of the multi-order diffraction structure.

[0078] In this embodiment, the feature size of the multi-order diffraction structure is set to 0.4 μm, and the height varies from 50 nm to 450 nm, with a total of 128 sampling points. The phase library constructed based on equation (5) is as follows: Figure 5 As shown, it can just cover the phase from 0 to 2π;

[0079] Step 6: Phase discrete encoding is completed by minimizing the phase residual, aiming to find the actual phase value that is closest to the ideal phase.

[0080] (6)

[0081] Step 7: Finally, discrete phase φ hybrid Mapping to the height distribution of a multi-order diffraction structure realizes the mapping from phase to physical structure, enabling the construction of multi-order diffraction lenses, such as... Figure 6 As shown, the multi-order diffraction structure is a block unit with a constant bottom width.

[0082] (7)

[0083] Where δ is the Kronecker function.

[0084] Based on the above-described off-axis bifocal multi-order diffractive lens construction process, this embodiment uses Rayleigh-Somurf diffraction and the time-domain finite difference method for verification.

[0085] Figure 7 (a) and Figure 7 (b) shows the light intensity distributions on the XZ and XY planes based on theoretical calculations. Figure 7 (c) and Figure 7 (d) shows the light intensity distribution of the XZ plane and the XY plane based on simulation. Both of them verify that the off-axis bifocal multi-order diffraction lens constructed by this technology can achieve a clear, high signal-to-noise ratio, and highly consistent bifocal focal spot at the target position, with a maximum off-axis angle deviation of only 1%.

[0086] This invention employs a spatial multiplexing mask that is not a typical image sampling point set, but rather a binary allocation carrier for the phases of the left and right off-axis focal wavefronts. Directly applying conventional blue noise sampling methods might disrupt the duty cycle relationship between the left and right wavefronts, leading to an imbalance in the energy distribution between the two focal points. Furthermore, the lens's effective aperture, off-axis hyperbolic phase, coherent diffraction propagation, and subsequent multi-order diffraction phase encoding all affect the final focal plane light field. Therefore, the blue noise method cannot simply replace a random mask; a constrained design is required for the off-axis dual-focal phase multiplexing process.

[0087] This invention does not simply apply the blue noise method to optical masks, but rather combines its low-frequency suppression characteristics with off-axis dual-focal phase channel allocation, energy duty cycle constraints, and focal plane light field quality requirements. While retaining the advantages of spatial multiplexing methods—simple structure, ease of manufacturing, and easy expansion—it can reduce low-frequency background noise caused by traditional multiplexing masks, improve the consistency of light intensity between the left and right dual focal points, and provide a more stable multiplexed phase basis for subsequent multi-order diffraction structure phase encoding.

[0088] Many specific details have been set forth in the foregoing description in order to provide a full understanding of the invention. However, the invention may also be implemented in other ways different from those described herein. Therefore, the scope of protection of the invention is not limited to the specific embodiments disclosed above.

[0089] The above are merely preferred embodiments of the present invention and are not intended to limit the present invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A design method for a spatially multiplexed off-axis bifocal multi-order diffractive lens based on blue noise optimization, characterized in that, The steps are as follows: Step 1: Obtain the ideal off-axis hyperbolic phase distribution; Define the off-axis angles of the left and right foci as θ respectively. L and θ R Substituting these values ​​into equation (1), we obtain the ideal off-axis hyperbolic phase distribution φ at the left and right foci. L and φ R ; （1） Where θ is the off-axis angle, k = 2*π / λ is the wave number; the coordinates x and y are the coordinate distributions of the lens plane, determined by R and w; Step 2: Perform blue noise optimization on the initial mask M0 with a given duty cycle and effective aperture constraint; Treating the blue noise optimization of the mask as a spatial distribution optimization problem, and using a Gaussian kernel G... σ For M0 convolution, the local density function D(x,y) is defined as: （2） Where σ is the local statistical scale; M0∈ {0,1} represents the distribution of the two wavefront phases in the mask space; If the duty cycle of one of the wavefronts is ρ, then ∑M = ρN must be satisfied. Where N is the total number of samples within the effective aperture, and D(x,y) is the degree of spatial clustering of the mask in the neighborhood. A larger D(x,y) corresponds to dense local pixels, and vice versa. Step 3: By iteratively swapping pixel positions in high-density and low-density regions, local clustering is gradually reduced while maintaining the duty cycle, so that the spatial clustering degree D(x,y) of the mask in the neighborhood is close to the target duty cycle ρ; thus obtaining a spatially multiplexed mask M that satisfies the statistical characteristics of blue noise. Blue The objective function to be optimized is: （3） Where Ω represents the set of spatial locations; Step 4, based on the spatial multiplexing mask M Blue The ideal off-axis hyperbolic phase φ of the left and right focal points obtained in step (1) is used. L and φ R The off-axis dual-focus reset phase is obtained, and the discrete phase mapping formula is: （4） Step 5: Calculate the phase shift distribution applied at different heights of the multi-order diffraction structure, and construct a phase library that is height- and phase-dependent. （5） Where n(λ) is the refractive index of the material at the working wavelength; i is the height index of the multi-order diffraction structure; Step 6: Complete phase discrete encoding by minimizing the phase residual. （6） Step 7, discrete phase φ hybrid Mapped to the height distribution of a multi-order diffraction structure, a multi-order diffraction lens is constructed through the mapping of phase to physical structure: （7） Where δ is the Kronecker function.

2. The design method for a spatially multiplexed off-axis bifocal multi-order diffractive lens based on blue noise optimization according to claim 1, characterized in that, The phase library in step 5 covers phases from 0 to 2π.

3. The design method for a spatially multiplexed off-axis bifocal multi-order diffractive lens based on blue noise optimization according to claim 1 or 2, characterized in that, In step 7, the multi-order diffraction structure is a block unit with a constant bottom width.

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