Vector vortex light field mode decomposition method based on light intensity distribution and clear formula

Abstract

The invention discloses a vector vortex light field mode decomposition method based on light intensity distribution and a clear formula. The method comprises the following steps: firstly, determining a mode order of a vector vortex light field to be measured, and selecting an orthogonal complete mode base capable of representing the light field; secondly, acquiring light intensity distribution of a plurality of polarization components of the vector vortex light field by using an intensity detector; and finally, the intensity values of the specific sampling points in the light intensity distribution are substituted into the provided clear formula, so that the amplitude coefficient and the phase coefficient of each mode in the light field can be recovered, and the electric field distribution of the light field can be reconstructed. According to the method, the mode coefficient and the electric field distribution can be accurately recovered from the light intensity distribution of each polarization component of the vector vortex light field, and the applicability is very high.

Description

Technical Field

[0001] The present invention relates to a method for decomposing the vector vortex optical field mode based on the light intensity distribution and explicit formula, belonging to the fields of structured light field, integrated optics and optical communication. Background Art

[0002] Vector vortex beams (VVBs) are an important class of vector structured beams, including common circularly polarized orbital angular momentum (CP-OAM) modes, azimuthally and radially polarized cylindrical vector (CV) beams, and higher-order CV beams. In recent years, VVBs have shown great potential in many applications such as high-capacity fiber optic communication, turbulence-resistant free-space communication, quantum information processing, light-matter interaction, super-resolution imaging, etc. In addition to the classical visible and infrared wavelengths, it has also attracted extensive attention from researchers in various electromagnetic wave bands from microwave to terahertz. Due to the requirements of these applications, the technology for precisely generating and controlling VVBs is rapidly developing. In addition to traditional free-space devices such as q-plates and spatial light modulators (SLMs), researchers have also developed devices such as microring resonators, on-chip waveguides, all-fiber devices, metasurfaces, and solid-state lasers, and have made great efforts for the integrated generation scheme of VVBs.

[0003] In this context, the general characterization technology of VVBs, as a key technology, has not been solved, and it is very important for establishing a standardized device fabrication and performance evaluation process, and further promoting these devices to practical applications. In addition, in information applications, consistent and stable standards are needed to evaluate the changes of VVBs at the various generation device ends and after transmission platforms such as optical fibers. The common models of VVBs are the higher-order Poincaré sphere and the hybrid-order Poincaré sphere, which can be expressed as the coherent superposition of four eigenmodes (such as CP-OAM modes). The problem of characterizing the weighting coefficients (amplitude coefficients and phase coefficients) of each mode in the four-mode superposition state is called vector mode decomposition. Once the weighting coefficients of each mode are determined, the complete electric field distribution of the optical field can be determined, and the accurate characterization of VVBs can be achieved. Vector mode decomposition also corresponds to another important problem, that is, signal demodulation when each mode is used as an independent channel.

[0004] Stokes polarization method is a simple and straightforward tool for characterizing vector optical fields. By measuring the intensity distributions of different polarization components of a light beam using a quarter-wave plate and a linear polarizer, one can obtain the polarization distribution of a two-mode superposition state on a single Poincaré sphere and the weighting coefficients of these two modes. Recently, polarization-separation metasurfaces have provided a compact and fast solution for the Stokes polarization method. However, the Stokes polarization method cannot directly characterize more general four-mode superposition states, which poses a non-negligible technological gap because four-mode superposition states are important for applications such as improving spectral efficiency in high-capacity communication networks and constructing hyperdimensional spin-orbit coupling states in a complete four-dimensional Hilbert space. In particular, for important platforms such as optical fibers or on-chip waveguides that generate and transmit VVB, the four-mode superposition states of the higher-order Poincaré sphere correspond to four degenerate modes that are strictly coaxial within the same mode group, have close propagation constants, and are easily coupled to each other. Independent manipulation of different mode groups is already common in fiber mode-division multiplexing systems, but due to the overly close propagation constants, the degenerate modes will couple to each other, and complex and cost-effective digital signal processing is still required to solve this problem. Therefore, the vector mode decomposition method for quadruple degenerate modes is very attractive for both device characterization and signal demodulation.

[0005] Currently, the main method for implementing vector mode decomposition is the neural network. After training a neural network with simulation or experimental data, similar to the Stokes polarization method, as long as the intensity distributions of different polarization components of the light beam are collected and input into the neural network, the mode coefficients can be obtained. In 2020, Wang L et al. completed mode basis classification through a convolutional neural network (CNN) based on two polarization components, that is, determining which mode basis an approximately pure mode belongs to (Wang L, Ruan Z, Wang H, et al. Laser & Photonics Reviews, 2020, 14(11): 2000249.). Since 2022, Yan B, Kim H et al. have explored the use of neural networks to achieve vector mode decomposition of arbitrary four-mode superposition states through a series of simulation works (Yan B, Zhang J, Wang M, et al. Optics & Laser Technology, 2022, 154: 108287.; Kim H. Optics and Lasers in Engineering, 2023, 160: 107310.). In 2023, Hou M et al. achieved the decomposition and adaptive control of the first-order mode through a CNN based on three polarization components in experiments (Hou M, Xu M, Xu J, et al. Nanophotonics, 2023, 12(15): 3165 - 3177.). However, neural networks usually need to be trained for specific mode types using a large amount of data, which is very time-consuming. Even a trained model requires a prediction time of several milliseconds to several seconds, which is not suitable for the high-speed demodulation requirements in the field of optical communication. In addition, currently, the mode decomposition of arbitrary four-mode superposition states has only been achieved for low-order (first-order to second-order) modes in optical fibers, and the generality for higher-order modes on other platforms (such as on-chip waveguides, metasurfaces, microring resonators) has not been proven. So far, a general vector mode decomposition method based on the light intensity distribution but without neural network training and based on explicit mathematical formulas has not been reported. Summary of the Invention

[0006] The object of the present invention is to solve the problems of limited applicable range and slow speed of existing vector mode decomposition methods, and propose a vector vortex optical field mode decomposition method based on light intensity distribution and explicit formulas. This novel method only needs to collect the light intensity distributions of different polarization components of the optical field and substitute specific sampling points into the proposed explicit formulas, then the amplitude and phase coefficients of each mode can be obtained at one time, thereby reconstructing the complete electric field distribution of the vector vortex beam. The proposed method has the advantages of simplicity, clarity, easy implementation, high speed, high precision, and anti-noise, and is an important candidate method for accurately characterizing the performance of vector vortex optical field generation devices and realizing signal demodulation at the information receiving end.

[0007] The technical solution adopted by the present invention is as follows:

[0008] A method for decomposing the vector vortex optical field mode based on the light intensity distribution and explicit formulas, the method comprising:

[0009] Step 1, determine the mode order of the vector vortex optical field to be measured, and select an orthogonal complete eigenmode basis that can characterize the optical field;

[0010] Step 2, use an intensity detector to collect the light intensity distributions of the respective polarization components of the vector vortex optical field;

[0011] Step 3, substitute the intensity values of the set sampling points in these light intensity distributions into a set of non-iterative explicit formulas proposed, that is, first recover the two-dimensional Jones vectors under the respective polarization components in a fixed linear polarization orbital angular momentum mode basis, then analyze the phase relationships of the respective two-dimensional Jones vectors, so as to analyze a complete four-dimensional Jones vector, and then use the conversion relationship between different eigenmode bases to recover the amplitude coefficients and phase coefficients of each mode in the optical field in the eigenmode basis to be measured, and reconstruct the complete electric field distribution of the optical field.

[0012] Among them, the vector vortex optical field refers to an optical field having electric field components in more than one direction, and its complete electric field distribution can be expressed as the coherent superposition of several orthogonal eigenmodes, such as circular polarization orbital angular momentum modes, linear polarization orbital angular momentum modes, linear polarization modes, and cylindrical vector modes, etc., under the weighting of arbitrary amplitude coefficients and phase coefficients.

[0013] Among them, the orthogonal complete eigenmode basis refers to a set of orthogonal eigenmodes that compose the vector vortex optical field to be measured, are calculated by the wave equation or actually measured, are orthogonal to each other, and are power-normalized.

[0014] Among them, the intensity detector refers to various detectors that can record the light intensity values, including but not limited to cameras with a charge-coupled device (CCD) or complementary metal-oxide-semiconductor (CMOS) as the photosensitive element, various photodetectors (PD) and their arrays.

[0015] Among them, the polarization component refers to the electric field component of the optical field after passing through different polarization elements, and the polarization elements include but not limited to linear polarizers, combinations of wave plates and linear polarizers, polarization beam splitters, and polarization separation metasurfaces.

[0016] Among them, the set sampling points refer to a series of sampling points with definite spatial positions that can be directly extracted from the light intensity distribution.

[0017] Among them, the non-iterative explicit formula refers to a series of non-iterative mathematical formulas that can explicitly derive each mode coefficient from the intensity values of sampling points without any special optimization algorithms. It is not limited to the programming languages and platforms using this formula, such as MATLAB, Python, or FPGA, nor is it limited to the specific algorithms for solving using this formula.

[0018] Advantages and positive effects of the present invention:

[0019] The method proposed by the present invention overcomes the problems of limited applicable range and slow speed of existing vector mode decomposition methods. It has the advantages of being simple and clear, fast, easy to implement, high-precision, and noise-resistant, and has excellent practical application value in characterizing vector vortex beam generation devices and signal demodulation, which can promote the development of structured light fields and their applications. Description of the drawings

[0020] Figure 1 It is a brief flowchart of the vector vortex light field mode decomposition method based on light intensity distribution and explicit formula proposed by the present invention.

[0021] Figure 2 It is a schematic diagram of the vector mode decomposition algorithm and the calculation results of the 1st order mode in the specific implementation. ψ = ψ0: Intensity distribution of the linearly polarized component in the ψ0 direction, such as setting the transmission angle of the polarizer to ψ0. Q + P: Intensity distribution obtained by first placing a quarter-wave plate and then a polarizer, similar to measuring the left-handed circularly polarized component. VMD: Vector modal decomposition. Arg.: Phase distribution. Real: Real part. Imag: Imaginary part. Amp.: Amplitude coefficient. Phase: Phase coefficient. Truth: True value. Sol.: Obtained solution.

[0022] Figure 3 It is the result of applying the vector mode decomposition algorithm to other types of modes in the specific implementation. Amp.: Amplitude coefficient. Phase: Phase coefficient. Truth: True value. Rec.: Recovered value. Specific implementation

[0023] The following further illustrates the present invention by taking the vector mode decomposition of a vector vortex light field composed of several modes as an example in combination with the drawings. The drawings are only for illustrative purposes and do not limit the applicable range of the present invention.

[0024] For the vector vortex light field mode decomposition method based on light intensity distribution and explicit formula, the brief process is shown in Figure 1. First, determine the mode order of the vector vortex optical field to be measured and select an orthonormal complete mode basis that can characterize this optical field; second, use an intensity detector to collect the intensity distributions of multiple polarization components of this vector vortex optical field; finally, substitute the intensity values at specific sampling points in these intensity distributions into the proposed explicit formula, and the amplitude coefficients and phase coefficients of each mode in this optical field can be recovered, and the electric field distribution of this optical field can be reconstructed.

[0025] The mode order of a four-mode superposition optical field can usually be conveniently determined by observing the number of lobes in the mode pattern. For example, the l-th order mode has 2l lobes, and in practical applications, it is usually known which order of mode is generated by a certain device. The vector vortex beam received by a two-dimensional intensity detector can be represented by four orthonormal complete mode bases: the CP-OAM mode, the CV mode, the linearly polarized (LP) mode, and the linearly polarized orbital angular momentum (LP-OAM) mode. Since different mode bases are orthonormal and can be converted into each other, the same two-dimensional optical field can be represented by any set of mode bases. The choice of which mode basis depends more on the actual use. For example, for an optical fiber, the CV mode is the most accurate eigenmode, and it is more convenient to expand the detected optical field in the CV mode basis for further analysis. For a rectangular waveguide, the most accurate one is the LP mode basis. Below, without restricting the angular order of the fiber mode, the formula from the intensity distribution to the mode coefficients will be derived through the most easily analyzable LP-OAM mode basis, and the analysis of any mode will be realized through the formula of mode conversion.

[0026] First, in the LP-OAM mode basis, the vector vortex optical field can be represented by Equation (1), where l1 and l2 represent the angular orders of each mode, m is the radial order, and both can take any integer value. C k is a complex number, |C k | represents the amplitude coefficient of each mode, and arg(C k ) represents the phase coefficient of each mode, including the phase term e i(wt-kz) related to the frequency and propagation of the light beam, and the initial phase The superscript "LO" indicates that these coefficients are obtained by expanding in the LP-OAM mode basis. The two elements of each column vector in the formula correspond to the electric fields in the x and y directions respectively.

[0027]

[0028] For the sake of clarity and convenience in derivation, we only focus on the optical field at a certain radius r0 here, and write C k together with the value of the radial field function F(r0) (RFF) of each mode at r0 as D k, as shown in Equation (2). The RFF is related to the boundary conditions of the waveguide or the type of mode in free space. For example, Laguerre-Gaussian beams, Bessel-Gaussian beams, and perfect vortex beams have different RFFs. D k is a four-dimensional Jones vector (4D-JV). When the RFF is known, solving for the mode coefficients is equivalent to solving for the 4D-JV in the LP-OAM mode basis.

[0029] We first derive the mathematical form of the linearly polarized component of the beam, that is, left-multiplying by a Jones matrix, as shown in Equation (3), which represents a polarizer with a polarization angle of ψ.

[0030]

[0031] is a linearly polarized scalar field and can be represented by a two-dimensional Jones vector (2D-JV) Next, the 4D-JV is determined by solving the 2D-JV of each polarization component. The process and results are as Figure 2 shown.

[0032] Denote Then the intensity distribution of the linearly polarized component in the ψ direction is as in Equation (4), where Δl = (l2 - l1).

[0033]

[0034] Denote as the phase reference. Next, there are two sampling methods to choose from.

[0035] 1. DOT scheme: Obtain the intensity values of three sampling points at known positions, that is,

[0036] where θ1 = 2k1π / Δl, θ2 = (π / 2 + 2k2π) / Δl, θ3 = (π + 2k3π) / Δl, and k1, k2, and k3 are arbitrary integers. Obtain the intermediate quantities

[0037]

[0038] 2. FT scheme: Perform a fast Fourier transform (FFT) on such an annular sampling sequence, and take three Fourier coefficients where real represents the real part and imag represents the imaginary part, and obtain

[0039] Intermediate quantities

[0040]

[0041] After obtaining c0, c1, and c2 using any one of the methods, the remaining calculation processes are exactly the same. First, can be uniquely determined by the four-quadrant arctangent of c2 and c1, and then we obtain

[0042]

[0043] Although 2D-JV has two solutions, that is, there is ambiguity in the solutions. To solve this ambiguity, we will first assume that the correct solution has been determined, and then through further derivation, give a deterministic index to find the correct solution. If the correct solution has been determined, when ψ = 0, we obtain When ψ = π / 2, we obtain However, they all use the first element as the phase reference. The phase relationships of the elements in 4D-JV are also important, so we need to analyze and the phase difference Δα between them. To analyze this phase difference, take another ψ = ψ0 (for example, ψ0 = 3π / 4), and we obtain two equivalent vectors G rec and G pred , as shown in Equation (9). G rec is the 2D-JV recovered from by following the steps of Equations (4) to (8) above, and there are also two sets of possible solutions. G pred is the 2D-JV predicted based on 4D-JV at ψ = ψ0 according to Equation (3).

[0044]

[0045] According to G rec = G pred , Δα can be solved using the cosine theorem, as shown in Equations (10 - 11).

[0046]

[0047] It can be seen that the three 2D-JV and all have two sets of possible solutions, and if a solution is given for each 2D-JV, then Δα also has two possible solutions.

[0048] To solve the ambiguity of all the above solutions, we establish an index.

[0049]

[0050] Here, G k represents the k-th element in G. If the ambiguity of all solutions is solved, ∈ Gis a very small number close to 0. The first 2D-JV cannot be determined using the existing information; however, for its two solutions, and the ambiguity of Δα can be simply resolved by finding when ∈ G is the smallest, that is, a can completely determine a a and a Δα. From this, we obtain two sets of conjugate solutions for the 4D-JV.

[0051] If the CV mode basis (as shown in Figure 2 ) and the LP mode basis are selected, the conjugate solution problem is not important because the CV or LP mode amplitude coefficients corresponding to the conjugate solutions are exactly the same, except for the sign of the phase.

[0052] However, for some applications, precise phase retrieval is essential. To solve the phase ambiguity, it is not enough to use only the linearly polarized components. Similar to measuring the left-handed circularly polarized component, for we place a quarter-wave plate with its fast axis at a 45° angle to the x-axis, followed by a linear polarizer in the x direction. As can be seen from Figure 2 , the two sets of conjugate solutions correspond to different intensity distributions, and only the correct solution is consistent with the actual intensity distribution. Considering that image reconstruction is often time-consuming, we can add a step for derivation and judgment according to the previous method, as shown in Eqs. (13 - 14). We still calculate ∈ G through Eq. (12), and the minimized ∈ G indicates that the correct solution has been found.

[0053]

[0054] So far, the 4D-JV, that is, the coefficients of each mode, under the LP-OAM mode basis have been completely determined.

[0055] Next, we show how to achieve the analysis of any mode through mode conversion. The vector vortex optical field can be expanded in the CP-OAM mode basis, as shown in Eq. (15)

[0056]

[0057] When l1 = -l and l2 = l, two other orthonormal complete mode bases, the CV and LP mode bases, can be obtained, as shown in Eqs. (16 - 17).

[0058]

[0059] There is a conversion relationship between the 4D-JV in the equivalent orthonormal complete mode bases, based on and can be obtained

[0060]

[0061] As described above, we have determined the 4D-JV in the LP-OAM mode basis, and the weighting coefficients of the CP-OAM, LP, or CV modes can be determined by simply using Eqs. (18-19).

[0062] When deriving the above method, no restrictions are imposed on l1, l2, and m, nor is the form of the radial field function restricted. Therefore, this method has the generality for various modes that satisfy different boundary conditions. As Figure 3 shown, the vector vortex optical field is generated in the corresponding mode basis shown in the figure. After solving the 4D-JV in the LP-OAM mode basis, the weighting coefficients in these mode bases are obtained using the transformation relations in Eq. (18-20). Figure 3 (a-b) show the modes in a step-index fiber, which are very similar to the Laguerre-Gaussian modes in a graded-index fiber or free space. Figure 3 (a) corresponds to the high-order Poincaré sphere scenario, i.e., four degenerate modes (l1 = -2, l2 = 2, and m = 4) in the same CV module; Figure 3 (b) corresponds to the mixed-order Poincaré sphere scenario (l1 = -1, l2 = 2, and m = 1). Similarly, this method is also fully applicable to other circularly symmetric modes, such as Bessel-Gaussian beams and perfect vortex beams. Figure 3 (c) shows the results of the first-order LP mode (l1 = -1, l2 = 1, and m = 1) in a rectangular waveguide. Here, a sampling method based on the shape of the rectangular waveguide is proposed, as Figure 3 shown by the dashed line in the first image of (c). First, the OAM mode is superimposed using the LP mode, and its intensity distribution is calculated; then, a circle of points with equal intensity is addressed, and their coordinates are the coordinates of the sampling points. This method has higher accuracy than simple rectangular sampling because on this circle of sampling points, the OAM mode conforms to a complex exponential function, while the LP mode conforms to a trigonometric function, so our above derivation is applicable. Cylindrical waveguides and rectangular waveguides represent two extreme cases because a circle can be considered a polygon with an infinite number of sides. Therefore, similar sampling and calculation methods can be applied to waveguides of other shapes.

[0063] The examples described herein are only a few examples of the usage scenarios of the present invention, and do not limit the types of devices, mode types, mode radial and angular functions, operating bands, etc., nor are they limited to the programming software and specific solution algorithms using the proposed formulas. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included within the protection scope of the present invention.

Claims

1. A method for decomposing the vector vortex optical field mode based on the light intensity distribution and an explicit formula, characterized in that It includes the following steps: Step 1: Determine the mode order of the vector vortex optical field to be measured, and select an orthogonal and complete eigenmode basis that can characterize the optical field; Step 2: Use an intensity detector to collect the intensity distributions of the polarization components of the vector vortex optical field; Step 3: Substitute the intensity values of the set sampling points in these intensity distributions into a set of proposed non-iterative explicit formulas. First, recover the two-dimensional Jones vectors under each polarization component in a fixed linearly polarized orbital angular momentum mode basis, then analyze the phase relationships of the two-dimensional Jones vectors, so as to analyze a complete four-dimensional Jones vector. Then, use the transformation relationship between different eigenmode bases to recover the amplitude coefficients and phase coefficients of each mode in the optical field in the eigenmode basis to be measured, and reconstruct the complete electric field distribution of the optical field.

2. The vector vortex optical field mode decomposition method based on light intensity distribution and explicit formula according to claim 1, characterized in that: The vector vortex optical field refers to an optical field with electric field components in more than one direction, and its complete electric field distribution can be expressed as the coherent superposition of several orthogonal eigenmodes, including but not limited to circularly polarized orbital angular momentum modes, linearly polarized orbital angular momentum modes, linearly polarized modes, and cylindrical vector modes, under the weighting of arbitrary amplitude coefficients and phase coefficients.

3. The vector vortex optical field mode decomposition method based on light intensity distribution and explicit formula according to claim 1, characterized in that: The orthogonal and complete eigenmode basis refers to a set of orthogonal eigenmodes that compose the vector vortex optical field to be measured, are calculated by the wave equation or actually measured, are orthogonal to each other, and are power-normalized.

4. The vector vortex optical field mode decomposition method based on light intensity distribution and explicit formula according to claim 1, characterized in that: The intensity detector refers to various detectors that can record the intensity values of the optical field, including but not limited to cameras with charge-coupled device (CCD) or complementary metal oxide semiconductor (CMOS) as photosensitive elements, various photodetectors (PD) and their arrays.

5. The vector vortex optical field mode decomposition method based on light intensity distribution and explicit formula according to claim 1, characterized in that: The polarization component refers to the electric field component of the optical field after passing through different polarization elements, and the polarization elements include but not limited to linear polarizers, combinations of wave plates and linear polarizers, polarization beam splitters, and polarization separation metasurfaces.

6. The vector vortex optical field mode decomposition method based on light intensity distribution and explicit formula according to claim 1, characterized in that: The set sampling points refer to a series of sampling points with definite spatial positions that can be directly extracted from the intensity distribution.

7. The vector vortex optical field mode decomposition method based on light intensity distribution and explicit formula according to claim 1, characterized in that: The non-iterative explicit formula refers to a series of non-iterative mathematical formulas that can clearly deduce the mode coefficients from the sampling point intensity values without any special optimization algorithms, not limited to the programming languages and platforms using this formula, including but not limited to MATLAB, Python, or FPGA, nor limited to the specific algorithms for solving using this formula.

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