A method for determining a single-vortex phase topological charge and a position of a singularity

Abstract

The application provides a method for determining single singular point vortex phase topological charge and singular point position. Step one: a variable size non-circular symmetric diaphragm and a 4f system are added at the rear end of a to-be-measured light field; step two: a camera is moved along an optical axis to collect two light intensity images, and an axial differential of the light intensity is calculated; step three: a singular point position is determined based on a phase correlation iteration method; and step four: a vortex phase topological charge is iteratively calculated. The application can reduce the complexity of a traditional system for determining singular point vortex phase topological charge and singular point position, reduce the introduction of phase errors, and reduce system cost. In addition, the application only needs to collect two near-field light intensity images to determine single singular point vortex phase topological charge and singular point position.

Description

Technical Field

[0001] This invention relates to the field of phase retrieval and quantitative phase imaging technology in optical measurement, and in particular to a method for determining the topological charge of a single singular vortex phase and the location of the singularity. Background Technology

[0002] Phase retrieval is a crucial topic in optical measurement and imaging technology, playing a vital role in both biomedical and industrial inspection fields. Looking back at the progress of optical measurement over the past half-century, since cameras can directly measure the amplitude / intensity of a light wave field but not its phase distribution, the process of recovering the phase from the intensity distribution can be considered a mathematical "inverse problem"—the phase retrieval problem. The intensity transmission equation method is a typical phase retrieval method. It is a second-order elliptic partial differential equation that describes the quantitative relationship between the change in light intensity along the optical axis and the phase of the light wave in a plane perpendicular to the optical axis. That is, given the axial differential of the light intensity and the known intensity distribution, phase information can be directly obtained by numerically solving the intensity transmission equation.

[0003] However, algorithms for obtaining phase using the light intensity transport equation generally require the introduction of a Teague auxiliary function, followed by simplification by converting the light intensity transport equation into two standard Poisson equations to obtain the phase distribution. However, the Teague auxiliary function implicitly assumes that the transverse energy flow field of light intensity is conservative. This assumption is generally not valid, so the Teague auxiliary function inevitably introduces a solution error, also known as phase difference, causing traditional methods to fail to provide an exact solution to the light intensity transport equation. The phase error is particularly pronounced at the boundaries. Therefore, to address the phase error in the solution of the light intensity transport equation and the potential lack of a solution for non-uniform light intensity, a boundary error-free solution method is implemented using constants and iterative methods (a boundary error-free solution method for the light intensity transport equation under non-uniform light intensity, CN201711484824.0).

[0004] However, when the light intensity contains minimum values, zero points, or phase vortices, the solution methods for the light intensity transmission equations mentioned above are difficult to handle, and this problem itself has been a highly controversial issue in history. This is because a vortex beam is a special light field with a certain topological charge and a phase singularity, which produces a hollow phenomenon in the light intensity distribution. Therefore, recovering the phase distribution of a vortex with a phase singularity has always faced a huge challenge. In 2001, Allen et al. (Allen LJ, Faulkner HML, Nugent KA, et al. Phase retrieval from images in the presence of first-order vortices[J]. Physical Review E, 2001, 63(3).) proposed an iterative phase retrieval method to solve this problem. It requires recording at least three light intensity distribution images at different propagation distances and focuses on the phase retrieval problem of first-order vortices. However, in 2013, Lubk et al. (Lubk A, Guzzinati G, F, et al. Transport of Intensity Phase Retrieval of Arbitrary Wave Fields Including Vortices[J]. Physical Review Letters, 2013, 111(17).) pointed out some errors in the derivation by Allen et al. and proposed a new "exhaustive" method to eliminate the light intensity blurring phenomenon caused by phase vortices. However, in the case of singularities in the light field, a strict solution to the linear elliptic partial differential equation TIE can be obtained, thereby achieving phase reconstruction. However, this generally requires specific boundary conditions, which can be obtained by cutting from each isolated point with zero light intensity to the boundary, so that these regions can be easily connected, thereby obtaining the boundary conditions. In actual experiments, this method requires not only acquiring light intensity information for phase recovery and providing specific boundary conditions, but also acquiring defocused images over a larger area to determine topological charge information. Therefore, based on the near-field light intensity map, how to quickly realize the recovery of the single singularity vortex light field (topological charge, position) is a problem to be solved. Summary of the Invention

[0005] To address the current technical challenges of recovering the vortex singularity position, vortex phase topological charge (including rotation direction), and large image acquisition range in vortex light fields based on the light intensity equation, a method for determining the vortex phase topological charge and singularity position of a single singularity is proposed.

[0006] The inventive concept of this invention is as follows: a method for determining the topological charge of a single-singularity vortex phase and the location of the singularity. First, a variable-size non-circular symmetric aperture and a 4f system are added to the rear end of the light field to be measured. Second, the camera is moved axially by a certain distance, and one light intensity image is acquired before and after the movement, along with one focused image and one defocused image. Then, the axial differential of the light intensity is calculated. Next, the singularity location is determined using the phase correlation ergodic method, and the topological charge of the vortex light field is iteratively calculated. Finally, based on the extended light intensity transmission equation, the vortex light field containing the singularity is ultimately recovered.

[0007] To achieve the above-mentioned objectives, the present invention employs the following technical solution: a method for recovering the topological charge and singularity position of a single singularity vortex phase, comprising the following steps:

[0008] 1.1. Add a variable-size non-circular symmetric aperture and a 4f system to the rear end of the optical field to be measured;

[0009] 1.2. Move the camera axially a distance dz, and acquire one light intensity image before and after the movement, which will be the focused image I. f and an out-of-focus image I o Then calculate the axial differential I of the light intensity. e ′ xp ;

[0010] 1.3 Determine the location of the singularity using the phase correlation traversal method;

[0011] 1.4. Iteratively calculate the topological charge of the vortex light field.

[0012] The specific process of step 1.1 is as follows:

[0013] A variable-size non-circular symmetric aperture and a 4f system (consisting of two convex lenses) are added to the rear end of the light field to be measured. The order and position of their addition in front of the camera receiving surface are as follows: aperture, two convex lenses. The aperture is placed on the front focal plane of the first convex lens, the front focal plane of the second convex lens coincides with the rear focal plane of the first convex lens, and the camera receiving surface coincides with the rear focal plane of the second convex lens.

[0014] The specific process of step 1.3 is as follows:

[0015] 2.1 Using the focused image I acquired in step 1.2 f For light intensity, at different positions (x v ,y v The phase of a standard vortex with a topological charge of 1 is used to synthesize the optical field information on the focusing surface. Where I... f The pixel size is M×N, where M and N are both positive integers, (x v ,y vLet x be the coordinates of the discretized image captured by the camera, where x is a positive integer and x is a negative integer. v ≤M, y v ≤N.

[0016] 2.2. The light field information generated in step 2.1 corresponding to different positions is propagated using the angular spectrum method over a distance of dz to obtain the corresponding defocused light field information. Then, the axial differential of the light intensity at the corresponding different positions is calculated.

[0017] 2.3 Axial differential distribution calculated at different locations The axial differential I in step 1.2 e ′ xp The cross-power spectrum is calculated as follows:

[0018]

[0019] 2.4. Traverse all positions to obtain M×N numbers. Value, the coordinates of the position corresponding to the maximum value (x) v ,y v () is the location of the singularity of the vortex light field to be measured.

[0020] The specific process of step 1.4 is as follows:

[0021] 3.1. Based on the singularity position (x) of the vortex light field determined in step 1.3 v ,y v ), generating standard vortex phases of topological charges of different integer orders (generally, the topological charge values ​​are set to 11 integers from -5 to 5);

[0022] 3.2 Using the focused image I acquired in step 1.2 f Let the light intensity be denoted by , and the standard vortex phase of different topological charges in step 3.1 be denoted by , to synthesize the light field information on the focusing surface. This light field information is then propagated using the angular spectrum method over a distance dz to obtain the corresponding defocused light field information. Multiple axial differentials of light intensity corresponding to different topological charges are calculated using the light field information on both the focusing and defocused surfaces.

[0023] 3.3 Calculate the multiple axial differential distributions in step 3.2 and the axial differential I of the light intensity obtained from the experiment in step 1.2. e ′ xp The root mean square error between these errors. The topological charge corresponding to the minimum value of these root mean square errors is the topological charge to be measured.

[0024] Compared with the prior art, the present invention has the following significant advantages: (1) It only requires the addition of 3 simple optical elements to meet the hardware requirements for determining the key parameters of the single singularity vortex phase, without the need for optical elements such as cylindrical lenses, thereby reducing the complexity and cost of the system and reducing the introduction of phase error to a certain extent. (2) Based on the hard boundary introduced by the aperture, only two near-field light intensity images are needed to accurately give the singularity position of the single singularity vortex phase, without the need to acquire images over a large distance. (3) Only two near-field light intensity images need to be acquired, without the need for far-field images to assist in determining the topological charge of the vortex phase. (4) The images for determining the singularity position and topological charge of the vortex phase can be shared.

[0025] The present invention will now be described in further detail with reference to the accompanying drawings. Attached Figure Description

[0026] The accompanying drawings are provided to further illustrate the invention and form part of the specification.

[0027] Figure 1 This is a flowchart illustrating the overall process for recovering the topological charge and singularity position of a single singularity vortex phase.

[0028] Figure 2 This is a block diagram of the verification experimental apparatus of the present invention.

[0029] Figure 3 This diagram illustrates the specific process of determining the singularity position in the method for recovering the topological charge and singularity position of a single singularity vortex phase.

[0030] Figure 4 This diagram illustrates the specific process of determining the topological charge in the method for recovering the topological charge and singularity position of a single singularity vortex phase. Detailed Implementation

[0031] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be described in detail below with reference to the accompanying drawings. Of course, the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0032] This invention provides a method for recovering the topological charge and singularity position of a single singularity vortex phase, applicable to situations requiring the acquisition of vortex phase distribution. Based on this invention, the determination of the vortex phase position and topological charge of a single singularity can be achieved by acquiring only focused and near-field defocused images without increasing the number of images acquired.

[0033] Combination Figure 1 The specific process of the method for recovering the topological charge and singularity position of a single singularity vortex phase proposed in this invention is as follows:

[0034] Step one: This method requires adding a variable-sized non-circular symmetric aperture and a 4f system (consisting of two convex lenses) to the rear end of the original system structure. To verify the effectiveness of this invention and to provide a more specific description of the aforementioned system rear end structure, combined with... Figure 2 The following description is provided. The laser, polarizer, spatial light modulator, aperture 1, convex lens 1, aperture 2, and convex lens 2 constitute a vortex light field generation system for the traditional computational hologram-spatial light modulator method. In this specific embodiment, aperture 1 is rectangular. Based on this system, a verification device for this method was constructed, which involves sequentially adding an aperture and two convex lenses (aperture 3, convex lens 3, and convex lens 4) in front of the camera receiving surface. The aperture is placed on the front focal plane of convex lens 3, the front focal plane of convex lens 4 coincides with the rear focal plane of convex lens 3, and the camera receiving surface coincides with the rear focal plane of convex lens 4.

[0035] Step two, data acquisition: Move the camera away from the lens along the optical axis by a distance dz (the distance can be selected from 1 to 50 μm). Acquire one intensity image before and after the movement. The acquired focused image is denoted as I. f An out-of-focus image is denoted as I. o Then calculate the actual light intensity axial differential I. e ′ xp :

[0036]

[0037] Step three: The above two steps completed data collection and preliminary data processing. Combined with... Figure 3 The specific process of determining the location of the singularity using the phase correlation traversal method will be described in detail.

[0038] First, combined Figure 3 The first box in the image is the focused image I acquired in step two. f For light intensity, at different positions (x v ,y v The phase of a standard vortex with a topological charge of 1 is used to synthesize the optical field information on the focusing surface. Where (x...) v ,y v ) represents the coordinates of the discretized image, all of which are positive integers.

[0039] Assuming the pixel size of the acquired image is M×N, that is, the number of pixels in the horizontal and vertical directions are M and N respectively, corresponding to different coordinate positions (x, y, ..., N). v ,y v The generation process of a standard vortex phase with a topological charge of 1 is as follows (a total of M×N phases are generated). value), x v ,y v We need to traverse 1 to M and 1 to N respectively:

[0040]

[0041] in, θ = arctan[(yy v ) / (xx v ]], (x,y) represents the image coordinate variables, each corresponding to a (x v ,y v Location information, where x and y are integers between 1 and M and between 1 and N, respectively. It is a (1,p)-order generalized Laguerre polynomial, where the parameter p≥0 is the radial index; in this invention, p takes the special case of 0. W0 represents the radius of the beam waist at the focal point (in the case of digital discretization of the acquired image, it is assigned a value based on the camera target size and pixel size, for example, 26.72μm). λ is the wavelength of light, angle() represents the phase angle of the complex matrix in radians, and its value ranges from -π to π. z takes the value of 0 here.

[0042] The complex amplitude information of the optical field on the synthesized focusing plane is (there are a total of M×N):

[0043]

[0044] Then, combine Figure 3 The second box in the image contains the light field information corresponding to different positions in the previous step. Using angular spectral propagation with a propagation distance of dz, the corresponding defocused light field information is obtained. And calculate the axial differential of the light intensity at different locations.

[0045] The specific generation process is as follows:

[0046]

[0047] in, f x and f y According to the definition of spatial frequency, these represent the spatial frequencies in the x-axis and y-axis directions, respectively.

[0048] Combination Figure 3 The third box in the image allows you to obtain specific values ​​for different positions (x). v ,y v The axial differential of the light intensity at point ) is:

[0049]

[0050] Next, based on the axial differential distribution calculated at different locations... The axial differential I′ obtained in step two exp The cross-power spectrum is calculated as follows:

[0051]

[0052] Finally, combining Figure 3 The fourth box in the image corresponds to the pixel size of the captured image, so it is necessary to traverse M×N positions, which corresponds to M×N... Values, select the position coordinates corresponding to the maximum value among these values, i.e. (x v ,y v The value is (x) max ,y max The corresponding time) The value is the largest, (x) max ,y max Then the position of the phase singularity of the vortex to be measured is the position of the vortex to be measured.

[0053] Step four, based on obtaining the phase singularity location, still based on the two acquired images, namely I f and I o , combined Figure 4 The process of iteratively calculating the topological charge (topological charge) of the vortex light field will be given in detail.

[0054] First, based on the singularity position (x) of the vortex light field determined in step three. max ,y max Generate standard vortex phases for different topological loads:

[0055]

[0056] in, θ = arctan[(yy max ) / (xx max ]], (x,y) represents the image coordinate variables, each corresponding to a (x v ,y v Location information, where x and y are integers between 1 and M and between 1 and N, respectively. It is a (m,p) order generalized Laguerre polynomial. In this invention, p is 0, and the parameter m is the azimuth indicator, i.e., the topological charge of the vortex phase. Here, the value of m can be set independently, generally iterating through 11 integers between -5 and 5. This range can be adjusted empirically. W0 represents the radius of the beam waist at the focal point (in the case of digital discretization of the acquired image, it is assigned according to the size of the camera target surface and the pixel size, for example, 26.72μm). The values ​​of W(z), z0, and k are consistent with those in step three, δ 0m The Kronecker delta function (i.e., δ when m = 0) 0m=1, and the rest are 0). angle() represents the phase angle of a complex matrix in radians, and its value is from -π to π. z is 0 here.

[0057] Then, using the focused image I acquired in step one... f For light intensity, different standard vortex phases For phase, synthesize the light field information on the focal plane. This light field information is propagated using the angular spectrum method over a distance of dz to obtain the corresponding defocused light field information.

[0058] Calculate the axial differentials of light intensities for different topological charges using light field information from the focusing and defocusing surfaces:

[0059]

[0060] Finally, calculate the different axial differential distributions I′ m The axial differential I of the light intensity obtained from the experiment in step two. e ′ xp The root mean square errors between multiple values ​​are denoted as RMSE for the corresponding m-values. m .

[0061]

[0062] The normalized root mean square error is:

[0063]

[0064] Here, max() and min() represent taking the maximum and minimum values, respectively. The order m corresponding to the minimum value in the normalized root mean square error is selected, and this value is the topological load to be measured. The sign of this topological load indicates the rotation direction of the vortex.

[0065] The parameters described above are merely embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for determining the topological charge and singularity position of a single singularity vortex phase, characterized in that... Includes the following steps: Step 1: Add a variable-size non-circular symmetric aperture and a 4f system to the rear end of the light field to be measured; Step 2: Move the camera a distance along the optical axis One light intensity image was captured before and after the movement, which were used as the focus image. and out-of-focus images Then calculate the axial differential of the light intensity. ; Step 3: Determine the location of the singularity using the phase correlation traversal method; Step 4: Iteratively calculate the topological charge of the vortex light field; In step one, the 4f system consists of two convex lenses. The aperture stop is placed on the front focal plane of the first convex lens, the front focal plane of the second convex lens coincides with the rear focal plane of the first convex lens, and the camera receiving surface coincides with the rear focal plane of the second convex lens. Step three involves determining the singularity location using the phase correlation ergodic method, including the following steps: S31. Using the focused image acquired in step two. Light intensity, at different locations The phase of a standard vortex with a topological charge of 1 is used to synthesize the optical field information on the focusing surface, where... The coordinates of the discretized image are all positive integers. We need to traverse 1~M and 1~N respectively. , These represent the number of pixels horizontally and vertically in the captured image, respectively. The light field information generated in S32 and S31 at different corresponding positions is propagated using the angular spectrum method, with a propagation distance of... This allows us to obtain the corresponding defocused light field information and calculate the axial differential of the light intensity at different locations. ; S33. Axial differential distributions calculated at different locations The axial differential in step two The cross-power spectrum is calculated as follows: ； in, Represents the Fourier transform; S34. Traverse all positions and calculate... indivual Value, the coordinates of the position corresponding to the maximum value. It is the location of the singularity of the vortex light field to be measured; Step four involves iteratively calculating the topological charge of the vortex optical field, including the following steps: S41. Based on the location of the singularity of the vortex light field determined in step three. Generate standard vortex phases of topological charges of different integer orders; S42. Using the focused image acquired in step one The light intensity is given by the different standard vortex phases in step S41, and the light field information on the focusing plane is synthesized. This light field information is then propagated using the angular spectrum method, with a propagation distance of [missing information]. Obtain the corresponding defocused light field information; use the light field information on the focusing and defocusing surfaces to calculate the axial differential of the light intensity for different topological charges; S43. Calculate the multiple axial differential distributions in step S42 and the axial differentials of the light intensity obtained from the experiment in step two. The root mean square error between these values ​​is used to select the topological load value corresponding to the minimum value among these root mean square errors as the final topological load to be determined.

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