An integer resolution method and system based on a vortex beam

Abstract

The application belongs to the technical field of integer factorization, and particularly relates to an integer factorization method and system based on a vortex beam, which comprises the following steps: setting an integer to be factorized as a mode index of a vortex beam, adjusting an azimuth angle of a pinhole in a digital microlens array device according to a factor to be tried; sequentially irradiating the digital microlens array device and a first thin lens with the vortex beam, and capturing an intensity distribution diagram at a back focal plane of the first thin lens by using a CCD camera; performing normalization processing on the intensity distribution diagram to obtain a normalized optical axis intensity value; and judging whether the factor to be tried is a factor of the integer to be factorized according to the optical axis intensity value. The integer factor and non-factor can be quickly distinguished by judging whether the factor to be tried is the factor of the integer to be factorized according to the optical axis intensity value.

Description

Technical Field

[0001] This invention belongs to the field of integer factorization technology, and particularly relates to an integer factorization method and system based on vortex beams. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] Integer factorization originated in mathematics but has attracted widespread interest from physicists who have studied the relationship between number theory and complex physical systems. Integer factorization, especially of large numbers, remains a challenging problem, but this difficulty often leads to high security in information encoding, cryptography, all-optical machine learning, and other applications. To date, various methods have been developed to achieve integer factorization, including quantum algorithms, quantum annealing, variational algorithms, and Gaussian sums. Gaussian sums are widely used in various methods, including the optical Talbot effect, Bose-Einstein condensation, nuclear magnetic resonance (NMR), cold atoms, and interferometers. Purely classical optical methods primarily include the optical Talbot effect and optical interferometry.

[0004] In practice, due to limited energy, the optical Talbot effect is distorted by optical diffraction, which sets an upper limit on the number to be factored. Perka et al. achieved a factorization with a maximum value of 27 in free-space propagation. For optical interferometry, multiple sources are typically used to achieve the Gaussian sum, with one source producing one term in the sum. This technique places extremely high demands on the precise position or phase control of multiple light sources, especially for large integer factorization. Summary of the Invention

[0005] To overcome the shortcomings of the prior art, the present invention provides an integer decomposition method and system based on vortex beams.

[0006] To achieve the above objectives, one or more embodiments of the present invention provide the following technical solutions:

[0007] The first aspect of this invention provides an integer decomposition method based on a vortex beam, comprising:

[0008] Set the integer to be decomposed as the mode index of the vortex beam, and adjust the azimuth angle of the pinhole in the digital microlens array device according to the factor to be tried.

[0009] A vortex beam is sequentially irradiated onto the digital microlens array device and the first thin lens, and a CCD camera is used to capture the light intensity distribution at the back focal plane of the first thin lens.

[0010] The light intensity distribution map is normalized to obtain the normalized optical axis light intensity value;

[0011] Determine whether the factor to be tried is a factor of the integer to be factored based on the light intensity value of the optical axis.

[0012] A second aspect of the present invention provides an integer factorization system based on a vortex beam, comprising:

[0013] Vortex beam generation module, used for vortex beams with mode exponents that are integers to be decomposed;

[0014] The light intensity distribution map acquisition module is used to capture the light intensity distribution map at the back focal plane of the first thin lens using a CCD camera after the vortex beam sequentially illuminates the digital microlens array device and the first thin lens.

[0015] The normalization module is used to normalize the light intensity distribution map to obtain the normalized optical axis light intensity value.

[0016] The judgment module is used to determine whether the factor to be tried is a factor of the integer to be decomposed based on the light intensity value of the optical axis.

[0017] The above one or more technical solutions have the following beneficial effects:

[0018] This invention sets the mode index of a vortex beam to be factored into an integer, uses the position of the pinhole in a pinhole sieve to determine the magnitude of the factor, and after being modulated by the pinhole sieve, the vortex beam is focused by a lens to the back focal plane. Factors and non-factors are distinguished by measuring the on-axis light intensity. This technique is simple and fast, and holds promise for applications in information coding and cryptography.

[0019] Advantages of additional aspects of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0020] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0021] Figure 1 This is a flowchart of the method in the first embodiment.

[0022] Figure 2 This is a diagram showing the pinhole distribution of the first embodiment.

[0023] Figure 3 (a), (b), and (c) in the figure are light intensity distribution diagrams taken under different trial factors p in the first embodiment, respectively.

[0024] Figure 4 In the diagrams (a) and (b), the factor decomposition results of the pattern indices l=30 and l=50 in the first embodiment are shown respectively.

[0025] In the figure, 1 is a laser, 2 is a liquid crystal spatial light modulator, 3 is a second thin lens, 4 is a pinhole plate, 5 is a third thin lens, 6 is a digital microlens array device, 7 is a first thin lens, and 8 is a CCD camera. Detailed Implementation

[0026] Example 1

[0027] like Figure 1 As shown, this embodiment discloses an integer decomposition method based on vortex beams, including:

[0028] Step 1: Set the integer to be decomposed as the mode index of the vortex beam, and adjust the azimuth angle of the pinhole in the digital microlens array device according to the factor to be tried.

[0029] Step 2: Illuminate the digital microlens array device and the first thin lens in sequence with the vortex beam, and use a CCD camera to capture the light intensity distribution at the back focal plane of the first thin lens.

[0030] Step 3: Normalize the light intensity distribution map to obtain the normalized optical axis light intensity value;

[0031] Step 4: Determine whether the factor to be tried is a factor of the integer to be factored based on the light intensity value of the optical axis.

[0032] In this invention, the integer to be factored is set as the mode index l of the incident vortex beam, while the factor to be tried is selected as the parameter p that determines the pinhole position. When p is a true factor of l, the light intensity value on the optical axis can reach its maximum. If we normalize the on-axis light intensity value, the on-axis light intensity value is 1 only when p is a factor of l. To ensure a high degree of distinction between factors and non-factors, the number of pinholes in this invention needs to satisfy... This will compress all non-factors to the threshold. the following.

[0033] In step 1, a hologram of the vortex beam is loaded into the liquid crystal spatial light modulator using digital holography. The laser emits a fully coherent beam that illuminates the liquid crystal spatial light modulator. The beam emitted from the liquid crystal spatial light modulator is then processed by a 4f imaging system to obtain the vortex beam.

[0034] The 4f imaging system includes: a second thin lens, a pinhole plate, and a third thin lens; a liquid crystal spatial light modulator is placed at the front focal plane of the second thin lens, the pinhole plate is placed at the rear focal plane of the second thin lens and is also at the front focal plane of the third thin lens; a digital microlens array device is placed at the rear focal plane of the third thin lens and is also at the front focal plane of the first thin lens, and a CCD camera is placed at the rear focal plane of the first thin lens.

[0035] The +1st or -1st order diffracted beam emitted by the 4f imaging system is the desired vortex beam. The +1st or -1st order beam can be filtered out using a 4f imaging system with pinholes placed on the spectral surface. The factor p to be tested can be determined by designing a pinhole sieve.

[0036] Pinhole sieve Figure 2 As shown, white represents light transmission, while black represents opacity. The transmittance function of the pinhole sieve is a binary function, which can be implemented using a digital microlens array device. Therefore, the generated vortex beam is irradiated onto the digital microlens array device, which loads a digital pinhole sieve. The beam is then focused by a thin lens onto the back focal plane, and a CCD camera is used for imaging. The digital microlens array device can be controlled by writing a program in MATLAB, continuously changing the pinhole sieve to continuously change the magnitude of the trial factor p.

[0037] Step 2 specifically includes:

[0038] First, assuming a Laguerre-Gaussian beam with a radial exponent of 0 as a vortex beam, its electric field can be described as follows:

[0039]

[0040] Where l is the mode index of the vortex beam, and ω0 is the beam waist size. Here are the cylindrical coordinates at the light source.

[0041] To achieve integer factorization, the transmittance function for the pinhole sieve is set as follows:

[0042]

[0043] Where M represents the number of pinholes, Let represent the initial phase, δ be the Delta function, and p be the azimuth angle determining the pinhole. Considering practical realities, the Delta function will be replaced by a circular hole of diameter d in the experiment. Where r... m =r0+m 2 / p•d represents the offset of the m-th pinhole from the center.

[0044] The vortex beam, after being modulated by a pinhole sieve, is focused by a thin lens with a focal length of f. The pinhole sieve is placed at the front focal plane of the thin lens, and the electric field at the rear focal plane can be calculated using the Collins diffraction integral formula.

[0045]

[0046] in

[0047] Substituting formulas (1) and (2) into (3), we get:

[0048]

[0049] Assume r0 >> m 2 If / p·d, then the light intensity value on the optical axis can be simplified to:

[0050]

[0051] in This represents the angle of the m-th pinhole around the center.

[0052] function It can be simplified to:

[0053]

[0054] The above formula is the Gauss sum formula, when yes If p is a factor of l, then each term in the Gaussian term equals 1, so the sum equals 1. Otherwise, the function... The value oscillates rapidly and takes very small values.

[0055] In step 3, the light intensity distribution map is normalized to obtain the normalized optical axis light intensity value, including: CCD-measured light intensity grayscale image, MATLAB reads the image and converts it into a matrix, and the matrix is ​​divided by its maximum value to achieve the light intensity normalization process.

[0056] In step 4, the factor to be tried is determined to be a factor of the integer to be decomposed based on the optical axis intensity value, including: if the optical axis intensity value is 1, then the factor to be tried is a factor of the integer to be decomposed; otherwise, the factor to be tried is a non-factor of the integer to be decomposed.

[0057] In this embodiment, the pinhole screen parameters of the digital microlens array device are set as follows: wavelength λ = 532nm, focal length of the three thin lenses is f = 400mm, diameter of the circular hole d = 0.04mm, distance of the first pinhole from the optical axis: r0 = 6mm, and number of pinholes M = 7.

[0058] like Figure 3 As shown in (a), (b), and (c), these are light intensity distribution diagrams at the focal plane under different trial factors p with a decomposition number l = 30. The cross symbol in the diagram indicates the position of the optical axis center.

[0059] When p = 5 and p = 10, which are factors of l, the normalized optic axis intensity value is 1. However, when p = 4, it is not a factor of l, and its optic axis intensity value is not 1. This can be easily and quickly distinguished simply by the magnitude of the optic axis intensity value.

[0060] like Figure 4As shown in (a) and (b), when factoring integers l = 30 and l = 53, where error bars represent the absolute difference between theoretical and experimental results, the results show that for both l = 30 and l = 53, as long as it is a factor of l, the light intensity value on the optical axis (using...) (Indicates) are all at the threshold Above, the factors other than l are all at the threshold. under.

[0061] Example 2

[0062] This embodiment discloses an integer factorization system based on vortex beams, including:

[0063] Vortex beam generation module, used for vortex beams with mode exponents that are integers to be decomposed;

[0064] The light intensity distribution map acquisition module is used to capture the light intensity distribution map at the back focal plane of the first thin lens using a CCD camera after the vortex beam sequentially illuminates the digital microlens array device and the first thin lens.

[0065] The normalization module is used to normalize the light intensity distribution map to obtain the normalized optical axis light intensity value.

[0066] The judgment module is used to determine whether the factor to be tried is a factor of the integer to be decomposed based on the light intensity value of the optical axis.

[0067] Furthermore, the vortex beam generation module includes: a laser, a liquid crystal spatial light modulator, and a 4f imaging system;

[0068] A hologram of a vortex beam is loaded into a liquid crystal spatial light modulator using digital holography. A fully coherent beam emitted from a laser is used to illuminate the liquid crystal spatial light modulator. The emitted beam from the liquid crystal spatial light modulator is then processed by a 4f imaging system to obtain a vortex beam.

[0069] Furthermore, the 4f imaging system includes: a second thin lens, a pinhole plate, and a third thin lens; the liquid crystal spatial light modulator is placed at the front focal plane of the second thin lens, and the pinhole plate is placed at the rear focal plane of the second thin lens and at the front focal plane of the third thin lens.

[0070] The emitted beam from the liquid crystal spatial light modulator passes through a second thin lens, a pinhole plate, and a third thin lens in sequence, and is then filtered to produce a +1 or -1 order diffracted beam, i.e., a vortex beam.

[0071] Furthermore, the determination of whether the factor to be tried is a factor of the integer to be decomposed is based on the optical axis intensity value, including: if the optical axis intensity value is 1, then the factor to be tried is a factor of the integer to be decomposed; otherwise, the factor to be tried is a non-factor of the integer to be decomposed.

[0072] Those skilled in the art will understand that the modules or steps of the present invention described above can be implemented using general-purpose computer devices. Optionally, they can be implemented using computer-executable program code, thereby allowing them to be stored in a storage device for execution by a computer device, or they can be fabricated as separate integrated circuit modules, or multiple modules or steps can be fabricated as a single integrated circuit module. The present invention is not limited to any particular combination of hardware and software.

[0073] While the specific embodiments of the present invention have been described above in conjunction with the accompanying drawings, this is not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of the present invention are still within the scope of protection of the present invention.

Claims

1. An integer factorization method based on vortex beams, characterized in that, include: Set the integer to be decomposed as the mode index of the vortex beam, and adjust the azimuth angle of the pinhole in the digital microlens array device according to the factor to be tried. A vortex beam is sequentially irradiated onto the digital microlens array device and the first thin lens, and a CCD camera is used to capture the light intensity distribution at the back focal plane of the first thin lens. The light intensity distribution map is normalized to obtain the normalized optical axis light intensity value; Determine whether the factor to be tried is a factor of the integer to be factored based on the light intensity value along the optical axis; The transmittance function for the pinhole sieve is set as follows: in, Indicates the number of pinholes. Indicates the initial phase. It is the Delta function. It determines the azimuth angle of the pinhole. Let be the cylindrical coordinates at the light source. In the experiment, the Delta function will be determined by the diameter. The round hole is used as a substitute, among which Indicates the first The offset of each pinhole from the center; After being blocked and modulated by a pinhole sieve, the vortex beam has a focal length of A thin lens is used for focusing, in which a pinhole sieve is placed at the front focal plane of the thin lens, and the electric field at the rear focal plane is calculated using the Collins diffraction integral formula: in ; Assumption Then the light intensity value on the optical axis can be simplified to: in Indicates the first The angle of each pinhole around the center, function It can be simplified to: The above formula is the Gauss sum formula, when yes Factor, or yes When the factor is 1, then each term in the Gaussian term is equal to 1, so the sum is equal to 1.

2. The integer factorization method based on vortex beams as described in claim 1, characterized in that, The number of pinholes in the digital microlens array device Mode index of vortex beam The relationship is: .

3. The integer factorization method based on vortex beams as described in claim 1, characterized in that, The step of determining whether the factor to be tried is a factor of the integer to be decomposed based on the optical axis intensity value includes: if the optical axis intensity value is 1, then the factor to be tried is a factor of the integer to be decomposed; otherwise, the factor to be tried is a non-factor of the integer to be decomposed.

4. The integer factorization method based on vortex beams as described in claim 1, characterized in that, A hologram of a vortex beam is loaded into a liquid crystal spatial light modulator using digital holography. A fully coherent beam emitted from a laser illuminates the liquid crystal spatial light modulator, and the beam emitted from the liquid crystal spatial light modulator is then processed by a 4f imaging system to obtain a vortex beam.

5. The integer factorization method based on a vortex beam as described in claim 4, characterized in that, The 4f imaging system includes: a second thin lens, a pinhole plate, and a third thin lens; the liquid crystal spatial light modulator is placed at the front focal plane of the second thin lens, and the pinhole plate is placed at the rear focal plane of the second thin lens and is also at the front focal plane of the third thin lens. The emitted beam from the liquid crystal spatial light modulator passes through a second thin lens, a pinhole plate, and a third thin lens in sequence, and is then filtered to produce a +1 or -1 order diffracted beam, i.e., a vortex beam.

6. An integer factorization system based on vortex beams, employing the integer factorization method based on vortex beams as described in any one of claims 1-5, characterized in that, include: Vortex beam generation module, used for vortex beams with mode exponents that are integers to be decomposed; The light intensity distribution map acquisition module is used to capture the light intensity distribution map at the back focal plane of the first thin lens using a CCD camera after the vortex beam sequentially illuminates the digital microlens array device and the first thin lens. The normalization module is used to normalize the light intensity distribution map to obtain the normalized optical axis light intensity value. The judgment module is used to determine whether the factor to be tried is a factor of the integer to be decomposed based on the light intensity value of the optical axis.

7. The integer factorization system based on a vortex beam as described in claim 6, characterized in that, The vortex beam generation module includes: a laser, a liquid crystal spatial light modulator, and a 4f imaging system; A hologram of a vortex beam is loaded into a liquid crystal spatial light modulator using digital holography. A fully coherent beam emitted from a laser is used to illuminate the liquid crystal spatial light modulator. The emitted beam from the liquid crystal spatial light modulator is then processed by a 4f imaging system to obtain a vortex beam.

8. The integer factorization system based on a vortex beam as described in claim 7, characterized in that, The 4f imaging system includes: a second thin lens, a pinhole plate, and a third thin lens; the liquid crystal spatial light modulator is placed at the front focal plane of the second thin lens, and the pinhole plate is placed at the rear focal plane of the second thin lens and is also at the front focal plane of the third thin lens. The emitted beam from the liquid crystal spatial light modulator passes through a second thin lens, a pinhole plate, and a third thin lens in sequence, and is then filtered to produce a +1 or -1 order diffracted beam, i.e., a vortex beam.

9. The integer factorization system based on a vortex beam as described in claim 6, characterized in that, The number of pinholes in the digital microlens array device Mode index of vortex beam The relationship is: .

10. The integer factorization system based on a vortex beam as described in claim 6, characterized in that, The step of determining whether the factor to be tried is a factor of the integer to be decomposed based on the optical axis intensity value includes: if the optical axis intensity value is 1, then the factor to be tried is a factor of the integer to be decomposed; otherwise, the factor to be tried is a non-factor of the integer to be decomposed.

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