A method for generating a symmetrical butterfly beam and its application in imaging

Abstract

The present application relates to a method for generating a symmetrical butterfly beam and its application in imaging. According to the amplitude and phase distribution of the beam, it is named a symmetrical Gaussian butterfly beam. The transmission characteristics of the beam are analyzed, and a symmetrical Gaussian butterfly beam is generated by the product of the Gaussian distribution function and the two-dimensional butterfly integral. The diffraction characteristics of the beam are theoretically derived and analyzed, and numerical simulation studies are performed on it. When the Gaussian light carrying image information is irradiated on the spatial light modulator loaded with the symmetrical butterfly beam phase diagram, it shows the characteristics of random imaging at different diffraction distances. Therefore, it has potential application value in image security. The present application generates a symmetrical butterfly beam by the product of the Gaussian term and the two-dimensional butterfly integral. The symmetrical butterfly beam is used for the transmission of image signals. Through this application, the regulation and application of the symmetrical butterfly beam is realized.

Description

Technical Field

[0001] The present invention belongs to the field of optical technology, and in particular relates to a method for generating a symmetrical butterfly beam and applying the same in imaging. Background Art

[0002] In the field of optics, laser beams with mutation functions have attracted great interest in both theory and application. Mutation functions can generally be described by seven basic mutations: fold, cusp, swallowtail, butterfly, hyperbolic cord, elliptical cord, and parabolic cord. Among these different types of beams, the manipulation of their symmetric intensity distribution has become a research hotspot in recent years. Most current research involves Airy beams and Pierce beams. In 2010 and 2013, symmetric circular Airy beams and two-dimensional symmetric Airy beams were realized in cylindrical coordinates and Cartesian coordinates, respectively. In 2021, researchers turned their attention to relatively higher-order mutation beams, namely Pierce-Gaussian beams and swallowtail-Gaussian beams, and conducted in-depth studies on their symmetric structures, resulting in symmetric Pierce and swallowtail beams. As the order of the mutation function increases, the difficulty of manipulating the beam's light field distribution gradually increases, and the complexity increases accordingly.

[0003] Researchers are committed to obtaining more high-dimensional diffraction mutation beams with adjustable properties and stable light field structures to control the focusing distance and focusing intensity of the beam, thereby realizing applications in many optical fields. High-order diffraction mutation beams can change the light field structure by manipulating their state variables. As the order of the symmetric diffraction beam increases, the light field structure it exhibits becomes richer, but the difficulty of generating the corresponding beam experimentally also increases. Therefore, it is of great significance to propose an effective method to realize high-dimensional rectangular symmetric diffraction beams with a distribution function of 6 orders or above, and to utilize their unique symmetric intensity light field distribution and propagation characteristics to realize their information transmission and imaging. An effective solution has not yet been proposed for this. Summary of the Invention

[0004] To solve the above problems, the present invention discloses a method for generating a symmetrical butterfly beam and applying the same in imaging, comprising the following steps:

[0005] Step 1: The two-dimensional light field distribution of the symmetric butterfly beam is derived based on the butterfly integral formula theory, and the diffraction characteristics of the symmetric butterfly beam during propagation are studied through numerical simulation;

[0006] The initial light field distribution of the symmetric butterfly beam is:

[0007]

[0008] Where w0 is the beam width of the Gaussian beam, b n (n=1,2) is the distribution factor, pn (n=1,2) is a constant, and SBu(·) is a variant of the butterfly integral: x, y, a1, a2 are dimensionless coordinates in space;

[0009] Step 2: The two-dimensional light field distribution of the symmetric butterfly beam interferes with the plane wave to obtain the CGH of the symmetric butterfly beam;

[0010] Step 3: By adjusting the constant p and introducing the vortex phase and astigmatism phase, the symmetric butterfly beam is controlled;

[0011] Step 4: The Gaussian light carrying the image information passes through a spatial light modulator loaded with the phase information of the symmetric butterfly beam to obtain image information at different distances, realizing the application of the symmetric butterfly beam in imaging.

[0012] Furthermore, the light field expressions of the symmetrical butterfly beam at different diffraction distances are: Through the Fresnel diffraction integral formula, the analytical expressions of the light field of the symmetrical butterfly beam at different diffraction distances can be obtained, namely:

[0013]

[0014] Where z is the diffraction distance, i is the imaginary unit, is the wave number, λ is the wavelength, and by derivation, we can further obtain

[0015] Where θ = kw0 2 exp(ikz),q(z)=iz+z R , During the propagation of the symmetrical butterfly beam, its light field gradually splits from a bright spot into four independent main lobes; the total energy gradually transfers to the four main lobes as the propagation distance increases.

[0016] Furthermore, the focusing intensity and focusing distance can be controlled by adjusting the constant p of the symmetrical light beam.

[0017] Furthermore, in step 3, a vortex phase and an astigmatism phase are introduced on the main lobe of the symmetric butterfly beam to generate an astigmatic symmetric butterfly vortex beam, and its topological charge information is determined according to the diffraction intensity pattern during the propagation process.

[0018] In step 4, by numerically simulating the process of diffraction after Gaussian light carrying image information illuminates the phase pattern of the symmetric butterfly beam, image information at different distances is obtained, thereby realizing the application of the symmetric butterfly beam.

[0019] u(x,y,0,0)=SBu(x,p,0,0)SBu(y,p,0,0),

[0020] Where p is a constant and BBu(·) is a variant of the butterfly integral: x, y, a1, a2 are dimensionless coordinates in space.

[0021] Working principle of the present invention:

[0022] A symmetric butterfly beam is generated by interfering a simulated symmetric butterfly beam with a plane wave using a phase modulation pattern generated on a spatial light modulator. This phase modulation pattern is then used to modulate a Gaussian beam, resulting in a symmetric butterfly beam. Numerical calculations reveal the intensity distribution of the butterfly beam at different cross-sections during transmission. At a specific distance, the beam undergoes self-focusing, becoming a bright line with the highest energy concentration. Due to its significant autofocusing and lateral self-acceleration properties, the beam deforms during propagation. As the transmission distance increases, the cross-sectional light field distribution of the beam changes, with the light field gradually splitting from a single bright spot into four independent main lobes.

[0023] Beneficial effects of the present invention: The beneficial effects of the above-mentioned symmetrical butterfly beam control and imaging system are as follows:

[0024] First, the butterfly beam's focusing ability is easier to control due to its higher order.

[0025] Secondly, after focusing, the butterfly beam diverges in a rectangular, symmetrical distribution, changing the information it transmits. This provides the opportunity for information conversion during transmission. By adjusting the dimensionless distribution factor in the above equation, the distribution and focal length of the rectangular light field can be altered, offering potential applications in imaging and optical encryption.

[0026] Third, the butterfly beam can also introduce vortex and astigmatism phases in the main lobe of the rectangular light intensity, which can manifest topological charge characteristics along the trajectory during propagation. This new type of symmetric beam opens up other possibilities for applications in particle capture and atmospheric transport. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 Schematic diagram of the flow of the control system of the symmetrical butterfly beam according to Embodiment 1, Embodiment 2 and Embodiment 3 of the present invention;

[0028] Figure 2 This is a kinoform diagram loaded on a spatial light modulator according to embodiment 1 of the present invention;

[0029] Figure 3 The side view profile of the symmetrical butterfly beam and the intensity diagram and energy flow changes during the propagation process of Example 1 of the present invention are shown;

[0030] Figure 4 This is a diagram of the propagation intensity of a symmetrical butterfly beam according to Example 1 of the present invention;

[0031] Figure 5 Graphs of the propagation intensity of symmetrical butterfly beams with different p values ​​controlled by embodiment 2 of the present invention;

[0032] Figure 6 (a) Shows the intensity diagram of symmetric butterfly vortex beams with topological charges of 2, 4, 6, 8, -2, -4, -6, and -8, and the inset shows the corresponding phases; Figure 6 (b) shows the diffraction intensity diagram of the astigmatic butterfly vortex beam after diffraction

[0033] Figure 7 Schematic diagram of imaging the letter “r” by a symmetrical butterfly beam at different distances according to Example 3 of the present invention.

[0034] Figure 8 This is a system diagram of this embodiment. DETAILED DESCRIPTION

[0035] The present invention will be further described below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are intended only to illustrate the present invention and are not intended to limit the scope of the present invention. It should be noted that the terms "front," "rear," "left," "right," "up," and "down" used in the following description refer to directions in the accompanying drawings, and the terms "inward" and "outward" refer to directions toward or away from the geometric center of a particular component, respectively.

[0036] like Figure 8 As shown, this embodiment discloses a system for generating and controlling a symmetrical butterfly beam, including a semiconductor solid-state laser, a beam expander collimator, a spatial light modulator, a beam splitter prism, a 4f optical imaging system, and an electrically coupled device. The system controls the phase of the incident beam to generate a symmetrical butterfly beam.

[0037] a laser for generating a Gaussian beam;

[0038] Beam expansion collimator, which collimates and expands the Gaussian beam;

[0039] Input image, used to verify the effectiveness of the system for image signal transmission;

[0040] The beam splitter prism is located between the collimating beam expander and the spatial light modulator, and is used to split the collimated and expanded Gaussian light beam, and transmit a part of the light to the spatial light modulator.

[0041] The spatial light modulator is used to load the phase pattern of the symmetric butterfly beam and phase modulate the Gaussian beam.

[0042] In a 4f optical imaging system, the reflected beam modulated by the spatial light modulator passes through the first lens, undergoing a Fourier transform to obtain a spectrum. This information is then superimposed on the symmetrical butterfly beam spectrum. An aperture is used to filter the signal to obtain a positive first order, and a second lens performs an inverse Fourier transform on the obtained order information.

[0043] A charge-coupled device is used to receive the initial light field and the light field information after diffraction at different distances as well as the transmitted image information.

[0044] Furthermore, the beam expander and collimator includes a microscope objective lens and a lens. The aperture is placed between the microscope objective lens and the lens to adjust the size of the light spot incident on the spatial light modulator as needed.

[0045] Furthermore, the reflective spatial light modulator is used to load the kinoform pattern to generate the symmetrical butterfly light beam.

[0046] Furthermore, the 4f optical imaging system includes two lenses and an aperture. The light beam reflected after modulation by the reflective spatial light modulator is Fourier transformed through the first lens to obtain a spectrum surface; the aperture is placed on the spectrum surface to obtain the positive first order of the spectrum surface, the distance between the second lens and the aperture is its focal length, and the obtained order information is inverse Fourier transformed.

[0047] Furthermore, the charge coupled device is used to receive diffraction intensity patterns of symmetrical butterfly light beams at different distances or image information transmitted by the system.

[0048] Furthermore, the hologram is obtained by interfering the two-dimensional field distribution of the symmetrical butterfly beam with the plane wave, and the light field distribution function of the butterfly beam is

[0049] u(x,y,0,0)=SBu(x,p,0,0)SBu(y,p,0,0)

[0050] The derivation process of the light field expression of the symmetric butterfly beam described in this application at different diffraction distances is as follows:

[0051] The definition of the butterfly integral is According to Fourier transform, its spectrum distribution is

[0052]

[0053] k x , k y is the variable in the x0, y0 direction of the spectrum, p n (n=1,2) is a constant used to control the symmetric butterfly beam. The spectral distribution is inverse Fourier transformed and multiplied by the Gaussian term to obtain the initial field distribution of the symmetric butterfly beam:

[0054]

[0055] Through the Fresnel diffraction integral formula, the analytical expression of the light field of the symmetrical butterfly light at different diffraction distances can be obtained, namely:

[0056]

[0057] Where z is the diffraction distance, i is the imaginary unit, is the wave number, λ is the wavelength, and by derivation, we can further obtain

[0058]

[0059] Where θ = kw0 2 exp(ikz),q(z)=iz+z R ,

[0060] In this embodiment, the interference between the initial light field of a symmetrical butterfly beam and a plane wave with parameters set as w0 = 3 mm, b1 = b2 = 0.07, and p1 = p2 = 0 is simulated to obtain a phase hologram, namely Figure 2 , and loaded onto the spatial light modulator, the outgoing beam is a symmetrical butterfly beam. The side view profile of the symmetrical butterfly beam is further simulated. Figure 3 (a) and the intensity diagram during the propagation process, namely Figure 3 (b1-b4). The diffraction distance is set to: z = 0, z = 0.036z R ,z=0.098z R ,z=0.14z R Its light field gradually splits from a bright spot into four independent main lobes. In addition, based on the Poynting vector principle, the energy flow of the symmetrical butterfly beam at different diffraction distances is calculated. It can also be clearly seen from the energy flow diagram that with the change of diffraction distance, the main lobe energy gradually transfers to the four main lobes, and finally the energy is dispersed to the four main lobes. Figure 3 (c1-c4). From the light intensity distribution and energy flow diagram in the figure, the symmetry of the butterfly symmetric beam can be clearly shown. In order to verify the effectiveness of the method, we will use the experimental results, namely Figure 3 (d1-d4) are compared with the simulation results, which verifies the feasibility of this method.

[0061] Example 2

[0062] like Figure 4 Based on the system of Example 1, this embodiment modulates the phase hologram through a spatial light modulator to control the focusing intensity of the symmetrical butterfly light beam.

[0063] Methods for controlling the focusing intensity of a symmetric butterfly beam include:

[0064] By changing the constant p and setting the other parameters the same as in Example 1, a new phase hologram with set parameters is obtained, and a new symmetrical butterfly beam is obtained by irradiating the spatial light modulator with Gaussian light. Figure 5 The following plots show the propagation intensity of a symmetric butterfly beam at different diffraction distances for different p values. Analysis of the plots shows that adjusting the p value can change the focusing intensity. The focusing intensity increases with increasing p value, and the focusing distance increases. This demonstrates the tunability of the symmetric butterfly beam.

[0065] Introducing vortex phase on the main lobe of a symmetric butterfly beam in is the azimuth angle, generating a symmetrical butterfly vortex beam, and then quoting the astigmatism phase ψ=a[(x 2 +y 2 )cos(2α)+2xysin(2α)], thus generating an astigmatic butterfly vortex beam, where the parameter is set to the astigmatism constant a=1.01×10 7 , astigmatism angle During the propagation process, the topological charge characteristics of the symmetric butterfly vortex beam can be reflected along the trajectory. Figure 6 (a) Shows the intensity diagram of symmetric butterfly vortex beams with topological charges of 2, 4, 6, 8, -2, -4, -6, and -8, and the inset shows their corresponding phases. Figure 6 (b) shows the diffraction intensity pattern of an astigmatic butterfly vortex beam after 40 cm of diffraction. The sign and magnitude of the topological charge can be determined from the alternating black and white stripes in the diffraction pattern. The number of dark stripes is equal to the topological charge. The positions of the dark stripes are marked with white lines in the figure, and the dark stripes are numbered ① and ②. The tilt of the stripes indicates the sign of the topological charge. For an astigmatic butterfly vortex beam with a positive topological charge, the diffraction pattern is tilted 45 degrees clockwise relative to the y-axis. For a negative topological charge, the diffraction pattern is tilted 45 degrees counterclockwise relative to the y-axis.

[0066] Example 3

[0067] On the basis of Example 1 and Example 2, the image information at different distances is obtained by numerically simulating the process of diffraction of Gaussian light carrying image information after passing through the phase pattern of the symmetrical butterfly beam. Figure 7 As shown in FIG. 1 , the letter r shows that the imaging information is transformed at different diffraction distances. This clarifies that the method designed in this application has the ability to transmit information and form images, and has great potential in applications such as optical encryption.

[0068] This application uses this system to achieve the control and imaging of symmetrical butterfly beams, which can well control the focusing intensity and distance of the beam and achieve imaging. The system structure is simple and easy to operate.

[0069] The technical means disclosed in the solution of the present invention are not limited to the technical means disclosed in the above-mentioned embodiment, but also include technical solutions composed of any combination of the above technical features.

Claims

1. A method for generating a symmetrical butterfly beam and applying it in imaging, characterized in that: The following steps are included: Step 1: The two-dimensional light field distribution of the symmetric butterfly beam is derived based on the butterfly integral formula theory, and the diffraction characteristics of the symmetric butterfly beam during propagation are studied through numerical simulation; The initial light field distribution of the symmetric butterfly beam is: Where w0 is the beam width of the Gaussian beam, b n , n=1,2; is the distribution factor, p n , n=1,2 are constants, SBu(·) is a variant of the butterfly integral: x, y, a1, a2 are dimensionless coordinates in space; Step 2: The two-dimensional light field distribution of the symmetric butterfly beam interferes with the plane wave to obtain the CGH of the symmetric butterfly beam; Step 3: By adjusting the constant p and introducing the vortex phase and astigmatism phase, the symmetric butterfly beam is controlled; Step 4: The Gaussian light carrying the image information passes through a spatial light modulator loaded with a symmetric butterfly beam phase hologram to obtain image information at different distances, realizing the application of symmetric butterfly beams in imaging.

2. The method for generating a symmetrical butterfly beam and applying the same in imaging according to claim 1, characterized in that: Through the Fresnel diffraction integral formula, the analytical expression of the light field of the symmetrical butterfly light at different diffraction distances can be obtained. The light field expression at different diffraction distances is: Where z is the diffraction distance, i is the imaginary unit, is the wave number, λ is the wavelength, and by derivation, we can further obtain Where θ = kw0 2 exp(ikz),q(z)=iz+z R , ξ=1,2, During the propagation of the symmetrical butterfly beam, its light field gradually splits from a bright spot into four independent main lobes; the total energy of the beam gradually disperses to the four main lobes.

3. The method for generating a symmetrical butterfly beam and applying the same in imaging according to claim 1, characterized in that: By introducing the constant p, the focusing intensity and focusing distance can be regulated.

4. The method for generating a symmetrical butterfly beam and applying the same in imaging according to claim 1, characterized in that: In step 3, a vortex phase and an astigmatism phase are introduced on the main lobe of the symmetric butterfly beam, thereby generating an astigmatic symmetric butterfly vortex beam, and its topological charge information is determined according to the diffraction intensity pattern during the propagation process.

5. The method for generating a symmetrical butterfly beam and applying the same in imaging according to claim 1, characterized in that: In step 4, by numerically simulating the process in which Gaussian light carrying image information illuminates the kinoform diagram of the symmetric butterfly beam and then diffracts, image information at different distances is obtained, thereby realizing the application of the symmetric butterfly beam.

Images