Method and system for generating a diffraction-free, partially coherent Pearcey beam

Description

2. They find numerous applications in volumetric imaging, ghost imaging, particle trapping, photon-correlated holography, and factorization. Partially coherent diffraction-free beams are attracting increasing interest due to the combined intrinsic advantages of diffraction-free beam factorization and partially coherent beams. 5. In recent years, the degree of coherence, as a degree of freedom specific to partially coherent beams, has generated much research ranging from fundamental physics to practical applications. In particular, the degree of coherence has been studied as an information carrier for image encryption. By introducing randomness, partially coherent beams make it possible to achieve information encryption up to 10 high capacity. However, partially coherent beams with low coherence are sensitive to optical diffraction effects, which leads to a degradation of all degrees of freedom, including coherence, during propagation. Consequently, theEncrypted information is inevitably altered. The solution to this problem relies on the construction of diffraction-free partially coherent beams. Moreover, the previously proposed partially coherent Pearcey beams exhibit a self-acceleration phenomenon inducing a shift during propagation, which is unfavorable for information transmission and optical encryption applications. Content of the invention 20 As a remedy to the aforementioned problems, the present invention proposes a method and a system for generating a partially coherent, diffraction-free Pearcey beam. Based on the principle of incoherent superposition, the invention makes it possible to construct a partially coherent, diffraction-free Pearcey beam, whose properties, such as intensity and degree of coherence, remain unchanged during 25 free-space propagation. According to certain embodiments, a first aspect of the invention proposes a system for generating a partially coherent, diffraction-free Pearcey beam, putting inimplements the following technical solution: A system for generating a partially coherent Pearcey beam without diffraction, comprising: A laser, the beam emitted by the laser passing through a half-wave plate to reach a beam expander, then, after expansion by the expander, passing through a beam splitter before being directed to a spatial light modulator; The spatial light modulator being configured to modulate the beam after splitting using the cross spectral density function in order to generate a modulated beam; An optical imaging system, the optical imaging system being configured to select the first-order diffraction spot of the modulated beam; A CCD camera, the CCD camera being intended to record the first-order diffraction pattern.10 Furthermore, the optical imaging system4f includes a first thin lens, a diaphragm, and a second thin lens. Furthermore, the spatial light modulator is configured to modulate the beam afterdivision using the cross spectral density function, more precisely that: Use the cross spectral density function to obtain pseudo-modes15 corresponding to different position points in the reciprocal plane of the beam after division; The pseudo-modes corresponding to different points in the reciprocal plane are generated as holograms based on a complex amplitude modulation algorithm.20 Furthermore, the complex amplitude modulation algorithm is defined as follows: ={Arg[(,)]+2}; Where the amplitude is obtained by numerical inversion; 1() = |(,)|, where 1 denotes the first-order Bessel function of the first kind; "Arg" denotes the phase of the function; denotes the frequency of the network; (,) is an arbitrary kernel function; = (×, ×),, ∈ [0,] denotes the position of the sampling points in the reciprocal plane; =(,) and = (⊥,∥) represent respectively the vector positions in the source plane and the reciprocal plane. According to certain embodiments, a second aspect of the invention proposes a methoddegeneration of a partially coherent diffraction-free Pearcey beam, putting into effect the following technical solution: A process for generating a partially coherent diffraction-free Pearcey beam, based on the system according to the first aspect, comprising: The beam emitted by the laser passes through a half-wave plate to reach a beam expander, then, after expansion, passes through a beam splitter before being directed to a spatial light modulator; The spatial light modulator is repeatedly refreshed to modulate the beam after division, with pseudo-modes corresponding to different points in the reciprocal plane being obtained based on the cross-spectral density function, and corresponding holograms being generated;10 The beam is processed by an optical imaging system4fa to select the first-order diffraction pattern corresponding to each pseudo-mode; A CCD camera records said first-order diffraction patterns of eachpseudo-mode, and the pseudo-modes are superimposed in order to form a partially coherent Pearcey beam.15 Furthermore, the step of modulating the beam after division by repeated refreshing of the spatial light modulators, obtaining pseudo-modes in the reciprocal plane and generating the corresponding holograms includes: The construction of a power spectral density function and an arbitrary kernel function using a Gaussian function and a cosine function;20 The introduction of dimensionless coordinates and difference coordinates in order to transform the mathematical form of said functions; The determination, on the basis of the transformed functions, of the cross spectral density function for a given point; The use of a Gaussian aperture as a beam truncation function to determine the truncated cross-spectral density function for a given point; the repeated refreshing of the spatial light modulators to modulate the beam after division, and the obtaining of pseudo-modes corresponding to different points in the planereciprocal from said truncated function. Based on a complex amplitude modulation algorithm, the pseudo-modes 30 BE2026 / 7289 5 at different points of the reciprocal plane are converted into holograms. Furthermore, the determination of the truncated cross-spectral density function for a given point is defined as follows: (+,−)=(−2+ 2−2− 2 4 )×(+,−); Where= 0 and 0 represent the width of the truncation function; + and −5 denote the power spectral density function and the arbitrary kernel function transformed using dimensionless coordinates and difference coordinates; denotes the truncated cross-spectral density. Furthermore, the step of recording, using a CCD camera, the first-order diffraction spots of each pseudo-mode and superimposing the pseudo-modes in order to deform a partially coherent Pearcey beam is defined as follows: (1,2)=∑() ∗(,1)(,2), ; Where() is a discrete power spectral density function representing the weight of the modes; (,2) is the truncated mode; ∗(,2) is the pseudo-mode;=(×,×),,∈[0,] denotes the position of the sampling points in the reciprocal plane; 1 and 2 are the coordinates in the source plane. Furthermore, the truncated mode is defined as follows: (,)=(− 2 20 2)(,); Where=(×,×),,∈[0,] denotes the position of the sampling points in the reciprocal plane; d denotes the sampling interval; (,)20 is an arbitrary kernel function;=(,) and=(⊥,∥) represent respectively the vector positions in the source plane and the reciprocal plane; denotes the size of the Gaussian truncation. Furthermore, the first-order diffraction pattern contains the pseudo-mode information.25 Compared to the prior art, the beneficial effects of the present invention are as follows: The present invention is based on the principle of incoherent superposition to construct a partially coherent diffraction-free Pearcey beam. The properties of this beam, BE2026 / 7289 6, such as intensity and degree of coherence, remain unchanged during free-space propagation, thus conferring a strong robustness to the beam, particularly with respect to thediffraction effects and other disturbances during transmission. In particular, the intensity and degree of coherence of this type of beam remain constant during propagation. Even under practical conditions, a truncated version retains a remarkable capacity for resistance to optical diffraction. These beams inherently possess two significant advantages: low coherence and diffraction-free propagation. The low coherence guarantees the robustness of the beam in harsh environments, while the diffraction-free property allows certain degrees of freedom, such as coherence and intensity, to be exploited as information carriers for long-distance transmission. Consequently, partially coherent, diffraction-free Pearcey beams have strong application potential in fields such as Optical communications, information transmission, and optical imaging, particularly in challenging environments. 15. Description of figuresThe accompanying figures, which form an integral part of the present invention, are intended to facilitate understanding. The illustrative embodiments and their descriptions serve to explain the invention and should not be interpreted as unduly limiting its scope. Figure 1 shows a structural diagram of a system for generating a partially coherent diffraction-free Pearcey beam according to a first embodiment of the invention; Figure 2 shows a flowchart of a method for generating a partially coherent diffraction-free Pearcey beam according to a second embodiment of the invention; Figure 3 shows the normalized intensity of a truncated partially coherent diffraction-free Pearcey beam during free-space propagation according to the second embodiment; Figure 4 represents the intensity and degree of coherence of a partially coherent Pearcey beam 30 BE2026 / 7289 7 without truncated diffraction during free-space propagation according to the second realization method;Figure 5 represents the intensity similarity (upper axis) and the degree of coherence (lower axis) of a partially coherent, diffraction-free Pearcey beam during its propagation in free space according to the second embodiment; Figure 6 represents the theoretical and experimental results of the intensity distribution of a partially coherent, diffraction-free Pearcey beam during its propagation in free space according to the second embodiment. Specific embodiments The present invention is described in more detail below with reference to the figures and embodiments. It should be noted that the following detailed descriptions are given for illustrative purposes only and are intended to provide a better understanding of the invention. Unless otherwise indicated, all technical and scientific terms used in this document have the same meaning as that generally understood by a person skilled in the art in the relevant technical field. It should also be noted that the terms used here are intended solely to describespecific embodiments and are not intended to limit the invention. Unless otherwise explicitly stated in the context, singular forms also include plural forms.20 In addition, the terms "comprising" and / or "including" indicate the presence of features, steps, operations, devices, components and / or combinations thereof. In the absence of conflict, the different embodiments of the invention and their characteristics can be combined. 25 Embodiment 1 As illustrated in Figure 1, the present embodiment proposes a system for generating a partially coherent, diffraction-free Pearcey beam, comprising: A laser; the beam emitted by the laser passing through a half-wave plate to reach a beam expander; then, after expansion by the expander, passing through a beam splitter 30 BE2026 / 7289 8 before being directed to a spatial light modulator; The spatial light modulator being configured to modulate the beam after splittingusing the cross-spectral density function to generate a modulated beam; An optical imaging system 4f, the optical imaging system 4f being configured to select the first-order diffraction pattern of the modulated beam; A CCD camera, the CCD camera being intended to record the first-order diffraction pattern. The optical imaging system 4f comprises a first thin lens, a diaphragm, and a second thin lens. The spatial light modulator modulates the beam after splitting using the cross-spectral density function 10, more precisely that: Use the cross-spectral density function to obtain pseudo-modes corresponding to different position points in the reciprocal plane of the beam after splitting; Based on a complex amplitude modulation algorithm, the pseudo-modes15 at the different points of the reciprocal plane are converted into holograms. The complex amplitude modulation algorithm is defined as follows: ={Arg[(,)]+2}; Wherethe amplitudeBisobtainedbynumericalinversion;1()=|(,)|,where1denotes the first-order Bessel function of the first kind; "Arg" denotes the phase 20 of the function; denotes the frequency of the network; (,) is an arbitrary kernel function; = (×, ×),, ∈ [0,] denotes the position of the sampling points in the reciprocal plane; = (,) and = (⊥, ∥) represent respectively the vector positions in the source plane and the reciprocal plane. The optical device is as illustrated in Figure 1. A beam with a wavelength of 25,632.8 nm is emitted by a helium-neon laser, passes through a half-wave plate, and is then broadened by a beam expander. The orientation of the half-wave plate is adjusted to horizontally align the polarization of the incident beam in order to obtain an optimal response from the spatial light modulator (SLM). After passing through a beam splitter, the beam is directed to a pure phase-modulated liquid crystal spatial light modulator (SLM). Using a complex amplitude modulation encoding algorithm, each pseudo-mode is designed as a hologram and loaded onto the SLM. After reflection on theSLMetleseparateurbelaisceau, lebeaisceau travers un system d'imagerie optique 4f , composé de deux lentilles identités et d'une diaphragme. Ce system 4f secommet de sélection et defiltre la piste diffraction de première ordre. La piste diffraction de première ordre 5 con le s'il ... The spatial light modulator (SLM), based on a complex amplitude modulation algorithm, allows each mode to be encoded in a form usable by the SLM for beam modulation, namely as holograms loaded onto the SLM. Based on this algorithm, the phase representation of the electric field on the SLM is 15to encode is given by: ={Arg[(,)]+2}(1); Where the amplitude is obtained by numerical inversion, 1()=|(,)|, 1 denoting the first-order Bessel function of the first kind, "Arg" representing the phase of the function, and the frequency of the grating. 20 Implementation 2 As illustrated in Figure 2, the present implementation proposes a method for generating a partially coherent, diffraction-free Pearcey beam, based on the system described in implementation 1, comprising: The beam emitted by the laser passes through a half-wave plate to reach a beam expander 25, then, after expansion, passes through a beam splitter before being directed towards a spatial light modulator; The spatial light modulator is repeatedly refreshed to modulate the beam after division; based on the crossed spectral density function, pseudo-modes corresponding to different points in the reciprocal plane are obtained, and 30 BE2026 / 7289 10 corresponding holograms are generated; The beam is then processed by an optical imaging system4fa to select theFirst-order diffraction spots corresponding to each pseudo-mode; A CCD camera records said first-order diffraction spots of each pseudo-mode, and the pseudo-modes are superimposed to form a partially coherent Pearcey beam. The beam modulation step after repeated refresh division of the spatial light modulators, based on the crossed spectral density function to obtain pseudo-modes in the reciprocal plane and generate corresponding holograms, more precisely comprises: The construction of a power spectral density function and an arbitrary kernel function using a Gaussian function and a cosine function; The introduction of dimensionless coordinates and difference coordinates to transform the mathematical form of said functions; The determination, based on transformed functions, of the crossed spectral density function for a given point; The use of a Gaussian aperture as a beam truncation function in order toto determine the truncated cross-spectral density function for a given point; The repeated refreshing of the spatial light modulators to modulate the beam after division, and the obtaining of pseudo-modes corresponding to different points of the reciprocal plane from said truncated function; The generation of holograms from the pseudo-modes corresponding to the different points of the reciprocal plane on the basis of a complex amplitude modulation algorithm. In the domain of spatial frequencies, the statistical properties of partially coherent light are described by the cross-spectral density function, defined as follows: (1,2)=∫()*(,1)(,2) 2(1); Where () represents the power spectral density function, and (,) is an arbitrary kernel function; = (,) and = (⊥,∥) respectively denote the vector positions in the source plane and in the reciprocal plane. In order to construct a partially coherent diffraction-free Pearcey beam, the following form is adopted: () = 2 2 (− 1 2 2 2)(2); (,) = ( 1 2 4⊥ 4+ 1 2 4∥ 4+ 1 2 ⊥+ 1 2 ∥)(3);Where the coherence width and the beam width are represented respectively. Dimensionless coordinates and difference coordinates are introduced as follows: +=(+,+)= (1+2) 2 ,−=(−,−)= (1−2) (4); Substituting equations (2) for (4) in equation (1), the crossed spectral density function for a given point can be rewritten as follows: (+,−)=(+)+(−)(5);10 (+)= 4 [(+, 2 )(+, 2 )](6); (−)=(− − 2 8 )(7); Where (⋅,⋅) denotes the Pearcey function, defined by (,)=∫[(4+ +∞ −∞ 2+)], and = 2 2 determines the global coherence of the source beam. The cross spectral density function depends only on the parameter .(+) represents a background modulation function, whose characteristics are described by (−). The propagation behavior of partially coherent light can be studied using the Fresnel diffraction integral, as follows: (1,2)= 1 22 ∬(1,2)[− 2 (1 2−21⋅1+1 2)][ 2 (2 2−22⋅2+ 2 2)]21 22(8);20 Substituting equations (5) for (7) in equation (8), then changing to coordinatesdimensionless and differenceless, a direct integration allows us to demonstrate that the cross spectral density function of the partially coherent diffraction-free Pearcey beams is independent of the propagation distance z, namely (+, −,) = (+, −, = 0). This clearly demonstrates that this beam is not affected by optical diffraction.25 However, this beam has an infinite extent, implying infinite energy, which is not physically realizable. To solve this problem, a Gaussian aperture BE2026 / 7289 12 is introduced as the truncation function of the beam. The crossed spectral density function of the truncated beam is then rewritten as the truncated crossed spectral density function for a given point, defined as follows: (+,−)=(−2+ 2−2− 2 4 )×(+,−)(9); Where = 0 and 0 represents the width of the truncation function.5 Its evolutionary behavior is illustrated in Figures 3 and 4, the width of the truncated Gaussian profile being fixed at 0 = 10 mm. The total propagation distance is 100 m, going from the left plane to the right plane. The results clearly show that the intensity (Figure 3) and theThe degree of coherence (Figure 4) remains practically unchanged throughout propagation. Each figure presents three planes corresponding to propagation distances z = 0.50 and 10⁻¹⁰ m. In order to quantify the ability of the water beam to resist optical diffraction, a similarity measure is introduced to evaluate the similarity between the intensity (or coherence) distributions at the receiving plane and the source plane. Similarity is defined as follows: () = [∬((x) × 0⁻²)]² ∬(x)² ∬(x)² ∬(x)² ,(10);15 Where (x) denotes the intensity or coherence distribution at propagation distance z, and (x) that at the source plane. Its value is within the interval [0, 1], a higher value indicating better resistance to diffraction. As illustrated in Figure 5, the similarity curves of intensity (upper axis) and degree of coherence (lower axis) are plotted. Although the curves decrease slightly with increasing propagation distance, the similarity remains approximately 0.98 at z = 100 m, demonstrating a strong resistance to diffraction. Furthermore, the influence of theThe global coherence parameter on the similarity of intensity and coherence was studied. The results show that, apart from a slight variation in the degree of coherence, these quantities are generally little affected by this parameter. This indicates that even for low coherence, the constructed beam retains a significant capacity to resist diffraction, thus resolving the classic problem according to which low coherence generally leads to increased diffraction and a more pronounced distortion of the light field. In order to further study the properties of the partially coherent Pearcey beam BE2026 / 7289 13 without truncated diffraction and to facilitate its experimental realization, the principle of superposition of pseudo-modes is adopted to represent this type of beam. The first-order diffraction spots of each pseudo-mode are recorded using a CCD camera, then superimposed to form a partially coherent Pearcey beam, according to the following expression: 5(1,2) = ∑(x,1)(x,2)(x,1); Where (x,1)(x,2)(x,1); is a discrete power spectral density function representing theweightofmodes;(,2)isthetruncatedmode(orpseudo-mode),defined as follows: (,)=(− 2 20 2)(,)(12); where = (×, ×), ∈ [0,] represents the position of the sampling points in the reciprocal plane, d being the sampling interval, and N the total number of sampling points in the horizontal and vertical directions. Since free space constitutes a linear system, the beam propagation behavior can be evaluated by inconsistent superposition of all pseudo-modes propagated up to the reception plane, in accordance with the theory of light wave propagation. The experimental results agree well with the theoretical results, demonstrating the feasibility of the proposed solution. Moreover, the beam propagation behavior in free space has been studied experimentally, as illustrated in Figure 6. The results show that the intensity of the beam does not vary during propagation. The experimental results are in good agreement with the theoretical predictions, confirming that the beam at 20The partially coherent construct exhibits strong resistance to optical diffraction. The parameters used are λ = 1, with propagation distances z = 0.5m, 1m and 1.5m. In this document, the design of a partially coherent, diffraction-free Pearcey beam is proposed theoretically and validated experimentally. The intensity and degree of coherence of this type of beam remain constant during propagation. Even under practical conditions, the truncated version retains a high resistance to optical diffraction. These beams offer two major advantages: low coherence and diffraction-free propagation. The low coherence ensures the robustness of the beam in harsh environments, while the diffraction-free property allows BE2026 / 7289 14 to exploit degrees of freedom such as coherence and intensity as information carriers for long-distance transmission. Consequently, these beams present strong application potential in the fields of optical communications,the transmission of information and optical imaging, particularly in difficult environments.5 Although the present invention has been described in detail with reference to