A method and system for generating a non-diffracting partially coherent pecton beam

Abstract

The application belongs to the field of light field regulation, and provides a non-diffraction partial coherent Poynting beam generation method and system, which comprises the following steps: the light beam emitted by a laser passes through a half-wave plate to reach a beam expander, and after being expanded by the beam expander, the laser irradiates onto a spatial light modulator; the laser beam is modulated by repeatedly refreshing the spatial light modulator, the pseudo modes of the laser beam at different position points on the reciprocal plane are obtained based on a cross-density spectrum function, and a corresponding holographic picture is generated; after being processed by a 4f optical imaging system, the first-order diffraction spots corresponding to each pseudo mode are screened out; the first-order diffraction spots of each pseudo mode are recorded by a CCD camera, and each pseudo mode is superimposed to form a partial coherent Poynting beam. The method has robustness in the transmission process of the light beam, that is, the influence of diffraction and other disturbances in the transmission process is resisted.

Description

Technical Field

[0001] This invention belongs to the field of optical field manipulation technology, specifically relating to a method and system for generating partially coherent Pierce beams without diffraction. 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] Partially coherent beams possess stable statistical properties, meaning they allow for phase and intensity fluctuations within the beam and over time. Their inherent ability to suppress beam interference and reduce optical noise ensures reliable performance in speckle-free optical imaging, super-resolution optical holography, and high-performance photonic computing applications. Furthermore, based on optical coherence theory, partially coherent beams can be viewed as an incoherent superposition of multiple modes. These beams are renowned for their inherent adaptability to environmental perturbations, as the effects of perturbations on individual modes tend to cancel each other out. This property makes partially coherent beams particularly suitable for critical applications such as optical communication, far-field optical imaging, and optical encryption and decryption in harsh environments.

[0004] As exact solutions to the homogeneous Helmholtz equations, diffractive beams retain their shape invariant as they propagate in free space. Since the pioneering work of Dunning et al., these beams have attracted considerable interest. According to the Babinski principle, all such beams are naturally unaffected by distortion. They have wide applications in volumetric imaging, ghost imaging, particle trapping, photon correlation holography, and factorization. Factorization of partially coherent diffractive beams and the partial coherent beams, both possessing inherent advantages, are receiving increasing attention.

[0005] In recent years, coherence, a unique degree of freedom of partially coherent beams, has sparked extensive research activity ranging from fundamental physics to valuable applications. Notably, coherence has been explored as an information carrier for image encryption; by introducing randomness, partially coherent beams have achieved high-capacity information encryption. However, partially coherent beams with low coherence are susceptible to optical diffraction, leading to the deterioration of all degrees of freedom, including coherence, during propagation. Therefore, encrypted information is inevitably compromised. The fundamental idea to overcome this challenge lies in how to construct partially coherent, diffraction-free beams. Previously constructed partially coherent Pierce beams, due to their self-accelerating nature, experience beam deflection during transmission, which negatively impacts optical encryption of information transmission. Summary of the Invention

[0006] To address the aforementioned issues, this invention proposes a method and system for generating a partially coherent Pierce beam without diffraction. This invention utilizes the principle of incoherent superposition to construct a partially coherent, diffraction-free Pierce beam, the intensity, coherence, and other properties of which remain unchanged during propagation in free space.

[0007] According to some embodiments, a first aspect of the present invention provides a diffraction-free partially coherent Pierce beam generation system, employing the following technical solution:

[0008] A diffraction-free partially coherent Pierce beam generation system includes:

[0009] A laser, the beam emitted by the laser passes through a half-wave plate to a beam expander, and after being expanded by the beam expander, the laser beam illuminates a spatial light modulator.

[0010] The spatial light modulator modulates the laser beam using a cross density spectral function to generate a modulated beam.

[0011] A 4f optical imaging system, wherein the 4f optical imaging system selects the first-order diffraction spot from the modulated beam;

[0012] A CCD camera, which is used to record first-order diffraction spots.

[0013] Furthermore, the 4f optical imaging system includes a first thin lens, an aperture stop, and a second thin lens.

[0014] Furthermore, the spatial light modulator modulates the laser beam using a cross-density spectral function, specifically as follows:

[0015] The pseudo-mode of the laser beam at different locations on the reciprocal plane is obtained using the cross density spectral function;

[0016] Holographic images are generated from pseudo-patterns at different locations on the reciprocal lattice plane based on a complex amplitude modulation algorithm.

[0017] Furthermore, the complex amplitude modulation algorithm specifically includes:

[0018] ;

[0019] The amplitude B was obtained through numerical inversion: ,in, "Arg" represents a first-order Bessel function of the first kind; "Arg" indicates taking the phase of the function. Indicates the grating frequency. For any kernel function, Indicates the position of the sampling point on the inverse plane. and These are the vector positions in the source plane and the reciprocal plane, respectively.

[0020] According to some embodiments, a second aspect of the present invention provides a diffraction-free partially coherent Pierce beam generation method, employing the following technical solution:

[0021] A diffraction-free partially coherent Pierce beam generation method, based on a diffraction-free partially coherent Pierce beam generation system as described in the first embodiment, includes:

[0022] The laser beam emitted by the laser passes through a half-wave plate and reaches the beam expander. After being expanded by the beam expander, the laser beam illuminates the spatial light modulator.

[0023] After repeatedly refreshing the spatial light modulator to modulate the laser beam, the pseudo-mode of the laser beam at different positions on the reciprocal plane is obtained based on the cross density spectrum function, and the corresponding holographic image is generated.

[0024] After processing by the 4f optical imaging system, the first-order diffraction spot corresponding to each pseudo-mode is selected.

[0025] The first-order diffraction spots of each pseudo-mode are recorded using a CCD camera, and the pseudo-modes are superimposed to form a partially coherent Pierce beam.

[0026] Furthermore, the laser beam is modulated by the repeatedly refreshed spatial light modulator, and the pseudo-modes of the laser beam at different positions on the reciprocal plane are obtained based on the cross-density spectral function, and corresponding holographic images are generated, specifically as follows:

[0027] Power spectral density function and arbitrary kernel function are constructed using Gaussian and cosine functions;

[0028] By introducing dimensionless sum and difference coordinates, the mathematical forms of the power spectral density function and arbitrary kernel function are transformed;

[0029] Based on the transformed power spectral density function and arbitrary kernel function, determine the cross spectral density function at a single location point;

[0030] By using Gaussian aperture as the cutoff function of the laser beam, the cutoff cross spectral density function at a single location point is determined.

[0031] The spatial light modulator is repeatedly refreshed to modulate the laser beam, and the pseudo-mode of the laser beam at different positions on the reciprocal plane is obtained by using the truncated cross spectral density function at a single position point.

[0032] Holographic images are generated from pseudo-patterns at different locations on the reciprocal lattice plane based on a complex amplitude modulation algorithm.

[0033] Furthermore, the truncated cross spectral density function for determining a single location point is specifically as follows:

[0034] ;

[0035] in, , This represents the width of the truncation function, where... and This represents the power spectral density function and arbitrary kernel function using dimensionless sum-difference coordinate transformation. This indicates a truncated cross spectrum.

[0036] Furthermore, the method of using a CCD camera to record the first-order diffraction spot of each pseudo-mode and superimposing each pseudo-mode to form a partially coherent Pierce beam specifically involves:

[0037] ;

[0038] in, It is the discrete power spectral density function, representing the mode weights. It is truncation mode. It's a pseudo-mode. Indicates the position of the sampling point on the inverse plane. and The coordinates are on the source plane.

[0039] Furthermore, the truncation mode is specifically as follows:

[0040] ;

[0041] in, The sampling point is represented by its position on the inverse plane, and d represents the sampling space. For any kernel function, and These are the vector positions in the source plane and the reciprocal plane, respectively. This indicates the width of the truncation function.

[0042] Furthermore, the first-order diffraction spot contains pseudo-mode information.

[0043] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0044] This invention utilizes the principle of incoherent superposition to construct a partially coherent, diffraction-free Pierce beam. The intensity, coherence, and other properties of this beam remain unchanged during propagation in free space, exhibiting robustness during transmission, i.e., resisting the effects of diffraction and other disturbances. The intensity and coherence of such beams remain constant during propagation. Even in practical scenarios, the truncated version retains significant resistance to optical diffraction. These beams naturally possess two significant advantages: low coherence and diffraction-free propagation. Low coherence ensures the beam's robustness in harsh environments, while the diffraction-free characteristic allows for the use of certain degrees of freedom, such as coherence and intensity, as an information carrier for long-distance information transmission. Therefore, diffraction-free partially coherent Pierce beams have enormous application potential in optical communication, information transmission, and optical imaging, especially in harsh environments. Attached Figure Description

[0045] 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.

[0046] Figure 1 This is a structural diagram of a diffraction-free partially coherent Pierce beam generation system according to Embodiment 1 of the present invention;

[0047] Figure 2 This is a flowchart of a diffraction-free partially coherent Pierce beam generation method according to Embodiment 2 of the present invention;

[0048] Figure 3 It is the normalized intensity of the truncated, non-diffractive coherent Pierce beam propagating in free space in Embodiment 2 of the present invention;

[0049] Figure 4 It is the sum and coherence of the truncated, non-diffractive coherent Pierce beam propagating in free space in Embodiment 2 of the present invention;

[0050] Figure 5 The intensity similarity (upper coordinate axis) and coherence (lower coordinate axis) of the truncated, non-diffractive coherent Pierce beam propagating in free space in Embodiment 2 of the present invention.

[0051] Figure 6 This is the theoretical and experimental result of the intensity distribution of the non-diffraction partially coherent Pierce beam propagating in free space in Embodiment 2 of the present invention. Detailed Implementation

[0052] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0053] It should be noted that the following detailed description is illustrative and intended to provide further explanation of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0054] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the scope of exemplary embodiments according to the invention. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0055] Where there is no conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.

[0056] Example 1

[0057] like Figure 1 As shown, this embodiment provides a diffraction-free partially coherent Pierce beam generation system, comprising:

[0058] A laser, the beam emitted by the laser passes through a half-wave plate to a beam expander, and after being expanded by the beam expander, the laser beam illuminates a spatial light modulator.

[0059] The spatial light modulator modulates the laser beam using a cross density spectral function to generate a modulated beam.

[0060] A 4f optical imaging system, wherein the 4f optical imaging system selects the first-order diffraction spot from the modulated beam;

[0061] A CCD camera, which is used to record first-order diffraction spots.

[0062] The 4f optical imaging system includes a first thin lens, an aperture stop, and a second thin lens.

[0063] The spatial light modulator modulates the laser beam using a cross density spectral function, specifically:

[0064] The pseudo-mode of the laser beam at different locations on the reciprocal plane is obtained using the cross density spectral function;

[0065] Holographic images are generated from pseudo-patterns at different locations on the reciprocal lattice plane based on a complex amplitude modulation algorithm.

[0066] The complex amplitude modulation algorithm is specifically as follows:

[0067] ;

[0068] The amplitude B was obtained through numerical inversion: ,in, "Arg" represents a first-order Bessel function of the first kind; "Arg" indicates taking the phase of the function. Indicates the grating frequency. For any kernel function, Indicates the position of the sampling point on the inverse plane. and These are the vector positions in the source plane and the reciprocal plane, respectively.

[0069] Optical devices such as Figure 1 As shown. Wavelength is The beam is emitted from a helium-neon laser, passes through a half-wave plate, and is expanded by a beam expander. The direction of the half-wave plate is adjusted to horizontally align the polarization direction of the emitted beam for optimal SLM response. After beam expansion, the beam illuminates a pure-phase liquid crystal spatial light modulator (SLM), and a complex amplitude modulation coding algorithm is used to design each pseudo-mode as a hologram and load it onto the SLM. After reflection by the SLM, the beam passes through an improved 4f optical imaging system consisting of two identical lenses and an aperture. The 4f optical imaging system is used to selectively filter out the first-order diffraction spots, which contain information necessary for this embodiment; the other diffraction orders are not needed. An aperture located in the frequency plane is used to select positive or negative first-order diffraction spots; the selected first-order diffraction spots contain information about each pseudo-mode. Each pseudo-mode is recorded by a CCD camera. Later, the pseudo-modes recorded by the CCD are superimposed to form a partially coherent Pierce beam.

[0070] Spatial light modulators utilize complex amplitude modulation (SLM) algorithms to program each mode into a form that the SLM can use to control the beam, which is essentially a hologram loaded onto the SLM.

[0071] Based on the complex amplitude modulation algorithm, the SLM phase corresponding to the electric field to be encoded is represented as:

[0072] (1);

[0073] The amplitude B was obtained through numerical inversion: ,in, "Arg" represents a first-order Bessel function of the first kind; "Arg" indicates taking the phase of the function. This indicates the grating frequency.

[0074] Example 2

[0075] like Figure 2 As shown, this embodiment provides a diffraction-free partially coherent Pierce beam generation method, based on a diffraction-free partially coherent Pierce beam generation system as described in Embodiment 1, comprising:

[0076] The laser beam emitted by the laser passes through a half-wave plate and reaches the beam expander. After being expanded by the beam expander, the laser beam illuminates the spatial light modulator.

[0077] After repeatedly refreshing the spatial light modulator to modulate the laser beam, the pseudo-mode of the laser beam at different positions on the reciprocal plane is obtained based on the cross density spectrum function, and the corresponding holographic image is generated.

[0078] After processing by the 4f optical imaging system, the first-order diffraction spot corresponding to each pseudo-mode is selected.

[0079] The first-order diffraction spots of each pseudo-mode are recorded using a CCD camera, and the pseudo-modes are superimposed to form a partially coherent Pierce beam.

[0080] The laser beam is modulated by a repeatedly refreshed spatial light modulator. Based on the cross-density spectral function, the pseudo-modes of the laser beam at different positions on the reciprocal plane are obtained, and corresponding holographic images are generated. Specifically:

[0081] Power spectral density function and arbitrary kernel function are constructed using Gaussian and cosine functions;

[0082] By introducing dimensionless sum and difference coordinates, the mathematical forms of the power spectral density function and arbitrary kernel function are transformed;

[0083] Based on the transformed power spectral density function and arbitrary kernel function, determine the cross spectral density function at a single location point;

[0084] Using the Gaussian aperture as the cutoff function of the beam, the cutoff cross spectral density function at a single location point is determined;

[0085] The spatial light modulator is repeatedly refreshed to modulate the laser beam, and the pseudo-mode of the laser beam at different positions on the reciprocal plane is obtained by using the truncated cross spectral density function at a single position point.

[0086] Holographic images are generated from pseudo-patterns at different locations on the reciprocal lattice plane based on a complex amplitude modulation algorithm.

[0087] In the spatial frequency domain, the statistical properties of partially coherent light are characterized by the cross-spectral density function. Its content is as follows:

[0088] (1);

[0089] in, Represents the power spectral density function. For any kernel function. and These are the vector positions in the source plane and the reciprocal plane, respectively. To construct a diffraction-free, partially coherent Pierce beam, we adopt the following form:

[0090] (2);

[0091] (3);

[0092] in and These represent the coherence width and beam width, respectively. Here we introduce dimensionless sum and difference coordinates.

[0093] (4);

[0094] Substituting equations (2)-(4) into equation (1), the cross spectral density function of the desired beam at a single location point can be rewritten as follows:

[0095] (5);

[0096] here:

[0097] (6);

[0098] (7);

[0099] in It is the Pierce function, defined as , This determines the global coherence of the light source beam. The cross-spectral density function depends on only one parameter. . It is a concavity / convexity function on the background, the latter being characterized by The propagation behavior of partially coherent light can be studied using Fresnel diffraction integrals, as shown below:

[0100] (8);

[0101] Substituting equation (5-7) into equation (8) and converting it to dimensionless sum and difference coordinates, through direct integration, we prove that the cross-spectral function of the diffraction-free partially coherent Pierce beam is independent of the propagation distance z, i.e. This clearly demonstrates that this beam of light is unaffected by optical diffraction.

[0102] However, this beam exhibits an infinite outline, symbolizing infinite energy, which is physically impossible. To address this issue, a Gaussian aperture is introduced as the beam cutoff function. The truncated version of the cross spectral density function is then rewritten as a truncated cross spectral density function at a single location point, specifically:

[0103] (9);

[0104] in , This indicates the width of the truncation function.

[0105] Its evolutionary behavior is as follows Figure 3 and Figure 4 As shown, the width of the truncation function is set to The total propagation distance is 100m, spanning from the left plane to the right plane. The results vividly demonstrate the intensity (e.g., Figure 3 (as shown) and coherence (as shown) Figure 4 (As shown) remains almost constant throughout the propagation. Each figure shows three planes, corresponding to propagation distances at z=0, 50m, and 100m, respectively.

[0106] To quantitatively describe the ability of a light beam to resist optical diffraction, we introduce a similarity metric to evaluate the similarity of the intensity (or coherence) distributions of the receiving and source planes. The similarity metric is defined as follows:

[0107] (10);

[0108] in, This represents the coherence or intensity distribution at a propagation distance z. This represents the coherence or intensity distribution at the source plane. Its value is in the range [0, 1], and the larger the value, the stronger the immunity to light diffraction.

[0109] like Figure 5 As shown, similarity curves for intensity (upper axis) and coherence (lower axis) are plotted. It can be observed that although the curve decreases slightly with increasing propagation distance, the similarity remains around 0.98 at a propagation distance z=100m, demonstrating the beam's strong resistance to optical diffraction. Furthermore, the effect of the only varying parameter, global coherence, on the similarity of intensity and coherence was investigated. The results show that, apart from a slight change in coherence, both are largely unaffected by this parameter. This indicates that even with low coherence, the constructed beam still possesses significant diffraction resistance. This effectively addresses the traditional dilemma that lower beam coherence typically leads to greater diffraction and more pronounced optical field distortion.

[0110] To further investigate the beam characteristics of truncated, non-diffractive coherent Pearcey beams and to facilitate experimental setup, the pseudomode superposition principle is used to represent such beams.

[0111] The first-order diffraction spots of each pseudo-mode are recorded using a CCD camera, and the pseudo-modes are superimposed to form a partially coherent Pierce beam, specifically:

[0112] (11);

[0113] in, It is the discrete power spectral density function, representing the mode weights. It is a truncation mode (or pseudo-mode), defined as:

[0114] (12);

[0115] here Let d represent the position of the sampling point on the inverse plane, d represent the sampling space, and N represent the total number of sampling points along the horizontal or vertical direction. Since free space is a linear system, the propagation behavior of a truncated, non-diffractive, partially coherent Pearcey beam can be evaluated by using the theory of light wave propagation and through the incoherent superposition of all pseudomodes propagating to the receiving plane.

[0116] The experimental results agree well with the theoretical results, proving the feasibility of the proposed method. Furthermore, we experimentally investigated the propagation behavior of the beam in free space, and the relevant results are as follows: Figure 6 As shown. Experimental results show that the beam intensity does not change with beam propagation. The experimental results agree well with theoretical predictions, proving that the partially coherent beam we constructed has strong resistance to optical diffraction. The relevant parameters were set as follows: =1, from left to right, the transmission distance z = 0.5m, 1m, 1.5m.

[0117] This paper theoretically proposes and experimentally advances the customization of diffraction-free partially coherent Pierce beams. The intensity and coherence of such beams remain constant during propagation. Even in practical scenarios, the truncated version retains significant resistance to optical diffraction. These beams naturally possess two significant advantages: low coherence and diffraction-free propagation. Low coherence ensures the beam's robustness in harsh environments, while the diffraction-free characteristic allows for the use of certain degrees of freedom, such as coherence and intensity, as information carriers for long-distance information transmission. Therefore, diffraction-free partially coherent Pierce beams have enormous application potential in optical communication, information transmission, and optical imaging, especially in harsh environments.

[0118] 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. A diffractionless, partially coherent, Poynting beam generating system characterized by, The application relates to a laser beam modulation method and device. The laser beam emitted by a laser passes through a half-wave plate to reach an expander, and the laser irradiates a spatial light modulator after being expanded by the expander; The spatial light modulator modulates the laser beam by using a cross-density spectrum function to generate a modulated light beam, specifically as follows: Pseudo modes of different position points of the laser beam on a reciprocal plane are obtained by using the cross-density spectrum function, specifically as follows: A Gaussian function and a cosine function are used to construct a power spectrum density function and an arbitrary kernel function; Dimensionless and difference coordinates are introduced to convert the mathematical forms of the power spectrum density function and the arbitrary kernel function; A cross-spectrum density function of a single position point is determined based on the converted power spectrum density function and the arbitrary kernel function; A Gaussian aperture is used as a truncation function of the laser beam to determine a truncated cross-spectrum density function of the single position point; The pseudo modes of the different position points of the laser beam on the reciprocal plane are obtained by repeatedly refreshing the spatial light modulator to modulate the laser beam and by using the truncated cross-spectrum density function of the single position point; A holographic picture is generated by using a complex amplitude regulation algorithm based on the pseudo modes of the different position points on the reciprocal plane; A 4f optical imaging system selects a first-order diffraction spot of the modulated light beam. A CCD camera is used to record the first-order diffraction spot.

2. A diffractionless partially coherent Pearcey beam generation system as claimed in claim 1, wherein, The 4f optical imaging system comprises a first thin lens, an aperture and a second thin lens.

3. A diffractionless partially coherent Pearcey beam generation system as claimed in claim 1, wherein, The complex amplitude regulation algorithm is specifically as follows: ； where B amplitude is obtained by numerical inversion, where, denotes the first kind of first order Bessel function; "Arg" denotes the phase of the function; denotes the grating frequency, is an arbitrary kernel function, denotes the position of the sampling point on the back plane, d denotes the sampling space, and are the vector positions in the source plane and the back plane, respectively.

4. A method for generating a non-diffracting partially coherent Pearcey beam based on a system for generating a non-diffracting partially coherent Pearcey beam according to any one of claims 1-3, characterized in that, The laser beam emitted by a laser passes through a half-wave plate to reach an expander, and the laser irradiates a spatial light modulator after being expanded by the expander; After the spatial light modulator is repeatedly refreshed to modulate the laser beam, pseudo modes of different position points of the laser beam on a reciprocal plane are obtained based on a cross-density spectrum function, and corresponding holographic pictures are generated, specifically as follows: A Gaussian function and a cosine function are used to construct a power spectrum density function and an arbitrary kernel function; Dimensionless and difference coordinates are introduced to convert the mathematical forms of the power spectrum density function and the arbitrary kernel function; A cross-spectrum density function of a single position point is determined based on the converted power spectrum density function and the arbitrary kernel function; A Gaussian aperture is used as a truncation function of the laser beam to determine a truncated cross-spectrum density function of the single position point; The pseudo modes of the different position points of the laser beam on the reciprocal plane are obtained by repeatedly refreshing the spatial light modulator to modulate the laser beam and by using the truncated cross-spectrum density function of the single position point; A holographic picture is generated by using a complex amplitude regulation algorithm based on the pseudo modes of the different position points on the reciprocal plane; After being processed by a 4f optical imaging system, a first-order diffraction spot corresponding to each pseudo mode is selected; A CCD camera is used to record the first-order diffraction spot of each pseudo mode, and each pseudo mode is superimposed to form a partially coherent Poynting beam. The determination of the truncated cross-spectrum density function of the single position point is specifically as follows:

5. A method of generating a non-diffracting partially coherent Pearcey beam as claimed in claim 4, wherein, The CCD camera is used to record the first-order diffraction spot of each pseudo mode, and each pseudo mode is superimposed to form a partially coherent Poynting beam, specifically as follows: ； wherein, , denotes the beam width, denotes the width of the truncation function, wherein, and denotes the power spectral density function and an arbitrary kernel function transformed with dimensionless and difference coordinates, denotes the truncated cross spectrum.

6. A method of generating a non-diffracting partially coherent Pearcey beam as claimed in claim 4, wherein, The truncated mode is specifically as follows: ； wherein, is a discrete power spectral density function, representing the mode weights, is a truncated mode, is a pseudo mode, denotes the position of the sample point on the inverse plane, d denotes the sampling space, and are the coordinates on the source plane.

7. A method of generating a non-diffracting partially coherent Pearcey beam as claimed in claim 6, wherein, The first-order diffraction spot contains information of the pseudo mode. ； where denotes the position of the sample point on the inverse plane, d denotes the sampling space, is an arbitrary kernel function, and are the vector positions in the source plane and the reciprocal plane, respectively, denotes the width of the truncation function.

8. A method of generating a non-diffracting partially coherent Pearcey beam as claimed in claim 4, wherein, ​

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