A method for propagating Hermitian elliptical vortex beams in a gradient refractive index medium

Abstract

This invention discloses a method for propagating a Hermitian elliptical vortex beam in a gradient refractive index medium, comprising: designing a controllable anomalous Hermitian elliptical vortex beam on an initial plane; constructing an electric field expression for the controllable anomalous Hermitian elliptical vortex beam at any propagation distance based on the Collins formula and the controllable anomalous Hermitian elliptical vortex beam; constructing an electric field expression for the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium based on the electric field expression for the controllable anomalous Hermitian elliptical vortex beam at any propagation distance, thereby obtaining the light intensity; adjusting the controllable anomalous Hermitian elliptical vortex beam and the gradient refractive index coefficient in the gradient refractive index medium to obtain different light intensity distributions of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium, thereby obtaining light spots of different shapes and spatial sizes. This invention can realize the propagation of the beam in a gradient refractive index medium and achieve effective control of different light intensity distribution shapes and spatial sizes of the beam.

Description

Technical Field

[0001] This invention relates to the field of optical technology, and in particular to a method for transmitting Hermitian elliptical vortex beams in a gradient refractive index medium. Background Technology

[0002] With the continuous development of modern technology, the transmission and manipulation of laser beams have attracted widespread attention, and the transmission of lasers in various media (such as uniaxial crystals, nonlocal media, and gradient refractive index media) has become a very active topic. Among these, gradient refractive index media refer to non-uniform media with a spatially gradient distribution of refractive index, thereby enabling light manipulation. Since the introduction of gradient refractive index media, the influence of their non-uniform refractive index on light transmission has become a widely studied topic. Currently, gradient refractive index media have shown broad application value in endoscope systems, optical information processing, optical lenses, and optical communication devices.

[0003] Beams with non-Gaussian intensity distributions can exhibit unique intensity profiles. Among these, anomalously hollow beams, which can possess ring-shaped distributions and central intensities, have garnered significant attention in fields such as optical tweezers and nonlinear optics. Building upon anomalously hollow beams, Hermitian functions have been introduced into optical field models in recent years, leading optical researchers to propose anomalously hollow Hermitian vortex beams (Optical and Quantum Electronics, 57, 2, 2025) and controllable anomalously hollow Hermitian vortex beams (Optical and Quantum Electronics, 56, 1653, 2024). This increases the dimensionality of optical field manipulation and enriches the controllable distribution patterns. However, the vortex phases in these beams exhibit circular symmetry, limiting beam manipulation. Furthermore, research on beam transmission manipulation primarily focuses on controlling beam parameters, and studies on the influence of transmission media on optical field manipulation mainly utilize environments such as circularly symmetric vortex phases and free space. Therefore, gradient refractive index media still present certain limitations in controlling controllable anomalous vortex beams. Summary of the Invention

[0004] This invention provides a method for propagating Hermitian elliptical vortex beams in a gradient refractive index medium, thereby overcoming the technical problem of uncontrollable and unadjustable beam shape propagation in a gradient refractive index medium.

[0005] To achieve the above objectives, the technical solution of the present invention is as follows:

[0006] A method for propagating a Hermitian elliptical vortex beam in a gradient refractive index medium includes:

[0007] S1: Expression of the electric field of a controllable anomalous Hermitian elliptical vortex beam on the initial plane based on the design of Hermitian elliptical vortex beam;

[0008] S2: Introducing the Collins formula, and constructing the electric field expression for the controllable anomalous Hermitian elliptical vortex beam at any transmission distance based on the Collins formula and the controllable anomalous Hermitian elliptical vortex beam;

[0009] S3: Introduce a gradient refractive index medium. Based on the electric field expression of the gradient refractive index medium and the controllable anomalous Hermitian elliptical vortex beam at any transmission distance, construct the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium.

[0010] S4: Obtain the light intensity of a controllable anomalous Hermitian elliptical vortex beam at any position z in a gradient refractive index medium based on the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in a gradient refractive index medium.

[0011] S5: Adjust the controllable anomalous Hermitian elliptical vortex beam and the gradient refractive index coefficient in the gradient refractive index medium to obtain different light intensity distributions of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium, thereby obtaining light spots of different shapes and spatial sizes, and realizing the transmission of the Hermitian elliptical vortex beam in the gradient refractive index medium.

[0012] Furthermore, based on the design of a controllable anomalous Hermitian elliptical vortex beam electric field expression on the initial plane, the following is provided:

[0013] S11. The electric field expression of the controllable anomalous Hermitian elliptical vortex beam in the initial light field at the initial plane z = 0 is shown in Equation (1).

[0014]

[0015] Where E0(x0,y0) is the electric field expression at the initial plane z=0; x0 is the coordinate value of the x-axis in the rectangular coordinate system, y0 is the coordinate value of the y-axis in the rectangular coordinate system, and C0 is a constant; a, c x and c y w is the controllability coefficient of the design. 0x w is the beam waist width of the Gaussian light along the x-axis. 0y H represents the beam waist width of the Gaussian beam on the y-axis, i is the imaginary unit, M is the topological charge number of the designed elliptical vortex phase; s H is an s-order Hermitian polynomial. n Let be an nth-order Hermitian polynomial, as shown in formula (2).

[0016]

[0017] In the formula, t is the coefficient of the Hermitian polynomial, which is a non-negative integer, and l is an intermediate variable for summation.

[0018] Furthermore, the Collins formula is introduced, and an electric field expression for an arbitrary transmission distance of a controllable anomalous Hermitian elliptical vortex beam is constructed based on the Collins formula and the controllable anomalous Hermitian elliptical vortex beam, including:

[0019] By introducing the Collins formula and substituting equation (1) into the Collins formula, and through integration, we obtain the electric field expression for the controllable anomalous Hermitian elliptical vortex beam at any transmission distance z, as shown in equation (3).

[0020]

[0021] Where x and y represent the x and y coordinates in a Cartesian coordinate system at a transmission distance z. k represents the wave number, λ is the wavelength, m is the numerical value of the topological charge of the elliptical vortex phase, and l x l y B and D are constants, and B and D are two matrix elements in the Collins formula; E1, E2, and E3 are the three components of the electric field expression, as shown in formulas (4)-(6).

[0022]

[0023] In the formula, p x and p y These are intermediate variables in the calculation process, as shown in formulas (7) and (8).

[0024]

[0025]

[0026] In the formula, A is one of the matrix elements in Collins' formula.

[0027] Furthermore, a gradient refractive index medium is introduced. Based on the electric field expression for the gradient refractive index medium and the controllable anomalous Hermitian elliptical vortex beam over arbitrary propagation distances, an electric field expression for the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium is constructed, including:

[0028] S31. Introduce a gradient refractive index medium and obtain the expression for the matrix element in the Collins formula in the gradient refractive index medium, as shown in formula (9).

[0029]

[0030] β is the coefficient of the gradient refractive index medium; z is the transmission distance;

[0031] S32. Substituting formula (9) into formulas (3)-(8), we obtain the electric field expression for the controllable anomalous elliptical vortex beam in a gradient refractive index medium, as shown in formula (10).

[0032]

[0033] Among them, E g (x,y,z) is the electric field expression for a controllable anomalous elliptical vortex beam in a gradient refractive index medium; E 1g E 2g E 3g The three components of the electric field expression in a gradient refractive index medium are shown in equations (11)-(13).

[0034]

[0035]

[0036] p xg and p yg These are intermediate variables in the calculation process, as shown in formulas (14) and (15).

[0037]

[0038] Furthermore, the light intensity of the controllable anomalous Hermitian elliptical vortex beam at any position z in the gradient refractive index medium is obtained based on the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium, as shown in Equation (16).

[0039] I g (x,y,z)=|E g (x,y,z)| 2 (16)

[0040] In the formula, I g (x,y,z) represents the light intensity of a controllable anomalous Hermitian elliptical vortex beam at any position z in a gradient refractive index medium.

[0041] Furthermore, by adjusting the controllable anomalous Hermitian elliptical vortex beam and the gradient refractive index coefficient in the gradient refractive index medium, different intensity distributions of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium are obtained, thereby generating light spots of different shapes and spatial sizes, realizing the transmission of the Hermitian elliptical vortex beam in the gradient refractive index medium, including:

[0042] Set the controllability coefficients a and c of the controllable anomalous Hermitian elliptical vortex beam. x , C y By combining the gradient refractive index coefficient β, light intensity distributions with different spatial distributions and sizes can be obtained, thereby obtaining light spots with different shapes and spatial sizes, and realizing the transmission of Hermitian elliptical vortex beams in gradient refractive index media.

[0043] Beneficial effects: This invention provides a method for transmitting a Hermitian elliptical vortex beam in a gradient refractive index medium. It constructs a controllable anomalous Hermitian elliptical vortex beam and, by jointly selecting the parameters of the controllable anomalous Hermitian elliptical vortex beam and the parameters of the gradient refractive index medium, realizes the transmission of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium, and achieves effective control over the different intensity distribution patterns and spatial dimensions of the controllable anomalous Hermitian elliptical vortex beam. Attached Figure Description

[0044] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0045] Figure 1 This is a flowchart of a method for realizing the transmission of a controllable anomalous Hermitian elliptical vortex beam in a gradient refractive index medium provided by the present invention;

[0046] Figure 2 Intensity distribution diagram of a controllable anomalous Hermitian elliptical vortex beam at plane z=0 in an embodiment of the present invention;

[0047] Figure 3 Intensity distribution diagram of a controllable anomalous Hermitian elliptical vortex beam in a gradient refractive index medium in an embodiment of the present invention;

[0048] Figure 4 The light intensity distribution diagram of a controllable anomalous Hermitian elliptical vortex beam in media with different gradient refractive indices at a distance of z = 2m in this embodiment of the invention. Detailed Implementation

[0049] To make the objectives, technical solutions, and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention without making any creative efforts shall fall within the scope of protection of the present invention.

[0050] This embodiment provides a method for propagating Hermitian elliptical vortex beams in a gradient refractive index medium, such as... Figure 1 As shown, including:

[0051] S1: Expression of the electric field of a controllable anomalous Hermitian elliptical vortex beam on the initial plane based on the design of Hermitian elliptical vortex beam;

[0052] S2: Introducing the Collins formula, and constructing the electric field expression for the controllable anomalous Hermitian elliptical vortex beam at any transmission distance based on the Collins formula and the controllable anomalous Hermitian elliptical vortex beam;

[0053] S3: Introduce a gradient refractive index medium. Based on the electric field expression of the gradient refractive index medium and the controllable anomalous Hermitian elliptical vortex beam at any transmission distance, construct the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium.

[0054] S4: Obtain the light intensity of a controllable anomalous Hermitian elliptical vortex beam at any position z in a gradient refractive index medium based on the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in a gradient refractive index medium.

[0055] S5: Adjust the controllable anomalous Hermitian elliptical vortex beam and the gradient refractive index coefficient in the gradient refractive index medium to obtain different light intensity distributions of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium, thereby obtaining light spots of different shapes and spatial sizes, and realizing the transmission of the Hermitian elliptical vortex beam in the gradient refractive index medium.

[0056] Specifically, firstly, based on the design of the Hermitian elliptical vortex beam, a controllable anomalous Hermitian elliptical vortex beam electric field expression on the initial plane is designed, which can transform the traditional vortex beam into a designable anomalous characteristic, enabling dynamic optical field modulation, and allowing the beam to propagate in a gradient refractive index medium.

[0057] Secondly, the Collins formula is introduced, and an electric field expression for the controllable anomalous Hermitian elliptical vortex beam at any transmission distance is constructed based on the Collins formula and the controllable anomalous Hermitian elliptical vortex beam. By introducing the controllable anomalous Hermitian elliptical vortex beam into the Collins formula, the electric field expression for the transmission of the beam is obtained, which provides a basis for obtaining the electric field expression of the beam in the gradient refractive index medium.

[0058] Furthermore, a gradient refractive index medium is introduced. Based on the electric field expression of the gradient refractive index medium and the controllable anomalous Hermitian elliptical vortex beam at any transmission distance, an electric field expression for the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium is constructed. Based on the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium, the light intensity of the controllable anomalous Hermitian elliptical vortex beam at any position z in the gradient refractive index medium is obtained. The obtained electric field expression is the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium. Based on this electric field expression, the basic characteristics of the light field can be analyzed, providing a basis for subsequent control of the light intensity of the beam in the gradient refractive index medium. By adjusting the beam parameters and the gradient refractive index parameters, the intensity distribution pattern and spatial size of the beam can be controlled.

[0059] Finally, by adjusting the controllable anomalous Hermitian elliptical vortex beam and the gradient refractive index coefficient in the gradient refractive index medium, different intensity distributions of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium are obtained, thereby obtaining light spots of different shapes and spatial sizes, and realizing the transmission of the Hermitian elliptical vortex beam in the gradient refractive index medium.

[0060] In a specific embodiment, the scheme for designing a controllable anomalous Hermitian elliptical vortex beam electric field expression on the initial plane based on the Hermitian elliptical vortex beam is as follows:

[0061] The electric field expression for the controllable anomalous Hermitian elliptical vortex beam in the initial light field at the initial plane z = 0 is shown in Equation (17).

[0062]

[0063] Where E0(x0,y0) is the electric field expression at the initial plane z=0; x0 is the coordinate value of the x-axis in the rectangular coordinate system, y0 is the coordinate value of the y-axis in the rectangular coordinate system, and C0 is a constant; a, c x and c y w is the controllability coefficient of the design. 0x w is the beam waist width of the Gaussian light along the x-axis. 0y H represents the beam waist width of the Gaussian beam on the y-axis, i is the imaginary unit, M is the topological charge number of the designed elliptical vortex phase; s For, H n It is a Hermitian polynomial, as shown in formula (18).

[0064]

[0065] In the formula, t is the coefficient of the Hermitian polynomial, which is a non-negative integer, and l is an intermediate variable for summation.

[0066] The vortex phases in existing anomalous hollow Hermitian vortex beams and controllable anomalous hollow Hermitian vortex beams exhibit circular symmetry, maintaining their intensity distribution symmetry throughout propagation. However, when a refractive index gradient exists in the medium, different parts of the beam experience different phase delays, resulting only in simple beam focusing. This approach fails to simultaneously achieve controllable beam modulation in terms of intensity distribution symmetry and spatial dimensions. Therefore, this solution introduces an elliptical vortex phase into the controllable anomalous Hermitian beam to construct a novel elliptical vortex beam, namely the controllable anomalous Hermitian elliptical vortex beam of this invention. This transforms the traditional vortex beam into a designable anomalous beam, enabling dynamic optical field modulation and controllable propagation of the beam in gradient refractive index media.

[0067] In a specific embodiment, the Collins formula is introduced, and the scheme for constructing the electric field expression for an arbitrary transmission distance of a controllable anomalous Hermitian elliptical vortex beam based on the Collins formula and the controllable anomalous Hermitian elliptical vortex beam is as follows:

[0068] Introducing the Collins formula, as shown in formula (19),

[0069]

[0070] Substituting equation (17) into the Collins equation and performing integration, we obtain the electric field expression for the controllable anomalous Hermitian elliptical vortex beam at any transmission distance z, as shown in equation (20).

[0071]

[0072] Where x and y represent the x-axis and y-axis coordinates in a Cartesian coordinate system at a transmission distance z. k represents the wave number, λ is the wavelength, m is the numerical value of the topological charge of the elliptical vortex phase, and l x l y B and D are constants, and B and D are two matrix elements in the Collins formula; E1, E2, and E3 are the three components of the electric field expression, as shown in formulas (21)-(23).

[0073]

[0074] In the formula, p x and p y These are intermediate variables in the calculation process, as shown in formulas (24) and (25).

[0075]

[0076]

[0077] In the formula, A is one of the matrix elements in Collins' formula.

[0078] In this scheme, by introducing a controllable anomalous Hermitian elliptical vortex beam into the Collins formula, the electric field expression for the propagation of the beam is obtained, which provides a basis for subsequently obtaining the electric field expression of the beam in a gradient refractive index medium.

[0079] In a specific embodiment, a gradient refractive index medium is introduced. Based on the electric field expression of the gradient refractive index medium and the controllable anomalous Hermitian elliptical vortex beam over any propagation distance, the scheme for constructing the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium is as follows:

[0080] S31. Introducing a gradient refractive index medium, we obtain the expression for the matrix elements in the Collins formula in the gradient refractive index medium, as shown in formula (26).

[0081]

[0082] β is the coefficient of the gradient refractive index medium; z is the transmission distance;

[0083] S22. Substituting formula (26) into formulas (20)-(25), we obtain the electric field expression for a controllable anomalous elliptical vortex beam in a gradient refractive index medium, as shown in formula (27).

[0084]

[0085] Among them, E g (x,y,z) is the electric field expression for a controllable anomalous elliptical vortex beam in a gradient refractive index medium; E 1g E 2g E 3g The three components of the electric field expression in a gradient refractive index medium are shown in equations (28)-(30).

[0086]

[0087] p xg and p yg These are intermediate variables used in the intermediate calculation process, as shown in formulas (31) and (32).

[0088]

[0089] In this scheme, the obtained electric field expression is the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in a gradient refractive index medium. Based on this electric field expression, the basic characteristics of the light field can be analyzed, providing a basis for subsequent control of the light intensity of the beam in the gradient refractive index medium.

[0090] In a specific embodiment, the light intensity of the controllable anomalous Hermitian elliptical vortex beam at any position z in the gradient refractive index medium is obtained based on the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium, as shown in formula (33).

[0091] I g (x,y,z)=|E g (x,y,z)| 2 (33)

[0092] In the formula, I g (x,y,z) represents the light intensity of a controllable anomalous Hermitian elliptical vortex beam at any position z in a gradient refractive index medium.

[0093] In this scheme, the intensity expression of a controllable anomalous Hermitian elliptical vortex beam in a gradient refractive index medium can be obtained. By adjusting the beam parameters and gradient refractive index parameters, the intensity distribution pattern and spatial size of the beam can be controlled.

[0094] In a specific embodiment, by adjusting the controllable anomalous Hermitian elliptical vortex beam and the gradient refractive index coefficient in the gradient refractive index medium, different intensity distributions of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium are obtained, thereby obtaining light spots of different shapes and spatial sizes. The scheme for realizing the transmission of the Hermitian elliptical vortex beam in the gradient refractive index medium is as follows:

[0095] Set the controllability coefficients a and c of the controllable anomalous Hermitian elliptical vortex beam. x c y By combining the gradient refractive index coefficient β, light intensity distributions with different spatial distributions and sizes can be obtained, thereby obtaining light spots with different shapes and spatial sizes, and realizing the transmission of Hermitian elliptical vortex beams in gradient refractive index media.

[0096] Because a controllable anomalous Hermitian elliptical vortex beam has multiple controllable parameters, the intensity distribution at the initial plane z=0 can be modulated by adjusting these parameters. The influence of multiple parameters on the controllable anomalous Hermitian elliptical vortex beam at the initial plane z=0 is analyzed in detail below. The characteristic parameters of the beam are... Figure 2 The subgraph is set as follows: Figure 2 In (a), a=2, c x =c y =8, M=1, n=s=1, w 0x =w 0y =2mm; Figure 2 In (b), a = 2, c x =c y =8, M=1, n=s=1, w 0x =1.5mm, w 0y =2mm; Figure 2 In (c), a = 2, c x =c y =8, M=1, n=s=1, w 0x =2mm, w 0y =1.5mm; Figure 2 In (d), a = 2, c x =1,c y =10, M=2, n=s=2, w 0x =2mm, w 0y =1.5mm; Figure 2 In (e), a = 2, c x =c y =8, M=1, n=2, s=0, w 0x =2mm, w0y =1.5mm; Figure 2 In (f), a = -2, c x =c y =8, M=2, n=s=0, w 0x =2mm, w 0y =1.5mm. When the w of the controllable anomalous Hermitian elliptical vortex beam 0x =w 0y When n = s, the light intensity of the beam exhibits a symmetrical petal-like distribution, such as... Figure 2 (a). If the light source w 0x <w 0y The light intensity distribution decreases along the x-axis, such as... Figure 2 (b); Conversely, if w 0x >w 0y The light intensity distribution decreases along the y-axis, such as... Figure 2 (c). With c x and c y With this adjustment, the symmetry of the light intensity distribution will also be lost, such as... Figure 2 (d). When n=2 and s=0 are selected, the beam will appear as two lobes along the x-axis, as shown below. Figure 2 (e) When n = s = 0, the beam will become an elliptical hollow beam. Therefore, by selecting different parameters, the beam can obtain different intensity distributions.

[0097] Figure 3 This is a light intensity distribution diagram of a controllable anomalous Hermitian elliptical vortex beam in a gradient refractive index medium at different transmission distances, with the parameters selected as: a = 2, c x =1,c y =8, M=1, n=2, s=1, w 0x =1.5mm, w 0y =2mm, λ=800nm, β=2, and introduce L=2πβ. Figure 3 The transmission distances in (a)-(f) are z = 0.15L, z = 0.25L, z = 0.5L, z = 0.7L, z = 0.75L, and z = 0.9L, respectively. When a controllable anomalous Hermitian elliptical vortex beam propagates in a gradient refractive index medium, the spatial dimensions of the light intensity distribution exhibit periodic changes with increasing transmission distance. Within a certain transmission distance, the light intensity distribution remains unchanged (e.g., ...). Figure 3 In (a), (c), (d), (f), however, changes in the spatial size period of the beam can cause focusing changes. Figure 3 (b) and (e)). Therefore, by controlling different transmission distances, the beam can achieve different spatial dimensions.

[0098] Figure 4This is a light intensity distribution diagram of a controllable anomalous Hermitian elliptical vortex beam in media with different refractive index gradients at z = 2m, with parameters selected as: a = 1, c x =8, c y =8, M=1, n=s=2, w 0x =1.5mm, w 0y =2mm, λ=800nm, z=2m. Figure 4 In (a)-(d), β is β=1, β=1.5, β=2, and β=3, respectively. When a controllable anomalous Hermitian elliptical vortex beam propagates in media with different gradient refractive indices, at a fixed propagation distance z=2m, the spatial size of the beam will be affected by β, and the spatial size of the beam can be reduced (e.g., ...). Figure 4 (b) or broaden ( Figure 4 (d) Therefore, for the same transmission distance z, the beam can achieve different spatial dimensions by setting β.

[0099] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for propagating a Hermitian elliptical vortex beam in a gradient refractive index medium, characterized in that, include: S1: The electric field expression of a controllable anomalous Hermitian elliptical vortex beam on the initial plane based on the design of a Hermitian elliptical vortex beam is as follows: The electric field expression for the controllable anomalous Hermitian elliptical vortex beam in the initial light field at the initial plane z=0 is shown in Equation (1). （1） in, Here is the expression for the electric field at the initial plane z=0; In a rectangular coordinate system The coordinate values ​​of the axis. In a rectangular coordinate system The coordinate values ​​of the axis. It is a constant; , and The controllability coefficient of the design; For Gaussian light in The waist width on the axis, For Gaussian light in The waist width on the axis, The imaginary unit, The topological charge number for the designed elliptical vortex phase; It is an s-order Hermitian polynomial. for The Hermitian polynomial of order 1 is shown in formula (2). （2） In the formula, denoted as the coefficients of the Hermitian polynomial, and denoted as a non-negative integer. For the intermediate variable in the summation; S2: Introducing the Collins formula, and constructing the electric field expression for the controllable anomalous Hermitian elliptical vortex beam at any transmission distance based on the Collins formula and the controllable anomalous Hermitian elliptical vortex beam; S3: Introducing a gradient refractive index medium, and based on the gradient refractive index coefficient of the gradient refractive index medium and the electric field expression of the controllable anomalous Hermitian elliptical vortex beam at any transmission distance, constructing the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium. S4: Obtain the light intensity of a controllable anomalous Hermitian elliptical vortex beam at any position z in a gradient refractive index medium based on the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in a gradient refractive index medium. S5: Adjusting the controllability coefficient of a controllable anomalous Hermitian elliptical vortex beam in a gradient refractive index medium , , By combining the gradient refractive index coefficient, we can obtain different intensity distributions of controllable anomalous Hermitian elliptical vortex beams in gradient refractive index media, thereby obtaining light spots of different shapes and spatial sizes, and realizing the transmission of Hermitian elliptical vortex beams in gradient refractive index media.

2. The method for propagating a Hermitian elliptical vortex beam in a gradient refractive index medium according to claim 1, characterized in that, By introducing the Collins formula, and based on the Collins formula and the controllable anomalous Hermitian elliptical vortex beam, an electric field expression for the controllable anomalous Hermitian elliptical vortex beam at any propagation distance is constructed, including: By introducing the Collins formula and substituting equation (1) into the Collins formula, and through integration, we obtain the electric field expression for the controllable anomalous Hermitian elliptical vortex beam at any transmission distance z, as shown in equation (3). (3) Where x and y represent the x-axis and y-axis coordinates in a Cartesian coordinate system at a transmission distance z. , Indicates wave number, Where is the wavelength, and m is the numerical value of the topological charge of the elliptical vortex phase. and These are two matrix elements in Collins' formula; , and The three components of the electric field expression are shown in equations (4)-(6). （4） （5） （6） In the formula, and These are intermediate variables in the calculation process, as shown in formulas (7) and (8). （7） （8） In the formula, This is one of the matrix elements in Collins' formula.

3. The method for propagating a Hermitian elliptical vortex beam in a gradient refractive index medium according to claim 2, characterized in that, Introducing a gradient refractive index medium, and based on the gradient refractive index coefficient and the electric field expression for a controllable anomalous Hermitian elliptical vortex beam over arbitrary propagation distances in the gradient refractive index medium, an electric field expression for a controllable anomalous Hermitian elliptical vortex beam in a gradient refractive index medium is constructed, including: S31. Introduce a gradient refractive index medium and obtain the expression for the matrix element in the Collins formula in the gradient refractive index medium, as shown in formula (9). （9） is the gradient refractive index coefficient of the gradient refractive index medium; For transmission distance; S32. Substituting formula (9) into formulas (3)-(8), we obtain the electric field expression for the controllable anomalous elliptical vortex beam in the gradient refractive index medium, as shown in formula (10). （10） in, Here is the electric field expression for a controllable anomalous elliptical vortex beam in a gradient refractive index medium. , , The three components of the electric field expression in a gradient refractive index medium are shown in equations (11)-(13). （11） （12） （13） and These are intermediate variables in the calculation process, as shown in formulas (14) and (15). （14） （15）。 4. The method for propagating a Hermitian elliptical vortex beam in a gradient refractive index medium according to claim 3, characterized in that, The intensity of a controllable anomalous Hermitian elliptical vortex beam at any position z in a gradient refractive index medium is obtained based on the electric field expression of the controllable anomalous Hermitian elliptical vortex beam in the gradient refractive index medium, as shown in formula (16). （16） In the formula, The intensity of a controllable anomalous Hermitian elliptical vortex beam at any position z in a gradient refractive index medium.

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