A method for preparing a vector hermite-laguerre gaussian beam based on collinear interference

Abstract

This invention relates to a method for preparing vector Hermitian-Laguerre Gaussian beams based on collinear interference. Combining the relationship between the Poincaré sphere and the Jones matrix, and based on the vortex light field generated by the complete solutions of two sets of Helmholtz equations, a method for preparing vector Hermitian-Laguerre Gaussian beams based on collinear interference is proposed. Unlike the commonly used single-mode vector light field preparation, this method, based on a phase-type spatial light modulator, a polarization beam splitter, and a half-wave plate, realizes the preparation of a dual-mode vector Hermitian-Laguerre Gaussian beam, reconstructs the polarization distribution on its intensity cross section, and studies the influence of mode difference on the polarization distribution with transmission distance. Compared with traditional single-mode vector light fields, this method achieves good results in preparing vector light fields with different modes. This method belongs to the category of light field manipulation and can be applied to the design of optical polarization elements.

Description

Technical Field

[0001] This invention relates to a method for preparing vector Hermitian-Laguerre Gaussian beams based on collinear interference. Unlike the commonly used single-mode vector light field preparation, this method utilizes two phase-type spatial light modulators, combining the relationship between the Poincaré sphere and the Jones matrix, and generating vortex light fields based on complete solutions to two sets of Helmholtz equations. This achieves the preparation of a dual-mode vector Hermitian-Laguerre Gaussian beam. Based on a polarization beam splitter and a half-wave plate, it enables the preparation of vector Hermitian-Laguerre Gaussian light fields composed of arbitrary modes. The polarization distribution on the intensity cross section is reconstructed, and the influence of mode difference with transmission distance on the polarization distribution is investigated. Compared to traditional single-mode vector light fields, this method achieves excellent results in preparing vector light fields with different modes. This method falls under the category of light field manipulation and can be applied to the design of optical polarization elements. Background Technology

[0002] The phenomenon of vortices in optical fields was first discovered by Boivin, Dow, and Wolf in 1967 near the focal plane of a lens group. In 1973, Bryngdahl first explored experimental methods for preparing vortexes. In 1979, Vaughan and Willets successfully prepared vortexes using continuous lasers. In 1990, Yu and Bazgenov V first successfully prepared vortexes using a grating method. In 1992, L. Allen discovered vortexes with phase factors under paraxial conditions. The vortex beam possesses orbital angular momentum, where l is the topological charge of the vortex beam's orbital angular momentum. The azimuth angle; each photon carries The orbital angular momentum, To reduce Planck's constant, this angular phase factor indicates that during the propagation of vortex light, if the beam propagates for one cycle, the wavefront rotates exactly one revolution around the optical axis, and the phase changes accordingly by 2πl.

[0003] Structured light fields and their shaping techniques, including the manipulation of various properties of light, have rapidly become an important branch of modern optics. Vortex beams, as the most common type of structured light field, have attracted widespread attention from researchers due to their phase singularity, topological charge, and hollow intensity distribution, and are widely used in fields such as optical tweezers, optical wrenches, and high-order quantum entanglement.

[0004] The fabrication of vortex beams is fundamental to vortex beam research. Under laboratory conditions, the spatial light modulator method is a commonly used approach. The spatial light modulator modulates the light wave by controlling an electric field to induce changes in the spatial phase or amplitude of the image on a liquid crystal display, thereby incorporating specific information into the light wave. A holographic pattern of vortex beams is fabricated using complex amplitude modulation techniques and loaded onto a spatial light modulator. When the spatial light modulator is illuminated with a linearly polarized Gaussian beam, the emitted light is the vortex beam.

[0005] In recent years, due to the discovery of vector beams, a novel type of structured beam, polarization vortexes have gradually become a hot research topic in the field of optical field manipulation. Vector vortex beams carry spin angular momentum (SAM). By simultaneously manipulating the degrees of freedom of both SAM and OAM, vector vortex beams can be obtained. These beams possess characteristics of both phase vortices and polarization vortices. The polarization states of a vector vortex beam exhibit anisotropic distribution across its intensity cross-section. This is formed by the superposition of two orthogonal scalar optical fields and their corresponding orthogonal polarization states. Based on this principle, any vector vortex optical field can be successfully prepared, such as vector Laguerre-Gaussian beams, vector Bessel beams, and vector Innes-Gaussian beams. These vector vortex optical fields have broad application prospects in laser communication, rotational detection, optical trapping, and high-resolution imaging.

[0006] To broaden its application prospects, vortex beam modes with stable transmission characteristics have attracted great attention. Based on the Helmholtz equations, three complete solutions for the Laguerre-Gaussian, Innes-Gaussian, and Hermite-Gaussian modes have been calculated in polar, elliptical, and Cartesian coordinate systems, respectively. These modes together form a complete orthogonal basis in Hilbert space, and any paraxial optical field can be characterized by the complex superposition of these scalar modes. Summary of the Invention

[0007] The technical problem solved by this invention is as follows: Unlike the commonly used single-mode vector light field preparation, this method, based on two phase-type spatial light modulators and combining the relationship between the Poincaré sphere and the Jones matrix, utilizes vortex light fields generated by complete solutions under two sets of Helmholtz equations to achieve the preparation of a dual-mode vector Hermitian-Laguerre Gaussian beam. Based on a polarization beam splitter and a half-wave plate, it achieves the preparation of vector Hermitian-Laguerre Gaussian light fields composed of arbitrary modes, reconstructs the polarization distribution on its intensity cross section, and studies the influence of mode difference on the polarization distribution with transmission distance. Compared with traditional single-mode vector light fields, this method achieves excellent results in the preparation of vector light fields with different modes.

[0008] The technical solution of this invention is:

[0009] This invention relates to a method for preparing a vector Hermit-Laguerre Gaussian beam based on collinear interference, which mainly includes the following steps:

[0010] (1) Complex vector patterns are represented by mathematical expressions, representing amplitude, phase, and polarization, and are characterized as follows: in, and The amplitude and phase information of the two scalar vortex beams were characterized, respectively. This represents the position in the horizontal coordinate system (x, y) or (r, φ), where z is the transmission distance. and The right-hand and left-hand circular polarization components are represented, respectively. θ represents a weighting factor with a value range of [0, π / 2], and 2α represents the phase difference between the two optical fields, where α ∈ [0, π]. To generate the Hermit-Laguerre (HLG) beam mentioned above, we define... and These are LG mode and HG mode respectively. The formula is then converted to... in and HG n,m (x, y, z) represent the mutually orthogonal LG and HG modes, respectively.

[0011] (2) The collinear interference optical path design is based on the derivation of the Jones matrix. The horizontal linear polarization can be characterized as [1, 0]T. Then, the polarization angle of the beam after passing through the half-wave plate is set to become in The polarization state of the beam is transformed into The beam is split into a horizontal polarization component by a polarizing beam splitter. and vertical polarization component In the experiment, the complex amplitude information of the light field was controlled by two spatial light modulators. The complex amplitude information loaded on the two spatial light modulators were as follows: and

[0012] HG n，m (ρ, z), where and HG n，m (ρ, z) represent the mutually orthogonal Laguerreotype and Hermitian-Gaussian modes, respectively. Since the spatial light modulator can only control horizontally polarized light, a half-wave plate with an angle of 45° to the fast axis needs to be added to the vertically polarized optical path to control the vertical polarization component. Converted to horizontal polarization component After introducing complex amplitude information, the two scalar light fields can be characterized as and Due to the design of collinear interference, It will again pass through a half-wave plate at a 45° angle to the fast axis and become After passing through a polarizing beam splitter again, the horizontal and vertical polarizations are collinearly combined. Then, through a quarter-wave plate with an angle of 45° to the fast axis, the horizontal and vertical polarizations are converted into left-handed and right-handed circular polarizations, respectively, thus realizing the preparation of a vector Hermite-Laguerre beam.

[0013] The principle of this invention is:

[0014] We primarily characterize complex vector patterns using mathematical expressions, which represent a superposition of amplitude, phase, and polarization, as shown in the following formula:

[0015]

[0016] in, and The amplitude and phase information of the two scalar vortex beams were characterized, respectively. This represents the position in the horizontal coordinate system (x, y) or (r, φ), where z is the transmission distance. and The right-hand and left-hand circular polarization components are represented, respectively. θ represents a weighting factor with a value range of [0, π / 2], and 2α represents the phase difference between the two optical fields, where α ∈ [0, π]. To generate the HLG beam mentioned above, we define... and These are LG mode and HG mode, respectively. In this case, Eq.1 can be represented as:

[0017]

[0018] in and HG n,m (x, y, z) represent the mutually orthogonal LG and HG modes, respectively. The expression is:

[0019]

[0020] in Characterized by the Laguerre polynomial, ω0 represents the optical waist radius when z = 0, and ω(z) represents the optical waist radius under different transmission distances z, which can be expressed as: Where z R It represents the length of Ruili.

[0021] HG n，m The expression for (x, y, z) is:

[0022]

[0023] Where n and m represent the horizontal and vertical exponents of the HG pattern, respectively, H m (·) and Hn (·) are the m-th and n-th order Hermitian polynomials, respectively, where k represents the wave vector, expressed as k = 2π / λ, and λ is the wavelength of the beam.

[0024] To generate the complex vector patterns expressed in Eq.2, we must ensure that the two scalar light fields are orthogonal to each other. and HG n，m The (x, y, z) patterns are all complete solutions to the Helmholtz equations, so they can be transformed into each other. The mode can be represented as HG n，m The linear superposition of (x, y, z) patterns consists of:

[0025]

[0026] Where n and m satisfy n = (2p + |l| + l) / 2 and m = (2p + |l| - l) / 2, and N is represented as NL in LG mode and HG mode respectively. G =2p+|l| and N HG =n+m, the coefficient b(n,m,k) is expressed as:

[0027]

[0028] Orthogonality is guaranteed when the number of modes N is different, but when the number of modes N is the same, some special cases need to be considered. Based on Eq.5 and Eq.6, we can obtain the known... The modal requirements of linear superposition of HG n，m The (x, y, z) modes, and the remaining HG modes under the condition of a fixed number of modes N, are the orthogonal terms we need. In other words, as long as any HG... n，m The (x, y, z) pattern does not appear in the linear superposition formula shown in Eq. 5, and any All patterns can be orthogonal to it. Based on this, we have achieved a basic theoretical verification of generating vector HLG beams.

[0029] The Poincaré sphere is widely used as an intuitive method for characterizing the polarization state of a vector light field. Its coordinate position is determined by the latitude and longitude on the sphere. Based on Eq.2 above, we introduce two variables: a weighting factor θ and a phase difference α. These two variables can be used to characterize the latitude and longitude information on the sphere, respectively. By transforming θ and α, we can achieve traversal of any position on the Poincaré sphere. By further changing the scalar light field mode information at the north and south poles of the Poincaré sphere, we can achieve the characterization of any light field polarization state distribution.

[0030] Since the polarization distribution of the laser beam is relatively random, it first needs to pass through a Glan prism to obtain a relatively pure horizontal linear polarization, which can be represented as [1, 0] based on the Jones matrix. T Then we set the polarization angle of the beam after passing through the half-wave plate to become in The polarization state of the beam is transformed into The beam is split into a horizontal polarization component by the PBS. and vertical polarization component In our experiment, we used two SLMs to control the complex amplitude information of the optical field. The complex amplitude information loaded on the two SLMs were as follows: and HG n，m (ρ, z), since SLM can only control horizontally polarized light, a half-wave plate with an angle of 45° to the fast axis needs to be added to the vertically polarized light path to control the vertical polarization component. Converted to horizontal polarization component After introducing complex amplitude information, the two scalar light fields can be characterized as and Due to the design of collinear interference, It will again pass through a half-wave plate at a 45° angle to the fast axis and become After passing through the PBS again, the horizontal and vertical polarizations are collinearly combined. Then, a quarter-wave plate with an angle of 45° to the fast axis is used to convert the horizontal and vertical polarizations into left-handed and right-handed circular polarizations, respectively, thus preparing the vector HLG beam. The expression for the vector light field at this point is:

[0031]

[0032] Comparing Eq.2 with Eq.7 reveals the polarization angle of the beam. There is a one-to-one correspondence between the weighting factor θ in the Poincaré sphere and the following:

[0033]

[0034] Through Eq.8, we can see that the latitude on the Poincaré sphere can be transformed by rotating the angle of the half-wave plate. By further changing the phase difference of the holograms loaded on the two SLMs, the longitude can be controlled, and finally, the vector HLG beam at any position on the surface of the Poincaré sphere can be prepared.

[0035] Based on Eq.3 and Eq.4 for LG and HG beams, we find that both optical fields contain a phase term that varies with the transmission distance z, where it is assumed that...

[0036]

[0037] This means that there is a phase difference between the LG beam and the HG beam that changes with the transmission distance and the optical field mode. Subtracting the two formulas in Eq. 9 yields...

[0038]

[0039] Where we set N LG =2p+|l|, N HG =m+n, then ΔN=NL G -N HG It can be seen that ΔN and Δψ are proportional. When ΔN = 0, there is no phase difference between the LG beam and the HG beam, and the polarization state of the vector HLG beam will not change with the transmission distance. However, when ΔN ≠ 0, the polarization state of the vector HLG beam will change with the transmission distance.

[0040] Compared with existing solutions, the main advantages of this invention are:

[0041] (1) Currently, most generated vector beams are composed of a single mode. This scheme achieves the preparation of a multimodal vector light field by superimposing two different modes - the Laguerre Gaussian mode and the Hermetic Gaussian mode.

[0042] (2) Reduce costs and save space. Common interference optical paths generally use Mach-Zehnder interferometers, Sagnac interferometers, etc. Such experimental optical paths will affect the beam quality due to interference instability. Using this method, the beams can be superimposed on the same path to form a collinear interference experimental device. The optical path is simple and stable.

[0043] (3) It is highly flexible. Based on two holograms, different complex amplitude information of the Laguerre Gaussian beam and the Hermit Gaussian beam are loaded respectively. The parameters can be flexibly adjusted according to the needs to realize the preparation of vector Hermit-Laguerre beams with different parameters. Attached Figure Description

[0044] Figure 1 A flowchart illustrating the application of a vector Hermit-Laguerre Gaussian beam preparation based on collinear interference;

[0045] Figure 2 A diagram of the experimental setup for preparing a vector Hermitian-Laguerre Gaussian beam;

[0046] Figure 3 The polarization state reconstruction results of the vector Hermite-Laguerre Gaussian beam with parameters set to l=0, p=1, m=n=1, ΔN=0 and transmission distances of z=0mm, 300mm, and 750mm on the equator of the Poincaré sphere.

[0047] Figure 4The polarization state reconstruction results of the vector Hermite-Laguerre Gaussian beam with parameters set to l=1, p=1, m=2, n=5, ΔN=4 on the equator of the Poincaré sphere and transmission distances of z=0mm, 300mm, and 750mm;

[0048] Figure 5 Let the position on the surface of the Poincaré sphere be (3π / 4, 0), (π / 4, π / 2), with parameters as follows:

[0049] The results of polarization state reconstruction of vector Hermitian-Laguerre Gaussian beams with l=1, p=1, m=2, and n=1; Detailed Implementation

[0050] This invention is based on the stable transmission characteristics of the complete solution of the Helmholtz equations. The experimental objective is to control the polarization angle of two modes of light fields using a half-wave plate and a polarizer. The subject of the experiment is a spatial light modulator, and the specific implementation steps are as follows:

[0051] The He-Ne laser emits a Gaussian beam with a wavelength of 633 nm. First, it passes through a Glan prism to obtain horizontally linearly polarized light. Then, it passes through a 4f system consisting of a filter system and lens L1 to achieve beam expansion and collimation. Next, it passes through a half-wave plate to adjust the polarization angle of the linearly polarized light. A PBS (Bipolarized Beam Filter) is used to separate the horizontal and vertical polarization components; the horizontally polarized light passes directly, while the vertically polarized light is deflected downwards by 90°. In the experiment, a SLM (Simplified Laser Lens) is used to control the complex amplitude of the optical field. Two SLMs in the optical path are loaded with holograms of the LG (Low-Growth) and HG (High-Growth) beams, respectively. As mentioned above, the SLM can only control the horizontally polarized light; therefore, the horizontally polarized component after passing through the PBS is directly modulated by the SLM. To prepare an HG beam, the vertical polarization component first needs to be converted to horizontal polarization by passing through a half-wave plate at a 45° angle to the fast axis. Then, it is converted to an LG beam and passed through the same half-wave plate to restore vertical polarization. The horizontally polarized HG beam and the vertically polarized LG beam are then collinearly superimposed using a PBS. Afterward, the BS separates the controlled light field, and the beam passes through a quarter-wave plate at a 45° angle to the fast axis to convert the horizontally and vertically polarized light into left-handed and right-handed circularly polarized light, respectively, thus preparing the vector HLG beam. The quarter-wave plate and polarizer within the dashed lines are used to measure Stokes parameters and detect the polarization distribution of the vector light field. The CCD is used to observe the intensity distribution image of the light field.

[0052] We selected several sets of orthogonal LG and HG beam solutions with different parameters and studied the polarization distribution of the synthesized vector HLG beam under varying transmission distance z. We selected a position on the equator of the Poincaré sphere with parameters θ = π / 4 and α = 0. First, we simulated and experimentally verified the polarization distribution of a vector Hermitian-Laguerre Gaussian beam with l = 0, p = 1, m = n = 1 under transmission distances of z = 0, 300, and 750 mm. The results are as follows: Figure 3 As shown, we calculated the polarization ellipse distribution on the optical field intensity cross section using Stokes parameters. It can be seen that the intensity of the vector HLG beam undergoes some distortion with increasing transmission distance z, while the polarization distribution remains unchanged. Subsequently, simulations and experimental verifications were performed on the polarization distribution of a vector Hermitian-Laguerre Gaussian beam with l=1, p=1, m=2, n=5 at transmission distances z=0, 300, and 750 mm. The results are as follows. Figure 4 As shown, the polarization distribution of the vector HLG beam rotates counterclockwise with increasing transmission distance z. Furthermore, we fabricated hybrid polarized Hermitian-Laguerre Gaussian beams at positions (3π / 4, 0) and (π / 4, π / 2) on a sphere, with parameters set to l = 1, p = 1, m = 2, and n = 1, as follows... Figure 5 As shown, the prepared vector beam has a relatively uniform intensity distribution and good quality.

[0053] Through experimental verification and theoretical derivation, it can be seen that this method can realize the preparation of Hermite-Laguerre Gaussian beams with arbitrary polarization distribution. For example, vector Hermite-Laguerre Gaussian beams with position parameters l=0, p=1, m=n=1 and l=2, p=1, m=n=2 are selected at special points on the equator of the Poincaré sphere, with transmission distances of z=0, 300, and 750 mm, and mixed polarization Hermite-Laguerre Gaussian beams with parameters l=1, p=1, m=2, and n=1 are selected at positions (3π / 4, 0) and (π / 4, π / 2) on the sphere. Experimental verification shows that this method can realize the control of Hermite-Laguerre vector light fields with arbitrary multi-parameters.

[0054] The contents not described in detail in this invention are existing technologies known to those skilled in the art.

Claims

1. A method for preparing a vector Hermit-Laguerre Gaussian beam based on collinear interference, characterized in that: To investigate the polarization characteristics of vector light fields under complex modes, a vector Hermitian-Laguerre-Gaussian beam was prepared and its polarization state was reproduced based on Laguerre-Gaussian and Hermitian-Gaussian beams. A linear superposition relationship exists between the Laguerre-Gaussian and Hermitian-Gaussian modes. The mutually orthogonal solutions of the two modes under arbitrary parameters were calculated. Based on a collinear interference device constructed using a spatial light modulator and a half-wave plate, a vector Hermitian-Laguerre-Gaussian beam with arbitrary modes was prepared.

2. The method for preparing a vector Hermit-Laguerre Gaussian beam based on collinear interference according to claim 1, characterized in that: To generate complex vector modes, we must ensure that the two scalar optical fields are orthogonal. The Laguerreotype and Hermetic modes, as complete solutions to the Helmholtz equations, can be interconverted. The mode numbers of the Laguerreotype and Hermetic modes are expressed as N. LG =2p+|l| and N HG =n+m, where N LG N HG Let N represent the number of modes, l represent the topological charge number, p represent the radial number of nodes, and n and m represent the exponents of the Hermitian mode in the transverse and longitudinal directions, respectively. LG N HG When the modes are not equal, orthogonality is certain. However, when the number of modes is equal, the remaining Hermetic modes, which are the linear superposition of Hermetic modes required to form the Laguerreotype modes, are the required orthogonal terms under the condition that the number of modes N remains unchanged.

3. The method for preparing a vector Hermit-Laguerre Gaussian beam based on collinear interference according to claims 1 and 2, characterized in that: Based on the formula for the number of modes, the modal difference can be expressed as ΔN = N LG -N HG By introducing the transmission term in the optical field, we can obtain the phase difference that changes with transmission distance and beam mode as Δψ=ΔNtan -1 (z / z R ), where z represents the transmission distance, z R Representing the Rayleigh distance, when ΔN = 0, the polarization distribution of the vector Hermit-Laguerre beam does not change with the transmission distance z. When ΔN ≠ 0, the phase difference Δψ will eventually gradually approach a certain value as the transmission distance z changes. This means that the polarization distribution on the optical field cross section will rotate counterclockwise at most.

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