Method for jointly regulating and controlling spin Hall effect of circular Airy beam by using uniaxial crystal and angular modulation

Abstract

The invention relates to the field of structured light field regulation and control and anisotropic medium light propagation, and discloses a method for regulating and controlling a spin Hall effect of a circular Airy beam by using uniaxial crystal and angular modulation. The method comprises the following steps: constructing a circular Airy beam with a self-focusing characteristic; angular phase modulation is applied to the system, so that the circular symmetry of the system is broken; the modulated light beam enters a uniaxial crystal to be propagated, and due to spin-orbit coupling in the crystal, a spin Hall effect is generated: a left-handed circular polarization component and a right-handed circular polarization component are gradually separated from each other and move along the positive axis and the negative axis respectively; after a light beam is emitted out of the uniaxial crystal, two focuses which are spatially separated and opposite in spinning direction are formed on a self-focusing focal plane. The device and the method can be used for structured light beam regulation and control, spin-orbit coupling and polarized light field modulation research.

Description

Technical Field

[0001] This invention relates to the field of structured light field manipulation and light propagation in anisotropic media, specifically to a method for controlling a circular Airy beam by applying angular phase modulation to break its circular symmetry, enabling it to propagate in a uniaxial crystal and forming two spatially separated focal points with opposite spin directions at the focal plane. Background Technology

[0002] Angular momentum is an important physical quantity describing the properties of an optical field, mainly including spin angular momentum, intrinsic orbital angular momentum, and extrinsic orbital angular momentum. Spin angular momentum can generate spin-orbit coupling effects with both intrinsic and extrinsic orbital angular momentum. When spin angular momentum couples with intrinsic orbital angular momentum, optical vortices can be generated; while its interaction with extrinsic orbital angular momentum leads to a reverse displacement in the transverse direction of optical field components of different circularly polarized states, i.e., the photon spin Hall effect. [AT O'Neil, I. MacVicar, L. Allen, et al., "Intrinsic and Extrinsic Nature of the Orbital Angular Momentum of a Light Beam," Phys. Rev. Lett. 88, 053601(2002)]

[0003] A uniaxial crystal is a system capable of generating spin-orbit coupling, thereby producing the photonic spin Hall effect. Existing techniques for generating the Hall effect through uniaxial crystals typically employ a Gaussian beam incident on the crystal, breaking the system's symmetry through methods such as tilted incidence [KY Bliokh, CT Samlan, C. Prajapati, et al., "Spin-Hall effect and circular birefringence of a uniaxial crystal plate," Optica 3, 1039–1047(2016)] or grafting different helical phases [H. Liu and L. Yuan, "Controlling the spin Halleffect of grafted vortex beams propagating in uniaxial crystal," Opt. Express31, 10434–10448 (2023)]. However, the Gaussian beam input in these methods is a divergent beam; as it passes through the uniaxial crystal, the optical field expands and its energy attenuates during propagation. Therefore, it is necessary to develop a new optical field manipulation mechanism to overcome the limitations of existing methods. A circular Airy beam is a beam that can spontaneously converge as it travels without relying on any lens, and can obtain a strongly focused spot on its self-focusing focal plane.This type of beam has shown significant application value in fields such as photobullet generation [P. Panagiotopoulos, DG Papazoglou, A. Couairon, et al., "Sharply autofocused ring-Airy beams transforming into non-linear intenselight bullets," Nat. Commun. 4, 2622 (2013)], particle manipulation [Y. Jiang, Z. Cao, H. Shao, et al., "Trapping two types of particles by modified circular Airybeams," Opt. Express 24, 18072–18081 (2016)], and micro / nano fabrication [T. Satoru, "Nanooptical measurement for next-generation nano / micro manufacturing based on localized light energy control," in Proc.SPIE, 2020), 115230G]. Therefore, it is necessary to construct a uniaxial crystal propagation system with a circular Airy beam as the incident light field to realize the generation of the photon spin Hall effect. Summary of the Invention

[0004] This invention provides a method for controlling the spin Hall effect of a circular Airy beam using a uniaxial crystal and angular modulation, in order to solve the problem of energy attenuation of Gaussian beams with transmission distance in the prior art.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A method for controlling the spin Hall effect of a circular Airy beam using a uniaxial crystal and angular modulation includes the following steps:

[0007] Step 1: Construct a circular Airy beam. Construct a circular Airy beam with self-focusing properties, which can spontaneously converge during transmission without relying on any lens, and obtain a strongly focused spot on the self-focusing focal plane.

[0008] Step 2: Apply angular phase modulation. An angular phase modulation is applied to the circular Airy beam before it is incident. The phase modulation function is the azimuth angle. The function is used to break the circular symmetry of the beam and broaden its orbital angular momentum spectrum, so that the initial 0th order circular Airy beam becomes a superposition state of orbital angular momentum containing higher order orbital angular momentum components.

[0009] Step 3: Incident on the uniaxial crystal. A circular Airy beam, with angular phase modulation applied, is incident perpendicularly along the optical axis of the uniaxial crystal, allowing it to propagate within the crystal. The uniaxial crystal is an anisotropic crystal with a unique optical axis, and its refractive index satisfies the following: the ordinary ray refractive index is... The refractive index of the light is ; .

[0010] Step 4: Generating the Spin Hall Effect. During propagation within the uniaxial crystal, the modulated beam undergoes spin-orbit coupling, generating the spin Hall effect: the left-hand circularly polarized component and the right-hand circularly polarized component gradually separate from each other, each moving along a positive orbital path. Axis and negative Move along the axis.

[0011] Step 5: Forming spin-separated focal points at the focal plane. After the beam exits the uniaxial crystal, two spatially separated focal points with opposite spin directions are formed at its self-focusing focal plane.

[0012] Furthermore, the specific modulation of the angularly modulated circular Airy beam propagating in a uniaxial crystal to produce the spin Hall effect is as follows:

[0013] The field distribution of the circular Airy beam on the initial plane in step 1 is expressed as follows: The radius of the initial planar circular Airy beam mm, scaling factor mm, attenuation coefficient .

[0014] Furthermore, let the angular phase modulation function described in step 2 be: ,in: , The modulation parameters are given. The angular modulation function can be expanded using a Fourier series as follows: ,in It is the expansion coefficient.

[0015] Furthermore, the propagation process of the modulated laser beam incident on the uniaxial crystal in step 3 is theoretically derived to obtain its transmission expression through the uniaxial crystal. When the laser beam propagates in the uniaxial crystal, the electric field obeys the equation: ,in, For vacuum wave vectors, Let be the dielectric tensor of the uniaxial crystal. When the propagation direction of the light beam is aligned with the optical axis of the crystal, we can obtain: ,in: For ordinary light refractive index, The refractive index is the refractive index of light.

[0016] Furthermore, consider the light field along... The optical axis propagates in the direction of the axial direction and is aligned with the optical axis of the crystal. The length of the crystal is... The incident end is Plane. Under the paraxial approximation, let and Let the left-hand and right-hand circularly polarized components of the light field be, when That is, when a beam of light propagates inside a uniaxial crystal, the expression for the light field is: In the formula: ,in: for Bessel function of order 1, Let be the transverse wave vector, and: Furthermore, considering that the uniaxial crystal is LiNbO3, its o-ray and e-ray refractive indices are respectively... , .

[0017] Furthermore, substituting the angular expansion coefficients into... In the middle, it can be rewritten as ,in: The analytical expression for the Hankel transform of a circular Airy beam can be approximated as: .

[0018] Subsequently, considering continuous boundary conditions at the output end face of the crystal, we can obtain the appropriate... At that time, the expression for the light field is: ,in: .

[0019] In traditional methods for generating spin Hall effect modulation using uniaxial crystals, Gaussian beams are typically used as input beams. However, the light field expands and energy attenuates during propagation. In contrast, this invention uses a circular Airy beam as the input beam. Its unique self-focusing properties effectively avoid the beam expansion and energy attenuation problems that occur during propagation. This provides a feasible technical solution for the photonic spin Hall effect based on self-focusing beams, and its application prospects are significant, extending to fields such as optical micromanipulation. Attached Figure Description

[0020] Figure 1 (a) is the phase distribution diagram of the modulation function. Figure 1 (b) is the orbital angular momentum spectrum of the circular Airy beam after angular modulation.

[0021] Figure 2The transmission of the modulated linearly polarized circular Airy beam through a uniaxial crystal is shown. (a) Normalized intensity distribution of the xz section; (b1)-(e3) Normalized intensity and phase distribution of the left-hand circularly polarized component ((b1)-(e1)), right-hand circularly polarized component ((b2)-(e2)), and total optical field ((b3)-(e3)) in the xy section at different transmission distances, where: (b1)-(b3) corresponds to z=0mm; (c1)-(c3) corresponds to z=25mm; (d1)-(d3) corresponds to z=32mm; (e1)-(e3) corresponds to z=37.5mm.

[0022] Figure 3 A schematic diagram illustrating the process of controlling the spin Hall effect of a circular Airy beam using a uniaxial crystal and angular modulation.

[0023] Figure 4 The diagram shows the transverse energy flux distribution of the left-hand circularly polarized component and the right-hand circularly polarized component at different transmission distances. (a1)-(a2) correspond to z=31mm; (b1)-(b2) correspond to z=32mm; (c1)-(c2) correspond to z=35mm. Detailed Implementation

[0024] The present invention will be further described below with reference to specific drawings and embodiments, but the present invention is not limited to the following embodiments.

[0025] Please refer to the following first. Figure 1 , Figure 1 These are the phase distribution diagram of the angular modulation function and the orbital angular momentum spectrum of the rounded Airy beam, respectively. Let the modulation function... Parameters in , Calculate the phase distribution diagram of the modulation function; it can be seen that it is about An axisymmetric phase structure can break the circular symmetry of a circular Airy beam. This angular modulation function broadens the angular momentum spectrum of the circular Airy beam; the broadened orbital angular momentum spectrum can be expressed by the formula... Calculations show that after the circular Airy beam is modulated by the angular function, most of the energy of the optical field is concentrated in... On the orbital angular momentum mode of the order.

[0026] Example: Example and Figure 2 The initial planar circular Airy beam's radius matches. mm, scaling factor mm, attenuation coefficient The length of a uniaxial crystal is mm, the incident light wavelength is 632.8 nm, and the refractive indices of the o-ray and e-ray are respectively , The angularly modulated beam is incident on the interior of the uniaxial crystal along a direction parallel to the optical axis of the uniaxial crystal and propagates there. It can be observed that... In the mm plane, due to angular modulation, the circular symmetry of the initial intensity distribution of the rounded Airy beam is broken, resulting in a "crescent moon" shape. At a distance of mm, due to spin-orbit coupling in the crystal, the centroid of the intensity of the left-hand circularly polarized component tends towards the right-hand side. The intensity centroid of the right-hand circularly polarized component tends towards the negative axis direction. Along the axial direction, the two gradually separate from each other. After the beam exits, it focuses on the self-focusing focal plane. At a distance of mm, the left-hand circularly polarized component and the right-hand circularly polarized component focus independently, forming two independent focal points with opposite spins, thus generating the photon spin Hall effect.

[0027] The method for generating the photonic Hall effect described in the above embodiments can be used Figure 3 The flowchart shown is used to summarize the process.

[0028] Figure 4 The transverse energy flux distributions of the left-hand and right-hand circularly polarized components are presented at different propagation distances. To understand the formation mechanism of the photon spin Hall effect, the transverse energy flux distributions of the left-hand and right-hand circularly polarized components at different propagation distances before, at, and after the focal plane were calculated. The results show that, before the focal plane, the transverse energy flux of the left-hand circularly polarized component is mainly directed towards the front. The axial direction converges, while the right-hand circularly polarized component points towards the negative direction. The photons converge along the axial direction, forming two spatially separated spin foci at the focal plane. Upon reaching the focal plane, the transverse energy flows of the two components begin to diverge in opposite directions. In the region behind the focal point, the energy flow divergence further intensifies, and the focal points of the two circularly polarized components gradually expand and tend to overlap. This result is consistent with numerical simulations of light intensity distribution. This indicates that the formation of the self-focusing focal plane photon spin Hall effect originates from the directional distribution of the transverse energy flow.

[0029] In summary, this invention achieves spin separation at the focal point on the self-focusing focal plane by utilizing a uniaxial crystal and angular modulation to jointly control the circular Airy beam, generating a spin Hall effect. Furthermore, the spin Hall effect can be controlled by modulation parameters, verifying the feasibility and stability of the method of this invention.

[0030] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings, but are not limited to the described embodiments. It should be noted that various changes, modifications, or substitutions made by those skilled in the art within the scope of the principles of the present invention are still within the protection scope of the present invention.

Claims

1. A method for controlling the spin Hall effect of a circular Airy beam using a uniaxial crystal and angular modulation, comprising the following steps: Step 1: Construct a circular Airy beam with self-focusing properties as the incident beam; Step 2: Before the circular Airy beam is incident, an angular phase modulation function is applied to it to cause the circular symmetry of the circular Airy beam to be broken; Step 3: Inject the angularly phase-modulated circular Airy beam parallel to the optical axis of the uniaxial crystal, allowing it to propagate inside the uniaxial crystal; Step 4: The modulated beam undergoes spin-orbit coupling in the uniaxial crystal, resulting in the spin Hall effect: the left-hand circularly polarized component and the right-hand circularly polarized component gradually separate from each other, moving along the positive and negative axes respectively. Axis and negative Move along the axis; Step 5: After passing through the uniaxial crystal, under the effect of the self-focusing property of the circular Airy beam, the two spin components separated by the uniaxial crystal will spontaneously converge to form two separation focal points with opposite spins on the focal plane.

2. The method for controlling the spin Hall effect of a circular Airy beam using a uniaxial crystal and angular modulation as described in claim 1, characterized in that: The complex amplitude expression of the circular Airy beam on the initial plane in step 1 is: ,in It is a constant. , , and These represent the Airy function, initial radius, radial scaling factor, and attenuation parameter of the circular Airy beam.

3. The method for controlling the spin Hall effect of a circular Airy beam using a uniaxial crystal and angular modulation as described in claim 1, characterized in that: The angular phase modulation function is the azimuth angle. The function.

4. The angular phase modulation function according to claim 3, characterized in that: The angular phase modulation function is expressed as follows: ,in , These are the modulation parameters.

5. The method for controlling the spin Hall effect of a circular Airy beam using a uniaxial crystal and angular modulation as described in claim 1, characterized in that: The ordinary refractive index of the uniaxial crystal is The refractive index of light is Furthermore, the beam propagation direction is consistent with the optical axis direction of the uniaxial crystal.

6. The method for controlling the spin Hall effect of a circular Airy beam using a uniaxial crystal and angular modulation as described in claim 1, characterized in that: The method is based on paraxial propagation theory, substituting the initial electric field into... In the equation, among which, Given the vacuum wave vector, analyze the propagation of the modulated circular Airy beam in a uniaxial crystal. Axial propagation characteristics: The incident end is Plane, when That is, when a light beam propagates inside a uniaxial crystal, the expression for the light field propagation is: ， In the formula: ; when At that time, the expression for optical field transmission is: ， In the formula: ； in: for Bessel function of order 1, The transverse wave vector is given, and the length of the crystal is given. , For the refractive index of o, Let be the refractive index of the e-ray.

7. The optical field transmission expression according to claim 6, characterized in that: During transmission, the left-hand circularly polarized component and the right-hand circularly polarized component converge in opposite lateral directions, thus forming a spatially separated focal point with opposite spin directions on the focal plane.

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