A method for realizing regulation and control of nonlinear Kerr medium trajectory transmission of a circular Airy beam

Abstract

The application discloses a kind of self-accelerating circular airy light beams in nonlinear Kerr medium trajectory control method, this method includes: introducing circular airy light beam in nonlinear Kerr medium;By the phase control function of self-accelerating circular airy light beam trajectory analytical derivation declared in the application;Phase control function is combined with circular airy light beam, and further adjustment of relevant parameters can obtain circular airy light beam along target trajectory transmission.In addition, among numerous nonlinear optical effects, Kerr effect is concerned due to its unique properties, that is, the refractive index of the medium will change with the change of light intensity.This phenomenon not only has important significance in theory, but also shows extensive potential value in practical application.The application can effectively control the propagation trajectory of circular airy light beam in nonlinear Kerr medium by saving time and material cost, improving work efficiency.

Description

Technical Field

[0001] This invention relates to the field of optics, and more specifically to a method for trajectory control of a circular Airy beam in a nonlinear Kerr medium. Background Technology

[0002] Since the nonlinear Kerr effect was first revealed, its significant influence on the dependence of refractive index on light intensity has become a widely studied topic in the field of nonlinear optics. Nonlinear Kerr media refer to a special class of media whose refractive index changes with the intensity of light passing through it; this phenomenon is called the Kerr effect, an instantaneous nonlinear interaction associated with nonlinear electronic polarization. The Kerr effect is not only theoretically significant in optical science and engineering but also demonstrates broad potential value in practical applications.

[0003] The generation, application, and control of circular Airy beams are a widely studied area of ​​interest in optics. These beams possess the characteristic of abruptly concentrating energy in front of the focal point while maintaining a low intensity profile. Based on these unique properties, circular Airy beams have shown broad application potential in fields such as biomedical therapy and optical micromanipulation.

[0004] In subsequent studies, scholars proposed numerous methods for controlling the trajectory of circular Airy beams. However, these studies have certain limitations in trajectory control, namely, all research on beam trajectory control is limited to free space and parabolic potential media environments. Summary of the Invention

[0005] In view of this, in order to solve the above-mentioned problems in the prior art, the present invention proposes a method for trajectory control of a circular Airy beam in a nonlinear Kerr medium.

[0006] The present invention solves the above problems through the following technical means:

[0007] A method for controlling the propagation of a circular Airy beam along a nonlinear Kerr medium trajectory includes the following steps:

[0008] In a nonlinear Kerr medium transport system, a circular Airy beam is introduced and incorporated into the Schrödinger wave equation to describe the path of the circular Airy beam. Propagation characteristics in the axial direction;

[0009] Given a circular Airy beam with a quadratic trajectory, the beam can propagate along this predetermined path;

[0010] Calculate the phase modulation function of a circular Airy beam ;

[0011] Phase modulation function The input wavefront is obtained by multiplying it by the initial field of the beam. ;

[0012] The target beam under phase modulation propagates along a predetermined optical trajectory in a nonlinear Kerr medium.

[0013] Furthermore, the specific modulation of the trajectory control of the circular Airy beam in the nonlinear Kerr medium is as follows:

[0014] Introducing the initial electric field expression for a circular Airy beam ,in As the attenuation factor, , The translation factor is... , The radius of the main circle Airy beam ring ;

[0015] In mathematical expressions, the initial input Substituting into the Schrödinger wave equation: In the formula For linear refractive index, the nonlinear coefficient of Kerr medium , Let the wave number be in free space. For wavelength, , It is the longitudinal propagation distance. ,in , , Represents the normalized horizontal coordinate. Define the Gaussian beam width. The critical collapse power of a circular Airy beam in a nonlinear Kerr medium is .

[0016] Given a beam with a quadratic trajectory in a nonlinear Kerr medium: the beam trajectory is... , Let be the curvature of the quadratic curve trajectory. Controlling the center position of the parabola's axis of symmetry, parameters Control the displacement of the beam, substitute... Calculated :Z={{\frac {1} {2{t}^{2}}[{\gamma}^{2}+2ct-2tx+{[{(2tx-2ct-{\gamma}^{2})}^{2}-4{t}^{2}({x}^{2}+{y}^{2}-2xc+{c}^{2})]}^{\frac {1} {2}}]\}}^{\frac {1} {2}} ,in , , , and It is the derivative of the parametric equation, here let .

[0017] Analytical Derivation of Trajectory Control and Phase Control Function :

[0018] because ,

[0019] P(Z)=\frac {1} {2}\int ^{Z}_{0} {{{[f'(\zeta )]}^{2}+{[g'(\zeta )]}^{2}-{[\frac {R(\zeta )} {\zeta}]}^{2}\} d\zeta} , Q(x,y)=\frac {1} {2}{\int ^{Z}_{0} {{{[f'(\zeta )]}^{2}+{[g'(\zeta )]}^{2}-{(\frac {R} {\zeta})}^{2}\} d\zeta -\frac {{(fx)}^{2}+{(gy)}^{2}} {Z}}\} ,

[0020] The calculated Substitute the phase modulation function After simplification, the expression for the phase modulation function is: .

[0021] Phase modulation function The phase-modulated input wavefront is obtained by combining it with the initial electric field expression of the circular Airy beam. : ,in Let be the amplitude of the electric field.

[0022] This invention employs a method for calculating the phase modulation function of a circular Airy beam in a nonlinear Kerr medium transmission system. Obtain the phase-modulated input wavefront This method saves costs, improves efficiency, and allows for flexible control of the trajectory of the circular Airy beam. Attached Figure Description

[0023] To make the technical solutions in the embodiments of the present invention clearer, the accompanying drawings involved in the description of the embodiments will be described in detail. It must be noted that the provided drawings only represent several embodiments of the present invention. For those skilled in the art, other possible representations of the drawings can be derived from these drawings without requiring creative work.

[0024] Figure 1 The trajectory of a circular Airy beam propagating in a nonlinear Kerr medium without phase function modulation. This invention provides the initial light of the circular Airy beam obtained through calculation and simulation. For the phase of the beam, and This is a cross-sectional view of the light beam.

[0025] Figure 2 This invention obtains the trajectory of a circular Airy beam propagating along a parabola in a nonlinear Kerr medium through calculation and simulation. The phase of the beam is shown in the figure. The transverse cross-section of the beam obtained after phase modulation is shown in the figure. As shown.

[0026] Figure 3 This invention obtains the trajectory of a circular Airy beam propagating along a parabola in a nonlinear Kerr medium through calculation and simulation. The phase of the beam is shown in the figure. The transverse cross-section of the beam obtained after phase modulation is shown in the figure. As shown.

[0027] Figure 4 This is a flowchart illustrating the trajectory of a circular Airy beam along a quadratic curve in a transmission system under a nonlinear Kerr medium, as described in this invention. Implementation

[0028] To provide a clearer and more explicit explanation of the above-mentioned objectives, features, and advantages of the present invention, this document will describe the technical solutions of the present invention in detail with reference to the accompanying drawings and related embodiments. It should be noted that the described embodiments represent only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0029] It should be noted that the specific modulation methods for trajectory control of the self-accelerating beam in complex media described in this invention are all as follows:

[0030] Introducing the initial electric field expression for a circular Airy beam ,in As the attenuation factor, , The translation factor is... , The radius of the main circle Airy beam ring ;

[0031] In mathematical expressions, the initial input Substituting into the Schrödinger wave equation: In the formula For linear refractive index, the nonlinear coefficient of Kerr medium , Let the wave number be in free space. For wavelength, , It is the longitudinal propagation distance. ,in , , Represents the normalized horizontal coordinate. Define the Gaussian beam width. The critical collapse power of a circular Airy beam in a nonlinear Kerr medium is .

[0032] Given a beam with a quadratic trajectory in a nonlinear Kerr medium: the beam trajectory is... , Let be the curvature of the quadratic curve trajectory. Controlling the center position of the parabola's axis of symmetry, parameters Control the displacement of the beam, substitute... Calculated :Z={{\frac {1} {2{t}^{2}}[{\gamma}^{2}+2ct-2tx+{[{(2tx-2ct-{\gamma}^{2})}^{2}-4{t}^{2}({x}^{2}+{y}^{2}-2xc+{c}^{2})]}^{\frac {1} {2}}]\}}^{\frac {1} {2}} ,in , , , and It is the derivative of the parametric equation, here let .

[0033] Furthermore, the trajectory control and phase control functions are analytically derived. :because , P(Z)=\frac {1} {2}\int ^{Z}_{0} {{{[f'(\zeta )]}^{2}+{[g'(\zeta )]}^{2}-{[\frac {R(\zeta )} {\zeta}]}^{2}\} d\zeta} , Q(x,y)=\frac {1} {2}{\int ^{Z}_{0} {{{[f'(\zeta )]}^{2}+{[g'(\zeta )]}^{2}-{(\frac {R} {\zeta})}^{2}\} d\zeta -\frac {{(fx)}^{2}+{(gy)}^{2}} {Z}}\} , calculate Substitute the phase modulation function After simplification, the expression for the phase modulation function is: .

[0034] Phase modulation function The phase-modulated input wavefront is obtained by combining it with the initial electric field expression of the circular Airy beam. : ,in Let be the amplitude of the electric field.

[0035] Example 1

[0036] Example 1 and Figure 1 The uncontrolled circular Airy beam was observed to generate filamentary (breathing) transport states in a nonlinear Kerr medium, forming a breathing-like structure.

[0037] In a nonlinear Kerr medium transport system, a circular Airy beam is introduced and incorporated into the Schrödinger wave equation to describe the path of the circular Airy beam. Propagation characteristics in the axial direction;

[0038] Given The circular Airy beam introduced in the design has a quadratic trajectory, which allows the beam to propagate along this predetermined path.

[0039] according to The given preset trajectory is used to calculate the phase modulation function of the circular Airy beam. ;

[0040] The phase modulation function obtained in The input wavefront is obtained by multiplying it by the initial field of the beam. ;

[0041] The target beam under phase modulation propagates along a predetermined optical trajectory in a nonlinear Kerr medium.

[0042] Example 2

[0043] Example 2 and Figure 2 Match, here the trajectory parameters are taken as , Input power It is possible to obtain, such as Figure 2 The trajectory shown is that the circular Airy beam travels downward along a parabola. As it propagates in a nonlinear Kerr medium, the beam is affected by diffraction, but it can still ensure that the main part propagates downward along the specified trajectory, demonstrating a certain degree of anti-diffraction capability.

[0044] Example 3

[0045] Example 3 and Figure 3 Match, here the trajectory parameters are taken as , Input power It is possible to obtain, such as Figure 3 The trajectory shown is that the circular Airy beam travels upward along a parabola. As it propagates in a nonlinear Kerr medium, the beam is affected by diffraction, but it can still ensure that the main part propagates upward along the specified trajectory, demonstrating a certain degree of anti-diffraction capability.

[0046] Since the two embodiments use similar generation methods, the production process for off-axis optical bottles can be used. Figure 4 To summarize.

[0047] Specifically, this invention combines a circular Airy beam with a phase modulation function in a nonlinear Kerr medium. By combining these methods, the light beam is able to propagate along a predetermined quadratic curve trajectory during its propagation, and its trajectory is analyzed in depth.

[0048] This invention employs a method for controlling the trajectory of a circular Airy beam. By using a phase control function, the trajectory of a circular Airy beam in a nonlinear Kerr medium transmission system can be well controlled.

[0049] This invention only selects the above-described embodiments and describes them in detail, including specific parameters and formulas, but this should not be construed as limiting the scope of this invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the principles of this invention, and these all fall within the scope of protection of this invention.

Claims

1. A method for realizing the regulation of the trajectory transmission of a circular Airy beam in a nonlinear Kerr medium, comprising the following steps: Under the nonlinear Kerr medium transmission system, a circular Airy beam is introduced, and it is included in the Schrödinger wave equation to describe the propagation characteristics of the circular Airy beam along the Axial direction; A quadratic curve trajectory is given to the circular Airy beam, so that the beam can propagate along this path; Computing a circular Airy beam phase control function ; Phase control function Multiplication with the beam initial field results in the input wavefront ; The target light beam is transmitted along a predetermined optical trajectory in a nonlinear Kerr medium under phase modulation. The specific modulation of the trajectory of the circular Airy beam in the nonlinear Kerr medium is: Expression of the initial electric field of the introduced circular Airy beam where is the attenuation factor, , is the translation factor, , is the radius of the main circular Airy beam ring ; In mathematical expression, the initial input is substituted into the Schrödinger wave equation: , wherein is the linear refractive index, the nonlinear coefficient of the Kerr medium , is the wave number in free space, is the wavelength, , is the normalized longitudinal propagation distance, is the Rayleigh length , , denotes the normalized transverse coordinate, is the Gaussian beam width, defined as The critical collapse power of the circular Airy beam in the nonlinear Kerr medium is ; A quadratic curve trajectory is given to a light beam in a nonlinear Kerr medium: the light beam trajectory is , the curvature of the quadratic curve trajectory is the center position of the parabolic symmetry axis is controlled by the parameter the displacement of the light beam is substituted by the calculation result is : where , , , and are the derivatives of the parametric equation, and here let ; Analytically deriving trajectory control phase control functions : Because , , , the calculated is substituted into the phase control function , and the expression form of the phase control function is obtained after arrangement and simplification: ; The phase control function Combining the initial electric field expression of the circular Airy beam : , where is the amplitude of the electric field.

Images