Airy pulse time domain evolution trajectory regulation and control method based on liquid crystal orientation design

Abstract

The invention provides an Airy pulse time domain evolution trajectory regulation and control method based on liquid crystal orientation design, and relates to the technical field of ultrafast optics. The method comprises the following steps: firstly, constructing a theoretical model of Airy pulse transmission in a liquid crystal medium, and establishing a nonlinear mapping relation between a liquid crystal molecular orientation angle and group velocity dispersion and a nonlinear mapping relation between the liquid crystal molecular orientation angle and third-order dispersion; establishing a polynomial function relationship between the dispersion parameter and the trigonometric function term of the liquid crystal molecular orientation angle by utilizing numerical fitting based on the mapping relationship; designing a spatial distribution rule of liquid crystal molecules along a propagation direction according to the target time domain trajectory, and determining dispersion distribution in a liquid crystal medium by establishing linear correlation between trigonometric function terms and propagation distances; and finally, carrying out orientation arrangement on the liquid crystal molecules. The spatial orientation gradient of the liquid crystal is designed, a dynamic dispersion environment is constructed on a transmission path, and accurate linear regulation and control of self-acceleration, self-deceleration and track bending degree of the Airy pulse are achieved.

Description

Technical Field

[0001] This invention relates to the field of ultrafast optics and light field manipulation technology, and in particular to a method for controlling the temporal evolution trajectory of Airy pulses based on liquid crystal orientation design. Background Technology

[0002] Airy pulses, as finite-energy light packets with diffraction-free, self-healing, and self-accelerating properties, have shown significant application potential in cutting-edge fields such as photonic bullet generation, particle manipulation, and plasma channel construction. In practical optical system applications, to meet specific signal synchronization, delay control, or energy transport requirements, it is often necessary to precisely tailor and customize the temporal evolution behavior of Airy pulses. The parabolic temporal trajectory of an Airy pulse is mainly determined by its initial spectral phase distribution and the dispersion characteristics of the transmission medium. In conventional isotropic media, the dispersion coefficient is usually constant, which means that once an Airy pulse is generated, its evolution trajectory is locked, making it difficult to adapt to complex and variable transmission environments.

[0003] Existing optical pulse dispersion modulation techniques mainly rely on passive devices such as grating pairs and prism pairs. Once these devices are fabricated, their dispersion structure is physically fixed, typically providing only uniform or periodic dispersion distributions, making it difficult to construct a continuously varying dispersion gradient environment along the light propagation path. Although the nonlinear effects in optical fibers can alter the pulse's dynamic behavior, this usually requires extremely high peak power from the incident light to excite the medium's nonlinear response. High-power operation not only increases system energy consumption but also easily leads to side effects such as spectral broadening, waveform splitting, or signal noise, affecting signal integrity.

[0004] Liquid crystal materials, due to their high optical birefringence and tunability, possess the physical basis for constructing non-uniform refractive index fields, making them ideal candidate media for miniaturized dispersion management. However, the orientation angle of liquid crystal molecules exhibits an extremely complex nonlinear dependence on the group velocity dispersion and third-order dispersion experienced by light waves. This poses significant theoretical challenges to directly utilizing liquid crystals for precise linearized trajectory inverse design. Currently, there is a lack of a systematic method that can transform these complex nonlinear physical quantities into simple geometric design parameters, thereby enabling on-demand customization of the time-domain behavior of finite-energy Airy pulses. Summary of the Invention

[0005] The purpose of this invention is to provide a method for controlling the time-domain evolution trajectory of Airy pulses based on liquid crystal orientation design, so as to solve the problems pointed out in the background art.

[0006] This invention provides a method for controlling the time-domain evolution trajectory of Airy pulses based on liquid crystal orientation design, comprising the following steps: Step S1: Construct a theoretical model of Airy pulse propagation in liquid crystal medium, and establish a nonlinear mapping relationship between the orientation angle of liquid crystal molecules and the group velocity dispersion and third-order dispersion of liquid crystal medium. Step S2: Based on the nonlinear mapping relationship, a polynomial function relationship is established between the group velocity dispersion and the third-order dispersion and the trigonometric function terms of the orientation angle of the liquid crystal molecules, respectively, using a numerical fitting method; Step S3: Based on the target time-domain evolution trajectory of the Airy pulse, design the spatial distribution law of the orientation angle of liquid crystal molecules along the direction of light wave propagation. By establishing the linear relationship between the trigonometric function terms and the propagation distance, determine the dispersion distribution of the liquid crystal medium along the propagation direction. Step S4: The liquid crystal molecules in the liquid crystal cell are oriented and arranged according to the spatial distribution law, so that the incident Airy pulse is dynamically dispersed and modulated when passing through the liquid crystal medium, thereby changing its temporal evolution trajectory.

[0007] Optionally, in step S1, the transmission of the Airy pulse in the liquid crystal medium follows a dispersion-dominated wave equation: ; in, For the electric field envelope, For transmission distance, For time, The imaginary unit, For group velocity dispersion, It is a third-order dispersion; The orientation angle of the liquid crystal molecules Corresponding effective refractive index satisfy: ; in, The ordinary light refractive index of the liquid crystal is... It is an unusual refractive index.

[0008] Optionally, the group velocity dispersion and third-order dispersion Determined by the following formula: ; in, Angular frequency, wavenumber From the formula Sure, It is the speed of light.

[0009] Optionally, in step S2, the polynomial function relationship includes a fitted expression for the first initial state and the second initial state: For the first initial state: ; ; For the second initial state: ; ; in, The orientation angle of the liquid crystal molecules; , respectively, are the fitting coefficients for group velocity dispersion. These are the fitting coefficients for the third-order dispersion.

[0010] Optionally, in step S3, the linear association is established with respect to the first initial state, specifically satisfying: ; in, For transmission distance, The slope parameter is used to control the dispersion distribution; the first initial state corresponds to the liquid crystal molecule orientation angle. Follow The axis gradually changes from 0 degrees to 90 degrees.

[0011] Optionally, in step S3, the linear association is established with respect to the second initial state, specifically satisfying: ; in, For transmission distance, The slope parameter is used to control the dispersion distribution; the second initial state corresponds to the liquid crystal molecule orientation angle. Follow The axis gradually changes from 90 degrees to 0 degrees.

[0012] Optionally, the propagation distance in the linear association with dimensionless distance The relationship is: ; in, To normalize the propagation distance, it is defined as follows: , The initial pulse width, The group velocity dispersion value at the input terminal.

[0013] Optionally, in step S4, the frequency domain electric field of the modulated Airy pulse... Represented as: ; in, The frequency domain distribution of the initial electric field. For frequency domain variables, and These are the phase modulation parameters.

[0014] Optionally, the phase modulation parameters and Defined by the following formula: ; ; in, as well as These are the fitting coefficients mentioned above.

[0015] Optionally, in step S4, the method for aligning the liquid crystal molecules includes any one of the following: rubbing alignment, digital micromirror device optical alignment, or laser direct writing alignment.

[0016] The present invention has achieved the following beneficial effects: This invention effectively solves the nonlinear design challenge faced in time-domain shaping of optical pulses using liquid crystal media by constructing a precise mapping model between the microscopic orientation of liquid crystal molecules and macroscopic dispersion parameters. This method simplifies complex physical dependencies into polynomial functions that are easy to calculate in engineering using numerical fitting techniques. This allows designers to construct a specific dispersion gradient distribution within the liquid crystal medium by adjusting a single linear slope parameter, thereby precisely canceling or enhancing the phase accumulation of the Airy pulse itself. This enables precise control over the time-domain evolution trajectory of self-accelerating and self-decelerating Airy pulses, i.e., free adjustment of their acceleration and trajectory curvature. Compared to control methods that rely on nonlinear effects, this invention, based on the principle of linear birefringence, is suitable for low light intensity conditions and does not introduce additional spectral noise, ensuring signal transmission quality. Furthermore, the underlying liquid crystal device has a mature fabrication process and a compact structure, providing a practical technical solution for the integration of ultrafast optical signal processing systems.

[0017] The features and advantages of the present invention will be set forth in the following description, and will be apparent in part from the description, or may be learned by practicing the invention. The objects and other advantages of the invention may be realized and obtained by means of the structures particularly pointed out in the written description and the accompanying drawings.

[0018] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0019] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used in conjunction with embodiments of the invention to explain the invention and do not constitute a limitation thereof. In the drawings: Figure 1This is a schematic diagram illustrating the temporal evolution trajectory of an Airy pulse propagating in liquid crystal media with different dispersion distributions, as described in an embodiment of the present invention. Figure 2 The diagram below is a schematic diagram of the liquid crystal molecule orientation design in an embodiment of the present invention, wherein: (a) is a schematic diagram of the liquid crystal molecule arrangement in the first initial state, (b) is a comparison diagram of the numerical fitting and theoretical calculation of the dispersion coefficient in the first initial state, (c) is a spatial distribution diagram of the liquid crystal molecule orientation angle in the first initial state, (d) is a schematic diagram of the liquid crystal molecule arrangement in the second initial state, (e) is a comparison diagram of the numerical fitting and theoretical calculation of the dispersion coefficient in the second initial state, and (f) is a spatial distribution diagram of the liquid crystal molecule orientation angle in the second initial state; Figure 3 The image shows the time-domain evolution trajectory of the Airy pulse obtained based on theoretical derivation in this embodiment of the invention, where: (a)-(d) correspond to the trajectories of the self-decelerating Airy pulse under different conditions, and (e)-(h) correspond to the trajectories of the self-accelerating Airy pulse under different conditions; Figure 4 The figure shows the simulation results of the time-domain evolution of the self-decelerating Airy pulse in the first initial state in the embodiments of the present invention (including cases without considering and considering the third-order dispersion). Figure 5 The figure shows the simulation results of the time-domain evolution of the self-decelerating Airy pulse in the second initial state in the embodiments of the present invention (including cases without considering and considering the third-order dispersion). Figure 6 The figure shows the simulation results of the time-domain evolution of the self-accelerating Airy pulse in the first initial state in the embodiments of the present invention (including cases without considering and considering the third-order dispersion). Figure 7 The figure shows the simulation results of the time-domain evolution of the self-accelerating Airy pulse in the second initial state in the embodiment of the present invention (including cases where the third-order dispersion is not considered and cases where it is considered). Detailed Implementation

[0020] The preferred embodiments of the present invention will be described below with reference to the accompanying drawings. It should be understood that the preferred embodiments described herein are for illustration and explanation only and are not intended to limit the present invention.

[0021] This invention provides a method for controlling the temporal evolution trajectory of Airy pulses based on liquid crystal orientation design. Addressing the technical challenge of flexibly, accurately, and linearly controlling the temporal evolution trajectory of Airy pulses in existing technologies, this method creatively introduces liquid crystal (LC) as the dispersion control medium. By deeply exploring the intrinsic physical relationship between the optical birefringence properties of liquid crystal materials and the dispersion effect of light pulses, this invention establishes a complete mapping system from microscopic liquid crystal molecule orientation design to macroscopic pulse temporal trajectory control. This method utilizes the specific spatial distribution of liquid crystal molecule orientation angles to construct dynamically changing group velocity dispersion (GVD) and third-order dispersion (TOD) distributions along the propagation direction. Thus, without introducing nonlinear side effects, it achieves self-acceleration, self-deceleration, and arbitrary trajectory customization of finite-energy Airy pulses. A schematic diagram of its principle is shown below. Figure 1 As shown.

[0022] Referring to the method flow of this invention, the control method mainly includes the following core steps: Construct a physical model of Airy pulse propagation and its dispersion mapping relationship; A fitting function between the dispersion coefficient and the orientation parameter was established based on numerical analysis; Design the spatial distribution pattern of liquid crystal orientation along the propagation direction; And based on this principle, liquid crystal orientation is implemented to achieve pulse modulation.

[0023] In the first stage of implementing the technical solution of this invention, it is necessary to first construct a rigorous physical model to describe the transmission behavior of Airy pulses in a non-uniform liquid crystal medium.

[0024] In a specific embodiment of the present invention, the Airy pulse refers to a finite-energy optical packet with an Airy function envelope characteristic in the time domain. It is well known that an ideal Airy packet has infinite energy, which is physically impossible; therefore, exponentially truncated finite-energy Airy pulses are used in practical applications. This pulse exhibits unique self-acceleration, self-healing, and non-dispersive (under certain conditions) characteristics during transmission. To describe the evolution of this pulse in a liquid crystal medium, the expression for the total electric field is first defined.

[0025] Assuming the light wave along Propagation along the axial direction, its total electric field This can be expressed as the time-varying electric field envelope. With carrier phase factor The product form is: ; in, Indicates the effect of propagation distance and time The slowly varying envelope function of a changing electric field; It represents the physical propagation distance along the optical axis, and its unit is usually meter (m) or centimeter (cm). This represents the time variable in the moving coordinate system, and its unit is seconds (s). The wave number at the center frequency is expressed in radians per meter (rad / m). The imaginary unit satisfies It is used to characterize the phase orthogonality in wave processes.

[0026] Under linear propagation conditions that neglect nonlinear effects of the medium (such as Kerr nonlinearity, stimulated Raman scattering, etc.) and external influences, the propagation of light pulses in isotropic or equivalent anisotropic media as described in this embodiment is mainly dominated by dispersion effects. In this case, the electric field envelope... The evolution in liquid crystal media strictly follows the dispersion-dominated wave equation. This equation is the linear form of the generalized nonlinear Schrödinger equation, and its specific mathematical expression is: ; The following is a detailed explanation of each term in the wave equation and its physical meaning: First item : Describes the electric field envelope as a function of propagation distance The rate of change of complex amplitude reflects the waveform evolution accumulation process caused by diffraction or dispersion.

[0027] Second item This describes the effect of second-order dispersion on pulse waveforms. The parameters are... Defined as the group velocity dispersion coefficient (GVD), its physical dimension is typically square seconds per meter. ). The magnitude of this value determines the degree of difference in velocity among different frequency components in the pulse spectrum, directly leading to the broadening or compression of the pulse in the time domain. When When the dispersion is normal, it is called normal dispersion; when the dispersion is normal, it is called normal dispersion. When this occurs, it is called anomalous dispersion. In this invention, by adjusting... The spatial distribution of the airy pulse can control the acceleration and deceleration behavior of the main lobe.

[0028] Third item This describes the effect of third-order dispersion on pulse waveforms. The parameters are... Defined as the third-order dispersion coefficient (TOD), its physical dimension is typically cubic seconds per meter (m). ). It mainly causes asymmetric distortion of the pulse waveform, resulting in an oscillating tail structure at the pulse edge. For ultrashort pulses (such as femtosecond pulses) or Airy pulses, the third-order dispersion is a non-negligible higher-order term, which determines the bending direction and energy flow characteristics of the Airy pulse tail.

[0029] To solve and control the aforementioned wave equation, the dispersion coefficient must be clearly defined. and The quantitative relationship between dispersion coefficient and the physical properties of the medium. According to physical optics theory, the dispersion coefficient is determined by the wavenumber. diagonal frequency Definition of higher-order derivatives: ; ; in, Angular frequency is the frequency of a light wave, measured in radians per second (rad / s). Wave number... The refractive index of the medium itself determines this. In this invention, the medium used is liquid crystal. For most nematic liquid crystals (such as the E7 liquid crystal material preferred in this embodiment), they exhibit uniaxial crystal behavior and significant optical birefringence. The wavenumber of light propagating in it... It can be represented as: ; In this formula, the parameter It specifically refers to the velocity of light in a vacuum, and its value is approximately .parameter It represents the effective refractive index experienced by light waves in a liquid crystal medium.

[0030] Therefore, the core issue becomes determining the effective refractive index. The specific form of liquid crystal materials. Liquid crystal materials are anisotropic, and their optical properties are mainly determined by the ordinary refractive index. ) and extraordinary refractive index, (Description). It is worth noting that both refractive indices are themselves frequency-dependent. This frequency dependence, a function of the refractive index, is the root cause of material dispersion. For typical nematic liquid crystals (such as E7), their refractive index in the visible and near-infrared bands typically follows the Cauchy dispersion formula: ; ; in, For the corresponding angular frequency wavelength ( ); as well as The Cauchy coefficient is determined by the specific chemical composition of the liquid crystal material. In this invention, the orientation of the liquid crystal molecules can be artificially controlled.

[0031] When a beam of linearly polarized light is incident on a liquid crystal medium, and there is an angle between its polarization direction and the director (optical axis direction) of the liquid crystal molecules... At that time, according to the refractive index ellipsoid theory in crystal optics, the effective refractive index experienced by this light wave (when propagating as an unusual e-ray) is... The following relationship must be satisfied: ; This formula clearly reveals the effective refractive index. It is the orientation angle The function of . Due to the difference in birefringence of liquid crystal molecules. Typically, the value is relatively large (e.g., between 0.2 and 0.3 for E7 LCDs), and this is achieved by changing the orientation angle. (From 0 degrees to 90 degrees), the effective refractive index and its frequency response characteristics (i.e., dispersion characteristics) can be significantly changed.

[0032] In summary, through the above theoretical derivation, this invention establishes a clear physical logic chain: the orientation angle of liquid crystal molecules. Effective refractive index Wave number Group velocity dispersion and third-order dispersion This chain demonstrates that by designing and controlling the spatial orientation angular distribution of liquid crystal molecules... This allows the desired dispersion distribution pattern to be constructed along the propagation path, thereby enabling manipulation of the time-domain evolution of the Airy pulse.

[0033] After clarifying the physical model, the embodiments of the present invention enter the crucial second stage: establishing a mathematical model that facilitates engineering design and inverse inversion. Although the above formulas can accurately calculate the dispersion value at any angle, directly applying formulas containing square roots and trigonometric functions... Substituting the expression into the wavenumber formula and then performing second and third derivatives with respect to frequency yields an exceptionally complex analytical expression containing numerous nested trigonometric and frequency terms. Such a complex analytical expression is extremely difficult to apply in practical waveform design and device fabrication due to its very low computational efficiency and its inability to intuitively reflect the control principles.

[0034] To address this technical challenge, this invention creatively proposes a simplified processing method based on numerical fitting. The core idea of ​​this method is to traverse the orientation angles using numerical calculation methods. The range of variation, to obtain a series of precise and Data points are used to find a simple functional relationship between the dispersion coefficient and the trigonometric function terms of the orientation angle.

[0035] In practical implementation, this invention establishes fitting models for two typical initial liquid crystal alignment states. These two states encompass the two basic trends of dispersion increasing and decreasing with spatial position, satisfying different requirements for self-acceleration and self-deceleration control. The specific liquid crystal alignment design and dispersion fitting results are as follows: Figure 2 As shown.

[0036] The first initial state corresponds to the liquid crystal molecule orientation vector changing from parallel to the light wave vector ( ) gradually changes to be perpendicular to the light wave vector ( The process of ), in this state, the trigonometric function terms It monotonically increases from 0 to 1.

[0037] Through the analysis of a large amount of numerical calculation data, this invention discovers group velocity dispersion. and third-order dispersion Follow The changes exhibit an excellent quadratic polynomial pattern. Therefore, this invention establishes the following fitting expression: For group velocity dispersion : ; For third-order dispersion : ; In the two formulas above, The orientation angle of the liquid crystal molecules; Right now , representing the square of the trigonometric function term.

[0038] : Represents group velocity dispersion about The quadratic fitting coefficients reflect the nonlinear curvature of the dispersion variation.

[0039] : Represents group velocity dispersion about The first-order fitting coefficient reflects the linear slope of the dispersion change.

[0040] : Represents group velocity dispersion The constant term fitting coefficient, whose physical meaning is when The group velocity dispersion value during (i.e., pure ordinary light transmission).

[0041] : Representing the third-order dispersion about The fitting coefficients for the quadratic, linear, and constant terms.

[0042] These fitting coefficients ( The specific value of the coefficient of determination depends entirely on the Cauchy coefficient of the selected liquid crystal material (such as E7) at a specific center wavelength. Through least squares methods or other optimization algorithms, extremely high-precision fitting results can be obtained. Typically greater than 0.999). For example, for an E7 LCD, at a specific wavelength, the fitted curve closely matches the theoretically calculated value over the entire range. The ranges almost completely overlap. This simplified model transforms complex differential and derivative operations into simple algebraic operations, greatly reducing the computational load.

[0043] The second initial state corresponds to the liquid crystal molecule pointing vector moving from a direction perpendicular to the light wave vector ( ) gradually changes to be parallel to the light wave vector ( The process of ). In this state, The value decreases from 1 to 0. To describe this trend and utilize mathematical symmetry, this invention employs trigonometric identities. ,choose It is used as an independent variable in modeling.

[0044] At this point, the fitting expression transforms into: For group velocity dispersion : ; For third-order dispersion : ; In the above formula, the independent variable is changed to The process of its increase from 0 to 1 corresponds to the orientation angle. The physical process of decreasing from 90 degrees to 0 degrees. Coefficient. and The physical meaning is similar to that described above, but the specific numerical value needs to be determined based on... The results are obtained by refitting or transforming the coordinates based on the baseline. This approach can uniformly handle both dispersion increase and dispersion decrease control modes, providing great flexibility for subsequent spatial distribution design.

[0045] After establishing the dispersion coefficient and orientation angle After understanding the algebraic relationships, the core step of this invention lies in designing the liquid crystal molecules along the direction of light propagation (i.e., The spatial distribution pattern of the axis. Traditional dispersion compensation devices (such as fiber gratings) usually have a fixed dispersion structure, while this invention aims to utilize the reconfigurability of liquid crystals to construct a dispersion gradient medium.

[0046] To achieve precise control over the Airy pulse evolution trajectory and to ensure that the control law possesses good analytical properties, this invention establishes a trigonometric function term ( or ) and transmission distance The linear correlation between them. It transforms complex angle control into a linear function of distance, causing the dispersion coefficient to move along... The axes exhibit a regular quadratic function distribution, thereby inducing specific phase accumulation.

[0047] For the first initial state, the orientation of the liquid crystal molecules in this embodiment is designed to satisfy the following linear relationship: ; in, This represents the actual propagation distance of the Airy pulse in the liquid crystal medium, with its value ranging from 0 to the effective length of the liquid crystal cell. This is a crucial control parameter introduced in this invention, defined as the orientation gradient parameter or dispersion control slope, with its physical dimension being reciprocal ( ). ).

[0048] The physical meaning of this formula is: the orientation angle of liquid crystal molecules along the propagation path. according to The pattern of continuous change is as follows.

[0049] when hour, The liquid crystal molecules are parallel to the optical axis.

[0050] along with The increase, It increases non-linearly.

[0051] when hour, The liquid crystal molecules are perpendicular to the optical axis.

[0052] By changing the parameters The size of the liquid crystal molecules can control how fast they rotate.

[0053] Larger This value means that within a very short distance, the liquid crystal molecules complete the deflection from parallel to perpendicular, resulting in the dispersion coefficient. and along The axis changes drastically, thus imposing significant phase modulation on the Airy pulse, causing its time-domain trajectory to bend sharply.

[0054] smaller A value of 0 indicates a gradual change in dispersion and a weaker modulation effect on pulses.

[0055] This uses a single parameter The method for regulating the entire time-domain evolution process greatly simplifies the complexity of system design.

[0056] For the second initial state, the orientation of the liquid crystal molecules in this embodiment is designed to satisfy the following linear relationship: ; This formula corresponds to the orientation angle of liquid crystal molecules. according to The regular changes.

[0057] when hour, (Notice The square of 0).

[0058] This design causes the dispersion coefficient to exhibit a gradient change along the propagation direction that is opposite to or complementary to that of the first state. It is often used to achieve a modulation effect opposite to that of the first state (for example, if the first state causes self-deceleration, the second state may lead to enhanced self-acceleration).

[0059] To make the design scheme of the present invention universal and independent of specific pulse width or absolute distance values, this embodiment introduces a dimensionless coordinate system.

[0060] Define normalized propagation distance for: ; in, The initial pulse width of the Airy pulse (e.g., full width at half maximum), in seconds; For the liquid crystal medium at the incident end ( The initial group velocity dispersion value at (). It is a characteristic length used in nonlinear optics and ultrafast optics to measure the significance of dispersion effects.

[0061] Define dimensionless distance for: ; Based on this, the above linear relationship can be rewritten in a dimensionless form: ; Substituting this relationship into the fitting formula in step S2, we can obtain the dispersion coefficient as a function of dimensionless distance. The functional relationship. For example, for group velocity dispersion. : ; This indicates that, in a dimensionless coordinate system, group velocity dispersion manifests as a distance... The quadratic function. This conclusion is crucial for the subsequent solution of the wave equation because it allows the variable-coefficient differential equation to be transformed into an integrable form, thereby obtaining an analytical solution.

[0062] After clarifying the dispersion parameters and After determining the spatial distribution of the pulse, in order to predict the control effect and guide device design, it is necessary to solve the wave equation to obtain the pulse evolution solution. In this embodiment, the analytical solution of the Airy pulse in a non-uniform liquid crystal medium is derived using frequency domain analysis.

[0063] Performing a Fourier transform on the aforementioned wave equation transforms the time-domain partial differential equation into a frequency-domain ordinary differential equation. Since the dispersion parameter varies with distance... (or The frequency domain solution is the initial spectrum multiplied by a phase factor that accumulates with distance. After rigorous integral derivation, the frequency domain electric field of the modulated Airy pulse is... It can be represented as: ; This formula precisely describes the phase accumulation process of a pulse in the frequency domain. Wherein, The frequency domain electric field distribution of the initial Airy pulse. This refers to the relative angular frequency. The exponent term in the formula... and These are the core phase modulation parameters in this invention, representing the cumulative effects of second-order and third-order dispersion along the propagation path.

[0064] It is a group of velocity dispersions Half of the integral along the propagation path (corresponding to the coefficient 1 / 2 in the wave equation). Combining the aforementioned fitting formula and linear mapping relationship, by... Perform integration to derive the parameters. The specific expression is as follows: ; In the denominator Specifically referring to the constant term fitting coefficient in the fitting formula of Example 2 In molecules This also corresponds to the coefficients of the quadratic, linear, and constant terms in the fitting formula. This expression indicates that the phase parameter... It is the distance of transmission. It is a cubic function. This means that by changing the distance... (Or by changing the control parameters) To change (At the physical scale), the second-order spectral phase of the pulse can be nonlinearly controlled.

[0065] Similarly, the phase modulation parameters caused by third-order dispersion Through the The integral is obtained, and its expression is: ; in, These are the fitting coefficients for the third-order dispersion. These parameters collectively determine the morphological evolution of the Airy pulse in the time domain.

[0066] For a self-decelerating Airy pulse, its initial time-domain electric field It is usually set as follows: ; in For Airy functions, This is the cutoff factor (attenuation constant) used to ensure finite pulse energy. It is obtained through Fourier transform. Then, by substituting the above frequency domain evolution equation and then by inverse Fourier transform, the time domain waveform of the pulse at any position can be obtained.

[0067] According to group delay theory, the temporal position (i.e., time domain trajectory) of the pulse main lobe is primarily determined by the first derivative of the spectral phase with respect to frequency. This invention introduces an additional phase term. This will introduce a time domain that is different from the time domain. Proportional time displacement. Due to It is distance a cubic function (and subject to) (Value modulation), which means that the time-domain trajectory of the pulse will contain The cube and square components. Through fine adjustment... The value can change the weight of these components, thereby straightening or even reversing the originally parabolic acceleration trajectory of the Airy pulse, achieving a dynamic switch from self-acceleration to self-deceleration.

[0068] The frequency domain electric field expression of the Airy pulse in a non-uniform liquid crystal medium was obtained through the aforementioned embodiments. Subsequently, this invention further solves for its analytical expression in the time domain using the inverse Fourier transform. This step is crucial for rigorously proving the control mechanism described in this invention mathematically, and also provides engineers with a theoretical tool for quickly predicting pulse behavior without time-consuming simulations.

[0069] For the self-decelerating Airy pulse defined above, its frequency domain expression is... It contains two phase factors: one is the initial cubic phase of the Airy pulse itself (corresponding to its self-acceleration / deceleration characteristics), and the other is the additional phase modulation introduced by the liquid crystal medium. In order to solve the time-domain solution ,right Perform the inverse Fourier transform integral: ; In one specific embodiment of the present invention, the neglect of third-order dispersion (i.e.) is first considered. In the case of [missing information], we only consider the effect of the non-uniform distribution of group velocity dispersion (GVD) on the pulse. Using the integral definition of the Airy function, i.e. Through rigorous integral variable substitution and complex function derivation, the time-domain envelope expression of the self-decelerating Airy pulse in a non-uniform liquid crystal medium can be obtained: ; This formula reveals that the pulse envelope still retains the Airy function. The form has changed, but its independent variable has changed significantly. In the formula... For normalized moving time coordinates, To normalize the propagation distance, This is the cutoff factor.

[0070] Based on the aforementioned time-domain wavefunction, this invention extracts the temporal evolution trajectory of the pulse's main lobe center. The position of the main lobe of an Airy pulse is determined by the real part of the Airy function being zero (or an extreme point). According to the above formula, the temporal trajectory of the pulse's main lobe can be extracted. The following relationship must be satisfied: ; Please note that here It is not a constant, but rather a distance as defined in the preceding embodiments. cubic function To avoid symbol redundancy, we will use [symbol name missing] consistently here. The constant term fitting coefficients represent the group velocity dispersion. Substituting the specific expression into the above equation, we obtain the crucial time-domain evolution trajectory control equation: ; After sorting, we get: ; This equation clearly shows the time-domain location of the pulse. No longer just distance parabolic function ( This is the standard trajectory of a traditional Airy pulse in a homogeneous medium, but it is superimposed with a fitting coefficient of liquid crystal orientation. and gradient parameters (Implication in) In the definition, that is The higher-order polynomial terms jointly determined by )

[0071] By changing the liquid crystal alignment gradient It can change physical distance The proportional relationship, thus in Positive or negative offsets are superimposed on the item, or even introduced. and This means that not only can the acceleration of the pulse (quadratic coefficient) be changed, but also jerk (Jerk, cubic coefficient) can be introduced, thereby achieving arbitrary customization of the pulse curvature. For example, when the deceleration effect provided by the liquid crystal is stronger than the inherent acceleration effect of the Airy pulse, the overall trajectory of the pulse will be reversed.

[0072] When further considering the third-order dispersion (TOD, i.e.) When this happens, the analytical solution becomes more complex. At this point, parameters are introduced. As a correction factor for the coefficients of the cubic term in the frequency domain, the final trajectory equation is corrected as follows: ; in It is determined by the third-order dispersion coefficient The determined perturbation term. This result shows that although third-order dispersion introduces waveform distortion (such as tailing oscillations), the perturbation term proposed in this invention based on second-order dispersion... The dominant regulatory mechanism remains in a leading position, ensuring the robustness of the regulatory measures.

[0073] For a self-accelerating Airy pulse, its initial spectral distribution has a phase conjugate relationship with that of a self-decelerating pulse (i.e., the spectral phase signs are opposite). Applying the same inverse Fourier transform method, its time-domain trajectory equation can be obtained: ; Note the difference in the symbols here. This indicates that the same liquid crystal dispersion structure has diametrically opposed effects on self-accelerating and self-decelerating pulses. If the liquid crystal structure provides... If the value is positive with increasing distance, it will increase the time delay of the self-decelerating pulse (making it slower) while decreasing the time delay of the self-accelerating pulse (counteracting its original acceleration trend). This differential control characteristic is another major technical highlight of this invention, enabling the same liquid crystal device to exhibit selective control functions for different types of Airy pulses, with different control parameters calculated through theoretical formulas. The time-domain trajectory below is as follows Figure 3 As shown.

[0074] To verify the accuracy of the above analytical derivation and further explore the impulse behavior under complex conditions that are difficult to handle with analytical solutions, this embodiment employs a step-by-step Fourier transform method for full numerical simulation. This simulation is not only a means of theoretical verification but also strong evidence of the practical feasibility of the technical solution of this invention.

[0075] The split-step Fourier method is a standard numerical method for solving the nonlinear Schrödinger equation and its linear variants (such as the wave equation described in this case). Its core idea is to consider the medium along the propagation direction. Divide into several tiny steps. Within each step, it is assumed that the dispersion and diffraction effects can act independently.

[0076] The specific implementation steps are as follows: The first step is to initialize the parameters. Set the initial pulse width (full width at half maximum, FWHM) of the Airy pulse to... Femtosecond (fs), cutoff factor The center wavelength of the light wave was chosen to be 800nm ​​(a typical wavelength for Ti: sapphire lasers).

[0077] The second step is to set the parameters of the liquid crystal material. A typical nematic liquid crystal, E7, is selected, with a typical refractive index... unusual light refractive index The value was obtained using Cauchy's formula at 800 nm. Input group velocity dispersion value. Determined based on the initial orientation state.

[0078] The third step is mesh generation. The total propagation distance (e.g., a physical length of 2cm) is divided into... to A step size is used to ensure numerical convergence accuracy. The time window is set to cover the pulse main lobe and long tail, for example... The number of sampling points is set to 4096 to prevent spectral aliasing.

[0079] The fourth step is iterative calculation. For the current electric field... Performing a Fast Fourier Transform (FFT) yields... Apply a dispersion operator in the frequency domain ,in and It is the spatial distribution function designed according to the present invention ( or In the current The points are calculated in real time; an inverse fast Fourier transform (IFFT) is performed to return them to the time domain; the distance is updated. Repeat the above steps until the endpoint is reached.

[0080] In the first initial state, liquid crystal molecules move from a position parallel to the wave vector ( Gradually change to vertical ( Detailed simulations were performed for both cases with and without third-order dispersion.

[0081] No third-order dispersion case (GVD only): Simulation results show that, with the adjustment parameters from Increase to The trajectory of the self-decelerating Airy pulse changed significantly, such as Figure 4 As shown.

[0082] when In a uniform medium, the pulse follows a standard parabolic trajectory, exhibiting typical self-deceleration behavior.

[0083] when At this time, the gradient dispersion introduced by the liquid crystal layer begins to take effect, the curvature of the pulse trajectory decreases, and the deceleration trend slows down.

[0084] when At this point, the main lobe trajectory of the pulse becomes noticeably straighter, and even shows a tendency to bend in the opposite direction at the end. This indicates that the dispersion gradient introduced by the liquid crystal effectively counteracts the self-deceleration effect of the Airy pulse portion, achieving strong intervention in the time-domain evolution.

[0085] For self-accelerating Airy pulses, such as Figure 6 As shown, add This value leads to a greater curvature of its trajectory (faster acceleration). This is because the positive dispersion accumulation provided by the liquid crystal layer is superimposed with the accelerating phase of the pulse itself, thus enhancing the acceleration effect.

[0086] Case including third-order dispersion (GVD+TOD): Introduction Subsequently, simulation images showed some subtle asymmetric oscillations at the edge of the main lobe of the pulse, and a slight change in the energy distribution of the trailing phase. However, from the overall trajectory, the motion path of the main lobe center was similar to that without... The timing is highly coincident. This strongly proves the basis proposed in this invention. The dominant control strategy remains effective in real-world physical systems with high-order dispersion interference. Superimposing the trajectory curves (solid lines) calculated analytically onto the simulated contour maps (pseudo-color maps) shows extremely high agreement (error less than 1%) across the entire propagation distance. Only in... When the value is very large and close to the end of propagation, due to the theoretical derivation... The use of initial value approximation resulted in slight deviations, but this does not affect the effectiveness of the overall regulation pattern.

[0087] In the second initial state, liquid crystal molecules move from vertical ( Gradually change to parallel ( At this point, the direction of the dispersion gradient is opposite to that in the first state. Simulation results show that: Self-decelerating Airy pulse As the value increases, its deceleration trend is further amplified, and the trajectory curves more sharply, such as... Figure 5 As shown.

[0088] Self-accelerating Airy pulse As the value increases, its accelerating trend is suppressed, and the trajectory tends to flatten, such as... Figure 7 As shown.

[0089] This result verifies that the present invention can be achieved by simply flipping the orientation boundary conditions of the liquid crystal cell (from...). Become This allows for the reverse control of pulse dynamics, greatly enhancing the functional flexibility of the device. A comprehensive comparison of the two scenarios leads to the conclusion that the parameters... The effects of self-deceleration and self-acceleration pulses are complementary, and this characteristic provides a theoretical basis for constructing multifunctional dispersion control devices.

[0090] In practical implementation, the fabrication process for achieving the specific spatial orientation distribution of liquid crystal molecules described above can employ mature technologies in this field. Depending on the different requirements for orientation accuracy and resolution, the specific implementation methods that can be selected include, but are not limited to: a light-controlled orientation method based on a digital micromirror device (DMD) capable of achieving micrometer-level high-precision nonlinear orientation; a variable-angle friction orientation method suitable for large-area gradually varying gradient designs; and a laser direct-write orientation method suitable for ultra-high resolution complex structures. Those skilled in the art only need to ensure that the liquid crystal molecules within the liquid crystal cell strictly conform to the design of this invention. The spatial distribution pattern can be followed. As for the specific substrate cleaning, alignment agent coating, cell pressing and liquid crystal injection and other conventional device packaging processes, they can all be achieved using conventional methods in this field, and will not be elaborated here.

[0091] In summary, the Airy pulse time-domain evolution trajectory control method based on liquid crystal orientation design provided by this invention has the following outstanding substantive features and significant progress: This invention is based entirely on the linear refractive index modulation of liquid crystals, without relying on high-intensity nonlinear effects. Therefore, it is applicable to various light intensity ranges from single-photon levels to high energy levels, and does not introduce additional nonlinear noise or frequency conversion, thus ensuring the purity of the signal.

[0092] By leveraging the designability of liquid crystal orientation, the complex problem of dispersion control is simplified to a single parameter. The geometric design. By changing The value can be continuously controlled from self-accelerating enhancement to self-decelerating reversal within the same material system.

[0093] The fabrication process provided by this invention is based on mature photo-alignment technology, does not require expensive electron beam exposure equipment, has the potential for low-cost, large-area fabrication, and is easy to promote and apply.

[0094] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A method for controlling the time-domain evolution trajectory of Airy pulses based on liquid crystal orientation design, characterized in that, Includes the following steps: Step S1: Construct a theoretical model of Airy pulse propagation in liquid crystal medium, and establish a nonlinear mapping relationship between the orientation angle of liquid crystal molecules and the group velocity dispersion and third-order dispersion of liquid crystal medium. Step S2: Based on the nonlinear mapping relationship, a polynomial function relationship is established between the group velocity dispersion and the third-order dispersion and the trigonometric function terms of the orientation angle of the liquid crystal molecules, respectively, using a numerical fitting method; Step S3: Based on the target time-domain evolution trajectory of the Airy pulse, design the spatial distribution law of the orientation angle of liquid crystal molecules along the direction of light wave propagation. By establishing the linear relationship between the trigonometric function terms and the propagation distance, determine the dispersion distribution of the liquid crystal medium along the propagation direction. Step S4: The liquid crystal molecules in the liquid crystal cell are oriented and arranged according to the spatial distribution law, so that the incident Airy pulse is dynamically dispersed and modulated when passing through the liquid crystal medium, thereby changing its temporal evolution trajectory.

2. The method for controlling the time-domain evolution trajectory of Airy pulses based on liquid crystal orientation design according to claim 1, characterized in that, In step S1, the transmission of the Airy pulse in the liquid crystal medium follows a dispersion-dominated wave equation: ； in, For the electric field envelope, For transmission distance, For time, The imaginary unit, For group velocity dispersion, It is a third-order dispersion; The orientation angle of the liquid crystal molecules Corresponding effective refractive index satisfy: ； in, The ordinary light refractive index of the liquid crystal is... It is an unusual refractive index.

3. The method for controlling the time-domain evolution trajectory of Airy pulses based on liquid crystal orientation design according to claim 2, characterized in that, The group velocity dispersion and third-order dispersion Determined by the following formula: ； in, Angular frequency, wavenumber From the formula Sure, It is the speed of light.

4. The method for controlling the time-domain evolution trajectory of Airy pulses based on liquid crystal orientation design according to claim 1, characterized in that, In step S2, the polynomial function relationship includes a fitted expression for the first initial state and the second initial state: For the first initial state: ； ； For the second initial state: ； ； in, The orientation angle of the liquid crystal molecules; , respectively, are the fitting coefficients for group velocity dispersion. These are the fitting coefficients for the third-order dispersion.

5. The method for controlling the time-domain evolution trajectory of Airy pulses based on liquid crystal orientation design according to claim 4, characterized in that, In step S3, the linear association is established for the first initial state, specifically satisfying: ； in, For transmission distance, The slope parameter is used to control the dispersion distribution; the first initial state corresponds to the liquid crystal molecule orientation angle. Follow The axis gradually changes from 0 degrees to 90 degrees.

6. The method for controlling the time-domain evolution trajectory of Airy pulses based on liquid crystal orientation design according to claim 4, characterized in that, In step S3, the linear association is established with respect to the second initial state, specifically satisfying: ； in, For transmission distance, The slope parameter is used to control the dispersion distribution; the second initial state corresponds to the liquid crystal molecule orientation angle. Follow The axis gradually changes from 90 degrees to 0 degrees.

7. A method for controlling the time-domain evolution trajectory of Airy pulses based on liquid crystal orientation design according to claim 5 or 6, characterized in that, Propagation distance in the linear correlation with dimensionless distance The relationship is: ； in, To normalize the propagation distance, it is defined as follows: , The initial pulse width, The group velocity dispersion value at the input terminal.

8. The method for controlling the time-domain evolution trajectory of Airy pulses based on liquid crystal orientation design according to claim 7, characterized in that, In step S4, the frequency domain electric field of the modulated Airy pulse Represented as: ； in, The frequency domain distribution of the initial electric field. For frequency domain variables, and These are the phase modulation parameters.

9. The method for controlling the time-domain evolution trajectory of Airy pulses based on liquid crystal orientation design according to claim 8, characterized in that, The phase modulation parameters and Defined by the following formula: ； ； in, as well as These are the fitting coefficients mentioned above.

10. The method for controlling the time-domain evolution trajectory of Airy pulses based on liquid crystal orientation design according to claim 1, characterized in that, In step S4, the method for aligning liquid crystal molecules includes any one of the following: rubbing alignment, digital micromirror device optical alignment, or laser direct writing alignment.

Images