A method of controlling the movement of a focused spatiotemporal wave packet in three-dimensional space

Abstract

The application discloses a method for controlling movement of a focused space-time wave packet in a three-dimensional space, comprising the following steps: S1, obtaining spherical aberration and distortion by theoretically analyzing primary aberration, so that the focal field moves along the optical axis direction of the focusing space and the focal field is offset along the x-axis and y-axis of the focusing space; S2, pre-splitting the incident wave packet and introducing spherical aberration function and distortion function into the incident wave packet; S3, focusing the incident wave packet by an objective lens to obtain a focused space-time vortex wave packet which can move in the three-dimensional space; and S4, changing the related coefficients in the spherical aberration function and the distortion function to realize arbitrary adjustment of the spatial position of the focused space-time wave packet in the three-dimensional space. According to the application, spherical aberration and distortion are comprehensively used to make the focused space-time wave packet move in any direction in the three-dimensional space, and the application has potential application in the fields of optical trapping, laser processing, light and matter interaction and the like.

Description

Technical Field

[0001] This invention relates to the field of micro-nano optics, and in particular to a method for controlling the movement of a focusing spatiotemporal wave packet in three-dimensional space. Background Technology

[0002] Photonic angular momentum is an important research area in nanophotonics, with wide applications in high-capacity, ultra-high-speed optical communication, quantum information processing, optical tweezers, super-resolution microscopy, and spin-orbit coupling. In the spatial domain, the orbital angular momentum of an optical field can be represented by a vortex wavefront. It means that among them It is the azimuth angle within the cross-section of the beam, and l is called the topological charge. With the rapid development of optical field manipulation theory and technology, people's understanding of photon angular momentum is also constantly deepening. A large number of theories and experiments have proven the existence of non-zero transverse spin angular momentum, and by introducing phase changes in the time domain, pure transverse orbital angular momentum can be obtained, and the resulting vortex beam rotates around an axis perpendicular to its propagation direction.

[0003] In 1856, German mathematician Seidel analyzed the sources of five lens aberrations, known as the Seidel Five Aberrations. Spherical aberration refers to the optical phenomenon where parallel light rays passing through the edge of a lens have a focal point closer to the lens, while rays passing through the center have a focal point farther away, preventing the light from converging at an ideal focal point. Distortion, on the other hand, occurs because the transverse magnification of the principal ray at the same position is not constant. Refraction, scattering, and reflection occur when light passes through lenses and other optical systems, leading to a decrease in image quality. Distortion is related to the field of view of the imaging system. While it does not affect image sharpness, it causes bending and distortion of the image on the focal plane. The Zernike polynomial has a good correspondence with classical aberrations (such as astigmatism, coma, and spherical aberration), allowing for rapid classification and quantification of wavefront aberrations. A comprehensive and accurate description of wavefront aberrations allows for effective analysis of the focused light field affected by these aberrations. By adjusting the aberration coefficients in the aberration function, different degrees of aberration can be represented. MAGonzález-Galicia et al. calculated the phase changes caused by lens aberrations using the Seidel aberration theory of thin lenses. They analyzed the effects of aberrations such as spherical aberration, coma, astigmatism, field curvature, and distortion on ultrashort pulse focusing under uniform illumination. Rakesh Kumar Singh et al. used vector Debye integrals to study the effects of primary aberrations on tight focusing of vector Laguerre-Gaussian beams, presenting a series of new phenomena regarding the polarization singularity index and phase changes of the focused field of Laguerre-Gaussian beams caused by aberration factors. Currently, there are many studies on the effects of aberrations on tight focusing fields, but few studies utilize aberrations containing phase information as a method for controlling the optical field, and there is also a lack of research on using aberrations to move the three-dimensional position of the spatiotemporal wave packet in strong focusing. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the purpose of this invention is to provide a method for controlling the movement of a focused spatiotemporal wave packet in three-dimensional space. This method enables the movement of the focused spatiotemporal wave packet within the focusing region, enriching the methods for controlling the spatiotemporal light field. It can be widely applied in fields such as light-matter interaction, optical information processing, data storage, and information encryption. To achieve the above-mentioned objectives and other advantages of this invention, a method for controlling the movement of a focused spatiotemporal wave packet in three-dimensional space is provided, comprising:

[0005] S1. Through theoretical analysis of primary aberrations, it is found that spherical aberration causes the focal field to shift along the optical axis of the focusing space, and distortion causes the focal field to shift along the x-axis and y-axis of the focusing space.

[0006] S2. By pre-splitting the incident wave packet, and simultaneously introducing spherical aberration function and distortion function into the incident wave packet;

[0007] S3. Focus the incident wave packet through the objective lens to obtain a focused spatiotemporal wave packet that can move in three-dimensional space;

[0008] S4. By changing the correlation coefficients in the spherical aberration function and the distortion function, the spatial position of the focused spatiotemporal wave packet can be arbitrarily adjusted in three-dimensional space.

[0009] This method involves moving a focused spatiotemporal wave packet in three-dimensional space under objective lens focusing. Based on the influence of optical aberrations on the spatial position of the focused spatiotemporal wave packet, spherical aberration and distortion are introduced to achieve three-dimensional movement of the spatiotemporal wave packet with lateral orbital angular momentum within the focusing space of a compact focusing system. Spherical aberration is used to move the focused spatiotemporal wave packet along the optical axis, while distortion is used to move the focused spatiotemporal wave packet along the x-axis and y-axis.

[0010] Compared with existing technologies, the advantages of this invention are: it provides a three-dimensional spatial position movement method for focused spatiotemporal wave packets generated under strong focusing conditions. There is a strong correlation between the three aberration coefficients and their corresponding offsets, and they are independent of each other, providing a convenient and efficient method for controlling the position of spatiotemporal vortex light fields with pure lateral orbital angular momentum. It has enormous application potential in fields such as microscopic imaging, optical tweezers, and laser processing.

[0011] This invention has minimal impact on the spatiotemporal spiral phase of the spatiotemporal focal field, and aberration factors do not affect the topological charge number of the focusing spatiotemporal wave packet. It can preserve the spatiotemporal spiral phase while moving the focusing spatiotemporal wave packet.

[0012] This invention has strong scalability. The method proposed in this invention can be applied to spatiotemporal optical fields across the entire wavelength range. Attached Figure Description

[0013] Figure 1 This is a schematic diagram of the method for moving and focusing spatiotemporal wave packets in three-dimensional space according to the present invention;

[0014] Figure 2 The diagram shows the intensity (a1)-(c1) and phase distribution (a2)-(c2) of the focused spatiotemporal wave packet at different positions on the optical axis under different spherical aberration coefficients in this invention, as well as the fitting curve (d) of the spherical aberration coefficient and the offset Δz; the spherical aberration coefficients in (a)-(c) are 0.3λ, 0λ, and -0.3λ, respectively; the isosurface plot of the focused spatiotemporal wave packet at 0.3 times the peak intensity is also given in (a1)-(c1); and the enlarged diagram of the nonlinear part of the relationship between the displacement and the spherical aberration coefficient is also given in (d).

[0015] Figure 3The following are the intensity (a1)-(b1) and phase distribution (a2)-(b2) of the focusing spatiotemporal wave packet focused at different positions on the focal plane under different x-distortion coefficients in this invention, as well as the fitting curve (c) of the x-distortion coefficient and the offset Δx; the x-distortion coefficients in (a) and (b) are 0.5λ and 1λ, respectively, and the contour plot of the focusing spatiotemporal wave packet at 0.3 times the peak intensity is also given in (a1) and (b1);

[0016] Figure 4 The following are the intensity (a1)-(b1) and phase distribution (a2)-(b2) of the focusing spatiotemporal wave packet focused at different positions on the focal plane under different y distortion coefficients in this invention, as well as the fitting curve (c) of the y distortion coefficient and the offset Δy; the y distortion coefficients in (a) and (b) are 0.5λ and 1λ, respectively, and the contour plot of the focusing spatiotemporal wave packet at 0.3 times the peak intensity is also given in (a1) and (b1);

[0017] Figure 5 The graph shows the relationship between the aberration coefficients and the three-dimensional displacements Δx, Δy, and Δz of the focusing spatiotemporal wave packet when the three aberrations are combined. Detailed Implementation

[0018] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] Reference Figure 1 A method for controlling the movement of a focusing spatiotemporal wave packet in three-dimensional space includes the following steps:

[0020] S1. Through theoretical analysis of primary aberrations, it is found that spherical aberration causes the focal field to shift along the optical axis of the focusing space, and distortion causes the focal field to shift along the x-axis and y-axis of the focusing space.

[0021] Furthermore, the aberration function is described using Zernike polynomials. The Zernike polynomials under Born and Wolf exponents can be written as:

[0022]

[0023] Where n and m represent the degree of the radial polynomial and the frequency of the angular function, respectively. It is the azimuth angle measured counterclockwise from the positive x-axis. ρ is the normalized radial coordinate, 0 ≤ ρ ≤ 1. Radial polynomial Represented as:

[0024]

[0025] During focusing, sinθ = ρsinα, where θ is the ray cone angle during focusing, α is the maximum value of the ray cone angle, and α = arcsin(NA), where NA is the numerical aperture of the objective lens. In the Zernike polynomial representation of aberrations, the functional expressions for primary spherical aberration, x-distortion, and y-distortion can be obtained as follows:

[0026]

[0027] It can be seen that the x-distortion and y-distortion are orthogonal to each other, and the real number A s A dx and A dy The intensity of an aberration is represented by its coefficient, called the aberration coefficient. Since spherical aberration causes focus shift at the focal point, the spherical aberration function can be used to address this. This allows the focal field to be moved along the optical axis in the focusing space; simultaneously, the x-distortion function can be used. and y distortion function This causes the focal field to shift along the x and y axes of the focusing space. Spherical aberration and distortion affect the wavefront of the incident wave packet, introducing additional phase factors into the incident wave packet. for:

[0028]

[0029] Where k = 2π / λ is the wave number, and λ is the center wavelength of the incident wave packet.

[0030] S2. The incident wave packet is pre-splitting, and spherical aberration and distortion functions are introduced into the incident wave packet simultaneously. Further, the incident wave packet needs to be pre-splitting to overcome spatiotemporal astigmatism during focusing, and then an additional phase factor is applied. Apply to the pre-splitting incident wave packet.

[0031] S3. Focus the incident wave packet through the objective lens to obtain a focused spatiotemporal wave packet that can move in three-dimensional space; establish a three-dimensional coordinate system in the focusing space with the focal point of the incident wave packet after focusing by the objective lens when there is no aberration as the origin, the direction of light field propagation is the z direction, and the plane passing through the focal point and orthogonal to the optical axis is the xy plane.

[0032] S4. By changing the correlation coefficients in the spherical aberration function and distortion function, the spatial position of the focused spatiotemporal wave packet can be arbitrarily adjusted in three-dimensional space. In the three-dimensional focusing space, the relationship between the spherical aberration coefficient and the displacement of the focused spatiotemporal wave packet along the z-direction is determined; the relationship between the x-distortion coefficient and the displacement of the focused spatiotemporal wave packet along the x-direction is determined; and the relationship between the y-distortion coefficient and the displacement of the focused spatiotemporal wave packet along the y-direction is determined. The method for determining the above relationships is as follows: when the incident wave packet is only affected by spherical aberration, the position of the focused spatiotemporal wave packet in the propagation direction is found, and its distance from the origin of the focusing space is the offset Δz corresponding to the spherical aberration coefficient; when only x-distortion or y-distortion factors are involved, the focused spatiotemporal wave packet will exhibit offset along the x-axis or y-axis. The offset Δx or Δy of the focused spatiotemporal wave packet along the x-axis or y-axis under different distortion coefficients is analyzed and obtained.

[0033] Example 1

[0034] like Figure 1 As shown, after an incident wave packet containing aberrations is focused by an objective lens, the three-dimensional spatial position of the focused spatiotemporal wave packet can be moved within the focusing space by changing the corresponding aberration coefficients. The intensity and phase distribution of the focused spatiotemporal wave packet in the spatiotemporal domain can be calculated using Richard Wolf vector diffraction integrals. Here, the specific implementation of the technical solution is illustrated using the movement of the focused spatiotemporal wave packet in the xyz direction as an example, including the following steps:

[0035] Step 1: Use primary ball difference x distortion and Y distortion The three-dimensional spatial position of the focusing spatiotemporal wave packet is controlled by the function expressions shown in equations (3)-(5), and the additional phase brought by these three aberrations to the incident wave packet is shown in equation (6).

[0036] Step 2: Taking the generation of a spacetime vortex carrying a transverse orbital angular momentum of -1 with a topological charge number in the focusing space as an example, the required incident wave packet after pre-splitting processing is:

[0037]

[0038] Among them, w p Let w be the waist radius of the incident wave packet in the spatial domain. t The incident wave packet's intensity decreases to 1 / e of its peak intensity in the time domain. 2 The half-pulse width at that time. Introducing spherical aberration and distortion functions into the incident wave packet, the incident wave packet then becomes...

[0039] Step 3: To obtain the focused spatiotemporal wave packet with three-dimensional positional movement in the focused space, the field distribution of the focused spatiotemporal wave packet is calculated using Richard Wolf vector diffraction integral:

[0040]

[0041] Where B(θ) is the apodization function of the objective lens, and here we use a sinusoidal objective lens. In this embodiment, NA is 0.9.

[0042] By setting the x-distortion coefficient and y-distortion coefficient to 0λ, the movement of the focusing spatiotemporal wave packet is analyzed when only spherical aberration exists. Figure 2 (a) shows that when the spherical aberration coefficient is 0.3λ, the focusing spatiotemporal wave packet is shifted by -0.5λ along the z-axis relative to the origin of the coordinate system in the focusing space. Figure 2 (b) is the focusing spatiotemporal wave packet when there is no spherical aberration (i.e., the spherical aberration coefficient is 0λ), at which point the wave packet has no offset in the focusing space; Figure 2 (c) indicates that when the spherical aberration coefficient is -0.3λ, the focusing spatiotemporal wave packet is shifted by 0.5λ along the z-axis relative to the origin of the coordinate system in the focusing space. Figure 2 (a2)-2(c2) presents the phase distribution of the focusing spacetime wave packet in the spacetime domain under three different conditions. It can be seen that during the movement along the z-axis, the focusing spacetime wave packet always carries a transverse orbital angular momentum with a topological charge of +1 in its xt plane. Furthermore, a summary analysis of more spherical aberration coefficients yields... Figure 2 (d) Offset Δz and spherical aberration coefficient A s The relationship curve. When the spherical aberration coefficient |A s When | is greater than 0.1λ, the offset Δz and the spherical aberration coefficient A s It exhibits a good linear relationship. Specifically, Δz = -1.5A s +0.05, A s ∈[-1,-0.1); Δz=-1.5A s -0.05, A s ∈(0.1,1). When the spherical aberration coefficient is in the range of [-0.1λ,0.1λ], the above linear relationship no longer holds, and instead exhibits nonlinear characteristics.

[0043] By setting the spherical aberration coefficient and the y-distortion coefficient to 0λ, we analyze the movement of the focusing spatiotemporal wave packet when only x-distortion exists. Figure 3 (a) shows that when the x-distortion coefficient is 0.5λ, the focusing spatiotemporal wave packet moves 0.56λ in the positive x-axis direction in the focusing space; Figure 3 (b) shows that when the x distortion coefficient is 1λ, the focusing spatiotemporal wave packet moves 1.12λ in the positive x-axis direction in the focusing space. Figure 3(a2) and 3(b2) present the phase distribution of the focusing spacetime wave packet in the spacetime domain for these two cases. It can be seen that during the movement along the x-axis, the focusing spacetime wave packet always carries a transverse orbital angular momentum with a topological charge of +1 in its xt plane. Furthermore, a summary analysis of more x-distortion coefficients yields... Figure 3 (c) Offset Δx and x distortion coefficient A dx The relationship curve is shown. It can be seen that the offset distance Δx of the focused spatiotemporal wave packet changes with the x-distortion coefficient A. dx The increase is due to the increase of [a], and they are approximately linearly related, satisfying Δx = p1A. dx +p2, where p1 = 1.107, p2 = 1.994 × 10 -17 .

[0044] By setting the spherical aberration coefficient and x-distortion coefficient to 0λ, we analyze the movement of the focusing spatiotemporal wave packet when only y-distortion exists. Figure 4 (a) shows that when the y distortion coefficient is 0.5λ, the focusing spatiotemporal wave packet moves 0.56λ in the positive y-axis direction within the focusing space; Figure 4 (b) shows that when the y distortion coefficient is 1λ, the focusing spatiotemporal wave packet moves 1.12λ in the positive y-axis direction within the focusing space. Figure 4 (a2) and 4(b2) present the phase distribution of the focusing spacetime wave packet in the spacetime domain for these two cases. It can be seen that during the movement along the y-axis, the focusing spacetime wave packet always carries a transverse orbital angular momentum with a topological charge of +1 in its xt plane. Furthermore, a summary analysis of more y-distortion coefficients yields... Figure 4 (c) Offset Δy and y distortion coefficient A dy The relationship curve is shown below. At this point, the offset Δy and the y-distortion coefficient A are... dy Similarly, the relationship is approximately linear; specifically, we can obtain Δy = p3A. dy +p4, where p3 = 1.107, p4 = 1.994 × 10 -17 .

[0045] Based on the above-obtained correspondence between a single aberration and the focusing spatiotemporal wave packet offset, in order for the focusing spatiotemporal wave packet to offset simultaneously in the x, y, and z directions in the focusing space, the three aberrations need to work together. Figure (5) shows the correspondence between the three-dimensional spatial offset of the focusing spatiotemporal wave packet and each coefficient when the three aberrations exist simultaneously, indicating that the focusing spatiotemporal wave packet is offset in the x, y, and z directions.

[0046] In summary, after the incident wave packet with additional spherical aberration, x-distortion, and y-distortion is focused by the objective lens, it is possible to achieve arbitrary movement of the focused spatiotemporal wave packet in three dimensions within the focused space, while maintaining the lateral orbital angular momentum and topological charge of the focused spatiotemporal wave packet unchanged during the movement.

[0047] The number of devices and processing scale described herein are for simplification of the invention. Applications, modifications, and variations of this invention will be readily apparent to those skilled in the art. Although embodiments of the invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. It can be applied to various fields suitable for this invention, and further modifications can be readily implemented by those skilled in the art. Therefore, without departing from the general concept defined by the claims and their equivalents, this invention is not limited to the specific details and illustrations shown and described herein.

Claims

1. A method for controlling the movement of a focusing spatiotemporal wave packet in three-dimensional space, characterized in that, include: S1. Through theoretical analysis of primary aberrations, it is found that spherical aberration causes the focal field to shift along the optical axis in the focusing space, and distortion causes the focal field to shift along the x and y axes of the focusing space. The aberration functions are described by Zernike polynomials, and Zernike polynomials expressed by Born and Wolf exponents are used. In the aberrations expressed by Zernike polynomials, the functions of primary spherical aberration, x-distortion, and y-distortion are obtained. In the aberrations expressed by Zernike polynomials, it is found that x-distortion and y-distortion are orthogonal to each other, and spherical aberration and distortion affect the wavefront of the incident wave packet, thus introducing an additional phase factor to the incident wave packet. The Zernike polynomial under Born and Wolf exponents can be written as: Where n and m represent the degree of the radial polynomial and the frequency of the angular function, respectively. It is the azimuth angle measured counterclockwise from the positive x-axis. ρ is the normalized radial coordinate, 0 ≤ ρ ≤ 1; radial polynomial Represented as: During the focusing process, sinθ=ρsinα, θ is the light cone angle during the focusing process, α is the maximum value of the light cone angle, α=arcsin(NA), NA is the numerical aperture of the objective lens; In the Zernike polynomial representation of aberrations, the functional expressions for primary spherical aberration, x-distortion, and y-distortion can be obtained as follows: It can be seen that the x-distortion and y-distortion are orthogonal to each other, and the real number A s A dx and A dy These are the spherical aberration coefficient, x-distortion coefficient, and y-distortion coefficient, representing the intensity of the aberration. Since spherical aberration causes focal shift at the focal point, the spherical aberration function is used... This allows the focal field to be moved along the optical axis in the focusing space; simultaneously, the x-distortion function can be used. and y distortion function This causes the focal field to shift along the x and y axes of the focusing space; spherical aberration and distortion affect the wavefront of the incident wave packet, introducing an additional phase factor to the incident wave packet: Where k = 2π / λ is the wave number, and λ is the center wavelength of the incident wave packet; S2. By pre-splitting the incident wave packet, and simultaneously introducing the spherical aberration function and distortion function into the incident wave packet; the incident wave packet needs to be pre-splitting, and then an additional phase factor is applied to the pre-splitting incident wave packet. S3. Focus the incident wave packet through the objective lens to obtain a focused spatiotemporal wave packet that can move in three-dimensional space; S4. By changing the correlation coefficients in the spherical aberration function and the distortion function, the spatial position of the focused spatiotemporal wave packet can be arbitrarily adjusted in three-dimensional space.

2. The method for controlling the movement of a focusing spatiotemporal wave packet in three-dimensional space as described in claim 1, characterized in that, In step S3, a three-dimensional coordinate system is established in the focusing space with the focal point of the incident wave packet after it is focused by the objective lens when there is no aberration as the origin. The direction of light field propagation is the z direction, and the plane that passes through the focal point and is orthogonal to the optical axis is the xy plane.

3. The method for controlling the movement of a focusing spatiotemporal wave packet in three-dimensional space as described in claim 1, characterized in that, In step S4, in the three-dimensional focusing space, the relationship between the spherical aberration coefficient and the displacement of the focusing spatiotemporal wave packet along the z direction is determined, the relationship between the x distortion coefficient and the displacement of the focusing spatiotemporal wave packet along the x direction is determined, and the relationship between the y distortion coefficient and the displacement of the focusing spatiotemporal wave packet along the y direction is determined.

4. The method for controlling the movement of a focusing spatiotemporal wave packet in three-dimensional space as described in claim 3, characterized in that, When the incident wave packet is only affected by spherical aberration, the position of the focusing spatiotemporal wave packet in the propagation direction is found, and the distance between it and the origin of the focusing space is the offset Δz corresponding to that spherical aberration coefficient.

5. The method for controlling the movement of a focusing spatiotemporal wave packet in three-dimensional space as described in claim 4, characterized in that, When only x-distortion or y-distortion factors are involved, the focusing spatiotemporal wave packet exhibits a shift along the x-axis or y-axis. Analysis reveals the shift Δx or Δy of the focusing spatiotemporal wave packet along the x-axis or y-axis under different distortion coefficients.

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