Engineering

Jian Chen; Jie Zhao; Xi Shen; Dewei Mo; Cheng-Wei Qiu; Qiwen Zhan

doi:10.1016/j.eng.2024.09.015

Engineering ›› 2025, Vol. 45 ›› Issue (2) :48 -56. DOI: 10.1016/j.eng.2024.09.015

Research Subwavelength Optics—Article

research-article

Engineering

Jian Chen ^a^,^b^,^c
, Jie Zhao ^a
, Xi Shen ^a
, Dewei Mo ^a
, Cheng-Wei Qiu ^c^,^*
, Qiwen Zhan ^a^,^b^,^*

Author information +

History +

PDF (2771KB)

Abstract

While spin–orbit interaction has been extensively studied, few investigations have reported on the interaction between orbital angular momenta (OAMs). In this work, we study a new type of orbit–orbit coupling between the longitudinal OAM and the transverse OAM carried by a three-dimensional (3D) spatiotemporal optical vortex (STOV) in the process of tight focusing. The 3D STOV possesses orthogonal OAMs in the x–y, t–x, and y–t planes, and is preconditioned to overcome the spatiotemporal astigmatism effect. x, y, and t are the axes in the spatiotemporal domain. The corresponding focused wavepacket is calculated by employing the Debye diffraction theory, showing that a phase singularity ring is generated by the interactions among the transverse and longitudinal vortices in the highly confined STOV. The Fourier-transform decomposition of the Debye integral is employed to analyze the mechanism of the orbit–orbit interaction. This is the first revelation of coupling between the longitudinal OAM and the transverse OAM, paving the way for potential applications in optical trapping, laser machining, nonlinear light–matter interactions, and more.

Graphical abstract

Keywords

Spatiotemporal optical vortex / Orbital angular momentum / Momentum interaction / Highly confined wavepacket / Diffraction

Cite this article

Download citation ▾

Jian Chen, Jie Zhao, Xi Shen, Dewei Mo, Cheng-Wei Qiu, Qiwen Zhan. Engineering. Engineering, 2025, 45(2): 48-56 DOI:10.1016/j.eng.2024.09.015

登录浏览全文

4963

注册一个新账户忘记密码

1. Introduction

Orbital angular momentum (OAM) is an important degree of freedom of light [1] that describes optical fields with a helical wavefront and provides a treasure trove of mode bases for tailoring structured light [2]. It has diverse applications, including sensing [3], quantum information processing [4], [5], optical tweezers [6], [7], [8], optical topology [9], [10], [11], [12], microscopy [13], metrology [14], and high-speed optical communications with large capacity [15], [16]. Over a considerable period of time, dominant investigations have focused on the longitudinal OAM, which makes the energy flow of a vortex beam circulate around its propagation axis [17]. Furthermore, by introducing a spiral phase in the spatiotemporal domain, a transverse OAM can be created, which drives the spatiotemporal optical vortex (STOV) to rotate around an axis perpendicular to the propagation direction of the beam [18]. Such novel STOVs have been experimentally demonstrated via nonlinear interaction between a high energy pulsed laser and air [19] and by using a linear modulation method based on a pulse shaper [20], [21], [22].

The emergence of the transverse OAM has sparked rapidly growing research interest due to its potential for greatly expanding the degrees of freedom. More recently, an STOV was experimentally generated by employing a simple resonant diffractive grating to form a phase vortex in the frequency-momentum space [23]. The topological darkness phenomenon of photonic crystal slabs was utilized to imprint spatiotemporal phase singularities onto reflected ultrashort pulses [24]. A spatiotemporal diffractive deep neural network was proposed for implementing an STOV multiplexer, enabling the transformation of spatially separated Gaussian beams into STOV wavepackets with adjustable topological charges (TCs) [25]. An STOV was also created in plasma-based terahertz emission stimulated by a two-color vortex field with broken cylindrical symmetry [26]. Tailoring the coherency of STOVs and spatiotemporal dislocation curves was demonstrated in partially coherent pulsed beams via coherent-mode representation and Fourier transforms [27]. Second harmonic generation in lithium niobite nonlinear photonic crystals was exploited to convert the fundamental wave of linearly chirped Gaussian pulse into second harmonic STOV pulse [28]. In addition, the second [29], [30], third [31], and high [32] harmonic generation of STOVs were investigated to confirm the transverse OAM conservation law. A single-frame method based on spatially resolved spectral interferometry was developed to quantitatively characterize the TC numbers, OAM helicity, pulse dispersion, and beam divergence of STOV pulses [33]. Fractal STOV in microwaves was achieved based on an analog spatiotemporal differentiator formed by an asymmetrical metasurface [34]. Propagation of STOVs in free space [35], [36], [37] and linear dispersive media [38], [39] was also implemented to reveal the evolution of their diffraction patterns. Moreover, spatiotemporal vortices with arbitrarily oriented OAMs were generated by employing a compact device with a topological defect in the wavevector-frequency spectra of the transmission coefficient [40]. The interaction of STOVs with atomic targets was studied by dissecting the spatially resolved photoelectron energy spectra [41]. Spin–orbit coupling between the transverse spin angular momentum (SAM) and transverse OAM [42] or between the longitudinal SAM and transverse OAM [43] was analyzed in vectorial STOVs.

Although diverse investigations on STOVs have been carried out, few reports have been published on the interaction between longitudinal and transverse OAMs. In this work, we study a new type of orbit–orbit coupling between longitudinal and transverse OAMs caused by the tight focusing of a three-dimensional (3D) STOV. The 3D STOV exhibits three orthogonal OAMs in the x–y, x–t, and y–t planes, and is preconditioned to mitigate the spatiotemporal astigmatism effect. x, y, and t are the axes in the spatiotemporal domain. The resulting focused wavepacket is computed using the Richards–Wolf vectorial diffraction theory, revealing the generation of a phase singularity ring due to the interactions among the transverse and longitudinal vortices within the highly confined STOV. The presented results provide an avenue for tailoring the phase singularity trace in the 3D spatiotemporal domain based on orbit–orbit interaction and may find potential applications in optical trapping, optical topology, laser machining, optical information processing, and more.

2. 3D STOV

Without loss of generality, a 3D STOV carrying transverse OAM with TC of −1 in the x–t plane and +1 in the y–t plane, and longitudinal OAM with TC of +2 in the x–y plane can be given as follows:

(1)

\begin{matrix} E (x, y, t) = \frac{\sqrt{2 (x^{2} + y^{2})}}{w} (\frac{x}{w} + \frac{j t}{w_{t}}) (\frac{y}{w} + \frac{j t}{w_{t}}) \times \\ \exp (- \frac{x^{2} + y^{2}}{w^{2}} - \frac{t^{2}}{w_{t}^{2}}) \exp (j 2 ϕ) \end{matrix}

where

E (x, y, t)

is the complex amplitude distribution of the 3D STOV, j is the imaginary unit,

w

is the waist radius of the Gaussian profile in the spatial domain,

w_{t}

is the pulse half-width at

1 / e^{2}

of the maximum intensity of the wavepacket in the temporal domain, and

ϕ = \tan^{- 1} (y / x)

is the azimuthal angle in the x–y plane. The intensity and phase distributions of the 3D non-preconditioned STOV are shown in Figs. 1(a) and (b), respectively. It can be seen that three orthogonal phase singularity traces exist in the wavepacket; moreover, due to the coupling among the three orthogonal OAMs, the phase distribution on each principal plane does not change continuously from −π to π when rotating around the singularity for one cycle. For example, the complex amplitude of the wavepacket on the t–x plane is reduced to the following:

(2)

E (x, 0, t) = \frac{\sqrt{2} |x|}{w} (\frac{x}{w} + \frac{j t}{w_{t}}) (\frac{j t}{w_{t}}) \exp (- \frac{x^{2}}{w^{2}} - \frac{t^{2}}{w_{t}^{2}})

Since the t-axis is the horizontal axis of the t–x plane, while the x-axis is the vertical axis, the term

x / w + j t / w_{t}

can be rewritten as

j (t / w_{t} - j x / w)

, which gives rise to a spiral phase with TC of −1 in the t–x plane. The term

j t / w_{t}

will bring an additional phase of π/2 for

t > 0

and an additional phase of −π/2 for

t < 0

. Considering that the phase is taken modulo 2π, the combined effect of the above two terms results in a four-quadrant phase distribution in the t–x plane. In the counterclockwise direction, the phase variations in the first to fourth quadrants are respectively in the range of [π, π/2], [−π/2, −π], [π, π/2], and [−π/2, −π]. Similarly, it is possible to figure out the formation mechanism of the phase distributions in the y–t and x–y planes. The detailed evolutions of the phase distributions in slices along the t-, x-, and y-axes are demonstrated in Appendix A Video S1. However, the coupling among these three orthogonal OAMs will not change the TC of the wavepacket along each dimension. The TC of the wavepacket (L) can be calculated by employing the following equation:

(3)

L = \frac{〈r \times Im (E^{*} \cdot (\nabla) E)〉}{〈I〉}

where

〈\cdot \cdot \cdot〉 = ∭ \cdot \cdot \cdot d V

r = (x, y, t)

is the position vector in the 3D STOV,

Im (∙)

is the imaginary part of the complex results, and

I = {|E|}^{2}

is the intensity distribution of the 3D STOV.

L

contains

L_{t}

L_{x}

, and

L_{y}

, which represent the OAM along the t-, x-, and y-axes, respectively. The TCs of the non-preconditioned wavepacket are evaluated to be

L_{t} = 1.998

L_{x} = 0.999

, and

L_{y} = - 0.999

. All are very close to the corresponding preset values.

To overcome the spatiotemporal astigmatism during the tight focusing process, the 3D STOV is preconditioned as the following:

(4)

\begin{matrix} E_{p} (x, y, t) = 2 j \frac{\sqrt{2 (x^{2} + y^{2})}}{w} (\frac{x}{w} + \frac{t}{w_{t}}) (\frac{y}{w} + \frac{t}{w_{t}}) \times \\ \exp (- \frac{x^{2} + y^{2}}{w^{2}} - \frac{t^{2}}{w_{t}^{2}}) \exp (j 2 ϕ) \end{matrix}

where

E_{p} (x, y, t)

is the complex amplitude distribution of the preconditioned wavepacket.

The spatiotemporal distributions of the preconditioned wavepacket on the three principal planes are shown in Figs. 1(c) and (d). From the isosurface in Fig. 1(c), it can be seen that the preconditioned wavepacket is reduced to four detached parts. The phase distributions in the t–x and y–t planes are binarized to π/2 in the orange regions and −π/2 in the blue regions, while the phase distribution in the x–y plane is divided into eight sectors with variation ranges of [π/2, π] in the red gradient sectors and [−π, −π/2] in the blue gradient sectors. The detailed evolutions of the phase distributions in slices of the preconditioned wavepacket along the t-, x-, and y-axes are demonstrated in Appendix A Video S2. According to Eq. (3), the TC of the longitudinal OAM of the preconditioned wavepacket is calculated to be

L_{pt} = 1.998

, which is consistent with the counterpart of the non-preconditioned STOV. Thus, the preconditioning process does not affect the longitudinal OAM and just collapses the transverse OAM in the incident wavepacket.

3. Tight focusing of the 3D STOV

The schematic diagram to focus the preconditioned wavepacket is illustrated in Fig. 2. For the numerical calculation of the focused wavepacket, each temporal slice of the incident wavepacket is assumed to be focused at the conjugate temporal position within the focal space. This assumption is credible for the commonly used pulsed light, whose pulse width is usually longer than 20 fs [44]. Chromatic and other aberrations are also ignored. Based on the Debye integral, the highly confined wavepacket on the focal plane can be calculated as follows:

(5)

E_{f} (r_{f}, φ, z_{f}, t) = \int_{0}^{α} \int_{0}^{2 π} E_{Ω} (θ, ϕ, t) \times e^{- j k [r_{f} \sin θ \cos (ϕ - φ) + z_{f} \cos θ]} \sin θ d θ d ϕ

where

α

represents the maximum convergence angle determined by the numerical aperture (NA) of the objective lens,

r_{f} = \sqrt{x_{f}^{2} + y_{f}^{2}}

is the radial distrance of the observation point

Q (r_{f}, φ)

on the focal plane,

φ = \tan^{- 1} (y_{f} / x_{f})

is the azimuthal angle in the focal plane, k = 2π/λ is the wavenumber, λ is the central wavelength of the incident wavepacket, θ is the convergence angle during the focusing of the wavepacket,

E_{f} (r_{f}, φ, z_{f}, t)

is the complex amplitude distribution of the focused wavepacket,

x_{f}

y_{f}

, and

z_{f}

are the Cartesian coordinates in the focal region of the objective lens, and

E_{Ω} (θ, ϕ, t)

is the refracted wavepacket on the spherical surface Ω. Here, an aplanatic objective lens adhering to the sine condition is employed, whose pupil apodization function is

\sqrt{\cos θ}

. Considering the transformation from Cartesian coordinates to spherical coordinates,

E_{Ω} (θ, ϕ, t)

can be written as follows:

(6)

\begin{matrix} E_{Ω} (θ, ϕ, t) = 2 j \frac{\sqrt{2} f \sin θ}{w} (\frac{f \sin θ \cos ϕ}{w} + \frac{t}{w_{t}}) (\frac{f \sin θ \sin ϕ}{w} + \frac{t}{w_{t}}) \times \\ \exp (- \frac{f^{2} \sin^{2} θ}{w^{2}} - \frac{t^{2}}{w_{t}^{2}}) \exp (j 2 ϕ) \sqrt{\cos θ} \end{matrix}

where

f

is the focal length of the objective lens. For the following simulations, we will analyze the focused spatiotemporal wavepacket on the focal plane, so we set

z_{f} = 0

in the integral.

In fact, the Debye integral can be regarded as the Fourier transform of the weighted refracted field [45]. Thus, to reveal the orbit–orbit interaction in the focused wavepacket, we can rewrite Eq. (5) as follows:

(7)

E_{f} (r_{f}, φ, z_{f}, t) = \frac{1}{k^{2}} F \{E_{Ω} (θ, ϕ, t) \cdot \exp (j k z_{f} \cos θ) / \cos θ\}

where

F \{\cdot\}

is the Fourier transform operator.

Since we ignore the temporal coupling in the wavepacket during the tight focusing process, according to Eq. (7), it can be seen that each temporal slice of the focused wavepacket is the Fourier transform of the corresponding temporal slice of the incident wavepacket. As mentioned above, to obtain the wavepacket on the focal plane,

z_{f}

is set to 0. Hence, the weighted refracted field in Eq. (7) is reduced to the following:

(8)

\begin{matrix} E_{WR Ω} (θ, ϕ, t) = (\frac{f \sin θ \cos ϕ}{w} + \frac{t}{w_{t}}) \times (\frac{f \sin θ \sin ϕ}{w} + \frac{t}{w_{t}}) \times \\ 2 j \frac{\sqrt{2} f \sin θ}{w} \exp (- \frac{f^{2} \sin^{2} θ}{w^{2}} - \frac{t^{2}}{w_{t}^{2}}) \exp (j 2 ϕ) / \sqrt{\cos θ} \end{matrix}

where

E_{WR Ω} (θ, ϕ, t)

represents the weighted refracted wavepacket and is divided into three terms. The first term,

f \sin θ \cos ϕ / w + t / w_{t}

, gives rise to the transverse OAM along the y-axis in the focused wavepacket; the second term,

f \sin θ \sin ϕ / w + t / w_{t}

, produces the transverse OAM along the x-axis in the focused wavepacket; and the third term,

2 j \sqrt{2} f \sin θ / w \cdot \exp (- f^{2} \sin^{2} θ / w^{2} - t^{2} / w_{t}^{2}) \exp (j 2 ϕ) / \sqrt{\cos θ}

, generates the longitudinal OAM in the focused wavepacket. Based on the convolution property of the Fourier transform, the focused wavepacket can also be obtained as follows:

(9)

E_{f} (r_{f}, φ, 0, t) = \frac{1}{k^{2}} {FT}_{1} * {FT}_{2} * {FT}_{3}

where the Fourier transforms

{FT}_{1}

{FT}_{2}

, and

{FT}_{3}

can be expressed as follows:

(10)

\begin{matrix} {FT}_{1} = F \{\frac{f \sin θ \cos ϕ}{w} + \frac{t}{w_{t}}\} \\ = F \{\frac{f \sin θ \cos ϕ}{w}\} + \frac{t}{w_{t}} \frac{2 π R_{0} J_{1} (R_{0} r_{f})}{r_{f}} \end{matrix}

(11)

\begin{matrix} {FT}_{2} = F \{\frac{f \sin θ \sin ϕ}{w} + \frac{t}{w_{t}}\} \\ = F \{\frac{f \sin θ \sin ϕ}{w}\} + \frac{t}{w_{t}} \frac{2 π R_{0} J_{1} (R_{0} r_{f})}{r_{f}} \end{matrix}

(12)

\begin{matrix} {FT}_{3} = F \{2 j \frac{\sqrt{2} f \sin θ}{w} \exp (- \frac{f^{2} \sin^{2} θ}{w^{2}} - \frac{t^{2}}{w_{t}^{2}}) \exp (j 2 ϕ) / \sqrt{\cos θ}\} \\ = \exp (- \frac{t^{2}}{w_{t}^{2}}) \cdot F \{2 j \frac{\sqrt{2} f \sin θ}{w} \exp (- \frac{f^{2} \sin^{2} θ}{w^{2}}) \exp (j 2 ϕ) / \sqrt{\cos θ}\} \end{matrix}

where

R_{0}

is the pupil radius of the objective lens and

J_{1} (\cdot)

is the Bessel function of the first kind of order 1. According to Eqs. (10), (11), it is clear that the value of t has a significant effect on

{FT}_{1}

and

{FT}_{2}

, since it superimposes a weighted pattern

2 π t R_{0} J_{1} (R_{0} r_{f}) / (w_{t} \cdot r_{f})

on them. Around the center of the incident wavepacket, the value of t is small, so the superimposed pattern

2 π t R_{0} J_{1} (R_{0} r_{f}) / (w_{t} \cdot r_{f})

can be ignored in

{FT}_{1}

and

{FT}_{2}

; however, as the temporal slice is away from the center of incident wavepacket, the absolute value of t is increased, leading to the superimposed pattern gradually becoming dominant in

{FT}_{1}

and

{FT}_{2}

. On the other hand, based on Eq. (12), the value of t only affects the amplitude of

{FT}_{3}

and has no impact on its phase distribution. The focused wavepacket results from the convolution of

{FT}_{1}

{FT}_{2}

, and

{FT}_{3}

; therefore, the parameter t plays a key role in the orbit–orbit interaction in the focused wavepacket. More detailed explanations are given in the next section.

4. Results of orbit–orbit interaction

In the simulations, we set

w = 0.5

w_{t} = 0.5

, and

R_{0} = 2

; the NA of the objective lens is 0.9, hence

f = R_{0} / NA = 2.22

. Based on the Debye integral, the simulation results of the highly confined wavepacket on the focal plane are demonstrated in Fig. 3. According to the intensity slice on the x–y plane shown in Fig. 3(a), a rectangular dark ring exists inside the focused wavepacket. The intensities of the focused field exhibit an S-shaped distribution on both the t–x and y–t planes, implying that both slices contain at least two phase singularities. The phase slices in Fig. 3(b) confirm this point. In the t–x plane, two spatiotemporal phase singularities are observed in the phase slice, with the phase varying clockwise from −π to π around each singularity, indicating that each singularity represents a transverse OAM with TC of –1. In contrast, the phase varies counterclockwise from −π to π around each of the two spatiotemporal phase singularities observed in the phase slice on the y–t plane, indicating that each singularity denotes a transverse OAM with TC of +1. The phase distribution of the slice in the x–y plane is very complicated, with a rectangular dark area in the center, since the phase singularity trace lies in this plane. Leveraging the zero-intensity distribution within the focused wavepacket, we can extract the spatiotemporal structure of the phase singularity trace, as shown in Fig. 3(c). The phase singularity trace in the center of the focused wavepacket is obviously rectangular, indicating that there exists strong orbit–orbit interaction among the transverse and longitudinal OAMs in this area. The phase singularities marked by black circles in Fig. 3(b) are located on the phase singularity ring shown in Fig. 3(c). The phase singularity trace in the head and end of the focused wavepacket is almost along the t-axis, demonstrating that the orbit–orbit interaction in these areas is much weaker—even vanished. The OAMs along the t-, x-, and y-axes are evaluated to be 1.996, 1.093, –1.093, respectively. These are very close to the preset values, indicating that the orbit–orbit coupling has a minor effect on the OAM of the whole focused wavepacket.

For a detailed analysis of the orbit–orbit coupling in the focused wavepacket, three representative temporal slices (i.e., t = 0 a.u., t = 0.5 a.u., and t = 1 a.u.) of the incident and focused wavepackets are selected to demonstrate the convolution of the Fourier transforms given in Eq. (9). For the central temporal slice of the incident wavepacket (t = 0 a.u.), its Fourier decomposition is shown in Fig. 4. At this moment, the first term of Eq. (8) is reduced to

4.44 \sin θ \cos ϕ

, whose Fourier transform

{FT}_{1}

exhibits two lobes along the x-axis, as shown in Fig. 4(a-i). The binary phase distribution of

{FT}_{1}

is shown in Fig. 4(a-ii). As shown in Figs. 4(c-i) and (c-ii), the Fourier transform

{FT}_{3}

of the third term of Eq. (8) gives rise to a vortex carrying longitudinal OAM with TC of 2. When

{FT}_{1}

is convolved with

{FT}_{3}

{FT}_{1}

acts like two impulses to shift the longitudinal vortex along the x-axis toward both the left and right directions. On the other hand, the second term of Eq. (8) is reduced to

4.44 \sin θ \sin ϕ

, whose Fourier transform

{FT}_{2}

exhibits two lobes along the y-axis, as shown in Fig. 4(b-i). The binary phase distribution of

{FT}_{2}

is shown in Fig. 4(b-ii). When convolved with

{FT}_{3}

{FT}_{2}

acts like two impulses to shift the longitudinal vortex along the y-axis in both the upward and downward directions. According to Eq. (9), the central temporal slice of the focused wavepacket results from the convolution of

{FT}_{1}

{FT}_{2}

, and

{FT}_{3}

. As shown in Figs. 4(d-i) and (d-ii), the longitudinal vortex is simultaneously shifted along the x- and y-axes toward four directions, producing a slice of the focused wavepacket with complicated intensity and phase distributions. Moreover, it is clear that Figs. 4(d-i) and (d-ii) are similar to their counterparts shown in the x–y plane of Figs. 3(a) and (b), indicating the validity of the convolution method in analyzing the formation of the focused wavepacket.

For the temporal slice of the incident wavepacket at t = 0.5 a.u., the complex field distributions of

{FT}_{1}

and

{FT}_{2}

are shown in Figs. 5(a) and (b), respectively. Since the superimposed pattern

4 π J_{1} (2 r_{f}) / r_{f}

is comparable to

F \{4.44 \sin θ \cos ϕ\}

, the intensity distribution of

{FT}_{1}

is elongated along the x-axis. Similarly, the intensity distribution of

{FT}_{2}

is elongated along the y-axis. As shown in Figs. 5(c-i) and (c-ii),

{FT}_{3}

still gives rise to a second-order longitudinal vortex, albeit with a lower intensity due to the larger value of t. Now,

{FT}_{1}

and

{FT}_{2}

can be regarded as two complex window functions. When they are convolved with

{FT}_{3}

, the resulting complex field distributions of the focused wavepacket’s temporal slice (shown in Figs. 5(d-i) and (d-ii)) are simpler than those of the central temporal slice. That is, the orbit–orbit coupling in the temporal slice at t = 0.5 a.u. is weaker than that in the central temporal slice.

When the incident wavepacket is temporally sliced at t = 1 a.u., the complex field distributions of

{FT}_{1}

and

{FT}_{2}

are shown in Figs. 6(a) and (b), respectively. Since the superimposed pattern

8 π J_{1} (2 r_{f}) / r_{f}

is almost completely dominant in both

{FT}_{1}

and

{FT}_{2}

, the Jinc function makes both

{FT}_{1}

and

{FT}_{2}

exhibit regular distributions of intensity and phase. At this point,

{FT}_{3}

still generates a second-order longitudinal vortex, but the intensity is further reduced, as shown in Figs. 6(c-i) and (c-ii). The convolution of

{FT}_{1}

{FT}_{2}

, and

{FT}_{3}

produces the corresponding temporal slice of the focused wavepacket, as shown in Figs. 6(d-i) and (d-ii). It is evident that the phase distribution of the focused wavepacket’s temporal slice closely resembles the phase distribution of the second-order longitudinal vortex shown in Fig. 6(c-ii), exhibiting a minor distortion at the beam’s periphery. Therefore, the orbit–orbit coupling within this temporal slice is further weakened to be negligible. As the value of t increases further, the dominance of the superimposed pattern

8 π t J_{1} (2 r_{f}) / r_{f}

will be further enhanced, making the complex field distribution of the focused wavepacket’s temporal slice closer to that of a second-order longitudinal vortex. In addition, the evolution of the orbit–orbit coupling exhibits similar behavior in the negative t range. Thus, strong orbit–orbit coupling exists in the center of the focused wavepacket, which is weakened significantly and can be considered negligible at the head and end.

To further reveal the phase singularity evolution inside the focused wavepacket, we rotate the slices around the t-, x-, and y-axis, respectively. The results corresponding to the slices with normal vectors of (π/6, π/2), (π/3, π/2), (2π/3, π/2), and (5π/6, π/2) are shown in Figs. 7(a)–(d), respectively. More detailed evolutions of the intensity and phase distribution as the slices are rotated around the t-axis are provided in Appendix A Video S3. As shown in Figs. 7(a-i)–(d-i), the intensity distribution evolves from S-shape to X-shape. In Figs. 7(a-ii)–(d-ii), the phase singularities on the ring trace are marked with black circles. Since the ring trace is parallel to the x–y plane, the positions (

t

x_{f}

y_{f}

) of the marked singularities change in different slices. More specifically, the eight marked singularities in Figs. 7(a-ii)–(d-ii) are located at (0.08 a.u., 0.47λ, –0.27λ), (–0.08 a.u., –0.47λ, 0.27λ), (0.05 a.u., 0.27λ, –0.47λ), (–0.05 a.u., –0.27λ, 0.47λ), (0.05 a.u., 0.31λ, 0.54λ), (–0.05 a.u., –0.31λ, –0.54λ), (–0.04 a.u., 0.52λ, 0.3λ), and (0.04 a.u., –0.52λ, –0.3λ). It can be found that the marked singularities in each slice are positioned symmetrically about the origin of the spatiotemporal coordinates.

The results corresponding to the slices with normal vectors of (π/6, 0), (π/3, 0), (2π/3, 0), and (5π/6, 0) are demonstrated in Figs. 8(a)–(d), respectively. More detailed evolutions of the intensity and phase distribution as the slices are rotated around the x-axis are provided in Appendix A Video S4. Unlike the case of rotating around the t-axis, the rotation of the slice around the x-axis will always sample the same pair of phase singularities on the ring trace. The intensity distribution evolves from circular shape to S-shape as the slice is rotated around the x-axis. The two fixed phase singularities in each slice (marked by black circles in Figs. 8(a-ii)–(d-ii)) are located at (0 a.u., –0.56λ, 0λ) and (0 a.u., 0.56λ, 0λ), which are distributed along the x-axis.

The intensity and phase distributions corresponding to the slices with normal vectors of (π/2, π/6), (π/2, π/3), (π/2, 2π/3), and (π/2, 5π/6) are depicted in Figs. 9(a)–(d), respectively. More details on the evolutions of the intensity and phase distributions as the slices are rotated around the y-axis are provided in Appendix A Video S5. Similar to the case of rotation around the x-axis, the intensity distribution evolves from circular shape to S-shape as the slice is rotated around the y-axis. The slices also contain a fixed pair of phase singularities on the ring trace, as marked in Figs. 9(a-ii)–(d-ii). The two marked phase singularities in each slice are located at (0 a.u., 0λ, 0.56λ) and (0 a.u., 0λ, −0.56λ), which are distributed along the y-axis.

5. Conclusions

In summary, we revealed the orbit–orbit interaction among the longitudinal and transverse OAMs in the tight focusing of 3D STOV. The spatiotemporal distribution of the highly confined wavepacket on the focal plane of the objective lens was obtained by utilizing the Debye integral. In the central region of the focused wavepacket, a ring-shaped trace of phase singularity is formed as a result of the orbit–orbit interaction. The phase singularity ring resembles a photonic toroidal vortex [46] but has a rectangular shape. Such a novel orbit–orbit interaction provides an avenue for tailoring the OAM distribution in the spatiotemporal domain to form exotic topological structures of phase singularity and may find potential applications in particle trapping, optical tweezer, laser machining, optical topology, light–matter interactions, toroidal electrodynamics, and more.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (12274299 and 92050202) and the Shanghai Science and Technology Committee (22QA1406600).

Compliance with ethics guidelines

Jian Chen, Jie Zhao, Xi Shen, Dewei Mo, Cheng-Wei Qiu, and Qiwen Zhan declare that they have no conflict of interest or financial conflicts to disclose.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.eng.2024.09.015.

References

Publishing order | Descend order by publishing year | Descend order by cited within

[1]	Allen L, Beijersbergen MW, Spreeuw RJC, Woerdman JP.Orbital angular momentum of light and the transformation of Laguerre–Gaussian laser modes.Phys Rev A 1992; 45(11):8185-8189.

[2]	Yao AM, Padgett MJ.Orbital angular momentum: origins, behavior and applications.Adv Opt Photonics 2011; 3(2):161-204.

[3]	Lavery MPJ, Speirits FC, Barnett SM, Padgett MJ.Detection of a spinning object using light’s orbital angular momentum.Science 2013; 341(6145):537-540.

[4]	Erhard M, Fickler R, Krenn M, Zeilinger A.Twisted photons: new quantum perspectives in high dimensions.Light Sci Appl 2018; 7(3):17146.

[5]	Molina-Terriza G, Torres JP, Torner L.Twisted photons.Nat Phys 2007; 3(5):305-310.

[6]	Padgett M, Bowman R.Tweezers with a twist.Nat Photonics 2011; 5(6):343-348.

[7]	Chen J, Wan C, Zhan Q.Engineering photonic angular momentum with structrured light: a review.Adv Photonics 2021; 3(6):064001.

[8]	Paterson L, MacDonald MP, Arlt J, Sibbett W, Bryant PE, Dholakia K.Controlled rotation of optically trapped microscopic particles.Science 2001; 292(5518):912-914.

[9]	Shen Y, Martínez EC, Rosales-Guzmán C.Generation of optical skyrmions with tunable topological textures.ACS Photonics 2022; 9(1):296-303.

[10]	Gao S, Speirits FC, Castellucci F, Franke-Arnold S, Barnett SM, Götte JB.Paraxial skyrmionic beams.Phys Rev A 2020; 102(5):053513.

[11]	Lin W, Ota Y, Arakawa Y, Iwamoto S.Microcavity-based generation of full Poincaré beams with arbitrary skyrmion numbers.Phys Rev Res 2021; 3(2):023055.

[12]	Sugic D, Droop R, Otte E, Ehrmanntraut D, Nori F, Ruostekoski J, et al.Particle-like topologies in light.Nat Commun 2021; 12:6785.

[13]	Yan L, Gregg P, Karimi E, Rubano A, Marrucci L, Boyd R, et al.Q-plate enabled spectrally diverse orbital-angular-momentum conversion for stimulated emission depletion microscopy.Optica 2015; 2(10):900-903.

[14]	Belmonte A, Rosales-Guzmán C, Torres JP.Measurement of flow vorticity with helical beams of light.Optica 2015; 2(11):1002-1005.

[15]	Wang J, Yang J, Fazal IM, Ahmed N, Yan Y, Huang H, et al.Terabit free-space data transmission employing orbital angular momentum multiplexing.Nat Photonics 2012; 6(7):488-496.

[16]	Bozinovic N, Yue Y, Ren Y, Tur M, Kristensen P, Huang H, et al.Terabit-scale orbital angular momentum mode division multiplexing in fibers.Science 2013; 340(6140):1545-1548.

[17]	Shen Y, Wang X, Xie Z, Min C, Fu X, Liu Q, et al.Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities.Light Sci Appl 2019; 8:90.

[18]	Bliokh KY, Nori F.Spatiotemporal vortex beams and angular momentum.Phys Rev A 2012; 86(3):033824.

[19]	Jhajj N, Larkin I, Rosenthal EW, Zahedpour S, Wahlstrand JK, Milchberg HM.Spatiotemporal optical vortices.Phys Rev X 2016; 6(3):031037.

[20]	Chong A, Wan C, Chen J, Zhan Q.Generation of spatiotemporal optical vortices with controllable transverse orbital angular momentum.Nat Photonics 2020; 14(6):350-354.

[21]	Chen J, Lu K, Cao Q, Wan C, Hu H, Zhan Q.Automated close loop system for three-dimensional characterization of spatiotemporal optical vortex.Front Phys 2021; 9:633922.

[22]	Hancock SW, Zahedpour S, Goffin A, Milchberg HM.Free-space propagation of spatiotemporal optical vortices.Optica 2019; 6(12):1547-1553.

[23]	Che Z, Liu W, Ye J, Shi L, Chan CT, Zi J.Generation of spatiotemporal vortex pulses by resonant diffractive grating.Phys Rev Lett 2024; 132(4):044001.

[24]	Liu W, Wang J, Tang Y, Wang X, Zhao X, Shi L, et al.Exploiting topological darkness in photonic crystal slabs for spatiotemporal vortex generation.Nano Lett 2024; 24(3):943-949.

[25]	Zhou J, Pu H, Yan J.Spatiotemporal diffractive deep neural networks.Opt Express 2024; 32(2):1864-1877.

[26]	Wang S, Bai Y, Li N, Liu P.Generation of terahertz spatiotemporal optical vortices with frequency-dependent orbital angular momentum.Opt Express 2023; 31(10):16267-16280.

[27]	Ding C, Liang C, Horoshko D, Korotkova O, Pan L, Liu Z.Method for generating spatiotemporal coherency vortices and spatiotemporal dislocation curves.Opt Express 2024; 32(1):609-624.

[28]	Liu S, Zhang X, Chen H, Xie H, Yang S, Zhu S, et al.Generation of spatiotemporal vortices in nonlinear photonic crystals.Opt Lett 2023; 48(22):5951-5954.

[29]	Gui G, Brooks NJ, Kapteyn HC, Murnane MM, Liao C.Second-harmonic generation and the conservation of spatiotemporal orbital angular momentum of light.Nat Photonics 2021; 15(8):608-613.

[30]	Hancock SW, Zahedpour S, Milchberg HM.Second-harmonic generation of spatiotemporal optical vortices and conservation of orbital angular momentum.Optica 2021; 8(5):594-597.

[31]	Wang H, Chen YY, Zhang X, Shen B.Generation and periodic evolution of third harmonics carrying transverse orbital angular momentum in air-plasma filaments.Opt Express 2023; 31(22):36810-36823.

[32]	Fang Y, Lu S, Liu Y.Controlling photon transverse orbital angular momentum in high harmonic generation.Phys Rev Lett 2021; 127(27):273901.

[33]	Gui G, Brooks NJ, Wang B, Kapteyn HC, Murnane MM, Liao CT.Single-frame characterization of ultrafast pulses with spatiotemporal orbital angular momentum.ACS Photonics 2022; 9(8):2802-2808.

[34]	Zhou Y, Zhan J, Xu Z, Shao Y, Wang Y, Dang Y, et al.Electromagnetic spatiotemporal differentiation meta-devices.Laser Photonics Rev 2023; 17(11):2300182.

[35]	Porras MA.Propagation of higher-order spatiotemporal vortices.Opt Lett 2023; 48(2):367-370.

[36]	Huang S, Wang P, Shen X, Liu J, Li R.Diffraction properties of light with transverse orbital angular momentum.Optica 2022; 9(5):469-472.

[37]	Huang S, Wang P, Shen X, Liu J.Properties of the generation and propagation of spatiotemporal optical vortices.Opt Express 2021; 29(17):26995-27003.

[38]	Hyde IV MW, Porras MA.Propagation of spatiotemporal optical vortex beams in linear, second-order dispersive media.Phys Rev A 2023; 108(1):013519.

[39]	Hancock SW, Zahedpour S, Milchberg HM.Mode structure and orbital angular momentum of spatiotemporal optical vortex pulses.Phys Rev Lett 2021; 127(19):193901.

[40]	Wang H, Guo C, Jin W, Song AY, Fan S.Engineering arbitrarily oriented spatiotemporal optical vortices using transmission nodal lines.Optica 2021; 8(7):966-971.

[41]	Chen Y, Zhou Y, Li M, Liu K, Ciappina MF, Lu P.Atomic photoionization by spatiotemporal optical vortex pulses.Phys Rev A 2023; 107(3):033112.

[42]	Bliokh KY.Spatiotemporal vortex pulses: angular momenta and spin–orbit interaction.Phys Rev Lett 2021; 126(24):243601.

[43]	Chen J, Yu L, Wan C, Zhan Q.Spin–orbit coupling within tightly focused circularly polarized spatiotemporal vortex wavepacket.ACS Photonics 2022; 9(3):793-799.

[44]	Rui G, Yang B, Ying X, Gu B, Cui Y, Zhan Q.Numerical modeling for the characteristics study of a focusing ultrashort spatiotemporal optical vortex.Opt Express 2022; 30(21):37314-37322.

[45]	Leutenegger M, Rao R, Leitgeb RA, Lasser T.Fast focus field calculations.Opt Express 2006; 14(23):11277-11291.

[46]	Wan C, Cao Q, Chen J, Chong A, Zhan Q.Toroidal vortices of light.Nat Photonics 2022; 16(7):519-522.

RIGHTS & PERMISSIONS

THE AUTHOR

PDF (2771KB)

Part of a collection:

Supplementary files

supplementary data

5617

Accesses

Citation

Detail

Sections

Recommended

Journal home

Browse

Online first

Latest issue

All volumes and issues

Collections

Authors & reviewers

Guidelines for authors

Call for papers

Editorial policy

Copyright & license

Ethical requirements

Download templates

About the journal

Aims & scope

Description

Editorial board

Abstracting / Indexing

Contact us

中文版

Abstract

Graphical abstract

Keywords

Cite this article

1. Introduction

2. 3D STOV

3. Tight focusing of the 3D STOV

4. Results of orbit–orbit interaction

5. Conclusions

Acknowledgments

Compliance with ethics guidelines

Appendix A. Supplementary data

References

RIGHTS & PERMISSIONS