Spatiotemporal optical vortex combs

Domain: physics
Published: 2025-11-26
Editorial outcome: major_revision

Abstract
Single-packet spatiotemporal optical vortices (STOVs) with transverse orbital angular momentum (OAM) have recently garnered significant attentions. Nevertheless, more advanced applications require diverse transverse OAM modes and multiple STOV wave packets, forming what we term a STOV comb. Analogous to the frequency combs, the STOV comb is a sequence of STOV teeth with each tooth carrying the customized transverse OAM mode. In this work, method for generating the customizable STOV combs and application of STOV combs in information transmission are demonstrated. Theoretical modeling and experimental generation of STOV comb with multiple teeth and diverse topological charges (TCs) is carried out. The strategies for adjusting the TC difference and the spatiotemporal interval between the adjacent STOV teeth are demonstrated and the generation of irregular STOV combs is realized. Besides, the two-channel information encoding and transmission is realized with the dual-tooth STOV combs. Our work generates customizable STOV combs, paving the way for advanced applications of STOV fields.

Introduction
Spatiotemporal optical vortices (STOVs) are characterized by transverse orbital angular momentum (OAM), energy circulation, and phase winding within the spatiotemporal domain, garnering considerable interests in recent years. The theoretical studies of STOVs began earlier and their experimental realization has been achieved in recent years. Both generation and characterization methods, along with the fundamental properties of STOVs, have been extensively explored in theory and in experiment.

In recent years, novel STOV structures such as cylindrical vector STOVs, Bessel STOVs, and toroidal light vortices, have been developed. These advancements have leaded to emerging applications like spatiotemporal differentiators. Notably, Huang et al. recently demonstrated the STOV string, the single wave packet containing multiple STOVs with customizable topological charge (TC) arrangements. The STOV string exhibits separated singularities and multiple OAM modes in the time domain, enabling STOV string-based communication. However, for the spatiotemporal vortex string, numerous STOVs and various transverse OAM modes are limited within a single wave packet, which cause the overlap and interact of different STOVs or multiple OAM modes. Moreover, advancements in STOV structures have thus far been limited to single wave packet, whereas applications in information transmission demand diverse transverse OAM modes and high-density STOV wave packets in the time domain. Addressing these requirements, the concept of a STOV comb emerges as a promising solution. Analogous to optical frequency combs, optical vortex combs, and the pulse bursts, a STOV comb comprises multiple STOV wave packets, namely the STOV teeth, and each tooth carries particular transverse OAM mode. This configuration offers an ultrafast sequence of independent STOV wave packets with distinct transverse OAM modes, potentially advancing the development and application of STOV-based technologies.

In this work, we demonstrate a unique method for the realization of STOV combs, where the teeth are separated with fixed spatiotemporal interval and each tooth carries specific transverse OAM mode. To generate the STOV comb, a specially designed phase is applied to the Fourier plane of a two-dimensional (2D) pulse shaper. Both the theoretical simulation and experimental realization of the STOV comb are executed. The simulation results and the experimental results show similar patterns. Furthermore, we demonstrate strategies for adjusting the TC difference and the spatiotemporal interval between adjacent STOV teeth, both theoretically and experimentally. Additionally, we realize generating STOV comb with arbitrary TC sequence. The proposed method generates customizable STOV combs with adjustable parameters and thus paves the way for versatile applications of STOV combs in advanced optical technologies. In addition, the two-channel information encoding and transmission with the dual-tooth STOV combs is demonstrated in this work.

Results
The system for generating and characterizing the STOV combs, as shown in Fig. 1a, is built. The output of the femtosecond oscillator is split into two parts with a 90:10 beam splitter. The stronger transmitted part is utilized to generate the STOV combs and the weaker reflected part serves as the reference pulses for measuring the STOV combs. The STOV combs are measured with the interference method, which requires that the duration of the reference pulses is much shorter than the duration of the STOV pulses. Therefore, the unchirped transmitted pulses are stretched in time domain first with the pulse stretcher composed of two transmission gratings and a reflecting mirror (see Supplementary Material S1) and a negative group delay dispersion (GDD) of ~-443860 fs2 is introduced. The chirped pulses are sent to the 2D pulse shaper which is composed of the transmission grating, the cylindrical lens and the spatial light modulator (SLM). The dispersion direction of the transmission grating is in the x direction. The designed phase is loaded to the SLM to modulate the incident pulses into the STOV combs. The designed phase is placed in the Fourier plane of the 2D pulse shaper and is expressed as

T(x,y) = exp(ik\gamma x + im_k \theta)

where x and y are the spatial coordinates perpendicular to the beam propagation direction, a and b are the start number and the end number of the teeth in the STOV comb, \gamma is the parameter for adjusting the spatiotemporal interval between adjacent teeth, the sequence of m_k from m_a to m_b is the TCs of the teeth in the STOV comb, \theta is the azimuthal angle. The phase item of exp(ik\gamma x) realizes the spatiotemporal shift of the kth STOV tooth and the item of exp(im_k \theta) realizes distributing vortex phase with TC of m_k to the kth STOV tooth.

An example of designed phase is shown in Fig. 1b, which is calculated with Eq. (1) and with parameters of a=0, b=3, \gamma=\pi \times 10^4 m^{-1}, m_k=[0, 1, 2, 3]. Figure 1c shows the difference between generating single STOV wave packet and generating multiple STOV wave packets from the Gauss broadband pulse. When generating multiple STOV wave packets, single Gaussian broadband pulse is divided into multiple wave packets and different transverse OAM modes are distributed to the multiple wave packets, forming a STOV comb with multiple teeth. The designed phase is utilized for realizing this purpose and the effect of the designed phase is shown in Fig. 1d. In Fig. 1d, a STOV comb, where the teeth are separated with fixed spatiotemporal interval and individual tooth carries specific transverse OAM mode, is generated. Notably, a reflective SLM is utilized in experiment. To show the evolution of light field more directly, a transmissive configuration is utilized in Fig. 1a and Fig. 1d.

To characterizing the STOV comb, the reference pulses are expanded with a beam expander composed of two spherical lenses. The STOV comb and the reference pulses are combined using a 50:50 beam splitter. The temporal delay between the teeth in the STOV comb and the reference pulses is controlled with the computer-controlled delay stage in the reference path. Time-resolved spatial fringe patterns, resulting from the interference of the teeth and reference pulses at varying temporal delays, are collected through subsequently adjusting the computer-controlled delay stage at fixed step. Through analyzing the spatial fringe patterns, the reconstruction of the spatiotemporal structures of teeth in the STOV comb can be realized. And the reconstruction of the STOV comb can be realized through arranging the teeth with fixed spatiotemporal interval. More experimental details are given in Materials and Methods.

In this section, the STOV combs with nine teeth are simulated with the femtosecond Fourier-transform-limited (FTL) pulses and generated in experiment with the chirped pulses. For the simulated STOV comb, the parameters of the designed phase, named a, b, \gamma and m_k in the Eq. (1), are chosen as -4, 4, \pi \times 10^4 m^{-1} and [-4, -3, -2, -1, 0, 1, 2, 3, 4], respectively. This parameter setting allows for the realization of the STOV comb, containing nine teeth. The isosurface plot of the simulated STOV comb is shown in Fig. 2a. The STOV comb with nine teeth is realized and the TCs of these nine teeth are -4, -3, -2, -1, 0, 1, 2, 3, 4, from left to right in order. The nine teeth are separated in the time domain with fixed interval of ~0.5 ps and arranged along the X-axis with regular interval of ~1.0 mm (see Supplementary Material S2).

In the simulation result, the neighboring teeth in the STOV comb overlap partially in space as the interval along the X-axis between the neighboring teeth is ~1 mm, smaller than the laser spot size. This overlap becomes problematic when measuring the spatiotemporal structure of the chirped STOV comb. As consequence, the parameter \gamma of the designed phase used in experiment is set bigger to avoid this overlap. A designed phase with parameters of a, b, \gamma, m_k being -4, 4, 3\pi \times 10^4 m^{-1}, [-4, -3, -2, -1, 0, 1, 2, 3, 4], is utilized. With this designed phase, the STOV comb with nine teeth and with TCs of -4, -3, -2, -1, 0, 1, 2, 3, 4, is generated and measured. The temporal interval and the spatial interval along X-axis between the neighboring teeth are adjusted to ~1.5 ps and ~3.0 mm. And the isosurface plot of the measured STOV comb is shown in Fig. 2b with the temporal interval between the neighboring teeth magnified 15 times for better exhibiting the structure of every tooth. Compared with the simulated STOV comb in Fig. 2a, the STOV comb in Fig. 2b shows bigger temporal interval and spatial interval as the designed phase is acquired with a bigger parameter \gamma and the teeth shows larger pulse duration as the STOV comb is generated with the chirped pulses. In Fig. 2a and b, the projections of the teeth at the x-t and y-t panel are also shown with blue-gray gradient color and the intensity decreases as the blue color turns into the gray color. The results measured in experiments show similar patterns with the simulated results. In Fig. 2a, b, the profiles of the STOV teeth tend to increasing in the spatiotemporal domain as their TCs diverging from zero. The STOV teeth with positive TCs and the STOV teeth with negative TCs possess opposite orientations at the y-t panel. In the STOV teeth, the zero intensity regions, caused by the vortex phase in the y-t panel, are observed.

The isosurface plot of STOV teeth with TCs of -3 and 3 in the simulated STOV comb are shown in Fig. 2c, e to show the detailed profile of single simulated STOV tooth. Three zero-intensity regions are observed in both STOV teeth. Figure 2d is the intensity distribution at the x-t panel (the surface diagram) and in the time domain (the blue curve) of the simulated STOV comb. In Fig. 2d, the peak intensity of every tooth in the STOV comb is observed. The corresponding experimental results of Fig. 2c-e are shown in Fig. 2f-h. In Fig. 2g, the temporal interval between the neighboring teeth is magnified by 15 times. Three zero-intensity regions are observed in Fig. 2f, h. As the STOV teeth occupy larger region in the spatiotemporal domain and their intensity becomes dispersed as their TCs diverges from zero, the peak intensity of the STOV teeth in the STOV comb declines as their TCs diverges from zero, which is verified both in Fig. 2d, g. The decline of peak intensity, as the TCs of the STOV teeth diverge from zero, is also observed in Fig. 2g and the tendency of intensity in Fig. 2g is similar to Fig. 2d.

The phase information of the teeth in the measured STOV comb is analyzed as well. The experimentally measured teeth with TCs of -3, -1, 0, 2, 4, are picked out for phase study. The vortex phase lies in the y-t panel of the STOV teeth. To extract the phase information, the central slice lying in the y-t panel of the STOV tooth is picked out and five slices are chosen from these five teeth. The intensity information and the phase information located at these five slices is calculated. The intensity distributions located at these five slices are shown in Fig. 3a–e and the corresponding phases of these five slices are shown in Fig. 3f–j. As shown in Fig. 3f–j, the number of phase singularities in the phases of the STOV teeth equals to the absolute value of the corresponding TCs. For the high-order STOV teeth, the multiple phase singularities are dislocated in the y-t panel, namely the space-time plane. This phenomenon is caused by the spatiotemporal astigmatism, namely the mismatch between temporal dispersion and spatial diffraction. The STOV comb with nine teeth simulated with the chirped pulses and parameters close to the experiment, is also realized (see Supplementary Material S3). The dislocated phase singularities, induced by the spatiotemporal astigmatism, are also observed in the phases in the y-t panel of the high-order STOV teeth simulated with the chirped pulses. And this phase distortion can cause the decrease in the purity of OAM modes and can induce the degradation of the high-order STOV teeth into multiple low-order STOVs. Thus, the distorted phases in the high-order STOV teeth, introduced by the spatiotemporal astigmatism, impose extra constraint on the maximal TC of the STOV comb. The spatiotemporal astigmatism, as well as the constraint on the maximal TC, can be eliminated with careful management of the GDD of chirped pulses.

The STOV comb simulated with the femtosecond FTL pulses, the STOV combs simulated with the chirped pulses, and the STOV comb generated in experiment with the chirped pulses show similar patterns, such as the spatiotemporal structures of the teeth, the several zero-intensity regions in the high-order STOV teeth. Some differences are also observed between these STOV combs, mainly in the intensity profiles of the STOV teeth, especially in the y-t panel. The differences between the STOV combs simulated with the femtosecond FTL pulses and the STOV combs simulated with the chirped pulses or generated in experiment with the chirped pulses, are induced by the different GDD between the femtosecond FTL pulses and the chirped pulses. The differences between the STOV combs simulated with the chirped pulses and the STOV comb generated in experiment with the chirped pulses, are induced by the slight differences in parameters and by the factors like fluctuations or noises during the experiment.

In this section, the approach for adjusting the characters of the STOV comb is discussed. Through adjusting the parameters of the designed phase, the TC difference, the temporal interval and the spatial interval along X-axis between adjacent teeth, named \Delta TC, \Delta t and \Delta x, respectively, are adjusted.

For simplicity of demonstration, STOV combs with less teeth are simulated and experimentally generated. In Eq. (1), \gamma is the parameter for adjusting the spatiotemporal interval between the adjacent teeth and the sequence of m_k from m_a to m_b is the TC sequence of the teeth in the STOV comb. Hence, two designed phases, with parameters of a=0, b=2, \gamma=\pi \times 10^4 m^{-1}, m_k=[0, 1, 2] and parameters of a=0, b=2, \gamma=1.2\pi \times 10^4 m^{-1}, m_k=[0, 2, 4] respectively, are utilized for simulation and are numbered the 1st phase and the 2nd phase. Both the 1st phase and the 2nd phase allow for generating a STOV comb with three teeth. The isosurface plot of the simulated STOV comb with the 1st phase is shown in Fig. 4a. Three teeth with the TCs of 0, 1, 2, are realized. The \Delta t and the \Delta x between the neighboring teeth are ~0.5 ps and ~1.0 mm. Figure 4b is the intensity distribution at the x-t panel (the surface diagram) and in the time domain (the red curve) of the STOV comb in Fig. 4a. The isosurface plot of the simulated STOV comb with the 2nd phase is shown in Fig. 4c. Three teeth with the TCs of 0, 2, 4, are realized. The \Delta t and the \Delta x between the neighboring teeth are ~0.6 ps and ~1.2 mm. Figure 4d is the intensity distribution at the x-t panel (the surface diagram) and in the time domain (the red curve) of the STOV comb in Fig. 4c. Through adjusting m_k from [0, 1, 2] to [0, 2, 4], namely the interval of m_k from 1 to 2, the \Delta TC between adjacent teeth is adjusted from 1 to 2. Through increasing the parameter \gamma by 1.2 times, the \Delta t and the \Delta x between the neighboring teeth are increased by 1.2 times.

This control strategy is also verified in experiment. Two designed phases, with parameters of a=0, b=2, \gamma=5\pi \times 10^4 m^{-1}, m_k=[0, 1, 2] and parameters of a=0, b=2, \gamma=6\pi \times 10^4 m^{-1}, m_k=[0, 2, 4] respectively, are utilized in experiment. The isosurface plot of the generated STOV combs with these two designed phases are shown in Fig. 4e and Fig. 4g, respectively. Figure 4f and Fig. 4h are the intensity distribution at the x-t panel (the surface diagram) and in the time domain (the red curve) of the STOV combs in Fig. 4e and Fig. 4g. For better visualization, the \Delta t between the neighboring teeth in Fig. 4e–h is magnified 10 times. Similar to the simulation, through adjusting the interval of m_k from 1 to 2, the \Delta TC between the adjacent teeth is adjusted from 1 to 2. The \Delta t and the \Delta x between the neighboring teeth are increased by 1.2 times through increasing the parameter \gamma by 1.2 times. Noticeably, both the temporal interval \Delta t and the spatial interval along X-axis \Delta x between the neighboring teeth are bound to the parameter \gamma. Besides, other parameters of the 2D pulse shaper, such as the focus length of the cylindrical lens and the spatial frequency of the grating, are also effective for adjusting the spatiotemporal interval between the adjacent teeth (see Supplementary Material S4). The cylindrical lens with shorter focus length realizes shorter spatiotemporal interval. The transmission grating with lower spatial frequency realizes shorter temporal interval without changing the spatial interval.

In this section, generation of arbitrary STOV comb is discussed. In the arbitrary STOV comb, the spatiotemporal interval between the adjacent teeth and the TC sequence can be irregular. To realize arbitrary STOV comb, the parameter \gamma in Eq. (1) is set to a sequence instead of a fixed value and the Eq. (1) can be expressed as

T(x,y) = \sum_{k=a}^{b} exp(ik\gamma_k x + im_k \theta)

Through setting the sequence of \gamma_k and the sequence of m_k with irregular numerical vectors, specific spatiotemporal intervals and given TC sequence can be set to the STOV comb and the arbitrary STOV comb can be realized. To verify this method, the sequence of \gamma_k and the sequence of m_k are set to the irregular vectors of [1.8\pi \times 10^4, 0, 1.6\pi \times 10^4, 2.4\pi \times 10^4] m^{-1} and [-2, 0, 3, -1], respectively. With a=-1 and b=2, a specifically designed phase is calculated and a STOV comb with 4 teeth is simulated.

The isosurface plot of the simulated STOV comb is shown in Fig. 5a. Figure 5b is the intensity distribution at the x-t panel (the surface diagram) and in the time domain (the red curve) of the STOV comb in Fig. 5a. The isosurface plot of the simulated STOV comb with the 2nd phase is shown in Fig. 4c. Three teeth with the TCs of 0, 2, 4, are realized. The \Delta t and the \Delta x between the neighboring teeth are ~0.6 ps and ~1.2 mm. Figure 4d is the intensity distribution at the x-t panel (the surface diagram) and in the time domain (the red curve) of the STOV comb in Fig. 4c. Through adjusting m_k from [0, 1, 2] to [0, 2, 4], namely the interval of m_k from 1 to 2, the \Delta TC between adjacent teeth is adjusted from 1 to 2. Through increasing the parameter \gamma by 1.2 times, the \Delta t and the \Delta x between the neighboring teeth are increased by 1.2 times.

This control strategy is also verified in experiment. Two designed phases, with parameters of a=0, b=2, \gamma=5\pi \times 10^4 m^{-1}, m_k=[0, 1, 2] and parameters of a=0, b=2, \gamma=6\pi \times 10^4 m^{-1}, m_k=[0, 2, 4] respectively, are utilized in experiment. The isosurface plot of the generated STOV combs with these two designed phases are shown in Fig. 4e and Fig. 4g, respectively. Figure 4f and Fig. 4h are the intensity distribution at the x-t panel (the surface diagram) and in the time domain (the red curve) of the STOV combs in Fig. 4e and Fig. 4g. For better visualization, the \Delta t between the neighboring teeth in Fig. 4e–h is magnified 10 times. Similar to the simulation, through adjusting the interval of m_k from 1 to 2, the \Delta TC between the adjacent teeth is adjusted from 1 to 2. The \Delta t and the \Delta x between the neighboring teeth are increased by 1.2 times through increasing the parameter \gamma by 1.2 times. Noticeably, both the temporal interval \Delta t and the spatial interval along X-axis \Delta x between the neighboring teeth are bound to the parameter \gamma. Besides, other parameters of the 2D pulse shaper, such as the focus length of the cylindrical lens and the spatial frequency of the grating, are also effective for adjusting the spatiotemporal interval between the adjacent teeth (see Supplementary Material S4). The cylindrical lens with shorter focus length realizes shorter spatiotemporal interval. The transmission grating with lower spatial frequency realizes shorter temporal interval without changing the spatial interval.

In this section, generation of arbitrary STOV comb is discussed. In the arbitrary STOV comb, the spatiotemporal interval between the adjacent teeth and the TC sequence can be irregular. To realize arbitrary STOV comb, the parameter \gamma in Eq. (1) is set to a sequence instead of a fixed value and the Eq. (1) can be expressed as

T(x,y) = \sum_{k=a}^{b} exp(ik\gamma_k x + im_k \theta)

Through setting the sequence of \gamma_k and the sequence of m_k with irregular numerical vectors, specific spatiotemporal intervals and given TC sequence can be set to the STOV comb and the arbitrary STOV comb can be realized. To verify this method, the sequence of \gamma_k and the sequence of m_k are set to the irregular vectors of [1.8\pi \times 10^4, 0, 1.6\pi \times 10^4, 2.4\pi \times 10^4] m^{-1} and [-2, 0, 3, -1], respectively. With a=-1 and b=2, a specifically designed phase is calculated and a STOV comb with 4 teeth is simulated.

The isosurface plot of the simulated STOV comb is shown in Fig. 5a. Figure 5b is the intensity distribution at the x-t panel (the surface diagram) and in the time domain (the red curve) of the STOV comb in Fig. 5a. A STOV comb with 4 teeth is realized and the TC sequence of the STOV comb is set to an irregular sequence of [-2, 0, 3, -1]. Meanwhile, the spatiotemporal interval the STOV comb in Fig. 5a is irregular. The temporal intervals and the spatial interval along X-axis between the adjacent teeth are approximately adjusted to 0.9 ps, 0.8 ps, 0.4 ps and 1.8 mm, 1.6 mm, 0.8 mm, respectively. Besides, another specifically designed phase is also calculated with parameters of a=-1, b=2, \gamma_k=[9\pi \times 10^4, 0, 8\pi \times 10^4, 12\pi \times 10^4] m^{-1} and m_k=[-2, 0, 3, -1]. And this designed phase is utilized for generating arbitrary STOV comb in experiment. A STOV comb with 4 teeth is generated and measured in experiment. And the isosurface plot of the measured STOV comb is shown in Fig. 5c. Figure 5d is the intensity distribution at the x-t panel (the surface diagram) and in the blue curve) of the STOV comb in Fig. 5c. The temporal interval between the neighboring teeth in Fig. 5c, d is magnified 10 times for better visualization. Similarly, a STOV comb with 4 teeth is realized and the TC sequence of the STOV comb is set to an irregular sequence of [-2, 0, 3, -1]. The spatiotemporal interval the STOV comb in Fig. 5c is irregular. The temporal intervals and the spatial intervals along X-axis between the adjacent teeth are approximately adjusted to 4.5 ps, 4.0 ps, 2.0 ps and 9.0 mm, 8.0 mm, 4.0 mm, respectively. Apparently, the arbitrary STOV comb, with irregular spatiotemporal intervals and TC sequence, is available through setting the parameters of \gamma_k and m_k to the irregular vectors.

In addition, the designed STOV combs, sequences of teeth with alternant TC sequence of [-1, 1, -2, 2, -3, 3, -4, 4, -5, 5, 0] and irregular TC sequence of [1, 4, 1, 4, 2, 1, 3, 5, 6, 2, 3], are theoretically simulated as well (see Supplementary Material S5). Other properties of the STOV comb, like the spectral property or the propagation property, are also analyzed. The multiple teeth in the STOV comb share identical spectrum in theory. The spectra of a STOV comb containing five teeth are measured in experiment and the measured spectra of multiple teeth are almost identical (see Supplementary Material S6). The propagation property of the STOV comb after different propagation distances and after being focused with a spherical lens is simulated (see Supplementary Material S7). Meanwhile, the teeth amount in the STOV comb is extensible. Through expanding the range between a and b in Eq. (1), the designed phase for the STOV comb with more teeth can be calculated and the STOV comb with more teeth can be simulated in theory or realized in experiment. But the number of teeth in the STOV comb is limited due to the spatiotemporal interval between the adjacent teeth. Generating, transmitting, or measuring the STOV comb with more teeth requires optimized conditions in experiment to accommodate more teeth. The femtosecond laser with lower repetition frequency, optical elements with bigger size, and the SLM with smaller pixel size can be utilized to accommodate more teeth in a STOV comb (see Supplementary Material S8). Furthermore, a shorter spatiotemporal interval is conducive to realizing a STOV comb with more teeth. The shorter spatiotemporal interval can be achieved through adjusting the parameters of the 2D pulse shaper, such as designed phase with smaller parameter \gamma, cylindrical lens with shorter focus length. Notable, in the 2D pulse shaper, the transmission grating with lower spatial frequency realizes shorter temporal interval without changing the spatial interval. But the shorter spatiotemporal interval may cause the overlap of teeth in time domain or in the space domain. Shorter pulse duration and smaller laser spot are beneficial for addressing this question. Besides, the distorted phases in the high-order STOV teeth, cause by the spatiotemporal astigmatism, impose extra constraint on the maximal TC of the STOV comb as the phase distortion can cause the decrease in the purity of OAM modes and can induce the degradation of the high-order STOV teeth into multiple low-order STOVs. But this constraint can be solved with careful dispersion management or the introduction of special STOV wave packets named the degradation-free spatiotemporal Bessel vortices.

The longitudinal OAM has been widely applied in optical communication and the sole research about STOV-based optical communication was realized with the STOV string. Likewise, the STOV combs offer strong ability in information encoding and transmission. And the STOV combs show potential in high-capacity communication as every tooth in the STOV comb can act as an individual communication channel in the information communication. Inspired by the researches about optical communication with longitudinal OAM and STOV string, two-channel information encoding and decoding is demonstrated with the dual-tooth STOV combs in this work, to verify the ability of the STOV combs in information encoding and transmission.

The experimental system is shown in Fig. 6a. The output of the femtosecond oscillator passes through the pulse stretcher composed of two transmission gratings and the reflecting mirror, and the 2D pulse shaper composed of the transmission grating, the cylindrical lens and the SLM, to generate the dual-tooth STOV combs. Through designing the phases uploaded to the SLM, the information is dynamically encoded in the dual-tooth STOV combs. And the information is decoded with the diffraction method. To realized decoding, the dual-tooth STOV combs are dispersed with the transmission grating and collimate with a cylindrical lens. Afterwards, the diffraction patterns of the dual-tooth STOV combs are collected with a CCD camera.

Through dynamically encoding, the dynamic dual-tooth STOV combs are capable of two-channel information transmission with each tooth being an individual channel. Two binary images, as shown in Fig. 6b and both with 28x14 pixels, are transmitted with the dual-tooth STOV combs simultaneously. The pictures are transmitted with a pixel-by-pixel manner. Both pictures contain two types of pixels, the black pixel and the white pixel. Therefore, there are four kinds of pixel combinations when transmitting these two images pixel by pixel, namely the combination of two black pixels, the combination of black pixel and white pixel, the combination of white pixel, and black pixel, and the combination of two white pixels. Using binary encoding, the four kinds of pixel combinations are encoded as (0,0), (0,1), (1,0), and (1,1), as shown in Fig. 6c. These codes are utilized for designing the phases uploaded to the SLM. In Eq. (1), the parameter m_k is set as [0,0], [0,1], [1,0], or [1,1] according to the pixel combination and the other parameters are set as a=0, b=1, and \gamma=5\pi \times 10^4 m^{-1}. According to the pixel combinations, the designed phases are dynamically uploaded to the SLM and four types of dual-tooth STOV combs are generated. And four types of diffraction patterns, as shown in Fig. 6d, are recorded with the CCD camera which correspond to four kinds of pixel combinations in Fig. 6c respectively. The restoration of information is realized through identifying the diffraction patterns. For example, the 36th pixel, namely the 8th pixel in the second pixel line, in the first picture is a black pixel and the 36th pixel in the second picture is a white pixel. Therefore, the 36th piece of information corresponds to the combination of black pixel and white pixel and is encoded as (0,1). The designed phase uploaded to the SLM is calculated with parameters of a=0, b=1, \gamma=5\pi \times 10^4 m^{-1}, m_k=[0, 1]. A dual-tooth STOV comb is generated, in which the tooth at 0th order is a Gaussian wave packet and the tooth at 1st order is a STOV wave packet with TC of 1. The diffraction pattern turns out as the second frame in Fig. 6d and the pixels are identifying as the combination of black pixel and white pixel.

With this dynamic encoding and decoding method, the two images in Fig. 6b are transmitted successfully. In this information encoding and transmission process, the diffraction patterns is detected with a CCD camera, which realized the decoding and transverse OAM recognition in the spatial domain. As the transverse OAM locates at spatiotemporal domain, the decoding and transverse OAM recognition can be realized in the temporal domain (see Supplementary Material S9).

In this work, the method for generating a unique STOV field named the STOV combs is demonstrated in this work. The STOV comb contains multiple STOV wave packets, namely the STOV teeth, with customizable transverse OAM modes. The 2D pulse shaper equipped with the designed phase is utilized to generate the STOV combs. The method for generating STOV comb is verified both theoretically and experimentally. The STOV comb containing nine teeth with TCs ranging from -4 to 4 is simulated in theory. And the STOV comb containing nine teeth with TCs ranging from -4 to 4 is also generated in experiment. The experimental results show similar patterns with the simulated results. The opposite orientations at the y-t panel between the STOV teeth with positive TCs and the STOV teeth with negative TCs are observed. The zero-intensity regions in the STOV teeth are observed as well and the number of the zero-intensity region equals the TC of the STOV tooth. The method proposed in this work offers flexible ability in adjusting the characters of the STOV comb. The TC difference between adjacent teeth is adjusted from 1 to 2 through adjusting the interval of m_k. The temporal interval and the spatial interval along X-axis between the neighboring teeth are increased by 1.2 times through adjusting the parameter \gamma. Besides, the potential for generating arbitrary STOV comb, the STOV comb with irregular spatiotemporal interval and irregular TC sequence, is verified. The 4-tooth arbitrary STOV comb with TC sequence of [-2, 0, 3, -1], the temporal intervals of 0.9 ps, 0.8 ps, 0.4 ps, and the spatial intervals along X-axis of 1.8 mm, 1.6 mm, 0.8 mm, is theoretically simulated. The 4-tooth arbitrary STOV comb with TC sequence of [-2, 0, 3, -1], the temporal intervals of 4.5 ps, 4 ps, 2 ps, and the spatial intervals along X-axis of 9 mm, 8 mm, 4 mm, is experimentally measured. The amount of the teeth in the STOV comb is extensible but limited by the experimental conditions. In aspect of application, the two-channel information encoding and transmission with the dual-tooth STOV combs is demonstrated and two binary images are transmitted synchronously with the dual-tooth STOV combs.

The STOV combs realized in this work offer several unique properties over other light structures such as the optical frequency combs. The optical frequency combs are sequences of pulses in the time domain, corresponding to a sequence of equally spaced spectral lines in the frequency domain. And the optical vortex comb is an optical frequency comb with each spectral lines of the comb carrying a distinct longitudinal OAM mode. The optical vortex comb carrying up to 50 OAM modes has been realized and its characterization is simpler than the STOV comb. For an optical vortex comb, the singularity of the longitudinal OAM locates at the spatial domain, which have been applied for the high-resolution spatial sensing and imaging. In contrast, the STOV combs carry the transverse OAM. As the singularity of the transverse OAM locates at spatiotemporal domain, it would offer the potential for sensing and imaging applications where both spatial and temporal resolution are needed. The STOV string is a wave packet carrying numerous STOVs, up to 28 STOVs as recently reported, with customizable TC arrangements and transverse OAM modes. The TC arrangements and the transverse OAM modes of STOV string can be well recognized from diffraction patterns. However, as numerous STOVs in the STOV string are confined in single wave packet, the STOVs may overlap and interact with each other. Whereas, a STOV comb with nine teeth has been realized in this work. The STOV comb is a sequence of multiple STOV wave packets, with each STOV wave packet carrying the customizable transverse OAM mode. In a STOV comb, each STOV wave packet can be independently controlled and thus the overlapping between the transverse OAM modes is avoided. Several aspects about the STOV comb remain to be improved, such as increasing the number of teeth in the STOV comb or eliminating the spatiotemporal astigmatism in the STOV comb. Besides, the teeth in the STOV comb can be designed as other types of STOV wave packets, such as the degradation-free spatiotemporal Bessel vortices which are beneficial for avoiding the degradation of the high-order STOV wave packets.

Methods
In simulation, the broadband pulses used for generating the STOV comb is expressed as

E(x,y,\omega) = exp(-(x^2+y^2)/a^2) exp(-(\omega-\omega_c)^2/b^2) exp(i \beta/2 (\omega-\omega_c)^2)

where x and y are the spatial coordinates perpendicular to the beam propagation direction, \omega_c is the optical carrier frequency, \omega_c=2\pi c/\lambda_c, c is the light speed in vacuum, \lambda_c is the central wavelength, a and b are the beam waist radius of the Gaussian profile in the spatial domain and frequency domain respectively, and \beta is the GDD of the broadband pulses. For \beta=0 fs^2, the broadband pulses are the femtosecond FTL pulses. For \beta \neq 0 fs^2, the broadband pulses are the chirped pulses. With this designed phase, the broadband pulses are modulated to the STOV combs. In theory, the STOV combs are simulated with the diffraction rules (see Supplementary Material S10).

According to the property of Fourier transform between time domain and frequency domain, the method for generating a shift in time domain can be expressed

E(x,y,t+t_0) = FFT(E(x,y,\omega) exp(i\omega t_0))

where t_0 is the shift value in time domain and FFT denotes the Fourier transform between time domain and frequency domain. The shift in time domain is introduced by multiplying a fixed phase in frequency domain. To generate the transverse OAM, an extra phase is also needed and the optical field of single tooth in frequency domain can be expressed

E_{tooth}(x,y,\omega) = E_{in}(x,y,\omega) exp(i\omega t_0) exp(im\theta)

where m is the topological charge and \theta is the azimuthal angle. And the optical field of the STOV comb is the sum of the optical field of every tooth and can be expressed as

E_{comb}(x,y,\omega) = \sum_{k=a}^{b} E_{in}(x,y,\omega) exp(i\omega_k t_0) exp(im_k \theta)
= E_{in}(x,y,\omega) \sum_{k=a}^{b} exp(i\omega_k t_0 + im_k \theta)

where a and b are the start number and the end number of the teeth in the STOV comb, the sequence of m_k from m_a to m_b is the topological charges of the teeth in the STOV comb. The approximation of linear spectral dispersion is suitable for the grating-based pulse shapers. In the Fourier plane of the two-dimensional pulse shaper, the relationship between the x-coordinate and the frequency components can be expressed as

x = \xi(\omega-\omega_c)

where \xi is the linear coefficient between the x-coordinate and the frequency components and is determined by the period of the grating and the focal length of the cylindrical lens. And the Eq. (6) can be written as

E_{comb}(x,y,\omega) = E_{in}(x,y,\omega) \sum_{k=a}^{b} exp(ik t_0 x/\xi + ik t_0 \omega_c + im_k \theta)

where the item of exp(ik t_0 \omega_c) is a constant item and can be ignored. The Eq. (8) can be written as

E_{comb}(x,y,\omega) = E_{in}(x,y,\omega) \sum_{k=a}^{b} exp(ik \gamma x + im_k \theta)

and the Eq. (9) contains the phase required for generating the STOV comb. The phase distribution of the designed phase plate required for generating the STOV comb can be written as

T(x,y) = \sum_{k=a}^{b} exp(ik \gamma x + im_k \theta)

where \gamma is the parameter for adjusting the temporal interval between the adjacent STOV teeth. The phase item of exp(ik \gamma x) realizes the temporal shift between adjacent STOV teeth and is also the function of the x-coordinate. Therefore, the teeth in the STOV comb are also shifted with fixed interval along x-axis. The designed phase is converted to a phase hologram with the existing method and the hologram is calculated as

T(x,y) = angle(exp(i \times angle(T) + \pi sinc(1 - |T|/max(|T|)) + sinc(1 - 1/max(|T|))))

where angle is the operation for calculating the phase angle, T is the specially designed phase T(x,y), and \alpha is the total length of a few pixels in the spatial light modulation.

The output of the femtosecond oscillator (Coherent, Mira-900, ~200fs, 80MHz) is split into two parts with a 90:10 beam splitter. The transmitted part is utilized to generate the STOV combs and the reflected part serves as the reference pulses for measuring the STOV combs. The measurement of the STOV combs requires that the duration of the reference pulses is much shorter than the duration of the STOV pulses. Therefore, the unchirped transmitted pulses pass through the pulse stretcher composed of two transmission gratings (Thorlabs, GP1012L, 1200 lines/mm) and the reflecting mirror first. And the unchirped transmitted pulses is stretched to ~8ps after linear chirping. And then, the chirped pulses pass through the 2D pulse shaper. The 2D pulse shaper is composed of the transmission grating (Thorlabs, GP2012L, 1200 lines/mm), the cylindrical lens and the SLM (Meadowlark Optics, E-19x12-500-1200-HDM8, 1920x1200, 8\mu m x 8\mu m pixel). As the designed phase is loaded to the spatial light modulator and the chirped pulses evolve into the chirped STOV combs after passing the 2D pulse shaper.

The STOV combs and the reference pulses are overlapped with a 50:50 beam splitter. And the self-referenced interferometry method is realized with the experimental system, which measures the intensity information and the phase information of the STOV comb simultaneously (see Supplementary Material S11). When measuring the STOV comb, the computer-controlled delay stage (Thorlabs, LTS300C/M) in the reference path is utilized for realizing the temporal delay between the STOV comb and the reference pulses. The time-resolved spatial fringe patterns due to the interference of the chirped teeth and the reference pulses at different temporal delays are collected with a CCD camera (Thorlabs, CS505MU) and are analyzed to reconstruct the structures of the teeth in the STOV comb. When acquiring the spatial fringe patterns, the delay stage is moved at fixed step of 10\mu m, corresponding to a time-resolved step of 66.7fs. And 400 frames are collected for single tooth from the spatial fringe pattern appearing to vanishing.

Besides, the 2D pulse shaper, composed of the transmission grating, the cylindrical lens and the SLM, is a 4f pulse shaper system. The distance between the grating and the cylindrical lens and the distance between the cylindrical lens and the SLM are equal to the focal length of the cylindrical lens. The focal length of the cylindrical lens utilized in simulation and in experiment is 0.25m. The spatial frequency of the grating utilized in simulation and in experiment is 1200 lines/mm.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Please find attached review report.", "reviewer_2": "The manuscript by Feng et al. presents an innovative method for generating a unique spatiotemporal structured light field, coined as \"STOV combs.\" This innovative approach enables the creation of ultrafast sequences of independent STOV wave packets, each customizable in transverse Orbital Angular Momentum (OAM) modes, with adjustable intervals and scalable numbers of adjacent STOV teeth. The authors have successfully validated their theoretical models with experimental evidence, demonstrating a robust correlation between the simulations and empirical results. The developed STOV comb supports a diverse range of transverse OAM modes and multiple wave packets, indicating substantial potential for advanced applications in OAM-based information transmission, high-resolution imaging, and sensing. This manuscript constitutes a significant contribution to the field and merits consideration for publication in Nature Communications. However, to enhance the manuscript's clarity and impact, I recommend addressing the following aspects:\n1. The manuscript occasionally estimates parameters, such as \"The Δt and the Δx between neighboring teeth are ~0.5 ps and ~1 mm,\" respectively. These estimates also appear in other sections without sufficient methodological details. Clarification on how these parameters were estimated, their precision, and potential error sources would enhance the manuscript's rigor.\n2. Given that diffraction characteristics of STOV are important aspects for their applications, a discussion on the diffraction properties of the more complex STOV comb, particularly its stability during propagation in air, would enrich the manuscript significantly.\n3. In the section \"Adjusting the Characters of the STOV Comb,\" customization of the STOV comb's characteristics is achieved by modulating the parameters of the designed phase, such as the Topological Charge (TC) difference and the spatiotemporal intervals between adjacent teeth. The manuscript could benefit from a discussion on how other components of the two-dimensional pulse shaper, such as the grating period and the focal length of the cylindrical lens, influence the STOV comb's characteristics, particularly the spatiotemporal intervals.\n4. In the \"Materials and Methods\" section, the authors detail using an SLM (Meadowlark Optics, E-19×12-500-1200-HDM8, 1920×1200, 8 μm×8 μm pixel) to construct the 2D pulse shaper. Given the 8-bit depth phase modulation capability of the SLM, it would be insightful if the authors could discuss potential discrepancies between the quasi-continuous theoretical phase and the experimentally implemented phase, and how these might affect the resultant STOV comb.\n5. The figures generally compare experimental results with theoretical simulations, yet Figure 3 only depicts direct intensity information, phase information, and topological charges without corresponding theoretical results. Including theoretical data in Figure 3 would substantiate the experimental observations and strengthen the manuscript's validity.\n6. The conclusion mentions that \"STOV combs offer strong potential in various applications like information encryption or information carrier.\" If possible, it would be beneficial for the authors to elaborate on the practical implementation of one of these applications, perhaps by constructing a theoretical demonstration that utilizes the STOV combs.", "reviewer_3": "Response Letter\nDear reviewers,\nWe would like to express our sincere gratitude for your valuable and constructive comments.\nYour insightful suggestions have been instrumental in enriching the depth and improving the\noverall quality of our manuscript.\nIn response, we have carefully reviewed and thoughtfully incorporated the reviewers’\nrecommendations. For the main common concerns, we have made the following key revisions:\n1. We have added a comparison between STOV combs and other related light structures,\nincluding optical frequency combs, optical vortex combs, and optical spatiotemporal vortex\nstrings. This addition highlights the key differences in their properties, advantages, and\npotential applications, thereby clarifying the unique features of STOV combs.\n2. We have included a theoretical analysis and discussion on the maximum number of teeth\nthat can be generated within a single STOV comb. This section also addresses the primary\nlimiting factors and explores possible solutions to overcome them.\n3. We have added an experimental demonstration showcasing the use of STOV combs for\noptical information encoding, further emphasizing the potential of STOV combs for\npractical applications.\nIn addition to these major improvements, we have addressed all specific concerns raised by the\nreviewers on a point-by-point basis. This includes supplementary theoretical analyses and new\nexperimental results. The main manuscript and supplementary materials have been\ncomprehensively revised accordingly as well, with all changes highlighted.\nWe believe that these revisions have significantly improved the clarity, rigor, and impact of the\nwork. Please find our detailed, point-by-point responses to each of the reviewers’ comments\nbelow.\nBest regards,\nFu Feng (on behalf of all authors)\nDetailed Responses to the Reviewers\nWe would like to kindly remind that, the questions raised by reviewers are In Blue, our\nresponses are In Black, and corresponding revision on the manuscript and supplementary\nmaterials are enclosed in the Blue Box .\nResponses to Reviewer #1\nThe authors present a method able to generate spatiotemporal optical vortex (STOV) combs,\nwherein each tooth carries a specific orbital angular momentum (OAM). The spatiotemporal\nintervals of comb teeth and the sequence of topological charges (TCs) can be flexibly controlled\nby a specific 2D phase design. The structure light enables an ultrafast sequence of independent\nSTOV wave packets with distinct transverse OAM modes, thereby potentially enhancing the\napplications of STOV beams. While both experimental and simulation results are clearly\ndemonstrated, there is insufficient discussion regarding the advantages, limitations, and\npotential applications of this light in comparison to other vortex beams. Additionally, the\nauthors need to address the following issues before I can recommend publication.\n1\n1. The STOV comb comprises multiple STOV wave packets, with each wave packet carrying\na specific transverse OAM. This characteristic represents its primary advantage. However, both\noptical vortex combs and spatiotemporal vortex strings can also contain multiple OAM states\nwithin a single beam. What advantages do the STOV combs offer in comparison to the two\ntypes of vortex light? Additionally, what are their potential applications? It is essential to\nenhance the discussions and comparisons from these aspects.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Dear reviewers,\nWe would like to express our sincere gratitude for your valuable and constructive comments.\nYour insightful suggestions have been instrumental in enriching the depth and improving the\noverall quality of our manuscript.\nIn response, we have carefully reviewed and thoughtfully incorporated the reviewers’\nrecommendations. For the main common concerns, we have made the following key revisions:\n1. We have added a comparison between STOV combs and other related light structures,\nincluding optical frequency combs, optical vortex combs, and optical spatiotemporal vortex\nstrings. This addition highlights the key differences in their properties, advantages, and\npotential applications, thereby clarifying the unique features of STOV combs.\n2. We have included a theoretical analysis and discussion on the maximum number of teeth\nthat can be generated within a single STOV comb. This section also addresses the primary\nlimiting factors and explores possible solutions to overcome them.\n3. We have added an experimental demonstration showcasing the use of STOV combs for\noptical information encoding, further emphasizing the potential of STOV combs for\npractical applications.\nIn addition to these major improvements, we have addressed all specific concerns raised by the\nreviewers on a point-by-point basis. This includes supplementary theoretical analyses and new\nexperimental results. The main manuscript and supplementary materials have been\ncomprehensively revised accordingly as well, with all changes highlighted.\nWe believe that these revisions have significantly improved the clarity, rigor, and impact of the\nwork. Please find our detailed, point-by-point responses to each of the reviewers’ comments\nbelow.\nBest regards,\nFu Feng (on behalf of all authors)\nDetailed Responses to the Reviewers\nWe would like to kindly remind that, the questions raised by reviewers are In Blue, our\nresponses are In Black, and corresponding revision on the manuscript and supplementary\nmaterials are enclosed in the Blue Box .\nResponses to Reviewer #1\nThe authors present a method able to generate spatiotemporal optical vortex (STOV) combs,\nwherein each tooth carries a specific orbital angular momentum (OAM). The spatiotemporal\nintervals of comb teeth and the sequence of topological charges (TCs) can be flexibly controlled\nby a specific 2D phase design. The structure light enables an ultrafast sequence of independent\nSTOV wave packets with distinct transverse OAM modes, thereby potentially enhancing the\napplications of STOV beams. While both experimental and simulation results are clearly\ndemonstrated, there is insufficient discussion regarding the advantages, limitations, and\npotential applications of this light in comparison to other vortex beams. Additionally, the\nauthors need to address the following issues before I can recommend publication.\n1\n1. The STOV comb comprises multiple STOV wave packets, with each wave packet carrying\na specific transverse OAM. This characteristic represents its primary advantage. However, both\noptical vortex combs and spatiotemporal vortex strings can also contain multiple OAM states\nwithin a single beam. What advantages do the STOV combs offer in comparison to the two\ntypes of vortex light? Additionally, what are their potential applications? It is essential to\nenhance the discussions and comparisons from these aspects.\nResponse: We are appreciative for this constructive comment.\nThe optical vortex combs, shown in Fig. R1a, is an optical frequency comb with each\ntooth(frequency) of the comb carrying a distinct longitudinal OAM mode with its phase\nwinding in the spatial domain, i.e. the x-y panel. The spatiotemporal vortex string, shown in\nFig. R1b, is a single ultrafast wave packet carrying numerous vortices with various transverse\nOAM modes. In contrast, the STOV comb proposed by this manuscript, shown in Fig. R1c, is\nan ultrafast sequence of independent STOV wave packets, with each wave packet carrying a\nspecific transverse OAM (spatiotemporal OAM mode) with its phase winding in the\nspatiotemporal domain, i.e. the x-t panel or the y-t panel.\nFig. R1. The comparison of the optical vortex comb, the STOV string, and the STOV comb. a The optical\nvortex comb (cited from Chen B, Zhou Y, Liu Y, et al. Integrated optical vortex microcomb[J]. Nature Photonics,\n2024, 18(6): 625-631). b Four 3-STOV strings with different topological charge (TC) arrangements (cited from\nHuang S, Li Z, Li J, et al. Spatiotemporal vortex strings[J]. Science Advances, 2024, 10(19): eadn6206). c The\nsimulated STOV comb with nine teeth.\ndifference in the OAM type. The optical vortex comb carries the longitudinal OAMs, which is\nlocated at the spatial domain. The STOV comb carries the transverse OAMs, which is located\nat the spatiotemporal domain.\nThe advantage of the STOV comb over the optical vortex comb is that the STOV comb offers\na new dimension for detecting the OAM and thus offers the potential for high-speed detection\nof the OAM. More precisely, the detection of the longitudinal OAM generally requires detecting\nthe interference fringes or the diffraction patterns with two-dimensional detector like a CCD\ncamera. However, for STOV combs, due to the spatiotemporal property of transverse OAM, it\nis possible to detect the OAM in the time domain with the single pixel detector like the\nphotodiode which can offer higher acquisition speed and can be beneficial for scenarios where\nfaster detections are needed. Besides, for normal OAM frequency combs, as the singularity of\nthe longitudinal OAM locates at the spatial domain, it has been applied for the high-resolution\nspatial sensing and imaging. However, for a STOV combs, as the singularity of the transverse\nOAM locates at the spatiotemporal domain, it would offer the potential for sensing and imaging\napplications where both spatial and temporal resolution are required.\nIn aspect of spatiotemporal vortex string. Both the spatiotemporal vortex string and the\nSTOV comb can carry multiple transverse OAM modes. The primary difference between the\nspatiotemporal vortex string and the STOV comb is the number of the wave packets. For\nspatiotemporal vortex string, various transverse OAM states are limited within a single wave\npacket and unavoidably overlap with each other. Therefore, STOV string suffers from low\nOAM purity and cross-talk between singularities. In contrast, the STOV comb in our\nmanuscript contains multiple STOV wave packets, with each STOV wave packet carrying one\nspecific and arbitrarily modulable transverse OAM state. Compared with the STOV string, the\nSTOV comb avoids overlap of different transverse OAM states, as well as the cross-talk\nbetween singularities and offers high OAM purity, which would be beneficial for applications\nsuch as information encoding etc...\nThe advantages and potential applications of the STOV combs are discussed in the\nConclusion section and the modification is shown below (marked as yellow).\nIn order to further demonstrate the application of STOV combs, we have performed an\nadditional experiment on two-channel information encoding and transmission with dual-tooth\nSTOV. The decoding process is realized with two strategies, namely the decoding with the\ndiffraction patterns of the dual-tooth STOV combs and the decoding with the temporal\nwaveform of the dual-tooth STOV combs.\nIn aspect of decoding with the diffraction patterns, the experimental system for the two-\nchannel information encoding and transmission with the dual-tooth STOV combs is shown in\nFig. R2a. The output beam from the femtosecond oscillator passes through the pulse stretcher\nand the 2D pulse shaper to generate the dual-tooth STOV combs. Through designing the phases\nuploaded to the SLM, the information is dynamically encoded in the dual-tooth STOV combs.\nAnd the information is decoded with the diffraction method. To realize decoding, the dual-tooth\nSTOV combs are dispersed with the transmission grating and are collimate with a cylindrical\nlens. The diffraction patterns of the dual-tooth STOV combs are recorded with a CCD camera.\nThe CCD camera is place at the focal plane of the cylindrical lens.\nFig. R2. Two-channel information encoding and decoding with dual-tooth STOV combs. a Schematic of the\nsystem for two-channel information encoding and decoding. BS: beam splitter, M: mirror, G: grating, CL: cylindrical\nlens, SLM: spatial light modulator, CCD: charge coupled device array. b The two binary images with 28×14 pixels\nfor transmitting. c The four kinds of pixel combinations and encoding. d The four types of diffraction patterns\nrecorded with the CCD camera corresponding to the four kinds of pixel combinations in (c).\nTwo binary images shown in Fig. R2b, with 28×14 pixels, are transmitted with the dual-tooth\nSTOV combs, independently and simultaneously. The pictures are synchronously transmitted\nwith a pixel-by-pixel manner. Both pictures contain two values ‘0’ and ‘1’, namely the black\npixel and white pixel. Therefore, there are four kinds of recombination when transmitting these\ntwo images pixel by pixel as shown in Fig. R2c, namely the combination of two black pixels,\nthe combination of black pixel and white pixel, the combination of white pixel and black pixel\nand the combination of two white pixels. Using binary encoding, the four kinds of pixel\ncombinations are encoded as (0, 0), (0, 1), (1, 0), and (1, 1). These codes are utilized for\ndesigning the phases uploaded to the SLM. The parameter m , for designing the phase, isset as\nk\n[0, 0], [0, 1], [1, 0], or [1, 1] according to the pixel combination and the other parameters are\nset as a=0, b=1, and =5π×104 m-1. According to the pixel combination, the designed phases\nare dynamically uploaded to the SLM and four types of dual-tooth STOV combs are generated.\nThe corresponding diffraction patterns, recorded with the CCD camera, are shown in Fig. R2d,\nwhich correspond to four kinds of pixel combinations in Fig. R2c respectively. The restoration of\ninformation is realized through identifying the diffraction patterns. For example, the 36th\npixel, namely the 8th pixel in the second line, is black and white for the first picture and second\npicture respectively. Therefore, the 36th piece of information corresponds to the combination of\nblack pixel and white pixel and is encoded as (0,1). The designed phase uploaded to the SLM\nis calculated with parameters of a=0, b=1, =5π×104 m-1, m =[0, 1]. A dual-tooth STOV comb\nk\nis generated, in which the first tooth is a Gaussian wave packet and the second tooth is a STOV\nwave packet with TC of 1. The diffraction pattern turns out as the second frame in Fig. R2d and\nthe pixels are identifying as the combination of black pixel and white pixel. By repeating\nthe same process, the two images are transmitted successfully. Moreover, the number of images\ntransmitted simultaneously can be increased by increasing the number of teeth carried by the\nSTOV combs easily.\nThis part is supplemented in section of “Information encoding and transmission with STOV\ncombs” in the manuscript from line 329 to line 384. The partial modification in the manuscript\nis shown below (marked as yellow).\nAnd some claims in the abstract, introduction and conclusion are also modified and the\nmodifications are shown below (marked as yellow).\nAs we have stated previously, an important property of STOV combs is its transverse\nOAM modes which offers the potential for detecting the OAM and decoding in the time domain.\nThe experimental system for decoding in the temporal domain is shown in Fig. R3a. The\ngeneration of STOV combs and encoding process remains unchanged with Fig. R2a. One tooth\nin the dual-tooth STOV comb is picked out with a reflecting mirror and recombined with\nanother tooth through a beam splitter after an extra time delay. The recombined beam is focused\nto a high-speed photodiode (Thorlabs, DXM30BF, 30GHz) to realize photovoltaic conversion\nand the temporal waveform is recorded with an oscilloscope (Keysight, DSAV164A, 16GHz).\nThe two binary images for transmitting are shown in Fig. R3b and the encoding method is\nshown in Fig. R3c. In the decoding process, as the picked-out tooth goes through a longer\noptical path compared with the other tooth, two teeth in the dual-tooth STOV combs turn out\nas two peaks in the temporal waveforms acquired with the oscilloscope and the intensity of the\ntwo peaks is utilized for decoding. Four types of temporal waveforms are acquired with the\noscilloscope as shown in Fig. R3d, which correspond to four kinds of pixel combinations in\nFig. R6c respectively. It can be observed that, the tooth of Gaussian wave packet turns out as a\n5\nstronger peak and the tooth of STOV wave packet with topological charge of 1 turns out as a\nweak peak in the temporal waveforms. The threshold value, for distinguishing the Gaussian\nwave packet and the STOV wave packet with topological charge of 1, is located at ~1.5 V. The\nrestoration of information is realized through identifying the intensity of two peaks in the\ntemporal waveforms. For example, the 36th pixel, namely the 8th pixel in the second pixel line,\nin the first picture is a black pixel and the 36th pixel in the second picture is a white pixel.\nTherefore, the 36th piece of information corresponds to the combination of black pixel and white\npixel and is encoded as (0,1). The designed phase uploaded to the SLM is calculated with\nparameters of a=0, b=1, =5π×104 m-1, m =[0, 1]. A dual-tooth STOV comb is generated, in\nk\nwhich the first tooth is a Gaussian wave packet and the second tooth is a STOV wave packet\nwith TC of 1. The temporal waveform turns out as the second frame in Fig. R3d and the pixels\nare identifying as the combination of black pixel and white pixel. With this dynamic encoding\nand decoding method, the two images in Fig. R3b are transmitted successfully.\nFig. R3. Two-channel information encoding and decoding with dual-tooth STOV combs. a Schematic of the\nsystem for two-channel information encoding and decoding. BS: beam splitter, M: mirror, G: grating, CL: cylindrical\nlens, SLM: spatial light modulator, CCD: charge coupled device array. b The two binary images with 28×14 pixels\nfor transmitting. c The four kinds of pixel combinations and encoding. d The four types of temporal waveforms\nacquired with the oscilloscope corresponding to the four kinds of pixel combinations in (c).\nExcept for decoding with the intensity information of the peaks, the decoding process can be\nachieved via analyzing the profile of the peaks as the waveforms of wave packets with different\nTCs show different temporal profiles, which would further facilitate the decoding process.\nFig. R4. The intensity isosurface and temporal intensity profile of STOV teeth with different TCs. a-c The\nintensity isosurface of the Gaussian wave packet, the STOV wave packet with TC of 1, the STOV wave packet with\nTC of 2. d-e The waveforms of the wave packets in (a-c).\npacket with TC of 1, and the STOV wave packet with TC of 2, are shown in Figs. R4a-c,\nrespectively. Figs. R4d-e are the corresponding waveforms of the wave packets in Figs. R4a-c.\nThe Gaussian wave packet is single-peak waveform, the STOV wave packet with TC of 1 is\ndual-peak waveform, and the STOV wave packet with TC of 2 is triple-peak waveform.\nTherefore, the profile features, namely the number of peaks in the temporal waveform, can be\nutilized for decoding and transverse OAM recognition. Nevertheless, it worth mentioning that\nrecording and resolving the peaks requires superior hardware and complicated experimental\nsystem. First, strong chirp is required to stretch the peaks, which makes resolving the peaks\nsimpler. Meanwhile, faster photodiodes and oscilloscopes, 50 GHz or higher, are required to\nrecord the peaks. In addition, the femtosecond oscillator with ultra-wide band spectrum is\npreferred for the realization of chirping and decoding.\nThis part is supplemented in section of “S9. Information encoding with STOV combs and\ndecoding in the temporal domain” in the supplementary material. The partial modification is\nshown below.\nThe corresponding modification in the manuscript is shown below (marked as yellow).\n2. The STOV comb shown in this manuscript is similar to the recent work (X. Liu et al.\n“Ultrafast bursts of tailored spatiotemporal vortex pulses,” arXiv:2407.19747). It may be\nnecessary to clarify whether the research in this manuscript was performed independently,\nhighlight the distinctions between the two studies, and appropriately cite this paper.\nResponse: We thank the reviewer for this kind reminder.\nAs the referenced manuscript was not published at the time of our study, we were not aware\nof its existence during the course of our work. As suggested, we have gone through the\nreferenced manuscript carefully, several distinctions exist between the two studies.\nFirstly, in methodology, in our work, diffraction theory is employed to simulate the STOV\ncombs and details of how to modulate the STOV topological values of each tooth and how to\ncontrol the spatiotemporal distance between teeth are given. In contrast, the referenced\nmanuscript elaborates a method called spatiotemporal holography for designing the phase\npattern. Details of the theoretical designing approach are not given (which we assume is due to\nits current status as a preprint).\nSecondly, in our manuscript, the STOV wave packets in a STOV comb are separated with\nequal spatiotemporal intervals, namely, the STOV wave packets are separated in both space and\ntime domain. In the referenced manuscript, the STOV wave packets in a spatiotemporal vortex\n7\nburst are separated only in time and is not separated spatially. A direct comparison of the\nspatiotemporal vortex burst and the STOV comb is shown in Fig. R5.\nFig. R5. The comparison of the spatiotemporal vortex burst and the STOV comb. a The spatiotemporal vortex\nburst (cited from Liu X, Liang C, Cao Q, et al. Ultrafast bursts of tailored spatiotemporal vortex pulses[J]. arXiv\npreprint arXiv:2407.19747, 2024). b The simulated STOV comb with nine teeth.\nThe mentioned paper is cited in our work. The corresponding revisions in the manuscript are\nshown below (marked as yellow).\n3. The authors think that the teeth amount in the STOV comb can be infinitely increased in\ntheory. I disagree with this claim. As shown in this study, spatiotemporal intervals of comb teeth\nhave to be designed carefully to prevent the spatiotemporal overlap among adjacent teeth. Apart\nfrom the spatial limitation of detector, the teeth amount is also restricted by some factors such\nas the pulse duration, spatiotemporal intervals, and TC values. The authors should point out\nthese limitations and evaluate the maximum feasible number of teeth from a technical\nstandpoint.\nResponse: Thanks a lot for raising this constructive recommendation.\nWe claimed that the teeth amount in the STOV comb can be infinitely increased under an\nideal assumption, in which the power of the laser pulses are unlimited, the performance of the\noptical elements or devices are ideal, and so on. Obviously, this deal assumption is unrealistic\nin experiment. In the following section, we give out a detailed analysis of the maximum number\nof teeth that can be generated and its limiting factors.\nIn the STOV comb, the teeth are distributed with equal spatiotemporal intervals, which is\nimportant for determining the maximum feasible number of teeth. The designed phase for\ngenerating the STOV comb is expressed as\n8\nb\nTx,yexpikxim (1)\nk\nka\nAssuming that the phase is calculated with an appropriate parameter  and the temporal\ninterval and the spatial interval of the generated STOV comb is Δt and Δx respectively.\nThe pulse duration and the size of laser spot play equal role in limiting the teeth amount\nin a STOV comb. As the teeth in a STOV comb are separated in the space-time domain with\nequal spatiotemporal interval, the temporal interval between the adjacent teeth should exceed\nthe pulse duration of single tooth or the spatial interval between the adjacent teeth should exceed\nthe laser spot of single teeth. Shorter pulse duration and smaller laser spot allows for a shorter\nspatiotemporal interval and more teeth can be accommodated in a STOV comb. As the pulse\nduration gets longer or the laser spot gets bigger, longer spatiotemporal interval is required to\navoid the spatiotemporal overlap between the adjacent teeth and the maximum number of teeth\nin the STOV comb should decrease.\nThe spatiotemporal interval between the teeth in the STOV comb is also a key factor for\nevaluating the maximum feasible number of teeth in the STOV comb. To generate a STOV\ncomb with more teeth, the spatiotemporal interval should be set shorter. The factors associated\nwith the spatiotemporal interval are the parameter  for calculating the designed phase, the\nfocus length of the cylindrical lens and spatial frequency of the transmission grating in the 2D\npulse shaper. A smaller  can realize shorter spatiotemporal interval. The cylindrical lens\nwith shorter focus length can also realize shorter spatiotemporal interval. The transmission\ngrating with lower spatial frequency can be utilized to achieve shorter temporal interval, which\ndoes not affect the spatial interval. But the shorter spatiotemporal interval may cause the overlap\nof teeth in time domain or in the space domain. Shorter pulse duration and smaller laser spot is\nhelpful for addressing this question.\nFig. R6. The limitations to the maximum feasible number of teeth. a The spatial intensity distribution of the\nSTOV tooth with topological charge of 2. b The spatial intensity distribution of the STOV tooth with topological\ncharge of 4. c Limitation from the size of the reflecting mirror to the maximum feasible number of teeth. d Limitation\nfrom the pixel size of the SLM to the maximum feasible number of teeth.\nof the STOV teeth along Y-axis increases as their TCs diverging from zero. Fig. R6a and Fig.\nR6b are the spatial intensity distribution of the STOV teeth with topological charges of 2 and\n4, recorded with a CCD camera. The length of the STOV tooth with topological charge of 4\nalong Y-axis almost reaches the width of the CCD along Y-axis as labelled with the orange\nrectangle in Fig. R6b. STOV teeth with topological charge over 4 or under -4 reach the\ndimensional limit of the CCD, which causes difficulty in the measurement of the STOV comb.\nSmaller laser spot is helpful for addressing this difficulty.\nEvaluating the maximum number of teeth in this STOV comb can be evaluated in the\nfollowing aspects.\nFirst, in the time domain, the repetition frequency F of the femtosecond laser utilized for\ngenerating the STOV comb, limits the maximum number of teeth. The time interval between\nadjacent femtosecond pulses is 1/F. To avoid the overlap of teeth generated from the adjacent\nfemtosecond pulses, the number of teeth generating from single femtosecond pulse should be\ncontrolled below\n1Ft\n. Therefore, the maximum number of teeth in the time domain can\nbe calculated as N 1Ft .\nt\nSecond, in the space domain, the size of the optical elements, mainly the circular reflecting\nmirror, limits the maximum number of teeth. As shown in Fig. R6c, when the generated STOV\ncomb transmits in free space and is reflected by the mirror, the multiple teeth occupy a certain\namount of space. Assuming that the mirror, with diameter of D, is standardly placed with an\nangle of 4 with the incident STOV comb and the mirror is aligned with the STOV comb to\ncover teeth as many as possible, the effective size of the mirror is\nDcos4\n, as shown in Fig.\nR6c a with M'. Therefore, the maximum number of teeth in the space domain can be calculated\nas N Dcos4 x.\ns\nThird, another limitation to the teeth amount is the pixel size of the SLM. The phase for\ngenerating the kth STOV tooth in equation (1) is calculated as\nT x,yexpikxim\nk k\n(2)\nexpikxexpim\nk\nWhen k 0, the phase item of expikx realizes the modulation of the wave vector and\ncause the kth STOV tooth deviates from the 0th STOV tooth with a certain angle when exits from\nthe SLM as shown in Fig. R6d. As the absolute value of k increases, the absolute value of k\nincreases and the deviation angle between the kth STOV tooth deviates and the 0th STOV tooth\ngets bigger. However, size of the square pixel in the SLM limits the maximum of the absolute\nvalue of k. For the square pixel with size of d, the corresponding spatial frequency is 1d\nand the maximal absolute value of k is limited not exceeding d. Therefore, when k 0,\nthe maximum number of teeth is calculated as d. Similarly, when k 0, the maximum\nnumber of teeth is also calculated as d. Considering the 0th STOV tooth, the maximum\nnumber of teeth limited by the pixel size of the SLM can be calculated asN 2d1.\nSLM\nOverall, the maximum feasible number of teeth should be the minimum of N , N , and\nt s\nN . Taking the experimental condition for measuring the STOV comb with nine teeth as an\nSLM\nexample, the repetition frequency of the femtosecond laser is 80 MHz, the parameter  for\ncalculating the designed phase is 3π×104 m-1, the diameter of the mirrors is 50 mm, the 2D pulse\nshaper is composed of the transmission grating (spatial frequency 1200 lines/mm), the\ncylindrical lens (focal length 250 mm), and the SLM (8 μm×8 μm pixel). This experimental\nconfiguration generates STOV comb with temporal interval of ~1.5 ps and the spatial interval\n10\nof ~3.0 mm. With the parameters above, the maximum number of teeth in the time domain is\ncalculated as N 1 Ft1  801061.51012 8333. The maximum number of teeth in\nt\nthe space domain is calculated as N Dcos4 x50cos4 3.011. The maximum\ns\nfeasible number of teeth in a STOV comb limited by the pixel size of the SLM is calculated as\nN 2d12  3π1048106 19 . And the round down method is utilized in\nSLM\nthese calculations. Therefore, the maximum feasible number of teeth under this experimental\ncondition is evaluated as 9 as we have measured in our study.\nThe claim that the teeth amount in the STOV comb is extensible is tempered. Under the\nabove calculation and analysis, the factors limiting the teeth amount has been claimed in the\nmanuscript in detail. The modification in the manuscript is shown below (marked as yellow).\nThe evaluation of the maximum feasible number of teeth is supplemented in section of “S8. Factors limiting the maximum feasible number of teeth in a STOV comb” in the supplementary\nmaterial. The partial modification is shown below.\n4. The duration of each wave packet is approximately 200 fs in the simulation results, whereas\nthe experimental duration is on the order of picoseconds. Could this discrepancy be attributed\nto the fact that the simulation did not account for the pulse chirp? I suggest conducting\nsimulations using experimental parameters to facilitate a quantitative comparison with\nexperimental data. Additionally, a brief description about the simulation should be included in\nthe methods section. Apart from the difference in wave packet durations, are there any\ndifference between the STOV combs generated by chirped pulses and those produced by\nFourier-transform limited pulses?\nWe would like to first clarify that there is a difference between the parameters used in the\nexperiments and those used in the simulations, due to two main reasons. First, in the experiment,\nthe pulse was chirped to the picosecond scale to enable measurement of the spatiotemporal\nstructure using an interferometric method. This method requires the measured pulse to be\nsignificantly broader than the reference pulse (i.e., an unchirped laser pulse) in time domain.\nSecond, in the simulation, a unchirped pulse was used to reduce the computational time and\nresource requirements. The duration of the unchirped femtosecond reference pulses is ~200 fs,\nwhereas the duration of the chirped pulses for generating the STOV combs is ~16 ps. The\nduration is stretched by ~80 times in the time domain. Simulation with the experimental\nparameters would take large computational time and resource requirements as large data size,\nmainly in the temporal domain, is involved due to the chirping process.\nThe discrepancy between the simulated duration and the experimental duration is attributed\nto the fact that the simulation did not account for the pulse chirp. Under this suggestion,\nsimulation of the STOV comb with experimental parameters is added to support the\nexperimental results in the manuscript.\nThe intensity isosurface of STOV comb with nine teeth, simulated with experimental\nparameters and measured experimentally, is shown in Fig. R7a and Fig. R7b, respectively. It\ncan be observed that the STOV comb simulated with experimental parameters is similar with\nthe STOV comb measured in experiment. And the STOV comb simulated with experimental\nparameters is also similar with the STOV comb simulated with the Fourier-transform limited\npulses (shown in Fig. R1c and Fig. R5b).\nFig. R7. The comparison of STOV comb simulated with experimental parameters and measured in\nexperiment. a The intensity isosurface of STOV comb with nine teeth simulated with experimental parameters. b\nThe intensity isosurface of STOV comb with nine teeth measured in experiment. (The temporal interval of the STOV\n12\ncombs in (a, b) is amplified magnified 15 times for better visualization.)\nMeanwhile, the phase information of the teeth in the simulated STOV comb in Fig. R7a is\nalso analyzed. The teeth with TCs of -3, -1, 0, 2, 4, in the simulated STOV comb are picked out\nfor phase study.\nThe intensity information and the phase information located at central slices of the five tooth\nis calculated. The intensity distributions at these five slices are shown in Figs. R8a-e and the\nphases of these five slices are shown in Figs. R8f-j. In Figs. R8f-j, green arrows are utilized to\nlabel the direction of phase eddying. Judging from the phases shown in Figs. R8f-j, it is obvious\nthat the corresponding TCs are -3, -1, 0, 2, 4, respectively. Compared with the experimental\nresults in the manuscript, the simulated results show similar pattern with the experimental\nresults both in intensity and in phase.\nFig. R8. The intensity distribution at the y-t panel of central slices of teeth with TCs of -3, -1, 0, 2, 4 in Fig.\nR7a and the corresponding phases. a-e The intensity distribution of specific slices at y-t panel of teeth with TCs\nof -3, -1, 0, 2, 4 in Fig. R7a. f-j, The phases of slices in (a-e).\nTo conclude, the STOV combs simulated with Fourier-transform limited pulses and chirped\npulsed show no significant differences between each other. We supplement simulations using\nexperimental parameters in the supplementary material to facilitate a quantitative comparison\nwith experimental data. This part is supplemented in section of “S3. Simulation of the STOV\ncombs with experimental parameters” in the supplementary material. The partial modification is\nshown below.\nThe corresponding modification in the manuscript is shown below (marked as yellow).\nIn the manuscript, a brief description about the simulation is supplemented in the methods\nsection from line 430 to line 469. The partial modification in the manuscript is shown below\n(marked as yellow).\n13\n5. a) The authors claimed that “the isosurface plot of the measured STOV comb is shown in Fig.\n2b with the temporal interval between the neighboring teeth magnified 15 times” in the main\ntext, but 10× amplification is given in the caption. The inconsistent should be corrected.\nb) In the caption of Fig. 4, “…in Fig. 4c and Fig. 4f is amplified magnified 10 times”. Fig. 4c\nshould be Fig. 4e.\nc) The description such as “the simulation result and the experimental result agree well with each other” and “Experimental result shows good agreement with the simulation” is not\ninappropriate, due to quantitative differences between simulation results and experimental\nresults.\nResponse: We are quite thankful for these corrections.\na) We have confirmed the experimental results. 10× amplification in the caption is corrected to\n15× amplification.\nb) In the caption of Fig. 4, “…in Fig. 4c and Fig. 4f is amplified magnified 10 times”. Fig. 4c\nhas been corrected to Fig. 4e.\nc) The description such as “the simulation result and the experimental result agree well with each other” and “Experimental result shows good agreement with the simulation” has been\ntempered.\nResponses to Reviewer #2\nThe manuscript by Feng et al. presents an innovative method for generating a unique\nspatiotemporal structured light field, coined as \"STOV combs.\" This innovative approach\nenables the creation of ultrafast sequences of independent STOV wave packets, each\ncustomizable in transverse Orbital Angular Momentum (OAM) modes, with adjustable\nintervals and scalable numbers of adjacent STOV teeth. The authors have successfully validated\ntheir theoretical models with experimental evidence, demonstrating a robust correlation\n14\nbetween the simulations and empirical results. The developed STOV comb supports a diverse\nrange of transverse OAM modes and multiple wave packets, indicating substantial potential for\nadvanced applications in OAM-based information transmission, high-resolution imaging, and\nsensing. This manuscript constitutes a significant contribution to the field and merits\nconsideration for publication in Nature Communications. However, to enhance the manuscript's\nclarity and impact, I recommend addressing the following aspects:\n1. The manuscript occasionally estimates parameters, such as \"The Δt and the Δx between\nneighboring teeth are ~0.5 ps and ~1 mm,\" respectively. These estimates also appear in other\nsections without sufficient methodological details. Clarification on how these parameters were\nestimated, their precision, and potential error sources would enhance the manuscript's rigor.\nResponse: We appreciate the reviewer sincerely for appraising our manuscript professionally\nand carefully. Meanwhile, this constructive comment is fully adopted.\nTo determine the spatial interval and the temporal interval between the adjacent STOV teeth,\nthe central coordinate of every STOV tooth in the x-t panel is calculated. Taking the STOV\ntooth with topological charge of 1 in the simulated STOV comb for example, the method for\ncalculating the central coordinate of this STOV tooth is detailed in the following. The simulated\nSTOV tooth with topological charge of 1 is shown in Fig. R9a. Its intensity profile along time\ncoordinate is shown in Fig. R3b with orange dotted line. Through interpolation, a quasi-\ncontinuous profile is obtained as shown in Fig. R9b with blue line. Thereby, the full wave half\nmaximum (FWHM) of the quasi-continuous profile is estimated, labelled with two asterisks in\nFig. R9b. The temporal center of the STOV tooth is acquired by calculating the average of the\ntime coordinates of the two asterisks. Similarly, the spatial center along X-axis is also calculated\nwith this method, as shown in Fig. R9c. The nine temporal centers and the nine spatial centers\nare shown in Table.R1.\nTable. R1. The temporal center and the spatial center along X-axis of the nine STOV teeth in the simulated\nSTOV comb. TC: topological charge. T: the temporal center (ps). S: the spatial center along X-axis (mm).\nTC -4 -3 -2 -1 0 1 2 3 4\nT -2.037 -1.526 -1.015 -0.505 0.005 0.515 1.025 1.536 2.046\nS -4.040 -3.033 -2.031 -1.033 -0.033 0.969 1.968 2.969 3.975\nWith the data in Table.R1, the temporal interval and the spatial interval between the adjacent\nSTOV teeth is calculated. As the STOV comb contains nine teeth, there eight temporal intervals\nand spatial interval. Through averaging, the temporal interval and the spatial interval turn out\nas 0.5105±0.0003 ps and 1.0021±0.0033 mm, respectively.\nIn addition, this method is also utilized for determine the temporal interval and the spatial\ninterval between the adjacent teeth in the STOV comb measured in experiment. The central\ncoordinates in the x-t panel of STOV teeth are calculated. As an example, the calculation of\nSTOV tooth with TC of 1 is shown in Figs. R9d-e. The temporal interval and the spatial interval\nin the STOV comb measured in experiment are determined, which are calculated as\n1.5113±0.0291 ps and 2.9967±0.0886 mm, respectively.\nIn conclusion, the temporal interval of the simulated comb is calculated as 0.5105 ps with\nprecision of 0.0003 ps and the spatial interval of the simulated comb turns out as 1.0021 mm\nwith precision of 0.0033 mm. The errors are mainly caused by the asymmetric structure of the\nin Fig. R9a and can cause error when determining the central coordinate in the x-t panel of the\nSTOV teeth. The temporal interval of the comb measured in experiment is calculated as 1.5113\nps with precision of 0.0291 ps and the spatial interval of the simulated comb turns out as 2.9967\nmm with precision of 0.0886 mm. The errors are also caused by the asymmetric structure of the\nSTOV teeth. Besides, the deviation of the optical path, especially the deviation of lase spot and\nSLM, would make the asymmetric structure more prominent and the error would get bigger.\nAccording to the calculation above, the spatial interval and the temporal interval of the\nsimulated comb with nine teeth are claimed as ~1.0 mm and ~0.5 ps and the spatial interval and\nthe temporal interval of the comb measured in experiment with nine teeth are claimed as ~3.0\nmm and ~1.5 ps in the manuscript.\nFig. R9. The method for calculating the center in the x-t panel of STOV teeth. a The intensity isosurface of\nSTOV tooth with topological charge of 1 in the simulated STOV comb. b Method for calculating the spatial center\nalong X-axis of the STOV tooth in (a). c Method for calculating the temporal center of the STOV tooth in (a). d The\nintensity isosurface of STOV tooth with topological charge of 1 in the experimental STOV comb. e Method for\ncalculating the spatial center along X-axis of the STOV tooth in (d). f Method for calculating the temporal center of\nthe STOV tooth in (d).\nThis part is supplemented in section of “S2. Calculation of the spatiotemporal interval\nbetween the adjacent STOV teeth” in the supplementary material. The partial modification is\nshown below.\nThe corresponding modification in the manuscript is shown below (marked as yellow).\n2. Given that diffraction characteristics of STOV are important aspects for their applications, a\n16\ndiscussion on the diffraction properties of the more complex STOV comb, particularly its\nstability during propagation in air, would enrich the manuscript significantly.\nResponse: Many thanks for the valuable reminder.\nWe agree with the reviewer that the diffraction properties of the STOV comb, particularly its\nstability during propagation in air, are important. Under this suggestion, the STOV comb after\npropagation of 1 m and 5 m is simulated. Besides, the circumstance in which the STOV comb\nis focused with a spherical lens is also simulated.\nFig. R10. The intensity isosurface of STOV comb after different propagation distances. a The intensity\nisosurface of STOV comb after propagation distance of 1 m. b The intensity isosurface of STOV comb after\npropagation distance of 5 m.\nFig. R11. The intensity isosurface of the STOV comb after focused by a spherical lens. The intensity isosurface\nof the comb after focused by a spherical lens. a The intensity isosurface of STOV comb at 0.2f after the spherical\nlens. b The intensity isosurface of STOV comb at 0.5f after the spherical lens. c The intensity isosurface of STOV\ncomb at the focal plane the spherical lens.\nThe spatiotemporal intensity profiles of STOV comb with nine teeth, experiencing\npropagation distances of 1 m and 5 m after generation, are simulated and shown in Fig. R10a\nand Fig. R10b respectively. As the propagation distance increases, the intensity distribution of\nSTOV comb sprawls out in the y-t panel and this situation is more obvious as the absolute value\nof topological charge gets larger.\nBesides, the diffraction property of the STOV comb after being focused with a spherical lens\nis also studied. In the simulation, the STOV comb is focused with a spherical lens and the\nspherical lens, with focal length f of 0.25 m. The spatiotemporal intensity profiles of STOV\ncomb with nine teeth are calculated at several places after the spherical lens and distances from\nthe lens are 0.2f, 0.5f, and f, respectively. The simulated spatiotemporal intensity profiles are\nshown in Figs. R11a-c respectively. The spatiotemporal intensity profiles of the STOV comb\nare focused in the spatial domain and the spatiotemporal intensity profile evolution of every\ntooth is observed. It can be observed that, after passing through the lens, the STOV combs\nexperience a clear focusing effect and the topological charge value of each tooth are maintained\nduring the focusing process.\nThis part is supplemented in section of “S7. Diffraction property of the STOV combs” in the\nsupplementary material. The partial modification is shown below.\nThe corresponding modification in the manuscript is shown below (marked as yellow).\n3. In the section \"Adjusting the Characters of the STOV Comb,\" customization of the STOV\ncomb's characteristics is achieved by modulating the parameters of the designed phase, such as\nthe Topological Charge (TC) difference and the spatiotemporal intervals between adjacent teeth.\nThe manuscript could benefit from a discussion on how other components of the two-\ndimensional pulse shaper, such as the grating period and the focal length of the cylindrical lens,\ninfluence the STOV comb's characteristics, particularly the spatiotemporal intervals.\nResponse: Thanks deeply for this suggestion.\nThe grating and the cylindrical lens are two important elements for building the 2D pulse\n18\nshaper. Changing the spatial frequency or period of the grating and changing the focal length\nof the cylindrical lens are also effective for adjusting the spatiotemporal interval in the STOV\ncomb. And this is verified with the simulation results.\nTo verifying the relationship between the spatial frequency or period of grating and the\nspatiotemporal interval in the STOV comb, the STOV combs with five teeth and TC sequence\nof [-2, -1, 0, 1, 2] are simulated with identical designed phase, cylindrical lens but different\ngratings. The periods of the gratings are set as 1/200 mm, 1/400 mm, 1/600 mm, 1/800 mm,\n1/1000 mm, 1/1200 mm, and 1/1400 mm, which correspond to the spatial frequencies of 200\nmm-1, 400 mm-1, 600 mm-1, 800 mm-1, 1000 mm-1, 1200 mm-1, 1400 mm-1.\nAs the periods of the gratings are set as 1/600 mm, 1/800 mm, 1/1200 mm, namely the spatial\nfrequencies set as 600 mm-1, 800 mm-1, 1200 mm-1, the intensity isosurface of the simulated\nSTOV combs are shown in Figs. R12a-c. The spatial intervals and the temporal intervals\nbetween the adjacent STOV teeth of these seven simulated STOV combs are analyzed. After\ncalculation, the variation of the spatial intervals versus the spatial frequency and the variation\nof the spatial intervals versus the spatial frequency are shown in Fig. R12d and Fig. R12e with\norange marks. The linear function, namely ykxb, are utilized to fit the discrete data. The\nspatial interval remains unchanged as the spatial frequency of the grating changes (k 0). The\ntemporal interval and the spatial frequency of the grating shows standard linear relationship and\nthe temporal interval increases as the spatial frequency of the grating increases.\nFig. R12. The effect of the spatial frequency of grating on the spatiotemporal interval of the STOV comb. a\nThe intensity isosurface of the simulated STOV comb as the spatial frequencies of the grating set as 600 mm-1. b\nThe intensity isosurface of the simulated STOV comb as the spatial frequencies of the grating set as 800 mm-1. c The\nintensity isosurface of the simulated STOV comb as the spatial frequencies of the grating set as 1200 mm-1. d The\nvariation of the spatial intervals versus the spatial frequency of grating and the linear function fitting. e The variation\nof the temporal intervals versus the spatial frequency of grating and the linear function fitting.\nIn conclusion, changing the spatial frequency or period of the grating does not affect the\n19\nspatial interval in the STOV comb. The temporal interval increases as the spatial frequency of\nthe grating increases in a standard linear relationship.\nIn aspect of the relationship between the focal length of the cylindrical lens and the\nspatiotemporal interval in the STOV comb, the STOV combs with five teeth and TC sequence\nof [-2, -1, 0, 1, 2] are simulated with identical designed phase, grating but different cylindrical\nlindrical\nlenses. And the focal lengths of the cylindrical lenses are 0.05 m, 0.10 m, 0.15 m, 0.20 m,\n0.25\nm, 0.30 m, 0.35m, respectively. As the focal lengths of the cylindrical lenses are set as 0.15 m,\n0.25 m and 0.30 m, the intensity isosurface of the simulated STOV combs are shown in Figs.\nR13a-c. The spatial intervals and the temporal intervals between the adjacent STOV teeth in\nthese seven STOV combs are analyzed. The variation of the spatial intervals versus the focal\nlengths is shown in Fig. R13d with orange marks. Similarly, the variation of the spatial intervals\nversus the focal lengths is shown in Fig. R13e with orange marks. To determine the relationship\nbetween the spatial interval and the focal length and the relationship between the temporal\ninterval and the focal length, the linear function, namely ykxb , are utilized to fit the\ndiscrete data. The spatial interval and the focal length, as well as the temporal interval and the\nfocal length, shows standard linear relationship, as shown in Fig. R13d and Fig. R13e.\nFig. R13. The effect of the focal length of cylindrical lens on the spatiotemporal interval of the STOV comb.\na The intensity isosurface of the simulated STOV comb as the focal length of the cylindrical lens set as 0.15 m. b\nThe intensity isosurface of the simulated STOV comb as the focal length of the cylindrical lens set as 0.25 m. c The\nintensity isosurface of the simulated STOV comb as the focal length of the cylindrical lens set as 0.30 m. d The\nvariation of the spatial intervals versus the focal lengths and the linear function fitting. e The variation of the temporal\nintervals versus the focal lengths and the linear function fitting.\nIn conclusion, the spatial interval the temporal interval increase as the focal length of the\ncylindrical lens increases in a standard linear relationship.\nThis part is supplemented in section of “S4. Adjusting the intervals of the STOV combs with\ngrating and cylindrical lens” in the supplementary material. The partial modification is shown\n20\nbelow.\nThe corresponding modification in the manuscript is shown below (marked as yellow).\n4. In the \"Materials and Methods\" section, the authors detail using an SLM (Meadowlark Optics,\nE-19×12-500-1200-HDM8, 1920×1200, 8 μm×8 μm pixel) to construct the 2D pulse shaper.\nGiven the 8-bit depth phase modulation capability of the SLM, it would be insightful if the\nauthors could discuss potential discrepancies between the quasi-continuous theoretical phase\nand the experimentally implemented phase, and how these might affect the resultant STOV\ncomb.\nResponse: We appreciate this precious suggestion.\nWe have supplemented the simulation of the STOV comb at 8-bit depth designed phase. The\n8-bit depth designed phase, identical to the experimentally implemented phase, is calculated\nfrom the quasi-continuous theoretical phase through 8-bit depth processing algorithm. The\nquasi-continuous theoretical phase and the corresponding 8-bit depth designed phase for\ngenerating STOV comb with 5 teeth, with parameters of a=-2, b=2, =5π104 m-1, 𝑚\nk\n=[-2, -\n1, 0, 1, 2], are calculated.\nTwo STOV combs are simulated under these two phases. The STOV comb simulated with\nthe quasi-continuous theoretical phase is shown in Fig. R14a and the five teeth are shown in\nFigs. R14b-f separately. The STOV comb simulated with the 8-bit depth phase is shown in Fig.\nR14g and the five teeth are shown in Figs. R14h-l separately. The STOV comb in Fig. R14a\nand the STOV comb in Fig. R14g are quite similar except for some minor details, as labelled\nwith green ellipses in Fig. R14b and Fig. R14h. Therefore, the 8-bit depth processing of the\nSLM shows little effect to the STOV comb.\na The STOV comb simulated with the quasi-continuous theoretical phase. b-f The five teeth in Fig. R14a. g The\nSTOV comb simulated with the 8-bit depth phase. h-l The five teeth in Fig. R14g.\n5. The figures generally compare experimental results with theoretical simulations, yet Figure\n3 only depicts direct intensity information, phase information, and topological charges without\ncorresponding theoretical results. Including theoretical data in Figure 3 would substantiate the\nexperimental observations and strengthen the manuscript's validity.\nResponse: We are grateful to the reviewer for this reminder. The suggestion is adopted frankly.\nThe theoretical results associated with Fig. 3 in the manuscript is calculated with the STOV\ncomb simulated with experimental parameters. The comb with nine teeth is simulated with\nexperimental parameters as shown in Fig. R15. The phase information of the teeth in the\nsimulated STOV comb is analyzed and substantiates the experimental results in Fig. 3 in the\nmanuscript. The teeth with TCs of -3, -1, 0, 2, 4, in the simulated STOV comb are picked out\nfor phase study.\nTo extract the phase information, central slice lying in the y-t panel of the STOV tooth is\npicked out and five slices are chosen from these five teeth. The intensity information and the\nphase information located at these five slices is calculated. The intensity distributions at these\nfive slices are shown in Figs. R16a-e and the phases of these five slices are shown in Figs.\nR16f-j. In Figs. R16f-j, green arrows are utilized to label the direction of phase eddying.\nJudging from the phases shown in Figs. R16f-j, it is obvious that the corresponding TCs are -3, -1, 0, 2,\n4, respectively. The simulated results in Fig. R16 shows similar pattern with the experimental\nresults in Fig. 3 in the manuscript.\ncomb is amplified magnified 15 times for better visualization.)\nFig. R16. The intensity distribution at the y-t panel of central slices of teeth with TCs of -3, -1, 0, 2, 4 in Fig.\nR15 and the corresponding phases. a-e The intensity distribution of specific slices at y-t panel of teeth with TCs\nof -3, -1, 0, 2, 4 in Fig. R15a. f-j The phases of slices in (a-e).\nThis part is supplemented in section of “S3. Simulation of the STOV combs with\nexperimental parameters” in the supplementary material. The partial modification is shown\nbelow.\nThe corresponding modification in the manuscript is shown below (marked as yellow).\n6. The conclusion mentions that \"STOV combs offer strong potential in various applications\nlike information encryption or information carrier.\" If possible, it would be beneficial for the\nauthors to elaborate on the practical implementation of one of these applications, perhaps by\nconstructing a theoretical demonstration that utilizes the STOV combs.\nResponse: We appreciate the reviewer for giving this constructive comment.\nexperiment on two-channel information encoding and transmission with dual-tooth STOV.\nThe decoding process is realized with two strategies, namely the decoding with the\ndiffraction patterns of the dual-tooth STOV combs and the decoding with the temporal\nwaveform of the dual-tooth STOV combs.\nIn aspect of decoding with the diffraction patterns, the experimental system for the two-\nchannel information encoding and transmission with the dual-tooth STOV combs is shown in\nFig. R17a. The output beam from the femtosecond oscillator passes through the pulse stretcher\nand the 2D pulse shaper to generate the dual-tooth STOV combs. Through designing the phases\nuploaded to the SLM, the information is dynamically encoded in the dual-tooth STOV combs.\nAnd the information is decoded with the diffraction method. To realize decoding, the dual-tooth\nSTOV combs are dispersed with the transmission grating and are collimate with a cylindrical\nlens. The diffraction patterns of the dual-tooth STOV combs are recorded with a CCD camera.\nThe CCD camera is place at the focal plane of the cylindrical lens.\nFig. R17. Two-channel information encoding and decoding with dual-tooth STOV combs. a Schematic of the\nsystem for two-channel information encoding and decoding. BS: beam splitter, M: mirror, G: grating, CL: cylindrical\nlens, SLM: spatial light modulator, CCD: charge coupled device array. b The two binary images with 28×14 pixels\nfor transmitting. c The four kinds of pixel combinations and encoding. d The four types of diffraction patterns\nrecorded with the CCD camera corresponding to the four kinds of pixel combinations in (c).\nTwo binary images shown in Fig. R17b, with 28×14 pixels, are transmitted with the dual-\ntooth STOV combs, independently and simultaneously. The pictures are synchronously\ntransmitted with a pixel-by-pixel manner. Both pictures contain two values ‘0’ and ‘1’, namely\nthe black pixel and white pixel. Therefore, there are four kinds of recombination when\ntransmitting these two images pixel by pixel as shown in Fig. R17c, namely the combination\nof two black pixels, the combination of black pixel and white pixel, the combination of white\npixel and black pixel and the combination of two white pixels. Using binary encoding, the four\nkinds of pixel combinations are encoded as (0, 0), (0, 1), (1, 0), and (1, 1). These codes are\nutilized for designing the phases uploaded to the SLM. The parameter m , for designing the\nk\nphase,isset to [0, 0], [0, 1], [1, 0], or [1, 1] according to the pixel combination and the other\nparameters areset as a=0, b=1, and =5π×104 m-1. According to the pixel combination, the\ndesigned phases are dynamically uploaded to the SLM and four types of dual-tooth STOV\ncombs are generated. The corresponding diffraction patterns are shown in Fig. R17d, which\ncorrespond to four kinds of pixel combinations in Fig. R17c respectively. The restoration of\ninformation is realized through identifying the diffraction patterns. For example, the 36th pixel,\n24\nnamely the 8th pixel in the second line, is black and white for the first picture and second picture\nrespectively. Therefore, the 36th piece of information corresponds to the combination of black\npixel and white pixel and is encoded as (0,1). The designed phase uploaded to the SLM is\ncalculated with parameters of a=0, b=1, =5π×104 m-1, m =[0, 1]. A dual-tooth STOV comb\nk\nis generated, in which the first tooth is a Gaussian wave packet and the second tooth is a STOV\nwave packet with TC of 1. The diffraction pattern turns out as the second frame in Fig. R17d\nand the pixels are identifying as the combination of black pixel and white pixel. By repeating\nthe same process, the two images are transmitted successfully. Moreover, the number of images\ntransmitted simultaneously can be increased by increasing the number of teeth carried by the\nSTOV combs easily.\nThis part is supplemented in section of “Information encoding and transmission with STOV\ncombs” in the manuscript from line 329 to line 384. The partial modification in the manuscript\nis shown below (marked as yellow).\nAnd some claims in the abstract, introduction and conclusion are also modified and the\nmodifications are shown below (marked as yellow).\nAs we have stated previously, an important property of STOV combs is its transverse\nOAM modes which offers the potential for detecting the OAM and decoding in the time domain.\nThe experimental system for decoding in the temporal domain is shown in Fig. R18a. The\ngeneration of STOV combs and encoding process remains unchanged with Fig. R17a. One\ntooth in the dual-tooth STOV comb is picked out with a reflecting mirror and recombined with\nanother tooth through a beam splitter after an extra time delay. The recombined beam is focused\nto a high-speed photodiode (Thorlabs, DXM30BF, 30GHz) to realize photovoltaic conversion\nand the temporal waveform is recorded with an oscilloscope (Keysight, DSAV164A, 16GHz).\nThe two binary images for transmitting are shown in Fig. R18b and the encoding method is\nshown in Fig. R18c. In the decoding process, as the picked-out tooth goes through a longer\noptical path compared with the other tooth, two teeth in the dual-tooth STOV combs turn out\nas two peaks in the temporal waveforms recorded with the oscilloscope and the intensity of the\n25\ntwo peaks is utilized for decoding. Four types of temporal waveforms are acquired with the\noscilloscope as shown in Fig. R18d, which correspond to four kinds of pixel combinations in\nFig. R6c respectively. It can be observed that, the tooth of Gaussian wave packet turns out as a\nstronger peak and the tooth of STOV wave packet with topological charge of 1 turns out as a\nweak peak in the temporal waveforms. The threshold value, for distinguishing the Gaussian\nwave packet and the STOV wave packet with topological charge of 1, is located at ~1.5 V. The\nrestoration of information is realized through identifying the intensity of two peaks in the\ntemporal waveforms. For example, the 36th pixel, namely the 8th pixel in the second pixel line,\nin the first picture is a black pixel and the 36th pixel in the second picture is a white pixel.\nTherefore, the 36th piece of information corresponds to the combination of black pixel and white\npixel and is encoded as (0,1). The designed phase uploaded to the SLM is calculated with\nparameters of a=0, b=1, =5π×104 m-1, m =[0, 1]. A dual-tooth STOV comb is generated, in\nk\nwhich the first tooth is a Gaussian wave packet and the second tooth is a STOV wave packet\nwith TC of 1. The temporal waveform turns out as the second frame in Fig. R18d and the pixels\nare identifying as the combination of black pixel and white pixel. With this dynamic encoding\nand decoding method, the two images in Fig. R18b are transmitted successfully.\nFig. R18. Two-channel information encoding and decoding with dual-tooth STOV combs. a Schematic of the\nsystem for two-channel information encoding and decoding. BS: beam splitter, M: mirror, G: grating, CL: cylindrical\nlens, SLM: spatial light modulator, CCD: charge coupled device array. b The two binary images with 28×14 pixels\nfor transmitting. c The four kinds of pixel combinations and encoding. d The four types of temporal waveforms\nacquired with the oscilloscope corresponding to the four kinds of pixel combinations in (c).\nExcept for decoding with the intensity information of the peaks, the decoding process can be\nachieved via analyzing the profile of the peaks as the waveforms of wave packets with different\nTCs show different temporal profiles, which would further facilitate the decoding process.\nThrough simulation, the intensity isosurface of the Gaussian wave packet, the STOV wave\npacket with TC of 1, and the STOV wave packet with TC of 2, are shown in Figs. R19a-c,\nrespectively. Figs. R19d-e are the corresponding waveforms of the wave packets in Figs. R19a-\nc. The Gaussian wave packet is single-peak waveform, the STOV wave packet with TC of 1 is\ndual-peak waveform, and the STOV wave packet with TC of 2 is triple-peak waveform.\nTherefore, the profile features, namely the number of peaks in the temporal waveform, can be\nutilized for decoding and transverse OAM recognition. Nevertheless, it worth mentioning that\nrecording and resolving the peaks requires superior hardware and complicated experimental\nsystem. First, strong chirp is required to stretch the peaks, which makes resolving the peaks\nsimpler. Meanwhile, faster photodiodes and oscilloscopes, 50 GHz or higher, are required to\nrecord the peaks. In addition, the femtosecond oscillator with ultra-wide band spectrum is\npreferred for the realization of chirping and decoding.\nintensity isosurface of the Gaussian wave packet, the STOV wave packet with TC of 1, the STOV wave packet with\nTC of 2. d-e The waveforms of the wave packets in (a-c).\nThis part is supplemented in section of “S9. Information encoding with STOV combs and\ndecoding in the temporal domain” in the supplementary material. The partial modification is\nshown below.\nThe corresponding modification in the manuscript is shown below (marked as yellow).\nResponses to Reviewer #3\nThe authors reported the Spatiotemporal optical vortex combs, which introduce a approach to\ngenerating and manipulating spatiotemporal optical vortex (STOV) combs contain multiple\nSTOV wave packets with different transverse OAM modes. However, the authors still need to\nconsider the following points:\n1. The intervals of the STOV comb in the time domain are separated with a fixed interval of\napproximately 0.5ps (repetition rate ~2 THz) and other temporal intervals. Compared to the\nrepetition rate of the laser used in the experiment (80 MHz), there is a significant difference.\nThis indicates that the generated STOV comb is not achieved through line-by-line modulation\nof the laser, suggesting that the modulation efficiency of the laser is extremely low. How did\nthe authors address this issue? Additionally, how does the repetition rate affect the temporal\ncoherence of the comb? What are the limits of repetition rate tuning, and how do these depend\non experimental constraints?\nResponse: We appreciate the reviewer deeply for assessing our manuscript professionally\nand responsibly. Besides, we are thankful to this constructive comment.\nIn aspect of Line-by-line modulation and modulation efficiency:\nIn order to answer the question, we would like to first clarify the difference between an\noptical frequency comb and an STOV comb. An optical frequency comb is a light source whose\nspectrum consists of a series of discrete, equally spaced frequency lines, resembling the teeth\nof a comb, at time domain, it is a series of pulses and each pulse contains one specific spectral\nline. In contrast, an STOV comb comprises multiple STOV wave packets, namely the STOV\nteeth, and each tooth carries particular transverse OAM mode. It worth pointing out that the\nbiggest difference between an STOV comb and an optical frequency comb lies on the fact\nthat for an STOV comb the spectrum of each wave packet is identical. In other words, no\nfrequency components are eliminated during the modulation process. And we have verified this\nin experiment. The system for measuring the spectrum of each STOV teeth is shown in Fig.\nR20a.\nFig. R20. The spectra property of the STOV comb. a Schematic of system for measuring the spectrum of every\nSTOV teeth in the STOV comb. BS: beam splitter, M: mirror, G: grating, CL: cylindrical lens, L: lens. b The spectra\nof the pulses from the femtosecond oscillator. c The spectra of the STOV wave packets in the STOV comb with five\nteeth.\nThe diaphragm, the lens and the fiber of the spectrograph are fixed on a translation stage and\nthrough moving the translation stage, the STOV teeth are picked out in sequence and their\nspectra are measured independently. The spectra of the pulses from the femtosecond oscillator\nis shown in Fig. R20b. STOV comb with five teeth is generated in experiment and the measured\nspectra of these five teeth are shown in Fig. R20c. The measured spectra of these five STOV\nteeth and the spectra of the femtosecond laser itself are almost identical, verifying that the\nmultiple STOV teeth in the STOV comb share identical spectrum.\nThe line-by-line modulation of the laser refers to a modulation method that disperse different\nfrequency components of a laser beam to different spatial positions and modulate each\nfrequency independently. One typical line-by-line shaper is shown in Fig. R21. It can be seen\nthat a pulse shaper is utilized to disperse the frequency component of incident beam to different\nposition and phase modulation element is placed at the intermediate focal plan to perform\nindependent frequency modulation of the beam. In our experiment, as shown Fig. 1a in the\nmain manuscript, a pulse shaper is also used for generating STOV comb. As consequence, the\nmain hardware used in the manuscript is identical to the ones used to perform line-by-line\nmodulation of laser.\nFig. R21. The significance and implementation of line-by-line pulse shaping (cited from Huang, Chen‐Bin,\net al. \"Spectral line‐by‐line shaping for optical and microwave arbitrary waveform generations.\" Laser & Photonics\nReviews 2.4 (2008): 227-248.)\nHowever, there are differences between the unit for generating the STOV comb and the line-\nby-line shaper. Generally, the line-by-line pulse shaping combines pulse shaping and frequency\ncombs, as shown in Fig. R21. The source for the line-by-line modulation is the frequency comb,\nnamely separated spectral lines. The spectral lines are dispersed and separated by the line-by-\nline shaper and then the intensity and phase of each individual spectral line is independently\nmodulated. In contrast, the source for generating the STOV comb is a femtosecond laser with\ncontinuous broadband spectrum. Although the femtosecond pulses are dispersed with a 1D\ngrating and collimated with cylindrical lens, single pixel column in the SLM still modulates a\nshort fragment of continuous frequency components, rather than single frequency component.\nIn this aspect, we agree with the reviewer that the STOV comb is not achieved through line-by-\nline modulation. As consequence, the modulation efficiency is also reduced.\nTo enhance the modulation efficiency, several strategies can be applied. When build the 2D\npulse shaper, the grating with higher spatial frequency, the cylindrical lens with longer focal\nlength, and the SLM with smaller pixel size can be utilized to increase the modulation efficiency.\nMore precisely, gratings with higher spatial frequency is capable of dispersing the femtosecond\npulse with higher resolution. Similarly, the cylindrical lens with longer focal length can realize\nthe same effect. And the SLM with smaller pixel size can modulate narrower spectral lines.\nIn aspect of temporal coherence of the STOV comb\nAs the STOV teeth has the same spectra as the incident femtosecond pulse utilized for\ngenerating the STOV comb. And the phase differences between different tooth are stable after\nmodulation, the temporal coherence is thus identical to the temporal coherence of the incident\nfemtosecond laser and determined by its spectrum width, stability etc…\nIn aspect of repetition rate, the repetition rate itself would not affect the temporal coherence\nof a femtosecond source, but the phase noise and timing jitter may degrade the temporal\ncoherence of the source and so does the temporal coherence of the generated STOV comb.\nAgain, it would be determined by laser itself.\nIn aspect of repetition rate tuning, one thing to consider when generating the STOV comb is\navoiding the spatiotemporal overlap between the adjacent teeth. And this determines the upper\nlimit of the repetition rate. The principle is that the spatiotemporal interval between the adjacent\nteeth in the STOV comb should be large enough so that the adjacent teeth in the STOV comb\nis separated distinctly. And this principle requires that the temporal interval between the\nadjacent teeth in the STOV comb should be greater than the pulse duration of the STOV teeth\nor the spatial interval between the adjacent teeth in the STOV comb should be greater than the\nspot diameters of the STOV teeth along the X-axis direction. One of these two conditions is\nsatisfied, the spatiotemporal overlap between the adjacent teeth can be avoid. Generally, the\ntemporal interval between the adjacent teeth in the STOV comb should be set twice the pulse\nduration of the STOV teeth or the temporal interval between should be set twice the pulse\nduration of the STOV teeth. Therefore, the upper limit of the repetition rate depends on the\npulse duration of the STOV teeth and the spot diameters of the STOV teeth along the X-axis\ndirection. The lower limit of repetition rate, namely the largest temporal interval between the\nneighboring teeth, is mainly determined by the pixel size of the SLM. The SLM utilized in\nexperiment offers pixel size of 8 μm which determines a largest  achievable in experiment.\nThe largest  is 1.25π×105 m-1. the corresponding temporal interval and spatial interval\nbetween the neighboring teeth are ~6.25 ps and ~12.5 mm, which corresponds to a repetition\nrate of 160 GHz. Besides, the temporal interval can be adjusted with the grating and the\ncylindrical lens for building the 2D pulse shaper and the temporal interval shows a standard\nlinear relationship with the spatial frequency of the grating and the focal length of the\ncylindrical lens. Through adopting grating with higher spatial frequency and cylindrical lens\nwith longer focal length, the temporal interval can increase and the lower limit of repetition rate\ncan decline further. Therefore, the lower limit of repetition rate depends on three factors, namely\nthe pixel size of the SLM, the spatial frequency of the grating and the focal length of the\ncylindrical lens.\n2. The experiment only generated a fixed number of STOV comb teeth (e.g., 9 teeth) and did\nnot demonstrate more complex or larger-scale STOV comb structures. Compared to optical\nfrequency combs [Laser Photonics Rev. 16(1), 2100184 (2022).] or optical vortex combs [Nat.\nPhotonics 18(6), 625-631 (2024).], the unique properties or characteristics of STOV combs\nwere not discussed. The paper does not explore new features of STOV combs in their generation,\npropagation, or interaction, nor does it propose new physical mechanisms or principles. What\nare the specific advantages or novel properties of STOV combs? How could these properties\nbe leveraged in applications or further research?\nResponse: Thanks very much for this constructive suggestion and the suggestion is well taken.\nLarger-scale STOV comb structures\nSimulation of STOV combs with teeth over 9 teeth can be realized by adjusting the\nparameters of the designed phase. For instance, a STOV comb with 17 teeth is simulated as\nshown in Fig.R22a and its temporal intensity profile is shown in Fig.R22b. It is also possible\nto model STOV combs with more teeth as long as the computational capacity is enough. In\nexperiment, a STOV comb with 9 teeth is realized. For a STOV comb with fixed spatiotemporal\n30\ninterval, namely the temporal interval of Δt and the spatial interval of Δx, the maximum number\nof teeth in this STOV comb is mainly limited by three factors.\nFirst, in the time domain, the repetition frequency F of the femtosecond laser utilized for\ngenerating the STOV comb, limits the maximum number of teeth. The time interval between\nadjacent femtosecond pulses is 1/F. To avoid the overlap of teeth generated from the adjacent\nfemtosecond pulses, the number of teeth generating from single femtosecond pulse should be\ncontrolled below\n1Ft\n. Therefore, the maximum number of teeth in the time domain can\nbe calculated as N 1Ft .\nt\nSecond, in the space domain, the size of the optical elements, mainly the circular reflecting\nmirror, limits the maximum number of teeth. As shown in Fig. R22c, when the generated STOV\ncomb transmits in free space and is reflected by the mirror, the multiple teeth occupy a certain\namount of space. Assuming that the mirror, with diameter of D, is standardly placed with an\nangle of 4 with the incident STOV comb and the mirror is aligned with the STOV comb to\ncover teeth as many as possible, the effective size of the mirror is\nDcos4\n, as shown in Fig.\nR6c a with M'. Therefore, the maximum number of teeth in the space domain can be calculated\nas N Dcos4 x.\ns\nThird, another limitation to the teeth amount is the pixel size of the SLM. The phase for\ngenerating the kth STOV tooth in equation (1) is calculated as\nT x,yexpikxim\nk k\n(2)\nexpikxexpim\nk\nWhen k 0, the phase item of expikx realizes the modulation of the wave vector and\ncause the kth STOV tooth deviates from the 0th STOV tooth with a certain angle when exits from\nthe SLM as shown in Fig. R22d. As the absolute value of k increases, the absolute value of k\nincreases and the deviation angle between the kth STOV tooth deviates and the 0th STOV tooth\ngets bigger. However, size of the square pixel in the SLM limits the maximum of the absolute\nvalue of k. For the square pixel with size of d, the corresponding spatial frequency is 1d\nand the maximal absolute value of k is limited not exceeding d. Therefore, when k 0,\nthe maximum number of teeth is calculated as d. Similarly, when k 0, the maximum\nnumber of teeth is also calculated as d. Considering the 0th STOV tooth, the maximum\nnumber of teeth limited by the pixel size of the SLM can be calculated asN 2d1.\nSLM\nOverall, the maximum feasible number of teeth should be the minimum of N , N , and\nt s\nN . Taking the experimental condition for measuring the STOV comb with nine teeth as an\nSLM\nexample, the repetition frequency of the femtosecond laser is 80 MHz, the parameter  for\ncalculating the designed phase is 3π×104 m-1, the diameter of the mirrors is 50 mm, the 2D pulse\nshaper is composed of the transmission grating (spatial frequency 1200 lines/mm), the\ncylindrical lens (focal length 250 mm), and the SLM (8 μm×8 μm pixel). This experimental\nconfiguration generates STOV comb with temporal interval of ~1.5 ps and the spatial interval\nof ~3.0 mm. With the parameters above, the maximum number of teeth in the time domain is\ncalculated as N 1 Ft1  801061.51012 8333. The maximum number of teeth in\nt\nthe space domain is calculated as N Dcos4 x50cos4 3.011. The maximum\ns\nfeasible number of teeth in a STOV comb limited by the pixel size of the SLM is calculated as\nN 2d12  3π1048106 19 . And the round down method is utilized in\nSLM\nthese calculations. Therefore, the maximum feasible number of teeth under this experimental\ncondition is evaluated as 9 as we have measured in our study.\nwith 17 teeth. b The temporal intensity profile of the simulated STOV comb with 17 teeth. c Limitation from the\nsize of the reflecting mirror to the maximum feasible number of teeth. d Limitation from the pixel size of the SLM\nto the maximum feasible number of teeth. e The spatial intensity distribution of the STOV tooth with topological\ncharge of 2. f The spatial intensity distribution of the STOV tooth with topological charge of 4.\nThe spatiotemporal interval between the teeth in the STOV comb is a key factor for\nevaluating the maximum feasible number of teeth in the STOV comb. To generate a STOV\ncomb with more teeth, the spatiotemporal interval should be set shorter. The factors associated\nwith the spatiotemporal interval are the parameter  for calculating the designed phase, the\nfocus length of the cylindrical lens and spatial frequency of the transmission grating in the 2D\npulse shaper. A smaller  is helpful for realizing shorter spatiotemporal interval, which also\nrelieves the limitation from the pixel size of the SLM. The cylindrical lens with shorter focus\nlength can also realize shorter spatiotemporal interval. The transmission grating with lower\nspatial frequency can be utilized to achieve shorter temporal interval, which doesn’t affect the\nspatial interval. But the shorter spatiotemporal interval may cause the overlap of teeth in time\ndomain or in the space domain. Strategy like, shorter pulse duration and smaller laser spot, is\nhelpful for addressing this question.\nprofiles of the STOV teeth along Y-axis increases as their TCs diverging from zero. Fig. R22e\nand Fig. R22f are the spatial intensity distribution of the STOV teeth with topological charges\nof 2 and 4, recorded with a CCD camera. the length of the STOV tooth with topological charge\nof 4 along Y-axis almost reaches the width of the CCD along Y-axis as labelled with the orange\nrectangle in Fig. R22f. STOV teeth with topological charge over 4 or under -4 reach the\ndimensional limit of the CCD, which causes difficulty in the measurement of the STOV comb.\nThe potential methods for extending the number of teeth in experiment is discussed in the\nmanuscript. The modification in the manuscript is shown below (marked as yellow).\nSpecific advantages or novel properties of STOV combs and potential applications\nIn aspect of difference between optical frequency combs and STOV comb. An optical\nfrequency combs are a series of discrete optical frequencies with equal frequency interval.\nCompared to the optical frequency combs, the STOV combs are a series of discrete transverse\nOAM modes with equal spatiotemporal interval. Moreover, for an optical frequency comb, each\ntooth carries only one frequency component, in contrast, for STOV combs, each tooth carries\none distinct transvers OAM mode but the spectrum is identical for different teeth, this is also\none of the biggest differences between optical frequency comb and STOV comb.\nThe optical vortex comb is an optical frequency comb with each frequency of the comb\ncarrying a distinct longitudinal OAM mode (each tooth carries a distinct longitudinal OAM\nmode, and one frequency component). Namely, the phase winding of the optical vortex comb\nis located at the spatial domain, i.e. the x-y panel. However, the STOV comb is an ultrafast\nsequence of independent STOV wave packets, with each wave packet carrying a specific\ntransverse OAM. First, the most obvious difference between the optical vortex comb and the\nSTOV comb is the difference of the OAM type. The optical vortex comb carries the longitudinal\nOAM, which is located at the spatial domain, i.e. the x-y panel. the STOV comb carries the\ntransverse OAM, which is located at the spatiotemporal domain, i.e. the x-t panel or y-t panel.\nAs the longitudinal OAM is located at the x-y panel, the detection of the longitudinal OAM\nrequires detecting the interference fringe pattern diffraction pattern of the optical vortex with\nthe two-dimensional detector like the CCD. However, the transverse OAM is located at the x-t\n33\npanel or y-t panel. Due to the spatiotemporal property of transverse OAM, it is feasible to detect\nthe OAM in the time domain with the single pixel detector like the photodiode. Therefore, the\nadvantage of the STOV comb over the optical vortex comb is that the STOV comb offers a new\ndimension for detecting the OAM and offers the potential for temporal high-speed detection of\nthe OAM. Besides, the singularity of the longitudinal OAM locates at the spatial domain, which\nhave been applied for the spatial super-resolution sensing, like the super-resolution imaging.\nThe singularity of the transverse OAM locates at the spatiotemporal domain, which offers the\npotential for the super-resolution sensing both in the spatial domain and the temporal domain.\nThe specific advantages and novel properties, as well as the potential applications of the\nSTOV combs are discussed in the Conclusion section and the modification is shown below\n(marked as yellow).\nFeatures of STOV combs in propagation and being focused\nThe diffraction properties of the STOV comb, particularly its stability and intensity evolution\nduring propagation in air, are important. The STOV comb after propagation of 1 m and 5 m is\nsimulated. Besides, the circumstance in which the STOV comb is focused with a spherical lens\nis also simulated.\nFig. R23. The intensity isosurface of STOV comb after different propagation distances. a The intensity\nisosurface of STOV comb after propagation distance of 1 m. b The intensity isosurface of STOV comb after\npropagation distance of 5 m.\nThe spatiotemporal intensity profiles of STOV comb with nine teeth, experiencing\npropagation distances of 1 m and 5 m after generation, are simulated and shown in Fig. R23a\nand Fig. R23b respectively. As the propagation distance increases, the intensity distribution of\nSTOV comb sprawls out in the y-t panel and this situation is more obvious as the absolute value\nof topological charge gets larger.\nBesides, the diffraction property of the STOV comb after being focused with a spherical lens\nis also studied. In the simulation, the STOV comb is focused with a spherical lens and the\nspherical lens, with focal length f of 0.25 m. The spatiotemporal intensity profiles of STOV\ncomb with nine teeth are calculated at several places after the spherical lens and distances from\nthe lens are 0.2f, 0.5f, and f, respectively. The simulated spatiotemporal intensity profiles are\nshown in Figs. R24a-c respectively. The spatiotemporal intensity profiles of the STOV comb\nare focused in the spatial domain and the spatiotemporal intensity profile evolution of every\ntooth is observed. It can be observed that, after passing through the lens, the STOV combs\nexperience a clear focusing effect and the topological charge value of each tooth are maintained\nduring the focusing process.\nFig. R24. The intensity isosurface of the STOV comb after focused by a spherical lens. The intensity isosurface\nof the comb after focused by a spherical lens. a The intensity isosurface of STOV comb at 0.2f after the spherical\nlens. b The intensity isosurface of STOV comb at 0.5f after the spherical lens. c The intensity isosurface of STOV\ncomb at the focal plane the spherical lens.\nThis part is supplemented in section of “S7. Diffraction property of the STOV combs” in the\nsupplementary material. The partial modification is shown below.\n3. The primary improvement in the study lies in the flexibility of the designed phase profile is\napplied to the Fourier plane of a two-dimensional (2D) pulse shaper, which enables the\ngeneration of more complex STOV comb structures, such as irregular spatiotemporal intervals\nand topological charge distributions. However, this improvement is more of a methodological\noptimization rather than a proposal of entirely new theoretical concepts or groundbreaking\nexperimental results. Compared to the Spatiotemporal vortex strings [Sci. Adv. 10, 6206\n(2024)], which demonstrated practical applications for data transmission based on STOV\nstrings, the discussion about application scenarios for Spatiotemporal optical vortex combs in\nthe article is rather conceptual, lacking specific experimental or application case support. To\nstrengthen the work, the authors could include experimental demonstrations of practical\napplications, such as data transmission or information encoding using STOV combs.\nBy providing more detailed theoretical analysis, robust experimental validation, and a deeper\nexploration of applications and challenges, the impact of the work could be enhanced.\nResponse: We thank the reviewer for raising this comment and the comment is adopted very\nwell.\nAs shown in Fig. R1, the primary difference between the spatiotemporal vortex string and\nthe STOV comb is the number of the wave packets. For spatiotemporal vortex string, various\ntransverse OAM states are limited within a single wave packet and unavoidably overlap with\neach other. Therefore, STOV string suffers from low OAM purity and cross-talk between\nsingularities. In contrast, the STOV comb in our manuscript contains multiple STOV wave\npackets, with each STOV wave packet carrying one specific and arbitrarily modulable\ntransverse OAM state. Compared with the STOV string, the STOV comb avoids overlap of\ndifferent transverse OAM states, as well as the cross-talk between singularities and offers high\nOAM purity, which would be beneficial for applications such as information encoding etc...\nIn order to further demonstrate the application of STOV combs, we have performed an\nadditional experiment on two-channel information encoding and transmission with dual-tooth\nSTOV. The decoding process is realized with two strategies, namely the decoding with the\ndiffraction patterns of the dual-tooth STOV combs and the decoding with the temporal\nwaveform of the dual-tooth STOV combs.\nIn aspect of decoding with the diffraction patterns, the experimental system for the two-\nchannel information encoding and transmission with the dual-tooth STOV combs is shown in\nFig. R25a. The output beam from the femtosecond oscillator passes through the pulse stretcher\nand the 2D pulse shaper to generate the dual-tooth STOV combs. Through designing the phases\nuploaded to the SLM, the information is dynamically encoded in the dual-tooth STOV combs.\nAnd the information is decoded with the diffraction method. To realize decoding, the dual-tooth\nSTOV combs are dispersed with the transmission grating and are collimate with a cylindrical\nlens. The diffraction patterns of the dual-tooth STOV combs are recorded with a CCD camera.\nThe CCD camera is place at the focal plane of the cylindrical lens.\nFig. R25. Two-channel information encoding and decoding with dual-tooth STOV combs. a Schematic of the\nsystem for two-channel information encoding and decoding. BS: beam splitter, M: mirror, G: grating, CL: cylindrical\nlens, SLM: spatial light modulator, CCD: charge coupled device array. b The two binary images with 28×14 pixels\nfor transmitting. c The four kinds of pixel combinations and encoding. d The four types of diffraction patterns\nrecorded with the CCD camera corresponding to the four kinds of pixel combinations in (c).\nTwo binary images shown in Fig. R25b, with 28×14 pixels, are transmitted with the dual-\ntooth STOV combs, independently and simultaneously. The pictures are synchronously\ntransmitted with a pixel-by-pixel manner. Both pictures contain two values ‘0’ and ‘1’, namely\nthe black pixel and white pixel. Therefore, there are four kinds of recombination when\ntransmitting these two images pixel by pixel as shown in Fig. R25c, namely the combination\nof two black pixels, the combination of black pixel and white pixel, the combination of white\npixel and black pixel and the combination of two white pixels. Using binary encoding, the four\nkinds of pixel combinations are encoded as (0, 0), (0, 1), (1, 0), and (1, 1). These codes are\nutilized for designing the phases uploaded to the SLM. The parameter m , for designing the\nk\nphase,isset to [0, 0], [0, 1], [1, 0], or [1, 1] according to the pixel combination and the other\nparameters areset as a=0, b=1, and=5π×104 m-1. According to the pixel combination, the\ndesigned phases are dynamically uploaded to the SLM and four types of dual-tooth STOV\ncombs are generated. The corresponding diffraction patterns are shown in Fig. R25d, which\ncorrespond to four kinds of pixel combinations in Fig. R25c respectively. The restoration of\ninformation is realized through identifying the diffraction patterns. For example, the 36th pixel,\n37\nnamely the 8th pixel in the second line, is black and white for the first picture and second picture\nrespectively. Therefore, the 36th piece of information corresponds to the combination of black\npixel and white pixel and is encoded as (0,1). The designed phase uploaded to the SLM is\ncalculated with parameters of a=0, b=1, =5π×104 m-1, m =[0, 1]. A dual-tooth STOV comb\nk\nis generated, in which the first tooth is a Gaussian wave packet and the second tooth is a STOV\nwave packet with TC of 1. The diffraction pattern turns out as the second frame in Fig. R25d\nand the pixels are identifying as the combination of black pixel and white pixel. By repeating\nthe same process, the two images are transmitted successfully. Moreover, the number of images\ntransmitted simultaneously can be increased by increasing the number of teeth carried by the\nSTOV combs easily.\nThis part is supplemented in section of “Information encoding and transmission with STOV\ncombs” in the manuscript from line 329 to line 384. The partial modification in the manuscript\nis shown below (marked as yellow).\nAnd some claims in the abstract, introduction and conclusion are also modified and the\nmodifications are shown below (marked as yellow).\nAs we have stated previously, an important property of STOV combs is its transverse\nOAM modes which offers the potential for detecting the OAM and decoding in the time domain.\nThe experimental system for decoding in the temporal domain is shown in Fig. R26a. The\ngeneration of STOV combs and encoding process remains unchanged with Fig. R25a. One\ntooth in the dual-tooth STOV comb is picked out with a reflecting mirror and is recombined\nwith another tooth through a beam splitter after an extra time delay. The recombined beam is\nfocused to a high-speed photodiode (Thorlabs, DXM30BF, 30GHz) to realize photovoltaic\nconversion and the temporal waveform is recorded with an oscilloscope (Keysight,\nDSAV164A, 16GHz).\nshown in Fig. R26c. In the decoding process, as the picked-out tooth goes through a longer\noptical path compared with the other tooth, two teeth in the dual-tooth STOV combs turn out\nas two peaks in the temporal waveforms acquired with the oscilloscope and the intensity of the\ntwo peaks is utilized for decoding. Four types of temporal waveforms are acquired with the\noscilloscope as shown in Fig. R26d, which correspond to four kinds of pixel combinations in\nFig. R6c respectively. It can be observed that, the tooth of Gaussian wave packet turns out as a\nstronger peak and the tooth of STOV wave packet with topological charge of 1 turns out as a\nweak peak in the temporal waveforms. The threshold value, for distinguishing the Gaussian\nwave packet and the STOV wave packet with topological charge of 1, is located at ~1.5 V. The\nrestoration of information is realized through identifying the intensity of two peaks in the\ntemporal waveforms. For example, the 36th pixel, namely the 8th pixel in the second pixel line,\nin the first picture is a black pixel and the 36th pixel in the second picture is a white pixel.\nTherefore, the 36th piece of information corresponds to the combination of black pixel and white\npixel and is encoded as (0,1). The designed phase uploaded to the SLM is calculated with\nparameters of a=0, b=1, =5π×104 m-1, m =[0, 1]. A dual-tooth STOV comb is generated, in\nk\nwhich the first tooth is a Gaussian wave packet and the second tooth is a STOV wave packet\nwith TC of 1. The temporal waveform turns out as the second frame in Fig. R26d and the pixels\nare identifying as the combination of black pixel and white pixel. With this dynamic encoding\nand decoding method, the two images in Fig. R26b are transmitted successfully.\nFig. R26. Two-channel information encoding and decoding with dual-tooth STOV combs. a Schematic of the\nsystem for two-channel information encoding and decoding. BS: beam splitter, M: mirror, G: grating, CL: cylindrical\nlens, SLM: spatial light modulator, CCD: charge coupled device array. b The two binary images with 28×14 pixels\nfor transmitting. c The four kinds of pixel combinations and encoding. d The four types of temporal waveforms\nacquired with the oscilloscope corresponding to the four kinds of pixel combinations in (c).\nExcept for decoding with the intensity information of the peaks, the decoding process can be\nachieved via analyzing the profile of the peaks as the waveforms of wave packets with different\nTCs show different temporal profiles, which would further facilitate the decoding process.\nintensity isosurface of the Gaussian wave packet, the STOV wave packet with TC of 1, the STOV wave packet with\nTC of 2. d-e The waveforms of the wave packets in (a-c).\nThrough simulation, the intensity isosurface of the Gaussian wave packet, the STOV wave\npacket with TC of 1, and the STOV wave packet with TC of 2, are shown in Figs. R27a-c,\nrespectively. Figs. R27d-e are the corresponding waveforms of the wave packets in Figs. R27a-\nc. The Gaussian wave packet is single-peak waveform, the STOV wave packet with TC of 1 is\ndual-peak waveform, and the STOV wave packet with TC of 2 is triple-peak waveform.\nTherefore, the profile features, namely the number of peaks in the temporal waveform, can be\nutilized for decoding and transverse OAM recognition. Nevertheless, it worth mentioning that\nrecording and resolving the peaks requires superior hardware and complicated experimental\nsystem. First, strong chirp is required to stretch the peaks, which makes resolving the peaks\nsimpler. Meanwhile, faster photodiodes and oscilloscopes, 50 GHz or higher, are required to\nrecord the peaks. In addition, the femtosecond oscillator with ultra-wide band spectrum is\npreferred for the realization of chirping and decoding.\nThis part is supplemented in section of “S9. Information encoding with STOV combs and\ndecoding in the temporal domain” in the supplementary material. The partial modification is\nshown below.\nThe corresponding modification in the manuscript is shown below (marked as yellow).\n40"}, {"round": 2, "reviewer_comments": {"reviewer_1": "Please find attached review report.", "reviewer_3": "The authors have provided reasonable and systematic responses to the previously raised questions. By explicitly\ndistinguishing the STOV comb from conventional frequency combs and similar structures, and by substantiating their claims\nwith additional experiments and detailed analysis, the authors have effectively addressed the major concerns. The authors’\npoint-by-point reply demonstrates a deep understanding of both the technical challenges and the broader scientific context.\nThe revisions and supplementary materials meet the standards and expectations of Nature Communications.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts."}, "author_response": "Response Letter\nDear reviewers,\nWe would like to express our sincere gratitude for your valuable and constructive comments.\nYour insightful suggestions have been instrumental in enriching the depth and improving the\noverall quality of our manuscript.\nIn response, we have carefully reviewed and thoughtfully incorporated the recommendations.\nThe major changes are:\n1. We have revised the comparison between our proposed STOV combs and other light\nstructures. The respective advantages of these structured light field, as well as the\ncomplementarity among them, are now discussed in greater depth.\n2. As recommended, we have added a note in the main text to value the work of\narXiv:2407.19747 and clarify the independence of the two works.\n3. The distortions of higher-order STOV wave packets, in phase and in intensity, have also\nbeen discussed. This phenomenon arises due to the spatiotemporal astigmatism. We have\nfurther provided extra discussions on strategies for its mitigation.\n4. The differences between the STOV comb simulated with femtosecond Fourier-transform-\nlimited pulses, the STOV comb simulated with chirped pulses, and the STOV comb\ngenerated with chirped pulses in experiment, as well as the sources to these differences, are\nanalyzed.\nWe have replied to the concerns raised by the reviewers on a point-by-point basis. This includes\nsupplementary theoretical analyses and new discussions. The main manuscript and the\nsupplementary materials have been revised accordingly as well, with all changes highlighted.\nBest regards,\nFu Feng (on behalf of all authors)\nDetailed Responses to the Reviewers\nWe would like to kindly remind that, the questions raised by reviewers are In Blue, our\nresponses are In Black, and the corresponding revision on the manuscript and supplementary\nmaterials are highlighted.\nResponses to Reviewer #1\nThe authors have modified this manuscript and added an application demonstration. While the\nauthors have made efforts to address all raised concerns, several issues remain insufficiently\nresolved. The overall quality of the current manuscript does not reach the publication standards\nof Nature Communications. The following issues should be carefully considered and fully\naddressed, and the rigor of the language should be strengthened prior to consideration for\npublication.\n1. The authors have added a comparison between STOV comb, optical vortex comb, and STOV\nstring. However, these comparisons are neither comprehensive nor impartial. The comparison\nmainly emphasizes the advantages of STOV comb without objectively analyzing its potential\ndrawbacks. To avoid misleading, the similarities and differences of these light fields should be\ndiscussed more comprehensively and objectively, instead of emphasizing the advantages of\nSTOV comb one-sidedly.\nlimitations. For instance, although optical vortex comb does not carry transverse OAM, it can\ncarry more OAM modes (Ref. 34), and its characterization is simpler than STOV. The STOV\nstring can carry numerous OAM modes within one wave packet. Although adjacent STOVs\nmay interact with each other, the TC arrangement of STOV string can be well recognized from\ndiffraction patterns (Ref. 32). The main difference between STOV string and STOV comb lies\nin that the STOV comb divides one pulse into several sub-pulses (teeth), each of which carries\none STOV. Thus, each STOV wavepacket can be independently controlled. However, the\nmaximum available teeth number of STOV comb in this study is nine, which is less than that\nof vortex comb and STOV string.\nThe authors claim that “due to the spatiotemporal property of transverse OAM, it is possible to\ndetect the OAM in the time domain with the single pixel detector like the photodiode”. The\nquestion is whether a single-pixel detector can recognize such a short pulse sequence. In the S9\nsection of supplemental, an extra delay line is added to realized decoding in the temporal\ndomain. Therefore, it might not be appropriate to consider it as an advantage. The description\nthat \"STOV string suffers from low OAM purity and cross-talk between singularities after long-\ndistance propagation\" also lacks sufficient scientific basis and will cause misunderstanding.\nThe relevant description in the introduction and discussion should be removed. Also, the claim\nthat the proposed STOV comb possesses high OAM purity is subjective and even questionable\n(see Comment 3), which should be corrected.\nResponse: Thanks a lot for raising this constructive recommendation.\nThe comparison between STOV comb, optical vortex comb, and STOV string is reorganized.\nThe advantages and the potential drawbacks of these light structures are both discussed. The\nclaim that “due to the spatiotemporal property of transverse OAM, it is possible to detect the\nOAM in the time domain with the single pixel detector like the photodiode”, has been removed.\nThe description that \"STOV string suffers from low OAM purity and cross-talk between\nsingularities after long-distance propagation\", and the similar description has been removed as\nwell. The claim that the proposed STOV comb possesses high OAM purity, has been removed\nto improve the rigor of the manuscript.\nThe revised comparison between STOV comb, optical vortex comb, and STOV string is stated\nin the manuscript from line 441 to line 468 and is shown below.\nThe STOV combs realized in this work offer several new properties over other light\nstructures such as the optical frequency combs, the optical vortex combs, and the optical\nspatiotemporal vortex strings. The optical frequency comb is a series of equally spaced\npulses in the time domain, corresponding to a sequence of equally spaced spectral lines\nin the frequency domain. And the optical vortex comb is an optical frequency comb\nwith each spectral lines of the comb carrying a distinct longitudinal OAM mode. The\noptical vortex comb carrying up to 50 OAM modes has been realized33 and its\ncharacterization is simpler than the STOV comb. For an optical vortex comb, the\nsingularity of the longitudinal OAM locates at the spatial domain, which have been\napplied for the high-resolution spatial sensing and imaging. In contrast, the STOV\ncombs carry the transverse OAM. As the singularity of the transverse OAM locates at\n2\nthe spatiotemporal domain, it would offer the potential for sensing and imaging\napplications where both spatial and temporal resolution are needed. The STOV string\nis a wave packet carrying numerous STOVs, up to 28 STOVs as recently reported31,\nwith customizable TC arrangements and transverse OAM modes. The TC arrangements\nand the transverse OAM modes of STOV string can be well recognized from diffraction\npatterns. However, as numerous STOVs in the STOV string are confined in single wave\npacket, the STOVs may overlap and interact with each other. Whereas, a STOV comb\nwith nine teeth has been realized in this work. The STOV comb is a sequence of\nmultiple STOV wave packets, with each STOV wave packet carrying the customizable\ntransverse OAM mode. In a STOV comb, each STOV wave packet can be\nindependently controlled and thus the overlapping between the transverse OAM modes\nis avoided. Several aspects about the STOV comb remain to be improved, such as\nincreasing the number of teeth in the STOV comb or eliminating the spatiotemporal\nastigmatism in the STOV comb. Besides, the teeth in the STOV comb can be designed\nas other types of STOV wave packets, such as the degradation-free spatiotemporal\nBessel vortices which are beneficial for avoiding the degradation of the high-order\nSTOV wave packets.\n2. The idea of this manuscript and the paper (arXiv:2407.19747) is similar. The authors claim\nthat they were not aware of its existence during the course of their work. I think the relevant\nclaim should be given in the main text to demonstrate the independence of two studies.\nResponse: We are appreciative for this constructive comment.\nA new paragraph named Note is added in the manuscript from line 469 to line 472 to give the\nrelevant claim. The added paragraph is shown below.\nNote: During the peer review process, an independent study that explored the light structure came\ninto our sight, which was reported as an online preprint (https://doi.org/10.48550/arXiv.2407.19747).\nWe would like to clarify that the two works were performed independently.\n3. The authors compared their results with that in arXiv:2407.19747 in Fig. R6. The comparison\nshows that the iso-intensity profiles of the STOV burst in arXiv are similar to the previously\nreported results, whereas the wavepackets of the STOV comb in this study are distorted. The\nspiral phases of STOV comb also appear distorted. As shown in Fig. 3, the phase singularities\nof STOV comb are dislocated in both time and space. Especially, for the cases of -3 and 4, the\nphase structures exhibit an obvious deviation from the spiral phase of a high-order STOV. The\nphase distortion probably results in a decrease in the OAM mode purity (Nature Photonics 15,\n608 (2021)). Also, it may lead to the degradation of a high-order STOV into multiple low-order\nSTOVs. From the phase in Fig. 3, it seems difficult to recognize the high-order STOV. The\nauthors need to explain this phase distortion and discuss its influences on the OAM purity as\nwell as the measured TC of STOV in the main text. If this influence cannot be neglected, it\nimposes a substantial constraint on the maximum TC that can be effectively utilized by this\nmethod. If the deterioration stems from the inherent flaws of method itself, it should be clearly\npointed out.\nWe would like to explain why the phase singularities are dislocated at first and then discuss the\npossible approaches for eliminating the influence of this phenomenon.\n(1) The reasons for the phase singularities dislocation and the distorted wave packets in\nexperiment\nThe dislocated phase singularities and the distorted wave packets, observed in the measured\nhigh-order (absolute value of topological charge over 1) STOV teeth, can be explained from\nthe following aspects.\nFirstly, in Fig. 3, it can be observed that the number of phase singularities of the STOV teeth\nequals to the absolute value of the corresponding topological charges. For the high-order STOV\nteeth, the multiple phase singularities are dislocated in the space-time plane. This phenomenon\nis caused by the spatiotemporal astigmatism, namely the mismatch between temporal dispersion\nand spatial diffraction [Liu, X., Cao, Q., Zhang, N., Chong, A., Cai, Y., & Zhan, Q. (2024).\nSpatiotemporal optical vortices with controllable radial and azimuthal quantum\nnumbers. Nature Communications, 15, 5435.] [Huang, S., Wang, P., Shen, X., & Liu, J. (2021).\nProperties of the generation and propagation of spatiotemporal optical vortices. Optics\nExpress, 29(17), 26995-27003.], [Hancock, S. W., Zahedpour, S., Goffin, A., & Milchberg, H.\nM. (2019). Free-space propagation of spatiotemporal optical vortices. Optica, 6(12), 1547-\n1553.]. Due to the spatiotemporal astigmatism, the phases of the high-order STOV teeth exhibit\nan obvious deviation from the standard high-order spiral phases. The dislocated phase\nsingularities in the high-order STOV teeth are also observed in our simulated results (Fig. S5\nin the supplementary materials). Besides, the phases under influence of the spatiotemporal\nastigmatism have been reported in other works, as shown in panel (a-e) in Fig. R1 which are\ncited from Hancock, S. W., Zahedpour, S., Goffin, A., & Milchberg, H. M. (2019). Free-space\npropagation of spatiotemporal optical vortices. Optica, 6(12), 1547-1553.\nSecondly, the phases, as well as the intensities, of the measured high-order STOV teeth, are\nextracted from the interferometric fringes, whereas the simulated results are calculated directly.\nFactors affecting the interferometric fringes will affect the measured phases, as well as the\nmeasured intensities. These factors include the detecting noise of the CCD, laser power\nfluctuation, small optical path deviation, fluctuations of the electronic components like spatial\nlight modulator, and so on. Under the influence of these factors, the phases and the intensity\nprofiles of the measured STOV teeth, exhibit some deformation from the simulated results.\n(2) The methods for eliminating the influence of the distorted phases\nWe agree with the reviewer that the phase distortion can result in a decreased OAM mode purity\nand it can lead to the degradation of a high-order STOV into multiple low-order STOVs. This\ninfluence is non-negligible and imposes constraint on the maximal TC of the STOV comb.\nThe influence of the distorted phase, as well as the constraint on the maximal TC of the STOV comb,\ncan be compensated through the following approaches.\nAs mentioned previously, the spatiotemporal astigmatism is caused by the mismatch between\ntemporal dispersion and spatial diffraction. Therefore, the spatiotemporal astigmatism can be\neliminated with the careful dispersion management. Through careful dispersion management,\nHancock et al. realized the STOV wave packet without the spatiotemporal astigmatism in\nsimulation. And the dislocation of phase singularities is removed. The relevant result is shown\nin panel (f) in Fig. R1.\nMore\nprecisely, a group delay dispersion (GDD) of (700)2/4ln(2) fs2 is introduced to the femtosecond\npulse. And this chirped pulse is utilized for simulating the STOV comb. The isosurface plot of\nthe simulated STOV comb is shown in Fig. R2a. The high-order STOV teeth in Fig. R2a show\nstructures with single hole. The intensity and the phase of the central slice lying in the y-t panel\nof the STOV tooth with TC of 4 is shown in Fig. R2b and Fig. R2c, respectively. It can be\nobserved that the STOV tooth with TC of 4 shows single hole and single singularity, which\nindicates that the spatiotemporal astigmatism is eliminated with careful dispersion management.\nBesides, it is feasible to replace the conventional STOV teeth in the STOV comb with the\nspecially designed STOV wave packets named the degradation-free spatiotemporal Bessel\nvortices to avoid degradation of higher order STOVs. The degradation-free spatiotemporal\nBessel vortices are time-diffraction-free transverse OAM beams which inherently resist\ndegradation. Notably, the spatiotemporal Bessel vortices with TCs as large as 100 and -100,\nhave been realized experimentally [Chen, W., Zhang, W., Liu, Y., Meng, F. C., Dudley, J. M., &\nLu, Y. Q. (2022). Time diffraction-free transverse orbital angular momentum beams. Nature\nCommunications, 13, 4021]. Therefore, for a STOV comb with teeth of the degradation-free\nspatiotemporal Bessel vortices, the constraint on the maximal TC, imposed by the\ndegradation of the high-order STOVs, can be overcome.\ncomb simulated with the chirped pulses with GDD of (700)2/4ln(2) fs2. b The intensity of the central slice lying in\nthe y-t panel of the STOV tooth with TC of 4. c The phase of the central slice lying in the y-t panel of the STOV\ntooth with TC of 4.\nThe modification in the manuscript\nThe resulting constraint on the maximum TC is non-negligible and has been explicitly pointed\nout in the main manuscript from line 193 to line 207, which says\nAs shown in Figs. 3f-j, the number of phase singularities in the phases of the STOV\nteeth equals to the absolute value of the corresponding TCs. For the high-order STOV\nteeth, the multiple phase singularities are dislocated in the y-t panel, namely the space-\ntime plane. This phenomenon is caused by the spatiotemporal astigmatism, namely the\nmismatch between temporal dispersion and spatial diffraction35. The STOV comb with\nnine teeth simulated with the chirped pulses and parameters close to the experiment, is\nalso realized (see Supplementary Material S3). The dislocated phase singularities,\ninduced by the spatiotemporal astigmatism, are also observed in the phases in the y-t\npanel of the high-order STOV teeth simulated with the chirped pulses. And this phase\ndistortion can cause the decrease in the purity of OAM modes and can induce the\ndegradation of the high-order STOV teeth into multiple low-order STOVs. Thus, the\ndistorted phases in the high-order STOV teeth, introduced by the spatiotemporal\nastigmatism, impose extra constraint on the maximal TC of the STOV comb. The\nspatiotemporal astigmatism, as well as the constraint on the maximal TC, can be\neliminated with careful management of the GDD of chirped pulses21,35.\nand from line 348 to line 354, which says\nBesides, the distorted phases in the high-order STOV teeth, cause by the spatiotemporal\nastigmatism, impose extra constraint on the maximal TC of the STOV comb as the\nphase distortion can cause the decrease in the purity of OAM modes and can induce the\ndegradation of the high-order STOV teeth into multiple low-order STOVs. But this\nconstraint can be solved with careful dispersion management or the introduction of\nspecial STOV wave packets named the degradation-free spatiotemporal Bessel\nvortices37.\n35. X. Liu, Q. Cao, N. Zhang, A. Chong, Y. Cai, Q. Zhan, Spatiotemporal optical vortices with\ncontrollable radial and azimuthal quantum numbers. Nat. Commun. 15, 5435 (2024).\n37. W. Chen, W. Zhang, Y. Liu, F. C. Meng, J. M. Dudley, Y. Q. Lu, Time diffraction-free transverse\norbital angular momentum beams. Nat. Commun. 13, 4021 (2022).\nThe modification in the supplemental material\nThe resulting constraint on the maximum TC has also been pointed out in the supplemental\nmaterial from line 220 to line 228, which says\nThe dislocated phase singularities, induced by the spatiotemporal astigmatism, are also\nobserved in the phases in the y-t panel of the high-order STOV teeth simulated with the\nchirped pulses. And this phase distortion can cause the decrease in the purity of OAM\nmodes and can induce the degradation of the high-order STOV teeth into multiple low-\norder STOVs. Thus, the distorted phases in the high-order STOV teeth, introduced by\nthe spatiotemporal astigmatism, impose extra constraint on the maximal TC of the\nSTOV comb. The spatiotemporal astigmatism, as well as the constraint on the maximal\nTC, can be eliminated with careful management of the GDD of chirped pulses.\nand from 427 to line 459, which says\nBesides, for the high-order STOV teeth, the corresponding phases are distorted and the\nphase singularities are dislocated in the y-t panel, namely the space-time plane. The\nwave packets of the high-order STOV teeth are also distorted. For example, in the\nSTOV comb in Fig. S4b, the intensity isosurface of the STOV tooth with TC of 3 shows\na structure with 3 holes and the phase of the central slice at y-t panel of this STOV tooth\nshows three dislocated phase singularities. This distorted phenomenon is caused by the\nspatiotemporal astigmatism, namely the mismatch between temporal dispersion and\nspatial diffraction. The phase distortion can cause the decrease in the purity of OAM\nmodes and can induce the degradation of the high-order STOV teeth into multiple low-\norder STOVs. Thus, the distorted phases in the high-order STOV teeth, introduced by\nthe spatiotemporal astigmatism, impose extra constraint on the maximal TC of the\nSTOV comb. However, the spatiotemporal astigmatism, as well as the constraint on the\nmaximal TC, can be eliminated with careful dispersion4,5. In our case, the chirped pulses\nwith GDD of (700)2/4ln(2) are utilized for simulation and the STOV comb without the\nspatiotemporal astigmatism, as shown in Fig. S12c, is realized. For the STOV tooth\nwith TC of 4, the intensity and the phase of the central slice lying in the y-t panel is\nshown in Fig. S12d and Fig. S12e, respectively. The high-order STOV teeth in Fig.\nS12c show structures with single hole and the phase in Fig. S12e shows single phase\nsingularity.\nof the reflecting mirror. b Limitation from the pixel size of the SLM. c The STOV comb simulated\nwith the chirped pulses with GDD of (700)2/4ln(2) fs2. d The intensity of the central slice lying in\nthe y-t panel of the STOV tooth with TC of 4 in (c). e The intensity of the central slice lying in the\ny-t panel of the STOV tooth with TC of 4 in (c).\nIn addition, it is feasible to replace the conventional STOV teeth in the STOV comb\nwith the special STOV wave packets named the degradation-free spatiotemporal Bessel\nvortices6. The degradation-free spatiotemporal Bessel vortices are time-diffraction-free\ntransverse OAM beams and inherently resist degradation. Notably, the spatiotemporal\nBessel vortices with topological charges as large as 100 and -100 have been realized\nexperimentally. Therefore, for a STOV comb with teeth of the degradation-free\nspatiotemporal Bessel vortices, the constraint on the maximal TC, imposed by the\ndegradation of the high-order STOVs, can be overcome.\n4. The simulation results obtained with experimental parameters and the simulation results from\nthe FFT limit pulse and experimental results show obvious differences, particularly, in the y-t\nplane. These differences should be pointed out and explained in both main text and\nsupplemental. The relevant descriptions in the main text (lines 199-200) and supplemental\nshould be modified.\nThe reasons for the discrepancies between the simulated results and the measured results\nAs mentioned previously, the GDD of the pulses utilized for generating STOV comb shows\nremarkable impact to the spatiotemporal astigmatism. The spatiotemporal astigmatism\ninfluences the intensity distributions and the phases of the STOV teeth distinctly, especially in\nthe y-t panel. To figure out the factors induced the discrepancies between the simulated results\nand the measured results, the parameters, especially the GDD, for the experiment and the\nsimulation are compared.\nTable. R1. The parameters for the simulation with femtosecond pulses, the simulation with chirped pulses and\nthe experiment.\nParameters for simulation with femtosecond FTL pulses\nPulse duration 100 fs @FWHW\nParameters of femtosecond Diameter of laser spot 1.44 mm @FWHW\nlaser Spectral bandwidth 4 nm\nGroup delay dispersion 0 fs2\nParameters of 2D pulse Spatial frequency of grating 1200 lines/mm\nshaper Focal length of cylindrical lens 0.25 m\nPropagation distance after generation 0.4 m\nParameters for simulation with chirped pulses\nPulse duration 8 ps @FWHW\nParameters of femtosecond Diameter of laser spot 1.44 mm @FWHW\nlaser Spectral bandwidth 4 nm\nGroup delay dispersion -443860 fs2\nParameters of 2D pulse Spatial frequency of grating 1200 lines/mm\nshaper Focal length of cylindrical lens 0.25 m\nPropagation distance after generation 0.4 m\nParameters for experiment\nPulse duration ~8 ps @FWHW\nParameters of femtosecond Diameter of laser spot ~1.4 mm @FWHW\nlaser Spectral bandwidth ~4 nm @FWHW\nGroup delay dispersion ~-443860 fs2\nParameters of 2D pulse Spatial frequency of grating 1200 lines/mm\nshaper Focal length of cylindrical lens 0.25 m\nPropagation distance after generation ~0.4 m\nIn previous results, a conventional femtosecond pulses with spectral bandwidth of 10 nm (795-\n805 nm) are utilized for simulation, whereas the broadband pulses in experiment show spectral\nbandwidth of ~4 nm (798-802 nm). As the spectral bandwidth is wider in the simulation, a\nsmaller GDD is introduced to stretch the broadband pulses to ~8 ps when simulating with the\nchirped pulses. Besides, the propagation distance of the STOV comb after generation also\n9\ndiffers between the simulation and the experiment. These differences in the parameters induced\nthe discrepancies between the simulated results and the measured results. To weaken the\ndiscrepancies induced by the deviation in parameters, the simulations are re-executed with the\nreset parameters. The reset parameters for the simulation with the femtosecond Fourier-\ntransform-limited (FTL) pulses, the simulation with the chirped pulses and the experiment are\nshown in Table R1.\nFor the re-executed simulation, the incident broadband pulses are expressed as\nEx,y,exp    x2  y2 a2  exp   c 2 b2  exp    i  2  c 2    (R1)\nwhere x and y are the spatial coordinates perpendicular to the beam propagation direction,\nω =2πc/λ is the angular frequency, where c is the light speed in vacuum, λ is the central\nc c c\nwavelength, a and b are the beam waist radius of the Gaussian profile in the spatial domain and\nfrequency domain respectively, and β is the GDD of the broadband pulse. For β=0 fs2, the\nincident light field is the femtosecond FTL pulses. For β≠0 fs2, the incident light field is the\nchirped pulses. And the GDD β is calculated as\n3L\n (R2)\n2c2d2cos3\nout\nwhere λ is the central wavelength of the broadband pulses, L is the distance of the\ngrating pair in the pulse stretcher, c is the light speed in vacuum, d is the period of the\ngrating pair, and θ is the diffraction angle of the central wavelength ray with respect\nout\nto the grating normal. In the experiment of this work, λ is 800 nm, L is ~0.55 m, c is 3×108 m/s, d is 1/1200\nmm and θ is set as ~57° for the introduce of big GDD and for stretching the\nout\nfemtosecond pulses effectively. The GDD β in the experiment is calculated to be about -443860 fs2 and the GDD β is set as -\n443860 fs2 when simulating the STOV comb with the chirped pulses. The time profile of the\nchirped pulses with GDD of -443860 fs2 and spectral bandwidth of 4 nm is shown in Fig. R3\nwith red curve. The time profile of the chirped pulses measured in experiment is shown in Fig.\nR3 with blue curve. It can be observed that the two temporal profiles in Fig. R3 are similar.\nFig. R3. The time profile of the chirped pulses with GDD of -443860 fs2 & spectral bandwidth of 4 nm (red)\nand the time profile of the chirped pulses measured in experiment (blue)\nWith the parameters in Table. R1, the STOV comb simulated with the femtosecond FTL pulses,\nthe STOV comb simulated with the chirped pulses and the STOV comb generated in experiment,\nare shown in Figs.R4a-c, respectively. The STOV combs in Figs.R4a-c show some differences\nin intensity profiles and these differences can be explained from the following aspects.\nThe STOV comb in Fig. R4a is simulated with the femtosecond FTL pulses with the GDD of 0\nfs2. The STOV comb in Fig. R4b is simulated with the chirped pulses with the GDD of -443860\nfs2. And the STOV comb in Fig. R4c is experimentally generated with the chirped pulses with\n10\nthe GDD of ~-443860 fs2. The STOV comb in Fig. R4a and the STOV comb in Fig. R4b & Fig.\nR4c are different in intensity distributions, as well as in pulse durations. These differences are\ninduced by the different GDD between the femtosecond FTL pulses and the chirped pulses. The\neffect of the GDD to the intensity distribution of the STOV wave packet has been researched\nin [Cao, Q., Chen, J., Lu, K., Wan, C., Chong, A., & Zhan, Q. (2021). Sculpturing\nspatiotemporal wavepackets with chirped pulses. Photonics Research, 9(11), 2261-2264].\nDifferences between STOV combs in Fig. R4b and Fig. R4c are also observed in aspect of\nintensity distributions, which can be attributed to the following reasons. Firstly, the parameters\nin simulation and in experiment are still slightly different as the parameters in simulation are\nestimated from the experiment, especially the GDD of the chirped pulses. The differences in\nparameters can cause certain differences in the profile of the STOV comb. Secondly, the\nsimulated STOV comb in Fig. R4b is calculated directly, whereas the STOV comb in Fig. R4c\nis extracted from the interferometric fringes. Factors affecting the interferometric fringes will\naffect the measured phases, as well as the measured intensities. These factors include the\ndetecting noise of the CCD, laser power fluctuation, small optical path deviation, fluctuations\nof the electronic components like SLM, and so on. Under the influence of these factors, the\nintensity profile of the STOV comb measured in experiment exhibits some deformations from\nthe intensity profile of the simulated STOV comb.\nFig. R4. The intensity isosurfaces of the STOV comb simulated with the TTL limit pulses, the STOV comb\nsimulated with the chirped pulses, and the STOV comb generated with the chirped pulses in experiment. a\nThe intensity isosurface of the STOV comb simulated with the TTL limit pulses. b The intensity isosurface of the\nSTOV comb simulated with the chirped pulses. c The intensity isosurface of the STOV comb generated with the\nchirped pulses in experiment. (The temporal interval of the STOV combs in (b, c) is magnified 15 times for better\nvisualization.)\nAlthough there are some differences between the STOV combs in Fig. R4b and Fig. R4c,\nthe\nSTOV combs in Fig. R4b and Fig. R4c show similar patterns as they are generated with close\nparameters. The intensity profiles and phases distribution of the central slices of the STOV teeth\nwith TCs of -3, -1, 0, 2, 4 in Fig. R4b and Fig. R4c are compared below. The intensity profiles\nand phases of the central slices of the simulated STOV teeth with TCs of -3, -1, 0, 2, 4 are\nshown in Fig. R5. And the intensity profiles and phases of the central slices of the STOV teeth\nwith TCs of -3, -1, 0, 2, 4 generated in experiment are shown in Fig. R6. It can be observed that\nthe results simulated with the chirped pulses and the results generated in experiment with the\nchirped pulses show similar pattern.\nwith TCs of -3, -1, 0, 2, 4 and the corresponding phases. a-e The intensity distribution of central slices at y-t panel\nof teeth (simulated with the chirped pulses) with TCs of -3, -1, 0, 2, 4. f-j, The phases of slices in (a-e).\nFig. R6. The intensity distribution at the y-t panel of central slices of teeth (measured in experiment) with TCs\nof -3, -1, 0, 2, 4 and the corresponding phases. a-e, The intensity distribution of central slices at y-t panel of teeth\n(measured in experiment) with TCs of -3, -1, 0, 2, 4. f-j, The phases of slices in (a-e).\nThe modification in the manuscript\nThe differences between the STOV comb simulated with the femtosecond FTL pulses,\nthe\nSTOV comb simulated with the chirped pulses and the STOV comb generated in experiment,\nhave been pointed out and explained in the manuscript from line 213 to line 225, which says\nThe STOV comb simulated with the femtosecond FTL pulses, the STOV comb\nsimulated with the chirped pulses, and the STOV comb generated in experiment with\nthe chirped pulses show similar patterns, such as the spatiotemporal structures of the\nteeth, the several zero-intensity regions in the high-order STOV teeth. Some differences\nare also observed between these STOV combs, mainly in the intensity profiles of the\nSTOV teeth, especially in the y-t panel. The differences between the STOV comb\n12\nsimulated with the femtosecond FTL pulses and the STOV comb simulated with the\nchirped pulses or generated in experiment with the chirped pulses, are induced by the\ndifferent GDDs of the femtosecond FTL pulses and the chirped pulses36. The\ndifferences between the STOV comb simulated with the chirped pulses and the STOV\ncomb generated in experiment with the chirped pulses, are induced by the slight\ndifferences in parameters and by the factors like fluctuations or noises during the\nexperiment.\nA reference is increased and is listed below.\n36. Q. Cao, J. Chen, K. Lu, C. Wan, A. Chong, Q. Zhan, Sculpturing spatiotemporal wavepackets with\nchirped pulses. Photonics Research 9(11), 2261-2264 (2021).\nThe modification in the supplemental material\nThe section named “S3. Simulation of the STOV combs with experimental parameters” in the\nsupplemental material is reorganized and the differences between the STOV comb simulated\nwith the femtosecond FTL pulses, the STOV comb simulated with the chirped pulses and the\nSTOV comb generated in experiment, have been pointed out and explained. The reorganized\nsection named “S3. Simulation of the STOV combs with experimental parameters” is shown\nbelow.\nS3. Simulation of the STOV combs with experimental parameters\nThe STOV comb simulated with parameters close to the experiment is also realized.\nThe broadband pulses for simulation are expressed as3\nEx,y,exp  x2  y2 a2exp2 b2exp  i  2  (S1)\n   c    2 c  \nwhere x and y are the spatial coordinates perpendicular to the beam propagation\ndirection, ω is the optical carrier frequency, ω =2πc/λ , c is the light speed in vacuum,\nc c c\nλ is the central wavelength, a and b are the beam waist radius of the Gaussian profile\nc\nin the spatial domain and frequency domain respectively, and β is the group delay\ndispersion (GDD) of the broadband pulse. For β=0 fs2, the incident light field is the\nfemtosecond Fourier-transform-limited (FTL) pulses. For β≠0 fs2, the incident light\nfield is the chirped pulses. And the GDD β is calculated as\n3L\n (S2)\n2c2d2cos3\nout\nwhere λ is the central wavelength of the broadband pulses, L is the distance of the\ngrating pair in the pulse stretcher, c is the light speed in vacuum, d is the period of the\ngrating pair, and θ is the diffraction angle of the central wavelength ray with respect\nout\nto the grating normal.\nof 4 nm (red) and time profile of the chirped pulses measured in experiment (blue)\nIn the experiment of this work, λ is 800 nm, L is ~0.55 m, c is 3×108 m/s, d is 1/1200\nmm and θ is set as ~57° for the introduce of big GDD and for stretching the\nout\nfemtosecond pulses effectively. The GDD β of the chirped pulses in the experiment is\ncalculated as around -443860 fs2. Thus, the GDD β is set as -443860 fs2 when simulating\nthe STOV comb with chirped pulses. The time profile of the chirped pulses with GDD\nof -443860 fs2 and spectral bandwidth of 4 nm is shown in Fig. S3 with red curve. The\ntime profile of the chirped pulses measured in experiment is shown in Fig. S3 with blue\ncurve. The FWHM of these two profiles is close.\nThe intensity isosurfaces of the STOV comb simulated with the femtosecond FTL\npulses, the STOV comb simulated with the chirped pulses, and the STOV comb\ngenerated with the chirped pulses in experiment are shown in Figs. S4a-c, respectively.\nThe temporal intervals between the neighboring teeth in the STOV combs in Fig. S4b\nand Fig. S4c are magnified 15 times for better visualization.\nFig. S4. The intensity isosurfaces of the STOV comb simulated with the femtosecond FTL\n14\npulses, the STOV comb simulated with the chirped pulses, and the STOV comb generated\nwith the chirped pulses in experiment. a The intensity isosurface of the STOV comb simulated with the femtosecond FTL\npulses. b The intensity isosurface of the STOV comb simulated with the chirped pulses. c The intensity isosurface of the\nSTOV comb generated with the chirped pulses in experiment. (The temporal intervals between the neighboring teeth in the\nSTOV combs in (b, c) are magnified 15 times for better visualization.)\nThe STOV combs in Fig. S4 show some differences. Firstly, the GDD of the pulses for\ngenerating the STOV wave packets show strong influence to the intensity profiles of\nthe STOV wave packets, as the GDD of the pulses determines the level of the\nspatiotemporal astigmatism in the STOV wave packets. The STOV comb in Fig. S4a is\nsimulated with the femtosecond FTL pulses with the GDD of 0 fs2. The STOV comb in\nFig. S4b is simulated with the chirped pulses with the GDD of -443860 fs2. And the\nSTOV comb in Fig. S4c is generated with the chirped pulses with the GDD of ~-443860\nfs2. Therefore, the differences between the STOV comb in Fig. S4a and the STOV\ncombs in Fig. S4b and Fig. S4c are induced by the different GDD of the femtosecond\nFTL pulses and the chirped pulses. Secondly, the STOV combs in Fig. S4b and Fig.\nS4c are generated with close parameters. The slight differences in parameters can also\ncause certain differences in the profile of the STOV comb. Besides, the simulated\nSTOV comb in Fig. S4b are calculated directly, whereas the STOV comb in Fig. S4c is\nextracted from the interferometric fringes. Factors affecting the interferometric fringes\nwill affect the measured STOV comb. These factors include the detecting noise of the\nCCD, laser power fluctuation, small optical path deviation, fluctuations of the\nelectronic components like SLM, and so on. Under the influence of these factors, the\nintensity profile of the measured STOV comb, exhibits some deformation from the\nintensity profile of the simulated STOV comb.\nFig. S5. The intensity distribution at the y-t panel of central slices of teeth (simulated with the\nchirped pulses) with TCs of -3, -1, 0, 2, 4 and the corresponding phases. a-e The intensity\ndistribution of central slices at y-t panel of teeth (simulated with the chirped pulses) with TCs of -3,\n-1, 0, 2, 4. f-j, The phases of slices in (a-e).\nAlthough there are some differences between the STOV combs in Fig. S4b and Fig.\nS4c, the STOV combs in Fig. S4b and Fig. S4c show similar patterns as they are\ngenerated with close parameters. To further verify this statement, the phase information\nof the teeth in the simulated STOV comb is analyzed. The teeth with TCs of -3, -1, 0,\n2, 4, in the STOV comb in Fig. S4b are picked out for phase study. To extract the phase\n15\ninformation, the central slice lying in the y-t panel of the STOV tooth is picked out and\nfive slices are chosen from these five teeth. The intensity information and the phase\ninformation located at these five slices is acquired. The intensity distributions at these\nfive slices are shown in Figs. S5a-e and the phases of these five slices are shown in\nFigs. S5f-j. Compared with the experimental results (Fig. 3) in the manuscript, the\nsimulated results show similar patterns with these experimental results both in intensity\nprofiles and in phase profiles. The dislocated phase singularities, induced by the\nspatiotemporal astigmatism, are also observed in the phases in the y-t panel of the high-\norder STOV teeth simulated with the chirped pulses. And this phase distortion can cause\nthe decrease in the purity of OAM modes and can induce the degradation of the high-\norder STOV teeth into multiple low-order STOVs. Thus, the distorted phases in the\nhigh-order STOV teeth, introduced by the spatiotemporal astigmatism, impose extra\nconstraint on the maximal TC of the STOV comb. The spatiotemporal astigmatism, as\nwell as the constraint on the maximal TC, can be eliminated with careful management\nof the GDD of chirped pulses.\n5. The authors have added application demonstrations, which enhances the significance of this\nstudy. However, prior studies have already demonstrated the applications of spatial vortex light\nand STOV in optical communications. The contributions of these previous works should be\nappropriately acknowledged in the section of application demonstration.\nResponse: We are appreciative for this constructive comment.\nThe contributions of these previous works, relevant to the applications of spatial vortex light\nand STOV in optical communications, is acknowledged and some representative works are\ncited in the section of application demonstration. The relevant description is given in the\nmanuscript from line 356 to line 358, which says\nThe longitudinal OAM has been widely applied in optical communication38,39 and the sole\nresearch about STOV-based optical communication was realized with the STOV string31.\nand from line 365 to ine 366, which says\nInspired by the researches about optical communication with longitudinal OAM and STOV\nstring, two-channel information encoding and decoding is demonstrated...\nTwo references are increased and are listed below.\n38. J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar,\nM. Tur, A. E. Willner, Terabit free-space data transmission employing orbital angular\nmomentum multiplexing. Nat. Photonics 6(7): 488-496 (2012).\n39. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, S.\nRamachandran, Terabit-scale orbital angular momentum mode division multiplexing in\nfibers. Science 340(6140): 1545-1548 (2013).\n16\n6. In the S7 section of Supplemental, the displayed results show the propagation properties\nrather than the diffraction properties. The authors claim that the diffraction property of the\nSTOV comb after being focused with a spherical lens is studied. Why are lenses instead of\ngratings to investigate diffraction characteristics?\nResponse: We thank the reviewer for this kind reminder.\nWe are sorry for confusing the propagation property and the diffraction property. As\ndemonstrated in [Huang, S., Wang, P., Shen, X., & Liu, J. (2021). Properties of the generation\nand propagation of spatiotemporal optical vortices. Optics Express, 29(17), 26995-27003],\nthe\nSTOVs propagate from near-field to far-field through a focus lens. Therefore, the property of\nthe STOV comb after being focused with a spherical lens is also the propagation property of\nthe STOV comb. The phrase “diffraction properties” in the S7 section of the supplemental\nmaterial has been altered with the phrase “propagation properties”. After modification, the\nphrase “propagation properties” appears in line 38, line 345, line 346, and line 357, in the\nsupplemental material, which says\n“S7. Propagation property of the STOV combs” (in line 38)\n“S7. Propagation property of the STOV combs” (line 345)\n“The propagation property of the STOV comb is important.” (line 346)\n“Besides, the propagation property of the STOV comb after being focused with a spherical lens\nis also studied.” (line 357)\n7. (a) The manuscript needs further polishing to enhance the fluency and logic of the language.\nThe rigor of description should be improved. Some exaggerated, subjective descriptions should\nbe corrected. The sentence in lines 311-313 is unclear. It is recommended to rewrite these\nsentences.\nResponse: We are quite thankful for these corrections.\nThe authors have gone through the manuscript carefully and modifications have been made to\nenhance the fluency and logic of the language, to improve the rigor of description, and to correct\nthe exaggerated, subjective descriptions.\nThe description about the spatiotemporal vortex string in the Introduction part of the previous\nmanuscript that “However, despite their ability to support various transverse OAM modes,\nSTOV string suffers from low OAM purity and cross-talk between singularities after long-\ndistance propagation.” is modified as “However, for the spatiotemporal vortex string, numerous\nSTOVs and various transverse OAM modes are limited within a single wave packet, which\ncauses the overlap and interact of different STOVs or multiple OAM modes.” in lines 49-52 in\nthe revised manuscript.\nSome descriptions like “obvious” or “distinct”, which tend to be exaggerated or subjective,\nis\naltered with “observed” or is removed. The relevant modifications are listed below.\nThe description that “In the STOV teeth, the zero intensity regions, caused by the vortex phase\nin the y-t panel, are distinct.” in lines 162-163 in the previous manuscript is modified as “In the\nobserved.” in lines 162-163 in the revised manuscript.\nThe description that “In Fig. 2d, the peak intensity of every tooth in the STOV comb is distinct.”\nin line 176 in the previous manuscript is modified as “In Fig. 2d, the peak intensity of every\ntooth in the STOV comb is observed.” in lines176-177 in the revised manuscript.\nThe description that “Judging from the phases shown in Figs. 3f-j, it’s obvious that the\ncorresponding TCs are -3, -1, 0, 2, 4, respectively.” in lines 196-198 in the previous manuscript,\nis removed.\nThe description that “Obviously, through adjusting m from [0, 1, 2] to [0, 2, 4], namely the\nk\ninterval of m from 1 to 2, the ΔTC between adjacent teeth is adjusted from 1 to 2.” in lines\nk\n237-239 in the previous manuscript is modified as “Through adjusting m from [0, 1, 2] to [0,\nk\n2, 4], namely the interval of m from 1 to 2, the ΔTC between adjacent teeth is adjusted from 1\nk\nto 2.” in lines 247-248 in the revised manuscript.\nThe description that “It’s obvious that the shift in time domain can be introduced by multiplying\na fixed phase in frequency domain.” in lines 435-436 in the previous manuscript is modified as\n“The shift in time domain is introduced by multiplying a fixed phase in frequency domain.” in\nlines 479-480 in the revised manuscript.\nA few small modifications are also made to improve the manuscript.\nThe description that “Through increasing the parameter  by 1.2 times. the Δt and the Δx\nbetween the neighboring teeth are increased by 1.2 times.” is modified as “Through increasing\nthe parameter  by 1.2 times, the Δt and the Δx between the neighboring teeth are increased\nby 1.2 times.” in line 249 in the revised manuscript.\nThe description that “Shorter pulse duration and smaller laser spot is beneficial for addressing\nthis question.” is modified as “Shorter pulse duration and smaller laser spot are beneficial for\naddressing this question.” in line 347 in the revised manuscript.\nMore descriptions about the parameters are given in the manuscript to improve the rigor of\ndescription.\nThe description that “In this section, the STOV combs with nine teeth are simulated and\ngenerated in experiment.” is modified as “In this section, the STOV combs with nine teeth are\nsimulated with the femtosecond Fourier-transform-limited (FTL) pulses and generated in\nexperiment with the chirped pulses.” in lines 131-133 in the revised manuscript.\nThe description that “Therefore, the unchirped transmitted pulses are stretched in time domain\nfirst with the pulse stretcher composed of two transmission gratings and a reflecting mirror (see\nSupplementary Material S1).” is modified as “Therefore, the unchirped transmitted pulses are\nstretched in time domain first with the pulse stretcher composed of two transmission gratings\n18\nand a reflecting mirror (see Supplementary Material S1) and a negative group delay dispersion\n(GDD) of ~-443860 fs2 is introduced.” in lines 85-86 in the revised manuscript.\nThe sentence that “Through expanding the range between a and b the equation (1), the designed\nphase for the STOV comb more teeth is calculated and the STOV comb more teeth can be\nsimulated in theory or realized in experiment.” in lines 311-313 in the previous manuscript is\nrewritten as “ Through expanding the range between a and b in equation (1), the designed phase\nfor the STOV comb with more teeth can be calculated and the STOV comb with more teeth can\nbe simulated in theory or realized in experiment.” in lines 331-334 in the revised manuscript.\n(b) As pointed out in the first-round report, the description such as “the simulation result and\nthe experimental result agree well with each other” and “Experimental result shows good\nagreement with the simulation” is not inappropriate. However, these statements have not been\ncorrected (see lines 70, 157, 393, etc.).\nResponse: We thank the reviewer for this kind reminder.\nThe description such as “the simulation result and the experimental result agree well with each\nother” and “Experimental result shows good agreement with the simulation” is corrected and\ntempered.\nThe description that “Experimental result shows good agreement with the simulation.” in line\n70 in the previous manuscript is rewritten as “The simulation results and the experimental\nresults show similar patterns” in line 67 in the revised manuscript.\nThe description that “From Fig. 2a and Fig. 2b, it can be figured out that the simulation result\nand the experimental result agree well with each other.” in line 157 in the previous manuscript\nis rewritten as “The results measured in experiment show similar patterns with the simulated\nresults.” in line 158 in the revised manuscript.\nThe description that “The experimental result shows good agreement with the simulation result.”\nin line 393 in the previous manuscript is rewritten as “The experimental results show similar\npatterns with the simulated results.” in line 421 in the revised manuscript.\n(c) The phrase \"amplified magnified\" appears repeatedly in the main text as well as the\nsupplemental.\nResponse: We thank the reviewer for this kind reminder.\nThe \"amplified magnified\" is altered with the word \"magnified\" both in the manuscript and the\nsupplemental material.\nThe modifications in the revised manuscript appear in line 171, line 279, line 298, and line 314,\nwhich says\n“The temporal intervals between the neighboring teeth in (b, g) are magnified 15 times for\nbetter visualization.” (line 171)\n“The temporal intervals between the neighboring teeth in (e-h) are magnified 10 times for better\n19\nvisualization.” (line 279)\n“The temporal intervals between the neighboring teeth in (c, d) are magnified 10 times for\nbetter visualization.” (line 298)\n“The temporal interval between the neighboring teeth in Fig. 5c and Fig. 5d is magnified 10\ntimes for better visualization.” (line 314)\nThese modifications in the revised supplemental material appear in line 176, line 184, which\nsays\n“The temporal intervals between the neighboring teeth in the STOV combs in Fig. S4b and Fig.\nS4c are magnified 15 times for better visualization.” (line 176)\n“The temporal intervals between the neighboring teeth in the STOV combs in (b, c) are\nmagnified 15 times for better visualization.” (line 184)\n(d) Some references are repeated in the list of references.\nResponse: We thank the reviewer for this kind reminder.\nThe list of references is overall checked. The 9th reference and the 23rd reference are repeated.\nThe 23rd reference is deleted and the corresponding citation in the main text is altered\nThe previous references from 21-24 is\n21. S. W. Hancock, S. Zahedpour, A. Goffin, H.M. Milchberg, Free-space propagation of\nspatiotemporal optical vortices. Optica 6, 1547-1553 (2019)\n22. G. Gui, N. J. Brooks, B. Wang, H. C. Kapteyn, M. M. Murnane, C.T. Liao, Single-frame\ncharacterization of ultrafast pulses with spatiotemporal orbital angular momentum. ACS\nPhotonics 9, 2802-2808 (2022).\n23. S. Huang, P. Wang, X. Shen, J. Liu, R. Li, Diffraction properties of light with transverse\norbital angular momentum. Optica 9(5), 469-472 (2022).\n24. C. Li, S. Wang, X. Li, Spatiotemporal pulse weaving scalar optical hopfions. Light Sci.\nAppl. 12(1), 54 (2023).\nAfter modification, the references from 21-24 is\n21. S. W. Hancock, S. Zahedpour, A. Goffin, H.M. Milchberg, Free-space propagation of\nspatiotemporal optical vortices. Optica 6, 1547-1553 (2019)\n22. G. Gui, N. J. Brooks, B. Wang, H. C. Kapteyn, M. M. Murnane, C.T. Liao, Single-frame\ncharacterization of ultrafast pulses with spatiotemporal orbital angular momentum. ACS\nPhotonics 9, 2802-2808 (2022).\n23. C. Li, S. Wang, X. Li, Spatiotemporal pulse weaving scalar optical hopfions. Light Sci.\nAppl. 12(1), 54 (2023).\n24. M. W. Hyde IV, Twisted spatiotemporal optical vortex random fields. IEEE Photonics J.\n13(2), 1-16 (2021).\nThe relevant citation is also modified. The previous citation that “Both generation and\ncharacterization methods, along with the fundamental properties of STOVs, have been\nextensively explored in theory and in experiment.4-23” is modified as “Both generation and\ncharacterization methods, along with the fundamental properties of STOVs, have been\nextensively explored in theory and in experiment.4-22” in lines 41-42 in the revised manuscript.\n20\n(e) The axis labels are missing in Fig. 3 and Fig. S4, and it is better that the phase is shown\nfrom -𝜋 to 𝜋. It should point out the simulation parameters of Fig. S4 (chirp or FFT limit pulses?)\nResponse: We thank the reviewer for this kind reminder.\nThe axis labels are added in Fig. 3 and Fig. S4. The phase is shown from -π to π. The simulation\nparameters of Fig. S4 is pointed out.\nThe modifications (lines 208-212) in the revised manuscript are shown below.\nFig. 3 The intensity distribution at the y-t panel of central slices of teeth (experimentally generated\nwith chirped pulses) with TCs of -3, -1, 0, 2, 4 in Fig. 2b and the corresponding phases. a-e, The\nintensity distribution of central slices at y-t panel of teeth with TCs of -3, -1, 0, 2, 4 in Fig. 2b. f-j, The\nphases of slices in (a-e).\nThe modifications in the revised supplemental material are shown below.\nFig. S5. The intensity distribution at the y-t panel of central slices of teeth (simulated with the\nchirped pulses) with TCs of -3, -1, 0, 2, 4 and the corresponding phases. a-e The intensity\ndistribution of central slices at y-t panel of teeth (simulated with the chirped pulses) with TCs of -3,\n-1, 0, 2, 4. f-j, The phases of slices in (a-e).\nBy the way, in this round response, as well as the first-round response, several researches are\ncited and the relevant results are shown here. But we don’t mean to give any comments on these\nworks or results. The sole purpose for citing these works is to address the concerns of the\nreviewers. We pay respects to these excellent works, which greatly enriched our cognition about\nthe STOV. Meanwhile, we are sorry for our negligence in the first-round response, like\nconfusing the propagation and diffraction. Fortunately, the negligence has been modified\naccording to the suggestions of the reviewers. Thanks again for these professional suggestions.\nDear reviewers,\nWe would like to express our gratitude for your valuable and constructive comments. The main\nmanuscript and the supplementary materials have been revised carefully according to the\ncomments.\nDetailed Responses to the Reviewers\nResponses to Reviewer #1\nThe authors have carefully revised their manuscript and addressed my concerns. I would like\nto recommend the current manuscript to be published. Some typos in the Supplementary\nMaterial should be corrected, e.g., line 169, \"fs2\", line 439, \"careful dispersion\".\nResponse: Thanks a lot for raising this valuable recommendation.\nThe \"fs2\" in line 169 in the Supplementary Material is corrected as \"fs2\".\nThe \" careful dispersion\" in line 439 in the Supplementary Material is corrected as \" careful\ndispersion management\".\n1"}, {"round": 3, "reviewer_comments": {"reviewer_1": "The authors have carefully revised their manuscript and addressed my concerns. I would like to recommend the current manuscript to be published. Some typos in the Supplementary Material should be corrected, e.g., line 169, \"fs2\", line 439, \"careful dispersion\"."}, "author_response": "Response Letter\nDear reviewers,\nWe would like to express our gratitude for your valuable and constructive comments. The main\nmanuscript and the supplementary materials have been revised carefully according to the\ncomments.\nDetailed Responses to the Reviewers\nResponses to Reviewer #1\nThe authors have carefully revised their manuscript and addressed my concerns. I would like\nto recommend the current manuscript to be published. Some typos in the Supplementary\nMaterial should be corrected, e.g., line 169, \"fs2\", line 439, \"careful dispersion\".\nResponse: Thanks a lot for raising this valuable recommendation.\nThe \"fs2\" in line 169 in the Supplementary Material is corrected as \"fs2\".\nThe \" careful dispersion\" in line 439 in the Supplementary Material is corrected as \" careful\ndispersion management\".\n1"}]

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