1. Introduction

The optical vortex, characterized by a helical phase wavefront and doughnut-shaped intensity profile, is an electromagnetic field structure with orbital angular momentum (OAM). Allen et al. [1] revealed that vortex beams carried the OAM of lħ per photon associated with the helical phase front of exp(ilθ), where θ is the azimuthal angle, l is the topological charge, and ħ is the reduced Planck’s constant [1–3]. Owing to these unique properties, optical vortices have attracted significant attention and supported substantial applications in both classical and quantum systems such as optical tweezers [4,5], super-resolution microscopy [6,7], astronomy [8], free-space optical communication [9,10], and quantum optical information processing [11,12].

Recently, optical vortices and ultrashort pulses have been combined to produce spatiotemporal optical vortices or ultrashort vortex pulses [13–15]. These vortices have drawn considerable attention not only owing to their properties of the wide spectrum bandwidth and short pulse duration, but also to the related discoveries such as tilted OAMs and toroidal structures [16–18], underpinning novel approaches to the manipulations of light fields. Specifically, because of their broad bandwidth and pulsed OAM, ultrashort vortex pulses are useful in optical communications to increase channel and information capacities either by multidimensionally structuring light in spatial and temporal domains or multiplexing OAM states with different orientations [19,20]. Additionally, tunable OAM states of ultrashort pulse duration have also contributed to progress in wider area of light-matter interactions: with these ultrashort OAM pulses, microscopic objects are monitored and manipulated via the gradient force of the intensity profile [21,22]; by use of the tilted OAMs, sculpting special photocurrent distributions in a semiconductor have been realized [23,24]; and with extreme ultraviolet OAM pulses, imaging and manipulating the magnetic and topological excitations of quantum matter have been performed [25,26].

A significant fundamental issue for ultrashort vortex pulses is to generate the controllable OAM states which provide an important degree of freedom for light field manipulations and the applications involved [27]. Methods commonly used for generating optical vortices, such as spiral phase plates (SPPs) [28,29]; computer-generated holograms (CGHs) [30]; and spatial light modulators (SLMs) [31], are used to produce ultrashort vortex pulses; and in these methods, the OAM states are transformed into ultrashort pulses by using phase-encoding elements. Additionally, spiral multi-pinhole plates, which work on the principle of using the spatial distributions of spirals to modulate the light field [32,33], have been introduced as an important alternative method to produce ultrashort vortex pulses. Essentially, using these plates avoids practical issues such as topological charge dispersion owing to the high wavelength sensitivity of SPPs [28], inevitable extensive angular dispersions introduced by CGHs [34], and the low damage threshold of SLMs. Multi-pinhole plates are damage-resistant, medium-dispersion-free, and easy-to-handle, making them suitable for generating ultrashort vortex pulses [33]. Furthermore, with the significant design principle, the method of the spiral plates has demonstrated its importance and capability in tailoring OAM states and has paved the way toward generating OAM-carrying matter waves, such as electron vortices [35] and surface plasmon polariton vortices [36].

In our previous research, we investigated ultrashort vortex pulses based on spiral multi-pinhole structures, including a single spiral for basic vortex generation [33] and multiple segmented spiral arrays for high-order vortex generation [37]. Here, we employed the improved spiral structure of the rotationally interleaved multispiral to conduct a further study on ultrashort vortex pulses. Although the interleaved spirals of the azimuthally symmetric rotation have been applied in studies on the OAM spectrum in monochromatic waves [38] and the stimulations of electron vortex beams [35], they have not been used in research on spatiotemporal ultrashort vortex pulses. Based on the rotationally interleaved multispiral, we aim to achieve a comprehensive and systematic theoretical analysis of spirals that modulate ultrashort pulses. Moreover, the spatiotemporal and evolutionary properties of ultrashort vortex pulses still need to be explored thoroughly for further applications and investigations.

In this study, rotationally interleaved multispirals were used to generate spatiotemporal optical vortices in ultrashort laser pulses. Theoretically, we provided an elaborate analysis on the multispiral modulation of an ultrashort pulse in a deconstructive process; this involved the three steps of the single spiral modulating the monochromatic wave, the multispiral modulating the monochromatic wave, and the multispiral modulating the polychromatic waves to form spatiotemporal optical vortices. Based on numerical simulations, we investigated the ultrashort vortex fields generated by different spiral structures; and comparatively analyzed their intensities, phases, power flows, and OAM states. Furthermore, we fabricated various multispiral plates using a femtosecond laser fabrication system and generated the corresponding ultrashort vortex pulses with matching topological charges. Our results demonstrated that the rotationally interleaved multispiral structure was effective and flexible for generating ultrashort vortex pulses.

2. Theoretical analyses

According to the basic principle of electromagnetic theory, an ultrashort laser pulse is a superposition of a series of monochromatic waves, as the pulse in time domain is decomposed by Fourier transform into components in frequency domain. The theoretical analyses, for understanding the rotationally interleaved multispiral modulation of ultrashort pulses and acquiring the mathematical expression of the spatiotemporal ultrashort vortices, involve three steps. First, the ultrashort pulse is deconstructed into monochromatic waves, and the multispiral is deconstructed as a single spiral to investigate the wave modulations. Then, the spirals are constructed into a complete rotationally interleaved multispiral to analyze the manipulation of the monochromatic field. Finally, the series of monochromatic components are added back into polychromatic waves to study the spatiotemporal vortex, where the polychromatic waves specify the field of the ultrashort pulses.

2.1 Modulation of single spiral in monochromatic field

The generation of optical vortices was first considered regarding the modulation of a monochromatic wave by a single spiral. The principle of modulating the transmitted light field is based on the helical phase front introduced by the increase in the spiral radius versus the azimuthal angle. The geometric structure of the single spiral is indicated by the red dotted line in Fig. 1(a) and 2(D) drawing in Fig. 1(b). The single spiral consists of N holes and can be mathematically expressed as

For the light source at point n, the complex amplitude is given by:

In Eq. (4), the spiral expression ρ_n is inputted to R_n. The small quantity of radial variation (lzλ/2πρ₁)φ_n compared with the diffraction distance z can be reasonably ignored, which has been widely applied in existing literature to analyze the function of spiral modulation [39,40].

 

Fig. 1. (a) Generation of ultrashort vortex pulses by the rotationally interleaved multispiral. The red dotted circle represents a single spiral structure, and the combined white and red dots characterize the structure of the multispiral, as illustrated in (b) and (c), respectively. Each spiral consists of N points with constant azimuthal increments and radial variations, which introduce the phase change of 6π in azimuth equaling to the topological charge of l = 3. The orthometric yellow lines and blue lines in (b) and (c) show the coordinate axes and circles as references, respectively.

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Fig. 2. (a) Off-axis distances versus the topological charge. The black and red points denote the single spiral and multispiral, respectively. (b) Intensity and phase distributions of the optical vortices with the topological charge of l = 3, obtained by the theoretical calculation and numerical simulation, respectively. (c) Pulse shape and phase distribution of the ultrashort vortex, with the amplitude representing the normalized Re(u) and the colors indicating the phase Im(u).

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Based on the Jacobi–Anger expansion

For light field transmission, the pinhole plate acts as the N-light subsource, and the integral of Eq. (2) is given in the form of the summation of the N terms of exp(ikR_n):

For the summation term $\sum\nolimits_n^N {(\ldots )}$ of the exponential function, it is known that [35]:

The condition l = p needs to be satisfied for widening the scope of this discussion; therefore, Eq. (7) can be written as

The single spiral method has been applied extensively to many physical fields involving the manipulation of optical vortices and OAM states [41,42]. However, the doughnut-shaped intensity profile of the generated vortex always deforms and finally breaks down when the topological charge goes to a higher order [38]. This phenomenon can be attributed to the structural characteristics of the asymmetry of the single spiral. As indicated by the spiral expression in Eq. (1), the spiral gap is governed by the (lzλ/2πρ₁)φ_n term, which determines the asymmetry of the spiral. As the topological charge l increases, the enlarged spiral gap causes the spiral to deviate from the central symmetry, possibly resulting in the failure of the spiral to form the doughnut-shaped intensity profile. The theoretical explanation is that the value of the enlarged spiral gap is no longer negligible and this prevents the integral in Eq. (4) from being a single Bessel function.

2.2 Modulation of multispiral in monochromatic field

To avoid the deterioration of the doughnut intensity profile and improve the quality of the vortex field with high-order topological charges, we employed a multispiral to remodel the spiral arrangement. The structure of the multispiral should satisfy two requirements: conforming to the principle of phase modulation and maintaining geometric symmetry. Accordingly, the multispiral has been designed as an interleaved arrangement of single spirals with azimuthally symmetric rotation, as shown in Figs. 1(a) and (c). Here, we introduced two integer parameters, M and m, to signify the mode of the multispiral structure, where M is the number of spirals, and m refers to the introduced phase difference of m × 2π between the radial adjacent spirals. Each M spiral with the same initial radius was successively rotated around the center with an equal angle interval of 2π/M, and the rotated spirals were interleaved into the multispiral of the M multiple symmetries, referred to as the rotationally interleaved multispiral. In Fig. 1(c), the multispiral of (M, m) = (3, 1) is used as a representative to demonstrate the structure with three (M = 3) spirals and one (m = 1) 2π radial phase difference. If the multispiral was designed to introduce a 2πl-phase variation, the parameters would have the relationship 2πl = M × 2πm, meaning that the superposition of M times 2πm-phase modulations would form a total phase change of 2πl.

Since the radial phase changes between the adjacent spirals maintained the 2πm-phase differences, the linear azimuthal phase modulation of each spiral was not influenced by the other spirals. Moreover, the rotationally interleaved multispiral greatly improved the symmetry distribution of the structure around the center. Here, we introduced the centroid of the spiral area as C = ∑ρ_n/N and employed the concept of off-axis distance to represent the asymmetry of the single-spiral structure [43]; the off-axis distance is defined as the length between the centroid and the origin of the coordinates. Figure 2(a) shows the off-axis distances of the two types of spirals versus the topological charge l, with the black solid squares and red solid circles representing the single spirals and multispirals, respectively. Each value of l corresponded to a single-spiral structure and a multispiral structure of (M, m) = (l, 1). The off-axis distance of the single spiral increased significantly with l. In contrast, the off-axis distance of the multispiral remained close to zero, indicating the greatly improved symmetry of the geometrical structure of the multispiral.

Next, we analyzed the generation of an optical vortex with a multispiral under the illumination of a monochromatic wave. The multispiral can be mathematically expressed as

Then, the complex field u(r, θ) at the observation plane arising from the combination of all the light sources can be written as

By substituting the expression φ_sn of the multispiral in Eq. (10) into the last factor in Eq. (12), exp[i(l-p)φ_sn], and using the results of Eq. (8), the summation of s and n in the above equation can be simplified as [35]

The above equation shows that the corresponding terms contribute to the light field u(r, θ) only when l = p. Therefore, for the multispiral, u(r, θ) is given as

In Eq. (14), the Bessel function J_l (kρ₁r/z) and the exponential function exp(ilθ) signify an optical vortex with a topological charge of l. The analyses showed that rotationally interleaved multispirals could effectively modulate monochromatic waves and generate an optical vortex. This meant that multispirals could enable the same azimuthal phase modulations that a single spiral could and that the improved symmetry of the structure could provide a way to generate optical vortices with better quality.

2.3 Modulation of multispiral in polychromatic field

We now extended the analyses to generate optical vortices using a multispiral in polychromatic waves and investigate the properties of spatiotemporal vortex pulses. Considering that the temporal ultrashort pulse is the Fourier transform of its spectral domain, the properties of ultrashort pulses can be discussed based on the superposition of the light fields in a series of single frequencies. Here, u(ω,r,θ) is used to denote the monochromatic field of a frequency ω with topological charge l(ω). For the polychromatic field of the ultrashort pulse, the topological charges carried by the field of the broadband spectrum are approximately linearly varied with ω, l(ω) = l+η(ω-ω₀), where ω₀ is the central frequency and η = Δl/Δω is the coefficient with spectral width Δω and corresponding mode variation Δl [33,44]. For all frequencies in the entire range of polychromatic waves, the topological charge of the ultrashort vortex pulse takes an approximate value of l [33]. We focused on the formation and evolution of ultrashort vortex pulses in this section. Accordingly, we adopted this approximate value to facilitate the analyses and ignore the influences of topological charges on spatiotemporal properties; the pulsed field can be considered to have a topological charge of l.

Using ω/c = 2π/λ to replace λ with frequency ω in Eq. (14) and introducing the parameters τ′, A, and B as $\tau ^{\prime} = \frac{1}{c}\left( {z + \frac{{{r^2} + \rho_1^2}}{{2z}}} \right)$, $A = \frac{{{i^{l - 1}}MN}}{{2\pi cz}}$, and $B = \frac{{{\rho _1}r}}{{cz}}$, respectively, Eq. (14) can be written as

Equation (17) can be represented as

For the integral of the first term with the integrand of ω in Eq. (19), the calculation can be performed directly by setting μ=v = n and λ=0 in Eq. (20):

Comparing Eq. (22) with Eq. (21), Eq. (22) has a difference in the Gamma function Γ(m + n+½)/Γ(m + n + 1) and an additional constant value α^1/2. Since the Gamma function is only the coefficient of the hypergeometric function F{…}, to facilitate the analyses, the amplitude scaling can be approximated as a constant value based on Taylor expansion and Stirling’s formula [48]. Consequently, Eq. (22) yields the approximated expression for:

Substituting Eqs. (21) and (23) into Eq. (19), u(t,r,θ) can be written as

Using the relation I_n (z) = i^-nJ_n (iz) and applying the addition theorem again, the summation term in Eq. (24) can be simplified as follows:

Conclusively, the complex field of the spatiotemporal ultrashort vortex pulse is expressed as

Based on the theoretical results of Eq. (26), we calculated the light fields of ultrashort vortex pulses. The intensity and phase distributions of the calculated vortex fields are shown in the left panel of Fig. 2(b), and the temporal shape of the vortex pulse is shown in Fig. 2(c). In the right panel of Fig. 2(b), the numerical simulations of the vortex field based on the Huygens-Fresnel principle in Eq. (2), with an extension to the time domain for the ultrashort pulse, are also shown for comparison. The field distributions in Fig. 2(b) are the temporal vortices at τ′-t = 0, which are selected as the typical mode of ultrashort vortex pulses. A comparison between the two vortices in Fig. 2(b) shows that the intensity and phase images of the vortices, based on the theoretical results of Eq. (26), present perfectly symmetric distributions; this is because ideal conditions with reasonable approximations were used, and the small quantities in the theoretical analyses were ignored. However, although the intensity and phase distribution of the vortex field obtained using Eq. (2) in the right panel were not as perfect as those of the images in the left panel, this vortex field had a doughnut intensity profile with good quality and a uniformly varied helical phase front with topological charges of l = 3, evidently presenting the features of the optical vortex in space at the instant t = 0. The theoretical analyses have convincingly provided the correct conclusions about spatiotemporal vortex pulses.

Figure 2(c) shows the calculated results of the temporal intensity curve and the corresponding phase of the ultrashort vortex pulse at a point on the doughnut ring with the maximum intensity, with the normalized amplitude of the intensity representing the pulse shape, and the color of the curve denoting the phase distribution. The origin of the time axis, t = 0, was shifted to the temporal center of the pulse on the observation plane. In combination with the phase front of exp(ilθ), it could be deduced that the phase distribution temporally rotated within the pulse cycles carrying a 2πl variation. Based on the envelope of the intensity amplitude, the geometric shape of the ultrashort pulse indicated that the optical vortex had a temporal Gaussian-type distribution.

The above theoretical analyses demonstrated that the rotationally interleaved multispiral could effectively modulate ultrashort polychromatic waves to generate a spatiotemporal ultrashort vortex pulse. The generated vortex pulse carried a topological charge of l and possessed a doughnut-shaped intensity profile of J_l (…), while it still had a Gaussian-type pulse duration with a time delay of τ′.

3. Simulations and discussion

Based on Eq. (2) with an extension to the ultrashort pulse, we numerically performed simulations of the ultrashort vortex pulses generated by different spiral structures. Specifically, we chose single spirals and multispirals of (M, m) = (2, 2) and (4, 1) to generate ultrashort vortex fields. Particularly, we use the multispiral definition method to denote a single spiral as (M, m) = (1, 4) to intuitively show the spiral number (M = 1) and then introduced a phase variation of 2πm = 2πl/M (m = 4). The three spiral structures were designed to generate vortices with the same topological charge of l = 4 for comparatively analyzing the modulation effects between the spirals. The five columns in Fig. 3 show the spiral structure, intensity distribution, phase distribution, power flow, and momentum. The ultrashort vortex fields were all derived from the pulses at t = 0, with the same time calibrations, as shown in Figs. 2(b)–2(c).

 

Fig. 3. (a)–(c) show three vortices with l = 4 generated by three spiral structures of (M, m) = (1, 4), (2, 2), and (4, 1). The five columns are the spiral structure, intensity distribution, phase distribution, power flow, and momentum. The backgrounds in the fourth and fifth columns represent the corresponding intensity and phase distributions, respectively. The directions and lengths of the arrows indicate orientation and strength, respectively. (d) and (e) show the variations in the power flow and OAM, respectively, versus the azimuthal angle. The data have been obtained from the vortex fields of (a)–(c) and extracted from the positions marked by dashed circular lines, as shown in the inset figures. Blue, red, and green lines represent the corresponding vortex fields.

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In Fig. 3, either the structure or the intensity and phase distributions indicate that the generated ultrashort vortex pulses carry the same topological charges of l = 4. The intensity images of the three vortices presented doughnut-shaped distributions, and the azimuthal phase variations were 8π around the vortex core. The deteriorations of the doughnut intensity profile and the rotational phase distribution in Fig. 3(a) as being asymmetric with respect to the center was clearly visualized. The images of the intensity and phase in Figs. 3(b)–3(c) show improved circularly symmetric distributions, particularly the field in Fig. 3(c). The results demonstrated that the extra spirals in the multispirals did not change the phase modulation, and they greatly improved the quality of distributions in the ultrashort vortex pulses.

For further analysis, we calculated the power flow and momentum based on the intensity and phase data of the above three vortices. According to the classical theory of electromagnetic waves, the power flow J(r) can be written as

The results of the power flow J(r) and momentum ∇_r φ(r) are shown in the fourth and fifth columns in Fig. 3, respectively, with the corresponding intensity and phase distributions as the backgrounds, where the lengths of the arrows denote the strengths of the corresponding variables; the clockwise rotations of the arrows indicate the helicity of the vortices. Significantly, as the number of spirals increased, the distributions of the arrows gradually become symmetric and well-ordered, as illustrated in Figs. 3(a)–3(c), especially for the arrows in the fields of Fig. 3(c) with a greatly improved symmetric distribution.

We azimuthally extracted the data of the power flow J(r) and momentum ∇_r φ(r) around the vortex cores from the three vortices to intuitively show the field distributions because the fluctuations in the data, as represented by the amplitude variations of the arrows, directly reflected the symmetric properties. The dashed circular lines in the inset images in Figs. 3(d)–3(e) indicate the locations of the extracted data; the positions are the circles at the maximum values of the doughnut-shaped intensity profiles. The curves in blue, red, and green represent the data of the three vortex fields, as shown in Figs. 3 (d) and (e). Figure 3(d) presents the normalized power flow J(r) in the azimuth, and Fig. 3(e) shows the OAM L_z(r) obtained from further calculations of ∇_r φ(r) based on Eq. (29). In Fig. 3(d), the power flow of the vortex field obtained by the single spiral fluctuates more than that under the other two cases of the multispirals. The same phenomenon also occurs in Fig. 3(e), wherein the momenta generated by the multispirals have comparatively stable variations around the vertical axis of l = 4, while the blue line presents a wide range of variation with the values of L_z(r),

The above results demonstrated that multispirals with a rotationally interleaved structure could effectively generate optical vortices with improved symmetric field distributions. The multispirals not only enabled the phase modulation to introduce the helical phases and form ultrashort vortex pulses, but also contributed to vortex fields, especially high-order ones, with circularly symmetric distributions. These results agreed with the theoretical analyses and demonstrated that using the rotationally interleaved multispiral structure was an effective method for generating ultrashort vortex pulses.

4. Experimental setup and results

In the experiment, we set up a Mach–Zehnder-type interferometer to generate ultrashort vortex pulses using multispirals, as shown in Fig. 4. A Ti:sapphire oscillator (Synergy Pro, Femtolasers Produktions) was used as the femtosecond laser source, with a central wavelength of 800 nm, repetition rate of 75 MHz, and average power of 500 mW. The measured pulse duration of the laser was 12 fs. Before generating the vortex pulses and measuring the modulated beams, the incident pulse from the ultrashort laser was initially measured by the spectral phase interferometry for direct electric-field reconstruction (SPIDER) (FC Spider, APE), using which the spectral amplitudes and phases of the ultrashort pulse were obtained [49,50]. A beam splitter (OA037, Femtolasers Produktions) splits the incident laser beam into the two arms of the interferometer. In one arm, where the beam was regarded as the object beam, a homemade pinhole plate was coaxially embedded and used to modulate the light field. In the other arm, with the beam regarded as the reference beam, a subsystem comprising four reflecting mirrors and a piezoelectric nano-translation stage (PI E-516) was constructed to control the time delay. By precisely controlling the translations of the piezoelectric stage to scan the time delay, the generated ultrashort vortex pulses could interfere spatially with the reference beam. The piezoelectric stage was controlled to achieve a spatial stepping precision of 150 nm, while the pulse duration of the double optical path introduced by the stage was temporally equal to 1 fs. Accordingly, a charge-coupled device (CCD) (Cascade 1 K, Princeton Instruments) recorded the interference patterns at equal time intervals by gradually changing the optical path difference between the two beams. The collected series of interference patterns represented ultrashort pulses at different times. Based on the spatially resolved Fourier transform spectrometry measurements, the field information of the ultrashort vortices was retrieved from the interferograms by calibrating the reference pulse measured by the SPIDER [49].

 

Fig. 4. (a) Schematic of experimental setup. BS1–3 and M1–2 are the optical components of the beam splitters and reflecting mirrors. SPIDER and CCD measure the spectral information and capture images. The time delay system comprises a piezoelectric stage and four reflecting mirrors. The pinhole plate carries the multispiral and consists of N = 90 pinholes per spiral; the pinhole radius is approximately 20 µm, the initial radius of the spirals 0.5 mm, and the propagation distance is z = 200 mm. The enlarged plate is a typical three-spiral structure with a mode of (M = 3, m = 1). (b) Photograph of the practical experimental setup. (c) The normalized spectrum intensity and the phase distributions of the input pulse. (d) Temporal profiles of the input pulse and the output pulse.

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A pinhole plate with multispiral was fabricated using a focused femtosecond laser (Spitfire, Spectra-Physics, with a central wavelength of 800 nm, repetition rate of 1 kHz, pulse duration of 35 fs, and average power of 3.2 W) punching pinholes on aluminum foils. Based on the ultrahigh-power laser source and optoelectronic components, including the focusing lens, translation stages, power meter, and electronic shutter, we built a fabricating system to create pinhole plates. By interconnecting all the components with computer programs, this fabrication system could operate automatically and had a working accuracy of approximately 1 µm. Consequently, using the system, pinhole plates could be conveniently and efficiently fabricated with satisfactory accuracy in the laboratory.

The fabricated pinhole plates had N = 90 pinholes per spiral, a pinhole radius of ρ_hole= 20 µm, and spiral initial radius of ρ₁= 0.5 mm. In fact, in the selection of the parameters of the multispiral pinhole plates, the related factors had been taken into comprehensive consideration. The pinhole radius was determined by the fabrication condition and the diffraction effect. If the pinholes were too small, the etching would tend to be unstable and difficult to control; while bigger pinholes would reduce the Airy disk of diffraction and thereby diminish the phase modulation effect. Furthermore, the pinhole radius together with the spiral’s perimeter determines the pinhole number. With N being 90, the space of the pinholes was reasonable for fabrication and the transmitting energy power was also moderate. Besides, the arrangements of the spirals and pinholes were determined based on Eq. (10). Each spiral was successively rotated around the center with an equal angle interval of 2π/M, and when the starting position of the first spiral was set at φ=0, the successive adjacent spiral was started at the rotated angle 2π/M. With the azimuthal angle increasing, the radius of the spiral varied evenly with an increment, while the radial distance between two adjacent spirals reduced, and it was deduced to be lzλ/Mρ₁ at φ=0. The inset of the pinhole plate in Fig. 4 shows the structure of a typical three-spiral with a mode of (M, m) = (3, 1). Though the parameter selection seemed to be somewhat complex, in practice, as long as the parameter conditions for generating the vortices were mathematically satisfied, the parameters are not limited too strictly.

To demonstrate the spectrum properties of the input pulse, we measured the reference pulse by the SPIDER, and the normalized spectrum intensity and phase distribution are shown in Fig. 4(c), respectively. Simultaneously, the curve of the temporal intensity profile of the reference pulse measured by the SPIDER is shown in Fig. 4(d), which gives the pulse duration of 12 fs. Moreover, to give information of the pulse duration of the output pulse, we extracted the temporal intensities from the interferograms of the reference pulse with different time delays and the vortex profiles produced by multispiral plate (1, 1), whereas the data in the profiles were taken at points on the maximum-intensity ring of the vortex doughnut. The curve of the temporal intensity profile of the output pulse is obtained and is also shown in Fig. 4(d). The output vortex pulse exhibits a broadened pulse duration of about 19 fs.

We fabricated various pinhole plates with different spiral structures to generate ultrashort vortex pulses. Figures 5(a)–5(f) show the multispiral plates with structural modes of (M, m) = (2, 1), (3, 1), (5, 1), (2, 2), (3, 2), and (4, 2), respectively. The spiral structures in Figs. 5(a)–5(c) have different spiral numbers of M = 2, 3, and 5, with the same radial phase difference of 1 × 2π (m = 1) between adjacent spirals. Accordingly, the spiral structures in Figs. 5(d)–5(f) carry different spiral numbers of M = 2, 3, and 4 with the same radial phase difference of 2 × 2π (m = 2). The corresponding intensity and phase distributions of the ultrashort vortices generated by these plates are presented in the second, third, fifth, and sixth columns. The ultrashort vortices were all obtained at t = 0, being the same time as the theoretical and simulated results.

 

Fig. 5. Experimental results of the ultrashort vortices generated by different multispirals. Structures of the multispirals with modes of (M, m) = (a) (2, 1); (b) (3, 1); and (c) (5, 1) with the corresponding intensity and phase distributions in the second and third columns; and (M, m) = (d) (2, 2); (e) (3, 2); and (f) (4, 2) with the corresponding intensity and phase distributions in the fifth and sixth columns.

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All the intensities in Fig. 5 exhibit doughnut-shaped distributions. Although some intensity distributions deviated from circular symmetry, the central areas of the vortex cores were still surrounded by doughnut-like intensity profiles. However, non-uniform phase distributions appeared around the vortex cores, especially in the case of high-order vortices of l = 6 and 8. The phase variations along the closed path on the doughnut-shaped profile had values of 2πl, corresponding to the topological charges defined by the closed-loop integrals $l = \frac{1}{{2\pi }}\oint {{\nabla _{\boldsymbol{r}}}\varphi (\boldsymbol{r} )d\boldsymbol{r}}$ [51], where l = 2, 3, 5, 4, 6, and 8 in Figs. 5(a)–5(f), respectively. The phase variations or topological charges of the generated vortices were consistent with the phase modulation designs of the corresponding multispirals. The asymmetrical and non-uniform intensity and phase distributions mainly resulted from the inevitable experimental inaccuracies produced when measuring and fabricating the pinhole plates. It is well known that high-order vortices are unstable and easily split into single-charge vortices around the vortex core owing to experimental perturbations [52]. Additionally, the optical vortices generated by the spirals also included other topological orders at the vortex cores, whereas the main mode with the topological charge of l dominated, and the additional vortices had smaller weights in the compound modes [38,40]. These factors led to the formation of disturbing singularities in the cores, resulting in asymmetrical distributions. Despite these undesirable singularities, the experimental results on the generated ultrashort vortices were essentially consistent with the theoretical predictions and demonstrated that the multispiral could effectively and flexibly generate ultrashort vortex pulses.

5. Discussion and conclusion

It would be interesting to notice that the diffraction of the laser pulse by the small circular pinhole on the plate will affect the parameters of the spatial and temporal parameters of the generated vortices. The wavefield diffracted by a pinhole can be regarded as the Airy disk, and it has a much larger size than the doughnut vortex size on the observation plane. For waves diffracted by any two pinholes on the diameter of a spiral, such an effect enables the waves to well overlap for superposition. When the wavelength is λ₀= 800 nm, at points of the doughnut ring of maximum intensity, the waves from the two pinholes are in-phase and constructively superposed. With all such pinhole pairs taken into account, the bright doughnut is formed. When the wavelength λ is different from λ₀= 800 nm, at points on the same doughnut ring, the light waves from the two pinholes are not in-phase, but they have a phase delay depending on the wavelength. Thus, it can be deduced that the spectra of the wavefield at a point in the observation plane acquire wavelength-dependent phase delays; this effect leads to the temporal chirp of the vortex pulse and to the broadened pulse with larger durations, as shown in Fig. 4(d). Additionally, for wavelength λ larger and smaller than λ₀= 800 nm, the radii of doughnut rings of maximum intensity are different from that of wavelength 800 nm. This indicates that the light fields of different wavelengths have doughnut rings with different sizes, resulting in the superposed field being broadened in the doughnut width. This can be regarded as the spatial chirping of the output pulses.

As an important parameter crucial for practical applications, the power-transmitting efficiency of the multispiral plate should be estimated. It can be calculated simply by taking the ratio of the light-transmitting area to the effective illuminating area, while the former is the total area of all the pinholes, and the latter is taken as the area of the circle enclosing the outmost point of the multispiral. Thus, the power-transmitting efficiency η_eff is written as ${\eta _{\textrm{eff}}} = MN\rho _{\textrm{hole}}^2/{({\rho _{\textrm{hole}}} + {\rho _N})^2}$, with ρ_hole the radius of the pinhole and ρ_N the maximum radius of the mutilspiral. Taking the typical multispiral plate with the spiral number M = 3 as the example, the transmitted efficiency is calculated to be η_eff =4.93%.

It is noted that in our previous work, the structure of the spiral array was proposed to modulate the ultrashort pulses of high order vortices, where the spiral array was composed of identical segmented spirals, exhibiting a quasi-circle arrangement in a single loop. The rotationally interleaved multispirals for the generation of the ultrashort vortices in this work is an advance over previous work. By interleaving multiple complete spirals with azimuthal rotation, the structure of the multispirals possesses multi-fold rotational symmetries, and it is demonstrated to be able to generate vortex fields with greatly improved circular symmetry and uniform distributions. Moreover, the multispirals promoted significantly the transmitted efficiency of the pinhole plate with multiple times increases, which could provide a foundation for exploring a broader range of applications.

In conclusion, we propose a rotationally interleaved multispiral to generate optical vortices in ultrashort laser pulses. Theoretically, we systematically and comprehensively analyzed multispirals that modulated ultrashort pulses and formed ultrashort vortices in a deconstructive process. Based on the numerical simulations, we further discussed and compared the vortex fields generated by different spiral structures and found that the fields formed by the multispirals had greatly improved distributions with circular symmetry and uniformity. Moreover, we experimentally demonstrated that multispirals could effectively modulate ultrashort pulses and flexibly produce vortices with various topological charges. The results and analyses demonstrated that the structures of the multispirals allowed them to generate ultrashort vortex pulses, not only enabling the azimuthal phase modulations to form helical phase front, but also contributing to vortex fields, especially high-order vortex fields, with circularly symmetric distributions. The study of rotationally interleaved multispirals in the ultrashort vortex field provided a significant view for understanding and manipulating vortex pulses. The ability to control vortex fields with specific topological charges or OAM states is significant for potential applications and extensive explorations, including other OAM-based systems, such as transverse OAMs, toroidal vortices, and Möbius strips. Although this study was performed with ultrashort optical pulses, it is not confined to the optical regime and can be used as reference for research on electron, plasmonic, neutron, and other matter vortices.