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Abstract
Spatiotemporal optical vortex (STOV) light is a new type of vortex light with transverse orbital angular momentum (OAM) which is different from conventional spatial vortex light. Understanding the properties of STOV are meaningful before STOV are applied. We present a theoretical study on the generation and propagation of spatiotemporal vortices step by step based on diffraction theory. The properties of the output pulses with different topological charges generated using 4 f pulse shaper in both the near-field and the far-field are analyzed. Using spiral phase mask, the intensity profiles of the output pulses immediately after the 4 f pulse shaper are of multi-lobe structures. With energies circulating around the phase singularity in the space-time plane, energy coupling occurs between the spatial and temporal domains in the wave packets during propagation, then the intensity profiles evolve into multi-hole shapes, and the holes tend to be merged for higher order STOV. The conservation of OAM in the space-time domain is shown clearly. The profiles of the output pulses in the near-field form donut rectangle shapes using π-step mask, and in the far-field, they split into a multi-lobe structure. The rules of the generation and evolution of STOV are revealed. The results demonstrate the physical properties of the STOV and the generation and propagation processes directly and clearly. It provides a guidance on the application of STOV.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical vortex with phase singularity and intensity null at the center of a laser beam has been proved to possess orbital angular momentum (OAM) [1]. It has wide applications and has attracted mass attentions [2–5]. Traditionally, the optical vortex is in the spatial region with OAM parallel or antiparallel to the light propagation direction, namely longitudinal OAM. Spatiotemporal optical vortex (STOV) light as a new type of vortex light, of which the OAM is transverse to the light propagation direction, namely transverse OAM. And the phase singularity and the amplitude null are in the space-time plane. The concept of STOV was proposed in 2005 [6], and spatiotemporal Bessel beams were obtained and analyzed about ten years ago [7]. In 2012, the OAM of STOV was analyzed [8]. Recently, STOV with transverse OAM was observed in femtosecond filamentaion where arrested self-focusing collapse occurred [9]. After that, the generations of STOV using 4 f pulse shaper system were proposed [10,11]. Moreover, the conservation of OAM of STOV was successfully analyzed through frequency-double of the fundamental STOV light [12]. The STOV lights open a new way for understanding the properties of light and pay a novel way for manipulating photons and may have important or special potential applications.
At present, the STOV pulses were mainly generated using 4 f pulse shaper [10–12]. STOV pulses with topological charge l = 1, 8 and l = 1, 2 were generated and analyzed in Refs. [10] and [11], respectively. Recently, the mode structure of STOV pulse was analyzed and the evolution of the first order STOV is shown in Ref. [13]. However, the evolution results of higher order STOV pulses were not shown. The properties and the rules of the generation and evolution of STOV pulses with different topological charges still need to be explored clearly and deeply. Since phase singularity is in the space-time plane, energy coupling in this plane also, then the properties of STOV are different from those of the conventional spatial vortex lights. There are some interesting questions, such as, is it possible to generate a higher order STOV with only one intensity null the same as that in conventional spatial vortex [14] based on current experimental conditions where 4 f pulse shaper, spiral phase mask and π-step phase mask are usually used? How does the multi-hole structure of higher topological STOV in the far-field form?
Here, we demonstrate a theoretical study on the generation and evolution of the STOVs with different topological charges, which are generated using a typical 4 f pulse shaper [15], step by step based on diffraction theory [16]. The rules of the generation and propagation of the STOV pulses carrying OAM with different topological charges are revealed clearly. The simulation results show that, by using spiral phase masks, STOV with topological charge l = n, n + 1 lobes can be generated in the near-field after the pulse shaper. In far-field, the beam profile of the generated STOV owns n holes, with the increasing of topological charge, the holes are merged, and the beam profile tends to split into two parts. For π-step phase mask, the beam profiles of the generated l = ±1 STOV in the near-field shown special donut rectangle shapes instead of ring shapes, and then they evolve into two lobe structure profiles in the far-field. The conservation of OAM in space-time domain and the energy circulation in the space-time plane (y-t plane) during propagation are shown clearly. High order STOV pulses, such as l = 2 STOV, are generated in the near-field also, which have not been reported. Based on the simulation, a π−0-π phase mask is hopeful to be used to generate this kind of STOV pulses.
The STOV pulses generated using 4 f pulse shaper are theoretically studied step by step, which demonstrates the generation and propagation of STOV directly and clearly. It helps us understand the physical properties of the generated STOV and guides the application of STOV lights.
2. Principle
The spatiotemporal profiles of the generated STOV pulses using pulse shaper are calculated step by step. The schematic of the 4 f pulse shaper system for the generation of STOV is shown in Fig. 1. For simplicity, Fourier-transform-limited pulses with Gaussian spatial profile and Gaussian spectral distribution are selected as the input ultrafast field, and the wave packet can be expressed as
where ${w_c}$ is the carrier frequency, ${w_c} = 2\pi c/{\lambda _c}$, c is vacuum light velocity, ${\lambda _c}$ is the central wavelength. a, b are the waist radii of the Gaussian profile in the space domain (x-y plane) and spectrum domain, respectively. Since the coefficients in the function and the below ones will not affect the results of the profiles of the STOV pulses, so they are omitted here. After it is diffracted by the first grating, the first order field immediately after the grating can be written as [17,18] where $\beta = \cos ({\theta _{i0}})/\cos ({\theta _{d0}})$, $\gamma = 2\pi /{w_c}d\cos ({\theta _{d0}})$, ${\theta _{i0}}$ and ${\theta _{d0}}$ are the incident angle and diffracted angle of the central wavelength ray with respect to the grating normal, respectively. d is the periodicity of the grating. All the x, y axis used here are normal to the beam propagation directions of the central wavelength. The x axis are in the dispersion plane of the grating, the directions of y axis are parallel to the direction of the grating grooves, namely normal to the dispersion plane.Based on the angular spectrum propagation diffraction theory, the field ${E_2}({x_2},{y_2},w)$ after propagating distance of z in free space can be calculated as [16,19]
where $H({f_x},{f_y},w) = \exp [ikz - i\pi \lambda z({f_x}^2 + {f_y}^2)]$ is the transfer function in free space. k is the wave vector. FT and IFT denote the spatial Fourier transform and inverse Fourier transform. fx, fy are spatial frequencies in x and y directions, respectively. In the 4 f pulse shaper system, the distance z between grating and cylindrical lens is equal to the focal length of the cylindrical lens f.The field after cylindrical lens can be written as [20]
then ${E_4}({x_4},{y_4},w)$ propagates to the Fourier plane with distance of f. The field before the Fourier plane is ${E_5}({x_5},{y_5},w)$. Phase mask is inserted in the Fourier plane. After the phase mask the field can be expressed as where $T(x,y)$ is the transmittance function of the phase mask. For the spiral phase mask $T(x,y){\rm{ = }}\exp ( - il\phi )$, l is the topological charge value, $\phi$ is the azimuthal angle and $\phi {\rm{ = ta}}{{\rm{n}}^{{\rm{ - }}1}}{\rm{(}}y/x)$.The propagation after the phase mask is the same as discussed above. The function for calculating the field right after the second grating is the same as that of the first grating except that the incident angle and diffracted angle here are equal to the diffracted angle and incident angle onto the first grating, respectively. The spatiotemporal profiles of the STOV in the far-field of the pulse shaper is calculated at the focal plane of a focus lens with a focal length of 1 m.
The field in the time domain is the inverse Fourier transform of the field in the spectrum domain, then the intensity in the time domain can be calculated as
The simulation results are demonstrated in the next section.
3. Simulation results and discussion
In the simulations, input pulse fields with the Fourier-transform-limited pulse duration of about 270 fs are used, the central wavelength is 800 nm, the gratings are of 1200 lines per mm, the focal lengths of the cylindrical focus lenses are 300 mm. The distances between the gratings and cylindrical lenses are 300 mm. Both the two waist radii of the Gaussian spatial profile in the x and y directions are about 0.5 mm. And spiral phase mask and π-step phase mask are used to generate STOV pulses with different topological charges. The schematic of the 4 f pulse shaper system for the generation of STOV is shown in Fig. 1.
3.1 STOV generation using a spiral phase mask
The spatiotemporal intensity profiles of the generated pulses with different topological charges in the near-field and far-field of the output of the pulse shaper are shown here in Fig. 2, respectively. Spiral phase masks with l = 1, 2, 3, 4, 6 (2π, 4π, 6π, 8π, 12π, respectively) are used in the simulations. As we can see that, the isosurfaces of the intensity distributions in the near-field are of multi-lobe structures oriented in the diagonal direction with l + 1 lobes for l spiral phase mask. Isosurfaces here and the following ones all at fifteen percent of the maximum intensity. In the far-field, for l = 1, there is one intensity null in the middle of the intensity profile which shows a little ellipse, and a 2π phase winding of the field is shown clearly at the last of column 1 in Fig. 2. As it is shown that the energy distribution around the singularity is not very uniform for l = 1 STOV, it is a little weak at the place where there exists a gap in the near-field, as shown in the third photo of column 1 in Fig. 2. As it is known that, for conventional spatial vortex light with l higher than 1, such as 2 or 3, there is an intensity null with larger diameter in the middle of the profile [14]. It is interesting that, for the generated STOV with l higher than 1, there are l number of intensity nulls in the profile instead of a larger hole. And each hole corresponding to 2π phase winding, namely l = 1. At first sight, it seems a higher order STOV evolves into several l = 1 STOV lights, which is possible to occur for higher order spatial vortex lights [21]. But as shown in the near-field, immediately after the second grating, multi-lobe structures are produced instead of forming a higher order STOV with one intensity null. The multi-lobe structure evolves into the multi-hole structure. With the increasing of topological charge, the holes are merged and the profile tends to form a two lobe shape in the far field. Since the photons own transverse OAM, energies will flow around the phase singularity in the space-time plane, energy coupling occur between the spatial and temporal domains in the wave packets during propagation, and with phase evolution during diffraction, the multi-lobe structure evolves into multi-hole structure, which will be shown in Fig. 3. The formation of multi-hole structure is mainly due to that the spiral phase mask is not an ideal phase mask for generating STOV using a 4 f pulse shaper system, namely the beam profile in the Fourier plane of the pulse shaper does not match the profile of the phase mask [10]. The effect is not obvious for low order STOV, but it becomes prominent for high order STOV. It is hard to generate a l > 1 STOV with only one intensity null at the center based on current experimental conditions with 4 f pulse shaper, spiral phase mask and π-step phase mask.
Fig. 1. Schematic of the 4 f pulse shaper system for the generation of STOV. The two inset photos in the middle are π-step phase mask (left) and spiral phase mask (right), respectively. The thick arrows represent the propagation directions of the central wavelength and the thin arrows represent the x, y axis directions in different calculated planes.
Fig. 2. Outputs of pulse shaper, the five columns are the results when spiral phase mask with l = 1, 2, 3, 4, 6 are used respectively. Rows 1 and 2 are the isosurfaces of the intensity distributions in the y-t plane in the near-field and far-field when the five spiral phase masks are inserted in the Fourier-plane of the pulse shaper, respectively. Row 3 shows the far-field intensities profiles of the generated STOV pulses on the y-t planes. Row 4 shows the phase windings on the y-t planes.
Fig. 3. Evolutions of the spatiotemporal intensity profiles of the generated STOV pulses during propagation when spiral phase masks are used. Propagation distance are 0 m, 0.5 m, 0.6 m, 0.8 m, 1 m after a 1 m focus lens. Rows 1 and 2 are the results of l = 1 and l = −1 respectively. Rows 3 and 4 are the results of l = 2 and l = −2 respectively. Near-field is immediately after the pulse shaper (z = 0f) and far-field is at the focus of the lens (z = 1f).
Although the intensity profiles of higher order STOVs in the far-field are merged at the center and multi-hole structures are obtained, and even the profiles split into two lobes, the phase circulations are still maintained for high order STOVs, which originates from the conservation of OAM. The total OAM are conserved and shown clearly here. The profiles of the corresponding negative topological charges in the near- and far-fields are nearly the same with those of the positive topological charge, with their structures located at another diagonal direction. Such as the profiles of l = −1, −2 shown in row 2 and row 4 in Fig. 3, respectively.
The evolutions of the STOV with l = ±1 and l = ±2 during propagation are shown in Fig. 3, respectively. The output pulses of the pulse shaper propagate from near-field to far-field through a 1 m focus lens. The spatiotemporal intensity profiles are calculated at several places after the lens, distances from the lens are z = 0, 0.5f, 0.6f, 0.8f, 1f, respectively. z = 0, 1f represent the near- and far-fields respectively. The multi-lobe structure evolves into an annular shape during propagation is shown clearly. Energies flow around the phase singularity in the space-time plane during propagation then donut shape and multi-hole shape beams are formed in the far-field. The positions of the holes of l = ±2 STOV are rotated during propagation also. The evolution of the profile of the spatiotemporal coupling pulse (STCP) depends on the evolution of the phase of the STCP. Since there is not nonlinear optical process involved in the generation and propagation of the STCP, the phase of the STCP evolves during propagation. In the near-filed, the phase of the STCP is not a spiral phase, so the STOV does not occur, with the STCP propagates to the far-field, the phase of the STCP evolves into a spiral phase then STOV is formed. The profiles of the STCP at the places before the formation of STOV depend on the phases of the STCP at that moment. The spatiotemporal intensity profiles of STOV with positive and negative topological charges are nearly the same with the structure orientations located at different diagonal directions. The evolutions show the formations of donut shape and multi-hole shape of STOVs directly and clearly.
3.2 STOV generation using a π-step phase mask
As we can see that, using a spiral phase mask in the Fourier plane, the STOV demonstrate a multi-lobe structure in the near-field and a multi-hole shape (considering low order topological charge situation) in the far field. The trend is reversed when π-step phase mask is used. The output pulse profiles of the pulse shaper when the π-step phase mask is rotated to different angles are shown in Fig. 4. Rows 1 and 3 are the profiles in the near- and far-field when the π-step phase mask is rotated clockwise (positive angle), rows 2 and 4 are the profiles in the near-field and far-field when the phase mask is rotated counterclockwise (negative angle), respectively. The angles are with respect to the dispersion direction of the grating. In order to obtain a donut shape STOV, the π-step phase mask needs to be rotated to an appropriate angle. In the simulation we find that, without rotating the π-step phase mask, two separate lobes parallel to each other and parallel to the time axis are generated. When the rotated angle is about 25°, a donut shape STOV is generated. The profile in the near-field, instead of forming a ring shape, is a rectangle shape with an intensity null at the center. It also results from that the phase of the rotated π-step phase mask does not match the beam profile in the Fourier plane of the pulse shaper [10]. In the far-field, two inclined lobes are obtained. The spatiotemporal intensity profiles of positive and negative angles are nearly the same just with the profiles located along different diagonal orientations.
Fig. 4. Spatiotemporal intensity profiles of the output of the pulse shaper in the near-field (row 1 and row 2) and far-field (row 3 and row 4) with the π-step phase mask rotated at different angles. Rows 1 and 3 are rotated clockwise (positive angle), rows 2 and 4 are rotated counterclockwise (negative angle), respectively.
The propagations of the STOV pulses when the phase mask is rotated to ±25° are shown in Fig. 5. The donut rectangle STOVs split into two peg-top lobes gradually during propagation. The evolution of the generated STOV using π-step phase mask shows the energy coupling between the space domain and time domain in the wave packet. The profiles of the STCP generated using π-step phase mask at different places also depend on the phases of the STCP at that moment. The evolution trends are nearly the same for the two STOVs with opposite topological charges.
Fig. 5. Evolutions of the spatiotemporal intensity profiles of the generated STOV pulses during propagation when π-step phase mask is rotated to ±25° respectively. Propagation distance are 0 m, 0.5 m, 0.6 m, 0.8 m, 1 m after a 1 m focus lens. Rows 1 and 2 are the results of +25° (l = −1) and −25° (l = 1) respectively. Near-field is immediately after the pulse shaper (z = 0f) and far-field is at the focus of the lens (z = 1f).
As discussed above, by rotating a π-step phase mask to an appropriate angle, it is possible to generated a STOV with l = 1. Then an interesting question arises. Is it possible to generate a higher order STOV using phase mask in the near-field? Since for some applications, higher order, such as l = 2, donut shape STOVs in the near-field may be need. And till now, donut shape higher order STOVs in the near-field still need to be explored. We simulate the generation of an l = 2 STOV in the near-filed by using a multi-step phase mask, which likes π-step phase mask, but with π-0-π phase distribution. When a π-0-π phase mask is inserted in the Fourier-plane of the pulse shaper, the numerical calculation results are shown in Fig. 6. The width of the 0 phase gap is about 0.375 mm. With changing the width of the 0 phase gap which defined as separation, the energy distribution in the STOV pulse is changed. In order to obtain a STOV pulse with uniform energy distribution, the separation used here is selected. And the phase mask is rotated to −25°. As a comparison, the results of π-step phase mask rotated to −25° (l = 1) is depicted in row 1 in Fig. 6. Row 2 is the results of π-0-π phase mask. As it can be seen that a two-hole structure is obtained in the near-field. The phase distribution in the y-t plane are shown in column 3 of row 2. As it can be seen that there are two 2π phase windings occur, which are shown by two red rings with arrows denote the phase increased directions. And the phase distribution of STOV generated using π-step phase mask shows one 2π phase winding in column 3 of row 1. The profile in the far-field shows a multi-lobe structure. 3 lobes are shown in column 4 of row 2. The intensity profiles in the near- and far- fields in the y-t plane are also shown in column 2 and column 5 of row 2, respectively. The evolution of the generated spatiotemporal coupling pulses during propagation are shown in row 3 in Fig. 6. With the same trends of l = 1 STOV pulses generated using π-step phase mask, the multi-hole structure evolves into multi-lobe shape. It is hopeful to be an l = 2 STOV in the near-field.
Fig. 6. Results of the output of the pulse shaper using π-0-π phase mask (row 2). Columns 1 and 4 are the intensity profiles in the near-field and far-field, and the corresponding intensity distribution in the y-t plane are shown in columns 2 and 5, respectively, the phase distribution in the near-field are shown in column 3. As a comparison, the corresponding results of the STOV (l = 1) generated with π-step phase mask rotated to −25° is depicted in row 1. Row 3 shows the evolution of the generated spatiotemporal coupling pulses. The evolution profiles are calculated when propagation distance are 0 m, 0.5 m, 0.6 m, 0.8 m, 1m after a 1 m focus lens. The insets in the first column of rows 1 and 2 show the phase masks used respectively.
4. Conclusion
The properties of the generation and evolution of STOV, generated using 4 f pulse shaper system, are theoretically studied based on diffraction theory. The rules of the generation and evolution of STOV using both spiral phase makes and π-step phase mask are revealed clearly. Due to the energy coupling in the space domain and time domain in the wave packet, and with diffraction, the STOV pulses show unique behaviors. Such as multi-lobe structure in near-field can evolve into multi-hole structure in the far-field during propagation, vice versa. The simulation shows the generation characteristics of STOV in the pulse shaper and their propagation properties in free space directly and clearly, and shows the conservation of OAM in space-time domain clearly. The model used in the simulation can be used to analyze the affects of each component used in experiments and can be used to design 4 f pulse shaper system for generating STOV in other spectral regions, such as in middle-infrared and far-infrared ranges. It helps us understand the physical properties of the generated STOV more deeply and guides the application of STOV.
Funding
National Natural Science Foundation of China (NSFC) (61527821, 61521093, 61905257, U1930115); Instrument Developing Project (YZ201538); Strategic Priority Research Program (XDB160106) of the Chinese Academy of Sciences (CAS); Shanghai Municipal Science and Technology Major Project (2017SHZDZX02); Shanghai Municipal Natural Science Foundation of China (20ZR1464500).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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