## Abstract

The spatiotemporal optical vortex (STOV) is unique, owing to its phase singularity in the space–time domain, and it can carry transverse orbital angular momentum (OAM). Diffraction is a fundamental wave phenomenon that is well known for conventional light; however, studies on the diffraction of light with transverse OAM are limited. Furthermore, methods that enable the fast detection of STOVs are lacking. Here, we theoretically and experimentally study the diffraction behaviors of STOVs, which are different from those of conventional light. The diffraction patterns of STOV pulses that are diffracted by a grating exhibit multilobe structures with a gap number that corresponds to the topological charge. The diffraction rules of STOVs are also revealed. An approach for the fast detection of STOVs is provided using their special diffraction properties. This method has potential applications in fields that require fast STOV recognition, such as STOV-based optical communications.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

An intrinsic property of optical vortex beams is the orbital angular momentum (OAM) carried by photons. The OAM orientations can be parallel or orthogonal to the propagation direction of the light beam and are classified as longitudinal and transverse OAM, respectively. A conventional spatial optical vortex beam with a phase singularity in the spatial plane that possesses longitudinal OAM has been demonstrated and widely applied [1–6]. A spatiotemporal optical vortex (STOV) is a novel optical vortex with a phase singularity in the space-time domain, and it holds transverse OAM [7–9]. Recently, STOVs were observed in experiments [10] and could be generated using a ${{4}}f$ pulse shaper system [11,12]. Furthermore, significant properties of STOVs were analyzed, such as the conservation of transverse OAM in nonlinear optical processes, second harmonic generation [13,14], the generation, propagation, and polarized properties of STOVs [11,15–20], the angular momenta and spin-orbit interaction of STOVs [21,22], and transverse shifts and time delays of STOVs that are reflected and refracted at a planar interface [23]. Moreover, technologies required for the measurement of STOVs have also been presented, such as transient-grating single-shot supercontinuum spectral interferometry [11,24] and interference methods [12]. Transverse spin angular momentum has enabled important applications [25–27]. Since the magnitude of transverse OAM can have a controllable larger value, the STOV beams will open a new area of vortex beam implementations and have unique and wide applications. Understanding the physical properties of STOVs, such as diffraction, is meaningful for theoretical studies as well as for the practical application of STOVs.

Diffraction is a fundamental phenomenon of a light wave and is well known for a conventional polychrome light beam. However, there are few studies on the diffraction properties of STOV light. For a conventional light beam without spatiotemporal coupling, the spectra are diffracted into one continuous line. However, for a STOV light beam with a phase singularity in the space–time domain, energy is coupled between the space and time domains, which may affect the spectral intensity distribution in the wave packet of a STOV. Transverse OAM makes fundamental optical phenomena such as diffraction unique in contrast to conventional light beams. Here, we theoretically and experimentally analyze the diffraction properties of STOV pulses generated using a ${{4}}f$ pulse shaper system. Interestingly, the diffraction patterns have multilobe structures. For a STOV with topological charge $l = {{n}}$, the diffraction pattern has ${{n}} + {{1}}$ lobes. Additionally, there are n gaps in the diffraction structure, and each gap corresponds to a ${{2}}\pi$ phase winding or topological charge $l = {{1}}$. Using this diffraction property, STOV pulses with different topological charges can be directly and rapidly recognized using one camera and a grating.

The STOV pulses are generated using a conventional ${{4}}f$ pulse shaper system (see Fig. 1), where the distances between the gratings (1200 line/mm, blaze wavelength 750 nm) and cylindrical lenses ($f = {{300}}\;{\rm{mm}}$) and those between the phase mask and cylindrical lenses are the same, and they are equal to the focal length of the cylindrical lens. The output pulses after the second grating are focused using a spherical lens ($f = {{1}}\;{\rm{m}}$), the STOV pulses are obtained in the focal plane, and the STOV generator is the same as that used in [11]. A grating (the third grating) with the same optical parameters as those used in the STOV generator is placed in the focal plane of the spherical lens to diffract the generated STOV pulses. Another cylindrical lens ($f = {{300}}\;{\rm{mm}}$) is positioned 300 mm after the third grating, and a camera is placed in the focal plane of this cylindrical lens, which is marked as the observed plane, to measure the diffraction patterns of the STOVs. Femtosecond pulses with broad spectral bandwidth from a mode-locked Ti:sapphire laser are used as input pulses for the ${{4}}f$ pulse shaper system. The spectra of the input pulses can comprise multiple spectral bandwidths that are tailored using different spectral filters.

The intensity profiles and diffraction patterns of the ideal STOV pulses in the $x \!- \!y$ plane are theoretically analyzed here. For simplicity, the field of the pulse generated from the mode-locked Ti:sapphire laser is described as a Gaussian pulse, which has Gaussian profiles in both the spatial and temporal domains with waist radii of $a$ and $b$ respectively. The field can be written as

The Fourier-transform-limited pulse duration of the Gaussian pulse in the calculation is approximately 200 fs. Then, the wave packet of an ideal STOV pulse with topological charge $l$ ($l\; \ne \;{{0}}$) can be expressed as [11]

When $l = {{0}}$, the STOV pulse turns into a Gaussian pulse in both the space and time domains. The field of the STOV in the frequency domain can be obtained by the Fourier transform of ${E_1}({x_1},{y_1},t)$ as

After the STOV is diffracted by a grating, the first-order field immediately after the grating can be expressed as [28]

where $\gamma = 2\pi /{w_c}d\cos ({\theta _{d0}})$, $\beta = \cos ({\theta _{i0}})/\cos ({\theta _{d0}})$, and ${\theta _{d0}}$ and ${\theta _{i0}}$ are the diffracted angle and incident angle of the central frequency (${w_c}$) ray, respectively. ${w_c} = 2\pi c/{\lambda _c}$, where $c$ is the velocity of light in a vacuum, and ${\lambda _c}$ is the central wavelength. In both the simulation and experiment, ${\lambda _c}$ is set at 760 nm. Then, ${E_2}({x_2},{y_2},w)$ propagates to the cylindrical lens over a distance of $z = {{300}}\;{\rm{mm}}$. The field before the cylindrical lens can be written asThen, the electrical field continuously propagates to the observed plane (OP), and the field in the OP is marked as ${E_5}({x_5},{y_5},w)$. The spatial-intensity profile of the STOV pulse in the focal plane of the spherical lens and the diffraction pattern of the STOV in the OP can be obtained as

To show the universality of the diffraction properties of STOV pulses, the intensity profiles and diffraction patterns of STOVs with $l = {{\pm 1}}$ and $l = {{\pm 4}}$ are shown as an example, and the results are shown in Fig. 2. Diffraction patterns of STOVs with $l = {{\pm 1}}$, ${{\pm 2}}$, ${{\pm 3}}$, ${{\pm 4}}$ are included in Supplement 1. The intensity profiles of ideal STOVs with $l = {{\pm 1}}$ and $l = {{\pm 4}}$ in the $x\! - \!y$ plane are shown in the first row of Fig. 2, and the corresponding diffraction patterns are shown in the third row. The top row is marked as the first row. The intensity profiles (diffraction patterns) in the $x\! - \!y$ plane are the sums of $I(x,y,w)$ along the $w$-dimension. As observed in row 1 of Fig. 2, the intensity profiles of the STOVs do not have a Gaussian spatial distribution. Instead, they appear as two circles that overlap each other, and the gap between the two circles increases with an increasing topological charge. As shown in row 3 of Fig. 2, for ideal STOV pulses with $l = {{n}}$, the diffraction patterns exhibit special multilobe structures with ${{n}} + {{1}}$ lobes and $n$ gaps. The dislocated lobes are parallel to each other, and the two lobes at each end possess higher energies, which become more pronounced for higher-order STOVs. The gap number is the same as the topological charge; hence, one gap corresponds to one topological charge or a ${{2}}\pi$ phase winding of the STOV. Moreover, the helicity of the STOV can be obtained clearly from the orientation of the multilobe structure.

The measured results of the spatial-intensity profiles of the STOVs in the focal plane of the spherical lens and their corresponding diffraction patterns are shown in rows 2 and 4 of Fig. 2, respectively. For the spatial-intensity profiles of STOVs, the features of two overlapped circles with different separated distances are clearly shown. The diffraction patterns of STOVs with different topological charges also show multilobe structures. The diffraction rule for STOVs with different topological charges is consistent with the simulation result. The helicity of the STOV can also be identified from the orientation of the multilobe structure. All the obtained key structures and properties from the experimental results, such as the multilobe structures, gap numbers, and the two energetic head lobes, are in good agreement with the simulation results. Note that, the generation of STOVs is based on space-to-time mapping, where the phase mask is inserted in the Fourier plane ($x\! - \!y$ plane) to realize the STOV in the space-time plane ($y\! - \!t$ plane). Then, the generation of the diffraction pattern of the STOV can be considered as time-to-space mapping, where the phase circulation in the space-time domain ($y\! - \!t$ plane) is mapped to the intensity distribution in the space domain ($x\! - \!y$ plane). Hence, the diffraction is a decoding process, which means that the value of the topological charge can be obtained directly without any retrieval algorithm, thereby enabling the fast detection of the topological charges of STOVs. The fast recognition of STOVs may enable various applications of STOV pulses, such as STOV-based optical communications. The proposed method for STOV detection can be called the diffraction method (D-method).

The formation of the unique diffraction pattern of a STOV beam can be attributed to the transverse OAM-affected spectral intensity distribution (TOASID) in the wave packet of the STOV. The simulated 3D isosurface plots of the intensity profiles of the STOVs with $l = {{\pm 1}}$, ${{\pm 4}}$ in the $x\! - \!y\! - \!w$ domain are shown in the first column of Fig. 3. The isosurfaces are plotted at 15% of the maximum intensity. Then, the 3D isosurfaces are viewed in the three main planes; namely $x\! - \!y$, $y\! - \!w$, and $x\! - \!w$, as shown in columns 2, 3, and 4, respectively. As observed in column 1, the spectral intensity distributions of STOVs exhibit multilobe structures. When 3D STOVs are projected onto the $x\! - \!y$ plane, the lobes overlap each other, and it results in the generation of STOV intensity profiles, as shown in the first row of Fig. 2. Columns 3 and 4 show that lobes with different spectra actually separated in the $y$-direction, and they are parallel to each other. Additionally, the two head lobes of the structures possess more energies. After diffraction, different STOV lobes are diffracted to different locations in the $x$-direction and are dislocated in the $y$-direction; then, unique diffraction patterns are obtained.

From the measured results shown in Fig. 2, it can be observed that weak trails occur on the two energetic heads in both the intensity profiles of the STOVs and the diffraction patterns. These properties are shown again in rows 2 and 4 of Fig. 4, which are marked with red circles. The trails occur primarily because the STOV pulses generated from the ${{4}}f$ pulse shaper are not ideal since the phase masks in the Fourier plane do not match the beam profile in this plane [11]. As demonstrated in the results, STOV pulses generated from the ${{4}}f$ pulse shaper may have special spatiotemporal intensity profiles, such as elliptic intensity profiles with nonuniform energy distribution and multihole structures for higher-order STOVs; thus, they deviate from ideal STOV pulses [11,12,15], which can be observed in Fig. S1 in Supplement 1. This phenomenon is verified using the STOV pulses generated from the ${{4}}f$ pulse shaper system in the simulation; the method is the same as that used in our previous study [15]. The simulated intensity profiles of the STOVs and the corresponding diffraction patterns are shown in rows 1 and 3 of Fig. 4, respectively. Weak trails occur in both the simulated intensity profiles and the corresponding simulated diffraction patterns of the STOVs. The simulated results are in good agreement with the experimental results. It can also be observed that the two head lobes in the diffraction patterns exhibit a slight curve, which are indicated with two red dashed curves in column 4, row 3 in Fig. 4, which is another feature of a nonideal STOV. This feature becomes more apparent in STOVs with broader spectra, as shown in Fig. 5.

We verify if the diffraction rule is still suitable for STOVs with broader spectral bandwidths, which correspond to shorter pulse durations. Then, STOV pulses with a full width at half maximum spectral bandwidth of approximately 20 nm are diffracted using the same experimental setup. The diffraction patterns of the STOVs with $l = {{\pm 1}}$, ${{\pm 4}}$ are shown in Fig. 5. The key features of the STOV diffraction patterns are clearly demonstrated. Moreover, the two head lobes contain wide spectra and spread to a larger area in the $x$-direction. They also exhibit a slight curve, similar to the simulated results shown in row 3 of Fig. 4. The trends observed in the generated lobes and gaps are consistent with the experimental results obtained with a narrow spectral bandwidth. Furthermore, STOVs with different topological charges can be identified clearly, which shows that the proposed D-method is suitable for the fast detection of STOVs with different spectral bandwidths.

In summary, the diffraction properties of light with transverse OAM were demonstrated theoretically and experimentally. The diffraction rules of STOVs were presented. The unique features of the diffraction patterns of STOV pulses originate from the TOASID effect. The diffraction properties of STOV beams assist with understanding the physical properties of STOVs. This may enable special applications, such as the fast detection of the topological charges of STOVs, which is required for future applications of STOVs, such as STOV-based optical communications.

## Funding

Shanghai Municipal Natural Science Foundation of China (20ZR1464500); Chinese Academy of Sciences (Strategic Priority Research Program (XDB160106)); National Natural Science Foundation of China (61521093, 61527821, 61905257, U1930115).

## 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.

## Supplemental document

See Supplement 1 for supporting content.

## REFERENCES

**1. **L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A **45**, 8185 (1992). [CrossRef]

**2. **H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. **75**, 826 (1995). [CrossRef]

**3. **L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science **292**, 912 (2001). [CrossRef]

**4. **L. Yan, P. Gregg, E. Karimi, A. Rubano, L. Marrucci, R. Boyd, and S. Ramachandran, Optica **2**, 900 (2015). [CrossRef]

**5. **J. Chen, C. Wan, and Q. Zhan, Adv. Photon. **3**, 064001 (2021). [CrossRef]

**6. **S. Huang, P. Wang, X. Shen, J. Liu, and R. Li, Appl. Phys. Lett. **120**, 061102 (2022). [CrossRef]

**7. **A. P. Sukhorukov and V. V. Yangirova, Proc. SPIE **5949**, 594906 (2005). [CrossRef]

**8. **M. Dallaire, N. McCarthy, and P. Piché, Opt. Express **17**, 18148 (2009). [CrossRef]

**9. **K. Y. Bliokh and F. Nori, Phys. Rev. A **86**, 033824 (2012). [CrossRef]

**10. **N. Jhajj, I. Larkin, E. W. Rosenthal, S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, Phys. Rev. X **6**, 031037 (2016). [CrossRef]

**11. **S. W. Hancock, S. Zahedpour, A. Goffin, and H. M. Milchberg, Optica **6**, 1547 (2019). [CrossRef]

**12. **A. Chong, C. Wan, J. Chen, and Q. Zhan, Nat. Photonics **14**, 350 (2020). [CrossRef]

**13. **S. W. Hancock, S. Zahedpour, and H. M. Milchberg, Optica **8**, 594 (2021). [CrossRef]

**14. **G. Gui, N. J. Brooks, H. C. Kapteyn, M. M. Murnane, and C.-T. Liao, Nat. Photonics **15**, 608 (2021). [CrossRef]

**15. **S. Huang, P. Wang, X. Shen, and J. Liu, Opt. Express **29**, 26995 (2021). [CrossRef]

**16. **S. W. Hancock, S. Zahedpour, and H. M. Milchberg, Phys. Rev. Lett. **127**, 193901 (2021). [CrossRef]

**17. **Q. Cao, J. Chen, K. Lu, C. Wan, A. Chong, and Q. Zhan, Photon. Res. **9**, 2261 (2021). [CrossRef]

**18. **J. Chen, C. Wan, A. Chong, and Q. Zhan, Nanophotonics **10**, 4489 (2021). [CrossRef]

**19. **J. Chen, C. Wan, A. Chong, and Q. Zhan, Opt. Express **28**, 18472 (2020). [CrossRef]

**20. **C. Wan, J. Chen, A. Chong, and Q. Zhan, Sci. Bull. **65**, 1334 (2020). [CrossRef]

**21. **K. Y. Bliokh, Phys. Rev. Lett. **126**, 243601 (2021). [CrossRef]

**22. **J. Chen, L. Yu, C. Wan, and Q. Zhan, ACS Photon. **9**, 793 (2022). [CrossRef]

**23. **M. Mazanov, D. Sugic, M. A. Alonso, F. Nori, and K. Y. Bliokh, Nanophotonics **11**, 737 (2022). [CrossRef]

**24. **S. W. Hancock, S. Zahedpour, and H. M. Milchberg, Opt. Lett. **46**, 1013 (2021). [CrossRef]

**25. **A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, Phys. Rev. Lett. **103**, 100401 (2009). [CrossRef]

**26. **A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, Nat. Photonics **9**, 789 (2015). [CrossRef]

**27. **P. Banzer, M. Neugebauer, A. Aiello, C. Marquardt, N. Lindlein, T. Bauer, and G. Leuchs, J. Eur. Opt. Soc. Rap. Publ. **8**, 13032 (2013). [CrossRef]

**28. **O. E. Martinez, J. Opt. Soc. Am. B **3**, 929 (1986). [CrossRef]