Subwavelength focusing of a spatio-temporal wave packet with transverse orbital angular momentum

Jian Chen; Chenhao Wan; Chenhao Wan; Andy Chong; Andy Chong; Qiwen Zhan; Qiwen Zhan

doi:10.1364/OE.394428

1. Introduction

In the spatial domain, the orbital angular momentum (OAM) of light can be characterized by a spiral wavefront of $\exp ({ - jl\varphi } )$, where φ is the azimuthal angle in the transverse plane and l is an arbitrarily positive or negative integer number called topological charge [1,2]. This is referred to as the longitudinal OAM, since the energy of the resulted optical vortex circulates around the axis along the propagation direction of the beam. Taking advantage of its unique focusing properties and higher degrees of freedom, longitudinal OAM of light has been extensively studied in optical tweezers [3–6], astronomy [7], free-space communications [8,9], super resolution microscopy [10], quantum information processing [11–13], etc.

Several recent theoretical studies also showed that the transverse OAM could be formed through introducing the temporal variations of the phase [14,15]; namely creating the so-called spatio-temporal (ST) vortex, for which the electromagnetic energy circulates around an axis perpendicular to the propagation direction of the beam. Through a nonlinear interaction between an extremely high power pulse and air, optical field with a small fraction of the energy exhibiting the transverse OAM has been reported [16]. Recently we developed a linear method that can experimentally produce ST wave packet with transverse OAM in a controllable manner [17]. Subsequently, the propagation of spatio-temporal optical vortices in free space are demonstrated by using single-shot supercontinuum spectral interferometry to measure the space- and time-resolved envelope of ultrafast laser pulses [18]. ST wave packet affords new opportunities for controlling pulsed optical beams, including the potential for diffraction-free propagation across extended distances and the control of the group velocity of light in a variety of optical media [19,20]. These findings are expected to open many important applications that exploit the interactions of such ST wave packets with matters. The ability to strongly focus or localize such light-matter interaction will be critical for these applications. In this work, we study the strong focusing of ST wave packet through high numerical aperture (NA) objective lens. It is found that the ST vortex would collapse at the focus due to the “spatio-temporal astigmatism” in ST focusing. Based on this observation, using an idea similar to the cylindrical lens mode converter that was developed for the generation of spatial OAM [21], we propose and demonstrate the creation of subwavelength ST vortex with transverse OAM.

2. Collapsing of the focused ST vortex

Without loss of generality, we express the scalar incident ST vortex with topological charge of +1 in the coordinates traveling with the wave packet as

(1)$${E_{ + 1}}({x,y,t} )= ({x + it} )\exp [{ - ({{x^2} + {y^2} + {t^2}} )/{w^2}} ],$$

where w is the waist radius of the Gaussian profile. The Gaussian profile sizes in the transverse spatial domain (x-y plane) and the temporal domain (t) are denoted with the same w to simplify the discussion. The intensity and corresponding phase distributions of the incident ST vortex are projected on three orthogonal planes in the ST domain, as shown in Figs. 1(a) and 1(b), respectively. From them, we can see a donut shape with zero value in the center of the intensity distribution in x-t plane. It is noted that the intensity is normalized by the maximum value of the ST wave packet, and all the following ST beams will be processed in the same way. The phase in the x-t plane changes continuously from -π to π in the clockwise direction, while the binary phase of the blue area and orange area in the t-y plane is -π/2 and π/2, respectively; and the binary phase in the x-y plane is 0 (the green area) and π (the crimson area). In addition, the isosurface of the ST vortex in Fig. 1(a) visualizes this three dimensional wave packet, exhibiting a hole in the center around y-axis, which is consistent with phase singularity on the x-t plane. All the results confirm the incident beam is a ST vortex with topological charge of +1.

Fig. 1. Incident ST vortex and configuration for focusing calculation. (a) Intensity distributions of the incident ST vortex on the three orthogonal planes (i.e. x-t, y-t and x-y planes) in the ST domain; the red isosurface visualizes the three dimensional shape of the ST vortex at its half maximum intensity. (b) The corresponding phase distributions of the ST vortex on the three orthogonal planes. (c) Schematic of focusing an ST vortex with a high NA objective lens. In (a) and (b), the spatial sizes of the ST vortex are normalized to the NA of the lens, meanwhile its temporal width is normalized to the pulse width.

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Subsequently, we use the schematic illustrated in Fig. 1(c) to study the tight focusing of the ST vortex. In the calculation, spatio-temporal coupling is ignored thus each temporal slice of the incident field is focused onto its corresponding temporal slice within the focal volume. The diffracted field on the focal plane could be calculated with the Debye integral as [22–24]

(2)$${E_f}({{r_f},\;\phi ,\;{z_f},t} )= \int_0^\alpha {\int_0^{2\pi } {{E_\Omega }({\theta ,\;\varphi ,t} )\times {e^{ - jk[{{r_f}\sin \theta \cos ({\varphi - \phi } )+ {z_f}\cos \theta } ]}}\sin \theta d\theta d\varphi } } ,$$

where α is the convergence semiangle determined by the NA of the lens, ${r_f} = \sqrt {x_f^2 + y_f^2} $, $\phi = {\tan ^{ - 1}}({{{{y_f}} \mathord{\left/ {\vphantom {{{y_f}} {{x_f}}}} \right.} {{x_f}}}} )$, ${E_\Omega }({\theta ,\;\varphi ,t} )$ is the optical field on the spherical surface Ω. In this case, aplanatic objective lens that obeys the sine condition is used, for which the pupil apodization function is $\sqrt {\cos \theta } $. Then, together with transformation relationship from Cartesian coordinates to spherical coordinates, ${E_\Omega }({\theta ,\;\varphi ,t} )$ can be given by

(3)$${E_\Omega }({\theta ,\varphi ,t} )= ({\sin \theta \cos \varphi + it} )\exp [{ - ({{{\sin }^2}\theta + {t^2}} )/{w^2}} ]\sqrt {\cos \theta } ,$$

with the spatial sizes of the optical field normalized by the NA of the lens and temporal size normalized to the width of the pulse. For our calculations, we will examine the focused ST wave packet at the focal plane, hence z_f=0 in the integral.

Assuming ST vortex is focused by a lens with NA=0.9, and the waist radius of the ST vortex w is set to be 0.5 a.u. (also normalized to the NA of the lens), the simulation results of the optical field on the focal plane are presented in Fig. 2. It is obvious that the OAM collapses, since the intensity distributions on all the three planes split and the isosurface also splits. In addition, all the corresponding phase distributions on the three planes are binarized with the blue areas of -π and the orange areas of π. From Eq. (2), clearly the focusing power of the objective lens only applies to the spatial coordinates and temporal coordinate is not affected. In the ST domain (x-t plane), the objective lens performs an operation similar to a cylindrical lens. It is well known that a spatial OAM focusing through cylindrical lens will collapse, converting Laguerre-Gaussian (LG) modes into Hermite-Gaussian (HG) modes. Thus the focusing ST vortex through a high NA objective lens will experience similar spatio-temporal astigmatism and lead to splitting phenomenon described in Fig. 2.

Fig. 2. Focused optical wave packet on the focal plane of the high NA lens. (a) Intensity distributions of the focused wave packet on the three orthogonal planes in the ST domain; the red isosurface shows the three dimensional shape of the focused wave packet at thirty percent of its maximum intensity. (b) The corresponding phase distributions of the focused wave packet on the three orthogonal planes. In (a) and (b), the x- and y-axes are normalized by the wavelength of the incident beam; while t-axis is normalized to the pulse width.

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3. Generating the focused ST OAM

The analogy to the cylindrical lens function also leads to a solution to overcome the collapsing of the focused ST vortex. Cylindrical lens mode converter has been developed to generate spatial OAM through converting HG modes into LG modes [21], while HG modes can be treated as linear superposition of LG modes. An ST vortex with topological charge of -1, can be expressed as:

(4)$${E_{ - 1}}({x,y,t} )= ({x - it} )\exp [{ - ({{x^2} + {y^2} + {t^2}} )/{w^2}} ].$$

Superposition of the two ST vortices in Eqs. (1) and (4) leads to a new incident ST wave packet as

(5)$$\begin{aligned}E_t^{ + 1}({x,y,t} )&= {E_{ + 1}}({x,y,t} )+ i{E_{ - 1}}({x,y,t} )\\ &= ({1 + i} )({x + t} )\exp [{ - ({{x^2} + {y^2} + {t^2}} )/{w^2}} ]. \end{aligned}$$

The intensity distributions, isosurface and phase distributions of the new incident field are shown in Figs. 3(a) and 3(b), respectively. From these results, we can see that the new incident field splits in the ST domain, which is very different from the previous incident ST vortex shown in Figs. 1(a) and 1(b), especially on the x-t plane (the spiral phase is replaced by the binary phase distribution). This field distribution in the ST plane resembles a rotated HG₀₁ mode in the spatial domain. The phase on the three orthogonal planes is binarized with blue areas of -3π/4 and crimson areas of π/4, as shown in Fig. 3(b). Noting that the separation line between the two phase areas in x-t plane is rotated by -45 degree relative to the t-axis. In other words, the split ST wave packet should be rotated by 45 degree with regard to the t-axis in the x-t plane. This rotation plays a key role in producing the transverse OAM in the tightly focused ST field.

Fig. 3. The intensity distributions (a), isosurface at half-maximum intensity (a) and phase distributions (b) of the new input ST field. The intensity distributions (c), isosurface at fifty-five percent of the maximum intensity (c) and phase distributions (d) of the corresponding focused ST beam.

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Using the Debye integral to calculate the tightly focused field on the focal plane corresponding to the newly pre-conditioned incident ST wave packet, the calculated results are shown in Figs. 3(c) and 3(d). Compared with the results in Fig. 2, we can see that the intensity distribution in the x-t plane in Fig. 3(c) regains a donut shape with the hollow core along the y-axis. The full widths at half maximum (FWHM) along the x- and y-axis are found to be 0.53λ and 0.78λ, respectively. Meanwhile, the phase distribution on the x-t plane in Fig. 3(d) is recovered with continuously clockwise variation in the range of [-π, π], indicating the focused OAM has topological charge +1. The finite NA of the objective lens leads to slight bending of the phase transition line between -π and π in the x-t plane as shown in Fig. 3(d). However, this does not affect the topological charge associated with the spiraling phase. The phase distributions on the y-t and x-y plane in Fig. 3(d) are binarized into -3π/4 (mazarine area), π/4 (yellow area), -π/4 (azury area), and 3π/4 (red area), respectively, manifesting that there exist no OAM in the y-t and x-y domain. Clearly subwavelength ST focus with purely transverse OAM of topological charge +1 has been obtained.

Similarly, tightly focused ST vortex with purely transverse OAM of topological charge -1 can also be created, just by changing the incident ST wave packet to the following expression

(6)$$E_t^{ - 1}({x,y,t} )= ({1 + i} )({x - t} )\exp [{ - ({{x^2} + {y^2} + {t^2}} )/{w^2}} ].$$

As shown in Figs. 4(a) and 4(b), we can see that the above incident ST wave packet splits in the spatio-temporal domain, and the phase distributions are also binarized into -3π/4 (blue areas) and π/4 (crimson areas), which are similar to those in Figs. 3(a) and 3(b). The different point is that now the incident wave packet is rotated by -45 degree with respect to the t-axis in the x-t plane, which is orthogonal to that in Fig. 3(a). The intensity distributions, isosurface, and phase distributions of the corresponding tightly focused field on the focal plane of the lens are presented in Figs. 4(c) and 4(d), which is similar to those in Figs. 3(c) and 3(d). Likewise, the FWHMs of the focused field along the x and y directions remain to be 0.53λ and 0.78λ, respectively. However, the phase distribution on the x-t plane in Fig. 4(d) changes continuously from -π to π in the anti-clockwise direction, indicating that the subwavelength focused ST vortex carries transverse OAM of topological charge of -1.

Fig. 4. The intensity distributions (a), isosurface at half maximum intensity (a) and phase distributions (b) of the input ST field given by Eq. (6). The intensity distributions (c), isosurface at fifty-five percent of the maximum intensity (c) and phase distributions (d) of the corresponding focused ST beam.

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Subwavelength focused ST vortex with purely transverse OAM of higher topological charge can also be generated based on the HG mode decomposition. For example, the HG20 mode in the spatial domain with its principle axis at an angle of -45 degree relative to the x-axis can be decomposed into a set of LG modes as following [21]:

(7)$$\textrm{HG}_{20}^{dig} ={-} i({\textrm{L}{\textrm{G}_{02}} - \textrm{L}{\textrm{G}_{20}}} )- \textrm{L}{\textrm{G}_{11}}.$$

Hence, in a similar way, to produce subwavelength focused ST wave packet with transverse OAM with topological charge of -2, we can construct the incident ST wave packet as

(8)$$\begin{aligned}E_t^{ - 2}({x,y,t} )&={-} i[{E_{02}^{LG}({x,y,t} )- E_{20}^{LG}({x,y,t} )} ]- E_{11}^{LG}({x,y,t} )\\ &= [{2{{({x + t} )}^2}/{w^2} - 1} ]\exp [{ - ({{x^2} + {y^2} + {t^2}} )/{w^2}} ], \end{aligned}$$

where

(9)$$E_{02}^{LG}({x,y,t} )= [{{{({x + it} )}^2}/{w^2}} ]\exp [{ - ({{x^2} + {y^2} + {t^2}} )/{w^2}} ],$$

(10)$$E_{20}^{LG}({x,y,t} )= [{{{({x - it} )}^2}/{w^2}} ]\exp [{ - ({{x^2} + {y^2} + {t^2}} )/{w^2}} ],$$

(11)$$E_{11}^{LG}({x,y,t} )= [{ - 2({{x^2} + {t^2}} )/{w^2} + 1} ]\exp [{ - ({{x^2} + {y^2} + {t^2}} )/{w^2}} ].$$

The incident ST wave packet and the corresponding tightly focused field are demonstrated in Fig. 5. We can see that the incident wave packet resembles a HG20 mode in the x-t plane, whose principle axis is rotated by -45 degree with respect to the t-axis. And the phase distributions are binarized into 0 (blue areas) and π (crimson areas). From Fig. 5(c), we can find that the focused field exhibits donut shape in the x-t plane. The FWHMs of the focused field along the x- and y-axis are evaluated to be 0.62λ and 0.94λ, respectively. Moreover, the phase distribution on the x-t plane in Fig. 5(d) changes twice continuously from -π to π in anti-clockwise direction, manifesting topological charge of -2 for the transverse OAM. The phase distributions in the other two planes are binarized into 0 (green areas) and π (crimson areas).

Fig. 5. The intensity distributions with isosurface at forty percent of the maximum intensity (a), and phase distributions (b) of the input ST field given by Eq. (8). The intensity distributions with isosurface at half of the maximum intensity (c), and phase distributions (d) of the corresponding focused ST vortex with transverse OAM of topological charge -2.

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4. Conclusions

In conclusions, we studied the strong localization of ST vortex focusing through high NA objective lens. Due to the lack of temporal focusing, when such ST vortex is focused by a high NA lens, the focusing power only applies to one of the transverse spatial dimension for the ST vortex, effectively rendering a cylindrical lens focusing for the ST vortex. Such an ST astigmatism leads to the collapsing of the ST vortex phase structure at the focus. Through this study we add more understanding to ST vortex under focusing condition, revealing interesting similarities and distinctive differences with its spatial OAM counterparts. With the understanding of the physics and using the analogy to cylindrical lens mode converter, we presented a simple remedy by replacing the incident wave packet with a pre-conditioned wave packet that consists of linear superposition of ST vortices with different topological charge. Using the pre-conditioned ST wave packet, subwavelength ST vortex with transverse OAM can be created at the focal plane. The capability of producing strongly localized ST vortices may find numerous applications ranging from microscopy, plasma physics, laser machining, nonlinear light-matter interactions, quantum information processing, and so on. The methodology presented here also offers insights for an extensive class of ST wave packet focusing and propagation studies.

Funding

National Natural Science Foundation of China (61805142, 61875245); Science and Technology Commission of Shanghai Municipality (19060502500).

Disclosures

The authors declare no conflicts of interest.

References

1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]

2. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]

3. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011). [CrossRef]

4. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001). [CrossRef]

5. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]

6. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995). [CrossRef]

7. N. M. I. I. Elias, “Photon orbital angular momentum in astronomy,” Astron. Astrophys. 492(3), 883–922 (2008). [CrossRef]

8. J. Wang, J. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]

9. N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340(6140), 1545–1548 (2013). [CrossRef]

10. L. Yan, P. Gregg, E. Karimi, A. Rubano, L. Marrucci, R. Boyd, and S. Ramachandran, “Q-plate enabled spectrally diverse orbital-angular-momentum conversion for stimulated emission depletion microscopy,” Optica 2(10), 900–903 (2015). [CrossRef]

11. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001). [CrossRef]

12. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007). [CrossRef]

13. M. Padgett, “Light's twist,” Proc. R. Soc. A 470(2172), 20140633 (2014). [CrossRef]

14. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336(1605), 165–190 (1974). [CrossRef]

15. K. Y. Bliokh and F. Nori, “Spatiotemporal vortex beams and angular momentum,” Phys. Rev. A 86(3), 033824 (2012). [CrossRef]

16. N. Jhajj, I. Larkin, E. W. Rosenthal, S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, “Spatiotemporal optical vortices,” Phys. Rev. X 6(3), 031037 (2016). [CrossRef]

17. A. Chong, C. Wan, J. Chen, and Q. Zhan, “Generation of spatiotemporal optical vortices with controllable transverse orbital angular momentum,” Nat. Photonics 14(6), 350–354 (2020). [CrossRef] .

18. S. W. Hancock, S. Zahedpour, A. Goffin, and H. M. Milchberg, “Free-space propagation of spatiotemporal optical vortices,” Optica 6(12), 1547–1553 (2019). [CrossRef]

19. M. Yessenov, B. Bhaduri, H. E. Kondakci, and A. F. Abouraddy, “Weaving the rainbow: space-time optical wave packets,” Opt. Photonics News 30(5), 34–41 (2019). [CrossRef]

20. A. Sainte-Marie, O. Gobert, and F. Quéré, “Controlling the velocity of ultrashort light pulses in vacuum through spatio-temporal couplings,” Optica 4(10), 1298–1304 (2017). [CrossRef]

21. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]

22. E. Wolf, “Electromagnetic diffraction in optical systems-I. An integral representation of the image field,” Proc. R. Soc. Lond. A 253(1274), 349–357 (1959). [CrossRef]

23. B. Richards and E. Wolf, “Electromagnetic diffraction in optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A 253(1274), 358–379 (1959). [CrossRef]

24. L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229(1-6), 85–91 (2004). [CrossRef]

Subwavelength focusing of a spatio-temporal wave packet with transverse orbital angular momentum

Abstract

1. Introduction

2. Collapsing of the focused ST vortex

3. Generating the focused ST OAM

4. Conclusions

Funding

Disclosures

References

Cited By

Figures (5)

Equations (11)

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