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Abstract
The orbital angular momentum (OAM) of light has important applications in a variety of fields, including optical communication, quantum information, super-resolution microscopic imaging, particle trapping, and others. However, the temporal properties of OAM in ultrafast pulses and in the evolution process of spin-orbit coupling has yet to be revealed. In this work, we theoretically studied the spatiotemporal property of time-varying OAM in the tightly focused field of ultrafast light pulses. The focusing of an incident light pulse composed of two time-delayed femtosecond sub-pulses with the same OAM but orthogonal spin states is investigated, and the ultrafast dynamics of OAM variation during the focusing process driven by the spin-orbit coupling is visualized. Temporal properties of three typical examples, including formation, increase, and transformation of topological charge are investigated to reveal the non-uniform evolutions of phase singularities, local topological charges, self-torques, and time-varying OAM per photon. This work could deepen the understanding of spin-orbit coupling in time domain and promote many promising applications such as ultrafast OAM modulation, laser micromachining, high harmonic generation, and manipulation of molecules and nanostructures.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The optical vortex beam, carrying a unique spiral wavefront and orbital angular momentum (OAM), has widely been studied and applied in a variety of fields, such as optical communication [1–3], quantum information [4,5], super-resolution microscopic imaging [6], micro-/nano-particle trapping and manipulation [7,8]. As a combination of optical vortex beam and ultrafast light pulse, recently the ultrafast optical vortex pulses have attracted great interest of researchers all over the world. Since the ultrafast vortex pulses possess the characteristics of OAM, ultra-short pulse duration and ultra-high peak intensity simultaneously, the ultrafast vortex pulses have shown significant application prospects in the fields of generation of attosecond pulses with designed OAM [9–13], detection of internal phase motion within optical solitons [14], precise laser material processing [15,16], ultrafast nonlinear spectroscopy [17], and others. Therefore, it is of great importance to investigate and manipulate the spatial and temporal properties of the ultrafast vortex pulses.
In recent years, ultrafast vortex pulses have been generated by using various methods such as spiral phase plate [18,19], spiral multi-pinhole plate [20], and metasurfaces [21]. Many interesting spatiotemporal properties of the ultrafast vortex pulses have been investigated both theoretically and experimentally [22–27]. For example, the upper limit of the OAM carried by vortex pulse and the lower limit of pulse duration have been theoretically explained [22]. The space-time optical vortex with controllable transverse OAM is experimentally realized [26,27]. Dynamical inversion of the topological charge of an optical vortex has been observed [28]. In high harmonic generation, attosecond extreme-ultraviolet pulses with designed OAM have been generated [9]. However, in these previous works, the OAM carried by the vortex pulses is invariant during the time. Recently, time-varying OAM has been discovered in extreme-ultraviolet femtosecond vortex pulses, which arises in high harmonic generation excited by time-delayed vortex pulses with different topological charges [29]. Owing to the novel properties of self-torque and angular acceleration, the time-varying OAM beams have great importance in manipulating the magnetic [30], molecular [31], and quantum excitations [32,33] in both femtosecond time and nanometer spatial scales [34].
In this paper, we theoretically study the time-varying OAM in a tightly focused field of ultrafast light pulses driven by the spin-orbit coupling. It is well known that for a light beam carrying both spin angular momentum (SAM) and OAM, the interaction between SAM and OAM could occur in tight focusing of the beam [35], resulting in the time variation of the total OAM in the focal field. However, many spatiotemporal properties of the time-varying OAM in such spin-orbit interaction process have yet to be revealed. Here, similar to previous works with two time-delayed pulses [10,29,36,37], we consider the incident light pulse composed of two time-delayed femtosecond sub-pulses with same OAM but orthogonal SAM states, and theoretically analyze the ultrafast dynamics of the OAM variation of the pulse during the focusing process through a high-numerical-aperture (NA) objective lens. Three examples with different initial topological charges of OAM (l = 0, 1, 2) are studied to demonstrate three typical time-varying OAM processes respectively, including the formation/increase of topological charge and the negative-to-positive transformation of OAM. In these processes, the temporal evolution of phase singularities in the focal field is revealed, and a concept of local topological charge is defined to quantify the non-uniform change of OAM in beam cross section at different times. In addition, the concept of self-torque [29] is used here to describe the change rate of the local topological charge in time domain. The values of OAM and SAM per photon in the focal plane are calculated to deliver a deeper understanding of the evolution of OAM and SAM in the focal field. Finally, we consider the influence of the delay time between the two sub-pulses on the evolution of the focal field. We believe that this work could not only deepen the understanding of spin-orbit interaction of light especially in time domain, but also promote many promising applications such as ultrafast OAM modulation [29], laser micromachining [15,16], high harmonic generation [9], and manipulation of molecules and nanostructures [33,38].
2. Results and discussion
2.1 Schematic of time-varying OAM generation
The schematic diagram of this work is shown in Fig. 1, which illustrates the time-varying OAM wave packet of the tightly focused field. As shown in Fig. 1(a), the incident light pulse is composed of two time-delayed femtosecond sub-pulses of Laguerre-Gaussian $LG_p^l$ mode with left- and right-handed circular polarizations, respectively. The two sub-pulses have same pulse width and center wavelength, and partial overlap with a time delay $\Delta \tau$. Therefore, the polarization state of the combined pulse is varying over the time in the range of whole wave packet. We consider the radial factor p = 0 of the $LG_p^l$ mode in this work, the combined incident light pulse carries both the two orthogonal SAMs (RCP: σ + = 1, LCP: σ - = -1) and an initial OAM (with topological charge l). The Jones vector of the incident pulse is,
Fig. 1. Schematic of the time-varying OAM generation. The results are calculated based on the Eqs. (8)–(11). (a) An incident pulse composed of a left circularly polarized and a right circularly polarized sub-pulses, with a time delay and angular momentum (σ -, l) and (σ +, l), respectively. The incident pulse is then focused by a high-NA objective lens, to generate time-varying OAM in Ez component of the focal field as indicated in the inset (purple dashed line box) The inset of black dashed line box shows the schematic of the focusing coordinate system with all structure parameters. (b1-b3) The normalized amplitude distributions of Ex component in the focal plane (z = 0) at the moment t = -10 fs, t = 0 fs, t = 10 fs, respectively. (b4-b6) are similar as (b1-b3) except for Ey component. (b7-b9) are similar as (b1-b3) except for Ez component. (c1-c3) Corresponding phase distributions of the field in (b1-b3). (c4-c6) Corresponding phase distributions of the field in (b4-b6). (c7-c9) Corresponding phase distributions of the field in (b7-b9). For the calculations in (b-c), the parameters are chosen as: initial topological charge l = 0, center wavelength λ0 = 800 nm, pulse duration τ0 = 50 fs, NA = 0.9, Δt = 40 fs.
After being tightly focused by a high-NA objective lens, the phase vortex generated by SAM-to-OAM conversion mainly occurs in the longitudinal electric-field component (Ez) of the focal field [35], leading to the time-varying OAM in the focused pulse. Inset of Fig.1a schematically gives the evolution diagram of the wave-front of Ez component in the focal plane (xy plane with z = 0) as time progresses, where the total topological charge is changing from l1 to l2 (l1 = σ - + l, l2 = σ + + l) during the pulse focusing process.
In order to demonstrate the spin-orbit coupling effect, in Fig. 1(b) we show an example of the timing diagram of amplitude distribution of the three electric-field components (Ex, Ey, Ez) during the focusing process with a zero initial topological charge (l = 0), and the corresponding phase distributions at different times are presented in Fig. 1(c). Here the two sub-pulses have the same center wavelength λ0 = 800 nm and pulse duration τ0 = 50 fs (FWHM), and the numerical aperture of the objective is NA = 0.9. The high-NA aplanatic objective lens obeys the sine rule r = fsinθ. The delay time between the two sub-pulses is fixed at Δτ = 40 fs, so as to ensure the two sub-pulses have an overlap range in time domain, and the corresponding spatial delay distance between the sub-pulses is 15λ0. All the analytical equations are based on the Richards-Wolf vectorial diffraction theory, which the Debye integral in spectrum can be expressed as [39,40]:
Finally, the temporal field of all the components can be obtained by the Fourier-transform in spectrum as:
2.2 Spin-orbit coupling in the time-space domain
To illustrate the different variations of OAM in the time-space domain, we consider three different cases with initial topological charge l = 0, 1, and 2 for the incident pulse. Similar to the previous works [41,42], the spatiotemporal distributions of the component Re(Ez) in the lateral spatiotemporal plane (x-t plane and y-t plane with z = 0) are calculated by Eqs. (8-11) and shown in Fig. 2, which contain the space-time information such as phase- and pulse-front and thus could clearly demonstrate that the topological charge is changing as time passes. Here the carrier angular frequency of input pulse beam is multiplied by 0.2 for sake of clear observation [41,42], and the results of the optical fields are normalized relative to the maximum value in each figure.
Fig. 2. Spin-orbit coupling in the time-space domain of three different cases with initial topological charge l = 0, 1, and 2. (a)-(b) The spatiotemporal distributions of the Re(Ez) in the lateral spatiotemporal plane for the case of l = 0. (d)-(e) are similar as (a-b) except for the case of l = 1. (g)-(h) are similar as (a-b) except for the case of l = 2. (c), (f) and (i) Isosurface plots (at the value 0.3) of the focused optical field Re(Ez) for the three cases in the 3D spatiotemporal coordinate. All results of the optical fields are normalized relative to the maximum value in each figure, the carrier angular frequency of input pulse beam is multiplied by 0.2 for sake of clear observation. All other parameters are the same as in Fig. 1.
For the first case with initial l = 0 in the incident pulse, Fig. 2(a) and Fig. 2(b) show the spatiotemporal evolution of Re(Ez) in the x-t plane and y-t plane, respectively, the corresponding 3-dimentional (3D) isosurface of Re(Ez) is presented in Fig. 2(c). Since the Re(Ez) value is mainly distributed in the range of (-0.5, 0.5), we choose the value of 0.3 in 3D isosurface for a clear observation. Based on the spin-orbit coupling of the combined incident pulse shown in Fig. 1(a), in the time range from t = -50 fs to t = 0 fs, the conversion from σ - = -1 to l1 = -1 mainly occurs, while the conversion from σ + = 1 to l2 = 1 occurs in the range from t = 0 fs to t = 50 fs, resulting in the time-varing topological charge from l1 = -1 to l2 =1. As a result, the distribution of Re(Ez) is always odd symmetric with respect to the axis x = 0 (Fig. 2(a)) and y = 0 (Fig. 2(b)) as time passes, because both l1 = -1 and l2 = 1 cause a phase difference of π between the two symmetrical points. The 3D isosurface shows vortex with counterclockwise/clockwise rotation in the time range before/after the moment of t = 0 fs (Fig. 2(c)). At the moment t = 0 fs, the OAM vortex disappears (Fig. 2(c)) because of the offset between l1 = -1 and l2 = 1, but the Re(Ez) distribution is still odd symmetric with respect to the axis x = 0 (Fig. 2(a)) corresponding to the 0 and π phase regions shown in Fig. 1(c8).
For the second case with initial l = 1, Fig. 2(d) and Fig. 2(e) show the spatiotemporal distributions of Re(Ez) in the x-t plane and y-t plane, respectively. Due to the interaction between the initial OAM (l = 1) and SAM (σ - = -1, σ + = 1) of the incident pulse, the generated topological charge of Ez in the focal plane is l1 = σ - + l = 0 in the time range from t = -50 fs to t = 0 fs, and l2 = σ + + l = 2 from t = 0 fs to t = 50 fs. As a result, in Fig. 2(e) the central main lobe of Re(Ez) splits from one bright spot to two pieces as time passes, and the corresponding 3D isosurface (Fig. 2(f)) changes from a plane wave distribution (l1 = 0) to a clockwise vortex shape with two spiral arms (l2 = 2). Here the Re(Ez) distribution is always even symmetric in the spatiotemporal domain with respect to the axis x = 0 (Fig. 2(d)) and y = 0 (Fig. 2(e)), because the generated topological charges l1 = 0 and l2 = 2 cause a phase difference of 0 and 2π between the two symmetrical points, respectively.
The third case with initial l = 2 is studied in Figs. 2(g)–2(i). In this case, the topological charge of Ez in the focal plane is l1 = σ - + l = 1 from t = -50 fs to t = 0 fs, and l2 = σ + + l = 3 from t = 0 fs to t = 50 fs. Thus, the 3D isosurface (Fig. 2(i)) changes from a small vortex with single spiral arm (l1 = 1) to a larger vortex with three spiral arms (l2 = 3) as time passes, and the rotational direction keeps clockwise. Since here both values of l1 and l2 are odd, the Re(Ez) distribution is also odd symmetric with respect to the axis x = 0 (Fig. 2(g)) and y = 0 (Fig. 2(h)).
The above three cases represent three typical situations of OAM variation in time domain, including the negative-to-positive transformation of topological charge (Fig. 2(c)), the formation of vortex from a plane wave (Fig. 2(f)), and the increasement of topological charge (Fig. 2(i)). It can be found that, in all cases the temporal evolution of OAM during the tight focusing process follows the spin-orbit coupling principle (l1 = σ - + l, l2 = σ + + l).
2.3 Spatiotemporal evolution of singularity in focal plane
The temporal variation of OAM is usually accompanied by the evolution of phase singularities in the beam cross section. To deliver a deep understanding on the spatiotemporal evolution of singularities during the tight focusing process, Figs. 3–5 show the Ez amplitude, phase and real part distributions in the focal plane at different moments for the three cases (l = 0, 1, 2) studied in the above section, respectively.
Fig. 3. Spatiotemporal evolution of focused field Ez for the case of initial l = 0 in the incident pulses. (a1)-(a7) The amplitude distribution evolution of the Ez field. (b1)-(b7) The phase distribution evolution of the Ez. (c1)-(c7) The real part evolution of the Ez. The simulated distributions area is restricted by 2 μm × 2 μm. The time range of evolution is t = -30 fs to t = 30 fs. The amplitude and the real part of the Ez is normalized by the corresponding maximum over the entire time range.
Fig. 4. Spatiotemporal evolution of focused field Ez for the case of initial l = 1 in the incident pulses. (a1)-(a7) The amplitude distribution evolution of the Ez field. (b1)-(b7) The phase distribution evolution of the Ez. (c1)-(c7) The real part evolution of the Ez. The simulated distributions area is restricted by 2 μm × 2 μm. The time range of evolution is t = -30 fs to t = 30 fs. The amplitude and the real part of the Ez is normalized by the corresponding maximum over the entire time range.
Fig. 5. Spatiotemporal evolution of focused field Ez for the case of initial l = 2 in the incident pulses. (a1)-(a7) The amplitude distribution evolution of the Ez field. (b1)-(b7) The phase distribution evolution of the Ez. (c1)-(c7) The real part evolution of the Ez. The simulated distributions area is restricted by 2 μm × 2 μm. The time range of evolution is t = -30 fs to t = 30 fs. The amplitude and the real part of the Ez is normalized by the corresponding maximum over the entire time range.
In Fig. 3, for the case of topological charge varying from l1 = -1 to l2 = 1, it can be observed that the amplitude distribution is firstly a donut shape with a central dark singularity at the moment t = -30 fs (Fig. 3(a1)), then it gradually changes to a dipole-like shape at the focusing moment t = 0 fs (Fig. 3(a4)), finally reverts back to the donut shape from t = 0 fs to t = 30 fs with a singularity in the center (Figs. 3(a5)-(a7)). The corresponding phase distribution shows opposite spiral phase patterns (Figs. 3(b1)-(b3) and Figs. 3(b5)-(b7)) before and after the moment t = 0 fs, while the real part distribution (Figs. 3(c1)-(c7)) presents time symmetric patterns with respect to the moment t = 0 fs. In fact, both the phase and real part patterns are rotating as time passes driven by the OAM, and experience a counterclockwise-to-clockwise conversion in rotation during the focusing process (see Visualization 1), which is consistent with the evolution of the helical wavefront shown in Fig. 2(c).
For the case of topological charge varying from l1 = 0 to l2 = 2, the amplitude distribution (Figs. 4(a1)-(a7)) changes gradually from a bright focal spot to a donut shape from t = -30 fs to t = 30 fs, while the corresponding phase distribution (Figs. 4(b1)-(b7)) varies from a uniform phase pattern in the center to a spiral phase pattern. It can be observed that two singularities with topological charge l = 1 (indicated as red and blue circles in Fig. 4(b4)) originate from the dark ring region around the central main-lobe (Fig. 4(a4)), and then move closer to each other, until they combine together to form a single singularity in the center with a topological charge l = 2 at about t = 30 fs (Fig. 4(b7)), which visualize the formation process of topological charge from l1 = 0 to l2 = 2. In this process, the real part pattern (Figs. 4(c1)-(c7)) gradually converts from a concentric ring shape to a rotational fan-shaped pattern accompanying with the formation of vortex (see Visualization 2).
The similar evolution of singularities in the phase distribution appears in the case of topological charge increasing from l1 = 1 to l2 = 3 shown in Fig. 5. Besides the initial dark singularity in the center (Fig. 5(a1)) with topological charge l = 1 (Fig. 5(b1)), two additional singularities with l = 1 also originate from the first dark ring region (Fig. 5(a4)), and then the three singularities (indicated as red, green and blue circles in Fig. 5(b4)) get closer to each other until they combine together to produce a single singularity with l = 3 in the center (Fig. 5(b7)). In this process, the rotational real part pattern (Figs. 5(c1)-(c7)) varies from a two-sector pattern to a six-sector pattern (see Visualization 3), in accordance with the increase of topological charge from l1 = 1 to l2 = 3.
In order to verify the correctness of the above results calculated by the analytical equations, the focused optical field Ez are also simulated by the 3D finite-difference time-domain (FDTD) method (Lumerical FDTD Solutions) for comparison. The FDTD simulation parameters are chosen as: the 3D simulation area is 5μm×5μm×5μm, all boundary conditions are PML, the mesh size is 30 nm in xy plane and 100 nm in z-direction. All other parameters are chosen same as the analytical model. A xy-plane frequency domain monitor is used to record the focus field in frequency domain, thus the temporal field can be obtained by the Fourier-transform of the frequency-domain results. Figures 6(a1)-(a7), 6(b1)-(b7), and 6(c1)-(c7) show the FDTD results of Ez for the case of initial l = 0, whose evolution law with time is consistent with that of analytical results as Fig. 3. Figures 6(d1)-(d7) and 6(e1)-(e7) show the FDTD results of the phase distribution of Ez for the case of initial l = 1 and l = 2, respectively, which are also consistent with that of analytical results in Figs. 4 and 5. The slight difference between the analytical and FDTD methods stems from the limited mesh size and time step in FDTD simulation, which could not be infinitely small and thus affect the numerical accuracy. Due to the limited mesh size and time step obtained by monitor in FDTD simulation, we cannot get the results at the exact moment of -30 fs, … 30 fs as same as the analytical results. So the results of the panels in Fig. 6 are chosen at the nearest moment of -30 fs, … 30 fs as the corresponding analytical results in Figs. 3–5.
Fig. 6. Spatiotemporal evolution of focused field Ez obtained with FDTD simulation for the case of initial l = 0, l = 1 and l = 2, respectively, in the incident pulses. (a1)-(a7) The amplitude distribution evolution of the Ez field for the case of initial l = 0. (b1)-(b7) The phase distribution evolution of the Ez for the case of initial l = 0. (c1)-(c7) The real part evolution of the Ez for the case of initial l = 0. (d1)-(d7) are similar as (b1)-(b7) but for the case of initial l = 1. (e1)-(e7) are similar as (b1)-(b7) but for the case of initial l = 2. The simulated distributions area is restricted by 2 μm × 2 μm. The time at each subfigure is chosen at the nearest moment of the corresponding results in Figs. 3–5.
2.4 Local topological charge and self-torque in focal plane
The above results (Fig. 3 to Fig. 5) present the non-uniform evolution of spiral phase distribution in the focal field. In order to reveal the detailed non-uniform property of the spiral phase distribution evolution, we define a concept of local topological charge as ${l_{local}}(\mathrm{\rho },\varphi ,t) = \partial \psi (\mathrm{\rho },\varphi ,t)/\partial \varphi $, where φ is the azimuthal angle of polar coordinates, and $\psi (\mathrm{\rho },\varphi ,t)$ is the phase at any point (ρ, φ) in the focal plane and at the moment of t. Different to the previously studied local topological charge in radial direction [43], here the llocal indicates the phase gradient of each point in the focal field along the azimuthal direction, and also describes the number of cycles for phase changing from 0 to 2π. Since in optical vortices the phase usually changes along the azimuthal direction, so we focus on the phase change along the azimuthal direction in the focal plane.
For the first case with initial l = 0 in the incident pulse, the distribution of local topological charge in the focal plane is shown in Figs. 7(a1)-(a7). At the start and the end moments (t = -30 fs and t = 30 fs), the value of llocal in the center main-lobe is almost uniformly distributed and equals to the predicted topological charge of l1 = -1 and l2 = 1, respectively. However, in the time range around the focusing moment (from t = -20 fs to t = 20 fs), the value of llocal shows non-uniform variation especially near the phase jump region between 0 and π (Fig. 3(b4)), which first decreases less than l1 = -1 before the focusing moment (t = 0 fs) and then rapidly increases larger than l2 = 1 after the focusing moment, and llocal = 0 at the moment t = 0 fs corresponding to the phase distribution in Fig. 3(b4) without any spiral phase. To further investigate the time variation of llocal, we calculate the self-torque [29], which is here defined as $\xi = \partial {l_{local}}(\mathrm{\rho },\varphi ,t)/\partial t$ and refers to the inherent angular acceleration of the light beam. The calculated self-torque distribution shows time symmetric patterns (Figs. 7(b1)-(b7)) with respect to the moment t = 0 fs, and reaches the maximum value at t = 0 fs.
Fig. 7. Local topological charge and self-torque for the case of initial l = 0. (a1)-(a7) Local topological charge distribution of Ez field in the focal plane (z = 0) at the moment -30 fs, -20 fs, -10 fs, 0 fs, 10 fs, 20 fs, 30 fs. (b1)-(b7) The corresponding self-torque distribution. (c) The purple /green curve shows the variation of the local topological charge at point A (red diamond in (b4)) /point B (red triangle in (b4)) over time, respectively. (d) Shows the variation of the self-torque at point A/point B over time, respectively. The parameters are chosen as: the simulated area is restricted by 2 μm×2 μm. Δφ=π/106 is the azimuthal step for derivative. All other parameters are as in Fig. 1.
To deeply understand the non-uniform variations of local topological charge and self-torque, we select two example points A (red diamond) and B (red triangle) at different positions as indicated in Fig. 7(b4), and present the continuous variations of their local topological charges and self-torques in Fig. 7(c) and Fig. 7(d), respectively. It can be found that the llocal of point A increases monotonically from -1 to 1 with time (purple curve in Fig. 7(c)), and its corresponding self-torque curve is almost flat near zero (purple curve in Fig. 7(d)), indicating that the llocal at point A has a slow rate of change. In contrast, the llocal of point B shows a sharp jump in the vicinity of t = 0 fs (green curve in Fig. 7(c)) even breaking through the normal l = -1 to 1 range, and the corresponding self-torque (green curve in Fig. 7(d)) appears a peak value at t = 0 fs in accordance with the results in Fig. 7(b4). The different performance of the two points is because the point B is close to the phase jump boundary between 0 and π (Fig. 3(b4)) and shows a much larger phase gradient and higher change rate of local topological charge, but the point A is far away from any phase jump boundary or singularities and shows a smooth and slow change in local topological charge. At other time far away from the focusing moment (t = 0 fs), such as before t = -50 fs or after t = 50 fs, the llocal of points A and B are both almost constant at -1 and 1, respectively (Fig. 7(c)), and the corresponding self-torque is always near 0 (Fig. 7(d)), due to the fact that the two time-delayed sub-pulses do not overlap in these ranges.
For the other two cases with initial l = 1 and l = 2 in the incident pulse, the distributions of llocal (Figs. 8(a1)-(a7) and Figs. 8(b1)-(b7)) are uniform in most region of the focal field like the above-mentioned point A, except the positions close to the phase jump boundary and singularities where the phase gradient is sharp like the point B. The value of llocal in most region continuously increases from 0 to 2 (Figs. 8(a1)-(a7)) and from 1 to 3 (Figs. 8(b1)-(b7)) as the prediction of spin-orbit coupling. It is noted that at the moment t= 30 fs, the phase singularities are displaced off-axis, leading to the non-uniform distribution of llocal. As the time goes on, the phase singularities will move closer to each other, and the llocal. distribution in the focal plane will become more uniform. As shown in Figs. 8(c1)-(c7) and Figs. 8(d1)-(d7), the self-torque distributions of the two cases are very similar, both showing near zero values in most region except the positions close to phase jump boundary and singularities.
Fig. 8. Local topological charge and self-torque for the cases of initial l = 1 and 2. (a1)-(a7) Local topological charge distribution of Ez field in the focal plane (z = 0) for the case of l = 1 at the moment -30 fs, -20 fs, -10 fs, 0 fs, 10 fs, 20 fs, 30 fs. The corresponding self-torque is shown in (c1)-(c2). (b1)-(b7) are similar as (a1-a7) except for the case of l = 2. The corresponding self-torque is shown in (d1)-(d2). All parameters are as in Fig. 7.
The results in Fig. 7 and Fig. 8 demonstrate that the evolution of the llocal is non-uniform in the focal field, specifically, in most region the time variation of llocal is smooth and slow, but sharp and fast variation of llocal appears in the points close to the phase jump boundary and singularities. It is worth noting that the self-torque determines the angular acceleration of light beam, therefore when the self-torque is not zero, the phase and real part patterns of Ez will rotate at a varying angular velocity. In the time range far away from the focusing moment (such as before t= -50 fs and after t = 50 fs), the self-torque is near zero as well as the angular acceleration, and therefore the angular velocity of the rotation can be calculated as constant value of $- {\omega _0}/(l + {\sigma ^ - })$ or $- {\omega _0}/(l + {\sigma ^ + })$ [44].
2.5 Time-varying OAM and SAM per photon of the focused field
To deliver a deeper understanding of the evolution of OAM and SAM in the focal field, the values of OAM per photon () and SAM per photon () in the focal plane (z = 0) are calculated by the following equations [25,45]:
It is worth noting that the OAM per photon () is dependent on the above-studied local topological charge llocal. The numerator of Eq. (12) for < L > is actually proportional to the integral of OAM density vector L at each point of focal field, which is defined as [46]:
Its z-component can be calculated as:
Figure 9 shows the longitudinal (z-component) angular momentum (AM) per photon of the focused field (Fig. 9(a, c, e)) and the incident pulses (Fig. 9(b, d, f)) for the three cases studied above. It can be seen that, for the first case with initial l = 0, the OAM of the incident pulse keeps zero (Fig. 9(b)), but the OAM in the focal plane changes from about -0.5ħ to 0.5ħ (Fig. 9(a)) and shows opposite trend of SAM, which means a part of SAM converts to the OAM in the focal plane, corresponding to the results in Fig. 3. At the moment t = 0 fs, the OAM in the focal plane equals to zero (Fig. 9(a)), well agreeing with the result of llocal = 0 shown in Fig. 7(a4). The total angular momentum ( + ) of the focused field and the incident pulses are almost same, demonstrating the conservation of total angular momentum. For the other two cases with initial l = 1 and l = 2, the OAM of the incident pulse keeps constant (Figs. 9(d) and 9(f)), but the OAM of the focused field tends to increase from 0.5ħ to 1.5ħ (Fig. 9(c)) and from 1.5ħ to 2.5ħ (Fig. 9(e)), respectively, also due to the SAM-to-OAM conversion. SAM obviously changes over time as well [48]. From the time-varying OAM and SAM per photon of the focused field, we can conclude that the OAM in the focal field comes from SAM of the incident pulse, and the delay of the two orthogonal polarization sub-pulses leads to the formation/increase of OAM and the negative-to-positive transformation of OAM in the focal field.
Fig. 9. Angular momentum (AM) dynamics of tightly focused field (a, c, e) and the corresponding incident pulses (b, d, f). The three curves in each figure represent OAM per photon (red curves), SAM per photon (blue curves) and total AM per photon (black curves), respectively. (a)-(b) For the case of initial l = 0 in the incident pulse. (c)-(d) For the case of initial l = 1 in the incident pulse. (e)-(f) For the case of initial l = 2 in the incident pulse.
2.6 Influence of time delay between the two sub-pulses
The analysis in all above sections is based on a fixed time delay of 40 fs between the two sub-pulses, corresponding to the spatial delay distance of 15λ0. As a result, the polarization ellipticity of the combined incident pulse in the overlap range of the two sub-pulses will change with time. The polarization ellipticity is defined as the sine of twice the ellipticity angle [49]. Figure 10(a) illustrates the variation of the polarization ellipticity of the combined pulse in the pulse duration with different delay time (30.00fs, 40.67 fs, 41.33 fs, 42.00 fs, 50.00 fs), and the corresponding delay distance is 0.25λ0 + 11λ0, 0.25λ0 + 15λ0, 0.5λ0 + 15λ0, 0.75λ0 + 15λ0, 0.75λ0 + 18λ0, respectively. Because of the overlap of left-/right-circularly polarized sub-pulses in time domain, the generated transient polarization state changes from left-handed circular polarization to linear polarization, and finally to right-handed circular polarization. Thus, in all cases of Fig. 10(a) the polarization state is always linear (polarization ellipticity of 0) at the moment t = 0 fs (here the middle of the pulse envelope), while the circular polarization (polarization ellipticity of -1 and +1) appears at both ends of the pulse (t < -50 fs and t > 50 fs). If the change of delay distance is small such as within a λ0, the polarization ellipticity shows very similar results (as red, green, yellow curves in inset of Fig. 10(a)). The maximum change rate of polarization ellipticity appears at the moment t= 0 fs, and it gets a larger result at longer delay time (as the pink curve in Fig. 10(a)) because of the shorter overlap range.
Fig. 10. Influence of time delay between the two sub-pulses. (a) The variation of the polarization ellipticity of the combined pulse in the pulse duration for certain delay time (30 fs, 40.67 fs, 41.33 fs, 42 fs, 50 fs), the corresponding the delay distance is 0.25λ0 + 11λ0, 0.25λ0 + 15λ0, 0.5λ0 + 15λ0, 0.75λ0 + 15λ0 and 0.75λ0 + 18λ0, respectively. The inset shows the enlarged region of the blue dot. (b1)-(b3) show the phase distribution of the focal field at the moment t = -10 fs, t = 0 fs, t = 10 fs, for the case of initial l = 0 with the delay distance of 0.25λ0+15λ0. (b4)-(b6) are similar as (b1)-(b3) except for the delay distance of 0.5λ0 + 15λ0. (b7)-(b9) are similar as (b1)-(b3) except for the delay distance of 0.75λ0 + 15λ0. (c1)-(c3) are similar as (b1)-(b3) except for the delay distance of 0.25λ0 + 11λ0, and the corresponding spatiotemporal distribution of the Re(Ez) in the x-t plane is shown in (c4). (d1)-(d3) are similar as (b1)-(b3) except for the delay distance of 0.75λ0 + 18λ0, and the corresponding spatiotemporal distribution of the Re(Ez) in the x-t plane is shown in (d4). All other parameters are as in Fig. 1.
We further study the influence of the delay time (delay distance) on the spatiotemporal distribution of the focused field. Here we choose the case of initial topological charge l = 0 as example, and compare the Ez phase distribution of the focus field at the moments t = -10 fs, t = 0 fs and t = 10 fs with different delay times. Figures 10(b1)-(b9) show the cases of three delay distances (0.25λ0 + 15λ0, 0.5λ0 + 15λ0, 0.75λ0 + 15λ0) with a step of 0.25λ0, where the phase presents not helical distribution but binary distribution at the moment t = 0 fs and the difference of the binary phases is always π (Figs. 10(b2), (b5), (b8)), due to the linear polarization state at t = 0 fs in Fig. 10(a). The different rotation angles of these binary phase patterns stem from the phase difference caused by the delay distance. However, at the moment t = -10 fs and t = 10 fs, the phase still maintains a helical distribution (Figs. 10(b1), (b3), (b4), (b6), (b7), (b9)), similar to the cases shown in Fig. 3. The corresponding Ez amplitude and real part patterns are all similar as that in Fig. 3 except with a rotation of the whole patterns, thus not shown again. The results in Figs. 10(b1)-(b9) and Fig. 3 reveal the influence of the delay distance varying within a λ0, which mainly induces a rotation on the focal field patterns.
For the delay distance larger than a wavelength λ0, we show the spatiotemporal distributions of Ez phase and real part with the delay distances of 0.25λ0 + 11λ0 and 0.75λ0 + 18λ0 in Figs. 10(c1)-(c4) and Figs. 10(d1)-(d4), respectively. It can be observed that the phase patterns in Figs. 10(c1)-(c3) and Figs. 10(d1)-(d3) are almost the same as those in Figs. 10(b1)-(b3) and Figs. 10(b7)-(b9), respectively, indicating that the change of multiple λ0 in delay distance has little effect on the phase distribution; that is, the influence of delay distance is periodic. A similar periodic rotation effect has been reported in a previous work [50], where the ultrafast Rabi-rotating vortices have been studied and shown the time-varying linear and angular momenta. The Re(Ez) results in Figs. 10(c4) and (d4) are similar as that in Fig. 2(a), except that the whole pulse duration increases with longer delay distance. Based on the results in Fig. 10, it can be inferred that the change of the delay distance (delay time) between the sub-pulses mainly causes a rotation of the focal field patterns and the variation of whole pulse duration, and has little influence on the trend of spatiotemporal evolution of OAM in the tight focusing process. Although we only show the case of initial l = 0 as example, the other two cases with initial l = 1 and l = 2 follow the same rules when the delay distance (delay time) changes.
3. Conclusion
In this work, we theoretically investigate the properties of time-varying OAM of a femtosecond pulse in a tightly focused field. The incident pulse, composed of two time-delayed sub-pulses with same topological charge but orthogonal circular polarization, is considered to be strongly focused by a high NA objective lens. The results show that, the topological charge of the longitudinal field (Ez component) in the focal plane is varying in time domain, which is dominated by the spin-orbit coupling effect. Three typical time-varying OAM processes have been studied, including the formation/increase and the negative-to-positive transformation of topological charge. In the negative-to-positive transformation process (Fig. 3), there is no spiral phase at the focusing moment (t = 0), replaced by a 0-π binary phase distribution that can be rotated by changing the delay distance (Fig. 10(b)). In the process of the formation/increase of topological charge (Fig. 4 and Fig. 5), additional phase singularities with l = 1 are generated and then gradually merge into single singularity to form a high-order OAM. The non-uniform distribution and spatiotemporal evolution of the local topological charge and the self-torque (Fig. 7 and Fig. 8) are revealed, in which more severe OAM change (with the local topological charge even breaking through the normal -1 to 1 range in Fig. 7(c)) and faster angular acceleration (Fig. 7(d)) appear in the singularity/phase jump region due to the stronger phase gradient. The time-varying OAM and SAM per photon of the focused field are calculated and show opposite trends in variation, further demonstrating that OAM in the focal field comes from SAM of the incident pulse.
In our analytical model and FDTD simulation, the incident pulse is always considered as a spatial and temporal Gaussian pulse without dispersion. In fact, for the pulse duration of 50 fs or more, the pulse broadening caused by the focusing objective material is calculated as 1-2 fs or less [51,52], therefore the dispersion effect is ignored in our calculation and analysis. In general, if the pulse width is less than 30 fs, the dispersion caused by the materials of optical devices must be considered.
Although in this work we only theoretically study the time-varying OAM in the tight focusing field, the experimental measurement of time-varying OAM is possible to be achieved by employing a strong-field ionization experiment [53]. This work not only reveals the spatiotemporal properties of time-varying OAM in focal field, but also has potential applications in many fields including multiphoton absorption processes [54], controlling 3D photoelectron momentum distributions [55,56], optical centrifuge for molecules [57–59], optical control of lattice vibrations [60–62].
Funding
The Guangdong Major Project of Basic and Applied Basic Research (2020B0301030009); National Natural Science Foundation of China (62175157, 61935013, 61975128); Leading Talents Program of Guangdong Province (00201505); Natural Science Foundation of Guangdong Province (2019TQ05X750); Shenzhen Peacock Plan (KQTD20170330110444030); Shenzhen Science and Technology Innovation Program (ZDSYS201703031605029, JCYJ20180305125418079, JCYJ20180507182035270, JCYJ20210324120403011).
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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