Time-varying optical spin-orbit Hall effect in tightly focused femtosecond optical field

Mingchao Zhu; Shenggui Fu; Zhongsheng Man; Zhongsheng Man

doi:10.1364/OE.519223

1. Introduction

The Hall effect (HE) is a well-known electromagnetic induction phenomenon that describes the transverse displacement of electrons that occurs when a current passes through a magnetic field [1]. The HE has a wide range of applications in the electronics field [2–7]. Similarly, the optical HE is a novel phenomenon of light that causes transverse shifts in light beams to occur at different interfaces [8]. The optical HE is caused by optical spin-orbit interaction (SOI) and is dominated by the law of conservation of angular momentum [8–16]. This effect has attracted considerable attention because of its important or potentially important applications in optical devices [17–20], quantum optics [21–23], precision metrology [24,25], and material characterization [26–29].

The HE includes the spin HE and the orbital HE. The spin HE is described as the spatial separation of the different spins of the beam. Examples include the transverse spin separation of cylindrical vector beams and the controllable longitudinal spin separation realized via the Pancharatnam-Berry phase [30–33]. The orbital HE describes the effect of intrinsic orbital angular momentum (OAM) on the trajectory of the beam associated with the extrinsic OAM, where this effect leads to orbital splitting of the beam. For example, the orbital-orbit interaction of a beam that is reflected at an air-glass interface induces the orbital HE in the beam intensity distribution [34].

Recently, it was found that when the polarization state of a beam evolves adiabatically from the equator of the Poincaré sphere toward the poles, the inherent spin component and the orbital component of the beam become separated, i.e., the spin-orbit HE occurs [35]. Unlike the spin HE and the orbital HE, the spin-orbit HE only occurs between the spin angular momentum (SAM) and the OAM in the vector field. Use of the nano-interference method not only allows the SAM state and the OAM state to be measured on the sub-wavelength scale but also enables the spin-orbit HE to be measured and distinguished [36]. It has been verified theoretically that a radially polarized vortex beam can induce spin-orbit separation in a tightly focused system [37]. Unfortunately, although the spin-orbit HE has been researched, previous work in the literature has focused on static beams. It should be noted here that the time characteristics of light provide additional degrees of freedom for light field modulation. Examples include generation of a toroidal optical vortex [38], the time variation of the SOI [39,40], and the dynamic evolution of the optical force and the optical torque in ultrafast light fields [41]. The time characteristics are highly responsive to SOI [42,43]. As a special case of SOI, the spin-orbit HE on the femtosecond scale is rarely mentioned in the literature, and its dynamic changes over time have not been revealed to date.

In this paper, we present a theoretical study of the time-varying spin-orbit HE in a femtosecond light field in a tightly focused system. Using a combination of the Richards-Wolf diffraction integral and the Fourier transform method, we obtain explicit expressions for all components of the electric field within the focal volume of the tightly focused system. Calculations show that the focusing field has ultrafast characteristics of alternation between bright and dark states and a distorted phase distribution. The polarization distribution during the dynamic evolution of the focal field is also studied. Furthermore, the time-varying polarization’s topological behavior within the focal field is revealed using Stokes parameters. By calculating the angular momentum (AM) distribution of the focal field, it is determined that the spin-orbit HE of the femtosecond light field in the tightly focused system is dynamic, and the time phase factor makes an important contribution to the special characteristics of the focused light field. In addition, wider pulse widths and shorter central wavelength values have positive effects on the monochromaticity of the light. This time-dependent dynamic optical phenomenon has potential for use in particle manipulation applications.

2. Theoretical model

To study the spin-orbit HE of the femtosecond light field under tight focusing conditions, we assume that the incident pulse contains a Laguerre-Gaussian vortex mode and a Gaussian time envelope and is azimuthally polarized. The incident field can be described as follows [44–47]:

(1)$$P({r,\varphi ,\omega } )= A{\left( {\frac{{\sqrt 2 r}}{w}} \right)^{|l |}}\exp \left( { - \frac{{{r^2}}}{{{w^2}}}} \right)\exp ({il\varphi } )\exp \left[ { - \frac{{{\tau^2}{{({\omega - {\omega_0}} )}^2}}}{{8\ln 2}}} \right]\left[ {\begin{array}{c} {\textrm{cos}({\varphi \textrm{ + }{\mathrm{\pi } / 2}} ){\mathbf{e}_x}}\\ {\sin ({\varphi \textrm{ + }{\mathrm{\pi } / 2}} ){\mathbf{e}_y}} \end{array}} \right]$$

where A is a normalized constant. r and φ are the polar radius and azimuth, respectively. w and τ are the spatial width and the time width of the incident pulse, respectively. l is the number of topological charges. ω₀ and ω are the central angular frequency and the circular frequency of the carrier, respectively. Finally, e_x and e_y are the unit vectors along the x and y directions, respectively.

We assume here that the incident femtosecond laser pulse is tightly focused into a circularly symmetric focusing system, as illustrated in Fig. 1. According to the Richards-Wolf vector diffraction theory [48], the electric field for each spectral component in the focal region can be expressed as follows:

(2)$$\begin{aligned} &\left[ {\begin{array}{c} {{E_x}({r,\phi ,z,\omega } )}\\ {{E_y}({r,\phi ,z,\omega } )}\\ {{E_z}({r,\phi ,z,\omega } )} \end{array}} \right] ={-} \frac{{ikf}}{{2\mathrm{\pi }}}\int\limits_0^\alpha {\int\limits_0^{2\mathrm{\pi }} {\sqrt {\cos \theta } \exp \{{ik[{ - r\sin \theta \cos ({\varphi - \phi } )+ z\cos \theta } ]} \}\left[ {\begin{array}{c} { - \sin \varphi }\\ {\cos \varphi }\\ 0 \end{array}} \right]} } \\&{\left( {\frac{{\sqrt 2 \beta \sin \theta }}{{\sin \alpha }}} \right)^{|l |}}\exp \left( { - \frac{{{\beta^2}{{\sin }^2}\theta }}{{{{\sin }^2}\alpha }}} \right)\exp ({il\varphi } )\exp \left[ { - \frac{{{\tau^2}{{({\omega - {\omega_0}} )}^2}}}{{8\ln 2}}} \right]\sin \theta \textrm{d}\varphi \textrm{d}\theta \end{aligned}$$

where f is the focal length, c is the speed of light, and α is the maximum convergence angle of half; its value is equal to arcsin(NA/n), where NA is the numerical aperture and n is the refractive index of the image space; β is the ratio of the pupil radius to the beam waist. For the azimuthally polarized Laguerre-Gaussian input beam, the corresponding longitudinal polarization vector in the focal region is 0, so there is no longitudinal electric field component [49].

Fig. 1. Schematic illustration of the focusing system and the coordinate system used in the calculation, where the focal plane is located at z = 0.

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By performing a Fourier transform, each spectral component is coherently superimposed to give the electric field of the femtosecond pulse in the focal field region as follows [47]:

(3)$${\boldsymbol{E}_j}({r,\phi ,z,t} )= \frac{1}{{2\mathrm{\pi }}}\int\limits_0^\infty {{\boldsymbol{E}_j}({r,\phi ,z,\omega } )\exp ({ - i\omega t} )\textrm{d}\omega } \;\;\;({j = x,y,z} )$$

Unless otherwise specified, all numerical calculations in this work are performed using the following values: l = 1, λ = 800 nm, τ = 5 fs, f = 3 mm, NA = 0.95, n = 1, and β = 1.

3. Results and discussion

3.1 Bright-dark alternating distribution of the tightly focused field

Using the analytical model established in the previous section, we first study the intensity distribution within the focal plane (z = 0). Figure 2 illustrates the intensity distributions of the normalized radial field component ${|{{E_\rho }} |^2}$, the azimuthal field component ${|{{E_\phi }} |^2}$, and the total field ${|{{E_{total}}} |^2}$, respectively, at different instants. In general, the changes in ${|{{E_\rho }} |^2}$. ${|{{E_\phi }} |^2}$, and ${|{{E_{total}}} |^2}$ within the focal region with respect to time are very obvious. At any time, ${|{{E_\rho }} |^2}$ shows a hot spot [see Fig. 2(a1)–2(a5)]. ${|{{E_\phi }} |^2}$ maintains a double-ring distribution [see Fig. 2(b1)–2(b5)]. ${|{{E_{total}}} |^2}$ represents the numerical superposition of ${|{{E_\rho }} |^2}$ and ${|{{E_\phi }} |^2}$. The fields at the two conjugate time positions have the same intensity distributions. Specifically, when t ranges from −10 fs to 0 fs, the relative value of ${|{{E_\rho }} |^2}$ increases gradually, while that of ${|{{E_\phi }} |^2}$ decreases gradually. The intensity distribution of the dominant ${|{{E_{total}}} |^2}$. gradually transitions from ${|{{E_\phi }} |^2}$ to ${|{{E_\rho }} |^2}$, which is caused by mutual conversion of ${|{{E_\rho }} |^2}$ and ${|{{E_\phi }} |^2}$. In comparison to the period where t ranges from −10 fs to 0s, when t ranges from 0 fs to 10 fs, ${|{{E_\rho }} |^2}$, ${|{{E_\phi }} |^2}$, and ${|{{E_{total}}} |^2}$ show the opposite evolutionary process. We see that similar bright spots and optical cages appear in the focal field and annihilate alternately with time. This particular optical field evolution is dependent on the time phase factor, which theoretically can identify and trap particles with different refractive indices [50–52]. To illustrate these aspects in more detail, Visualization 1 provides a vivid demonstration of the dynamic evolution of the total field and its two field components.

Fig. 2. Intensity distributions within the focal plane at different time positions (i.e., t = 0 fs, ± 5 fs, and ±10 fs). The three rows from top to bottom represent the intensity distributions of (a1)-(a5) the radial field component, (b1)-(b5) the azimuthal field component, and (c1)-(c5) the total field, respectively. All strengths are normalized with respect to the maximum value at the corresponding time. Unless otherwise specified, the calculation range used in this work is always 8λ × 8λ.

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3.2 Distorted phase of the tightly focused field

The phase changes with the intensity within the focal field. Figure 3 shows the phase distributions of the radial field component and the azimuthal field component, as denoted by $arg[{{E_\rho }} ]$ and $arg[{{E_\phi }} ]$, respectively, at different moments. At the static time [see Fig. 3(a3) and 3(b3)], $arg[{{E_\rho }} ]$ remains constant along the radial direction, whereas $arg[{{E_\phi }} ]$ shows a π-phase jump. At non-static times [see Fig. 3, with the exception of Fig. 3(a3) and 3(b3)], both $arg[{{E_\rho }} ]$ and $arg[{{E_\phi }} ]$ are distorted. These components have opposite torsions at two conjugate times; this behavior is related to the time-assistant Gouy phase shift [53,54]. Visualization 2 shows the time evolution of both $arg[{{E_\rho }} ]$ and $arg[{{E_\phi }} ]$ in detail.

Fig. 3. Phase distributions in the focal plane at different time positions (t = 0 fs, ± 5 fs, and ± 10 fs). The two rows from top to bottom represent the phase distributions of (a) the radial field component and (b) the azimuthal field component, respectively.

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3.3 Polarization topological behavior of dynamic evolution in tightly focused field

Polarization plays an essential role in the interactions between light and matter. Overall, the time evolution of the polarization within the focus is very complex, and the orbital-to-spin conversion leads to the existence of a longitudinal polarization component. Simultaneously, we note that the center region always has a left-handed circular polarization at any time [see Fig. 4], which is called the C point, and thus the center of focus is always a hot spot [see Figs. 2(c1)-2(c5)]. Specifically, over the range from t = −10 fs to 0 fs, the ellipticity, the orientation, and the handedness of the local polarization ellipse in the outer region that is dominated by ${|{{E_\phi }} |^2}$.all change with time. Within the central region dominated by ${|{{E_\rho }} |^2}$, the ellipticity and the orientation change with time, but the handedness remains unchanged [see Fig. 4]. The handedness of the local polarization ellipse within the central region is related to the sign of the topological charge of the incident field [55], and is independent of time. The polarization evolution during the period from t = 0 fs to 10 fs within the focal plane has the opposite characteristics to that occurring during the period from t = −10 fs to 0 fs [see Fig. 4]. When t = 0, the polarization distribution in the central region, which is dominated by ${|{{E_\rho }} |^2}$, shows perfect cylindrical symmetry, and there is a hybrid distribution of linear, circular and elliptical polarization states [see Fig. 4(c)]. At the same time, the local polarization ellipse in the central region not only changes its handedness in the radial direction, but also changes its ellipticity and orientation; in contrast, it only changes its orientation in the azimuthal direction. In summary, a significant polarization evolution occurs along the time axis in the focal region, particularly in the peripheral region dominated by ${|{{E_\phi }} |^2}$. Visualization 3 provides a more specific illustration of the time evolution process of the polarization in the focusing field.

Fig. 4. Polarization distributions in the focal plane at different time positions (t = 0 fs, ± 5 fs, and ±10 fs), where green and red denote the left and right rotations of the polarization ellipses, respectively.

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The time evolution of the polarization will inevitably lead to changes in the topological structure. To reveal this dynamic polarization topological behavior, we refer to the normalized Stokes parameters S₁, S₂, and S₃ [56–58]:

(4)$${S_1}\textrm{ = }\frac{{{{|{{E_x}} |}^2} - {{|{{E_y}} |}^2}}}{{{{|{{E_x}} |}^2} + {{|{{E_y}} |}^2}}}$$

(5)$${S_2}\textrm{ = }\frac{{2\textrm{Re} [{E_x^ \ast {E_y}} ]}}{{{{|{{E_x}} |}^2} + {{|{{E_y}} |}^2}}}$$

(6)$${S_3}\textrm{ = }\frac{{2{\mathop{\rm Im}\nolimits} [{E_x^ \ast {E_y}} ]}}{{{{|{{E_x}} |}^2} + {{|{{E_y}} |}^2}}}$$

Figure 5 shows the contours for the normalized Stokes parameters S₁ = 0, S₂ = 0, and S₃ = 0 at different times on the focal plane. We know that the point where S₁ = S₂ = 0 is a C point (or C line); the polarization state on this point is a circular polarization, and the polarization state around the point is an elliptical polarization. S₃ = 0 represents the L line, and the handedness of the polarization ellipse on this line is uncertain. In other words, the polarization state on the L line is a linear polarization, and the state around the L line is elliptical polarization. We find that S₁ = 0 and S₂ = 0 form two twisted lines [see Figs. 5(a1)-5(a5), with the exception of Fig. 5(a3)], and these lines become distorted over time. We find that the two sets of lines always intersect at the center of the focus at all times, and that they also exhibit opposite distortions at two conjugate time positions. When located at the static time (t = 0), the lines are no longer distorted and form four straight lines that intersect at the center, along with several overlapping concentric circles [see Fig. 5(a3)]. In other words, the polarization state at the center of the focal plane is always a circular polarization (C point) and is independent of time. When t = 0, the circular polarization within the focal plane forms several circles (C lines), and these circles are surrounded by an elliptical polarization. When S₃ = 0 (L line), only circles appear, and the sizes and numbers of these circles vary over time [see Figs. 5(b1)-5(b5)]. Note that the location of the L line is the line where the SAM is zero, and the polarization ellipses on the two sides of the line have opposite handedness. The calculation results are consistent with those reported in previous research and thus provide strong theoretical support for further study of the AM of the focal field [57]. Visualization 4 shows the time evolution of the polarization topology within the focal field in detail.

Fig. 5. Theoretical calculation isolines for the normalized Stokes parameters at the different time positions (t = 0 fs, ± 5 fs, and ±10 fs). The two rows from top to bottom are the cases where (a1)-(a5) S₁ = S₂ = 0 and (b1)-(b5) S₃ = 0, respectively. The red solid lines, the blue dotted lines, and the green solid lines represent the results for S₁ = 0, S₂ = 0, and S₃ = 0, respectively.

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3.4 Time evolution of spin-orbit HE in tightly focused field

Next, we mainly study the AM in a femtosecond light field within the tight focusing system. The SAM and the OAM generated by the focal field can be expressed as follows [59,60]:

(7)$${\boldsymbol{J}_S} = \frac{{{\varepsilon _0}}}{{4{\omega _0}}}{\mathop{\rm Im}\nolimits} [{{\boldsymbol{E}^ \ast } \times \boldsymbol{E}} ]$$

(8)$${\boldsymbol{J}_O} = \frac{{{\varepsilon _0}}}{{4{\omega _0}}}\boldsymbol{r} \times {\mathop{\rm Im}\nolimits} [{{\boldsymbol{E}^ \ast } \cdot (\nabla )\boldsymbol{E}} ]$$

where ε₀ is the dielectric constant in a vacuum and a superscripted asterisk indicates a complex conjugate operation.

For the rotationally symmetrical light field, the transverse component of the AM density shows an odd-symmetric distribution on the focal plane, and thus only the longitudinal component contributes to the net AM of the field. The emergence of the longitudinal SAM illustrates an interesting optical process called orbital-spin conversion. Figure 6 depicts the longitudinal SAM and OAM normalized at different time on the focal plane. Along the entire timeline, the SAM distribution within the central region that is dominated by ${|{{E_\rho }} |^2}$ remains basically unchanged. When the topological charge is 1, this distribution is positive in the central region, and the sign is reversed along the radial direction [see Figs. 6(a1)-6(a5)], the OAM, which is always positive, presents as doughnut shapes, and it varies in spatial size with time [see Figs. 6(c1)-6(c5)]. When the topological charge of the incident beam changes from 1 to -1, even though the SAM and OAM densities are in opposite directions, the spin-orbit Hall effect always occurs [see Figs. 6(b1)-6(b5) and Figs. 6(d1)-6(d5)]. The SAM and the OAM at the two conjugate time positions have the same spatial distribution. In detail, from the time when t ranges from −10 fs to 0 fs, the relative value of the SAM in the peripheral region that is dominated by ${|{{E_\phi }} |^2}$ decreases continuously. The ellipticity of the polarization ellipse within the central region changes slightly, and its handedness remains unchanged [see Figs. 4(a)-4(c)], and thus the change in SAM in the region is not obvious. Simultaneously, the size of the OAM shown as the doughnut shape continues to decrease [see Figs. 6(c1)-6(c3) and 6(d1)-6(d3)]. The different AMs move closer to each other over time. In the period where t ranges from 0 fs to 10 fs, the two AMs on the focal plane show evolutionary trends that are opposite to those in the previous stage. In general, with the evolution of time, the peak distance of the SAM and the OAM become closer and farther away with time, which represents the dynamic performance of the optical spin-orbit HE under the tight focusing condition of the femtosecond light field. Visualization 5 shows the dynamic evolution of the spin-orbit HE under the tight focusing condition of the femtosecond light field in detail.

Fig. 6. AM distributions within the focal plane at different time positions (t = 0 fs, ± 5 fs, and ±10 fs). The four rows from top to bottom represent (a1)-(a5) the SAM and (c1)-(c5) the OAM when the topological charge of the incident beam is 1, and (b1)-(b5) the SAM and (d1)-(d5) the OAM when the topological charge of the incident beam is −1, respectively. Orange (positive) and blue (negative) distributions indicate the forward and reverse directions along the optical axis, respectively.

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To show the time evolution of the spin-orbit HE more intuitively here, we have drawn the one-dimensional linear distributions of the normalized longitudinal SAM and OAM along the x-axis direction at different times [see Figs. 7(a1)-7(a5)]. We find that the variation of the AM along the x-axis direction is very obvious, and that its distribution at the corresponding conjugate time position is the same. Additionally, the SAM at x = 0 is always positive and its distribution remains basically unchanged. The OAM is also always positive, and it gradually approaches and then moves away from the center over time, thus forming a time-varying spin-orbit HE. The evolution process is shown in detail in Visualization 6.

Fig. 7. One-dimensional AM distributions along the x-axis direction on the focal plane at the different time positions (t = 0 fs, ± 5 fs, and ±10 fs). The four lines from top to bottom represent different time parameters: (a1)-(a5) l = 1, τ = 5 fs and λ = 800 nm, (b1)-(b5) l = 1, τ = 15 fs and λ = 800 nm, (c1)-(c5) l = 1, τ = 5 fs and λ = 500 nm and (d1)-(d5) l = −1, τ = 5 fs, λ = 500 nm. The blue and orange lines denote the SAM and the OAM, respectively. The calculation size along the x-axis direction is 8λ.

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For a femtosecond optical field, the pulse width and the central wavelength will inevitably affect the AM distribution. To confirm this statement, we present several examples that are used to analyze the dependence of the AM distribution on the pulse width and the central wavelength. Figures 7(b1)-7(b5) show the normalized AM distribution along the x-axis direction in the focal plane at different times when τ = 15 fs. Both the SAM and the OAM remain basically unchanged along the entire time axis, and the dynamic spatial separation between the two is not obvious [see Figs. 7(b1)-7(b5)]; this indicates that the time-varying spin-orbit HE is dependent on the pulse width. The pulse width is inversely proportional to the spectral width, and thus for a wider pulse, the spin-orbit HE is similar to that for a monochromatic light field. The change process is shown in detail in Visualization 7. Furthermore, when the central wavelength of the incident light field is 500 nm, the SAM in the central region does not change significantly along the time axis, and although the relative value of the OAM changes continuously, the degree of separation between the two remains basically unchanged [see Figs. 7(c1)-7(c5)]. This occurs because, when the center wavelength of the pulse decreases, the center frequency will increase, and the monochromaticity of the light will then be enhanced; this means that the time evolution of the spin-orbit HE is no longer obvious. The change process is shown in detail in Visualization 8. Furthermore, it can be observed that the spin-orbit Hall effect is always present when the topological charge of the incident beam changes from 1 to -1 [see Figs. 7(d1)-7(d5)]. Visualization 9 shows this process in detail.

4. Conclusion

In summary, we have theoretically studied the spin-orbit HE of an azimuthally polarized femtosecond light field in a tightly focused system. Using the Richards-Wolf diffraction integral and the Fourier transform method, we obtained explicit expressions for all components of the electric field within the focal volume of the tightly focused system. The results show that the focusing field has ultrafast bright and dark alternation characteristics and a distorted phase distribution. In addition, we also studied the polarization distribution during the dynamic evolution of the focal field. Furthermore, the time-varying polarization topological behavior in the focal field is demonstrated using the Stokes parameters. Most importantly, by calculating the AM distribution of the focal field, we determined that the spin-orbit HE of the femtosecond light field within the tightly focused system is dynamic. The time phase factor is important for realization of these special characteristics of the focused light field. Moreover, wider pulse widths and shorter central wavelengths both have a positive effect on the monochromaticity of the light. The findings in the research reported here may have important and potentially important applications in optical manipulation, particularly for control of particle rotation on the femtosecond scale.

Funding

Natural Science Foundation of Shandong Province (ZR2020MA087, ZR2021YQ02); National Natural Science Foundation of China (12074224, 61975128).

Disclosures

The authors declare no conflicts of interest.

Data availability

All the initial data of figures that support the findings of this study are available from the corresponding author upon reasonable request.

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Time-varying optical spin-orbit Hall effect in tightly focused femtosecond optical field

Abstract

1. Introduction

2. Theoretical model

3. Results and discussion

3.1 Bright-dark alternating distribution of the tightly focused field

3.2 Distorted phase of the tightly focused field

3.3 Polarization topological behavior of dynamic evolution in tightly focused field

3.4 Time evolution of spin-orbit HE in tightly focused field

4. Conclusion

Funding

Disclosures

Data availability

References

Supplementary Material (9)

Data availability

Cited By

Figures (7)

Equations (8)

Optics Express

Name	Description
Visualization 1	Visualization 1 provides a vivid demonstration of the dynamic evolution of the total field and its two field components.
Visualization 2	Visualization 2 shows the time evolution of the phase distributions of the radial field component and the azimuthal field component in detail.
Visualization 3	Visualization 3 provides a more specific illustration of the time evolution process of the polarization in the focusing field.
Visualization 4	Visualization 4 shows the time evolution of the polarization topology within the focal field in detail.
Visualization 5	Visualization 5 shows the dynamic evolution of the spin-orbit Hall effect under the tight focusing condition of the femtosecond light field in detail.
Visualization 6	Visualization 6 is the linear distribution of time-varying angular momentum along the x-axis.
Visualization 7	Visualization 7 is the linear distribution of time-varying angular momentum along the x-axis when the pulse width is 15 fs.
Visualization 8	Visualization 8 shows the linear distribution of the time-varying angular momentum along the x-axis when the central wavelength is 500 nm.
Visualization 9	Visualization 9 shows the linear distribution of the time-varying angular momentum along the x-axis when the topological charge is -1 and the central wavelength is 500 nm.