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Abstract
Spin and orbital angular momenta are two of the most fundamental physical quantities that describe the complex dynamic behaviors of optical fields. A strong coupling between these two quantities leads to many intriguing spatial topological phenomena, where one remarkable example is the generation of a helicity-dependent optical vortex that converts spin to orbital degrees of freedom. The spin-to-orbit conversion occurs inherently in lots of optical processes and has attracted increasing attention due to its crucial applications in spin–orbit photonics. However, current researches in this area are mainly focused on the monochromatic optical fields whose temporal properties are naturally neglected. In this work, we demonstrate an intriguing temporal evolution of the spin-to-orbit conversion induced by tightly-focused femtosecond optical fields. The results indicate that the conversion in such a polychromatic focused field obviously depends on time. This temporal effect originates from the superposition of local fields at the focus with different frequencies and is sensitive to the settings of pulse width and central wavelength. This work can provide fundamental insights into the spin–orbit dynamics within ultrafast wave packets, and possesses the potential for applications in spin-controlled manipulations of light.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Light is an electromagnetic radiation whose dynamic behaviors can be physically characterized by a number of conserved quantities, such as energy, momentum, and angular momentum (AM) [1]. The AM of light is naturally divided into spin and orbital contributions, which are associated with the polarization and spatial degrees of freedom of light, respectively [2]. Since the seminal work by J. H. Poynting [3], it has been a consensus that a plane wave, if circularly polarized, carries an intrinsic spin angular momentum (SAM) of σħ per photon, where σ = ±1 is the helicity parameter and ħ is the reduced Plank’s constant. Unlike the photonic spin which is truly intrinsic, the orbital angular momentum (OAM) of light can be either intrinsic or extrinsic [4]. The intrinsic part is usually linked to a helical phase structure within optical vortices, while the extrinsic part is associated with the beam trajectory and depends on the choice of coordinate origin [5]. For a light beam carrying a phase factor of exp(iℓφ), it will possess an intrinsic OAM equivalent to ℓħ per photon when the topological charge ℓ takes non-zero values [6–8]. Generally, SAM and OAM are considered two separable and independent quantities when light beams propagate paraxially in free space or through homogeneous and isotropic media [9]. This separation, however, is not always true, because these two quantities can be strongly coupled in a more general case beyond the paraxial limitation or optical homogeneity [10–13].
Over the past 30 years, the spin–orbit coupling (SOC) of light has drawn numerous attention due to its fundamental and emerging applications in optical tweezing [14–16], imaging [17], metrology [18,19], and quantum entanglement [20,21]. As an intrinsic property of light, SOC widely exists in lots of basic optical processes, including light scattering [22], focusing [23–29], propagation in photonic crystal waveguides and multihelicoidal fibers [30–32], and reflection/deflection at sharp interfaces [33]. Different optical processes also result in different phenomena and, therefore, should be discussed under different frameworks. Here, we focus only on the spin-to-orbit conversion, a phenomenon that manifests as the generation of a helicity-dependent optical vortex in the output field [25,34]. The spin-to-orbit conversion plays an important role in light–matter interactions and has been extensively explored in various optical systems [34–38]. However, current researches in this area are mainly restricted to the monochromatic optical fields of which the temporal properties are usually neglected. According to the Fourier transform between time and frequency, femtosecond optical fields are inherently polychromatic with a very broad spectrum [39]. This property also makes it possible to sculpt light in the space-time domain by tailoring the amplitude and phase of spectral components [40–45]. Very recently, the spin–orbit interactions in femtosecond optical fields have been successfully detected by a photoionization experiment [46]. This novel approach has the potential to characterize the complex spatial structures of strong pulsed fields, but it still has challenges to provide a complete spatiotemporal analysis of coupled spin‒orbit phenomena due to the missing temporal information.
Therefore, in this work, we provide an analytical model to reveal the nontrivial temporal dynamics of the spin-to-orbit conversion induced by tightly focused circularly polarized femtosecond optical fields. Based on the Richards-Wolf diffraction integral and Fourier transform method, we investigate the dynamics of field distributions and polarization properties, and further calculate both local densities and mean (expectation) values of SAM and OAM. It shows that the conversion between SAM and OAM is time-dependent for optical pulses, which is in sharp contrast to monochromatic continuous waves. Furthermore, the effect of pulse width and central wavelength is numerically explored. The results indicate that the temporal evolutions of the SAM and OAM of the focused field are sensitive to the pulse width and the central wavelength.
2. Analytical theoretical model
In the focusing process of a circularly polarized light, there generates a conical k-vectors distribution accompanied by a geometric phase originating from variations of the wavevector direction [22]. This process also induces a spin-to-orbit conversion effect and produces a longitudinally polarized optical vortex in the output field, which is well-known in monochromatic continuous waves [47]. For an ultrafast wave packet, however, the AM conversion is a dynamic process, and the SAM and OAM of the focused field are expected to be time-varying. To prove this, we considered the incident field as a paraxial, circularly polarized, z-propagating femtosecond optical pulse, as shown in Fig. 1(a). Without loss of generality, the pulse is assumed to possess a Gaussian profile in both the spatial and temporal/spectral domains. Then, the incident field could be expressed in the spatial-spectral domain as [48]:
Fig. 1. Tight focusing of a circularly polarized femtosecond optical pulse. (a) Schematic illustration of the focusing system and coordinate system followed in our calculations, the focal plane is located at z = 0. (b)‒(d) Intensity and phase distributions in the focal plane at different temporal locations t = 0 fs, and ±20 fs. Shown are the intensity patterns of the transverse field component (top panels), the longitudinal field component (middle panels), and the phase distributions of the longitudinal field component (bottom panels). Since absolute values are irrelevant, all intensity patterns are normalized relative to the maximum in each panel. The instantaneous polarization states are also drawn on the intensity patterns in (b), where the left- and right-handed elliptical (or circular) polarizations are shown in green and orange, respectively. The other parameters are chosen as: λ0 = 800 nm, σ = + 1, w = 2.7 mm, τ = 10 fs, f = 3 mm, NA = 0.9, and n = 1. Both the intensity, polarization, and phase distributions exhibit a time-dependent morphology, see Visualization 1 for these dynamic evolutions.
When the incident pulse is focused by a high numerical-aperture (NA) objective lens, slight variations come up for the different spectral components due to their differences in wavevectors and amplitudes. The focal field corresponding to each spectral component can be calculated using the diffraction integrals developed by B. Richards and E. Wolf [49], which can be expressed in a Cartesian basis as:
Here, (ρ, ϕ, z) stands for the cylindrical coordinate system applied in the focal region, k = nω/c is the wavenumber, n is the refractive index of the medium, c is the light speed in a vacuum, f is the focal length, ${\tilde{E}_0}$ is the complex amplitude of each spectral component of the incident pulse, α = arcsin(NA/n) is the semi-aperture angle of the objective lens, and Jξ (·) is the ξ-th order Bessel function of the first kind. It can be seen clearly that the longitudinal field ${\tilde{E}_z}$ carries a helical phase with topological charge ℓ = + 1 or −1, depending on the helicity of the incident circular polarization, which is a direct consequence of the spin-to-orbit conversion and frequency independent.
The spatial-temporal E-field in the focal region is a superposition of electromagnetic oscillations with different frequencies over the bandwidth, so it can be obtained by the Fourier transform:
3. Results and discussion
3.1 Temporal evolution of the focal field and AM distributions
The intensity patterns of the transverse field component, the longitudinal field component, as well as the phase diagrams for the longitudinal field at three different times t = 0 fs and ± 20 fs are plotted in Figs. 1(b), 1(c), and 1(d), respectively. From these distributions, one can find that the transverse field component [Fig. 1(b1–b3)] appears as a hot spot at any moment, while the longitudinal counterpart [Fig. 1(c1–c3)] has a doughnut-shaped profile and exhibits a helical phase distribution [Fig. 1(d1–d3)] resulted from the spin-to-orbit conversion (σ = + 1 to ℓ = + 1, as predicted by Eq. (4)). Due to the excellent symmetry of the focusing system and the incident light, all intensity profiles are rotationally symmetric with respect to the z-axis. To better visualize temporal variations, these intensity plots are normalized relative to their own maximum values to eliminate the effects caused by changes in pulse strength over time. Evidently, the spatial size is time-varying for both transverse and longitudinal field components, which increases first from t = ‒20 fs to t = 0 fs and then decreases up to t = 20 fs. It is also noted that the fields share the same intensity profile at two conjugate temporal locations.
Furthermore, the polarization states (in-plane projections) are overlaying on the transverse field component to observe the evolving polarization during the focusing process, where the left- and right-handed polarizations are plotted in green and orange, respectively. Since the incident light field is left-handed circularly polarized without a helical phase before focusing, it carries only SAM (SAM = 1ħ per photon). After focusing, the input SAM will be partly transferred to the OAM of the focused field. Consequently, the polarization distributions at the mainlobe are dominated by the elliptical polarization states with left-handedness (0 < SAM < 1ħ), except for a C-point singularity at the center due to the absence of the longitudinal component Ez. From the time slices illustrated in Fig. 1(b1–b3), it can be seen that there is an obvious polarization evolution along the time-axis, especially in the outer region where Ez-field dominates. This implies that Ez-field plays an important role in polarization evolution. For the phase distributions of Ez component, as shown in Fig. 1(c1–c3), a standard helical phase without any distortion appears only at t = 0 fs, but at other times the phase becomes distorted and twisted (see Visualization 1 for the full dynamic process shown in Fig. 1).
The AM properties of light are generally associated with its polarization and spatial degrees of freedom. Therefore, the time-varying intensity, polarization, and phase morphologies shown in Fig. 1 will correspond to a time-varying AM distribution. Local AM density will provide an effective measure for this dynamics process. Here, referring to the expressions used in Ref. [40], the local SAM and OAM densities (in units of ħ) are defined as:
These distributions are shown in Fig. 2, where one can see how the two distinct AM forms exhibit different behaviors and evolve in the time domain. As shown, the distributions of the longitudinal SAM and OAM densities are both radially varied, exhibiting similar multiple-concentric-ring patterns. Meanwhile, the differences are also significant between the two quantities. On the one hand, the values of the local SAM densities [Fig. 2(a1–a3)] are strictly restricted by |Sz| ≤ 1ħ, but this does not hold for the OAM densities [Fig. 2(b1–b3)]. On the other hand, the distributions of the two AM densities are exactly complementary, which means that the SAM and OAM in such focusing geometry are converted only locally. The total AM density, as a sum of the two densities, is spatially homogenous and equal to the initial value of the input field, ruled by the conservation law. In addition to these spatial characteristics, the computed AM distributions also possess rich time-varying features (see Visualization 2 for a better understanding of the dynamics).
Fig. 2. AM distributions in the focal plane. (a) Local SAM densities and (b) OAM densities at different temporal locations t = 0 fs, and ±20 fs, calculated utilizing the electric field shown in Fig. 1. Sz and Lz represent the longitudinal components of the SAM and OAM densities, respectively. See Visualization 2 for the evolution of the local AM densities.
The mean values of SAM and OAM per photon are another two crucial parameters for evaluating such complex spin‒orbit dynamics, given by [9,40]:
These two equations indicate that the mean values represent in fact the spatially weighted averages of the local SAM and OAM densities defined before. They provide a more intuitive way to visualize the temporal variations of the AM distribution than a 2D density map. As depicted in Fig. 3(a), the mean values of SAM and OAM per photon vary continuously but oppositely over the time range from ‒20 fs to 20 fs, which leaves the summation invariant, as expected from the conservation rules. The distribution curves are symmetric with respect to t = 0 fs, and at this moment the conversion efficiency between SAM and OAM is the lowest ∼25%, but it increases to ∼50% at t = ±20 fs. It should be pointed out that in defocus planes (z ≠ 0), the SAM and OAM are also time-varying, but their distributions are no longer symmetric relative to the moment t = 0 fs. To confirm that the observed dynamic conversion only appears in a focused pulsed field, the same distribution curves of a focused continuous wave (CW) is also calculated and plotted in Fig. 3(b). Evidently, the conversion process is static (not varying in time) for the CW mode.
Fig. 3. Time-dependent mean AM per photon. (a) Mean values of the SAM (blue line) and OAM (orange line) per photon as a function of time t, ranging from ‒20 fs to 20 fs. (b) Same plots as (a) except that the incident field is a continuous wave.
Given the significant differences between the two cases, it is not difficult to provide a qualitative explanation for this temporal effect by comparing the fundamental properties of the pulse and CW mode. In fact, the essential difference between these two fields originates from the spectrum (i.e., the CW mode is monochromatic and the pulse is polychromatic). When a femtosecond optical pulse is tightly focused, different spectral components will also lead to different field distributions, where the spatial extension of a focused field decreases with increasing frequency. The instantaneous electric field in the focal region is actually described by the superposition of the Fourier spectral components. Due to the offsets in frequencies and spatial profiles, the superposition between these spectral components will induce variations to the instantaneous field, and affect the relative amplitudes and phases between the three orthogonal field components. Therefore, an evident time-varying morphology can be identified in intensity, polarization, and phase distributions from the time slices in Fig. 1. Some of the recent studies [50,51] have also demonstrated that when an optical field contains multiple frequencies, the motion of the field vector becomes more complicated and even exhibits a chaotic-like trajectory in the time domain. This time-dependent polarization will cause a SAM fluctuation in the focused field, which will also make the OAM change accordingly to satisfy the conservation law [see Fig. 3(a)].
3.2 Effect of pulse width and central wavelength on the temporal evolution
Based on the above analyses, the temporal effect of the spin-to-orbit conversion depends on the degree of monochromaticity of light, which is usually characterized by the ratio Δω/ω0. For a quasi-monochromatic light, the criterion meets the condition Δω/ω0 ≪ 1. Then, changing the pulse width or central wavelength of the incident pulse will inevitably affect the evolution of the AM distributions. To confirm this, the dependence of AM distributions on pulse width and central wavelength is analyzed in detail.
Figure 4 shows the temporal evolution of the local AM densities in the focal plane for the case of pulse width τ = 50 fs. Compared with the results in Fig. 2, the temporal evolution of the AM distributions becomes less obvious, indicating that the dynamic spin-to-orbit conversion depends strongly on the pulse width. As depicted in Figs. 4(a) and 4(b), the local SAM and OAM densities remain almost steady over the time range from t = ‒100 fs to t = 100 fs (see Visualization 3 for the evolution of the local AM densities). This difference is derived from the change in the monochromaticity of light. Since the spectrum width and pulse width are inversely proportional, the ratio Δω/ω0 is smaller for a wider pulse, and thus the evolution acts closer to the case for a monochromatic continuous wave. In contrast, when increasing the central wavelength of the pulse, the monochromaticity of light will get worse due to the decrease of central frequency ω0. This change is expected to induce dramatic temporal variations in the AM distributions. As proof, the local AM densities for the case of central wavelength λ0 = 1.5 µm are depicted in Fig. 5. Notably, a larger central wavelength results in more intuitive and significant dynamics, one then finds strongly time-varying features from the AM distributions (see also Visualization 4).
Fig. 4. Effect of pulse width on the AM distributions. (a) Local SAM densities and (b) OAM densities in the focal plane at different temporal locations t = 0 fs, and ±100 fs, for the case of pulse width τ = 50 fs. In order to record the full dynamic process, the time range in all calculations is ‒2τ to 2τ, which covers the whole temporal envelope of a single pulse. See Visualization 3 for the evolution of the local AM densities.
Fig. 5. Effect of central wavelength on the AM distributions. (a) Local SAM densities and (b) OAM densities in the focal plane at different temporal locations t = 0 fs, and ±20 fs, for the case of central wavelength λ0 = 1.5 µm. The pulse width is τ = 10 fs. See Visualization 4 for the evolution of the local AM densities.
The effect of pulse width and central wavelength on the temporal evolution can also be confirmed from the time-dependent mean AM per photon distributions. When the pulse width τ = 50 fs, the mean values of SAM and OAM remain almost unchanged along the time-axis, as shown in Fig. 6(a). However, for the other case where the central wavelength λ0 = 1.5 µm, remarkable temporal variations can be observed from the distributions illustrated in Fig. 6(c). To quantitatively analyze the effect, we take the fluctuation range of the mean AM as the index to evaluate the temporal variations. The dependence of the fluctuation ranges on pulse width is plotted in Fig. 6(b). It shows that as the pulse width increases from 10 fs to 100 fs, the fluctuation range of the mean AM decreases smoothly from 0.256ħ to 0.003ħ. The dependence of the fluctuation ranges on the central wavelength is also explored. As depicted in Fig. 6(d), when the central wavelength increases gradually from 0.8 µm to 1.8 µm, the fluctuation range of the mean AM changes accordingly from 0.256ħ to 0.312ħ, showing an opposite trend to Fig. 6(b). The results in Figs. 4‒6 reveal the dominant role of the monochromaticity of light in the temporal dynamics, which is self-consistent with the qualitative explanation made in the previous section.
Fig. 6. Manipulating the temporal evolution of the mean AM per photon. (a) Time-dependent mean values of the SAM and OAM per photon when the pulse width τ = 50 fs. (b) Dependence of the fluctuation ranges of the mean AM on the pulse width. (c) Time-dependent mean values of the SAM and OAM per photon when the central wavelength λ0 = 1.5 µm. (d) Dependence of the fluctuation ranges of the mean AM on the central wavelength.
It should also be emphasized that these results can be directly generalized to other SOC phenomena, e.g., spin Hall effect [19,52] and orbit-to-local-spin conversion [46,53]. Here, we consider the spin-to-orbit conversion just for the simplicity of theoretical description. As explained, the temporal effect demonstrated in this work is caused by the superposition of multiple spectral components with different spatial distributions. Such temporal effect should exist in a variety of optical systems requiring the participation of ultrafast wave packets, especially when the paraxial limitation and optical homogeneity are broken. Sculpting light in both space and time has gained increasing interest recently [39,54]. Motivated by recent advances, several research groups have attempted to manipulate the spatiotemporal structures of light and related SOC phenomena under tight focusing conditions. In the previous work [48], we demonstrated that a time-varying OAM can be formed by focusing two time-delayed femtosecond vortex pulses with orthogonal circular polarizations. A similar scheme, with two time-delayed vortex pulses carrying different OAMs, time-varying SOC phenomena can also be found in the latest research [55]. Although these two works have shown some time-varying structures of tightly focused structured optical pulses, the temporal variations of the focused field stem from the time-varying features of the incident field, which is completely different from the mechanism in this work.
4. Conclusion
In summary, we revisited the focusing behaviors of circularly polarized femtosecond optical fields from the perspective of temporal effect. Based on the Richards-Wolf diffraction integral and Fourier transform method, we explored the spatiotemporal field distributions and AM properties of the focused fields in detail, and demonstrated an intriguing temporal evolution manifested in the spin-to-orbit conversion. Unlike the AM conversion in CW mode which is almost static, the conversion process within a femtosecond optical field exhibits definite time-varying features. The temporal effect is derived from the superposition of multiple Fourier spectral components, where the discrepancies in frequencies and spatial profiles induce a time-dependent morphology to the resultant field distributions. Consequently, the dynamic evolution is affected by the monochromaticity of light and depends strongly on the pulse width and central wavelength. For a wider pulse width, the local AM densities as well as the mean values per photon remain almost steady along the time-axis, behaving similarly to a CW mode. However, by increasing the central wavelength, which brings a negative influence on the monochromaticity, a more intuitive and significant temporal evolution will be observed. Interestingly, although one can expect a dynamic AM distribution from a time-dependent morphology of the focused field, the total AM, as the sum of SAM and OAM, is always conserved (not varying with time). The results of our work provide a theoretical platform for investigations of the complex spin–orbit dynamics within ultrafast wave packets. Since the AM properties are highly important for light–matter interactions, the dynamic AM distributions may also find potential applications in particle manipulations and light shaping, especially on ultrafast time scales.
Funding
Guangdong Major Project of Basic and Applied Basic Research (2020B0301030009); National Natural Science Foundation of China (12074268, 61975128, 62105219, 62175157); Natural Science Foundation of Guangdong Province (2019TQ05X750); Shenzhen Peacock Plan (KQTD20170330110444030); Shenzhen Science and Technology Program (JCYJ20210324120403011, RCJC20210609103232046).
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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