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Abstract
Orbit-induced localized spin angular momentum (OILS) has recently garnered significant attention. This paper introduces periodic edge dislocation (PED) into the tight focusing system. The study delves into the tight focusing characteristics of the radially polarized vortex plane beam with PED, demonstrating that PED serves as a straightforward and effective means of manipulating OILS, especially when both the orbital angular momentum and the polarization of the incident beam are fixed. Our findings indicate that the longitudinal OILS reaches its maximum when the difference between the period of PED and the vortex topological charge is equal to 1. Conversely, when the difference is 0, the transverse OILS reaches its maximum, while the longitudinal OILS reaches its minimum. Similar patterns are also observed in linearly polarized vortex beams. This research proposes a simple and practical way to control OILS, contributing to our understanding of optical orbit-spin coupling.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Angular momentum (AM) is a fundamental property of photons, drawing widespread attention due to its distinctive interaction with matter. This property is divided into two main components: spin angular momentum (SAM) and orbital angular momentum (OAM). SAM is associated with the circular and elliptic polarizations of the beam [1], while OAM pertains to the helical wavefront of the beam [2]. The AM can be transferred to a particle when a light beam collides with it. SAM causes the particle to rotate around its own axis [3], whereas OAM leads to rotation around the optical axis [4]. Despite SAM and OAM represent distinct degrees of freedom for light beams, optical spin-orbit interaction can occur under specific conditions, including light-matter interaction in anisotropic media [5], structured media [6], and tight-focusing systems [7]. The well-established understanding is that SAM converts to OAM under tight focusing systems [8], while the reverse process, orbital-to-spin conversion (OSC), remains an open question [9]. Presently, research is concentrated on this question, with orbit-induced localized spin angular momentum (OILS) emerging as a prominent area of investigation. For instance, Han et al. and Yu et al. studied the OSC of fully coherent radially and linearly polarized beams, respectively, in a tight focusing setup, uncovering the occurrence of OILS of them [10,11], Wang et al. investigated the generation of OILS in partially coherent light and partially polarized light. It is pointed out that the reduction of spatial coherence leads to a monotonous decrease in OILS [12]. Additionally, the decrease in the polarization degree of the incident light results in varying degrees of weakening in both transverse and longitudinal OILS [13]. Though these researchers verified that OILS emerges in different polarized beams, a gap remains in methods for controlling OILS when both the OAM and the incident beam polarization are fixed.
Nye and Berry [14] introduced optical dislocation into wave theory, represented as wavefront dislocation or phase singularity. In the field of singularity optics [15], dislocations are classified into screw, edge, and mixed edge-screw dislocations [16]. A screw dislocation is characterized by a screw phase structure that varies around a central singularity. In contrast, an edge dislocation involves a $\pi$ phase jump in the wave phase along a line or ring in the transverse plane [17]. The application of optical dislocation extends to various domains such as astrometry [18], particle capture [4,19,20], optical communication [21,22], and 3D imaging [23]. Vortices, which simulate screw dislocations [17], for example, are widely used in particle trapping, quantum entanglement, and imaging applications [24]. Huang et al. utilized edge dislocation to enhance the self-healing capability of circular Pearcey beams [25]. However, it is noteworthy that, to the best of our knowledge, the study of edge dislocations and mixed edge-screw dislocations in tightly focused systems is relatively scarce.
In this article, we demonstrate that PED effectively regulates OILS without requiring changes to the OAM or polarizations of the incident beam. We begin by investigating the tight focusing characteristics of the radially polarized vortex plane beam with PED. Following that, we show that OILS appears in the radially polarized vortex plane beam with PED and examine how PED influences OILS. By modifying the PED phase, we observe regular changes in the OILS of the radially polarized vortex plane beam while maintaining a fixed vortex topological charge. Similar phenomena observed in the case of linear polarization demonstrate the method’s broad applicability. The alterations in OILS, stemming from the changes in PED, present a straightforward and practical approach to control OILS while keeping the orbital angular momentum and polarization of the incident beam constant.
2. Theoretical analyses of radially polarized vortex plane beam with PED
The electric field of radially polarized plane beams with $l$-order concentric vortex and the PED phase ($\exp (i\phi _n)$) can be expressed as:
Here $\phi$ is the azimuthal angle, and $l$ represents the topological charge. $\phi _{\rm {n}}$ is the PED phase. Essentially, this is equivalent to periodically introducing a phase shift of $\pi$ to the incident plane wave. The period of the PED phase is regulated by adjusting $n$, defined as the period number. The amplitude function $A_0$ is representative of the amplitude at the point where the beam enters the pupil of the objective lens, and it is set to $A_0=1$.
It’s clear that the initial phase of the incident field can be visualized as the superposition of the PED phase and vortex phase, as depicted in Fig. 1. In Fig. 1(a) (b) and (d), the phase of the incident field is shown when the period number $n$ is not equal to the vortex topological charge $l$. The phase distribution of the incident field appears relatively chaotic. However, when $n=l$, the phase of the incident field exhibits similarity to the vortex phase as shown in Fig. 1(c). Moreover, it should be noted that in Fig. 1(b) and (d), we present the phase of the incident field when $n=l+1$ and $n=l-1$, and their values are opposite.
Fig. 1. The initial phase superimposition of the PED and vortex. $n=5$, (a) $l=2$; (b) $l=4$; (c) $l=5$; (d) $l=6$.
The electric field of a tightly focused beam passing through a high numerical aperture (NA) lens can be calculated using the Richards-Wolf vector diffraction theory. Specifically, the three electric field components $(E_x, E_y, E_z)$ of the radially polarized plane beams carrying arbitrary vortex topological charge and PED can be determined. When a light beam is focused by a high NA objective lens that satisfies the sine approximation condition, the electric fields near the focus can be expressed as [26].
According to the definition of SAM density, it can be expressed as the following formula [27]:
In the process of interaction between light and matter, particles and media are generally non-magnetic [28] and we find the PED has similar regulation rules on electric spin and magnetic spin, so in this paper, we only show the regulation effect of PED on electric SAM density. We can write the electric field strength in three directions as ${E_j} = \left | {{E_j}} \right |\exp \left ( {i{\varphi _j}} \right )$, $j = x,y,z$. The SAM components of the three dimensions can then be written as follows [27]:
The relative ratios of these three constituents indicate the spinning properties of local electric fields, i.e., local spin states.
3. Focusing of radially polarized vortex plane beam with PED modulation
The investigation begins with an examination of the total intensity distribution at the focal plane with the PED phase and no vortex topological charge ($l=0$). Figure 2 illustrates the total intensity distribution at the focal plane for different period numbers $n$. Specifically, for period numbers $n$ of 1, 2, 6, and 9, the beam exhibits 2, 4, 12, and 18 lobes, respectively. This observation suggests that in the absence of a vortex topological charge, the optical field at the focal plane takes on a multi-lobe structure, with the number of lobes being twice the period number, $2n$. Additionally, as the period number $n$ increases, the radius of the annular lobes expands.
Fig. 2. The total intensity of the radially polarized plane beam with PED and no vortex topological charge ($l=0$) at the focal plane. (a)–(d) $n=1, 2, 6, 9$.
The cooperative modulation effects of the PED phase and vortex on the light field at the focal plane are investigated in Figs. 3. The total intensity at the focal plane, modulated by both PED and vortex, is shown for cases where $n=l$ and $n\neq l$. Taking $n=6$ as an example, in Figs. 3(a)–(h), corresponding to $l=2-9$, a ring inside and ring lobes outside appear after the introduction of the vortex phase compared with Fig. 2(c). In Figs. 3(a)–(e), as the topological charge $l$ changes from 0 to 6, the size of the strongest spots at the focal plane gradually shrinks. The number of outside lobes remains constant at $2n$. However, when $l$ changes from 6 to 9 in Figs. 3(e)–(h), the size of the strongest spots becomes larger. It is worth noting that when the period number $n$ and the number of the vortex topological charges $l$ are identical, the strongest intensity spots coalesce into one smallest point in the middle of the focus plane. As shown in Figs. 3(i)–(l) for $n=l=2, 5, 8, 9$, the strongest intensity points gather into one smallest point in the center of the focus plane. These characteristics could serve as a novel method for determining the vortex topological charge of incident light, presenting potential applications in optical communication.
Fig. 3. The total intensity of the radially polarized plane beam modulated by the PED and the vortices at the focal plane. (a)–(h) $n=6$, l = 2–9; (i)–(l) $n=l$.
4. Manipulation of orbit-induced localized spin angular momentum
To investigate the occurrence of OILS at the focal plane and assess how PED influences OILS, we must first prove that OILS can exist in this system. In Fig. 4, where we fix the period number $n=5$, the normalized intensity diagrams of the three-dimensional components of the electric field at the focal plane, along with the corresponding phase diagrams (inset), are shown for vortex topological charges $l=0$ and $2$, in Figs. 4(a1)–(a3) and Figs. 4(c1)–(c3), respectively. According to the Eq. (5), the SAM density distribution $\left ( {{S_x},{S_y},{S_z}} \right )$ can be calculated through the electric field, as shown in Figs. 4(b1)–(b3) for $l=0$ and Figs. 4(d1)–(d3) for $l=2$. Here, the electric field intensity is normalized to the maximum of the total intensity, and the SAM density is normalized to the maximum of $S_x$. These visualizations provide a foundation for understanding the OILS distribution at the focal plane and set the stage for further research on how PED influences this phenomenon.
Fig. 4. The normalized intensity, phase (insets), and the normalized SAM density profiles on the focal plane of the tightly focused radially polarized plane beam with the PED and the vortex. (a1)–(b3) $n=5, l=0$; (c1)–(d3) $n=5,l=2$.
In Fig. 4(b3), it is evident that when the incident beam carries no OAM ($l=0$), the density of the longitudinal SAM ($S_z$) is 0. In this case, a pure transverse SAM is present in the focus plane, characterized by an annular flap-like distribution, as shown in Figs. 4(b1)–(b2). On the other hand, when the incident light carries OAM, taking $l=2$ as an example, $S_z$ appears, and its magnitude is comparable to that of the transverse SAM, as illustrated in Fig. 4(d3). The inner ring of the $S_z$ distribution exhibits a continuous negative value, while the outer ring has discrete positive values. Simultaneously, the distribution of transverse SAM also undergoes changes in the focal region, as demonstrated in Figs. 4(d1)–(d2). Further, the numerical results show that the total SAM is zero, indicating that no additional SAM is transferred from the incident OAM. Instead, the incident OAM contributes to inducing the longitudinal SAM, a process known as orbit-induced localized SAM conversion [10,11].
Then, we studied the transverse polarization states on the focal plane for further analysis. Figure 5(a)–(d) present the polarization ellipticity distributions for $n=5$ with $l=0, 2, 4, 6$, respectively. The red line denotes linear vibration, whereas the blue (yellow) represents the right (left) circular polarization. In Fig. 5(a), where only the PED is modulated, all polarization states are linear. In contrast, in Fig. 5(b)–(d), when vortex phase modulation is introduced, local elliptical polarization states emerge. The change in the polarization ellipticity distribution confirms the presence of OILS. We further investigated the impact of the sign of $l$ and $n$ on the results. Just as shown in Fig. 6, it demonstrates one-dimensional distributions of the three normalized SAM density components along the y axis with $n=3,l=4$ [Fig. 6(a)] and $n=3,l=-4$ [Fig. 6(b)]. By comparison, we find the longitudinal SAM density ($S_z$) reverses from positive ($n=3,l=4$) to negative ($n=3,l=-4$) with its amplitude unchanging, whereas it has no influence on the distributions of $S_x$ and $S_y$. Regarding the sign of $n$, we found it has no impact on the distribution of SAM. These analyses provide further insights into the intricate modulation effects of PED and vortex on polarization states at the focal plane.
Fig. 5. Transverse polarization states on the focal plane with (a) $n=5$, $l=0$; (b) $n=5$, $l=2$; (c) $n=5$, $l=4$ and (d) $n=5$, $l=6$. The red line denotes linear vibration, the blue (yellow) represents the right (left) circular polarization, the background image illustrates the transverse component diagram of light intensity ($|E_x|^2+|E_y|^2$).
Fig. 6. Cross-sectional SAM density distributions of the x, y, and z components on the focal plane of the tightly focused beam with (a) $n=3,l=4$ and (b) $n=3,l=-4$. The insets are their initial phase distributions. Line scans along the y axis of the normalized SAM density distributions.
To find how the PED and the OAM change the OILS, in Figs. 7(a1)-(b2), the cross-sectional view along the y-direction of normalized $S_z$ and $S_x$ are presented, illustrating the change in $l$ with $n=0$ and $n=3$. Figures 7(a1) and (b1) depict $S_z$ and $S_x$ without the PED, where the longitudinal OILS is significantly weakened when the vortex topological charge $l>1$, and the transverse OILS is greatly weakened when $l>0$.
Fig. 7. Cross-sectional SAM density distributions of the (a) z and (b) x components on the focal plane with different vortex topological charge $l$. (1) $n=0$; (2) $n=3$. Line scans along the y axis of the normalized SAM density distributions. $S_z, S_x$ are normalized by the maximum values induced by different topological charges. (c1) Transverse polarization states on the focal plane when $S_z$ reaches to the maximum ($n=3,l=4$). (c2) Transverse polarization states on the focal plane when $S_x$ reaches to the maximum ($n=3,l=3$).
In Figs. 7(a2) and (b2), the PED is added, and the period number is $n=3$. As for $S_z$ in Fig. 7(a2), it can be observed that the maximum value of $S_z$ is achieved when $l=2$ and $l=4$, while the minimum value of $S_z$ occurs when $l=3$. As $l$ gradually approaches $n$ from 0, the magnitude of $S_z$ continuously increases, reaching a maximum value at $l=n\pm 1$, and the positions where the peak value occurs are consistently close to the center of the focal plane. This phenomenon indicates that the closer $l$ is approaching $n\pm 1$, the stronger $S_z$ localized at a smaller range. When $l$ is equal to $n$, the modulus of $S_z$ becomes very small in the range of $\pm \lambda$. It is worth noting that the direction of the modulus of $S_z$ is reversed for $l=n-1$ and $l=n+1$.
Regarding $S_x$, in Figs. 7(b2), when $l$ gradually approaches $n$ from 0, the absolute value of $S_x$ increases continuously, reaching a maximum value at $l=n$. Taking Fig. 7(b2) as an example, we can get the maximum value of $S_x$ when $l=3$. Moreover, from Figs. 7(a1)–(b2), the PED subtly influences the OILS, as indicated by the observed variations. It is possible to regulate OILS while preserving a fixed vortex topological charge by choosing the right PED with knowledge of the relationship between the period number ($n$) and the vortex topological charge ($l$) in OILS manipulation. These rules provide valuable insights for precisely and controllably adjusting the focussed beam’s OILS.
Furthermore, in Figs. 7(c1) and (c2), the transverse polarization states on the focal plane are illustrated when $S_z$ reaches its maximum ($n=3,l=4$) and when $S_x$ reaches its maximum ($n=3,l=3$). In the case of $S_z$ reaching its maximum ($n=3,l=4$), the polarization ellipse is visible at the center of the polarization diagram. On the other hand, when $S_x$ reaches its maximum ($n=3,l=3$), the polarization ellipse transforms into a linear shape in the center, indicating zero longitudinal spin in the center. These observations align with the calculated results of $S_z$. In the subsequent analysis, we concentrate on comparing OILS without PED and OILS with appropriate PED, while keeping the vortex topological charge fixed. Using $l=4$ as an example, we explore various PED configurations, and the normalized $S_z$ and $S_x$ of the radially polarized plane beam are presented in Figs. 8(a)-(b), line scans along the y-axis.
Fig. 8. The topological charge of vortices, $l=4$, cross-sectional SAM density distributions of (a) $S_z$; (b) $S_x$. Line scans along the y axis of the normalized SAM density distributions. $S_z, S_x$ are normalized by the maximum values induced by different period number.
Figure 8(a) illustrates that the modulus of $S_z$ with $n=3$ or $n=5$ is larger than that of $n=0$. As previously mentioned, the modulus of $S_z$ reaches a maximum value at $l=n\pm 1$, demonstrating that selecting $n=3$ or $n=5$ for $l=4$ enhances the OILS, achieving approximately five times in a narrower span compared with the case without PED. Conversely, when the number of periods $n$ equals to the topological charge $l$ of the vortex, i.e., $n=4$, the $S_z$ is smaller than that without the PED, indicating the suppression of OILS. Furthermore, the catalysis direction of $S_z$ can be controlled by adjusting the number of periods $n$. The catalysis direction of $S_z$ when $n=5$ is opposite to that when $n=3$ from Fig. 8(a). Similarly, in Fig. 8(b), choosing the number of periods $n$ equal to the topological charge $l$ of the vortex, i.e., $n=4$, can enhance the catalytic effect in $S_x$. Thus, we can control OILS by manipulating the dislocation’s phase.
Moreover, we extend our investigation to explore how PEDs control OILS in other polarizations. In Fig. 9, we maintain a fixed vortex topological charge $l=4$ while altering the incident plane beam to be linearly polarized. Figures 9(a)–(b) display the cross-sectional SAM density distributions of $S_z$ and $S_x$ of $y$ polarized plane beam with the PED. Similar patterns emerge regarding how PEDs control OILS in these cases. In Fig. 9(a), $S_z$ is strengthened by $n=3$ or $n=5$, whereas $S_x$ is strengthened and $S_z$ is weakened by $n=4$. Interestingly, the capability of PEDs to modulate OILS is applicable even in different polarizations. Therefore, the use of PEDs to alter OILS with various polarizations is a viable approach, offering flexibility in controlling OILS in different polarizations.
Fig. 9. The topological charge of vortices, $l=4$, the cross-sectional SAM density distributions of $y$ polarized plane beam with the PED (a) $S_z$; (b) $S_x$. Line scans along the y axis of the normalized SAM density distributions.
5. Conclusions
In conclusion, our study delves into the manipulation of OILS using PEDs. The introduction of suitable PEDs proves effective in enhancing or weakening the OILS of a beam while maintaining fixed OAM and polarization. Initially, we examine PED properties on a radially polarized vortex beam in a tightly focused system. The electric field at the focal plane with the PED phase exhibits a ring with $2n$-lobes, where $n$ is the number of periods in the additional PED phase. When $n$ matches with the number of topological charges of the vortex $l$, the lobes converge into a single spot. Consequently, the electric field components in the focal region can be comprehensively modulated by adjusting $n$ and $l$, laying the foundation for OILS manipulation using PEDs. Subsequently, we explore the SAM density distribution, observing that the strongest $S_z$ emerges when $n=l\pm 1$, while the weakest $S_z$ and the strongest $S_x$ occur when $n=l$. The sign of $l$ only affects the direction of $S_z$ without influencing its magnitude, and it has no impact on $S_x$ and $S_y$, and the sign of $n$ has no influence on the SAM. This realization makes it possible to manipulate OILS with a fixed vortex topological charge by using the proper PED phases. Notably, our further investigations reveal that PEDs can control OILS in linear polarization as well, meaning the adaptability of this method. Overall, our work deepens the understanding of OILS and provides a simple and practical approach to manipulate local SAM density in tightly focused systems.
Funding
Ministry of Science and Technology of the People's Republic of China (2022YFC2808203); Ministry of Education of the People's Republic of China (2016XZZX004-01, 2017QN81005); National Natural Science Foundation of China (11474254, 11804298).
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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