Spatiotemporal coupling induced controllable orientation of photonic orbital angular momentum at subwavelength scale

Zhaorui Zhang; Si Gu; Bing Gu; Bing Gu; Bingjue Li; Guanghao Rui; Guanghao Rui

doi:10.1364/OE.509594

1. Introduction

Contemporary scientific investigations have illuminated the dual nature of photons, revealing their capacity to convey both linear momentum and angular momentum. This angular momentum manifests in two distinct forms: spin angular momentum (SAM) and orbital angular momentum (OAM). Of particular interest is the latter, associated with the intriguing helical phase distribution recognized as optical vortices. These vortices exhibit notable characteristics such as phase singularities and zero intensity at their core, constituting well-established phenomena within the realm of modern singular optics. Notably, the pioneering work of Allen et al. [1] in 1992 marked the inception of optical vortex exploration. While research endeavors have predominantly concentrated on longitudinal vortices within the spatial domain—characterized by their alignment along the propagation axis of the beam—the longitudinal OAM of light has garnered substantial attention due to its distinctive focusing attributes and expanded degrees of freedom. The exploration of light bearing longitudinal OAM has traversed diverse domains, encompassing optical trapping [1], free-space communication [2], super-resolution microscopy [3], as well as quantum information processing [4].

Recent advancements in both theoretical and experimental investigations have unveiled an intriguing avenue: the incorporation of temporal phase variations as a means to engender transverse OAM [5]. This innovation gives rise to the emergence of spatiotemporal optical vortices (STOV), a captivating phenomenon where electromagnetic energy elegantly circumnavigates an axis orthogonal to the beam's propagation direction. Notably, researchers have taken strides in crafting transverse OAM STOV through judicious utilization of linear methodologies [6]. Subsequent exploration delved into the spatial and temporal attributes of ultrafast laser pulses, employing single-shot supercontinuum spectral interferometry to substantiate the propagation of space-time optical vortices within unobstructed space [7]. Related investigations have extended to encompass the intricate behaviors of STOV upon interacting with planar, isotropic interfaces—providing insight into their reflection, refraction, as well as the nuanced transverse (Hall effect) and Goos-Hänchen shifts experienced by the reflected and transmitted beams at these interfaces [8]. Furthermore, the generation of transverse OAM STOV has been achieved through the adept utilization of topological optical differentiators and space-time differentiators, offering a means to discern rapid fluctuations in pulse envelopes [9]. A novel methodology has also come to light, allowing for the meticulous study of STOV at subwavelength scales. By harnessing high numerical aperture objective lenses, researchers have harnessed the power to concentrate and generate pure transverse OAM STOV possessing spatial dimensions that transcend the classical diffraction limit [10]. In a symphony of physical intricacies, the interplay between longitudinal SAM and transverse OAM within highly confined circularly polarized STOV engenders elaborate spatiotemporal phase singularity architectures, auguring fresh paradigms for light-matter interactions replete with untapped effects and functionalities [11].

In addition to the longitudinal and transverse OAM, a captivating facet of photonic OAM has been reported recently: the potential for its intentional tilting in relation to the optical axis [14]. Such OAM tilting becomes a tangible prospect in scenarios involving a swiftly moving observer approaching relativistic velocities. Given the pivotal role of OAM in shaping light-matter interactions, there arises an imperative to explore viable methodologies for achieving precise control over both the magnitude and orientation of tilted OAM. This endeavor holds the promise of furnishing supplementary degrees of freedom with far-reaching implications for quantum communication and optical manipulation. In this context, a promising avenue has been unveiled—a scheme that harnesses the time-reversal methodology alongside vectorial diffraction theory to engender optical vortex fields characterized by OAM orientations of arbitrary angles. This innovation is effectuated within a 4Pi optical configuration and emerges as a result of structured light's malleability [12]. Another avenue, reported in recent literature, delves into the fusion of spatiotemporal vortices with spatial vortices within a unified wave packet [13]. This entwining imbues the wave packet with a tilted OAM, a quality modulated through judicious manipulation of both spatiotemporal and spatial topological charges. Recently, a novel STOV wavepacket with multiple phase singularities embedded in different space-time domains is proposed, which realizes the engineering of both magnitude and orientation of photonic OAM in space-time [14]. Despite their merits, these methodologies encounter certain limitations, notably confined controllable ranges that hinder their broad applicability. Moreover, the pursuit of tightly focused optical wave packets with tunable OAM at subwavelength scales remains an uncharted territory. Despite the strides made, the coupling of spatiotemporal vortices with spatial vortices has yet to conquer the challenge of finely controllable OAM within these confines. Thus, this research terrain beckons for innovative solutions that transcend current constraints and unlock the full potential of tilted OAM for multifaceted applications. In this work, a method to achieve controllable orientation of photonic OAM at subwavelength scale is proposed. When a wavepacket carrying both spatiotemporal and spatial vortices is focused by a high numerical aperture lens, the spatiotemporal coupling would lead to entangled and twisted vortex tunnels, exhibiting a complex three-dimensional spiral phase structure. Consequently, the orientation of spatial OAM deviates from the original optical axis direction. Through elaborately adjusting both topological charge and pulse width of the illumination, the orientation of the photonic OAM can be precisely tuned in three-dimensional space. These findings may have wide applications in optical tweezers, quantum communication etc.

2. Control of photonic OAM orientation in two-dimensional place

Without loss of generality, we can express an incident x-polarized STOV that carries both longitudinal (x-y plane) and transverse (x-z’ plane) OAM as follows:

(1)$${E_x}(x,y,z^{\prime}) = {(\frac{x}{{{w_p}}} + i{\mathop{\rm sgn}} ({l_1})\frac{y}{{{w_p}}})^{|{{l_1}} |}}{(\frac{x}{{{w_p}}} + i{\mathop{\rm sgn}} ({l_2})\frac{{z^{\prime}}}{{{w_t}}})^{|{{l_2}} |}}\exp \left( { - \frac{{{x^2} + {y^2}}}{{w_p^2}} - \frac{{z^{{\prime}2}}}{{w_t^2}}} \right)$$

where sgn is the sign function, w_p is the waist radius of the Gaussian profile in spatial domain, w_t is the pulse halfwidth at 1/e² of the maximum intensity of the wavepacket in temporal domain, l₁ and l₂ denote the topological charge of the spatial and spatiotemporal vortex embedded within the wavepacket, respectively. z’ = z – ct is the local frame coordinate, where c is the speed of light in vacuum. In this paper, for the convenience of the study, z is set to be zero. In this representation, the incident wave combines spatial and temporal characteristics, encapsulating both longitudinal and transverse OAM, and it is modulated by the Gaussian spatial profile and temporal pulse shape.

Figure 1(a) showcases a wavepacket that demonstrates intersecting spatiotemporal and spatial vortices. The intensity distributions are projected onto three orthogonal planes within the Cartesian coordinate system. In this representation, both spatial and spatiotemporal topological charges are assumed to be +1. Notably, the isosurface provides insight into the behavior of the spatiotemporal vortex, which showcases a twisting motion within the x-t plane. This motion culminates in the creation of an optical tunnel along the y-direction. Simultaneously, the spatial vortex undergoes rotation within the x-y plane, ultimately giving rise to the formation of an additional tunnel along the t-direction. The presence of these optical tunnels results in a decrease in nearby field strength, causing the electric field intensity in both the x-t plane and the x-y plane to exhibit a distinctive two-lobe distribution. In contrast, the y-t plane contains two orthogonal dual optical tunnels, leading the optical field to split into four lobes. Consequently, the optical field intensity in this scenario is noticeably diminished.

Fig. 1. (a) Intensity distribution of the wavepacket carrying spatiotemporal and spatial vortices (l₁ = 1, l₂ = 1). (b) Schematic of the apparatus for tight focusing of the spatiotemporal wavepacket. (c) Intensity distribution of the preprocessed incident field of the spatiotemporal wavepacket. Distribution of (d) total intensity, (e, f) x-component of the electric field intensity and the corresponding phase, (g, h) z-component of the electric field intensity and the corresponding phase of the focal field. Superimposed isosurfaces represent 2% of the peak intensity.

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The spatial structure inherent in photonic crystals and metamaterials significantly shapes the propagation of waves. Recently, attention has shifted towards structures that dynamically change over time on a scale comparable to the period of incident waves. To investigate the intricate interplay between optical fields and time-varying materials at a subwavelength scale, the generation of a tightly focused optical wavepacket with adjustable OAM has become imperative. However, when dealing with tightly focused wavepacket, a challenge arises. The phase structure of the spatiotemporal vortex becomes unstable due to the intense focusing achieved through a high numerical aperture objective. To counter this, it becomes essential to counteract the spatiotemporal astigmatism effect. This is accomplished by subjecting the incident wavepacket to a preprocessing step that involves the linear superposition of spatiotemporal vortices featuring different topological charges [10]. By adhering to the same procedural steps, the expression of the wavepacket, initially defined in Eq. (1), must undergo preprocessing. This is encapsulated in the modified expression:

(2)$${E_x}({x,y,z^{\prime}} )= (1 + i){(\frac{x}{{{w_p}}} + i{\mathop{\rm sgn}} ({l_1})\frac{y}{{{w_p}}})^{|{{l_1}} |}}{(\frac{x}{{{w_p}}} + {\mathop{\rm sgn}} ({l_2})\frac{{z^{\prime}}}{{{w_t}}})^{|{{l_2}} |}}\exp \left( { - \frac{{{x^2} + {y^2}}}{{w_p^2}} - \frac{{z^{{\prime}2}}}{{w_t^2}}} \right). $$

Using the scheme illustrated in Fig. 1(b), the electric field E_W after the refraction of the objective lens can be written in spherical coordinates:

(3)$$\begin{array}{c} {E_\varOmega }(\theta ,\varphi ,z^{\prime}) = (1 + i)(\frac{{\sin \theta \cos \varphi }}{{{w_p}}} + i{\mathop{\rm sgn}} ({l_1})\frac{{\sin \theta \sin \varphi }}{{{w_p}}})\\ (\frac{{\sin \theta \cos \varphi }}{{{w_p}}} + {\mathop{\rm sgn}} ({l_2})\frac{{z^{\prime}}}{{{w_t}}})\exp \left( { - \frac{{{{\sin }^2}\theta }}{{w_p^2}} - \frac{{z^{{\prime}2}}}{{w_t^2}}} \right) \end{array}. $$

Considering an objective lens obeys sine condition, the electric field in the focal region can be calculated with the Debye integral as:

(4)$${E_f}({r_f},\Phi ,z_f^{\prime}) = \int\limits_0^\textrm{a} {\int\limits_0^{2\pi } {P(\theta ,\varphi ){E_\varOmega }(\theta ,\varphi ,z^{\prime})\sqrt {\cos \theta } } } \times {e^{ - ik[{r_f}\sin \theta \cos (\varphi - \Phi ) + z_f^{\prime}\cos \theta ]}}\sin \theta d\theta d\varphi, $$

where a is the maximized focusing angle determined by the numerical aperture of the lens, and r_f = tan⁻¹(y_f/x_f). The polarization distribution P(θ,φ) of the refractive field will affect the structure of the focused wavepacket. In the case of x-polarized wavepacket, it can be expressed as:

(5)$$P = \left[ {\begin{array}{c} {({\cos \theta {{\cos }^2}\phi + {{\sin }^2}\phi } ){\textbf{e}_\textbf{x}}}\\ {({\cos \theta \sin \phi \cos \phi - \sin \phi \cos \phi } ){\textbf{e}_\textbf{y}}}\\ {({ - \sin \theta \cos \phi } ){\textbf{e}_{\textbf{z}^{\prime}}}} \end{array}} \right]. $$

In the subsequent calculation, the spatial dimensions of the incoming wavepacket have been normalized according to the lens's numerical aperture. We assume both the waist radius (w_p) and the pulse duration (w_t) of the vortex beam to be 0.5 a.u.. In Fig. 1(c), the intensity distribution of the incident field after preprocessing is showcased, projected onto three orthogonal planes. While the pattern of intensity splitting aligns with the original incident field (depicted in Fig. 1(a)), it now predominantly occupies the x-z’ plane. The distribution shifts from lobes to an approximately circular configuration. By examining the isosurface plots, it is evident that even though the incident field still maintains tunneling along the z’-axis at this stage, there is a division in the spatiotemporal domain. The previously complete toroidal light intensity distribution morphs into two lobes.

Utilizing Eq. (4), we have computed the corresponding focal field of the spatiotemporal optical field as given in Eq. (2), assuming a numerical aperture of 0.9 for the lens. It is worthy of noting that the pulse duration considered in this work is relatively large, therefore Eq. (4) is applicable to predict the characteristics of focal field. However, an accurate model of triple-integration must be adopted when the pulse width is less than 100 fs. In that case, the spatiotemporal vortex structure of the focal field would collapse since the contributions from undesired OAM modes become nontrivial, making it impossible to precisely control the orientation of photonic OAM [15]. Figures 1(e) and 1(g) vividly portray the intensity distributions of the x- and z’-components of the focal field, respectively. Given the x-polarized nature of the incident field, the y-component of the electric field within the focal region becomes notably weak. When a tightly focused wavepacket carries both spatiotemporal vortices and longitudinal spatial vortices, the interaction between these vortices leads to intricate twisting and distortion of the vortex tunnel. This intricate interplay gives rise to a complex three-dimensional spiral phase structure. The mutual influence between the two vortex types is distinctly revealed in the isosurface plots. As depicted in Fig. 1(e), the once orthogonal vortex tunnels undergo a tilt, resulting in two phase singularities within the x-z’ plane and the x-y plane, respectively. Additionally, a dumbbell-shaped intensity distribution manifests in the y-t plane. The phase distribution illustrated in Fig. 1(f) validates the characteristics of the aforementioned intensity patterns. On the y-z’ plane, the phase demonstrates a binary distribution of π/4 and −3π/4, while in the x-z’ and x-y planes, two singularities emerge in each scenario. Centered around each singularity point (x, z’(y)) = (±0.96π, 0), a helical phase distribution spanning the range of [-π, π] becomes apparent. The comprehensive intensity distribution of the focused field is presented in Fig. 1(d), featuring two distinct inclined vortex tunnels. The interaction between spatiotemporal and spatial vortices leads to a shift in the orientation of the spatial OAM from the original optical axis direction.

The average OAM of each photon is a vector with its three components pointing towards the x, y, and z’ directions [24]. The average OAM of photons within the wavepacket can be obtained through the following volume integral:

(6)$$\textbf{M} = \frac{{\int_{ - \infty }^\infty {\textbf{r} \times \textbf{g}dV} }}{{{\varepsilon _0}{\omega ^2}\int_{ - \infty }^\infty {{{|E |}^2}dV} }}$$

where r × g represents angular momentum density, ε₀ is the vacuum permittivity, and |E|² denotes the electric field intensity of the focal field. Angular momentum density is determined by the cross product of the position vector r and the linear momentum density g:

(7)$$\begin{array}{c} \textbf{r} \times \textbf{g} = \frac{{{\varepsilon _0}}}{2}[y( - i\omega {E^\ast }\frac{{\partial E}}{{\partial z^{\prime}}} + i\omega E\frac{{\partial {E^\ast }}}{{\partial z^{\prime}}} + 2\omega k{|E |^2})\\ - z^{\prime}( - i\omega {E^\ast }\frac{{\partial E}}{{\partial y}} + i\omega E\frac{{\partial {E^\ast }}}{{\partial y}})]\vec{\boldsymbol{x}}\\ + \frac{{{\varepsilon _0}}}{2}[z^{\prime}(c\frac{{\partial {E^\ast }}}{{\partial x}}\frac{{\partial E}}{{\partial z^{\prime}}} + c\frac{{\partial E}}{{\partial x}}\frac{{\partial {E^\ast }}}{{\partial z^{\prime}}} + i\omega E\frac{{\partial {E^\ast }}}{{\partial x}} - i\omega {E^\ast }\frac{{\partial E}}{{\partial x}})\\ - x( - i\omega {E^\ast }\frac{{\partial E}}{{\partial z^{\prime}}} + i\omega E\frac{{\partial {E^\ast }}}{{\partial z^{\prime}}} + 2\omega k{|E |^2})]\vec{\boldsymbol{y}}\\ + \frac{{{\varepsilon _0}}}{2}[x( - i\omega {E^\ast }\frac{{\partial E}}{{\partial y}} + i\omega E\frac{{\partial {E^\ast }}}{{\partial y}})\\ - y(c\frac{{\partial {E^\ast }}}{{\partial x}}\frac{{\partial E}}{{\partial z^{\prime}}} + c\frac{{\partial E}}{{\partial x}}\frac{{\partial {E^\ast }}}{{\partial z^{\prime}}} + i\omega E\frac{{\partial {E^\ast }}}{{\partial x}} - i\omega {E^\ast }\frac{{\partial E}}{{\partial x}})]\boldsymbol{\vec{z}^{\prime}} \end{array}. $$

Utilizing Eq. (6) and (7), we can ascertain all three components of photonic OAM. When both the spatial topological charge and the spatiotemporal topological charge are set to +1, the x-, y-, and z’- components of photonic OAM amount to 0, ℏ, and 0.8ℏ, respectively. Given the non-zero y-component of photonic OAM, the orientation of the OAM diverges from the initial optical axis direction, inducing a deflection within the y-z’ plane. The resultant deflection angle θ_f, relative to the y-axis, can be expressed as θ_f = tan^–1(M_z’/M_y) = 37.5°. These findings signify that the interplay between spatiotemporal and spatial vortices engenders photonic OAM within a wavepacket, characterized by a distinctive deflection angle. Moreover, we observe that the temporal extent, pulse duration (w_t), of the incident light field influences the deflection of photonic OAM within the focal field. Illustrated in Fig. 2(a), the distribution of intensity in the focal field varies with different pulse durations, maintaining parameters such as topological charge and waist radius in accordance with those in Fig. 1. Inferred from the intensity distribution across the y-z’ plane, the predominant axis orientation of the dumbbell-shaped focal field rotates in tandem with changes in w_t. This visual rotation elucidates the evolving orientation of photonic OAM in the y-z’ plane. Employing Eq. (7), we quantify the deflection angle θ_f of photonic OAM corresponding to distinct pulse durations. As depicted in Fig. 2(a), a consistent linear relationship exists between the deflection angle and the pulse duration. As the pulse duration increments from 0.1 to 1, the deflection angle steadily rises from 14° to 54°, with a sensitivity of 4° per 0.1 pulse duration unit. Similarly, manipulating the waist radius of the light field can modulate the deflection angle of photonic OAM within the focal field's y-t plane. Nevertheless, the relationship between these two parameters lacks clear regularity and thus is omitted from further discussion.

Fig. 2. (a) Dependence of deflection angle θ_f on pulse width of the incident spatiotemporal wavepacket. The corresponding intensity distributions of focal field are presented in the insets. (b)-(e) Intensity distributions of focal field for incident spatiotemporal wavepacket carrying spatiotemporal and spatial topological charges of different signs. Superimposed isosurfaces represent 2% of the peak intensity.

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Next, we delve into the influence of topological charges on the deflection angle of photonic OAM. Table 1 provides insights into the impact of topological charge sign variations on the components and orientation of photonic OAM when the absolute values of spatiotemporal and spatial topological charges are both set to 1. Figures 2(b)-(e) further elucidate the corresponding focal field intensity distributions under these conditions. When the signs of spatiotemporal and spatial topological charges align, the major axis of the focal field, resembling a dumbbell shape, spans across quadrants two and four of the y-t coordinate system. Conversely, when these topological charges carry opposite signs, the major axis of the focal field shifts to occupy quadrants one and three. This discussion underscores the connection between the focal field's distribution and photonic OAM. However, it primarily offers a general reference for OAM direction. Specifically, while altering the overall sign of the topological charge does not impact the intensity distribution of focal field, it distinctly determines the quadrant of the photonic OAM direction. As shown in Table 1, employing the four combinations of spatiotemporal and spatial topological charge signs allows comprehensive coverage of the y-t plane for photonic OAM direction. Therefore, the sign of the incident field's topological charge dictates the quadrant of the focal field OAM direction, while the absolute value of the topological charge enables versatile adjustments in the specific orientation of photonic OAM within these respective quadrants. Furthermore, varying the values of spatiotemporal/spatial topological charges results in corresponding increases/decreases in the orientation angle of photonic OAM within the y-t plane. Taking the reference deflection angle of 37.5°, which is generated when both spatiotemporal and spatial topological charges are set to 1, we observe that when spatiotemporal/spatial topological charges are increased to 2, the deflection angle similarly increases/decreases to 58.7° and 21.8°, respectively. Should both topological charges be increased to 2 simultaneously, the deflection angle will exhibit a slight increment to 38.6°. Consequently, through coordinated control of the pulse width of the incident field and the topological charges, we attain a flexible and extensive range of control over the orientation of photonic OAM within the y-t plane.

Table 1. The components of photonic OAM in the focal field and the deflection angle θ_f within the y-t plane when the incident field carries different spatiotemporal and spatial topological charges

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3. Three-dimensional control of photonic OAM orientation based on coupling of spatiotemporal dual vortices and spatial vortices

In this section, building upon the groundwork established in the preceding segment, we present a demonstration of how the interplay between spatiotemporal and spatial vortices can be harnessed to attain precise control over the orientation of photonic OAM within a three-dimensional spatial framework. We contemplate an x-polarized spatiotemporal wavepacket that not only carries longitudinal OAM within the x-y plane but also encompasses a pair of mutually orthogonal transverse OAM components within the x-z’ and y-z’ planes. The electric field governing this composite wavepacket finds expression as follows:

(8)$$\begin{array}{c} {E_x}(x,y,z^{\prime}) = {(\frac{x}{{{w_p}}} + i{\mathop{\rm sgn}} ({l_1})\frac{y}{{{w_p}}})^{|{{l_1}} |}}{(\frac{x}{{{w_p}}} + i{\mathop{\rm sgn}} ({l_2})\frac{{z^{\prime}}}{{{w_t}}})^{|{{l_2}} |}}\\ {(\frac{y}{{{w_p}}} + i{\mathop{\rm sgn}} ({l_3})\frac{{z^{\prime}}}{{{w_t}}})^{|{{l_3}} |}}\exp \left( { - \frac{{{x^2} + {y^2}}}{{w_p^2}} - \frac{{z^{{\prime}2}}}{{w_t^2}}} \right) \end{array}. $$

Assuming both spatiotemporal (l₂, l₃) and spatial (l₁) topological charges are established to be 1, the isosurface distribution depicted in Fig. 3(a) distinctly underscores the convergence of three mutually orthogonal optical pathways at the origin. This occurrence manifests in the emergence of symmetry within the intensity distribution observed across the trio of orthogonal planes. The incorporation of parameter l₃ introduces additional degrees of freedom, enabling a more intricate control over the OAM traits within the focal field.

Fig. 3. (a) Intensity distribution of the wavepacket carrying both dual spatiotemporal vortices and spatial vortex; (b) Intensity distribution of the preprocessed incident field of the spatiotemporal wavepacket; (c) Intensity distribution of the focal field. (d) Dependence of deflection angle θ_f and φ_f on pulse width of the incident spatiotemporal wavepacket. The corresponding intensity distributions of focal field are presented in the insets. Superimposed isosurfaces represent 2% of the peak intensity.

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Similarly, during the computation of the optical focal field, the incoming wavepacket will undergo prior processing as detailed below:

(9)$$\begin{array}{c} {E_x}(x,y,z^{\prime}) = 2i{(\frac{x}{{{w_p}}} + i{\mathop{\rm sgn}} ({l_1})\frac{y}{{{w_p}}})^{|{{l_1}} |}}{(\frac{x}{{{w_p}}} + {\mathop{\rm sgn}} ({l_2})\frac{{z^{\prime}}}{{{w_t}}})^{|{{l_2}} |}}\\ {(\frac{y}{{{w_p}}} + {\mathop{\rm sgn}} ({l_3})\frac{{z^{\prime}}}{{{w_t}}})^{|{{l_3}} |}}\exp \left( { - \frac{{{x^2} + {y^2}}}{{w_p^2}} - \frac{{z^{{\prime}2}}}{{w_t^2}}} \right) \end{array}. $$

After mitigation of spatiotemporal chromatic dispersion, the incident field's intensity becomes predominantly concentrated within the spatiotemporal plane (depicted in Fig. 3(b)), displaying a distinct hexagonal lobed pattern. Simultaneously, the spatial plane maintains its characteristic four-lobed structure. Upon precise focusing of the wavepacket, as outlined in Eq. (9), the interplay between the dual spatiotemporal vortices and the spatial vortex engenders a complex interwoven and distorted OAM within the optical tunnels. This intricate coupling leads to a profound breakdown of symmetry in the intensity distribution across the spatiotemporal plane (as seen in Fig. 3(c)). Through comprehensive calculations grounded in Eq. (6),(7), we have quantified the photonic OAM within the focal field, yielding (M_x, M_y, M_z’) = (–ℏ, ℏ, 1.4ℏ). In contrast to the findings in the previous section, the x-component of OAM is no longer negligible. This non-zero component leads to a divergence in the orientation of photonic OAM from the original optical axis direction. Consequently, deflections occur in the x-z’ and y-z’ planes, characterized by deflection angles θ_f = tan^–1(M_z’/M_y) = 54° and φ_f = tan^–1(M_z’/M_x) = 54°, respectively. This intriguing phenomenon is realized through the cross-coupling of spatial vortices and orthogonal spatiotemporal vortices, enabling orientation deflection of photonic OAM within the wavepacket, a departure from the confines of the y-z’ plane. Further insights emerge when considering the pulse duration (w_t) of the illuminating light field, which has a discernible impact on the deflection of photonic OAM across three-dimensional space. The findings illustrated in Fig. 3(d) underscore a robust linear relationship and similar sensitivity between pulse duration and deflection angle. Notably, each increment of 0.1 in pulse duration corresponds to an increase of approximately 4.5° in both θ_f and φ_f. The incorporation of dual spatiotemporal vortices enhances the versatility in steering the orientation of OAM within spatial coordinates. Illustrated in Fig. 4, altering the signs of both spatiotemporal and spatial topological charges induces a rotational transformation in inclined vortex tunnels, visually depicting changes in photonic OAM orientation. This manipulation of topological charge combinations bestows the ability to adjust the direction of OAM across the eight quadrants of the spatial coordinate system. Furthermore, maintaining consistent signs of topological charges while varying their absolute values fine-tunes the specific orientation of photonic OAM within corresponding quadrants. Empirical investigations reveal that both θ_f and φ_f diminish as the absolute value of the spatial topological charge |l₁| increases. Building on this insight, the spatiotemporal topological charge influences the relative magnitudes of θ_f and φ_f. As elucidated in Table 2, an escalated |l₂|/|l₃| ratio yields higher θ_f/φ_f values. Thus, by collectively manipulating pulse duration, spatiotemporal, and spatial topological charges, a broad spectrum of control over the orientation of photonic OAM within three-dimensional space can be methodically realized.

Fig. 4. The relationship between the signs of topological charge of the incident field carrying dual spatiotemporal vortices and a spatial vortex and the orientation quadrants of the photonic OAM within the focal field. Corresponding focal field intensity distributions are provided in the insets. Superimposed isosurfaces represent 2% of the peak intensity.

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Table 2. The components of photonic OAM in the focal field and the deflection angle θ_f and φ_f in three-dimensional space when the incident field carries different spatiotemporal and spatial topological charges

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4. Conclusion

In conclusion, we present a theoretical framework demonstrating the attainment of controlled orientation for photonic OAM by harnessing spatiotemporal coupling at subwavelength scales. By skillfully focusing a wavepacket containing both transverse spatiotemporal vortices and longitudinal spatial vortices through a high numerical aperture lens, the interplay between these components yields intricately entangled and twisted vortex tunnels. This dynamic interaction culminates in a divergence of the spatial OAM orientation from its initial optical axis. Our explorations underscore the substantial impact of optical topological charges on the direction and angle of inclination of vortex tunnels. Through judicious manipulation of the sign permutations of spatiotemporal and spatial topological charges, the orientation of photonic OAM within distinct quadrants can be meticulously tailored. Moreover, the absolute magnitude of the topological charge imparts the capacity to adjust the deflection angle of photonic OAM within corresponding quadrants. Additionally, we ascertain that the temporal profile of the incident field exerts discernible influence on the deflection angle of photonic OAM, where augmented pulse durations correlate with heightened deflection angles. This emergent paradigm of vortex light, engendered by the synergy of spatiotemporal interplay, affords unprecedented control over the orientation of photonic OAM, substantially enriching the spectrum of possibilities in OAM research. Recognizing the fundamental role of OAM in light-matter interactions, this manipulated OAM paradigm could be adeptly employed in constructing optical spanners endowed with customizable torque across arbitrary three-dimensional axes. These capabilities proffer a plethora of prospective applications encompassing optical tweezers, spin-orbit angular momentum coupling, quantum communication, and a horizon beyond.

Funding

National Natural Science Foundation of China (12274074, 12134013, 12074066, 92050202).

Acknowledgment

G. R. acknowledged the support by the Zhishan Young Scholar Program of Southeast University.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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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l₁	l₂	l₃	θ_f	φ_f
+1	+1	+1	54°	54°
+2	+1	+1	53°	51.3°
+3	+1	+1	26°	23.2°
+3	+2	+1	34.4°	28.6°
+3	+1	+2	24.8°	32.3°

l₁	l₂	l₃	θ_f	φ_f
+1	+1	+1	54°	54°
+2	+1	+1	53°	51.3°
+3	+1	+1	26°	23.2°
+3	+2	+1	34.4°	28.6°
+3	+1	+2	24.8°	32.3°

Spatiotemporal coupling induced controllable orientation of photonic orbital angular momentum at subwavelength scale

Abstract

1. Introduction

2. Control of photonic OAM orientation in two-dimensional place

3. Three-dimensional control of photonic OAM orientation based on coupling of spatiotemporal dual vortices and spatial vortices

4. Conclusion

Funding

Acknowledgment

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Tables (2)

Equations (9)

Optics Express

l₁	l₂	OAM_y	OAM_z’	θ_f
+1	+1	ℏ	0.8ℏ	37.5°
+1	−1	–ℏ	0.8ℏ	142.5°
−1	+1	ℏ	–0.8ℏ	322.5°
−1	−1	–ℏ	–0.8ℏ	217.5°
+2	+1	2ℏ	0.8ℏ	21.8°
+1	+2	ℏ	1.6ℏ	58.7°
+2	+2	2ℏ	1.6ℏ	38.6°