Structuring light with flows

Abstract

Structured light has profoundly advanced optical manipulation, processing and imaging. However, its practical deployment in free space is limited by the constrained solutions of Helmholtz equation, which are bound to fixed propagation laws. Here, we reframe structured light as optical flows through a hydrodynamic description beyond the conventional field formalism, achieving flexible light structuring in free space via streamline engineering. Within this framework, we demonstrate the on-demand generation of diverse families of beams, with tailored propagation dynamics, including Gaussian, Bessel, Airy and vortex beams, and introduce specialized modes that overcome complex propagation challenges. To validate the designed energy streamlines, we perform optical tweezers experiments, treated as analogous to fluid particle-tracking velocimetry, demonstrating potential for high-precision optofluidic manipulation. For free-space optical communication, we show how vortex modes with a tailored flow can improve channel capacity, resilience to turbulence and non-line-of-sight capability. The hydrodynamic framework reported here provides precise control over light in free space, opening avenues in optomechanics, optofluidics, imaging, metrology, and communications.

Introduction

The field of structured light has evolved since early investigations of Laguerre-Gaussian and Hermite-Gaussian modes in laser cavities¹. Advances in theoretical modeling, together with the development of optical devices, such as spatial light modulators (SLMs), have broadened the scope of structured light research well beyond fundamental optics. Diverse families of beams are obtained as exact solutions of the Helmholtz equation in different coordinate systems, including Gaussian modes²—such as Hermite–Gaussian, Laguerre–Gaussian and Ince–Gaussian beams—that typically exhibit self-similar diffraction, preserving their transverse intensity profiles up to an overall scaling factor. In contrast, propagation-invariant modes such as Bessel³, cosine⁴, Mathieu⁵ and parabolic⁶ beams are diffraction-free and self-healing, featuring reconstruction after perturbation. Airy beams, characterized by a caustic phase⁷, combine the non-diffractive nature of self-healing with parabolic acceleration, leading to a spectrum of paraxial and non-paraxial modes that show accelerating⁸ and autofocusing properties⁹. These structured light fields now find applications across diverse domains, including fundamental physics¹⁰, telecommunications¹¹, security systems¹², micromachining¹³, imaging¹⁴ and optical manipulation of biological specimens¹⁵. Notably, phase-structured light, particularly vortex beams carrying orbital angular momentum (OAM)¹⁶, enables the exertion of not only forces but also torques on microscopic objects¹.

Despite these advances, the established structured light modes, constrained by the limited forms of Helmholtz equation solutions, typically obey fixed propagation laws. For instance, Airy beams accelerate along fixed parabolic trajectories⁷, Gaussian modes diverge hyperbolically and uncontrollably², and vortex modes inherently undergo OAM-dependent radial expansion¹⁷. This limits the adaptability and functionality of structured light in real-world applications^18,19,20,21. Typical methods used to impose a chosen structure to light include ray^{22,23,24,25,26}, wave^{27,28,29,30,31} and Fourier^32,33,34 optics, generally aiming at shaping a beam into one of the specific solutions of Helmholtz equations, but these methods are generally incompatible with each other.

In this work, we present a hydrodynamic formulation of optics for flexibly shaping structured light, drawing on the analogy between optics and fluid dynamics. Departing from the static field formulation based on the solutions of Helmholtz equation, we introduce energy streamlines³⁵ as three-dimensional trajectories of the Poynting vector to represent structured beam as optical flows and provide an intuitive depiction of photon motion and energy transport (Fig. 1). Using hydrodynamic principles, we develop a versatile method for streamline engineering, to flexibly sculpt a variety of structured light modes on demand. This method enables the integration of hallmark features from established structured light families within a single framework, including adjustable self-similarity akin to Gaussian modes, diffractionless and self-healing properties characteristic of propagation-invariant modes, customizable trajectories inspired by self-accelerating modes, as well as the intrinsic OAM of vortex beams and controlled energy delivery to counteract attenuation (Figs. 2 and 3). Streamline engineering facilitates the creation of specialized structured modes designed to overcome specific propagation challenges. For instance, addressing the broadening simultaneously induced by diffraction and OAM inherent in conventional vortex modes^20,21, we construct a type of vortex beam—non-diffracting perfect vortex beam that simultaneously suppresses both broadening effects (Fig. 4). Analogously to particle-tracking velocimetry in fluid dynamics, optical tweezer experiments demonstrate that microspheres trapped in the light field follow the designed streamlines, validating the effectiveness of the streamline-engineering approach and suggesting potential applications in precision 3D optomechanical control (Fig. 5). By addressing propagation challenges in free-space, we show that flow-customized vortex modes can improve optical communications by increasing the number of independent communication channels, enhancing resilience to turbulence, and improving non-line-of-sight communication capabilities in the presence of obstacles (Figs. 6 and 7). This streamline-engineering toolkit provides an experimentally validated, adaptable platform for shaping structured light, unlocking possibilities in areas such as optomechanics, optofluidics, imaging, metrology and communications.

Fig. 1: Analogy between field and hydrodynamics formulations.

Static complex field representation of Laguerre–Gaussian (a) and Bessel (b) beams governed by the Helmholtz equation, in the wave picture; c, d Dynamic energy flow representation of (a), (b), in the hydrodynamics picture based on streamlines (red curves). Amp., amplitude; arb. u., arbitrary units.

Fig. 2: Sculpting streamlines for adjustable self-similarity and customized self-acceleration.

a, b Bessel beams with adjustable self-similarity, exhibiting self-shrinking a and self-stretching b radii; red curves indicate the energy streamlines along the vortex mainlobe. c, d Bessel beams with customizable self-accelerating dynamics, propagating along a parabolic c and a spiral d trajectory; the yellow dashed lines denote the optical axis. e–g Intensity maps of (a–c) in the y–z plane; red dashed curves denote the vortex mainlobes. h Three-dimensional intensity iso-surface corresponding to (d). The streamlines herein are directly drawn from the streamline function with the distributions of Poynting vector calculated in MATLAB, which are consistent with the analytical solutions from the hydrodynamic differential equations in this work. i–l The corresponding momentum-space angular-spectrum distributions of (a–d) calculated by four-step streamline-engineering approach in Methods. Amp., amplitude; Pha., phase; arb. u., arbitrary units.

Fig. 3: Manipulating streamline interactions for self-healing and tunable energy delivery.

a When the mainlobe of a Bessel beam is intercepted by an obstacle (white rectangle), its energy streamlines (shown in yellow) are truncated. The sidelobe streamlines (red curves) spontaneously flow into the mainlobe region, reconstructing the original streamline structure. b In free-space propagation, a standard Bessel beam exhibits no interaction between mainlobe and sidelobe streamlines. c By actively converging sidelobe streamlines into the mainlobe, the local spatial energy density is enhanced; blue-dashed curves highlight the primary region of energy concentration (energy cones). d–f Intensity maps in the y–z plane corresponding to (a–c), respectively. g Attenuated scattered path of a standard Bessel beam propagating in a lossy medium (milk suspension). h Propagating profiles of the standard Bessel beam in the lossy medium, with the normalized intensity on the mainlobe depicted as blue triangles in (j)), used to probe the attenuation curve (blue curve in (j)). i After adjusting the inflow rate of sidelobe streamlines, the main-lobe intensity remains invariant (blue circles in (j)) despite the medium’s attenuation. For a second lossy medium (red curve in (j)), the probing and dynamically compensated mainlobe intensities are shown by red triangles and red circles, respectively. j Attenuation curves for both media (blue and red) with corresponding normalized main-lobe intensities: triangles denote unmodified probing Bessel beams and circles denote dynamically compensated Bessel beams. arb. u., arbitrary units.

Fig. 4: Hydrodynamic depiction for vortex dynamics.

a Conventional vortex beams and b Perfect vortex beams showing a radius increase with both OAM (l =10 and 20) and propagation distance. The rotation speed of energy streamlines (red curves) is reduced over propagation and does not increase with OAM. c Non-diffracting Bessel beams feature a radius growing with OAM, but staying constant over propagation distance. The rotation speed of energy streamlines decreases as the OAM increases, but is invariant with propagation. d Iso-propagation vortex beams, showing a fixed radius for all OAM (yellow arrows) that expands with propagation distance. The rotation speed of energy streamlines increases with OAM and decreases as the beam propagates. e Non-diffracting perfect vortex beams (NDPVBs), featuring a radius invariant with both OAM and distance. The rotation speed of energy streamlines increases with OAM and remains constant during propagation. f y-z intensity maps and (g) x-y complex-amplitude distributions for NDPVBs from (e). The demonstration for streamline rotation speeds can be found in Supplementary Note 7. Amp., amplitude; arb. u., arbitrary units.

Fig. 5: Probing photonic streamline dynamics through optofluidic velocimetry.

a Schematic of the optical tweezers setup for photonic streamline mapping. Adapted from Yan, W. et al. Hydrodynamic Insight Drives Multimodal Light Field Dynamics via Streamline Engineering. Preprint at arXiv:2507.07928 (2025). Licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Changes were made. Key components: beam splitters (BS), phase-only spatial light modulator (SLM), and lens pair (L1-L2). The open-top microfluidic chamber, containing 2-μm diameter polystyrene microspheres in deionized water, was mounted on a Nano-LP200 piezo nanopositioner. Synchronized z-axis displacement and CMOS camera exposure enable 3D trajectory reconstruction of liquid-borne probing microspheres through coordinated scanning. b–e Reconstructed 3D microsphere trajectories (time-color-coded discrete points) in: b 22th-order NDPVB, c 38th-order NDPVB, d self-similar NDPVB with linear radial expansion, and e conventional perfect vortex beam. Red curves: Predesigned energy streamlines. The conventional beam (e) only maintains 2D perfect in a single transverse plane, particles escape and steering ceases when diffracting beyond this plane. f–i the CMOS camera frames (Supplementary Movie 3) of liquid-borne probing microspheres on z-scan liquid-surface.

Fig. 6: Enhanced capacity and mitigated modal scattering amidst atmospheric turbulence.

a In conventional free-space communication, the limited-size receiver (Rx) obstructs OAM modes due to their expansion with propagation distance (z) and OAM order; when the size of OAM modes expands larger than the receiver, the optical link fails. b In contrast, NDPVBs maintain an on-demand, invariant size with respect to both z and OAM, enabling robust transmission; and the optical link remains established for large OAM and long distance, even in the presence of obstacles (due to self-healing ability) or turbulence. a, b Adapted with permission from Ref.³³, © Optica Publishing Group. c Comparison of the number of independent spatial subchannels supported by various spatial multiplexing techniques, evaluated for normalized space-bandwidth product (S) ranging from 1 to 30. d–h Assessment of free-space propagation under atmospheric turbulence for OAM beams, Laguerre-Gaussian beams (LGBs, nonzero–radial-index), iso-propagation vortex beams (IPVBs), and non-diffracting perfect vortex beams (NDPVBs), respectively. The transmission distance is 1000 m with a turbulence strength of C²_n = 5 × 10⁻¹⁵. Each beam has a consistent waist size while varying its OAM values (l = 22, 26, 30, 34, 38). d Normalized intensity retained by each initiated mode at z = 1000 m; Bars represent the mean from 80 independent trials simulated with a modified von Kármán turbulence model. Error bars depict the range (minimum to maximum) across these trials. e–h Crosstalk matrices for LGBs, OAM modes, IPVBs, and NDPVBs, providing a quantitative assessment of modal stability across the transmission pathway. Simulation details: propagation distance mesh of 40 m, analysis area of 0.5 m × 0.5 m, and turbulence outer and inner scales of 300 m and 0.01 m, respectively. arb. u., arbitrary units. (See Supplementary Note 10 for further details).

Fig. 7: NDPVB-multiplexing NLOS image transmission with ultra-high color fidelity.

a, b Schematics of non-line-of-sight channels, with obstacles fully blocking the receiver and streamline-engineered NDPVBs with (a) customized self-accelerating trajectories or (b) adjustable self-similar expansion navigate around the obstacles to restore the communication. Tx, transmitter; Rx, receiver. a, b Adapted with permission from ref. ³³, © Optica Publishing Group. c Original true-color image (128 × 128 pixels, 24-bit depth, 2²⁴ possible colors). d Color histograms for the full image and its individual red, green and blue 8-bit channels, each mapped to bit-positions 1–24. Bit-stream mapping: 24 bits per pixel are encoded onto NDPVB modes with indices l = (−60, 60, −55, 55, …, −5, 5) (mode spacing = 5) and carried via (Case 1) self-accelerating or (Case 2) self-similar beams to bypass occlusions (red lines). e Reconstructed image after demultiplexing, showing negligible distortion. Bit-error rates: 2.19 × 10⁻⁴ (Case 1) and 8.90 × 10⁻⁵ (Case 2), both well under the 3.8 × 10⁻³ forward-error-correction threshold. Experimental intensity patterns of Case 1: f streamline-engineered beam self-bends to surmount a wall-shaped mask, (j) standard NDPVB is blocked. Corresponding x-y intensity maps are shown in (g) and (k), respectively. Experimental intensity patterns of Case 2: (h) beam with expanded self-similar radius clears a plate-shaped mask, (l)standard NDPVB fails. Measured x-y intensity distributions appear in (i) and (m). See Supplementary Movie 5 for details of (f–m). arb. u., arbitrary units. Retrieved image channels and sharp-edged occluder mask can be found in Supplementary Note 11.

Results

Optics-fluids analogy

The parallel between laser dynamics and fluid or superfluid behavior dates back to the early 1970s, when laser-physics equations were recast into the form of the complex Ginzburg–Landau equations³⁶. Since then, this hydrodynamic framework has provided deep insight into a host of phenomena³⁶—from superconductivity and superfluidity to Bose–Einstein condensation³⁷. This connection is further underpinned by the formal linkage between the time-dependent equations of hydrodynamics and the steady-state Helmholtz equation governing monochromatic waves^38,39. Motivated by these connections, researchers have probed the hydrodynamic character of optical fields, revealing a wealth of nonlinear behaviors—chaos, multi-stability, and even turbulence—predicted theoretically and confirmed experimentally in laser systems^{40,41,42,43,44,45,46,47}. In 1989, Coullet et al., inspired by fluid vortices, formally introduced the notion of optical vortices⁴⁸, catalyzing the modern exploration of structured light. Building on this hydrodynamic insight, we now broaden the optics–fluid analogy to structured light shaping through the lens of optical flows and streamlines.

In classical fluid dynamics, the Eulerian description characterizes the fluid’s motion via a velocity field v(R, t), where R = {x, y, z} denotes spatial position. By integrating this field with the hydrodynamic differential equations, one visualizes the flow using streamlines—trajectories whose tangent at each point aligns with the local velocity. Formally, if R(z, t) = {x(z, t), y(z, t), z}, then its evolution obeys

$$\frac{dx(z,t)}{{v}_{x}({{{\bf{R}}}}(z),t)}=\frac{dy(z,t)}{{v}_{y}({{{\bf{R}}}}(z),t)}=\frac{dz}{{v}_{z}({{{\bf{R}}}}(z),t)}$$

(1)

where v = {v_x, v_y, v_z}. Regions where streamlines cluster correspond to faster flow, whereas sparse regions signify slower motion. This formalism lays the groundwork for mapping optical-field evolution onto an analogous hydrodynamic picture.

In this analogy, a monochromatic light field can be treated as an steady-state unchanging flow field³⁵ (∂v/∂t = 0), where its velocity field is represented by the momentum distribution of light—quantified by the Poynting vector⁴⁹. Specifically, for a scalar wave ψ(R), the Poynting vector p(R) —the local expectation value of the momentum operator—is given by

$${{{\bf{p}}}}({{{\bf{R}}}})={\mbox{Im}}{\psi }^{\ast }({{{\bf{R}}}})\nabla \psi ({{{\bf{R}}}})={|\psi ({{{\bf{R}}}})|}^{2}\nabla {\mbox{arg}}[\psi ({{{\bf{R}}}})],$$

(2)

By substituting this momentum distribution (Poynting vectors) into Eq. (1), the energy streamlines of the optical field—roadmap of photon motion—can be derived, which is the streamlines of the Poynting vector. These trajectories, R(z) = {x(z), y(z), z} = {r(z), φ(z), z}, can be determined in Cartesian or cylindrical coordinates by solving the hydrodynamic differential equations³⁵:

$$\begin{array}{cc}dx(z)/dz={{p}}_{x}({{{\bf{R}}}}(z))/{{p}}_{z}({{{\bf{R}}}}(z)),& dy(z)/dz={{p}}_{y}({{{\bf{R}}}}(z))/{{p}}_{z}({{{\bf{R}}}}(z));\end{array}$$

(3-1)

$$\begin{array}{cc}dr(z)/dz={{p}}_{r}({{{\bf{R}}}}(z))/{{p}}_{z}({{{\bf{R}}}}(z)),& d\varphi (z)/dz={{p}}_{\varphi }({{{\bf{R}}}}(z))/[r(z){{p}}_{z}({{{\bf{R}}}}(z))];\end{array}$$

(3-2)

where p = {p_x, p_y, p_z} = {p_r, p_φ, p_z}. These energy streamlines offer an intuitive picture of light propagation, often likened to the Bohmian trajectories that describe experimentally measurable paths of average photon trajectories^35,50,51,52. In quantum physics, the trajectories of the Poynting vector in light (or quantum-mechanical waves) are described as streamlines in the Madelung hydrodynamic interpretation⁵³, which are later regarded as paths of quantum particles in the Bohm–de Broglie interpretation^54,55.

For Bessel vortex beams³⁵, the energy streamlines follow the helical trajectories

$${{{\bf{R}}}}(z)=\{r(z),\varphi (z),z\}=\left\{{r}_{0},{\varphi }_{0}+\frac{l}{{{r}_{0}}^{2}{k}_{z0}}z,z\right\},$$

(4)

where r₀ and φ₀ denote each streamline’s initial radial and azimuthal coordinates, k_z0 represents the longitudinal wavevector component of the plane waves that constitute the Bessel beam. These paths wrap indefinitely around a cylinder of initial radius r₀ with no radial divergence (Fig. 1d). In contrast, Laguerre–Gaussian beams yield streamlines of the form³⁵

$${{{\bf{R}}}}(\zeta )=\{\rho (\zeta ),\varphi (\zeta ),\zeta \}=\left\{{\rho }_{0}\sqrt{1+{\zeta }^{2}},{\varphi }_{0}+\frac{l}{{{\rho }_{0}}^{2}}\arctan \zeta,\zeta \right\},$$

(5)

in scaled coordinates $(r,\varphi,z)\equiv ({\omega }_{0}\rho,\varphi,k{\omega }_{0}^{2}\zeta )$, where k donates the wavenumber. Here, each trajectory traces a helix on a hyperboloid, exhibiting pronounced radial spreading (Fig. 1c). These helical energy streamlines vividly illustrate the intrinsic rotational flow characteristic of vortex beams, serving as the hydrodynamic manifestation⁵⁶ of optical OAM.

To describe the optical/fluid field completely, an infinite number of streamlines would be required. Areas with clustered streamlines indicate faster flow, while sparse areas suggest slower motion. Due to limited sampling and the need for clear visualizations, we use a finite number of streamlines and adjust their brightness to represent streamline density. This preserves the key physical insights while ensuring clarity and computational efficiency.

Reversing the perspective in Eqs. (2–5), we established a four-step streamline-engineering approach (See Methods) to tailor the energy flowing of structured light: 1. Streamline Configuration: Prescribe the desired trajectories R(z) to encode specific flowing/propagating dynamics; 2. Momentum-field Sculpting: Use the hydrodynamic equations [Eq. (3)] together with the fluid-continuity condition to compute the required momentum (‘velocity’) distribution p(R); 3. Angular-spectrum engineering: Devise an angular-spectrum distribution S(k_r, ϕ, k_z) in momentum-space that produces the target momentum distribution when transformed into real space; 4. Real-space beam construction: Implement an optical Fourier transform (via a lens) to convert S(k_r, ϕ, k_z) into the physical field ψ(R), realizing deterministic control over its free-space dynamics. Building on this streamline-engineering approach, we sculpted diverse structured light modes on demand within a single framework (Fig. 2 and Fig. 3) and constructed specialized vortex modes designed to overcome complex propagation challenges of conventional vortex modes (Fig. 4).

Freely shaping multi-modal structured light with flows

This streamline-engineering method enables the on-demand customization of the multi-modal dynamics of diverse structured light within a single framework, including adjustable self-similarity akin to Laguerre–Gaussian modes, diffraction-free and self-healing properties characteristic of Bessel modes, customizable trajectories inspired by self-accelerating modes, the intrinsic OAM-carrying nature of vortex beams, and controlled energy delivery to counteract environment attenuation. The Bessel mode offers simple analytical form and provide a simple and elegant streamline configuration: in Eq. (4), when l is zero, the energy streamlines are axial straight line; otherwise, these streamlines wrap indefinitely around a cylinder of initial radius with no radial divergence. Therefore, we employed the Bessel beam as a pedagogical case study, integrating additional controllable propagation features by engineering their streamlines.

In cylindrical coordinates, the Bessel beam’s streamlines (Eq. (4)) are reshaped from R(z) = {r_I, φ₀ + l/(r₀²k_z0)z, z} to {r_I(z), φ₀ + l/[r_I²(z)k_z0(z)]z, z}, where r_I is the radius of the beam’s innermost-ring/mainlobe (Region of Interest, ROI), thereby endowing the beam with a dynamically adjustable self-similar innermost-ring radius r_I(z) (Fig. 2a–b). In Cartesian coordinates, the original streamlines of Bessel beam R(z) = {r_Icosφ(z), r_Isinφ(z), z} are translated to {x_s(z)+r_Icosφ(z), y_s(z)+r_Isinφ(z), z}, where φ(z) = φ₀ + l/(r_I²k_z0)z, so that beam’s mainlobe follows an arbitrary three-dimensional trajectory (x_s(z), y_s (z)), thus imparting customizable self-acceleration to the Bessel vortex mode (Fig. 2c–d).

Bessel beams can be decomposed into two regions: the mainlobe (i.e., ROI), which carries the highest intensity and governs interactions with matter or detectors, and a series of surrounding sidelobes. The sidelobes sustain and regulate the dynamics of the mainlobe; together, they constitute the entire beam and account for its total energy. During free-space propagation, main- and sidelobe energy streamlines rotate along their respective cylindrical surfaces without mutual energy exchange (Eq. 4, Fig. 3b). When an obstacle intercepts and truncates the mainlobe streamlines, the sidelobe streamlines spontaneously flow into the mainlobe region upon further propagation, reconstructing both its intensity profile and streamline structure (Fig. 3a). In the absence of these supportive sidelobe streamlines in conventional vortex beams, such spontaneous self-healing cannot occur (see Supplementary Note 4 and Supplementary Movie 1 for details).

During self-healing, sidelobe streamlines converge into the mainlobe without tunability. By invoking the optical fluid continuity equation (Methods), these sidelobes can be recast as an energy reservoir for the vortex mainlobe. The rate of sidelobe-to-mainlobe streamline transfer can be actively controlled, thereby dynamically adjusting the mainlobe’s spatial energy density to satisfy application-specific intensity requirements. For instance, by directing nearly all sidelobe streamlines into the mainlobe, the local energy density of mainlobe associated with matter or detector interactions can be selectively enhanced (Fig. 3c), yielding improved energy efficiency and signal-to-noise ratios (a case study in Supplementary Note 5 demonstrates a threefold improvement, as well as corresponding experimental results of Supplementary Movie 2). Moreover, by tuning the sidelobe inflow rate, the mainlobe intensity can be held constant across media with different attenuation coefficients, thereby meeting the stringent requirements of imaging and sensing in complex environments under uniform structured-light illumination⁵⁷. In Fig. 3g–j, attenuation profiles were first measured using a fundamental Bessel beam in various decay media, and the corresponding optimal sidelobe convergence rates were determined to dynamically compensate for the mainlobe’s energy decay. (Experiment Setup in Supplementary Note 6)

In contrast to established structured light which follow fixed propagation laws and specialized techniques^{22,23,24,25,26,27,28,29,30,31,32,33,34} that only shapes specific type of structured light, the streamline-engineering approach offers a more flexible and versatile strategy for shaping multimodal structured light, potentially broadening its functional applications^15,18,19. While vortex beams serve as representative and pedagogical cases in our demonstrations due to their well-defined streamline configurations and scientific importance, the underlying methodology establishes a general framework that is broadly applicable to diverse structured light families, including both vortex and non-vortex modes.

On-demand creation of specialized structured vortex modes

Among structured light modes, vortex beams^16,48—exemplified by higher-order Laguerre–Gaussian modes— represent a distinct subclass whose unique dynamics present further challenges. Such beams carry OAM with a helical wavefront and central dark core, and have attracted growing interest in optical manipulation, communications, and quantum technologies⁵⁸. However, their practical deployment remains hindered by two coupled propagating defects: First, beams with higher OAM orders exhibit substantial radial expansion and divergence, posing challenges for spatial-division multiplexing, fiber transmission, and optical trapping^20,21. Second, conventional vortex fields are subject to diffraction-induced broadening over extended propagation distances, degrading their performance in long-range communication, sensing, imaging, and quantum information^59,60. For instance, the beam radius of a zero–radial-index Laguerre–Gaussian mode increases with both topological charge (l) and distance (z), following: r(|l | , z) = w₀sqrt(|l | +1)sqrt[1 + (z/z₀)²], where w₀ and z₀ denote the beam waist and Rayleigh distance (Fig. 4a: OAM- and distance-dependent propagation radius). Although specialized designs—including perfect vortex⁶¹, Bessel vortex³, and iso-propagation vortex beams^31,33—can mitigate one of these issues, none concurrently overcomes both to date. Although to certain extent perfect vortex beams^61,62 maintain a OAM-independent size in a single 2D transverse plane, their radius changes along propagation, as a function of both distance and OAM (Fig. 4b). Propagation-invariant intensity profiles can be achieved with Bessel beams³, yet their ring radius grows substantially with increasing OAM (Fig. 4c). Lastly, iso-propagation vortex beams^31,33 are engineered to keep a OAM-independent propagating radius, but they nonetheless undergo diffraction-induced broadening as they propagate (Fig. 4d). For concurrently overcoming both propagation limits, we turn to the hydrodynamic perspective to gain deeper insight into their coupled propagation dynamics.

From a hydrodynamic perspective, vortex dynamics are characterized by twisted flows and helical streamlines, with an angular rotation rate ω_z = dφ(z)/dz = φ’(z). This rotation rate is directly tied to the photon’s OAM⁵⁶ and influences optical forces and momentum transfer during light–matter interactions^35,50. However, the combined effects of diffraction-driven spreading and OAM-driven expansion cause ω_z to diminish as either the propagation distance or the OAM magnitude increases (Fig. 4a-d; detailed analysis in Supplementary Note 7). In other words, the energy rotation rate does not increase for higher optical OAM; instead, it slows down over propagation. This observation suggests a potential strategy: by synchronizing ω_z with the beam’s OAM and stabilizing it against diffraction-induced decay, beam’s dual broadening could be inversely suppressed. Based on this insight, we develop a new type of vortex beam by configuring the streamline geometry. Specifically, we modify the streamlines of a Bessel beam (Eq. (4)) from R(z) = {r_I, φ₀ + l/(r_I²k_z0)z, z} (Eq. (4)) to R(z) = {r_I, φ₀ + l/(r_I²k_zl)z, z}, where r_I is the radius of the innermost ring and k_zl ≈ sqrt[k²-((|l | +2)/r_I)²] is an OAM-customized longitudinal wavevector component that counteracts the OAM-driven expansion of the innermost rings (Supplementary Notes 1). This adjustment effectively anchors its radius r_I to be invariant with respect to both OAM ($l$) and propagation distance (z), and yields an energy rotation rate ω_z = l/(r_I²k_zl)≈ l/(r_I²k)∝l, which is synchronized with the beam’s OAM and remains constant during propagation, ensuring expected vortex dynamics (as shown in Fig. 4e). Following the four-step method (see Methods), the angular-spectrum distribution of these streamline-customized Bessel modes, with a physically feasible non-diffracting range z∈ (b-a, b + a), can be expressed as:

$$S({k}_{x},{k}_{y})={{ {\mathcal F} }}_{z}\{rect[(z-b)/2a]{e}^{i{k}_{zl}z}{e}^{il\varphi }\}={{{\rm{sinc}}}}(2a({k}_{z}-{k}_{zl})){e}^{-i{k}_{z}b}{e}^{il\varphi },$$

(6)

where the subscript “z” of “${\mathcal F}$” refers to the Fourier transform dimensions and k_x²+k_y²+k_z² = k² (Experiment validations in Supplementary Note 1). The resulting vortex modes exhibit OAM- and propagation-invariant mainlobe size (Fig. 4e-g).

Unlike conventional 2D “perfect” vortex beams, which maintain an OAM-independent radius only in a fixed transverse plane^31,61,62, these streamline-customized vortex modes overcome both diffraction and OAM-induced expansion during propagation. As a result, they can be classified as (propagation-invariant) Non-Diffracting Perfect Vortex Beams⁶³ (NDPVBs). The flexibility of the streamline-engineering approach also allows NDPVBs to be customized with multimodal dynamics (e.g., adjustable self-similarity, customizable trajectories, and controlled energy delivery in Figs. 2 and 3), expanding their potential applications in complex, real-world environments^18,19,20,21. In the subsequent sections, we will explore the use of optical tweezers to probe the engineered streamlines (Fig. 5) and examine the advantages of these streamline-customized vortex modes with flexible multi-modal dynamics, particularly in free-space optical communications (Figs. 6 and 7).

Probing streamline dynamics by optical tweezers

In classical particle-tracking velocimetry for fluid dynamics⁶⁴, tracer beads are injected into a flow and their trajectories are recorded by high-speed cameras to reconstruct the underlying streamlines. An analogous strategy was adopted here: microspheres were seeded into the shaped light field, and their motion was captured to map the beam’s streamlines. The primary aim of these experiments was to demonstrate that the particle trajectories indeed faithfully follow the streamlines prescribed by our hydrodynamic approach, thereby validating its predictive capability. The optical-tweezers platform (Fig. 5a) was built on an inverted confocal microscope (Nikon TE2000-U) equipped with a 4-f holographic beam-shaping stage housing a reflective spatial light modulator (Holoeye Leto). A continuous-wave laser (Coherent Verdi-V5) was phase-modulated with computer-generated holograms to generate the required multimode structured light. These fields interacted with polystyrene microspheres of 2 μm diameter suspended in de-ionized water inside a custom open-top chamber, which comprised an acrylic plate with a through-hole and a glass coverslip forming the base. The open-top geometry minimized hydrodynamic resistance and allowed unrestricted particle motion at the water surface. The chamber was mounted on a Nano-LP200 piezo nanopositioner that displaced the liquid-borne microspheres along the optical axis in synchrony with the camera exposure. This coordinated z-scan permitted three-dimensional bead trajectories to be reconstructed with high spatial accuracy.

Representative measurements (Fig. 5b–d) reveal that tracer-sphere trajectories in 22nd- and 38th-order NDPVBs, as well as in self-similar NDPVB with the linearly expanding radius, adhere closely to the predesigned energy-streamline contours. This close agreement between designed and measured trajectories provides direct experimental validation of customized optical momentum flows engineered using our framework. By contrast, conventional perfect-vortex beams (Fig. 4b) yield a 2D perfect ring only within a single transverse plane; once the beam diffracts off that plane, trapped beads escape and trapping ceases (Fig. 5e). In the NDPVB configuration, however, robust three-dimensional trapping and steering within OAM-invariant radii persist over the full axial scan (Fig. 5b, c), thereby validating the propagation-invariant, three-dimensional perfect vortex configuration. Looking ahead, the combination of smaller probe particles, faster volumetric imaging, and axial-view detection modules⁶⁵ could achieve improved resolution of energy streamlines. Looking forward, the capability to engineer optical momentum flows potentially establishes a foundational capability for controlling light-matter interactions in three dimensions. This flow-based approach could thus provide a new design tool for future studies in optomechanical control, potentially enabling applications such as microfluidic flow steering or directed transport that leverage tailored photon trajectories⁶⁶.

Boosting capacity and robustness in free-space communications: a case study

In addressing complex propagation challenges of free-space optical communication^20,21, streamline-engineered NDPVBs—whose transverse size does not vary with OAM order or propagation distance—can support a substantially larger number of mutually orthogonal sub-channels than conventional OAM modes (zero-radial-index Laguerre–Gaussian beams), potentially increasing information capacity (Fig. 6a–c). When considering the effects of atmospheric turbulence on long-haul free-space links and remote sensing and ranging scenarios, NDPVBs were found to experience exhibit relatively weaker and more uniform modal scattering than conventional vortex beams—an observation attributed to their constant, OAM-invariant beam profile over distance (Fig. 6d–h). By exploiting flexible multimodal dynamics of streamline-engineered framework to NDPVB (e.g., customizable self-similar and self-accelerating dynamics), a traditional line-of-sight free-space link can be extended to support robust non-line-of-sight scenarios (Fig. 7). Together, these features suggest a potential framework for next-generation OAM-based free-space optical systems^20,21 that combine higher capacity with turbulence resilience and flexible link geometries.

The advances in information acquisition and processing have underscored the critical role of multiplexing in scaling communication capacity⁶⁷. Optical multiplexing strategies exploiting polarization and wavelength degrees of freedom have extended system bandwidth^68,69. Among emerging approaches, spatial mode-division multiplexing—where orthogonal spatial modes act as independent channels—has garnered increasing attention^11,70. In a representative scheme, a free-space optical link combining Q orthogonal OAM modes, two polarization states, and T wavelengths yields an aggregate capacity of Q×2×T×100 Gbit/s when each channel carries 100 Gbit/s using quadrature phase-shift keying. This configuration enables petabit-per-second-scale throughput⁷¹, thereby substantially improving both capacity and spectral efficiency in free-space optical communication systems. Despite its promise, vortex-based spatial mode-division multiplexing encounters notable limitations during free-space propagation. Conventional vortex beams suffer from diffraction-induced spreading and OAM-dependent radial expansion—effects that intensify with increasing topological charge and propagation distance^17,20,21,59. As a result, the practical number of accessible spatial subchannels, Q, is restricted by both the finite aperture of the receiving optics and these propagation limitations^20,21 (Fig. 6a).

To enable a direct comparison among diverse spatial multiplexing schemes, the number of supported individual subchannels, Q, was evaluated as a function of the normalized space-bandwidth product⁷² S = πR₀×NA/λ, where R₀ and NA are the common aperture radius and numerical aperture of both transmitter and receiver, and λ is the wavelength. This dimensionless factor represents the maximum space–bandwidth product a beam can occupy relative to a fundamental Gaussian mode; only modes satisfying this criterion are transmitted, thereby fixing the subchannel count. The resulting approximations for each technique are⁷²: Q_OAM(S) ≈ 2floor[S] + 1 for conventional OAM mode multiplexing, Q_LG(S) ≈ 0.5floor[S](floor[S] + 1) for Laguerre-gaussian beam multiplexing, Q_HG(S) ≈ 0.5floor[S](floor[S] + 1) for Hermite-Gaussian beam multiplexing, Q_MIMO(S) ≈ round[0.9S²] for multi-input multi-output transmission, Q_IPVB(S) ≈ floor(189.16$\sqrt{S-0.066}$) for Iso-propagation vortex beam multiplexing³³, respectively.

By employing the NDPVB basis—which inherently mitigates both diffraction-induced spreading and OAM-driven radial expansion (Fig. 6b)—a substantially expanded set of transmission modes can be accessed beyond those supported by conventional spatial multiplexing, thereby potentially boosting overall link capacity. Exploiting the OAM- and distance-invariant beam profile, the number of NDPVB subchannels is found, via the approach of ref. ⁷², to scale as Q_NDPVB(S) ≈ floor(569.21$\sqrt{S}$-3) (see Supplementary Note 8 for details). For realistic free-space links with finite apertures and S < 30, NDPVB multiplexing outperforms the existing schemes in available subchannels, as shown in Fig. 6c. For example, in our proof-of-principle setup (S = 6.25, Supplementary Note 9), Q_NDPVB reaches 1419—versus Q_OAM = 13, Q_LG = Q_HG = 21, Q_MIMO = 35, and Q_IPVB = 471—an enhancement ranging from threefold to over a hundredfold. We further evaluated multi-vortex geometric (MVG) beam multiplexing⁷⁰, noting that for S < 30, Q_MVG  ≈ Q_LG and thus remains below Q_NDPVB. Although practical capacity also depends on mode-spacing choices and inter-channel crosstalk, a higher upper bound on Q directly translates into a larger usable mode set and, typically, potentially increased capacity^20,21.

Atmospheric turbulence, together with diffraction-induced spreading and OAM-dependent radial expansion, limits both capacity and range in free-space links^18,20,21, and modal scattering grows with increasing turbulence strength or beam size increase⁷³. As shown in Fig. 6d–h, streamline-engineered NDPVBs, whose profiles are invariant to both OAM and propagation distance, exhibit lower and more uniform modal scattering than conventional OAM modes, Laguerre-gaussian beams and iso-propagation vortex beams under the same turbulent conditions. This turbulence resilience cements NDPVBs as potential carriers for high-capacity, turbulence-robust free-space multiplexing⁷⁴ (See Supplementary Note 10 for details). In addition to compensating for medium attenuation (Fig. 3), the dynamics of controlled energy delivery to the NDPVBs’ mainlobes could enable probing of atmospheric turbulence strength along the propagation path, thereby potentially facilitating targeted mitigation of turbulent effects⁷⁵.

Free-space optical links—whose beam diameters are far smaller than those of radio waves—are prone to interruption by occluders, leading to severe signal loss or complete blackout. Conventional systems thus require an unobstructed line-of-sight (LOS) between transmitter and receiver (Fig. 6a, b), which restricts their use in complex environments. By contrast, streamline-engineered NDPVBs exploit the flexible dynamics of adjustable self-acceleration and self-similarity to steer around obstacles, extending robust transmission to non-line-of-sight (NLOS) scenarios under dynamic occlusions (Fig. 7a, b).

A high-dimensional NLOS proof-of-principle is illustrated in Fig. 7c–m: a 128 × 128-pixel true-color Mandrill image (24-bit depth) was split into its red, green and blue channels (8 bits each) and mapped onto 24 NDPVB modes with indices l = (−60, 60, −55, 55, …, −5, 5), using a mode spacing of 5 to suppress crosstalk. Two NLOS channel-engineering cases were tested—self-accelerating trajectory (Case 1) and self-similar radius (Case 2)—to navigate a dynamic occluder (red parts in Fig. 7f–i). After demultiplexing (Supplementary Note 9 and Supplementary Movie 4), the image was faithfully reconstructed with bit-error rates of 2.19×10⁻⁴ (Case 1) and 8.90×10⁻⁵ (Case 2), both well below the 3.8×10⁻³ forward-error-correction limit (Fig. 7e). In contrast, conventional vortex beams and standard NDPVBs without streamline-engineering multimodal dynamics failed under the same conditions (Fig. 7j–m). The proof-of-concept setup—when using a digital mirror device at 11 kHz—achieved a peak data rate of 24×11k = 0.264 Mbit/s. Although our demonstration employed a single occluder due to current modulation limitations of commercial spatial light modulators, future high-performance platforms (e. g., metasurface) should enable even more complex NLOS links. Overall, streamline-engineered NDPVBs with flexible dynamics marks them as one of promising carriers for next-generation free-space optical networks^18,20,21 that combine ultrahigh capacity, turbulence resilience and robust NLOS transmission.

Discussion

Historically, research in structured light has largely followed a bottom-up approach, focusing on fine-tuning the fundamental parameters of electromagnetic fields, thereby controlling the internal momentum flows to ultimately guide specific global characteristics of light. In contrast, our study suggests a transformative approach by directly acting on the internal optical flows—momentum density—and providing a holistic control over light. Moving beyond the traditional static field presentation in the wave picture to a dynamic flow presentation in hydrodynamic frame, this formulation, centered on energy streamlines, offers an intuitive roadmap of photon trajectories, encompassing free-space phenomena such as diffraction, OAM-induced expansion, self-healing, self-acceleration, and self-similarity within a unified hydrodynamic framework. (See Supplementary Note 12 for the comparison among ray-based (geometric optics), field-based (wave optics), and flow-based (hydrodynamic) descriptions).

By directly engineering these streamlines, we have established a versatile and experimentally validated platform for the on-demand shaping of structured light. This framework transcends the limitations of conventional modes, which are confined to fixed propagation laws imposed by specific solutions to the Helmholtz equation and incompatible shaping techniques^{22,23,24,25,26,27,28,29,30,31,32,33,34}. Our approach, therefore, provides not only fundamental insights into light-matter interactions^49,76 and free-space optics but also opens avenues for applications across multiple domains. For instance, the ability to prescribe optical momentum flows—validated here by optical-tweezer velocimetry—potentially establishes a new design tool for advanced optomechanical control, opening prospects for applications such as microfluidic steering where tailored photon trajectories could provide a pathway for exerting optical forces with high spatial specificity⁶⁶. Moreover, our case study in free-space communications demonstrates that flow-shaped beams can simultaneously enhance channel capacity, turbulence resilience, and non-line-of-sight capability, underscoring their potential in next-generation optical networks^18,20,21. While vortex beams serve as representative cases in our demonstrations, the underlying methodology establishes a general framework that is broadly applicable to diverse structured light families, including both vortex and non-vortex modes (See Supplementary Note 13 for non-vortex case).

Looking forward, arising from the intrinsic link between momentum flows and optical angular momentum, the streamline-based framework provides a foundation for studying OAM dynamics across diverse structured light modalities, including both vortex and non-vortex beams⁵⁶. Furthermore, the incorporation of both spin and orbital components in optical flows^18,49 offers a potential route to extend this approach to vectorial structured light, moving beyond the scalar regime explored in this work. Although formal similarities exist between statistical optics and stochastic hydrodynamics, establishing a physically consistent fluid-mechanical description of optical coherence remains an open challenge that invites further theoretical investigation. We anticipate that this hydrodynamic perspective can provide a unified foundation for future explorations in structured light design, with potential implications for imaging⁵⁷, manipulation¹⁵, communication^20,21, simulation of fluid phenomena via optical analogues⁷⁷, and other advanced photonic technologies.

Methods

Streamline-engineering approach for freely shaping structured light

The principle of the four-step streamline-engineering approach is outlined in Supplementary Table 1: 1. Streamline Configuration: Prescribe the desired trajectories R(z) to encode desired propagation behaviors of Structured Light. 2. Momentum-field Sculpting: Use the hydrodynamic equations [Eq. (3)] together with the fluid-continuity condition to compute the required momentum distribution p(R); 3. Angular-spectrum Engineering: Devise an angular-spectrum distribution S(k_r, ϕ, k_z) in momentum-space that produces the target momentum field when transformed into real space. 4. Real-space beam construction: Implement an optical Fourier transform (via a lens) to convert S(k_r, ϕ, k_z) into the physical field ψ(R), realizing the customized structured light.

The four rows of Supplementary Table 1 illustrate the structured light shaping with different propagation behaviors: (1) non-diffracting perfect vortex beams⁶³ with OAM- and z-invariant radius r_I (Supplementary Note 1), (2) adjustable self-similarity with dynamic radius r_I(z), (3) customized self-acceleration with arbitrary trajectory (x_s(z), y_s(z)), and (4) tunable energy delivery with adjusted energy density I(z). Detailed derivations for Supplementary Table 1 can be found in Supplementary Note 1, and the validations of sculpted momentum fields can be found in Supplementary Note 2 with Supplementary Movie 6. Experimental generation and tomography for customized structured light can be found in Supplementary Note 3. Notably, this method primarily constructs analytic streamline trajectories R(z), with the potential for extension to more complex, non-analytic streamline configurations (e.g., numerically-defined, unclosed-form expressions) in future developments.

Streamline interaction manipulation by fluid-continuity equation

The propagation of optical energy in a beam can be described by optical Fluid-Continuity Equation⁴⁹ (namely, Transport of Intensity Equation⁷⁸), read as:

$$\frac{\partial W(r,\varphi,z)}{\partial z}=-c({\nabla }_{\perp }{{{{\boldsymbol{p}}}}}_{\perp }(r,\varphi,z)),$$

(7)

where W(r, φ, z) is the energy density, p_⊥(r, φ, z) the transverse energy-flow (momentum) density, ∇_⊥ denotes the transverse divergence operator, and c is the velocity of light. This relationship indicates that the transverse energy-flow (momentum) density p_⊥(r, φ, z), influences the propagation of optical energy density W(r, φ, z). If one treats W as a fluid density field and p_⊥ as the corresponding transverse velocity field, Eq. (7) becomes exactly the continuity equation of incompressible-fluid dynamics, enforcing conservation of matter.

Bessel modes can decompose these beams into two regions: mainlobe (region of interest): the high-intensity region that interacts directly with matter or detectors; sidelobes: a concentric series of rings that sustain and regulate the mainlobe’s propagation. During free-space propagation, energy streamlines in these two regions {r, φ₀ + l/(r²k_z0)z, z}, circulate on distinct cylindrical surfaces without exchanging energy. However, by invoking Eq. (7), we can actively couple the sidelobes to the mainlobe as an on-demand energy reservoir.

Considering the ring area of the mainlobe at r = r_I, the derivative ∂W(r_I, φ, z)/∂z is directly proportional to ∂I(z)/∂z = I’(z). The divergence of transverse energy flow or momentum density, ∇_⊥p_⊥(r, φ, z), can be equated to the flux of transverse energy flow across the ring. Owing to the beam’s perfect axisymmetry, azimuthal flows circulate within the ring and do not contribute to net flux; only the radial component matters. Hence, the radial energy flow exhibits a direct correlation with -I’(z₀), donated as -CI’(z₀) with the integrated positive constant C (Raw 4, Supplementary Table 1). When I’(z)>0, the negative sign in radial energy flow indicates an inward radial flow: sidelobe streamlines converge into the mainlobe, replenishing and amplifying its energy (see Fig. 3c). When I’(z) <0, the flow reverses, allowing controlled depletion of the mainlobe back into the sidelobes.

By programming the spatial light modulator to impose a desired axial energy gradient I’(z), one gains continuous, bidirectional control over the sidelobe-to-mainlobe energy transfer rate. This tunability enables: 1. Enhanced local intensity—by funneling nearly all sidelobe energy inward, one can boost detector signal-to-noise and overall energy efficiency (Fig. 3c, a three-fold improvement is detailed in Supplementary Note 5). 2. Attenuation compensation—by matching the sidelobe streamlines inflow rate to medium-induced decay, the mainlobe’s intensity remains constant during propagating (Fig. 3g–j). In this way, the sidelobes serve not only as passive scaffolding for self-healing (Fig. 3a) but also as an actively addressable energy reservoir for precision-tailored structured-light applications⁵⁷ (Fig. 3g–j).

Overall, the presence of sidelobes in our streamlined beams is a fundamental physical requirement, not an arbitrary design choice. These sidelobes emerge as a necessary supporting structure to ensure that the entire optical field, when engineered for customized mainlobe dynamics, remains a valid and stable solution to the Helmholtz equation in free space. In our framework, they function as a programmable energy reservoir. Beyond their passive role in intrinsic processes like self-healing, we actively manipulate the energy flow from these sidelobes to the mainlobe—governed by the optical fluid-continuity equation—to achieve on-demand control over the axial intensity profile, enabling functionalities such as dynamic attenuation compensation and localized energy enhancement in Fig. 3.