Structuring light with flows

Domain: neuroscience
Published: 2026-02-04
Editorial outcome: minor_revision

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 introduces 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 auto-focusing 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 customized applications18–21. Typical methods used to impose a chosen structure to light include ray22–26, wave27–31 and Fourier32–34 optics, 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 modes20,21, 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, generally 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.

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 waves38,39. 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 systems40–47. 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

dx(z,t)/v_x(R(z),t) = dy(z,t)/v_y(R(z),t) = dz/v_z(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 (\partial v/\partial t=0), where its velocity field is represented by the momentum distribution of light—quantified by the Poynting vector49. Specifically, for a scalar wave \psi(R), the Poynting vector p(R)—the local expectation value of the momentum operator—is given by

p(R) = Im \psi*(R) \nabla \psi(R) = |\psi(R)|^2 \nabla arg[\psi(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), \phi(z), z}, can be determined in Cartesian or cylindrical coordinates by solving the hydrodynamic differential equations35:

dx(z)/dz = p_x(R(z))/p_z(R(z)), dy(z)/dz = p_y(R(z))/p_z(R(z)); (3-1)

dr(z)/dz = p_r(R(z))/p_z(R(z)), d\phi(z)/dz = p_\phi(R(z))/[r(z)p_z(R(z))]; (3-2)

where p={p_x, p_y, p_z}={p_r, p_\phi, 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 trajectories35,50–52. 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 interpretation54,55.

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

R(z) = {r(z), \phi(z), z} = {r_0, \phi_0 + l/(r_0^2 k_{z0})z, z}, (4)

where r_0 and \phi_0 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 r_0 with no radial divergence (Fig. 1d). In contrast, Laguerre–Gaussian beams yield streamlines of the form35

R(\zeta) = {\rho(\zeta), \phi(\zeta), \zeta} = {\rho_0 \sqrt{1+\zeta^2}, \phi_0 + l/\rho_0^2 arctan \zeta, \zeta}, (5)

in scaled coordinates (r, \phi, z) \propto (\omega \rho, \phi, k\omega^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, \phi, 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, \phi, k_z) into the physical field \psi(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 provides 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, \phi_0 + l/(r_0^2 k_{z0})z, z} to {r_I(z), \phi_0 + l/[r_I^2(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_I cos \phi(z), r_I sin \phi(z), z} are translated to {x_s(z)+r_I cos \phi(z), y_s(z)+r_I sin \phi(z), z}, where \phi(z)=\phi_0 + l/(r_I^2 k_{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 the 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 techniques22–34 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 applications15,18,19. 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 beams16,48—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 trapping20,21. 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 information59,60. 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) = w_0 \sqrt{|l|+1} \sqrt[1+(z/z_0)^2], where w_0 and z_0 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 beams31,33—can mitigate one of these issues, none concurrently overcomes both to date. Although to certain extent perfect vortex beams61,62 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 beams31,33 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 \omega_z = d\phi(z)/dz = \phi'(z). This rotation rate is directly tied to the photon’s OAM56 and influences optical forces and momentum transfer during light–matter interactions35,50. However, the combined effects of diffraction-driven spreading and OAM-driven expansion cause \omega_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 \omega_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, \phi_0 + l/(r_I^2 k_{z0})z, z} (Eq. (4)) to R(z) = {r_I, \phi_0 + l/(r_I^2 k_{zl})z, z}, where r_I is the radius of the innermost ring and k_{zl} \approx sqrt[k^2-((|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 \omega_z = l/(r_I^2 k_{zl}) \approx l/(r_I^2 k) \propto 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 \in (b-a, b+a), can be expressed as:

S(k_x, k_y) = F_z {rect[(z-b)/2a] e^{i k_{zl} z} e^{i l \phi}} = sinc(2a(k_z - k_{zl})) e^{-i k_z b} e^{i l \phi}, (6)

where the subscript “z” of “F” refer to the Fourier transform dimensions and k_x^2 + k_y^2 + k_z^2 = k^2 (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 plane31,61,62, 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 environments18–21. 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 multimodal 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 multimodal structured light. These fields interacted with polystyrenemicrospheres of 2 \mu m 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 communication20,21, streamline-engineered NDPVBs—whose transverse size does not vary with OAM order or propagation distance—can support a substantially larger number of mutually orthogonal subchannels than conventional OAM modes (zero-radial-index Laguerre–Gaussian beams), potentially increasing information capacity (Fig. 6a–c). When considering the effects of atmospheric turbulence, together with diffraction-induced spreading and OAM-dependent radial expansion, limits both capacity and range in free-space links18,20,21, 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 \times 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 \times 10^{-4} (Case 1) and 8.90 \times 10^{-5} (Case 2), both well below the 3.8 \times 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 \times 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 networks18,20,21 that combine ultrahigh capacity, turbulence resilience and robust NLOS transmission.

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, \phi, 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, \phi, k_z) into the physical field \psi(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:

\partial W(r, \phi, z) / \partial z = -c(\nabla_\perp \cdot p_\perp(r, \phi, z)), (7)

where W(r, \phi, z) is the energy density, p_\perp(r, \phi, z) the transverse energy-flow (momentum) density, \nabla_\perp denotes the transverse divergence operator, and c is the velocity of light. This relationship indicates that the transverse energy-flow (momentum) density p_\perp(r, \phi, z), influences the propagation of optical energy density W(r, \phi, z). If one treats W as a fluid density field and p_\perp 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_0, \phi_0 + l/(r_0^2 k_{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 \partial W(r, \phi, z)/\partial z is directly proportional to \partial I(z)/\partial z = I'(z). The divergence of transverse energy flow or momentum density, \nabla_\perp \cdot p_\perp(r, \phi, 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'(z), donated as -C I'(z) 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.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors presents a framework for structured light shaping by re-conceptualizing light propagation in terms of “optical flows”—a hydrodynamic analogy that describes energy transport via streamlines of the Poynting vector. The authors demonstrate both numerically and experimentally the generation of customizable optical fields. Further, they validated and exhibited the optical properties via serval examples in real application.\nIn my opinion, the work is interesting, and can advance the development of optofluidics, optical manipulation, and high-capacity information coding. Moreover, the experimental results support the idea presented. To the best of my knowledge, the idea is novel. However, there are some issues should be addressed before publication.\n1. The conceptualization of light as an \"optical flow\" offers a natural and intriguing extension of analogies between optics and fluid dynamics. In engineering optics, light fields are often described via geometric rays (e.g., in caustics) or through wave-optical field representations. It would be beneficial for interdisciplinary readers if the authors could more explicitly compare and contrast these three perspectives—ray-based (geometric optics), field-based (wave optics), and flow-based (hydrodynamic) descriptions—in terms of their characteristics, relationships, and respective domains of applicability. Such a discussion would help contextualize the advantages and limitations of the proposed hydrodynamic approach.\n2. The authors develop a four-step streamline-engineering method to tailor structured light, which enables the integration of hallmark features from various structured light families within a single framework. While this method is versatile, it would be helpful to clarify whether it is limited to constructing analytic streamline forms, or if it can also accommodate numerically-defined or more complex streamline configurations that may not have closed-form expressions.\n3. In Line 430, the authors note that the current experimental demonstration uses only a single occluder due to limitations of commercial spatial light modulators, and suggest that future platforms like metasurfaces could enable more complex NLOS links. Since the proposed method relies on Fourier-domain modulation, its performance is inherently constrained by the capabilities of the modulating device. For readers less familiar with Fourier optics, a more detailed explanation of the relationship between the four-step modulation process and the practical limitations of current spatial light modulators (e.g., spatial bandwidth) would be valuable.\n4. In Supplementary Note 10 (Fig. S15B), the NDPVB are shown to evolve into a single ring after 1000 m of propagation. It would be useful to know whether this single-ring structure persists at longer distances. Additionally, the crosstalk analysis in Fig. 6 is performed under a relatively moderate turbulence strength, C2n = 5×10-15. How would the performance of NDPVBs and other beam types change under stronger turbulence conditions? A brief discussion or supplementary analysis on this point would strengthen the robustness claims.\n5. In Line 555, the authors describe attenuation compensation achieved by matching the sidelobe streamlines’ inflow rate to the medium-induced decay, thereby maintaining constant mainlobe intensity during propagation (Figs. 3G–J). A relevant question arises: does this sidelobe inflow rate have an upper limit in practice? What factors govern or constrain this maximum achievable inflow rate, and how do system parameters—such as beam power, spatial structure, or modulator characteristics—affect this process? Clarifying these practical limitations would help readers better evaluate the feasibility and scalability of this compensation mechanism, particularly in applications requiring stable illumination such as light-sheet microscopy [Nylk et al., Sci. Adv. 2018; 4 : eaar4817].\n6. In line 59, the authors, the author make an analogy between optics and fluid mechanics. Is there a one-to-one correspondence between them? That is, the intensity, phase, polarization of lights have corresponding entities inn fluid mechanics.\n7. In line 229, the authors claim structured Vortex beams carrying OAM can be generated via Sculpting Streamlines. The authors need to explain what is the fundamental mechanism for generating OAM?\n8. It is suggested that the author should appropriately add the field phases which can better exhibit OAM. Additionally, are the topological charge of such structured vortex beam calculated or characterized by special approach?\n9. There are some minor mistake in spelling or singular/plural issues that should be only addressed before publication. For example, in line 52 “This limit the…”; in line 180 “Poynting vector caculated in Matlab”; in line 274 “vortex beam by cofiguring the streamline”.\nIn summary, I think the approach original and the results convincing. The ability to unify various types of structured beams within a single scheme is notable. This hydrodynamic insight into light behavior appears promising for further investigation. After the above-mentioned issues are adequately addressed, I commend the manuscript is published in Nature Communications.", "reviewer_2": "This is a revision of the manuscript entitled \"Structuring Light with Flows\" by Wenxiang Yan et al. The manuscript explores the analogy between hydrodynamic flows and structured light, aiming to engineer optical beams with potentially superior properties compared to traditional structured modes. The authors provide experimental evidence supported by theoretical arguments, complemented with applied examples. Overall, the work presents an ambitious and exhaustive investigation of this phenomenon, which is timely and relevant for the optics community. The idea is interesting and merits consideration; however, some aspects of the manuscript require clarification and improvement before publication.\n- Consider expanding the references connecting optics and fluid analogies. Including these would provide better context and strengthen the contribution. Consider adding, for example:\nOptical Rankine vortex and anomalous circulation of light. Physical review letters, 99(16), 163901 (2007).\nAn analogy strategy for transformation optics. New J. Phys. 16 063008 (2014).\n- It would be very helpful to show the transverse amplitude and phase distributions of the beams at the plane of the spatial light modulator. Including at least one exemplary digital hologram would further support reproducibility and aid readers in understanding the experimental implementation.\n- The manuscript would benefit from an elaboration on the orthogonality characteristics of the Non-diffracting perfect vortex beams (NDPVB).\n- While the proposed method is promising, the manuscript currently lacks a discussion on its limitations. For instance, under which experimental conditions or parameter ranges might the approach fail or become impractical?\n- The hydrodynamics-based picture is appealing, but how does the analogy account for polarization and coherence effects in structured light? A discussion on this point would substantially enhance the completeness of the work.\n- There appear to be some overlaps between the present manuscript and Ref. [64], which also shares some authors. In particular, please compare the first paragraph of Section 3.1 in Ref. [64] with lines 85–94 of the current manuscript, as well as Figure 4 in Ref. [64] with Figure 5 of the current manuscript. Could the authors clarify what is novel in the present work and explicitly differentiate it from Ref. [64]?\n- Some passages are difficult to follow due to wording. A careful revision of grammar, style, and typographical errors is recommended. As an example, in Fig. 2 the yellow text reads “Optica Axis”, but I believe it should be “Optical Axis”.\nThe manuscript addresses an interesting topic with potential impact. However, I cannot recommend publication of this work in its current form. Revisions are necessary to clarify novelty, improve presentation, and address the points raised above.", "reviewer_3": "Hydrodynamics described by the Navier-Stokes or Euler equations, describes the temporal evolution of fluid motion over time, capturing complex phenomena like velocity and pressure changes. The Helmholtz equation, however, is a time-independent, linear partial differential equation that represents a simplified version of the wave equation, often used to describe steady-state wave phenomena, such as the spatial distribution of monochromatic waves (waves of a single frequency). While connected —for instance, by taking the time-independent form of an acoustic wave equation derived from hydrodynamics—they represent different aspects of physics: hydrodynamics deals with the full, time-dependent evolution of fluids, while the Helmholtz equation models time-harmonic spatial solutions.\nThe Helmholtz equation may be derived from the time-dependent wave equation, itself related to the Euler equations of hydrodynamics (a simplified form of the Navier-Stokes equations). In essence, the Helmholtz equation is the steady-state (time-independent) version of the wave equation, which is a direct descendant of the more general fluid motion equations used in hydrodynamics. Som papers for links are here:\nH. White, P. Bailey, J. Lawrence, J. George, J. Vera, Physics Open, Volume 1,2019,\n100009\nPhilos Trans A Math Phys Eng Sci 2020 Jul 20;378(2177):20190233\nThe main novelty of this paper seems to be linking optical waves to hydrodynamics in the case of structured light over the above references and in doing the work makes beams that has hydrodynamic Depiction for Vortex Dynamics. This leads to long range beams with for example fixed radii of independent of azimuthal index propagating long distances (e.g. figure 4). The tweezers study seemed less convincing to me with statements such as “may enable three-dimensional optomechanical control via engineered photon trajectories, potentially extending to applications such as microfluidic flow steering, directed cell transport, and assembly of microscale structures”. Many of these can be performed already – what is new and what can be achieved specifically that cannot with normal beam shaping. The Communications area study and statements are perhaps more convincing in this regard.\nOverall, what this approach shows is vortex beams with fixed radii, limited diffraction and variable index. What is not discussed clearly is the role, need for side lobes in the generated profiles and how they effect each of the experiments above. Though of some merit, I am not convinced with the data supplied this is worthy of Nature Communications but is better suited to a more specialised journal"}, "author_response": "Response to reviewers (Manuscript ID: NCOMMS-25-66816)\nNovember 15, 2025\nWe sincerely thank the editor and all reviewers for their insightful and valuable feedback, which\nhas significantly contributed to enhancing the quality of our manuscript. Below, we address\neach comment from the reviewers, highlighting our responses and the corresponding changes\nmade to the revised manuscript. We hope that our revisions address all concerns raised by the\nreviewers, making our manuscript more suitable for publication in Nature Communications.\nBelow is a comprehensive response to the comments from all Reviewers.\nPlease note the following conventions used in our response:\n Comments from the reviewers are presented in italicized text.\n Our responses to the reviewers' comments are in blue text, with key points underlined to\nemphasize their importance in the Responses.\n Modifications made in the revised manuscript are highlighted in red.\nResponse to Reviewer 1: Page 2-19\nResponse to Reviewer 2: Page 20-29\nResponse to Reviewer 3: Page 30-42\n“The authors presents a framework for structured light shaping by re-conceptualizing light\npropagation in terms of “optical flows”—a hydrodynamic analogy that describes energy\ntransport via streamlines of the Poynting vector. The authors demonstrate both numerically\nand experimentally the generation of customizable optical fields. Further, they validated and\nexhibited the optical properties via serval examples in real application.”\n“In my opinion, the work is interesting, and can advance the development of optofluidics, optical manipulation, and high-capacity information coding. Moreover, the experimental\nresults support the idea presented. To the best of my knowledge, the idea is novel. However,\nthere are some issues should be addressed before publication.”\nResponse: We sincerely thank the reviewer for their positive assessment of our work,\nacknowledging its novelty and potential to advance fields such as optofluidics and optical\ncommunication. We are also grateful for their recognition that the experimental results\nsolidly support the proposed concept. We have carefully considered all the comments and\nprovide point-by-point responses below, hoping that the revisions will fully address their\nconcerns.\nComment 1: “The conceptualization of light as an \"optical flow\" offers a natural and\nintriguing extension of analogies between optics and fluid dynamics. In engineering optics,\nlight fields are often described via geometric rays (e.g., in caustics) or through wave-optical\nfield representations. It would be beneficial for interdisciplinary readers if the authors could\nmore explicitly compare and contrast these three perspectives—ray-based (geometric optics),\nfield-based (wave optics), and flow-based (hydrodynamic) descriptions—in terms of their\ncharacteristics, relationships, and respective domains of applicability. Such a discussion would\nhelp contextualize the advantages and limitations of the proposed hydrodynamic approach.\nResponse 1: We thank the reviewer for raising this insightful question regarding the\ncomparison of different perspectives for describing structured light. This comparison is\nbeneficial for interdisciplinary readers. As suggested, we employ the Bessel beam—a typical\nexample with a simple analytical form and well-known properties—to illustrate and contrast\nthe three perspectives: ray-based (geometric optics), field-based (wave optics), and flow-based\n(hydrodynamic) descriptions.\nbeam and (d-f) for high-order Bessel vortex beam.\n The Field-based (Wave Optics) Perspective in Fig. R1(a) and (d) provides the\nfundamental mathematical expression of the static light field, governed by the Helmholtz\nequation. For a Bessel beam, this yields a propagation-invariant three-dimensional solution\nas\n3\n( )   U\nB e sse\n(l r , , z ) = J\nl\nk\nr 0\nr e il e ik z0 z .\n The Ray-based (Geometric Optics) Perspective in Fig. R1(b) and (e) approximates the\nlight field as a set of rays. The caustics formed by the coalescence of these rays correspond\nto \"intensity singularities\" and predict the primary intensity pattern. For instance, a zero-\norder Bessel beam exhibits a straight axial caustic (its main lobe), while a higher-order\nvortex Bessel beam features a tubular caustic along the axis (its main lobe ring). The\ntrajectories of the geometric ray in Bessel beams can be expressed as [J. Opt. A: Pure Appl.\nOpt. 10, 035005 (2008)]\n \n(),(),= (−)2+1, +tan−1(−), .\n0 0 0\n The Flow-based (Hydrodynamic) Perspective in Fig. R1(c) and (f) utilizes three-\ndimensional trajectories of the momentum field, i.e., energy streamlines or optical flows,\nto depict the intrinsic dynamic flow of energy within the light field. For example, the\nstreamlines of a zero-order Bessel beam are straight lines along the axis, while those of a\nhigher-order vortex Bessel beam form helices on a cylindrical surface, clearly\ndemonstrating the energy circulation inherent to an optical vortex. Streamline\ntrajectories are denoted by\n    R ( z ) = r ( z ) , ( z ) , z =\n\nr\n0\n,\n0\n+\nr\n0\n2\nl\nk\nz 0\nz , z\n\n,\nThe relationships and contrasts among these perspectives can be summarized as follows: The\nstatic field representation of the wave picture serves as the foundational mathematical-physical\ndescription. The dynamic flow picture describes the internal energy transport and could be\napplied to the propagation control of electromagnetic waves in free space, including light,\nradar, microwaves, and THz waves. When a geometric optics approximation is applied to the\nenergy streamlines, the ray picture emerges [J. Opt. A: Pure Appl. Opt. 10, 035005 (2008)].\nThus, the geometric ray description is an approximation of the flow picture, offering advantages\nin computational speed for simplified engineering scenarios but with a more limited scope of\napplicability.\nIn summary, for free-space optics and structured light, the wave picture describes the specific\nstatic field distribution, the ray picture quickly outlines the major intensity patterns via caustics,\nand the flow picture offers a description of the dynamic laws governing energy flow,\nthereby integrating these aspects within a unified hydrodynamic framework. We believe\nthis discussion helps to clarify the context and potential of our proposed approach.\nIn response to this insight, we have incorporated the following detailed discussion into the\nrevised manuscript and Supplementary Note 12:\n“ (Line 465) Historically, optical research has largely followed a \"bottom-up\" approach, focusing on fine-\ntuning the fundamental parameters of electromagnetic fields, thereby interplaying internal flows to\nultimately guide specific external behaviors of light, as depicted by the green arrows in Fig. 8. In contrast, our\nstudy suggests a transformative approach by directly orchestrating the internal optical flows, paving the way\nfor more comprehensive and holistic control over light, as reflected by the blue arrows in Fig. 8. Moving\nbeyond the traditional static “field presentation” in wave picture to dynamic “flow presentation” in\nhydrodynamic picture, this framework, centered on energy streamlines, offers an intuitive roadmap of\nphoton trajectories, encompassing free-space phenomena such as diffraction, OAM-induced expansion, self-\nhealing, self-acceleration, and self-similarity within a unified hydrodynamic framework. (See Supplementary\nNote 12 for the comparison among ray-based (geometric optics), field-based (wave optics), and flow-based\n(hydrodynamic) descriptions)”\n“\nSupplementary Note 12: Comparison among Wave, Ray and\nHydrodynamic Picture\nGiven the simple analytical form and well-established properties of the Bessel beam, we use it as\na representative example to illustrate and compare the three perspectives: ray-based (geometric\noptics), field-based (wave optics), and flow-based (hydrodynamic) descriptions.\n The Field-based (Wave Optics) Perspective, shown in Figs. S17(a) and (d), provides the\nfundamental mathematical expression of the static light field governed by the Helmholtz\nequation. For a Bessel beam, this yields a propagation-invariant three-dimensional solution as\nU (r,,z)= J (k r)eileik z0 z. (S39)\nBessel l r0\n The Ray-based (Geometric Optics) Perspective, illustrated in Figs. S17 (b) and (e),\napproximates the light field as a collection of rays. The caustics formed by the coalescence of these\nrays correspond to \"intensity singularities\" and predict the primary intensity pattern. For\n4\ninstance, a zero-order Bessel beam exhibits a straight axial caustic (i.e., its main lobe), while a\nhigher-order vortex Bessel beam features a tubular caustic along the axis (i.e., its main lobe ring).\nThe trajectories of the geometric ray can be expressed as15\n \n(),(),= (−)2+1, +tan−1(−),. (S40)\n0 0 0\n The Flow-based (Hydrodynamic) Perspective, depicted in Figs. S17(c) and (f), utilizes three-\ndimensional trajectories of the momentum field, i.e., energy streamlines, to depict the intrinsic\ndynamic flow of energy within the light field. For example, the streamlines of a zero-order Bessel\nbeam are straight lines along the axis, while those of a higher-order vortex Bessel beam form\nhelices on a cylindrical surface, clearly demonstrating the energy circulation inherent to an optical\nvortex. Streamline trajectories are denoted by\n5\n    R ( z ) = r ( z ) , ( z ) , z =\n\nr\n0\n,\n0\n+\nr\n0\n2\nl\nk\nz0\nz , z\n\n, (S41)\nFig. S17. Comparison among wave, ray and hydrodynamic picture. (a-c) for 0-order Bessel\nbeam and (d-f) for high-order Bessel vortex beam.\nThe relationships and contrasts among these perspectives are as follows: The static field\nrepresentation of the wave picture serves as the foundational mathematical-physical description. The\ndynamic flow picture describes the internal energy transport and can be applied to the propagation\ncontrol of electromagnetic waves in free space, including light, radar, microwaves, and THz waves.\nWhen a geometric optics approximation is applied to the energy streamlines, the ray picture is\nrecovered15. Thus, the geometric ray description is an approximation of the flow picture, offering\nadvantages in computational speed for simplified engineering scenarios at the cost of limited\napplicability.\nIn summary, for free-space optics and structured light, the wave picture describes the specific\nstatic field distribution, the ray picture quickly outlines the major intensity patterns via caustics, and\nthe flow picture offers a description of the dynamic laws governing energy flow, thereby\nintegrating these aspects within a unified hydrodynamic framework.”\nComment 2:“The authors develop a four-step streamline-engineering method to tailor\nstructured light, which enables the integration of hallmark features from various structured\nlight families within a single framework. While this method is versatile, it would be helpful to\nclarify whether it is limited to constructing analytic streamline forms, or if it can also\naccommodate numerically-defined or more complex streamline configurations that may not\nhave closed-form expressions.”\nResponse 2: We thank the reviewer for raising this important point regarding the applicability\nof our streamline-engineering method. Our current framework primarily on constructing\nanalytic streamline trajectories, which already enable significant versatility for shaping\nstructured light. We fully agree that extending the method to accommodate more complex, non-\nanalytic streamline configurations—such as numerically-defined or unclosed-form\nexpressions—represents a promising direction for future work. In response, we have added a\nclarification in the Methods section to reflect this point:\n“ (Line 514) The four rows of Table 1 illustrate the structured light shaping with different\npropagation behaviors: (1) non-diffracting perfect vortex beams70 with OAM- and z-invariant radius r\nI\n, (2) adjustable self-similarity with dynamic radius r(z), (3) customized self-acceleration with arbitrary\nI\ntrajectory (x(z), y(z)), and (4) tunable energy delivery with adjusted energy density I(z). Detailed\ns s\nderivations for Table 1 can be found in Supplementary Note 1, and the validations of sculpted\nmomentum fields can be found in Supplementary Note 2 with Supplementary Movie 6. Notably, this\nmethod primarily constructs analytic streamline trajectories R(z), with the potential for extension\nto more complex, non-analytic streamline configurations (e.g., numerically-defined, unclosed-\nform expressions) in future developments. While vortex beams serve as representative cases in\nour demonstrations, the underlying methodology establishes a general framework that is broadly\napplicable to diverse structured light families, including both vortex and non-vortex modes (See\nSupplementary Note 13 for non-vortex case)”\nComment 3:“In Line 430, the authors note that the current experimental demonstration uses\nonly a single occluder due to limitations of commercial spatial light modulators, and suggest\nthat future platforms like metasurfaces could enable more complex NLOS links. Since the\nproposed method relies on Fourier-domain modulation, its performance is inherently\nconstrained by the capabilities of the modulating device. For readers less familiar with Fourier\noptics, a more detailed explanation of the relationship between the four-step modulation\nprocess and the practical limitations of current spatial light modulators (e.g., spatial bandwidth)\nwould be valuable.”\nResponse 3: We thank the reviewer for this constructive comment, which highlights the\n6\nimportance of clarifying the practical constraints imposed by spatial light modulators (SLMs)\nin our method.\nThe modulation capabilities of SLMs are fundamentally limited by their pixel pitch and panel\nsizes. Specifically, the maximum modulated transverse wavevector, k max and k max, are given\nx y\nby k max =1/δx and k max =1/δy, where δx and δy are the pixel sizes. When using an SLM in a\nx y\nlens focusing process to load the momentum-space angular-spectrum distribution S(k , k ), the\nx y\nmaximum values of k max and k max become k max = 0.5Lk/f and k max = 0.5L k/f, where L and L\nx y x x y y x y\nare the panel dimensions, k is the wavenumber, and f is the focal length of the focusing lens.\nWhen the designed streamline trajectories vary significantly—such as in cases involving\ncomplex paths for intricate occlusions or strong attenuation compensation—the resulting\nangular-spectrum distribution S(k , k ) contains higher spatial frequency components. If these\nx y\ncomponents exceed the SLM’s modulation limits (k max and k max), the experimental realization\nx y\nwill deviate from the design due to the loss of high-frequency content. Therefore, devices with\nfiner pixel resolution or larger active areas, such as metasurfaces with subwavelength-scale\npixels, may help support the generation of more complex optical flows in future\nimplementations.\nThe following explanation has been added to the Methods – Experimental Generation section\nwith revised Fig. 9 to aid readers less familiar with Fourier-domain modulation. This addition\naims to provide a clearer connection between the theoretical framework and current\nexperimental practicalities:\n“(Line 541) Generation. In Fig. 9a, a reflective phase-only spatial light modulator (SLM) imprinted\nwith computer-generated hologram patterns (Fig. 9c) transforms a collimated laser light wave into the\ncomplex field S(k, k) (Fig. 9b), with help of spatial filtering via a 4-F system consisting of two identical\nx y\nlenses L1 and L2, and an iris as well. The resulting field is responsible for customized structured light\nin the focal volume of lens L3. Notably, the maximum modulated transverse wavevector, kmax and kmax\nx y\n, are given by kmax = 0.5Lk/f and kmax = 0.5Lk/f, where L and L are the panel dimensions of\nx x y y x y\nSLM (Fig. 9c), k is the wavenumber, and f is the focal length of the focusing lens L3. When the\nconstructed streamline trajectories vary significantly — such as in more complex trajectories\ndesigned to handle intricate occlusions or compensate for attenuation in high-loss media —\nthe angular-spectrum distribution S(k, k) in Fig. 9b will include higher spatial frequency\nx y\ncomponents. These components may exceed the modulation limits of the SLM, causing a\nmismatch between the design and the device's capabilities, which results in degraded\nexperimental reproduction due to the loss of high-frequency components. Future platforms\nwith enhanced modulation capabilities, such as metasurfaces with subwavelength\nsubwavelength-scale pixels, could potentially overcome these limitations.\nExperiment setup; BS, beam splitter; SLM, phase-only spatial light modulator; L1-L5, lens; M, mirror;\nRAPM, right-angle prism mirror; HRPM, hollow roof prism mirror. b Target angular specturm from\nfour-step streamline-engineering approach are modulated to c the phase-only hologram by complex-\namplitude modulation and loaded on the SLM panel with the panel dimensions of L and L. ”\nx y\nComment 4(a): “In Supplementary Note 10 (Fig. S15B), the NDPVB are shown to evolve into\na single ring after 1000 m of propagation. It would be useful to know whether this single-ring\nstructure persists at longer distances.\nResponse 4(a): We thank the reviewer for raising this relevant point regarding the propagation\nbehavior of NDPVBs. The angular-spectrum expression of the NDPVB, given by Equation (6)\nin the main text, is:\nS(k ,k )=F {rect[(z-b)/2a]eik zl zeil}=sinc(2a(k −k ))e−ik z beil, (6) of main text\nx y z z zl\nHere, the rectangular function, rect[(z-b)/2a], defines the physically realizable non-diffracting\nrange as z (b-a, b+a). The angular-spectrum distribution, which takes the form of a ring-\nshaped sinc function, indicates that the NDPVB exhibits characteristics of a quasi-Bessel mode\nin the spatial domain. In the practical communication and turbulence studies presented in\nSection-“Boosting Capacity and Robustness in Free-Space Communications: a Case Study”,\nthe non-diffracting range is typically set to match the propagation distance to the detector—for\ninstance, z (0, 1000) meters in this case.\nDue to the conical wavevector structure inherent to Bessel-like modes and the continuous\nenergy transfer from sidelobes to the main lobe during propagation, the sidelobes that sustain\ndiffraction-free propagation are gradually depleted. Once the beam reaches the end of the\ndesignated non-diffracting range, these sidelobes are effectively exhausted. Beyond this point,\nthe beam can no longer maintain its propagation-invariant profile and naturally dissipates with\nan uncontrolled ring structure. To support transmission over longer distances, the non-\n8\ndiffracting range can be extended by adjusting the parameters in Equation (6) accordingly.\nThus, in Supplementary Note 10, the non-diffracting range was specifically set from 0 to 1000\nmeters to match the specific propagation scenario under investigation. For clarity, we have\nadded an explanatory note in the caption of Fig. S15, stating that the non-diffracting range is\nconfigured as 0 to 1000 meters.\n“Fig. S15. Assessing Free-space Propagation Amidst Atmospheric Turbulence. (A) Illustrate the\ncomplex field patterns and the beam sizes (indicated by dashed rectangles) of OAM beams, LG beams,\nLG-beam-driven iso-propagation vortex beams (IPVBs), and NDPVBs, respectively, at transmission\npoints; the intensity (Int) and phase are visualized by the colormap; Each beam has a consistent beam\nwaist while varying its OAM values (l = 22, 26, 30, 34, 38). (B) These observations are post a 1000 m\nfree-space transmission through atmospheric turbulence with a turbulence strength of C2n = 5×10-15.\nThe simulation details include a propagation distance mesh of 40 m, an analysis area of 0.5 m×0.5 m,\nand turbulence outer and inner scales of 300 m and 0.01 m, respectively. (C-F) Normalized intensity\nspectrums (averaged 80 trials) retained by each initiated mode with l = 22 at z = 1000 m, providing\nassessments of modal stability across the transmission pathway. The non-diffracting range of NDPVB is\nconfigured from 0 to 1000 meters.”\nComment 4(b): Additionally, the crosstalk analysis in Fig. 6 is performed under a relatively\nmoderate turbulence strength, C2n = 5×10-15. How would the performance of NDPVBs and\nother beam types change under stronger turbulence conditions? A brief discussion or\nsupplementary analysis on this point would strengthen the robustness claims.”\nResponse 4(b): We thank the reviewer for raising this important question regarding the\nperformance of NDPVBs under stronger turbulence conditions, which helps to further assess\nthe robustness of our approach.\nThe actual bulk turbulence channel affects both the amplitude and phase of structured light\n[Martin P. J. Lavery et al., Free-space propagation of high-dimensional structured optical fields\nin an urban environment. Sci. Adv.3, e1700552(2017)]. In our simulation, we employed the\nsplit-step beam propagation method [J. D. Schmidt, Numerical simulation of optical wave\npropagation: With examples in matlab,” (SPIE, 2010)], instead of the thin phase model. The\nsplit-step beam propagation method can effectively demonstrate the influence of real bulk\nturbulence on the complex amplitude of structured light, including phenomena such as vortex\nsplitting. These effects manifest in the mode spectrum as crosstalk and variations in the mean\norbital angular momentum [Lavery M P J. Vortex instability in turbulent free-space propagation.\nNew Journal of Physics, 2018, 20(4): 043023].\nAs the turbulence strength increases beyond the moderate level presented in Fig. 6 (e.g., C2\nn\n>5×10−15), all structured beams—including NDPVBs, conventional OAM modes, LG beams,\nand IPVBs—would experience stronger mode scattering over the same transmission distance.\nThis leads to a further reduction in the normalized intensity retained by each launched mode\n9\nand a corresponding increase in inter-modal crosstalk, resulting in greater off-diagonal\nspreading and reduced clarity in the crosstalk matrices. [A. Klug et al., \"The orbital angular\nmomentum of a turbulent atmosphere and its impact on propagating structured light fields, New\nJ. Phy. 23, 093012 (2021)].\nNotably, owing to their OAM- and propagation-distance-invariant intensity profile, NDPVBs\nare expected to maintain a comparative advantage in crosstalk uniformity and turbulence\nresilience over conventional vortex beams, even as absolute performance degrades with\nincreasing turbulence strength.\nIn response to the reviewer's suggestion, we have added a supplementary analysis in\nSupplementary Note 10 to discuss these expected trends under stronger turbulence conditions,\nwhich further supports the robustness claims associated with the proposed beams:\n“To explore the performance of different beams impacted by atmospheric turbulence, we utilized the\nmodified von Karman turbulence model13 in our simulation. We assumed a moderate turbulence level,\ncharacterized by an optical turbulence strength of C2 = 5×10-15 with a propagation distance mesh of ∆z\nn\n= 40 m, a simulation area size of 0.5 m × 0.5 m, and turbulence outer and inner scales of 300 m and 0.01\nm, respectively. The simulated results averaged 80 trials. The procedure and setup for experimental\nvalidation are consistent with ref.14 and are realized by imprinting both the turbulence transmission\nfunction and conjugated demultiplexing modes in the SLM2 in Fig. S13(a). As illutrated in Fig. S15 and\nFigs. S6D–S6H, streamline-engineered NDPVBs, whose profiles are invariant to both OAM and\npropagation distance, exhibit markedly lower and more uniform modal scattering than conventional\nOAM modes, Laguerre-Gaussian beams and iso-propagation vortex beams under the same turbulent\nconditions, standing as promising candidates for atmospheric optical communication multiplexing.\nAs the turbulence strength increases beyond the moderate level presented in Fig. 6 (e.g., C2 >\nn\n5×10−15), all structured beams—including NDPVBs, conventional OAM modes, LG beams, and\nIPVBs—would experience stronger mode scattering over the same transmission distance. This leads\nto a further reduction in the normalized intensity retained by each launched mode and a corresponding\nincrease in inter-modal crosstalk, resulting in greater off-diagonal spreading and reduced clarity in the\ncrosstalk matrices. Notably, owing to their OAM- and propagation-distance-invariant intensity profile,\nNDPVBs are expected to maintain a comparative advantage in crosstalk uniformity and turbulence\nresilience over conventional vortex beams, even as absolute performance degrades with increasing\nturbulence strength14.”\nComment 5: “In Line 555, the authors describe attenuation compensation achieved by\nmatching the sidelobe streamlines’ inflow rate to the medium-induced decay, thereby\nmaintaining constant mainlobe intensity during propagation (Figs. 3G–J). A relevant question\narises: does this sidelobe inflow rate have an upper limit in practice? What factors govern or\nconstrain this maximum achievable inflow rate, and how do system parameters—such as beam\npower, spatial structure, or modulator characteristics—affect this process? Clarifying these\npractical limitations would help readers better evaluate the feasibility and scalability of this\ncompensation mechanism, particularly in applications requiring stable illumination such as\n10\nlight-sheet microscopy [Nylk et al., Sci. Adv. 2018; 4 : eaar4817].”\nResponse 5: We appreciate the reviewer’s thoughtful comment regarding the practical limits of\nthe sidelobe inflow rate, which helps clarify key constraints in our attenuation compensation\napproach. Indeed, the maximum achievable inflow rate is fundamentally constrained by the\nmodulation capabilities of the spatial light modulator (SLM).\nSpecifically, when an SLM is used in a lens-based Fourier setup to encode the momentum-\nspace distribution S(k , k ), the maximum spatial frequencies that can be modulated are given\nx y\nby k max = 0.5Lk/f and k max = 0.5L k/f, where L and L denote the SLM panel dimensions, k is\nx x y y x y\nthe wavenumber, and f is the focal length of the Fourier lens. In cases where the designed\nstreamline trajectories vary significantly—such as when compensating for strong attenuation\nor navigating complex occlusions—the resulting angular spectrum contains higher spatial\nfrequency components. If these exceed the SLM’s modulation limits, the experimental\nreproduction degrades due to loss of high-frequency information. Crucially, a high sidelobe\ninflow rate necessitates rapid spatial redistribution of energy, which is precisely encoded in\nthese high-frequency components of the angular spectrum. Thereby, the SLM's spatial\nbandwidth fundamentally constrains the maximum sidelobe inflow rate achievable in practice.\nTo support higher inflow rates with more intricate energy trajectories, devices with finer pixel\nresolution or larger active areas—such as optical metasurfaces—could be employed in future\nimplementations, particularly in applications demanding precise intensity control such as light-\nsheet microscopy [Nylk et al., Sci. Adv. 2018; 4: eaar4817].\nTo clarify these practical limitations of current SLMs and their relationship with the inflow rate,\nwe have added a more detailed explanation in the Method 2 – Experimental Generation\nsection, accompanied by the revised Fig. 9:\n“(Line 541) Generation. In Fig. 9a, a reflective phase-only spatial light modulator (SLM) imprinted\nwith computer-generated hologram patterns (Fig. 9c) transforms a collimated laser light wave into the\ncomplex field S(k, k) (Fig. 9b), with help of spatial filtering via a 4-F system consisting of two identical\nx y\nlenses L1 and L2, and an iris as well. The resulting field is responsible for customized structured light\nin the focal volume of lens L3. Notably, the maximum modulated transverse wavevector, kmax and kmax\nx y\n, are given by kmax = 0.5Lk/f and kmax = 0.5Lk/f, where L and L are the panel dimensions of SLM (Fig.\nx x y y x y\n9c), k is the wavenumber, and f is the focal length of the focusing lens L3. When the constructed\nstreamline trajectories vary significantly — such as in more complex trajectories designed to handle\nintricate occlusions or compensate for attenuation in high-loss media — the angular-spectrum\ndistribution S(k, k) in Fig. 9b will include higher spatial frequency components. These components\nx y\nmay exceed the modulation limits of the SLM, causing a mismatch between the design and the device's\ncapabilities, which results in degraded experimental reproduction due to the loss of high-frequency\ncomponents. Future platforms with enhanced modulation capabilities, such as metasurfaces with\nsubwavelength subwavelength-scale pixels, could potentially overcome these limitations.\nExperiment setup; BS, beam splitter; SLM, phase-only spatial light modulator; L1-L5, lens; M, mirror;\nRAPM, right-angle prism mirror; HRPM, hollow roof prism mirror. b Target angular specturm from\nfour-step streamline-engineering approach are modulated to c the phase-only hologram by complex-\namplitude modulation and loaded on the SLM panel with the panel dimensions of L and L. ”\nx y\nComment 6: “In line 59, the authors make an analogy between optics and fluid mechanics. Is\nthere a one-to-one correspondence between them? That is, the intensity, phase, polarization of\nlights have corresponding entities in fluid mechanics.”\nResponse 6: We thank the reviewer for raising this interesting and fundamental question\nregarding the optics–fluid analogy. In the context of free-space structured light propagation,\nthe analogy primarily establishes a correspondence between the density and velocity fields in fluid\nmechanics and the energy (intensity) and momentum distributions in optics, respectively.\nSpecifically, the optical intensity corresponds to the fluid density, while the Poynting vector\ndistribution corresponds to the velocity field.\nAt the current stage, a comprehensive one-to-one mapping encompassing all optical properties\nhas not been fully established. For example, although future extensions may relate fluid angular\nmomentum to optical angular momentum density, clear counterparts for optical phase and\npolarization in a fluid-mechanical context remain undefined. We agree that further exploring\nthese aspects represents a promising research direction, which could enrich the hydrodynamic\ndescription of light and reveal new physical insights.\nAlthough polarization lacks a direct fluid analog, the streamline-engineering approach\nintroduced in this work may be extendable to vectorial structured light. The Poynting vector\ncomprises two components: an orbital component arising from spatial inhomogeneities and a\nspin component associated with polarization inhomogeneities. [Bekshaev, A., Bliokh, K. Y. &\nSoskin, M. Internal flows and energy circulation in light beams. J. Opt. 13, 053001 (2011).]\nutilizing only the orbital component of the optical flow. Looking forward, incorporating the\nspin flow component could provide a route for shaping vectorial structured light beyond the\nscalar regime, as noted in the Discussion section：\n“(Line 491). Looking forward, arising from the intrinsic link between momentum flows and optical\nangular momentum, the streamline-based framework provides a foundation for studying OAM\ndynamics across diverse structured light modalities, including both vortex and non-vortex beams64.\nFurthermore, the incorporation of both spin and orbital components in optical flows30,57,75,88 offers a\npotential route to extend this approach to vectorial structured light, moving beyond the scalar regime\nexplored in this work. Separately, although formal similarities exist between statistical optics and\nstochastic hydrodynamics, establishing a physically consistent fluid-mechanical description of optical\ncoherence remains an open challenge that invites further theoretical investigation. We anticipate that\nthis hydrodynamic perspective may provide a unified foundation for future explorations in structured\nlight design, with potential implications for imaging65, manipulation26, communication32,33, simulation\nof fluid phenomena via optical analogues89, and other advanced photonic technologies.”\nComment 7:“In line 229, the authors claim structured Vortex beams carrying OAM can be\ngenerated via Sculpting Streamlines. The authors need to explain what is the fundamental\nmechanism for generating OAM?”\nResponse 7: The reviewer raises a fundamental point. In our framework, the generation of\nOAM is intrinsically linked to the design of helical energy streamlines. Rather than invoking a\nnew generation mechanism, we provide a hydrodynamic perspective that reframes the\nestablished OAM characteristics in terms of customizable momentum flows. By sculpting\noptical flows, we reconfigure structured light fields, which operates on vortex beams whose\nOAM generation relies on well-established physical mechanisms.\nWithin this optical-flow-based framework, we have demonstrated the flexible tailoring of\nvarious structured light families, including Gaussian, propagation-invariant, vortex, and self-\naccelerating modes. Among these, vortex beams represent a well-established subclass of\nstructured light, widely studied over the past three decades. To provide an intuitive and\nrepresentative case, we employ the Bessel vortex mode (U (r,,z)= J (k r)eileik z0 z)—owing\nBessel l r0\nto its analytical simplicity and well-known properties—as a representative case for engineering.\nIts streamline configuration,\n13\n( z )  r ( z ) , ( z ) , z  r\n0\n,\n0 r\n0\n2\nl\nk\nz0\nz , z ,   R = =\n\n+\n\noffers an intuitive\nphysical picture: when the topological charge l = 0, the streamlines are straight lines along the\npropagation axis (Fig. R2(a)); when l ≠ 0, they form helices wrapping around a cylindrical\nsurface (Fig. R2(b)), directly reflecting the intrinsic energy circulation associated with optical\nvortices. We selected the Bessel vortex mode as a representative case due to its well-defined\nstructure and conceptual clarity, which facilitate reader comprehension. This choice allows us\nto clearly demonstrate the integration of additional controllable propagation features through\nflow sculpting, as shown in Figs. 2 and 3. It is important to note that our streamline-engineering\napproach is general and can be extended beyond this specific example to underpin a versatile\nframework applicable to diverse structured light families.\nFig. R2. The optical flows for 0-order Bessel beam (a) and high-order Bessel vortex beam (b).\nAs detailed in Section “On-demand Creation of Structured Vortex Modes” of the main text, we\nconstruct specific streamline trajectories to realize non-diffracting perfect vortex beams\n(NDPVBs), whose transverse profiles remain invariant with respect to both OAM and\npropagation distance. The angular-spectrum expression of the NDPVB, given by Equation (6),\nx\n,k\ny\n)\nz\n{ r e c t [ ( z - b ) / 2 a ]e ik z zl e il } s in c ( 2 a ( k\nz\nk\nzl\n) ) e ik bz e il ,   = F = − − (6) of main text\ntakes the form of a ring-shaped sinc function modulated by a vortex phase term eilϕ, indicating\nthat the resulting beam exhibits characteristics of a quasi-Bessel vortex mode in real space.\nExperimental results (Supplementary Fig. 4) further confirm that the generated NDPVB\nbelongs to this class of vortex modes and carries an OAM of lℏ per photon.\nBeyond the scope of the present study, the relationship between optical flow and angular\nmomentum is preliminarily addressed in a subsequent theoretical investigation (Ref. 64, arXiv\npreprint). Briefly, the angular momentum density j is given by the cross product r×p,\nwhere p denotes the momentum density. Since optical flows represent three-dimensional\ntrajectories of the momentum field, they inherently connect to the angular momentum of the\nlight. By combining the optical–fluid and an optical–mechanical analogies (the latter stemming\nfrom the structural similarity between the Helmholtz and Schrödinger equations), we have\nshown that the OAM carried by an optical flow can be derived from its streamline\ntrajectories R(z)={r(z),φ(z),z}:\nJ =kr2(z)(z) .\nz\nFor instance, substituting the streamlines of Laguerre–Gaussian or Bessel beams—given by\nEquations (4) and (5) in the main text (R(z)=r(z),(z),z=\n\nr , +\nl\nz,z\n\n, (4) of main text,\n 0 0 r2k \n0 z0\nand\n15\n( )  ( ) , ( ) , \n0\n1 2 ,\n0\nl\n0\n2\na r c t a n ,        \n\nR = =  + +  ， (5) of main text) —into this\nframework ( 2 ( ) ( )  J\nz\n=  k r z  z ) yields an OAM value of lℏ, consistent with classical\nelectromagnetic theory. More importantly, this framework also provides a basis for constructing\nintrinsic orbital angular momentum in non-vortex fields, which is a central focus of Ref. 64.\nTo improve clarity, we have added a brief explanatory note in the main text:\n“(Line 154)... This streamline-engineering method enables the on-demand customization of the\nmulti-modal dynamics of diverse structured light within a single framework, including adjustable self-\nsimilarity akin to Laguerre–Gaussian modes, diffraction-free and self-healing properties characteristic\nof Bessel modes, customizable trajectories inspired by self-accelerating modes, the intrinsic OAM-\ncarrying nature of vortex beams, and controlled energy delivery to counteract environment attenuation.\nThe Bessel mode offers simple analytical form and provide a simple and elegant streamline\nconfiguration: in Equation (4), when l is zero, the energy streamlines are axial straight line; otherwise,\nthese streamlines wrap indefinitely around a cylinder of initial radius with no radial divergence.\nTherefore, we employed the Bessel beam as a case study, integrating additional controllable\npropagation features by engineering their streamlines.”\n“(Line 491). ... Looking forward, arising from the intrinsic link between momentum flows and optical\nangular momentum, the streamline-based framework provides a foundation for studying OAM\ndynamics across diverse structured light modalities, including both vortex and non-vortex beams64.\nFurthermore, the incorporation of both spin and orbital components in optical flows30,57,75,88 offers a\npotential route to extend this approach to vectorial structured light, moving beyond the scalar regime\nexplored in this work. Separately, although formal similarities exist between statistical optics and\nstochastic hydrodynamics, establishing a physically consistent fluid-mechanical description of optical\ncoherence remains an open challenge that invites further theoretical investigation. We anticipate that\nthis hydrodynamic perspective may provide a unified foundation for future explorations in structured\nlight design, with potential implications for imaging65, manipulation26, communication32,33, simulation\nof fluid phenomena via optical analogues89, and other advanced photonic technologies.”\nComment 8(a):“It is suggested that the author should appropriately add the field phases which\ncan better exhibit OAM.\nComment 8(b): Additionally, are the topological charge of such structured vortex beam\ncalculated or characterized by special approach?”\nResponse 8(a) and 8(b): We thank the reviewer for these constructive comments regarding the\nphase representation and topological charge characterization of the structured vortex beams.\nIn response to Comment 8(a), we have added the corresponding complex-amplitude/phase\ndistributions in Fig. 2, Fig. 4 and Fig. S10 to better visualize the carried topological\ncharge/OAM:\n“Fig. 2. Sculpting streamlines for adjustable self-similarity and customized self-acceleration. a,\nb Bessel beams with adjustable self-similarity, exhibiting self-shrinking a and self-stretching b radii; red\ncurves indicate the energy streamlines along the vortex mainlobe. c, d Bessel beams with customizable\n16\nself-accelerating dynamics, propagating along a parabolic c and a spiral d trajectory. e–g Intensity maps\nof a–c in the y–z plane; red dashed lines denote the vortex mainlobes. h Three-dimensional intensity\niso-surface corresponding to d. The streamlines herein are directly drew from the “streamline” function\nwith the distributions of Poynting vector calculated in Matlab, which are consistent with the analytical\nsolutions from the hydrodynamic differential equations in this work. i-l The corresponding\nmomentum-space angular-spectrum distributions of a-d calculated by four-step streamline-\nengineering approach in Methods.”\n“Fig. 4. Hydrodynamic Depiction for Vortex Dynamics. (a) Conventional vortex beams and (b)\nPerfect vortex beams—radii increase with both OAM and distance; rotation speed of energy\nstreamlines (red curves) slow down over propagation and do not speed up with higher OAM. The\nyellow double-headed arrow in (b) indicates that perfect vortex fields merely maintain OAM-invariant\nradii in a single transverse plane. Here, 𝑙ℏ (2lħ) denotes the OAM per photon. (c) Non-diffracting Bessel\nbeams—radius grows with OAM but stays constant over distance; rotation speed of energy streamlines\ndecelerates as OAM increases but is invariant along propagation. (d) Iso-propagation vortex beams—\nradius is fixed for all OAM (yellow arrows) but expands with distance; rotation speed of energy\nstreamlines accelerates synchronously with OAM yet decreases as the beam propagates. (e) Non-\ndiffracting perfect vortex beams (NDPVBs)—radius is invariant of both OAM and distance (indicated\nby yellow arrows); rotation speed of energy streamlines accelerates synchronously with OAM and\nremains constant during propagation. (f) y-z intensity maps and (g) x-y complex-amplitude\ndistributions in plane of the NDPVBs from (e) with l = 10. The demonstration for streamline rotation\nspeeds can be found in Supplementary Note 7.”\n17\n“Fig. S10. Streamline-Engineered Energy Transfer in NDPVBs. a-d Uniform propagation\nconfiguration: a Momentum-space angular spectrum: White numerals indicate normalized total\nenergy (1.00), red curve shows k-axis intensity profile at k=0. b the x-z plane intensity pattern where\ny x\nblue-dotted curves identify the main area of energy concentration with the most concentrated location\nat z = a whose x-y plane intensity pattern is displayed in c; d the energy of the whole beam (the blue\ncurve) and of the area inside the aperture (the red curve) with the radius of R (the red part in b) which\n0\ncan only pass through the innermost ring, as the inset illustrated. e-h and i-l are the same as a-d but for\nlinearly and exponentially increased intensity configuration, respectively, with the predesigned peak at\nz =z, where the yellow curve in l depicts the energy of the area inside the aperture with a radius of\n1\n1.2R.”\ndetermined using conventional and well-established methods. Specifically, the phase structure\nof the beam is analyzed, and the topological charge is calculated from the azimuthal phase\ndependence, where a phase change of 2πl around the phase singularity corresponds to a charge\nof l. This is further confirmed experimentally using interferometric techniques, which are\ndetailed in Supplementary Fig. 4. For the Non-Diffracting Perfect Vortex Beams (NDPVBs)\npresented in this work, the embedded vortex phase term (eilϕ) in their angular-spectrum\nexpression (Eq. (6)),\nS(k ,k )=F {rect[(z-b)/2a]eik zl zeil}=sinc(2a(k −k ))e−ik z beil,(6) of main text\nx y z z zl\ndirectly defines the designed topological charge, and the experimental results (Supplementary\nFig. 4) verify the successful generation of beams with topological charge l and carrying lℏ of\nOAM per photon.\nComment 9:“There are some minor mistake in spelling or singular/plural issues that should\nbe only addressed before publication. For example, in line 52 “This limit the…”; in line 180\n“Poynting vector caculated in Matlab”; in line 274 “vortex beam by cofiguring the streamline”\ntypographical and grammatical errors. We have thoroughly reviewed the entire manuscript and\ncorrected all identified issues to ensure linguistic accuracy and consistency. The specific errors\nmentioned have been amended as follows: “This limit” has been changed to “This limits”;\n“caculated” to “calculated”; “cofiguring” to “configuring”. We appreciate the reviewer’s\nattention to detail, and we believe these revisions have improved the clarity and quality of the\nmanuscript.\nComment: In summary, I think the approach original and the results convincing. The ability\nto unify various types of structured beams within a single scheme is notable. This\nhydrodynamic insight into light behavior appears promising for further investigation. After\nthe above-mentioned issues are adequately addressed, I commend the manuscript is\npublished in Nature Communications.\nResponse: We are deeply grateful for the reviewer's supportive assessment—describing\nour approach as \"original,\" the results as \"convincing,\" and the unifying framework as\n\"notable.\" We are especially encouraged by their recognition of the promise this\nhydrodynamic perspective holds for future investigation. We thank the reviewer once\nmore for their insightful review and constructive suggestions, which have significantly\ncontributed to improving our manuscript. We have carefully addressed all the points\nraised in the comments above and believe that the revised version now meets the\nstandards for publication in Nature Communications.\n“This is a revision of the manuscript entitled \"Structuring Light with Flows\" by Wenxiang\nYan et al. The manuscript explores the analogy between hydrodynamic flows and structured\nlight, aiming to engineer optical beams with potentially superior properties compared to\ntraditional structured modes. The authors provide experimental evidence supported by\ntheoretical arguments, complemented with applied examples. Overall, the work presents an\nambitious and exhaustive investigation of this phenomenon, which is timely and relevant for\nthe optics community. The idea is interesting and merits consideration; however, some\naspects of the manuscript require clarification and improvement before publication.”\nResponse: We sincerely thank the reviewer for their thoughtful assessment of our\nmanuscript and their encouraging comments regarding the timeliness and relevance of\nour work to the optics community. We also appreciate their recognition of the ambitious\nscope and experimental-theoretical integration in our study. We have carefully considered\nall the points raised and have made corresponding revisions to improve the clarity and\nrigor of the manuscript. Below, we provide point-by-point responses to each comment,\nand we hope that the revised version now meets the high standards expected for\npublication.\nComment 1: “Consider expanding the references connecting optics and fluid analogies.\nIncluding these would provide better context and strengthen the contribution. Consider adding,\nfor example:\nOptical Rankine vortex and anomalous circulation of light. Physical review letters,\n99(16), 163901 (2007).\nAn analogy strategy for transformation optics. New J. Phys. 16 063008 (2014).”\nResponse 1: We are grateful to the reviewer for this valuable suggestion to strengthen the\nfoundational context of our work. In response, we have expanded the discussion of optics–fluid\nanalogies in the introduction and added the recommended references (now Refs. 54 and 55) in\nthe revised manuscript:\n“(Line 88) Motivated by these connections, researchers have probed the hydrodynamic character of\noptical fields, revealing a wealth of nonlinear behaviors—chaos, multistability, and even turbulence—\npredicted theoretically and confirmed experimentally in laser systems52–55. In 1989, Coullet et al.,\ninspired by fluid vortices, formally introduced the notion of optical vortices56, catalyzing the modern\nexploration of structured light. Building on this hydrodynamic insight, we now broaden the optics–fluid\nanalogy to structured light shaping through the lens of optical flows and streamlines.”\n“……\n53. Rosanov, N. N., Fedorov, S. V. & Shatsev, A. N. Curvilinear Motion of Multivortex Laser-\nSoliton Complexes with Strong and Weak Coupling. Phys. Rev. Lett. 95, 053903 (2005).\n54. Swartzlander, G. A. & Hernandez-Aranda, R. I. Optical Rankine Vortex and Anomalous\nCirculation of Light. Phys. Rev. Lett. 99, 163901 (2007).\n20\n55. Yao, K., Chen, H., Liu, Y. & Jiang, X. An analogy strategy for transformation optics. New J.\nPhys. 16, 063008 (2014).\n56. Coullet, P., Gil, L. & Rocca, F. Optical vortices. Optics Communications 73, 403–408 (1989).\n57. Bekshaev, A., Bliokh, K. Y. & Soskin, M. Internal flows and energy circulation in light beams.\nJ. Opt. 13, 053001 (2011).\n……”\nComment 2: “It would be very helpful to show the transverse amplitude and phase\ndistributions of the beams at the plane of the spatial light modulator. Including at least one\nexemplary digital hologram would further support reproducibility and aid readers in\nunderstanding the experimental implementation.”\nResponse 2: We thank the reviewer for this constructive suggestion, which helps improve the\nreproducibility and clarity of our experimental methodology. In response, we have enhanced\nthe experimental details in the following ways:\n1. As suggested, we now display the complex angular-spectrum distributions (i.e., the fields\nencoded onto the SLM) for the four representative cases in the revised Fig. 2.\n“Fig. 2. Sculpting streamlines for adjustable self-similarity and customized self-acceleration. a,\nb Bessel beams with adjustable self-similarity, exhibiting self-shrinking a and self-stretching b radii; red\ncurves indicate the energy streamlines along the vortex mainlobe. c, d Bessel beams with customizable\n21\nself-accelerating dynamics, propagating along a parabolic c and a spiral d trajectory. e–g Intensity maps\nof a–c in the y–z plane; red dashed lines denote the vortex mainlobes. h Three-dimensional intensity\niso-surface corresponding to d. The streamlines herein are directly drew from the “streamline” function\nwith the distributions of Poynting vector calculated in Matlab, which are consistent with the analytical\nsolutions from the hydrodynamic differential equations in this work. i-l The corresponding\nmomentum-space angular-spectrum distributions of a-d calculated by four-step streamline-\nengineering approach in Methods.”\n2. To facilitate experimental reproduction, we have added a detailed description of the\nmodulation process—from angular-spectrum distributions to phase-only holograms—in\nthe Methods section“Experimental Generation.”Additionally, Fig. 9 now provides an\nexemplary computer-generated hologram loaded on the SLM.\n“(Line 541) Generation. In Fig. 9a, a reflective phase-only spatial light modulator (SLM) imprinted\nwith computer-generated hologram patterns (Fig. 9c) transforms a collimated laser light wave into the\ncomplex field S(k, k) (Fig. 9b), with help of spatial filtering via a 4-F system consisting of two identical\nx y\nlenses L1 and L2, and an iris as well. The resulting field is responsible for customized structured light\nin the focal volume of lens L3. Notably, the maximum modulated transverse wavevector, kmax and kmax\nx y\n, are given by kmax = 0.5Lk/f and kmax = 0.5Lk/f, where L and L are the panel dimensions of\nx x y y x y\nSLM (Fig. 9c), k is the wavenumber, and f is the focal length of the focusing lens L3. When the\nconstructed streamline trajectories vary significantly — such as in more complex trajectories\ndesigned to handle intricate occlusions or compensate for attenuation in high-loss media —\nthe angular-spectrum distribution S(k, k) in Fig. 9b will include higher spatial frequency\nx y\ncomponents. These components may exceed the modulation limits of the SLM, causing a\nmismatch between the design and the device's capabilities, which results in degraded\nexperimental reproduction due to the loss of high-frequency components. Future platforms\nwith enhanced modulation capabilities, such as metasurfaces with subwavelength\nsubwavelength-scale pixels, could potentially overcome these limitations.\nExperiment setup; BS, beam splitter; SLM, phase-only spatial light modulator; L1-L5, lens; M, mirror;\nRAPM, right-angle prism mirror; HRPM, hollow roof prism mirror. b Target angular specturm from\nfour-step streamline-engineering approach are modulated to c the phase-only hologram by complex-\namplitude modulation and loaded on the SLM panel with the panel dimensions of L and L. ”\nx y\nThese additions provide readers with the necessary information to understand and reproduce\nour experimental approach. We appreciate the reviewer's input in helping us improve the clarity\nof our experimental methodology.\nComment 3: “The manuscript would benefit from an elaboration on the orthogonality\ncharacteristics of the Non-diffracting perfect vortex beams (NDPVB).”\nResponse 3: We appreciate the reviewer's insightful comment regarding the orthogonality\ncharacteristics of NDPVBs, which is indeed crucial for their application in multiplexing\nsystems.\nThe orthogonality of NDPVBs can be rigorously established through both analytical and\nexperimental approaches. Analytically, the angular-spectrum expression of NDPVBs given by\nEquation (6)，\n \nx\n, k\ny\n) = F\nz\n{ r e c t [ ( z - b ) / 2 a ]e ik z zl e il } = s in c ( 2 a ( k\nz\n− k\nzl\n) ) e − ik bz e il . (6) of main text\nreveals that modes with different topological charges are mathematically orthogonal：\nS (k ,k ) S (k ,k )\nl x y l x y\n1 2\n=[sinc(2a(k −k ))e−ik z beil 1 ][sinc(2a(k −k ))e−ik z beil 2 ]dk d\nz zl z zl r\n1 2\n=sinc(2a(k −k ))e−ik z bsinc(2a(k −k ))e−ik z bdk eil 1 eil 2 d\nz zl z zl r r\n1 2\n=(l −l )sinc(2a(k −k ))e−ik z bsinc(2a(k −k ))e−ik z bdk\n1 2 z zl z zl r\n1 2\n=0 whenl  l\n1 2\nExperimentally, the multiplexing results in Supplementary Fig. 13 have confirmed this\northogonality.\nTo address this point more comprehensively, we have enhanced the discussion in\nSupplementary Note 9 (\"Orthogonality and Transmission Matrix of NDPVB Multiplexing\") to\ninclude more detailed mathematical derivations:\n“Orthogonality and transmission matrix of NDPVB multiplexing. The angular spectra\nwith different topological charges are mathematically orthogonal:\nl1\n[ s\ns i n\n( l1\n, k ) S\nx y l2\ni n c ( 2 a ( k\nc ( 2 a ( k\nz\nl ) s i n\n2\n( k\nz\nk\nc (\n, k\nx y\nk\nzl1\n) ) \nzl1\n2 a ( k\n)\n) ) \ne\nz\ne\nik\nik b il e z 1\nb s i n c ( z\nk ) ) e\nzl1\n][ s i\n2 a (\nik bz\nn\nk\ns\nc (\nz\ni n\n2\nc\na ( k\nk\nzl2\n( 2 a\nz\n) ) \n( k\ne\nz\nk\nzl2\nik bz\nk\n) ) \nd k\nzl2\ne\nr\nik b e z\nil e 1\nil2\ne\n]\nil2\nd k\nd\nr\nd\nr\n  \n  \n\n=\n=\n=\n=\n\n\n0\n\nw\n−\nh e n\n\nl1 \n−\n−\nl2\n−\n−\n−\n−\n−\n−\n−\n−\n−\n− )\n−\n\n) e − ik bz d k\nr\n(S38)\nWhen combined with the normalized amplitude coefficient from Methods and Eq. (S11),\nA\nl\n= 3 3 .2 8 2 3 l + 4 .0 8 4 9 , NDPVBs construct the orthonormalized subchannels with the\nuniform power spectrum of OAM as\nxy\n{ A\nl1\nS\nl1\n( k\nx\n, k\ny\n) }\nxy\n{ A\nl2\nS\nl2\n( k\nx\n, k\ny\n) }\n(l1 ,l2 )\n.  F F = (S39)\nLet the transmitted and received signals at time t be…”\nThese additions provide readers with a complete understanding of the orthogonality properties\nthat make NDPVBs suitable for high-capacity communication.\nComment 4: “While the proposed method is promising, the manuscript currently lacks a\ndiscussion on its limitations. For instance, under which experimental conditions or parameter\nranges might the approach fail or become impractical?”\nResponse 4: We sincerely thank the reviewer for raising this important point regarding the\nlimitations of our approach. Providing a clear discussion of these boundaries will indeed help\nreaders better evaluate the practical scope and potential of our method. The current framework\nexhibits limitations in two key aspects that warrant consideration.\nTheoretical design: While our method effectively constructs analytic streamline trajectories\nthat already enable significant versatility and integration of diverse structured light features,\nits extension to more complex, non-analytic streamline configurations (e.g., numerically-\ndefined or without closed-form expressions) represents a promising direction for future\ndevelopment. Such advancement could further broaden the applicability of our approach to\neven more challenging beam-shaping scenarios. In response to this observation, we have\nincorporated a discussion of this aspect in the Methods section:\n“ (Line 514) The four rows of Table 1 illustrate the structured light shaping with different\npropagation behaviors: (1) non-diffracting perfect vortex beams70 with OAM- and z-invariant radius r\nI\n, (2) adjustable self-similarity with dynamic radius r(z), (3) customized self-acceleration with arbitrary\nI\ntrajectory (x(z), y(z)), and (4) tunable energy delivery with adjusted energy density I(z). Detailed\ns s\nderivations for Table 1 can be found in Supplementary Note 1, and the validations of sculpted\nmomentum fields can be found in Supplementary Note 2 with Supplementary Movie 6. Notably, this\nmethod primarily constructs analytic streamline trajectories R(z), with the potential for extension\nto more complex, non-analytic streamline configurations (e.g., numerically-defined, unclosed-\nform expressions) in future developments. While vortex beams serve as representative cases in\nour demonstrations, the underlying methodology establishes a general framework that is broadly\napplicable to diverse structured light families, including both vortex and non-vortex modes (See\nSupplementary Note 13 for non-vortex case).”\nExperimental implementation: Practical constraints are primarily imposed by the limited\nspatial bandwidth of current spatial light modulators (SLMs). The finite pixel pitch and\npanel dimensions fundamentally limit the maximum spatial frequencies that can be encoded.\nIn details, the maximum transverse wavevectors that can be modulated are given by k max =1/δx\nx\nand k max =1/δy, where δx and δy are the pixel sizes. When using an SLM in a lens focusing\ny\nprocess to load the momentum-space angular-spectrum distribution S(k , k ) (Fig. 9),\nthe\nx y\nmaximum transverse wavevectors become k max = 0.5Lk/f and k max = 0.5L k/f, where L and L\nx x y y x y\nare the panel dimensions, k is the wavenumber, and f is the focal length of the focusing lens.\nSpecifically, when implementing complex streamline trajectories—such as those required for\nintricate occlusion scenarios or strong attenuation compensation—the resulting angular-\nspectrum distribution S(k , k ) contains higher spatial frequency components that may exceed\nx y\nthe SLM's modulation limits (k max and k max). This mismatch can lead to deviations between the\nx y\ndesigned and experimentally realized beams. Future platforms with enhanced modulation\ncapabilities, such as metasurfaces with subwavelength-scale pixels, could potentially overcome\nthese limitations.\nTo address these limitations and potential avenues for future improvement, we have added a\ndedicated discussion in the Methods– Experimental Generation section. These additions\nprovide readers with a comprehensive understanding of both current capabilities and future\npossibilities:\n25\n“(Line 541) Generation. In Fig. 9a, a reflective phase-only spatial light modulator (SLM) imprinted with\ncomputer-generated hologram patterns (Fig. 9c) transforms a collimated laser light wave into the\ncomplex field S(k, k) (Fig. 9b), with help of spatial filtering via a 4-F system consisting of two identical\nx y\nlenses L1 and L2, and an iris as well. The resulting field is responsible for customized structured light\nin the focal volume of lens L3. Notably, the maximum modulated transverse wavevector, kmax and kmax\nx y\n, are given by kmax = 0.5Lk/f and kmax = 0.5Lk/f, where L and L are the panel dimensions of SLM (Fig.\nx x y y x y\n9c), k is the wavenumber, and f is the focal length of the focusing lens L3. When the constructed\nstreamline trajectories vary significantly — such as in more complex trajectories designed to handle\nintricate occlusions or compensate for attenuation in high-loss media — the angular-spectrum\ndistribution S(k, k) in Fig. 9b will include higher spatial frequency components. These components\nx y\nmay exceed the modulation limits of the SLM, causing a mismatch between the design and the device's\ncapabilities, which results in degraded experimental reproduction due to the loss of high-frequency\ncomponents. Future platforms with enhanced modulation capabilities, such as metasurfaces with\nsubwavelength subwavelength-scale pixels, could potentially overcome these limitations.\nFig. 9. Experimental setup for Generation and Tomography of customized structured light. a\nExperiment setup; BS, beam splitter; SLM, phase-only spatial light modulator; L1-L5, lens; M, mirror;\nRAPM, right-angle prism mirror; HRPM, hollow roof prism mirror. b Target angular specturm from\nfour-step streamline-engineering approach are modulated to c the phase-only hologram by complex-\namplitude modulation and loaded on the SLM panel with the panel dimensions of L and L. ”\nx y\nComment 5: “The hydrodynamics-based picture is appealing, but how does the analogy\naccount for polarization and coherence effects in structured light? A discussion on this point\nwould substantially enhance the completeness of the work.”\nResponse 5: We thank the reviewer for raising this insightful question regarding the extension\nof the hydrodynamic analogy to polarization and coherence—two important aspects that would\nfurther enrich the framework.\nthe density and velocity fields in fluid mechanics and the optical intensity and momentum\ndistributions (i.e., Poynting vector) in free-space structured light. A direct one-to-one mapping\nfor polarization and coherence has not yet been incorporated into this analogy.\nRegarding polarization, although it lacks a direct counterpart in conventional fluid\nmechanics, our streamline-engineering approach possesses inherent extensibility toward\nvectorial structured light. This is because the Poynting vector comprises both an orbital\ncomponent (related to spatial phase gradients) and a spin component (associated with\npolarization inhomogeneities) [Bekshaev, A., Bliokh, K. Y. & Soskin, M. Internal flows and\nenergy circulation in light beams. J. Opt. 13, 053001 (2011).]. In the present work, we focus on\nscalar light fields derived from the scalar Helmholtz equation, utilizing only the orbital flow for\nbeam shaping. Future incorporation of the spin component would provide a natural route to\nextend this hydrodynamic approach to the shaping of complex vectorial structured light. This\nexciting potential underscores the extensibility of our flow-based framework.\nThe description of optical coherence within a hydrodynamic picture presents a fascinating\nand open challenge. While certain formal similarities exist between statistical optics and\nstochastic hydrodynamics, a physically consistent fluid-mechanical description of optical\ncoherence is yet to be established. This would be a compelling opportunity for future theoretical\nwork at the intersection of optics and fluid dynamics. Addressing this challenge could unlock\nnew capabilities in controlling partially coherent structured light.\nWe fully agree that these aspects represent valuable directions for future research. In response\nto the reviewer's suggestion, we have added a paragraph in the Discussion section that outlines\nthe current scope of the analogy regarding polarization and coherence, while also highlighting\nthese promising avenues for future extension:\n“(Line 491). Looking forward, arising from the intrinsic link between momentum flows and optical\nangular momentum, the streamline-based framework provides a foundation for studying OAM\ndynamics across diverse structured light modalities, including both vortex and non-vortex beams64.\nFurthermore, the incorporation of both spin and orbital components in optical flows30,57,75,88 offers a\npotential route to extend this approach to vectorial structured light, moving beyond the scalar regime\nexplored in this work. Separately, although formal similarities exist between statistical optics and\nstochastic hydrodynamics, establishing a physically consistent fluid-mechanical description of optical\ncoherence remains an open challenge that invites further theoretical investigation. We anticipate that\nthis hydrodynamic perspective may provide a unified foundation for future explorations in structured\nlight design, with potential implications for imaging65, manipulation26, communication32,33, simulation\nof fluid phenomena via optical analogues89, and other advanced photonic technologies.”\nComment 6: “There appear to be some overlaps between the present manuscript and Ref. [64],\nwhich also shares some authors. In particular, please compare the first paragraph of Section\n3.1 in Ref. [64] with lines 85–94 of the current manuscript, as well as Figure 4 in Ref. [64] with\nFigure 5 of the current manuscript. Could the authors clarify what is novel in the present work\n27\nand explicitly differentiate it from Ref. [64]?”\nResponse 6: We thank the reviewer for raising this important point regarding the relationship\nbetween the present manuscript and our related theoretical preprint (Ref. 64, arXiv). We are\npleased to clarify the distinct contributions and chronological relationship between these studies.\nThe present manuscript, \"Structuring Light with Flows,\" establishes the foundational concept\nand a general four-step methodology for sculpting structured light via engineered optical\nflows. It introduces the core framework of flow-based light shaping, supported by\nexperimental demonstrations spanning multiple structured-light families. This methodology\nforms the essential basis for subsequent developments in flexibly designing and generating\nversatile structured fields.\nRef. 64 represents a subsequent theoretical investigation that builds upon the foundational\nframework established in the present work, \"Structuring Light with Flows.\" The present work\nintroduces the core concept of sculpting structured light through optical flows and presents\na systematic four-step method.\nIn our follow-up work (Ref. 64, initially titled \"Vortex-free intrinsic OAM\"), we explore the\ntheoretical relationship between optical flow and angular momentum in greater depth.\nSpecifically, we examine how angular momentum density (j = r × p) connects to optical flow\ntrajectories and demonstrate that optical flows can induce orbital angular momentum\n(OAM), particularly in non-vortex fields. This investigation of flow-induced vortex-free\nOAM constitutes the central focus of Ref. 64.\nThe key distinctions between the two works are summarized as follows:\n The present work establishes the fundamental streamline-engineering framework for\nstructuring optical flows.\n Ref. 64 builds upon this foundation to specifically induce and manipulate the OAM\nin non-vortex fields.\nAlthough both studies employ similar optical tweezers platforms as they originate from the\nsame laboratory infrastructure, they validate fundamentally different physical phenomena:\n Figure 5 in the present work demonstrates that microspheres trace the pre-designed\nstreamlines in structured vortex beams, validating our flow-shaping approach.\n Figure 4 in Ref. 64 verifies the existence of intrinsic OAM in specifically designed non-\nvortex fields through mechanical transfer from photonic OAM to macroscopic rotation.\nTo ensure clarity for readers, we have revised the relevant descriptions in the current manuscript\nto more clearly delineate the relationship between these works:\n“(Line 491). Looking forward, arising from the intrinsic link between momentum flows and optical\nangular momentum, the streamline-based framework provides a foundation for studying OAM\ndynamics across diverse structured light modalities, including both vortex and non-vortex beams64.\nFurthermore, the incorporation of both spin and orbital components in optical flows30,57,75,88 offers a\n28\npotential route to extend this approach to vectorial structured light, moving beyond the scalar regime\nexplored in this work. Separately, although formal similarities exist between statistical optics and\nstochastic hydrodynamics, establishing a physically consistent fluid-mechanical description of optical\ncoherence remains an open challenge that invites further theoretical investigation. We anticipate that\nthis hydrodynamic perspective may provide a unified foundation for future explorations in structured\nlight design, with potential implications for imaging65, manipulation26, communication32,33, simulation\nof fluid phenomena via optical analogues89, and other advanced photonic technologies.”\nIt should be noted that the current Ref. 64 arXiv version is an early-stage manuscript that has\nnot been submitted for publication and will undergo substantial development before being\nconsidered as an independent, subsequent submission. We will ensure that any future\nsubmission based on the Ref. 64 preprint undergoes major revisions in its introductory and\nmethodological descriptions to clearly establish its relationship to and distinct contributions\nfrom the present work.\nIn summary, the present manuscript represents the foundational study, while Ref. 64 explores a\nspecific theoretical extension of this framework. We hope this clarification adequately\naddresses the reviewer's concerns.\nComment 7: “ Some passages are difficult to follow due to wording. A careful revision of\ngrammar, style, and typographical errors is recommended. As an example, in Fig. 2 the yellow\ntext reads “Optica Axis”, but I believe it should be “Optical Axis”.”\nResponse 7: We thank the reviewer for their careful reading and for highlighting issues related\nto wording and typographical errors. We have conducted a thorough language editing of the\nentire manuscript to improve readability, correct grammatical inaccuracies, and ensure\nconsistency in style. Specifically, the text “Optica Axis” in Fig. 2 has been corrected to “Optical\nAxis” as suggested. We appreciate the reviewer’s attention to these details, and we believe the\nmanuscript has been enhanced in clarity and presentation as a result of these revisions.\nComment : The manuscript addresses an interesting topic with potential impact. However, I\ncannot recommend publication of this work in its current form. Revisions are necessary to\nclarify novelty, improve presentation, and address the points raised above.\nResponse : We are grateful for the reviewer's overall positive assessment of our work's\ninterest and potential impact, and we sincerely thank them for their valuable feedback\nthroughout the review process. The constructive comments and suggestions have been\ninstrumental in helping us enhance the clarity, strengthen the presentation, and better\narticulate the novelty of our study. We have carefully addressed each of the raised points\nin the detailed responses above and have implemented corresponding revisions\nthroughout the manuscript. We hope that the revised version now meets the journal's\nstandards for publication.\n“Hydrodynamics described by the Navier-Stokes or Euler equations, describes the temporal\nevolution of fluid motion over time, capturing complex phenomena like velocity and pressure\nchanges. The Helmholtz equation, however, is a time-independent, linear partial differential\nequation that represents a simplified version of the wave equation, often used to describe\nsteady-state wave phenomena, such as the spatial distribution of monochromatic waves (waves\nof a single frequency). While connected —for instance, by taking the time-independent form of\nan acoustic wave equation derived from hydrodynamics—they represent different aspects of\nphysics: hydrodynamics deals with the full, time-dependent evolution of fluids, while the\nHelmholtz equation models time-harmonic spatial solutions.\nThe Helmholtz equation may be derived from the time-dependent wave equation, itself related\nto the Euler equations of hydrodynamics (a simplified form of the Navier-Stokes equations). In\nessence, the Helmholtz equation is the steady-state (time-independent) version of the wave\nequation, which is a direct descendant of the more general fluid motion equations used in\nhydrodynamics. Some papers for links are here:\nH. White, P. Bailey, J. Lawrence, J. George, J. Vera, Physics Open, Volume 1,2019, 100009\nPhilos Trans A Math Phys Eng Sci 2020 Jul 20;378(2177):20190233 ”\nThe main novelty of this paper seems to be linking optical waves to hydrodynamics in the case\nof structured light over the above references and in doing the work makes beams that has\nhydrodynamic Depiction for Vortex Dynamics. This leads to long range beams with for example\nfixed radii of independent of azimuthal index propagating long distances (e.g. figure 4).\nThe tweezers study seemed less convincing to me with statements such as “may enable three-\ndimensional optomechanical control via engineered photon trajectories, potentially extending\nto applications such as microfluidic flow steering, directed cell transport, and assembly of\nmicroscale structures”. Many of these can be performed already – what is new and what can\nbe achieved specifically that cannot with normal beam shaping. The Communications area\nstudy and statements are perhaps more convincing in this regard.\nOverall, what this approach shows is vortex beams with fixed radii, limited diffraction and\nvariable index. What is not discussed clearly is the role, need for side lobes in the generated\nprofiles and how they effect each of the experiments above. Though of some merit, I am not\nconvinced with the data supplied this is worthy of Nature Communications but is better suited\nto a more specialised journal.”\nResponse from Authors:\nWe sincerely thank Reviewer 3 for their thorough evaluation and constructive comments on our\nmanuscript. The reviewer's feedback encompasses several important aspects: (1) insightful\ntheoretical background on wave-fluid analogies, (2) technical questions regarding the optical\ntweezer experiments and the functional role of sidelobes, and (3) an assessment of our work's\n30\nscope that appears to focus primarily on Vortex Dynamics. We appreciate this feedback and\nrecognize that our original presentation may not have sufficiently clarified the generality of our\napproach.\nTo address this, we first provide a concise restatement of our core contribution—a general\nhydrodynamic framework for structuring light beyond specific mode families—and clarify why\nvortex beams served as illustrative cases. Importantly, we have now supplemented the\nmanuscript with additional examples involving non-vortex beams to further demonstrate the\nmethod's broad applicability.\nWe then provide detailed point-by-point responses to all technical questions raised. We\nsincerely hope that these clarifications, together with the corresponding revisions in the\nmanuscript, will fully address the reviewer's concerns and more clearly demonstrate the\nsignificance of our work.\n1. Background and Challenges in Free-Space Optics\nStructured light has enabled significant advances across multiple fields, including fundamental\nphysics, telecommunications, security systems, micromachining, and optical manipulation of\nbiological specimens [Nat. Photon.15, 253–262 (2021); Light Sci. Appl. 11, 205 (2022); J. Opt.\n25, 103001 (2023);]. However, its application in free-space optics faces constraints due to the\nlimited forms of solutions to the Helmholtz equation—such as Gaussian, propagation-\ninvariant, accelerating, and vortex modes (Table R1)—each bound to fixed and incompatible\npropagation laws:\n Fixed propagation dynamics: Gaussian families diverge hyperbolically and\nuncontrollably, Airy beams are confined to parabolic trajectories, and vortex modes\ninherently undergo OAM-dependent radial expansion, limiting their adaptability in various\npractical scenarios.\n Incompatible shaping techniques: Existing methods based on ray, wave, or Fourier\noptics [Ref. 34-46 in maintext] typically manipulate individual mode families (i.e., specific\nHelmholtz solutions) and lack a unified framework capable of integrating diverse\nstructured modes or addressing complex propagation challenges.\nTo expand the functionality of structured light in free-space optics, more versatile theoretical\nand methodological approaches are needed—ones that can overcome the limitations\ninherent in traditional field solutions of the Helmholtz equation. This challenge represents\nan ongoing research direction in the field.\nequation solutions, typically obey fixed and incompatible propagation laws. [Nat. Photon.15,\n253–262 (2021)]\n2. Core Advances: A Hydrodynamic Framework for Structured Light\nTo address the challenges outlined above, we have developed a hydrodynamic approach that\nprovides a unified methodology for structuring light beyond the constraints of traditional mode-\nspecific solutions. Our key contributions include:\n2.1 Complementary Framework: From Field to Flow Descriptions\nBuilding upon established optics-fluid analogies, we introduce \"optical flows\" as a\ncomplementary perspective for conceptualizing structured light. The streamlines of these\nflows—defined by Poynting-vector trajectories—offer a hydrodynamic representation that\nworks alongside conventional field descriptions. This flow description provides an intuitive and\nunified roadmap of energy transport, as exemplified by Bessel and Laguerre-Gaussian vortex\nbeams in Fig. 1, with applicability extending across different structured light families.\n2.2 General Streamline-Engineering Methodology\nUsing hydrodynamic principles, we develop a versatile streamline-engineering method to\nflexibly sculpt diverse structured light modes on demand. This approach enables the on-demand\ncustomization of the multi-modal dynamics of diverse structured light within a unified\nframework, including (as shown in Figs. 2–3):\n Adjustable self-similarity (Gaussian-like)\n Diffraction-free and self-healing properties (Bessel-like)\n Customizable trajectories (Airy-like)\n Intrinsic OAM (vortex-like)\n Controlled energy delivery to counteract environment attenuation\npropagation challenges, such as structured vortex beams (Non-diffracting Perfect Vortex Beams)\nthat simultaneously suppress both diffraction- and OAM-induced broadening (Fig. 4). While\nvortex beams are used as representative examples due to analytical clarity of their streamlines\n(Figs. 1-3), the approach is broadly applicable to diverse structured light families, including\nboth vortex and non-vortex modes.\n2.3 Experimental Validation via Optofluidic Velocimetry\nUsing optical tweezers in a manner analogous to particle-tracking velocimetry in fluid\ndynamics, we experimentally verified that microscopic particles faithfully follow pre-designed\nenergy streamlines (Fig. 5). These results provide direct validation of our streamline-\nengineering approach and confirm our capability to control momentum flows in structured light.\n2.4 Application Demonstration in Free-Space Communications\nTo demonstrate the practical utility of our hydrodynamic approach, we applied the\nstreamline-engineering framework to free-space optical communications—a domain where\nconventional structured light modes face interconnected propagation constraints, including\nOAM-dependent divergence, diffraction broadening, turbulence scattering, and limited non-\nline-of-sight capability [Optica, 2, 357–365 (2015); J. Opt. 18, 074014 (2016).]. Unlike\nconventional methods that typically optimize only one propagation property at a time, our flow-\nbased shaping enables simultaneous, multidimensional control across multiple free-space\nbehaviors within a unified design framework. The mutli-modal, highly-customized beams\nexhibit integrated performance enhancements in channel capacity, turbulence resilience, and\nnon-line-of-sight transmission (Figs. 6-7), illustrating how our general flow-based control\nframework can concurrently address complex, real-world propagation challenges that lie\nbeyond the reach of mode-specific solutions.\n3. Clarification on the Use of Vortex Beams as Representative Examples\nWe thank the reviewer for their observation regarding the prominent use of vortex beams in our\ndemonstrations and are pleased to clarify the rationale for this illustrative choice, which stems\nfrom both scientific and pedagogical considerations.\nVortex beams represent one of the most extensively studied subclasses of structured light, with\nwell-established applications across multiple domains including optical manipulation,\ncommunications, and quantum technologies. As direct optical analogs to well-known fluid\nvortices, these beams provide a natural bridge between hydrodynamics and optics. More\nimportantly, their streamline configurations offer exceptional conceptual clarity for\ndemonstrating our flow-based framework. Taking the analytically tractable Bessel vortex mode\nas an example, its streamline behavior follows R(z) = {r , φ + l/(r 2k )z, z}.\n0 0 0 z0\nreveals a fundamental transition: when l = 0, streamlines form straight axial paths (Fig. R2a)\nwithout diffraction-induced divergence (non-diffraction); when l ≠ 0, they evolve into\nhelices wrapping around cylindrical surfaces (Fig. R2b), directly visualizing the intrinsic\nenergy circulation (vortex).\nFig. R2. The optical flows for 0-order Bessel beam (a) and high-order Bessel vortex beam (b).\nThis hydrodynamic representation—as contrasted with conventional field depictions in Fig. 1\nof main text—uniquely captures the rotational energy transport characteristic of vortex beams,\nan aspect not readily apparent in traditional wave-optical descriptions. The clear physical\nintuition provided by this vortex-based illustration makes it particularly effective for\ndemonstrating the insights offered by our flow-based perspective compared to other structured\nlight families.\nBased on these well-understood vortex configurations, we designed pedagogical examples to\ndemonstrate how our streamline-engineering approach enables the integration of multimodal\npropagation features within a unified framework. To facilitate reader comprehension, we show\nhow straightforward modifications of fundamental vortex streamlines can produce distinct\npropagation behaviors: radial evolution adjustments create Gaussian-like divergence with\ncontrollable self-similarity (Figs. 2a, 2b), while spatial displacement of trajectories generates\ncustomizable Airy-like self-acceleration (Figs. 2c, 2d). These intentionally simplified examples\nserve to intuitively convey the core principles of flow-based shaping.\nThis systematic progression from fundamental streamlines to complex beam designs\ndemonstrates our methodology's general applicability, which extends naturally to non-vortex\nbeam configurations. For instance, even simpler non-vortex systems—such as fundamental\nBessel beams with straight streamlines (R(z) = {x , y , z}, Fig. R3a)—can be engineered to\n0 0\ncreate customized accelerating beams (R(z) = {x +x(z), y +y(z), z}, Figs. R3b, R3c) with\n0 s 0 s\narbitrary trajectories (x(z), y(z)). This extension to non-vortex beams confirms the framework's\ns s\nversatility beyond vortex-specific applications, pointing to its potential applicability across\ndiverse structured light families.\nspirally accelerating beam and trefoilly accelerating beams. d-f and g-i the corresponding\nwave picture and accelerating trajectories.\nTo better convey the generality of our methodology beyond specific vortex beams, we have:\n Enhanced the description of the framework's general applicability throughout the main text:\n“(Line 154) This streamline-engineering method enables the on-demand customization of the multi-\nmodal dynamics of diverse structured light within a single framework, including adjustable self-\nsimilarity akin to Laguerre–Gaussian modes, diffraction-free and self-healing properties characteristic\nof Bessel modes, customizable trajectories inspired by self-accelerating modes, the intrinsic OAM-\ncarrying nature of vortex beams, and controlled energy delivery to counteract environment attenuation.\nThe Bessel mode offers simple analytical form and provide a simple and elegant streamline\nconfiguration: in Equation (4), when l is zero, the energy streamlines are axial straight line; otherwise,\nthese streamlines wrap indefinitely around a cylinder of initial radius with no radial divergence.\nTherefore, we employed the Bessel beam as a case study, integrating additional controllable\npropagation features by engineering their streamlines.”\n“(Line 208) In contrast to established structured light which follow fixed propagation laws and\nspecialized techniques34–46 that only shapes specific type of structured light, the streamline-engineering\napproach offers a more flexible and versatile strategy for shaping multimodal structured light,\npotentially broadening its functional applications26,30,31,66. While vortex beams serve as representative\ncases in our demonstrations due to their well-defined streamline configurations and scientific\nimportance, the underlying methodology establishes a general framework that is broadly\napplicable to diverse structured light families, including both vortex and non-vortex modes (See\nSupplementary Note 13 for non-vortex case).”\n35\n“(Line 514)The four rows of Table 1 illustrate the structured light shaping with different propagation\nbehaviors: (1) non-diffracting perfect vortex beams70 with OAM- and z-invariant radius r, (2)\nI\nadjustable self-similarity with dynamic radius r(z), (3) customized self-acceleration with arbitrary\nI\ntrajectory (x(z), y(z)), and (4) tunable energy delivery with adjusted energy density I(z). Detailed\ns s\nderivations for Table 1 can be found in Supplementary Note 1, and the validations of sculpted\nmomentum fields can be found in Supplementary Note 2 with Supplementary Movie 6. Notably, this\nmethod primarily constructs analytic streamline trajectories R(z), with the potential for extension\nto more complex, non-analytic streamline configurations (e.g., numerically-defined, unclosed-\nform expressions) in future developments. While vortex beams serve as representative cases in\nour demonstrations, the underlying methodology establishes a general framework that is broadly\napplicable to diverse structured light families, including both vortex and non-vortex modes (See\nSupplementary Note 13 for non-vortex case).”\n Included additional case of non-vortex beam shaping in the Supplementary Note 13：\n“Supplementary Note 13: Non-vortex Case of Streamline-Engineering\nApproach.\nWhile vortex beams serve as representative cases in our demonstrations, the underlying methodology\nestablishes a general framework that is broadly applicable to diverse structured light families, including\nboth vortex and non-vortex modes. For instance, simpler non-vortex systems—such as fundamental\nBessel beams with straight streamlines (R(z) = {x, y, z}, Supplementary Fig. 18a)—can be engineered\n0 0\nto create customized accelerating beams (R(z) = {x+x(z), y+y(z), z}, Supplementary Figs. 18b and 18c)\n0 s 0 s\nwith arbitrary trajectories (x(z), y(z)). This extension to non-vortex cases confirms the framework's\ns s\nversatility beyond vortex-specific applications, pointing to its potential applicability across diverse\nstructured light families.\nBessel beams, spirally accelerating beam and trefoilly accelerating beams. d-f and g-i the corresponding\nwave picture and accelerating trajectories.”\nIn light of the comprehensive clarification above, we hope to have addressed the reviewer's\nprimary concern regarding the perceived focus on vortex-specific applications. Our work\nestablishes a general hydrodynamic framework for structured light that transcends individual\nmode families. Vortex beams are used as particularly effective illustrative examples due to\ntheir conceptual clarity and physical relevance—not as boundaries of the methodology. The\ncore innovation lies in introducing a unified flow-based approach that enables simultaneous\ncontrol over multiple propagation dimensions—a capability fundamentally beyond the reach of\nconventional mode-specific shaping techniques.\nBuilding upon this clarified foundation, we now provide detailed point-by-point responses\nbelow to address the reviewer's specific technical questions regarding the optical tweezer\nexperiments and the functional roles of sidelobes, with corresponding revisions implemented\nthroughout our manuscript.\n4. Point-by-Point Responses\nComment 1: “Hydrodynamics described by the Navier-Stokes or Euler equations, describes\nthe temporal evolution of fluid motion over time, capturing complex phenomena like velocity\nand pressure changes. The Helmholtz equation, however, is a time-independent, linear partial\ndifferential equation that represents a simplified version of the wave equation, often used to\ndescribe steady-state wave phenomena, such as the spatial distribution of monochromatic\nwaves (waves of a single frequency). While connected —for instance, by taking the time-\nindependent form of an acoustic wave equation derived from hydrodynamics—they represent\ndifferent aspects of physics: hydrodynamics deals with the full, time-dependent evolution of\nfluids, while the Helmholtz equation models time-harmonic spatial solutions.\nThe Helmholtz equation may be derived from the time-dependent wave equation, itself\nrelated to the Euler equations of hydrodynamics (a simplified form of the Navier-Stokes\nequations). In essence, the Helmholtz equation is the steady-state (time-independent) version\nof the wave equation, which is a direct descendant of the more general fluid motion equations\nused in hydrodynamics. Some papers for links are here:\nH. White, P. Bailey, J. Lawrence, J. George, J. Vera, Physics Open, Volume 1,2019, 100009\nPhilos Trans A Math Phys Eng Sci 2020 Jul 20;378(2177):20190233 ”\nResponse 1: We sincerely thank the reviewer for their insightful explanation and the valuable\nreferences. This connection between the governing equations is, in fact, the foundational\nanalogy upon which our \"optical flow\" concept is built. The key insight we exploit is that for\nsteady-state flows and monochromatic optical fields, the streamline representation provides a\nunified descriptive framework. We have incorporated the suggested references (now Refs. 50\n37\nand 51) to more clearly articulate this foundational premise in the revised manuscript.\n“(Line 82) The parallel between laser dynamics and fluid or superfluid behavior dates back to the early\n1970s, when laser-physics equations were recast into the form of the complex Ginzburg–Landau\nequations48. Since then, this hydrodynamic framework has provided deep insight into a host of\nphenomena48—from superconductivity and superfluidity to Bose–Einstein condensation49. This\nconnection is further underpinned by the formal linkage between the time-dependent equations of\nhydrodynamics and the steady-state Helmholtz equation governing monochromatic waves50-51.\nMotivated by these connections, researchers have probed the hydrodynamic character of optical fields,\nrevealing a wealth of nonlinear behaviors—chaos, multistability, and even turbulence—predicted\ntheoretically and confirmed experimentally in laser systems52–55. In 1989, Coullet et al., inspired by fluid\nvortices, formally introduced the notion of optical vortices56, catalyzing the modern exploration of\nstructured light. Building on this hydrodynamic insight, we now broaden the optics–fluid analogy to\nstructured light shaping through the lens of optical flows and streamlines.”\n“……\n47. Berry, M. V. & McDonald, K. T. Exact and geometrical optics energy trajectories in twisted\nbeams. J. Opt. A: Pure Appl. Opt. 10, 035005 (2008).\n48. Graham, R. & Haken, H. Laserlight - first example of a second-order phase transition far away\nfrom thermal equilibrium. Z. Physik 237, 31–46 (1970).\n49. Aranson, I. S. & Kramer, L. The world of the complex Ginzburg-Landau equation. Rev. Mod.\nPhys. 74, 99–143 (2002).\n50. White, H., Bailey, P., Lawrence, J., George, J. & Vera, J. A discussion on a dynamic vacuum\nmodel: Derivation of Helmholtz equation from Schrödinger equation. Phys. Open 1, 100009 (2019).\n51. Rousseaux, G. & Kellay, H. Classical hydrodynamics for analogue space–times: open channel\nflows and thin films. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 378, 20190233 (2020).\n52. Brambilla, M. et al. Transverse laser patterns. II. Variational principle for pattern selection,\nspatial multistability, and laser hydrodynamics. Phys. Rev. A 43, 5114–5120 (1991).\n……”\nThis addition strengthens the theoretical foundation of our work and more clearly acknowledges\nrelevant prior developments in relating fluid dynamics to wave phenomena. We are grateful for\nthe reviewer's input, which has helped enhance the scholarly context of our study.\nComment 2: “ The main novelty of this paper seems to be linking optical waves to\nhydrodynamics in the case of structured light over the above references and in doing the work\nmakes beams that has hydrodynamic Depiction for Vortex Dynamics. This leads to long range\nbeams with for example fixed radii of independent of azimuthal index propagating long\ndistances (e.g. figure 4)…\nOverall, what this approach shows is vortex beams with fixed radii, limited diffraction and\nvariable index…”\nresponse, the primary novelty of our work extends beyond the creation of vortex beams with\nfixed radii and limited diffraction. It lies in establishing a general hydrodynamic framework for\nstructuring light that enables flexible, multimodal shaping across diverse beam families—\nincluding both vortex and non-vortex—within a unified approach. To better demonstrate this\nbreadth, we have supplemented the manuscript with additional examples of non-vortex beam\nengineering (Supplementary Note 13) and enhanced textual clarifications throughout (e.g., in\nthe Introduction and Methods) to emphasize the methodology's applicability beyond vortex\ndynamics. We hope these revisions more clearly convey the unifying and broad applicability of\nour flow-engineering methodology.\nComment 3: “ The tweezers study seemed less convincing to me with statements such as “may\nenable three-dimensional optomechanical control via engineered photon trajectories,\npotentially extending to applications such as microfluidic flow steering, directed cell transport,\nand assembly of microscale structures”. Many of these can be performed already – what is new\nand what can be achieved specifically that cannot with normal beam shaping. The\nCommunications area study and statements are perhaps more convincing in this regard.”\nResponse 3: We appreciate the reviewer's thoughtful comment regarding the optical tweezer\nexperiments and apologize for any overstatement in our original wording. The primary goal of\nthis tweezer study was to provide the experimental validation that we can directly engineer\nand measure the 3D trajectories of the optical momentum flows (i.e., energy streamlines).\nThe novelty lies not in the manipulation tasks themselves, but in the design framework.\nConventional beam shaping controls the field amplitude and phase (the \"what\"), from which\nthe momentum flow is a derived consequence. Our method directly prescribes the momentum\nflow (the \"how\"), from which the field is synthesized. This inversion of the design process—\nengineering the flow first—is what is new and was confirmed by the tracer particles faithfully\nfollowing our pre-designed streamlines (Fig. 5).\nWe fully agree that exploring specific application advantages is a crucial future direction.\nAccordingly, we have revised the manuscript to frame these results as a foundational validation\nof this new design capability and to discuss potential applications in a more prospective manner.\n“ (Line 304) In classical particle-tracking velocimetry for fluid dynamics76, tracer beads are injected\ninto a flow and their trajectories are recorded by high-speed cameras to reconstruct the underlying\nstreamlines. An analogous strategy was adopted here: microspheres were seeded into the shaped light\nfield, and their motion was captured to map the beam’s “streamlines”. The primary aim of these\nexperiments was to demonstrate that the particle trajectories indeed faithfully follow the streamlines\nprescribed by our hydrodynamic approach, thereby validating its predictive capability. The optical-\ntweezers77 platform (Fig. 5a) was built on an inverted confocal microscope (Nikon TE2000-U)\nequipped with a 4-f holographic beam-shaping stage housing a reflective spatial light modulator\n39\n(Holoeye Leto). A continuous-wave laser (Coherent Verdi-V5) was phase-modulated with computer-\ngenerated holograms to generate the required multimode structured light. These fields interacted with\npolystyrene microspheres of 2 μm diameter suspended in de-ionized water inside a custom open-top\nchamber, which comprised an acrylic plate with a through-hole and a glass coverslip forming the base.\nThe open-top geometry minimised hydrodynamic resistance and allowed unrestricted particle motion at\nthe water surface. The chamber was mounted on a Nano-LP200 piezo nanopositioner that displaced the\nliquid-borne microspheres along the optical axis in synchrony with the camera exposure. This coordinated\nz-scan permitted three-dimensional bead trajectories to be reconstructed with high spatial accuracy.\nRepresentative measurements (Figs. 5b–d) reveal that tracer-sphere trajectories in 22nd- and 38th-\norder NDPVBs, as well as in self-similar NDPVB with the linearly expanding radius, adhere closely to\nthe predesigned energy-streamline contours. This close agreement between designed and measured\ntrajectories provides direct experimental validation of customized optical momentum flows\nengineered using our framework. By contrast, conventional perfect-vortex beams (Fig. 4b) yield a 2D\n“perfect” ring only within a single transverse plane; once the beam diffracts off that plane, trapped beads\nescape and trapping ceases (Fig. 5e). In the NDPVB configuration, however, robust three-dimensional\ntrapping and steering within OAM-invariant radii persist over the full axial scan (Figs. 5b and 5c),\nthereby validating the propagation-invariant, three-dimensional perfect vortex configuration. Looking\nahead, the combination of smaller probe particles, faster volumetric imaging, and axial-view detection\nmodules78 could achieve improved resolution of energy streamlines. Looking forward, the capability to\nengineer optical momentum flows potentially establishes a foundational capability for controlling light-\nmatter interactions in three dimensions. This flow-based approach could thus provide a new design\ntool for future studies in optomechanical control, potentially enabling applications such as microfluidic\nflow steering or directed transport that leverage tailored photon trajectories79.”\n“(Line 479) By directly engineering these streamlines, we have established a versatile and\nexperimentally validated platform for the on-demand shaping of structured light. This framework\ntranscends the limitations of conventional modes, which are confined to fixed and incompatible\npropagation laws imposed by specific solutions to the Helmholtz equation. Our approach therefore\nprovides not only fundamental insights into light-matter interactions57,75,88 and free-space optics but\nalso opens avenues for applications across multiple domains. For instance, the ability to prescribe\noptical momentum flows—validated here by optical-tweezer velocimetry—potentially establishes a\nnew design tool for advanced optomechanical control, opening prospects for applications such as\nmicrofluidic steering where tailored photon trajectories could provide a pathway for exerting optical\nforces with high spatial specificity79. Moreover, our case study in free-space communications\ndemonstrates that flow-shaped beams can simultaneously enhance channel capacity, turbulence\nresilience, and non-line-of-sight capability, underscoring their potential in next-generation optical\nnetworks30,32,33.”\nComment 4: What is not discussed clearly is the role, need for side lobes in the generated\n40\nprofiles and how they effect each of the experiments above.”\nResponse 4: We thank the reviewer for this insightful question, which allows us to clarify a\nfundamental aspect of our flow-shaping framework: the physical necessity and functional\nroles of sidelobes. Their presence and behavior are not arbitrary but stem from satisfying the\nglobal constraints of the Helmholtz equation when customized flow dynamics are imposed on\na specific region of the beam.\nPhysical Origin of Sidelobes: Helmholtz Equation Constraints\nWhile our flow-engineering approach offers significant flexibility in structuring light, the\nresulting beams must still satisfy the fundamental constraints of the Helmholtz equation in free\nspace. Our four-step method—from streamline configuration to real-space beam construction—\nensures this physical consistency by enforcing the momentum-space relationship\n41\nk\nz\n= k 2 − k 2x − k 2y . Crucially, the designed streamlines primarily prescribe the dynamics of the\nregion of interest (ROI) (i.e., the main lobe). The accompanying sidelobes naturally emerge in\nnon-ROI regions as a necessary physical structure to ensure that the entire field constitutes a\nvalid, stable solution to the Helmholtz equation. In this framework, the sidelobes act as\nthe supporting physical structure that enables the customized propagation of the main lobe.\nThis foundational role leads to specific, distinct functions across our experiments, as governed\nby the optical \"fluid-continuity equation\" (Method 3):\n1. In Self-Healing Experiments (Figs. 3a): Here, the sidelobes serve as a passive energy\nreservoir. When the mainlobe is obstructed, the intrinsic diffraction dynamics cause the\nsidelobe streamlines to spontaneously flow into and reconstruct the damaged region. The\npresence of these intact sidelobes is a prerequisite for the self-healing process. As a\ncounterexample, we have added a demonstration in Supplementary Note 4 showing that a\nconventional vortex beam without supportive sidelobes (e.g., an OAM mode with zero\nradial index) fails to self-heal after impairment.\n2. In Tunable Energy Delivery & Attenuation Compensation (Figs. 3c-j): In this active\ncontrol scenario, the sidelobes function as a dynamic energy reservoir. By prescriptively\nmanipulating the radial energy flow from the sidelobes to the mainlobe (i.e., the \"inflow\nrate\"), we can directly control the mainlobe's axial intensity profile, I(z). This mechanism\nenables intensity enhancement or precise compensation for energy decay in lossy media,\nmaintaining a constant mainlobe intensity under propagation.\n3. In Non-Diffracting Perfect Vortex Beams (NDPVBs, Fig. 4): For beams designed to be\npropagation-invariant, the sidelobes provide the necessary energy source to fuel the\nconical wavevector structure that suppresses diffraction. A trade-off is that their energy is\ngradually depleted over the designated non-diffracting range. This is why the NDPVB in\nour communication case study evolves into a single ring at the end of its 1000m range\n(Supplementary Fig. 15b), as the sustaining energy from the sidelobes has been consumed.\nIn summary, sidelobes are not merely an incidental feature but constitute a programmable\n\"energy reservoir\" and supporting structure that is physically essential for realizing the\ndesigned optical flows. We have revised the manuscript (Method 3) to incorporate this\nclarification, emphasizing that the sidelobes enable the main lobe's customized dynamics by\nensuring global compliance with wave physics:\n“(Line 598) Overall, the presence of sidelobes in our streamlined beams is a fundamental physical\nrequirement, not an arbitrary design choice. These sidelobes emerge as a necessary supporting\nstructure to ensure that the entire optical field, when engineered for customized mainlobe dynamics,\nremains a valid and stable solution to the Helmholtz equation in free space. In our framework, they\nfunction as a programmable energy reservoir. Beyond their passive role in intrinsic processes like self-\nhealing, we actively manipulate the energy flow from these sidelobes to the mainlobe—governed by\nthe optical fluid-continuity equation—to achieve on-demand control over the axial intensity profile,\nenabling functionalities such as dynamic attenuation compensation and localized energy enhancement\nin Fig. 3.”\nIn summary, we have addressed all raised concerns as follows: First, we have strengthened the\ntheoretical foundation by incorporating the suggested references, more firmly grounding the\noptics-fluid analogy. Second, we have clarified that our core contribution is a general\nhydrodynamic framework for structuring light, with vortex beams serving as powerful\nillustrative cases due to their conceptual clarity; we have further demonstrated this generality\nby including non-vortex examples in the supplement. Third, we have refined the description of\nthe optical tweezers experiment to better highlight its role in validating a new design\nframework—direct flow engineering—rather than claiming immediate application supremacy.\nFinally, we have provided a detailed explanation of the critical and multifaceted role of\nsidelobes as a programmable energy reservoir across our experiments.\nOverall Response and Conclusion:\nWe sincerely thank all the reviewers once again for their thorough and constructive feedback，\nwhich has been invaluable in helping us refine our presentation and more clearly articulate the\nscope and significance of our work.\nThe revisions have enabled us to present this unified flow-based framework with greater clarity,\nunderscoring its significance as an advance in free-space optics. This methodology transcends\nthe limitations of conventional mode-specific shaping, opening new avenues for controlling\nlight propagation in complex scenarios. Since the core concept of momentum-flow engineering\nis rooted in general wave physics, it holds promise for other wave-based domains like radio-\nfrequency and terahertz systems. This concrete bridge between optics, fluid dynamics, and\ncommunication technologies underscores the work's interdisciplinary nature, which aligns\nwell with the broad scope of Nature Communications.\nWe believe the substantial revisions, undertaken in direct response to the reviewer's comments,\nhave strengthened the manuscript and addressed the key concerns raised. We hope the revised\nversion now meets the journal’s high standards for publication.\n42"}, {"round": 2, "reviewer_comments": {"reviewer_1": "My concerns have been addressed in the revisions,and I recommend this work is published in Nature Communications.", "reviewer_2": "In my opinion the manuscript has been significantly improved and is now suitable for publication in Nature Communications.", "reviewer_3": "The authors have performed a convincing revision of the paper and answered the questions posed. I am pleased to support publication in Nature Communications."}, "author_response": ""}]

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