1. INTRODUCTION

Vortices manifest ubiquitously in nature, ranging from quantum vortices in liquid nitrogen to spiral formations in galaxies within the Milky Way. These formations are observable at both macroscopic and structured electromagnetic scales. The pivotal research by Allen et al. in 1992 highlighted vortex fields’ inherent ability to carry orbital angular momentum (OAM), positioning them as a cornerstone in light field studies [1]. This unique characteristic has catalyzed significant advancements in diverse areas including high-dimensional classical and quantum information communication [2–4], optical manipulation of microparticles [5,6], precision optical measurements [7,8], advanced optical imaging [9,10], and processing [11,12].

However, a ubiquitous challenge with vortex fields is the inherent diffraction-induced enlargement in beam size and divergence concomitant with the increase in the mode index, and notably, the OAM index in vortex beams. Figure 1(a) elucidates this behavior, showcasing a subset of Laguerre-Gaussian (LG) beams characterized by a null radial index (i.e., OAM modes). This intrinsic diffraction-driven, OAM-dependent propagation restricts the versatile potential of vortex fields. In optical communication, for instance, the progressive increase in beam size and divergence with OAM necessitates larger receivers for accommodating more modes, thereby limiting the spatial multiplexing capacity, especially when bound by realistically sized receivers [13–16]. Figure 1(b) aptly illustrates this limitation. Notably, while efforts have been made to navigate this impediment, conventional “perfect” vortex beams primarily maintain an OAM-independent size around a focal plane, subsequently degrading during propagation due to the OAM-dependent divergence [17,18], as displayed in Fig. S3 in Supplement 1. To date, vortex beams that exhibit an OAM-independent size and divergence remain unreported.

 

Fig. 1. Comparative analysis of conventional OAM modes versus IPVs. (a) Traditional OAM modes showcase OAM-dependent size and divergence, with each color representing a unique OAM order $l$. In contrast, IPVs (shown by the orange curve) manifest OAM-independent size and divergence, where ${M_{\rm{IPV}}}$ is derived from the square root of the selected global quality factor $M_{\rm{IPV}}^2$. (b) Receivers with limited size obstruct high-order OAM modes due to their increasing beam size and divergence. However, (c) schematically showcases how IPVs, regardless of $l$, can easily traverse through, owing to their OAM-independent, customizable propagation characteristics, and maintain maximum structure even after disturbance or obstacles. (d), (e) Complex field patterns for OAM modes, LG-beam-driven IPVs, and IPVB-driven IPVs at transmission and reception points, respectively, following free-space transmission (e.g., $z = {1000}\;{\rm m}$). These patterns are shown with identical beam waist sizes but varying OAM orders. Notably, the transverse scales between (d) and (e) are adjusted to enhance observation clarity after free-space propagating expansion. $M_{\rm{IPV}}^2$, $M_{\rm{OAM}}^2$, and $M_{\rm{LG}}^2$ are the quality factors of IPVs, OAM modes, and LG beams, respectively.

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Our investigation has revealed that the innermost rings of LG beams exhibit unique dynamic transmission properties: they display minimal size and divergence, both of which, with accurate radial index configurations, can remain OAM-independent during propagation. We designate these innermost rings, which exhibit OAM-independent propagation, as iso-propagation vortices (IPVs; the “iso-” prefix means “same” or “equal,” reflecting their consistent propagation characteristics across all OAM values), as portrayed in the second rows of Figs. 1(d) and 1(e) and depicted by the orange-dashed rectangle. However, these IPVs necessitate conveyance via LG beams for effective free-space transmission, potentially leading to sidelobe energy inefficiencies in certain applications. By employing the energy redistribution mechanism, strategically redistributing and reversing the radial internal flow of LG beams from outward to inward, and subsequently reconstructing the IPVs in momentum space, we can engineer iso-propagation vortex beams (IPVBs). These beams not only benefit from independent free-space transmission with OAM-independent propagation but also significantly reduce sidelobe energy wastage, as illustrated in the third row of Fig. 1(e). Moreover, the propagation intensity and trajectory of IPVBs are modifiable. Due to their OAM-independent propagation coupled with the reversed, inward radial energy flows, IPVBs demonstrate exceptional resilience across diverse environments—from free space to complex media. They retain their structural integrity post-damage, maintain consistent intensity across various lossy environments, and manifest reduced modal scattering in atmospheric turbulence, as visualized artistically in Fig. 1(c). Such attributes make IPVBs compelling for applications spanning imaging, microscopy, optical communication, metrology, quantum information processing, and light-matter interactions. For instance, in optical communication, our case study suggests that the IPVB basis, with its OAM-independent propagation, facilitates a broader range of data channels. This enhances information capacity, surpassing traditional spatial multiplexing methods.

2. RESULT

A. Iso-Propagation Vortices: Achieving OAM-Independent Size and Divergence

The complex amplitude distribution of a normalized LG beam carrying OAM of $l{ћ}$ per photon within the cylindrical coordinate system $(r,\;\varphi ,\;z)$ can be described by

Note that in many OAM-related systems, OAM modes are normally represented by ${{\rm LG}_{l,0}}(r,\;\varphi, z)$, namely, belonging to a subset of LG beams where $p = {0}$, as depicted in Fig. 1.

In pursuit of a free-space optical (FSO) signal carrier whose size and divergence are independent of the OAM, let us turn our attention towards studying the characteristics of the innermost ring (i.e., the brightest ring) of LG beams. Upon evaluating Eq. (1), an analytical solution for the size of the innermost ring of the LG beam can be derived as (see Note 1 in Supplement 1 for details)

The subscript “IR” refers to the innermost ring. The divergence angle for this innermost ring is represented by

Consequently, we can define a “quality factor,” $M_{\rm{IR}}^2$, for LG beams in terms of their innermost-ring radius, instead of their root-mean-squared radius. This quality factor can be formulated as

Given a global quality factor as $M_{\rm{IR}}^2$, the radial index of the beam can be accordingly determined through the relation $p(l) = {\rm round}({({0.5|}l| + {1})^2}/{2}M_{\rm{IR}}^2 - {0.5}({\rm |}l| + {1}))$, where round (·) denotes rounding to the nearest integer. According to Eqs. (3)–(5), these innermost rings with mode indices $(l,\;p(l))$, endowed with the global quality factor $M_{\rm{IR}}^2$, exhibit OAM-independent size and divergence during propagation, as depicted in the second rows of Figs. 1(d) and 1(e) and experimentally demonstrated in Fig. S2 in Supplement 1. Therefore, these innermost-ring vortices are termed iso-propagation vortices (IPVs), whose quality factor $M_{\rm{IPV}}^2$ is equal to $M_{\rm{IR}}^2$. This behavior contrasts with that of conventional “perfect” vortices, which tend to maintain a constant size only near the focal plane but exhibit OAM-dependent divergence during propagation [21,22], as illustrated in Fig. S3 in Supplement 1.

Furthermore, Eqs. (3)–(5) highlight that these parameters—$M_{\rm{IR}}^2(l,\;p)$, ${\rho _{\rm{IR}}}(z)$, and ${\theta _{\rm{IR}}}$—parameters increase with $|l|$ but decrease with $p$. This contrasts with the LG modes, including OAM modes, where corresponding parameters increase with both $|l|$ and $p$. Therefore, for a given value of $l$, the parameters of the IPVs are smaller than those of the LG modes. This distinction underscores the unique propagation characteristics of the IPV, highlighting its potential advantages in applications where higher values of $l$ are beneficial. As demonstrated in Fig. S4 in Supplement 1, the comparison between $M_{\rm{LG}}^2(l,\;p)$ and $M_{\rm{IR}}^2(l,\;p)$ for the initial 10,000 orders highlights a dispersion range of one to 300 for $M_{\rm{LG}}^2(l,\;p)$ while $M_{\rm{IR}}^2(l,\;p)$ primarily falls between zero and 20. Thus, the transmission properties of IPVs—including their reduced size, divergence, and quality factor—surpass those of the conventional LG beam.

Although IPVs are recognized for their OAM-independent propagation and enhanced transmission dynamics, they inherently rely on LG beams for their conveyance, as illustrated in the second rows of Figs. 1(d) and 1(e). While the innermost-ring-based IPVs primarily govern the interaction with external sensors or matter, this dependency can be restrictive, especially when considering practical applications such as optical communication, micromanipulation, manufacturing, and light-matter interactions, where sidelobe energy may not be fully harnessed, leading to potential energy inefficiencies. To mitigate these issues and enable IPVs to transmit independently in free space with minimized sidelobe energy wastage, we propose a two-step approach.

 

Fig. 2. Unveiling the inward energy flow in IPVBs. (a) Intensity map of a LG beam in the $x - z$ plane, characterized by $l = {19}$ and $p = {12}$, and (b) $z$-directional intensity profile for the innermost circle. (c)–(h) Similar patterns as (a) and (b) but are specific to IPVBs with varied intensity profiles along the $z$-axis: (c), (d) exponential intensification [$I(z) = \exp ({10}z){\rm rect}(z/{2}{z_0})$], (e), (f) constant intensity [$I(z) = {\rm rect}(z/{2}{z_0})$], and (g), (h) linear increase [$I(z)\; = {5}z{\rm rect}(z/{2}{z_0})$]. Red dashed lines in (a) and (c) mark positions at $z = {0}$, ${z_{1\:}} = {0.5}{z_0}$ and ${z_{2\:}} = \;{z_0}$ with ${z_{0\:}} = {0.15}\;{\rm m}$. Blue dotted curves in (c) identify the main area of energy concentration; the inset in (d) displays the $x - y$ plane intensity pattern at ${z_2}$, illustrating that the energy of the innermost ring (enclosed by the red circle) exceeds 85% of the energy of the total beam. The inset in (f) presents the complex angular spectrum of (e). (i)–(k) Radial patterns of the LG beam and the exponentially intensified energy-flow-inward IPVB at $z = {0}$, ${z_1}$, and ${z_2}$ respectively; red-dashed boxes highlight the sidelobes of LG beams and IPVBs, with the brown and cyan arrows indicating the expansion and contraction of sidelobes of the LG beam and the IPVB, respectively. (l) Synchronized divergence of the innermost ring size of the LG beam and the IPVB, diverging synchronously along the $z$-direction. (m)–(p), (q)–(t) Radial energy flows [19] (indicated by red arrows) for the LG beam and IPVB in (i)–(k), with brown (outward) and cyan (inward) arrows marking the direction of energy flow in the sidelobes of the LG beam and the IPVB, respectively. Note that, when $z\; \gt {z_2}$ and outside the effective propagation range of $z \in \;(- {z_0},\;{z_0})$, the beam will dissipate without the inward energy flow and thus there is no arrow indicating energy flows in (t).

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Energy flow, a classical dimension describing light fields, provides a natural and efficient pathway for delving into their most intimate and deep features, a principle that holds especially true in the context of vortex light fields [19,23–27], where it facilitates an in-depth exploration of unique energy transmission characteristics. The energy flows are calculated by the complex field distributions (${\boldsymbol E}$), following from the cycle-average Poynting vector [19], given as ${\boldsymbol p} = {{{\varepsilon _0}} / {({2\omega})}}{\mathop{\rm Im}\nolimits} [{{{\boldsymbol E}^*} \times ({\nabla \times {\boldsymbol E}})}]$, where ${\varepsilon _0}$ and $\omega$ are the vacuum permittivity and circular frequency, respectively. The energy redistribution mechanism elucidates that for conventional LG beams or OAM modes, the radial energy flow, described by $I(x,\;y,\;z)kr/R(z)$ (Ref. [23]), is consistently outward for $z\; \gt \;{0}$, inevitably causing beam expansion and intensity fading during propagation as $I(z) = {1/}{w^2}(z)$, where ${I^\prime}(z) \lt {0}$, as displayed in Figs. 2(a), 2(b) and 2(m)–2(p). In contrast, for IPVBs, the exterior sidelobes can serve as an “energy reservoir” that regulates energy fluctuation for the innermost ring by the radial energy flow, according to the continuity equation [19] (see details in Section 4.B). An inward radial energy flow with ${I^\prime}(z) \gt {0}$ (i.e., growth model for intensity redistribution) redirects energy inwards to the innermost ring from external sidelobes, with the inverse process for ${I^\prime}(z) \lt {0}$ corresponding to the outward radial energy flow like conventional LG beams or OAM modes. Therefore, by implementing ${I^\prime}(z) \gt {0}$, we redistributed and reversed the radial energy flow of sidelobes (from conventional outward direction to inward direction), contracting the beam dimension and enhancing energy density during propagation, as depicted in Figs. 2(i)–2(k) and 2(m)–2(t). As a result, the innermost rings of IPVBs harness a significant proportion of the energy [${\gt}{85}\%$ in Fig. 2(d)], minimizing potential sidelobe energy inefficiencies.

Accounting for the Beer-Lambert Law, where light intensity exhibits exponential decay in a medium, we adopt an exponential growth model for intensity redistributing as a representative example in Fig. 2. Energy-flow-reversed IPVBs, endowed with OAM-independent propagation and high spatial energy density, emerge as potent tools for applications in imaging, microscopy, optical communication, metrology, quantum information processing, and light-matter interactions.

B. Resilience across Diverse Media

Owing to their radial energy flow reversal, IPVBs exhibit outstanding stability in various media, ranging from free space to complex environments. These beams maintain their structure even after damage, consistently sustain their intensity in lossy media, and display diminished and uniform modal scattering amidst atmospheric turbulence.

C. Enhanced Capacity in Optical Communication: a Case Study

In the ongoing quest for higher capacity in information capturing and processing, recent investigations have revealed significant insights [32]. Optical multiplexing, which employs degrees of freedom like polarization and wavelength, has bolstered the capacity of communication systems [33–35]. A burgeoning approach in this area is spatial mode-division multiplexing, wherein orthogonal spatial modes serve as distinct communication channels [36–41]. For instance, a free-space optical link that amalgamates spatial mode-division multiplexing with $Q$ orthogonal modes, polarization-division multiplexing with two polarization states, and wavelength-division multiplexing with $T$ wavelengths can achieve a significant capacity increase. When encoded with 100 Gbit/s quadrature phase-shift keying data, the cumulative capacity attains $Q \times {2} \times T \times 100\;{\rm Gbit/s}$. This magnitude can amplify several Pbit/s (Ref. [39]), thereby greatly enhancing capacity and spectral efficiency in FSO communications. Nevertheless, spatial mode-division multiplexing faces challenges, particularly the increase in beam size and divergence due to diffraction, which becomes more pronounced with higher mode indices [13–16]. This necessitates larger receivers to handle higher capacities, which may be impractical due to size constraints. These limitations are exemplified in Fig. 1(b).

 

Fig. 3. Highlighting the resilience of energy-flow-reversed IPVBs. (a), (b) Self-healing property: an IPVB is blocked by a square obstacle at $z = - {150}\;{\rm mm}$ (the obstructed area and energy are 19.5% and 20.9%, respectively), with (a) experimental intensity maps at different $z$-axial locations (see Visualization 4 for an evolution of the IPVB in the $z$ direction) and (b) simulated transversal energy flows, calculated from the cycle-averaged Poynting vector [19], where PCC is the Pearson correlation coefficient of innermost rings and the red arrows indicate the value and direction of each flow (see Visualization 5 for the visualization of energy flow evolution). The sharp-edged square obstacle is produced as masks via the process of photoetching chrome patterns on a glass substrate (displayed in Fig. S5 in Supplement 1). Attenuation-compensating property: $x - y$ plane intensity maps of (c) LG beam and (d) attenuation-compensated [i.e., $I(z) = \exp ({3.5}z)$] IPVB in the lossy medium 1 [milk suspension with a measured attenuation curve $\exp (- {3.5}z)$ plotted with the blue-dashed curve in (e)] at different $z$-axial locations. The red font digits on the maps depict the normalized intensity value on the innermost rings of the LG beam and attenuation-compensated IPVBs, which are plotted in (e) with the blue triangles and blue circles, respectively. The red triangles and red circles in (e) represent the measured intensity profiles of the LG beam and the attenuation-compensated [i.e., $I(z) = \exp ({7.2}z)$] IPVB in another milk suspension (medium 2) with a measured attenuation curve $\exp (- {7.2}z)$ (red-dashed curve).

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Fig. 4. Assessing free-space propagation amidst atmospheric turbulence. (a)–(c) Complex field patterns and beam sizes (indicated by dashed rectangles) of OAM beams, LG beams, and IPVBs, respectively, wherein the intensity (Int) and phase are visualized by the colormap. Each beam has a consistent beam waist while varying its OAM values ($l = {25}$, 26, 30, 34, 38). These observations are post a 1000 m free-space transmission through atmospheric turbulence with a turbulence strength of $C_n^2 = {5} \times {{10}^{- 15}}$. The simulation details include a propagation distance mesh of 40 m, an analysis area of ${0.5}\;{\rm m} \times {0.5}\;{\rm m}$, and turbulence outer and inner scales of 300 m and 0.01 m, respectively. (d) Normalized intensity that each initiated mode retains [as illustrated in Fig. 1(d)] at $z = {1000}\;{\rm m}$. (e)–(g) Crosstalk matrices for LG beams, OAM modes, and IPVBs, providing assessments of modal stability across the transmission pathway.

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Utilizing the IPVB basis, which has OAM-independent propagation traits, permits access to a broader range of subchannels beyond traditional methods, thereby amplifying the information capacity for viable free-space optical systems. To validate this, we tested the performance of IPVBs in a direct line-of-sight FSO communication system (see Supplement 1 Note 6 for the details of our experimental setup). The system quality factor, $S$, can be determined following the procedure of Ref. [42], and is expressed as ($\pi {R_0} \times {\rm NA}/\lambda$), where ${R_0}$ and NA denote the aperture radius and numerical aperture of both transmitter and receiver circular apertures, while $\lambda$ denotes the wavelength. This factor allows beams with a quality factor ${M^2}$ less than $S$ to traverse this system, thereby defining the number of possible transmission subchannels or the $Q$ value. For instance, when employing the LG beam for multiplexing, the accessible subchannels are computed as ${Q_{\text{LG}}}(S) \approx {0.5\,{\rm floor}}[S]({\rm floor}[S] + {1})$, based on the resolution of $M_{\rm{LG}}^2(l,\;p)\; = |l| + {2}p + {1} \le S$. Similarly, the number of subchannels for conventional OAM mode multiplexing, Hermite-Gaussian beam multiplexing, and multi-input multi-output transmission are denoted as ${Q_{\rm{OAM}}}{(S)} \approx 2\,{\rm floor}[S] + {1}$, ${Q_{\rm{HG}}}(S) = {Q_{\rm{LG}}} \approx 0.5\,{\rm floor}[S]({\rm floor}[S] + {1})$, and ${Q_{\rm{MIMO}}}(S) = \;{\rm round}[{0.9}S]$, respectively [42]. The quality factor of our experimental system is computed to be 6.25, and accordingly, the respective $Q$ values in Laguerre-Gaussian mode, Hermite-Gaussian mode, OAM mode, and multi-input multi-output transmission mode are 13, 21, 21, and 35. In contrast, using IPVB for spatial multiplexing transmission, the $Q$ value turns out to be 105, which represents an enhancement span from 300% to 808% when juxtaposed with the above-mentioned transmission protocols. For further details and discussion, refer to Supplement 1 Note 6.

Figure 5 offers an exemplification, highlighting a high-dimensional IPVB-multiplexed transmission of a classic painting, “Self Portrait of Vincent van Gogh.” The painting, reproduced in true color, was digitally partitioned into its red, green, and blue (RGB) layers, as illustrated in Figs. 5(a) and 5(b). Each color layer was encoded with an 8-bit depth, and the color distribution histograms are delineated in Fig. 5(d). The data for each pixel was transmitted using 24-bit IPVB multiplexing, achieving high color accuracy. The bits from 1 to 24 corresponded to mode indices $l = [{52},- {50},{48},- {46},{44},- {42},{40},- {38},{32},- {30},\;{28},- {26},\;{24},\def\LDeqbreak{}- {22},{20},- {18},{16},- {14},{12},- {10},{8},- {6},{4},- {2}]$ with a quality factor $M_{\rm{IPV}}^2 = {5.6}$. After receiving and deciphering the data through the experimental setup delineated in Supplement 1 Note 6, the true color image was reconstructed with remarkable color fidelity, demonstrating a bit error rate (BER) of ${1.83} \times {{10}^{- 4}}$. This BER is significantly lower than the forward error correction limit of ${3.8} \times {{10}^{- 3}}$, as presented in Fig. 5(c). In this experiment, the peak data transmission rate reached ${24} \times {11}\;{\rm k} = {0.264}\;{\rm Mbit/s}$, leveraging digital mirror devices operating at an 11 kHz refresh rate. Additionally, Visualization 7 illustrates the experimental demultiplexing intensity patterns for 256 decimal numbers (i.e., 0, 1, 2, …, 255).

 

Fig. 5. Achieving image transmission using high-dimensional IPVB-multiplexing boasting exceptional color fidelity. (a) Iconic “Starry Night” by Vincent van Gogh, rendered in true colors, encapsulating 24 bits of color depth and showcasing ${{2}^{24}}$ distinct colors across its ${128} \times {128}$ pixels. (b) Dissecting the three RGB layers from (a), detailing the encoding from bits 1 to 24. (c) Received true color image post-recovery, with an ${\rm error}\;{\rm rate} = {1.83} \times {{10}^{- 4}}$ (the green pixels indicate the incorrect data received). (d) Color distribution histograms of the true color image with 24-bit color depth and the RGB layers with 8-bit color depth.

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To furnish a more holistic comparison of subchannels across diverse spatial multiplexing techniques, we evaluated the number of possible transmission subchannels, $Q$, for different values of the system quality factor, $S$. We found that ${Q_{\rm{IPVB}}}$ can be estimated using the formula ${Q_{\rm{IPVB}}} \approx \sqrt {3.578 \times {{10}^4}S - 2367}$ (refer to Supplement 1 Note 7 for details). Figure 6 demonstrates that in practical free-space optical systems with finite receivers and when $S \lt {30}$, IPVB multiplexing, owing to its OAM-independent propagation characteristics, offers more subchannels compared to conventional spatial multiplexing methods [42]. Our investigation also covers multi-vortex geometric (MVG) beam multiplexing [38], a newer concept in structured light research. For systems where $S \lt {30}$, ${Q_{\rm{MVG}}}$ for MVG beam multiplexing is similar to ${Q_{\rm{LG}}}$, and thus remains lower than ${Q_{\rm{IPVB}}}$.

3. DISCUSSION AND CONCLUSION

In the annals of optical research, the favored “bottom-up” approach has been a mainstay. This strategy has always leaned heavily on the meticulous calibration of fundamental parameters, aiming to interplay the internal flows and finally guide external behaviors of light, as depicted by the green arrows in Fig. 7. However, our study introduces a nuanced shift from this traditional stance. Instead of solely adjusting foundational elements, we delve deeper, turning our attention to the intricate core of internal flows, particularly the electromagnetic momentum density or the Poynting vector in momentum space. By deftly redistributing the radial internal flows within LG beams, we have realized IPVBs, distinguished by their OAM-independent propagation and robust transmission dynamics. This refined methodology, with its emphasis on directly modulating internal flows, suggests a more comprehensive approach to the understanding and manipulation of light. While rethinking the bedrock principles of optical field design, our research illuminates an enriched path, hinting at the potential for a more harmonized control over light’s behavior and its intrinsic parameters. This concept is embodied by the blue arrows in Fig. 7, pointing to prospective avenues for future explorations in the optical domain.

 

Fig. 6. Examination of the independent spatial subchannels count for various spatial multiplexing techniques, ranging from $S = {1}$ to 30.

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Fig. 7. Transformative shift in optical field control approaches.

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In this investigation, we have delved into the nuanced realm of vortex beams, highlighting the novel characteristics of the iso-propagation vortex beams (IPVBs). A defining aspect of these beams is their redistributed and reversed radial energy flows, enabling them to exhibit OAM-independent propagation dynamics. This reversal of internal energy flow allows IPVBs to navigate challenges that conventional beams often face, showcasing adaptability across a spectrum of environments, from free space to complex conditions. One inherent trait of IPVBs is their stability of structure and intensity. This stems from their unique inward energy flows, which render them with improved resilience in various media. Furthermore, they exhibit a diminished and more uniform modal scattering in atmospheric turbulence due to their OAM-independent propagation coupling with reversed inward energy flows. Consequently, these characteristics hint at the potential utilities of IPVBs in diverse optical applications, from imaging and microscopy to advanced communication channels. Our case study within the realm of optical communication provided insights into how IPVBs encompass a broader spectrum of data channels, potentially enhancing the information bandwidth, compared to some traditional spatial multiplexing techniques.

The approach of delving into the beams to modulate their internal flows represents a nuanced perspective in optical field design. The emphasis here is on the energy-flow reversal, which, in our findings, showcased promising results. However, while these results shed light on the potential of direct internal flow manipulation in optics, further studies and validations would be instrumental in understanding the broader implications and applications in the evolving domain of optics and photonics.

4. METHOD

A. Generation of IPVB

A monochromatic light field $U(x,\;y,\;z)$, propagating along the $z$-axis, can be represented by its angular spectrum $S({k_x},\;{k_y},\;{k_z})$ in the momentum space as

By substituting Eq. (8) into Eq. (7) and reconstructing IPVs in the momentum space, we can formulate IPVBs with their angular spectrums expressed as

Besides the spiral phase $l \phi$, there exist two other types of characteristic phases in Eq. (8); the Gouy phase, ${-}(l + {2}p(l) + {1}){\tan ^{- 1}}(z/{z_0})$, corresponds to an excess delay of the wavefront, and the curvature phase, $k{({r_{\rm{IR}}}(z))^2}/({2}\;R(z))$, is responsible for wavefront bending. In most scenarios, the curvature phase is a relatively slowly varying function compared with the Gouy phase, with its rate in variation at least one to two orders of magnitude slower. For simplicity, we ignore the curvature phase in our approach. The Gouy phase, which introduces a $z$-dependent phase delay, significantly affects both the longitudinal wavevectors and, subsequently, the radial wavevector, thereby causing beam divergence, as being exemplified in LG beams. This relationship highlights the critical influence of the Gouy phase on the dynamics of beam propagation. Nevertheless, our proposed iso-propagation vortices scheme offers an effective method to manage this influence.

Since the intensity of IPVBs still tends to fade out due to the possible scattering or absorption in lossy media, naturally, it becomes appealing to control and compensate for the intensity of IPVBs along the propagation direction. By setting

Equation (11) governs the dynamics of the beam’s innermost ring (i.e., IPV), dictating its propagation characteristics. The sidelobes, serving as cooperative variables, facilitate this process through an energy exchange mechanism with the innermost ring, mediated by radial energy flows. This relationship underscores the integral role of sidelobe distribution in maintaining the propagation integrity of the innermost ring. In an unbounded domain $z \;\in \;(- \infty ,\;\infty)$, IPVBs, as derived from Eq. (11), would theoretically exhibit infinite power, mirroring the idealistic nature of Bessel and Airy beams. To circumvent this, akin to the practical adaptation of Bessel and Airy beams through energy and propagation distance limitation, IPVBs can also be rendered experimentally viable. Implementing a finite window, such as a rectangular function ${\rm rect}((z - a)/{2}b)$, confines the beam within a manageable propagation range $z \;\in \;(a - b,\;a + b)$, facilitating its generation in laboratory settings. This approach ensures that IPVBs, while inspired by ideal models, can be realized with finite energy and specific propagation limits, thus making them suitable for experimental investigation and application.

After Berry and Balazs theoretically predicted Airy beams in 1979, the self-accelerating beams have stimulated substantial research interest and have found a variety of applications, including particle manipulation and propelling [43], bending surface plasmons and electrons [44], curved plasma generation [45], light-sheet microscopy [46], single-molecule imaging [47], and others. There is little doubt that IPVBs with arbitrarily self-accelerating capability provide more versatility and can also have significant advantages in many fields. According to the Fourier phase-shifting theorem [48], the angular spectrum of the self-accelerating IPVBs with customized trajectories, expressed in terms of the displacement vector ${\boldsymbol d}(z) = g(z){\boldsymbol {\hat x}} + h(z){\boldsymbol {\hat y}} = (g(z),h(z))$, can be expressed as

 

Fig. 8. Self-accelerating Airy-beam-like IPVB with the parabolic trajectory. (a) Simulation result of the parabolic IPVB with ${\boldsymbol d}(z) = - {1.6} \times {{10}^{- 4}}\;[{0},\;{({5}z)^2}]$ in which the white dotted line serves as a reference for the optical axis and the yellow dotted curve depicts the center of self-accelerating parabolic IPVB. (b) Schematic of using self-accelerating IPVB with customizable trajectories in non-line-of-sight links where receivers are masked by obstacles. (c) Experimental setup for generating and detecting IPVBs with controllable self-accelerating trajectories and intensity profiles. BS, beam splitter; SLM, phase-only spatial light modulator; L1–L5, lenses; M, mirror; RAPM, right-angle prism mirror; HRPM, hollow roof prism mirror; CMOS, complementary metal oxide semiconductor camera.

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An example of parabolic Airy-beam-like IPVB is displayed in Fig. 8(a) and Visualization 6. IPVBs with customizable trajectories also have the potential to craft reliable space links transitioning from line-of-sight to non-line-of-sight channels [49,50], where receivers are masked by obstacles, as schemed in Fig. 8(b).

These IPVBs can be optically generated by a lens-focusing process [48]: an incident light field distribution ${S_{\rm{IPVB}}}({k_r},\;\phi ,\;{k_z})$ in the front focal plane of the lens (focal length $f$) will be transformed into IPVB in the focal field. It should be noted that in the real-space coordinate system, the radial wavenumber ${k_r}$ must be converted into the radial coordinate in the incident plane (i.e., the front focal plane) according to the relation ${k_{r\:}} = \;kr/f$. IPVBs can also be generated in real space without a lens focusing process by imprinting ${{\cal F} _{\textit{xy}}}[{S_{\rm{IPVB}}}({k_r},\;\phi ,\;{k_z})]$ directly on the spatial light modulator. The experiment setup is depicted in Fig. 8(c), which is complemented by a photograph of the actual experimental apparatus provided in Supplement 1 Note 9. A reflective SLM (Holoeye GAEA-2, 3.7 um pixel pitch, ${3840} \times {2160}$) imprinted with computer-generated hologram patterns transforms a collimated laser light wave into the complex field ${S_{\rm{IPVB}}}({k_r},\;\phi ,\;{k_z})$, with help of spatial filtering via a 4-F system consisting of lenses L1 and L2, and an iris as well. The resulting field is responsible for the IPVB in the focal volume of lens L3. A delay line, consisting of a right-angle and a hollow-roof prism mirrors and a translation stage, enables the different cross-sections of IPVB s to be imaged on a complementary metal oxide semiconductor (CMOS) camera (Dhyana 400BSI, 6.5 um pixel pitch, ${2048} \times {2040}$) after a relay 4-F system consisting of two lenses (L4 and L5). The combination of the delay line and the relay system enables us to record intensity cross sections at different $z$-axial locations before and after the focal plane of the lens L3.

B. Interpretation for Radial Energy Flow Reversal

Based on the continuity equation [19] or transport of intensity equation [51], the transverse energy flow, denoted as ${{\boldsymbol p}_ \bot}(r,\;\varphi ,\;z)$, influences the propagation characteristics of beam intensity. This relationship can be mathematically formulated as