Caustic Beamforming in Wireless Systems

• Caustic Beamforming is a beam-control method that uses controlled diffraction to form curved energy envelopes with self-bending, self-healing, and near-field non-diffracting properties.
• It leverages mathematical frameworks such as Helmholtz eigenmodes, catastrophe theory, and generalized Snell’s law to design beams that overcome blockage and optimize energy distribution.
• Implementation options range from full digital phased arrays to phase-only and amplitude-only surfaces, enabling secure, efficient, and robust solutions for next-generation wireless networks.

Caustic beamforming (CB) is a beam-control paradigm in which controlled diffraction is used to concentrate electromagnetic energy along prescribed curved trajectories or regions, rather than at a single far-field angle or focal point. In wireless systems, it is presented as an alternative to conventional steering and beamfocusing when blockage is frequent and propagation is highly dynamic. Its representative propagation behaviors are self-bending, self-healing, and near-field non-diffracting transport, and the same caustic concept also appears in other wave systems, including non-reciprocal spin-wave beamforming in anisotropic magnonic media (Liu et al., 10 Jun 2026, Wagle et al., 2024).

1. Conceptual scope and distinction from steering and focusing

Conventional beamforming techniques primarily steer energy along desired directions or focus it at specific locations. In the radiative near-field regime, this distinction becomes especially important. For an ELAA with physical aperture $D$ operating at sub-millimeter wavelengths $\lambda$, the Rayleigh distance is

$z_R=\frac{2D^2}{\lambda},$

so users can remain in a region where spherical-wavefront effects dominate over long distances. In that setting, conventional angular steering designs a planar wavefront to maximize gain in a given direction and has no range DoF, whereas beamfocusing exploits spherical curvature to focus energy to a single point $\mathbf q(t)=[x_t,y_t]^T$ and must be reconfigured at discrete anchor points along a trajectory, with peak array gain $M$ but a switching overhead $\tau_s$ per update (Wang et al., 8 Jun 2026).

CB instead sculpts a continuous curved intensity envelope that tangentially follows a desired path. In the wireless formulation, it eliminates beam-switching overhead by a one-time setting of a non-uniform phase profile across the aperture, while distributing beamforming gain over the trajectory rather than concentrating it at a single point. This makes it a distinct operating mode rather than a minor variant of steering or focusing. A common misconception is to treat CB as curved steering; the caustic construction is more specific, because the beam maximum is associated with an envelope of rays or wavefronts rather than a single departure angle or a single focus (Liu et al., 10 Jun 2026, Wang et al., 8 Jun 2026).

2. Mathematical formulations

The wireless theory of CB is organized into three complementary frameworks. The first starts from the homogeneous Helmholtz equation

$$\nabla^2 E(x,y,z)+k^2E(x,y,z)=0,\qquad k=\frac{2\pi}{\lambda}.$$

In cylindrical coordinates $(\rho,\phi,z)$, this yields non-diffracting Bessel beams,

$$E(\rho,z)=J_0(k_p\rho)e^{i k_z z},\qquad k_p^2+k_z^2=k^2,$$

whose transverse profile $J_0(k_p\rho)$ remains invariant along $\lambda$0. In elliptical or parabolic cylindrical coordinates, separable Mathieu and Weber functions yield elliptic and parabolic non-diffracting beams, respectively (Liu et al., 10 Jun 2026).

The second framework is catastrophe-theory caustics. In geometrical optics, rays emanating from an aperture have local directions set by a phase gradient, and a caustic is the envelope of a family of rays where adjacent rays become tangent, producing intensity singularities. The stationary-phase points $\lambda$1 satisfy

$\lambda$2

and the caustic is specified by the envelope condition

$\lambda$3

Representative cases include the Airy beam as a fold catastrophe, Pearcey as a cusp, and swallowtail and umbilic caustics as higher-order catastrophes. Under the paraxial approximation, the Airy main lobe follows

$\lambda$4

The third framework is generalized Snell’s-law or Eikonal design. If an aperture coordinate is $\lambda$5, a phase profile $\lambda$6 can be imposed such that the local ray angle $\lambda$7 satisfies

$\lambda$8

To force the main lobe to trace an arbitrary curve $\lambda$9, one sets $z_R=\frac{2D^2}{\lambda},$0, giving

$z_R=\frac{2D^2}{\lambda},$1

and integration yields the phase mask that directs energy tangent to the prescribed caustic (Liu et al., 10 Jun 2026).

A broader wave-physics definition appears in work on caustic-like spin-wave beams. There, a caustic is the envelope of a family of wave-fronts or ray trajectories where nearby rays concentrate, and in anisotropic media ray directions are determined by the group-velocity vector $z_R=\frac{2D^2}{\lambda},$2. In two dimensions, a fold caustic occurs at an inflection point of the isofrequency contour satisfying

$z_R=\frac{2D^2}{\lambda},$3

In the spin-wave setting, the Kalinikos–Slavin formalism supplies the dispersion of the fundamental thickness mode, and a near-field diffraction model expresses the steady-state response as an inverse Fourier transform involving the dynamic susceptibility tensor $z_R=\frac{2D^2}{\lambda},$4 and the excitation-field spectrum $z_R=\frac{2D^2}{\lambda},$5. This suggests that CB is not restricted to free-space wireless propagation, but is one realization of a more general envelope-of-rays design principle in dispersive wave systems (Wagle et al., 2024).

3. Beam classes and characteristic propagation

CB can be classified by mathematical origin. Helmholtz eigenmodes include Bessel beams with rectilinear propagation and concentric-ring transverse pattern, Mathieu beams with straight trajectories and elliptical arcs, and Weber beams with straight trajectories and parabolic-arc profiles. Catastrophe-theory beams include Airy beams with parabolic self-bending and transverse Airy-function lobes, Pearcey beams with an auto-focusing cusp shape, and swallowtail and umbilic beams with more complex curves and multiple cusps. Eikonal or generalized-Snell’s-law caustics allow arbitrary trajectories $z_R=\frac{2D^2}{\lambda},$6, with a transverse field that locally resembles an Airy-like beam (Liu et al., 10 Jun 2026).

The defining propagation properties are self-bending, self-healing, and near-field non-diffracting behavior. For self-bending, the physical interpretation is that the main intensity lobe emerges from coherent interference of rays whose tangent directions change across the aperture, so the constructive-interference locus moves along a curved path. For an Airy beam under paraxial scalar theory,

$z_R=\frac{2D^2}{\lambda},$7

with normalized coordinates $z_R=\frac{2D^2}{\lambda},$8 and $z_R=\frac{2D^2}{\lambda},$9. The main lobe peaks at $\mathbf q(t)=[x_t,y_t]^T$0, so $\mathbf q(t)=[x_t,y_t]^T$1. For self-healing, when a portion of the beam is obstructed near $\mathbf q(t)=[x_t,y_t]^T$2, the remaining unobstructed wavefront continues to interfere downstream and reconstruct the original caustic envelope at $\mathbf q(t)=[x_t,y_t]^T$3. In the simulation reported for an Airy beam, a transverse obstruction at $\mathbf q(t)=[x_t,y_t]^T$4 is followed by full-lobe recovery at $\mathbf q(t)=[x_t,y_t]^T$5. For near-field non-diffracting behavior, Bessel beams satisfy

$\mathbf q(t)=[x_t,y_t]^T$6

and propagate as $\mathbf q(t)=[x_t,y_t]^T$7 while preserving transverse shape. In practice, however, truncated approximations maintain only quasi-non-diffracting behavior over a finite depth of field; “non-diffracting” is therefore an ideal modal property and a finite-range approximation in hardware realizations (Liu et al., 10 Jun 2026).

4. Secure and blockage-resilient wireless operation

The principal wireless use cases described for CB are physical layer security and service stability in 6G-like near-field scenarios. In the security case study, the transmitter is a 256-element ULA at $\mathbf q(t)=[x_t,y_t]^T$8 with half-$\mathbf q(t)=[x_t,y_t]^T$9 spacing, the legitimate user is at $M$0, and the eavesdropper region is a circle of radius $M$1 around $M$2. Conventional far-field steering and near-field focusing cannot avoid leakage into that uncertainty region. A caustic-beam design based on a cubic-phase or Eikonal phase mask bends the main lobe around the entire region, and at transmit power $M$3 the worst-case secrecy rate improves by more than $M$4 compared with steering or focusing. In the service-stability case study, a $M$5 aperture at $M$6 serves a user moving either along broadside with $M$7 or along a parabolic path. With spectral efficiency versus user displacement $M$8 as metric, a focusing beam peaks at $M$9 and shows rapid roll-off beyond $\tau_s$0, whereas Bessel and Mathieu non-diffracting beams remain within $\tau_s$1 of peak over $\tau_s$2, and an Airy self-bending beam maintains stable spectral efficiency along its parabolic trajectory. The same paper also identifies embodied AI and ISAC settings, such as mobile robots and drones, where non-diffracting and self-healing caustic beams can maintain link quality without constant re-training (Liu et al., 10 Jun 2026).

A more specialized secure near-field construction partitions the transmit aperture into caustic and focusing subarrays. Let

$\tau_s$3

and define the uncertain eavesdropper region

$\tau_s$4

If the ray from $\tau_s$5 toward the eavesdropper is

$\tau_s$6

then the caustic subarray is

$\tau_s$7

and the focusing subarray is $\tau_s$8. On $\tau_s$9, the phase is conventional focusing,

$$\nabla^2 E(x,y,z)+k^2E(x,y,z)=0,\qquad k=\frac{2\pi}{\lambda}.$$0

On $$\nabla^2 E(x,y,z)+k^2E(x,y,z)=0,\qquad k=\frac{2\pi}{\lambda}.$$1, the phase-only caustic profile can be written in compact form using

$$\nabla^2 E(x,y,z)+k^2E(x,y,z)=0,\qquad k=\frac{2\pi}{\lambda}.$$2

as

$$\nabla^2 E(x,y,z)+k^2E(x,y,z)=0,\qquad k=\frac{2\pi}{\lambda}.$$3

The resulting sampled array weights satisfy $$\nabla^2 E(x,y,z)+k^2E(x,y,z)=0,\qquad k=\frac{2\pi}{\lambda}.$$4 with $$\nabla^2 E(x,y,z)+k^2E(x,y,z)=0,\qquad k=\frac{2\pi}{\lambda}.$$5. In simulations with $$\nabla^2 E(x,y,z)+k^2E(x,y,z)=0,\qquad k=\frac{2\pi}{\lambda}.$$6, $$\nabla^2 E(x,y,z)+k^2E(x,y,z)=0,\qquad k=\frac{2\pi}{\lambda}.$$7, half-$$\nabla^2 E(x,y,z)+k^2E(x,y,z)=0,\qquad k=\frac{2\pi}{\lambda}.$$8 spacing, $$\nabla^2 E(x,y,z)+k^2E(x,y,z)=0,\qquad k=\frac{2\pi}{\lambda}.$$9, $(\rho,\phi,z)$0, and $(\rho,\phi,z)$1, this scheme achieves up to an $(\rho,\phi,z)$2 reduction of the worst-case eavesdropping rate, a mean secrecy-rate gap greater than $(\rho,\phi,z)$3 at $(\rho,\phi,z)$4, and closed-form runtime below $(\rho,\phi,z)$5 for $(\rho,\phi,z)$6, while ADMM and S-procedure baselines require at least about $(\rho,\phi,z)$7. The paper interprets the robustness physically: the rays from $(\rho,\phi,z)$8 are tangent to a curve that avoids $(\rho,\phi,z)$9, so leakage remains small for all admissible eavesdropper positions (Liu et al., 25 Mar 2026).

5. Mobility, throughput, and paradigm selection in the near field

For near-field mobile communications, CB has been analyzed against pointwise beamfocusing through an Airy-beam model. For a parabola $$E(\rho,z)=J_0(k_p\rho)e^{i k_z z},\qquad k_p^2+k_z^2=k^2,$$0, the aperture phase profile is

$$E(\rho,z)=J_0(k_p\rho)e^{i k_z z},\qquad k_p^2+k_z^2=k^2,$$1

With a ULA of $$E(\rho,z)=J_0(k_p\rho)e^{i k_z z},\qquad k_p^2+k_z^2=k^2,$$2 elements at positions $$E(\rho,z)=J_0(k_p\rho)e^{i k_z z},\qquad k_p^2+k_z^2=k^2,$$3 and a user at $$E(\rho,z)=J_0(k_p\rho)e^{i k_z z},\qquad k_p^2+k_z^2=k^2,$$4, the near-field channel is

$$E(\rho,z)=J_0(k_p\rho)e^{i k_z z},\qquad k_p^2+k_z^2=k^2,$$5

and the CB weight vector is

$$E(\rho,z)=J_0(k_p\rho)e^{i k_z z},\qquad k_p^2+k_z^2=k^2,$$6

The instantaneous received SNR is

$$E(\rho,z)=J_0(k_p\rho)e^{i k_z z},\qquad k_p^2+k_z^2=k^2,$$7

Using the stationary phase method, the closed-form SNR along the trajectory becomes

$$E(\rho,z)=J_0(k_p\rho)e^{i k_z z},\qquad k_p^2+k_z^2=k^2,$$8

where $$E(\rho,z)=J_0(k_p\rho)e^{i k_z z},\qquad k_p^2+k_z^2=k^2,$$9. The associated continuous throughput is expressed in closed form in terms of the trajectory arc-length

$J_0(k_p\rho)$0

with

$J_0(k_p\rho)$1

as

$J_0(k_p\rho)$2

A central result is that $J_0(k_p\rho)$3 contains no explicit user speed $J_0(k_p\rho)$4: once the Airy-caustic intensity envelope is established, the SNR at each point on the path is fixed, and the time-averaged log-rate is invariant to traversal speed (Wang et al., 8 Jun 2026).

The comparison with discrete beamfocusing formalizes the spatial-gain versus temporal-overhead trade-off. The approximate beamfocusing throughput is

$J_0(k_p\rho)$5

with geometric factor $J_0(k_p\rho)$6 under the paraxial approximation. Solving $J_0(k_p\rho)$7 yields the switching-overhead threshold

$J_0(k_p\rho)$8

The maximum tolerable $J_0(k_p\rho)$9 is obtained by choosing the dwell time $\lambda$00 that maximizes this concave expression, and the asymptotic result

$\lambda$01

shows that at extremely high frequencies the allowable reconfiguration time vanishes. In the numerical example with $\lambda$02, $\lambda$03, $\lambda$04, $\lambda$05, and parabola $\lambda$06, beamfocusing outperforms CB for small overhead $\lambda$07, whereas CB dominates for larger overhead or higher frequencies. The design guidance given is correspondingly geometric and hardware-aware: the curvature $\lambda$08 should match the expected user path, larger $\lambda$09 boosts peak CB gain through $\lambda$10 while extending coverage, and higher $\lambda$11 worsens beamfocusing defocusing through $\lambda$12 and drives $\lambda$13 toward zero (Wang et al., 8 Jun 2026).

6. Realization architectures, related wave systems, and open problems

Practical CB realizations are divided into three hardware classes. Full amplitude-and-phase digital phased arrays assign each element an amplitude gain $\lambda$14 and phase $\lambda$15, enabling arbitrary caustic profiles, multi-beam superposition, and slot-level reconfiguration, at the cost of the highest power, cost, and complexity. Phase-only surfaces include 3D-printed dielectric plates with static continuous masks, RIS implementations using PIN diodes or varactors with quantized phase control such as $\lambda$16–$\lambda$17 bits and reconfiguration on $\lambda$18–$\lambda$19 scales, and liquid-crystal metasurfaces in the THz range with continuous phase but slower reconfiguration on $\lambda$20–$\lambda$21 scales. Amplitude-only or binary surfaces include leaky-wave antennas, holographic surfaces based on a pre-computed interference hologram $\lambda$22, and on-off switch metasurfaces with binary coding. The reported trade-off is systematic: phase plates approximate the desired parabolic Airy trajectory but exhibit increased transverse sidelobes relative to full-amplitude synthesis, while amplitude-only or binary implementations capture the main caustic bend with pronounced distortion and reduced gain. The open challenges listed for wireless CB are diffraction-aware channel modeling and acquisition, especially matrix-friendly incorporation of boundary diffraction such as the Uniform Theory of Diffraction into MIMO models; universal CB algorithms for 3D obstacles, multi-user demands, and dynamic environments, including physics-informed neural networks that map 3D scene embeddings to aperture amplitude and phase profiles; and impairment mitigation for phase quantization, amplitude coupling, beam squint in wideband arrays, and mutual coupling, with CAPAs, TTD networks, and joint array-metasurface calibration identified as emerging solutions (Liu et al., 10 Jun 2026).

Related work shows that caustic beamforming is not confined to free-space radio links. D. Wagle et al. study non-reciprocal caustic-like spin-wave beams in a YIG film, using the anisotropic dispersion and non-reciprocity of the magnonic system together with a nano-constricted rf waveguide. Their near-field diffraction model computes the magnetization response from $\lambda$23 and the antenna-field spectrum, and spatially resolved micro-focused BLS agrees with both the model and MuMax3 over frequencies from $\lambda$24 to $\lambda$25 and fields from $\lambda$26 to $\lambda$27. In the Damon–Eshbach configuration, two caustic beams appear at approximately $\lambda$28 from the bias field on only one side; in the BVW configuration, a single beam appears at $\lambda$29 depending on field sign. Beam angles predicted from the group-velocity direction match BLS within $\lambda$30, divergence is sub-$\lambda$31, and the effective attenuation length exceeds $\lambda$32. The paper presents this as a geometry-agnostic toolbox for on-chip magnonic steering, reservoir computing, holography, neuromorphic networks, and low-loss interconnects, and also notes extensions to plasmonics, phononics, and photonics in anisotropic media. This suggests that the caustic formulation of beam control is best understood as a general wave-engineering framework whose wireless form is one prominent instantiation rather than its only domain (Wagle et al., 2024).