Phase-Engineered Caustics

• Phase-engineered caustics are optical structures defined by a tailored phase profile that forces beam trajectories along predetermined, mathematically designed paths.
• They employ stationary-phase analysis and inverse-design techniques via diffractive optics, metasurfaces, and spatial light modulators to achieve complex envelope shaping.
• These methods enable advanced applications including beam routing, optical trapping, and structured illumination across micro- and nanoscale domains.

Phase-engineered caustics are optical or wavefield intensity structures whose envelope—the caustic—follows a prescribed spatial trajectory, realized through precise control of the input field's phase profile. By employing stationary-phase or envelope-based design principles, alongside advanced phase and amplitude modulation (via planar diffractive optics, metasurfaces, spatial light modulators, or binary/plasmonic masks), complex caustic trajectories can be embedded into propagating beams, wave packets, or even plasmonic surface polaritons. These methods enable the shaping of both the macroscopic trajectory (global caustic path) and the microscopic intensity or phase texture (local caustic structure), providing exceptional versatility for beam routing, structured illumination, and wave-matter interaction at both micro- and nanoscale domains.

1. Theoretical Foundations and Caustic Design Principles

A caustic is mathematically defined as the envelope of a family of rays or paths, equivalently emerging as singularities in the mapping between input and output coordinates under wave propagation. In the wave-optical framework, caustics correspond to loci of coalescing stationary-phase points in integrals describing field evolution—typically via the Rayleigh–Sommerfeld or Huygens–Fresnel propagators. Explicitly, the stationary-phase condition applied to the wavefield integral yields the caustic surface as the locus where the phase's first and second derivatives simultaneously vanish.

Engineering an arbitrary caustic trajectory requires imposing a tailored spatial phase (and sometimes amplitude) distribution on the input plane so that the field's rays/guided modes/grating diffracted beams converge constructively along the desired path. The required phase gradient at each point governs the local emission direction, ensuring tangency of the local wavevector to the prescribed envelope. This concept underpins a broad set of inverse-design strategies—both in real and Fourier domains—and admits analytic solutions for many canonical curves (parabola, circle, exponential, polynomial, etc.), alongside numerical integration for fully arbitrary paths (Epstein et al., 2013, Zhou et al., 2023, Zannotti et al., 2019).

2. Core Methodologies and Implementation Regimes

Approaches to phase-engineered caustics fall into several broad categories, each exploiting the unique affordances of the underlying physical platform:

• Planar/Fourier-domain phase engineering: The field is engineered via its angular spectrum, often by solving for the spectral phase that ensures the constructive superposition of plane waves along the prescribed caustic. For translation-invariant beams, spectral methods yield propagation-invariant caustics with arbitrary transverse shape, provided the map between spectral angle and caustic parameter remains single-valued (Zannotti et al., 2019).
• Inverse ray mapping and compensation phase: In the direct-space framework, each input coordinate maps to a specific direction or ray, with the phase profile derived from the inverse mapping of the desired caustic envelope (path length difference, eikonal relations). The compensation-phase technique in metasurfaces generalizes this to arbitrary prescribed trajectories and even morphing in-plane caustic structures along propagation (Zhou et al., 2023).
• Segmented/superposed caustics for non-convex paths: For non-convex or multi-valued caustic curves (such as helical or Lissajous trajectories), a segment-wise design is implemented: the curve is divided into convex intervals, each realized by its own phase prescription, and then coherently superposed in the input field's spectrum. This enables complex, self-intersecting, or multi-valued paths inaccessible to convex-only designs (Wen et al., 2016).
• Binary phase gratings and momentum matching: For surface plasmon polaritons (SPPs), the requisite in-plane phase and missing momentum are embedded in a 2D binary grating; the sign–of–cosine binarization ensures phase transfer and efficient SPP launching along arbitrary caustic trajectories (Epstein et al., 2013).

Methodology	Representative Platform	Capabilities
Fourier-space phase	SLM, metasurface, microoptics	General caustic path; propagation-invariance
Real-space phase	Plasmonic binary mask, SLM	Arbitrary trajectories, amplitude fine-tuning
Segmented/superposed	Microoptics, SLM, free-space	Non-convex/multi-valued caustics
PB phase/OAM superposition	SLM (SU(2) beams)	Geometric-phase shifting, dynamic caustics

3. Canonical Platforms: Plasmonics, Metasurfaces, and Structured Beams

Plasmonic Caustics

The generation of self-accelerating SPP beams along arbitrary caustics is achieved by encoding the required transverse phase gradient into a binary grating mask, which ensures both phase matching and momentum compensation. The canonical phase law for a target caustic $y = f(z)$ is:

$$\frac{d\phi_i(y)}{dy} = k\,\frac{f'[z(y)]}{\sqrt{1+\left[f'[z(y)]\right]^2}}$$

with analytic forms readily available under paraxial conditions for polynomial and exponential $f(z)$ (Epstein et al., 2013). Experimental implementations use carefully controlled Ag thin films and near-field scanning optical microscopy to verify caustic steering with sub-diffraction precision.

Metasurface and Micro-optical Implementations

Arbitrary caustic engineering in free space is achieved using metasurfaces patterned via two-photon polymerization lithography. The compensation-phase approach prescribes a phase mask:

$\varphi(x',y') = k\left[L(x',y')-F(x',y')\right]$

where $L$ is the path length from meta-atom to caustic, and $F$ a reference wavefront. Polymer nanofin meta-atoms encode the local phase (via in-plane rotation) and amplitude (via height modulation), enabling high-fidelity rendering of 1D and 2D caustic structures with subwavelength resolution over propagation lengths exceeding centimeters (Zhou et al., 2023).

Structured Propagation-invariant Fields

Generalization of propagation-invariant beams (Bessel, Airy, Mathieu) to arbitrary caustic envelopes leverages stationary-phase analysis in angular (spectral) space. The approach yields both spectral-ODE and real-space “Bessel pencil” constructions, the latter allowing uniform intensity along arbitrary curves and even arbitrary letters or open curves (Zannotti et al., 2019). SU(2) beams enable structuring caustics via OAM superpositions modulated for Pancharatnam–Berry phases, with the envelope following Lissajous or higher-order parametric trajectories (Li et al., 17 Nov 2025).

4. Nontrivial Geometries, Non-convex and Dynamic Caustics

The superposition caustic approach allows the embedding of non-convex or topologically intricate caustic trajectories (e.g., elliptical helices, knotted paths) by decomposing the total path into monotonic segments and constructing the corresponding amplitude–phase spectra segment-wise. By ensuring energy localization and constructive interference predominantly in the local segment's caustic zone, side lobes and cross-talk are minimized (Wen et al., 2016).

Dynamically reconfigurable caustics have been demonstrated on ultrasonically modulated liquid surfaces, where a Digital Twin framework iteratively optimizes the spatial phase of a 2D acoustic transducer array to shape the convective fluid interface in real time. The refracted wavefront thereby sculpts a dynamic caustic on the projection plane, with compensation for system drift, fluid dynamics, and experimental deviation, enabling caustic imaging and animation at up to 10 kHz in practical bench-top systems (Nagakura et al., 22 May 2025).

5. Phase-Engineered Caustics in Non-Hermitian and Lattice Systems

In non-Hermitian photonic lattices, such as $\mathcal{PT}$-symmetric dimer arrays, phase-engineering enables the routing of energy along prescribed caustic paths with amplification or attenuation controlled by the gain/loss parameter. Notably, this platform enables the realization of aberration-free focal spots with highly tunable intensity at arbitrary propagation distances, due to the enhanced density of states and singular interference phenomena at exceptional points. The explicit stationary-phase design computes the required input phase mask for arbitrary caustic or focal trajectories directly from the lattice's dispersion relation (Bender et al., 2015).

6. Applications and Performance Characteristics

Phase-engineered caustics underpin a variety of advanced photonic and plasmonic technologies:

• Beam shaping and routing: Construction of arbitrary self-accelerating beams, curved light sheets, and transport of high-intensity lobes along complex paths.
• Optical manipulation and trapping: Sculpted caustics as guides or trapping rails for microscopic particles, including dynamically switchable traps (Li et al., 17 Nov 2025).
• Nonlinear and structured illumination microscopy: Caustic structured beams extend depth of field and enable curved focal planes for advanced imaging modalities (Zhou et al., 2023).
• On-chip plasmonic circuits: Routing SPPs along designer trajectories for compact guiding and nanoscale information processing (Epstein et al., 2013).
• Free-space communication: Encoding data via caustic signatures (shape, rotation rate) using OAM and phase geometry multiplexing (Li et al., 17 Nov 2025).
• Real-time and adaptive displays: Dynamic caustic patterning with fluidic or optomechanical platforms for reconfigurable interactive applications (Nagakura et al., 22 May 2025).

Performance is dictated by platform-specific parameters: fabrication resolution and quantization for metasurfaces, grating periodicity and groove depth for plasmonic masks, and phase modulation speed and range for SLMs or PATs. Design trade-offs involve efficiency (grating cycles, mode overlap), caustic fidelity (mask quantization, aberration correction), and dynamic bandwidth (update rates, latency in feedback-optimized systems).

7. Limitations, Extensions, and Outlook

Several constraints stem from practical considerations: spectral overlap (leading to side lobes in superposed segments), finite phase quantization and amplitude control (pixelation, feature size), and the inherent softness or lower contrast of dynamic liquid-based caustics relative to solid-state devices. Methods to mitigate these limitations include amplitude weighting in segmented designs, use of higher-resolution lithography for mask fabrication, adaptive optics for aberration compensation, and incorporation of supplementary degrees of freedom (such as amplitude, polarization, or geometric phase).

Extensions include the design of truly three-dimensional non-separable caustic trajectories (knots, links in full 3D), grayscale or multilevel caustics via dual-modulation, GPU-accelerated or neural surrogates for rapid optimization in dynamic systems, and translation to other wave domains (acoustic, electron, and matter-wave caustics). The unification of caustic stationary-phase theory with phase-engineered mask or metasurface design, validated by precise experimental implementation across multiple physical platforms, continues to expand the reach and practical impact of phase-engineered caustics across optics, condensed matter, and wave physics (Zhou et al., 2023, Wen et al., 2016, Zannotti et al., 2019, Li et al., 17 Nov 2025, Nagakura et al., 22 May 2025, Epstein et al., 2013, Bender et al., 2015).