Partially Coherent Airy Beams

• Partially coherent Airy beams are structured optical fields that combine the self-accelerating and diffraction-free properties of deterministic Airy beams with the statistical nature of partially coherent light.
• They are characterized using cross-spectral density and Wigner phase-space analysis to quantify the interplay between spatial incoherence, finite-energy truncation, and beam dynamics.
• Applications in imaging and free-space communications leverage these beams’ non-diffracting, self-healing, and noise-resistant features for enhanced performance in turbulent environments.

Partially coherent Airy beams constitute a class of structured optical fields synthesizing the diffraction-free and self-accelerating character of deterministic Airy beams with the statistical properties of partially coherent light. By introducing spatial incoherence and finite energy, these beams exhibit propagation behaviors and structural robustness not observed in their fully coherent counterparts. The field is characterized by the use of cross-spectral density, Wigner phase-space analysis, and caustic theory to fully capture the interplay between coherence, finite-energy truncation, and beam dynamics.

1. Mathematical Formulation and Core Concepts

The defining mathematical object for partially coherent Airy beams is the cross-spectral density (CSD), $W(x_1, x_2; z)$, which encodes both spatial intensity and mutual coherence. For the ideal, fully coherent Airy beam, the field amplitude at $z=0$ is

$$\psi_0(x,0) = Ai\left(\frac{x}{x_0}\right) \exp\left(a\, \frac{x}{x_0}\right)$$

with scale parameter $x_0$ and apodization $a>0$ ensuring finite energy. Partial coherence is introduced by compiling the field as an ensemble of randomly shifted or modulated Airy modes, yielding

$$W(x_1, x_2, 0) = \left\langle \psi^*(x_1, 0)\, \psi(x_2, 0)\right\rangle$$

which can equivalently be written as a Mercer expansion or as an average over random realizations (Martínez-Herrero et al., 2022, Sanz et al., 2024).

In the spectral or angular domain, partial coherence is implemented by drawing the displacements or spectral jitters ($\lambda$, $\Delta K$) from appropriate random distributions. For example, a beam composed of mutually incoherent Airy modes displaced by random $\lambda$ with Gaussian kernel $P(\lambda)$ yields a CSD of the form

$z=0$0

(Martínez-Herrero et al., 2022, Ponomarenko et al., 16 Jan 2026).

Analytical progress is obtained by moving to phase space: the Wigner distribution function (WDF) for these beams is a convolution of the coherent Airy WDF with the source’s eigenvalue-noise spectrum, for instance,

$z=0$1

where $z=0$2 encodes the coherence bandwidth (Ponomarenko et al., 16 Jan 2026).

2. Impact of Coherence and Truncation: Finite-Energy and Propagation Ranges

Realistic sources require both partial coherence and finite total energy. Energy finiteness is imposed by apodizing the Airy profile, typically via exponential or Gaussian envelopes in real or spectral domains, e.g.,

$z=0$3

or

$z=0$4

with the corresponding CSDs implementing correlations among shifted Airy components (Sanz et al., 2024).

Propagation invariance and self-acceleration are maintained only within a critical propagation distance $z=0$5 set by the spatial coherence length and the tail truncation scale. For prototypical CSDs, the overlap (fidelity) between the Airy-like profile at $z=0$6 and the ideal profile decays exponentially,

$z=0$7

with coherence/truncation-determined range $z=0$8 (Martínez-Herrero et al., 2022). Beyond this range, the distinctive Airy beam features—sharp non-diffracting lobes and parabolic trajectory—fade due to decoherence and finite power.

In the Wigner picture, momentum cutoffs $z=0$9 enforce finite energy and yield finite spatial width without impacting the accelerating trajectory (Ponomarenko et al., 16 Jan 2026). The critical propagation distances in various models obey scaling laws:

• For ABIBs on incoherent backgrounds: $$\psi_0(x,0) = Ai\left(\frac{x}{x_0}\right) \exp\left(a\, \frac{x}{x_0}\right)$$0, $$\psi_0(x,0) = Ai\left(\frac{x}{x_0}\right) \exp\left(a\, \frac{x}{x_0}\right)$$1, where $$\psi_0(x,0) = Ai\left(\frac{x}{x_0}\right) \exp\left(a\, \frac{x}{x_0}\right)$$2 is coherence length, $$\psi_0(x,0) = Ai\left(\frac{x}{x_0}\right) \exp\left(a\, \frac{x}{x_0}\right)$$3 is aperture size. Shorter coherence or tighter truncation reduces the robust propagation regime (Hajati et al., 2021).
• For CSD-based beams: the propagation-invariant window scales as the root of the CSD kernel’s parameter controlling spatial-frequency spread (Martínez-Herrero et al., 2022).

3. Beam Structure: Shape Invariance, Acceleration, and Smoothing

A central result is that self-acceleration and shape invariance survive partial coherence and even total incoherence, provided the underlying random mixtures retain the Airy structure (Martínez-Herrero et al., 2022, Ponomarenko et al., 16 Jan 2026, Sanz et al., 2024). This is reflected in the intensity and mutual coherence: $$\psi_0(x,0) = Ai\left(\frac{x}{x_0}\right) \exp\left(a\, \frac{x}{x_0}\right)$$4 indicating that the main-lobe’s parabolic trajectory is preserved for all coherence levels in infinite-energy models (Martínez-Herrero et al., 2022).

However, side-lobe oscillations are strongly suppressed as coherence decreases. In the Wigner formalism, reduction in the coherence parameter $$\psi_0(x,0) = Ai\left(\frac{x}{x_0}\right) \exp\left(a\, \frac{x}{x_0}\right)$$5 results in exponential damping of negative fringes in the phase-space distribution and a broadening (smoothing) of the main lobe, with the left (rear) tail transitioning toward monotonic decrease (Ponomarenko et al., 16 Jan 2026). In the flux-trajectory picture, loss of coherence truncates the plateau region of the velocity field, introducing trajectory dispersion and ultimately causing spatial smearing (Sanz et al., 2024).

4. Self-Healing, Nonlinear Effects, and Caustic Theory

Caustic (stationary-phase) analysis reveals that partially coherent Airy beams retain the ability to self-heal after encountering obstacles, with the process being coherence-dependent (Zhang et al., 2023). The self-healing distance $$\psi_0(x,0) = Ai\left(\frac{x}{x_0}\right) \exp\left(a\, \frac{x}{x_0}\right)$$6 increases with decreasing coherence length $$\psi_0(x,0) = Ai\left(\frac{x}{x_0}\right) \exp\left(a\, \frac{x}{x_0}\right)$$7, but the maximum profile similarity upon healing also increases, i.e., highly incoherent Airy beams recover more fully, but over a longer propagation range.

The self-accelerating caustic remains a parabola $$\psi_0(x,0) = Ai\left(\frac{x}{x_0}\right) \exp\left(a\, \frac{x}{x_0}\right)$$8 in both coherent and partially coherent regimes, derived directly from phase-space or ray-optics arguments (Zhang et al., 2023). Stationary-phase evaluation of multimode field integrals—each mode containing a random spectral displacement—quantifies the evolution of intensity and modal structure.

Nonlinear propagation in Kerr-type or saturable media introduces additional complexity, as both the nonlinearity ($$\psi_0(x,0) = Ai\left(\frac{x}{x_0}\right) \exp\left(a\, \frac{x}{x_0}\right)$$9) and partial coherence ($x_0$0) modify the beam profile (Chen et al., 2023). Notably, partial incoherence can pre-diffract the beam, mitigating the deleterious effects of self-focusing nonlinearity, while in self-defocusing regimes incoherence suppresses anomalous diffraction. This balancing is captured by exponential factors of opposite sign in the beam-quality metric

$x_0$1

where the optimal coherence length maximizes robustness against nonlinear and incoherent distortions.

5. Phase-Space and Flux-Trajectory Formalisms

The Wigner distribution function (WDF) gives a unified description bridging spatial, spectral, and statistical domains (Ponomarenko et al., 16 Jan 2026). The WDF for a partially coherent Airy beam encompasses all coherence regimes,

$x_0$2

with intensity and mutual coherence retrieved via momentum integrations and Fourier inversion, respectively.

Generalized flux-trajectory methods, as developed by Sanz & Martínez-Herrero, compute the local velocity field and “mean” trajectories from the CSD (Sanz et al., 2024). These trajectories flow according to

$x_0$3

where $x_0$4 is derived from intensity and flux calculated from the ensemble or modal decomposition. In the plateau region, parabolic trajectories correspond to perfect self-acceleration. Diminishing coherence destroys the plateau and leads to spatial broadening, enabling direct prediction of where acceleration and shape invariance break down.

6. Applications and Optimization Strategies

Practical applications leverage the intrinsic robustness imparted by partial coherence. In imaging and free-space optical communications, partially coherent Airy beams suppress speckle and mitigate scintillation, enabling more stable transmission through random or turbulent media (Hajati et al., 2021, Chen et al., 2023). The engineering of the spatial coherence length and truncation scale provides tunable trade-offs between propagation-invariant range, beam-shape retention, and resistance to noise or distortion.

Caustic self-healing, quantified experimentally and in simulation, plays a central role in scenarios requiring recovery of the main lobe after obstruction, relevant for encoding, encryption, and resilient beam delivery. Nonlinear propagation optimization is possible by tuning coherence to balance between pre-diffraction and nonlinear distortion (Chen et al., 2023).

Optimization guidelines for laboratory and applied use include:

• Setting truncation factor ($x_0$5) small enough to retain nondiffracting properties while matching the system aperture (Zhang et al., 2023).
• Selecting coherence width ($x_0$6) in the 100–200 μm range to balance side-lobe suppression, self-healing strength, and beam width over tens of millimeters.
• Using multiplexed phase masks and rotating diffusers to synthesize targeted modal ensembles in experimental platforms (Zhang et al., 2023, Chen et al., 2023).

7. Extensions, Quantum Analogy, and Generalizations

The paraxial wave equation’s mathematical identity with the Schrödinger equation underpins a direct analogy: the CSD of a partially coherent Airy beam maps onto a quantum density matrix, and environmental decoherence or truncation correspond to spatial decoherence or apodization in matter waves (Martínez-Herrero et al., 2022). As a result, all concepts—statistical mixtures, overlap fidelity, trajectory drift—are exportable to the non-relativistic quantum domain.

Generalized flux-trajectory and Wigner methods extend beyond Airy beams, providing a template for structured partially coherent fields including Bessel, vortex, and higher-order Gaussian beams. Such approaches are mathematically homologous to open-system quantum trajectory theory, further reinforcing the cross-disciplinary utility of these analytical frameworks (Sanz et al., 2024).

---

Major references for all results: (Hajati et al., 2021, Martínez-Herrero et al., 2022, Chen et al., 2023, Ponomarenko et al., 16 Jan 2026, Zhang et al., 2023, Sanz et al., 2024).