Annular Channel Eigenmodes (ACEs)

• Annular Channel Eigenmodes (ACEs) are rigorously derived optical beams that maximize energy confinement within annular regions while carrying orbital angular momentum.
• They are formulated via a Hermitian eigenvalue problem using Hankel transform discretization to optimize spatial and spectral properties for minimal mode overlap.
• Both simulation and experimental results show that ACEs achieve up to -30 dB adjacent-channel crosstalk, supporting high-fidelity spatial-division multiplexing in OAM communications.

Annular Channel Eigenmodes (ACEs) are a rigorously derived class of orbital angular momentum (OAM)–bearing optical beams, defined as the optimal band-limited solutions for maximizing energy confinement within prescribed annular regions (channels) of the focal plane. ACEs are constructed to physically isolate optical energy in spatially distinct channels and serve as an orthonormal mode basis with minimal spatial overlap, enabling high-density, low-crosstalk OAM spatial-division multiplexing. Their formal definition arises from a Hermitian eigenvalue problem linking energy maximization within an annulus to the spectral band-limiting imposed by a finite circular pupil, resulting in beams whose spatial and spectral properties are systematically optimized. Both numerical simulation and experimental measurements confirm that ACEs significantly suppress modal crosstalk compared to conventional perfect optical vortices (POVs), with scalability governed by channel geometry, underlying physics, and device constraints (Li et al., 7 Nov 2025).

1. Mathematical Formulation of ACEs

The formulation of ACEs begins with the objective of maximizing the fraction of optical energy confined to an annular region of the focal plane, subject to spatial frequency (band) limitations imposed by the optical system. Let $U(r,\phi)$ denote the scalar optical field in the focal plane, and $P(p,\theta)$ the complex amplitude in the pupil plane, band-limited to radius $p_0$. The forward mapping is given by the two-dimensional Fourier (Hankel) transform, $U = H P$.

Energy contained within an annulus $R_1 \leq r \leq R_2$ is defined as:

$$E_{\text{annulus}} = \iint_{R_1 \leq r \leq R_2} |U(r, \phi)|^2\, r\, dr\, d\phi$$

The total transmitted energy is:

$$E_{\text{total}} = \iint_{p \leq p_0} |P(p, \theta)|^2\, p\, dp\, d\theta$$

Maximizing the ratio $\eta = E_{\text{annulus}} / E_{\text{total}}$ leads to a Rayleigh quotient of a Hermitian operator $M = H^\dagger S H$, where $S$ is a diagonal selection operator (indicator for the annulus). Thus, the core problem reduces to the eigenvalue problem:

$M P_n = \lambda_n P_n$

where the eigenvalue $\lambda_n \in [0, 1]$ quantifies the fraction of mode $n$'s total energy contained within the target annulus. The eigenvector $P_0$ associated with the largest $\lambda_0$ prescribes the optimal beam for energy confinement.

The kernel for this eigenproblem is:

$$K((r, \phi), (r', \phi')) = \int_{p \leq p_0} e^{i p r \cos(\phi - \theta)} e^{-i p r' \cos(\phi' - \theta)} p\, dp\, d\theta$$

which is Hermitian and band-limited in $p$. The eigenmodes $\psi_n(r, \phi)$ exhibit normalized energy concentration in the annulus given by $\lambda_n$:

$$\lambda_n = \frac{\int_{R_1}^{R_2} \int_0^{2\pi} |\psi_n(r,\phi)|^2 r dr d\phi }{\int_0^\infty \int_0^{2\pi} |\psi_n(r,\phi)|^2 r dr d\phi}$$

By construction, $\lambda_n \rightarrow 1$ signals nearly perfect energy confinement.

2. Numerical Construction and Mode Properties

Numerically, the continuous problem is discretized by sampling $p$ and $r$ on appropriate quadrature nodes (e.g., $N_p$ Gauss–Legendre nodes for $p$ and $N_r$ points for $r$). The Hankel transform becomes a matrix $H_{kj} = J_\ell(p_j r_k) p_j w_j$, parameterized by the OAM order $\ell$. The selection matrix $S$ is diagonal, indicating inclusion in the annular region. The Hermitian matrix $M = H^\top S H$ (size $N_p \times N_p$) is diagonalized by standard eigendecomposition algorithms, yielding eigenvectors $P_n$ (in the pupil plane) and associated eigenvalues $\lambda_n$. Back-transforming $P_n$ yields spatial profiles $R_n(r)$ and focal-plane modes:

$\psi_n(r, \phi) = R_n(r)\, e^{i \ell \phi}$

Each annulus can be independently assigned a topological charge $\ell$. The eigenmodes form an orthonormal basis in both the pupil and focal planes; modes belonging to non-overlapping annuli are nearly orthogonal in space as well as angular momentum.

3. Crosstalk Metrics, Scaling, and Comparative Analysis

Modal crosstalk is quantified by the coefficient:

$$T_{m,n} = \left| \iint \psi_m^*(r, \phi) \psi_n(r, \phi) r dr d\phi \right|^2$$

For spatial-division multiplexing, lower $T_{m,n}$ indicates reduced energy leakage between modes (channels). Simulated studies with identical system parameters (numerical aperture NA = 0.0175, wavelength $\lambda=532$ nm, six equal-width annuli in $r \in [140\,\mu\text{m}, 380\,\mu\text{m}]$) demonstrate that Gaussian-enveloped POVs exhibit typical adjacent-channel crosstalk of approximately $-16$ dB, while ACEs achieve $-30$ dB or better.

ACEs’ confinement sharpens with increasing annular width $\Delta r = R_2 - R_1$: the adjacent-channel crosstalk scales as

$$T_\text{adjacent}(\text{ACEs}) \approx A\,\exp(-\alpha \Delta r)$$

with $\alpha \approx 0.2\,\mu\text{m}^{-1}$ (system-dependent), implying each $10\,\mu$m width extension yields $\sim$2 dB additional suppression. By contrast, POV crosstalk saturates with ring width due to unavoidable sidelobes originating from hard spectro-spatial truncation. ACEs, by virtue of their natural apodization, avoid these spectral artifacts.

4. Simulation Parameters and Outputs

Typical simulation parameters include a wavelength $\lambda = 532$ nm, NA = 0.0175 (governing the effective bandlimit), and six annular channels spanning $140\,\mu$m to $380\,\mu$m. Discretization involves approximately $N_p \approx 200$ radial samples (pupil space) and $N_r \approx 500$ samples (focal plane). Key outputs are crosstalk matrices $T_{m,n}$ (visualized as color-maps), radial intensity profiles (demonstrating steep signal roll-off beyond channel edges for ACEs), and SNR vs. channel-width curves (with linear growth for ACEs and plateau for POVs).

Mode Class	Typical Adjacent Crosstalk	Energy Confinement ($\eta$)
Gaussian-POVs	$\sim$–16 dB	86.8%
ACEs	$\sim$–30 dB	>90%

These results highlight the exponential scaling and absolute crosstalk suppression achievable by ACEs under practical optical constraints.

5. Experimental Implementation and Measurement

Experimentally, ACEs are generated using a 532 nm laser, a collimation system, and a phase-only spatial light modulator (SLM). The optimal pupil phase profile $P_\text{ACEs}(p, \theta)$ is encoded via a checkerboard algorithm. Fourier transformation is implemented with an $f=250$ mm lens, and first-order diffraction is isolated and projected onto a CCD through a 4f system.

For quantitative assessment, annular-masked detectors in the focal plane register power $P_{ji}$ for transmitted mode $i$ and detection channel $j$. Experimentally measured crosstalk entries are calculated as $XT_{ji} = P_{ji}/P_{ii}$, and energy confinement as $\eta_i = P_{ii}/(\sum_j P_{ji})$. Under these schemes, ACEs exhibit mean off-diagonal crosstalk below –13 dB (versus –11 dB for POVs) and average energy confinement exceeding 90% (a 36% reduction in spillover compared to POVs, measured as $1/(1-\text{average spillover})$).

6. Physical Implications and Limits for OAM Communications

The consequences for OAM-based communications are substantial. ACEs’ higher channel isolation (–30 dB compared to –16 dB for POVs) permits denser spatial-division multiplexing and reduced bit-error rates. The smooth apodization minimizes sensitivity to diffraction and system misalignments and yields improved Gouy phase (GV) tolerance relative to Laguerre–Gaussian (LG) modal bases. System designers can flexibly choose channel radii and widths to match device resolution and SLM characteristics. Limitations include demands on the spatial resolution of the SLM to encode the complex amplitude and phase of the ACEs, as well as the computational complexity of solving large-scale Hermitian eigenproblems as the number of channels increases.

In summary, Annular Channel Eigenmodes constitute an optimal orthonormal mode basis for maximizing spatial energy confinement, enabling robust suppression of OAM modal crosstalk, and facilitating high-fidelity, high-density optical communication systems in both simulated and experimental regimes (Li et al., 7 Nov 2025).