Holographic Mode Decomposition

• Holographic mode decomposition is a method that represents physical fields as a superposition of spatial, spectral, or quasi-particle modes to fully characterize amplitude and phase.
• It employs holographic techniques and advanced statistical reconstruction, including detector tomography and machine-learning methods, to enhance mode selectivity and precision.
• Key applications span high-capacity optical communications, adaptive optics, and quantum systems, driving innovations in secure data channels and emergent bulk dynamics.

Holographic mode decomposition is a set of methodologies for analyzing, encoding, and reconstructing optical, condensed matter, or field-theoretical data in terms of spatial, spectral, or quasi-particle modes, with holographic techniques supplying the means to disentangle, multiplex, and fully characterize both the amplitude and phase structure of physical systems. In modern research, the term spans spatial mode detection in optics, holographic multiplexing protocols for communication, spectral analysis in wavefront sensing, renormalization group decomposition via singular value analysis, and the paper of collective excitations in holographic fluid and superfluid models. Central to these approaches are concepts of mode selectivity, statistical reconstruction, and mapping of emergent phenomena across auxiliary "bulk" dimensions.

1. Principles of Holographic Mode Decomposition

The essence of holographic mode decomposition is to represent a physical field (optical, quantum, or classical) as a superposition of basis modes, and to extract from measurements or reconstructions the weights, phases, and correlations of those modes. In canonical optical applications, spatial holograms or digital phase masks are employed to project an incoming field onto spatial mode filters (e.g., Hermite–Gaussian, Laguerre–Gaussian, or Bessel–Gaussian) that separate channels by their distinct modal symmetry, topological charge, or propagation invariant.

Mathematically, the process involves expressing the field $E(x, y)$ or $U(x, y, z)$ as

$$U(x, y, z) = \sum_{n, m} c_{nm} \text{HG}_{nm}(x, y, z; q),$$

where $c_{nm}$ are generally complex coefficients encoding amplitude and phase. In holographic approaches, calibration and decomposition require knowledge of the mode overlap probabilities, often benchmarked against ideal analytical expressions (see, e.g., displaced Gaussian overlap formulas), and the physical implementation of holograms as phase or amplitude-phase masks is crucial to the fidelity of modal separation.

The holographic paradigm is extended, in more abstract settings, to renormalization-group analyses, with singular value decomposition (SVD) mapping data onto scale-dependent layers that may be interpreted as "bulk modes" of an emergent holographic space (Matsueda, 2016).

2. Methodologies and Modal Tomography

A rigorous foundation for holographic mode decomposition is provided by statistical reconstruction of the detector's positive operator-valued measure (POVM) elements (Bobrov et al., 2014). Here, a holographic spatial mode detector is modeled as a black-box device ideally projecting onto $|\psi_n\rangle\langle\psi_n|$, while real devices exhibit non-idealities captured by a generalized POVM,

$$\widetilde{\pi}_n = \sum_{k,p} \theta_{k,p}^{(n)} |\psi_k\rangle \langle\psi_p|,$$

where the coefficients $\theta_{k,p}^{(n)}$ encode imperfections such as mode crosstalk and leakage. By sending a calibrated set of displaced Gaussian beams into the detector,

$$\phi_0(x-d_i) = \sqrt{\frac{2}{\pi w^2}} \exp\left[-\frac{(x-d_i)^2}{w^2}\right],$$

the response probabilities are recorded and a constrained optimization (“detector tomography”) is performed with positivity and normalization constraints: $$\min_{\Pi} \lVert P - F \Pi \rVert,\quad \Pi_n \geq 0,\quad \sum_n \Pi_n = 1.$$ This reconstructs the full statistical POVM, allowing validation of the detector's mode selectivity. Experimental results demonstrate that amplitude–phase holograms outperform phase-only masks (similarity $S_\text{amp} \approx 0.73$ vs $S_\phi \approx 0.19$), at the cost of efficiency trade-offs.

In advanced implementations, convolutional neural networks (CNNs) trained on heterodyne image data can directly extract modal coefficients, including phases, providing an approach that is more robust to beam centering errors and with higher accuracy than overlap integral-based methods (Schiworski et al., 2021).

3. Holographic Multiplexing and Mode Expansion in Optical Communications

Holographic mode decomposition is foundational in mode division multiplexing (MDM), enabling the use of both the radial ($p$) and azimuthal ($\ell$) degrees of freedom in Laguerre–Gaussian beams for high-capacity free-space and fiber-optic communication systems (Trichili et al., 2016). Computer-generated holograms (CGHs) encode amplitude and phase information, and through spatial carrier multiplexing, hundreds of distinct modes can be spatially separated. The field,

$$\text{LG}_{p\ell}(r, \varphi) = \sqrt{\frac{2p!}{\pi w_0^2 (p+|\ell|)!}} \left(\frac{\sqrt{2} r}{w_0}\right)^{|\ell|} L_p^{(|\ell|)}\left(\frac{2r^2}{w_0^2}\right) \exp\left(-\frac{r^2}{w_0^2}\right) \exp(i \ell \varphi),$$

serves as the modal basis, with channel capacity further increased through wavelength independence, polarization diversity, and radial mode selection.

Wavenumber-division multiplexing (WDM) in holographic line-of-sight MIMO setups generalizes the mode decomposition to spatially continuous channels, representing transmit currents and receiver fields in the Fourier basis and analyzing interference, spectral efficiency, and hardware complexity for various digital processing architectures (Sanguinetti et al., 2021).

4. Physical Implementations and Advancements

Substantial precision gains in holographic mode decomposition have been achieved through the deployment of metasurfaces (Jones et al., 2021). These nanostructured diffractive elements, with pixel periods on the order of $\sim$360 nm, encode phase maps via the geometric Pancharatnam–Berry phase, dramatically reducing mode cross-coupling compared to conventional SLMs. With system-level calibration, metasurface-enhanced sensors achieve a mode weight fluctuation measurement of $6 \times 10^{-7}/\sqrt{\text{Hz}}$ at 80 Hz, surpassing SLM-based decomposition by over three orders of magnitude. Noise analyses identify optimal aperture size, electronic noise limits, and strategies to mitigate low-frequency optical drift.

HoloTile CGH methods further advance modalities by tiling phase-only sub-holograms with PSF-shaping phase profiles, deliberately confining the spatial extent of each frequency component in the Fourier domain. The overall SLM phase is constructed as

$$\varphi_{\text{SLM}} = \varphi_{\text{tiles}} + \varphi_{\text{PSF}} + \varphi_{\text{lens}},$$

permitting near-arbitrary shaping of the output profile, including top-hat, disk, ring, and vortex patterns, thereby reducing speckle and improving reconstruction fidelity. This decoupled control over mode decomposition enables parallel applications in optical manipulation, printing, quantum communication, and multiplexed mode analysis (Glückstad et al., 2023).

Multi-mode Bessel–Gaussian (MBG) holography exploits superposed ring and spiral phase structures with stackable mode parameters (axial prism $a$, OAM order $l$) for selective reconstruction. Each mode combination functions as an independent channel or encryption key, reconstructed only when the incident beam has matching negative parameters, expanding security and multiplexing capabilities (Yuan et al., 7 May 2024).

5. Holographic Mode Decomposition in Many-Body and Quantum Systems

In condensed matter and quantum systems, holographic mode decomposition is closely linked to renormalization group flows and information-theoretic representations. The continuous SVD (CSVD) of data arrays, combined with an inverse Mellin transformation, decomposes snapshots of critical systems (e.g., 2D Ising model) into scale-dependent layers,

$$M(x, y) = \int_0^\infty dz\,U(z, x) \sqrt{\lambda(z)} V(z, y),$$

with $\lambda(z) \sim z^{-\Delta}$ encoding anomalous dimensions. The reconstructed correlator,

$$\rho_\eta(l) = \int_0^\infty dz\,\mathcal{R}(l, z) z^{\eta-1},$$

is inverted via Bromwich integral, yielding components $\mathcal{R}(l, z)$ associated with finite correlation length $(z \sim 1/\xi)$. The aggregated structure parallels holographic renormalization, with bulk-like integration measures and IR cutoff warp factors echoing AdS/CFT geometries (Matsueda, 2016).

6. Mode Decomposition in Holographic Fluids and Superfluids

Collective excitations and symmetry breaking in holographic fluids and superfluids are elucidated via mode decomposition analyses of quasinormal spectra (Zhong et al., 2022, Zhao et al., 2023). In charged fluids with broken translations, the transverse channel reveals both a diffusive mode (with dispersion $\omega = -i D_\perp k^2$) and a frozen transverse Goldstone mode ($\omega = 0$), root causes stemming from the absence of shear modulus. Weak explicit breaking introduces pinning frequencies and relaxation rates, characterized hydrodynamically by

$$(\bar{\Omega}-i\omega)(\Gamma-i\omega) + \omega_0^2=0,$$

admitting solutions with mass gaps and diffusive-to-sound crossovers. In superfluid phase transitions, the decomposition of unstable quasinormal modes determines a critical length scale ($l_c = 2\pi/k_c$) for spinodal decomposition, paralleling Cahn–Hilliard diffusion but rooted in holographic dynamical instabilities. Four stages—exponential growth, formation of nonlinear structures, bubble coalescence, and final relaxation—characterize the temporal evolution, with dynamical heterogeneity quantified via local chemical potential variance.

7. Applications, Implications, and Outlook

Holographic mode decomposition underpins high-capacity multiplexed communication, adaptive optics, precision metrology, quantum information, and new forms of computational imaging. The ability to robustly encode, reconstruct, and analyze mode content with phase fidelity enables secure data channels, error-resilient wavefront sensing, and advanced material processing. In theoretical physics, mode decomposition serves as a bridge between network models, RG flows, and emergent “bulk” dynamics. A plausible implication is the continued convergence of holographic paradigms in physical engineering and mathematical modeling, leveraging spatial, spectral, and dynamical degrees of freedom to achieve modal control and characterization with unprecedented precision.

The future trajectory for holographic mode decomposition involves further integration of metasurface technology, machine learning-enhanced tomography, high-dimensional OAM/Bessel–Gaussian multiplexing, and expanded applications in non-equilibrium quantum systems, with ongoing methodological developments optimizing for efficiency, selectivity, and resilience against cross-coupling and noise.