Super-Poynting Vector in Advanced Energy Flow

• Super-Poynting Vector is a generalized form of the classical Poynting vector, incorporating additional degrees of freedom to more accurately represent energy flows in complex media.
• It decomposes energy currents into radiative and reactive components, facilitating precise modeling in anisotropic, metamaterial, and gravitational environments.
• Studies leverage the Super-Poynting Vector to reveal topological features such as skyrmion structures and supermomentum phenomena, aiding advances in structured light applications.

The Super-Poynting Vector is a generalization or extension of the classical Poynting vector, designed to capture additional nuances and topological or dynamical features of energy flow in electromagnetism, structured light, general relativity, fluid models, metamaterials, and related systems. While the standard Poynting vector S = E × H (or E × B in Gaussian units) describes the local electromagnetic energy current, numerous works show that it can be insufficient, ambiguous, or require augmentation—via separation into radiative and reactive components, inclusion of additional degrees of freedom, or adaptation to non-electromagnetic or gravitational fields. Studies of structured light, singular optics, complex media, and even general relativistic spacetimes have all motivated or necessitated “super” versions of the Poynting vector for more complete descriptions of physical energy transfer.

1. Definition and Motivation

The notion of a Super-Poynting Vector arises whenever the classical Poynting vector fails to fully account for the structure, speed, or directionality of energy flow in electromagnetic or analog contexts. Several motivations and definitions have emerged:

• In anisotropic, magnetic, or complex media, the standard S may include non-propagating “pseudo-power” terms or require inclusion of magnetization currents (Wang, 2015, Sergeev et al., 2022).
• In time-varying or pulsed fields, the Poynting vector must be separated into radiative and reactive contributions, with proper energy flux represented by a modified or “super” vector omitting the non-propagating (stored) parts (Xiao, 2019, Xiao, 2021).
• In structured optical fields or beams carrying angular momentum, local momentum densities derived from phase gradients (canonical momentum) can lead to “supermomentum” and “backflow”—phenomena the Poynting vector does not directly describe (Ghosh et al., 16 Feb 2024, Radożycki, 15 May 2025).
• In general relativity, the super-Poynting vector is defined as the energy current derived from the Bel–Robinson tensor, playing a role analogous to the electromagnetic Poynting vector but related to the flow of gravitational super-energy (Wylleman et al., 2020, Vargas-Rodríguez et al., 2022).
• Topologically nontrivial energy flows such as skyrmion-like structures or vortex skeletons in the Poynting field are also often classified under the “Super-Poynting” rubric (Wang et al., 2023, Alagashev et al., 2022).

2. Fundamental Properties and Key Formulations

The Super-Poynting Vector admits several forms, tailored to specific physical contexts.

• Electromagnetic decomposition:

In time-dependent or spatially structured fields, the energy density U(x), Poynting vector S(x), local inertia density I(x), and reactive energy density R(x) satisfy (Kaiser, 2011):

$$I(x) = \frac{1}{c^2} \sqrt{U(x)^2 - |S(x)|^2}, \quad R(x) = c^2 I(x) = \sqrt{U(x)^2 - |S(x)|^2}, \quad v(x) = \frac{c S(x)}{U(x)}.$$

The Super-Poynting Vector may be identified as the full, instantaneous v = (c S)/U rather than a period averaged value.

• Decomposition in anisotropic media:

For plane waves in lossless, anisotropic media (Wang, 2015):

$$S = S_{\text{power}} + S_{pseu}, \qquad S_{\text{power}} = v_{\mathrm{ph}}^2 (D \times B) = W_{\mathrm{em}} v_{\mathrm{ph}},$$

with "pseudo-power" S_{pseu} having zero divergence and no net energy flow.

• Separation of radiative/reactive flux:

For pulse radiation,

$$S_{\text{rad}} = E \times H - \frac{1}{2}(H \times A + \varphi D),$$

where S_{\text{rad}} transports only radiative energy, omitting the non-propagating part (Xiao, 2021).

• In metamaterials or magnetic media:

$P = \frac{c}{4\pi} [E \times (H - 4\pi M)],$

where the M × E term compensates bulk/surface energy transport without contributing to entropy or heat flux (Sergeev et al., 2022).

• Gravitational superenergy:

The super-Poynting vector in general relativity (via the Bel–Robinson tensor T_{abcd}) for an observer with 4-velocity u^a is (Wylleman et al., 2020, Vargas-Rodríguez et al., 2022):

$\mathcal{P}^a = -h^{ab} T_{b c d e} u^c u^d u^e,$

with h^{ab} projecting onto the rest space of u^a.

3. Super-Poynting Vector in Structured Light and Singular Optics

In the context of structured beams (e.g., vortex beams, skyrmionic fields), the Super-Poynting Vector is used to capture both topological and superoscillatory features:

• Poynting and canonical momentum distinction:

The canonical momentum density

$$\mathbf{P} = \frac{1}{2\omega} \mathrm{Im}[ \mathbf{E}^* \cdot (\nabla) \mathbf{E} ]$$

differs significantly from the Poynting vector, especially near phase singularities in optical vortices (Ghosh et al., 16 Feb 2024). Near these points, canonical momentum can exceed the optical wavenumber (“supermomentum”) and even show “backflow,” both manifestations of superoscillation.

• Topological Poynting structures:

Skyrmion-like energy flows have been observed experimentally in the focal region of counter-propagating vortex beams (Wang et al., 2023). The Poynting vector field exhibits a transition from Néel to Bloch to Néel skyrmion types, governed by the spatial distribution of vortex phase singularities in the electromagnetic components. The skyrmion number is

$$N_{\rm sk} = \frac{1}{4\pi} \iint \mathbf{e}_p \cdot \left( \partial_x \mathbf{e}_p \times \partial_y \mathbf{e}_p \right) dx dy,$$

where $\mathbf{e}_p$ is the local unit vector along the Poynting field.

• Spin and orbital decomposition:

Especially in microstructured optical fibers, the Poynting vector is decomposed into spin and orbital parts:

$$\mathbf{S} = \mathbf{P}_{\rm spin} + \mathbf{P}_{\rm orb}$$

with

$$\mathbf{P}_{\rm orb} = C\,\mathrm{Im}\{\mathbf{E}^* \cdot \nabla \mathbf{E}\},\quad \mathbf{P}_{\rm spin} = C \nabla \times \mathrm{Im}(\mathbf{E}^* \times \mathbf{E})$$

where C is a normalization constant (Alagashev et al., 2022).

4. Implications and Applications

The generalization and careful parsing of the energy flow captured by the Super-Poynting Vector has concrete implications.

• Energy transfer, storage, and the distinction between radiative/reactive components:

Especially in antenna theory, structured light, and radiating systems, the explicit separation of propagating and stored (reactive) energies avoids ambiguities and unphysical results such as infinite stored energies or surface-dependent radiative power (Xiao, 2019, Xiao, 2021).

• Near-field effects and optical manipulation:

Engineering fields with specific super-Poynting vector structures allows for tailored energy flow, controlled angular momentum transfer, and enhanced optical trapping or torque—relying on the manipulation of phase singularities, polarization, and local momentum densities (Julián-Macías et al., 17 Oct 2024, Wang et al., 2023, Ghosh et al., 16 Feb 2024).

• Non-Hermitian and complex media:

In PT-symmetric or gain/loss structures, the instantaneous (complex) Poynting vector enables the correct identification of stopped or fast light, in situations where the time-averaged (real) vector can only predict positive energy flux (Zhang et al., 2022).

• General relativity and gravitational analogues:

The super-Poynting vector is fundamental in understanding energy transport in the vacuum gravitational field, especially in Petrov type D spacetimes and in identifying principal observers measuring vanishing energy flux in both EM and gravitational sectors (Wylleman et al., 2020, Vargas-Rodríguez et al., 2022).

5. Physical Constraints, Controversies, and Cautions

• Physical observability and invariance:

Alternate definitions or decompositions of the Poynting vector (e.g., in axion electrodynamics, or via the inclusion of pseudo-power terms) do not alter measurable physical quantities as long as the definitions are used consistently. Inconsistent adoption can yield spurious enhancements or unphysical predictions (e.g., apparent increases in measurable voltage or power) (Zhou, 2022, Wang, 2015).

• Superluminal energy flow and special relativity:

In anisotropic media, improper inclusion of pseudo-power terms can lead to calculated energy velocities exceeding c, in direct violation of relativity. The physically correct energy current—the “super-Poynting vector” in this sense—must always conform to relativistic bounds (Wang, 2015).

• Entropy and energy transport in quantum and magnetic systems:

In thermomagnetic phenomena, bulk magnetization currents contribute to the energy (Poynting) flux, but these are dissipationless and carry no entropy; thus, they are excluded from the heat (entropy) current relevant for Nernst or thermoelectric effects (Sergeev et al., 2022).

6. Tables: Representative Super-Poynting Vector Contexts

Physical Context	Super-Poynting Vector (Key Ingredients)	Reference arXiv ID
Anisotropic Media	S = S_power + S_pseu; only S_power carries true energy	(Wang, 2015)
Time-varying EM Field	S_mod = S + (reactive energy terms/subtracted)	(Xiao, 2019, Xiao, 2021)
Magnetic/Meta-Materials	P_total = (c/4π)[E × (H – 4πM)]	(Sergeev et al., 2022)
Singular/Topological Optics	P_skyrmion: Skyrmion number, phase singularity induced structure	(Wang et al., 2023, Alagashev et al., 2022)
Gravitational Superenergy	Super-Poynting: P^a = -h^{ab} T_{b c d e} u^c u^d u^e (Bel–Robinson)	(Wylleman et al., 2020, Vargas-Rodríguez et al., 2022)
Canonical Momentum	P_canonical ≠ P_Poynting, governs supermomentum/backflow	(Ghosh et al., 16 Feb 2024, Radożycki, 15 May 2025)

7. Future Directions

• Engineering Topological Energy Flows:

Custom design of field structures for robust, localized energy delivery, leveraging phase singularities and skyrmion-type Poynting topologies (Wang et al., 2023).

• Advanced optical manipulation:

Use of superoscillatory, backflow, and supermomentum features for the precise control of forces on nanoscale objects and in quantum weak-measurement schemes (Ghosh et al., 16 Feb 2024, Radożycki, 15 May 2025).

• Extension to Nonlinear and Quantum Regimes:

Generalization of super-Poynting constructs to account for nonlinear media, entangled fields, or situations with well-defined quantized energy currents.

• Relativistic and Gravitational Applications:

Further paper of principal observer congruences, expansion of super-Poynting algorithms in numerical relativity, gravitational wave localization, and astrophysical context (Vargas-Rodríguez et al., 2022, Wylleman et al., 2020).

---

In summary, the Super-Poynting Vector is not a single universal substitute for the canonical S = E × H, but a family of constructs and decompositions, each tailored to reveal the precise and sometimes topologically or dynamically nontrivial features of energy flow in advanced electromagnetic, optical, hydrodynamic, and relativistic systems. Its proper application requires attention to physical context, adherence to conservation principles, and—crucially—consistency with measurable quantities and fundamental physical laws.