Tri-Polarized Spectrum in 6G Communications

• Tri-polarized spectrum is a framework that treats three mutually orthogonal polarization dimensions as independent degrees of freedom, enabling simultaneous transmission of independent data streams.
• Full-vector electromagnetic modeling leverages tri-polarized antennas and continuous-aperture sensing to jointly process spatial and polarization modes for improved direction-of-arrival estimation and capacity.
• Implementations in holographic MIMO and near-field systems demonstrate enhanced spectral efficiency, interference suppression, and multiplexing gains compared to dual-polarized systems.

Searching arXiv for papers on tri-polarized spectrum, tri-polarized communications, and related polarization frameworks. Tri-polarized spectrum denotes a representation, exploitation, or processing framework in which three mutually orthogonal polarization dimensions are treated as concurrent informational degrees of freedom rather than as auxiliary descriptors. In electromagnetic communications, this typically refers to the use of antennas or continuous apertures supporting three orthogonal polarization states, often associated with the radial, azimuth, and elevation directions in spherical coordinates, or equivalently with three Cartesian axes, so that independent data streams, channel modes, or sensing signatures can be carried across all three polarization dimensions (Yang et al., 7 Jan 2025). In signal processing and polarimetry, related three-component formulations treat the full vector polarization state as intrinsically three-dimensional rather than reducible to a two-component projection (Lilly, 2011, Quinn, 2014). Across these domains, the central technical theme is that the “spectrum” is no longer purely spatial or purely frequency-domain: it is a joint spatial–polarization structure whose identifiability, capacity, and statistical behavior depend on full vector-field modeling.

1. Electromagnetic definition and conceptual scope

In the 6G continuous-space communication setting, tri-polarization refers to the use of antennas supporting three mutually orthogonal polarization states, often associated with the radial, azimuth, and elevation directions in spherical coordinates: $r$, $\theta$, and $\varphi$ (Yang et al., 7 Jan 2025). This enables the simultaneous transmission and reception of independent data streams across all three polarization dimensions, promising a theoretical tripling of spatial degrees of freedom over single-polarized systems (Yang et al., 7 Jan 2025). The same paper places tri-polarization within a broader 3D continuous-space context in which both base stations and users can move continuously in three-dimensional space and channel state information in 3D continuous-space becomes crucial for future 6G and beyond-6G systems (Yang et al., 7 Jan 2025).

In holographic MIMO, the term is used operationally for systems in which each patch antenna-element independently processes and radiates or receives signals in all three orthogonal polarization states $x$, $y$, and $z$ (Wei et al., 2023). The transmitter and receiver are both equipped with holographic multiple-input multiple-output surfaces comprising compact sub-wavelength tri-polarized patch antennas, and the resulting channel is inherently a $3 \times 3$ polarization-coupled operator rather than a scalar propagation law (Wei et al., 2022, Wei et al., 2023).

In continuous-aperture sensing, the notion becomes explicitly spectral. A tri-polarized continuous aperture array measures electromagnetic field components along all three orthogonal axes $x$, $y$, and $z$, and the resulting processing exploits both self- and cross-covariances of tri-polarized signals to construct a tri-polarized spectrum for joint direction-of-arrival and attitude estimation (Si et al., 2 Oct 2025). Here the “spectrum” is a subspace functional formed from all polarization combinations rather than a single-polarization pseudospectrum.

A broader interpretive implication is that the term “tri-polarized spectrum” does not designate a single universal mathematical object across fields. In communications it often denotes the exploitable set of spatial–polarization modes of a channel; in array processing it denotes a joint subspace spectrum built from nine covariance and cross-covariance operators; in three-component signal analysis it refers more generally to the spectral content averaged over three signal components and tied to polarization geometry (Lilly, 2011). This suggests that the common invariant is not a specific formula but a full-vector treatment of three orthogonal polarization dimensions.

2. Field-theoretic channel modeling in 3D continuous space

The most explicit electromagnetic formulation is given by the 3D continuous-space channel model for tri-polarized multi-user communications (Yang et al., 7 Jan 2025). The dyadic Green’s function $\theta$0 models the relation between 3D current sources at the transmitter and the resulting electromagnetic fields at the receiver:

$\theta$1

Here $\theta$2 is a vector current density, and the dyadic Green’s function fully captures polarization coupling including near-field and far-field terms (Yang et al., 7 Jan 2025). The same framework explicitly incorporates scatterers and spherical wavefronts, with scattered fields calculated using the method of moments with high accuracy (Yang et al., 7 Jan 2025).

A central modeling step is the decomposition of the dyadic Green’s function using spherical wave functions. The paper expands both currents and fields in terms of vector spherical harmonics:

$\theta$3

where $\theta$4 correspond to TE and TM modes, and the combination of spherical harmonics covers all polarization directions (Yang et al., 7 Jan 2025). The dyadic Green’s function is decomposed as

$\theta$5

and the source current and electric field admit SVD-linked expansions

$\theta$6

$\theta$7

with $\theta$8 and $\theta$9 normalized spherical wave functions and $\varphi$0 singular values from the radiation operator’s SVD (Yang et al., 7 Jan 2025).

At the receiver, tri-polarization appears as a weighted combination of the three orthogonal field components at each user position:

$\varphi$1

where the weights correspond to tri-polarized antenna gains or combining coefficients (Yang et al., 7 Jan 2025). This model supports maximum polarization diversity because it retains the full vector current and field interactions and allows the expansion coefficients to be optimized for all three polarization directions (Yang et al., 7 Jan 2025).

Near-field holographic MIMO papers adopt the same dyadic Green’s function basis but in discretized surface form. The electric field at a receive point $\varphi$2 due to a surface current $\varphi$3 is written as

$\varphi$4

with

$\varphi$5

and

$\varphi$6

(Wei et al., 2023). The full channel matrix is block-partitioned by polarization:

$\varphi$7

so that co-polar and cross-polar sub-channels are both explicit (Wei et al., 2023, Wei et al., 2022). This matrix form is one of the clearest operational realizations of the tri-polarized spectrum in communications: the exploitable modes are distributed across all nine polarization couplings.

3. Subspace spectra, covariance structure, and continuous apertures

In continuous-aperture sensing, the tri-polarized spectrum is formulated as a generalized MUSIC-type functional defined over all self- and cross-covariances of the three polarization components (Si et al., 2 Oct 2025). The spatially continuous received signal at snapshot $\varphi$8 is modeled as

$\varphi$9

and each sample point on the continuous aperture measures all three polarization components:

$x$0

(Si et al., 2 Oct 2025).

For each polarization pair $x$1, the covariance or cross-covariance is

$x$2

with sample estimate

$x$3

(Si et al., 2 Oct 2025). Since there are $x$4 combinations, the method uses nine covariance and cross-covariance matrices rather than a single covariance matrix. Each such matrix supports a signal–noise subspace decomposition, and the corresponding noise subspace $x$5 satisfies the generalized orthogonality condition

$x$6

(Si et al., 2 Oct 2025).

The tri-polarized MUSIC spectrum is then defined as

$x$7

where $x$8 is the sampled steering vector and $x$9 is the discretized noise subspace basis for the $y$0 polarization pair (Si et al., 2 Oct 2025). The peaks of this joint spectrum across the $y$1 grid provide the estimated directions of arrival.

Because the array is spatially continuous, direct eigendecomposition of infinite-dimensional covariance operators is intractable. The paper therefore develops an equivalent continuous-discrete transformation in which the received aperture is approximated by infinitesimal non-overlapping regions and the continuous inner products are approximated using Gauss-Legendre quadrature:

$y$2

(Si et al., 2 Oct 2025). This enables practical eigendecomposition and tri-polarized spectrum calculation.

The significance of this construction is that it operationalizes polarization diversity as redundancy across nine subspace constraints. The paper states that only the tri-polarized spectrum reveals both target DOAs in a scenario where single-polarized spectra miss one due to orientation misalignment, and that product-form fusion sharpens peaks and reduces sidelobe effects (Si et al., 2 Oct 2025). A plausible implication is that tri-polarized spectral fusion is not merely a robustness enhancement but an identifiability mechanism whenever target orientation interacts strongly with aperture response.

4. Capacity, degrees of freedom, and multiplexing limits

The tri-polarized spectrum has a direct capacity interpretation in communication systems because polarization enlarges the set of independent spatial–electromagnetic modes. In the 3D continuous-space multi-user model, the single-user channel capacity is expressed as

$y$3

where the modal index $y$4 runs over all spatial–polarization modes (Yang et al., 7 Jan 2025). For the multi-user case with scattering, the capacity is

$y$5

and both $y$6 and $y$7 include all polarization field components (Yang et al., 7 Jan 2025). The paper states that with tri-polarized antennas, the number of independent spatial channels can triple compared to uni-polarization, and that channel capacities can be increased roughly threefold when moving from uni- to tri-polarized antenna systems, especially in rich-scattering and near-field environments (Yang et al., 7 Jan 2025).

A closely related near-field holographic analysis examines a uniform linear array whose elements are three infinitesimal dipoles transmitting different signals in the three spatial dimensions, with a receiver consisting of a single element with three orthogonal infinitesimal dipoles (Mestre et al., 2024). In the holographic limit, the channel for each dipole triplet is

$y$8

and the global Gram matrix converges to a $y$9 Hermitian matrix $z$0 that characterizes the available spatial eigenmodes and depends explicitly on receiver position (Mestre et al., 2024). For the fully polarized case $z$1, the eigenvalues are given in closed form:

$z$2

(Mestre et al., 2024). The number of available spatial streams depends on received SNR and receiver position, but the use of three orthogonal polarizations at the transmitter guarantees the almost universal availability of two spatial streams, whereas the use of only two polarizations results in a more extensive region where maximum multiplexing gain is available (Mestre et al., 2024). The same work states that tri-polarized ULA guarantees at least two spatial streams everywhere if the received SNR at a broadside reference point exceeds $z$3 dB (Mestre et al., 2024).

Near-field tri-polarized holographic MIMO surfaces reach similar conclusions from a system perspective. One study reports that channel capacity in tri-polarized HMIMOS can almost achieve $z$4 times gain compared with dual-polarized HMIMOS and $z$5 times compared with conventional HMIMOS (Wei et al., 2022). Another states that triple polarization is exploited for multi-user holographic MIMO systems “aiming at capacity boosting without enlarging the antenna array size” (Wei et al., 2023). These formulations treat polarization as a way to extract additional degrees of freedom when dense arrays are already approaching aperture-limited spatial saturation.

Setting	Tri-polarized role	Reported effect
3D continuous-space multi-user communications (Yang et al., 7 Jan 2025)	Three orthogonal polarization dimensions in full EM channel	DoF can triple compared to uni-polarization
Holographic ULA in the continuous limit (Mestre et al., 2024)	Three orthogonal infinitesimal dipoles per element	Almost universal availability of two spatial streams
Near-field TP HMIMOS (Wei et al., 2022)	Full $z$6 polarization exploitation	Almost $z$7 times gain over DP HMIMOS; $z$8 times over conventional HMIMOS

Taken together, these results indicate that tri-polarized spectrum should be understood as an eigenmode resource that is constrained jointly by aperture, geometry, receiver position, SNR, and polarization coupling rather than by element count alone. This is consistent with the statement that the ultimate limitation is array aperture, not element count, in the holographic limit (Mestre et al., 2024).

5. Correlation, scattering, and statistical behavior

A tri-polarized formulation changes not only rank and capacity but also the statistical structure of channels and measurements. In the 3D continuous-space communication model, simulation results show that transmit power, apertures, scatterers, and sample intervals have significant impacts on statistical properties and channel capacities (Yang et al., 7 Jan 2025). The presence of tri-polarization and scatterers lowers temporal autocorrelation function values faster and yields more uncorrelated branches in spatial cross-correlation functions, reflecting richer multipath and polarization diversity (Yang et al., 7 Jan 2025). The same study reports that ACF and CCF decrease with scattering (Yang et al., 7 Jan 2025).

In tri-polarized HMIMOS, the theoretical correlation analysis is conducted using the imaginary part of the dyadic Green’s function (Wei et al., 2023). The transmitter correlation factor is stated to increase with reduced patch spacing and with user distance in the near field, and users far from the transmitting surface experience higher correlation than those closer within the near-field regime, resulting in lower channel capacity (Wei et al., 2022). These works also emphasize that cross-polarization channel components are nonnegligible and cannot be ignored for performance-optimal design (Wei et al., 2023).

A separate but related formulation appears in three-component oscillation analysis. For a real trivariate process

$z$9

the analytic signal

$3 \times 3$0

defines a unique complex 3-vector whose polarization state is represented as an instantaneous ellipse in three dimensions (Lilly, 2011). The aggregate spectrum is

$3 \times 3$1

and the mean frequency and second central moment are defined from this spectrum (Lilly, 2011). The paper states that the first few moments of the spectrum, averaged over the three signal components, are intimately linked to the rates of change of the ellipse parameters (Lilly, 2011). In that setting, the trivariate instantaneous bandwidth contains five contributions: amplitude modulation, deformation, in-plane precession, and two out-of-plane effects (Lilly, 2011).

This broader signal-processing perspective matters because it shows that a tri-polarized spectrum need not be interpreted solely as a communications-channel object. It can also denote a spectral description whose moments encode the time variation of three-dimensional polarization geometry. A plausible implication is that statistical descriptors for tri-polarized communication channels and those for trivariate oscillatory signals may be more closely related than standard scalar or dual-polarized models suggest.

6. Precoding, diversity tradeoffs, and implementation regimes

Practical exploitation of a tri-polarized spectrum requires processing architectures that manage cross-polarization coupling rather than assuming it away. In tri-polarized HMIMOS, a user-cluster-based precoding scheme assigns users to one of three polarizations, which is easy to implement, but reduces the system’s diversity (Wei et al., 2023). In the near-field surface formulation, the same principle is described as partitioning the user set into three disjoint clusters $3 \times 3$2, $3 \times 3$3, and $3 \times 3$4, with each user assigned one fixed polarization, thereby eliminating cross-polarization interference at the expense of reducing polarization diversity by a factor of three (Wei et al., 2022).

A more complete strategy is the two-layer precoding scheme proposed for near-field tri-polarized HMIMOS (Wei et al., 2022). The first layer uses Gaussian elimination to find a precoder in the null space of the matrix collecting all cross-polarization blocks, enforcing

$3 \times 3$5

and thereby removing cross-polarization interference (Wei et al., 2022). The second layer performs block diagonalization to eliminate inter-user interference within each co-polarized channel (Wei et al., 2022). This design is stated to realize higher spectral efficiency than other schemes without sacrificing diversity when combined with two-layer power allocation (Wei et al., 2022).

Power allocation itself interacts with polarization asymmetry. Because the effective ranks and gains of the three polarizations differ, especially as the $3 \times 3$6-polarization decays with distance, the paper distinguishes several strategies, including full utilization of all three polarizations (Wei et al., 2022). It explicitly states that three polarizations should all be employed and shows

$3 \times 3$7

with $3 \times 3$8 (Wei et al., 2022).

The literature also records an important implementation limit: although densely packed arrays approach the holographic limit, making the array denser with the same length does not increase the number of spatial degrees of freedom past the limit set by aperture and polarization diversity (Mestre et al., 2024). Thus, tri-polarized spectrum utilization is not simply an argument for denser sampling; it is a prescription for exploiting vector electromagnetic structure within aperture-constrained systems.

7. Related three-dimensional polarization frameworks and terminological caution

The phrase “tri-polarized spectrum” appears in several domains with partially overlapping meanings, and some caution is therefore required. In solar and heliospheric polarimetry, a symmetric three-polarizer measurement and representation system, denoted $3 \times 3$9, is used to derive $x$0 or Stokes parameters (Deforest et al., 2021). However, the supplied data explicitly notes that there is no information in the provided paper content on the three-polarizer system’s mathematical framework or its application to Stokes or $x$1 parameters, and the extended explanation is drawn from standard literature rather than from the paper content itself (Deforest et al., 2021). It should therefore not be conflated directly with the communications or CAPA use of tri-polarized spectrum.

In optical metasurfaces, a non-interleaved TiO$x$2 metasurface encodes three distinct phase profiles into three orthogonal polarization bases with almost zero crosstalk (Yueqiang et al., 2019). Each metasurface pixel can encode up to three completely independent phase profiles in the same spatial location, and with RGB wavelength multiplexing, nine independent information pieces can be encoded (Yueqiang et al., 2019). This is a polarization-channel multiplexing result rather than a communication-theoretic spectral analysis, but it shares the same structural principle: three orthogonal polarization channels act as parallel information carriers.

In statistical polarization analysis, the full Stokes vector $x$3 is treated on the three-dimensional Poincaré sphere, where circular and linear polarization are not statistically independent (Quinn, 2014). The paper derives a three-dimensional sampling distribution

$x$4

in spherical coordinates and presents a higher-dimensional generalization of the Rice distribution (Quinn, 2014). This framework is not a tri-polarized spectrum in the array-processing sense, but it is another example of why lower-dimensional polarization models can be misleading when the full three-component state is observable.

The principal misconception across these literatures is that adding a third polarization is merely a straightforward extension of dual-polarization or a bookkeeping convenience. The cited works instead show three distinct consequences. First, in communications and holographic arrays, the third polarization changes the available eigenmode structure and multiplexing regions (Mestre et al., 2024, Yang et al., 7 Jan 2025). Second, in continuous-aperture sensing, it changes the subspace geometry by introducing nine covariance and cross-covariance relations (Si et al., 2 Oct 2025). Third, in statistical polarization analysis, it changes the probability law itself by coupling quantities that are treated as separable in lower-dimensional formalisms (Quinn, 2014).

Taken together, these results establish tri-polarized spectrum as a genuinely three-dimensional vector-field concept. Its defining property is not merely the presence of three channels, but the explicit exploitation of the coupling, geometry, and statistical structure induced by those channels across propagation, estimation, and information transfer (Yang et al., 7 Jan 2025, Si et al., 2 Oct 2025, Mestre et al., 2024).