TAM Vectorial Holography

• TAM vectorial holography is a structured-light approach that unifies spin (polarization) and orbital angular momentum to achieve complete vector field synthesis.
• It employs a symmetry framework using su(2) algebra to construct TAM eigenstates and coherent-state fields, enabling continuous control of spatial and polarization modes.
• Metasurface implementations and nonlinear extensions facilitate multiplexed holographic imaging and advanced applications in communications, microscopy, and chiral sensing.

Searching arXiv for recent and foundational papers on TAM vectorial holography and TAM/coherent-state structured light. Total Angular Momentum (TAM) vectorial holography denotes a class of structured-light and metasurface schemes in which the relevant optical degree of freedom is the total angular momentum of light, combining spin angular momentum (SAM, associated with polarization) and orbital angular momentum (OAM, associated with spatial mode structure), rather than treating polarization and spatial phase as separable design channels. In this setting, holography is not limited to scalar phase reconstruction: it targets the full vectorial field, including amplitude, phase, and polarization, and can map distinct TAM input states to distinct holographic outputs. Recent work places this program on an explicitly symmetry-based footing by exploiting the shared $su(2)$ structure of SAM and OAM, constructing TAM eigenfields and TAM coherent-state fields as superpositions of circular polarization and Laguerre–Gaussian beams, and extending these ideas to metasurfaces and nonlinear flat optics for multiplexing, beam synthesis, and information encoding (Aguirre-Olivas et al., 5 Mar 2026).

1. Symmetry framework and definition of TAM

The basic kinematic identity is

$\mathbf{J}=\mathbf{L}+\mathbf{S},$

with $\mathbf{J}$ the total angular momentum, $\mathbf{L}$ the orbital contribution, and $\mathbf{S}$ the spin contribution. In the optical context used in TAM vectorial holography, SAM is identified with polarization, while OAM is identified with helical or otherwise structured spatial modes. A generic TAM state can therefore be indexed by a polarization state and an OAM index, and orthogonality can arise from either degree of freedom (Jung et al., 29 Sep 2025).

A central conceptual move in the TAM literature is to reframe light-matter and field-structure problems in terms of symmetries, especially total angular momentum and helicity, instead of relying exclusively on the traditional SAM/OAM split. In a symmetry-based treatment, helicity is

$$\Lambda=\frac{\mathbf{J}\cdot\mathbf{P}}{|\mathbf{P}|},$$

with $\mathbf{P}$ the linear momentum operator, and it acts as the generator of electromagnetic duality transformations (Fernandez-Corbaton et al., 2012). This viewpoint avoids the fundamental separability problem of SAM and OAM for transverse Maxwell fields and emphasizes $J_z$ and $\Lambda$ as the physically meaningful quantities in free-space electromagnetism. A common misconception is that “spin-to-orbit conversion” is a single generic mechanism. The symmetry-based account distinguishes at least two cases: in focusing, the relevant mechanism is breaking of transverse translational symmetry, whereas in scattering it is breaking of electromagnetic duality symmetry (Fernandez-Corbaton et al., 2012). This distinction matters for holography because any device designed to encode, preserve, or convert TAM must be assessed relative to the actual conserved or broken symmetries of the optical system.

Within this broader symmetry program, TAM vectorial holography may be understood as the synthesis and readout of vectorial optical fields in a basis adapted to $\mathbf{J}$, rather than to scalar phase alone or to an a priori separation of polarization and OAM. This suggests a more robust language for structured-light design, especially when nonparaxiality, helicity selectivity, or joint polarization–spatial control are central.

2. TAM eigenfields and coherent-state field construction

A recent formalization of TAM-structured paraxial fields begins from the observation that both SAM and OAM can be described under the $\mathbf{J}=\mathbf{L}+\mathbf{S},$0 Lie algebra. In that construction, SAM is represented by the spin-$\mathbf{J}=\mathbf{L}+\mathbf{S},$1 basis of left- and right-circular polarization, while Laguerre–Gaussian beams are mapped to spin-$\mathbf{J}=\mathbf{L}+\mathbf{S},$2 subspaces with

$\mathbf{J}=\mathbf{L}+\mathbf{S},$3

where $\mathbf{J}=\mathbf{L}+\mathbf{S},$4 is the radial index and $\mathbf{J}=\mathbf{L}+\mathbf{S},$5 the azimuthal index. TAM then follows from $\mathbf{J}=\mathbf{L}+\mathbf{S},$6 addition, with allowed total quantum numbers

$\mathbf{J}=\mathbf{L}+\mathbf{S},$7

This supplies a common algebraic language for joint control of polarization and spatial mode structure (Aguirre-Olivas et al., 5 Mar 2026).

TAM eigenstates are built by Clebsch–Gordan superpositions,

$\mathbf{J}=\mathbf{L}+\mathbf{S},$8

with coefficients

$\mathbf{J}=\mathbf{L}+\mathbf{S},$9

In the optical realization, the corresponding vector field is

$\mathbf{J}$0

where $\mathbf{J}$1 denotes circular polarization and $\mathbf{J}$2 a Laguerre–Gaussian mode. All components belong to the same modal number $\mathbf{J}$3, which preserves coherent spatial structure and propagation behavior (Aguirre-Olivas et al., 5 Mar 2026).

The coherent-state extension is the key step for continuous TAM control. For a fixed $\mathbf{J}$4, the TAM coherent-state field is

$\mathbf{J}$5

with coefficients $\mathbf{J}$6 determined by the $\mathbf{J}$7 coherent-state construction and a single complex parameter $\mathbf{J}$8. In this framework, $\mathbf{J}$9 continuously interpolates the spatial and polarization content with periodicity $\mathbf{L}$0, while the phase $\mathbf{L}$1 rigidly rotates the field structure by $\mathbf{L}$2 about the optical axis, with $\mathbf{L}$3 periodicity (Aguirre-Olivas et al., 5 Mar 2026). The stated significance is that a single complex parameter controls both polarization and spatial degrees of freedom while preserving the underlying angular-momentum symmetry.

3. Vector-beam content and paraxial propagation behavior

One of the more concrete optical consequences of the TAM construction is that familiar cylindrical vector beams appear as explicit TAM eigenstates. For $\mathbf{L}$4 and integer $\mathbf{L}$5, the formalism yields orthonormal vector beams with radial and azimuthal polarization distributions: $\mathbf{L}$6 Using the relations

$\mathbf{L}$7

these superpositions reduce to radial and azimuthal polarization patterns (Aguirre-Olivas et al., 5 Mar 2026). Thus, cylindrical vector beams are not auxiliary constructions external to TAM theory; they are embedded directly in the TAM basis.

The same work states that changing $\mathbf{L}$8 modifies the degree of mixing between polarization and spatial modes, continuously transforming the beam from a pure circularly polarized Laguerre–Gaussian beam to a radial or azimuthal polarization vector beam, while varying $\mathbf{L}$9 rotates the entire polarization and modal structure about the propagation axis (Aguirre-Olivas et al., 5 Mar 2026). Under paraxial propagation, these fields maintain their polarization and intensity distributions up to a scaling factor. Within the scope of vectorial holography, this is operationally important because it implies that the encoded joint polarization–spatial structure is not immediately degraded by propagation in the paraxial regime.

A plausible implication is that TAM coherent-state parameterization provides a compact control manifold for vectorial hologram design: rather than independently tuning separate polarization and OAM channels, one may navigate a symmetry-constrained family of fields with a single complex control parameter. That inference follows directly from the stated continuous and symmetry-preserving coupling of the two degrees of freedom.

4. Metasurface realizations and multiplexed TAM holography

The most direct realization of TAM vectorial holography in flat optics is a bi-layer metasurface architecture that overcomes the chirality limitations of single-layer designs. In that scheme, TAM is explicitly defined as the combination of SAM and OAM, and the device enables “true polarization-OAM multiplexing,” allowing independent generation of vectorial holographic images for each orthogonal TAM input state (Jung et al., 29 Sep 2025). The reason bi-layer structure matters is that cascading two layers extends the accessible Jones matrices from unitary symmetric matrices to arbitrary unitary matrices, thereby allowing independent control of anisotropy and chirality.

The device-level formalism is Jones-matrix based. A transmissive transformation is assigned at each spatial sample point so that each TAM state $\mathbf{S}$0 yields a target output vector field. The paper gives both a general layered decomposition

$\mathbf{S}$1

and a practical implementation in which the top layer acts as a local half-wave plate (Jung et al., 29 Sep 2025). The four independently tunable parameters are the two layer orientations and the phase delays associated with the bottom-layer elliptical nanopost axes.

The reported demonstrations include a single metasurface encoding four independent TAM states, corresponding to two polarizations and two OAM values, each reconstructing a different vectorial image. The experimental characterization is described as showing high fidelity, with correlation and polarization cosine similarity of approximately $\mathbf{S}$2, and low crosstalk between channels (Jung et al., 29 Sep 2025). The same platform is stated to extend to vector-beam inputs represented on a higher-order Poincaré sphere and to bidirectional TAM vectorial holography, where opposite illumination directions yield different mappings, reaching 24 total images on one sample (Jung et al., 29 Sep 2025).

This bi-layer work is naturally connected to earlier multichannel vectorial holography based on birefringent metasurfaces. In that earlier platform, a unitary symmetric Jones matrix at each pixel was engineered as

$\mathbf{S}$3

supporting multiple polarization channels, quantified phase relations such as

$\mathbf{S}$4

and a total of twelve polarization channels with negligible cross-talk (Zhao et al., 2019). That work framed the concept as relevant to arbitrary spin-to-angular-momentum conversion and vectorial holographic display. Relative to it, the bi-layer TAM architecture adds the specific capacity to assign independent vectorial holograms to orthogonal polarization–OAM states, rather than to polarization channels alone (Jung et al., 29 Sep 2025).

5. Nonlinear TAM addition and structured-light generation

TAM vectorial holography is not restricted to linear optics. A nonlinear extension has been demonstrated in third-harmonic generation from a single thin film of amorphous silicon, where the experimentally relevant conserved quantity is the total angular momentum projection on the optical axis together with helicity (Menshikov et al., 2024). In this system, a tightly focused femtosecond pump at $\mathbf{S}$5 generates third-harmonic light at $\mathbf{S}$6, and the pump polarization is varied with a quarter-wave plate from linear through elliptical to nearly circular states.

For an isotropic, cylindrically symmetric film, the addition rule is

$\mathbf{S}$7

which reduces in the reported experiment to

$\mathbf{S}$8

because $\mathbf{S}$9 and $$\Lambda=\frac{\mathbf{J}\cdot\mathbf{P}}{|\mathbf{P}|},$$0 for the isotropic case (Menshikov et al., 2024). A right-circularly polarized pump with TAM projection $$\Lambda=\frac{\mathbf{J}\cdot\mathbf{P}}{|\mathbf{P}|},$$1 therefore yields third-harmonic light with projection $$\Lambda=\frac{\mathbf{J}\cdot\mathbf{P}}{|\mathbf{P}|},$$2. When the pump contains both right- and left-circular components, the allowed third-harmonic TAM projections become

$$\Lambda=\frac{\mathbf{J}\cdot\mathbf{P}}{|\mathbf{P}|},$$3

with weights controlled by the pump ellipticity (Menshikov et al., 2024).

The nonlinear polarization source is written as

$$\Lambda=\frac{\mathbf{J}\cdot\mathbf{P}}{|\mathbf{P}|},$$4

and, for a pure right-circular input,

$$\Lambda=\frac{\mathbf{J}\cdot\mathbf{P}}{|\mathbf{P}|},$$5

making the TAM tripling explicit (Menshikov et al., 2024). The paper further states that the shape of the third-harmonic light is highly sensitive to the polarization state of the pump and that simulations reproduce the measured patterns quantitatively. Within the language of holography, this establishes a route to nonlinear vectorial holography in which amplitude, phase, and polarization structure emerge from nonlinear TAM addition rather than only from linear phase engineering.

6. Conceptual limits, measurement issues, and interpretive cautions

A recurring issue in TAM-related optics is the status of SAM and OAM as separate observables. Several works warn against a naive interpretation. One result is that canonical SAM and OAM for the single photon are mutually compatible but unsharp quantum observables and therefore require POVMs for their joint measurements, whereas a non-canonical decomposition gives sharp observables described by PVMs but makes SAM and OAM mutually incompatible (Motta et al., 2018). Another formulation emphasizes that neither the commonly used spin nor orbital operators is, in general, a true generator of rotations for electromagnetic fields, and that total angular momentum and helicity remain the relevant symmetry generators (Fernandez-Corbaton et al., 2013). These points do not invalidate SAM/OAM engineering in practice, but they delimit its physical interpretation, especially outside the paraxial regime.

Measurement can also be confounded by intrinsic–extrinsic distinctions. In fractional vortex beams, the intrinsic OAM matches the generating topological charge $$\Lambda=\frac{\mathbf{J}\cdot\mathbf{P}}{|\mathbf{P}|},$$6, but the total OAM need not match $$\Lambda=\frac{\mathbf{J}\cdot\mathbf{P}}{|\mathbf{P}|},$$7 because of an extrinsic contribution associated with topologically structured darkness generated at phase discontinuities. The paper decomposes the field into an intrinsic component and a structured-darkness component, attributes the latter to evanescent waves at the phase discontinuity, and shows that careful cylindrical-lens orientation is needed to separate intrinsic from total OAM experimentally (Alperin et al., 2017). For TAM vectorial holography, the direct implication stated in that work is that phase discontinuities can disrupt the desired OAM encoding and create mismatches between designed and actual TAM content.

There is also a field-structure issue in nonparaxial optics. A fully vectorial treatment shows that longitudinal and transverse, spin and orbital angular-momentum components can all become significant in arbitrary superposition states, and that orthogonal superpositions can produce spin-orbit shift from longitudinal SAM to OAM, especially under nonparaxial conditions (Fang et al., 2019). This suggests that any high-NA or strongly vectorial TAM hologram must be analyzed with the full vectorial field, not only with scalar phase maps.

Taken together, these results qualify several common simplifications. TAM vectorial holography is not merely OAM holography with polarization labels added; it sits at the intersection of symmetry, transversality, helicity, and vectorial field synthesis. Its most rigorous formulations therefore privilege $$\Lambda=\frac{\mathbf{J}\cdot\mathbf{P}}{|\mathbf{P}|},$$8 and helicity, while its practical implementations must account for measurement unsharpness, extrinsic angular-momentum artifacts, and nonparaxial vector-field effects (Fernandez-Corbaton et al., 2012).

7. Applications and research directions

The current literature assigns TAM vectorial holography to several application classes. The coherent-state TAM framework identifies high-capacity classical and quantum communications, optical trapping, precision microscopy, spin-orbit photonics, quantum state engineering, and studies of classical entanglement and geometric representations such as the Poincaré sphere and Majorana constellation as natural targets (Aguirre-Olivas et al., 5 Mar 2026). The multichannel metasurface literature emphasizes high-capacity and secure information processing, polarization-controlled display, and optical encryption (Zhao et al., 2019, Jung et al., 29 Sep 2025). The nonlinear flat-optics work adds structured-light generation and detection in the nonlinear regime, including polarization-sensitive third-harmonic pattern formation and potential use in polarization detection and chiral sensing (Menshikov et al., 2024).

A consistent theme across these directions is multiplexing by orthogonality. OAM offers an unbounded set of orthogonal modes, while polarization offers a compact but fully vectorial control manifold. TAM vectorial holography combines both, and recent metasurface implementations explicitly map different orthogonal TAM input states to different arbitrary vectorial outputs (Jung et al., 29 Sep 2025). In parallel, the coherent-state program indicates that one can continuously traverse families of such states with a single complex control parameter while preserving symmetry (Aguirre-Olivas et al., 5 Mar 2026).

A plausible implication is that future TAM holographic systems will increasingly merge three strands now visible separately in the literature: symmetry-grounded state construction, full Jones-matrix or multi-layer flat-optics implementation, and nonlinear TAM addition. The available evidence already supports that convergence: TAM coherent-state fields provide the state space, bi-layer metasurfaces provide arbitrary linear vectorial mapping, and nonlinear flat optics provides harmonic-state generation governed by TAM projection and helicity.