Liquid-Crystal Tunable Elliptical Retarder Array

• The liquid-crystal tunable elliptical retarder array is a programmable platform that synthesizes spatially varying pure retarder fields via cascaded LC-SLMs.
• It encodes skyrmionic topology by extracting an S²-valued axis geometry from Mueller-matrix decomposition, enabling precise topological measurement.
• Robust pixel-level control and noise tolerance tests demonstrate stable skyrmion-charge retention, paving the way for topological optical data storage.

A liquid-crystal-based tunable elliptical retarder array is a programmable optical platform that synthesizes a spatially varying retarder field whose local anisotropy is controlled pixel by pixel. In the formulation demonstrated for topological encoding, the platform is implemented as a reconfigurable arbitrary retarder array built from cascaded liquid-crystal spatial light modulators (LC-SLMs), and its operative object is not merely an intensity or phase pattern but a pure-retarder Mueller-matrix field whose local axis geometry defines an $S^2$-valued anisotropy texture (Zhang et al., 22 Aug 2025). In this setting, the array functions as structured matter for light-matter interaction, enabling direct encoding, measurement, and perturbation testing of skyrmionic topology. Within the broader liquid-crystal retarder lineage, it is closely related to earlier space-variant liquid-crystal Pancharatnam–Berry optical elements that implemented tunable retarder-like transformations in selected two-dimensional orbital-angular-momentum subspaces (Slussarenko et al., 2012).

1. Definition and optical formalism

At the polarization-optics level, the platform realizes a pure retarder Mueller matrix field. For an incident Stokes vector $S_{\mathrm{in}}$, the output is written as

$S_{\mathrm{out}} = M S_{\mathrm{in}}.$

For a pure retarder, the Mueller matrix reduces to

$$M = \begin{pmatrix} m_{11} & 0 \ 0 & R \end{pmatrix},$$

with $R\in SO(3)$ acting as a rotation on the Poincaré sphere (Zhang et al., 22 Aug 2025). In this representation, a three-dimensional rotation is specified by a rotation axis and a rotation angle, which correspond physically to the retarder’s fast-axis geometry and retardance.

The paper defines the axis field as a normalized $S^2$-valued vector,

$$A(r,\theta)= \begin{bmatrix} A_1(r,\theta)\ A_2(r,\theta)\ A_3(r,\theta) \end{bmatrix}, \qquad A_1^2 + A_2^2 + A_3^2 = 1.$$

This projected axis field is extracted from the retarder’s Mueller-matrix decomposition and is termed the axis geometry. The axis geometry is the physically meaningful carrier of topology in the demonstrated system.

This formulation is significant because it separates the full optical response from the topologically relevant degree of freedom. The resulting $S^2$-valued field is not a conventional liquid-crystal director field; the skyrmions are instead encoded into the optical anisotropy by selecting a distinguished axis, which is explicitly stated to be fundamentally different from the more commonly known skyrmions formed by director fields in liquid crystals (Zhang et al., 22 Aug 2025).

2. Array architecture and programmable control

The tunable elliptical retarder array is built by cascading multiple LC-SLM layers, each layer acting as a linear retarder with a controllable retardance and fast-axis orientation. By cascading three such layers, the platform synthesizes an effective retarder whose net response is generally an elliptical retarder, so that the axis geometry can vary over the full sphere rather than being restricted to purely linear anisotropy (Zhang et al., 22 Aug 2025).

The array is described as pixel-level controllable and reconfigurable. Its control variables are the phase patterns loaded onto the LC-SLMs; in the methods, the axis-geometry field is written as

$A = A(\theta_1,\theta_2,\theta_3),$

where $\theta_1(r,\theta)$, $S_{\mathrm{in}}$0, and $S_{\mathrm{in}}$1 are the phase patterns applied to the three modulators. The local retarder state is therefore determined jointly by the three layer settings. In the simulation and perturbation model, the retardance of each constituent layer is identified as

$S_{\mathrm{in}}$2

The architectural importance of this arrangement is that the optical anisotropy field becomes programmable rather than fixed. A plausible implication is that the term “array” refers not only to spatial multiplexing across pixels but also to the synthesis of a field of locally distinct elliptical retarder states, each recovered through polarimetric decomposition rather than inferred from a nominal drive signal alone.

3. Encoding of axis geometry and skyrmion textures

The encoding principle begins with a target $S_{\mathrm{in}}$3-valued axis field $S_{\mathrm{in}}$4, which is then implemented through the LC-SLM cascade so that the effective Mueller-matrix decomposition reproduces the desired axis geometry (Zhang et al., 22 Aug 2025). The paper gives the design

$S_{\mathrm{in}}$5

$S_{\mathrm{in}}$6

$S_{\mathrm{in}}$7

with

$S_{\mathrm{in}}$8

This parameterization fixes both the radial profile and the angular winding. The boundary conditions are explicit: $S_{\mathrm{in}}$9 places the field at one pole at the center, and $S_{\mathrm{out}} = M S_{\mathrm{in}}.$0 places it at the opposite pole at the boundary. The angular term $S_{\mathrm{out}} = M S_{\mathrm{in}}.$1 sets the skyrmion number target $S_{\mathrm{out}} = M S_{\mathrm{in}}.$2. For $S_{\mathrm{out}} = M S_{\mathrm{in}}.$3, the field is a standard Néel-type skyrmion, while other choices produce Bloch-type, higher-order, and multi-textured variants.

The topological charge is evaluated through

$S_{\mathrm{out}} = M S_{\mathrm{in}}.$4

Experimentally, the measured axis-geometry fields closely matched the designed ones, and the skyrmion numbers agreed with theory for six showcased structures: Bloch-type skyrmion, Néel-type skyrmion, skyrmion lattice, skyrmion bag, meron lattice, and higher-order skyrmion (Zhang et al., 22 Aug 2025).

The practical workflow is stated explicitly: encode the target pattern into the LC-SLM phase maps, measure the Mueller matrix of the resulting structured medium, decompose the Mueller matrix to recover the axis geometry, and compute the skyrmion number from the recovered $S_{\mathrm{out}} = M S_{\mathrm{in}}.$5-field. In this sense, the array is both an overview engine and a measurable topological carrier.

4. Measurement, perturbation tolerance, and topological readout

Topological robustness was tested by introducing artificial Gaussian noise into the retardance or phase patterns of the LC-SLM layers. The perturbation model adds a random variable

$S_{\mathrm{out}} = M S_{\mathrm{in}}.$6

to the SLM phase settings, with the noise standard deviation swept from

$S_{\mathrm{out}} = M S_{\mathrm{in}}.$7

in increments of $S_{\mathrm{out}} = M S_{\mathrm{in}}.$8 (Zhang et al., 22 Aug 2025). Quantification was performed by generating 120 independent noise samples across all noise levels, using six repeated trials per noise level, measuring the skyrmion number for each realization, and reporting the mean and standard deviation.

Three regimes are identified. In the low-noise regime, $S_{\mathrm{out}} = M S_{\mathrm{in}}.$9, $$M = \begin{pmatrix} m_{11} & 0 \ 0 & R \end{pmatrix},$$0 and the topology is stable. In the intermediate regime, the skyrmion number begins to fluctuate. In the high-noise regime, $$M = \begin{pmatrix} m_{11} & 0 \ 0 & R \end{pmatrix},$$1, the topology collapses and the encoded charge is lost (Zhang et al., 22 Aug 2025). The paper further gives a homotopy-based preservation criterion: if the perturbed field differs everywhere from the ideal one by less than 1 in norm, the skyrmion number is preserved, which is translated into an angular tolerance of up to $$M = \begin{pmatrix} m_{11} & 0 \ 0 & R \end{pmatrix},$$2 everywhere without changing the underlying topological charge.

Robustness is confirmed both quantitatively and visually. The quantitative signature is skyrmion-number stability under perturbation. The visual signature is the sequence of recovered axis-field maps, which show that the internal texture distorts as noise increases, while the boundary conditions remain intact in the protected regime; once boundary integrity is lost, the skyrmion collapses. This makes the readout intrinsically topological rather than purely morphological. The information-storage demonstration is framed accordingly: the platform is optically readable, reconfigurable, and topologically protected, and the skyrmion-bag ASCII demo “SKYRME” stores data through the measured skyrmion numbers of four enclosed skyrmions (Zhang et al., 22 Aug 2025).

5. Relation to earlier liquid-crystal retarder analogues in OAM space

A direct antecedent of the array concept is the tunable liquid-crystal Pancharatnam–Berry optical element developed as a spatial-mode converter for the orbital angular momentum of light. That device is a birefringent liquid-crystal plate with uniform phase retardation $$M = \begin{pmatrix} m_{11} & 0 \ 0 & R \end{pmatrix},$$3 and spatially varying optical-axis orientation $$M = \begin{pmatrix} m_{11} & 0 \ 0 & R \end{pmatrix},$$4, so its phase action is geometric rather than thickness-based (Slussarenko et al., 2012). In the circular-polarization basis, part of the field remains in the same spin state with amplitude $$M = \begin{pmatrix} m_{11} & 0 \ 0 & R \end{pmatrix},$$5, while the converted part flips handedness and acquires a geometric phase $$M = \begin{pmatrix} m_{11} & 0 \ 0 & R \end{pmatrix},$$6; maximum conversion occurs at the half-wave condition $$M = \begin{pmatrix} m_{11} & 0 \ 0 & R \end{pmatrix},$$7.

In the two-dimensional OAM subspace $$M = \begin{pmatrix} m_{11} & 0 \ 0 & R \end{pmatrix},$$8, the device was designed to convert pure OAM eigenmodes into equal-weight superpositions of opposite-handed eigenmodes and vice versa, thereby simulating, for specific input states, the behavior of a $$M = \begin{pmatrix} m_{11} & 0 \ 0 & R \end{pmatrix},$$9 phase retarder analogous to a quarter-wave plate in polarization space (Slussarenko et al., 2012). Two specific variants were defined: a mode converter for $R\in SO(3)$0, which performs the OAM-basis conversion, and a mode generator for $R\in SO(3)$1, which produces the same final modes directly from a Gaussian input.

The relevance of this earlier work to tunable elliptical retarder arrays is conceptual and technological. Conceptually, it established that a liquid-crystal device with spatially varying anisotropy and electrically tunable retardation can implement retarder-like transformations in a reduced optical state space. Technologically, it showed tunability through liquid-crystal birefringence controlled by an external AC electric field, using nematic liquid crystals, photoalignment with SD1 azo-dye, ITO electrodes, an approximately $R\in SO(3)$2 cell gap, E7 nematic filling, and UV inscription with a $R\in SO(3)$3 He–Cd laser (Slussarenko et al., 2012). This suggests a lineage in which liquid-crystal spatial structuring evolves from mode-specific geometric-phase conversion to pixel-programmable synthesis of arbitrary axis-geometry fields.

6. Limitations, misconceptions, and practical constraints

The most immediate misconception is that the demonstrated platform is already a monolithic true elliptical retarder device. The paper is explicit that the implementation is a compound synthetic platform assembled from cascaded LC-SLM layers rather than a native single-element elliptical retarder (Zhang et al., 22 Aug 2025). This matters for stability, integration, and system complexity.

A second misconception is that topological protection removes all sensitivity to optical or mechanical imperfections. The reported robustness is conditional: thermal drift and mechanical vibrations were present in addition to the injected Gaussian noise, topology is protected only while boundary conditions remain intact, and sufficiently strong perturbations destroy the charge (Zhang et al., 22 Aug 2025). The protected regime is therefore bounded rather than absolute.

A related limitation concerns the distinction between exact unitary retarders and approximate or state-selective analogues. In the earlier OAM-mode-converter work, the quarter-wave-plate analogy is explicitly partial. The device is phase-only and polarization dependent; for selected spin-orbit input states it produces the desired superposition, but for other basis states the output does not remain within the intended two-dimensional OAM subspace (Slussarenko et al., 2012). Because the element is phase-only, the output also acquires higher odd OAM orders, with more than 80% of the power remaining in the target subspace and the rest treated as losses or filtered out.

For the retarder-array platform, broader practical constraints are also stated directly: LC-based systems can be limited in modulation speed; fully rewritable, high-density, and truly compact devices will likely require improved materials and architectures; and spatial confinement and device complexity remain obstacles (Zhang et al., 22 Aug 2025). The current system is therefore best understood as a proof of concept for topological optical information storage and programmable anisotropy engineering, rather than as a finished memory technology.

These limitations do not diminish the conceptual scope of the device class. They instead delimit its present status: a liquid-crystal platform in which local retardance and axis geometry are programmable, recoverable by Mueller polarimetry, and usable as carriers of nontrivial topology, with antecedents in earlier liquid-crystal retarder analogues for structured-light mode control (Zhang et al., 22 Aug 2025, Slussarenko et al., 2012).