Optical framed knots as information carriers

The study explores how three-dimensional topological features in structured light—specifically knots formed in polarization fields—can move beyond fundamental demonstrations to practical information encoding. Prior work in singular optics created diverse topologies (Möbius strips, ribbons, knots) but typically treated them as 2D transverse modes rather than 3D objects characterized by robust topological invariants. This limits their integration with topological quantum information methods that rely on braid representations. The research question is whether optical polarization fields can host framed knots whose topological invariants (e.g., twisting of a framing field) can be measured and used as robust information carriers. The study introduces a construct of framed knots in optical polarization fields, demonstrates their experimental generation and characterization, and devises a protocol leveraging braid/framing invariants and prime factorization to encode and decode information.

The paper situates its contribution within singular optics and topological photonics. It references generation and applications of optical beams with singularities, including orbital angular momentum modes in classical and quantum communications. It reviews prior demonstrations of knotted structures: intensity nulls in scalar optical fields, bichromatic fields, electromagnetic field lines, and polarization singularities (C-lines). It highlights braid representations as foundational in topological quantum information and notes prior optical knots have been employed analogously to simple singular beams rather than as full 3D topological objects. It also cites mathematical frameworks for knotted fields and stereographic projections used to convert braids to knots, and techniques for polarization tomography and holographic field synthesis.

Concept and definition: The authors define framed knots in optical polarization fields via C-lines—curves of pure circular polarization in monochromatic fields. Around a C-line, surrounding polarization ellipses exhibit a rotation by integer multiples of π along closed contours. The framing vector field is defined by the axis of the adjacent polarization ellipse that is perpendicular to the C-line’s tangent, ensuring a framing integer equal to the number of 2π twists around the knot. In cases of ambiguity, the framing is chosen to maintain continuity with minimal twisting. Field construction: A knotted optical field E^k is constructed with a circularly polarized component possessing knotted phase singularities and an accompanying longitudinal component ensuring compliance with Maxwell’s equations. Superposition with an oppositely helical plane wave shapes the resulting C-line into the knot determined by the singularities of E^k. In paraxial beams (the experimental focus), the longitudinal component is negligible, so the C-line aligns with the knotted vortices regardless of plane-wave amplitude. Braid representation and projection: Knots are related to braids via Alexander’s theorem. The authors represent braids as zeros of a complex field in coordinates (u,v), where u = x + i y and v = exp(i h). A stereographic projection maps the braid in (x,y,h) to a knot in cylindrical coordinates (ρ, φ, z) using a standard mapping (u and v expressed rationally in ρ and z), effectively connecting the braid ends and embedding the knot on a torus. This projection underpins prior optical knot constructions and is used here to align properties of framed knots with their braid representations. Unwrapping and twisting extraction: A coordinate transformation cuts the knot along a chosen azimuth and unwraps it, mapping φ to h while preserving the local orientation of the knot’s frame. In the unwrapped representation, each strand’s twisting angle (azimuthal orientation of the ribbon in the frame aligned to the unwrapped tangent) is extracted along the strand, enabling determination of the number of half-twists per strand. Prime encoding scheme: Using the strands of a framed braid, the number of half-twists per strand d_k is assigned to strands that exhibit twists, each with a designated prime p_k. The total number of half-twists M_A = Σ_k d_k is a topological invariant of the framed knot/knotted ribbon. The scheme defines β(α) from the braid’s structure and constructs N_{αβ}(M_A) = Π_k p_k^{d_k}, so that prime factorization of N_{αβ} recovers the twist counts d_k. Protocol: Alice selects α, determines the framed braid representation of her knotted ribbon K_A, allocates d_k among strands (respecting M_A), assigns primes p_k, computes β, and sends K_A and (α, β) to Bob. Bob measures the knot, extracts M_A and the strand twist data via unwrapping, computes N_{αβ}(M_A), and factors it to recover d_k according to a pre-agreed strand-to-prime convention. Experimental generation and measurement: The team implements paraxial-knotted C-lines using a folded Sagnac interferometer. An 810-nm laser is split into orthogonally polarized components, each modulated by a phase-only spatial light modulator that encodes both amplitude and phase. One component is shaped into a knotted vortex field (right-handed circular component), while the other forms a broad Gaussian (left-handed component), effectively acting as the plane-wave perturbation. After coherent recombination, the resulting field contains the framed knotted C-line. The knot and its frame are reconstructed via polarization tomography using quarter- and half-wave plates, a polarizing beam splitter, and a CMOS camera. Specific holograms generate fields for trefoil and cinquefoil knots using analytically designed scalar fields Ψ(ρ, φ) derived from stereographic-projection methods, with Gaussian apodization and chosen parameters (trefoil: a = 1, b = 0.5, s = 1.2; cinquefoil: a = 0.5, b = 0.24, s = 0.65). The measured knots are then unwrapped via the coordinate transformation to extract twisting angles per strand and compared with theoretical predictions.

• Defined and realized framed knots in optical polarization fields by assigning a physically measurable framing vector to knotted C-lines and linking its total twist to a topological invariant.
• Developed a prime-encoding protocol that maps the number of half-twists per braid strand to the prime factorization of an integer N_{αβ}, enabling information encoding in the topological structure of framed braids/knots. An illustrative three-strand example yields N = 518,400 with factorization corresponding to twist counts across strands.
• Experimentally generated paraxial-framed trefoil and cinquefoil knots using a folded Sagnac interferometer and SLM holography. Theoretical field designs used parameters: trefoil (a = 1, b = 0.5, s = 1.2) and cinquefoil (a = 0.5, b = 0.24, s = 0.65).
• Polarization tomography reconstructed the knotted C-lines and their frames. Measured structures showed good qualitative agreement with theory, with minor perturbations near knot extremities (birth/annihilation points of C-lines).
• Unwrapped representations allowed extraction of twisting angles along strands; both strands exhibited the same total number of half-twists, matching theoretical expectations. While individual strand twists in the unwrapped view may deviate from exact integer multiples of π due to the cut location, the summed twisting over strands matches integer multiples as required by the closed topology.
• Demonstrated a practical pathway to encode and decode topologically protected information using optical-framed knots combined with polarization tomography and coordinate unwrapping.

The results address the central goal of harnessing 3D topological invariants in structured light for information encoding. By defining a measurable framing for C-line knots and linking strand-wise half-twist counts to prime factorization, the work bridges singular optics with braid-based notions central to topological information processing. The experimental trefoil and cinquefoil demonstrations validate that the framing structure can be generated, measured, and mapped to braid parameters despite practical imperfections, supporting the feasibility of robust, topology-assisted encoding. The encoding scheme leverages the robustness of topological invariants (total twist M_A and the framed braid structure) against continuous perturbations, offering stability advantages. The discussion outlines extensions to non-paraxial regimes using tight focusing to strengthen longitudinal polarization control and reduce longitudinal extent, potentially enabling compact, wavelength-scale knotted structures with improved framing control. Practically, a transmission and decoding pipeline is described: encode via interferometric holography, transport the field, and decode via polarization tomography and unwrapping. While single-plane reconstructions are conceivable, aberrations and setup imperfections motivate the more direct tomographic approach adopted here.

The paper introduces a construct for framed knots in optical polarization fields based on knotted C-lines, and demonstrates their experimental generation, characterization, and use as information carriers via a prime factorization-based encoding scheme. It shows how measurable framing twist can be extracted and related to braid representations, enabling topologically protected encoding and decoding. Future work includes extending to non-paraxial optical knots for enhanced longitudinal control and compactness, developing more sophisticated polarization tomography for tightly focused beams, and exploring applications in secure classical/quantum communications and topological information processing leveraging braid-based protocols.

• The demonstrations are in the paraxial regime; non-paraxial implementations (which could offer stronger longitudinal control and shorter axial extent) remain future work.
• Experimental imperfections (e.g., optical aberrations and interferometer-induced errors) cause minor perturbations, especially near knot extremities, and complicate single-plane reconstructions of the full 3D field.
• The unwrapping procedure can yield non-integer half-twist counts per strand due to the cut position; only the total twist across strands is strictly quantized without further conventions.
• Decoding requires full polarization tomography over multiple planes and a pre-agreed convention linking primes to braid strands, adding complexity and coordination.
• Longitudinal extent of paraxial knots can be large, making compact integration more challenging compared to tightly focused non-paraxial implementations.