Berry and Dennis (2003) gave a complete theory of the polarization singularities of optically anisotropic crystals that are simultaneously birefringent, dichroic (absorbing), and chiral (optically active): singular axes (non-Hermitian degeneracies where the two refractive indices coincide), C points (directions of purely circular polarization, pinned by the 'haunting theorem' as chirality varies), and L lines (directions of purely linear polarization), all displayed as calculable structures in the interference figures of thick crystal slabs. The theory predicts that these effects are extremely delicate — they require optically thick, strongly absorbing crystals, in which the transmitted intensity is reduced by many orders of magnitude — and the paper closed by stating that 'the observation and systematic exploration of these effects is a challenge for experimental optics', while noting that the analysis relied on a simplified model of a 15-real-parameter crystal space. This problem asks for the experimental realization: the direct observation and mapping of the predicted singularity landscape in real birefringent dichroic chiral (and the limiting dichroic non-chiral) crystals, including the delicate thick-crystal interference phenomena, together with an exploration of the full parameter space beyond the local model. No experiment has yet reported these observations, although singular axes have been measured in a weakly absorbing, field-chiral biaxial crystal.

Background

In crystal optics, a homogeneous anisotropic crystal supports, for each propagation direction, two eigenpolarizations with generally different refractive indices. 'Birefringent' means these indices differ with direction (biaxial crystals have two optic axes along which the indices degenerate); 'dichroic' means the crystal absorbs the two eigenpolarizations differently, making the constitutive matrix complex; 'chiral' (optically active, gyrotropic) means the crystal lacks mirror symmetry and rotates the plane of polarization, represented by an optical-activity (gyration) tensor. When all three effects coexist, the governing matrix is neither real, symmetric, nor Hermitian, and the eigenpolarizations are non-orthogonal.

Berry and Dennis (2003) developed a formalism — projection of the sphere of propagation directions onto the stereographic plane — yielding explicit formulas for the refractive indices and polarizations of such crystals, and classified three types of polarization singularity in direction space. First, singular axes: dichroism splits each optic axis into a pair of degeneracies of a non-Hermitian matrix (exceptional points, where eigenvalues and eigenvectors coalesce), which approach and re-coalesce as chirality increases; these act as square-root branch points of the interference fringes. Second, C points: directions where an eigenpolarization is purely circular; each carries a topological index (the number of rotations of the axes of the surrounding polarization ellipses on a closed circuit, divided by 2π2\pi). In absorbing chiral crystals all C points have index +1/2+1/2 and, by the 'haunting theorem', remain fixed in direction space while the singular axes migrate and coalesce as chirality varies; in absorbing non-chiral crystals the C points coincide with the singular axes and have the 'unprecedented' index +1/4+1/4 (a quarter rotation of the crossed pair of ellipse axes). Third, L lines: loci of purely linear polarization separating regions of right- and left-handedness. A local model (linearizing the governing polynomials near an axis) captures the essential behavior. All these structures appear directly in interference figures (conoscopic patterns, obtained by sending a divergent cone of light through a crystal slab between polarizer and analyser) of the slab: the strings of interference zeros emanating from singular axes, the C points appearing as extinguished dark spots among those zeros, and the L lines as dark spots that move as the polarizer is rotated. An early experimental confirmation exists only for the dichroic non-chiral singular axis itself, in an absorbing biaxial mineral (iolite) experiment of the 1950s recounted by Berry and Dennis; the delicate chiral effects have no experimental counterpart. Related theory has since been extended to bianisotropic (magneto-electric) media (Berry 2005), to polarization evolution near a single non-Hermitian degeneracy (Berry 2011), to the splitting of optic-axis cones in gyrotropic crystals with dichroism (Merkulov 2014), and consolidated in a review of exceptional points in anisotropic crystals including interfaces (Sturm 2024). Experimental polarization-singularity optics itself is well developed: spatially resolved Stokes polarimetry (measurement, point by point, of the four Stokes parameters that specify the polarization state) has mapped C lines and L surfaces in transparent, non-chiral birefringent crystals (Flossmann et al. 2005; Schwarz et al. 2009), and polarization singularities near non-Hermitian degeneracies are intensively studied in the momentum space of photonic-crystal slabs (Chen et al. 2021) — but none of these platforms contains dichroism together with natural chirality in a bulk crystal. The full crystal parameter space is 15 real dimensions (the real and imaginary parts of the anisotropy and gyration tensors), with pleochroism (wavelength dependence of the coefficients) providing a physically convenient one-dimensional path through it; Berry and Dennis argued that anisotropic chirality introduces no essentially different phenomena, but this was argued from a simplified model rather than a systematic exploration.

Problem Statement

Observe and systematically explore, in real crystals, the polarization singularities predicted by the theory of birefringent dichroic chiral crystals (Berry and Dennis 2003), and determine whether the simplified model on which the predictions rest captures the full phenomenology. Concretely, the problem comprises two linked parts that together form a single objective, both of which are required — bringing the non-Hermitian singularity structure of absorbing chiral crystal optics under experimental control:

  1. Experimental observation. Using interference figures (conoscopic observation of a crystal slab between polarizer and analyser combinations, or equivalent spatially resolved Stokes polarimetry, i.e. point-by-point measurement of the four Stokes parameters specifying the polarization state), observe in a crystal that is simultaneously biaxial, dichroic, and chiral: (a) the singular axes — the pairs of non-Hermitian degeneracies into which each optic axis splits under dichroism, and their approach, coalescence, and disappearance as chirality is increased (including the helical migration geometry); (b) the C points — the directions of purely circular polarization, fixed by the haunting theorem while the singular axes migrate, with their +1/2+1/2 topological index in the chiral case and, in the dichroic non-chiral limit, their coincidence with the singular axes and 'unprecedented' +1/4+1/4 index (quarter-rotation of the crossed polarization-ellipse pair); (c) the L lines — the loci of purely linear polarization separating regions of opposite polarization handedness, detectable as dark spots moving when the polarizer is rotated. This includes the experimentally delicate thick-crystal regime in which the crystal is optically thick enough that several strings of interference zeros separate a singular axis from the neighboring C points, so that the predicted dark C-point spots are embedded among interference zeros and absorption suppresses the transmitted intensity by many orders of magnitude — the regime the source paper explicitly identified as the challenge for experimental optics.

  2. Systematic exploration of the parameter space. Extend the exploration beyond the local model and the representative two-parameter (absorption A, chirality Γ) plane used in the theory: characterize how the singularity landscape changes across the 15-real-dimensional crystal parameter space (the real and imaginary parts of the anisotropy and gyration tensors, including pleochroism, i.e. the wavelength dependence of these coefficients, and anisotropic — direction-dependent — chirality), and establish whether qualitatively new polarization-singularity phenomena appear beyond those of the simplified model, testing the source's argument that no essentially different phenomena are expected. The exploration must be systematic, and the resulting singularity landscape — including the anisotropic (direction-dependent) chirality — must be argued to be representative of the full 15-dimensional space. The natural completion boundary is an experimentally grounded account of the singularity landscape of these crystals, sufficiently systematic that the predicted phenomena (singular-axis geometry and migration, C-point indices and haunting behavior, L-line structure, and the interference-zero strings of the thick-crystal regime) are directly observed and mapped, together with a determination of whether the local model's phenomenology is generic over the full parameter space; an obstacle-quantification argument is accepted in place of observation only for a specific phenomenon that is shown to be unobservable in principle in every accessible crystal class, and only together with the observational results for the remaining phenomena.

References

  1. M. V. Berry and M. R. Dennis, 'The optical singularities of birefringent dichroic chiral crystals', Proceedings of the Royal Society of London Series A 459, 1261–1292 (2003). DOI: 10.1098/rspa.2003.1155, https://doi.org/10.1098/rspa.2003.1155
  2. F. Flossmann, U. T. Schwarz, M. Maier and M. R. Dennis, 'Polarization Singularities from Unfolding an Optical Vortex through a Birefringent Crystal', Physical Review Letters 95, 253901 (2005). DOI: 10.1103/PhysRevLett.95.253901, https://doi.org/10.1103/PhysRevLett.95.253901
  3. U. T. Schwarz, F. Flossmann and M. R. Dennis, 'Topology of Generic Polarization Singularities in Birefringent Crystals', Topologica 2, 006 (2009). DOI: 10.3731/topologica.2.006, https://doi.org/10.3731/topologica.2.006
  4. C. T. Samlan and Nirmal K. Viswanathan, 'Chiral dynamics of exceptional points in weakly absorbing biaxial crystal', Optics Letters 43, 3538–3541 (2018). DOI: 10.1364/OL.43.003538, https://doi.org/10.1364/OL.43.003538
  5. C. T. Samlan and Nirmal K. Viswanathan, 'Field-controllable Spin-Hall Effect of Light in Optical Crystals: A Conoscopic Mueller Matrix Analysis', Scientific Reports 8 (2018). DOI: 10.1038/s41598-018-20402-4, https://doi.org/10.1038/s41598-018-20402-4
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  9. Chris Sturm, 'Electromagnetic Waves in Crystals: The Presence of Exceptional Points', Advanced Photonics Research 5, 2300235 (2024). DOI: 10.1002/adpr.202300235, https://doi.org/10.1002/adpr.202300235
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  11. Florence Grenapin, Alessio D'Errico and Ebrahim Karimi, 'Spin–orbit coupling induced by ascorbic acid crystals', Nanophotonics 12, 2789–2795 (2023). DOI: 10.1515/nanoph-2022-0502, https://doi.org/10.1515/nanoph-2022-0502
  12. A. Brenier, A. Majchrowski and E. Michalski, 'Chirality versus dichroism: Competition and role in conical diffraction displayed from the Nd:Bi2ZnOB2O6 acentric biaxial laser crystal', Optical Materials 72, 813–820 (2017). DOI: 10.1016/j.optmat.2017.07.032, https://doi.org/10.1016/j.optmat.2017.07.032
  13. Weijin Chen, Qingdong Yang, Yuntian Chen and Wei Liu, 'Evolution and global charge conservation for polarization singularities emerging from non-Hermitian degeneracies', Proceedings of the National Academy of Sciences 118, e2019578118 (2021). DOI: 10.1073/pnas.2019578118, https://doi.org/10.1073/pnas.2019578118
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