Guiding light via geometric phases

Journal name:
Nature Photonics
Year published:
Published online

All known methods for transverse confinement and guidance of light rely on modification of the refractive index, that is, on the scalar properties of electromagnetic radiation1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. Here, we disclose the concept of a dielectric waveguide that exploits vectorial spin–orbit interactions of light and the resulting geometric phases12, 13, 14, 15, 16, 17. The approach relies on the use of anisotropic media with an optic axis that lies orthogonal to the propagation direction but is spatially modulated, so that the refractive index remains constant everywhere. A spin-controlled cumulative phase distortion is imposed on the beam, balancing diffraction for a specific polarization. As well as theoretical analysis, we present an experimental demonstration of the guidance using a series of discrete geometric-phase lenses made from liquid crystal. Our findings show that geometric phases may determine the optical guiding behaviour well beyond a Rayleigh length, paving the way to a new class of photonic devices. The concept is applicable to the whole electromagnetic spectrum.

At a glance


  1. Geometric phase.
    Figure 1: Geometric phase.

    a, Reference xyz system with orientation of the optic axis u and corresponding ordinary/extraordinary (o/e) field directions; the angle θ between u and the y axis varies from point to point. b, Geometric phase acquired by a plane wave, CP at the input, propagating along z in a transversely homogeneous medium with θ = π/4 as a function of birefringence retardation δ(z), relative to the case with θ = 0. The geometric phase sign is fixed by the CP input handedness (blue and red lines). If θ is uniform along z (dashed lines), the geometric phase reaches a maximum (in the example π/2) when δ = π and then decreases to zero for δ = 2π. If angle θ is suddenly inverted at δ = π (solid lines), the phase grows monotonically. If θ is sinusoidally modulated along z (solid lines with circles), the phase increases monotonically at a slightly lower rate than in the previous case.

  2. Concept of the geometric-phase waveguide.
    Figure 2: Concept of the geometric-phase waveguide.

    a, Three-dimensional illustration of a continuously modulated geometric-phase waveguide. The orientation of the optic axis is longitudinally sinusoidal and transversely Gaussian. We sketch nine sections within a modulation period, with the black rods representing the optic axis and the colours corresponding to θ; the guided light beam is represented as a red arrow. b, Geometric-phase accumulation across the beam profile versus propagation in the plane-wave limit (that is, without diffraction), corresponding to a (blue solid line) and in the limit of an optic axis that is unmodulated along z (red dashed line). Here the maximum θ is π/4.

  3. Theory and simulations.
    Figure 3: Theory and simulations.

    a,b, Effective photonic potential versus x and maximum θ angle Γ0 (we assumed a Gaussian distribution for the transverse orientation by setting Γ = Γ0exp(−x2/wD2)) perceived by the defocused (a; LCP input) and confined (b; RCP input) waves, respectively. The terms proportional to Γ2 are taken into account (see Methods). c, Corresponding fundamental guided mode: field amplitude versus x and Γ0. df, FDTD simulations for Γ0 = 15° when the input beam is LCP (d) and RCP (e). The colour scale gives the local light intensity, and the red lines indicate the beam radius evolution for a homogeneous medium, that is, for ordinary diffraction. In f, the evolution of the confined beam polarization state within a modulation period is shown. Here λ = 1 μm, no = 1.5, ne = 1.7, σ(z) is sinusoidal and the transverse distribution has wD = 5 μm.

  4. Apparatus.
    Figure 4: Apparatus.

    a, Experimental set-up. Five equally spaced electrically tuned GPLs form a discrete-element geometric-phase waveguide. A 532 nm continuous-wave Gaussian beam is adjusted in transverse size with a telescope to match the fundamental mode of the waveguide. It is then circularly polarized with a quarter-wave plate (QWP) and launched into the waveguide. Beam profiles at various intermediate positions z along the propagation and at the output were acquired by a movable CCD camera and used to reconstruct the mode parameters of the beam after each GPL. b, Distribution of the optic axis and corresponding photograph of a GPL imaged between crossed polarizers; dark fringes correspond to regions where the optic axis is aligned parallel to one of the polarizers.

  5. Experiment.
    Figure 5: Experiment.

    ac, Data-reconstructed beam evolution for the following cases: guided mode, obtained for RCP input (a); divergent beam, obtained for LCP input (b); free-space diffracting beam for the same input parameters (c). The colour scale gives the local light intensity. Horizontal lines and L1–L5 labels indicate the GPL positions within the discrete sequence (dashed lines mark removed GPLs). d, Beam radius versus z in the guided case. Dots are measurement data, blue solid lines are the corresponding Gaussian-beam fits between subsequent GPLs, blue shaded areas indicate confidence regions at one standard deviation, the black dashed line is the theoretical prediction from the ABCD method, accounting for the Gaussian-beam imperfections (quantified by the M2 parameter), and the dashed green line corresponds to the ideal Gaussian beam case, with M2 = 1. Vertical dashed lines mark the GPL positions. e, Measured light intensity profiles (I versus x,y) at the input plane of each GPL for the guided case. Scale bar, 400 μm.


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Author information

  1. These authors contributed equally to this work

    • Sergei Slussarenko &
    • Alessandro Alberucci


  1. Dipartimento di Fisica, Università di Napoli Federico II, Complesso Universitario di Monte S. Angelo, via Cintia, 80126 Napoli, Italy

    • Sergei Slussarenko,
    • Bruno Piccirillo,
    • Enrico Santamato &
    • Lorenzo Marrucci
  2. Centre for Quantum Dynamics and Centre for Quantum Computation and Communication Technology, Griffith University, Brisbane, Queensland 4111, Australia

    • Sergei Slussarenko
  3. Nonlinear Optics and OptoElectronics Lab, University Roma Tre, I-00146 Rome, Italy

    • Alessandro Alberucci &
    • Gaetano Assanto
  4. Optics Lab, Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland

    • Alessandro Alberucci &
    • Gaetano Assanto
  5. Centro de Física do Porto, Faculdade de Ciências, Universidade do Porto, 4169-007 Porto, Portugal

    • Chandroth P. Jisha
  6. Consiglio Nazionale delle Ricerche, Institute for Complex Systems (ISC), Via dei Taurini 19, 00185 Rome, Italy

    • Gaetano Assanto
  7. Consiglio Nazionale delle Ricerche, Institute of Applied Science & Intelligent Systems (ISASI), Via Campi Flegrei 34, 80078 Pozzuoli (NA), Italy

    • Lorenzo Marrucci


This work was jointly conceived by A.A., C.P.J., G.A. and L.M. S.S. designed and carried out the experiment, with the help and supervision of B.P., E.S. and L.M. A.A. and C.P.J. developed the theory and performed the numerical simulations, with the help and supervision of L.M. and G.A. All authors discussed the results and contributed to the manuscript.

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