### Helical Bloch waves

Understanding the physics of light propagation in t-PCF is quite challenging, because the natural coordinate system—helicoidal—is non-orthogonal. This led us to introduce a new concept: helical Bloch waves. The optical Bloch waves of any untwisted periodic structure are mathematically described by the product of a periodic function (with periodicities that match the structure) and a term representing the phase progression of the Bloch wave **[Russell (1986)]**. A convenient physical picture for the modes guided in a t-PCF can be constructed by generalising Bloch's theorem so that the azimuthally periodic function follows the twist. At any given value of *z*, the periodic function will repeat at angular intervals 360°/*N*, where *N* is the number of times the structure repeats over one complete revolution. The Bloch waves can then be calculated analytically using an expansion in terms of azimuthal harmonics of OAM order. Substituting this field Ansatz into Maxwell's equations allows the dispersion relation to be derived and the properties of the modes to be investigated **[Xi (2014)]**. To explore the properties of helical Bloch waves, we fabricated a t-PCF with a ring of six solid glass "satellite" cores around its axis. This structure supports 6 non-degenerate helical Bloch modes with different values of orbital angular momentum, in both left and right circularly polarised states **[Russell (2017)****]**. To determine the OAM order of the modes guided through the t-PCF, the output was superimposed on to a divergent Gaussian beam and the resulting fringe pattern imaged using a CCD camera. The single- and double-spiral interference patterns below confirm that the fibre generates optical vortices and preserves the magnitude and sign of the OAM for all four modes. Similar experiments carried out at multiple wavelengths and for fibres up to 50 m long have confirmed that the t-PCFs preserve the magnitude and sign of the OAM.