Terahertz orbital angular momentum modes with flexible twisted hollow core antiresonant fiber

THz radiation is more and more commonplace in research laboratories as well as in everyday life, with applications ranging from body scanners at airport security to short range wireless communications. In the optical domain, waveguides and other devices to manipulate radiation are well established. This is not yet the case in the THz regime because of the strong interaction of THz radiation with matter, leading to absorption, and the millimeter size of the wavelength and therefore of the required waveguides. We propose the use of a new material, polyurethane, for waveguides that allows high flexibility, overcoming the problem that large sizes otherwise result in rigid structures. With this material we realize antiresonant hollow-core waveguides and we use the flexibility of the material to mechanically twist the waveguide in a tunable and reversible manner, with twist periods as short as tens of wavelengths. Twisting the waveguide, we demonstrate the generation of modes carrying orbital angular momentum. We use THz time domain spectroscopy to measure and clearly visualize the vortex nature of the mode, which is difficult in the optical domain. The proposed waveguide is a new platform offering new perspectives for THz guidance and particularly mode manipulation. The demonstrated ability to generate modes with orbital angular momentum within a waveguide, in a controllable manner, will be beneficial to both fundamental, e.g. matter-radiation interaction, and applied, e.g. THz telecommunications, advances of THz research and technology. Moreover, this platform is not limited to the THz domain and could be scaled for other electromagnetic wavelengths.


INTRODUCTION
The rapidly increasing availability of sources and detectors has brought the field of THz radiation, also known as millimeter waves, from an interesting physics regime to the real world, with applications in security, telecommunications, characterization of biological and solid-state matter, and many others. 1,2 These various applications require, on top of generating and detecting the desired radiation, means of transporting and manipulating the fields.
In the context of guiding THz radiation, there has been a large effort in realizing waveguides which are at the same time small, flexible and low loss. The size of the waveguides is constrained by the wavelength to be guided and therefore THz waveguides must have millimeter to centimeter sized cross sections, which makes them very rigid. Furthermore, the material providing the confinement strongly affects the loss of the waveguides; unlike at optical wavelengths, there is no material with very low loss in the THz region, so that the use of hollow waveguides is necessary. A variety of types of hollow waveguides have been studied for THz, which may be separated into metallic, dielectric and hybrid waveguides (including metamaterial waveguides). 3,4,5,6,7,8,9 Metallic hollow waveguides can have subwavelength cross-section and have relatively low loss. Those characteristics arguably make them the current best option for THz waveguides. Metamaterial hollow waveguides offer the potential for control of the radiation, 5 can be subwavelength 6 and potentially low loss. 7 However, both complexity of fabrication and the relative lack of research in this field leaves doubts about their real potential. Dielectric THz hollow waveguides are the scaled version of their more successful optical counterpart and have been investigated in the form of Kagome structures, 10,11 simple capillaries, 12 antiresonant structures, 13 and several other variations. 8,9 Although some of these have the possibility of achieving excellent guidance, the relationship between loss and size has meant that realizations of this approach are large rigid structures that were of little practical use. Leaving aside the rigidity issue temporarily, there is one class of hollow dielectric waveguides that deserves more detailed consideration, mostly because of the greatly expanding recent interest and results in the optical domain: antiresonant fibers.
Antiresonant fibers 14 have been, in the last five years, dominating the panorama of hollow core fibers. The reason is that a very simple structure, i.e. a circular array of capillaries, allows guidance of very large bandwidths with very low loss almost independently of the absorption of the material used for the structure. Moreover, by tuning the few parameters of this structure it is possible to obtain single mode operation in large cores 15 and to tune the 3 bending loss of the structure. 16 In the context of THz, the realization of such fibers allows reducing the overall structure size compared to bandgap or Kagome structures, controlling the modal properties and further reducing the waveguide loss with any material. Although interesting, this is not sufficient unless the waveguide can be made usable in practical applications. The enabling factor behind this work is the use of a novel material for waveguides, namely polyurethane, the Young's modulus of which is 2 to 3 orders of magnitude lower than conventional waveguide dielectrics and allows even centimeter sized tubes, rods and structures to be bent with radius 10 times the diameter. This means a 1 cm diameter fiber can be bent in a 10 cm radius circle (an example is shown in Fig. S1 of the supplementary material). This unique property allows the fabrication of waveguides with large cross sections that are still flexible, and allows for further mechanical manipulations of the fibers. The specific manipulation we will exploit in this paper is twisting of the structure in a controlled and reversible way.
The idea of twisting fiber structures is very interesting because it enables creating modes with an orbital angular momentum (OAM) 17,18,19 and controlling the state of polarization and the fiber's optical activity. 20,21 It should be kept in mind that in order to obtain any effect, the fiber to be twisted cannot be cylindrically symmetric. There have been interesting recent advances in this research by twisting photonic crystal fibers (PCFs). So far, there have been two ways to give PCFs a twist: by post processing with a CO2 laser 17,18 or by twisting the preform during fiber drawing. 18,22,23 Both methods result in a permanent twist of the fiber and the twist occurs at high temperatures where the material viscosity is low. When in a solid state, mechanical, and therefore reversible, twist of silica fibers cannot be used to produce twist rates high enough to achieve the desired properties, 17,18,19,20,21,22,23 because of the inherent material stiffness. Moreover, even when fabricated with a permanent deformation, these twisted structures have been realized only in the optical domain and only theoretically suggested in the THz, once more because of the combination of size and necessary deformation required to achieve structures showing interesting properties.
The interest behind being able to realize modes that have orbital angular momentum derives from the plethora of applications this class of modes has found in 25 years of investigation, 24,25,26,27 including but not limited to, imaging and microscopy, 28 quantum photonics and entangled states, 29 optical tweezers, 30 and optical communications. 31 Modes carry orbital angular momentum when their phase structure is rotating. Assuming an time dependence, they have an optical vortex along their axis and an azimuthal phase dependence of the form − ℓ , with ℓ being a positive or negative integer and the angular coordinate, and they carry a momentum of ℓℏ per photon. The helical phase front and the azimuthal 2πℓ phase change require a phase singularity in the middle 4 which leads to a central intensity null. 27 Because of the large interest involved, there have been many proposals on how to generate OAM modes, including phase plates, 32 diffractive elements using holograms 33 or spatial light modulators, 34 mode conversion with cylindrical lenses, 24 metasurfaces, 35,36 microbend gratings in fibers 37,38 and twisted PCFs. 18 The generation of OAM for THz radiation has been done similarly, by using some of the techniques also used for optical beams. 39,40,41,42,43 However, the ability to generate OAM within a waveguide in a tunable manner is very appealing both in the optical and especially in the THz domain where this combination was not possible before.
In this paper we report the realization of a simple, hollow core antiresonant waveguide, which is flexible and opens up a new class of THz waveguides. The flexibility of the waveguide is used to twist the fiber along its axis and to create THz vortex modes possessing orbital angular momentum, as represented in Fig

FLEXIBLE HOLLOW CORE WAVEGUIDE
Polyurethane (PU) has a Young's modulus orders of magnitude lower compared to silica and can withstand elongations of 600%, 44 making it effectively a rubber-like elastic material. The ability to fiber-draw PU and the possibilities to apply its property to elastically deform have been recently demonstrated in the realization of a tunable metamaterial. 45 These mechanical properties make PU a perfect candidate for realizing a structure that can be twisted mechanically. The choice of a hollow core structure is due to both the poor optical properties of polyurethane and to the potential of this structure to be scaled at any wavelength independently from the material properties, in particular loss. As already mentioned, of the class of hollow core structures, the antiresonant tube lattice structure is the simplest and the most obvious choice. The proposed fiber is realized by six PU tubes arranged 5 in a circle, as shown in the inset Fig. 2(a) and the schematics in Figs. 1 and 3(a). The tubes are commercially available with large outer sizes and are fiber-drawn to about 3 mm in outer diameter. The fiber structure is held together by gluing the tubes to a plastic disc on each end while the middle is free standing. The thickness of the disc is 5 mm and a hole was drilled to size to accommodate the tube lattice structure. While such a structure can be drawn to smaller dimensions if necessary, the structure was suitable without the necessity of further scaling for testing the principle in the THz regime. The resulting fiber core has a 3 mm diameter, and the capillaries have thickness of about =400 µm. Using phase resolved propagation measurements in 3 and 5 mm PU samples we calculated a refractive index of =1.6 at 0.3 THz. This yields resonances at frequency multiples of =300 GHz, calculated by using the analytical expression for antiresonant fibers with an air-core: 14 where is the speed of light.
The fiber was characterized in a THz time domain spectroscopy system. 2  when comparing the transmission between the various sample lengths, the coupling efficiency played a significant role and a comparison between them leads to errors larger than the actual values. This is also due to the fiber being multimoded in a good part of the transmission window. As a reference, we measured ~10 dB loss including coupling and propagation around 0.8 THz for the 10 cm fiber. Simulations of the structure using a commercial finite element solver (COMSOL) show minimum propagation loss approaching 0.1 dB/m for the fundamental mode ( Fig. 2(b)).
For information about the loss of high order modes see the relevant section in the supplementary material. The simulated structure has core diameter of 3.2 mm and capillaries with 3 mm diameter, 400 µm wall thickness and refractive index of 1.6. The loss of PU is set to a constant value of 1 dB/cm for all frequencies which is the measured value of the bulk material at 0.5 THz. While such low simulated losses demonstrate the potential of this fiber, they are only found over a relatively narrow band and assume a perfect structure. It is instead reasonable to consider the 6 lowest loss to be around 1-2 dB/m for the fundamental mode in the transmission bands of interest to this investigation. Note that loss could be further reduced with thinner and more widely spaced capillaries. Effective single mode behavior could be achieved by carefully choosing the ratio of the capillaries and core diameter 15 or by twisting the structure. 23 In order to measure the modal content, near-field raster scan measurements were performed at the output end of the waveguide. The great advantage of THz TDS is that it allows us to acquire information on the intensity, electric field and phase of the modes and also it allows analyzing their time evolution. With this technique, the amplitude of the THz electric field (not its envelope or intensity) is directly measured as a function of the time delay between the pump and probe pulses generating and detecting the broadband THz pulses. Therefore, the time delay allows sampling of the THz electric field in time. The measured amplitude of the electric field can be mathematically processed with a Fourier transformation to obtain the complex electric field as a function of frequency. Accessing the complex field is necessary for obtaining full information about modes/fields where the phase of the electric field is a characteristic feature, as it is for modes carrying OAM. A schematic of the measurement system is shown in

DISCUSSION -NUMERICAL INVESTIGATION
The field generated and measured clearly has OAM, but the appearance of it is not that of a clean integer OAM mode, for which the phase singularity would be centered, the intensity profile symmetric, and the phase profile radially uniform.
In order to better understand the process of OAM mode generation in this fiber, finite elements numerical simulations using a helical coordinate system transformation 17,18,21 were performed. In all the simulations reported in the paper a mode is considered to be guided in the core when more than 50% of the square of the electric field is contained in the circle inscribed within the capillaries. In the lower frequency transmission bands (0.2 and 0.5 THz) the fiber only supports one core mode, which does not have OAM in the twist rates range used in the experiments.
Higher twist allows modes carrying OAM to be supported, but are quite lossy. This explains why we do not measure OAM modes in the lower bands. In the 0.6-0.9 THz transmission band, the straight fiber supports multiple guided modes, which allows guidance of OAM modes if properly excited, as also previously numerically reported for a similar structure at a similar frequency. 46 When twisted with 10 cm period, the fiber still supports the fundamental mode, although the degeneracy in polarization is lifted, as well as other modes all carrying OAM. It should be remembered that in circularly symmetric optical fibers there are 4 possible modes that can carry OAM for ℓ=1, consisting of the LP11 manifold: TE01, TM01, HE21 odd , HE21 even . 38 Modes with similar electric field distributions to all those modes are supported in the fiber. Figure 8 shows effective refractive indices of the modes in the fiber at 0.72 THz as a function of twist and some of the modes' profiles and phase. For information on the propagation loss of these modes see the relevant section in the supplementary material. Applying twist removes the degeneracy in the fundamental mode and also in the LP11-like modes. The LP11 manifold mode splits in 4 non-degenerate modes with a similar distribution to the ones carrying OAM in a standard optical fiber. It is, however, the hybrid (HE21 like) modes that carry OAM in this case and the two hybrid modes have opposite rotation direction. At even higher twist rates, some of the modes became very lossy and disappear, and modes with higher order OAM are guided. It should be noted that apparent missing data points arise from coupling of the modes with modes propagating in the outer tubes and therefore having less than 50% of the field square in the core. At a twist period of 10 cm, the one used in the experiments, there are 3 modes guided: the ℓ=0, and the hybrid modes with ℓ=1 and ℓ=2. Moreover, at about this twist rate, the fundamental mode and the OAM mode of order 1 have a refractive index crossing, allowing for efficient coupling from one mode to the other. With an effective index differences in the order of 10 -3 in this frequency band, the coupling length between two modes is in the order of 10 cm, even if very little perturbation is considered. This is consistent with the experiment. Moreover, the structure of the fiber and the twist are not perfect, therefore increasing the coupling between modes. It should be noted that, for these conditions, the ℓ=1 mode is also the mode with the lowest loss (see also Fig. S3 (2) where ( ) = (0)[( 2 + 2 )/ 2 ] 1/2 , (0) is the beam waist, the Rayleigh range, the radial order (number of radial nodes in the intensity distribution, set to 0 in our calculation), = 2 ⁄ the wave vector, the wavelength, = 2 ⁄ the wave angular frequency, the speed of light, 0 the arrival time of the maximum of the pulse, and Δ the full width half maximum pulse duration. It should be noted that the minus sign in − ( − +ℓ ) is used for consistency with the mathematical formalism used in the data analysis and numerical simulations. Figure 9 shows the result of adding the ℓ=0 and ℓ=1 modes with a one to one contribution in electric field, as defined in Eq. (2), which gives a very close match to our measurement. Although the result obtained by summing the numerically calculated modes ( Fig. 9(a)) is, as expected, more similar to the measurements than the analytical model ( Fig. 9(b)), the two results are quite close, indicating that Laguerre-Gaussian modes are a good and mathematically simpler approximation for the system. Confirming the presence of the two modes does not explain whether the OAM carrying mode is generated by directional coupling or already excited at the fiber input. We calculated the mode overlap integral between the excitation and the two modes (for details see the relevant section of the supplementary material) and confirmed that it is unlikely that the ℓ=1 mode could be sufficiently excited at the input in our experiment. This, along with the modal crossing at the measured twist rate, suggests that the OAM mode is indeed the result of mode conversion in the fiber. The time evolution of the electric field component measured (Fig. 6) provides some additional clues as to the modal composition of the fields: positive and negative lobes are spiraling, and visible along the entire spiral.
Because of the single polarization measurement, this hints toward the measurement of a field that is not purely radially or azimuthally polarized, but hybrid, which is in agreement with the simulation. Measurement of the complementary linear polarization could help elucidate the exact modal decomposition further.

CONCLUSION
In conclusion, we have reported a new flexible hollow waveguide for the THz regime. The waveguide flexibility allowed mechanical twisting to generate modes with OAM in a broad bandwidth (0.6-0.9 THz), starting from a simple Gaussian beam with linear polarization. The extreme flexibility allows to explore twist periods of order several tens of wavelengths, which is difficult in other implementations. The unique features of THz TDS allowed us to measure and therefore visualize the vortex nature of OAM of light. This innovative platform has the potential for changing the panorama of THz waveguides and allowing for a wide range of mode manipulation options within the waveguide. In addition to providing new perspectives for delivery of THz radiation, for generation of OAM or both, the results here reported give access to new physics in the field of twisted fibers by using THz TDS to obtain information on the the interaction of radiation and the structure. Examples are: measuring the split in degeneracy 15 between left and right OAM modes; and measuring phase and group velocity by looking at the velocity of rotation of the electric field and the intensity of the mode, respectively.
Preliminary simulations show that separating the capillaries and therefore the index of the core mode and the capillaries' modes, will result in cleaner and lower loss OAM modes. Some future investigation and applications of this technology include manipulating the polarization: from simple rotation to changing the polarization state; generation of higher order OAM, by higher twist rates or by slightly changing the parameters of the structure.
Indeed the trasmission bands where the untwisted fiber supports only the fundamental mode, with a certain twist rate start supporting high order modes with OAM, the order of which is determined by the structure.

SUPPLEMETARY MATERIAL
See supplementary materials for information about: "fiber flexibility", "transmission of the twisted fiber", "loss of the high order modes" and "mode overlap integral and coupling". The data used for the results here reported are openly available with DOI: 10.5281/zenodo.1164309.