## Abstract

Spiral twisting offers additional opportunities for controlling the loss, dispersion, and polarization state of light in optical fibers with noncircular guiding cores. Here, we report an effect that appears in continuously twisted photonic crystal fiber. Guided by the helical lattice of hollow channels, cladding light is forced to follow a spiral path. This diverts a fraction of the axial momentum flow into the azimuthal direction, leading to the formation of discrete orbital angular momentum states at wavelengths that scale linearly with the twist rate. Core-guided light phase-matches topologically to these leaky states, causing a series of dips in the transmitted spectrum. Twisted photonic crystal fiber has potential applications in, for example, band-rejection filters and dispersion control.

The effect of twisting on the propagation of light in different kinds of optical fibers has been explored for polarization control (*1*–*3*), long-period grating couplers (*4*–*7*), and elimination of higher-order modes from fiber lasers (*8*). We study twisted solid-core photonic crystal fiber (PCF), a type of microstructured fiber in which a regular hexagonal lattice of hollow channels is disposed symmetrically around a central glass core (*9*). When continuously twisted, PCF exhibits a series of dips in its transmission spectrum, an effect that has been previously attributed to some form of grating-diffraction (*10*, *11*). We show that it can be explained by the formation in the cladding of leaky orbital resonances that couple to the core mode, causing transmission loss.

The helical fibers (Fig. 1) were fabricated by rigidly fixing one end of a length of solid-core PCF while mounting the other end at the center of a motorized rotation stage. The PCF had a hexagonal array of hollow channels of diameter *d* ~0.9 μm and spacing Λ ≈ 2.9 μm, yielding *d*/Λ < 0.4, which makes it “endlessly single-mode”; that is, it supports only one transverse mode at all wavelengths (*12*). A scanning electron micrograph of its microstructure is shown in the Fig. 2A inset. A permanent twist was produced by rotating the stage while scanning a focused CO_{2} laser beam along the fiber with a steering mirror fixed to a precision motorized linear translation stage. The laser power was chosen to heat the fiber to the glass-softening temperature.

Linearly polarized light from a supercontinuum source was coupled into the samples, and the transmitted power spectra were measured by using an optical spectrum analyzer. The results for two 6-mm-long pieces of PCF with twist rates of α = 10.8 and 13.6 rad/mm (helical pitch *L* = 581 and 461 μm) show a pattern of distinct transmission dips (Fig. 2) that shifts toward longer wavelength at higher twist rates. The data points from a series of measurements at different twist rates are plotted in Fig. 3A for the four strongest resonances in Fig. 2. Despite such a complex and dispersive system, the dip wavelengths scale linearly with twist rate, each orbital resonance having a different slope, and the lines pass through zero wavelength at zero twist rate.

A comparison of the experimental results with transmission spectra modeled by using a helicoidal coordinate transformation (supplementary text section S1) shows excellent agreement (Fig. 2) considering that no fitting parameters were used. Almost-identical numerical spectra are obtained for right-circular (RC) and left-circular (LC) polarization states (we define LC as spinning in the same direction as the spiral, which is taken to be clockwise looking in the direction of propagation). In Fig. 3, C to E, the axial (*S _{z}*) and azimuthal (

*S*

_{az}) Poynting vector distributions are plotted for the four modes. They are ring-shaped,

*S*

_{az}flowing in the direction of the spiral in each case.

*S*

_{az}has the same sign in both core and cladding for the RC-polarized case, whereas the signs are opposite for the LC case. In Fig. 3F, the polarization states in the vicinity of the core are shown for resonance a in the RC case and for resonances b and d in the LC case (α = 13.6 rad/mm); they are almost perfectly circular in both core and cladding in all cases (for more details, see supplementary text section S2).

The hexagonal cladding structure in the untwisted PCF cladding supports a fundamental “space-filling” mode (SM) whose axial Poynting vector, *S*_{SM}, points precisely along the fiber axis (*12*). This Bloch mode, which occurs at the Γ point at the center of the Brillouin zone, has an effective axial refractive index that is greater than 1, with the result that field antinodes form in the glass strands, trapped laterally by total internal reflection at the glass-air interfaces. When the fiber is gently twisted (αΛ << 1), these field antinodes are forced to follow a helical path around the core. This permits *S*_{SM} to point partially into the transverse plane of the fiber, the local angle between the fiber axis and the helical path being ϕ = sin^{−1}[αρ/(1 + α^{2}ρ^{2})^{0.5}] ≈ αρ at radius ρ. This leads to a component of momentum flow in the azimuthal direction (Fig. 1B), creating orbital angular momentum (OAM) and causing orbital resonances to form for certain combinations of radius, twist rate, and wavelength. Taking *n*_{az} ≈ αρ*n*_{SM}), multiplied by the circumference, should equal a multiple of 2π. This leads to the relationship*l* is an integer representing the order of the resonance, λ is the vacuum wavelength, and α twists clockwise in the +*z* direction. Fitting to the experimental results [and the finite-element (FE) modeling] reveals that the product *n*_{SM}ρ^{2} is a constant for the system. The resonant wavelengths in Eq. 1 scale with the reciprocal of *L*, which distinguishes the effect from grating-related phenomena, when the resonant wavelength would scale with *L* (apart from slight deviations because of dispersion).

Equation 1 also requires that the orbital mode order should be proportional to the reciprocal wavelength at a fixed twist rate and that λ → ∞ at *l* = 0. Applying this condition to the experimental data in Fig. 3A and selecting values of *l* so that each line consistently goes through the point 1/λ = 0, *l* = 0 produce the plot in Fig. 3B, which shows that the orders of successive resonances (a, b, c, and d in Fig. 2A) run from *l* = 5 to *l* = 8. To obtain these excellent fits (accurate to within ±2.5%), we set *n*_{SM}ρ^{2} equal to 54.6 μm^{2}.

The LC- and RC-polarized cladding modes predicted by FE modeling yield almost identical transmission spectra, but do they carry the same angular momentum? By analogy with Laguerre-Gaussian beams (*13*, *14*), one would expect, in the paraxial approximation,*J* is the total axial flux of angular momentum and *P* is the axial energy flux. The integrals are evaluated over the entire cladding mode (ρ > Λ). It turns out that *J*/*P*, evaluated from the FE results, takes values that are quite close to integral multiples of λ/2π as predicted by Eq. 2, being smaller by 2 for the LC case as expected from the opposite spin angular momentum. The results are summarized in Fig. 4. The values of *l* in Fig. 4 are derived from the fit to Eq. 2 and represent the OAM order, and σ is taken to be ±1 depending on whether the mode is RC or LC polarized. The total angular momentum order is then *l* + σ, which does not yield perfect agreement with the expression in Eq. 2, although the trend is correct. We attribute this to a breakdown in the paraxial approximation (*15*).

A further aspect of the helical system is that the effective axial refractive indices of the orbital resonances and the core mode are topologically affected by the helical path (Fig. 5A). In the RC case, the effective axial refractive indices of both cladding and core (denoted by *n*) ≈ 4 mm. Moving away from synchronism, the modes split strongly into cladding-ring–like and corelike modes, the interaction becoming weaker as the wavelength detunes from the anticrossing point and the dephasing increases. The axial Poynting vector distributions of the modes illustrate this coupling phenomenon (Fig. 5B).

Figure 5A suggests that the transmission should exhibit some polarization dependence in the vicinity of the resonances, but this is not seen in the experimental results. We attribute this to fabrication-related nonuniformities in pitch, which would need to be kept below 0.05% to observe the small (2.5 nm) splitting between resonances in Fig. 5. In addition to their fundamental scientific interest as guided wave devices in “twisted space,” helical PCF has many potential applications in wavelength filtering and dispersion control. The thermal postprocessing technique also allows structures with axially varying twist rates to be fabricated, providing further flexibility in design.

## Supplementary Materials

www.sciencemag.org/cgi/content/full/337/6093/446/DC1

Materials and Methods

Figs. S1 and S2