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Excitation of Orbital Angular Momentum Resonances in Helically Twisted Photonic Crystal Fiber

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Science  27 Jul 2012:
Vol. 337, Issue 6093, pp. 446-449
DOI: 10.1126/science.1223824

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 (13), long-period grating couplers (47), 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 CO2 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.

Fig. 1

(A) Perspective view of one period L (= 2π/α, where α is the twist rate) of a helical PCF with a negative twist (i.e., anticlockwise looking in the direction of propagation, +z). The blue tubes represent the hollow channels passing through the glass, and the solid glass core is formed by a missing channel at the center. For clarity, the twist rate is greatly exaggerated in this sketch (L/Λ is of order 100 in the experiments, where Λ is interhole spacing). (B) Local axial and azimuthal components of the refractive index nSM of the fundamental space-filling Bloch mode in the twisted cladding. The angle ϕ between the fiber axis and the channels increases with radius ρ, following the approximate relationship ϕ ≈ αρ.

Fig. 2

Measured (solid curve) and calculated (dotted curve) transmission spectra of PCFs with twist rates α of (A) 10.8 rad/mm and (B) 13.6 rad/mm. The letters a, b, c, and d mark the strongest individual cladding resonances, which track linearly to longer wavelength as the twist rate increases (Fig. 3A). [(A) inset] Scanning electron micrograph of the PCF structure.

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.

Fig. 3

(A) Twist rate plotted versus vacuum wavelength for the four resonances a, b, c, and d in Fig. 2. The solid circles are the experimental measurements, and the lines fit both to FE modeling and to a simple theory based on azimuthal resonances. (B) Reciprocal optical wavelength at resonance in units of μm−1 plotted against mode order for the experimental data (solid points). From top to bottom, the values of L are 581, 509, 461, 415, and 341 μm, yielding twist rates of 10.8, 12.3, 13.6, 15.1, and 18.4 rad/mm. The lines are plotted by using Eq. 2 with ρ2nSM = 54.6 μm2. (C) The Sz distributions (very slightly off-resonance to enhance the visibility of the cladding fields) for the four resonances at a twist rate of 13.6 rad/mm [dashed horizontal line in (A)]. For compactness, only one quadrant is shown per mode; the patterns have perfect sixfold symmetry. (D and E) The Saz distributions for the RC- and LC-polarized versions of the four modes; blue shading indicates clockwise power flow. (F) Polarization state of the transverse electric field.

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 (Sz) and azimuthal (Saz) Poynting vector distributions are plotted for the four modes. They are ring-shaped, Saz flowing in the direction of the spiral in each case. Saz 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, SSM, 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 SSM 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 + α2ρ2)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 nSM as the refractive index vector of the SM mode along the helical path, the orbital resonance condition can be readily derived by requiring that the azimuthal component of the wave vector (proportional to the refractive index naz ≈ αρnSM), multiplied by the circumference, should equal a multiple of 2π. This leads to the relationshipρ(nSM×α)=nSMρ2α=lλ2π (1)where 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 nSMρ2 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 nSMρ2 equal to 54.6 μm2.

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,JP=modeρSaz(ρ,ϕ)ρdρdϕmodeSz(ρ,ϕ)ρdρdϕ=(l+σ)λ2π (2)where 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).

Fig. 4

Angular momentum (AM) orders predicted by the fits in Fig. 2 (l ± 1) at each of the four resonances for α = 13.6 rad/mm (almost identical results are obtained at other twist rates), compared with J 2π/Pλ evaluated from the FE results.

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˜co and n˜SM) rise in the twisted structure, the cladding index increasing by a larger factor as expected from the topology, and an anticrossing appearing at ~793 nm. In the LC case, however, n˜co actually falls in the twisted structure while n˜SM rises, although by a much smaller amount than in the RC case. As a result, an anticrossing forms at a lower value of index, centered at ~795.5 nm. At the center of the anticrossings, even and odd modes exist, each with identical loss and a coupling length of λ/(2δn) ≈ 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).

Fig. 5

Numerical modeling of the behavior in the vicinity of anticrossing c at ~794 nm (α = 13.6 rad/mm, L = 461 μm). (A) (Top) Calculated dispersion of the modal refractive indices for LC- and RC-polarized modes. The indices nco (core mode) and nSM (space-filling cladding mode) refer to the untwisted case. The splitting at the center of each anticrossing corresponds to a coupling length of ~4 mm. (Bottom) Loss spectra in dB/mm. At the center of both LC and RC anticrossings, the loss of even and odd modes at this wavelength is ~2.3 dB/mm. (B) Distributions of axial Poynting vector at the six points (1 to 6), illustrating the coupling between orbital cladding resonance and core mode in the vicinity of the anticrossings. These distributions are almost identical for equivalent points in the LC case.

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

References (1618)

References and Notes

  1. Acknowledgments: The authors wish to thank M. Padgett and S. Barnett for helpful comments on the analysis in the manuscript and S. Burger from JCMwave GmbH for assistance with the finite element modeling.
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