We demonstrate experimentally and theoretically that a nanoscale hollow channel placed centrally in the solid-glass core of a photonic crystal fiber strongly enhances the cylindrical birefringence (the modal index difference between radially and azimuthally polarized modes). Furthermore, it causes a large split in group velocity and group velocity dispersion. We show analytically that all three parameters can be varied over a wide range by tuning the diameters of the nanobore and the core.
© 2010 Optical Society of America
Cylindrically polarized beams and guided modes—in which the electric field is azimuthal or radial—are finding applications in, for example, high-resolution microscopy , focusing of plasmons [2, 3], optical trapping , fiber lasers , quantum optics [6, 7], atom optics [8, 9] and laser machining . Recent papers have reported the excitation and propagation of such modes in hollow-core photonic crystal fiber (PCF) [11, 12, 13], in standard fiber with a W-doping profile , and in all solid photonic bandgap fibers . Cylindrically polarized beams can also be generated by nonlinear conversion in a few-moded PCF . In all these cases, the radial and the azimuthal modes exhibit similar dispersion and are almost degenerate , allowing them to couple under the influence of bends and other perturbations.
Here we report a PCF whose submicrometer solid-glass core contains a central wide hollow “nanobore.” In such fibers, there is strong electric field enhancement in the vicinity of the nanobore when the Gaussian-like linearly polarized fundamental mode is excited . We show here that the nanobore has the additional effect of greatly increasing the difference in modal index, group velocity, and group velocity dispersion (GVD) between radial and azimuthal modes. As a result, it is possible to maintain radial and azimuthal polarization effectively against bends and perturbations, just as a conventional high-birefringence fiber maintains modes linearly polarized along its eigenaxes.
The nanobore fiber was fabricated using the conventional silica stack-and-draw procedure . To avoid structural distortions during the large reduction in scale, the draw-down was carried out in three stages. The first draw resulted in a cane with an outer diameter of , a core diameter of , and a hollow channel in diameter [Fig. 1a]. This was then drawn down to a core diameter of containing a hollow channel [Fig. 1b]. During the final draw, surface-tension-induced nanobore shrinkage was controlled by judicious use of pressure. Figures 1c, 1d show high- resolution scanning electron micrographs (SEMs) of the resulting fiber structure. The radii of the core and nanobore are and and were found to remain constant to within over lengths of several meters. The air-filling fraction in the cladding region was estimated from SEM images to be .
The setup for optical characterization is shown in Fig. 2a. A tunable CW Ti:sapphire laser was used in the measurement of the near-field mode irradiance profiles (Fig. 3), and a PCF-based supercontinuum (SC) source, emitting from 480 to in a single transverse mode, was used for the group- delay measurements (Fig. 4) . Radially and azimuthally polarized modes were generated from a linearly polarized beam using a commercial liquid-crystal-based polarization converter  [see Fig. 2b]. A tunable liquid-crystal phase plate (not shown) introduced a π-phase shift in half of the beam, as required to generate a doughnut mode. For each wavelength setting, the control voltage of the liquid-crystal plate was reoptimized. An additional twisted-nematic liquid-crystal cell was used to switch the input polarization between vertical and horizontal, yielding an output polarization state that is either radial or azimuthal without need to change the coupling [see Fig. 2b]. An intensity asymmetry of 20–40% was observed in the input irradiance profiles. Although the orientation of the electric field varies spatially, the polarization state is locally linear, as illustrated in Fig. 2b.
To measure the group delay of the modes, the cylindrically polarized beam was coupled into a Mach–Zehnder interferometer , one arm of which included a long length of nanobore PCF. Using a objective, launch efficiencies up to 23% were obtained. The limited launch efficiencies are likely caused by the asymmetric input profile as well as the very small core size. The reference beam of the interferometer passed through a set of identical objectives (to ensure balanced dispersion between the arms) and a computer-controlled delay line (path length difference ) and was then recombined with the PCF signal. The combined beam was then passed, via a grating monochromator, to a photodiode, and the signal monitored as a function of . The short coherence length of the SC light ensures well- defined, narrow fringes (typical width ), allowing accurate measurement of the group delay.
4. EXPERIMENTAL RESULTS
4A. Mode Irradiance Profiles
Output irradiance profiles were obtained by imaging the near-field at the fiber end face on to a CCD camera. Note that not all of the subwavelength fine features are resolvable in the image. A doughnut-shaped irradiance distribution was observed for both modes [Figs. 3a, 3b]. To analyze the polarization state, output irradiance profiles were measured for different orientations of polarizer P2 [see Figs. 3c, 3d, 3e, 3f]. Some asymmetry is apparent in the profiles, which we attribute to a slightly eccentric nanobore position and core ellipticity. The data in Figs. 3c, 3d, 3e, 3f were combined to extract the spatial polarization profiles shown in Figs. 3g, 3h, which match the input profiles generated by the polarization converter, demonstrating that both radially and azimuthally polarized modes are maintained after propagation in the nanobore PCF.
4B. Dispersion Measurement
Using the setup described in Section 3, a clean azimuthally polarized mode could be launched over the broad wavelength range , resulting in a group-delay curve (Fig. 4, upper panel) that is continuous and smooth. The GVD, obtained by taking the derivative of a polynomial fit to the group delay, is plotted in the lower panel in Fig. 4. A zero dispersion wavelength (ZDW) for this mode is found at . The radially polarized mode could only be observed in two slightly narrower wavelength ranges, and , with a ZDW at . Its rather discontinuous group-delay curve suggests the existence of two separate modes and the presence of intermodal coupling (see Section 5). Interestingly, over the wavelength range , the azimuthally polarized mode has anomalous dispersion, whereas the radially polarized mode has normal dispersion—quite different from the case of a conventional linearly birefringent fiber, when the dispersion is very similar for both polarizations.
The optical properties of the cylindrically polarized modes were analyzed using both a quasi-analytical step-index model (based on a perfect circular strand with a nanobore at its center) and a finite-element approach based on the actual PCF structure.
5A. Quasi-Analytical Step-Index Model
In this model, the core is approximated by a circular central hollow bore (refractive index ) and a circular annulus of fused silica [index from ]. The refractive index of the outer medium, which, in the actual fiber, consists of a network of thin glass membranes, is approximated as
Using the standard approach outlined in many textbooks for circular-cylindrical structures (e.g., Snyder and Love  or Yeh et al. ), Maxwell’s equations can be solved in their full vectorial form using Bessel functions and the effective refractive indices and transverse field distributions of the guided modes calculated. The azimuthal and radial modes of the nanobore PCF are closely related to the and modes of a step-index fiber, with nonzero field components for the azimuthal and for the radial. As a result, the order of the relevant Bessel function for the z-field components is zero and the problem reduces to two matrices, one each for the azimuthal and the radial mode. Using this approach, the geometry of the structure was tuned so as to reproduce the measured GVD of the cylindrically polarized modes (Fig. 4). The best agreement was obtained for bore radius, core radius and , quite close to those of the PCF used in the experiments. The calculated mode irradiance distribution (not shown) as well as the spatial polarization distribution also agree well with the experimental data in Fig. 3.
The phase indices of the azimuthal and radial modes are and , yielding a cylindrical birefringence of 0.032, which translates to a beat length of at wavelength. In contrast, the cylindrical birefringence in a standard SMF28 fiber, designed to be single-mode at but operated at , is or four orders of magnitude smaller than in the nanobore PCF.
5B. Finite-Element Calculations
Finite-element (FE) calculations were performed (using COMSOL Multiphysics) on a perfectly six-fold-symmetric structure with dimensions identical to those in the actual fiber, including the Sellmeier expansion for the refractive index of silica . Figure 5 shows the resulting radial and azimuthal mode Poynting vector profiles at . Note that the Poynting vector of the radially polarized mode is discontinuous at both the inner and outer core boundaries (TM boundary conditions), whereas the azimuthal Poynting vector is continuous (TE boundary conditions). The phase indices are and , yielding a cylindrical birefringence of 0.020, slightly smaller than predicted by the step-index model. A third mode with a phase index of 1.222, related to the hybrid mode in standard step-index fibers, is also present Fig. 5c.
The Poynting vector profiles of the modal fields, calcu lated after passing through a linear polarizer, are shown in Figs. 5d, 5e, 5f, 5g, 5h, 5i. The profiles compare well with the measured ones [Figs. 3c, 3d, 3e, 3f] for orthogonal orientations of the polarizer. Calculated vector electric field plots are shown in Figs. 5j, 5k, 5l and also compare well with the measurements [Figs. 3g, 3h]. Although the Poynting vector distribution and the phase index of the mode are very similar to those of the radially polarized mode, the vector field distributions are markedly different.
5C. Group Velocity Dispersion
The GVD curves calculated using the FE code (solid curves) and the step-index model (dashed curves) agree well with the experimental data for the azimuthal mode; see Fig. 4, lower panel. Reasonable agreement is also found for the radial mode in the range, however, for theory and experiment diverge strongly, the experimental data fitting more closely to the GVD curves for the -like mode. The inset in Fig. 4 (lower panel) shows that the dispersion curves of the radial and the -like modes cross at . Strong intermodal coupling results, which causes the measured dispersion to be a complicated mixture of that of the two modes, meaning that the radial mode dispersion cannot be cleanly measured in the range .
5D. Tunable Birefringence
The cylindrical birefringence can be tuned by varying the nanobore radius a, as illustrated in Fig. 6, where the step- index model was used with . Since the core radius is quite small and the outer refractive index step [using Eq. (1), this has the value 0.37 at a wavelength of ] is large, is significant (0.028 at ) even when . It is some 2.5 times larger at for a nanobore radius of . The beat length can be as short as and of course can be tuned to larger values by decreasing a. The inset in Fig. 6 shows the effective indices of the radial and azimuthal modes, calculated for and . Figure 7 shows the dependence of on normalized core radius for different ratios . It can be seen that depends strongly on the ratio .
5E. Tunable Dispersion
The ZDWs move to a shorter wavelength as a increases, as seen in Fig. 8 for three PCFs with and nanobore radii , 100, and . The radial GVD is more strongly affected by the presence of the nanobore than the azimuthal. For the radial GVD remains normal for all wavelengths (i.e., there is no ZDW). The great difference in GVD tunability between the azimuthal and radial modes means that the positions of their ZDWs can be almost independently controlled (see Fig. 9), which could be of interest in nonlinear applications.
6. CONCLUSIONS AND PERSPECTIVE
Nanobore PCF effectively maintains azimuthal and radial polarization states. In addition, the phase indices, group velocities, and GVD of the orthogonal azimuthal and radial modes are strongly different and can be widely tuned by varying the core and nanobore radii. A simple quasi-analytical step-index model provides convenient approximate solutions for the modal indices and field profiles, which compare well with FE simulations of the actual fiber structure. These unique characteristics are interesting for many applications; for example, the azimuthal mode in the range of anomalous dispersion has been used for nonlinear quantum squeezing, permitting for the first time creation of a continuous-variable hybrid-entangled state . The nanobore can be filled with other materials, such as metals or nonlinear glasses, e.g., tellurite [25, 26]. Since the nanobore fiber is polarization maintaining over a broad wavelength range, it could be used to transmit broadband radially polarized modes with high fidelity along extended curved paths, with applications in, e.g., high-resolution microscopy systems .
We thank Silke Rammler for help in fabricating the fiber, Namvar Jahanmer for the scanning electron micrographs, and Juan Carlos Loredo Rosillo for help with computer-aided design drawings used for finite-element-method simulations.
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