1. INTRODUCTION

Cylindrically polarized beams and guided modes—in which the electric field is azimuthal or radial—are finding applications in, for example, high-resolution microscopy [1], focusing of plasmons [2, 3], optical trapping [4], fiber lasers [5], quantum optics [6, 7], atom optics [8, 9] and laser machining [10]. 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 [14], and in all solid photonic bandgap fibers [15]. Cylindrically polarized beams can also be generated by nonlinear conversion in a few-moded PCF [16]. In all these cases, the radial and the azimuthal modes exhibit similar dispersion and are almost degenerate [16], 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 $180\mathrm{nm}$ 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 [17]. 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.

2. FIBER

The nanobore fiber was fabricated using the conventional silica stack-and-draw procedure [18]. 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 $1.8\mathrm{mm}$, a core diameter of $170\mathrm{\mu m}$, and a hollow channel $96\mathrm{\mu m}$ in diameter [Fig. 1a]. This was then drawn down to a core diameter of $31\mathrm{\mu m}$ containing a $16\mathrm{\mu m}$ 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 $465\mathrm{nm}$ and $90\mathrm{nm}$ and were found to remain constant to within $\pm 5\mathrm{nm}$ over lengths of several meters. The air-filling fraction in the cladding region was estimated from SEM images to be $F=0.86$.

3. SETUP

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 $1750\mathrm{nm}$ in a single transverse mode, was used for the group- delay measurements (Fig. 4) [19]. Radially and azimuthally polarized modes were generated from a linearly polarized beam using a commercial liquid-crystal-based polarization converter [20] [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 [21], one arm of which included a $41\mathrm{cm}$ long length of nanobore PCF. Using a $60\times 0.85\mathrm{NA}$ 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 $60\times 0.85\mathrm{NA}$ objectives (to ensure balanced dispersion between the arms) and a computer-controlled delay line (path length difference $\mathrm{\Delta }x$) 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 $\mathrm{\Delta }x$. The short coherence length of the SC light ensures well- defined, narrow fringes (typical width $200\mathrm{\mu m}$), 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 $550–1060\mathrm{nm}$, 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 $900\mathrm{nm}$. The radially polarized mode could only be observed in two slightly narrower wavelength ranges, $700–860\mathrm{nm}$ and $900–1050\mathrm{nm}$, with a ZDW at $720\mathrm{nm}$. 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 $720–900\mathrm{nm}$, 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.

5. THEORY

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 $n_b$) and a circular annulus of fused silica [index $n_s(\lambda )$ from [22]]. 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 [23] or Yeh et al. [24]), 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 ${\mathrm{TE}}_{01}$ and ${\mathrm{TM}}_{01}$ modes of a step-index fiber, with nonzero field components $(E_{\phi },H_r,H_z)$ for the azimuthal and $(E_r,E_z,H_{\phi })$ 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 $4\times 4$ 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 $88\mathrm{nm}$ bore radius, $540\mathrm{nm}$ core radius and $F=0.84$, 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 $n_r=1.243$ and $n_a=1.275$, yielding a cylindrical birefringence of 0.032, which translates to a beat length of $26\mathrm{\mu m}$ at $820\mathrm{nm}$ wavelength. In contrast, the cylindrical birefringence in a standard SMF28 fiber, designed to be single-mode at $1.55\mathrm{\mu m}$ but operated at $820\mathrm{nm}$, is $4.4\times {10}^{-6}$ 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 [22]. Figure 5 shows the resulting radial and azimuthal mode Poynting vector profiles at $820\mathrm{nm}$. 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 $n_r=1.219$ and $n_a=1.239$, 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 ${\mathrm{HE}}_{21}$ 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 ${\mathrm{HE}}_{21}$ 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 $700–860\mathrm{nm}$ range, however, for $\lambda >860\mathrm{nm}$ theory and experiment diverge strongly, the experimental data fitting more closely to the GVD curves for the ${\mathrm{HE}}_{21}$-like mode. The inset in Fig. 4 (lower panel) shows that the dispersion curves of the radial and the ${\mathrm{HE}}_{21}$-like modes cross at $877\mathrm{nm}$. 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 $860–900\mathrm{nm}$.

5D. Tunable Birefringence

The cylindrical birefringence $B_c$ can be tuned by varying the nanobore radius a, as illustrated in Fig. 6, where the step- index model was used with $b=540\mathrm{nm}$. 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 $820\mathrm{nm}$] is large, $B_c$ is significant (0.028 at $930\mathrm{nm}$) even when $a=0$. It is some 2.5 times larger at $810\mathrm{nm}$ for a nanobore radius of $250\mathrm{nm}$. The beat length can be as short as $11\mathrm{\mu m}$ 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 $a=0$ and $250\mathrm{nm}$. Figure 7 shows the dependence of $B_c$ on normalized core radius $b/\lambda$ for different ratios $a/b$. It can be seen that $B_c$ depends strongly on the ratio $a/b$.

5E. Tunable Dispersion

The ZDWs move to a shorter wavelength as a increases, as seen in Fig. 8 for three PCFs with $b=540\mathrm{nm}$ and nanobore radii $a=0$, 100, and $200\mathrm{nm}$. The radial GVD is more strongly affected by the presence of the nanobore than the azimuthal. For $a>195\mathrm{nm}$ 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 [7]. 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 [27].

ACKNOWLEDGMENTS

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.

 

Fig. 1 Optical microscope images of (a) the first and (b) the second cane. The diameter of the hole in the core was reduced from $96\mathrm{\mu m}$ to $16\mathrm{\mu m}$. (c) SEM of the cladding microstructure of the fused silica nanobore PCF. (d) SEM of the fiber core. The core has diameter $930\mathrm{nm}$ with a $180\mathrm{nm}$ wide nanobore located in its center.

Download Full Size | PDF

 

Fig. 2 (a) Experimental setup: light from a PCF-based SC source passes through a polarizer (P1) and a polarization converter [20] that converts the polarization state to either radial or azimuthal. This beam is coupled into a Mach–Zehnder interferometer. One arm includes $41\mathrm{cm}$ of the nanobore PCF shown in Figs. 1c, 1d. The fiber arm and reference arm are recombined, pass a grating monochromator (GM) and the interference signal is measured by detector (D). A CCD camera and a polarizer (P2) are used to analyze the polarization state of the beam exiting the fiber. (b) Operating principle of the polarization converter using a liquid-crystal cell with alignment layers linearly rubbed on one side and circularly on the other side (from [20]).

Download Full Size | PDF

 

Fig. 3 (a)–(f) Normalized Poynting vector profiles at $\lambda =820\mathrm{nm}$ for radially (top row) and azimuthally (bottom row) polarized modes obtained by imaging a $2.1\times 2.1{\mathrm{\mu m}}^2$ region of the near field at the fiber end face onto a CCD camera. (c)–(f) Poynting vector profiles measured for orthogonal orientations of the polarizer (P2) (linear gray scale identical in each image). (g), (h) Vector plots of the measured electric field orientations in (a) and (b).

Download Full Size | PDF

 

Fig. 4 (top panel) Experimentally determined relative group delay of azimuthal and radial modes as a function of wavelength; (lower panel) measured (symbols) and calculated (curves) GVD of the azimuthal (black), radial (red), and ${\mathrm{HE}}_{21}$ modes (blue). The shaded area indicates anomalous dispersion. The solid curves are the results of FE simulations, and the dashed curves are based on the analytical step-index model. (inset) Detail of the region where the effective phase indices of the radial and ${\mathrm{HE}}_{21}$ modes match.

Download Full Size | PDF

 

Fig. 5 (a)–(c) FE calculations of the normalized modal Poynting vector profiles at $\lambda =820\mathrm{nm}$ for radial (top row) and azimuthal (middle row) modes and the ${\mathrm{HE}}_{21}$ higher-order mode (bottom row). All three modes show a doughnut-shaped profile. (d)–(i) Simulated profiles for orthogonal positions of the linear polarizer (P2). (j)–(l) Vector electric field distributions.

Download Full Size | PDF

 

Fig. 6 Spectral dependence of cylindrical birefringence $B_c$ for different nanobore radii a. Core radius and air-filling fraction are $540\mathrm{nm}$ and 0.84. Inset shows the phase indices of the azimuthal (solid curves) and radial (dashed curves) modes for $a=0$ (black) and $250\mathrm{nm}$ (green).

Download Full Size | PDF

 

Fig. 7 Calculated cylindrical birefringence versus normalized frequency for different values of the ratio $a/b$ (step-index model). Core radius and air-filling fraction are $540\mathrm{nm}$ and 0.84. The silica refractive index is kept constant at $n_s=1.45$. The low frequency edge of each curve (near $b/\lambda =0.4$) corresponds to the cut-off frequency of the radial mode.

Download Full Size | PDF

 

Fig. 8 Calculated GVD curves for radial and azimuthal modes (step-index model). Core radius and air-filling fraction are $540\mathrm{nm}$ and 0.84. Data are plotted for three different values of a.

Download Full Size | PDF

 

Fig. 9 Calculated relative change of the second (long wavelength) ZDW (${\lambda }_{\mathrm{ZDW}}$) as function of nanobore radius for the azimuthal (black) and radial (red) modes (step-index model). Curves have been normalized to the value of ${\lambda }_{\mathrm{ZDW}}$ at $a=0$. Inset shows the absolute values of ${\lambda }_{\mathrm{ZDW}}$. Core radius and air-filling fraction are $540\mathrm{nm}$ and 0.84. For $a<153\mathrm{nm}$ (dashed line), ${\lambda }_{\mathrm{ZDW}}$ for the azimuthal mode decreases by a maximum of 2% ($\sim 17\mathrm{nm}$), while ${\lambda }_{\mathrm{ZDW}}$ for the radial mode decreases by a maximum of 14% ($\sim 100\mathrm{nm}$).

Download Full Size | PDF

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef]   [PubMed]  

2. W. Chen and Q. Zhang, “Realization of an evanescent Bessel beam via surface plasmon interference excited by a radially polarized beam,” Opt. Lett. 34, 722–724 (2009). [CrossRef]   [PubMed]  

3. T. Lan, J. He, and C. Tien, “Versatile excitation of localized surface plasmon polaritons via spatially modulated polarized focus,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2010), paper QMD5.

4. M. Michihata, T. Hayashi, and Y. Takaya, “Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radially polarized beam,” Appl. Opt. 48, 6143–6151 (2009). [CrossRef]   [PubMed]  

5. D. Lin, K. Xia, J. Li, R. Li, K. Ueda, G. Li, and X. Li, “Efficient, high-power, and radially polarized fiber laser,” Opt. Lett. 35, 2290–2292 (2010). [CrossRef]   [PubMed]  

6. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, “Remote preparation of single-photon “hybrid” entangled and vector-polarization states,” Phys. Rev. Lett. 105, 030407 (2010). [CrossRef]   [PubMed]  

7. C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Förtsch, D. Elser, U. L. Andersen, C. Marquardt, P. St. J. Russell, and G. Leuchs, “Hybrid-entanglement in continuous variable systems,” http://arxiv.org/abs/1007.1322.

8. T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78, 4713–4716 (1997). [CrossRef]  

9. J. Yin and Y. Zhu, “Doughnut-beam-induced Sisyphus cooling in atomic guiding and collimation,” J. Opt. Soc. Am. B 15, 25–33 (1998). [CrossRef]  

10. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32, 1455–1461 (1999). [CrossRef]  

11. A. A. Ishaaya, B. Shim, C. J. Hensley, S. Schrauth, A. L. Gaeta, and K. W. Koch, “Efficient excitation of polarization vortices in a photonic bandgap fiber with ultrashort laser pulses,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2008), paper CThV3. [PubMed]  

12. T. G. Euser, G. Whyte, M. Scharrer, J. S. Y. Chen, A. Abdolvand, J. Nold, C. F. Kaminski, and P. St. J. Russell, “Dynamic control of higher-order modes in hollow-core photonic crystal fibers,” Opt. Express 16, 17972–17981 (2008). [CrossRef]   [PubMed]  

13. A. Ishaaya, C. J. Hensley, B. Shim, S. Schrauth, K. W. Koch, and A. L. Gaeta, “Highly-efficient coupling of linearly- and radially-polarized femtosecond pulses in hollow-core photonic band-gap fibers,” Opt. Express 17, 18630–18637 (2009). [CrossRef]  

14. S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34, 2525–2527 (2009). [CrossRef]   [PubMed]  

15. V. Pureur, J. C. Knight, and B. T. Kuhlmey, “Higher order guided mode propagation in solid-core photonic bandgap fibers,” Opt. Express 18, 8906–8915 (2010). [CrossRef]   [PubMed]  

16. M. L. Hu, C. Y. Wang, Y. J. Song, Y. F. Li, L. Chai, E. E. Serebryannikov, and A. M. Zheltikov, “A hollow beam from a holey fiber,” Opt. Express 14, 4128–4134 (2006). [CrossRef]   [PubMed]  

17. G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nat. Photon. 1, 115–118 (2007). [CrossRef]  

18. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006). [CrossRef]  

19. W. Wadsworth, N. Joly, J. Knight, T. Birks, F. Biancalana, and P. St. J. Russell, “Supercontinuum and four-wave mixing with Q-switched pulses in endlessly single-mode photonic crystal fibres,” Opt. Express 12, 299–309 (2004). [CrossRef]   [PubMed]  

20. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948–1950 (1996). [CrossRef]   [PubMed]  

21. H.-T. Shang, “Chromatic dispersion measurement by white-light interferometry on metre-length single-mode optical fibres,” Electron. Lett. 17, 603–605 (1981). [CrossRef]  

22. J. W. Fleming, “Dispersion in GeO2 –SiO2 glasses,” Appl. Opt. 23, 4486–4493 (1984). [CrossRef]   [PubMed]  

23. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

24. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978). [CrossRef]  

25. H. W. Lee, M. A. Schmidt, H. K. Tyagi, L. P. Semprere, and P. St. J. Russell, “Polarization-dependent coupling to plasmon modes on submicron gold wire in photonic crystal fiber,” Appl. Phys. Lett. 93, 111102 (2008). [CrossRef]  

26. M. A. Schmidt, N. Granzow, N. Da, M. Peng, L. Wondraczek, and P. St. J. Russell, “All-solid bandgap guiding in tellurite-filled silica photonic crystal fibers,” Opt. Lett. 34, 1946–1948 (2009). [CrossRef]   [PubMed]  

27. S. W. Hell and J. Wichmann, “Breaking the diffraction resolution limit by stimulated emission: stimulated-emission-depletion fluorescence microscopy,” Opt. Lett. 19, 780–782 (1994). [CrossRef]   [PubMed]