We demonstrate excitation of the azimuthally-polarized TE01 cylindrical waveguide mode in hollow glass and metal waveguides with 780 nm light. Experimentally, we demonstrate formation of the vectorial vortex beams, and measure attenuation lengths of the TE01 mode in hollow optical fibers with diameters of 50–100 microns. By silver-coating the inner walls of the dielectric fibers, we demonstrate a ≈ 200% increase in the attenuation length.
© 2010 Optical Society of America
Atom guides formed by hollow-core optical fibers have potential applications in atomic sources  and nonlinear optics . They may also enable compact designs for atom interferometers, which have recently shown excellent promise for inertial sensing . The fundamental guiding mechanism relies on the AC Stark shift, whereby atom confinement depends on the sign of the detuning of the guiding light from atomic resonance. An atom in a red-detuned laser field is attracted to high intensity, while one in a blue-detuned field is attracted to low intensity.
These two options have led to a number of schemes for elastic atom guidance in fibers, each with certain benefits and drawbacks. Renn et al.  used a red-detuned beam in small-diameter hollow-core fibers to guide hot atoms. The beam propagated in a fundamental, grazing incidence (EH11) mode, shown schematically in Fig. 1a, and has the advantage of being speckle-free with easy alignment. The high intensity confinement, however, leads to both larger energy level shifts and increased spontaneous photon scattering rates. Blue-detuned atom guidance using evanescent fields (Fig. 1b) has also been used to guide both hot [5, 6] and cold  atoms. This offers low light propagation losses, but suffers from speckle due to the multimode propagation and light scattering at the fiber entrance. Fatemi et al.  developed an efficient light coupling technique to eliminate light scattering. However, the use of evanescent fields is generally inefficient, since the evanescent field is only a small fraction of the peak field, and must overcome the attractive van der Waals forces between the atom and the glass surface. Pilloff  suggested a hollow optical fiber with a metal-coated outer surface to enhance evanescent fields, but to the best of our knowledge such fibers have not been fabricated.
More recently, groups have begun using photonic crystal fibers (PCFs) to guide both hot and cold atoms with red-detuned light for nonlinear optical experiments at low light levels. These fibers have smaller diameters than the atom guiding experiments described above. Takekoshi et al.  demonstrated guiding of hot 87Rb atoms through PCF with core diameter of 9μm. Bajcsy et al.  used a 7μm core PCF for few-photon optical switching in 87Rb. However, for many experiments the guiding light must be switched off so that it does not affect the results, but because of the small mode-field diameter, this “off” time must be short to prevent atoms from leaving the beam and sticking to the fiber walls. Using blue-detuned light may alleviate this, because the atoms are already confined to the dark region so that the guiding light need not be shut off.
A technique that circumvents these issues was suggested by Wang et al. , who presented a theoretical treatment of guiding parameters using the TE01 hollow mode in a hollow metal waveguide. This mode has a harmonic intensity null on the optical axis, so that the benefits of dark-field guiding are maintained, and because it does not rely on evanescent fields, laser power is efficiently used for guiding. Experimentally, however, TE01 mode guidance in hollow optical fibers has been restricted to IR wavelengths in fibers with core diameters ≥ 300μm [12-15]. Furthermore, while significant efforts have been made to reduce losses in fused silica waveguides with metal- or dielectric wall coatings, to the best of our knowledge these efforts have also only been demonstrated in fibers with core diameters ≥ 250μm [16-20].
In this paper, we propose and experimentally demonstrate excitation of the TE01 mode with 780 nm light in small-diameter (50–100 μm) hollow fibers for atom guiding. The technique is demonstrated for both bare and metal-coated fused silica waveguides with diameters between 50–100μm. The metal coating is particularly beneficial to the TE01 mode, which is the lowest loss mode of a cylindrical metal waveguide . We achieve a ≈ 200% increase in the attenuation length using the silver-coated fibers.
The propagation characteristics of the modes of hollow metal and hollow dielectric waveguides have been considered extensively in the literature . The TE01 mode is shown schematically in Fig. 1c. Its low loss property is due to the azimuthal polarization, which causes the field to be zero at the metal boundary. The field profile is defined mathematically in the core region by:
where ρ and θ are cylindrical coordinates, a is the fiber core radius, and Jn(x) are the nth order Bessel functions of the first kind. For a hollow glass fiber, the cladding region has a higher index than the core, resulting in lossy propagation, with an exponential dependence of the intensity on the propagation distance z:
where α is the attenuation constant of the electric field. Propagation losses in a hollow dielectric fiber scale with λ 2/a 3 and are calculated following the prescription in Marcatili . For TEnm:
and for EHnm modes:
where ν is the refractive index of the fiber and unm, the mth zeroes of J n-1(x), are 2.405 for EH11 and 3.832 for TE01 . For the fused silica hollow fibers discussed here, with a = 25μm, 2α= 25.9m -1 or 1/(2α)= 3.86cm. For the same fiber, the EH11 grazing incidence mode used in red-detuned guiding (e.g. Ref. ) has 1/(2α) = 6.0 cm at 780 nm. For a metallic fiber, the attenuation is modified substantially due to the complex index of refraction, which generally results in greatly reduced attenuation for a perfect coating.
We consider the optical potential for 87Rb atoms. In a linearly polarized laser field, this potential is:
where Γ = 2π×6.1 MHz is the linewidth, Is = 1.6 mW/cm2 is the saturation intensity, Δ is the detuning, and ΔLS = 2π× 7.1 THz (14.7 nm) is the fine-structure splitting. Eq. 5 is valid for Δ ≪ ω, where ω is the trap laser frequency. The cross section in Fig. 2 shows the intensity as a function of radial coordinate for 1 mW input power, and the resulting potential depth, for Δ = 500 GHz (1 nm). The potential energy axis is in units of the Doppler temperature, TD = h̄Γ/(2kB), which is approximately the temperature of atoms cooled in a magnetooptical trap. For comparison, the grazing incidence EH11 mode intensity profile is also shown. A full theoretical treatment of TE01 interaction with atoms is given in Ref. .
The TE01 and EH11 modes are significantly smaller than the core size. For the TE01 mode, the maximum is located at ρ = 0.48a, so the atoms are far from the wall, unlike evanescent field atom guidance in which the atoms approach subwavelength distances from the fiber wall. This mode is also smaller than the EH11 mode: The full-width at half-maximum (FWHM) of the TE01 mode is approximately half of the FWHM of the EH11 mode. For atoms, especially Bose-Einstein condensates, the relevant parameter is the trap frequency of the harmonic portion of the trap, indicated in Fig. 2. For the same input power, the smaller diameter of the TE01 mode results in tighter confinement, even though the peak intensity is lower. The radial trap frequencies for the two modes in Fig. 2 are ω TE01 = 8.1 kHz and ω EH11= 5.6 kHz.
We note that higher order modes, such as the EH13 mode, have maxima at larger radii, with a more anharmonic profile that should allow higher atom flux and even lower photon scattering rates than the TE01 mode [22-24]. To lowest order, the modes EHm,n have a radial intensity dependence of ρ 2n. Such higher order modes have been excited using subwavelength gratings at 10.6μm wavelength for 300μm diameter hollow metal waveguides .
3. Experimental Setup
To create the TE01 beams for fiber coupling, we begin with the prescription described in Ref.  and shown in Fig. 3 for producing azimuthally-polarized fields. Briefly, a nematic liquid crystal spatial light modulator (SLM) is illuminated with a Gaussian laser beam polarized at 45 degrees. The nematic SLM introduces a phase delay only for the vertically polarized component, so that a π phase delay applied in two opposite quadrants changes those polarizations to -45 degrees. By passing this beam through a subsequent π-phase step that affects all polarizations on the bottom half of the beam, a quadrant approximation to an azimuthally-polarized beam is created:
where w 0 is the 1/e 2 beam radius of the intensity of the input Gaussian beam, sgn(x) is x/∣x∣ for x = 0 (0 for x = 0), and E 0 is the incident maximum electric field strength. The π phase step was produced by depositing 390 nm of SiN on half of a glass substrate.
By symmetry we need only consider the x-polarization component for y > 0 in Eq. 7:
where C 1(ω 0), C 2(a) are normalization constants for E in TE01 and E TE01, respectively. The integral (Eq. 8) is maximized numerically with respect to ω 0 for a = 25μm to determine the mode purity, ∣A∣2. The maximum overlap of E in TE01 with E TE01 is 0.83, when ω 0 = 0.77a, resulting in a mode purity of 69%. As we will discuss below, mode purity can be increased prior to fiber coupling at the expense of power throughput by using traditional spatial filtering through a pinhole. This works well because the π-phase plate makes the TE00 mode contribution negligible, and the TE01/TM01/EH21 modes are the first higher order modes. Additionally, the Bessel function profile of the TE01 mode matches well to the transmitted field through a pinhole. Such pre-filtering is important for atom optics to minimize the effects of scattered light.
The overlap integral can be made arbitrarily close to 1.0 with a subsequent aperture to eliminate diffraction rings (Fig. 4a). By adjusting the diameter of the aperture to the diameter of the first null in the beam, the effective mode purity is increased. Of course, this reduces the overall throughput, but results in a beam of excellent purity for fiber launching with minimal scatter. The combination of the pinhole and aperture effectively results in cylindrically symmetric filtering in both the Fourier and image planes. In Fig. 4b, we plot the overlap integral and power throughput with respect to pinhole size (normalized to the unmodified, focused Gaussian waist, ωf = fλ/π ω 0, where f is the lens focal length) to determine optimal parameters for coupling into a cylindrical waveguide. As is shown in Fig. 4b, the overlap integral can be made arbitrarily close to 1.0 at the expense of throughput. When the pinhole size is a factor of ≈ 2.4 larger than ωf , the overlap integral is ≈0.96, with ≈80% transmission, which we have used in this paper as a compromise between mode purity and transmitted power. Given a small core size, even inefficient techniques for generating azimuthally polarized beams may still be acceptable, along with this type of spatial filtering, because the potential can be very deep with only modest propagating power. Because of the spatially inhomogeneous polarization profile, care must be taken when relaying the beam, as all reflections will contain both s- and p-polarizations whose relative phase is important. In our case, we relayed the beam from the SLM to the hollow fiber using two standard dielectric mirrors, but these mirrors were tested carefully to ensure that the beam polarization was not modified. We note that, in general, dielectric mirrors will impart different phases to the s- and p-polarizations, and we saw significant variation even among mirrors with the same part number.
In this section, we show input and output mode profiles, and report measured attenuation lengths in a fiber with core diameter of 50μm (a = 25μm). To couple E in TE01 into the fibers, the hollow optical fibers are first laid flat in aluminum V-grooves. After the aperture in Fig. 4, a triplet lens is used to focus the beam onto the hollow fiber. In practice, the focused hollow beam size is controlled by adjusting the aperture size so that the diameter of the first intensity null equals the fiber diameter, as prescribed by Eq. 1. This is less efficient than finding the optimum location and magnification of the triplet lens, but is easier to align and is sufficient for demonstration purposes.
Figure 5 (top panel) shows the input and output mode images after propagation through a fiber length of 11 cm. To show that the polarization is azimuthal, we also recorded the output beam after passing through a polarizer at three orientations. Additionally, we show an interferogram recorded by interfering a vertically polarized reference beam with the output mode. This interferogram shows the expected π phase shift across the beam as a lateral shift of the fringe pattern by half a period. Interference contrast occurs maximally on the left and right sides of the beam, where the beam is vertically polarized, and reduces toward the top and bottom where the beam is horizontally polarized. Beam cross sections are also shown for the input beam, and outputs of the 3.0 cm and 8.0 cm fibers.
The throughput was measured by imaging the transmitted beam onto a CCD camera and integrating only the contribution from the fiber core. This is to discriminate between scattered and cladding light from the light within the core. If this is not done, fictitiously high throughputs are measured. The input beam is also imaged and measured. For the 3.0, 8.0, and 11.0 cm long fibers, we measured throughputs of 39%, 8.1%, and 3.2%, giving 2α ≈ 0.31 cm-1, or a 1/e attenuation length of 3.2 cm (1.35 dB/cm) using the exponential law of Eq. 2. We note that the input coupling efficiency is approximately constant since our starting mode is unchanged and hence drops out of our estimation of α. Our results are consistent with a coupling efficiency greater than 95%. We compare these results to the theoretical value of attenuation from Eq. 3, which gives 1/(2α) = 3.86 cm (1.12 dB/cm), in relatively good agreement with our results. When the fiber diameter is increased to 100μm, we achieve a transmission of 58% in a 12 cm long fiber, which corresponds to an attenuation length of 21.9 cm (0.20 dB/cm). This is in agreement with the expected increase by the cube of the radius. We also measured the attenuation length of the EH11 mode. For the 50μm-diameter fiber, we measured a 1/e attenuation length of 5.5 cm, in agreement with the theoretical value of 6.0 cm from Eq. 4.
5. Silvered fibers
For some experiments, the short attenuation length of the 50μm fiber may be acceptable, but increasing the attenuation length is likely advantageous for most experiments. For instance, longer atom guiding distances or interaction lengths may be required, or the potential depth may need to be kept constant over the fiber length. Furthermore, metallic waveguides have lower bend losses than uncoated hollow dielectric waveguides  The attenuation constants can be increased by coating the inner walls of the fiber, as has been demonstrated for fibers with large core diameters (> 250μm) for increased EH11 mode or multimode throughput [16–19]. Most efforts have been restricted to far IR transmission, although Mohebbi  demonstrated EH11 propagation in a silver-coated 250-micron-diameter guide at 800 nm, and 150 micron rectangular waveguides have been demonstrated with short pulses at 850 nm .
As stated earlier, the TE01 mode is the lowest loss mode of a cylindrical metal waveguide. With a silver cylindrical waveguide (n = 0.143 - 5.13i), we find from Eq. 3 that the attenuation length in a 50 μm diameter fiber should increase by over two orders of magnitude to 6.7 m (0.006 dB/cm) for a perfect coating with no scatter. We note that for the EH11 grazing incidence mode, a perfect silver coating should increase the attenuation length to 1.2m (0.036 dB/cm).
We have used a wet chemistry, metal layering technique to increase the attenuation length, to be described in detail elsewhere. Briefly, the 50um-diameter-fiber was silanized with 10% (by volume) of aminopropyltriethoxysilane (Fisher Scientific) in methanol by a method described in Ref. . The silanized fiber was coated by carboxyl-terminated gold nanoparticles (8 nm in diameter) which were prepared using thioctic acid and poly(ethylene glycol) (molecular weight ≈ 600) containing carboxyl-terminated ligands as described in Ref. . Then the nanoparticle-coated fiber was treated by a Tollens reaction , which generates smooth silver films on gold nanoparticle arrays, and has recently been used for enhancing Raman scattering by single molecules. An atomic force microscope (AFM) image of a silver-coated glass slide using this technique is shown in Fig. 6a. We achieve a surface roughness of ≈ 11 nm RMS on the glass slide, which compares favorably to other results [16, 20], and we measured a reflectivity at normal incidence at 780 nm of 85%; a thick, scatter-free silver coating should achieve 96%. Because of the small core diameter of the fiber, the reactant solutions were drawn into the fiber by capillary action instead of being pumped through, as is typically done for larger fibers. For the metallized fiber, the mode breakdown is the same as a dielectric fiber far from the cutoff wavelength, but the losses are substantially reduced, especially for the TE01 mode. We note that most silver-coated fibers employ a halogenized coating as well [16–18], which enhances robustness and prevents tarnishing of the silver layer. We have not used any dielectric or polymer coatings. For atom optics applications, such coatings may not be necessary because the fibers are under vacuum, and the fibers can be placed in an inert atmosphere until installed. We have found that unprotected coatings typically tarnish over a period of a few days in air.
Figure 6b shows the output of a typical silvered fiber. The attenuation length has increased by 170% to ≈8.1 cm. At the output of a 12.3 cm fiber, more than 20% percent of the light remains guided. For comparison, we also measured the EH11 attenuation length in the silvered fiber. We observed a 95% increase compared with the uncoated fiber. The attenuation length is still well below the theoretical limit probably due to surface roughness or to a coating that is too thin. We cannot directly measure the surface roughness or thickness on the interior of the fiber. The coating dynamics may be very different in a small-bore fiber than on a glass slide, and we are currently exploring coating variables.
6. Additional considerations
Coupling atoms from a cold atom source or thermal reservoir should be possible using similar techniques as used in prior experiments. Furthermore, the mode exiting the fiber continues to propagate as a hollow mode, because it is very closely matched to the TEM* 01 doughnut mode, which is a solution of the free-space vector Helmholtz equation. For the 50μm-diameter fiber considered here, the beam divergence is approximately 16 mrad so that the beam diameter is approximately 600 microns at a distance of 2 cm from the fiber tip. Thus, the generated TE01 mode itself can be used as the funnel for loading the fibers from a typical cold atom source. These experiments are currently under investigation in our lab.
Ultimately, the choice of guiding technique depends strongly on the desired experiment, but some general guidelines for consideration are listed here. TE01 mode atom guidance, and blue-detuned guidance in general, are advantageous for situations in which energy level shifts and photon scattering need to be reduced (See Refs. [22, 23] for reviews of optical dipole traps). For blue traps, this is generally achieved using box-like, anharmonic potentials [22–24]. The TE01 intensity profile is harmonic, and while this results in a softer potential, the scattering rate, and therefore the average intensity sampled by the atom, is still substantially reduced. We recently demonstrated this in a harmonic, blue-detuned trap, in which a reduction of 50 in scattering rate was observed . The benefit also dependes on potential depth. As shown in Ref. , blue-detuning is generally advantageous when the optical potential is much greater than the kinetic energy. Furthermore, if a dark, harmonic waveguide potential is desired, then TE01 blue-detuned beam propagation is favorable.
One should also consider whether optical propagation losses can be tolerated. If the experiment benefits from a potential that is independent of propagation distance, then metal-coated fibers using TE01 beam propagation are advantageous, especially for fiber diameters ≤ 50μm (due to the 1/a 3 loss). We note also that metal coated fibers have substantially better bend loss characteristics than uncoated fibers . For situations in which incoherent atom transport over short distances is desired, red-detuned EH11 guidance in uncoated fibers is the most practical to implement.
We have demonstrated the generation and propagation of pure TE01 hollow beams in hollow core optical fibers. These beams are suitable for atom guidance, and have advantages over propagation using red-detuned beams by keeping the atoms in the dark region of the beam. We have measured attenuation lengths in the fibers, and shown an increase in the attenuation length of almost 200% by silver coating the inner walls of the hollow fibers. This work was funded by the Office of Naval Research.
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