1. Introduction

Over the last few years, hollow-core optical fibers (HOF) have been proposed as viable solution in a wide range of applications, spreading from optical communication technology [1,2], to sensing [3] and spectroscopy [4]. In the beginning, hollow photonic bandgap fibers have drawn much attention due to the capability of guiding the light along the central air hole region [5]. Then, other more conventional guidance mechanisms were explored [2,6] and the HOF described in this paper further exploits its potential by guiding the light based on conventional refractive index difference between the ring core and surrounding cladding. This type of HOF has some unique advantages; for example, it is inherently compatible with standard single-mode fiber (SMF) by adiabatic mode transformation and fiber devices can be fabricated by filling the hollow region with proper materials. Moreover, high birefringence can be obtained by engineering the shape of the ring core and the central air hole, endowing a new degree of freedom in fiber design. This allows optical waveguides with novel and unusual birefringence properties, such as, for example, a differential group delay with monotonic wavelength dependence [7]. Actually, in comparison to prior elliptical hollow fibers, the proposed structure can take advantage of its intrinsic polarization dependent, and wavelength dependent mode coupling of the dual-core waveguide structure for flexible differential group delay (DGD) control in the spectral domain.

The elliptical hollow-core optical fiber (EHOF) described in this paper, has been fabricated accordingly to the procedure introduced in [8], and briefly described here for completeness. The first stage of the process consists in grinding two parallel planes along a silica tube (see Fig. 1 ); then, the rest of deposited glass area including the germanosilicate core ring, was fabricated by conventional modified chemical vapor deposition. Afterwards, the preform is partly collapsed, so that, owing to surface tension, the external cross section recovers an almost circular shape. Finally, the fiber is drawn while keeping proper pressure inside the hollow.

 

Fig. 1 Basic description of the EHOF preform manufacturing (figure not in scale).

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The scanning electron microscope (SEM) image of the final fiber cross section is shown in Figs. 2(a) and 2(b), where the central air hole, the doped core and the pure silica cladding appear as black, light-gray and dark-gray areas, respectively. Quite interesting, while the air hole retained a more or less rounded shape, the doped ring became lanceolate. More in details, the air hole is elliptical with major and minor axes about 9.2 and 8.1 μm, respectively, whereas the lanceolate core is about 0.2 μm larger than the air hole, along the minor axis, and almost 3 times longer, along the major one. The entire fiber section is slightly elliptical, with major and minor axes that measure approximately 125 and 110 μm, respectively. The concentration of GeO₂ in doped regions, measured with an electron microprobe analysis system with wavelength-dispersive spectrometry [9], is about 16.5%/mol. Despite of the eccentric core shape, fusion splicing with common single mode fiber was easily carried out with standard splicer without any particular care with insertion losses less than 2 dB. This figure can be improved by an adiabatic tapering.

 

Fig. 2 (a) SEM picture of the cross section of the EHOF. (b) Detail of the doped region. (c) Garnet crystal and adjacent pressure shadows within a fine-grained dark matrix. Garnet diameter measures 2.5 cm.

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The lanceolate shape of the doped area has an evident analogy in natural rock microstructures called “pressure shadows”, represented in Fig. 2(c). In fact, many deformed rocks contain sites with a deviant mineralogy and texture, which can be interpreted as an effect of rearrangement of material by local dilatation and precipitation during deformation [10]. Such “dilatation sites” can be isolated and elongated (veins) or can be flanking rigid objects (pressure shadows). Actually, when flattening occurs in rocks that contain rigid larger grains (porphyroblasts), zones of relatively low pressure develop on the sides of the crystals that are normal to the maximum compressive stress, and material transported in solution may precipitate in the “shadow” of porphyroblasts [10].

In the specific case of the EHOF, the role of the rigid grain is played by the central hole or, more specifically, by the surface tension at the inner surface. Actually, as a result of the surface tensions acting on the outer and inner surfaces, a pressure shadow is created in which the germanium dioxide diffuses, to form the lanceolate doped region.

2. Experimental characterization

An experimental analysis of the guidance properties of the EHOF has been performed with the setup shown in Fig. 3(a) . An SMF patch-cord carrying the power of a CW external cavity tunable laser (ECL) was air-coupled to the EHOF using a high precision XY-stage. The field transmitted by the EHOF was then observed with an analog infrared camera.

 

Fig. 3 Experimental setups: (a) Characterization of modal properties (ECL: external cavity laser; SMF: single mode fiber; Vidicon: infrared camera). (b) Characterization of polarization properties (PC: polarization controller; LP: linear polarizer; PD: photodiode).

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Figure 4(a) shows the images of the transmitted field at 1550 nm for three different input coupling conditions: images (i) and (ii) refer to the case in which the core of the input SMF was aligned with only one of the two peaked doped regions at a time, whereas image (iii) refers to the case in which the SMF was centered on the EHOF, so that both regions were illuminated. To verify if the two peaked doped regions are optically coupled, the input alignment was set so to excite only one of them; then, the wavelength was varied between 1500 and 1600 nm and, correspondingly, no power transfer between the regions was observed at the output. This allows us to conclude that, at least in the observed spectral band, the two peaked doped regions are uncoupled and the EHOF practically behaves as a dual-core optical fiber for a launching condition that ensures the excitation of only one of peaked regions. For this reason, hereinafter the two peaked regions will be called “cores”.

 

Fig. 4 (a) Images of output field at 1550 nm for different alignment of the injecting SMF taken with the infrared camera. (b) Solid red curves: normalized transmittance of the first core and second core for a given input SOP. Dashed blue curves: theoretical best fit. (c) Differential group delay per unit length of the first (blue) and second (red) core. Solid lines refer to experiments and dashed lines to numerical simulations.

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Due to the geometrical asymmetry of each core, we may expect them to be highly birefringent. Polarization properties have been investigated with the wavelength scanning method [11] whose setup is shown in Fig. 3(b). A polarization controller (PC) controls the input state of polarization (SOP) of the probe signal, which is air-coupled to only one core of the EHOF at a time. The emerging signal is passed through a rotating linear polarizer (LP) and the transmitted power is measured by a photodiode (PD) for different orientations of the LP. Five different input SOPs have been used in the experiment and, for each of them, the output polarized power was measured for two orthogonal orientations of the LP, so to compensate for scalar losses. These measurements were then repeated at different wavelengths.

Actually, let A_θ(f) be the transmitted power measured when the LP is rotated by an angle θ; then, the normalized transmittance of the EHOF reads T(f) = [1 + (A₀-A_π/2)/(A₀ + A_π/2)]/2. These normalized transmittance have been calculated from experimental data and then fitted with the function S(f) = {1 + a₀ sin[2πL(δτ₀ + δτ₁Δf)Δf + ϕ₀]}/2, where Δf = f-f₀, f₀ = 193.55 THz (λ₀ = 1550nm), L = 2.97m (EHOF length) and parameters a₀, δτ₀, δτ₁ and ϕ₀ have been determined by least squares method. Note that the term δτ(Δf) = δτ₀ + δτ₁Δf represents the PMD coefficient (differential group delay per unit length [12], ) of the fiber; accordingly, δτ₀ is the PMD coefficient at 193.55 THz (1550 nm) and δτ₁ is the PMD coefficient slope, i.e. δτ₁ = d δτ/df.

As an example, solid red curves of Fig. 4(b) represent the measured normalized transmittance, T(f), as a function of frequency, for a given input SOP, and for each of the two cores; blue dashed curves are the corresponding best fitting functions S(f).

The oscillatory behavior testifies the propagation of two polarization modes in each core. Note also that the beating frequency is different for the two cores but chirped in both cases, meaning that the two cores are slightly different, but both have a frequency dependent DGD. This is quantitatively confirmed by the best fitting parameters; namely, δτ₀ is 0.32 and 0.21 ps/m for the first and the second core, respectively, while δτ₁ is 0.009 and 0.012 ps/m/THz, respectively. The corresponding PMD coefficients are shown in Fig. 4(c), as a function of wavelength, with blue (first core) and red (second core) solid curves. As one can see, the gap between the two experimental curves is about 0.1 ps/m at 1500 nm and broadens as the wavelength increases. This can be attributed to slightly different geometrical shape of the cores (see Fig. 2(b)), likely originated by a slight asymmetry in the preform grinding. Despite the different absolute value, the DGD of two cores behaves similarly; actually, in both cases it decreases quite rapidly with wavelength and is almost halved in 100 nm. An analogous unusual behavior was previously observed only for a photonic band gap fiber [13], which however has a much more complex cobweb cladding and a strongly asymmetric hollow core.

3. Numerical simulations

The modal properties of the EHOF pointed out by the experimental analysis have been confirmed also by numerical simulation. To this aim, we considered a model accounting for the actual shape of the lanceolate core and including also the material birefringence induced by thermal stress during the drawing process [14–16]. To determine this birefringence, we used the plane strain approximation to calculate the stress distribution, σ_x, σ_y and σ_z, on the cross section. This enables the calculation of the local anisotropic refractive indexes as n_i = n₀ + C₂σ_i + C₁(σ_j + σ_k), where C₁ and C₂ are the stress-optic coefficients; n₀ is the refractive index of the unstressed material (calculated using Sellmeier equations) and (i,j,k) is a cyclic permutation of (x,y,z).

The cross section used for simulation has been drawn according to the SEM image shown in Figs. 2(a) and 2(b), and the GeO₂ concentration has been set constant and equal to 16.5%/mol over the whole lanceolate core. The stress-optic coefficients are assumed to be C₁ = −4.19 × 10⁻¹² m²/N and C₂ = −0.69 × 10⁻¹² m²/N and their wavelength dependences are disregarded [15]. The thermal expansion coefficients of core and cladding are α_core = 1.736 × 10⁻⁶ K⁻¹ and α_clad = 5.766 × 10⁻⁷ K⁻¹, respectively, while the Young’s modulus and Poisson's ratio are E_core = 62.02 × 10⁹ Pa and E_clad = 72.45 × 10⁹ Pa, and ν_core = 0.170 and ν_clad = 0.165, respectively. Stress distribution has been calculated assuming that the initial temperature of the fiber is 1020°C and the final working temperature is 20°C [16,17]. The resulting material birefringence, n_x−n_y, is shown in Fig. 5(a) . Eventually, mode analysis of the EHOF has been carried out with finite element method [18].

 

Fig. 5 Numerical results at λ = 1550 nm: (a) Material birefringence, n_x−n_y, induced by thermal stress and calculated with plain strain approximation. (b) Normalized power density of the four lowest modes. The arrows refer to the corresponding electric field vectors.

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Numerical simulation confirmed that in the range 1500~1600 nm only four modes propagate, and their normalized power density at 1550 nm is shown in Fig. 5(b). As observed in the experimental analysis, the fields are mostly confined in the two cores, each of which supports two orthogonal polarized modes.

Figure 6(a) shows the effective refractive indexes, n_eff, of the modes as a function of wavelength, and Fig. 6(b) represents the birefringence B = n_eff,x-n_eff,y of each core (blue circles refer to the first core, whereas red triangles to the second one). The DGD per unit length has been calculated as δτ = (B − λ dB/dλ)/c [19], and the result is shown with dashed lines in Fig. 4(c). As one can note, numerical simulations and experimental results have a qualitative agreement, for both have the same monotonic dependence on wavelength; however, numerical simulations overestimated the DGD of the guided modes to result in an off-set in Fig. 4(c). This might be attributed to assumption of uniform GeO₂ concentration over the lanceolate core in the simulation.

 

Fig. 6 Numerical results: (a) Effective index vs. wavelength for the 4 propagating modes. (b) Birefringence of the two cores.

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4. Conclusions

An elliptical hollow fiber with dual-core guidance properties has been investigated, both experimentally and numerically. Measurements and simulations show that the cores are not coupled and two modes with orthogonal polarization propagate in each of them. The cores exhibit fairly high DGD per unit length, decreasing almost linearly with wavelength.

As an example of application, this EHOF can be used as an integrated Mach-Zender interferometer, in which the two cores play the role of the two arms [20]. This design has an improved stability, because “common mode” noise acts equally on both cores and thus is effectively rejected. Furthermore, the possibility of filling the hollow with proper material and the peculiar birefringence properties, suggests that this kind of fiber may be used as a basic building block of highly sensitive polarimetric fiber sensors.