Hybrid transmission bands and large birefringence in hollow-core anti-resonant fibers

Wei Ding; Ying-Ying Wang

doi:10.1364/OE.23.021165

1. Introduction

Recently developed low attenuation [1–3 ], low bending-loss [4], broad working-bandwidth [5], and high damage-threshold [6,7 ] hollow-core anti-resonant fibers (HC-ARFs) with negative curvature (or hypocycloid) core-surrounds and a variety of cladding structures [1,2,8–10 ] may prove to be a promising workbench for many important optical applications ranging from gas-based nonlinear optics [5,11–13 ], ultra-intense pulse delivery/compression [6,7 ] to low-latency data transmission [14,15 ], precise interferometric sensing [16] and quantum optics [17]. Though encouraging performances have been demonstrated in laboratory environments, HC-ARFs still confront difficulties when marching into real applications, probably owing to their poor modal control properties.

Fiber’s modal control includes two aspects, i.e. the transverse modal control and the polarization control. For a HC-ARF, the transverse modal control can be achieved by properly designing nested air holes in the cladding region to resonantly out-couple higher-order modes [14]. On the other hand, till now, in HC-ARFs, effective polarization control is yet to be demonstrated. In order to advance environmental stability and lifetime for real applications, many optical systems urgently need polarization maintaining capability, i.e. birefringence [16–19 ].

Achieving large birefringence in a hollow-core fiber (HCF) is always challenging because of the lack of mechanical stress. In a well-confined core mode of a HCF, light tends not to overlap with glass, making it difficult to attain substantial birefringence from any intentional geometric asymmetry [20]. In a hollow-core photonic bandgap fiber (HC-PBGF), this issue can be solved by deliberately exploiting the anti-crossing effect between the core mode and the surface mode. A combination of large birefringence (Δn ~3 × 10⁻⁴) and low loss (10 dB/km @ 1530 nm) has been achieved in sacrifice of fiber’s transmission bandwidth (<10 nm) [21]. In ARFs, however, the broadband transmission requirement restrains the utilization of the anti-crossing effect. Conventional methods of introducing geometric asymmetries, like using an elliptical core or varying thickness of the core-surround, have been attempted [22–25 ] but are far from success. New techniques are in demand.

In search for new routes for large birefringence, one has to go to the origins of light guidance in HC-ARFs. However, it is deemed that the guidance mechanism of this fiber has yet to be fully elucidated [11]. In contrast, another type of hollow-core fiber, i.e. PBGF, borrows the concept and analytical tools from the field of solid-state physics [12,26,27 ] and exploits the photonic bandgap effect of the two-dimensional (2D) periodic claddings [28,29 ] with translational symmetry. As a result, optimum performances of PBGFs are predicted in hexagonal lattice structures with a large cladding layer number. In ARFs, a commonly accepted viewpoint is that light guidance is related to the anti-resonant reflection of glancing incidence light impinging on the glass webs [30,31 ]. However, this simple picture of anti-resonant reflecting optical waveguide (ARROW) [30,31 ] only considers individual elements of glass walls and ignores other factors like the cladding arrangements, thus cannot explain many subtleties of ARFs such as ultralow confinement loss, spiky peaks in transmission spectra, etc. Moreover, in ARFs, the translational symmetry and large number of cladding layers seem not essential [32] and high-performance ARFs can be realized in very simple geometries with only 1-2 cladding layers [2,33,34 ]. These results imply that to analyze density of state (DOS) of the cladding lattice like PBGFs do may mislead our understanding to ARFs and analysis on other structural symmetries rather than the translational one may be more helpful.

Bearing above considerations, we have proposed a semi-analytical method to calculate confinement loss spectra of single-wall HC-ARFs [35]. We pointed out that the attenuation coefficient can be derived from the overall energy leakage in the transverse plane, which can be integrated by using a 2D Green’s theorem. Meanwhile, the phase of the electric field at fiber’s outermost boundary tends to be locked at a fixed value (i.e. forming a wavefront). This phase-locking feature can be exploited to simplify a 2D waveguide problem to a series of 1D slab under a very coarse geometry transformation. After achieving approximate agreements with numerical simulations across wide wavelength range, we further tested extending our semi-analytical model to more complicated ARFs [24]. Instead of focusing on the lattice periodicity, i.e. the translational symmetry, we use the picture of multi-layered dielectric structures surrounding an air core [36].

To demonstrate the power of our analytical calculation method, which should allow one to foresee more than numerical method does, in this paper, with regard to the polarization properties, we demonstrate a HC-ARF design procedure guided by our analytical model. We start from a single-wall toy ARF with desired properties and then extend to an experimentally realizable fiber structure, so-called multi-layered ARF. A new type of ARROW transmission band is engineered in a controllable way, allowing HC-ARF to possess low loss, broad transmission bandwidth and large birefringence simultaneously.

2. Calculation and analysis

In [35], we pointed out that the phase of the electric field at the outermost boundary of a single-wall ARF tends to be locked at a certain value determined by the order of the ARROW band. This feature lowers the complexity of estimating the complex value of the electric field at those positions and leads to a semi-analytical model to quantitatively calculate attenuation coefficient. However, in [35], our treatments are limited in the frequency regions of normal ARROW bands. In this paper, we extend our calculations out of these normal bands, so that different segments of glass walls could have different thicknesses and may correspond to different ARROW band orders.

Inset of Fig. 1(a) illustrates a square-shape single-wall ARF toy structure having two different glass thicknesses. In the horizontal direction, the inscribe radius (a ₁) and the glass thickness (t ₁) are 9.76 μm and 0.67 μm respectively. While, in the vertical direction, these geometric parameters become a ₂ = 9.76 μm and t ₂ = 0.54 μm. The refractive indices of air and glass are n ₁ = 1 and n ₂ = 1.45 respectively. According to the ARROW principle, the frequency range of the m ^th-order transmission band can be expressed as,

\frac{c \cdot (m - 1)}{2 t_{1, 2} \cdot \sqrt{n_{2}^{2} - n_{1}^{2}}} < f < \frac{c \cdot m}{2 t_{1, 2} \cdot \sqrt{n_{2}^{2} - n_{1}^{2}}}, m = 1, 2, \dots

where c is the velocity of light in vacuum. The two glass thicknesses, t ₁ and t ₂, give rise to two sets of ARROW resonant frequencies. Consequently, Fig. 1(a) plots two types of ARROW bands. The frequency regions where both t ₁ and t ₂ shares the same value of m are the normal bands marked in cyan, while those frequency regions where t ₁ and t ₂ result in different m’s (e.g. the frequency region 218-265 THz corresponds to m = 2 for t ₂ = 0.67 μm and m = 1 for t ₂ = 0.54 μm) give rise to a new type of ARROW band in magenta color which, to the best of our knowledge, has yet to be explored. In this paper, we name it as hybrid ARROW band.

Fig. 1 (a) Simulated loss and effective index spectra of the fundamental core mode in the two polarizations for a square-shape single-wall ARF (depicted in the inset). For the horizontal polarization, the amplitude and phase distributions of the x-component of the electric field are plotted at the frequencies of (b) 174 THz and (c) 243 THz respectively. (d) Simulated and (e) modeled differences of the real (Δn^r, black curves) and the imaginary (Δnⁱ, red curves) parts of the effective modal indices between the two polarizations. Blue circles in (e) are the results of Eq. (2a) where the integration is only carried out in the frequency regions marked by gray.

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Figure 1(a) plots the simulated confinement loss and effective index spectra in two polarizations by using a finite-element mode solver (Comsol Multiphysics with optimized mesh and perfectly matched layer configurations [14,37,38 ]). The simulations are implemented for the fundamental core mode, i.e. the HE₁₁-like leaky mode. Based on the definition of Eq. (1), we observe light transmission in both normal and hybrid bands. However, for a frequency in one normal band, e.g. 174 THz, the phases of the electric fields at all the outermost boundaries are locked to a single value of −180° [Fig. 1(b)], whereas, for a frequency in one hybrid band, e.g. 243 THz, the locked phases are different for the glass walls having different thicknesses [Fig. 1(c)]. Within this hybrid band, the vertical (horizontal) glass walls are embodied with the 2nd-order (1st-order) ARROW band nature, which forces the phase of the electric field at the outermost boundary to be locked to 0° (−180°). Note that, in the phase diagrams of Figs. 1(b,c), there are some fast oscillatory components in the vicinity of glass walls. These have been recently recognized to have some Fano-resonance origins and will not significantly influence the baseline of the attenuation spectra [39]. In the following, we ignore these fast oscillatory field components.

Altogether, Fig. 1(a) tells us hybrid ARROW bands could guide light and their phase- locking characters slightly differ with those of normal ARROW bands. Incorporating these new phase-locking characters into our semi-analytical model, we calculate the structure of Fig. 1(a). Excluding the Fano-resonance related spiky features, the modeled loss and effective index spectra across all the normal and hybrid bands (raw data not shown for simplicity) agree well with the simulated results in the accuracy of <5%.

Now, we focus on the polarization properties of the above structure. Figures 1(d) and 1(e) show the simulated and the modeled results of the differences of the real and the imaginary parts of the effective modal indices for two polarizations. Large birefringence of ~10⁻⁴ in the 1st hybrid band is observed in both two diagrams. To apprehend this, we turn to the Kramers- Kronig relation [40,41 ], which is believed to universally exist between loss and dispersion (i.e. imaginary and real parts of effective modal index) of waveguide. Although rigorous derivative of the K-K relation for waveguide needs to take into account all the modes and all the inter-modal field overlaps [41] (more importantly, such a derivative may be not applicable to leaky mode), we still empirically write out the K-K integration formulae as follows,

Δ n^{r} (ω) \approx \frac{2}{π} \int_{0}^{+ \infty} \frac{Δ n^{i} (ω') \cdot ω'}{ω'^{2} - ω^{2}} d ω'

Δ n^{i} (ω) \approx - \frac{2}{π ω} \int_{0}^{+ \infty} \frac{Δ n^{r} (ω') \cdot ω'^{2}}{ω'^{2} - ω^{2}} d ω'

Here,

Δ n^{r} \equiv n^{r}_{V} - n^{r}_{H}

and

Δ n^{i} \equiv n^{i}_{V} - n^{i}_{H}

represent the birefringence and the polarization- dependent loss (PDL), respectively. For both of them, our semi-analytical model can give out estimated values even in the frequency regions very close to ARROW resonances, which facilitates the integrals in Eq. (2) . In contrast, numerically simulating effective modal index in the frequency regions close to resonances is very difficult, even though not impossible. In Fig. 1(e), we carry out Eq. (2a) and integrate the red curve, i.e.

Δ n^{i} (ω')

, only in the frequency regions marked by gray, where Δnⁱ is notably non-zero. The integral results are presented in Fig. 1(e) as blue circles, qualitatively fitting with the black curve. Note that, in order to avoid mathematical ambiguities, our integration omits the frequency regions where the denominator

ω'^{2} - ω^{2}

could be zero.

The K-K relation indicates that the large birefringence in the 1st hybrid band originates from the shape of the PDL spectrum. Along the red curve in Fig. 1(e), we see alternatively spaced dips and peaks (e.g. the ones around 215 THz and 267 THz). Their positions coincide with the low and high frequency edges of the hybrid bands, and their depth and height are nearly equal. Via the K-K integrations [Eq. (2a)], these features of the PDL spectrum result in large (small) birefringence in hybrid (normal) transmission bands because the integral contributions from the dips and the peaks have the same (opposite) signs in hybrid (normal) bands. Using the K-K relation, we interpret why the largest birefringence appears in the 1st hybrid band. More importantly, Fig. 1(e) predicts birefringence (~10⁻⁴) much larger than those in the normal bands of ARF [22–25 ]. The mechanism underneath this finding not only differs with solid fibers [42] but also differs with HC-PBGFs [20,21 ]. For solid fibers, mechanical stress is usually the dominant method to produce birefringence; for HC-PBGFs, anti-crossing effect between core mode and surface mode plays the role to produce birefringence in a narrow band; while here, for a HC-ARF, hybrid ARROW band does the work.

To intuitively understand above birefringence characteristics, we can also analyze the transverse field confinement of the core mode. It is known that the effective index of the core mode decreases as the transverse field confinement tightens. In the case of no transverse field confinement, the core mode will be degraded to a plane wave with an effective index of 1. In Fig. 2 , we fix the inscribed core radius (9.76 μm) and the working frequency (243 THz), and analyze the field distributions of the core mode in different ARROW bands by varying the glass wall thickness. In Figs. 2(a) and 2(b), the glass thickness T1/T2 is set to be one and two times of t ₁/t ₂ respectively, and the ARF stays in the 1st and 2nd hybrid band at the working frequency. In Figs. 2(c) and 2(d), T1/T2 is set to be 0.5 and 1.5 times of t ₁/t ₂ respectively, and the ARF is in the 1st and 2nd normal band. For both vertical and horizontal polarizations of the fundamental core mode, we plot normalized electric field amplitudes along two line cuts [the red and the blue lines depicted in Figs. 2(e) and 2(f)]. Standing waves with nodes (zero points) are distributed in the inward side of the outermost boundary. The number of the nodes equals to the order of the standing waves, i.e. the order of the ARROW band. For example, in Fig. 2(a), for both polarizations, the glass wall T1/T2 exhibits the 2nd-order/1st-order ARROW band nature with 2/1 nodes of standing wave. Obviously, because of these nodes, the glass wall T1 pushes the modal field toward the fiber axis (O) much harder than the glass wall T2. Hence, the vertical glass walls play the major role for field confinement. On the other hand, comparing the two blue curves in Fig. 2(a), the glass wall T1 exerts a tighter field confinement when the core mode is in the horizontal polarization. It is because a glass slab applies the tighter field confinement to a p-polarization wavelet than to an s-polarization wavelet [43]. As a result, the horizontal polarization core mode has a smaller effective index than the vertical one, leading to notable birefringence in the 1st hybrid band. We can also see the distinction of the transverse field confinements in two polarizations from the 2D contour plots [Figs. 2(e) and 2(f)]. In a similar way, by analyzing the transverse field confinement in Fig. 2(b) for the 2nd hybrid band, one can also draw a conclusion that $Δ n^{r} = n^{r}_{V} - n^{r}_{H}$ is positive but smaller than that in the 1st hybrid band. On the other hand, in Figs. 2(c) and 2(d), i.e. the normal bands, the thinner glass wall T2 pushes the modal field toward the fiber axis a little bit harder than the thicker glass wall T1. This explains why the birefringence Δn^r becomes negative and much weaker than those in hybrid bands. All these analyses match well with Fig. 1. Additionally, a similar analysis can have an implication that a square-shape ARF, where the distinction of different glass walls is maximized under two polarizations, is preferred for large birefringence.

Fig. 2 Simulated electric field amplitudes along two line cuts (depicted in the inset). All the data are normalized by the values in the fiber axis (O). For each polarization of the core mode, only the major field component is plotted. The descriptions of (a) the 1st and (b) 2nd hybrid bands and (c) the 1st and (d) 2nd normal bands are given in the text. t ₁ = 0.67 µm, and t ₂ = 0.54 µm. Linear-scale contour diagrams (e) and (f) show the field amplitude distributions in the two polarizations for the case (a). The farthest contour lines stand for 0.2 times of the peak values.

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Using the aforementioned two analysis approaches, we can elucidate the situations of the conventional means of producing birefringence in ARFs by introducing geometric asymmetries. Figure 3 calculates and simulates two typical single-wall ARFs incorporating with geometric asymmetries [22–25 ]. In the first case [Figs. 3(a) and 3(b)], a circular core with the radius of 9.76 µm is surrounded by a glass wall having a continuously varying thickness from t ₁ = 0.67 µm to t ₂ = 0.54 µm. It is seen that the hybrid transmission bands disappear because all the wavelengths in these regions hit some resonance corresponding to a certain glass thickness. In the second case [Figs. 3(c) and 3(d)], an elliptical core (9.76 µm × 8.74 µm) is surrounded by a uniformly-thick glass wall, which only has normal ARROW bands. Therefore, in a conventional ARF structure, large birefringence is difficult to attain because of the lack of the hybrid transmission bands.

Fig. 3 (a,c) Modeled and (b,d) simulated losses and effective indices under two polarizations. Geometries of two single-wall HC-ARFs are depicted in the insets.

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3. Experimentally realizable structure

Based on above analyses, we conclude that in order to obtain large birefringence in a realistic HC-ARF, it is better to use the 1st hybrid band and a square-shaped core. Considering practical fabrication restrictions, Fig. 4(a) presents our first fiber design, where one layer of curved glass wall having two different thicknesses is arranged with the 4-fold symmetry. All the glass webs are connected to an annular outer jacket and do not touch with each other [14,38,44 ]. After verifying that the 1st hybrid band and large birefringence (~10⁻⁴) are both preserved, in Fig. 4(b), we add two more layers of glass webs into this structure to strengthen light confinement, forming a multi-layered ARF. The rings r ₁/r ₂/r ₃ in Fig. 4(b) are concentric circles. Our design concept partly coincides with Bragg fiber [36] and a recently proposed nested ARF [14]. In all the three structures, no translational symmetry is exploited and light confinement enhances as the layer number increases. However, different with Bragg fiber, our structure adopts a negative curvature core, and, different with the nested ARF, our fiber uses multi-layered surrounds. As a result, our multi-layered ARF is more realistic than Bragg fiber in the context of single glass material, and our specially-designed glass surrounds could preserve the hybrid ARROW bands. Figure 4(c) shows the simulated loss and effective index spectra of the geometry in Fig. 4(b). The 1st hybrid band appears with a relatively low attenuation (the lowest value: ~0.34 dB/m @ 1190 nm for the horizontal polarization). The phase-locking features are manifested in Figs. 4(d) and 4(e) for the normal and the hybrid band respectively. Large birefringence (~10⁻⁴) is presented in a broad wavelength range of 60 nm [fractional bandwidth of ~5 percent, Fig. 4(f)], much wider than the HC-PBGF in [21] (<1 percent, <10 nm). Note that for our multi-layered ARF the transmission properties in the normal bands (e.g. ~10 dB/km in the 1st normal band and <0.1 dB/km in the 2nd normal band) are comparable with the nested ARFs in [14]. The confinement loss for higher order modes (HOM) in the 1st hybrid band has also been simulated and shows a level of 10 dB/m for the lowest HOM, indicating that this proposed fiber has a good single-modedness property.

Fig. 4 Schematics of (a) a realistic single-wall ARF and (b) a realistic triple-layered ARF. a = 9.76 μm, R = 27.76 μm, t ₁/t ₂ = 0.67/0.54 μm, and r ₁/r ₂/r ₃ = 23/17/11 µm. (c) Simulated loss and effective index spectra of the structure (b). For the vertical polarization, the phase distributions of E_y are plotted at the frequencies of (d) 174 THz and (e) 250 THz. (f) Simulated birefringence. (g) A stack design suited to fabricate the ARF of (b).

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Practical applications of HC-ARFs may require different levels of birefringence, transmission bandwidth and attenuation. Indeed, these three properties can be compromised by adjusting structural parameters. By altering the glass wall thicknesses t ₁ and t ₂, a balance between the birefringence and transmission bandwidth can be achieved. A smaller core size results in a larger birefringence, however, in sacrifice of transmission loss. The number of rings, the ring curvature and the distance between rings also give the degree of freedom in adjusting the fiber performances and the structure complexities.

With regard to fiber fabrication, our design should be conceivable. With the advances of fiber fabrication skills, especially those used in hypocycloid-core Kagome fibers [1] and negative curvature fibers [2], drawing the multi-layered ARF should be feasible, at least in a relatively large scale for mid-infrared light guidance. In Fig. 4(g), we give a possible stack design for this purpose. Different sizes of rods and capillary tubes constitute a pattern alike Fig. 4(b). This pattern is easy to maintain in the stack-cane process. In the cane-fiber step, careful control to the pressurization and vacuum lines inside the canes and relatively cold drawing with high tension is preferred to maintain the curvy structures.

4. Conclusion

In conclusion, we identify, for the first time to our best knowledge, hybrid ARROW transmission bands, where differently-thick glass walls stay in different ARROW band orders and lock the phase of the electric field at the outermost boundary to different values. By empirically applying the Kramer-Kronig relations for waveguides and analyzing the mode field confinement, we interpret why large birefringence (~10⁻⁴) appears in the 1st hybrid band. Furthermore, we manage to preserve hybrid ARROW bands and large birefringence in a realistic ARF design. In this procedure, in order to lower confinement loss, we adopt the design concept of multi-layered dielectric structure like Bragg fiber does. Our final multi-layered ARF design simultaneously possesses a ~1 dB/m attenuation coefficient (~0.34 dB/m lowest), a ~10⁻⁴ birefringence, and a broad bandwidth of ~60 nm. This is the first time our semi-analytical model links to a realistic ARF structure having practical applications.

In conjunction with our previous works [24,35 ], we preliminarily present a new design guideline for ARFs based on a semi-analytical modeling method. Instead of exploiting the translational symmetry in the cladding region, like what has been done in PBGFs, we focus on the symmetry in the normal directions of glass webs. Based on our studies, we can reasonably infer that, for an ARF, in order to improve optical properties, engineering core-surround from the inside out to construct a multi-layered dielectric structure may be more efficient than solely modifying its cladding.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61275044, 61377098, and 11204366), the Instrument Developing Project of the Chinese Academy of Sciences (No. YZ201346), the Beijing National Science Foundation (No. 4142006), and the State Key Laboratory of Advanced Optical Communication Systems Networks, China.

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Hybrid transmission bands and large birefringence in hollow-core anti-resonant fibers

Abstract

1. Introduction

2. Calculation and analysis

3. Experimentally realizable structure

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (3)

Optics Express