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In this work, dual-core antiresonant hollow core fibers (AR-HCFs) are numerically demonstrated, based on our knowledge, for the first time. Two fiber structures are proposed. One is a composite of two single-core nested nodeless AR-HCFs, exhibiting low confinement loss and a circular mode profile in each core. The other has a relatively simple structure, with a whole elliptical outer jacket, presenting a uniform and wide transmission band. The modal couplings of the dual-core AR-HCFs rely on a unique mechanism that transfers power through the air. The core separation and the gap between the two cores influence the modal coupling strength. With proper designs, both of the dual-core fibers can have low phase birefringence and short modal coupling lengths of several centimeters.
© 2016 Optical Society of America
1. Introduction
Hollow-core optical fibers have attracted a great deal of interest over the past decade within fiber optics, due to their advantages in terms of high power and ultra-short laser pulse delivery [1], pulse compression [2], mid-infrared (mid-IR) transmission [3, 4] and THz guidance [5]. Two major categories of single-material hollow-core fibers have been studied so far. One is the hollow-core photonic bandgap fiber (HC-PBGF) [6], which guides light in the air core using a periodic cladding structure, showing a photonic band gap. The other is the hollow core antiresonant fiber (HC-ARF) that exhibits much broader transmission bandwidth and much lower core-mode overlap with the cladding, which means smaller nonlinear effects and higher material damage threshold. Over the last few years, the core wall shape of the HC-ARF with negative curvature has been considered helpful to reduce transmission loss [7]. Recently a new version has emerged, consisting of very simple cladding of one single layer or nested untouched capillaries, which can further reduce the leakage loss [8, 9].
On the other hand, fiber based elements are important research subjects for integrations in fiber laser sources and fiber laser deliver systems. Among them, dual-core fibers, as the basis of fiber couplers, are the most fundamental fiber elements for splitting or combining powers. Previously, HC-PBGF based dual-core fibers were theoretically and experimentally studied [10–12]. However, dual-core HC-ARFs has not appeared so far to our knowledge. Considering the HC-ARFs advantages over other fibers, a HC-ARF based fiber coupler will be a promising trend to deal with high-energy laser pulses in future all-fiber ultrafast laser systems.
In this work, the initial difficulties hindering realization of the HC-ARF based couplers are qualitatively analyzed. Then, we propose and numerically investigate two types of dual-core HC-ARF structures. One is a composite of two single-core nested nodeless HC-ARFs, which exhibits low confinement loss and preserves the circular shape of supermode fields in each core. The other has a relatively simple structure, where the whole outer jacket is an elliptical tube, and presents a more uniform and wide transmission band in the near-infrared region. The unique modal coupling mechanism of the dual-core HC-ARFs that thoroughly occurs in air, named “modal leakage guidance”, is different from other dual-core fibers. The influence of two critical geometric parameters on the modal coupling strength is studied. It is demonstrated that with proper designs, both of the dual-core HC-ARFs can exhibit low phase birefringence and short modal coupling length.
2. Fiber designs and numerical simulations
It is well known that typical fiber couplers rely on interactions between core-mode evanescent fields. Specifically, two cores are placed sufficiently close so that part of the individual core modal fields will overlap each other through the cladding, leading to power transmission from one core to the other. Therefore, the overlap between the core mode and the cladding mode fields is necessary for modal couplings. For the HC-PBGFs, even though most of the core modal field is confined by the photonic bandgap, a small fraction of this field still enters the solid-state cladding materials via the modal couplings between the core and the surface modes in the core-cladding interfaces. Therefore, the modal field overlap of two close cores is still possible. However, this coupling mechanism becomes invalid for the HC-ARFs. Figure 1(a) shows the structure of a nested nodeless HC-ARF [8]. The hollow fiber core is formed by the surroundings of a ring of eight equally-spaced untouched large-diameter capillaries. From the “ARROW” model [13], modal fields with wavelengths of
where t is the capillary thickness, n is the material refractive index (RI), and m is a positive integer, will be in resonance with the cladding capillary and localized within the capillary materials. For other non-resonant wavelengths, core mode can almost not exist in the capillary material, and is reflected back by the core-cladding interface. Thus, the core mode is formed due to the cladding confinement effect, as shown in Fig. 1(b). The field intensity contour visually depicts the inwards expelling effect on the core modal field by the cladding capillaries. The nested smaller silica tube in each capillary and the so-called nodeless structure can both enhance the inhibition of modal couplings between the core and the cladding modes and thereby reduce the leakage loss. Though the tight confinement on the core mode brings an extremely low transmission loss in theory, it constitutes an obstacle for field diffusion outwards. Furthermore, instead of the dense lattice structures as in the HC-PBGF cladding, a single layer of large-diameter capillaries prevents the physical approaching of the two cores, and the intensity of the core mode field nearby the fiber jacket is too low.
Fig. 1 (a) The structure of a nested nodeless HC-ARF. (b) Simulated core modal field profile and the electric field contour plot. The field intensity values increase from blue to red.
To achieve modal couplings, a straightforward solution is to deform the outer structure of two identical single-core HC-ARF as shown in Fig. 1(a), and then approximate the two cores, as shown in Fig. 2(a) (“composite circular dual-core fiber” for short). In each fiber component, two adjacent cladding capillaries are removed and part of the jacket at the same side is sliced out. When these two fiber components are spliced together by side, the hollow core regions are conjoined, resulting in modal coupling. Given that modal coupling depends on the modal field leakage through the gaps between the two cores, this coupling mechanism is named modal leakage guidance. Notice that in all the other fiber couplers (including the HC-PBGF coupler in [12]), the core mode field must enter the cladding materials to transfer the power. However, in virtue of the modal leakage guidance here, the core mode field in the dual-core ARF is thoroughly localized in the air, and not involved in the cladding. Otherwise, the advantage of the HC-ARFs, i.e., small modal overlap between core mode and cladding regions would shrink and the transmission loss level would increase rapidly, which is against the demands for high-intensity laser applications.
Fig. 2 (a) The proposed composite circular dual-core HC-ARF structure; The simulated y-polarized (b) antisymmetric (odd) and (c) symmetric (even) supermode field profiles. The field intensity values increase from blue to red.
Similar to other dual-core fibers, modal coupling turns one individual core mode into a pair of supermodes, one even (symmetric) and the other odd (antisymmetric). As the structure symmetry of an individual fiber is broken, each core mode is no longer degenerated, but split into two orthogonal polarizations. Thus, at least four supermodes are involved in the dual-core fiber. The shortest distance at which input power is totally transferred from one core to the other, called the coupling length Lc, is inversely proportional to the coupling strength, as
Here, β is the propagation constant of a supermode. Then, the phase birefringence (PB) of the dual-core fiber can be evaluated by the difference between Lc,x and Lc,y, asFigure 2(b) shows the y-polarized antisymmetric supermode profile, with one positive and one negative intensity peaks that are located in the two cores. Figure 2(c) shows the y-polarized symmetric supermode profile, in which both cores are located in the positive intensity peaks. From the coupled-mode theory, the supermode profile in one core must be different from that of a single-core fiber, or else no coupling would occur. However, if the difference is large, the modal matching efficiency between couplers and single-core fibers would be low, so a tradeoff is needed. To evaluate this difference, the normalized electric field intensity within the circular domain with half fiber core diameter (surrounded by the dashed in Fig. 2(b) and 2(c)) are integrated for the single-core fiber and the dual-core fiber, respectively. As the case in Fig. 2, 59% fundamental mode electric field distributes within the circular domain for the single-core fiber, and 58% for the dual-core fiber. Such a small distinction embodies the dual-core fiber mode profile basically remains circular, which is attributed to the two additional tubes to reduce the gap between the two cores, added in the middle of the dual-core fiber, one up and one down, each touching adjacent tubes for physical holding, as shown in Fig. 2.The properties of the dual-core HC-ARFs are calculated using the software COMSOL Multiphysics, based on the 2D full-vector finite-element method with perfectly matched layers (PML). In the calculations, the RIs of air and silica are 1 and 1.45, respectively. Dual-core and single-core fibers share the same geometric parameters as follows: the fiber core diameter is Dcore = 30 μm, the diameters of the outer and inner capillary diameters are Dcl1 = 15.8 μm and Dcl1 = 8.3 μm, and they have identical thickness of t = 0.55 μm. The separation between the adjacent outer capillaries is d1 = 5t = 2.75 μm and the fiber jacket thickness is T = 10 μm.
In dual-core fiber designs, the most relevant parameters are the separation between the two cores, dcore, and the gap between the additional tubes, d2, both of which significantly influence Lc. Figure 3 shows the x/y-polarized Lc dependence on dcore (normalized to Dcore) at d2 = d1, d2 = 2d1, d2 = 3d1 and d2 = 4d1, for composite circular dual-core HC-ARF. The coupling length increases almost linearly with the core separation, which coincides with the general rule of modal couplings. However, the slope decreases with the enlargement of additional-tube gaps, indicating that the gap expansion tends to dominate the modal leakage guidance. On the other hand, Lc,y is longer than Lc,x and the difference between them is relatively large with small gaps. However, this difference becomes smaller as the gap expands, and even Lc,y turns to be shorter than Lc,x, as shown in the inset plot of Fig. 3. This is reasonable since the modal leakage channel lies along the x-direction. Thus, the x-polarized field is more readily transversely guided than the y-polarized field with small gaps. Nevertheless, with the gap enlargement, the structure asymmetry along the y-direction becomes more important. Then the y-polarized field guidance through the channel is dramatically enhanced, exhibiting a rapid increase of the modal coupling strength, or a decrease of the modal coupling length.
Fig. 3 The coupling length of the composite circular dual-core fiber (Lc,x and Lc,y) dependence on the normalized core separation dcore/Dcore with different additional tube gaps. The inset plot shows the local magnification for d2 = 4d1, with Lc,x and Lc,y very close.
Next, the dependence of the confinement loss and coupling length on wavelengths are calculated with dcore/Dcore = 1.1 and d2 = 4d1. The x/y-average confinement loss spectrum of the composite circular dual-core HC-ARF also presents a shape similar to the single-core NC-HCF, as shown in Fig. 4(a). In the short wavelength region, dual-core and single-core fiber have identical loss levels. However, as the wavelength increases, the loss curve of the dual-core fiber decreases to the minimum around 1.5 μm, and then increases earlier than the single-core fiber. In the long wavelength region, the loss increase for the dual-core fiber is partly because the tight confinement on an individual core mode is loosened by the modal leakage guidance and partly because the touching effect between the additional tubes with adjacent cladding tubes results a small amount of field scratches the cladding tubes.
Fig. 4 (a) The average confinement loss spectra of the two proposed dual-core HC-ARF with dcore/Dcore = 1.1 and d2 = 4d1, in comparison with that of the single-core HC-ARF. (b) The spectra of Lc,x and Lc,y (left axis), as well as the PB (right axis) for the composite circular dual-core fiber and (c) for the elliptical dual-core fiber.
Figure 4(b) gives the spectra of the coupling lengths in x/y-polarizations for the composite circular dual-core fiber. As the wavelength increases, the coupling length curves rise up until around 1.45 μm, and then fall down. In fact, this trend is the opposite of the confinement loss spectrum given in Fig. 4(a). On the other hand, low loss means tight confinement on individual core modes, which in the meanwhile weakens the modal leakage guidance between the two cores, and vice versa.
Regarding modern fiber fabrication techniques, besides the relatively complicated composite circular dual-core fiber, we propose a simple elliptical dual-core HC-ARF structure, as shown in Fig. 5(a). The whole outer jacket is a big elliptical capillary, and two small elliptical tubes arranged in the middle can bisect the capillary, forming two separated cores. Very recently, a single-core HC-ARF with elliptical cladding tubes was proposed in Ref [14]. In fact, elliptical cladding tubes have already been fabricated [15]. Calculations reveal that the design of this dual-core fiber structure follows the rules of the former proposed one. In short, the closer the cores or the larger the core gaps, the smaller the coupling length. In Fig. 5(a), the half major axes of the big and small elliptical tubes are a1 = 55 μm, a2 = 17.25 μm, and the half minor axes are b1 = 40 μm, b2 = 8.25 μm. To simplify, the other untouched cladding tubes are circular, with identical diameters of Dcl = 22 μm. The core diameter is still 30 μm, with dcore/Dcore = 1.1 and d2 = 4d1 = 11 μm. As shown in Figs. 5(b) and 5(c), the mode profile in each core is not as circular as in the former dual-core fiber, because the surrounded cladding tubes are not all rings and are no longer approximately centrosymmetric distributed. Within the low confinement-loss wavelengths, the loss-level for the elliptical dual-core fiber is also higher than the former, as shown in Fig. 5(a). One reason is that nested cladding tubes were not used, and the other is the negative curvature of the ellipse is not as strong as a circle when tangential to the core region. However, the coupling length for both proposed dual-core fibers can be as short as several centimeters, as shown in Figs. 4(b) and 4(c). Further, the elliptical structure even exhibits shorter coupling lengths than the composite circular one, which means that loss discrimination can be neglected in future power coupling applications. Moreover, in the wavelength range from 1.3 μm to 2.0 μm, the loss spectrum of the elliptical dual-core fiber is more uniform than the other dual-core fiber. In the long wavelength region, the loss level is even lower than the single-core fiber. This property supports wide-spectrum laser applications, e.g., the femtosecond laser delivery.
Fig. 5 (a) The proposed elliptical dual-core HC-ARF structure. The simulated y-polarized antisymmetric (odd) (b) and symmetric (even) (c) supermode field profiles. The field intensity values increase from blue to red.
Figures 4(b) and 4(c) give the PB spectra for the two types of dual-core fibers, according to Eq. (3). The PB values are always very small, in the order of 10−6. Thus, in theory the modal couplings of these two dual-core fibers are not sensitive to input polarization. Low PB can reduce the azimuthal alignment requirements for connecting the dual-core fibers with single-core fibers, the details of which are in the scope of further research.
3. Conclusion
In summary, we proposed two types of dual-core HC-ARFs. One is a composite of two single-core nested nodeless with lower confinement loss and more circular mode profile in each core. The other has a relatively simple structure that has an elliptical capillary as the outer jacket. The modal couplings rely on the mechanism of leakage modal guidance that thoroughly occurs in air. The modal coupling length is mainly affected by the core separations and the gaps between additional tubes. The confinement loss of the dual-core fibers are higher than that of a single-core fiber, but the elliptical dual-core fiber has a more uniform and wide transmission band. With proper designs, the phase birefringence can be low, and the modal coupling lengths for both dual-core fiber types can be as short as several centimeters, thus favoring the realization of the HC-ARF coupler in practice.
Funding
Innovation Program of Academy of Opto-Electronics (AOE) of Chinese Academy of Science (CAS) (Y50B14A13Y); Chinese Innovative Talent Promotion Plans for Innovation Teams in Priority Fields (2014RA4051); International S&T Cooperation Projects (2014DFR50600); Project of CAS Interdisciplinary Innovation Team.
Acknowledgments
The authors are very grateful to Andrey Pryamikov for his useful discussion and constructive suggestions.
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