1. Introduction

In fiber optics, a topic of current great interest lies in one type of hollow-core fiber where light is guided by anti-resonant reflection from arrays of glass walls, i.e. anti-resonant reflecting optical waveguide (ARROW) principle [1]. Compared to its bandgap counterpart, this light guidance mechanism allows hollow-core anti-resonant fiber (HARF) simultaneously possessing broad transmission bands, low transmission loss, high laser damage threshold, and mainly single-mode operation, paving the way for plenty of interdisciplinary applications ranging from highly efficient laser-matter interaction [2–4], ultra-intense pulse delivery [5,6], few-cycle pulse generation [7,8] to chemical sensing [9,10], bio-photonics [11] and quantum optics [12]. By altering the core shape from nearly-circular to hypocycloidal [13] or negative curved [14], the transmission performance of HARF has been continuously breaking its record in recent years that beats hollow-core bandgap fiber (HC-PBGF) in many optical regions, e.g. the visible (130 dB/km [6,15]) and the mid-infrared (34 dB/km [16]). Despite these encouraging breakthroughs, for most applications other than telecommunication, the importance of further reducing transmission loss has become trivial because in most cases only tens of meters of fiber are used. Flexibility, on the other hand, is more urgently demanded. In some extent a fiber that can be mechanically and optically bent to a relatively small radius determines whether HARF can march into commercialization and real-world applications. Regretfully, unlike HC-PBGF [17] and solid core fibers, HARF, usually having a large core size and a simple cladding structure, loses its outstanding performances of both low attenuation and broad bandwidth when bent to a small radius [18].

Though the guidance mechanism of HARF has yet to be fully elucidated [2,19–21], several attempts have already been made to reduce the macro-bending loss of HARF. In Kagome type fiber, experiment has shown that by increasing the cladding layer number from 3 to 4 the bending loss can be dramatically reduced to ~0.2 dB/m with a bending radius of 5 cm and wavelength of 1500-1700 nm (denoted as R_b = 5 cm, λ = 1500-1700 nm, same below) [22]. However, when the authors tried to fit their experiment with simulation, their numerical calculation did not show much discrepancy between the cladding layer number of 2 and 4. A nodeless (or knot-free) structure, first demonstrated in [23], is proved to be a promising avenue towards low bending loss HARF [24]. Eliminating the knot-induced Fano resonances, the coupling strength between the core mode and the cladding mode can be efficiently suppressed even in a bent state [24]. Very lately, nodeless HARF with one layer of 6 or 7 tubes have been realized [25–27] for different applications. Particularly in [25] a remarkably low bending loss is observed in the long wavelength side (LWS) of the transmission band (<0.5 dB/m at R_b = 3 cm, λ = 1064 nm). Similar result is also reported in our recent work [28]. However, it is also shown that the fiber’s transmission band shrinks severely because of the huge bending loss in the short wavelength side (SWS). This phenomenon, also being observed in many other HARFs [6,16,22,24], seems to imply that the SWS of the transmission band of HARF is unsuitable for light guidance under bending. Many attractive applications relevant to broad bandwidth thus can only be envisioned in a straight HARF or at a large bending radius.

To gain a physical insight into the rapidly increased bending loss in the SWS, an analytical model has been built by Setti et al [29] where, similar to many other works [30–32], they attribute the main reason of the band edge shift in the SWS to the coupling between core mode and tube airy modes, whose effective indices are phase matched in a tight bending condition. Possible solution was suggested by decreasing the cladding hole size (e.g. increasing the tube number from 8 to 9) to lower the effective indices of the tube airy modes. This is obviously not an ideal strategy since small tube size leads to high leakage loss and multimode guidance. In many publications [21,25,33], the tube number of 6-8 has been proved as the optimal choice for HARFs.

Since the bending-induced resonances are difficult to eliminate in a HARF, in this work, rather than suppressing these resonances, we aim to explore the feasibility of light guidance between these resonances in the SWS in a bent fiber. We demonstrate that, in a nodeless fiber with optimized number of tubes and very thin glass thickness, the position of these new resonances, i.e. the loss peaks, can be engineered to affect only a certain portion of spectral region while the SWS still exhibits broadband light guidance with acceptable bending loss. This phenomenon has not been explicitly identified before owing to much thicker glass walls [24], non-optimized tube number [25] and touched geometry [16], albeit clue has been occasionally provided [34]. More importantly, our state-of-the-art fabrication technique allows us to not only confirm but also systematically investigate the bending loss behavior in both SWS and LWS via a series of HARFs containing 6 untouched tubes with very thin glass thickness (410 nm). These high performance fibers exhibits broad transmission window from 850 nm to >1700 nm (limited by light source and testing equipment) with low attenuation at the level of 100 dB/km and low bending loss in the LWS of 0.2 dB/m at R_b = 5 cm, λ = 1550 nm. In the SWS, one of these HARFs achieved a fairly broad bandwidth of 110 THz in the sub-band with acceptable bending loss of 4.5 dB/m at R_b = 2 cm. This is, to the best of our knowledge, the first time to conceive the concept of light guidance in the SWS under tight bending and to demonstrate its feasibility both in theory and in experiment. The capability to engineer a bent HARF for its transmission properties provides a new approach for light guidance in demanded spectrum windows for many interesting applications.

This paper is structured as follows. In Section 2, we study the tube airy mode under bending state and use an analytical method to explore the feasibility of light guidance in the SWS under bending. Section 3 adopts numerical simulation and an asymmetric fiber structure to investigate the effect of inter-tube gaps. The couplings between the core mode and the tube airy modes are also illuminated. The core message of this paper lies in Section 4 where, following our theoretical prediction, we fabricate a series of HARFs with varied tube and core dimensions and explicitly demonstrate sub transmission bands in the SWS under tight fiber bending. Conclusions are drawn in Section 5.

2. Design of HARF under bending

As depicted in Fig. 1(a), a hollow core with inscribed diameter of D is surrounded by one layer of tubes with the number of N and the inner diameter of d. Firstly, the nodeless geometry eliminates the Fano resonances [35] and allows each tube to be treated as an isolated one. Secondly, we temporarily ignore the influence of the inter-tube gaps and leave it to be discussed in the next section. Thirdly, we adopt a very thin glass thickness of t = 410 nm (experimentally realizable as shown in Section 4) and achieve a broad ARROW bandwidth. For example, the 1st-order ARROW band, where our fiber design is implemented, spread from 860 nm ($\approx 2t\sqrt{n_2{}^2-n_1{}^2}$) to >2.7 μm [21]. Here, n₁ = 1 and n₂ = 1.45 are the refractive indices of air and silica respectively. Moreover, because of the very thin glass thickness, the shift of the cut-off frequencies of the cladding dielectric modes, i.e. the edges of the ARROW bands, is minor to consider even under tight bending [29]. Under above three prerequisites, inside the 1st-order ARROW band, the light guidance will be mainly affected by phase matching (or resonance) between the core mode (the fundamental one) and the cladding airy modes. The purpose of this work is therefore to find out optimal parameters of a HARF in terms of its light guidance under bending condition.

 

Fig. 1 Nodeless HARF under bending. (a) Schematics of the fiber bending configuration of Type I and the effect of the conformal mapping on the refractive index. (b,c) Dispersion curves of the equivalent air core capillary (thick black line) and the tubes (dashed lines for dielectric modes and thin solid lines for airy modes) under the simplified model. The notations of the mode types (HE, EH, LP) and the mode indices (μ, ν) are in accord with Ref [21]. The tubes in the outermost side (b) and in the second outermost position (c) under bending are respectively calculated with D = 40 μm, d = 24 μm, t = 410 nm, and N = 7. Numerical simulation to the core mode of the actual fiber (holy squares) agrees reasonably with the simplified modeling. (d,e) The maximum frequency spans (ΔF) in the SWS as a function of the bending radius and the tube diameter for the (d) Type I and (e) Type II fiber bending configurations.

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In the straight state, a transfer matrix approach in cylindrical coordinates [36] can precisely calculate the dispersion curves $n_{eff}(F)$ of an equivalent air core capillary and an individual tube as a function of the normalized frequency$F\equiv 2t\sqrt{n_2{}^2-n_1{}^2}/\lambda$. The equivalent air core capillary has an infinite thick glass wall while the tube has a glass thickness of 410 nm. Apart from the ARROW resonances, the dispersion curve of the equivalent capillary matches well with that of the core mode of the HARF [21].

In the case of fiber bending, a conformal mapping technique [37] can be used with the refractive indices in the cross-section multiplied by $e^{x/R_b}$ with R_b the bending radius and the bending direction in the –x axis. Throughout this work, we fix the polarization to be in the y direction and leave the discussion of polarization dependence elsewhere. Additionally, like done in [29], inside each tube, the conformal factor can be simplified to be a uniform value of $\eta \equiv e^{\overline{x}/R_b}$ with $\overline{x}$ the average x-coordinate. With this simplification, it can be analytically derived that, if $n_{eff}(F)$ stands for the dispersion function of a straight waveguide, $\eta \cdot n_{eff}(\eta \cdot F)$ is the one under bending (see the Appendix). Once $\overline{x}$ > 0 and η > 1, the raised dispersion curves of the tube airy modes may cross with that of the core mode and cause new resonances and loss peaks [29,31]. Our aim is to design a bent HARF having a resonance-free spectral range as broad as possible in the SWS of the 1st-order ARROW band.

We define a fiber bending configuration as shown in Fig. 1(a) to be type I, where one inter-tube gap lies on the + x axis. Considering the phase matching between the core mode and the airy modes of the outermost tubes with the maximum $\overline{x}$, a series of intersections (marked by F11, F12, etc., caused by the phase matching with the tube airy modes of LP₀₁, LP₁₁ etc. respectively) appear in the region of F < 1 [Fig. 1(b)]. More intersections (F21, F22, etc.) can also be found via the phase matching between the core mode and the airy modes in for the second outermost tubes [see Fig. 1(c)]. For a straight fiber, all these intersections stay near the edge of the ARROW band and some of them (e.g. F11 and F21) overlap with each other. While in a bent state these resonances will shift along the frequency axis and split apart. We define the maximum separation between these intersection points to be the frequency span (ΔF) in the SWS of the ARROW band, inside which no resonance exists. The variation of ΔF with the tube number N, the tube diameter d, and the bending radius R_b are plotted in Fig. 1(d). Notice that in calculation we use the precise transfer matrix method [36] rather than the Marcatili’s approximate formula [38] because the latter omits the tube thickness therefore the influence of the ARROW bands to the tube airy modes. In Fig. 1(d), it is manifest that, when N decreases from 8 to 6, the maximum frequency span (ΔF) of the SWS sub-band doubles. In this process, on one hand the allowed maximum tube size of a nodeless HARF increases, on the other hand the separation angle between adjacent tubes increases as well. Both factors raise the possibility of finding a broad continuous frequency interval between those new resonances (corresponding to F11, F12, F21, F22, etc.,) in a 6-tube fiber. We do not further reduce the tube number because high performance HARF should occur among N = 6,7,8 [21,25,33]. In Fig. 1(c), it is also seen that, as the core diameter D decreases from 40 μm to 32 μm, the broad SWS sub-band could be obtained in a wider range of bending radius.

Figure 1(e) calculates another fiber bending configuration (type II) with one tube on the + x axis. Similar conclusions of Fig. 1(d) can be drawn. Under bending, a broad SWS sub-band can be most possibly obtained in the 6-tube fiber and this sub-band may occupy a significant proportion of the ARROW window. In practice, it is actually difficult to ascertain the fiber bending configuration. However, the results of Figs. 1(d) and 1(e) imply that, once an identical bending radius is maintained along one piece of fiber, whatever the local fiber bending configuration, there is a high chance to realize light guidance in the SWS of the ARROW band for a 6-tube fiber. In contrast, for an 8-tube fiber, because of the small ΔF and possible variation of the transmission sub-window with the fiber bending configuration, there is little chance for light guidance in the SWS in a bent fiber.

3. Influence of inter-tube gaps

In a nodeless HARF, inter-tube gaps are crucial components of the cladding and their influence to optical performance should be quantified. We use a numerical method, i.e. a finite-element mode solver (Comsol Multiphysics) together with optimized mesh size and perfectly matched layer [39], to calculate a bent fiber. The conformal mapping technique is implemented.

Solely investigating the effect of inter-tube gaps is not straightforward because it is usually mixed with the effects of the core and the tube sizes. We hereby adopt an asymmetric fiber geometry where we fix the core diameter D, the tube diameter d, positions of most tubes and only allow the position of tube T2 to vary, as shown in Fig. 2(a). In such a geometry, the variants only include gaps G1 and G2 and the position of tube T2, which can be measured by the gap separation distance g [see Fig. 2(a)]. According to previous analysis, the shift of the position of tube T2 will give rise to shift of the loss peak due to the change of the conformal factor η and can be easily identified. Under a bending radius of 3 cm, Figs. 2(b) and 2(d)-2(f) show this effect. On the other hand, in the magenta and the red curves of Fig. 2(b), we find a new class of loss peak which originates from the resonance inside gap G1 [Figs. 2(g) and 2(h)]. In Fig. 2(b), no explicit feature indicates resonance inside gap G2.

 

Fig. 2 Effect of inter-tube gaps. (a) Schematics of asymmetric fiber structure with the position of T2 and the gap separation g varying. The bending direction is in the –x axis, the polarization is in the y direction and the whole structure is mirror symmetric with the x coordinate. D = 40 μm, d = 24 μm, and t = 410 nm. (b) Simulated bending loss spectra at R_b = 3 cm with g = 3.3/5.3/7.59/10.22/13.28 μm, corresponding to the azimuthal angle of T2 to be 51°/55°/60°/65°/72° respectively. (c-h) Distributions of the z component of the normalized Poynting vector in the transverse plane at various loss peaks in (b). (c) stands for the common loss peak (λ ~1640 nm) of the resonance inside tube T1, (d-f) are those inside tube T2 for g = 3.3/5.3/7.59 μm respectively, and (g,h) are those inside gap G1 for g = 10.22/13.28 μm respectively.

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Examining the loss peaks in Fig. 2(b) in more detail, we plot their field distributions. For g = 3.3/5.3/7.59 μm, using the method presented in Section 2, the loss peaks at wavelengths of 1243 nm, 1162 nm and 1035nm can be accurately identified to be the couplings between the core mode and the tube airy modes in T2. This result also verifies the analytical model in Section 2 and hints the possibility of finding sub transmission bands in the SWS. As the tube T2 shifts toward the –x direction, the light guidance between the resonances of T1 [the black square in Fig. 2(b)] and T2 is slightly improved in terms of both bandwidth and attenuation, while the LWS sub-band stays nearly unchanged. For these three g’s, inter-tube gaps have an ignorable effect on the transmission properties.

When g increases to the value of 10.22 μm, the loss peak due to the tube T2 moves out of the spectrum. A new resonance occurs inside the gap G1 and is manifest at λ = 1090 nm. At a larger g of 13.28 μm, this resonance severely affects the whole loss spectrum. Not only a bigger loss peak occurs at λ = 1220 nm but also the baseline of the loss spectrum rises to >2 dB/m in the LWS and >10 dB/m in the SWS. In Fig. 2(h), we can see an explicit connection of the field patterns of the core mode and the gap resonance. We hereby conclude that in order to avoid severe influence of inter-tube gaps the gap separation should be kept in a region of g < 10.22 μm (or g/D < 0.25, this ratio is in some extent independent of d).

4. Fabrication and characterization

Above analyses predict broadband light guidance in the SWS under certain conditions (Fig. 1). However, such a prediction is made with many approximations, e.g. uniform conformal factor, fixed fiber bending configuration, etc. In this section, we will use the experimental approach to verify this conjecture. Figures 3(a)-3(c) show the scanning electronic microscope (SEM) images of a series of HARFs with 6 untouched tubes drawn from similar canes using stack-and-draw technique. The glass thicknesses are measured to be roughly 410 nm. Finely adjusting pressurization during fiber drawing, we can precisely control and produce different ratios of tube diameter (d) to core diameter (D). Fiber #1 [Fig. 3(a)] has an outer diameter (OD) of 125 μm, a tube diameter d of 18 μm, a gap separation g of 11 μm and a core diameter D of 40 μm (d/D = 0.45). The structural parameters of Fiber #2 [Fig. 3(b)] are OD = 145 μm, D = 40 μm, d = 24 μm and g = 8 μm (d/D = 0.6). In Fiber #3 [Fig. 3(c)], we managed to fabricate a fiber with a smaller core diameter and larger ratio of d/D (OD = 125 μm, D = 32 μm, d = 22 μm, g = 5 μm and d/D = 0.69). Here, all the gap separations are greater than zero, ensuring a nodeless geometry, and the largest ratio of gap separation to core diameter (g/D) is 0.275, near the acceptable region to ignore the resonance inside the gap (see Section 3). In a straight state (or at a large bending radius, e.g. R_b = 16 cm), three fibers behave similarly and exhibit a broad transmission band from 850 nm to >1700 nm. Actually, the measured transmission bandwidth (greater than 170 THz) is limited by our characterization facilities in the long wavelength region. Simulations prove that the light guidance can extend to the wavelength of 2700 nm [Fig. 2(b)], implying a transmission bandwidth of ~240 THz. Inside such a broad transmission window, even one portion in its SWS, which may have light transmission function under fiber bending according to our above analysis, could contain a great amount of spectral resource. This is the context of our study to a bent HARF. Using the cut-back method, the measured transmission loss of our fiber in the straight state is at the level of 100 dB/km (see the Appendix), worse than the record of 17 dB/km for HARFs [40], but being reasonably enough for many applications. Our fibers are not specially designed for single modeness but single-mode operation can be easily achieved in both straight and bent states once the in-coupling condition is adjusted. These characteristics are similar to those reported in [25, 26], thus will not be the main focus of this study.

 

Fig. 3 Characterizations of 3 nodeless HARFs under bending. (a-c) SEM images of Fiber #1, #2, #3 respectively having different tube sizes and core radii as indicated. All the SEM images have the same scale bar. (d-f) Transmission spectra of the three HARFs under different bending radii. The legend in each graph denotes bending radius in cm × number of turns.

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Optical characterizations of the bent fibers are performed by bending a 10 m long fiber to radii of 16 cm, 5 cm, 4 cm, 3 cm and 2 cm respectively with various turns. A supercontinuum source (400-1800 nm) and an optical spectrum analyzer (OSA) (600-1700 nm) are used. The transmission spectra of Fiber #1-#3 are shown in Figs. 3(d)-3(f). Taking the spectra at R_b = 16 cm as the references, mimicking a fiber in the straight state, the bending loss can be acquired for different tight bending radii. For prudence, each measurement at one tight bending radius, i.e. R_b = 2,3,4,5 cm, has been followed by a measurement at R_b = 16 cm to ensure that the fiber can be released and recovered to the initial status. The measured bending loss spectra are shown in Figs. 4(a)-4(c). Both Figs. 3(d)-3(f) and Figs. 4(a)-4(c) indicate that different structural dimensions of HARFs behave in different manners under tight bending. Firstly, in the LWS of the transmission window, a bending loss of ~0.2 dB/m at R_b = 5 cm is found in both Fibers #1 and #2, similar to the result in [25] (0.5 dB/m at R_b = 3 cm, λ = 1064 nm) and represents one of the lowest bending loss value for HARF. Fiber #1, with a small d/D, shows a typical bending loss spectrum of a HARF where the transmission window keeps shrinking toward the longer wavelength as R_b decreases, and the bending loss at a fixed wavelength approximately follows the trend of $R_b{}^{-2}$, predicted by the Marcatili and Schmeltzer’s model [38]. Considering the very long wavelength side of the transmission window for Fibers #2 and #3, similar behavior with Fiber #1 can be observed.

 

Fig. 4 (a)-(c) Measured bending loss spectra of Fiber #1, #2, #3. (d-i) Simulated bending loss spectra under the bending configuration of (d-f) Type I and (g-i) Type II.

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When we consider the SWS, while the light guidance in Fiber #1 gradually disappears under tight bending, the other two fibers, especially Fiber #3, preserve certain extent of light guidance, which is very interesting and may hint practical applications. In Fiber #3, with a quite large d/D = 0.69, it is clearly seen that the wavelength region from 1150 nm to 1250 nm [the gray region in Fig. 3(f)] has lost light guidance at R_b = 5 cm, but the newly-emerged sub-band helps it to partly recover light guidance at tighter bending. This sub-band keeps shifting towards the longer wavelength [the dashed arrow in Fig. 3(f)]. In some situations, the newly-emerged sub-band is even wider than HC-PBGFs and many HARFs operating in the 2nd-order ARROW band [18]. For example, the magenta curve in Fig. 3(f) shows a very broad sub-band from 980 nm to 1520 nm (~110 THz) at R_b = 2 cm. For Fiber #2, the general situation is between Fiber #1 and #3, but two sub-bands are observed. Altogether, Fibers #2 and #3 demonstrate a unique tailoring capability of the sub transmission band, which enables an unprecedented light guidance in the SWS under tight fiber bending. In contrast, Fiber #1 still complies with the conventional design principle and therefore will unavoidably lose light guidance in the SWS under tight bending. According to the picture presented in Fig. 1, the resonances induced by bending only influence specific range of wavelength, and the remainder spectral regions still allow broad light guidance. Engineering d, D, and R_b provides a new degree of freedom to optimize optical performances in a bent HARF.

Figures 4(b) and 4(c) also show that, for Fiber #2, the bending losses in the SWS are at the level of 9 dB/m, 12 dB/m, 5 dB/m and 20 dB/m for R_b = 5 cm, 4 cm, 3 cm and 2 cm respectively, and, for Fiber #3, these figures drop to 2.5 dB/m, 5.5 dB/m, 2.5 dB/m, and 4.5 dB/m respectively. Sometimes, these bending losses are even lower than those in the LWS for a same fiber. We want to remind readers that in a conventional negative curvature HARF with nodes the typical bending loss in the LWS is 2.5 dB/m at R_b = 6 cm, λ = 1064 nm [18], and 3.18 dB/m at R_b = 2.5 cm, λ = 3200 nm [16]. Fiber #3 exhibits a novel tailoring capability together with outstanding characteristics in terms of both bandwidth and bending loss.

We further carry out numerical simulations by using practical structural parameters in both Type I and Type II fiber bending configurations. Figures 4(d)-4(i) verify an excellent agreement with the experiment. The trend and the general level of the bending loss of Fiber #1 in the LWS with the decrease of R_b are perfectly corroborated. For Fiber #2 and #3, the emergence of the SWS sub-band is also observed in simulation. Although the bending configuration is unknown in our measurements and will affect the precise resonance positions and bending loss values, we still find a general match between experiment and simulation, especially for the bending configuration Type I. A relatively broader bandwidth and lower bending loss in the simulation comparing with the experiment is probably due to the non-idealities in fabrication and measurement. We believe it should be possible to improve the experimental results by inspecting and carefully controlling the bending configuration using X-ray tomography technique [41].

5. Conclusions

In conclusion, state-of-the-art nodeless HARFs have been demonstrated with outstanding optical performance on the aspects of transmission bandwidth and bending loss. On one hand, a broad and flat light transmission from 850 to >1700 nm (within the 1st-order ARROW band) beats most reported HARFs. On the other hand, the first observation of sub transmission bands in the SWS and the in-depth investigation show the tailorability of the transmission band in a tightly bent fiber. At a particular bending radius, optimum performance can be engineered in terms of transmission bandwidth, single modeness and attenuation via carefully tailoring the fiber parameters. Our HARF design method, taking into account both the effects of dielectric mode and airy mode of the cladding tube, will offer a versatile tool in situations where fiber packing is crucial. Such a nodeless HARF is a promising workbench for real-world applications including high power pulse compressor, gas-based nonlinear optics, interferometric sensing, bio-photonics and quantum optics.

Appendix

1. Relationship between dispersion functions

Given a source-free dielectric waveguide having an arbitrary cross-sectional refractive index profile $\{n_l\left(A_l\right), l=1,2,3\dots \}$, at any frequency ${\omega }_0$, one can express its mode, whatever guided mode, radiation mode, or leaky mode, as $\{E\left(r\right)e^{i(\beta z-{\omega }_{0}t)}, H(r)e^{i(\beta z-{\omega }_{0}t)}\}$. Here, the waveguide axis is in the z direction, β is the propagation constant, and A_l represents an area element on the x-y plane. Except different field convergence behaviors at infinity, all the modes should satisfy Maxwell’s equations and boundary continuity conditions.

It is not difficult to prove that, when the refractive index profile changes to $\{\eta \cdot n_l\left(A_l\right)\}$ with η being a constant, the field distribution $\{E\left(r\right)e^{i(\beta z-\frac{{\omega }_0}{\eta }t)}, \eta \cdot H(r)e^{i(\beta z-\frac{{\omega }_0}{\eta }t)}\}$ meets both Maxwell’s equations and boundary continuity conditions as well, so that representing one waveguide mode. This means, at the new frequency ${\omega }_0/\eta$, the waveguide mode has the same propagation constant β, i.e. the conclusion drawn in Section 2.

2. Fiber attenuation measurement

Figure 5 shows the attenuation spectrum of fiber #2 measured by a cut-back method from 80 m to 10 m. A supercontinuum source is butt-coupled to the fiber and an OSA (600-1700 nm) is used for characterization. The fiber exhibits an attenuation figure of 100 dB/km from 1000 nm to >1700 nm.

 

Fig. 5 Attenuation spectrum of Fiber #2 (insert) measured using cut-back method from 80m to 10m fiber.

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61377098, 61275044 and 61575218), the Beijing National Science Foundation (No. 4142006) and the Instrument Developing Project of the Chinese Academy of Sciences (No. YZ201346).

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