Wideband low confinement loss anti-resonant hollow core fiber with nested U-shape tube

: We propose a nested U-shape tube anti-resonant hollow core fiber (UARF) that can effectively reduce the confinement loss (CL) as well as the loss oscillations. The key parameters of UARF have been optimized via numerical analysis. Simulation results show that the CL of proposed UARF is lower than 0.01dB/km over a 550 nm operational bandwidth range from 1.3 µm to 1.85 µm. This CL is nearly one order of magnitude lower than the nested anti-resonant nodeless fiber (NANF). Moreover, the loss ratio between higher-order modes to the fundamental mode is verified to be more than 100,000 over a ultrawide bandwidth of 1000 nm, which indicates its excellent single mode performance. The tolerance towards the structure deformation of UARF has been evaluated for the purpose of practical fiber fabrication. Thus, the proposed UARF has potential application in large capacity data transmission, nonlinear optics, gas sensing and so on.


Introduction
Hollow core fibers (HCFs) by guiding the light over the air core have been extensively investigated, due to the advantages of small transmission delay, less Rayleigh scattering, and small nonlinear coefficient [1][2][3][4][5][6][7][8].Meanwhile, the applications of HCF have been comprehensively explored for high peak power pulse delivery [9,10], gas sensing and nonlinear optics [11][12][13], high energy pulse compression [14], mid-IR radiation delivery [15], high harmonic generation [16].Besides, the intrinsic features of HCF enables its application in large capacity data transmission with the wavelength ranging from short band to long wavelength band (such as L-band to 2 micro meter) [17][18][19][20].According to the light guidance mechanism, HCFs can be divided into two categories [21].One category is photonic bandgap fiber (PBGF) based on the photonic bandgap effect [22], where the light at specific frequency can be guided over the hollow core.PBGF is promising for the HCF related applications, but faces great challenges in terms of operational bandwidth and fabrication technique [21].The other category is anti-resonant fiber (ARF) relying on a combination of antiresonance and the inhibited coupling effect, to obtain a reduction of transmission attenuation and an enhancement of operational bandwidth [21,[23][24][25][26].
Generally, reducing the transmission loss while increasing the operational bandwidth at the same time is quite challenging.The total loss of a HCF include surface scattering loss (SSL), the leakage loss (namely, the confinement loss, CL) and the bending loss (BL).Among the three losses, the SSL and CL are the most key concerns in current theoretical researches [21].As for the PBGF, the main loss comes from the SSL [27], which is difficult to overcome due to the frozen effect arising in thermodynamic fluctuations of surface capillary waves [28].The lowest loss of PBGF was experimentally reported of 1.72 dB/km [29].Alternatively, as for the ARF, the loss is mainly referred to the CL determined by the power leakage from the core [21].Meanwhile, the SSL of ARF is typically lower than that of PBGF because of the anti-resonance feature.Therefore, ARF is a potential candidate to simultaneously achieve both ultra-low loss and wide operational bandwidth [6,23].In recent years, some pioneering work on the ARF design has been carried out.In 2017, the loss of 7.7dB/km for the single-ring ARF was characterized [30].To further reduce the CL, the most promising way is to change the ARF structure with additional anti-resonant layers.In 2018, Gao experimentally reported a conjoined-tube based ARF including two D-shaped air holes, resulting to a minimum loss of 2 dB/km at 1512 nm [31].On the other hand, the nested anti-resonant nodeless fiber (NANF) attracted great attentions with its loss reduction from 1.3 dB/km to 0.28 dB/km, after a joint optimization of both the structural parameters and the fabrication process [7,[32][33][34].According to the light [34].Furthermore, there also exist lots of theoretical investigations about novel ARF structures, for the purpose of further CL reduction [35][36][37][38][39].According to the light [39].For example, double negative curvature anti-resonant fibers (D-ARF) were proposed with the CL lower than 0.1 dB/km over a wavelength range of 320 nm (from 1.28 µm to 1.85 µm) [39] or with a loss ratio of 100,000 between higher-order modes and the fundamental mode [37].However, the additional use of glass structures will inevitably create more glass connections and thus enhance the Fano resonance significantly, resulting in a non-flat loss spectrum [21,40,41].Due to the influence of glass nodes, the operational bandwidth such as D-ARF is narrower in comparison with single-ring ARF or NANF.Moreover, the fluctuation of the loss spectrum of ARF induced by the Fano resonance is more significant in the long wavelength range [40], which causes the difficulty to potential applications such as the new wavelength band optical communications.Therefore, it is desired to design a kind of fiber structure to enhance the operational wavelength range with ultra-low loss and low loss oscillations.
In this paper, we propose a novel nested U-shape tube structure of ARF (UARF) to reduce the CL and decrease the loss fluctuation induced by the Fano resonance, as well as enlarge the low CL operational bandwidth.After parameters optimization, we put forward a theoretical model of 5-tube UARF and realize an operational bandwidth over 550 nm span from 1.3 µm to 1.85 µm, with the CL lower than 0.01 dB/km.Meanwhile, the loss fluctuation between maximum and minimum CLs is less than 0.03 dB/km over the wavelength range from 1.2 µm to 2.2 µm.Our proposed UARF guarantees a significant performance improvement, in terms of both the CL and operational bandwidth, in comparison with existing ARF.Additionally, the loss ratio between higher-order modes and the fundamental mode is more than 100,000 for the proposed UARF.Those results indicate that the proposed UARF is a promising candidate for the HCF involved applications.

Nested U-shaped HCF structure
The cross-section of the proposed UARF is shown in Fig. 1.The cladding includes a unit of circular tube and a U-shape glass structure, which can be considered as a circular tube nested with a U-shape structure.From a geometrical view, the nested U-shape structure can be regarded as a combination of one semicircular layer and two straight layers, which are parallel to each other and tangent to the semicircular layer.The geometrical parameters of the UARF are denoted as the core radius R core , the tube gap g, the glass thickness t, the tube radius R, the radial separation z between the tube and the U-shape structure, the semicircular radius r, and the distance d between parallel layers of U-shape structure.Please note that, d is equal to 2r, and R is not equal to z + r in our design.
It has been verified that the CL of ARF is related to the power leakage from air core to the cladding [23].Changing the cladding structure of ARF with additional anti-resonant layer is an effective solution to reduce the power leakage in tubes.In addition, we identify that, the surrounding power leakage can be effectively suppressed, when anti-resonant layers are close to each other even though they are along the radial direction of the fiber.As shown in Fig. 1, the parallel glass layers of U-shape structure form a gap while the connection nodes are sufficiently far away from the fiber core [21], which is the key point to reduce both the CL and the Fano resonance induced loss fluctuation simultaneously.
In order to verify the feasibility of reducing the CL with the U-shape structure, the variation of power density of cladding boundary (partially from −30°to 30°) of NANF and UARF has been, respectively, shown in Fig. 2. The power density can reveal the power leakage of different fiber structures.The core diameter, the glass thickness, and the tube gap of two fibers are the same, and we also utilize the same number of anti-resonant layers for those two fibers for the purpose of a fair comparison.The insets of (i) and (ii) show the cross-section of corresponding fiber structures (red-mark is the partially calculated boundary), respectively.We can observe that, the power density of proposed UARF is generally lower than that of NANF, and the maximum power density of UARF is found less than 10 −8 W/m 2 .Therefore, we can conclude that the U-shape structure is effective to reduce the CL in comparison with the nested structure.
Then numerical investigations have been carried out by using the COMSOL Multiphysics based on finite element method (FEM).In order to obtain accurate simulation results, we set the maximum element size of glass and air regions to λ/6 and λ/4, respectively [21].The minimum element number of glass regions is 5.In addition, we set a small penetration of t/2 of all cladding tubes into the jacket layer and smaller nested tubes into the outer tubes based on the practical consideration [38,42].A perfectly matched layer (PML) is placed outside the jacket layer to enclose the simulation area, to obtain the imaginary part of the mode eigenvalue to calculate the CL of fiber [35].The thickness of the jacket layer and PML are set as 3 µm and 10 µm respectively.The CL can be calculated [37] where n imag is the imaginary part of effective refractive index, λ is the free space wavelength.
The glass thickness t is determined by the anti-resonance condition [23,35] where n glass and n air are refractive indexs of glass and air, respectively.M (=1,2,3. . . ) represents the order of antiresonance, and M = 1 is chosen in our investigation for the purpose of the For the NANF, the radial separation is optimized to 15.45 µm with the radius of the small nested tube of 8.1 µm.For our proposed fiber, the corresponding one is optimized to 7 µm with the radius of the small semicircular of 5.83 µm.
operational bandwidth enhancement.The SSL can be estimated after calibration [21,27] where F is the power overlap between the fundamental mode and the silica boundary, and the calibration factor η is 300 when λ 0 is 1.55 µm [21].Besides, the random microbend loss (RML) caused by the coupling between fundamental mode and LP 11 mode can be calculated [37,43] where β 0 is the fundamental mode propagation constant and ∆β 01 is the difference of fundamental mode propagation constant and LP 11 mode group propagation constant.C(∆β 01 ) is the power spectral density at ∆β 01 , ⟨0|x|0⟩ and ⟨0|x|1⟩ are the spot radius of fundamental mode and LP11 mode respectively.
To have a better understanding of the loss contributions of the ARF, we have done the theoretical analysis as shown in Fig. 3.The basic parameters of proposed UARF and the referred NANF are kept the same, i.e., the core radius R core = 20 µm, the tube gap g = 3.3 µm, the glass thickness t = 0.5 µm, and the tube radius R = 24 µm.The other optimized parameters of NANF and UARF are, z = 10.08 µm, r = 10.32 µm and r N = 12 µm.The total loss performances of the UARF and NANF have been shown as the scatter points with red and yellow color respectively.And the SSL (green dot-dashed line) of UARF is almost equal to the one of NANF.As can be seen from the figure, the SSL of NANF is the dominant loss and slightly higher than CL in the shorter wavelength range (< 1.51 µm).For the longer wavelength range (> 1.51 µm), the CL becomes the dominant loss contribution.This increase of CL is originated from the power leakage that caused by the wavelength away from the anti-resonance condition.Therefore, the CL should be further reduced to meet the demands on low total loss large capacity optical transmissions toward to the long wavelength range (L-band to 2 µm).It is obviously that the proposed UARF has the reduced total loss (red scattered points) in the entire 1000 nm wavelength range with a comparison of that of NANF.Meanwhile, the optimized CL is nearly one order of magnitude lower than that of NANF, which resulted in the total loss of UARF being lower than 0.1 dB/km in a wide wavelength range (from 1.51 µm to 2.2 µm).

Parameter optimization of nested U-shaped structure
According to the basic analysis above, and to investigate the CL performance for our proposed fiber structure, we first carry out numerical simulations to obtain the relationship between the semicircular radius r or the radial separation z and CL.We consider two cases for our proposed fiber with 5-tube and 6-tube, respectively, which are two typical arrangements for the ARF design.In our simulations, some parameters are fixed for all fibers, i.e., the core radius R core = 20 µm, the tube gap g = 3.3 µm, and the glass thickness t = 0.5 µm.Then the tube radius R = 24 µm for 5-tube and R = 16.2 µm for 6-tube can be derived from the geometric relationship.
The CL with respect to the variation of semicircular radius r and radial separation z are illustrated in Fig. 4(a) and (b), respectively, when the operation wavelength is 1.55 µm.Once we set the initial value of z = 0.42R in Fig. 4(a), we observe that the semicircular radius r has a great impact on the CL.Under the condition of 0.4 > r/R > 0.65, the CL of LP 01 modes (solid lines) reduces dramatically as r decreases.When 0.25 < r/R < 0.4 is satisfied, the CL has an erratic fluctuation, which is inferred from the Fano resonance arising in the closed parallel layers [44].It's worth noting that, as for the 5-tube configuration, the CL of LP 11 mode (red dashed line) reaches a peak value at r/R = 0.43, which is caused by the coupling of LP 11 mode and the cladding tube mode.As for the effect of the radial separation z, as shown in Fig. 4(b), the CL of LP 01 mode (solid lines) is independent on z/R, when z/R is at the range of 0.3 to 0.6.Additionally, we find that 5-tube case has lower CL of LP 01 mode and higher CL of LP 11 mode than that of 6-tube case.For the 5-tube UARF, we take two values of semicircular radius into account, relating to two specific fiber configurations in Fig. 4(a).Type I configuration has the highest CL of LP 11 mode, under the condition of r = 0.43R (10.32 µm).Type II configuration has the lowest CL of LP 01 mode when r = 0.33R (7.92 µm) is satisfied.Due to the independence of the radial separation on the CL of LP 01 mode within the range, we choose z = 0.42R for Type I and Type II fiber.
Figure 5 shows the loss spectra of proposed fibers (Type I and Type II), the NANF [21] and the D-ARF [37], for the ease of loss performance comparison.The insets of Fig. 5 are LP 01 mode field distribution of (i) NANF, (ii) D-ARF, (iii) Type I of UARF and (iv) Type II of UARF.As shown in Fig. 5, Type I has a wide operational bandwidth with the CL lower than 0.01 dB/km from 1.3 µm to 1.85 µm and a minimum CL of 0.0056 dB/km at 1.55 µm, which is significantly better than that of NANF and D-ARF.Moreover, within the whole wavelength range from 1.2 µm to 2.2 µm, the difference of the loss spectra between maximum CL and minimum CL is lower than 0.03 dB/km, indicating a flat CL performance for our proposed UARF.For Type II UARF, the minimum CL of LP 01 mode is even lower than 0.001 dB/km because of the small semicircular radius r.However, its loss spectrum is seriously fluctuated due to the Fano resonance.Therefore, we select Type I (R = 24 µm, r = 10.32 µm and z = 12 µm) as the proposed fiber design for subsequent discussions.Obviously, the UARF shows the lower CL with the same operational bandwidth comparable to NANF.It means that the proposed UARF has the lower transmission loss and wider operational bandwidth.The corresponding fiber structures and LP 01 mode field distributions are shown at the right-side, the core radius, tube gap and glass thickness of all these fibers are same, while the other parameters of D-ARF are the same as [37].

Higher-order mode extinction ratio
For the HCF-based data transmission, the single-mode operation is one of the important requirements.Such feature can be characterized by the loss ratio between the higher-order modes to the fundamental mode, which also called as higher order mode extinction ratio (HOMER) [37,38].In order to obtain robust single-mode operation, we can suppress the higher-order mode for the ease of the HOMER improvement.Figure 6(a) shows the effect of semicircular radius r on effective refractive index of each mode.Since the core radius is fixed, the effective refractive index of LP 01 mode and LP 11 mode are almost constant.On the contrary, the effective refractive index of the cladding mode (CM) increases with r, due to the deformation of U-shaped air cavity.When r/R is closed to 0.43, the effective refractive index of the CM matches that of LP 11 mode, corresponding to the CL peak of LP 11 mode in Fig. 4(a).As shown in Fig. 6(b), we have also calculated the effective refractive index difference between the CM and two LP modes of proposed Type I fiber.The refractive index difference of LP 11 mode and the CM is almost zero over a wide wavelength range, indicating that LP 11 mode has been effectively suppressed.Based on the optimal parameters of U-shape structure, the CL and HOMER curves with respect to the operation wavelength are presented in Fig. 6(c).During the CL calculation, three modes are investigated.In Fig. 6(c), the loss performances of LP 11 mode and LP 21 mode are obtained to determine the lowest CL and the HOMER.We can observe that the proposed fiber has a HOMER more than 100,000 (blue dashed line) over the wavelength range from 1.2 µm to 2.2 µm.Compared with the D-ARF [37], our proposed UARF not only achieves a similar loss performance, but also ensures an excellent single-mode operation over much wider wavelength range.The insets of Fig. 6 are the field distributions of (i) LP 21 mode, (ii) LP 11 mode, and (iii) LP 01 mode.Obviously, part of higher-order modes couples with CM.Finally, the comparison of confinement loss, operational bandwidth and HOMER between our proposed fiber and other typical reports has been shown in Table 1.The proposed fiber has realized a confinement loss of <0.01 dB/km over 550 nm operational bandwidth and the HOMER of >100,000 over 1000 nm range, which shows a significant improvement than other designed fibers theoretically.

Fabrication tolerance
Although the proposed UARF has characteristics of low CL, wide operational bandwidth and single-mode operation, the tolerance of the fiber fabrication is also pivotal for practical applications.During the fabrication of practical ARFs, the geometrical deviations are unavoidable [45].
Initially, we consider that the semicircular of U-shape is deformed due to imperfect air pressure.Figure 7(a) shows the impact of fabrication deformation on the CL performance, when the semicircular of the proposed fiber is collapsed into the unexpected ellipse.We define the radial direction of the ellipse as the axis a, and half of the distance between parallel layer as the axis b.Assuming that the axis b of the ellipse and radial separation does not change, the extent of ellipse can be defined by a/b (ideally a/b = 1).It can be seen that the change of a/b has little impact on the CL.When a/b > 1.1, the CL even slightly decreases with the increment of a/b.Therefore, the high tolerance of a/b variation from 0.6 to 1.4 is anticipated.Next, we consider that the structure of parallel layer cannot be strictly maintained.Figure 7(b) shows the influence on the loss spectrum when the parallel layer of U-shape structure is varied.We set θ as the angle that layers deflect from the horizontal (ideally θ = 0°), and assume that all structures are deflected at the same angle for the ease of discussion.As shown in Fig. 7(b), the variation of θ has significant impact on the loss spectrum, because the gap of U-shape structure cannot be effectively maintained.When θ = 15°is set, although the CL level of proposed fiber is obviously higher than that of θ = 0°, it is still lower than that of NANF (gray dashed line).Meanwhile, based on the simulation results, the CL performance will not be degraded when θ is a negative value.Thus, the tolerance towards the deflection angle θ for our proposed URAF can be reached at least 15°under the desired CL target.

Conclusion
We have proposed a nested U-shape tube anti-resonant HCF to mitigate the influence of the Fano resonance on the confinement loss and then to realize low loss ultrawide operational wavelength.After parameters optimization, the confinement loss of proposed HCF is lower than 0.01dB/km over 550nm bandwidth ranging from 1.3 µm to 1.85 µm.Moreover, more than 100,000 of the higher-order mode extinction ratio is obtained over a wide wavelength range from 1.2 µm to 2.2 µm, which shows the excellent wideband single-mode performance.The fabrication tolerance of proposed fiber is also investigated by taking the deformation of U-shape into account.We identify that the confinement loss is insensitive to the semicircular of U-shape but more sensitive to the parallel layer deflection angle.From a viewpoint of practical fabrication, it is recommended to keep the deflection angle less than 15°to maintain the performance of low loss ultrawide operational bandwidth.The results show that the proposed UARF has the potential applications in data transmission, nonlinear optics, gas sensing etc.

Fig. 2 .
Fig.2.Calculated power density at the cladding (red-mark of structure cross-section) of NANF and the proposed UARF.The insets show the fiber structures and profiles of fundamental mode.For those two fibers, the core radius, the glass thickness, the tube gap, and the tube radius are chosen the same as 20 µm, 0.5 µm, 3.3 µm, and 16.2 µm, respectively.For the NANF, the radial separation is optimized to 15.45 µm with the radius of the small nested tube of 8.1 µm.For our proposed fiber, the corresponding one is optimized to 7 µm with the radius of the small semicircular of 5.83 µm.

Fig. 3 .
Fig. 3. Calculated the losses of UARF and NANF.These scattered points are total loss of corresponding fiber.The dot-dashed line is SSL of UARF, almost identical to that of NANF.

Fig. 4 .
Fig. 4. Calculated confinement loss variation with (a) semicircular radius r under the condition of z = 0.42R, and (b) radial separation z under the condition of r = 0.5R.The tube radius R = 24 µm for 5-tube (red lines) and R = 16.2 µm for 6-tube (blue lines), and wavelength at 1.55 µm.

Fig. 5 .
Fig. 5. Calculated loss spectra of the proposed fiber (Type I and Type II) and other fibers.The corresponding fiber structures and LP 01 mode field distributions are shown at the right-side, the core radius, tube gap and glass thickness of all these fibers are same, while the other parameters of D-ARF are the same as[37].

Fig. 6 .
Fig. 6.(a) The effective refractive index of LP 01 mode (red), LP 11 mode (green) and the CM (gray dashed) with a variation of semicircular radius r. (b)The difference of the CM and LP mode with the variation of wavelength.(c) Loss spectra of LP 01 mode, LP 11 mode and LP 21 mode, as well as the HOMER (dot-dashed line in right axis), mode field distributions of LP 01 mode, LP 11 mode and LP 21 mode are at the right-side.The parameters of proposed fiber (b) and (c) are R = 24 µm, r = 10.32 µm and z = 12 µm.

Fig. 7 .
Fig. 7. (a) Calculated confinement loss with the variation of semicircular deformed into an ellipse that defined by a/b, and the insets shows a/b = 0.6 and a/b = 1.4,(b) the loss spectra with a variation of parallel layer deflection θ.