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Abstract
Hollow-core nested anti-resonant nodeless fibers (HC-NANFs) exhibit great performance in low loss and large bandwidth. Large core sizes are usually used to reduce confinement losses, but meanwhile, bring side effects such as high bending and coupling losses. This study proposes a small-core HC-NANF with a relatively low confinement loss. Semi-circular tubes (SCTs) are added to constitute the core boundary and reduce the fiber-core radius (R). Double NANFs tubes and single-ring tubes are added inside the SCTs to reduce loss. Simulation results show that the optimized structure with R of 5 µm has confinement loss and total loss of 0.687 dB/km and 4.27 dB/km at 1.55 µm, respectively. The bending loss is less than 10 dB/km at 1.4 ∼ 1.6 µm with a bending radius of 10 mm. The direct coupling loss with standard single mode fiber is greatly reduced to ∼ 0.125 dB compared to other HC-NANFs. The modified structure of HC-NANFs also shows a large bandwidth, effective single-mode operation, potentially high birefringence performance, and remarkable robustness of the optimized structure parameters, making it suitable for short-haul applications in laser-based gas sensing, miniaturized fiber sensing, etc.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Hollow-core antiresonant fibers (HC-ARFs) based on the antiresonant reflecting optical waveguide (ARROW) [1] model is a novel type of hollow-core fibers. Different types of HC-ARFs have been demonstrated, such as single-ring HC-ARFs [2], conjoined-tube fibers (CTFs) [3], and hollow-core nested antiresonant nodeless fibers (HC-NANFs) [4,5]. HC-NANFs exhibit many advantages, including low loss, large bandwidth (BW), low nonlinearity, high damage threshold, and ultralow latency, owing to their special structure and guidance mechanism [4]. Recently, the lowest loss of HC-NANFs with one nested ring was reported: 0.22 dB/km at 1625nm [6]. A loss of 0.174 dB/km was also demonstrated by HC-NANF with double nested rings, namely DNANF [7]. This performance metric continues its rapid reduction and is expected to break the loss limit of conventional standard single-mode fibers (SSMFs).
HC-ARFs confine light through the reflection of silica-air interfaces, the ARROW effect, and the negative curvature effect [8,9]. The dominating loss of HC-ARFs is the confinement loss (CL), which is in proportion to the imaginary part of the effective index neff of the fundamental mode (FM) [10]. HC-ARFs with large cores can significantly decrease neff and hence the CL. In practice, fiber-core radius R is usually in the range of 13 to 34 µm [2,3,6,11,12]. The reported realistic HC-ARF with the smallest core is a hollow-core negative-curvature fiber for UV guidance with R ∼ 7.5 µm and CL ∼ 0.13 dB/m at 300 nm [13]. The tapering technique is also used to fabricate short HC-ARFs with a small core [14] and the tapered HC-ARF with the smallest core has R ∼ 2.9 µm [15].
A large core size causes high micro-bending loss (MBL) and macro-bending loss, which is also called bending loss (BL). The MBL increases with a shorter wavelength, and it is the dominating loss at short wavelengths in HC-NANFs [16,17]. The BL is approximately 0.2 dB/m when the bending radius Rc is ∼ 2 cm [12]. This limits the miniaturized application of HC-NANFs. In addition, the very large direct coupling loss between an HC-NANF with a large core size and an SSMF with R ∼ 4.1 µm is unacceptable. Many low-loss coupling methods between HC-ARFs and SSMFs, such as the fiber mode field adapter inserted method [18], fiber tapering method [19], and fiber reverse-tapering method [20], have been developed to introduce HC-ARFs in fiber systems. However, they are inconvenient for miniaturized applications because of their complex structure, relatively high coupling loss, and low structural strength. Hence, a large core size greatly limits the bending and coupling performance and miniaturized application of HC-NANFs.
Semi-circular tubes (SCTs) constitute a part of circular tubes that have been used as the nested elements [21] or as the core of HC-ARFs with nested tube groups N = 3 or 4 [22,23]. SCTs could give rise to the ARROW effect, negative curvature effect, or high birefringence (Hi-Bi) in HC-ARFs [24,25]. Furthermore, the radius of the microstructure region radius Rm is free to adjust when SCTs constitute the core. Another similar designed structure is a semi-elliptical tube (SET), which presents a better negative curvature effect [26]. However, compared with SETs, SCTs are more practicable. The fabrication method of SCTs was discussed in [24] and a highly birefringent HC-ARF with SCTs has been fabricated recently [25], which has a combination of phase birefringence of 9.1 × 10−5, a minimum loss of 185 dB/km, a bandwidth of 133 nm, and effective single-mode operation. Inspired by this structure, SCTs could be added in DNANFs as the core boundary to increase the number of silica rings m and decrease Rm to a suitable value. The design freedom is also greatly improved in this structure, enabling the design of relatively low-loss small-core HC-NANFs.
In this paper, we propose a small-core HC-NANF with SCTs. The value of the designed R is 5 µm. The optimized fiber shows excellent loss performance. The simulation results show that CL ∼ 0.687 dB/km and the total loss (TL) ∼ 4.27 dB/km @ 1.55 µm. The central wavelength is approximately 1.375 µm and the lowest TL is 3.88 dB/km @ 1.4 µm. The BL is lower than 10 dB/km at 1.4 ∼ 1.6 µm when Rc is 10 mm. The direct coupling loss with SSMF is ∼ 0.125 dB. The spectral and modal performances are also excellent. The TL is lower than 5 dB/km @ 1.225 ∼ 1.6 µm, and the 3 dB-BW is ∼ 475 nm, extending from 1.15 µm to 1.625 µm. The minimum higher-order mode extinction ratio (HOMER) is higher than 40 dB. Hi-Bi ∼ 10−4 and low TL < 10 dB/km are achieved simultaneously by changing the thickness of SCTs. In summary, the proposed HC-NANF with SCTs has low propagation loss, low BL, low coupling loss, large BW, effective single-mode, and achievable Hi-Bi performance. The optimized structure parameters have great robustness. It is expected to expand the potential short-haul application range of HC-NANFs in laser-based gas sensing, miniaturized fiber sensing, etc.
2. Universal method of reducing R or loss of HC-ARFs
To obtain low-loss small-core HC-ARFs, three universal methods are revisited. The loss includes CL, BL, and direct coupling loss. The first two methods are suitable for reducing R or CL and BL. The last method is suitable for reducing R or direct coupling loss. In addition, the CL has an inverse correlation with R while the BL and direct coupling loss have a positive correlation with R.
2.1 Increasing number of silica rings m of HC-ARFs
For the ideal HC-ARF structure shown in Fig. 1(a), the ideal HC-ARF structure is formed by periodical concentric silica antiresonant rings (SARRs) and air antiresonant rings (AARRs) surrounding the air core [27,28]. The thicknesses of SARRs and AARRs are silica and air antiresonant thicknesses t and h, respectively. The Rm = R + m • (t + h) for the ideal HC-ARF structure. There is no supporting structure in AARRs to support SARRs in the ideal HC-ARF structure, so this structure is impractical. According to the theoretical formula of CL for the ideal HC-ARF structure in [8] and [27], CL decreases rapidly as (1/R)2m+3. The negative curvature effect is not considered because it is absent in an ideal HC-ARF structure. Thus, increasing m could enhance the localization of light to reduce R or CL. For realistic HC-ARFs, CL ∼ (1/R)2m+3 is no longer valid and CL becomes proportional to (1/R)4 for single-ring HC-ARFs [29] and (1/R)8 for HC-NANFs [4]. There are many structures designed to increase m, but few of them are practical. As shown in Fig. 1(b), for CTFs, m is increased according to the number of glass bars, with one bar increasing m by 1. The 1-bar CTF has been fabricated and the 2-bar CTF has been designed [3,30]. For HC-NANFs, m is increased according to the number of nested rings. In this case, one nested ring increases m by 2. The NANF and DNANF have been designed and fabricated [4,6,7]. Increasing the number of nested rings is more efficient than increasing the number of glass bars. However, the value of Rm is fixed when the number of nested rings is increased for HC-NANFs.
Fig. 1. (a) Transverse section of the ideal HC-ARF structure with m = 2. (b) Schematic illustration of the single-ring HC-ARF, NANF, DNANF, 1-bar CTF, and 2-bar CTF.
2.2 Increasing radius of the microstructure region Rm of HC-ARFs
The second method is based on increasing Rm to keep the core away from the nested point, which is the main leakage channel of HC-NANFs [6,12]. This method has been mostly ignored before and was proposed recently [6].
The geometrical structure and parameters of single-ring HC-ARFs with N = 6 are shown in Fig. 2(a), where r is the radius of the tube, g is the gap between two adjacent tubes, P is the nested point, and R is the radius of the internally tangent circle at the core region with negative curvature boundary. The Rm expressed by Eq. (1) [17,29] increases as N decreases when R and g are fixed. For example, the loss of 5 nested tubes HC-NANF (5T-NANF) with R ∼ 17.25 µm, Rm ∼ 54.4 µm, and N = 5 is lower than the loss of 6 nested tubes HC-NANF (6T-NANF) with the same R ∼ 17.25 µm, smaller Rm ∼ 43 µm and N = 6 [6], as shown in Fig. 2(b). Therefore, the loss could be reduced by increasing Rm, namely, decrease N. However, Rm has an upper limit since the fiber strength becomes weaker with larger Rm, so the value of Rm should be suitable.
Fig. 2. (a) A transverse section of a single-ring HC-ARFs with N = 6. (b) Schematic illustration of the 6T-NANF and 5T-NANF with the same R.
2.3 Optimizing mode fields of FMs of HC-ARFs
Direct coupling loss of HC-ARF is influenced by the mode fields of FMs, which is decided by the R and N. In this study, ignoring the significant mismatch of numerical aperture (NA), the angular alignment error, the gap between interfaces, the reflection of interfaces, and the structural deformation in practice, the direct coupling loss is evaluated by Eq. (2) considering only the overlap of coupling modes [31]. E1 and E2 are the vector transverse electric field distributions of the two coupled modes, respectively. According to Eq. (2), the value of coupling loss is influenced by the shape of two coupling modes of electric field distributions.
To optimize R or N through optimization of the coupling loss, two coupling situations were considered, as shown in Fig. 3. One is the coupling between Gaussian beams with different beam waist ω and the FM of the fiber under test (FUT). Another is the coupling between the FM of a step-index solid core fiber (SCF) with a different core radius RSCF and the FM of the FUT. The core refractive index of SCF was set as ncore = 1.4457 while the cladding refractive index of SCF was set as ncladding = 1.4378. These values are the same as those of an SSMF and indeed it becomes an SSMF when RSCF = 4.1 µm.
Fig. 3. Schematic illustration of the two coupling cases. Top: coupling between Gaussian beam and the FM of the FUT; bottom: coupling between the FM of an SCF and that of the FUT.
3. Design method and progress of HC-NANF with SCTs
3.1 Basic geometrical structure
The basic geometrical structure of HC-NANF with SCTs is designed through theoretical analysis and simulation. Parameters of this structure include m, N, and the original structure parameters, i.e., ti, hi, g, and R.
3.1.1 Number of silica rings m
The relationship between R and CL of ideal HC-ARF structure for different values of m is shown in Fig. 4. The validity and accuracy of the model of the ideal HC-ARF structure were thus verified because the solid lines and points are nearly coincident. According to the results in Fig. 4, CL rapidly decreases with the increase of R and m. When R is smaller than 6 µm, CL is around the sub-dB/km range only if m ≥ 6, with Rm approximately equal to 30 µm. If the negative curvature effect is not considered, a tube is equivalent to two SARRs and one AARR [3,8]. Based on this equivalent relationship, the DNANF approximately has six SARRs, which coincides with the ideal HC-ARF structure with m = 6, but it has only three AARRs, which are fewer than those of the ideal HC-ARF structure with m = 6. According to the above approximate estimation, the DNANF could not have CL < 1 dB/km when R is smaller than 6 µm. Thus, it is necessary to add additional structures, such as SCTs in DNANF, to increase m and reduce CL.
Fig. 4. Simulated dependence of the CLs of the ideal HC-ARF structure on R when m = 2, 3, 4, 5, 6, 7. The solid lines are the theoretically calculated results, where λ = 1.55 µm. The points are the simulated results according to the finite element model of the ideal HC-ARF structure, where Z = 0. The dashed lines are the calculated results for Rm = 20, 30, and 40 µm.
3.1.2 Number of nested tube groups N
For the large-core HC-ARFs including the successfully fabricated HC-NANFs, N is usually 5, 6, 10, and 12 [6,12,32]. This study shows that the range of N in small-core HC-NANFs is extremely limited, even no suitable value of N can be set in the unmodified HC-NANFs structure. Two factors are considered next.
One factor is the limit of the achievable geometrical structure of HC-ARFs with a ring wall, including single-ring HC-ARFs and HC-NANFs. The relationships in Eq. (1) are also valid in HC-NANFs. According to the calculated results, Rm ≤ 20.95 µm when N ≥ 4 if R is 5 µm and g is 2.4 µm. The high loss of this structure is unavoidable because the small Rm causes a strong leak of light from the nested point. In addition, when N = 3, Rm = 51.73 µm, which is a value close to the general radius of fiber ∼ 62.5 µm. The value of Rm is too large so this structure has poor strength. Thus, the structure of HC-NANF with N = 3 should be modified.
Another factor is the coupling loss, decided by the mode fields of FMs for different values of N. In two cases shown in Fig. 3, the coupling losses change with ω and RSCF for different values of N. The absolute value of the coupling loss is influenced by N and CL simultaneously. Therefore, the value of N could not be optimized by comparing the absolute values of coupling losses for different values of N. The values of ω0 and RSCF-0 at the minimum coupling loss are only related to the shape of E1 and E2, which is decided by N, R, and g. The values of ω0 and RSCF-0 for an SSMF, single-ring HC-ARFs with different values of N, and the ideal HC-ARF with m = 1 are shown in Table 1.
Table 1. Values of ω0 and RSCF-0 for different fiber designs
The values of ω0 and RSCF-0 increase as N decreases when R is 5 µm and g is 2.4 µm. The core shape of the ideal HC-ARF is similar to that of a single-ring HC-ARF with N = ∞. Note that the single-ring HC-ARF with N = 3 has the same coupling characteristics as the SSMF because of their similar values of ω0 and RSCF-0. Given that the values of ω0 and RSCF-0 increase with R and g, they would reach values close to those of the SSMF for the single-ring HC-ARF with N ≥ 4 when R and g are increased. However, increasing R and g is inconsistent with the small-core design. Hence, the optimum value of N is 3.
Considering the two factors above, N was set to 3, forcing the structure of the DNANF with N = 3 to be further modified.
3.1.3 Original structure parameters: ti, hi, g, and R
Considering that Rm is too large for traditional DNANFs when N = 3, SCTs were added to constitute the core boundary. The value of Rm could then be adjusted in a considerably large range. Simulation results show that the space on either hand of DNANFs tubes leads to a light leak and mode coupling between FMs and cladding modes, which is reduced by adding extra single-ring tubes to the structure, as shown in Fig. 5.
Fig. 5. A transverse section of the basic geometrical structure of an HC-NANF with SCTs. The structure in the blue dashed square represents DNANFs tubes, the structure in the red dashed circular arc-shaped box is an SCT, and the structure in the green dashed square is a single-ring tube.
The structure parameters of an HC-NANF with SCTs are R, g, ti, and hi (i = 1, 2, 3, 4, 5), as shown in Fig. 5, where g is the gap between two adjacent SCTs, ti (i = 1, 2, 3, 4, 5) are the thickness of SCTs, DNANFs tubes, and single-ring tubes, respectively, hi (i = 1, 2, 3, 4) are the thickness of air regions along the vertical direction, and h5 is the inner diameter of the single-ring tubes. The values of the original structure parameters ti and hi were set to first order antiresonant thicknesses of 0.372 µm and 3.25 µm respectively to reduce Rm. The original value of g was set to 3 µm, which is approximately the common value of g in other HC-NANFs. Besides, the value of R was optimized to reduce the coupling loss between the SSMF and HC-NANF with SCTs. According to the second coupling case in Fig. 3, the FUT was set to an HC-NANF with SCTs presenting different values of R, while the SCF with RSCF = 4.1 µm was set to be the SSMF. The dependence of coupling loss on R from the simulation is shown in Fig. 6. The coupling loss reaches its minimum when R is approximately 5 µm. Therefore, R = 5 µm was chosen as the optimum value. Note that the Rm = R + h1 + h2 + h3 + h4 + t1 + 2 • (t2 + t3 + t4) is approximately equal to 20.6 µm at first if the nested depth of the DNANFs tubes is ignored.
3.2 Optimization and tolerance analysis of structure parameters ti, hi, and g
3.2.1 Optimization of structure parameters ti, hi, and g
The CL is usually the dominant loss in HC-ARFs. According to the simulation results, the CL of the original HC-NANF with SCTs is approximately 415 dB/km, which is mainly caused by the mode coupling between the FM and cladding modes, and the small Rm. The structure parameters were thus further optimized to decrease CL.
The number of structure parameters is eleven, and it is difficult to achieve global optimal values through traditional single-parameter scanning methods. A simple parameter optimization method, called the “inside-out loop optimization method”, is used to obtain the optimum parameter values for the HC-NANF with SCTs. The process followed in this method is presented in Fig. 7(a). The structure of the HC-NANF with SCTs is described by a vector of structure parameters X = [x1, x2, …, x11], whose elements represent the inside-out structure parameters g, t1, h1, t2, h2, t3, h3, t4, h4, t5, and h5 respectively. The i-th element xi is optimized by parameter scanning in a proper range. When CLi = CLio reaches the minimum in this range at xi = xio, the parameter xi is optimized to xio, and xi+1 is then optimized with the same process. When i = 11, this loop is over. If CL1 - CL11 < 0.05 dB/km, the parameter optimization process is finished and the vector of optimum structure parameters is achieved. Otherwise, the loop is executed continually.
Fig. 7. (a) Process of the parameter optimization method, called “inside-out loop optimization method”. Simulation results show the dependence of the (b) CL of the FM and (c) Rm on xi for the HC-NANF with SCTs in different loops. (d) A transverse section of the optimized structure and (e) FM electric field distributions of the HC-NANF with SCTs.
After six loops, the CL reduces from 415.51 dB/km to 0.687 dB/km and stays nearly constant at the end, as shown in Fig. 7(b). The CL reduces rapidly in the first two loops and breaks the 1-dB/km limit in the third loop. The CL reduces slowly at the last three loops. The reduction of CL is only 0.0138 dB/km at the last loop. Similarly, the value of Rm increases from 20.6 µm to 27.66 µm, and keeps nearly invariable at the end, as shown in Fig. 7(c). The value of Rm increases rapidly in the first loop, fluctuates in the second, third, and fourth loops, and stabilizes in the fifth and sixth loops. The CL reduces rapidly with the increase of Rm at the first loop. Therefore, according to this analysis, the CL could be greatly reduced by increasing Rm.
The final optimized structure parameters X = [g, t1, h1, t2, h2, t3, h3, t4, h4, t5, h5] is [2.4, 0.45, 3.1, 0.3525, 5.3, 0.365, 4.75, 0.41, 6.8, 0.5, 6.8]. The optimized structure and the FM electric field distributions of the HC-NANF with SCTs are shown in Figs. 7(d) and (e), respectively.
3.2.2 Tolerance analysis of structure parameters ti, hi, and g
It is difficult to fabricate the HC-NANF with SCTs with accurate values of the optimum structure parameters. Therefore, tolerance analysis of these structure parameters is necessary. The tolerable value of CL was set to 1 dB/km. The simulation results of the various ranges of these parameters, including the negative variation percentage (NVP), positive variation percentage (PVP), and total variation percentage (TVP), compared with the optimum values, are shown in Table 2. The results show the excellent robustness of these structure parameters. Their TVP is higher than 20% in all cases. The tolerance of g is the highest, with a TVP of 85%. The tolerances of t1, h1, h3, and t5 are also high, with a TVP between 40% to 60%. The tolerance of h4 is the lowest, with a TVP of 23%. In addition, TL < 4.7 dB/km in the tolerance range is achieved for all these structure parameters except for t1.
Table 2. Results of tolerance analysis of structure parameters
4. Analysis of fiber performance
4.1 Loss
4.1.1 Propagation loss in straight fiber
The loss of HC-ARFs includes CL, surface scattering loss (SSL), MBL, and absorption loss. Simulation results show that 99.88% of the mode power of the FM is in the air, so the absorption loss is negligible. The MBL is the main loss at a short wavelength in large-core HC-NANFs. The theory of the microbending effect [33] indicates the MBL is in proportion to R 2·C(Δβ), where C(Δβ) is the power spectral density. When C(Δβ) = 1/Δβ 2, MBL ∼ R 2/Δβ 2, namely MBL ∼ R 2/(nLP01-nLP11)2, where nLP01 and nLP11 are the effective refractive index of the LP01 and LP11 modes, respectively. Therefore, the predicted MBL of the HC-NANF with SCTs is estimated to be much lower than the MBL of large-core HC-NANFs, and it is ignored in this study.
The SSL is calculated by η (λc) • F [34,35], where η (λc) = (1.55 µm / λc)2 • η (1.55 µm) is the scale factor for the fiber with central wavelength λc, which is approximately 1.375 µm; η (1.55 µm) is usually 300 [34]. Thus, η (λc) is approximately 381. F is the normalized electric field intensity at the interfaces. The SSL and TL of the HC-NANF with SCTs were calculated to be 3.58 dB/km and 4.27 dB/km at 1.55 µm. Thus, the SSL is the dominating loss of the small-core HC-NANF with SCTs. This is because F increases when R decreases.
It is worth mentioning that this method of SSL calculation comes from research on SSL of hollow-core photonic-bandgap fibers (HC-PBFs), where η (1.55 µm) = 300 is an empirical value of the 7-cell HC-PBF [34]. Note that η is influenced by the different core sizes and the fabrication configuration. For example, η (1.5 µm) is approximately equal to 201 for the 19-cell HC-PBF [36]. The core boundary thickness tc of HC-NANFs is thicker than tc of HC-PBFs, which usually means lower surface roughness [37] and lower SSL. Besides, the ARROW effect could also reduce E0 at the core boundary. Thus, realistic values of SSL of HC-NANFs are usually lower than the values calculated by this method [12,38]. In this study, the SSL calculated by this method is reserved. We suppose that realistic values of SSL and TL of HC-NANF with SCTs would be lower.
4.1.2 Bending loss
HC-NANFs usually present high BL because of their large core and low m [39]. The high BL greatly limits the application of HC-NANFs. Here, we investigate the bending characteristics of the HC-NANF with SCTs. The BL is calculated using the standard conformal transformation [40], where the bent structure is transformed into its equivalent straight structure with equivalent refractive index profile, n′, defined by Eq. (3), with Rc as the radius of curvature and x is the transverse distance from the center of the fiber. The structure deformation from structure bending is ignored. The change of n induced by stress-optic effects is also ignored because most of the light propagates in the air.
Owing to the C3v symmetric structure of the HC-NANF with SCTs, the BL changes with a period of 60° in bending directions. BL spectra in three typical bending directions, namely 0°, 30°, and −30°, were simulated for different values of Rc, as shown in Fig. 8. The BL is lower than 10 dB/km @ Rc > 8 mm at 1.55 µm in these bending directions. The redshift of the short-wavelength boundary happens when Rc decreases, especially in 30° bending directions. However, the BL is still lower than 10 dB/km @ Rc = 10 mm in 1.4 ∼ 1.6 µm. By contrast, the BL of 7-cell HC-PBFs is extremely low, with no appreciable drop in transmission observed until the fiber breaks [41]. The BL of the SSMF is 5 ∼ 15 dB/m @ Rc = 10 mm according to the theoretical analysis and the experiments [42]. Therefore, the HC-NANF with SCTs features remarkable broadband bending characteristics.
Fig. 8. (a) Schematic illustration of three typical bending directions. Simulated dependence of the BL spectra of the HC-NANF with SCTs in (b) 0°, (c) 30°, and (d) −30° bending directions.
4.1.3 Coupling loss
It is common to couple the HC-NANF with SCTs with a Gaussian beam or SSMF in practical application. Here, we investigate the coupling loss of HC-NANF with SCTs compared with an SSMF and a commercial 7-cell HC-PBF free of surface modes, whose geometrical structures and FM intensity distributions are shown in Fig. 9(a).
Fig. 9. (a) Geometrical structures and FM intensity distributions of HC-NANF with SCTs, 7-cell HC-PBF, and SSMF. Simulated dependence of the coupling loss on (b) ω and (c) RSCF.
The coupling model in Fig. 3 was employed to calculate the coupling loss for different values of ω and RSCF. For the coupling of the Gaussian beam, as shown in Fig. 9(b), the HC-NANF with SCTs and the SSMF present minimum coupling losses of 0.139 dB and 0.012 dB, respectively, both at ω = 4.375 µm. The 7-cell HC-PBF has a minimum coupling loss of 0.135 dB at ω = 3.875 µm. Concerning the coupling of the SCF, as shown in Fig. 9(c), the HC-NANF with SCTs has a minimum coupling loss of 0.125 dB at RSCF = 4.1 µm. The coupling loss between the HC-NANF with SCTs and the SSMF is 0.125 dB. In contrast, the 7-cell HC-PBF has a coupling loss of 0.15 dB with the SSMF and a minimum coupling loss of 0.11 dB at RSCF = 3.375 µm. Therefore, the coupling loss between the SSMF and the HC-NANF with SCTs is lower than the coupling loss between the SSMF and the 7-cell HC-PBF. The reason is that the size of FM intensity distributions of the SSMF is similar to that of the HC-NANF with SCTs and larger than that of the 7-cell HC-PBF, as shown in Fig. 9(a). In addition, the 7-cell HC-PBF has a lower minimum coupling loss with Gaussian beam and SCF. This is because the shape of the 7-cell HC-PBF FM presents a greater similarity with a circle compared with the shape of the FM of the HC-NANF with SCTs.
4.2 Spectra and modal characteristics
The loss spectra of the FM and LP11 modes of the HC-NANF with SCTs are shown in Fig. 10(a). Although the structure of the HC-NANF with SCTs is optimized at 1.55 µm, λc is approximately equal to 1.375 µm and the lowest TL is ∼ 3.88 dB/km @ 1.4 µm. Note that TL < 5 dB/km is achieved between 1.225 µm and 1.6 µm. The 3 dB-BW is ∼ 475 nm, extending from 1.15 µm to 1.625 µm. The higher-order modes with lower loss are LP11 modes. Note that CLsLP11 is higher than 105 dB/km in Fig. 10(a). The values of neff for the FM and LP11 modes, and HOMER for different values of λ, are shown in Fig. 10(b), where HOMER is 48.4 dB at 1.55 µm and higher than 40 dB in propagation loss spectra. Therefore, the HC-NANF with SCTs has broad BW and effective single-mode characteristics.
Fig. 10. (a) Loss spectra of the FM and LP11 modes in the HC-NANF with SCTs. (b) Simulated dependence of neff for the FM and LP11 modes, and HOMER, on λ. The inset shows the electric field distributions of LP11 modes in the HC-NANF with SCTs.
4.3 Birefringence
The Hi-Bi of HC-ARFs is usually obtained by appropriately modifying tc along one direction while leaving it unchanged in the orthogonal direction with the antiresonance value [24,25,43]. This method is also used to obtain Hi-Bi HC-NANFs with SCTs. Given that N = 3 in the HC-NANF with SCTs, there are two achievable Hi-Bi structures. As shown in Fig. 11(a) and (b), one is obtained by changing t1 for tp in one SCT, whereas the other is obtained by changing t1 for tp in the other two SCTs.
Fig. 11. Two achievable Hi-Bi structures changing t1 for tp in (a) one SCT and (b) the other two SCTs of the HC-NANF with SCTs. Simulated dependence of the (c) FM loss and (d) Δn on tp in two structures. “x” and “y” in the legend refer to the polarization (dominant vectorial direction of the E-field of the FMs).
The birefringence characteristics of these two structures are simulated in the first- and second-order antiresonance regions, as shown in Fig. 11(c) and (d). The results show that the birefringence Δn increases with the FM loss. The value Δn ∼ 10−4 is realized at tp = ∼ 0.325, 0.575, 1.075, and 1.35 µm while the FM loss is lower than 10 dB/km. In addition, the HC-NANF with SCTs has broadband Hi-Bi. For example, when tp is approximately equal to 0.575 µm in Fig. 11(a), the HC-NANF with SCTs has Δn > 10−4 at 1.35 ∼ 1.64 µm while the FM loss is lower than 10 dB/km, as shown in Fig. 12.
Fig. 12. Simulated dependence of the loss of FMs and Δn on λ in the HC-NANF with SCTs with tp ∼ 0.575 µm in Fig. 11(a).
5. Comparison between HC-NANF with SCTs, 7-cell HC-PBF, and large-core HC-NANFs
In the field of hollow-core fibers (HCFs), the relatively small core 7-cell HC-PBFs with high loss and large-core HC-NANFs with low loss have been fabricated and applied successfully. It is meaningful to compare the designed HC-NANF with SCTs with these two successful HCFs.
From a transmission point of view, as shown in Table 3, the core diameter D of the HC-NANF with SCTs decreases a factor of 2.8 ∼ 3.8 compared with large-core HC-NANFs. The minimum loss of the HC-NANF with SCTs is lower than that of the 7-cell HC-PBF with a similar core size. The BL and direct coupling loss with SSMF of the HC-NANF with SCTs extremely reduced compared with large-core HC-NANFs. The maximum BW of the HC-NANF with SCTs is lower than that of the large-core HC-NANFs but higher than that of the 7-cell HC-PBF. In addition, the HC-NANF with SCTs presents effective single-mode characteristics. Therefore, the HC-NANF with SCTs has remarkable transmission characteristics and application potential.
Table 3. Comparison of transmission characteristics between the HC-NANF with SCTs, 7-cell HC-PBF, and large-core HC-NANFs
From a structural point of view, as shown in Fig. 13, the 7-cell HC-PBF has more than 250 capillaries with a thickness of 100 nm. In comparison, the large-core HC-NANF usually has 10 ∼ 12 capillaries with thickness > 300 nm. Similarly, the HC-NANF with SCTs has 15 capillaries and 3 SCTs with thickness > 300 nm. Thus, it keeps the structural advantage of large-core HC-NANFs, which present few thick capillaries. It is also beneficial to the fabrication.
6. Conclusions
In this study, we demonstrate a small-core HC-NANF with SCTs. The complete design method and progress are shown. The results of characteristics analysis and comparison show excellent performance.
Three universal methods of reducing R or loss of HC-ARFs were discussed by theoretical and geometrical analysis. Some examples were also described to verify these methods. The CL of HC-ARFs could be reduced by increasing m or Rm. Optimizing mode fields of FMs by changing R and N could reduce coupling loss of HC-ARFs.
The design method and progress of HC-NANF with SCTs were shown. The basic geometrical structure of the HC-NANF with SCTs was determined after adding SCTs and single-ring tubes to the DNANFs. The optimization and tolerance analysis of the structure parameters ti, hi, and g were conducted. CL ∼ 0.687 dB/km @ 1.55 µm was realized through the parameter optimization method. The tolerance of the optimized structure parameters was analyzed and the results show that these optimized structure parameters have great robustness.
The fiber performance in terms of loss, including the propagation loss in straight fiber, BL and coupling loss, and spectra and modal characteristics, were simulated and analyzed in detail. The results show low propagation loss, low BL, low coupling loss, broad BW, and effective single-mode characteristics for the HC-NANF with SCTs. In addition, the possibility of broadband Hi-Bi ∼ 10−4 was realized by adjusting the value of t1.
The notable transmission and realizable structure characteristics of the HC-NANF with SCTs were demonstrated in contrast with 7-cell HC-PBFs and large-core HC-NANFs. The fabrication of HC-NANF with SCTs is challenging to traditional fiber draws even if HC-ARF with SCTs has been fabricated recently. Some new fabrication techniques and/or materials such as 3D printing might be used to realize this relatively complicated structure. The proposed HC-NANF with SCTs is expected to expand the potential short-haul application range of HC-NANFs in laser-based gas sensing, miniaturized fiber sensing, etc.
Funding
National Natural Science Foundation of China (61935002).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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