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Abstract
Stable generation and propagation of ultrafast high-order mode beams has become an important research direction. A core diameter not more than 10 times the wavelength is regarded as the upper limit for single mode transmission. However, a high-power laser requires a core diameter 20 to 40 times the wavelength to achieve high-power and stable output, which exceeds the design limit of the traditional fiber. In this paper, a novel 37-cell hollow core photonic bandgap fiber (HC-PBF) that only supports pure TE01 mode over a bandwidth of 50 nm with the lowest loss of 0.127 dB/km is proposed. The HC-PBF has a core diameter of more than 40 μm. Single mode guidance is achieved by adjusting the lattice size in a particular of the cladding. The best single mode performance with a loss ratio as high as 150,000 between TE01 mode and other modes with minimum loss is obtained. The fiber also has low bend-loss and thus can be coiled to a small bend radius of 1 cm having 1.6 dB/km bend loss. The tunability of the single-mode window and the manufacturing feasibility of the proposed fiber are also discussed.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, beams with polarization singularities have attracted considerable attention in the fields of optical communication [1,2], particle trapping [3,4], high-dimensional quantum entanglement [5,6], high resolution imaging [7], and material processing [8]. Fiber lasers with stable high-order mode (HOM) pulse output have become an important research direction due to the high peak power and annular intensity distribution [9,10]. Despite the steady development of laser sources and processing technology, system technology is generally considered to be the limiting factor in today's industrial applications. This includes beam delivery, beam steering and shaping. By introducing flexible fiber-based beam delivery, the application of high-power lasers in industry has made great progress. The fiber used for high-power laser delivery needs to have the advantages of good flexibility, large mode field, high safety, and bending resistance. At present, solid-core fibers (SCFs) are mostly used in laser delivery [11], Quartz glass is used to control the refractive index of the core [12,13]. Unfortunately, silica-core-based fiber is only capable of nanojoule pulse energy in the ultrashort pulses regime due to its low material damage threshold, high dispersion, and other nonlinear effects. If the energy density per area of the fiber cross-section exceeds the threshold value, and the output beam will become unstable. Therefore, a core diameter of 20-40 times the wavelength is required to meet the high-power output requirements [14], which exceeds the design limit of SCFs. The application of high-power lasers can be made simple by the introduction of hollow core fibers (HCFs) [15]. Because the gas wrapped in the core will not be optically damaged under high intensity [16], it can stably guide pulses with small dispersion and high peak power over a considerable distance, and it also has the ability to transmit mid-infrared lasers compared to SCFs [17].
Several different types of HCFs have been developed. Among them, HCFs based on inhibited-coupling (IC) guidance mechanism have been well applied in high-energy laser transmission [18,19], because the overlap of the guided mode with silica cladding has been drastically reduced [20]. The transmission of milli-Joule pulse energy with intensities nearing petawatts per square centimeter has been demonstrated in Kagome hollow core fiber [19]. IC-HCFs are characterized by broadband transmission and large core size, make them ideal candidates for high power delivery in many cases. However, these fibers may be limited in performance due to bend sensitivity, thus limiting their prospects for applications that require a tighter bending radius or a compact footprint. Of course, there have been some fiber designs that seek to obtain low bending loss while not giving up the flexibility and splicing capabilities of the fiber [17,21], but these designs still have a lot of room for improvement. Hollow core photonic bandgap fibers (HC-PBFs) have more robust bend performance [22] and are ideal candidate to overcome limitation of a compact high-power laser system [23,24]. Although, they usually have smaller mode field diameter than IC-HCFs, considerable large core sizes can still be achieved by removing air holes near the core. In addition, abundant structural parameters can provide sufficient choices for mode management of HC-PBFs [25]. One method is to realize the filtering of the HOMs by precisely controlling the resonant coupling between the fundamental mode of the secondary core in the cladding and the HOMs in the core. Due to the existence of residual surface mode, single-mode transmission can only be achieved in a small wavelength range [26]. A fractal photonic bandgap fiber based on scale invariance has been proposed in recent years, and it has been confirmed that single-mode transmission is possible for fibers with a core diameter of more than 35 times the wavelength [14].
As a waveguide structure that carries laser transmission, HC-PBF perfectly combines the advantages of fiber lasers with compact structure, high conversion efficiency, good beam quality, and gas lasers with flexible wavelength selection, narrow laser lines, high damage thresholds, weak nonlinear effects. This provides a new idea for solving the technical bottlenecks encountered by traditional lasers in terms of high-energy delivery, wavelength expansion, and spectral line control, especially in the application of mid-infrared lasers, which has huge development potential.
In this paper, we have fully exploited the air hole tunable advantage of the HC-PBF in the cladding, and achieved ultra-low loss single-mode transmission in 37-cell HC-PBF for the first time. Based on the power loss shape of different guided modes, cladding defects are introduced in 12 power loss directions to achieve filtering of all modes except TE01 within a bandwidth exceeding 50 nm. Without any polarization controller, excellent single-mode performance with a loss ratio as high as 150,000 times can be obtained. The single-mode window can be moved in multiple wavelength ranges by linearly stretching fiber structure in the scale to wavelength, which means that this fiber possesses an excellent tunability. We demonstrated the typical characteristics of the fiber when lattice period is 6 μm. TE01 has a loss of less than 0.5 dB/km in the range of 1786 nm-1840 nm. The dispersion is lower than 10 ps.nm-1.km-1, the mode field area is greater than 825 µm2, and the nonlinear coefficients is less than 1.7×10−4 w-1/km. The fiber has excellent bending resistance characteristics, even if the bending radius reaches 1 cm, the loss is only 1.6 dB/km. Furthermore, the filtering effect with a complete fiber structure is also shown in comparison, and the fabrication feasibility of proposed fiber is discussed.
2. Fiber structure and performance
Simulations were performed using COMSOL Multiphysics with an optimized mesh size and a perfectly matched layer. The cross-section and structural parameters of the proposed hollow core fiber is displayed in Fig. 1. The structure of photonic crystal in the cladding is formed by a triangular arrangement of hexagonal air holes with the lattice constant Λ = 6 µm. The large air core is introduced into the HC-PBF by removing 37 air holes from the cladding structure. The diameter of cladding air hole is d0 = 0.98Λ, the hexagonal air hole circular angle diameter is dc = 0.2Λ, the core radius is R = 20 µm, and the thickness of silica ring around the air core is tcore = 370 nm. The key to single-mode transmission is to optimize the lattice structure in 12 directions in the cladding. Blue and pink lattices are used to suppress HE or EH modes, and yellow lattices are used to suppress higher-order TE modes except TE01. Among them, the yellow lattice needs to keep the core wall thickness constant during scaling to ensure low loss of TE01 (i.e., the scaling center needs to be set in the middle of the core wall). The corresponding diameters are: d1 = 0.93Λ, d2 = 0.89Λ, d3 = 0.96Λ. The background material of the designed fiber is pure silica, and the refractive index at different wavelengths is determined by the Sellmeier equation.
It is difficult to generate pure TE01 mode in traditional SCF. Two difficulties need to be overcome: (1) destroy the degeneracy of the same order modes, making TE01 loss lower than other modes; (2) suppress fundamental mode. It is generally believed that the mode coupling of HCF is much weaker than that of SCF, because the extremely low interaction of optical modes with the fiber cladding. Indeed, cross-coupling between the two orthogonally polarized modes in a HCF should only occur at the core/cladding interface, but the optical field strength at this interface is vanishingly small. Thus, the same reason that impedes the introduction of high birefringence into HCF also prevents intermodal coupling. This was proved in [27], even with a very small birefringence of $\Delta n \approx {10^{ - 7}}$, HCF designs can support the propagation of polarized light with an incredibly high polarization extinction ratio of up to 70 dB, meaning the power of unwanted polarization is well suppressed. In this paper, the propagation constant of TE01 is closest to HE21 mode, and their refractive index difference is greater than 1.0 × 10−6 in the single-mode window. In addition, the loss ratio of HE21 to TE01 is as high as 106, which can further eliminate the loss caused by mode coupling.
The polarization direction of the mode represents the direction of power loss. TE01 is an angular polarization mode, and the electric field direction at the core glass interface is always parallel to the interface. Therefore, introducing defects in the fiber cladding corresponds to the unwanted modes polarization directions, which can increase the limiting loss. Generally, the loss of HC-PBF mainly comes from the surface scattering loss (SSL), and the confinement loss (CL) occupies a very small component. In the proposed model, the SSL is guaranteed to remain unchanged, and filtering is implemented only by the CL.
The confinement loss is obtained from the imaginary part of the effective mode refractive index [28]:
The SSL was caused by surface capillary waves frozen into the fiber as it solidified, which is calculated based on the model after calibration [29,30]:
Theoretical models indicate a sharp increase in the number of core modes as the core diameter is enlarged. An ideal 37-cell HC-PBF structure is predicted to support as many as 80 modes, including degeneracies, according to the following equation [31]:
At first, we compare the transmission loss characteristics of TE01 and other modes (EH31, HE11, TE02, HE31) with minimal loss. The detailed information of the fiber filtering effect is shown in Fig. 2. The losses of more than 40 modes are calculated and the highest order is TE03. Among them, TE01 has a higher loss at the edge of the bandgap near 1786 nm, and then the loss gradually decreases as the wavelength moves. In the range of 1786 nm–1840 nm, the loss is always lower than 0.5 dB/km, and the lowest value is 0.127 dB/km at 1840 nm. In contrast, other modes always maintain high loss great than 100 dB/km in this range, and the loss ratio exceeds 1000. As the wavelength moves, the loss of TE02 gradually decreases, and the single-mode window is cut off. In Fig. 2(b), the CL and SSL of TE01 (solid line) and other minimum loss modes (dashed) are shown respectively. Obviously, the difference in scattering loss between them is very small, basically less than 2 dB/km, and the roughness of the glass surface has basically the same effect on them. The total loss difference comes from the CL.
Fig. 2. Loss contrast between TE01 and other modes with minimal loss: (a) total loss, (b) confinement loss and scattering loss.
Figure 3 shows some of the main characteristics of the designed fiber. The F factor (blue line) and dispersion (red line) curves in the single-mode operating range is plotted in Fig. 3(a). The F factor indirectly indicates the size of the scattering loss. In the single-mode window, the average value of FΛ is 0.004, indicating that the mode field has a very low overlap rate with the glass surface. And TE01 has very small dispersion characteristics. Among them, the dispersion value reaches 0 ps.nm-1.km-1 near 1798 nm and is always less than 10 ps.nm-1.km-1 in the single-mode window. Smaller dispersion can eliminate signal delay caused by fiber pulse broadening.
Fig. 3. Mode characteristics of TE01 in the designed fiber: (a) F factor (blue) and dispersion (red), (b) mode effective area (blue) and nonlinear coefficient (red).
Traditional SCFs generally have supercritical nonlinear effects in high-power laser systems, which limit the power increase of lasers. Nonlinearity is inversely proportional to the mode field area, so an effective way to overcome these problems is to use large mode field area fiber. Their calculation relationship is as follow:
where ${A_{_{\textrm{eff}}}}$ denotes the mode effective area and ${n_2}$ represents the nonlinear refractive index (≈4 × 10−23 m2/W in air [23]).Through a bandgap light guidance mechanism, light is confined in the hollow core, which enables a two to three orders of magnitude reduction in optical nonlinearities. Results of $\gamma$ and ${A_{_{\textrm{eff}}}}$ are given as functions of wavelength and shown in Fig. 3(b), the nonlinear coefficient is less than 1.7 × 10−4 w-1/km over the operation band. Since the mode field in the 37-cell HC-PBF overlaps with glass is only about 0.1% [24], in this fiber model, 99.7% of the power is focused in the core, we ignore the influence of mode overlap effect on nonlinear index of the fiber. In the range of 1786 nm–1840 nm, the mode effective area is 825 µm2–828 µm2, which decreases as the wavelength increases.
The electric fields and transversal power flows (Pt) of the first five modes (TE01, TE02, HE11, HE31, TE03) with the lowest loss at 1830 nm are shown in Fig. 4 to visualize the filtering effect of lattice modulation on modes. In general, it is relatively difficult to implement single-mode operation for large mode field fiber because of its numerous modes and insensitivity to fiber deformation. However, by observing the polarization direction of the mode, it can be found that all other modes have outward polarization direction, except angular polarized TE modes. According to these polarization directions, the lattice size can be adjusted in the cladding to filter the outward polarization modes, while only the angular polarized TE01 mode is retained. It can be seen from the Pt that the power of modes is along the lattice modulation direction leak into the cladding. TE02 and TE03 are also angular polarization modes, so it is necessary to specially modulate the filling rate of the innermost air holes of the cladding (d3 in Fig. 1) to achieve the filtering of TE02 and TE03. In addition, the core wall thickness needs to be uniform during the lattice modulation process, especially the core wall around the yellow lattice, otherwise the TE01 power will also be lost to the cladding.
Fig. 4. Electric fields (E), transversal power flows (Pt) of the first five modes (TE01, TE02, HE11, HE31, TE03) with the lowest loss at 1830nm.
Single-mode transmission can be achieved at any wavelength by linearly stretching the fiber structure, which means that this fiber structure possesses an excellent tunability. We choose Λ = 5 µm, 7 µm, 9 µm, and 10 µm as typical examples to analyze the single-mode window tunability of the designed fiber, corresponding to Figs. 5(a)–5(d). Among them, the core wall thickness has not changed, it is always 370 nm. The single-mode window will move toward longer wavelengths as the lattice constant increases. When Λ = 5 µm, the single-mode window is 1497 nm–1540 nm, and the minimum loss is 0.724 dB/km at 1530nm; when Λ = 7 µm, the single-mode window is 2095 nm–2135 nm, and the minimum loss is 0.124 dB/km at 2135 nm; when Λ = 9 µm, the single-mode window is 2640 nm–2720 nm, and the minimum loss is 0.067 dB/km at 2720 nm; when Λ = 10 µm, the single-mode window is 2920 nm–3000 nm, and the minimum loss is 0.049 dB/km at 3000 nm. This is likely to have most impact in the field of medicine, allowing for example the delivery of radiation at 2.94 µm as used in many surgical procedures due to the strong water absorption at this wavelength [32].
Fig. 5. when the designed fiber lattice period is (a) Λ = 5 µm, (b) Λ = 7 µm, (c) Λ = 9 µm and (d) Λ = 10 µm, the loss contrast between TE01 and other modes with minimal loss.
Since the HC-PBF has core diameter significantly than the operational wavelength, it is worth studying its macro-bending characteristics. In Eq. (5), a bent fiber is equivalent to a straight fiber with the same refractive index distribution. This approximation method is widely employed in hollow-core fibers [29]:
Where ${R_c}$ is the radius of curvature, x is the distance from the center of the waveguide, ${n_i}$ is the original refractive index distribution of the straight fiber.
Figure 6(a) shows the bending loss characteristics of TE01 mode under different bending radii (@1830nm). Obviously, the fiber has more robust bending characteristics than anti-resonant fiber [17]. And the loss will increase when the bending radius is about 3 cm, even if the bending radius is 1 cm, the loss of TE01 is only 1.6 dB/km. Fiber bending has little effect on scattering loss, mainly resulting in increased leakage loss. In Fig. 6(b), the mode field and power flow direction of TE01 and HE11 are shown when the fiber bending radius is 0.5 cm in x-direction.
Fig. 6. Bending characteristics: (a) TE01's bending loss is a function of bending radius @1830nm, (b) power flow of TE01 and HE11 when bending radius is 0.5 cm in x direction.
3. Compared with complete structure fiber
In order to better understand the mode filtering effect of cladding lattice modulation, a comparative fiber with a complete lattice structure is designed. Figure 7(a) shows the complete fiber structure. The color-marked lattices in Fig. 1 are not scaled here, they are the same size as other lattices, and the corresponding lattice diameter is: d0 = d1 = d2 = d3 = 0.98Λ. In addition, other structural parameters of the fiber are the same as those in Fig. 1. The model supports more than 40 modes. In Fig. 7(b), the loss curves of some modes (HE11, TE01, TE02, TE03) within the bandgap are plotted separately. The low loss curves of the guided modes are destroyed by cross-coupling of multiple surface modes, resulting in multiple loss peaks, which destroy the low loss useable bandwidth of the fiber. Among them, all guided modes are divided into wide low-loss windows, for example, the loss of TE01 is less than 1 dB/km in 1705 nm–1860 nm and 1915 nm–2055 nm. TE01 has a minimum loss of 0.0968 dB/km at 1810 nm. TE01 has unique low-loss characteristics, even lower than the fundamental mode, which is the same in 19-cell HC-PBF [33]. Obviously, the 37-cell HC-PBF has a very low loss close to that of a traditional silicon-core multimode fiber due to its large core. Even for the higher-order mode TE03, its loss is less than 5 dB/km in 1957 nm–2125 nm wavelength range.
Fig. 7. Fiber structure without scaling the lattice: (a) cross-section structure, (b) losses of HE11, TE01, TE02, TE03 in the bandgap range.
For many years, an effective way to reduce the transmission loss of HC-PBF is to increase the size of the core, but the large core structure will guide unwanted modes transmission in the fiber, which limits the application potential of HC-PBF. By comparing with the complete structure model, we can see that the selective filtering effect of the mode based on the specific direction of the cladding region lattice scaling is obvious, which can achieve the complete filtering of all the guide modes except TE01, and retain the ultra-low loss characteristics of TE01, realizing true single-mode transmission. Here the single-mode window is limited due to the coupling effect of the surface modes, optimizing the fiber structure can move the surface modes outside the band gap, thereby realizing a wider single-mode window.
4. Discussion
The proposed single-mode single-polarization HC-PBF has many unique optical properties due to its mode filtering effect and large air hole core. However, a point of practical importance is to ensure that the proposed fiber can be fabricated. Due to limited conditions, we are unable to draw and test the proposed fiber. All analyses in this article are based on simulation results. In order to provide a reference for the actual fabricated, here we quantitatively analyze the impact of the key structural parameters on the single-mode filtering effect. In Fig. 8(a), the loss comparison between TE01 and other modes with minimum loss under different core wall thickness is shown, the gray area is the manufacturing tolerance. In the range of 250 nm–450 nm, the loss of TE01 is always less than 0.5 dB/km, the loss ratio is always greater than 2000, and the single-mode filtering effect is well maintained, which provides us with a high manufacturing tolerance. As we all know, the core wall thickness is directly related to the surface mode, and high tolerance can provide enough space to optimize the surface mode to expand the single-mode window. In the fiber structure shown in Fig. 1, the yellow lattice structure (d3) is the key to the single-mode retention effect, which is used to filter higher-order TE modes. Therefore, the tolerance of this diameter is also analyzed. Figure 8(b) shows the loss comparison under different d3. In the range of 0.93Λ–0.97Λ, the loss of TE01 is always less than 0.7 dB/km, the loss ratio is always great than 1500. Among them, the blue marks in the figure are the ideal structure parameters of the proposed fiber.
Fig. 8. Manufacturing tolerance of key parameters: (a) loss comparison of different core wall thickness, (b) loss comparison of d3 in different modulation ratio. The blue marks are the ideal parameters of the proposed fiber.
Generally speaking, the drawing step of fiber includes: making a preform with designed microstructure pattern, and then drawing preform into a fiber form with fiber drawing equipment. The complex structure of HC-PBF determines that its fabrication is more complicated and costly than traditional fibers. At present, the drawing process of HC-PBF is relatively mature, and remarkable progress has been attained in recent years. As early as 2013, the 37-cell HC-PBF with 4.5 dB/km was drawn for the first time [34]. Potential manufacturing challenges for the designed fiber mainly include: (1) precisely control the size and position of certain air holes in the cladding, they are the key to pattern filtering; (2) precisely controlling the temperature and the drawing speed during the drawing process to prevent collapsing of the air holes. Although the designed fiber supports TE01 instead of HE11, we believe that its application prospects are still broad. Because the TE01 beam has a circular intensity distribution with a harmonic zero in the center and constant polarization before and after focusing, so it is suitable for long-distance propagation. The unique properties and low loss of TE01 make it an interesting mode for many applications, including material handling, atomic guidance, and high numerical aperture focusing.
5. Conclusion
In conclusion, we proposed and numerically simulated an ultra large core HC-PBF with only pure TE01 mode transmission. The 12 fixed directions lattices in the cladding are regularly scaled to filter out the modes with out of plane polarization; the innermost air holes in the cladding are scaled to filter out the higher-order TE modes except TE01. The simulation results show that in a core with diameter of more than 40 µm, TE01 has a loss of less than 0.5 dB/km in the range of 1786 nm–1840 nm, and the loss ratio of all other modes exceeds 1000 times in a single-mode window exceeding 50 nm. The dispersion is lower than 10 ps.nm-1.km-1 and mode field greater than 825 µm2. By stretching the fiber structure laterally in the wavelength range, the tunability of the single-mode window can be flexibly realized. Due to the influence of the coupling effect of the surface mode, the further expansion of single-mode window bandwidth is limited and the loss is creased. In addition, we compared the fiber with complete fiber structure model, and analyzed the filtering effect of lattice modulation on the multimode HC-PBF. Meanwhile, the fabrication challenges of the proposed HC-PBF also discussed. The proposed fiber could be a potential candidate for high-power laser delivery due to its large hollow-core structure, single-polarization sing-mode guidance, and high stability performance.
Funding
National Natural Science Foundation of China (61835006, 61775107).
Disclosures
The authors declare no conflicts of interest.
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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