Higher-Order Mode Suppression in Antiresonant Nodeless Hollow-Core Fibers

Negative curvature hollow-core fibers (NC-HCFs) are useful as gas sensors. We numerically analyze the single-mode performance of NC-HCFs. Both single-ring NC-HCFs and nested antiresonant fibers (NANFs) are investigated. When the size of the cladding tubes is properly designed, higher-order modes (HOMs) in the fiber core can be coupled with the cladding modes effectively and form high-loss supermodes. For the single-ring structure, we propose a novel NC-HCF with hybrid cladding tubes to enable suppression of the first two HOMs in the core simultaneously. For the nested structure, we find that cascaded coupling is necessary to maximize the loss of the HOMs in NANFs, and, as a result, NANFs with five nested tubes have an advantage in single-mode guidance performance. Moreover, a novel NANF with hybrid extended cladding tubes is proposed. In this kind of NANF, higher-order mode extinction ratios (HOMERs) of 105 and even 106 are obtained for the LP11 and LP21 modes, respectively, and a similar level of 105 for the LP02 modes. Good single-mode performance is maintained within a broad wavelength range. In addition, the loss of the LP01 modes in this kind of NANF is as low as 3.90 × 10−4 dB/m.


Introduction
Hollow-core (HC) fibers are optical fibers that can confine the main part of the electric field within a hollow core, along which gases or liquids can flow. Their hollow-core microstructure makes these fibers promising tools for optofluidic applications, for example, as gas sensors [1,2]. Meanwhile, the gas-filled HC fiber can be used as a nonlinear medium for supercontinuum [3][4][5] or high-order harmonic generation [6]. The simplest HC fiber is a capillary; however, its high confinement loss is an inherent problem [7][8][9]. Compared with capillaries, the loss of microstructure HC fibers is much lower. Generally, there are two families of microstructure HC fibers: photonic bandgap fibers (HC-PBGFs) and hollow-core antiresonant fibers (HC-ARFs).
An HC-PBGF has an air core in the center with a periodic structure cladding. The light propagating in the core is confined by the photonic bandgap of the periodic cladding structure. As reported in [10], the minimum confinement loss of HC-PBGFs can be as low as 1.2 dB/km. However, the bandwidth of HC-PBGFs is only around 10-30% of the central wavelength [11].
The confinement mechanism of HC-ARFs is different. A series of thin glass walls that function like Fabry-Perot resonators surround the air core in an HC-ARF. This mechanism provides a broad bandwidth for guiding light. In 2017, Wang et al. proposed a clear physical insight into the optical guidance mechanism in HC-ARFs based on a multi-layered model [12]. In the last few years, several types of HC-ARFs have been studied. The first type is Kagome fibers, which consist of multiple layers of thin glass walls and air holes [3]. Kagome fibers have a wide bandwidth that can reach where λ is the wavelength and n silica and n air are the refractive indices of silica and air, respectively. The core diameter D core = 20 × 1040 nm. To accurately calculate the confinement loss, a perfectly matched layer was utilized in the outmost boundary of the geometries. A Sellmeier equation was used to describe the dispersion for silica [31]. The material loss was neglected, since the material absorption is quite low in this wavelength range [32]. The refractive index of air was chosen to be 1. The wavelength in this simulation was 1040 nm if not specified otherwise. Firstly, we study the classical six-tube NC-HCF, whose geometry is shown in Figure 1a [15]. All cladding tubes surrounding the core have an equal diameter D tube . Figure 1b shows the simulated loss curves of the fundamental LP 01 (HE 11 ) modes and six HOMs (TE 01 , HE 21 , TM 01 , EH 11 , HE 31 , and HE 12 ) as a function of D tube /D core . The loss of the LP 11 group of modes (TE 01 , HE 21 , and TM 01 ) has a peak at D tube /D core = 0.68, which is the same as in [15]. When D tube /D core = 0.68, the effective refractive index (n eff ) of the LP 11 modes in the core is equal to that of the fundamental mode in the cladding tube, and, as a result, a high-loss supermode is formed. This could be explained by the equation [15]: where the fitting parameters are f core = 1.077 for the core and f tube = 0.991 for the cladding tube, and u lm is the m-th zero of the Bessel function J l . The cladding tube diameter has little influence on the loss of the LP 01 modes because the effective index of the fundamental core mode in the NC-HCF core is larger than that of the fundamental mode in the annular cladding tube when they have the same diameter [19]. Meanwhile, the effective index increases with the diameter of an annular tube but the cladding tubes always have a smaller diameter than the core. As a result, phase matching between the LP 01 and cladding tube modes occurs only when D tube is larger than D core . However, it cannot happen in an NC-HCF with six cladding tubes. The mode patterns of LP 01 and LP 11 modes when D tube /D core = 0.68 are shown in Figure 1c, where a supermode is clearly observed for the LP 11 modes. LP 21 (EH 11 and HE 31 ) and LP 02 (HE 12 ) modes both have two high-loss peaks. The first peak, at D tube /D core = 0.52 for the LP 21 modes and D tube /D core = 0.49 for the LP 02 modes, is a result of coupling between the respective HOM in the core and the fundamental mode of the cladding tubes. The second peak, which appears at D tube /D core = 0.81 for the LP 21 modes and D tube /D core = 0.77 for the LP 02 modes, results from the coupling with the LP 11 modes of the cladding tubes. The supermodes corresponding to the loss peaks in Figure 1b are also shown in Figure 1c. The first HOM that could be excited in an NC-HCF is the LP 11 modes, which means that applying cladding tubes with a diameter D tube = 0.68 D core in an NC-HCF could improve the single-mode guidance performance of the fiber. Although an NC-HCF with D tube /D core = 0.68 can suppress the LP 11 modes of the fiber, the LP 21 modes are not strongly suppressed. The loss of the EH 11 mode is only 17.1 dB/m when D tube = 0.68 D core . To effectively increase the loss of the LP 21 modes, we propose a novel NC-HCF with hybrid cladding tubes, as shown in Figure 2a. Three of the cladding tubes have diameter D tubeA = 0.68 D core , and the others have diameter D tubeB = 0.52 D core . Different cladding tubes are placed alternately around the fiber core. The inner boundary of the outer capillary is a polygon with rounded corners that closely fits the adjacent cladding tubes. From the analysis above, it is not surprising to see that both the LP 11 and LP 21 modes form high-loss supermodes, as verified in Figure 2b-d. The higher-order mode extinction ratio (HOMER) is commonly used to quantitatively describe the single-mode performance [23]. The HOMER is defined as the ratio of the loss of the higher-order core modes to that of the fundamental mode. For comparison, the loss and HOMER of the hybrid NC-HCF and those of the corresponding NC-HCF with uniform cladding tubes (D tube = 0.68 D core ) are shown in Figure 3. and LP02 (HE12) modes both have two high-loss peaks. The first peak, at Dtube/Dcore = 0.52 for the LP21 modes and Dtube/Dcore = 0.49 for the LP02 modes, is a result of coupling between the respective HOM in the core and the fundamental mode of the cladding tubes. The second peak, which appears at Dtube/Dcore = 0.81 for the LP21 modes and Dtube/Dcore = 0.77 for the LP02 modes, results from the coupling with the LP11 modes of the cladding tubes. The supermodes corresponding to the loss peaks in Figure 1b are also shown in Figure 1c.  The first HOM that could be excited in an NC-HCF is the LP11 modes, which means that applying cladding tubes with a diameter Dtube = 0.68 Dcore in an NC-HCF could improve the single-mode guidance performance of the fiber. Although an NC-HCF with Dtube/Dcore = 0.68 can suppress the LP11 modes of the fiber, the LP21 modes are not strongly suppressed. The loss of the EH11 mode is only 17.1 dB/m when Dtube = 0.68 Dcore. To effectively increase the loss of the LP21 modes, we propose a novel NC-HCF with hybrid cladding tubes, as shown in Figure 2a. Three of the cladding tubes have diameter DtubeA = 0.68 Dcore, and the others have diameter DtubeB = 0.52 Dcore. Different cladding tubes are placed alternately around the fiber core. The inner boundary of the outer capillary is a polygon with rounded corners that closely fits the adjacent cladding tubes. From the analysis above, it is not surprising to see that both the LP11 and LP21 modes form high-loss supermodes, as verified in Figure 2b-d. The higher-order mode extinction ratio (HOMER) is commonly used to quantitatively describe the single-mode performance [23]. The HOMER is defined as the ratio of the loss of the higher-order core modes to that of the fundamental mode. For comparison, the loss and HOMER of the hybrid NC-HCF and those of the corresponding NC-HCF with uniform cladding tubes (Dtube = 0.68 Dcore) are shown in Figure 3.
The losses and effective refractive indexes of the fiber for the two polarizations were calculated. There is no obvious difference in the fiber mode indices in the two polarizations. In both the vertical and horizontal polarizations in Figure 2a, the loss of the LP01 mode is 0.31 dB/m and the effective refractive index is 0.999374, which means that there is no birefringence in the hybrid NC-HCF. The outer capillary of the hybrid cladding tube NC-HCF leads to a higher loss for all the modes. Nevertheless, both the LP11 and LP21 modes can be coupled to the cladding modes effectively and form high-loss supermodes, which allow their HOMERs to reach a level of 1000. The HOMERs of the LP11 modes in the hybrid NC-HCF are close to those of the uniform structure. However, the HOMERs of the LP21 modes in the hybrid NC-HCF are notably larger than those of the uniform structure because phase matching for the LP21 modes is also achieved in the hybrid NC-HCF. Neither of the two fibers can achieve efficient phase matching for the LP02 mode, so they have close HOMER values. As a result of applying the hybrid cladding tubes, the fiber can suppress the LP11 and LP21 modes simultaneously.  Figure 2a, the loss of the LP 01 mode is 0.31 dB/m and the effective refractive index is 0.999374, which means that there is no birefringence in the hybrid NC-HCF. The outer capillary of the hybrid cladding tube NC-HCF leads to a higher loss for all the modes. Nevertheless, both the LP 11 and LP 21 modes can be coupled to the cladding modes effectively and form high-loss supermodes, which allow their HOMERs to reach a level of 1000. The HOMERs of the LP 11 modes in the hybrid NC-HCF are close to those of the uniform structure. However, the HOMERs of the LP 21 modes in the hybrid NC-HCF are notably larger than those of the uniform structure because phase matching for the LP 21 modes is also achieved in the hybrid NC-HCF. Neither of the two fibers can achieve efficient phase matching for the LP 02 mode, so they have close HOMER values. As a result of applying the hybrid cladding tubes, the fiber can suppress the LP 11 and LP 21 modes simultaneously.

Single-Mode Guidance Performance of NANFs
Compared with NC-HCFs, which have only one antiresonant layer, NANFs could reduce not only the confinement loss of the LP01 fundamental modes but also that of the HOMs. The geometry of a typical NANF structure, consisting of six nested tubes, is shown in Figure 4a. Compared with a single antiresonant layer NC-HCF, the phase-matching scenario for high-loss supermodes in an NANF is more complicated. Two cladding modes (CM1 and CM2) are typically involved in the phase matching. CM1 is located in the middle of the first and second cladding tubes, as shown in Figure 4b. CM2 is located inside the second cladding tube, as shown in Figure 4c. To explore the phase matching between the LP11 and CM1 modes, we keep the first cladding tube diameter at Dtube1 = 0.80 Dcore and change the second cladding tube diameter [29]. The loss of the LP11 modes as a function of Dtube2/Dcore is shown in Figure 5a and reaches a maximum at Dtube2/Dcore = 0.23. However, the loss of the LP01 modes increases as the value of Dtube2/Dcore is reduced and the second cladding tube prevents the LP11 supermode from reaching a high loss, as can be seen from the mode pattern in Figure 5c. When Dtube1 = 0.80 Dcore and Dtube2 = 0.23 Dcore, the loss of the LP01 modes is 1.47 × 10 −3 dB/m, which is quite high for NANFs, while the loss of the LP11 modes is no more than 5 dB/m.
It is more difficult to realize phase matching between the LP11 and CM2 modes. Because CM2 is a circular tube mode, the phase-matching condition for a high-loss supermode should be close to Dtube2 =

Single-Mode Guidance Performance of NANFs
Compared with NC-HCFs, which have only one antiresonant layer, NANFs could reduce not only the confinement loss of the LP 01 fundamental modes but also that of the HOMs. The geometry of a typical NANF structure, consisting of six nested tubes, is shown in Figure 4a. Compared with a single antiresonant layer NC-HCF, the phase-matching scenario for high-loss supermodes in an NANF is more complicated. Two cladding modes (CM1 and CM2) are typically involved in the phase matching. CM1 is located in the middle of the first and second cladding tubes, as shown in Figure 4b. CM2 is located inside the second cladding tube, as shown in Figure 4c.

Single-Mode Guidance Performance of NANFs
Compared with NC-HCFs, which have only one antiresonant layer, NANFs could reduce not only the confinement loss of the LP01 fundamental modes but also that of the HOMs. The geometry of a typical NANF structure, consisting of six nested tubes, is shown in Figure 4a. Compared with a single antiresonant layer NC-HCF, the phase-matching scenario for high-loss supermodes in an NANF is more complicated. Two cladding modes (CM1 and CM2) are typically involved in the phase matching. CM1 is located in the middle of the first and second cladding tubes, as shown in Figure 4b. CM2 is located inside the second cladding tube, as shown in Figure 4c. To explore the phase matching between the LP11 and CM1 modes, we keep the first cladding tube diameter at Dtube1 = 0.80 Dcore and change the second cladding tube diameter [29]. The loss of the LP11 modes as a function of Dtube2/Dcore is shown in Figure 5a and reaches a maximum at Dtube2/Dcore = 0.23. However, the loss of the LP01 modes increases as the value of Dtube2/Dcore is reduced and the second cladding tube prevents the LP11 supermode from reaching a high loss, as can be seen from the mode pattern in Figure 5c. When Dtube1 = 0.80 Dcore and Dtube2 = 0.23 Dcore, the loss of the LP01 modes is 1.47 × 10 −3 dB/m, which is quite high for NANFs, while the loss of the LP11 modes is no more than 5 dB/m.
It is more difficult to realize phase matching between the LP11 and CM2 modes. Because CM2 is a circular tube mode, the phase-matching condition for a high-loss supermode should be close to Dtube2 = 0.68 Dcore. We therefore sweep Dtube1 for Dtube2/ Dcore = 0.66, 0.67, 0.68, 0.69, and 0.70. The results are shown To explore the phase matching between the LP 11 and CM1 modes, we keep the first cladding tube diameter at D tube1 = 0.80 D core and change the second cladding tube diameter [29]. The loss of the LP 11 modes as a function of D tube2 /D core is shown in Figure 5a and reaches a maximum at D tube2 /D core = 0.23. However, the loss of the LP 01 modes increases as the value of D tube2 /D core is reduced and the second cladding tube prevents the LP 11 supermode from reaching a high loss, as can be seen from the mode pattern in Figure 5c. When D tube1 = 0.80 D core and D tube2 = 0.23 D core , the loss of the LP 01 modes is 1.47 × 10 −3 dB/m, which is quite high for NANFs, while the loss of the LP 11 modes is no more than 5 dB/m.
It is more difficult to realize phase matching between the LP 11 and CM2 modes. Because CM2 is a circular tube mode, the phase-matching condition for a high-loss supermode should be close to D tube2 = 0.68 D core . We therefore sweep D tube1 for D tube2 /D core = 0.66, 0.67, 0.68, 0.69, and 0.70. The results are shown in Figure 5b. The maximum loss of the LP 11 mode is 26.9 dB/m and appears at D tube1 = 0.80 D core and D tube2 = 0.68 D core , simply the smallest D tube1 with D tube2 = 0.68 D core in our sweep range. The mode pattern of this supermode is displayed in Figure 5d. The LP 01 mode loss of this structure is as high as 5.85 × 10 −3 dB/m. Whether the LP 11 modes are coupled with CM1 or CM2, their loss cannot be raised to more than 100 dB/m, which can be realized in NC-HCFs with one antiresonant layer. At the same time, the loss of the LP 01 mode is too high. It is not effective to improve the single-mode guidance performance by coupling the HOM only with one single cladding mode in an NANF if a low loss of the LP 01 mode is desired.  Figure 5d. The LP01 mode loss of this structure is as high as 5.85 × 10 −3 dB/m. Whether the LP11 modes are coupled with CM1 or CM2, their loss cannot be raised to more than 100 dB/m, which can be realized in NC-HCFs with one antiresonant layer. At the same time, the loss of the LP01 mode is too high. It is not effective to improve the single-mode guidance performance by coupling the HOM only with one single cladding mode in an NANF if a low loss of the LP01 mode is desired. To further increase the loss of the LP11 modes in the NANFs and meanwhile maintain a low loss of the LP01 modes, we turn to a structure with five nested tubes, as illustrated in Figure 6a. In this way, the area of the cladding modes becomes larger, and, as a result, the HOMs can be coupled with the cladding modes more strongly. A structure with fewer nested tubes could have a larger Dtube1, which makes it possible to realize phase matching between the LP11, CM1, and CM2 modes. The losses of the LP11 modes as a function of Dtube1/Dcore for Dtube2/Dcore = 0.66, 0.67, 0.68, 0.69, and 0.70 are shown in Figure  6b. An interesting result is that for each Dtube2, the maximum loss value appears when Dtube1/Dtube2 is close to 1.75. This result indicates that Dtube1/Dtube2 = 1.75 fulfils the phase-matching condition between CM1 and CM2. The mode loss is higher when Dtube2 = 0.69 Dcore. The maximum loss of the LP11 mode is 313.96 dB/m when Dtube1 = 1.21 Dcore and Dtube2 = 0.69 Dcore, and the mode pattern is shown in Figure 6d. The loss of the LP01 modes is only 3.64 × 10 −4 dB/m. Thus, the cascaded coupling between the LP11, CM1, and CM2 modes increases the LP11 mode loss significantly. However, the LP21 and LP02 modes have different phase-matching conditions. We then keep Dtube2 = 0.69 Dcore and sweep Dtube1 to study the loss of the other modes. As displayed in Figure 6c, similar to the case in a single tube layer cladding NC-HCF, both LP21 and LP02 modes have two high-loss peaks that correspond to different HOMs of CM1. To further increase the loss of the LP 11 modes in the NANFs and meanwhile maintain a low loss of the LP 01 modes, we turn to a structure with five nested tubes, as illustrated in Figure 6a. In this way, the area of the cladding modes becomes larger, and, as a result, the HOMs can be coupled with the cladding modes more strongly. A structure with fewer nested tubes could have a larger D tube1 , which makes it possible to realize phase matching between the LP 11 , CM1, and CM2 modes. The losses of the LP 11 modes as a function of D tube1 /D core for D tube2 /D core = 0.66, 0.67, 0.68, 0.69, and 0.70 are shown in Figure 6b. An interesting result is that for each D tube2 , the maximum loss value appears when D tube1 /D tube2 is close to 1.75. This result indicates that D tube1 /D tube2 = 1.75 fulfils the phase-matching condition between CM1 and CM2. The mode loss is higher when D tube2 = 0.69 D core . The maximum loss of the LP 11 mode is 313.96 dB/m when D tube1 = 1.21 D core and D tube2 = 0.69 D core , and the mode pattern is shown in Figure 6d. The loss of the LP 01 modes is only 3.64 × 10 −4 dB/m. Thus, the cascaded coupling between the LP 11 , CM1, and CM2 modes increases the LP 11 mode loss significantly. However, the LP 21 and LP 02 modes have different phase-matching conditions. We then keep D tube2 = 0.69 D core and sweep D tube1 to study the loss of the other modes. As displayed in Figure 6c, similar to the case in a single tube layer cladding NC-HCF, both LP 21 and LP 02 modes have two high-loss peaks that correspond to different HOMs of CM1. The high-loss peaks of the LP 21 modes appear at D tube1 = 1.17 D core and D tube1 = 1.27 D core , and the high-loss peaks of the LP 02 modes appear at D tube1 = 1.13 D core and D tube1 = 1.23 D core .

NANFs with Extended Cladding Tubes
As illuminated in Section 3, the cascaded coupling between the HOMs in the fiber core and the cladding modes can raise the loss of the HOMs by increasing the area of the cladding modes. It is impossible to realize the cascaded coupling in NANFs with six nested circular cladding tubes. It is also impossible to alternately place the hybrid claddings tubes in an NANF with five nested tubes. To

NANFs with Extended Cladding Tubes
As illuminated in Section 3, the cascaded coupling between the HOMs in the fiber core and the cladding modes can raise the loss of the HOMs by increasing the area of the cladding modes. It is impossible to realize the cascaded coupling in NANFs with six nested circular cladding tubes. It is also impossible to alternately place the hybrid claddings tubes in an NANF with five nested tubes. To increase the area of CM1 in an NANF with six nested tubes, we propose a novel NANF with extended cladding tubes with the geometry shown in Figure 7a. The first cladding tube is extended with a length of l. To simplify the design process, we keep D tube1 = 0.80 D core . The loss of the LP11 modes in the NANF with extended cladding tubes is investigated as a function of l when Dtube2/Dcore = 0.66, 0.67, 0.68, 0.69, and 0.70, as shown in Figure 7b. The maximum loss value is 343.7 dB/m when l = 9.4 × 1040 nm and Dtube2/Dcore = 0.69. On the other hand, the LP21 mode loss as a function of l when Dtube2/Dcore = 0.50, 0.51, 0.52, 0.53, and 0.54 is shown in Figure 7c. The maximum loss value is 1270.3 dB/m when l = 9.3 × 1040 nm and Dtube2/Dcore = 0.51. Interestingly, the loss peaks of these two HOMs are close to each other. The second-largest value of the LP21 mode loss is 1153.3 dB/m when l = 9.4 × 1040 nm and Dtube2/Dcore = 0.51. As a result, it is very easy to apply a hybrid cladding tube structure in NANFs with extended cladding tubes. The geometry of the NANF with hybrid extended cladding tubes is shown in Figure 8a. All of the first cladding tubes have an extended length l = 9.4 × 1040 nm. Three of the second cladding tubes have an inner diameter of Dtube2A = 0.69 Dcore and the others an inner diameter of Dtube2B = 0.51 Dcore. The nested cladding tubes are alternately arranged. The mode patterns of the LP01, LP11, and LP21 modes are shown in Figure 8b-d, respectively. As can be clearly observed, high-loss supermodes are realized for both the LP11 and LP21 modes.
As for the hybrid NC-HCF, the losses and effective refractive indexes of the fiber in the two polarizations were calculated. In the vertical polarization in Figure 8a, the LP01 mode loss is 3.90 × 10 −4 dB/m and is 4.05 × 10 −4 dB/m in the horizontal polarization. In both polarizations, the effective refractive index is 0.999363, which means that there is no birefringence in the NANF with hybrid extended cladding tubes. Then, we made a comparison between the two HOM-suppressed NANFs in Figure 9. One structure considered is an NANF with five nested tubes of parameters Dtube1 = 1.21 Dcore and Dtube2 = 0.69 Dcore, the circular tube NANF with the best single-mode guidance performance in Section 3. The other structure is an NANF with hybrid extended cladding tubes proposed in this Section. The LP01 mode loss of the NANF with five nested tubes is 3.64 × 10 −4 dB/m. The NANF with hybrid extended cladding tubes has a slightly higher LP01 mode loss of 3.90 × 10 −4 dB/m. The NANF with five nested  Figure 7c. The maximum loss value is 1270.3 dB/m when l = 9.3 × 1040 nm and D tube2 /D core = 0.51. Interestingly, the loss peaks of these two HOMs are close to each other. The second-largest value of the LP 21 mode loss is 1153.3 dB/m when l = 9.4 × 1040 nm and D tube2 /D core = 0.51. As a result, it is very easy to apply a hybrid cladding tube structure in NANFs with extended cladding tubes. The geometry of the NANF with hybrid extended cladding tubes is shown in Figure 8a. All of the first cladding tubes have an extended length l = 9.4 × 1040 nm. Three of the second cladding tubes have an inner diameter of D tube2A = 0.69 D core and the others an inner diameter of D tube2B = 0.51 D core . The nested cladding tubes are alternately arranged. The mode patterns of the LP 01 , LP 11 , and LP 21 modes are shown in Figure 8b-d, respectively. As can be clearly observed, high-loss supermodes are realized for both the LP 11 and LP 21 modes.
As for the hybrid NC-HCF, the losses and effective refractive indexes of the fiber in the two polarizations were calculated. In the vertical polarization in Figure 8a, the LP 01 mode loss is 3.90 × 10 −4 dB/m and is 4.05 × 10 −4 dB/m in the horizontal polarization. In both polarizations, the effective refractive index is 0.999363, which means that there is no birefringence in the NANF with hybrid extended cladding tubes. Then, we made a comparison between the two HOM-suppressed NANFs in Figure 9. One structure considered is an NANF with five nested tubes of parameters D tube1 = 1.21 D core and D tube2 = 0.69 D core , the circular tube NANF with the best single-mode guidance performance in Section 3. The other structure is an NANF with hybrid extended cladding tubes proposed in this Section. The LP 01 mode loss of the NANF with five nested tubes is 3.64 × 10 −4 dB/m. The NANF with hybrid extended cladding tubes has a slightly higher LP 01 mode loss of 3.90 × 10 −4 dB/m. The NANF with five nested tubes has a higher LP 11 mode loss. Although the NANF with five nested tubes has a HOMER of the level of 10 3 for the LP 21 modes, the NANF with hybrid extended cladding tubes can achieve a level of 10 5 or even 10 6 for the LP 21 modes because of its extremely high LP 21 mode loss. A HOMER level of 10 5 for the LP 20 modes is also obtained for the NANF with hybrid extended cladding tubes. Thus, the NANF with hybrid extended cladding tubes overall permits a more effective single-mode guidance performance.
Micromachines 2019, 10, x FOR PEER REVIEW 9 of 12 with hybrid extended cladding tubes overall permits a more effective single-mode guidance performance.  In the end, we verified the broadband guidance of the NANF with hybrid extended cladding tubes. The loss and the HOMER as a function of wavelength in the range 700-1300 nm are displayed in Figure  10. The losses increase at long wavelengths because the ratio between the core diameter and wavelength becomes lower. A relatively large core could reduce the loss. Similar to the nested elliptical cladding tube NANF in [23], the loss of the fundamental mode of the NANF with hybrid extended cladding with hybrid extended cladding tubes overall permits a more effective single-mode guidance performance.  In the end, we verified the broadband guidance of the NANF with hybrid extended cladding tubes. The loss and the HOMER as a function of wavelength in the range 700-1300 nm are displayed in Figure  10. The losses increase at long wavelengths because the ratio between the core diameter and wavelength becomes lower. A relatively large core could reduce the loss. Similar to the nested elliptical cladding tube NANF in [23], the loss of the fundamental mode of the NANF with hybrid extended cladding In the end, we verified the broadband guidance of the NANF with hybrid extended cladding tubes. The loss and the HOMER as a function of wavelength in the range 700-1300 nm are displayed in Figure 10. The losses increase at long wavelengths because the ratio between the core diameter and wavelength becomes lower. A relatively large core could reduce the loss. Similar to the nested elliptical cladding tube NANF in [23], the loss of the fundamental mode of the NANF with hybrid extended cladding tubes rises quickly at long wavelengths. As a result, the HOMERs are reduced at long wavelengths. Phase matching for the LP 11 and LP 21 modes are achieved in the wavelength range, so the HOMERs are higher than 10 4 at most wavelengths. At shorter wavelengths, the HOMERs are typically above 10 5 .

Conclusions
In conclusion, we have numerically analyzed the single-mode guidance performance of NC-HCFs. The loss of HOMs can be increased by enabling phase matching with cladding modes. To improve the LP21 mode loss of single tube layer NC-HCFs, we propose a novel NC-HCF with hybrid cladding tubes. Both the LP11 and LP21 modes have an HOMER level of 1000 in the hybrid NC-HCF. Phase matching between the LP11 and cladding modes in NANFs is also explored. The cascaded coupling between the LP11, CM1, and CM2 modes improves the LP11 mode loss. High-loss supermodes could be realized in an NANF with five instead of six nested tubes. A novel NANF with hybrid extended cladding tubes is proposed, where not only the LP11 but also the LP21 modes are suppressed while the loss of the LP01 modes is only 3.90 × 10 −4 dB/m. All of the HOMERs calculated at 1040 nm are larger than 10 5 . Phase matching for high-loss supermodes is realized within a broad band. We believe that the results presented here are of value for further NC-HCF designs and the proposed structures will enable novel applications.

Conclusions
In conclusion, we have numerically analyzed the single-mode guidance performance of NC-HCFs. The loss of HOMs can be increased by enabling phase matching with cladding modes. To improve the LP 21 mode loss of single tube layer NC-HCFs, we propose a novel NC-HCF with hybrid cladding tubes. Both the LP 11 and LP 21 modes have an HOMER level of 1000 in the hybrid NC-HCF. Phase matching between the LP 11 and cladding modes in NANFs is also explored. The cascaded coupling between the LP 11 , CM1, and CM2 modes improves the LP 11 mode loss. High-loss supermodes could be realized in an NANF with five instead of six nested tubes. A novel NANF with hybrid extended cladding tubes is proposed, where not only the LP 11 but also the LP 21 modes are suppressed while the loss of the LP 01 modes is only 3.90 × 10 −4 dB/m. All of the HOMERs calculated at 1040 nm are larger than 10 5 . Phase matching for high-loss supermodes is realized within a broad band. We believe that the results presented here are of value for further NC-HCF designs and the proposed structures will enable novel applications.