Stress-Induced Polarization-Maintaining Large-Mode-Area Photonic Crystal Fibers With Deviation of the Single-Mode Transmission Band and Delocalization of Higher-Order Modes

The nonlinear effects and laser-induced optical and thermal damage in optical fibers, together with the limitations of beam quality and mode-field area, restrict the power scaling-up of single-mode output for developing high-power fiber lasers in the kilowatt and above range. The design of photonic crystal fibers (PCFs) with large mode areas is an effective way to address this problem. In this paper, the demands and challenges of designing very large-mode-area (VLMA-) PCFs are discussed, including the overall fiber structure design and property simulation, especially the precise definition of single-mode operating conditions of VLMA-PCFs. Finally, an advanced stress-induced polarization-maintaining, Yb-doped, PCF structure with a large mode area realized by introducing both leakage channels and higher order mode-filtering units is proposed and analyzed theoretically, for which a maximum core diameter of 101 <inline-formula><tex-math notation="LaTeX">$\boldsymbol{\mu}{\text{m}}$</tex-math></inline-formula> and single-mode field diameter of 76.33 <inline-formula><tex-math notation="LaTeX">$\boldsymbol{\mu}{\text{m}}$</tex-math></inline-formula> at 1064 <inline-formula><tex-math notation="LaTeX">$\text{nm}$</tex-math></inline-formula> and a birefringence value <inline-formula><tex-math notation="LaTeX">$\boldsymbol{> 10^{-4}}$</tex-math></inline-formula> orders of magnitude are achieved.

Stress-Induced Polarization-Maintaining Large-Mode-Area Photonic Crystal Fibers With Deviation of the Single-Mode Transmission Band and Delocalization of Higher-Order Modes Yuan Ma , Member, IEEE, Rui Wan , Huanhuan Yang , Yanfu Li , Chao Chen , and Pengfei Wang , Member, IEEE Abstract-The nonlinear effects and laser-induced optical and thermal damage in optical fibers, together with the limitations of beam quality and mode-field area, restrict the power scaling-up of single-mode output for developing high-power fiber lasers in the kilowatt and above range.The design of photonic crystal fibers (PCFs) with large mode areas is an effective way to address this problem.In this paper, the demands and challenges of designing very large-mode-area (VLMA-) PCFs are discussed, including the overall fiber structure design and property simulation, especially the precise definition of single-mode operating conditions of VLMA-PCFs.Finally, an advanced stress-induced polarizationmaintaining, Yb-doped, PCF structure with a large mode area realized by introducing both leakage channels and higher order mode-filtering units is proposed and analyzed theoretically, for which a maximum core diameter of 101 µm and single-mode field diameter of 76.33 µm at 1064 nm and a birefringence value > 10 −4 orders of magnitude are achieved.Index Terms-Finite element method, photonic crystal fibers, stress-induced birefringence, optical fiber amplifiers.

I. INTRODUCTION
T RADITIONAL double-clad fibers are limited by the reg- ulation of material doping and the difficulty of preparation [1], making them unable to satisfy the growing demand Yuan Ma was with the State Key Laboratory of Transient Optics and Photonics, Xi'an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences (CAS), Xi'an 710119, China.She is now with Information and Navigation College, Air Force Engineering University, Xi'an 710077, China (e-mail: mayuan18@mails.ucas.ac.cn).
Rui Wan and Chao Chen are with the State Key Laboratory of Transient Optics and Photonics, Xi'an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences (CAS), Xi'an 710119, China, and also with the Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China (e-mail: wanrui17@ mails.ucas.ac.cn; chenchao@opt.ac.cn).
Pengfei Wang is with the State Key Laboratory of Transient Optics and Photonics, Xi'an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences (CAS), Xi'an 710119, China (e-mail: pfwang@opt.ac.cn).
Digital Object Identifier 10.1109/JPHOT.2024.3395776 for high-power fiber laser gain output and high-power laser transmission.In 1991, Philip Russel proposed the introduction of two-dimensional photonic crystals of optical wavelength magnitude in optical fiber cladding glass [2], which transformed the material problem of optical fibers into a structural problem and introduced the concept of PCFs [3], [4], [5], [6], [7].PCFs have many excellent optical properties that are different from those of conventional fibers, such as endless single-mode [8], [9], [10], [11], [12], [13], low limiting loss [14], adjustable dispersion [15], high birefringence [16], and large mode area [17], [18].The distinctive optical attributes of PCF significantly exceed those of traditional optical fibers in functional capabilities, leading to their augmented application in emerging technological domains including high-power pulsed lasers [19], high-bandwidth fiber-optic communication systems [20], [21], [22], fiber-optic sensing, and systems characterized by pronounced nonlinear properties [23].PCFs can be used in lasers both as a direct gain medium and a medium for dispersion compensation and transmission of optical energy.
In the development of high-power fiber lasers, the increase in single-mode output power is the key to their development.It has been quite difficult to increase the power of a single laser more significantly due to physical mechanisms, such as nonlinear effects, optical and thermal damage, limitations in beam quality, mode field area, and fiber material properties [24].Compared to conventional optical fibers, PCF enables single-mode transmission with a large mode field area and significantly reduces the fiber's laser power density and nonlinear effects.Furthermore, it can also improve the damage threshold of the fiber material while ensuring the laser transmission quality, which provides an effective way to address the power enhancement of fiber lasers.
As the fiber laser gain medium changes from a conventional double-clad fiber to a PCF, the control of the laser output mode is no longer strictly limited by the refractive index of the fiber core and cladding material.In the computation of V-parameters, achieving precise calculations for the refractive index and effective core diameter of double-clad fibers with specialized microstructures proves challenging.However, the effective refractive index of the PCF cladding can be changed by adjusting the structural parameters of the fiber cladding.The effective core diameter of the PCF can be defined in several different ways [25], [26], such as 0.5Λ [25], Λ √ 5 [26], 0.64Λ [27] and Λ [6].Therefore, we introduce a core overlap factor to determine whether the fiber is used for single-mode transmission.
For conventional step-index single-mode fibers at 1064 nm, the maximum single-mode mode field diameter (MFD) is constrained to below 15 μm.To further increase their MFD under the single-mode transmission band, several measures can be considered for the optimization of fiber material and structure.The first is to suppress higher order modes (HOMs) by controlling the core or cladding doping distribution and reducing the HOM gain.For instance, in 2003, Siegman et al. proposed gain guided index anti-guided fiber [28] (GG-IAG), which has a large diameter core with a negative refractive index step from the cladding to the core combined with an adequately large gain coefficient in the core.The single-mode output is still guaranteed when the core diameter of the fiber is greater than 100 μm.The second category is mode conversion, which can be achieved by using HOM transmission or coupling HOMs from the core to other cores in the cladding [29].In 2014, Xiuquan Ma et al. increased the core size from 55 μm to 60 μm for effective single-mode, chiral-coupled, core (CCC) Ge-doped and Yb-doped, doubleclad fibers and experimentally demonstrated their robust singlemode performance [30].The third way to achieve the objective is by introducing HOM discrimination mechanisms and HOM leakage channels in the fiber.Examples include mode matching, bending mode selection, resonant filtering, etc. Mode matching excites only the fundamental mode of the fiber, and this approach can be used for all types of few-mode fibers, but as the mode field area increases, it becomes more difficult to excite only one mode operation.Bend selection is also not possible for rod fibers with large field diameters.The emergence of PCFs has given us a wealth of freedom in designing fiber structures.Therefore, the resonant filter module is introduced into the design of the PCF cladding structure, which allows resonant coupling of the HOMs in the fiber core to the filter structure in the cladding.This structure allows the HOMs to be delocalized, a major approach to achieving single-mode operation in fibers with large mode field areas.In 2011, NKT photonics designed an LMA-PCF with an HOM-filtering structure by adding distributed mode filter (DMF) to filter HOMs, and the fiber achieved single-mode operation at 1064 nm with an MFD of 59 μm [31].
In addition to the single-mode transmission property of LMF-PCFs, high birefringence endows them with priors over characteristic conventional polarization-maintaining (PM) fibers [32].PM-LMA-PCFs have the following advantages: larger mode birefringence that can be more than one order of magnitude higher than those of conventional PM fibers, higher freedom of structure design, high radiation resistance, and excellent characteristic of PCFs capable of single-mode, large-mode field area transmission.When a single-mode LMA-PCF was used to achieve higher power laser output, the transmitted laser was still vulnerable to polarizability degradation in some high-power experiments.This was mainly due to formation of defects in fiber fabrication process that affect the symmetry of the fiber structure, allowing two orthogonal polarization modes to couple during laser propagation, thus changing the polarization state of the transmitted laser light.PM-LMA-PCFs combine improved polarizability and large mode area and thus can ensure high output power, high beam quality, and polarization control [33].
PM-LMA-PCFs with different MFD are mainly used to obtain high birefringence values by introducing structural birefringence and stress-induced birefringence in the fibers.For PCFs with core diameters less than 15 μm, structural birefringence can be introduced by adjusting the size and shape of the air holes around the core, or by adjusting the core shape to change the refractive index distribution in both directions to reduce the symmetry of the PCF.The world's first PM-PCF was developed by this method by Ortigosa-Blanch at the University of Bath [34].This fiber has a MFD of 3.8 μm and can obtain a high birefringence value of 3.7 × 10 −3 at 1550 nm.In 2008 Grzegorz Golojuch at Wroclaw University of Technology induced structural birefringence by introducing small holes around the core to break the symmetry of the fiber cross-section.The fiber reached its birefringence of 1.0 × 10 −4 at 1300 nm with a MFD of 10 μm [35].With the gradual increase of the MFD, the structural birefringence value decreases rapidly, and it is difficult to introduce structural birefringence to have a large impact on the birefringence value of fibers with LMA-PCF.Another way to obtain high birefringence is by introducing stress-applied part (SAP) with different coefficients of thermal expansion into the fiber cladding, which produces stress birefringence due to the anisotropy of core materials induced by thermal stresses inside the fiber due to elasto-optical effect during fiber pulling.A Yb-doped single transverse mode rodtype PCF combining the advantages of low nonlinearity and polarization stability was reported by the University of Jena, Germany in 2008 [36].The fiber has a MFD of 54 μm and experimentally measured polarization greater than 85%, with the polarization maintaining capability coming from the SAP structure in its cladding.In 2019 NKT developed a PM-PCF with a 30 μm Yb-doped core, 250 μm pump cladding, and a SAP structure [37].the polarization extinction ratio >18 dB.In 2022, China's Anyang Laser has successfully developed a Yb-doped PM-VLMA-PCF with a MFD of 65 μm, which has a M 2 <1.1 at 1030 nm and a polarization extinction ratio >15 dB [38].LMA Yb-doped PM-PCFs can effectively relieve the gain pressure of the subsequent system of high-power fiber laser, and at the same time, improve the integration, reliability and availability of the system.Meanwhile, it can also realize pulse output with high polarization maintaining capability and high beam quality, effectively suppressing the amplitude-frequency effect caused by polarization mode dispersion and intermodal interference in the process of fiber amplification of high pulse energy, therefore, the LMA-PM-PCF has a wide range of applications in ultra-high-power lasers and femtosecond lasers.
Tailoring the microstructure of optical fibers is a unique way to improve the single-mode, output power of fiber lasers and optimize the polarization control device in fiber lasers.This design of fiber structure also helps reduce the impact on the output power and beam quality of the fiber laser limited by fiber nonlinear effects, optical damage and thermal damage.In this paper, a new fiber structure called polarisation-maintaining higher-order filtered PCF (PM-HOF-PCF) is designed to achieve a large mode field area and single-mode, as well as polarization-maintaining transmission.Use of core overlap factor as single-mode determination criterion is proposed, which is more accurate than V-parameter related single-mode determination method widely adopted for the conventional refractive index guided fibers.Our design of new PCF structure focuses on introducing the concepts of both leakage channels and resonant coupling in the cladding.The Ge rings in the designed fiber cladding are used as resonantfiltering modules.PCFs naturally have the property of HOM delocalization due to their mode-sieving guiding mechanism.When the characteristic solution of the cladding structure and the effective refractive index of the HOMs match, the HOMs of the fiber core are resonantly coupled to the cladding, making the HOMs of the fiber core delocalized.In addition, this structure can generate stress-induced, high birefringence value in the fiber by introducing materials with different thermal expansion coefficients in the cladding, which helps achieve fiber polarization maintenance.The above design is used to realize the single-mode operation of VLMA-PCFs.

II. NUMERICAL ANALYSIS METHODS
The optical guiding mechanism of a PCF can be regarded as a modified total internal reflection light guiding when the core of the fiber is replaced by a position with a missing air hole.Similar to the conventional step-index fiber, the normalized frequency parameter (V-parameter) of the PCF can be defined in (1) [9], [39], [40].
The cutoff-free single-mode transmission condition can be formulated as V ≤ π for PCFs with the core replacing one air-hole position [39], [40].The n F SM is the effective refractive index of the fundamental space-filling mode in the air-hole cladding.In PCF based on both hexagonal lattices and square lattices, the n F SM can be analytically computed most accurately from an equivalent circular unit cell having the same area as the original unit cell and the boundary conditions are perfect electric and perfect magnetic conductors for the electromagnetic field.ρ represents the effective radius of the fiber core, and for the PCFs studied, ρ has different definitions.As the number of air holes replaced by the core increases, there is no standardized definition of the core effective radius, therefore, some limitations appear when analyzing the single-mode operating conditions of the PCF using the V-parameter.It is more reasonable to determine the single-mode operating conditions of complex microstructured fibers by introducing a core overlap factor (Γ).
The expression for Γ is shown in (2), where hex is the cross-section of the core doping.Γ is a double integral of the normalized intensity over the core region [41], which describes how tightly the modes are confined in the fiber core and helps distinguish whether they are guided.The normalized intensity is given by (3), where P is the integral of the intensity over the entire fiber cross-section.Single-mode transmission can be assured based on the following criterion: 1) If the 1 st HOM (the one with the largest effective refractive index or propagation constant other than the LP0m mode) overlap factor Γ(LP 11) ≤ 20%, it requires the difference between the overlap factor of the FM and the ratio of the 1st HOM to be larger than 30% to achieve endless singlemode transmission.2) Otherwise, the 1 st HOM overlap factor Γ(LP 11)is >20%, and the fundamental mode overlap factor is higher than 50%.The difference between the overlap factor of the FM and the ratio of the 1 st HOM is larger than 30%.Such an overlap factor distribution can prove that single-mode transmission is possible under the specific conditions of the currently calculated fiber structure and transmission band.Compared to the normalized parameter method, the overlap factor method that does not require the definition of an effective core radius is more suitable for analyzing structures in which the core replaces multiple air holes, and the calculation results are more accurate.
The size of the mode field area of the fiber is calculated using (4) [8], [42]: where E is the transverse electric field component of the PCF.The wider the transverse electric field distribution of the PCF cross-section, the larger the PCF's MFD, which is an essential guide for design of a large-mode field PCF.The effective mode field diameter (MFD) is expressed as (5): The stresses inside the material change the refractive index of the optical waveguide through the Elasto-Optical effect, which affects the performance of the optical waveguide device.The refractive index of an optical waveguide affected by stress is given by the Elasto-Optical effect as shown in ( 6)- (8). ) where C 1 and C 2 are the first and second stress photoelastic coefficients, respectively, σ x and σ y is the positive stress in the X-and Y-axis direction, respectively, and N 0 is the refractive index of the original material containing no stress elements.The stress birefringence is obtained by solving (9).

III. STRUCTURAL DESIGN OF LARGE-MODE-FIELD POLARIZATION-MAINTAINING PCFS
The cross-section and refractive index distribution of the LMA microstructured Yb-doped fiber considered in this study are shown in Fig. 1.These two fiber structures share some common features like expanded fiber core (gray part of the structure) by replacing the 19 air-hole cells of the triangular lattice with Ybdoped rods and n core = 1.45 ± i × 10 −4 (i = 0.5, 1, 1.5), and i × 10 −4 represents the refractive index difference between the fiber core and the cladding silica substrate.PML with a thickness of 10 μm is used in the outermost layer so that waves incident on the layer are not reflected toward the interface.Fig. 1(a) shows a cross-sectional view of PM-PCF with a stress-induced, polarization-maintaining structure with a stomatal diameter of d = 1.2-5.9μm and a hole spacing of Λ = 20-23.75μm.The white part is a clad silica substrate with a refractive index of 1.4500, a thermal expansion coefficient of 5.50 × 10 −7 K −1 , and an inner cladding diameter of (8.5Λ-10) μm.The blue part is set to be air, and d/Λ ranges from 0.05-0.25.Boron-doped fused silica rods with thermal expansion coefficient of 2.54 × 10 −6 K −1 are introduced as stress elements (yellow section) in the optical waveguide.These stress elements having a diameter of 12-15 μm and refractive index of 1.4412 are distributed in a rectangular shape, replacing the multiple air holes on both sides of the fiber core.The relationship between the fiber structure and the properties of single mode and birefringence was analyzed by changing the structural and material parameters of the fiber.To make a fair comparison between these two different structure designs, the fiber-hole spacing Λ for PM-HOF-PCF is set to 20 μm.The HOMs are expected to be doubly filtered out by using Ge ring HOM filters in the high refractive index region of the leakage channel.The dual cladding structure of the fiber helps to limit the area ratio between the pump cladding and core diameter of the fiber, enhancing pump absorption to achieve higher power while maintaining singlemode transmission and prohibiting nonlinear effects.Fig. 1(b) and (d) show the refractive index profile of the corresponding fiber in Fig. 1(a) and (c).The gray and gray slash areas represent the refractive index of the Yb-doped fiber core and the undoped fused silica substrate, respectively.The refractive index of Boron-doped fused silica is indicated in orange.The green part presents the high refractive index of Ge-doped fused silica as HOM filters and the blue part for the air.

A. Finite Element Simulation for PM-PCF
In PCFs with larger diameters, stress induction can result in higher birefringence values than structure induction.The PM-PCF structure designed in this study achieves the purpose of increasing the core diameter and introducing high birefringence by increasing the number of air holes replaced by the solid core and adding stress elements with a coefficient of thermal expansion α different from the surrounding cladding material (fused silica) on the left and right sides of the core.The prefabricated boron-doped fused silica rods (boron rods) are prepared according to the thermal expansion coefficient of stress elements required for the design, and the doping molar ratio of B 2 O 3 is 23.5%.The fiber structure is optimized by analyzing the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.effect of different numbers and diameters of boron rods on the birefringence values they will impose at the fiber core.
Fig. 2 shows the distribution of PM-PCF' birefringence with the number and structure of stress elements.From Fig. 2, it can be seen that the birefringence value at the location of the air holes is 0. This is mainly due to the Poisson's ratio of air being zero, which will not be affected by the external force to produce elastic deformation, and therefore the air will not be affected by the effect of the elastic-optical effect to make a change in its refractive index, and the corresponding value of birefringence will also be zero.As the number of boron rods increases, the area of high birefringence region gradually increases, and the specific numerical analysis is shown in Fig. 3.The variation of N x − N y in the direction of the slow axis of the fiber cross section is illustrated in Fig. 3(a).N x and N y can be calculated by ( 6)-( 7), and all positive values in the core region represent that all N x is greater than N y in the core region.The maximum birefringence at the core position for introducing 6, 8, 20, 28, and 32 boron rods is 2.70 × 10 −4 , 2.90 × 10 −4 , 3.75 × 10 −4 , 3.97 × 10 −4 , and 3.25 × 10 −4 ,respectively.However, when the number of boron rods is 6 and 8, the birefringence values within the core region (the core diameter is 88 μm) are not uniformly distributed, and the birefringence value cannot reach 1 × 10 −4 in the region of less than 20 μm radius in the fiber core.
Fig. 3(b) and (c) depict the variation of the magnitude of the birefringence value in the slow-axis direction of the fiber core cross-section with the diameter and number of boron rods.Fig. 3(b) shows the slow-axis birefringence distribution along the fiber core for introducing 20 boron rods with diameters of 12 μm, 13 μm, 14 μm and 15 μm, respectively.For a boron rod with a diameter of 15 μm, the core birefringence values are in a range of 1.4 × 10 −4 − 6.5 × 10 −4 , and birefringence values on the order of 1 × 10 −4 are achieved.For a boron rod with a diameter of 14 μm, core birefringence values are in a range of 1.2 × 10 −4 − 5.8 × 10 −4 .As the boron rod diameter decreases, the birefringence values in some regions of the fiber core can no longer reach an order of 10 −4 .This can be derived by analyzing the birefringence value distribution for boron rod diameters of 12 and 13 μm.
The effect of boron rod diameter on the birefringence values for 20 and 32 boron rods is analyzed in Fig. 3(c).There is a demarcation point in the distribution of the birefringence values caused by the boron rods.In the region where the radius is less than 25 μm in the fiber core, the stress-induced birefringence due to introduction of 32 boron rods is higher than that of 20 boron rods.In the region where the core radius is greater than 25 μm, the opposite birefringence change tendency is observed.For PM-PCF, the stress field generated by both 20 and 32 boron rods can cause high average birefringence values.However, the introduction of excessive boron rods increases the contact surface area between the stress zone and the fused silica substrate, which is not conducive to fiber fabrication process optimization.By analyzing the relationship between the number and diameter of boron rods and the resultant birefringence, it is suggested that a sufficiently high and uniformly distributed birefringence can be introduced by 20 boron rods with a diameter of 15 μm.The above parameters of boron rods are used to reduce the variables in the fiber structural design in subsequent analysis of the fiber transmission characteristic.
Fig. 4 depicts the distribution of the effective refractive index and the core overlap factor of the fiber when the PM-PCF's core diameter is 88 μm and the refractive index difference (Δn) between the core and cladding is 5.0 × 10 −5 , 1.0 × 10 −4 and 1.5 × 10 −4 .The overlap factor of this structure is calculated in Fig. 4(f) for Δn = 5.0 × 10 −5 , which results in core overlap factors of 0.904 and 0.701 for LP01 and LP11-y at 1000 nm, respectively.It can be concluded that the single-mode transmission band is less than 1000 nm.As shown in the gray area in Fig. 4(d), the difference in the core overlap factor between  LP01 and LP11-y is greater than 30% with n core = 1.44990 and Δn = 1.0 × 10 −4 .Based on the single-mode determination, the y-polarization single-mode transmission band of the PM-PCF is within 1050-1300 nm, and the corresponding single-mode MFD is between 44.48 and 50.43 μm.For the situation that Δn is 1.5 × 10 −4 , as shown in the gray area in Fig. 4(b), the y-polarization single-mode transmission band is between 1300-1400 nm, and a single-mode MFD of 40.37-45.41μm can be achieved.From the above analysis, it can be concluded that as Δn increases, the single-mode transmission band tends to redshift.The PM-PCF can achieve single-mode transmission at 1064 nm when the n core is 1.44990.Furthermore, there is a monotonic increment of the MFD from 44.48 to 50.43 μm as the single-mode transmission wavelength moves toward the longwave direction, with a single-mode MFD of about 48.42 μm at 1064 nm.
This study focuses on analyzing the LP01-x, LP01-y, LP11-x and LP11-y modes of PCFs, as shown in Fig. 5.The calculation results show that the n ef f difference (structural birefringence value) of the two merged modes of LP01 is 2 × 10 −9 .This is mainly due to the large core diameter, the birefringence value caused by the anisotropic fiber structure is not enough to enable the fiber to achieve polarization-maintaining.The mode distribution, n ef f , and core overlap factor of the two modes LP01-x and LP01-y are basically the same.Therefore, only one of the two linearly polarized modes of LP01 is listed in this study.
A further increase in the MFD is achieved by increasing the aperture distance (Λ) from 20 μm to 23.09 μm and increasing the effective core diameter of the fiber (i.e., Yb doped core diameter) from 88 μm to 101.6 μm.The variation in the core overlap coefficient, effective refractive index and mode field area in the 1000-1500 nm range is calculated for the PM-PCF with a structure of Λ = 23.09μm and d/Λ = 0.1, as illustrated in Fig. 6.
The analysis shows that the single-mode transmission band is within 1250-1450 nm (Fig. 6(b)), corresponding to a singlemode MFD of 49.55 to 52.97 μm (Fig. 6(c)).Although the mode field area increases with increasing hole pitch, it results in a redshift of the single-mode transmission band, which makes the single-mode transmission unfeasible at 1053 nm and 1064 nm.The increase in the hole spacing leads to a decrease in the proportion of air, which ultimately leads to an increase in the effective refractive index of the cladding.The refractive index difference between the cladding and doped core is also affected.The study focuses on the single-mode characteristics of the emission wavelength of the fiber doped with rare-earth ions Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The analysis in Fig. 4 indicates that the single-mode transmission band gradually redshifts as Δn increases.Therefore, Δn can be reduced by increasing the refractive index of the core, thus enabling the single-mode transmission band shifted to short wavelength side.The fiber core overlap factors and the difference between the core overlap factors of the LP01 and LP11 are calculated for different core refractive indices (n core = 1.44990, 1.44991, 1.44992, 1.44993, 1.44994, and 1.44995), as shown in Fig. 7. Fig. 7(b) shows that the core overlap factor of the fiber's LP01 and LP11-y is only 7.4% and 4.9%, respectively, which cannot assure single-mode transmission at n core = 1.44990.As the refractive index of the core increases, the core overlap factor of the LP01 and LP11-y increases gradually, and ΔΓ increases as well.Fig. 7(c) shows that ΔΓ can be greater than 30% only for core refractive indices of 1.44993 and 1.44994, where ypolarization single-mode transmission is achieved (gray shaded part).The green point chart in Fig. 7(c) shows the FM field area decreases with the increase of the refractive index of the core when is n core < 1.44993.For n core = 1.44993, a minimum MFD of 49.04 μm and at same time a maximum ΔΓ is achieved.
It is theoretically possible to achieve a smaller refractive index difference by adjusting the ratio of air holes to hole spacing (d/Λ), which increases the mode field area and single-mode transmission bandwidth.To investigate the effect of d/Λ on single-mode transmission, the mode field distribution, effective refractive index, FM mode field area and core overlap factor of different modes of the fiber are calculated, as shown in Figs. 8  and 9.
Fig. 8(a), (c), and (e) shows the distribution of the effective refractive index (n ef f ) of the FM (LP01), 1 st HOM (LP11-x, LP11-y) and the fiber core for the different core refractive index structures.The effective refractive index of the FM is influenced by the size of the air hole in the cladding.Since the refractive index of air is lower than that of the cladding substrate, the n ef f decreases with increasing d/Λ as the air hole diameter increases while Λ remains constant.As the n core increases, the proportion of n ef f > n core gradually decreases.The n ef f of the FM is uniformly higher than the n ef f of the 1 st HOM.An inflection point occurs in the process of decreasing the n ef f of LP11-x when d/Λ increases.The position of this inflection point is closely related to the n core , which corresponds to the inflection points d/Λ = 0.16, 0.10, and 0.08 when the n core is 1.11990, 1.44993 and 1.44994, respectively.While d/Λ is smaller than the inflection point, the n ef f of LP11-x is closer to the n ef f of FM.On the contrary, the n ef f of LP11-x is nearer to the n ef f When the n core increases, the d/Λ ratio that enables singlemode transmission becomes smaller.The effective refractive index diagram shows that when the d/Λ ratio decreases, the air-hole size decreases, and Δn decreases.Therefore, singlemode transmission can be implemented by adjusting the d/Λ ratio to reduce the refractive index difference between the core and cladding layers.The PM-PCF has 20 boron rods of 15 μm diameter with different thermal expansion coefficients symmetrically distributed on both sides of the fiber core, and it introduces the stress in the fiber that allows the fiber to realize polarization maintenance.Therefore, this structure can achieve single-mode, polarization-maintaining transmission at 1064 nm.The maximum MFD can reach 52.81 μm where n core = 1.44993,Λ = 23.09μm, and d/Λ = 0.125.

B. Finite Element Simulation for PM-HOF-PCF
The PM-HOF-PCF (Fig. 1(c)) is based on the structure of PM-PCF (Fig. 1(a)) with further introduction of leakage channels and a higher-order mode filter structure.We tend to use the PM-HOF-PCF design to achieve single-mode transmission, a very large-mode-area, as well as polarization-maintaining characteristic.Hexagonal leakage channels and Ge-doped, HOMfiltering structures are introduced in the cladding to reduce the power occupation of HOMs in the core, which makes the HOM delocalization and double-filtered.The two-dimensional photonic crystal structure of the inner cladding of the double-clad fiber is maintained simultaneously.The finite element method solver is used to simulate the power overlap rate of the relevant modes in the core of the PM-HOF-PCF rod fiber design, as shown in Fig. 10.The single-mode transmission characteristics of the fiber are analyzed by calculating the core overlap factor, and the corresponding mode field distributions of the LP0m and LP11 at key wavelengths are illustrated.The mode variations are complicated because of the inclusion of multiple filter structures in the fiber and the large span of the calculated band.
The LP0m mode field in a band from 1000 nm to 1160 nm is shown in Fig. 10(a), which belongs to the LP03 mode (Γ is indicated by the green dash zone).Γ increases and then decreases with increasing wavelength.The decrease is mainly due to the transfer from the LP03 to LP02 mode.The LP02 mode appears at 1110 nm (mode field diagram is shown in Fig. 10(b), and Γ is shown in blue square).As Γ of the LP02 mode gradually increases, Γ of the LP03 mode gradually decreases to 0.07%.The LP02 mode is dominant at 1110-1310 nm, and Γ first increases and then decreases.In the 1300-1500 nm range, the LP02 mode is transformed into the LP01 mode, and Γ for the LP01 mode (shown in red square) gradually increases.Fig. 11 illustrates the specific process of LP02 mode to LP01 mode transition and the corresponding fiber core overlap factor change in the 1300-1350 nm.Γ(LP11-x) and Γ(LP11-y) are shown in gray (model field in Fig. 10(d), (e), and (f)) and orange triangle, respectively, in Fig. 12.The field profiles illustrate the working principle of the PM-HOF-PCF.In particular, the HOMs effectively couple out from the core to the cladding and thus effectively delocalized and transformed into the cladding modes.These modes are guided either in the resonator elements or inside the cladding Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.As can be seen from Fig. 10, the LP0m mode of PM-HOF-PCF fiber has two conversions at 1000-1500 nm, with the core overlap factor of LP03 mode gradually decreasing near 1150 nm, LP02 mode's core overlap factor gradually increases, and reaches a maximum value of 81.68% at 1250 nm.When wavelength increases, the fiber core overlap factor of LP02 gradually starts to decrease and converts to LP01 mode, and the specific conversion process can be seen in Fig. 11.The effective refractive index of LP02 is matched to the structural characteristic solution of the cladding and resonant coupling occurs at 1300 nm, so that LP02 modes are partially leaked from the Ge-doped mode filtering structure in the fiber cladding, except for some of the modes in the core.The overlap factor of LP02 core at 1300 nm is 64.17%, and with the increase of wavelength, its core overlap factor starts to decrease from 33.18% (1310 nm) to 10.49% (1320 nm), and finally decreases to 1.63% at 1350 nm, and most of the modes in the core are coupled with the germanium ring structure in the cladding and leaked.The core overlap factor of LP01 mode is the opposite of that of LP02 mode, as the wavelength increases, the mode field in the cladding gradually disappears, and the core overlap factor increases from 8.05% (1300 nm) to 79.38% (1350 nm), and the mode field is basically completely confined to propagate in the core at 1350 nm.
The difference in the core overlap factor (red and blue parts) and the variation in the MFD (gray part) of the LP0m and LP11 are illustrated in Fig. 12. Region I: X-polarization, singlemode transmission is achievable in the 1050 nm to 1110 nm band (shaded in green).The MFD is 64.29-76.33μm, and the maximum MFD is 76.33 μm at 1050 nm.Full single-mode transmission is achieved at 1100 nm (shaded in gray) with a MFD of 64.69 μm.Region II: Single-polarization, single-mode transmission can be achieved in the 1190-1250 nm band (shaded in green).It has a MFD of 56.98-57.16μm and a maximum MFD of 57.16 μm (1190 nm).The MFD corresponding to full singlemode transmission at 1200 nm is 56.91 μm (shaded in gray).Region III: As highlighted in green shade, the structure of the fiber can achieve single-polarization, single-mode transmission in the 1400-1450 nm band.The corresponding MFD is 56.18-56.31μm.The maximum MFD is 56.31 μm (at 1450 nm).At the 1400 nm, XY polarization can be single-mode transmission (gray-shaded part), and the MFD is 56.18 μm.From the analysis, the introduction of Ge ring in the PM-HOF-PCF type fiber can effectively make the HOMs in the core delocalized and achieve a higher ΔΓ at the same time, which is equivalent to a higher-order mode filtering structure.A larger single-polarized single-mode field diameter can be achieved by PM-HOF-PCF type instead of PM-PCF with the same structural and material parameters such as hole spacing Λ, d/Λ and core refractive index.

V. CONCLUSION
A thorough numerical analysis based on the finite-element method has been performed to study the effects of fiber structure, stress-induced and refractive index change on the modes which propagate in large-mode-area, double-cladding, Yb-doped, microstructured fibers.As fiber structure designs have become more complex, several difficulties have arisen in evaluating the modal content and consequently in maintaining the singlemode operation of the fiber.This paper proposes a method for PCF's single-mode determination criterion based mainly on evaluating the mode overlap in the doped core region, which is more accurate and has a wider application than the traditional V-parameter method.In terms of PM-PCF design, the stress region is introduced into the fiber structure, and the polarizationmaintaining characteristics of the fiber are studied by analyzing the birefringence distribution.PM-PCF's fiber core replaces 19 air hole positions, five layers of air holes and stress zones work together to form the fiber's cladding structure.To achieve precise modulation of the single-mode transmission band by the fiber structure, the effects of the fiber-hole spacing and d/Λ ratio on the single-mode transmission band are also analyzed.The design of a new type of PM-HOF-PCF structure with a large-mode-area, polarization-maintaining ability is proposed.By introducing a Ge ring structure with resonant filtering in the fiber cladding, the HOM is delocalized through resonant coupling with the HOM, and the single-polarized and single-mode transmission can be achieved in the range of 1050-1110 nm within the emission band of Y b 3+ ions.The single-mode field diameter for the proposed PM-HOF-PCF is 64.29-76.33μm, and the maximum MFD is 76.33 μm and the birefringence value is larger than 1.0 × 10 −4 .Compared with the PCF with the same structural parameters without the introduction of cladding higher-order filter structure, its single-mode MFD increases from 48.42 μm to 76.33 μm, and the increase in the MFD accounts for 36.6% of the total diameter.45641

Fig. 1 .
Fig. 1.(a) Cross-section and (b) refractive index of PM-PCF with boron rod stress zones and a solid fiber core replacing 19 air holes.(c) Cross-section and (d) refractive index of PM-HOF-PCF with Ge rings as HOM filters, boron rod stress element zones and a solid fiber core replacing 19 air holes.

Fig. 1 (
c) shows a schematic diagram of fiber PM-HOF-PCF.A leakage channel and an HOM filter are introduced to achieve single-mode, LMA-PM transmission.The HOM filter is a ring structure doped with germanium (green part), and the refractive index of the Ge ring n(GeO 2 ) = 1.45 + 23.5 × 10 −4 .The coefficient of thermal expansion of Ge-doped silica is 5.50 × 10 −7 K −1 .Thickness of the Ge-Doped ring in PM-HOF-PCF is (0.2 * Λ − d/2)μm.

Fig. 3 .
Fig. 3. Variation of PM-PCF (a) fiber cross-section Nx-Ny distribution in the slow-axis direction with the number of boron rods in the stress region.Birefringence distribution of PM-PCF in the slow-axis direction: (b) 20 boron rods with different diameters, and (c) 20 and 32 boron rods with set diameter of 13 and 14 µm, respectively.

Fig. 6 .
Fig. 6.The change in (a) the effective refractive index, (b) core overlap factor, and (c) MFD with different wavelengths.Refractive index of the core is 1.4499, and Λ = 23.09µm.

Fig. 7 .
Fig. 7. Core refractive index of 1.44990-1.44995at 1064 nm with Γ is 23.09 µm.The change of (a) effective refractive index, (b) core overlap factor, and (c) difference in core overlap factor between LP01 and LP11 (ΔΓ) and MFD, with the core refractive index.

Fig. 10 .
Fig. 10.Simulated core overlap factor of different modes in the PM-HOF-PCF with a fixed hole spacing of 20 µm and a n core of 1.44985.Insets illustrate the simulated electric field profiles of different modes at specific wavelengths.

Fig. 12 .
Fig. 12. Trend of ΔΓ and MFD with λ at 1000-1500 nm with a fixed Λ of 20 µm and n core of 1.44985.structure.The profile of the HOM (1400 nm) in Fig.10(e) clearly demonstrates this behavior.As can be seen from Fig.10, the LP0m mode of PM-HOF-PCF fiber has two conversions at 1000-1500 nm, with the core overlap factor of LP03 mode gradually decreasing near 1150 nm, LP02 mode's core overlap factor gradually increases, and reaches a maximum value of 81.68% at 1250 nm.When wavelength increases, the fiber core overlap factor of LP02 gradually starts to decrease and converts to LP01 mode, and the specific conversion process can be seen in Fig.11.The effective refractive index of LP02 is matched to the structural characteristic solution of the cladding and resonant coupling occurs at 1300 nm, so that LP02 modes are partially leaked from the Ge-doped mode filtering structure in the fiber cladding, except for some of the modes in the core.The overlap factor of LP02 core at 1300 nm is 64.17%, and with the increase of wavelength, its core overlap factor starts to decrease from 33.18% (1310 nm) to 10.49% (1320 nm), and finally decreases to 1.63% at 1350 nm, and most of the modes in the core are coupled with the germanium ring structure in the cladding and leaked.The core overlap factor of LP01 mode is the opposite of that of LP02 mode, as the wavelength increases, the mode field in the cladding gradually disappears, and the core overlap factor increases from 8.05% (1300 nm) to 79.38% (1350 nm), and the mode field is basically completely confined to propagate in the core at 1350 nm.The difference in the core overlap factor (red and blue parts) and the variation in the MFD (gray part) of the LP0m and LP11 are illustrated in Fig.12.Region I: X-polarization, singlemode transmission is achievable in the 1050 nm to 1110 nm band (shaded in green).The MFD is 64.29-76.33μm, and the maximum MFD is 76.33 μm at 1050 nm.Full single-mode transmission is achieved at 1100 nm (shaded in gray) with a MFD of 64.69 μm.Region II: Single-polarization, single-mode transmission can be achieved in the 1190-1250 nm band (shaded in green).It has a MFD of 56.98-57.16μm and a maximum MFD of 57.16 μm (1190 nm).The MFD corresponding to full singlemode transmission at 1200 nm is 56.91 μm (shaded in gray).Region III: As highlighted in green shade, the structure of the fiber can achieve single-polarization, single-mode transmission in the 1400-1450 nm band.The corresponding MFD is 56.18-56.31μm.The maximum MFD is 56.31 μm (at 1450 nm).At Manuscript received 10 March 2024; revised 13 April 2024; accepted 28 April 2024.Date of publication 1 May 2024; date of current version 27 May 2024.This work was supported by Youth Innovation Promotion Association, CAS under Grant 2017446.(Yuan Ma and Rui Wan contributed equally to this work.)(Corresponding author: Pengfei Wang.)