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  Panda-type bias-preserving few-mode fiber assisted by concentric circular stress zones

  Liu Yanan, Lijin, Yuan Xueguang, Zhang Yangan, Zhang Zhen'
State Key Laboratory of Information Photonics and Optical Communication, Beijing University of Posts and Telecommunications, Beijing 100876, China

Abstract


Abstract A panda-type bias-preserving few-mode fiber structure with concentric circular-shaped stress zones is proposed, which can stably support 10 transmission modes. With the introduction of a low refractive index stress region, the effective refractive index difference between higher-order modes is improved by nearly an order of magnitude, and the minimum effective refractive index difference between neighboring modes reaches 2 × 10 − 4 2 × 10 − 4 2xx10^(-4)2 \times 10^{-4} at 1550 nm, and is not lower than 1.8 × 10 − 4 1.8 × 10 − 4 1.8 xx10^(-4)1.8 \times 10^{-4} in the C-band. (Mode dispersion is not higher than | − 55.0219 | ps / ( nm ⋅ km | − 55.0219 | ps / nm ⋅ km |-55.0219|ps//(nm*km:}|-55.0219| \mathrm{ps} /\left(\mathrm{nm} \cdot \mathrm{km}\right. ), and the maximum bending loss is in the order of 10 − 7 dB / m 10 − 7 dB / m 10^(-7)dB//m10^{-7} \mathrm{~dB} / \mathrm{m} (bending radius ⩾ 9.5 cm ⩾ 9.5 cm >= 9.5cm\geqslant 9.5 \mathrm{~cm} ). The research results provide ideas for the design of short-haul, high-capacity optical fibers.


Keywords Fiber optics; Few-mode fiber; Space division multiplexing; Bias preserving; Simple merged mode

CCS TN29 Literature Symbol Code A DOI: 10.3788/AOS230674

  1 Introduction


The emergence of new data services and the rapid development of cloud computing technology have put forward an urgent demand for the enhancement of transmission capacity in optical interconnection networks. Traditional single-mode fiber is limited by the nonlinear Shannon limit, and it is difficult to improve the capacity. For this reason, time division multiplexing (TDM), wavelength division multiplexing (WDM), and space division multiplexing (SDM) technologies have been applied to optical fiber design, of which SDM technology has received widespread attention for its ability to reach the maximum capacity [ 1 − 3 ] [ 1 − 3 ] ^([1-3]){ }^{[1-3]} . Few-mode fiber is a typical device that uses space-division multiplexing, but the mode crosstalk that occurs during transmission in few-mode fiber is a major problem. To solve this problem, multiple-input multiple-output (MIMO) digital signal processing techniques are usually introduced at the receiver side [ 4 ] [ 4 ] ^([4]){ }^{[4]} , but as the number of transmitted modes increases, the complexity of the system grows nonlinearly, resulting in large power consumption [ 5 ] [ 5 ] ^([5]){ }^{[5]} . For this reason, it is important to suppress mode coupling from the root cause, maximize the separation of intrinsic modes, achieve low crosstalk between fiber-guided modes, and simplify the communication system.

Theoretical studies have shown that when the effective refractive index difference of the modes ( Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} ) is greater than 10 − 4 10 − 4 10^(-4)10^{-4} , the modes can be considered to be free of merging, and the energy coupling is effectively reduced [ 6 − 7 ] [ 6 − 7 ] ^([6-7]){ }^{[6-7]} . In order to improve the intermode Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} and expand the transmission capacity, the optical fibers used usually include elliptical core fiber [ 8 − 9 ] [ 8 − 9 ] ^([8-9]){ }^{[8-9]} , ring core fiber [ 10 − 11 ] [ 10 − 11 ] ^([10-11]){ }^{[10-11]} , panda-type bias-preserving few-mode fiber [ 12 − 14 ] [ 12 − 14 ] ^([12-14]){ }^{[12-14]} , etc. Zhao et al. proposed a bias-preserving few-mode fiber consisting of a central circular hole and an elliptical ring core, which can transmit 10 modes, and the bias-preserving few-mode fiber in [ 12 − 14 ] [ 12 − 14 ] ^([12-14]){ }^{[12-14]} is able to transmit 10 modes at [ 12 − 14 ] [ 12 − 14 ] ^([12-14]){ }^{[12-14]} wavelength Δ n eff > 1.32 × 10 − 4 Δ n eff  > 1.32 × 10 − 4 Deltan_("eff ") > 1.32 xx10^(-4)\Delta n_{\text {eff }}>1.32 \times 10^{-4} ; Xiao et al. proposed a bias-preserving few-mode fiber with four apertures around the core, which provides an idea for separating the higher-order merging modes, and reaches a minimum of 1.65 × 10 − 4 1.65 × 10 − 4 1.65 xx10^(-4)1.65 \times 10^{-4} at 1550 nm, and then proposed a bias-preserving few-mode fiber with elliptical apertures in the center of core. After that, a bias-preserving few-mode fiber with elliptical aperture in the center was proposed [ 17 ] [ 17 ] ^([17]){ }^{[17]} , which is Δ n eff > 1.93 × 10 − 4 Δ n eff  > 1.93 × 10 − 4 Deltan_("eff ") > 1.93 xx10^(-4)\Delta n_{\text {eff }}>1.93 \times 10^{-4} between adjacent modes at 1550 nm; Chen et al.

A few-mode fiber with air holes introduced in the center and around the core, the number of modes is increased and the fiber is Δ n eff > 1.25 × Δ n eff  > 1.25 × Deltan_("eff ") > 1.25 xx\Delta n_{\text {eff }}>1.25 \times 10 − 4 10 − 4 10^(-4)10^{-4} in the 1520 ∼ 1580 nm 1520 ∼ 1580 nm 1520∼1580nm1520 \sim 1580 \mathrm{~nm} band. The above structure balances the number of modes and the separation of merged modes, which improves the minimum Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} It is worth noting that there is a contradiction between the fundamental mode separation and the higher-order mode separation, and balancing the separation of the lower-order and higher-order merged modes is expected to further improve the minimum Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} Δ n eff > 1.25 × Δ n eff  > 1.25 × Deltan_("eff ") > 1.25 xx\Delta n_{\text {eff }}>1.25 \times Δ n eff > 1.25 × Δ n eff  > 1.25 × Deltan_("eff ") > 1.25 xx\Delta n_{\text {eff }}>1.25 \times 10 − 4 10 − 4 10^(-4)10^{-4}.

In this paper, a panda-type bias-preserving few-mode fiber with concentric circular stress zones is proposed, which is highlighted by the concentric circular stress zones around the elliptical annular core. With suitable optical parameters and structural dimensions, the fiber structure is capable of stably transmitting 10 modes in the C-band, and all the modes are no longer concatenated and maintain good polarization characteristics. In addition, the fiber has good bending resistance, relatively small mode dispersion, and good process robustness in fabrication. Compared with the fiber without concentric circular-shaped stress region with the same parameters, the stress region of the designed fiber has excellent ability to separate higher-order simple merged modes and mode constraints. Therefore, the structure has important research value in optical interconnection networks for short-distance high-capacity transmission.

  2 Fiber structure and principle analysis


The cross-section of the fiber structure is shown in Fig. 1. Figure 1(a) shows the overall cross-section of the fiber, the cladding diameter is D D DD , the setting is 125 μ m 125 μ m 125 mum125 \mu \mathrm{~m} , the two apertures are distributed in the x x xx axis and symmetric about the y y yy axis, the aperture radius is r r rr , the center of the y y yy axis away from the distance of Λ Λ Lambda\Lambda . Figure 1(b) shows an enlarged schematic of the core section, with the half-length and half-short axes of the elliptical annular core being b x b x b_(x)b_{x} and b y b y b_(y)b_{y} , respectively, and the half-length and half-short axes of the elliptical core being a x a x a_(x)a_{x} and a y a y a_(y)a_{y} , and the radius of the concentric circular stress region being R R RR . The material of the cladding and the elliptical core is pure silica ( SiO 2 ) SiO 2 (SiO_(2))\left(\mathrm{SiO}_{2}\right) ; the core is silica doped with germanium dioxide ( GeO 2 ) GeO 2 (GeO_(2))\left(\mathrm{GeO}_{2}\right) , and the doping rate (mole fraction) is 23.75 % 23.75 % 23.75%23.75 \% with reference to the literature [15-16, 19]; the concentric circular stress zone is R R RR in radius.

Corresponding author: *xzhang@bupt.edu.cn

The stress region is a silica material doped with boron trioxide ( B 2 O 3 ) B 2 O 3 (B_(2)O_(3))\left(\mathrm{B}_{2} \mathrm{O}_{3}\right) . At 1550 nm, the refractive indices of the above three parts of the material are 1.444, 1.478, and 1.428, respectively, and the refractive index of the porous part is 1. The refractive index of the material is 1.444, 1.478, and 1.428, respectively. Concentric circles

The shaped stress zone forms a refractive index depression between the core and the cladding, and the fiber is able to further separate the simple and merged modes by using the high refractive index contrast of the core-stress zone-cladding and the non-circular symmetry of the fiber structure [ 7 , 17 ] [ 7 , 17 ] ^([7,17]){ }^{[7,17]} .

Fig. 1 Schematic diagram of the fiber structure. (a) Schematic of the overall cross-section of the optical fiber; (b) Schematic of the enlarged geometry of the fiber core
Fig. 1 Schematic of fiber structure. (a) Overall cross section of optical fiber; (b) schematic of enlarged geometrical structure of the fiber core

By mode analysis, the mode effective area ( A eff A eff  A_("eff ")A_{\text {eff }} ) in a few-mode fiber can be expressed as
A eff = [ ∬ | E ( x , y ) | 2 d x d y ] 2 / ∬ | E ( x , y ) | 4 d x d y , ( 1 ) A eff  = ∬ | E ( x , y ) | 2 d x d y 2 / ∬ | E ( x , y ) | 4 d x d y , ( 1 ) A_("eff ")=[∬|E(x,y)|^(2)(d)x(d)y]^(2)//∬|E(x,y)|^(4)dxdy,(1)A_{\text {eff }}=\left[\iint|E(x, y)|^{2} \mathrm{~d} x \mathrm{~d} y\right]^{2} / \iint|E(x, y)|^{4} \mathrm{~d} x \mathrm{~d} y,(1)

Where: E ( x , y ) E ( x , y ) E(x,y)E(x, y) is the mode intensity distribution. The optical beam method is used to analyze the fiber bending resistance. When the fiber is bent, the bending can be simulated by the equivalent refractive index distribution equation, i.e.
n ′ ( x , y ) = n ( x , y ) × [ 1 + ( x × cos ⁡ θ + y × sin ⁡ θ ) / R eff ] n ′ ( x , y ) = n ( x , y ) × 1 + ( x × cos ⁡ θ + y × sin ⁡ θ ) / R eff  {:[n^(')(x","y)=n(x","y)xx],[[1+(x xx cos theta+y xx sin theta)//R_("eff ")]]:}\begin{gathered} n^{\prime}(x, y)=n(x, y) \times \\ {\left[1+(x \times \cos \theta+y \times \sin \theta) / R_{\text {eff }}\right]} \end{gathered}

where n ( x , y ) n ( x , y ) n(x,y)n(x, y) is the refractive index of the straight fiber, θ θ theta\theta is the bending angle with respect to the x x xx axis, and R eff R eff  R_("eff ")R_{\text {eff }} is the equivalent bending radius in quartz fibers R eff / R ≈ 1.28 [ 20 ] R eff / R ≈ 1.28 [ 20 ] R_(eff)//R~~1.28^([20])R_{\mathrm{eff}} / R \approx 1.28^{[20]} . In an equivalent straight fiber, the bending loss ( L b ) L b {:L_(b))\left.L_{\mathrm{b}}\right) ) is related to the imaginary part of the effective refractive index of the guiding mode ( n eff n eff  n_("eff ")n_{\text {eff }} ) and the wavelength ( λ λ lambda\lambda ) ( [ 21 ] [ 21 ] ^([21]){ }^{[21]} ), which is given by
L b = 40 × π × Im ( n eff ) / ( λ × ln ⁡ 10 ) L b = 40 × π × Im n eff / ( λ × ln ⁡ 10 ) L_(b)=40 xx pi xx Im(n_(eff))//(lambda xx ln 10)L_{\mathrm{b}}=40 \times \pi \times \operatorname{Im}\left(n_{\mathrm{eff}}\right) /(\lambda \times \ln 10)

  3 Parameter optimization design


The structural parameters of the elliptical core are designated as a x = 3.39 μ m 、 a y = a x = 3.39 μ m 、 a y = a_(x)=3.39 mum、a_(y)=a_{x}=3.39 \mu \mathrm{~m} 、 a_{y}=、 2.42 μ m 2.42 μ m 2.42 mum2.42 \mu \mathrm{~m} . The larger the radius of the air holes and the closer they are to the y y yy axis, the greater the effect on the modes, and through preliminary adjustments, the values of r r rr and Λ Λ Lambda\Lambda are set to 20 μ m 20 μ m 20 mum20 \mu \mathrm{~m} and 26.05 μ m 26.05 μ m 26.05 mum26.05 \mu \mathrm{~m} , respectively. In this paper, the core of the fiber is no longer a conventional circle, but an elliptical ring structure, which changes the structure of the fiber from circular symmetry to axial symmetry, which results in the direction of the electric field vectors of the transmission modes in the fiber converging to a certain direction, forming a distribution similar to that of the linearly polarized modes, and therefore all the modes discussed in this paper are named in the manner of the "LP".

Too many modes in a few-mode fiber will lead to serious inter-mode crosstalk problems, while a smaller number is not conducive to the improvement of fiber capacity, so the target number of fiber modes in this experiment is 10. The main factor affecting the number of modes is the core area, so the half-length axis ( b x ) b x (b_(x))\left(b_{x}\right) and the half-short axis ( b y ) b y (b_(y))\left(b_{y}\right) of the elliptical ring core are optimized first. In λ = 1550 nm 、 b y = λ = 1550 nm 、 b y = lambda=1550nm、b_(y)=\lambda=1550 \mathrm{~nm} 、 b_{y}=、

In the case of 3.37 μ m 、 R = 5.12 μ m 3.37 μ m 、 R = 5.12 μ m 3.37 mum、R=5.12 mum3.37 \mu \mathrm{~m} 、 R=5.12 \mu \mathrm{~m}、 , the variation curve of Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} with b x b x b_(x)b_{x} between mode n eff n eff  n_("eff ")n_{\text {eff }} and neighboring modes is shown in Fig. 2. When b x < 5 μ m b x < 5 μ m b_(x) < 5mumb_{x}<5 \mu \mathrm{~m} , the n eff n eff  n_("eff ")n_{\text {eff }} of modes L P 21 b x L P 21 b x LP_(21 b)^(x)L P_{21 b}^{x} and L P 21 b y L P 21 b y LP_(21 b)^(y)L P_{21 b}^{y} are lower than the cladding refractive index and the modes are cut off. The n eff n eff  n_("eff ")n_{\text {eff }} of the four highest-order modes are extremely close to each other near b x = 5.2 μ m b x = 5.2 μ m b_(x)=5.2 mumb_{x}=5.2 \mu \mathrm{~m} , and the modes interconvert to each other, resulting in very small values of Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} between their next-nearest neighboring modes around b x = 5.2 μ m b x = 5.2 μ m b_(x)=5.2 mumb_{x}=5.2 \mu \mathrm{~m} , as shown in Fig. 2(b); and the two fundamental modes are extremely close to each other around b x b x b_(x)b_{x} before Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} reaches the 4.8 μ m 4.8 μ m 4.8 mum4.8 \mu \mathrm{~m} level, as shown in Fig. 2(c). b14> after Δ n eff > 10 − 4 Δ n eff  > 10 − 4 Deltan_("eff ") > 10^(-4)\Delta n_{\text {eff }}>10^{-4} is realized; the Δ n eff Δ n eff Deltan_(eff)\Delta n_{\mathrm{eff}} between LP 01 x LP 01 x LP_(01)^(x)\mathrm{LP}_{01}^{x} and LP 11 a y LP 11 a y LP_(11a)^(y)\mathrm{LP}_{11 \mathrm{a}}^{y} shows a decreasing trend with the increase of b x b x b_(x)b_{x} , and the energy crosstalk between the two modes increases significantly especially after b x > 5.35 μ m b x > 5.35 μ m b_(x) > 5.35 mumb_{x}>5.35 \mu \mathrm{~m} . Extending the core half-axis size from 4.46 μ m 4.46 μ m 4.46 mum4.46 \mu \mathrm{~m} to 5.56 μ m 5.56 μ m 5.56 mum5.56 \mu \mathrm{~m} solves the problem of polarization crosstalk and spatial mode crosstalk in the second mode group and between LP 11 b y LP 11 b y LP_(11b)^(y)\mathrm{LP}_{11 \mathrm{~b}}^{y} and LP 21 a x LP 21 a x LP_(21a)^(x)\mathrm{LP}_{21 \mathrm{a}}^{x} . To ensure the number of modes and the effective separation of all modes, b x b x b_(x)b_{x} can be 5.18 μ m 5.18 μ m 5.18 mum5.18 \mu \mathrm{~m} or in the range 5.04 ∼ 5.04 ∼ 5.04∼5.04 \sim 5.14 μ m 5.14 μ m 5.14 mum5.14 \mu \mathrm{~m} and 5.28 ∼ 5.31 μ m 5.28 ∼ 5.31 μ m 5.28∼5.31 mum5.28 \sim 5.31 \mu \mathrm{~m} .

At 1550 nm, setting b x = 5.04 μ m 、 R = 5.12 μ m b x = 5.04 μ m 、 R = 5.12 μ m b_(x)=5.04 mum、R=5.12 mumb_{x}=5.04 \mu \mathrm{~m} 、 R=5.12 \mu \mathrm{~m}、 , b y b y b_(y)b_{y} is insufficient for 3.3 μ m 3.3 μ m 3.3 mum3.3 \mu \mathrm{~m} , causing LP 21 b x LP 21 b x LP_(21 b)^(x)\mathrm{LP}_{21 b}^{x} and LP 21 b y LP 21 b y LP_(21 b)^(y)\mathrm{LP}_{21 b}^{y} modes to leak into the cladding. Therefore, this experiment explores the effect of b y b y b_(y)b_{y} on the mode characteristics in the 3.3 ∼ 5 μ m 3.3 ∼ 5 μ m 3.3∼5mum3.3 \sim 5 \mu \mathrm{~m} range. As shown in Fig. 3, the half-length axis of the fiber core mainly affects the number of modes, and the half-short axis has a more significant effect on mode separation. As b y b y b_(y)b_{y} increases, the core ellipticity decreases, the fiber birefringence effect is weakened, and the n eff n eff  n_("eff ")n_{\text {eff }} of the two pairs of orthogonally polarized modes ( LP 01 y LP 01 y LP_(01)^(y)\mathrm{LP}_{01}^{y} and LP 01 x LP 01 x LP_(01)^(x)\mathrm{LP}_{01}^{x} , and LP 21 a x LP 21 a x LP_(21 a)^(x)\mathrm{LP}_{21 a}^{x} and L P 21 a y L P 21 a y LP_(21 a)^(y)L P_{21 a}^{y} ) at 3.94 μ m 3.94 μ m 3.94 mum3.94 \mu \mathrm{~m} are approximately equal, and the Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} of both modes decrease by about 95 % 95 % 95%95 \% . are reduced by about 95 % 95 % 95%95 \% , creating a serious problem of polarization simplicity. In addition, the transverse electric field distributions of LP 11 a x LP 11 a x LP_(11a)^(x)\mathrm{LP}_{11 \mathrm{a}}^{x} and L P 11 b x L P 11 b x LP_(11b)^(x)L \mathrm{P}_{11 \mathrm{~b}}^{x} or L P 21 a y L P 21 a y LP_(21 a)^(y)L \mathrm{P}_{21 a}^{y} and LP 21 b x LP 21 b x LP_(21b)^(x)\mathrm{LP}_{21 \mathrm{~b}}^{x} are basically only angularly different from each other π / 2 π / 2 pi//2\pi / 2 , and thus the Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} between the two pairs of modes varies dramatically during the process of approaching the circular symmetry of the fiber core, and the spatial merger is especially aggravated around b y = 4.2 μ m b y = 4.2 μ m b_(y)=4.2 mumb_{y}=4.2 \mu \mathrm{~m} . around the same time, the spatial simplicity is exacerbated. Except for the above case, the Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} between the remaining modes all meet the requirement of greater than 10 − 4 10 − 4 10^(-4)10^{-4} . In order to suppress the mode energy coupling and guarantee the transmission performance, b y b y b_(y)b_{y} can be selected in the range of 3.3 ∼ 3.65 μ m 3.3 ∼ 3.65 μ m 3.3∼3.65 mum3.3 \sim 3.65 \mu \mathrm{~m} and 4.3 ∼ 4.58 μ m 4.3 ∼ 4.58 μ m 4.3∼4.58 mum4.3 \sim 4.58 \mu \mathrm{~m} .

The proposed fiber structure enhances the Δ n eff Δ n eff Deltan_(eff)\Delta n_{\mathrm{eff}} between higher-order modes while ensuring the separation of fundamental modes, which in turn improves the separation of all concatenated modes. Concentric circular stress zones with low refractive index are introduced between the core and the cladding.


Fig. 2 Relationship between modes n eff n eff  n_("eff ")n_{\text {eff }} and neighboring modes Δ n eff Δ n eff Deltan_(eff)\Delta n_{\mathrm{eff}} and b x b x b_(x)b_{x} at 1550 nm. (a) variation of n eff n eff n_(eff)n_{\mathrm{eff}} with b x b x b_(x)b_{x} ; (b) variation of Δ n eff Δ n eff Deltan_(eff)\Delta n_{\mathrm{eff}} with b x b x b_(x)b_{x} .

Fig. 2 At 1550 nm , effective refractive index n eff n eff  n_("eff ")n_{\text {eff }} and effective refractive index difference Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} as a function of b x b x b_(x)b_{x} . (a) Variation of n eff n eff  n_("eff ")n_{\text {eff }} with b x b x b_(x)b_{x} ; (b) Variation of Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} with b x b x b_(x)b_{x}

Fig. 3 Relationship between modes n eff n eff  n_("eff ")n_{\text {eff }} and neighboring modes Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} and b y b y b_(y)b_{y} at 1550 nm. (a) variation of n eff n eff  n_("eff ")n_{\text {eff }} with b y b y b_(y)b_{y} ; (b) variation of Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} with b y b y b_(y)b_{y}

Fig. 3 At 1550 nm , effective refractive index n eff n eff  n_("eff ")n_{\text {eff }} and effective refractive index difference Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} as a function of b y b y b_(y)b_{y} . (a) Variation of n eff n eff  n_("eff ")n_{\text {eff }} with b y b y b_(y)b_{y} ; (b) Variation of Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} with b y b y b_(y)b_{y}

The problem can be solved effectively. As shown in Fig. 4(a) and (b), in the wavelength range of 1530-1570 nm, the n eff n eff  n_("eff ")n_{\text {eff }} curves of the higher-order modes of conventional fibers without this stress region are almost completely overlapped; under the same parameter settings, the introduction of a concentric stress region with a radius of 5.12 μ m 5.12 μ m 5.12 mum5.12 \mu \mathrm{~m} results in the effective separation of the higher-order modes of the fiber, and achieves a similar effect as that of the lower-order modes. The n eff n eff  n_("eff ")n_{\text {eff }} curves of the higher-order modes in the fiber are effectively separated, achieving similar effects as those of the lower-order modes. Comparison of Figs. 4(c) and (d) shows that the Δ n eff Δ n eff Deltan_(eff)\Delta n_{\mathrm{eff}} of LP 11 b x − LP 11 b y 、 LP 21 a x LP 11 b x − LP 11 b y 、 LP 21 a x LP_(11b)^(x)-LP_(11b)^(y)、LP_(21a)^(x)\mathrm{LP}_{11 \mathrm{~b}}^{x}-\mathrm{LP}_{11 \mathrm{~b}}^{y} 、 \mathrm{LP}_{21 \mathrm{a}}^{x}、 LP 21 a y 、 LP 21 a y − LP 21 b x LP 21 a y 、 LP 21 a y − LP 21 b x LP_(21a)^(y)、LP_(21a)^(y)-LP_(21b)^(x)\mathrm{LP}_{21 \mathrm{a}}^{y} 、 \mathrm{LP}_{21 \mathrm{a}}^{y}-\mathrm{LP}_{21 \mathrm{~b}}^{x}、 and LP 21 b x − LP 21 b y LP 21 b x − LP 21 b y LP_(21b)^(x)-LP_(21b)^(y)\mathrm{LP}_{21 \mathrm{~b}}^{x}-\mathrm{LP}_{21 \mathrm{~b}}^{y} are significantly improved. The above phenomenon is mainly due to the fact that the electric field spots of the L P 11 b x L P 11 b x LP_(11 b)^(x)L P_{11 b}^{x} and L P 11 b y 、 L P 21 a x L P 11 b y 、 L P 21 a x LP_(11 b)^(y)、LP_(21 a)^(x)L P_{11 b}^{y} 、 L P_{21 a}^{x}、 and LP P 21 a y P 21 a y P_(21 a)^(y)P_{21 a}^{y} , L P 21 b x L P 21 b x LP_(21 b)^(x)L P_{21 b}^{x} and L P 21 b y L P 21 b y LP_(21 b)^(y)L P_{21 b}^{y} modes are surrounded by the core and the overall distribution is nearly circular, which makes the concentric circular stress zone at the periphery of the core effective for the reconstruction of the breakup of the above mode clusters. In addition, the overall increase in Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} values between the proposed fiber modes compared to the fiber without concentric circular stress zones is not less than the minimum Δ n eff Δ n eff Deltan_(eff)\Delta n_{\mathrm{eff}} of 1.8 × 10 − 4 ( LP 11 a x 1.8 × 10 − 4 LP 11 a x 1.8 xx10^(-4)(LP_(11a)^(x):}1.8 \times 10^{-4}\left(\mathrm{LP}_{11 \mathrm{a}}^{x}\right. and LP 11 b x LP 11 b x LP_(11b)^(x)\mathrm{LP}_{11 \mathrm{~b}}^{x} and the Δ n eff Δ n eff Deltan_(eff)\Delta n_{\mathrm{eff}} values between LP 11 b y LP 11 b y LP_(11b)^(y)\mathrm{LP}_{11 \mathrm{~b}}^{y} and LP 21 a x LP 21 a x LP_(21a)^(x)\mathrm{LP}_{21 \mathrm{a}}^{x} are of the order of 10 − 3 10 − 3 10^(-3)10^{-3} . order of magnitude).
  Further investigation of the guide mold metrics with concentric circular stress zone radius

At 1550 nm wavelength, the radius of the concentric stress zone starts to increase from 5.04 μ m 5.04 μ m 5.04 mum5.04 \mu \mathrm{~m} by fixing b x = 5.04 μ m b x = 5.04 μ m b_(x)=5.04 mumb_{x}=5.04 \mu \mathrm{~m} and b y = 3.37 μ m b y = 3.37 μ m b_(y)=3.37 mumb_{y}=3.37 \mu \mathrm{~m} , i.e., the half-length axis size of the core is 5.04 μ m 5.04 μ m 5.04 mum5.04 \mu \mathrm{~m} . Figure 5(a) shows that as the size of the concentric stress zone increases, the individual modes n eff n eff  n_("eff ")n_{\text {eff }} gradually decrease, and when their radius is larger than 5.6 μ m 5.6 μ m 5.6 mum5.6 \mu \mathrm{~m} , the higher-order modes begin to leak into the cladding. Therefore, in order to maintain the number of modes that can be transmitted by the fiber, the radius must not exceed 5.6 μ m 5.6 μ m 5.6 mum5.6 \mu \mathrm{~m} . Optimizing the radius scanning of the concentric stress zones in the 5.04 ∼ 5.6 μ m 5.04 ∼ 5.6 μ m 5.04∼5.6 mum5.04 \sim 5.6 \mu \mathrm{~m} range in 0.019 μ m 0.019 μ m 0.019 mum0.019 \mu \mathrm{~m} steps, the data shows that the Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} between all adjacent modes is no less than 1.8 × 10 − 4 1.8 × 10 − 4 1.8 xx10^(-4)1.8 \times 10^{-4} , as shown in Fig. 5(b) ( LP 11 a x LP 11 a x LP_(11a)^(x)\mathrm{LP}_{11 \mathrm{a}}^{x} and LP 11 b x LP 11 b x LP_(11b)^(x)\mathrm{LP}_{11 \mathrm{~b}}^{x} and LP 11 b y LP 11 b y LP_(11b)^(y)\mathrm{LP}_{11 \mathrm{~b}}^{y} and LP 21 a x LP 21 a x LP_(21a)^(x)\mathrm{LP}_{21 \mathrm{a}}^{x} and LP 21 a x LP 21 a x LP_(21a)^(x)\mathrm{LP}_{21 \mathrm{a}}^{x} and LP 21 a x LP 21 a x LP_(21a)^(x)\mathrm{LP}_{21 \mathrm{a}}^{x} and LP 21 a x LP 21 a x LP_(21a)^(x)\mathrm{LP}_{21 \mathrm{a}}^{x} and LP 21 a x LP 21 a x LP_(21a)^(x)\mathrm{LP}_{21 \mathrm{a}}^{x}). b14> between Δ n eff Δ n eff Deltan_(eff)\Delta n_{\mathrm{eff}} values are much larger than 10 − 4 10 − 4 10^(-4)10^{-4} ). This suggests that the radius of the stress zone can be chosen to be in the range of 5. 04 ∼ 5.6 μ m 04 ∼ 5.6 μ m 04∼5.6 mum04 \sim 5.6 \mu \mathrm{~m} .

The fiber fabrication process may cause variations in the dimensions of the structure, and as mentioned above, the fiber structure is allowed to have a certain amount of error in the process under the condition that the difference in effective refractive indices of adjacent modes is greater than 10 − 4 10 − 4 10^(-4)10^{-4} .
(a)



→ − LP 01 y − LP 01 x ⊸ LP 01 x − LP 11 a y → LP 21 a x − LP 21 a y → − LP 01 y − LP 01 x ⊸ LP 01 x − LP 11 a y → LP 21 a x − LP 21 a y rarr-LP_(01)^(y)-LP_(01)^(x)quad⊸LP_(01)^(x)-LP_(11a)^(y)quad rarrLP_(21a)^(x)-LP_(21a)^(y)\rightarrow-\mathrm{LP}_{01}^{y}-\mathrm{LP}_{01}^{x} \quad \multimap \mathrm{LP}_{01}^{x}-\mathrm{LP}_{11 \mathrm{a}}^{y} \quad \rightarrow \mathrm{LP}_{21 \mathrm{a}}^{x}-\mathrm{LP}_{21 \mathrm{a}}^{y}
∽ − LP 11 a y − LP 11 a x ⊸ LP 21 a y − LP 21 b x ⊸ ⋆ LP 21 b x − LP 21 b y ∽ − LP 11 a y − LP 11 a x ⊸ LP 21 a y − LP 21 b x ⊸ ⋆ LP 21 b x − LP 21 b y ∽-LP_(11a)^(y)-LP_(11a)^(x)⊸LP_(21a)^(y)-LP_(21b)^(x)⊸***LP_(21b)^(x)-LP_(21b)^(y)\backsim-\mathrm{LP}_{11 \mathrm{a}}^{y}-\mathrm{LP}_{11 \mathrm{a}}^{x} \multimap \mathrm{LP}_{21 \mathrm{a}}^{y}-\mathrm{LP}_{21 \mathrm{~b}}^{x} \multimap \star \mathrm{LP}_{21 \mathrm{~b}}^{x}-\mathrm{LP}_{21 \mathrm{~b}}^{y}

Fig. 4 Relationship between Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} and λ λ lambda\lambda between modes n eff n eff  n_("eff ")n_{\text {eff }} and neighboring modes. (a) variation of n eff n eff  n_("eff ")n_{\text {eff }} with λ λ lambda\lambda in the absence of concentric stress zones; (b) variation of Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} with λ λ lambda\lambda in the presence of concentric stress zones n eff 随 λ 的变化；（c）无 n eff 随  λ  的变化；（c）无  n_("eff 随 "lambda" 的变化；（c）无 ")n_{\text {eff 随 } \lambda \text { 的变化；（c）无 }}随的变化；（）无 with concentric stress zones; ( d) variation of Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} with λ λ lambda\lambda in the presence of concentric stress zones

Fig. 4 Effective refractive index n eff n eff  n_("eff ")n_{\text {eff }} and effective refractive index difference Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} as a function of λ λ lambda\lambda . (a) Variation of n eff n eff  n_("eff ")n_{\text {eff }} with λ λ lambda\lambda without concentric-circular stress-applying region; (b) variation of n eff n eff  n_("eff ")n_{\text {eff }} with λ λ lambda\lambda with (b) variation of n eff n eff  n_("eff ")n_{\text {eff }} with λ λ lambda\lambda with concentric-circular stress-applying region; (c) variation of n eff n eff  n_("eff ")n_{\text {eff }} with λ λ lambda\lambda with concentric-circular stress-applying region; (d) variation of Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} with λ λ lambda\lambda with concentric-circular stressapplying region

Fig. 5 Relationship between modes n eff n eff n_(eff)n_{\mathrm{eff}} and neighboring modes Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} and R R RR at 1550 nm. (a) variation of n eff n eff n_(eff)n_{\mathrm{eff}} with R R RR ; (b) variation of Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} with R R RR

Fig. 5 At 1550 nm , effective refractive index n eff n eff  n_("eff ")n_{\text {eff }} and effective refractive index difference Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} as a function of R R RR . (a) Variation of n eff n eff  n_("eff ")n_{\text {eff }} with R R RR ; (b) Variation of Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} with R R RR

  4 Results and discussion


The structural dimensions of each part of the fiber are chosen as r = 20 μ m 、 Λ = r = 20 μ m 、 Λ = r=20 mum、Lambda=r=20 \mu \mathrm{~m} 、 \Lambda=、 26.05 μ m 、 a x = 3.39 μ m 、 a y = 2.42 μ m 、 b x = 5.04 μ m 26.05 μ m 、 a x = 3.39 μ m 、 a y = 2.42 μ m 、 b x = 5.04 μ m 26.05 mum、a_(x)=3.39 mum、a_(y)=2.42 mum、b_(x)=5.04 mum26.05 \mu \mathrm{~m} 、 a_{x}=3.39 \mu \mathrm{~m} 、 a_{y}=2.42 \mu \mathrm{~m} 、 b_{x}=5.04 \mu \mathrm{~m}、、、 , b y = 3.37 μ m b y = 3.37 μ m b_(y)=3.37 mumb_{y}=3.37 \mu \mathrm{~m} and R = 5.12 μ m R = 5.12 μ m R=5.12 mumR=5.12 \mu \mathrm{~m} . Mode analysis is performed at 1550 nm, and the simulation data show that 10 modes ( n eff n eff n_(eff)n_{\mathrm{eff}} of LP 01 y 、 LP 01 x 、 LP 11 v y LP 01 y 、 LP 01 x 、 LP 11 v y LP_(01)^(y)、LP_(01)^(x)、LP_(11v)^(y)\mathrm{LP}_{01}^{y} 、 \mathrm{LP}_{01}^{x} 、 \mathrm{LP}_{11 \mathrm{v}}^{y}、、 , LP 11 a x 、 LP 11 b x 、 LP 11 b y 、 LP 21 a x 、 LP 21 a y 、 LP 21 b x 、 LP 21 b y ) LP 11 a x 、 LP 11 b x 、 LP 11 b y 、 LP 21 a x 、 LP 21 a y 、 LP 21 b x 、 LP 21 b y {:LP_(11a)^(x)、LP_(11b)^(x)、LP_(11b)^(y)、LP_(21a)^(x)、LP_(21a)^(y)、LP_(21b)^(x)、LP_(21b)^(y))\left.\mathrm{LP}_{11 \mathrm{a}}^{x} 、 \mathrm{LP}_{11 \mathrm{~b}}^{x} 、 \mathrm{LP}_{11 \mathrm{~b}}^{y} 、 \mathrm{LP}_{21 \mathrm{a}}^{x} 、 \mathrm{LP}_{21 \mathrm{a}}^{y} 、 \mathrm{LP}_{21 \mathrm{~b}}^{x} 、 \mathrm{LP}_{21 \mathrm{~b}}^{y}\right)、、、、、、 ) satisfy the condition of axial transmission along the fiber, which are arranged in the descending order of n eff n eff  n_("eff ")n_{\text {eff }} to get the mode electric field diagrams shown in Fig. 6. The upward arrows in the mode field diagram show that the modes are well polarized. In addition, the mode effective area is also a key factor in the optical inter

The important index of the continuous transmission system. Under the above structural parameters, the mode effective area ( A eff ) A eff  {:A_("eff "))\left.A_{\text {eff }}\right) ) of the concentric stress zone-assisted fiber is kept in the range of 30 ∼ 30 ∼ 30∼30 \sim 50 mm 2 50 mm 2 50mm^(2)50 \mathrm{~mm}^{2} , which suppresses the nonlinear effect of the fiber to a certain extent, and the mode effective area can be further improved by increasing the core area. Table 1 contains detailed values of the effective refractive indices ( n eff ′ n eff  ′ (n_("eff ")^('):}\left(n_{\text {eff }}^{\prime}\right. and n eff ′ ′ ) n eff  ′ ′ {:n_("eff ")^(''))\left.n_{\text {eff }}^{\prime \prime}\right) for each mode and the effective refractive index difference between neighboring modes ( Δ n eff ′ Δ n eff  ′ Deltan_("eff ")^(')\Delta n_{\text {eff }}^{\prime} and Δ n eff ′ ′ Δ n eff  ′ ′ Deltan_("eff ")^('')\Delta n_{\text {eff }}^{\prime \prime} ), both for the case of A eff A eff  A_("eff ")A_{\text {eff }} and for the case of ( n eff ′ n eff  ′ (n_("eff ")^('):}\left(n_{\text {eff }}^{\prime}\right. and n eff ′ ′ ) n eff  ′ ′ {:n_("eff ")^(''))\left.n_{\text {eff }}^{\prime \prime}\right) with and without the concentric circular shaped stress region. A comparison of the data reveals that the difference between the neighboring modes of the LP 21 LP 21 LP_(21)\mathrm{LP}_{21} mode group after the introduction of the concentric circular-shaped stress zone

The analyses match, with Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} no less than 2 × 10 − 4 2 × 10 − 4 2xx10^(-4)2 \times 10^{-4} between neighboring modes.


(i) LP 21 b x LP 21 b x LP_(21b)^(x)\mathrm{LP}_{21 \mathrm{~b}}^{x} ; (j) LP P 21 b y P 21 b y P_(21b)^(y)\mathrm{P}_{21 \mathrm{~b}}^{y}

Fig. 6 Transverse electrical fields, amplitudes, and polarization directions of each mode at 1550 nm . (a) LP 01 y LP 01 y LP_(01)^(y)\mathrm{LP}_{01}^{y} ; (b) LP 01 x LP 01 x LP_(01)^(x)\mathrm{LP}_{01}^{x} ; © LP 11 y y LP 11 y y LP_(11y)^(y)\mathrm{LP}_{11 \mathrm{y}}^{y} ; (d) LP 11 a x LP 11 a x LP_(11a)^(x)\mathrm{LP}_{11 \mathrm{a}}^{x} ; (e) LP 11 b x LP 11 b x LP_(11b)^(x)\mathrm{LP}_{11 \mathrm{~b}}^{x} ; (f) LP 11 b y LP 11 b y LP_(11b)^(y)\mathrm{LP}_{11 \mathrm{~b}}^{y} ; (g) LP 21 a x LP 21 a x LP_(21a)^(x)\mathrm{LP}_{21 \mathrm{a}}^{x} ; (h) LP 21 a y LP 21 a y LP_(21a)^(y)\mathrm{LP}_{21 \mathrm{a}}^{y} ; (i) LP 21 b x ; LP 21 b x ; LP_(21b)^(x);\mathrm{LP}_{21 \mathrm{~b}}^{x} ; (j) LP 21 b y LP 21 b y LP_(21b)^(y)\mathrm{LP}_{21 \mathrm{~b}}^{y} .

Table 1 Specific Data for n eff 、 ′ n eff 、 ′ ′ 、 Δ n eff ′ 、 Δ n eff ′ ′ n eff  、 ′ n eff  、 ′ ′ 、 Δ n eff  ′ 、 Δ n eff  ′ ′ n_("eff "、)^(')n_("eff "、)^('')、Deltan_("eff ")^(')、Deltan_("eff ")^('')n_{\text {eff } 、}^{\prime} n_{\text {eff } 、}^{\prime \prime} 、 \Delta n_{\text {eff }}^{\prime} 、 \Delta n_{\text {eff }}^{\prime \prime}、、、、 and A eff A eff  A_("eff ")A_{\text {eff }}
Table 1 Data of n eff ′ , n eff ′ ′ , Δ n eff ′ , Δ n eff ′ ′ n eff  ′ , n eff  ′ ′ , Δ n eff  ′ , Δ n eff  ′ ′ n_("eff ")^('),n_("eff ")^(''),Deltan_("eff ")^('),Deltan_("eff ")^('')n_{\text {eff }}^{\prime}, n_{\text {eff }}^{\prime \prime}, \Delta n_{\text {eff }}^{\prime}, \Delta n_{\text {eff }}^{\prime \prime}, and A eff A eff  A_("eff ")A_{\text {eff }}
Mode n eff ′ n eff  ′ n_("eff ")^(')n_{\text {eff }}^{\prime} n eff ′ ′ n eff  ′ ′ n_("eff ")^('')n_{\text {eff }}^{\prime \prime}
Δ n eff ′ / Δ n eff  ′ / Deltan_("eff ")^(')//\Delta n_{\text {eff }}^{\prime} /
10 − 4 10 − 4 10^(-4)10^{-4}
Deltan_("eff ")^(')// 10^(-4)| $\Delta n_{\text {eff }}^{\prime} /$ | | :---: | | $10^{-4}$ |
Δ n eff ′ ′ / Δ n eff  ′ ′ / Deltan_("eff ")^('')//\Delta n_{\text {eff }}^{\prime \prime} /
10 − 4 10 − 4 10^(-4)10^{-4}
Deltan_("eff ")^('')// 10^(-4)| $\Delta n_{\text {eff }}^{\prime \prime} /$ | | :---: | | $10^{-4}$ |
A eff / A eff  / A_("eff ")//A_{\text {eff }} /
μ m 2 μ m 2 mum^(2)\mu \mathrm{m}^{2}
A_("eff ")// mum^(2)| $A_{\text {eff }} /$ | | :---: | | $\mu \mathrm{m}^{2}$ |
LP 01 y LP 01 y LP_(01)^(y)\mathrm{LP}_{01}^{y} 1.45512 1.45669 2.0 1.7 37
LP 01 x LP 01 x LP_(01)^(x)\mathrm{LP}_{01}^{x} 1.45492 1.45652 4.3 6.0 41
LP 11 a y LP 11 a y LP_(11a)^(y)\mathrm{LP}_{11 \mathrm{a}}^{y} 1.45449 1.45592 3.3 3.2 30
LP 11 a x LP 11 a x LP_(11a)^(x)\mathrm{LP}_{11 \mathrm{a}}^{x} 1.45416 1.45560 57.0 38.6 31
LP 11 b x LP 11 b x LP_(11b)^(x)\mathrm{LP}_{11 \mathrm{~b}}^{x} 1.44846 1.45174 4.7 3.2 46
LP 11 b y LP 11 b y LP_(11b)^(y)\mathrm{LP}_{11 \mathrm{~b}}^{y} 1.44799 1.45142 28.1 34.6 50
LP 21 a x LP 21 a x LP_(21a)^(x)\mathrm{LP}_{21 \mathrm{a}}^{x} 1.44518 1.44796 2.0 0.3 42
LP 21 a y LP 21 a y LP_(21a)^(y)\mathrm{LP}_{21 \mathrm{a}}^{y} 1.44498 1.44793 2.2 0.2 46
LP 21 b x LP 21 b x LP_(21b)^(x)\mathrm{LP}_{21 \mathrm{~b}}^{x} 1.44476 1.44791 2.0 0.9 41
LP 21 b y LP 21 b y LP_(21b)^(y)\mathrm{LP}_{21 \mathrm{~b}}^{y} 1.44456 1.44782 - - 42
Mode n_("eff ")^(') n_("eff ")^('') "Deltan_("eff ")^(')// 10^(-4)" "Deltan_("eff ")^('')// 10^(-4)" "A_("eff ")// mum^(2)" LP_(01)^(y) 1.45512 1.45669 2.0 1.7 37 LP_(01)^(x) 1.45492 1.45652 4.3 6.0 41 LP_(11a)^(y) 1.45449 1.45592 3.3 3.2 30 LP_(11a)^(x) 1.45416 1.45560 57.0 38.6 31 LP_(11b)^(x) 1.44846 1.45174 4.7 3.2 46 LP_(11b)^(y) 1.44799 1.45142 28.1 34.6 50 LP_(21a)^(x) 1.44518 1.44796 2.0 0.3 42 LP_(21a)^(y) 1.44498 1.44793 2.2 0.2 46 LP_(21b)^(x) 1.44476 1.44791 2.0 0.9 41 LP_(21b)^(y) 1.44456 1.44782 - - 42| Mode | $n_{\text {eff }}^{\prime}$ | $n_{\text {eff }}^{\prime \prime}$ | $\Delta n_{\text {eff }}^{\prime} /$ <br> $10^{-4}$ | $\Delta n_{\text {eff }}^{\prime \prime} /$ <br> $10^{-4}$ | $A_{\text {eff }} /$ <br> $\mu \mathrm{m}^{2}$ | | :---: | :---: | :---: | :---: | :---: | :---: | | $\mathrm{LP}_{01}^{y}$ | 1.45512 | 1.45669 | 2.0 | 1.7 | 37 | | $\mathrm{LP}_{01}^{x}$ | 1.45492 | 1.45652 | 4.3 | 6.0 | 41 | | $\mathrm{LP}_{11 \mathrm{a}}^{y}$ | 1.45449 | 1.45592 | 3.3 | 3.2 | 30 | | $\mathrm{LP}_{11 \mathrm{a}}^{x}$ | 1.45416 | 1.45560 | 57.0 | 38.6 | 31 | | $\mathrm{LP}_{11 \mathrm{~b}}^{x}$ | 1.44846 | 1.45174 | 4.7 | 3.2 | 46 | | $\mathrm{LP}_{11 \mathrm{~b}}^{y}$ | 1.44799 | 1.45142 | 28.1 | 34.6 | 50 | | $\mathrm{LP}_{21 \mathrm{a}}^{x}$ | 1.44518 | 1.44796 | 2.0 | 0.3 | 42 | | $\mathrm{LP}_{21 \mathrm{a}}^{y}$ | 1.44498 | 1.44793 | 2.2 | 0.2 | 46 | | $\mathrm{LP}_{21 \mathrm{~b}}^{x}$ | 1.44476 | 1.44791 | 2.0 | 0.9 | 41 | | $\mathrm{LP}_{21 \mathrm{~b}}^{y}$ | 1.44456 | 1.44782 | - | - | 42 |

After setting the above parameters, the effect of wavelength on the modes is investigated in the 1530 ∼ 1570 nm 1530 ∼ 1570 nm 1530∼1570nm1530 \sim 1570 \mathrm{~nm} wavelength range. Fig. 4 (b) shows the corresponding curves of mode n eff n eff  n_("eff ")n_{\text {eff }} as a function of wavelength, and it can be seen that the variation of each mode n eff n eff  n_("eff ")n_{\text {eff }} is smaller than that of 0.1 % 0.1 % 0.1%0.1 \% , and the variation trend is relatively flat. The wavelength-dependent curve of Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} between neighboring modes is shown in Fig. 4 (d), and the minimum Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} remains not lower than 1.8 × 10 − 4 1.8 × 10 − 4 1.8 xx10^(-4)1.8 \times 10^{-4} in this wavelength range. It can be seen that the fiber characteristics are stable with respect to the wavelength. At the wavelength of 1550 nm, Eqs. (2) and (3) are applied to analyze the bending characteristics of the optical fiber, and the simulation results show that

It is shown that when the bending radius along the x x xx and y y yy axes is not less than 9.5 cm, none of the 10 modes leaks into the cladding, and the bending-induced loss is small (up to the order of 10 − 7 dB / m 10 − 7 dB / m 10^(-7)dB//m10^{-7} \mathrm{~dB} / \mathrm{m} ). It can be seen that the introduction of concentric stress zones ensures the bending resistance of the fiber while effectively separating the modes. The dispersion characteristics of the fiber are shown in Fig. 7, and the value of mode dispersion (D) does not exceed |-55.0219| ps/(nm - km) in the wavelength range of 1530-1570 nm, and the mode dispersion can be further reduced by increasing the length of the half-short axis of the fiber core, etc., in order to make the fiber more suitable for transmission applications.

Fig. 7 Dispersion of 10 modes in C-band
Fig. 7 Dispersions of 10 modes in the C-band

On the basis of the proposed fiber structure, the concentric stress zones are replaced by square, elliptical, and rectangular stress zones, respectively. Through mode analysis, it is found that the number of modes never reaches 10 when the square stress region is introduced; when the elliptical stress region is introduced, the number of modes transmitted by the fiber in the C-band can be maintained at 10 only if the ellipticity is less than 1.006, when the structure is close to a circle; when the rectangular stress region is introduced, the number of modes in the

The minimum Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} at 1550 nm cannot be reached 2 × 10 − 4 2 × 10 − 4 2xx10^(-4)2 \times 10^{-4} . Therefore, the best performance of the proposed structure is obtained when concentric circular stress zones are introduced.

Table 2 lists the performance of few-mode fibers reported in recent years. It is found that the minimum Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} of the designed fiber at 1550 nm is significantly improved, and the minimum Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} at C-band is better than most of the structures, which has a great potential for the application in the short-haul high-capacity communication system.

  Table 2 Research data on few-mode fibers reported in recent years

Table 2 Research data of few-mode fiber reported in recent years
Ref. Mode number

Min. Δ n eff / 10 − 4 Δ n eff  / 10 − 4 Deltan_("eff ")//10^(-4)\Delta n_{\text {eff }} / 10^{-4} ( ( (( at 1550 nm ) ) ))
Min. Deltan_("eff ")//10^(-4) ( at 1550 nm)| Min. $\Delta n_{\text {eff }} / 10^{-4}$ | | :---: | | $($ at 1550 nm$)$ |

Min. Δ n eff / 10 − 4 Δ n eff  / 10 − 4 Deltan_("eff ")//10^(-4)\Delta n_{\text {eff }} / 10^{-4} (in wide band)
Min. Deltan_("eff ")//10^(-4) (in wide band)| Min. $\Delta n_{\text {eff }} / 10^{-4}$ | | :---: | | (in wide band) |

Dispersion / / /// ( ps ⋅ nm − 1 ⋅ km − 1 ) ps ⋅ nm − 1 ⋅ km − 1 (ps*nm^(-1)*km^(-1))\left(\mathrm{ps} \cdot \mathrm{nm}^{-1} \cdot \mathrm{~km}^{-1}\right)
Dispersion // (ps*nm^(-1)*km^(-1))| Dispersion $/$ | | :---: | | $\left(\mathrm{ps} \cdot \mathrm{nm}^{-1} \cdot \mathrm{~km}^{-1}\right)$ |
 Max. L b / ( dB ⋅ m − 1 ) L b / dB ⋅ m − 1 L_(b)//(dB*m^(-1))L_{\mathrm{b}} /\left(\mathrm{dB} \cdot \mathrm{m}^{-1}\right)
[ 10 ] [ 10 ] [10][10] 8 1.19    1.12 ( 1.12 ( 1.12(1.12( at C-band) -
10 − 5 ( 10 − 5 ( 10^(-5)(10^{-5}( at 1 cm ) ) ))
[ 14 ] [ 14 ] [14][14] 16 1    1 ( 1 ( 1(1( at C-band) ⩽ | 25 | ⩽ | 25 | <= |25|\leqslant|25|
1 ( 1 ( 1(1( at 8 mm ) ) ))
[ 15 ] [ 15 ] [15][15] 10 -    1.32 ( 1.32 ( 1.32(1.32( at C+L-band) < | 60 | < | 60 | < |60|<|60| -
[ 16 ] [ 16 ] [16][16] 10 1.65    1.52 ( 1.52 ( 1.52(1.52( at 1510 − 1630 nm ) 1510 − 1630 nm ) 1510-1630nm)1510-1630 \mathrm{~nm}) ⩽ | − 36 | ⩽ | − 36 | <= |-36|\leqslant|-36|
10 − 10 ( 10 − 10 ( 10^(-10)(10^{-10}( at 5 cm ) ) ))
[ 17 ] [ 17 ] [17][17] 10 1.93    1.8 ( 1.8 ( 1.8(1.8( at 1510 − 1630 nm ) 1510 − 1630 nm ) 1510-1630nm)1510-1630 \mathrm{~nm}) < | − 70 | < | − 70 | < |-70|<|-70|
10 − 8 ( 10 − 8 ( 10^(-8)(10^{-8}( at 3 cm ) ) ))
[ 18 ] [ 18 ] [18][18] 14 1.3    1.25 ( 1.25 ( 1.25(1.25( at 1520 − 1580 nm ) 1520 − 1580 nm ) 1520-1580nm)1520-1580 \mathrm{~nm}) - -
This paper 10 2
1.8 ( 1.8 ( 1.8(1.8( at C-band ) ) )) 		< | − 56 | < | − 56 | < |-56|<|-56|
10 − 7 ( 10 − 7 ( 10^(-7)(10^{-7}( at 9.5 cm ) ) ))
Ref. Mode number "Min. Deltan_("eff ")//10^(-4) ( at 1550 nm)" "Min. Deltan_("eff ")//10^(-4) (in wide band)" "Dispersion // (ps*nm^(-1)*km^(-1))" Max. L_(b)//(dB*m^(-1)) [10] 8 1.19 1.12( at C-band) - 10^(-5)( at 1 cm) [14] 16 1 1( at C-band) <= |25| 1( at 8 mm) [15] 10 - 1.32( at C+L-band) < |60| - [16] 10 1.65 1.52( at 1510-1630nm) <= |-36| 10^(-10)( at 5 cm) [17] 10 1.93 1.8( at 1510-1630nm) < |-70| 10^(-8)( at 3 cm) [18] 14 1.3 1.25( at 1520-1580nm) - - This paper 10 2 1.8( at C-band ) < |-56| 10^(-7)( at 9.5 cm)| Ref. | Mode number | Min. $\Delta n_{\text {eff }} / 10^{-4}$ <br> $($ at 1550 nm$)$ | Min. $\Delta n_{\text {eff }} / 10^{-4}$ <br> (in wide band) | Dispersion $/$ <br> $\left(\mathrm{ps} \cdot \mathrm{nm}^{-1} \cdot \mathrm{~km}^{-1}\right)$ | Max. $L_{\mathrm{b}} /\left(\mathrm{dB} \cdot \mathrm{m}^{-1}\right)$ | | :---: | :---: | :---: | :---: | :---: | :---: | | $[10]$ | 8 | 1.19 | $1.12($ at C-band) | - | $10^{-5}($ at 1 cm$)$ | | $[14]$ | 16 | 1 | $1($ at C-band) | $\leqslant\|25\|$ | $1($ at 8 mm$)$ | | $[15]$ | 10 | - | $1.32($ at C+L-band) | $<\|60\|$ | - | | $[16]$ | 10 | 1.65 | $1.52($ at $1510-1630 \mathrm{~nm})$ | $\leqslant\|-36\|$ | $10^{-10}($ at 5 cm$)$ | | $[17]$ | 10 | 1.93 | $1.8($ at $1510-1630 \mathrm{~nm})$ | $<\|-70\|$ | $10^{-8}($ at 3 cm$)$ | | $[18]$ | 14 | 1.3 | $1.25($ at $1520-1580 \mathrm{~nm})$ | - | - | | This paper | 10 | 2 | $1.8($ at C-band $)$ | $<\|-56\|$ | $10^{-7}($ at 9.5 cm$)$ |

  5 CONCLUSIONS


A concentric shaped stress region-assisted panda-type bias-preserving few-mode fiber is proposed, and the effects of elliptical ring core size, concentric shaped stress region size on modes n eff n eff  n_("eff ")n_{\text {eff }} and Δ n eff Δ n eff  Deltan_("eff ")\Delta n_{\text {eff }} are investigated. The results show that by optimizing the parameters, the structure can effectively separate the simple and merged modes transmitted in the fiber. The changes in mode properties before and after the introduction of the concentric circular-shaped stress region are analyzed, and it is demonstrated that the low-refractive-index stress region has the effect of significantly increasing the effective refractive-index difference between higher-order modes. The bending resistance of the fiber is better, and at the same time, the mode dispersion is smaller. In addition to this, by changing the core and the dimensions of each part, this fiber structure is expected to further increase the number of guiding modes, and the elements of concentric circular shaped stress region are also applicable to other fiber structure designs. This study has an important application value in future optical interconnect transmission systems and provides new ideas for the development and design of bias-preserving few-mode fibers.

  References


[1] Tu Jiajing, Li Zhaohui. A review of space-division multiplexing optical fiber research[J]. Journal of Optics,2021, 41(1): 0106003.
Tu J J, Li Z H. Review of space division multiplexing fibers[J]. Acta Optica Sinica, 2021, 41(1): 0106003.
[2] Puttnam B J, Rademacher G, Luís R S. Space-division multiplexing for optical fiber communications[J]. Optica, 2021, 8 (9): 1186-1203.

［3］ Pei Li, Wang Jianshuai, Zheng Jingjing, et al. Characteristics of space-division multiplexing fiber and its application research[J]. Infrared and Laser Engineering, 2018, 47(10): 1002001.
Pei L, Wang J S, Zheng J J, et al. Research on specialty and application of space-division-multiplexing fiber[J]. Infrared and Laser Engineering, 2018, 47(10): 1002001.
[4] Parmigiani F, Jung Y, Grüner-Nielsen L, et al. Elliptical core few mode fibers for multiple-input multiple output-free space division multiplexing transmission[J]. IEEE Photonics Technology Letters, 2017, 29(21): 1764-1767.
[5] Sillard P, Bigot-Astruc M, Molin D. Few-mode fibers for mode-division-multiplexed systems[J]. Journal of Lightwave Technology, 2014, 32(16): 2824-2829.
[6] Yaman F, Bai N, Zhu B Y, et al. Long distance transmission in few-mode fibers[J]. Optics Express, 2010, 18(12): 13250-13257.

[7] Zheng Jingjing, Song Yujing, Pei Li, et al. Design and characterization of rectangular auxiliary ring-core fiber for space division multiplexing[J]. Journal of Optics, 2022, 42(16): 1606006.
Zheng J J, Song Y J, Pei L, et al. Design and characteristics of rectangular auxiliary ring-core fiber for space division multiplexing[J]. Acta Optica Sinica, 2022, 42(16): 1606006.
[8] Corsi A, Chang J H, Rusch L A, et al. Design of highly elliptical core ten-mode fiber for space division multiplexing with 2×2 MIMO[J]. IEEE Photonics Journal, 2019, 11(2): 7202010.
[9] Gao Y, Li Y L, Li X, et al. An elliptical-core few-mode fiber with low loss and low crosstalk for the MIMO-FREE applications[J]. Frontiers in Physics, 2022, 9: 796549.
[10] Wang L X, LaRochelle S. Design of eight-mode polarizationmaintaining few-mode fiber for multiple-input multiple-outputfree spatial division multiplexing[J]. Optics Letters, 2015, 40 (24): 5846-5849.
[11] Song Y J, Zheng J J, Pei L, et al. Design and characteristics of ellipse-assisted ring-core fiber for space division multiplexing[J]. Optical Fiber Technology, 2022, 72: 102984.
[12] Yan H Z, Li S Y, Xie Z Y, et al. Design of PANDA ring-core fiber with 10 polarization-maintaining modes[J]. Photonics Research, 2016, 5(1): 1-5.
[13] Du Z Y, Wang C C, Li P X, et al. Fully degeneracy-lifted PANDA few-mode fiber based on the segmented ring-core[J]. Optik, 2022, 255: 168710.
[14] Cao Y, Zhao Y L, Yu X S, et al. Design and characterization of 16-mode PANDA polarization-maintaining few-mode ring-core fiber for spatial division multiplexing[J]. Optical Engineering, 2017, 56(11): 116102.
[15] Zhao J J, Tang M, Oh K, et al. Polarization-maintaining few mode fiber composed of a central circular-hole and an ellipticalring core[J]. Photonics Research, 2017, 5(3): 261-266.
[16] Xiao H, Li H S, Jian S S. Hole-assisted polarizationmaintaining few-mode fiber[J]. Optics & Laser Technology, 2018, 107: 162-168.
[17] Xiao H, Li H S, Wu B L, et al. Elliptical hollow-core optical fibers for polarization-maintaining few-mode guidance[J]. Optical Fiber Technology, 2019, 48: 7-11.
[18] Chen Y J, Wang J P, Chen H Y, et al. Design of few-mode optical fibers with air-hole structure for MIMO-less data transmission[J]. Optical Engineering, 2020, 59(11): 116103.

[19] Dai Y, Tian F, Wang QJ, et al. Design of low differential mode delay 4-LP few-mode fiber [J]. Journal of Optics, 2021, 41(21): 2106003.
Dai Y, Tian F, Wang Y J, et al. Design of 4-LP few-mode

Concentric-Circular Stress-Applying Region-Assisted Panda PolarizationMaintaining Few-Mode Fiber

Liu Yanan, Yan Xin, Yuan Xueguang, Zhang Yang'an, Zhang Xia*State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China

Abstract


Objective The emergence of new data services and the rapid development of cloud computing have put forward urgent demands for improving transmission capacity in optical interconnection networks. The capacity of traditional single-mode fibers is difficult to enhance due to the nonlinear Shannon limit. Space division multiplexing (SDM) technology has caught extensive attention for its ability to reach maximum capacity. Few-mode fibers are a typical design using SDM, and the mode crosstalk that occurs during the few-mode fiber transmission is a major problem. The multiple-input multiple- output (MIMO) digital signaling is a major problem. output (MIMO) digital signal processing technology is usually introduced at the receiver to solve this problem. However, as the number of guided modes increases, the system complexity will grow However, as the number of guided modes increases, the system complexity will grow nonlinearly, resulting in significant power consumption. simplified by suppressing the mode coupling from the root, which is separating the adjacent eigenmodes to the maximum extent. The polarization- maintaining few-mode fibers with special features is also a good solution. maintaining few-mode fibers with special structures can improve the capacity and ensure a higher mode separation degree. However, the previous fiber designs ignore the contradiction between the lower mode separation degree and the higher mode separation degree. designs ignore the contradiction between the lower-order mode separation and the higher-order mode separation. Thus, we take this as a breakthrough point to introduce concentric-circular stress-applying region, improving this contradiction relationship and increasing the effective refractive index difference between adjacent modes. Therefore, we take this as a breakthrough point to introduce concentric-circular stress-applying region, improving this contradiction relationship and increasing the effective refractive index difference between adjacent modes. Meanwhile, the polarization characteristics, mode effective area, wavelength dependence, mode dispersion, bending resistance, and other indicators are considered to yield better transmission performance and reliability.


Methods To adapt to the development of optical interconnection networks with short-distance and large-capacity transmission, we propose a concentric -The prominent feature of this fiber is that this concentric-circular region is set around the elliptical-ring core. The prominent feature of this fiber is that this concentric-circular region is set around the elliptical-ring core. Since the core mainly determines the guided mode number, parameter optimization is Since the core mainly determines the guided mode number, parameter optimization is first performed on the semi-major axis ( b x b x b_(x)b_{x} ) and semi-minor axis ( b y ) b y (b_(y))\left(b_{y}\right) of the elliptical-ring core. Subsequently, the concentric-circular stress region with a lower refractive index is also more suitable for the elliptical-ring core. stress region with a lower refractive index is introduced between the core and the cladding to improve the effective refractive index difference between the higher-order modes and ensure that the cladding has a lower refractive index. The concentric-circular stress region with a lower refractive index is introduced between the core and the cladding to improve the effective refractive index difference between the higher-order modes and ensure the separation of fundamental modes. Comparison conducted on optical fibers with the same parameters without Concentric-circular stress regions or stress regions of other shapes indicates that the concentriccircular stress region has an excellent ability to Additionally, frequency sweeping is conducted at 1530-1570 nm to investigate the modal wavelength dependency and mode dispersion of the fiber. Finally, the beam propagation method (BPM) is adopted to simulate the fiber bending, and the bending resistance is analyzed. Our study provides ideas for the design of short-term fiber bending and the design of short-term fiber bending. study provides ideas for the design of short-distance and large-capacity optical fibers.

Results and Discussions Through the design and optimization of the fiber (Fig. 1), the results show that the proposed optical fiber can transmit 10 modes of data (Fig. 6). stably (Fig. 6). The introduction of concentric-circular stress-applying region in the structure can enhance the effective refractive index difference between The introduction of concentric-circular stress-applying region in the structure can enhance the effective refractive index difference between higher-order modes by nearly an order of magnitude, balancing the lower-order mode separation and the higher-order mode separation (Fig. 4). The minimum effective refractive index difference between adjacent modes reaches 2 × 10 − 4 2 × 10 − 4 2xx10^(-4)2 \times 10^{-4} at 1550 nm (Table 1 ) and not less than 1.8 × 10 − 4 1.8 × 10 − 4 1.8 xx10^(-4)1.8 \times 10^{-4} over the C- band. At 1530-1570 nm, mode dispersion is not higher than | − 55.0219 | ps ⋅ nm − 1 ⋅ km − 1 | − 55.0219 | ps ⋅ nm − 1 ⋅ km − 1 |-55.0219|ps*nm^(-1)*km^(-1)|-55.0219| \mathrm{ps} \cdot \mathrm{nm}^{-1} \cdot \mathrm{~km}^{-1} (Fig. 7) and can be further reduced by increasing the semi-minor axis of the fiber core to better adapt to the short-distance range. In addition, the bending resistance of the fiber is analyzed. The results indicate that when the bending radius is no less than | − 55.0219 | ps ⋅ nm − 1 ⋅ km − 1 | − 55.0219 | ps ⋅ nm − 1 ⋅ km − 1 |-55.0219|ps*nm^(-1)*km^(-1)|-55.0219| \mathrm{ps} \cdot \mathrm{nm}^{-1} \cdot \mathrm{~km}^{-1} (Fig. 7), the bending resistance of the fiber can be further reduced by the semi-minor axis of the fiber core. The results indicate that when the bending radius is no less than 9.5 cm , none of the 10 modes will be leaked into the cladding and the maximum bending-induced loss is in the order of 10 − 7 dB / m 10 − 7 dB / m 10^(-7)dB//m10^{-7} \mathrm{~dB} / \mathrm{m} .

Conclusions We put forward a panda polarization-maintaining few-mode fiber with concentric-circular stress-applying region. The effects of elliptical-ring core size and concentric-circular stress region size on the effective refractive index of 10 modes and the effective refractive index difference between adjacent modes are studied. The effects of elliptical-ring core size and concentric-circular stress region size on the effective refractive index of 10 modes and the effective refractive index difference between adjacent modes are studied. Numerical results show that by optimizing the parameters, this structure can separate the degenerate modes transmitted in the fiber. The mode characteristic changes before and after the introduction of the concentric-circular stress region are analyzed. It is proven that this low refractive index stress region can significantly improve the effective refractive index difference between higherorder modes. It is proven that this low refractive index stress region can significantly improve the effective refractive index difference between higherorder modes. The bending resistance of the fiber is sound, with small mode dispersion. In addition, the fiber structure is expected to further increase the number of guided modes by changing the size of the core and other parts. The element of the concentric-circular stress-applying region is also suitable for designing other fiber structures. The element of the concentric-circular stress-applying region is also suitable for designing other fiber structures. provide a new idea for the development and design of polarization-maintaining few-mode fibers.
Key words fiber optics; few-mode fiber; space division multiplexing; polarization-maintaining; degenerate mode

  1. Received: 2023-03-14; Revised: 2023-04-26; Accepted: 2023-05-09; Web First: 2023-06-28

    Supported by the National Natural Science Foundation of China under Grant No. 61935003 and the State Key Laboratory of Information Photonics and Optical Communication of Beijing University of Posts and Telecommunications under Grant Nos. IPOC2022ZZ01, IPOC2022ZT02, IPOC2020ZZ01.