Hollow Hole Assist Type Optical Fiber and Design Method

The hole-assisted optical fiber design addresses the challenge of Rayleigh scattering loss by optimizing the core and cladding structure with a hexagonal close-packed hole arrangement, resulting in reduced loss and improved transmission capacity across the specified wavelength range.

JP7697527B2Active Publication Date: 2025-06-24NIPPON TELEGRAPH & TELEPHONE CORP
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Patent Information

Application Number
JP2023564330
Authority / Receiving Office
JP · JP
Patent Type
Patents
Current Assignee / Owner
Filing Date
2021-12-01
Publication Date
2025-06-24
Estimated Expiration
2041-12-01

AI Technical Summary

Technical Problem

Existing optical fibers face challenges in reducing Rayleigh scattering loss, especially in pure silica core structures, which limits the transmission capacity and efficiency of optical communication systems.

Method used

The development of a hole-assisted optical fiber with a single-mode step-index core structure and a 1-3 layer hexagonal close-packed hole-assisted structure, which optimizes the refractive index distribution and hole arrangement to reduce Rayleigh scattering loss.

Benefits of technology

This design effectively reduces Rayleigh scattering loss across wavelengths from 1260 nm to 1625 nm, enhancing the transmission capacity and efficiency of optical communication systems without the need for additional dopants.

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Abstract

In order to solve the above-mentioned problem, the purpose of the present invention is to provide the structure and design method of an optical fiber with which it is possible to reduce Rayleigh scattering loss. This hole-assisted optical fiber (HAF) includes small-diameter and large-diameter HAFs capable of only single-mode propagation at wavelengths of 1260-1625 nm. The HAF is characterized by comprising: a core region having a uniform refractive index distribution; a uniform cladding region surrounding the core region; and holes disposed in first to third layers (the numbers of holes of which are 6, 18, 36, respectively) in the cladding region so as to be hexagonal close-packed circumferentially except for the core region.
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Description

Technical Field

[0001] The present disclosure relates to a low-loss hole-assisted optical fiber and a method for designing the same.

Background Art

[0002] Currently, due to the development of new information technologies, traffic in optical fiber networks is increasing rapidly. The transmission capacity limit of existing single-mode fibers (SMF) is considered to be about 100 Tbps due to the influence of non-linear optical effects and fiber fuse. Therefore, a new high-capacity communication system is required for future applications.

[0003] In order to overcome the transmission capacity limit of existing SMF and realize a further high-capacity communication system, it is effective to expand the wavelength band used and increase the multiplexing number. In long-distance transmission, it is necessary to use a wavelength band with low transmission loss, and the available wavelength band is limited compared to short-distance networks such as access networks. Therefore, by reducing the loss of wavelength bands that cannot be used for long-distance transmission, the wavelength multiplexing number can be increased and the transmission capacity can be expanded.

[0004] However, since Rayleigh scattering loss, which is a factor of optical fiber loss, is inversely proportional to the fourth power of the wavelength, there is a problem that Rayleigh scattering loss increases on the short-wavelength side and transmission cannot be performed at the same relay interval as in the long-wavelength band. In addition, in optical fiber design, dopants such as fluorine and germanium dioxide are added to silica glass to control the refractive index distribution, but the dopants cause compositional fluctuations, which are factors of Rayleigh scattering loss.

[0005] Regarding the study of optical fibers aimed at reducing Rayleigh scattering loss, optical fibers using multi-component glass, fluorine-added optical fibers, etc. have been reported (see, for example, Non-Patent Document 1 and Non-Patent Document 2).

Prior Art Documents

Non-Patent Literature

[0006]

Non-Patent Literature 1

Non-Patent Literature 2

Summary of the Invention

Problems to be Solved by the Invention

[0007] Optical fibers using multi-component glass are expected to achieve low loss, and various studies have been conducted. However, due to microcrystal precipitation during manufacturing and the inclusion of OH groups, etc., it is considered difficult to achieve low loss. Therefore, currently, in order to reduce Rayleigh scattering loss, it is considered most effective to expand the overlapping area between the pure silica region of the optical fiber and the electric field distribution.

[0008] Generally, an optical fiber is composed of a core region with a high refractive index and a cladding region with a lower refractive index compared to the core region. It is common to control the refractive index distribution by adding a dopant such as germanium dioxide to the core region to increase the refractive index and forming the cladding region with pure silica glass. On the other hand, in order to reduce losses, there is also an optical fiber in which the core region is made of pure silica glass and the refractive index is reduced by adding fluorine to the cladding region to control the refractive index distribution.

[0009] However, even in a pure silica core structure, in order to comply with the existing SMF (ITU-T Recommendation G.652), it is necessary to reduce the core radius, which causes an increase in loss due to the fluorine cladding region. That is, there has been a problem that it is difficult to reduce the transmission loss due to Rayleigh scattering loss even in an optical fiber with a pure silica core structure.

[0010] Therefore, an object of the present invention is to provide a structure of an optical fiber and a design method thereof that can reduce Rayleigh scattering loss in order to solve the above problems.

Means for Solving the Problems

[0011] In order to achieve the above object, the optical fiber according to the present invention adopts a single-mode (step-index) core structure and a structure having a 1-3 layer hole-assisted structure.

[0012] Specifically, the optical fiber according to the present invention is a hole-assisted optical fiber, a core region having a uniform refractive index distribution with a diameter of 2a, a relative refractive index difference with respect to the core region of Δ, a uniform cladding region surrounding the core region, and holes arranged in one or a plurality of layers so as to be hexagonal close-packed in the circumferential direction excluding the core region in the cladding region, and is characterized by satisfying Condition A. [Condition A] The combination of the core radius a and the radius Rin of the inscribed circle inscribed in the innermost layer of the hole layer is The ratio ΔαR of the Rayleigh scattering loss αR of the hole-assisted optical fiber calculated by changing the core radius a and the radius Rin to the Rayleigh scattering loss αRs calculated by changing the core radius a for a normal optical fiber having the same structure as the hole-assisted optical fiber except for the absence of holes is in the region to the left of ΔαR = -0.01 dB / km when plotted on a graph with the core radius a and the radius Rin as axes.

[0013] By using a hole-assisted fiber (HAF) structure in which holes are provided around the core, it is possible to satisfy the bending loss condition of the SMF without newly adding a dopant. The optical fiber according to the present invention appropriately controls the refractive index distribution and the holes, and by using a combination of a small-diameter and a large-diameter core structure and the HAF structure, Rayleigh scattering can be reduced at wavelengths from 1260 nm to 1625 nm compared to existing SMFs. [Supplementary Explanation] When germanium dioxide is added to the core and the cladding is pure quartz, by forming a small-diameter core structure, the overlapping area of the electric field distribution and the pure quartz region (cladding) can be expanded, and the Rayleigh scattering loss can be reduced. When the core is pure quartz and fluorine is added to the cladding, by forming a large-diameter core structure, the electric field distribution can be confined within the pure quartz region (core) (the overlapping area with the fluorine-added cladding is reduced), and the Rayleigh scattering loss can be reduced.

[0014] For example, in the optical fiber according to the present invention, the six holes are arranged in one layer, the relative refractive index difference Δ is 0.25% or more and 0.4% or less, and the occupancy S defined by the formula (C1) is 0.42 or more and 0.54 or less, and in the graph of the combination, A1(3.23,20.00) A2(2.25,15.23) A3(2.14,14.26) A4(2.11,12.00) A5(2.14,10.45) A6(2.36,8.00) A7(5.32, 8.00) A8(5.05, 9.16) A9(4.70, 12.65) A10(4.55, 15.16) A11(4.57, 18.26) A12(4.50, 20.00) It is characterized by being inside a polygon with the above as vertices.

Number

[0015] For example, in the optical fiber according to the present invention, 18 of the said holes are arranged in two layers, the relative refractive index difference Δ is 0.20% or more and 0.35% or less, and the occupancy ratio S defined by formula (C2) is 0.15 or more and 0.25 or less, the said combination in the said graph, B1(2.79, 17.00) B2(2.70, 16.00) B3(2.27, 14.00) B4(2.18, 12.70) B5(2.18, 12.10) B6(2.32, 10.80) B7(2.29, 9.60) B8(2.54, 8.00) B9(5.21, 8.00) B10(4.96, 9.10) B11(4.79, 10.80) B12(4.77, 12.00) B13(4.66, 12.90) B14(4.66, 14.50) B15(4.32, 17.00) It is characterized by being inside a polygon with the above as vertices.

Number

[0016] For example, in the optical fiber according to the present invention, 36 of the holes are arranged in three layers. The relative refractive index difference Δ is 0.15% or more and 0.40% or less, and the occupancy S defined by the formula (C3) is 0.04 or more and 0.18 or less. The combination is in the graph. C1(2.45,14.00) C2(1.96,11.80) C3(1.86,11.57) C4(2.16,8.00) C5(5.52,8.00) C6(5.45,8.07) C7(5.23,9.14) C8(5.18,10.86) C9(4.50,12.64) C10(4.21,14.00) It is characterized by being inside a polygon having the above as vertices.

Number

[0017] Such an optical fiber is designed as follows. The design method is as follows. For any relative refractive index difference Δ and hole occupancy S, calculate the desired bending loss condition, the desired confinement loss condition, and the desired cut-off condition when changing the core radius a and the radius Rin of the inscribed circle inscribed in the innermost layer of the hole layer. Plot the bending loss condition, the confinement loss condition, and the cut-off condition on a graph with the core radius a and the radius Rin as axes. Calculate the Rayleigh scattering loss of the hole-assisted optical fiber when the core radius a and the radius Rin are changed. Calculating the Rayleigh scattering loss when the core radius a is changed for a normal optical fiber having the same structure as the hole-assisted optical fiber except for the absence of the holes; Plotting the ratio ΔαR of the Rayleigh scattering loss of the hole-assisted optical fiber to the Rayleigh scattering loss of the normal optical fiber on the graph; Detecting, on the graph, an overlapping region where the region on the right side of the bending loss condition or the confinement loss condition, the region on the left side of the cut-off condition, and the region on the left side of any of the ratios ΔαR overlap; and Setting the core radius a and the radius Rin included in the overlapping region as design values of the hole-assisted optical fiber.

[0018] Note that the above inventions can be combined as much as possible.

Effect of the Invention

[0019] The present invention can provide a structure of an optical fiber capable of reducing Rayleigh scattering loss and a design method thereof.

Brief Description of the Drawings

[0020]

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Mode for Carrying Out the Invention

[0021] Embodiments of the present invention will be described with reference to the accompanying drawings. The embodiments described below are examples of the present invention, and the present invention is not limited to the following embodiments. In the present specification and drawings, components having the same reference numerals indicate the same components as each other.

[0022] (Embodiment 1) Design a small-diameter and large-diameter HAF in which only a single mode can propagate at wavelengths from 1260 nm to 1625 nm. The HAF has a core region 11 with a uniform refractive index distribution, a uniform cladding region 12 surrounding the core region 11, and holes 13 arranged in one to three layers (number of holes: 6, 18, 36) in a hexagonal close-packed pattern in the circumferential direction excluding the core region 11 within the cladding region 12. As an example, the structure of a two-layer HAF is shown in FIG. 1. Here, a is the core radius, Δ is the relative refractive index difference, d is the hole diameter, Rin is the radius of the inscribed circle inscribed in the hole layer, Rout is the radius of the circumscribed circle circumscribing the hole layer, and S is the hole occupancy ratio. The hole occupancy ratio S is defined by the following formula.

Equation

[0023] First, fix the relative refractive index difference Δ and the hole occupancy ratio S, and search for the core radius a and the inscribed circle radius Rin that satisfy the bending loss condition, the confinement loss condition, and the cut-off condition. The calculation is performed by using the two-dimensional finite element method to calculate the loss from the propagation constant. Then, the bending loss and the leakage loss (confinement loss) are calculated from the propagation constant. The various conditions are shown below. (Condition 1) The bending loss condition is the region where the loss of the fundamental mode at a wavelength of 1625 nm and a bending radius of 30 mm is 0.5 dB / 100 turns or less. (Condition 2) The leakage loss condition is the region where the loss of the fundamental mode at a wavelength of 1625 nm is 0.001 dB / km or less. (Condition 3) The cut-off condition is the region where the loss of the first higher-order mode at a wavelength of 1260 nm is 1 dB / m or more.

[0024] As an example in Fig. 2, the curves of the bending loss condition (solid line), leakage loss (confined loss) condition (dashed line), and cut-off condition (dotted-dashed line) at a relative refractive index difference Δ = 0.25% and a hole occupancy ratio S = 0.42 in the case of a single-layer structure with germanium dioxide added to the core and pure silica as the cladding are shown. The cut-off wavelength is in the region to the left of the dotted-dashed line, and the bending loss and the confined loss can achieve the above conditions in the regions to the right of the solid line and the dashed line, respectively, for the HAF that can be realized. Therefore, in Fig. 2, the HAF that satisfies the desired characteristics can be realized in the region surrounded by the solid line and the dotted-dashed line.

[0025] Furthermore, the thin line in Fig. 2 indicates the reduction effect ΔαR of Rayleigh scattering loss. Here, ΔαR was derived by the following procedure. The Rayleigh scattering loss αR can be empirically expressed by Equation (2) from the overlapping area of the power distribution P(x, y) and the Rayleigh scattering coefficient distribution A(x, y).

Equation

[0026] The Rayleigh scattering coefficient distributions when adding fluorine to silica and when adding germanium dioxide can be empirically expressed by Equations (3) and (4) using the refractive index distribution. [Equation 3] A(x, y)=A0(1 + 0.41|Δ|) (3) [Equation 4] A(x, y)=A0(1 + 0.44|Δ|) (4) Here, A0 is the Rayleigh scattering coefficient of silica, which is 0.8 [dB / km·μm 4It is assumed that there is no Rayleigh scattering loss in the pore region, and A = 0 is set.

[0027] At this time, let the Rayleigh scattering loss in the HAF structure of the present invention be αR, the Rayleigh scattering loss of the existing optical fiber without using the HAF structure be αRs, and ΔαR is defined by Equation 5. [Equation 5] ΔαR = αR / αRs (5) Here, the calculation wavelength of ΔαR is 1310 nm, the core radius and Δ of the existing SMF are 4.5 μm and 0.35% respectively, and αRs is 0.3073 dB / km.

[0028] Generally, since an optical fiber transmission line is used with a length of several tens of kilometers or more, if the loss per unit length can be reduced by 0.01 dB / km or more, a significant improvement in the loss-to-noise ratio can be achieved. Therefore, in the HAF of the present invention, ΔαR ≤ -0.01 dB / km is set as the structural condition (Condition 4). In FIG. 2, the shaded area in the figure is a structure that can simultaneously satisfy all of Conditions 1 to 4.

[0029] That is, the optical fiber of the present embodiment is a hole-assisted optical fiber, a core region 11 having a uniform refractive index distribution with a diameter of 2a, a relative refractive index difference with the core region 11 of Δ, a uniform cladding region 12 surrounding the core region 11, and holes 13 arranged in one or more layers so as to be hexagonal close-packed in the circumferential direction excluding the core region 11 in the cladding region 12, and is characterized by satisfying Condition A. [Condition A] The combination of the core radius a and the radius Rin of the inscribed circle inscribed in the innermost layer of the hole layer is The ratio ΔηR of the Rayleigh scattering loss ΔηR of the hole-assisted optical fiber calculated by changing the core radius a and the radius Rin to the Rayleigh scattering loss ΔηRs calculated by changing the core radius a for a normal optical fiber having the same structure as the hole-assisted optical fiber except for the absence of the holes is in the region to the left of ΔηR = -0.01 dB / km when plotted on a graph with the core radius a and the radius Rin as axes.

[0030] (Example 1) In the case of a single-layer structure in which germanium dioxide is added to the core and the cladding is made of pure quartz, when S is changed between 0.42 and 0.54 and Δ is changed between 0.25% and 0.40%, the result of similarly deriving the design region is shown in FIG. 3. A structure that realizes desired characteristics is formed in the region surrounded by the solid line in the figure. Here, when Rin is 8 μm or less, the mode field diameter is also generally limited to 8 μm or less, and the connection loss with the existing SMF becomes apparent. For this reason, in the HAF of the present invention, Rin is set to 8 μm or more.

[0031] That is, in the HAF of the present embodiment, six of the holes are arranged in a single layer (N = 6), the relative refractive index difference Δ is 0.25% or more and 0.4% or less, and the occupancy S defined by Equation (1) is 0.42 or more and 0.54 or less, the combination is in the graph, A1(3.23,20.00) A2(2.25,15.23) A3(2.14,14.26) A4(2.11,12.00) A5(2.14,10.45) A6(2.36,8.00) A7(5.32,8.00) A8(5.05,9.16) A9(4.70,12.65) A10(4.55,15.16) A11(4.57,18.26) A12(4.50,20.00) It is characterized by being inside a polygon having [a certain point] as a vertex.

[0032] Therefore, in the single-layer structure HAF of the present invention, by setting Δ to 0.25 to 0.40% and S to 0.42 to 0.54, and Rin to 8 to 20 μm and a to 2.0 to 5.5 μm within a range, it becomes possible to realize desired characteristics. Note that the upper limit value of Rin is set to 20 μm as a value that enables creation of all pores within the cladding region for all S within the above range.

[0033] (Example 2) In the case of a two-layer structure in which germanium dioxide is added to the core and the cladding is made of pure quartz, while S is changed between 0.15 and 0.25 and Δ is changed between 0.20% and 0.35%, the result of similarly deriving the design region is shown in FIG. 4. The lower limit of Rin is the same as in Example 1. The upper limit of Rin is set to 17 μm as a value that enables creation of all pores within the cladding region for all S within the above range.

[0034] That is, in the HAF of this example, 18 of the above-mentioned pores are arranged in two layers (N = 18), the relative refractive index difference Δ is 0.20% or more and 0.35% or less, and the occupancy ratio S defined by formula (1) is 0.15 or more and 0.25 or less, the combination is in the graph, B1(2.79,17.00) B2(2.70,16.00) B3(2.27,14.00) B4(2.18,12.70) B5(2.18,12.10) B6(2.32,10.80) B7(2.29,9.60) B8(2.54,8.00) B9(5.21,8.00) B10(4.96,9.10) B11(4.79,10.80) B12(4.77,12.00) B13(4.66,12.90) B14(4.66, 14.50) B15(4.32, 17.00) It is characterized by being inside a polygon having the above as vertices.

[0035] (Example 3) In the case of a three-layer structure in which germanium dioxide is added to the core and the cladding is made of pure quartz, when S is changed between 0.04 and 0.18 and Δ is changed between 0.15% and 0.4%, the results of similarly deriving the design region are shown in FIG. 5. The lower limit of Rin is the same as in Example 1. The upper limit of Rin is set to 14 μm as a value at which all the holes can be formed in the cladding region for all S within the above range.

[0036] That is, in the HAF of this example, 36 of the above holes are arranged in three layers (N = 36), the relative refractive index difference Δ is 0.15% or more and 0.40% or less, and the occupancy S defined by Equation (1) is 0.04 or more and 0.18 or less, the combination is in the above graph, C1(2.45, 14.00) C2(1.96, 11.80) C3(1.86, 11.57) C4(2.16, 8.00) C5(5.52, 8.00) C6(5.45, 8.07) C7(5.23, 9.14) C8(5.18, 10.86) C9(4.50, 12.64) C10(4.21, 14.00) It is characterized by being inside a polygon having the above as vertices.

[0037] In the above embodiments, the HAF with germanium dioxide added to the core and pure quartz as the cladding was described. However, in the HAF of the present invention, instead of adding germanium to the core, the core may be made of pure quartz and fluorine may be added to the cladding. FIGS. 6 to 8 show a comparison between the design regions when germanium dioxide is added to the core and pure quartz is used as the cladding, and the design regions when the core is made of pure quartz and fluorine is added to the cladding.

[0038] (Example 4) FIG. 6 shows the design range (a, Rin) of a HAF having a pure silica core, in which the six holes are arranged in one layer (N = 6), the relative refractive index difference Δ is 0.25% or more and 0.4% or less, and the occupancy S defined by formula (1) is 0.42 or more and 0.54 or less. D1(3.50,20.00) D2(3.38,17.48) D3(2.54,14.65) D4(2.59,14.26) D5(2.50,10.00) D6(2.36,9.10) D7(2.46,8.00) D8(5.32,8.00) D9(5.04,9.03) D10(4.75,12.26) D11(4.64,12.90) D12(4.66,14.26) D13(4.55,15.03) D14(4.55,18.39) D15(4.45,19.10) D16(4.48,20.00) That is.

[0039] (Example 5) FIG. 7 shows the design range (a, Rin) of a HAF having a pure silica core, in which the 18 holes are arranged in two layers (N = 18), the relative refractive index difference Δ is 0.20% or more and 0.35% or less, and the occupancy S defined by formula (1) is 0.15 or more and 0.25 or less. E1(2.28,17.00) E2(2.82, 15.00) E3(2.63, 14.15) E4(2.54, 10.45) E5(2.70, 8.80) E6(2.61, 8.00) E7(5.25, 8.00) E8(4.91, 9.25) E9(4.77, 11.40) E10(4.86, 12.25) E11(4.63, 13.35) E12(4.66, 14.60) E13(4.36, 17.00) It is as follows.

[0040] (Example 6) Figure 8 shows a design range (a, Rin) of a HAF that has a pure quartz core, in which 36 of the holes are arranged in two layers (N = 36), the relative refractive index difference Δ is 0.15% or more and 0.40% or less, and the occupancy ratio S defined by formula (1) is 0.04 or more and 0.18 or less. F1(2.54, 14.00) F2(2.54, 13.7) F3(2.36, 12.57) F4(2.36, 10.11) F5(2.25, 9.64) F6(2.43, 8.39) F7(2.36, 8.00) F8(5.23, 8.00) F9(5.00, 9.00) F10(4.79, 11.00) F11(4.75, 11.64) F12(4.20, 14.00) It is as follows.

[0041] (Embodiment 2) Figure 9 is a diagram for explaining the design method of the HAF of Embodiment 1. This design method is Calculating desired bending loss conditions, desired confinement loss conditions, and desired cut-off conditions when varying the core radius a and the radius Rin of the inscribed circle inscribed in the innermost layer of the hole layer at any refractive index difference Δ and hole occupancy S (step S01). Plotting the bending loss conditions, the confinement loss conditions, and the cut-off conditions on a graph with the core radius a and the radius Rin as axes (step S02). Calculating the Rayleigh scattering loss of the hole-assisted optical fiber when varying the core radius a and the radius Rin (step S03). Calculating the Rayleigh scattering loss when varying the core radius a for a normal optical fiber having the same structure as the hole-assisted optical fiber except that there are no holes (step S04). Plotting the ratio ΔαR of the Rayleigh scattering loss of the hole-assisted optical fiber to the Rayleigh scattering loss of the normal optical fiber on the graph (step S05). Detecting, on the graph, an overlapping region where the region on the right side of the bending loss condition or the confinement loss condition, the region on the left side of the cut-off condition, and the region on the left side of any of the ratios ΔαR overlap (step S06), and Setting the core radius a and the radius Rin included in the overlapping region as the design values of the hole-assisted optical fiber (step S07). It is characterized by the above.

Explanation of Signs

[0042] 11: Core region 12: Clad region 13: Hole

Claims

【Claim 1】 A method for designing a hole-assisted optical fiber, wherein the hole-assisted optical fiber has a core region with a uniform refractive index distribution having a diameter of 2a, a relative refractive index difference with respect to the core region of Δ, a uniform cladding region surrounding the core region, and holes arranged in one or more layers so as to be hexagonal close-packed in the circumferential direction excluding the core region within the cladding region, The design method includes: Calculating desired bending loss conditions, desired confinement loss conditions, and desired cut-off conditions when changing the core radius a and the radius Rin of the inscribed circle inscribed in the innermost layer of the hole layers at an arbitrary relative refractive index difference Δ and hole occupancy S; Plotting the bending loss conditions, the confinement loss conditions, and the cut-off conditions on a graph with the core radius a and the radius Rin as axes; Calculating the Rayleigh scattering loss of the hole-assisted optical fiber when changing the core radius a and the radius Rin; Calculating the Rayleigh scattering loss when changing the core radius a for a normal optical fiber having the same structure as the hole-assisted optical fiber except for the absence of the holes; Plotting the ratio ΔαR of the Rayleigh scattering loss of the hole-assisted optical fiber to the Rayleigh scattering loss of the normal optical fiber on the graph; Detecting an overlapping region on the graph where the region on the right side of the bending loss condition or the confinement loss condition, the region on the left side of the cut-off condition, and the region on the left side of an arbitrary ratio ΔαR overlap; and Setting the core radius a and the radius Rin included in the overlapping region as the design values of the hole-assisted optical fiber. A design method characterized by the above.

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