JP7270913B2 - Method and apparatus for evaluating mode group delay characteristics of optical fiber - Google Patents

Description

The present disclosure relates to methods and apparatus for evaluating modal group delay characteristics of optical fibers.

Due to the spread of various Internet services, the amount of traffic flowing through a single optical fiber is rapidly increasing year by year. On the other hand, the transmission capacity that can be propagated through optical fibers is finite, and it is predicted that single-mode fibers (SMF), which are widely used today, will not be able to cope with future increases in traffic. In order to overcome this situation, spatial multiplexing transmission systems are attracting attention, and mode multiplexing transmission systems using optical fibers capable of propagating a plurality of modes (hereinafter referred to as few-mode fibers) are being studied.

In a mode multiplex transmission system using few-mode fibers, the mode group delay characteristic is one of the important characteristics. The reason for this is that the greater the group delay difference (DMD: Differential Mode Delay) between modes, the greater the amount of calculation required for digital signal processing to restore the transmission signal. Another reason is that the smaller the DMD, the higher the generation efficiency of Four Wave Mixing (FWM) between modes. Therefore, in order to design a mode-multiplexed transmission system, it is necessary to grasp the group delay characteristics over the entire length of a few-mode fiber and the local group delay characteristics.

N. Shibata, M.; Ohashi, R. Maruyama, andN. Kuwaki, "Measurement of differential group delay and chromatic dispersion for LP01 and LP11 modes of few-mode fibers with depressed claddings," Opt. Review, vol. 22, no. 1, pp. 65-70, 2015. S. Ohno, D. Iida, K. Toge, and T. Manabe, "High-resolution measurement of differential mode delay of few-mode fiber using phase reference technique for swept-frequency interferometry," O pt. Fiber Technol. , vol. 40, pp. 56-61, 2018. M. Ohashi et al. , "Longitudinal fiber parameter measurements of two-mode fiber links by using OTDR," ECOC2014, Th. 1.4.5, 2014. A. Nakamura et al. , "Effective mode field diameter for LP11 mode and its measurement technique," Photon. Technol. Lett. , vol. 28, no. 22, pp. 2553-2556, 2016. B. Brixner, "Refractive-Index Interpolation for Fused Silica,"J. Opt. Soc. Am. , vol. 57, no. 5, pp. 674-676, 1967. I. H. Malitson, "Interspecimen Comparison of the Refractive Index of Fused Silica," J. Am. Opt. Soc. Am. , vol. 55, no. 10, pp. 1205-1209, 1965. S. Kobayashi, N.; Shibata, S.; Shibata, and T. Izawa, ``Characteristics of Optical Fibers in Infrared Wavelength Region,'' REVIEW OF THE ELECTRICAL COMMUNICATION LABORATORIES, vol. 26, pp. 453-467, 1978. D. Krumbholz, E.; Brinkmeyer, andE. -G. Neumann, "Core/cladding power distribution, propagation constant, and group delay: Simple relation for power-law graded-index fibers," J. Am. Opt. Soc. Am. , vol. 70, no. 2, pp. 179-183, 1980. D. Gloge, "Weakly Guiding Fibers," Appl. Opt. vol. 10, no. 10, pp. 2252-2258, 1971. J. D. Love and C.I. D. Hussey, "Variational approximations for higher-order modes of weakly-guiding fibers," Optical and Quantum Electronics, vol. 16, pp. 41-48, 1984.

For example, Non-Patent Document 1 and Non-Patent Document 2 disclose a method of directly evaluating the average value and distribution characteristics of group delay characteristics in a few-mode fiber. On the other hand, when designing a transmission system using SMF, usually fiber parameters are used. Therefore, in order to design a mode-multiplexed transmission system using a few-mode fiber in the same manner as a transmission system using a conventional SMF, a technique for evaluating mode group delay characteristics from fiber parameters is required.

The present disclosure has been made focusing on the above circumstances, and a mode group delay characteristic evaluation method for evaluating the mode group delay characteristic of a few mode fiber using fiber parameters obtained by a simpler method than the group delay characteristic, and It aims at providing an evaluation apparatus.

In order to achieve the above object, the mode group delay characteristic evaluation method and evaluation apparatus according to the present invention provide a relative refractive index difference and an MFD (Mode The modal group delay characteristics were evaluated from fiber parameters such as Field Diameter.

Specifically, the mode group delay characteristic evaluation method according to the present disclosure includes:
A mode group delay characteristic evaluation method for evaluating the mode group delay characteristic of an optical fiber,
Approximate solution of modal group delay characteristic of optical fiber is derived using group refractive index of core, group refractive index of cladding, normalized frequency and normalized propagation constant when the refractive index difference between core and cladding of optical fiber is small. It is characterized by having a mode group delay calculation procedure for calculating the mode group delay characteristic of the propagation mode in the optical fiber under test based on the fact that it is possible.

Specifically, the mode group delay characteristic evaluation device according to the present disclosure is
A mode group delay characteristic evaluation device for evaluating the mode group delay characteristic of an optical fiber,
Approximate solution of modal group delay characteristic of optical fiber is derived using group refractive index of core, group refractive index of cladding, normalized frequency and normalized propagation constant when the refractive index difference between core and cladding of optical fiber is small. It is characterized by comprising mode group delay calculating means for calculating the mode group delay characteristic of the propagation mode in the optical fiber under test based on the fact that it is possible.

In the present disclosure, it is possible to provide a mode group delay characteristic evaluation method and an evaluation apparatus for evaluating the mode group delay characteristic of a few-mode fiber with an inexpensive device configuration compared to a method of direct measurement.

It is a figure explaining the mode group delay characteristic evaluation apparatus of 1st Embodiment. It is a figure explaining the mode group delay characteristic evaluation apparatus of 2nd Embodiment.

Hereinafter, embodiments of the present disclosure will be described in detail with reference to the drawings. Note that the present disclosure is not limited to the embodiments shown below. These implementation examples are merely illustrative, and the present disclosure can be implemented in various modified and improved forms based on the knowledge of those skilled in the art. In addition, in this specification and the drawings, constituent elements having the same reference numerals are the same as each other.

(First embodiment)
FIG. 1 is a diagram for explaining the mode group delay characteristic evaluation apparatus of this embodiment. This mode group delay characteristic evaluation apparatus is a mode group delay characteristic evaluation apparatus for evaluating the mode group delay characteristic of an optical fiber,
fiber parameter acquisition means 11 for acquiring the mode field radius, relative refractive index difference, core refractive index, cladding refractive index, core group refractive index, cladding group refractive index, and core radius of a desired propagation mode in the optical fiber under test;
electric field distribution approximating means 12 for obtaining an approximate electric field distribution of the propagation mode using the mode field radius and a higher-order Gaussian function;
The mode field radius, the relative refractive index difference, the core refractive index, the clad refractive index, the core group refractive index, the clad group refractive index, the core radius, and the electric field distribution approximation acquired by the fiber parameter acquisition means mode group delay calculation means 13 for calculating the mode group delay characteristic of the propagation mode in the optical fiber under test using the electric field distribution obtained by the means;
characterized by comprising

Here, the fiber parameter acquisition means 11 acquires information a1 on the mode order and mode field radius of the desired propagation mode, and the acquired relative refractive index difference, core refractive index, clad refractive index, core group refractive index, clad group Information a2 on the refractive index and core radius is output. The electric field distribution approximating means 12 receives the information a1 of the mode order and the mode field radius, calculates an approximate electric field distribution of the propagation mode using them and a higher-order Gaussian function, and outputs electric field distribution information a3. . The mode group delay calculating means 13 receives the information a2 and the information a3, calculates the mode group delay using the information and the mode group delay evaluation formula, and outputs the mode group delay information a4. The electric field distribution approximating means 12 may output the information a1, and the mode group delay calculating means 13 may further receive the information a1.

This mode group delay characteristic evaluation method is a mode group delay characteristic evaluation method for evaluating the mode group delay characteristic of an optical fiber,
a fiber parameter acquisition procedure for acquiring the mode field radius, relative refractive index difference, core refractive index, cladding refractive index, core group refractive index, cladding group refractive index, and core radius of the desired propagation mode in the optical fiber under test;
an electric field distribution approximation procedure for obtaining an approximate electric field distribution of the propagating mode using the mode field radius and a higher-order Gaussian function;
The mode field radius, the relative refractive index difference, the core refractive index, the clad refractive index, the core group refractive index, the clad group refractive index, the core radius, and the electric field distribution approximation obtained in the fiber parameter acquisition procedure a mode group delay calculation procedure for calculating a mode group delay characteristic of the propagation mode in the optical fiber under test using the electric field distribution obtained in the procedure;
characterized by performing

First, the fiber parameter acquisition means 11 acquires the mode field radius, the relative refractive index difference, the core refractive index, the clad refractive index, the core group refractive index, the clad group refractive index, and the core radius of the desired propagation mode in the optical fiber under test. do.

As means for obtaining the mode field radius, for example, NFP (Refractive Near Field Pattern) method, FFP (Far Field Pattern) method, VA (Vertical Aligned) method, etc. A method of obtaining the mode field radius from the light intensity distribution can be used.

Moreover, as means for obtaining the relative refractive index difference, core refractive index, and clad refractive index, a method for obtaining the refractive index profile of an optical fiber, such as the RNFP method, can be used. Thereby, the relative refractive index difference Δ(r) at the coordinate r in the radial direction and the relative refractive index difference _Δ0 at the center of the optical fiber can be obtained.

これらの値を用いることにより、被試験光ファイバ長手方向のモード群遅延特性を取得することも可能となる。 Also, as means for obtaining the mode field radius and the relative refractive index difference, for example, the optical time domain reflectometry method described in Non-Patent Documents 3 and 4 can be used. By using such a method, it is possible to acquire the mode field radius distribution w(z) and the relative refractive index difference distribution Δ(z) in the longitudinal direction (z direction) of the optical fiber under test. Furthermore, if the optical fiber is of a step type, the core radius a(z) can be obtained from the relative refractive index difference Δ(z) and the cutoff wavelength by the following equation.

By using these values, it is also possible to acquire the mode group delay characteristics in the longitudinal direction of the optical fiber under test.

ここで、ｎ_１（λ）は波長λでのコアの屈折率、ｎ_２（λ）は波長λでのクラッドの屈折率である。 [Refractive index, group refractive index calculation procedure details]
Next, a method for calculating the group refractive index will be described.
The group refractive indices N ₁ (λ) and N ₂ (λ) of the core and clad in the optical fiber are expressed by the following equations.

where n ₁ (λ) is the refractive index of the core at wavelength λ and n ₂ (λ) is the refractive index of the cladding at wavelength λ.

In order to obtain the value of the group refractive index, the wavelength dependence of the refractive index is necessary. The wavelength dependence of the refractive index can be obtained by using Sellmeier's relational expression as shown in the following equation (3).

Formula (3) is a formula showing the physical content of the wavelength dependence of the refractive index, where Ai corresponds to the ultraviolet and infrared absorption wavelengths, and Bi corresponds to the oscillator strength. It is known that the measured values match well in a wide wavelength range including Also, as shown in Non-Patent Document 5 and Non-Patent Document 6, it is known that the value of k can be sufficiently obtained by using up to three terms. The Sellmeier coefficients Ai and Bi of glasses used in optical fibers are shown in Non-Patent Document 7.

Therefore, the group refractive indices of the core and cladding are obtained by substituting equation (3) into equation (2). That is, the relative refractive index difference Δ of the optical fiber is obtained using the above-described method of measuring the relative refractive index difference Δ, and the coefficients Ai and Bi of the equation (3) corresponding to Δ are calculated from the materials constituting the optical fiber. By determining from the values, the group refractive index of the core and clad of the optical fiber can be derived.

Ｅ_ｌｍは周方向のモード次数がｌでありかつ径方向のモード次数がｍのときの電界分布、ｒ及びθは光ファイバの中心を原点とする極座標、ｎ（ｒ）は光ファイバの中心から距離ｒでの屈折率、ｋは真空中の波数、β_ｌｍはＬＰ_ｌｍモードの伝搬定数である。 [Details of modal group delay characteristic calculation procedure]
Next, the arithmetic processing for calculating the mode group delay characteristics will be described.
The wave equation in optical fiber can be described by the following equation.

_Elm is the electric field distribution when the mode order in the circumferential direction is l and the mode order in the radial direction is m, r and θ are polar coordinates with the origin at the center of the optical fiber, and n(r) is from the center of the optical fiber. is the refractive index at distance r, k is the wave number in vacuum, and β _lm is the propagation constant of the LP _lm mode.

Here, assuming that the solution of the wave equation of equation (4) is as follows:

Substituting equation (5) into equation (4) yields the following equation.

ここで、ａはコア半径を表し、ｎ_１及びｎ_２はそれぞれコア及びクラッドの屈折率を表す。 Let the refractive index distribution of the optical fiber be as follows.

where a represents the core radius and _n1 and _n2 represent the refractive indices of the core and cladding, respectively.

The normalized propagation constant b _lm of the LP _lm mode in an optical fiber is given by

ｋは波数を、Δは比屈折率差を表し次式で定義されている。

where v represents the normalized frequency and is defined by the following equation.

k represents the wave number, and Δ represents the relative refractive index difference, which are defined by the following equation.

Multiply equation (2) by _Elm and integrate over the cross-sectional area A of the optical fiber.

Equation (12) can be rearranged as follows.

Therefore, the normalized propagation constant b _lm can be expressed as follows.

Waveguide dispersion can be expressed by the following equation using normalized frequency v and normalized propagation constant _blm (see, for example, Non-Patent Document 8).

Therefore, d(vb _lm )/dv is obtained by the following equation.

ここで、cは光速を表し、β_ｌｍはＬＰ_ｌｍモードの伝搬定数を表す。 On the other hand, the group delay τ _lm of the LP _lm mode can be written as follows.

where c represents the speed of light and β _lm represents the propagation constant of the LP _lm mode.

In addition, the normalized propagation constant _blm is defined by equation (8). Using the weakly guiding approximation that the refractive index difference is small (see, for example, Non-Patent Document 9), the propagation constant β _lm can be described as follows.

Differentiating equation (18) with respect to wavenumber k gives

Also, the following approximation is used here.

Therefore, equation (19) can be approximated as follows.

Therefore, the group delay τ _lm can be approximated as follows.

ただし、β_ｌｍはＬＰ_ｌｍモードの伝搬定数、ｋは真空中の波数、ｃは光速、Δ（ｒ）は半径方向の座標ｒの比屈折率差、Δ_０は光ファイバ中心の比屈折率差、ｖは正規化周波数、Ａは光ファイバ断面積を表す。 Then, substituting d(vb _lm )/dv calculated above into equation (22), the group delay τ _lm of the LP _lm mode can be written as:

where β _lm is the propagation constant of the LP _lm mode, k is the wave number in vacuum, c is the speed of light, Δ(r) is the relative refractive index difference of the radial coordinate r, and Δ ₀ is the relative refractive index difference at the center of the optical fiber. , v is the normalized frequency, and A is the optical fiber cross-sectional area.

Therefore, by substituting the relative refractive index differences Δ(r) and Δ ₀ , the core group refractive index N ₁ , the cladding group refractive index N ₂ , the core radius a, and the electric field distribution E _lm into the above equation (23), , LP _lm mode group delay τ _lm can be obtained.

ｒ及びθは光ファイバの中心を原点とする極座標、ｗ_ｌｍはＬＰ_ｌｍモードのモードフィールド半径、Ｌ^ｌ _ｍ－１（ｘ）はラゲールの倍多項式である。 Here, in the electric field distribution approximation procedure, the electric field distribution approximating means 12 can derive the electric field distribution _Elm using a higher-order Gaussian function (see, for example, Non-Patent Document 10).

r and θ are polar coordinates with the origin at the center of the optical fiber, w _lm is the mode field radius of the LP _lm mode, and L ^lm ₋₁ (x) is the Laguerre double polynomial.

Therefore, in the mode group delay calculation procedure, the mode group delay calculation means uses the mode field radius w _lm obtained in the fiber parameter acquisition procedure, the relative refractive index differences Δ(r) and Δ ₀ , the core refractive index n ₁ , the cladding refractive index Using n ₂ , core group refractive index N ₁ , cladding group refractive index N ₂ , core radius a, and electric field distribution E _lm obtained by the electric field distribution approximation procedure, modal group delay characteristics of propagation mode LP _lm in the optical fiber under test can be calculated. Also, the DMD can be calculated from the difference in group delay characteristics of each mode.

ここで、Ｓｚは時間平均ポインティングベクトルの軸成分である。

として、式（２５）を以下のように書き換える。

The process of deriving Equation (15) is as follows.
The following equations are obtained from equations (1) and (4) in Non-Patent Document 8.

where Sz is the axis component of the time-averaged Poynting vector.

, the equation (25) is rewritten as follows.

Consider the left side of equation (27).

By substituting equation (28) into equation (27) and arranging, the desired equation (15) is obtained.

(Second embodiment)
FIG. 2 is a diagram for explaining the mode group delay characteristic evaluation device of this embodiment. The mode group delay characteristic evaluation apparatus of this embodiment includes fiber parameter acquisition means 11 and mode group delay calculation means 13 . By combining equations (23) and (24), an analytical solution for the modal group delay can be obtained.

For example, if the optical fiber under test has a stepped refractive index profile, Δ(r)=0. Therefore, the integral term in the optical fiber cross-sectional area in Equation (23) can be expressed by the following equation.

In the case of the LP ₀₁ mode, the electric field distribution E ₀₁ is expressed by the following equation from equation (24).

For the LP ₁₁ mode, from equation (24), the electric field distribution E ₁₁ is expressed by the following equation.

Therefore, the group delay τ ₀₁ of the LP ₀₁ mode shown in Equation (23) can be expressed by the following equation. The same is true for the group delay τ ₁₁ of the LP ₁₁ mode.

By substituting the relative refractive index difference, the group index, the core radius, and the mode field radius into the above equations (32) and (33), the LP ₀₁ mode and the LP ₁₁ mode of the optical fiber under test having a stepped refractive index profile can be obtained. The mode group delay characteristics can be obtained. Also, the DMD can be calculated from the difference in group delay characteristics of each mode.

Thus, in this embodiment, the fiber parameter acquisition means 11 acquires the relative refractive index difference, group refractive index, core radius, and mode field radius. Then, the modal group delay calculator 13 obtains the group delay τ _lm of the LP _lm mode using the analytical solution of the modal group delay obtained by combining equations (23) and (24). Although examples of LP ₀₁ and LP ₁₁ modes are shown in this embodiment, the present disclosure can be applied to any values of l and m.

The process of deriving equation (32) is as follows.
For the LP ₀₁ mode, equation (29) can be written as

Substituting equation (30) into the first term of equation (34) yields

Substituting equation (30) into the second term of equation (34) yields

By substituting equations (35) and (36) into equation (34) and applying it to equation (23), equation (32) can be derived.

Also, the process of deriving Equation (33) is as follows.
For the LP ₁₁ mode, equation (29) can be written as

となるように正規化する。

Since the final result is the same whether the function in the circumferential direction is cos θ or sin θ, the calculation proceeds with the case of cos θ as an example.

Normalize so that

Substituting equation (40) into the first term of equation (39) yields

Substituting equation (40) into the second term of equation (39) yields

Substituting equation (40) into the third term of equation (37) yields

By substituting equations (41), (42) and (43) into equation (37) and applying it to equation (23), equation (33) can be derived.

It should be noted that the present invention is not limited to the above-described embodiments, and various modifications can be made without departing from the gist of the present invention.

In short, the present invention is not limited to the above-described embodiments as they are, and can be embodied by modifying constituent elements without departing from the scope of the present invention at the implementation stage. Also, various inventions can be formed by appropriate combinations of the plurality of constituent elements disclosed in the above embodiments. For example, some components may be omitted from all components shown in the embodiments. Furthermore, constituent elements across different embodiments may be combined as appropriate.

The present disclosure can be applied to the information and communications industry.

11: Fiber parameter acquisition means 12: Electric field distribution approximation means 13: Mode group delay calculation means

Claims (8)

A mode group delay characteristic evaluation method for evaluating the mode group delay characteristic of an optical fiber,
a fiber parameter acquisition procedure for acquiring the mode field radius, relative refractive index difference, core refractive index, cladding refractive index, core group refractive index, cladding group refractive index, and core radius of the desired propagation mode in the optical fiber under test;
an electric field distribution approximation procedure for obtaining an approximate electric field distribution of the propagating mode using the mode field radius and a higher-order Gaussian function;
The mode field radius, the relative refractive index difference, the core refractive index, the clad refractive index, the core group refractive index, the clad group refractive index, the core radius, and the electric field distribution approximation obtained in the fiber parameter acquisition procedure a mode group delay calculation procedure for calculating a mode group delay characteristic of the propagation mode in the optical fiber under test using the electric field distribution obtained in the procedure ;
A method for evaluating mode group delay characteristics, comprising : In the electric field distribution approximation procedure, the approximate electric field distribution of the propagation mode is obtained using the following equation,
2. A method for evaluating mode group delay characteristics according to claim 1 .

where l and m are the mode orders in the circumferential and radial directions of the optical fiber under test, respectively, w _lm is the mode field radius of the LP _lm mode in the optical fiber under test, and L ^l _m−1 (x) is Laguerre and r and θ are polar coordinates with the origin at the center of the optical fiber under test. In the mode group delay calculation procedure, the mode group delay is calculated using the following equation,
2. A method for evaluating mode group delay characteristics according to claim 1 .

where _βlm is the propagation constant of the _LPlm mode, k is the wavenumber in vacuum, c is the speed of light, Δ(r) is the relative refractive index difference of the radial coordinate r, and _Δ0 is the center of the optical fiber under test. , v is the normalized frequency, and A is the cross-sectional area of the optical fiber under test. In the fiber parameter acquisition step, the longitudinal mode field radius distribution and the relative refractive index difference distribution of the optical fiber under test are acquired from backscattered light generated when the test light is incident on the optical fiber under test; Characterized by deriving the core radius from the relative refractive index difference and the cutoff wavelength obtained in advance,
2. A method for evaluating mode group delay characteristics according to claim 1 . A mode group delay characteristic evaluation method for evaluating the mode group delay characteristic of an optical fiber,
a fiber parameter acquisition procedure for acquiring the mode field radius, relative refractive index difference, core refractive index, cladding refractive index, core group refractive index, cladding group refractive index, and core radius of the desired propagation mode in the optical fiber under test;
A mode group that obtains an analytical solution of the mode group delay by combining the electric field distribution equation represented by the number C41 and the mode group delay equation represented by the number C42, and calculates the mode group delay using the analytical solution a delay calculation procedure ;
A method for evaluating mode group delay characteristics, comprising :

where l and m are the circumferential and radial mode orders of the optical fiber under test, w _lm is the mode field radius of the LP _lm mode, L ^l _m−1 (x) is Laguerre's double polynomial, r and θ is a polar coordinate whose origin is the center of the optical fiber under test.

where _βlm is the propagation constant of the _LPlm mode, k is the wavenumber in vacuum, c is the speed of light, Δ(r) is the relative refractive index difference of the radial coordinate r, and _Δ0 is the center of the optical fiber under test. , v is the normalized frequency, and A is the cross-sectional area of the optical fiber under test. Analytical solution of the group delay τ ₀₁ of the fundamental mode (LP ₀₁ mode) and the group delay τ ₁₁ of the first higher-order mode (LP ₁₁ mode) at the wavelength λ when the optical fiber under test has a stepped refractive index profile As, characterized by being calculated using the following formula,
6. A method for evaluating mode group delay characteristics according to claim 5.

A mode group delay characteristic evaluation device for evaluating the mode group delay characteristic of an optical fiber,
a fiber parameter acquiring means for acquiring a mode field radius, a relative refractive index difference, a core refractive index, a clad refractive index, a core group refractive index, a clad group refractive index, and a core radius of a desired propagation mode in the optical fiber under test;
electric field distribution approximating means for obtaining an approximate electric field distribution of the propagation mode using the mode field radius and a higher-order Gaussian function;
The mode field radius, the relative refractive index difference, the core refractive index, the clad refractive index, the core group refractive index, the clad group refractive index, the core radius, and the electric field distribution approximation acquired by the fiber parameter acquisition means mode group delay calculation means for calculating mode group delay characteristics of the propagation mode in the optical fiber under test using the electric field distribution obtained by the means ;
A mode group delay characteristic evaluation device comprising: A mode group delay characteristic evaluation method for evaluating the mode group delay characteristic of an optical fiber,
a fiber parameter acquiring means for acquiring a mode field radius, a relative refractive index difference, a core refractive index, a clad refractive index, a core group refractive index, a clad group refractive index, and a core radius of a desired propagation mode in the optical fiber under test;
A mode group that obtains an analytical solution of the mode group delay by combining the electric field distribution equation represented by the number C51 and the mode group delay equation represented by the number C52, and calculates the mode group delay using the analytical solution a delay calculation means;
A mode group delay characteristic evaluation device comprising:

where l and m are the mode orders in the circumferential and radial directions of the optical fiber under test, respectively; _lm is LP _lm mode field radius of the mode, L ^l _m-1 (x) is Laguerre's double polynomial, and r and θ are polar coordinates with the origin at the center of the optical fiber under test.

However, β _lm is LP _lm mode propagation constant, k is the wavenumber in vacuum, c is the speed of light, Δ(r) is the relative refractive index difference of the radial coordinate r, Δ ₀ is the relative refractive index difference at the center of the optical fiber under test, v is the normalized frequency, and A is the cross-sectional area of the optical fiber under test.

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