TEST METHOD AND TEST DEVICE FOR MODE FIELD DIAMETER

Abstract

The purpose of the present disclosure is to provide a mode field diameter test method and test device that enable acquisition of a mode field diameter for an arbitrary higher-order mode. The present disclosure is a mode field diameter test method including: a test light incidence procedure for selectively causing test light to be incident in a mode subject to measurement, on one end of an optical fiber 10 under test; a far-field pattern measurement procedure for measuring a far-field pattern of the mode subject to measurement, with respect to a divergence angle θ at the other end of the optical fiber under test, by a far-field scanning technique; and a mode field diameter calculation procedure for calculating, using an equation, a mode field diameter from information about incident mode orders in the test light incidence procedure and the far-field pattern measured in the far-field pattern measurement procedure.

Claims

1. A mode field diameter test method including: a test light incidence procedure for selectively causing test light to be incident in a mode subject to measurement, on one end of an optical fiber under test; a far-field pattern measurement procedure for measuring a far-field pattern of the mode subject to measurement, with respect to a divergence angle θ at the other end of the optical fiber under test, by a far-field scanning technique; and a mode field diameter calculation procedure for calculating a mode field diameter from information about incident mode orders in the test light incidence procedure and the far-field pattern measured in the far-field pattern measurement procedure, by using Equation (C1), $\left[\mathrm{Math}.C1\right]$ $\begin{array}{cc}\mathrm{MFD}=\sqrt{v+2\mu -1}\frac{\lambda }{\pi }\sqrt{2\frac{{\int }_0^{\pi /2}{.Math.F_{v\mu }\left(\theta \right).Math.}^2\sin \theta \cos \theta d\theta }{{\int }_0^{\pi /2}{.Math.F_{v\mu }\left(\theta \right).Math.}^2{\sin }^3\mathrm{\theta cos}\theta d\theta }}& \left(\mathrm{C1}\right)\end{array}$ where ν is the order in a circumferential direction, μ is the order in a radial direction, λ represents a wavelength of the test light, θ is the divergence angle, F.sub.νμ(θ) is an electric field distribution in a far field, and |F.sub.νμ(θ)|.sup.2 represents the far-field pattern.

2. A mode field diameter test method including: a test light incidence procedure for selectively causing test light to be incident in a mode subject to measurement, on one end of an optical fiber under test; a far-field pattern measurement procedure for measuring a far-field pattern of the mode subject to measurement, with respect to a divergence angle θ at the other end of the optical fiber under test, by a far-field scanning technique; a near-field pattern calculation procedure for calculating an electric field distribution in a near field from the far-field pattern measured in the far-field pattern measurement procedure; and a mode field diameter calculation procedure for calculating a mode field diameter from information about incident mode orders in the test light incidence procedure and near-field pattern calculated in the near-field pattern calculation procedure, by using Equation (C2), $\left[\mathrm{Math}.C2\right]$ $\begin{array}{cc}\mathrm{MFD}={\frac{2\sqrt{2}}{\sqrt{v+2\mu -1}}\left[\frac{{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)r^3\mathrm{dr}}{{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)\mathrm{rdr}}\right]}^{1/2}& \left(\mathrm{C2}\right)\end{array}$ where ν is the order in a circumferential direction, μ is the order in a radial direction, r is a radial coordinate in a cross section of the optical fiber, and E.sub.νμ(r) is the electric field distribution of an LP.sub.νμ mode in the near field.

3. A mode field diameter test method including: a test light incidence procedure for selectively causing test light to be incident in a mode subject to measurement, on one end of an optical fiber under test; a far-field pattern measurement procedure for measuring a far-field pattern of the mode subject to measurement, with respect to a divergence angle θ at the other end of the optical fiber under test, by a far-field scanning technique; a near-field pattern calculation procedure for calculating an electric field distribution in a near field from the far-field pattern measured in the far-field pattern measurement procedure; and a mode field diameter calculation procedure for calculating a mode field diameter from information about incident mode orders in the test light incidence procedure and near-field pattern calculated in the near-field pattern calculation procedure, by using Equation (C3), $\left[\mathrm{Math}.C3\right]$ $\begin{array}{cc}\mathrm{MFD}=2{\sqrt{2\left(2\mu -1\right)}\left[\frac{{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)\mathrm{rdr}}{{\int }_0^{\infty }{\left\{\frac{{\mathrm{dE}}_{v\mu }\left(r\right)}{\mathrm{dr}}\right\}}^2\mathrm{rdr}}\right]}^{1/2}& \left(\mathrm{C3}\right)\end{array}$ where ν is the order in a circumferential direction, μ is the order in a radial direction, r is a radial coordinate in a cross section of the optical fiber, and E.sub.νμ(r) is the electric field distribution of an LP.sub.νμ mode in the near field.

4. A mode field diameter test device comprising: test light incidence means for selectively causing test light to be incident in a mode subject to measurement, on one end of an optical fiber under test; far-field pattern measurement means for measuring a far-field pattern of the mode subject to measurement, with respect to a divergence angle θ at the other end of the optical fiber under test, by a far-field scanning technique; and mode field diameter calculation means for calculating, using a first equation, a mode field diameter from information about incident mode orders in the optical fiber under test by the test light incidence means and the far-field pattern measured by the far-field pattern measurement procedure, or mode field diameter calculation means for calculating an electric field distribution in a near field from the far-field pattern and further calculating a mode field diameter using a second or third equation.

5. The mode field diameter test device according to claim 4, wherein the first equation is Equation (C4), $\left[\mathrm{Math}.C4\right]$ $\begin{array}{cc}\mathrm{MFD}=\sqrt{v+2\mu -1}\frac{\lambda }{\pi }\sqrt{2\frac{{\int }_0^{\pi /2}{.Math.F_{v\mu }\left(\theta \right).Math.}^2\sin \theta \cos \theta d\theta }{{\int }_0^{\pi /2}{.Math.F_{v\mu }\left(\theta \right).Math.}^2{\sin }^3\mathrm{\theta cos}\theta d\theta }}& \left(\mathrm{C4}\right)\end{array}$ where ν is the order in a circumferential direction, μ is the order in a radial direction, λ represents a wavelength of the test light, θ is the divergence angle, F.sub.νμ(θ) is an electric field distribution in a far field, and |F.sub.νμ(θ)|.sup.2 represents the far-field pattern.

6. The mode field diameter test device according to claim 4, wherein the second equation is Equation (C5), $\left[\mathrm{Math}.C5\right]$ $\begin{array}{cc}\mathrm{MFD}={\frac{2\sqrt{2}}{\sqrt{v+2\mu -1}}\left[\frac{{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)r^3\mathrm{dr}}{{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)\mathrm{rdr}}\right]}^{1/2}& \left(\mathrm{C5}\right)\end{array}$ where ν is the order in a circumferential direction, μ is the order in a radial direction, r is a radial coordinate in a cross section of the optical fiber, and E.sub.νμ(r) is the electric field distribution of an LP.sub.νμ mode in the near field.

7. The mode field diameter test device according to claim 4, wherein the second equation is Equation (C6), $\left[\mathrm{Math}.C6\right]$ $\begin{array}{cc}\mathrm{MFD}=2{\sqrt{2\left(2\mu -1\right)}\left[\frac{{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)\mathrm{rdr}}{{\int }_0^{\infty }{\left\{\frac{{\mathrm{dE}}_{v\mu }\left(r\right)}{\mathrm{dr}}\right\}}^2\mathrm{rdr}}\right]}^{1/2}& \left(\mathrm{C6}\right)\end{array}$ where ν is the order in a circumferential direction, μ is the order in a radial direction, r is a radial coordinate in a cross section of the optical fiber, and E.sub.νμ(r) is the electric field distribution of an LP.sub.νμ mode in the near field.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0024] FIG. 1 is a diagram explaining a mode field diameter test method according to the present disclosure.

[0025] FIG. 2 is a diagram explaining a mode field diameter test device according to the present disclosure.

[0026] FIG. 3 is a diagram explaining a refractive index distribution.

[0027] FIG. 4 shows electric field distributions in the near field.

[0028] FIG. 5 shows far-field patterns.

[0029] FIG. 6 shows the summarized results of calculated mode field diameters.

[0030] FIG. 7 shows a refractive index distribution.

[0031] FIG. 8 shows electric field distributions in the near field.

[0032] FIG. 9 shows far-field patterns.

[0033] FIG. 10 shows the summarized results of calculated mode field diameters.

DESCRIPTION OF EMBODIMENTS

[0034] Hereinafter, embodiments of the present disclosure will be described in detail with reference to the drawings. The present disclosure is not limited to the following embodiments. These embodiments are merely examples, and the present disclosure can be implemented in various forms modified and improved based on the knowledge of those skilled in the art. In the present description and drawings, components labeled with the same reference signs mutually represent the same components.

[0035] FIG. 1 is a process diagram explaining a mode field diameter test method according to the present embodiment. The mode field diameter test method includes:

[0036] a test light incidence procedure S01 for selectively causing test light to be incident in a mode subject to measurement, on one end of an optical fiber under test;

[0037] a far-field pattern measurement procedure S02 for measuring a far-field pattern of the mode subject to measurement, with respect to a divergence angle θ at the other end of the optical fiber under test, by a far-field scanning (FFS) technique; and

[0038] a mode field diameter calculation procedure S03 for calculating a mode field diameter, by Equation (1), from the far-field pattern measured in the far-field pattern measurement procedure.

[0039] The incidence procedure S01 includes performing:

[0040] a generation step (S01a) for generating test light having a desired wavelength;

[0041] a mode conversion step (S01b) for converting the test light generated in the generation step into a mode subject to measurement; and

[0042] an incidence step (S01c) for causing the test light, which was converted into the mode subject to measurement in the mode conversion step, to be incident on one end of the optical fiber under test.

[0043] The measurement procedure S02 includes performing a step for measuring far-field intensity of the test light emitted from the far end of the optical fiber under test with respect to the divergence angle θ, and measuring a far-field intensity distribution of the mode subject to measurement.

[0044] The mode field diameter calculation procedure S03 includes performing:

[0045] a mode order acquisition step (S03a) for acquiring orders in a circumferential direction and in a radial direction of the mode incident on the optical fiber under test in the incidence procedure; and a step (S03b) for acquiring the mode field diameter from the mode orders being incident on the optical fiber under test in the incidence procedure, and the intensity distribution measured in the measurement procedure.

[0046] The details of the calculation of the mode field diameter will be described later.

[0047] FIG. 2 is a diagram explaining a configuration example of a mode field diameter test device according to the present embodiment. The mode field diameter test device includes:

[0048] test light incidence means for selectively causing test light to be incident in a mode subject to measurement, on one end of an optical fiber under test;

[0049] far-field pattern measurement means for measuring a far-field pattern of the mode subject to measurement, with respect to a divergence angle θ at the other end of the optical fiber under test, by a far-field scanning (FFS) technique; and

[0050] arithmetic operation means for calculating a mode field diameter by substituting information about the mode orders being incident on the optical fiber under test by the test light incidence means and the far-field pattern measured by the far-field pattern measurement means into Equation (1).

[0051] The test light incidence means includes a light source 11, and a mode converter 12.

[0052] The test light output from the light source 11 is converted into a mode subject to measurement by the mode converter 12, and is then made incident on an optical fiber 10 under test. For the light source 11, any light generating means capable of generating test light having a desired wavelength can be used, and examples are a fiber laser and a semiconductor laser diode. For the mode converter 12, any means capable of converting the mode of the test light can be used, and an example is a mode multiplexer including a directional coupler composed of a planar light wave circuit as described in Non-Patent Literature 1.

[0053] The far-field pattern measurement means includes an alignment device 13, a rotation stage 14, an optical receiver 15, a control unit 16, and an A/D (analog/digital) converter 17.

[0054] The test light emitted from the far end of the optical fiber 10 under test is incident on the optical receiver 15 disposed on the rotation stage 14, and is photoelectrically converted. An electrical signal from the optical receiver 15 is converted into digital data by the A/D converter 17. At this time, by rotating the position of the optical receiver 15 around the position of the test light emission end of the optical fiber 10 under test with the use of the rotation stage 14, the far-field intensity distribution (far-field pattern) of the mode subject to measurement with respect to the emission angle is obtained. The alignment device 13 can be used to locate the position of the test light emission end of the optical fiber 10 under test, at the center of the rotation stage 14. The control unit 16 can be used for controlling the rotation angle of the rotation stage 14, that is, the position of the optical receiver 15.

[0055] The control unit 16 may output, to the mode converter 12, a signal for controlling an incident mode on the optical fiber 10 under test.

[0056] The arithmetic operation means includes a signal processing unit 18.

[0057] The signal processing unit 18 acquires the orders in the circumferential direction and in the radial direction of the mode subject to measurement from the mode converter 12 or the control unit 16. The signal processing unit 18 performs arithmetic processing for calculating the mode field diameter from the obtained mode orders and the measured far-field pattern.

[0058] The control unit 16 and the signal processing unit 18 can also be realized by a computer and a program, and the program can be recorded in a recoding medium or provided through a network.

[0059] Hereinafter, the arithmetic processing for calculating the mode field diameter will be described.

[0060] With separation of variables in the radial direction and the circumferential direction, the electric field distribution of an arbitrary linearly polarized (LP) mode in an optical fiber can be expressed by the following equation.

[00003]$\begin{array}{cc}\left[\mathrm{Math}.4\right]& \\ E_{\mathrm{\nu \mu }}\left(r,\varphi \right)=E_{\mathrm{\nu \mu }}\left(r\right)\left\{\begin{array}{c}\cos \left(\mathrm{\nu \varphi }\right)\\ \sin \left(\mathrm{\nu \varphi }\right)\end{array}\right\}& \left(4\right)\end{array}$

[0061] where E.sub.νμ represents an electric field distribution in a near field of the LP mode where ν is the order in the circumferential direction and μ is the order in the radial direction, that is, the LP.sub.νμ mode. The parameters r and φ represent the radial coordinate and the azimuth, respectively, in a cross section of the optical fiber.

[0062] Here, a radial electric field distribution E.sub.νμ(r) is approximated by the following equation using the associated Laguerre polynomials (see, for example, Equation (9) in Non-Patent Literature 2).

[00004]$\begin{array}{cc}\left[\mathrm{Math}.5\right]& \\ E_{\mathrm{\nu \mu }}\left(r\right)=\sqrt{\frac{2\left(\mu -1\right)!}{\pi \left(\nu +\mu -1\right)!}}{\left(\frac{\sqrt{2}r}{w}\right)}^{\nu }\frac{1}{w}{L}_{\mu -1}^{\nu }\left(\frac{2r^2}{w^2}\right)\exp \left(-\frac{r^2}{w^2}\right)& \left(5\right)\end{array}$

[0063] Here, 2w represents the mode field diameter that is defined as the spot size when the electric field distribution in the optical fiber is approximated by a higher-order Gaussian mode.

L.sub.μ-1.sup.ν(x)  [Math. 5-1]

is an associated Laguerre polynomial, and is expressed by the following equation.

[00005]$\begin{array}{cc}\left[\mathrm{Math}.6\right]& \\ L_{\mu -1}^{\nu }\left(x\right)=\underset{k=0}{\overset{\nu }{.Math.}}{\left(1\right)}^k\frac{\left(\nu +\mu -1\right)!}{k!\left(\nu -k\right)!\left(k+\mu -1\right)!}{x}^k& \left(6\right)\end{array}$

[0064] At this time, an electric field distribution in a far field F.sub.νμ(0) can be expressed by the following equation (see, for example, Equation (6) in Non-Patent Literature 3).

[Math. 7]

F.sub.νμ(θ)=∫.sub.0.sup.∞E.sub.νμ(r)J.sub.ν(kr sin θ)rdr  (7)

where θ represents the divergence angle from the near field to the far field, k is the wave number of light in a vacuum, and J.sub.ν is the first-class Bessel function of order ν.

[0065] Here, by substituting Equations (5) and (7) into an equation of a mode field diameter defined by the second-order moment of the far-field pattern used conventionally for calculating the mode field diameter of a single-mode fiber (see, for example, Equation (3-2) in Non-Patent Document 4), the following relational equation is obtained.

[00006]$\left[\mathrm{Math}.8\right]$$\begin{array}{cc}\frac{\lambda }{\pi }\sqrt{2\frac{{\int }_0^{\pi /2}{.Math.F_{v\mu }\left(\theta \right).Math.}^2\sin \theta \cos \theta d\theta }{{\int }_0^{\pi /2}{.Math.F_{v\mu }\left(\theta \right).Math.}^2{\sin }^3\theta \cos \theta d\theta }}=\frac{2w}{\sqrt{v+2\mu -1}}& \left(8\right)\end{array}$

where λ represents the wavelength of the test light, and |F.sub.νμ(θ)|.sup.2 represents the far-field intensity distribution.

[0066] Therefore, the mode field diameter that is defined as the spot size can be calculated by substituting a far-field intensity distribution, which was measured by fixing the angle φ in the circumferential direction at an arbitrary constant value, into Equation (9) below.

[00007]$\left[\mathrm{Math}.9\right]$$\begin{array}{cc}\mathrm{MFD}=\sqrt{v+2\mu -1}\frac{\lambda }{\pi }\sqrt{2\frac{{\int }_0^{\pi /2}{.Math.F_{v\mu }\left(\theta \right).Math.}^2\sin \theta \cos \theta d\theta }{{\int }_0^{\pi /2}{.Math.F_{v\mu }\left(\theta \right).Math.}^2{\sin }^3\mathrm{\theta cos}\theta d\theta }}& \left(9\right)\end{array}$

[0067] Moreover, by substituting Equation (5) into definition equations of Petermann I and Petermann II (see, Equation (8) in Non-Patent Document 5 and Equation (3) in Non-Patent Document 6, respectively) used conventionally for calculating the mode field diameter of a single-mode fiber from the electric field distribution in the near field, the following relational equations are obtained.

[00008]$\left[\mathrm{Math}.10\right]$$\begin{array}{cc}2{\sqrt{2}\left[\frac{{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)r^3\mathrm{dr}}{{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)\mathrm{rdr}}\right]}^{1/2}=2w\sqrt{v+2\mu -1}& \left(10\right)\end{array}$$\left[\mathrm{Math}.11\right]$$\begin{array}{cc}2{\sqrt{2}\left[\frac{{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)\mathrm{rdr}}{{\int }_0^{\infty }{\left\{\frac{{\mathrm{dE}}_{v\mu }\left(r\right)}{\mathrm{dr}}\right\}}^2\mathrm{rdr}}\right]}^{1/2}=\frac{2w}{\sqrt{2\mu -1}}& \left(11\right)\end{array}$

[0068] Here, the electric field distribution in the near field can be calculated from the electric field distribution in the far field using the following equation (see, for example, Equation (6) in Non-Patent Literature 3).

[00009]$\left[\mathrm{Math}.12\right]$$\begin{array}{cc}{E}_{v\mu }\left(r\right)=\frac{k^2}{2}{\int }_0^{\infty }{F}_{v\mu }\left(\theta \right)J_v\left(\mathrm{kr}\sin \theta \right)\sin \left(2\theta \right)d\theta & \left(12\right)\end{array}$

[0069] Therefore, the mode field diameter that is defined as the spot size can also be calculated by substituting the far-field intensity distribution, which was measured by fixing the angle φ in the circumferential direction at an arbitrary constant value, into Equation (12) to calculate the electric field distribution in the near field, and further substituting the electric field distribution in the near field into Equation (13) or Equation (14) below.

[00010]$\left[\mathrm{Math}.13\right]$$\begin{array}{cc}\mathrm{MFD}={\frac{2\sqrt{2}}{\sqrt{v+2\mu -1}}\left[\frac{{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)r^3\mathrm{dr}}{{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)\mathrm{rdr}}\right]}^{1/2}& \left(13\right)\end{array}$$\left[\mathrm{Math}.14\right]$$\begin{array}{cc}\mathrm{MFD}=2{\sqrt{2\left(2\mu -1\right)}\left[\frac{{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)\mathrm{rdr}}{{\int }_0^{\infty }{\left\{\frac{{\mathrm{dE}}_{v\mu }\left(r\right)}{\mathrm{dr}}\right\}}^2\mathrm{rdr}}\right]}^{1/2}& \left(14\right)\end{array}$

Example 1

[0070] Numerical calculations were performed to confirm the validity of Equations (9), (13) and (14).

[0071] FIG. 3 is a diagram explaining the refractive index profile of an optical fiber used for the numerical calculations (optical fiber under test). The optical fiber under test had a graded index core having a shape parameter α of 1.98, and a low refractive index layer. Parameters of the core diameter, the relative refractive index difference, etc. are as shown in FIG. 3.

[0072] FIG. 4 shows the electric field distribution in the near field of the above-mentioned optical fiber. The solid line represents the result of calculations using a finite element method (FEM). The result in FIG. 4 was standardized so that the maximum value of the electric field was 1. The dotted line represents the result obtained by substituting w, which maximizes the value given by Equation (15) below, into Equation (5).

[00011]$\left[\mathrm{Math}.15\right]$$\begin{array}{cc}I=\frac{{\int }_0^{\infty }{g}_{v\mu -\mathrm{FEM}}\left(r\right)E_{v\mu }\left(r\right)\mathrm{rdr}}{{\int }_0^{\infty }{g}_{v\mu -\mathrm{FEM}}^2\left(r\right)\mathrm{rdr}{\int }_0^{\infty }{E}_{v\mu }^2\left(r\right)\mathrm{rdr}}& \left(15\right)\end{array}$

[0073] Here, g.sub.νμ-FEM(r) is the electric field distribution of the LP.sub.νμ mode calculated using the FEM. That is, the dotted line represents the result of approximating the electric field distribution calculated using the FEM by a higher-order Gaussian mode, and a value that is twice of w used for obtaining this result is a mode field diameter defined as a spot size to be obtained this time, and is a reference value for comparison.

[0074] FIG. 5 shows far-field patterns represented by the square of the absolute value of the electric field distribution in the far field obtained by substituting the result represented by the solid line in FIG. 4 into Equation (7).

[0075] FIG. 6 is a table collectively showing the above-mentioned reference value, the mode field diameter calculated by substituting the result represented by the solid line in FIG. 4 into Equations (13) and (14), and the mode field diameter calculated by substituting the result in FIG. 5 into Equation (9). It can be understood that the mode field diameters calculated by either of Equations agree well with the reference value.

[0076] Thus, it was confirmed that a desired mode field diameter can be obtained using any one of Equations (13), (14) and (9).

Example 2

[0077] In order to confirm the validity of Equations (9), (13) and (14), the mode field diameter of the optical fiber operating in a 2LP mode was measured and compared with the result obtained by numerical calculation.

[0078] FIG. 7 shows the refractive index distribution of the optical fiber used for the measurement. The solid line represents the measurement result. The broken line represents the result of approximation by an ideal step-type refractive index distribution for calculation.

[0079] FIG. 8 shows electric field distributions in the near field. The solid line represents the result of calculation using the FEM, based on the broken line in FIG. 7. The dotted line represents the result of approximating the electric field distribution, which was calculated using the FEM, by a higher-order Gaussian mode, and a value that is twice of w used to obtain this result is a mode field diameter defined as a spot size to be obtained this time, and is a reference value for comparison.

[0080] FIG. 9 shows far-field patterns. Plots represent measured values, and a solid line represents a far-field pattern expressed by the square of the absolute value of an electric field distribution in a far field obtained by substituting a solid line in FIG. 8 into Equation (7). The measured values agree well with the results predicted by numerical calculations.

[0081] FIG. 10 is a table collectively showing the above-mentioned reference value, and the mode field diameter calculated by substituting the result in FIG. 9 into Equation (9). It can be understood that the mode field diameter calculated by Equation (9) agrees well with the reference value.

[0082] Thus, it was confirmed that a desired mode field diameter can be obtained using Equation (9).

[0083] The following supplements the derivation process of Equation (10).

[0084] The numerator on the left side of Equation (10) is expressed by the following equation.

[00012]$\left[\mathrm{Math}.10-1\right]$${\int }_0^{\infty }\frac{2\left(\mu -1\right)!}{\pi \left(v+\mu -1\right)!}{\left(\frac{2r^2}{w^2}\right)}^v{\frac{1}{w^2}\left[L_{\mu -1}^v\left(\frac{2r^2}{w^2}\right)\right]}^2\exp \left(-\frac{2r^2}{w^2}\right)r^3\mathrm{dr}=\frac{2\left(\mu -1\right)!}{\pi \left(v+\mu -1\right)!}\frac{w^2}{8}{\int }_0^{\infty }{x^{v+1}\left[L_{p-1}^v\left(x\right)\right]}^2\exp \left(-x\right)\mathrm{dx}=\frac{2\left(\mu -1\right)!}{\pi \left(v+\mu -1\right)!}\frac{w^2}{8}\frac{\left(v+\mu -1\right)!}{\left(\mu -1\right)!}\left(v+2\mu -1\right)=\frac{w^2}{4\pi }\left(v+2\mu -1\right)$$\left[\mathrm{Math}.10-2\right]$$\mathrm{where}x=\frac{2r^2}{w^2}$

[0085] The denominator on the left side of Equation (10) is expressed by the following Equation.

[00013]$\left[\mathrm{Math}.10-3\right]$${\int }_0^{\infty }\frac{2\left(\mu -1\right)!}{\pi \left(v+\mu -1\right)!}{\left(\frac{2r^2}{w^2}\right)}^v{\frac{1}{w^2}\left[L_{\mu -1}^v\left(\frac{2r^2}{w^2}\right)\right]}^2\exp \left(-\frac{2r^2}{w^2}\right)\mathrm{rdr}=\frac{2\left(\mu -1\right)!}{\pi \left(v+\mu -1\right)!}\frac{1}{4}{\int }_0^{\infty }{x^v\left[L_{p-1}^v\left(x\right)\right]}^2\exp \left(-x\right)\mathrm{dx}=\frac{2\left(\mu -1\right)!}{\pi \left(v+\mu -1\right)!}\frac{1}{4}\frac{\left(v+\mu -1\right)!}{\left(\mu -1\right)!}=\frac{1}{2\pi }$

Thus, Equation (10) can be mathematically derived.

[0086] For Equation (8), let

ρ=k sin θ  [Math. 8-1]

then the equation can be rewritten as follows.

[00014]$\left[\mathrm{Math}.8-2\right]$$\frac{\lambda }{\pi }\sqrt{2\frac{{\int }_0^{\pi /2}{.Math.F_{v\mu }\left(\theta \right).Math.}^2\sin \theta \cos \theta d\theta }{{\int }_0^{\pi /2}{.Math.F_{v\mu }\left(\theta \right).Math.}^2{\sin }^3\theta \cos \theta d\theta }}=2{\sqrt{2}\left[\frac{{\int }_0^{\infty }{F}_{v\mu }^2\left(\rho \right)\rho d\rho }{{\int }_0^{\infty }{F}_{v\mu }^2\left(\rho \right){\rho }^{3}d\rho }\right]}^{1/2}$

Therefore, Equation (8) can be derived by performing arithmetic operations similar to Equation (10).

[0087] On the other hand, Equation (11) was derived using numerical analysis. More specifically, the relationship among w, ν, and μ was understood from the values obtained by:

[0088] step 1: specifying w, ν, and μ and substituting w, ν, and μ into Equation (5);

[0089] step 2: substituting the electric field distribution of Equation (5) into the left side of Equation (11); and step 3: changing w, ν, and μ and repeating steps 1 and 2.

[0090] Consequently, Equation (11) was obtained.

Another Embodiment

[0091] The present invention is not limited to the above-described embodiment, and can be implemented in various modified forms without departing from the scope of the gist of the invention. In short, this invention is not limited precisely to the above-described embodiment, and can be embodied by modifying the components in the implementation stage without departing from the scope of the gist.

[0092] Moreover, various inventions can be made by suitably combining a plurality of the components disclosed in the above-described embodiment. For example, some of the components may be omitted from the entire components described in the embodiment. Furthermore, components of different embodiments may be suitably combined.

INDUSTRIAL APPLICABILITY

[0093] The present disclosure is applicable to information and communication industries.

REFERENCE SIGNS LIST

[0094] 10: optical fiber under test [0095] 11: light source [0096] 12: mode converter [0097] 13: alignment device [0098] 14: rotation stage 14 [0099] 15: optical receiver [0100] 16: control unit [0101] 17: A/D converter [0102] 18: signal processing unit