Patent Publication Number: US-11391644-B2

Title: Optical fiber testing method and optical fiber testing device

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a 371 U.S. National Phase of International Application No. PCT/JP2019/023569 filed on Jun. 13, 2019, which claims priority to Japanese Application No. 2018-193530 filed on Oct. 12, 2018. The entire disclosures of the above applications are incorporated herein by reference. 
     TECHNICAL FIELD 
     The present disclosure relates to a test method and a test apparatus for measuring a mode dependent loss and an inter-modal crosstalk in a few-mode optical fiber. 
     BACKGROUND ART 
     With the increase in large-volume content such as movies and games, and the spread of smartphones, traffic amounts in optical fiber networks have been increasing every year. On the other hand, transmission capacity of single mode fiber that is currently used as a transmission medium is approaching its limit. As one technique for addressing future traffic increases, mode multiplexing transmission using a few-mode fiber has been focused on. In this technique, a mode dependent loss or an inter-modal crosstalk at a connection point of the few-mode fiber is one of important optical characteristics. 
     As an optical fiber test method, Optical Time Domain Reflectometry (hereinafter, referred to as OTDR) is renowned. The OTDR is a method and apparatus in which a pulsed test light is incident on an optical fiber under test (hereinafter, referred to as a FUT) to acquire distribution data (OTDR waveforms) based on intensities of a backscattered light of a Rayleigh scattered light originating from the test light pulse propagating within the optical fiber and a Fresnel reflected light, and a round trip time. This technique can be used to test optical properties of optical fibers. Non Patent Literature (NPL)  1  discloses a method for testing inter-modal crosstalk at a connection point of a few-mode fiber using an OTDR having multiple channels. 
     CITATION LIST 
     Non Patent Literature 
     
         
         NPL 1: M. Yoshida, et al., “Mode coupling measurement at a splice point between few-mode fibers using a synchronous multi-channel OTDR,” OFC2016, Th1J.4, 2016. 
         NPL 2: A. Nakamura et. al., “Effective mode field diameter for LP 11  mode and its measurement technique,” IEEE Photon. Technol. Lett., vol. 28, no. 22, pp. 2553-2556, 2016. 
       
    
     SUMMARY OF THE INVENTION 
     Technical Problem 
     However, the test method described in NPL 1 does not refer to a method for testing a mode dependent loss at a connection point. The present invention has been made in view of such circumstances, and has an object to provide an optical fiber test method and an optical fiber test apparatus for measuring a mode dependent loss and an inter-modal crosstalk in a fundamental mode and a first higher-order mode at a connection point of a few-mode fiber. 
     Means for Solving the Problem 
     In order to achieve the object described above, in the optical fiber test method and test apparatus according to the present invention, a mode dependent loss and an inter-modal crosstalk in a fundamental mode and a first higher-order mode at a connection point are calculated by using an approximation expression of an inter-modal coupling efficiency that is obtained in approximating electric field distributions of the fundamental mode and the first higher-order mode in a few-mode fiber by Gaussian function and Hermite Gaussian function. 
     Specifically, an optical fiber test method according to the present invention includes a light incident procedure that makes a test light pulse of a wavelength capable of propagating in a fundamental mode and a first higher-order mode be incident, in any one of the fundamental mode or the first higher-order mode, on one end of an optical fiber under test in which a plurality of the same type optical fibers are connected in series, a measurement procedure that measures an intensity distribution for a distance, from the one end, of each of a fundamental mode component and a first higher-order mode component of a return light of the test light pulse made incident in the light incident procedure, a transmittance ratio computation procedure that computes a ratio K of a transmittance of the first higher-order mode component to a transmittance of the fundamental mode component of the return light at the connection point of the optical fiber under test from the intensity distribution measured in the measurement procedure, and a calculation procedure in which by using a first mathematical equation and a second mathematical equation in a mathematical equation for finding a coupling efficiency between respective modes at a connection portion of an optical fiber, based on an electric field distribution of each mode in the optical fiber and an amount of axial displacement at the connection portion, the first mathematical equation being obtained by approximating the electric field distributions of the fundamental mode and the first higher-order mode in the optical fiber by Gaussian function and Hermite Gaussian function, the second mathematical equation being obtained by making simultaneous equations of the mathematical equation for finding transmittances of the respective modes from the coupling efficiency between the respective modes and the first mathematical equation, the ratio K of the transmittances computed in the transmittance ratio computation procedure is substituted into the second mathematical equation to calculate the amount of axial displacement, and the amount of axial displacement is substituted into the first mathematical equation to calculate a coupling efficiency η 01-01  between the fundamental modes, a coupling efficiency η 11-11  between the fundamental mode and the first higher-order mode group, and a coupling efficiency between η 11-11  the first high-order mode groups. 
     Further, an optical fiber test apparatus according to the present invention includes a light incident section that makes a test light pulse of a wavelength capable of propagating in a fundamental mode and a first higher-order mode be incident, in any one of the fundamental mode or the first higher-order mode, on one end of an optical fiber under test in which a plurality of the same type optical fibers are connected in series, a measurement section that measures an intensity distribution for a distance, from the one end, of each of a fundamental mode component and a first higher-order mode component of a return light of the test light pulse made incident by the light incident section, a transmittance ratio computation section that computes a ratio K of a transmittance of the first higher-order mode component to a transmittance of the fundamental mode component of the return light at the connection point of the optical fiber under test from the intensity distribution measured in the measurement section, and a calculation section in which by using a first mathematical equation and a second mathematical equation in a mathematical equation for finding a coupling efficiency between respective modes at a connection portion of an optical fiber, based on an electric field distribution of each mode in the optical fiber and an amount of axial displacement at the connection portion, the first mathematical equation being obtained by approximating the electric field distributions of the fundamental mode and the first higher-order mode in the optical fiber by Gaussian function and Hermite Gaussian function, the second mathematical equation being obtained by making simultaneous equations of the mathematical equation for finding transmittances of the respective modes from the coupling efficiency between the respective modes and the first mathematical equation, the ratio K of the transmittances computed in the transmittance ratio computation section is substituted into the second mathematical equation to calculate the amount of axial displacement, and the amount of axial displacement is substituted into the first mathematical equation to calculate a coupling efficiency η 01-01  between the fundamental modes, a coupling efficiency η 01-11  between the fundamental mode and the first higher-order mode group, and a coupling efficiency η 11-11  between the first high-order mode groups. 
     Here, a following Equation (C1) may be used as the second mathematical equation, and a following Equation (C2) may be used as the first mathematical equation, 
                     [       Math   .           ⁢   C     ⁢           ⁢   1     ]     ⁢                                       d   2       w   2       =     K   -         K   2     +     2   ⁢   K     -   2                 (     C   ⁢           ⁢   1     )                 [       Math   .           ⁢   C     ⁢           ⁢   2     ]     ⁢                                       η     01   -   01       =     exp   (     -       d   2       w   2         )       ⁢     
     ⁢       η     01   -   11       =       η     11   -   01       =         d   2       w   2       ⁢     exp   (     -       d   2       w   2         )           ⁢     
     ⁢       η     11   -   11       =       (     1   -       d   2       w   2       +       1   2     ⁢       d   4       w   4           )     ⁢     exp   (     -       d   2       w   2         )                 (     C   ⁢           ⁢   2     )               
where, w represents a mode field diameter of the fundamental mode and the first higher-order mode in the optical fiber under test, and d represents the amount of axial displacement.
 
     In the calculation procedure and the calculation section, further, logarithmic transformation may be performed on the coupling efficiency η 01-01  and the coupling efficiency to calculate a mode dependent loss and logarithmic transformation may be performed on the coupling efficiency η 01-11  to calculate an inter-modal crosstalk. 
     Effects of the Invention 
     The present invention can provide an optical fiber test method and an optical fiber test apparatus for measuring a mode dependent loss and an inter-modal crosstalk in a fundamental mode and a first higher-order mode at a connection point of a few-mode fiber. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a diagram for explaining an optical fiber test method according to the present invention. 
         FIG. 2  is a diagram for explaining an optical fiber test apparatus according to the present invention. 
         FIG. 3  is a diagram illustrating relationships between light intensity distributions in a fundamental mode and a first higher-order mode, and xy coordinates. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Hereinafter, embodiments of the present invention will be described with reference to the drawings. The embodiments described below are examples of the present invention, and the present invention is not limited to the following embodiments. In this specification and the drawings, constituent elements having the identical reference signs are assumed to be mutually the same. 
       FIG. 1  is a process chart for explaining an optical fiber test method according to the present embodiment. The optical fiber test method is characterized by performing a light incident procedure S 01 , a measurement procedure S 02 , an acquisition procedure S 03 , and a calculation procedure S 04  described below. The light incident procedure S 01  makes a test light pulse of a wavelength capable of propagating in a fundamental mode and a first higher-order mode be incident, in any one of the fundamental mode or the first higher-order mode, on one end of an optical fiber under test in which two of the same type optical fibers are connected in series. The measurement procedure S 02  measures an intensity distribution for a distance, from the one end, of each of a fundamental mode component and a first higher-order mode component of a return light of the test light pulse made incident in the light incident procedure S 01 . The acquisition procedure S 03  acquires a ratio K of a transmittance of the first higher-order mode component to a transmittance of the fundamental mode component of the return light at the connection point of the optical fiber under test from the intensity distribution measured in the measurement procedure S 02 . The calculation procedure S 04  substitutes the ratio K of the transmittances obtained in the acquisition procedure into Equation (20) to calculate a value (d 2 /w 2 ), substitutes the value (d 2 /w 2 ) into Equations (11) to (13) to calculate a coupling efficiency η 01-01  between the fundamental modes, a coupling efficiency η 01-11  between the fundamental mode and a first higher-order mode group, and a coupling efficiency between the first high-order mode groups (step S 04   a ), and further, performs logarithmic transformation on the coupling efficiency η 01-01  and the coupling efficiency η 11-11  to calculate a mode dependent loss and performs logarithmic transformation on the coupling efficiency η 01-11  to calculate an inter-modal crosstalk (step S 04   b ). 
     The light incident procedure S 01  performs: 
     a generation step of generating a test light pulse of a wavelength capable of propagating through an optical fiber under test in a fundamental mode and a first higher-order mode, and 
     
         
         an light incident step of making the test light pulse generated in the generation step be incident on one end of the optical fiber under test in any one of the fundamental mode or the first higher-order mode. 
       
    
     The measurement procedure S 02  performs:
     a mode demultiplexing step of dividing a return light of the test light pulse incident on one end of the optical fiber under test in the light incident step into the fundamental mode and the first higher-order mode, and   a light intensity acquisition step of photoelectric-converting each of the mode components of the return light divided in the mode demultiplexing step, and acquiring an intensity distribution for a distance, from one end of the optical fiber under test, of each of the mode components of the return light. Specifically, in the light incident procedure S 01  and the measurement procedure S 02 , backscattered light intensity distributions from one end of the optical fiber under test in the fundamental mode and the first higher-order mode are measured using the backscattered light measurement technique as described in NPL 2.   

     The acquisition procedure S 03  performs:
     a transmittance ratio acquisition step of acquiring a ratio of transmittances generated in the mode components of the return light at any position of the optical fiber under test from the intensity distribution of each of the mode components of the return light acquired in the light intensity acquisition step.   

     In the calculation procedure S 04  performs:
     an inter-modal coupling efficiency computation step (S 04   a ) of computing an inter-modal coupling efficiency at a connection point using an approximation expression from the ratio of the transmittances acquired in the transmittance ratio acquisition step, and α mode dependent loss and inter-modal crosstalk acquisition step (S 04   b ) of acquiring a mode dependent loss and an inter-modal crosstalk from the inter-modal coupling efficiency acquired in the inter-modal coupling efficiency computation step. Details for computing the inter-modal coupling efficiency, the mode dependent loss, and the inter-modal crosstalk will be described later.   

       FIG. 2  is a diagram for explaining a configuration example of an optical fiber test apparatus  101  according to the present embodiment. The optical fiber test apparatus  101  includes a light incident means, a measurement means, a transmittance ratio computation means, and a calculation means described below. The light incident means makes a test light pulse of a wavelength capable of propagating in a fundamental mode and a first higher-order mode be incident, in any one of the fundamental mode or the first higher-order mode, on one end of an optical fiber under test  10  in which two of the same type optical fibers are connected in series. The measurement means measures an intensity distribution for a distance, from the one end, of each of a fundamental mode component and a first higher-order mode component of a return light of the test light pulse made incident by the light incident means. The transmittance ratio computation means computes a ratio K of a transmittance of the first higher-order mode component to a transmittance of the fundamental mode component of the return light at the connection point of the optical fiber under test  10  from the intensity distribution measured by the measurement means. The computation means substitutes the ratio K of the transmittances computed by the transmittance ratio computation means into Equation (20) to calculate a value (d 2 /w 2 ), substitutes the value (d 2 /w 2 ) into Equations (11) to (13) to calculate a coupling efficiency η 01-01  between the fundamental modes, a coupling efficiency η 01-11  between the fundamental mode and a first higher-order mode group, and a coupling efficiency η 11-11  between the first high-order mode groups, and further, performs logarithmic transformation on the coupling efficiency η 01-01  and the coupling efficiency η 11-11  to calculate a mode dependent loss and performs logarithmic transformation on the coupling efficiency η 01-11  to calculate an inter-modal crosstalk. 
     The light incident means includes:
     a generating unit A generating a test light pulse of a wavelength capable of propagating through an optical fiber under test  10  in a fundamental mode and a first higher-order mode, and   a mode multiplexing/demultiplexing unit B making the test light pulse generated by the generating unit A be incident on the optical fiber under test  10  in any one of the fundamental mode or the first higher-order mode, and dividing a return light of the test light pulse into the fundamental mode and the first higher-order mode.   

     The measurement means includes:
     the mode multiplexing/demultiplexing unit B,   a light receiving unit C photoelectric-converting each of the mode components of the return light divided by the mode multiplexing/demultiplexing unit B, and   a signal processing unit  19 , of a calculation processing unit D, acquiring an intensity distribution for a distance, from one end of the optical fiber under test  10 , of each of the mode components of the return light, when the test light pulse is made incident on one end of the optical fiber under test  10  in any one of the fundamental mode or the first higher-order mode, based on an output signal from the light receiving unit C and converted into digital data.   

     The transmittance ratio computation means includes:
     a transmittance ratio computation unit  20 , of the calculation processing unit D, computing a ratio of transmittances generated at a connection point from the intensity distribution of each of the mode components of the return light.   The calculation means includes, of the calculation processing unit D:   an inter-modal coupling efficiency computation unit  21  computing an inter-modal coupling efficiency from the ratio of the transmittance computed by the transmittance ratio computation unit  20 , and   a mode dependent loss and inter-modal crosstalk computation unit  22  computing a mode dependent loss and an inter-modal crosstalk from the inter-modal coupling efficiency computed by the inter-modal coupling efficiency computation unit  21 .   

     The generating unit A includes a light source  11 , a pulse generator  12 , and a light intensity modulator  13 . The light source  11  can output a continuous light of a wavelength capable of propagating through the optical fiber under test  10  in the fundamental mode and the first higher-order mode, and the output continuous light is made into a pulse to be a test light pulse by the light intensity modulator  13  in accordance with a signal of the pulse generator  12 . The light intensity modulator  13  is an acoustic optical modulator provided with an acoustic optical switch configured to pulse-drive an acoustic optical element, for example. Note that the pulse generator  12  may output a trigger signal to the calculation processing unit D to determine a timing when to start the measurement of the backscattered light intensity distribution. 
     The mode multiplexing/demultiplexing unit B includes has an optical circulator  14  and a mode multiplexer/demultiplexer  15 . The test light pulse generated by the light intensity modulator  13  is incident on the mode multiplexer/demultiplexer  15  via the optical circulator  14 . The mode multiplexer/demultiplexer  15  is a mode multiplexer/demultiplexer provided with a directional coupler including a planar lightwave circuit, for example, as described in NPL 2. The test light pulse is incident on one end of the optical fiber under test  10  in any one of the fundamental mode or the first higher-order mode from the mode multiplexer/demultiplexer  15 . 
     When the test light pulse incident in any one of the fundamental mode or the first higher-order mode propagates through the optical fiber under test  10 , some of the test light pulses are coupled to those in a fundamental mode and a first higher-order mode propagating in a reverse direction by Rayleigh scattering, and become backscattered lights in the fundamental mode and the first higher-order mode, respectively. The backscattered lights are re-incident on the mode multiplexer/demultiplexer  15  as return light. At this time, the fundamental mode component and first higher-order mode component of the return light are divided by the mode multiplexer/demultiplexer  15 . 
     The light receiving unit C includes two optical receivers ( 16 ,  17 ). Among the return lights divided into each mode by the mode multiplexer/demultiplexer  15 , a mode component the same as the incident test light pulse is incident on the optical receiver  16  via the optical circulator  14 , and a mode component different from the incident test light pulse is incident on the optical receiver  17 , and those incident mode components are subjected to photoelectric conversion. 
     The calculation processing unit D includes an A/D (analog to digital) converter  18 , the signal processing unit  19 , the transmittance ratio computation unit  20 , the inter-modal coupling efficiency computation unit  21 , and the mode dependent loss and inter-modal crosstalk computation unit  22 . Electrical signals from the optical receivers  16  and  17  are converted to digital data by the A/D converter  18 . The digital data is input to the signal processing unit  19 . 
     The signal processing unit  19  acquires the intensity distribution for the fundamental mode and first higher-order mode components of the return light. Furthermore, the transmittance ratio computation unit  20  acquires the ratio of the transmittances of the fundamental mode and first higher-order mode components of the return light at the connection point in the intensity distribution. Then, the inter-modal coupling efficiency computation unit  21  performs calculation processing for computing the inter-modal coupling efficiency at the connection point. The mode dependent loss and inter-modal crosstalk computation unit  22  performs calculation processing for computing the mode dependent loss and the inter-modal crosstalk from the acquired inter-modal coupling efficiency. 
     The calculation processing unit D can be realized by a computer and a program, and the program can be recorded on a recording medium or provided through a network. 
     Hereinafter, the calculation processing for computing the inter-modal coupling efficiency, the mode dependent loss, and the inter-modal crosstalk will be described. 
     Electric field distributions of the fundamental mode and two orthogonal first higher-order modes in the optical fiber are approximated by the following Gaussian function and Hermite Gaussian function. 
     
       
         
           
             
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     Note that E 1  represents an electric field distribution in the fundamental mode (LP 01  mode), E 2  and E 3  represent electric field distributions of two orthogonal first higher-order modes (LP 11   a  and LP 11   b  modes, respectively), w represents a mode field diameter of the fundamental mode and the first higher-order mode, and x and y represent coordinates with the center of an optical fiber cross section being the origin.  FIG. 3  is a diagram illustrating relationships between the light intensity distributions in respective modes and the xy coordinates. 
     An inter-modal coupling efficiency ↓ mn  of the optical fiber under test in which two of the same type optical fibers are connected is expressed by the following equation. 
               [       Math   .           ⁢   M     ⁢           ⁢   2     ]     ⁢                                     η   mn     =              ∫     ∫         E   m     ⁡     (     x   ,   y     )       ⁢       E   n     ⁡     (       x   -     d   ⁢           ⁢   cos   ⁢           ⁢   θ       ,     y   -     d   ⁢           ⁢   sin   ⁢           ⁢   θ         )       ⁢   dxdy              2       ∫     ∫                E   m     ⁡     (     x   ,   y     )            2     ⁢   dxdy   ⁢     ∫     ∫                E   n     ⁡     (     x   ,   y     )            2     ⁢   dxdy                         (   4   )                 (   M2   )               
E m  and E n  represent an electric field distribution in a mode input to the connection portion and an electric field distribution in a mode output from the connection portion, respectively. That is, η mn  represents an efficiency of coupling from the mode of m to the mode of n at the connection portion. Additionally, d represents the amount of axial displacement at the connection point, and η represents an angle formed by the x-axis and an axial displacement direction. From Equations (1) to (4), the following equations are obtained.
 
     
       
         
           
             
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                   ) 
                 
               
             
           
         
       
     
     Here, two orthogonal first higher-order modes strongly couples during propagation, and so, are difficult to distinguish in actual measurement. Thus, two orthogonal first higher-order modes are collectively considered as a single first higher-order mode group. At this time, Equations (5) to (10) can be expressed as the following equations. 
               [       Math   .           ⁢   M     ⁢           ⁢   4     ]     ⁢                                     η     01   -   01       =     exp   (     -       d   2       w   2         )             (   11   )                 η     01   -   11       =       η     11   -   01       =         d   2       w   2       ⁢     exp   (     -       d   2       w   2         )                 (   12   )                 η     11   -   11       =       (     1   -       d   2       w   2       +       1   2     ⁢       d   4       w   4           )     ⁢     exp   (     -       d   2       w   2         )               (   13   )                 (   M4   )               
η 01-01  represents the coupling efficiency between the fundamental modes, η 01-11  and η 11-01  represent the coupling efficiencies between the fundamental mode and the first higher-order mode group, and η 11-11  represents the coupling efficiency between the first higher-order mode groups. This can eliminate the angle θ that represents the axis displacement direction.
 
     On the other hand, assuming that refractive indices are n 1  and n 2 , and mode field radii are w 1  and w 2 , of the optical fiber under test at an incident end (near end) side and a distal end side of the test light pulse, respectively, when the test light pulse is incident on the optical fiber under test in the fundamental mode, the transmittances at the connection point at the backscattered light intensity of the fundamental mode component and the first higher-order mode component are obtained by the following equations. 
               [       Math   .           ⁢   M     ⁢           ⁢   5     ]     ⁢                                     L   1     =         (         n   2     ⁢     w   2           n   1     ⁢     w   1         )     2     ⁢     (       η     01   -   01       +     η     01   -   11         )     ⁢     (       η     01   -   01       +     η     11   -   01         )               (   14   )                 L   2     =         (         n   2     ⁢     w   2           n   1     ⁢     w   1         )     2     ⁢     (       η     01   -   01       +     η     01   -   11         )     ⁢     (       η     01   -   11       +     η     11   -   11         )               (   15   )                 (   M5   )               
L 1  and L 2  represent the transmittances at the connection point at the backscattered light intensity of the fundamental mode component and the first higher-order mode component, respectively.
 
     Furthermore, when the test light pulse is incident on the optical fiber under test in the first higher-order mode, the transmittances at the connection point at the backscattered light intensity of the fundamental mode component and the first higher-order mode component are obtained by the following equations. 
               [       Math   .           ⁢   M     ⁢           ⁢   6     ]     ⁢                                     L   3     =         (         n   2     ⁢     w   2           n   1     ⁢     w   1         )     2     ⁢     (       η     11   -   01       +     η     11   -   11         )     ⁢     (       η     01   -   01       +     η     11   -   01         )               (   16   )                 L   4     =         (         n   2     ⁢     w   2           n   1     ⁢     w   1         )     2     ⁢     (       η     11   -   01       +     η     11   -   11         )     ⁢     (       η     01   -   11       +     η     11   -   11         )               (   17   )                 (   M6   )               
L 3  and L 4  represent the transmittances at the connection point at the backscattered light intensity of the fundamental mode component and the first higher-order mode component, respectively.
 
     From Equations (14) to (17), by taking a ratio of L 2  to L 1 , or a ratio of L 4  to L 3 , the ratio K of the transmittances can be expressed by the following equation. 
     
       
         
           
             
               [ 
               
                 
                   Math 
                   . 
                   
                       
                   
                   ⁢ 
                   M 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 7 
               
               ] 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         K 
                         = 
                         
                           
                             
                               L 
                               2 
                             
                             
                               L 
                               1 
                             
                           
                           = 
                           
                             
                               
                                 L 
                                 4 
                               
                               
                                 L 
                                 3 
                               
                             
                             = 
                             
                               
                                 
                                   η 
                                   
                                     01 
                                     - 
                                     11 
                                   
                                 
                                 + 
                                 
                                   η 
                                   
                                     11 
                                     - 
                                     11 
                                   
                                 
                               
                               
                                 
                                   η 
                                   
                                     01 
                                     - 
                                     01 
                                   
                                 
                                 + 
                                 
                                   η 
                                   
                                     11 
                                     - 
                                     01 
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         ( 
                         18 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   M7 
                   ) 
                 
               
             
           
         
       
     
     By taking the ratio of the transmittances in this way, a section dependent on the refractive index and mode field radius of the optical fiber under test can be eliminated, so it is possible to reduce (eliminate) the effect of a backscattered light capture rate difference due to unconformities in the refractive index and the mode field diameter between optical fibers that are connected in the optical fiber under test. 
     From Equations (11) to (13) and (18), the following equation is obtained. 
     
       
         
           
             
               [ 
               
                 
                   Math 
                   . 
                   
                       
                   
                   ⁢ 
                   M 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 8 
               
               ] 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   d 
                                   2 
                                 
                                 
                                   w 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             2 
                             ⁢ 
                             
                               K 
                               ( 
                               
                                 
                                   d 
                                   2 
                                 
                                 
                                   w 
                                   2 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               ( 
                               
                                 1 
                                 - 
                                 K 
                               
                               ) 
                             
                           
                         
                         = 
                         0 
                       
                     
                     
                       
                         ( 
                         19 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   M8 
                   ) 
                 
               
             
           
         
       
     
     Equation (19) has a multiple root when (d 2 /w 2 ) is √3−1, and there are two solutions in other conditions. Normally, in considering that the amount of axial displacement possibly generated at the connection point is 2 μm or less, and the mode field diameter of the optical fiber at the test wavelength is 4.68 μm or more, the solution of Equation (19) is as follows. 
     
       
         
           
             
               [ 
               
                 
                   Math 
                   . 
                   
                       
                   
                   ⁢ 
                   M 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 9 
               
               ] 
             
             ⁢ 
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             d 
                             2 
                           
                           
                             w 
                             2 
                           
                         
                         = 
                         
                           K 
                           - 
                           
                             
                               
                                 K 
                                 2 
                               
                               + 
                               
                                 2 
                                 ⁢ 
                                 K 
                               
                               - 
                               2 
                             
                           
                         
                       
                     
                     
                       
                         ( 
                         20 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   M9 
                   ) 
                 
               
             
           
         
       
     
     Thus, (d 2 /w 2 ) is computed from the obtained ratio K of the transmittances using Equation (20), and substituted into Equations (11) to (13) so that the inter-modal coupling efficiency can be calculated. 
     Furthermore, logarithmic transformation is performed on the inter-modal coupling efficiencies computed in accordance with the above scheme so that the mode dependent loss and the inter-modal crosstalk can be computed. 
     OTHER EMBODIMENTS 
     Note that the present invention is not limited to the above-described embodiments, and can be variously modified and implemented within the scope not departing from the gist of the present invention. In short, the present invention is not limited to the above-described embodiment as it is, and can be embodied with the components modified within the scope not departing from the gist thereof when implemented. For example, the calculation processing unit D can be realized by a computer and a program, and the program can be recorded on a recording medium or provided through a network. In the examples described above, a fiber in which two of the same type optical fibers are connected in series is described as an optical fiber under test, but the test can be performed using a fiber in which three or more of the same type optical fibers are connected in series as an optical fiber under test. 
     Furthermore, various inventions can be formed by appropriate combinations of a plurality of components disclosed in the above-described embodiments. For example, several components may be deleted from all of the components illustrated in the embodiments. Furthermore, components of different embodiments may be appropriately combined with each other. 
     REFERENCE SIGNS LIST 
     
         
           10 : Optical fiber under test 
           11 : Light source 
           12 : Pulse generator 
           13 : Light intensity modulator 
           14 : Optical circulator 
           15 : Mode multiplexer/demultiplexer 
           16 ,  17 : Optical receiver 
           18 : A/D converter 
           19 : Signal processing unit 
           20 : Transmittance ratio computation unit 
           21 : Inter-modal coupling efficiency computation unit 
           22 : Mode dependent loss and inter-modal crosstalk computation unit 
           101 : Optical pulse test apparatus