Patent Publication Number: US-2013243382-A1

Title: Gradient-index multimode optical fibers for optical fiber connectors

Description:
RELATED APPLICATIONS 
     This application claims the benefit of priority under 35 U.S.C. §119 of U.S. Provisional Application Ser. No. 61/610,123 filed on Mar. 13, 2012 the content of which is relied upon and incorporated by reference in its entirety. 
    
    
     FIELD 
     The present disclosure relates to optical fibers and optical fiber connectors, and in particular relates to gradient-index multimode optical fibers for optical fiber connectors. 
     BACKGROUND 
     Optical fiber connectors are used in a variety of telecommunications applications to connect one optical fiber to another, or to connect an optical fiber to a telecommunications device. Certain optical fiber connectors include a short section of single-mode optical fiber (SMF) called a stub fiber that interfaces with a field optical fiber within the connector. When a connector is operably connected (mated) to another connector, the stub fiber resides between the field fiber of its own connector and the stub fiber of the mating connector. 
     When all the optical fibers are aligned and otherwise matched in size and configuration, the light travels in the field and stub fibers in the lowest or fundamental mode, namely the LP 01  mode. However, a misalignment, a mismatch in the mode-field diameter (MFD) of the fibers, or a combination of these and other factors can cause light to travel in higher-order modes, such as the LP 11  mode for a short distance even though the fibers are SMFs. Thus, though an optical fiber may be designed to be an SMF, there are circumstances under which they operate as multimode optical fibers for short distances (&lt;5 cm). 
     Coherent light traveling in different guided modes takes different optical paths and can cause multi-path interference (MPI). MPI can cause light transmitted through the connector to have significant time-dependent fluctuations that are exacerbated by the use of off-the-shelf SMFs designed for long-haul telecommunications applications. MPI and the attendant power fluctuations are undesirable and degrade the performance of the telecommunications system in which the optical fiber connector is used. 
     SUMMARY 
     Gradient-index (also called graded-index) multimode optical fibers suitable for use in optical fiber connectors as stub fibers are disclosed herein. The gradient-index multimode optical fibers have a fundamental mode (LP 01 ) that substantially matches the mode field diameter of an SMF to reduce or minimize connector loss. In addition, the group index difference (i.e., the group delay) among the different guided modes is minimized to reduce MPI. 
     An aspect of the disclosure is a gradient-index, multimode optical fiber for use in an optical fiber connector having an operating wavelength λ. The optical fiber includes a core having a radius r 0 , with a cladding immediately surrounding the core. The core and cladding supporting a fundamental mode and at least one higher-order mode, the core and cladding defining a mode-field diameter MFD MM  and a relative refractive index profile Δ, wherein Δ is defined by the relationship: 
     
       
         
           
             
               Δ 
               = 
               
                 
                   Δ 
                   0 
                 
                  
                 
                   [ 
                   
                     1 
                     - 
                     
                       
                         ( 
                         
                           r 
                           
                             r 
                             0 
                           
                         
                         ) 
                       
                       α 
                     
                   
                   ] 
                 
               
             
             , 
           
         
       
     
     where r is a radial coordinate, Δ 0  is a maximum relative refractive index at r=0, and a is a profile parameter. Moreover, the core radius r 0  is in the range from 6 μm to 20 μm, Δ 0  is in the range from 0.4% to 2.5%, α is in the range from 1.9 and 4.1, and the mode-field diameter MFD MM  is between 8.2 μm and 9.7 μm when the operating wavelength of λ=1310 nm and is between 9.2 μm and 10.9 μm when the operating wavelength λ=1550 nm. 
     Another aspect of the disclosure is a gradient-index, multimode optical fiber for use in an optical fiber connector having an operating wavelength λ and a single-mode optical fiber (SMF) that has a mode-field diameter MFD SM  and a first group index difference Δn gSM . The gradient-index, multimode optical fiber includes a gradient-index core having a radius r 0  and a relative refractive index profile Δ with a maximum relative refractive index Δ 0 , and a cladding immediately surrounding the core. The cladding has a constant relative refractive index profile, and the core and cladding support multiple guided modes. The optical fiber also has a mode-field diameter MFD MM  that is substantially the same as the SMF mode-field diameter MFD SM , and a multimode group index difference Δn g  such that the ratio Δn gSM /Δn g  satisfies 
       2 ≦Δn   gSM   /Δn   g ≦300.
 
     Another aspect of the disclosure is an optical fiber connector. The connector includes a multimode stub fiber having a first end and that is operably supported by a first alignment member. The multimode stub fiber consists of a length L of the multimode optical fiber as described immediately above and as also described in more detail below. The connector also has a single-mode field optical fiber having a second end and that is operably supported by a second alignment member relative to the first alignment member such that the respective first and second ends of the multimode stub fiber and the single-mode field optical fiber are operably aligned and interfaced. 
     Another aspect of the disclosure is an optical fiber connector. The connector includes a single-mode field optical fiber having an end and a mode-field diameter MFD SM  and a group index difference Δn gSM . The connector also includes a first alignment member that operably supports the field optical fiber, a stub fiber having an end, and a second alignment member that operably supports the stub fiber so that the stub and field optical fiber ends are aligned and interfaced. The stub fiber includes of a length of a multimode optical fiber that comprises: a) a gradient-index core having a radius r 0  and a refractive index profile Δ with a maximum relative refractive index Δ 0 ; b) a cladding immediately surrounding the core, the cladding having a constant relative refractive index profile, wherein the core and cladding support multiple guided modes; c) a mode-field diameter MFD MM  that is substantially the same as the mode-field diameter MFD SM  of the field optical fiber; and d) a multimode group index difference Δn g  wherein 2≦Δn gSM /Δn g ≦300. 
     Another aspect of the disclosure is a gradient-index multimode stub fiber for use in an optical fiber connector at an operating wavelength and that has a single-mode field optical fiber with a group index difference Δn gSM  and a mode-field diameter. The multimode stub fiber has a core having a radius r 0  and a relative refractive index Δ with an α parameter in the range from 1.9 to 4.1, and a cladding immediately surrounding the core. The cladding has a constant relative refractive index. The core and cladding are configured to support multiple guided modes while having a group index difference Δn g  that satisfies 2≦Δn gSM /Δn g ≦300, and a mode-field diameter MFD MM  that satisfies 0.9≦MFD MM /MFD SM ≦1.1 at operating wavelengths λ of 1310 nm and 1550 nm. 
     Another aspect of the disclosure is an optical fiber connector that utilizes the stub fiber described immediately above. The connector has a first alignment member that operably supports the stub fiber, with the stub fiber having an end. The connector also includes a second alignment member that operably supports the field optical fiber such that the respective ends of the stub fiber and the single-mode field optical fiber are operably aligned and interfaced. 
     These and other aspects of the disclosure will be further understood and appreciated by those skilled in the art by reference to the following specification, claims and appended drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A more complete understanding of the present disclosure can be had by reference to the following Detailed Description when taken in conjunction with the accompanying drawings, where: 
         FIG. 1  is a schematic cross-sectional view of an example prior art optical fiber connector that employs a single-mode stub fiber; 
         FIG. 2  is a close-up side view of first and second stub fibers and a field fiber in a misaligned configuration that can arise when two optical fiber connectors like those in  FIG. 1  are operably mated; 
         FIG. 3  is a plot of the operating wavelength λ (microns) versus transmission (dB) and shows an example transmission efficiency curve for an example conventional stub fiber, with the peak-to-peak wavelength spacing Δλ of about 32 nm shown in the plot; 
         FIG. 4  shows a cross-sectional view of an example gradient-index multimode optical fiber according to the disclosure, along with the relative refractive index profile Δ; 
         FIG. 5  is a plot of the relative refractive index profile Δ as a function of radius r for three Design Examples DE 1 through DE 3 of the gradient-index multimode optical fiber according to the disclosure; 
         FIG. 6  plots the group index difference Δn g  versus the effective index n eff  and shows the group index difference at both 1310 nm and 1550 nm for a first Design Example; 
         FIG. 7  is the same plot as  FIG. 6  but for a second Design Example; 
         FIGS. 8A and 8B  are similar to  FIGS. 6 and 7 , but plot the group index difference Δn g  versus the effective index n eff  for operating wavelengths of 1310 nm and 1550 nm, respectively; 
         FIG. 9  is a plot similar to the plot of  FIG. 4  for four additional Design Examples DE 4 through DE 7 of the gradient-index multimode optical fiber according to the disclosure; and 
         FIG. 10  is similar to  FIG. 1  and shows an example of an optical fiber connector that uses the multimode gradient-index optical fiber of the present disclosure as the stub fiber. 
     
    
    
     DETAILED DESCRIPTION 
     Reference is now made in detail to various embodiments of the disclosure, examples of which are illustrated in the accompanying drawings. Whenever possible, the same or like reference numbers and symbols are used throughout the drawings to refer to the same or like parts. The drawings are not necessarily to scale, and one skilled in the art will recognize where the drawings have been simplified to illustrate the key aspects of the disclosure. 
     The claims as set forth below are incorporated into and constitute part of this Detailed Description. 
     The entire disclosure of any publication or patent document mentioned herein is incorporated by reference. 
     The symbol μm and the word “micron” are used interchangeably herein. 
     Mode field diameter or MFD is a measure of the spot size or beam width of light propagating in an optical fiber. The MFD is a function of the source wavelength, fiber core radius r and fiber refractive index profile. In an example, the mode field diameter MFD can be measured using the Peterman II method, where MFD=2w, and 
     
       
         
           
             
               
                 w 
                 2 
               
               = 
               
                 2 
                  
                 
                   
                     
                       ∫ 
                       0 
                       ∞ 
                     
                      
                     
                       
                         E 
                         2 
                       
                        
                       r 
                        
                       
                           
                       
                        
                       
                          
                         r 
                       
                     
                   
                   
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                        
                       
                         
                           ( 
                           
                             
                                
                               E 
                             
                             / 
                             
                                
                               r 
                             
                           
                           ) 
                         
                         2 
                       
                     
                      
                     
                         
                     
                   
                 
                  
                 r 
                  
                 
                     
                 
                  
                 
                    
                   r 
                 
               
             
             , 
           
         
       
     
     where E is the electric field distribution in the optical fiber and r is the radial coordinate of the optical fiber. The MFD of a single-mode optical fiber is denoted herein as MFD SM , while the MFD of the multimode optical fiber  100  of the present disclosure as described below is denoted MFD MM . 
     The parameter a (also called the “profile parameter”) as used herein relates to the relative refractive index Δ, which is in units of “%,” where r is the radius (radial coordinate), and which is defined by: 
     
       
         
           
             
               Δ 
               = 
               
                 
                   Δ 
                   0 
                 
                  
                 
                   [ 
                   
                     1 
                     - 
                     
                       
                         ( 
                         
                           r 
                           
                             r 
                             0 
                           
                         
                         ) 
                       
                       α 
                     
                   
                   ] 
                 
               
             
             , 
           
         
       
     
     where Δ 0  is the maximum relative refractive index, r 0  is the radius of the core, r is the radial coordinate, e.g., in the range 0≦r≦r F , wherein r=0 is the initial radial point of the profile, r F  is the outer diameter of the cladding (with the inner cladding radius being r 0 ), and α is the aforementioned profile parameter and is a real number exponent. For a step index profile, α is greater than or equal to 10. For a gradient-index profile, α is less than 10. The term “substantially parabolic” can be used to describe substantially parabolically shaped relative refractive index profiles with α=2, as well as profiles in which the curvature of the relative refractive index of the core varies slightly from α=2. It is noted here that different forms for the core radius r 0  and maximum relative refractive index Δ 0  can be used without affecting the fundamental definition of Δ. An example fiber  100  as discussed below has a substantially parabolic relative refractive index profile. 
     The limits on any ranges cited herein are considered to be inclusive and thus to lie within the range unless otherwise specified. 
     The term “dopant” as used herein refers to a substance that raises the relative refractive index of glass relative to pure undoped SiO 2 . One or more other substances that are not dopants may be present in a region of an optical fiber (e.g., the core) having a positive relative refractive index Δ. 
     The term “mode” is short for guided mode. A single-mode fiber as the term is used herein means an optical fiber designed to support only a single mode over a substantial length of the optical fiber (e.g., at least several meters) but that under certain circumstances can support multiple modes over short distances (e.g., tens of millimeters). A multimode optical fiber means an optical fiber designed to support the fundamental mode and at least one higher-order mode over a substantial length of the optical fiber. 
     The cutoff wavelength λ C  of a mode is the minimum wavelength beyond which a mode ceases to propagate in the optical fiber. The cutoff wavelength of a single-mode fiber is the minimum wavelength at which an optical fiber will support only one propagating mode. The cutoff wavelength λ C  of a single-mode fiber corresponds to the highest cutoff wavelength among the higher-order modes. Typically the highest cutoff wavelength λ C  corresponds to the cutoff wavelength of the LP 11  mode. A mathematical definition can be found in Single Mode Fiber Optics, Jeunhomme, pp. 39-44, Marcel Dekker, New York, 1990, wherein the theoretical fiber cutoff is described as the wavelength at which the mode propagation constant becomes equal to the plane wave propagation constant in the outer cladding. This theoretical wavelength is appropriate for an infinitely long, perfectly straight fiber that has no diameter variations. 
     The operating wavelength λ is the wavelength at which a particular optical fiber operates, with example first and second operating wavelengths being 1310 nm and 1550 nm, which are commonly used in telecommunications systems that include optical fiber connectors of the type disclosed herein. 
     The phrase “SMF-28e fiber” as used hereinbelow refers to a particular type of single-mode optical fiber made by Corning, Inc., of Corning, N.Y. The term “SMF-28e” is a registered trademark of Corning, Inc. 
       FIG. 1  is a schematic cross-sectional diagram of a conventional example optical fiber connector (“connector”)  10 , which is based generally on the Unicam® optical fiber connector from Corning, Inc., of Corning, N.Y. The connector  10  includes a stub fiber  20  that has opposite ends  22  and  24 , and that in an example has a length L in the range from 13 mm to 20 mm. The stub fiber  20  is supported by a stub alignment member  30  (e.g., a ferrule), and in an example the stub fiber is secured therein using, for example, an epoxy. The stub alignment member  30  has a straight face or tip  32 , which is factory polished so that the corresponding stub fiber end  22  is also polished. 
     The optical fiber connector  10  also includes a field optical fiber (“field fiber”)  40  that has an end  42  and that is operably supported by an alignment member  50 , e.g., a ferrule. The stub fiber  20  is optically coupled to field optical fiber  40  by aligning and interfacing the two fibers at their respective ends  24  and  42 . This is accomplished, for example, via a mechanical or fusion splice member  60  that includes an interior  64 , which in an example contains an index-matching material (e.g., a gel)  66 . 
       FIG. 1  also shows an end portion of a mating connector  10 ′ configured to mate with connector  10 . The mating connector  10 ′ can be a stub-fiber type of connector that includes an alignment member  30 ′ that supports a stub fiber  20 ′ having an end  22 ′ at a straight facet  32 ′. The mating connector  10 ′ can be also a regular connector built on a fiber jumper. 
     In prior-art types of connectors  10  as shown in  FIG. 1 , a standard single-mode optical fiber (SMF) is typically used for stub fiber  20 . However, as discussed above, SMFs are generally not strictly limited to single-mode operation at the operating wavelengths λ of 1310 nm or 1550 nm for a short fiber length of a few centimeters. Under certain conditions (when the SMF length is less than about 5 cm), higher-order modes can propagate in an SMF. Thus, in the discussion below, single mode (SM) fiber  20  is described as having certain attributes of a multimode optical fiber, such as a group index difference Δn gSM . 
     For standard SMFs of a few meters in length, the higher-order modes are completely attenuated and so are not observed. However, at lengths significantly shorter, such as those associated with stub fiber  20 , an SMF can carry significant power in the higher-order modes. Moreover, for standard SMFs, the group index difference Δn g  between the fundamental mode and the higher-order modes can be exceedingly large. As a consequence, the light traveling over the different optical paths in an SMF can interfere, giving rise to the aforementioned detrimental MPI. 
       FIG. 2  is a close-up view of SM stub fiber  20 ′ of connector  10 ′, SM stub fiber  20  of connector  10 , and field fiber  40  of connector  10  in a misaligned configuration that can arise when connectors  10 ′ and  10  of  FIG. 1  are mated. The three fibers shown in  FIG. 2  are optically coupled at interfaces I 1  and I 2 . In this configuration, stub fiber  20 ′ is the launching fiber and field fiber  40  is the receiving fiber. 
     The optical fibers  20 ′ and  20  of  FIGS. 1 and 2  are shown as being misaligned (offset) relative to one another at interface I 1 , while optical fibers  20  and  40  are shown as being misaligned relative to one another at interface  12 . These misalignments can and do happen in practice. 
     As shown in the lower half of  FIG. 2 , the fundamental mode LP 01  travels in stub fiber  20 ′ of connector  10 ′ toward stub fiber  20 . Because stub fibers  20  and  20 ′ are misaligned, the fundamental mode LP 01  excites a higher-order LP 11  mode to travel in stub fiber  20 , so that now both the fundamental mode LP 01  and the higher-order mode LP 11  travel in SM stub fiber  20 . When these two modes encounter the misaligned SM field fiber  40 , the fundamental LP 01  mode of stub fiber  20  couples mainly to the fundamental LP 01  mode of field fiber  40 . Some light of the fundamental LP 01  mode of stub fiber  20  couples also to the higher order LP 11  mode of field fiber  40 . Similarly, the higher order LP 11  mode of stub fiber  20  couples mainly to the higher order LP 11  of field fiber  40 . Some light of the higher order LP 11  mode of stub fiber  20  couples also to the fundamental LP 01  mode of field fiber  40 . After propagating a certain length in SM field fiber  40 , the LP 11  mode gets cut off and only the fundamental LP 01  mode travels therein. Light from the fundamental LP 01  mode ultimately gets detected, as shown by a photodetector  70  and a corresponding electrical signal S 70 . Because the LP 01  and LP 11  modes from stub fiber  20  have different phases, the power of the excited LP 01  mode in filed fiber  40  exhibits an oscillation behavior as a function of wavelength due to interference effects. The light associated with the higher order mode LP 11  is lost. 
     It is also noted that SM optical fibers that have been used as stub fibers  20  in conventional connectors  10  have been SMFs designed to meet ITU G.652 standards for long-distance transmission in telecommunications systems. However, a stub fiber operates over a decidedly shorter distance and so need not meet this particular standard. Consequently, the fiber  100  disclosed herein for use as a stub fiber is designed to optimize connector performance and is not constrained in performance due to limitations associated with off-the-shelf SMFs that were not designed for use as stub fibers. 
     Gradient-Index Multimode Optical Fiber 
     An aspect of the disclosure is directed to a gradient-index multimode optical fiber that has a MFD MM  substantially matched to the single-mode MFD SM  of a standard SMF. The gradient-index multimode optical fiber is configured to minimize the mode delay or group index difference Δn g  between the fundamental mode LP 01  and the higher-order modes. Because the MPI period is inversely proportional to the group index difference Δn g  between the fundamental mode and higher order modes, adverse MPI effects can be substantially reduced by reducing or minimizing the group index difference. The gradient-index multimode optical fiber disclosed herein can be used in a connector as a stub fiber to reduce insertion loss. In addition, it has the advantage that it can be manufactured using existing multimode fiber production processes. 
     With reference again to  FIG. 2 , the amount of optical power transmitted by field fiber  40  depends on the coupling or transmission efficiencies at the two optical fiber interfaces (joints) I 1  and I 2 . The transmission efficiencies are determined by the amount of misalignment (offset) between the two interfacing fibers, their orientation (angle) at the first and second interfaces, and the polarization of light traveling within the fibers. The amount of transmitted optical power also depends on the attenuation of higher-order modes in the fiber over the length of the fiber segment, and the delay between the different modes. A stub fiber can have a length L in the range from 10 mm to 20 mm, with 15 mm to 20 mm being typical. 
     The transmission efficiency η can be expressed mathematically as: 
     
       
         
           
             
               
                 
                   η 
                   = 
                   
                     
                       
                         η 
                         0101 
                         
                           ( 
                           1 
                           ) 
                         
                       
                        
                       
                         η 
                         0101 
                         
                           ( 
                           2 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           l 
                           , 
                           m 
                         
                       
                        
                       
                         
                           η 
                           
                             01 
                              
                             
                                 
                             
                              
                             lm 
                           
                           
                             ( 
                             1 
                             ) 
                           
                         
                          
                         
                           η 
                           
                             lm 
                              
                             
                                 
                             
                              
                             01 
                           
                           
                             ( 
                             2 
                             ) 
                           
                         
                          
                         
                            
                           
                             
                               - 
                               
                                 α 
                                 lm 
                               
                             
                              
                             L 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           l 
                           , 
                           m 
                         
                       
                        
                       
                         2 
                          
                         
                           
                             
                               η 
                               0101 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                              
                             
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                               0101 
                               
                                 ( 
                                 2 
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                                 01 
                                  
                                 
                                     
                                 
                                  
                                 lm 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                              
                             
                               η 
                               
                                 lm 
                                  
                                 
                                     
                                 
                                  
                                 01 
                               
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                         
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                            
                           
                             
                               - 
                               
                                 
                                   α 
                                   lm 
                                 
                                 2 
                               
                             
                              
                             L 
                           
                         
                          
                         cos 
                          
                         
                           
                             ( 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   lm 
                                 
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                                 L 
                               
                               λ 
                             
                             ) 
                           
                           . 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     In most cases the LP 11  mode is the dominant higher-order mode, in which case: 
     
       
         
           
             
               
                 
                   
                     η 
                     = 
                     
                       
                         
                           η 
                           0101 
                           
                             ( 
                             1 
                             ) 
                           
                         
                          
                         
                           η 
                           0101 
                           
                             ( 
                             2 
                             ) 
                           
                         
                       
                       + 
                       
                         
                           η 
                           0111 
                           
                             ( 
                             1 
                             ) 
                           
                         
                          
                         
                           η 
                           1101 
                           
                             ( 
                             2 
                             ) 
                           
                         
                          
                         
                            
                           
                             
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                               2 
                             
                              
                             
                                 
                             
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                               α 
                               11 
                             
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                             L 
                           
                         
                       
                       + 
                       
                         2 
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                               η 
                               0101 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
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                               0101 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
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                               0111 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                              
                             
                               η 
                               1101 
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                           
                         
                          
                         
                            
                           
                             
                               - 
                               
                                 
                                   α 
                                   11 
                                 
                                 2 
                               
                             
                              
                             L 
                           
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 Δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                   11 
                                 
                                  
                                 L 
                               
                               λ 
                             
                             ) 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where η 01 01   (1)  is the coupling coefficient of the LP 01  mode from the launching fiber to the stub fiber, η 01 01   (2)  is the coupling coefficient from the stub fiber to the receiving fiber, η 01 lm   (1)  is the coupling coefficient from the LP 01  mode to a higher-order mode LP lm , η lm 01   (2)  is the coupling coefficient from the LP lm  to the LP 01  mode at the second joint, Δn lm  is the effective index difference between the LP lm  mode and the LP 01  mode, λ is the operating wavelength of light from a coherent light source (not shown), and α lm  is the attenuation coefficient of the LP lm  mode and is not to be confused with the α parameter associated with the effective refractive index profile Δ. 
     From Eq. (2), the transmission efficiency fluctuation can be expressed as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                              
                             η 
                           
                           
                              
                             λ 
                           
                         
                         = 
                           
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                                         n 
                                         11 
                                       
                                     
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                                   1 
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                              
                             
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                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
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       FIG. 3  is a plot of operating λ wavelength (microns) versus transmission (dB) and shows an example transmission efficiency of an example conventional stub fiber, with the peak-to-peak wavelength spacing Δλ of about 32 nm as shown in the plot. From Eq. (3), the peak-to-peak wavelength spacing (i.e., the transmission fluctuation wavelength spacing) can be obtained by the relationship: 
       Δλ=λ 2 /(Δ n   g11   L ),  (4)
 
     where Δn g11  is the group index difference for the LP 11  mode versus the LP 01  mode and L is the length of the optical fiber. As a reference, for a stub fiber made of SMF-28e fiber, with a group index difference Δn g11 =0.004, an operating wavelength λ=1310 nm and a length L=13.3 mm yields the transmission fluctuation wavelength spacing Δλ of about 32 nm as shown in the plot (for L=15 mm, Δλ is about 29 nm). This size transmission fluctuation wavelength spacing Δλ is relative short and translates into a high probability of MPI occurring over the length of the stub fiber. 
     In example embodiments, fiber  100  disclosed herein has a length L in the range from 13 mm to 20 mm and has a transmission fluctuation wavelength spacing Δλ≧50 nm, and in another example Δλ≧100 nm, and in another example, Δλ≧200 nm, all at an operating wavelength of λ=1310 nm. 
     More generally, the expression in Eq. (4) can contain Δn g , which represents the group index difference between the fundamental mode LP 01  and an arbitrary higher-order mode. In many instances, the approximation Δn g11 ≈Δn g  is adequate, e.g., when the higher-order mode LP 11  is the dominant or sole higher-order mode. 
     Equations (3) and (4) show that there are three main factors that affect the fluctuation in the transmission efficiency: the loss of higher-order modes, the group index difference Δn g  and the coupling coefficients η at the fiber interfaces. The transmission fluctuation wavelength spacing Δλ can be made larger by increasing the loss of the higher-order modes, by reducing the group index difference Δn g  while optimizing the coupling coefficients, or by a combination of these effects. 
       FIG. 4  is a cross-sectional view of an example gradient-index multimode fiber (“fiber”)  100  according to one embodiment of the disclosure. Fiber  100  has a core region (“core”)  110  having a radius r 0  and a cladding region (“cladding”)  120  immediately surrounding the core. In an example, core  110  comprises pure silica glass (SiO 2 ) or silica glass with one or more dopants that increases the relative refractive index Δ in a gradient-index fashion to the maximum Δ 0  at r=0, i.e., on an optical fiber central axis Δ 0 , and that monotically decreases to a value Δ=0 at r=r 0 . Suitable dopants include GeO 2 , Al 2 O 3 , P 2 O 5 , TiO 2 , ZrO 2 , Nb 2 O 5 , and Ta 2 O 5 , as well as combinations thereof. An example cladding  120  has a uniform (i.e., constant) relative refractive index Δ=0 out to the cladding outer radius r F . 
     The fiber  100  disclosed herein has a relatively small group index difference Δn g  for reducing MPI, while the MFD is substantially matched to that of SMFs typically used in connectors. Thus, in an example, the key relative refractive index profile parameters (Δ 0 , α, r 0 ) are selected to minimize the group index difference Δn g  (which minimizes the delay between guided modes) and to substantially match the MFD to that of a typical SMF. In an example, fiber  100  has a group index difference Δn g  in the range from substantially 0 to 2×10 −4 . 
     In one example, Δ 0  is in the range from 0.4% to 2.5%, while in another example Δ 0  is the range from 0.5% to 2%, and in another example Δ 0  is in the range from 0.7% to 2%. Also in an example, the core size (radius) r 0  is in the range from 6 μm to 20 μm, while in another example r 0  is in the range from 7 μm to 16 μm, while in yet another example r 0  is in the range from 8 μm to 16 μm. Also in an example, the parameter α is in the range from 1.9 to 4.1, while in another example α is in the range from 1.9 to 2.5, while in yet another example α is in the range from 1.95 to 2.15. 
       FIG. 5A  is a plot of Δ(%) vs r (μm) for three different Design Examples DE 1 through DE 3 for fiber  100 . Table 1 below summarizes, for each Design Example, the parameters that minimize the group delay at both 1310 nm and 1550 nm, and that provide a MFD MM  that substantially matches the MFD SM  of typical single-mode launching and receiving fibers. For a typical SMF, the MFD SM  is between 8.2 μm and 9.7 μm at 1310 nm, and between 9.2 μm and 10.9 μm at 1550 nm. In the table below, λ C  is the cut-off wavelength of LP 11  mode. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 DESIGN PARAMETERS FOR DE 1 THROUGH DE 3 
               
            
           
           
               
               
               
               
               
            
               
                   
                 PARAMETER 
                 DE 1 
                 DE 2 
                 DE 3 
               
               
                   
                   
               
               
                   
                 Δ 0   
                 2% 
                 0.94% 
                 0.5% 
               
               
                   
                 r 0  (μm) 
                 15.2 
                 10.1 
                 7.1 
               
               
                   
                 α 
                 2 
                 2 
                 3.5 
               
               
                   
                 λ C  (μm) 
                 &gt;5000 
                 3547 
                 2078 
               
               
                   
                 MFD MM  @ 1310 nm 
                 9.2 
                 9.2 
                 9.5 
               
               
                   
                 MFD MM  @ 1550 nm 
                 10.1 
                 10.1 
                 10.4 
               
               
                   
                   
               
            
           
         
       
     
       FIG. 6  plots the group index difference Δn g  versus the effective index n eff  at an operating wavelength λ of 1310 nm and 1550 nm for the Design Example DE 1. Design Example DE 1 has a core radius r 0 =15.2 μm, a maximum relative refractive index Δ 0 =2%, and α=1.99. The MFD MM  is 9.2 μm at 1310 nm and 10.1 μm at 1550 nm. These values of MFD MM  are substantially the same as the nominal values of the MFD SM  of SMFs used as field fibers, which have MFDs of 9.2 μm and 10.4 μm at 1310 nm and 1550 nm, respectively. In an example of fiber  100 , at operating wavelengths λ of 1310 nm and 1550 nm, mode-field diameter MFD MM  satisfies 0.9≦MFD MM /MFD SM ≦0.1. 
     There are 9 mode groups at 1310 nm, and 8 mode groups at 1550 nm. The group index difference Δn g &lt;2×10 −4  for the first 6 mode groups, which is 20 times smaller than the group index difference Δn gSM  for a typical SMF (when it operates as a multimode fiber), can be achieved for the first 7 mode groups at both 1310 nm and 1550 nm wavelengths. This increases the peak-to-peak transmission fluctuation wavelength spacing Δλ from 29 nm to 580 nm. For the first two mode groups LP 01  and LP 11 , which are dominant modes for PI, the group index difference Δn g &lt;2×10 −5 , which is 200 times smaller than the group index difference Δn gSM  for a typical SMF. This means that MPI effects are much less likely to occur in fiber  100  than in a typical SMF. 
     Thus, in an example embodiment, the ratio Δn gSM /Δn g  of the SMF group index difference to the group index difference of fiber  100  is in the range 2≦Δn gSM /Δn g ≦300, with an exemplary value in this range being Δn gSM /Δn g =20, and 200. Thus, in an example, the transmission fluctuation wavelength spacing Δλ, which is inversely proportional to the group index difference (see Eq. 4, above), can be made larger by a factor of between 20 and 200. 
       FIG. 7  is the same plot as  FIG. 6 , but for Design Example DE 2. Design Example DE 2 has a core radius r 0 =10.1 μm, a maximum relative refractive index Δ 0 =0.94%, and α=2.01. The MFD MM  is 9.2 μm at 1310 nm and 10.1 μm at 1550 nm. There are 5 mode groups at 1310 nm, and 4 mode groups at 1550 nm. The group index Δn g &lt;2×10 −4  (which again is 20 times smaller than that for say SMF-28e fiber), can be achieved for the first 2 to 3 mode groups at both 1310 nm and 1550 nm wavelengths. This increases the peak-to-peak transmission fluctuation wavelength spacing Δλ from 29 nm to 580 nm in the aforementioned example stub fiber with a length L=15 mm, and thus mitigates the potential for adverse MPI effects. 
       FIG. 8A  and  FIG. 8B  are similar to  FIGS. 6 and 7  and show the group index difference Δn g  at both 1310 nm and 1550 nm, respectively, for Design Example DE 3. Design Example DE 3 has a core radius r 0 =7.1 μm, a maximum relative refractive index Δ 0 =0.5%, and α=3.5. As the core radius decreases, Δn g  depends on both α and r 0 , with the optimum value occurring when α is about 3.5. For a group index difference Δn g =3×10 −4 , the transmission fluctuation wavelength spacing Δλ is over 300 nm, more than 10 times larger than that for SMF-28e fiber. This substantially reduces the amount of MPI that can occur when fiber  100  is used as a stub fiber. 
       FIG. 9  is a plot similar to the plot of  FIG. 4  for four additional Design Examples DE 4 through DE 7 for fiber  100 . The four example fibers  100  were made using an outside vapor deposition process. The plot of  FIG. 9  shows that all four relative refractive index profiles have a maximum Δ (i.e., Δ 0 ) slightly below 2%, with a core radius of about 15 μm. Table 2 below shows the calculated values for the MFD MM  and Δn g  for each of the four Design Examples DE 4 through DE 7. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 DESIGN PARAMETERS FOR DE 4 THROUGH DE 7 
               
            
           
           
               
               
               
               
               
            
               
                   
                   
                   
                 Δn g  @ 
                 Δn g  @ 
               
               
                   
                 MFD MM  @ 
                 MFD MM  @ 
                 1310 nm 
                 1550 nm 
               
               
                   
                 1310 nm (mm) 
                 1550 nm (mm) 
                 (×10 −5 ) 
                 (×10 −5 ) 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                 DE 4 
                 9.2 
                 10.0 
                 4.9 
                 3.4 
               
               
                 DE 5 
                 9.3 
                 10.1 
                 4.3 
                 2.4 
               
               
                 DE 6 
                 9.2 
                 10.1 
                 3.8 
                 2.0 
               
               
                 DE 7 
                 9.1 
                 9.9 
                 5.3 
                 3.4 
               
               
                   
               
            
           
         
       
     
     The values for MFD MM  for Design Examples DE 4 through DE 7 at 1310 nm range from 9.1 to 9.3 mm, while the values for the MFD MM  at 1550 nm range from 9.9 to 10.1 mm. These MFD MM  values are very close to the MFD SM  value of a standard single mode fiber. The Δn g  for Design Examples DE 4 through DE 7 is below 5.3×10 −5  at 1310 nm, and below 3.4×10 −5  at 155 nm. 
       FIG. 10  is similar to  FIG. 1  and shows an example embodiment of a connector  200  similar to connector  10 , but that includes fiber  100  with opposite ends  102  and  104  serving as a multimode stub fiber. As discussed above, the use of fiber  100  as a stub fiber reduces the adverse effects of MPI. This is because fiber  100  is designed to be compatible with single-mode launch and receive fibers associated with connectors and intentionally supports multiple modes in a manner that reduces adverse MPI effects. 
     It will be apparent to those skilled in the art that various modifications to the preferred embodiment of the disclosure as described herein can be made without departing from the spirit or scope of the disclosure as defined in the appended claims. Thus, the disclosure covers the modifications and variations provided they come within the scope of the appended claims and the equivalents thereto.