Patent Publication Number: US-9846275-B2

Title: Quasi-single-mode optical fiber with a large effective area

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of priority under 35 U.S.C. §119 of U.S. Provisional Application Ser. No. 62/056,784 filed on Sep. 29, 2014, which is related to U.S. Provisional Patent Application Ser. No. 62/086,383, filed Dec. 2, 2014, and entitled “Optical transmission systems and methods using a QSM large-effective-area optical fiber,” which application is incorporated herein by reference. 
    
    
     FIELD 
     The present disclosure relates to optical fibers, and in particular relates to a quasi-single-mode optical fiber with a large effective area. 
     The entire disclosure of any publication or patent document mentioned herein is incorporated by reference, including the publication by I. Roudas, et al., “Comparison of analytical models for the nonlinear noise in dispersive coherent optical communications systems,” IEEE Photonics Conference, paper MG3.4, Bellevue, Wash., September 2013; the publication by Sui et al., “256 Gb/s PM-16-QAM Quasi-Single-Mode Transmission over 2600 km using Few-Mode Fiber with Multi-Path Interference Compensation,” OFC Conference, San Francisco, Calif., Mar. 9-13, 2014, Fiber Non-linearity Mitigation and Compensation (M3C) (ISBN: 978-1-55752-993-0); and the publication by S. J. Savory, “Digital filters for coherent optical receivers,” Optics Express, Vol. 16, No. 2, Jan. 21, 2008, pp. 804-818. 
     BACKGROUND 
     Optical fibers are used for a variety of applications, especially in long-haul, high-speed optical communications systems. Optical fibers have an optical waveguide structure that acts to confine light to within a central region of the fiber. One of the many benefits of optical fibers is their ability to carry a large number of optical signals in different channels, which provides for high data transmission rates and a large bandwidth. 
     The increasing demand for bandwidth and higher data transmission rates has resulted in optical fibers carrying more channels and higher amounts of optical power. At some point, however, the optical power carried by the optical fiber can give rise to non-linear effects that distort the optical signals and reduce the transmission capacity of the optical communications system. Consequently, there is a practical limit to how much optical power an optical fiber can carry. 
     Because the optical power is confined by the waveguide structure of the optical fiber, the intensity determines the severity of non-linear effects in the optical fiber. The intensity is defined as the amount of optical power in the guided light divided by the (cross-sectional) area over which the guided light is distributed. This area is referred to in the art as the “effective area” A eff  of the optical fiber. The effective area A eff  is calculated from the electromagnetic field distribution of the light traveling within the optical fiber using techniques and methods known in the art. 
     It is well-known that optical fibers with large effective areas A eff  are desirable in optical transmission systems because of their relatively high power threshold for nonlinear distortion impairments. The larger the effective area A eff , the lower the intensity and thus the less non-linear effects. Because of this feature, an optical fiber with a large effective area A eff  may be operated at higher optical powers, thereby increasing the optical signal-to-noise ratio (OSNR). 
     Unfortunately, the effective area A eff  of optical fibers cannot simply be increased without bound. The conventional wisdom in the art is that an effective area A eff  of about 150 μm 2  is the limit for a true single-mode fiber to maintain sufficient bend robustness, (i.e., reduced loss due to bending). In some cases, an effective area A eff  of 150 μm 2  may in fact already be too large for some bending-loss requirements. However, the bending loss of an optical fiber can be reduced by increasing the mode confinement and hence the cutoff wavelength of the optical fiber associated with single-mode operation. Increasing the effective area A eff  beyond present-day values would require raising the cutoff wavelength to be above the signal wavelength, thereby resulting in few-mode operation, which gives rise to undesirable optical transmission impairments such as modal dispersion and multipath interference (MPI). 
     Alternatives to increasing the effective area A eff  of the optical fiber to reduce adverse non-linear effects include decreasing the effective nonlinear index n 2 . The nonlinear physics of an optical fiber depends on the ratio n 2 /A eff . However, changing n 2  is difficult and the resulting effect is likely to be very small. Reducing the fiber attenuation is another alternative for better transmission performance. A lower fiber attenuation reduces the need for amplification and thus reduces the noise of the transmission link, which in turn reduces the required signal power for a given required OSNR. However, reducing the attenuation of the optical fiber impacts the optical fiber transmission system in a different way than by changing the effective area A eff , so that these two parameters cannot be exactly traded off. 
     What is needed therefore is a more robust type of large-effective-area optical fiber that reduces adverse non-linear effects while also having sufficiently small bending loss. 
     SUMMARY 
     An aspect of the disclosure is a QSM optical fiber. The QSM fiber includes a core having a centerline and an outer edge, with a peak refractive index n 0  on the centerline and a refractive index n 1  at the outer edge. A cladding section surrounds the core and has a first inner annular cladding region immediately adjacent the core. The core and cladding section support a fundamental mode LP 01  and a higher-order mode LP 11  and define: i) for the fundamental mode LP 01 : an effective area A eff &gt;170 μm 2  and an attenuation of no greater than 0.17 dB/km at 1530 nm; ii) for the higher-order mode LP 11 : an attenuation of at least 1.0 dB/km at 1530 nm; and iii) a bending loss of BL&lt;0.02 dB/turn at 1625 nm and for a bend diameter D B =60 mm. 
     In one example of the QSM fiber described above, the effective area A eff &gt;200 μm 2  at 1530 nm. 
     Another aspect of the disclosure is a QSM optical fiber that has a core having a centerline and a radius r 1  greater than 5 μm, with a peak refractive index n 0  on the centerline and a refractive index n 1  at the radius r 1 ; a cladding section surrounding the core, wherein the cladding section includes a first inner annular cladding region immediately adjacent the core with a minimum refractive index n 2 , a second inner annular cladding region immediately adjacent the first inner annular cladding region and having minimum refractive index n 3 , and a ring immediately adjacent the second inner annular cladding region and having a refractive index n R , wherein n 0 &gt;n 1 &gt;n 3 &gt;n 2  and n R &gt;n 3 &gt;n 2 ; wherein the core and cladding section support a fundamental mode LP 01  and a higher-order mode LP 11  and define: i) for the fundamental mode LP 01 : an effective area A eff &gt;150 μm 2  and an attenuation of no greater than 0.17 dB/km at 1530 nm; ii) for the higher-order mode LP 11 : an attenuation of at least 1.0 dB/km at 1530 nm; and iii) a bending loss of BL&lt;0.02 dB/turn at 1625 nm and for a bend diameter D B =60 mm. 
     In one example of the QSM fiber described immediately above, the effective area A eff &gt;170 μm 2 , while in another example, the effective area A eff &gt;200 μm 2 . In another example, n 1 &gt;n R , while in other examples n 1 =n R  and n 1 &lt;n R . 
     Another aspect of the disclosure is an optical transmission system that includes the QSM optical fiber as disclosed herein. The optical transmission system further includes an optical transmitter configured to emit light that defines an optical signal that carries information; an optical receiver optically coupled to the optical transmitter by the QSM fiber and configured to receive the light emitted by the optical transmitter and transmitted over the QSM optical fiber in the fundamental mode LP 01  and the higher-order mode LP 11  thereby giving rise to multipath interference (MPI), wherein the optical receiver generates an analog electrical signal from the received light; an analog-to-digital converter (ADC) that converts the analog electrical signal into a corresponding digital electrical signal; and a digital signal processor electrically connected to the ADC and configured to receive and process the digital electrical signal to mitigate the MPI and generate a processed digital signal representative of the optical signal from the optical transmitter. 
     Additional features and advantages are set forth in the Detailed Description that follows, and in part will be readily apparent to those skilled in the art from the description or recognized by practicing the embodiments as described in the written description and claims hereof, as well as the appended drawings. It is to be understood that both the foregoing general description and the following Detailed Description are merely exemplary, and are intended to provide an overview or framework to understand the nature and character of the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The accompanying drawings are included to provide a further understanding, and are incorporated in and constitute a part of this specification. The drawings illustrate one or more embodiment(s), and together with the Detailed Description serve to explain principles and operation of the various embodiments. As such, the disclosure will become more fully understood from the following Detailed Description, taken in conjunction with the accompanying Figures, in which: 
         FIG. 1  is a front elevated view of a section of quasi-single-mode (QSM) fiber as disclosed herein; 
         FIG. 2  is a plot of the refractive index n versus radius r that illustrates an example refractive index profile for an example of the QSM fiber of  FIG. 1 ; 
         FIG. 3  is similar to  FIG. 2  and illustrates an example refractive index profile for the QSM fiber that does not include the ring portion of the cladding section; 
         FIG. 4  is a close-up, cross-sectional view of the QSM fiber of  FIG. 1 , illustrating an example where the fiber includes an axial (longitudinal) refractive-index perturbation designed to provide substantial attenuation of the higher-order mode while not substantially attenuating the fundamental mode; 
         FIG. 5  is similar to  FIG. 2  and illustrates in a single plot three different refractive index profiles p 1 , p 2  and p 3  for example QSM fibers, wherein the inner cladding for the different profiles has different depths; 
         FIG. 6  is a plot of the bending loss BL (dB/turn) versus the bending diameter D B  (mm) for the three refractive index profiles p 1 , p 2  and p 3  of  FIG. 5 ; 
         FIG. 7  is a plot of the ratio of the measured outputted optical power P OUT  to the inputted optical power P IN  (P OUT /P IN ) versus the wavelength λ (nm) for the three different refractive index profiles p 1 , p 2  and p 3  of  FIG. 5 , wherein the plot is used to calculate the cutoff wavelength λ c  from multimode to single-mode operation; 
         FIGS. 8A and 8B  plot of the signal power distribution SP in arbitrary power units (a.p.u.) as a function of the differential mode delay or DMD (ns) in a two-mode (LP 01  and LP 11 ) QSM optical fiber for the case where there is negligible mean differential modal attenuation or DMA ( FIG. 8A ) and for the case where there is a high DMA ( FIG. 8B ), wherein the solid shows the total power, the dashed line shows the power in the fundamental mode LP 01 , and the dashed-dotted line shows the power in the higher-order mode LP 11 ; 
         FIGS. 9A and 9B  are plots of the effective DMD, denoted DMD Eff , versus the DMA (dB/km) for an example optical transmission system that employs an example of the QSM fiber disclosed herein, wherein for  FIG. 9A  the DMD units are nanoseconds whereas in  FIG. 9B  the DMD Eff  is in units of the tap (temporal) spacing τ of the signal processor; and 
         FIG. 10  is a schematic diagram of an optical transmission system that employs the QSM fiber as disclosed herein along with MPI compensation to recover the signal that travels in the fundamental LP 01  mode of the QSM 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. 
     Terminology 
     The term “relative refractive index,” as used herein in connection with the multimode fibers and fiber cores discussed below, is defined as:
 
Δ( r )=[ n ( r ) 2   −n   S   2 )]/(2 n   S   2 )
 
where n(r) is the refractive index at radius r, unless otherwise specified and n S  is the reference index. The relative refractive index is defined at the operating wavelength λ p . In another aspect, n S  is the index of undoped silica (SiO 2 ). The maximum index of the index profile is denoted n 0 , and in most cases, n 0 =n(0).
 
     As used herein, the relative refractive index is represented by Δ and its values are given in units of “%,” unless otherwise specified. In the discussion below, the reference index n REF  is that for pure silica. 
     The term “dopant” as used herein generally refers to a substance that changes the relative refractive index of glass relative to pure (undoped) SiO 2  unless otherwise indicated. 
     The term “mode” is short for a guided mode or optical mode. A “multimode” optical fiber means an optical fiber designed to support the fundamental guided mode and at least one higher-order guided mode over a substantial length of the optical fiber, such as 2 meters or longer. A “single-mode” optical fiber is an optical fiber designed to support a fundamental guided mode only over a substantial length of the optical fiber, such as 2 meters or longer. A “few mode” or “few-moded” optical fiber is an optical fiber designed to support a fundamental guided mode and one or two higher-order modes over a substantial length of the optical fiber, such as 2 meters or longer. A “quasi-single mode” fiber is distinguished from a “few-mode” fiber in that the former seeks to use only the fundamental mode to carry information while the latter seeks to use all of the few modes to carry information. 
     The term “cutoff” is used herein refers to the cutoff wavelength λ c  that defines the boundary for single-mode and multimode operation of an optical fiber, wherein single-mode operation of the fiber occurs for wavelengths λ&gt;λ c . The cutoff wavelength λ c  as the term is used herein can be measured by the standard 2 m fiber cutoff test, FOTP-80 (EIA-TIA-455-80), to yield the “fiber cutoff wavelength,” also known as the “2 m fiber cutoff” or “measured cutoff”. The FOTP-80 standard test is performed to either strip out the higher order modes using a controlled amount of bending, or to normalize the spectral response of the fiber to that of a multimode fiber. 
     For examples of the QSM fiber disclosed herein, the cutoff wavelength λ c &gt;1600 nm, or more preferably λ c &gt;1700 nm, or more preferably λ c &gt;1750 nm, or even more preferably λ c &gt;1800 nm. 
     The number of propagating modes and their characteristics in a cylindrically symmetric optical fiber with an arbitrary refractive index profile is obtained by solving the scalar wave equation (see for example T. A. Lenahan, “Calculation of modes in an optical fiber using a finite element method and EISPACK,” Bell Syst. Tech. J., vol. 62, no. 1, p. 2663, February 1983). The light traveling in an optical fiber is usually described (approximately) in terms of combinations of LP (linear polarization) modes. The LP 0p  modes with p&gt;0 have two polarization degrees of freedom and are two-fold degenerate. The LP mp  modes with m&gt;0, p&gt;0 have both two polarization and two spatial degrees of freedom. They are four-fold degenerate. In the discussion herein, polarization degeneracies are not counted when designating the number of LP modes propagating in the fiber. For example, an optical fiber in which only the LP 01  mode propagates is a single-mode fiber, even though the LP 01  mode has two possible polarizations. A few-mode (or “few moded”) optical fiber in which the L 01  and LP 11  modes propagate supports three spatial modes but nevertheless is referred herein as having two modes for ease of discussion. 
     As used herein, the “effective area” A eff  of an optical fiber is the cross-sectional area of the optical fiber through which light is propagated and is defined as: 
                 A   eff     =     2   ⁢           ⁢   π   ⁢           ⁢         (       ∫   0   ∞     ⁢       E   2     ⁢           ⁢   rdr       )     2         ∫   0   ∞     ⁢       E   4     ⁢           ⁢   rdr             ,         
where E is the electric field associated with light propagated in the fiber and r is the radius of the fiber. The effective area A eff  is determined at a wavelength of 1550 nm, unless otherwise specified.
 
     Macrobend performance of the example QSM fibers disclosed herein was determined according to FOTP-62 (IEC-60793-1-47) by wrapping 2 turns around a mandrel having a diameter D B (e.g., D B =60 mm) and measuring the increase in attenuation due to the bending using an encircled flux (EF) launch condition. 
     In the discussion below, any portion of the optical fiber that is not the core is considered part of the cladding, which can have multiple sections. In some of the Figures (e.g.,  FIG. 1  and  FIG. 4 ), the cladding is shown has having a limited radial extent (e.g., out to radius r g ) for ease of illustration even though the cladding in principle extends beyond this limit. 
     The C-band is defined as the wavelength range from 1530 nm to 1565 nm; The L-band is defined as the wavelength range from 1565 nm to 1625 nm; and the C+L wavelength band is defined as the wavelength range from 1530 nm to 1625 nm. 
     The limits on any ranges cited herein are considered to be inclusive and thus to lie within the stated range, unless otherwise specified. 
     QSM Optical Fiber 
       FIG. 1  is an elevated view of a section of a QSM fiber  10  as disclosed herein. The QSM fiber  10  has a body  11  configured as described below and includes a centerline  12  that runs longitudinally down the center of the QSM fiber. 
       FIG. 2  is a plot of the refractive index n versus radius r of QSM fiber  10  as measured from centerline  12 , illustrating an example refractive index configuration (profile) for the QSM fiber. The QSM fiber  10  has a central core (“core”)  20  with a cladding section  30  surrounding the core. In an example, core  20  is made primarily silica and preferably alkali doped, e.g., potassium doped silica. Core  20  is preferably substantially free, and preferably entirely free, of GeO 2 . Core  20  may also include fluorine as a dopant. 
     The cladding section  30  includes a number of regions, namely a first inner annular cladding region or “inner cladding”  32 , a second inner annular cladding region or “moat”  34  surrounding the inner cladding, and an annular outer cladding region or “ring” 38  surrounding moat  34 . The shape of the core  20  is approximately triangular, but can vary from a step profile to an alpha profile. The core  20  has an outer edge  21  at a radius r e , which can be considered the core radius, which in example is also equal to radius r 1 . In one example, the core radius r e  or r 1 &gt;5 μm, while in another example, r e  or r 1 &gt;7 μm. 
     In an example, neither the core  20  nor the cladding section  30  includes germanium. The different regions of cladding section  30  may be made of fluorine-doped silica. In an example, cladding section  30  is doped with fluorine while core  20  is doped with potassium. 
     The example refractive index profile of the example QSM fiber  10  of  FIG. 2  can be described by nine fiber parameters (P): Five refractive indices n 0 , n 1 , n 2 , n 3  and n R , and four radii r 1 , r 2 , r 3  and r R . The refractive index n 0  is the peak refractive index and occurs at r=0, i.e., on centerline  12  within core  20 . The refractive index n 1  represents the refractive index at the interface between the core  20  and the adjacent inner cladding  32 , i.e., at the core edge  21 , which in an example is associated with radius r e . The refractive index n 2  represents the minimum refractive index for inner cladding  32 . The refractive index n 3  represents the minimum refractive index for moat  34 . The refractive index n R  represents the refractive index of ring  38 . 
     In an example, the radius r 1  represents both the radius of core  20  and the inner radius of inner cladding  32 , while the radius r 2  represents the outer radius of the inner cladding. The radius r 3  represents the outer radius of moat  34 . The radius r R  represents the inner radius of ring  38 . The radius r g  represents the radius where ring  38  ends and the glass coating  39  of refractive index n g  that makes up the rest of the QSM fiber  10  begins. 
     In an example, the nine fiber parameters P are designed for a nominal glass radius r g =62.5 μm. Small adjustments, to especially the cladding parameters (r 3 , n 3 ) and ring parameters (n R , r R ) may be required if the fiber glass radius r g  is changed, which is optional for reducing bending loss. In  FIG. 1 , the core edge radius r e  is slightly smaller than the inner cladding radius r 1  due to shortcomings in the refractive index measurement. In the plot of  FIG. 5  discussed below, the transition from core  20  to inner cladding  32  is vertical so that r e =r 1 . 
     In an example embodiment of QSM fiber  10 , n 0 &gt;n 1 &gt;n 3 &gt;n 2 . In another example, n 1 &gt;n R , while another example n 1 ≦n R . Also in an example, n R &gt;n 3 &gt;n 2 . 
       FIG. 3  is similar to  FIG. 2  and illustrates an example refractive index profile for an example QSM fiber  10  wherein the cladding region  30  does not include the outer ring  38 . For the “no-ring” profile of  FIG. 3 , the inner radius of inner cladding  32  is denoted r i  and has an associated refractive index n i . The shape of the core  20  is approximately step-like in the example, but can vary from a step profile to an alpha profile. The small bumps bi and b 2  in the refractive index profile of  FIG. 3  are features arising from the expected draw stress distribution and are not critical to the design. As noted above, the inner radius r i  of inner cladding  32  can be equal to the radius r 1  of core  20 . 
     The QSM fiber  10  disclosed herein has a relatively large effective area A eff , which in one example is A eff &gt;150 μm 2 , while in another example is A eff &gt;170 μm 2 , while yet in another example is A eff &gt;200 μm 2 . The QSM fiber  10  is designed to be operated using only the fundamental mode LP 01  just as in single-mode fiber, while the one additional higher-order mode LP 11  is not used. The one additional higher-order mode LP 11  can impair the transmission of optical signals traveling in the QSM fiber unless appropriate MP-compensating digital signal processing is applied to the received (transmitted) signal. 
     In an example, the fundamental mode LP 01  has a fundamental-mode effective index, the higher-order mode LP 11  has a higher-order-mode effective index, and wherein a difference Δn eff  between the fundamental-mode effective index and the higher-order-mode effective index is |Δn eff |&gt;0.001 at a wavelength of 1550 nm. 
     Higher-Order-Mode Impairments 
     The main two impairments caused by the presence of the higher-order mode LP 11  in QSM fiber  10  are multipath interference (MPI) and excess loss (EL). An aspect of the disclosure includes using QSM fiber  10  for optical signal transmission while electronically mitigating MPI of the optical signal using digital signal processing techniques that are known in the art and as described in greater detail. The electronic mitigation of MPI effects enables the deployment of QSM fiber  10  in an optical transmission system. To this end, in an example, the aforementioned parameters P of QSM fiber  10  are substantially optimized, while the excess loss EL, which cannot be compensated, is substantially minimized (e.g., made substantially zero). This avoids having the excess loss EL reduce the benefit of having a relatively large effective area A eff  used to overcome detrimental non-linear effects, as explained above. 
     Because the higher-order mode LP 11  of QSM fiber  10  is undesirable and unused, the design and configuration of QSM fiber  10  is different than that for conventional few-mode optical fibers that seek to transmit information in the higher-order modes. In particular, because conventional few-mode optical fibers seek to utilize the information transmitted in the few higher-order modes, these modes need to have relatively low differential modal attenuation (DMA). As is explained in greater detail below, the QSM fiber  10  disclosed herein has relatively high DMA, i.e., the higher-order mode LP 11  is intentionally subjected to a relatively large attenuation to reduce the degree of optical transmission impairment caused by this higher-order mode. 
     Ideally, QSM fiber  10  would have a relatively large phase index difference between all supported modes to minimize mode-coupling, while at the same time having a small group index difference between all supported modes. This latter attribute minimizes the digital signal processing required to remove MPI artifacts from the received signal. Unfortunately, this is not possible in fibers with large effective area A eff . Qualitatively, this is because, for any mode, the group index (n g ) is related to phase index (or “effective index” n e ) as follows: 
               n   g     =       n   e     -     λ   ⁢       dn   e       d   ⁢           ⁢   λ                 
The difference in the group index n g  between two modes is therefore given by:
 
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               
                 n 
                 g 
               
             
             = 
             
               
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   n 
                   e 
                 
               
               - 
               
                 λ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 Δ 
                 ⁢ 
                 
                   
                     dn 
                     e 
                   
                   
                     d 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     λ 
                   
                 
               
             
           
         
       
     
     In the limit of very large effective area A eff , the wavelength dispersion of all modes approaches that of the bulk glass, in which case the last term in the equation for Δn g  vanishes so that Δn g ≈Δn e . Consequently, one cannot simultaneously have a low mode coupling (large Δn e ) and a small differential mode delay (DMD, small Δn g ). 
     In the QSM fiber  10  disclosed herein, low mode coupling is substantially preserved while, as noted above, the DMD is managed by intentionally designing the QSM fiber to have as much loss (i.e., a high DMA) as possible for the higher-order mode LP 11 . A high DMA reduces the number of equalizer taps (i.e., memory) required in the digital signal processor used for MPI compensation, thereby reducing system complexity, as described below. High DMA values also reduce the total MPI level, which may have an upper limit in terms of the efficacy of the MPI compensation digital signal processing. 
     In one example, the DMA for a wavelength of 1530 nm is DMA≧1.0 dB/km, while in another example, the DMA≧4.0 dB/km. Also in one example, the coupling coefficient CC between the fundamental mode LP 01  and the higher-order mode LP 11  at a wavelength of 1530 nm is CC&lt;0.002 km −1 , while in another example, the coupling coefficient CC&lt;0.001 km −1 . 
     One way to increase the DMA for the higher-order mode LP 11  is to shift the cutoff wavelength λ c  to its lowest possible value consistent with macrobend requirements. Another way is to make the higher-order modes lossy in a mode-selective way.  FIG. 4  is a close-up cross-sectional view of a portion of an example QSM fiber  10  that includes an axial (longitudinal) refractive-index perturbation  52 .  FIG. 4  includes a plot of refractive index n versus the axial distance z down the QSM fiber that illustrates an example form of the refractive-index perturbation having a constant period Λ. The refractive-index perturbation  52  is configured to increase the attenuation (DMA) of the higher-order mode LP 11  while not substantially increasing the attenuation of the fundamental mode LP 01 . In an example, refractive-index perturbation  52  is in the form of a long-period grating that substantially matches a difference in the effective indices of the higher-order mode LP 11  and a radiative cladding mode at the operating wavelength, i.e., a period Λ≈1/Δn, where Δn is the effective index difference between the higher-order mode LP 11  and the radiative cladding mode). 
     In an example, axial refractive-index perturbation  52  has a wavelength resonance and includes a non-constant (e.g., chirped) period Λ that serves to to widen the bandwidth of the resonance as compared to the constant period configuration. In an example, axial refractive-index perturbation  52  can be formed in QSM fiber  10  using known methods, such as laser irradiation. In an example, the axial refractive-index perturbation  52  can be formed as the fiber is being drawn, such as by irradiating the fiber with one or more lasers. In an example, the period Λ of the refractive-index perturbation is chosen such that there is substantially no resonant coupling of the LP 01  and LP 11  modes in the C+L bands, and in an example at a wavelength of 1530 nm. 
     The so-called “Gaussian Noise (GN)” model of optical transmission posits that the launch-power-optimized system Q-factor scales with the effective area A eff  as:
 
 Q   2   ∝A   eff   2/3  
 
so that increasing the effective area A eff  from 150 to 175 μm 2  increases Q 2  by about 11% or 0.45 dB. Increasing the effective area A eff  from 150 to 250 μm 2  increases Q 2  by 41% or 1.5 dB. An example simulation was carried out for an erbium-doped fiber-amplified (EDFA) polarization-multiplexed (PM)-16QAM (Quadrature Amplitude Modulation) optical transmission system having 80 channels, a 32 GHz (Nyquist) channel spacing, a 50 km span length, ideal (noise and distortion-free) transmitter and receivers and a QSM fiber  10  with span loss of 0.158 dB/km. The simulation shows that increasing the effective area A eff  from 150 μm 2  to 250 μm 2  increases the reach at 11.25 dB from 3000 km to 4000 km. Hence, while a 1.5 dB increase in optimal Q 2  seems small, it can lead to a significant reach improvement.
 
     This simulation suggests that with 50 km spans, increasing the effective area A eff  from 150 μm 2  to 250 μm 2  and increasing the span loss from 0.158 dB/km to 0.215 dB/km produces no net change in Q 2 . Hence the excess loss EL (i.e., the additional loss resulting from mode coupling above the intrinsic LP 01  attenuation) of just 0.057 dB/km can completely erase the advantage of the increase in effective area A eff . An excess loss EL of even 0.01 dB/km can decrease the reach of QSM fiber  10  with an effective area A eff =250 μm 2  by about 200 km. The advantage of large effective area fibers with an effective area A eff  of less than 250 μm 2  would likewise be reduced. 
     It was found through modeling that conventional refractive index profiles cannot achieve sufficiently large DMA and effective areas A eff  exceeding 175 μm 2  without also introducing excess macrobend loss. However, it was also found that the judicious addition of the ring  38  of increased refractive index n R  relative to the refractive index n c  of the outer cladding  34  can enhance the LP 11  mode coupling to the glass coating  39 , thereby increasing the DMA without significantly impacting bend performance. In this regard, the index n R  of the ring  38  must not exceed the effective index n eff  of the fundamental mode. In an example, ring  38  includes at least one absorbing dopant that contribute to the attenuation of the higher-order mode LP 11 . Examples of absorbing dopants include titanium or other transition metals. In another example, ring  38  does not include any absorbing dopants. In an example, ring  38  includes fluorine dopant, which is not an absorbing dopant. 
     Example QSM Fibers 
     Table 1 below sets forth example QSM fiber parameters P for three examples of QSM fiber  10 . In the Tables below, P stands for the given parameter, “MIN 1 ” and “MAX 1 ” stand for first example minimum and maximum values for the given parameter, “MIN 2 ” and “MAX 2 ” for second example minimum and maximum values for the given parameter, and “MIN 3 ” and “MAX 3 ” for third example minimum and maximum values for the given parameter. The parameters P in the following Tables are based on QSM fiber  10  having a nominal radius r g =62.5 μm. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 EXAMPLE 1 
               
            
           
           
               
               
               
               
               
               
               
            
               
                 P 
                 MIN0 
                 MAX0 
                 MIN1 
                 MAX1 
                 MIN2 
                 MAX2 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 n 0   
                 1.4430 
                 1.4450 
                 1.4436 
                 1.4448 
                 1.4438 
                 1.4447 
               
               
                 n 1   
                 1.4430 
                 1.4450 
                 1.4430 
                 1.4436 
                 1.4432 
                 1.4434 
               
               
                 n 2   
                 1.4400 
                 1.4430 
                 1.4406 
                 1.4419 
                 1.4408 
                 1.4415 
               
               
                 n 3   
                 1.4390 
                 1.4430 
                 1.4406 
                 1.4422 
                 1.4408 
                 1.4412 
               
               
                 r 1  [μm] 
                 5 
                 15 
                 7 
                 12 
                 8 
                 11 
               
               
                 r 2  [μm] 
                 25 
                 38 
                 28 
                 35 
                 31 
                 33 
               
               
                 r 3  [μm] 
                 40 
                 62.5 
                 45 
                 55 
                 48 
                 52 
               
               
                 r R  [μm] 
                 40 
                 62.5 
                 47 
                 57 
                 50 
                 54 
               
               
                   
               
            
           
         
       
     
     Table 2 below is an alternative representation of the refractive index data of Table 1. In Table 2, the refractive index change relative to pure silica is used. This refractive index change is represented by the relative refractive index Δ, which is given by 
               Δ   =         n   2     -     n   s   2         2   ⁢           ⁢     n   s   2           ,         
where n is the refractive index value from the tables above (at 1550 nm) and n S =1.444374, the refractive index of pure silica.
 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 EXAMPLE 1 USING Δ VALUES 
               
            
           
           
               
               
               
               
               
               
               
            
               
                 P 
                 MIN 
                 MAX 
                 MIN1 
                 MAX1 
                 MIN2 
                 MAX2 
               
               
                   
               
               
                 Δ 0   
                 −9.5082E−04 
                   4.3350E−04 
                 −5.3573E−04 
                   2.9498E−04 
                 −3.9733E−04 
                   2.2573E−04 
               
               
                 Δ 1   
                 −9.5082E−04 
                   4.3350E−04 
                 −9.5082E−04 
                 −5.3573E−04 
                 −8.1248E−04 
                 −6.7411E−04 
               
               
                 Δ 2   
                 −3.0237E−03 
                 −9.5082E−04 
                 −2.6095E−03 
                 −1.7114E−03 
                 −2.4714E−03 
                 −1.9878E−03 
               
               
                 Δ 3   
                 −3.7137E−03 
                 −9.5082E−04 
                 −2.6095E−03 
                 −1.5040E−03 
                 −2.4714E−03 
                 −2.1951E−03 
               
               
                 r 1  [μm] 
                 5 
                 15 
                 7 
                 12 
                 8 
                 11 
               
               
                 R 2  [μm] 
                 25 
                 38 
                 28 
                 35 
                 31 
                 33 
               
               
                 r 3  [μm] 
                 40 
                 62.5 
                 45 
                 55 
                 48 
                 52 
               
               
                 r R  [μm] 
                 40 
                 62.5 
                 47 
                 57 
                 50 
                 54 
               
               
                   
               
            
           
         
       
     
     The second example of QSM fiber  10  is set forth in Table 3 below and represents an example of the “no ring” configuration such as shown in  FIG. 3 . 
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 EXAMPLE (NO RING) 
               
            
           
           
               
               
               
               
               
               
               
            
               
                 P 
                 MIN 
                 MAX 
                 MIN1 
                 MAX1 
                 MIN2 
                 MAX2 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 n 0   
                 1.4435 
                 1.4445 
                 1.4437 
                 1.4443 
                 1.4438 
                 1.4442 
               
               
                 n 1   
                 1.4430 
                 1.4450 
                 1.4427 
                 1.4438 
                 1.4430 
                 1.4435 
               
               
                 n i   
                 1.4410 
                 1.4420 
                 1.4412 
                 1.4418 
                 1.4413 
                 1.4417 
               
               
                 n 2   
                 1.4397 
                 1.4413 
                 1.4400 
                 1.4412 
                 1.4402 
                 1.4409 
               
               
                 n 3   
                 1.4380 
                 1.4410 
                 1.4387 
                 1.4405 
                 1.4390 
                 1.4402 
               
               
                 r 1 [μm] 
                 5 
                 12 
                 5 
                 10 
                 6 
                 9 
               
               
                 r i  [μm] 
                 6 
                 13 
                 7 
                 12 
                 7 
                 11 
               
               
                 r 2  [μm] 
                 18 
                 33 
                 19 
                 29 
                 20 
                 25 
               
               
                   
               
            
           
         
       
     
     The third example of QSM fiber  10  is set forth in Table 4 below and represents another example of the “no ring” configuration. 
                     TABLE 4                  EXAMPLE 3 (NO RING)                                         P   MIN   MAX   MIN1   MAX1   MIN2   MAX2                                                 n 0     1.4435   1.4445   1.4437   1.4443   1.4438   1.4442       n 1     1.4425   1.4435   1.4426   1.4433   1.4427   1.4432       n i     1.4410   1.4420   1.4412   1.4418   1.4405   1.4409       n 2     1.4397   1.4413   1.4400   1.4412   1.4402   1.4409       n 3     1.4380   1.4410   1.4387   1.4405   1.4390   1.4402       r 1 [μm]   6   14   6   12   7   10       r i  [μm]   7   15   8   14   9   13       r 2  [μm]   25   35   20   34   25   30                    
QSM Properties of Example Profiles
 
       FIG. 5  is similar to  FIG. 2  and shows first, second and third example refractive index profiles p 1 , p 2  and p 3  (solid, dashed and dotted lines, respectively) for example QSM fibers  10 , wherein the different index profiles have different depths for inner cladding  32 .  FIG. 6  is a plot of the predicted bend loss BL (dB/turn) versus bend diameter D B  (mm) at a wavelength of 1625 nm as obtained using optical modeling. The three solid straight lines in  FIG. 5  are approximate upper bounds for the example profiles p 1 , p 2  and p 3  based on fitting the oscillation peaks. All three example profiles p 1 , p 2  and p 3  yield a bending loss BL&lt;5 mdB/turn at a bend diameter D B  of 60 mm. 
       FIG. 7  is a plot of the ratio of the measured outputted optical power P OUT  to the inputted optical power P IN  (P OUT /P IN ) versus the wavelength λ (nm) for the three different refractive index profiles p 1 , p 2  and p 3  of  FIG. 5 , wherein the plot is used to calculate the cutoff wavelength λ c  from multimode to single-mode operation. 
     Table 5 below summarizes the predicted optical properties of the three example profiles p 1 , p 2  and p 3  of  FIG. 5 . The values for the bending loss BL are obtained from the straight-line fits of  FIG. 6  while the cut-off wavelengths λ c  are estimated from the power trace plots of  FIG. 7 . The effective area A eff  is measured in μm 2  at λ=1550 nm. The straight fiber LP 11  mode cutoff wavelength λ c  is measured in nanometers (nm). The straight-fiber LP 11  mode radiative loss at 1550 nm is denoted RL and is measured in dB/km. The fundamental mode macro-bend loss BL is measured in dB/turn at λ=1625 nm and a bend diameter D B =60 mm. 
                     TABLE 5                  PREDICTED OPTICAL PROPERTIES FOR 3 EXAMPLE PROFILES                                 Profile   A eff  [μm 2 ]   λ c  [nm]   RL [dB/km]   BL [dB/turn]                                         p1   237   1800   11.4   1.9 × 10 −3         p2   232   1850   5.1   0.9 × 10 −3         p3   227   1885   2.3   0.6 × 10 −3                      
Relationship Between DMA and N T  
 
     One of the advantages of QSM fiber  10  is that it reduces the number of taps needed for the digital signal processor used for MPI compensation in an optical transmission system.  FIGS. 8A and 8B  plot the signal power distribution SP in arbitrary power units (a.p.u.) as a function of the DMD (ns) in a two-mode (LP 01  and LP 11 ) QSM fiber  10 . The solid shows the total power; the dashed line shows the power in the fundamental mode LP 01 ; the dashed-dotted line shows the power in the higher-order mode LP 11 . 
     In  FIG. 8A , there is negligible mean DMA, while in  FIG. 8B  here is a high mean DMA. All other fiber parameters P were kept the same, and in both cases the signal was launched into the fundamental mode LP 01  only. The amount of significantly delayed contributions (the tail of the dashed black line) is decreased as the DMA increases. This enables use of a QSM fiber  10  having a relatively large DMD with a digital signal processor having a reduced number N T  of taps as compared to conventional MPI compensation. 
     The amount of significantly delayed contributions (the tail of the dashed black line) is decreased as the DMA increases. This enables use of a QSM fiber  10  having a relatively large DMD with a digital signal processor having a reduced number N T  of taps as compared to conventional MPI compensation. 
       FIGS. 9A and 9B  plot the effective DMD, denoted DMD Eff , versus the DMA (dB/km) for an example optical transmission system that utilizes the QSM fiber  10  disclosed herein. In  FIG. 9A , the DMD Eff  has units of nanoseconds (ns) while in  FIG. 9B  the DMD Eff  is in units of the tap (temporal) spacing τ of the signal processor, wherein each tap has duration of 31.25 ps. The effective DMD is defined as the time interval that includes 99.95% of the interfering pulse energy and represents the amount of delay the digital signal processor needs to compensate for MPI. The calculations used to generate  FIGS. 9A and 9B  assume a DMD of 1 ns/km and a length L=100 km of QSM fiber  10 . 
     The plots of  FIGS. 9A and 9B  show the effect of the non-zero DMA on the number N T  of taps needed to compensate for the optical transmission impairment of the information-carrying optical signal traveling in the fundamental mode. The calculation of the required number N T  of equalizer taps is approximate. The calculation is based on a mean MPI compensation, so the results can be considered as establishing a lower bound on the number N T  of taps. 
     Optical Transmission System with QSM Fiber 
       FIG. 10  is a schematic diagram of an example optical transmission system (“system”)  100  that employs the QSM fiber  10  as disclosed herein. System  100  includes an optical transmitter  110 , a section of QSM fiber  10 , an optical receiver  130 , an analog-to-digital converter ADC electrically connected to the optical receiver and a digital signal processor DSP electrically connected to the analog-to-digital converter. Also optionally included in system  100  is a decision circuit  150  electrically connected to the digital signal processor DSP. 
     The digital signal processor DSP includes an MPI mitigation system  134  that in example includes a plurality of equalizer taps  136 . System  10  and in particular MPI mitigation system  134  is configured to perform electronic equalization of optical transmission impairments to the optical signal using methods known in the art. In one example, MPI mitigation system  134  includes four finite impulse response (FIR) filters in a butterfly structure (not shown), wherein each filter has a number of taps  136 , which are recursively adjusted based on a least-mean-square (LMS) algorithm. 
     The QSM fiber section  10  includes an input end  112  optically connected to optical transmitter  110  and an output end  114  optically connected to optical receiver  130 , thereby establishing an optical connection between the optical transmitter and the optical receiver. In an example, QSM fiber  10  includes an amplifier  160 , e.g., an EDFA. 
     In the operation of system  100 , transmitter  110  generates light  200  that defines an input analog optical signal OS that carries information only in the fundamental mode LP 01 . Light  200  enters the input end  112  of QSM fiber  10  and travels the length of the fiber to output end  114 . Most of light  200  travels in the fundamental mode (LP 01 ) while a portion of the light travels in the higher-order mode LP 11 . The light  200  is denoted as  200 ′ at the output end of QSM fiber  10  due the light having impairments described above by virtue of having traveled through QSM fiber  10 . 
     Optical receiver  130  receives light  200 ′ as emitted from the output end  114  of QSM fiber  10  and converts this light into a corresponding analog electrical signal SA. The analog electrical signal SA passes through analog-to-digital converter ADC, which forms therefrom a corresponding digital electrical signal SD. The digital electrical signal SD is then received by digital signal processor DSP, which performs digital processing of the digital electrical signal. In particular, the digital signal processor DSP is configured to perform equalization of MPI using MPI mitigation system  134  and the equalizer taps  136  therein based on techniques known in the art. The digital signal processor DSP outputs a processed digital electrical signal SDP that is representative (to within the limits of MPI mitigation system  134 ) of the initial optical signal OS generated by transmitter  110 . The processed digital electrical signal SDP, which includes the information originally encoded into optical system OS, continues downstream to be processed as needed (e.g., by a decision circuit  150 ) for the given application. 
     As noted above, the relatively high DMA of ≧1 dB/km or &gt;4 dB/km results in less complex digital signal processing, i.e., the number N T  of equalizer taps  136  is reduced as compared to conventional optical transmission systems that employ MPI compensation. Also, as noted above, high DMA values also reduce the total MPI level, which may have an upper limit in terms of the efficacy of the MPI compensation digital signal processing. 
     It will be apparent to those skilled in the art that various modifications to the preferred embodiments 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.