Few mode optical fiber links for space division multiplexing having trenched fibers with high leak losses for leaky modes and low bend losses

Abstract

The invention relates to an optical link comprising N optical fibers, with N?2. Each optical fiber comprises an optical core and an optical cladding surrounding the optical core, the optical core having a single ?.sub.i graded-index profile with ?.sub.i?1, and the optical core having a radius R.sub.1i, where i E [1; N] is an index designating said optical fiber. Said optical cladding comprises a region of depressed refractive index n.sub.trenchi, called a trench, surrounding the optical core. According to embodiments of the invention, for all optical fibers in said link, said optical core radius R.sub.1i and said length L.sub.i are chosen such that R.sub.1i?13.5 ?m and so as to satisfy a criterion C of quality. Thus, the invention provides a few-mode optical fiber link, which allow guiding an increased number of LP modes as compared to prior art FMF links, while reaching low Differential Mode Group Delay.

Claims

1. An optical link comprising N optical fibers, with N?2, N being an integer, each optical fiber comprising an optical core and an optical cladding surrounding the optical core, the optical core having a single ?.sub.i graded-index profile with ?.sub.i?1, x.sub.i being a non-dimensional parameter defining an index profile shape of the optical core, and the optical core having a radius R.sub.1i, expressed in microns (?m), and a maximal refractive index n.sub.0i, where i?[[1;N]] is an index designating said optical fiber, wherein the optical cladding having a refractive index n.sub.Cli at an outer edge of the optical cladding, wherein said optical cladding comprises a region of depressed refractive index n.sub.trenchi being a trench, surrounding the optical core, said trench having an inner radius R.sub.2i, with R.sub.2i?R.sub.1i, and an outer radius R.sub.3i, with R.sub.3i>R.sub.2i, said optical link having an average optical core radius R.sub.1link satisfying a criterion C of quality of optical communications defined by the following equation: $C=10.Math.\frac{\mathrm{Max}\left|{\mathrm{DMGD}}_{\mathrm{link}}\right|}{{\left(R_{1\mathrm{link}}^2.Math.{\mathrm{Dn}}_{1\mathrm{link}}\right)}^3}$ where DMGD.sub.link is the Differential Mode Group Delay between two guided modes in said optical link, where Max|DMGD.sub.link| is expressed in ps/km and is an absolute maximum value of Differential Mode Group Delay between any combination of guided modes in said optical link, where $R_{1\mathrm{link}}=\frac{\underset{i=1}{\overset{N}{.Math.}}{R}_{1i}{L}_i}{\underset{i=1}{\overset{N}{.Math.}}{L}_i}$ with L.sub.i a length of optical fiber i in said link, and where ${\mathrm{Dn}}_{1\mathrm{link}}=\frac{\underset{i=1}{\overset{N}{.Math.}}{\mathrm{Dn}}_{1i}{L}_i}{\underset{i=1}{\overset{N}{.Math.}}{L}_i}$ with Dn.sub.1i=n.sub.0i?n.sub.Cli is a core-cladding index difference for optical fiber t, at ?=?.sub.C, where ?.sub.c is a central transmission wavelength of an operating band for the optical fiber, and for at least one optical fiber i in said link, said optical core radius R.sub.1i is chosen such that R.sub.1i?13.5 ?m and for all optical fibers i?[[1;N] in said link, said length L.sub.i are chosen such that C?18, and wherein at least one of said optical fibers has trench parameters 55?1000.Math.|(R.sub.3i?R.sub.2i).Math.Dn.sub.3i.Math.(R.sub.1i.sup.2.Math.Dn.sub.1i)|?150 with R.sub.1i, R.sub.2i, and R.sub.3i each being expressed in microns (?m), and where Dn.sub.3i=n.sub.trenchi?n.sub.Cli is the trench-cladding index difference at ?=?.sub.C.

2. The optical link according to claim 1, wherein Dn.sub.3??3.Math.10.sup.3.

3. The optical link according to claim 1, wherein at least one of said optical fibers has an optical core radius R.sub.1i and an ?.sub.i-value of said graded index profile such that: $C=10.Math.\frac{\mathrm{Max}\left|{\mathrm{DMGD}}_i\right|}{{\left(R_{1i}^2.Math.{\mathrm{Dn}}_{1i}\right)}^3}?18$ where DMGD.sub.i is the Differential Mode Group Delay between two guided modes in said optical fiber, where Max|DMGD.sub.i| is the absolute maximum value of DMGD between any combination of guided modes in said optical fiber, and where Dn.sub.1i?n.sub.0i?n.sub.Cli is the core-cladding index difference at ?=?.sub.C for said optical fiber.

4. The optical link according to claim 3, wherein said optical fiber is R.sub.1i?20 ?m.

5. The optical link according to claim 1, wherein the optical link guides 4 to 16 LP modes.

6. The optical link according to claim 1, wherein the optical link guides 6 to 16 LP modes.

7. The optical link according to claim 1, wherein for all optical fibers i?[[1;N]] in said link, said lengths L.sub.i are chosen so as to minimize Max|DMGD.sub.link| on said link.

8. The optical link according to claim 1, wherein at least two optical fibers in said link have DMGD.sub.i showing opposite signs for at least one mode guided by said optical fibers, where DMGD.sub.i is the Differential Mode Group Delay between said one mode and any other guided mode in optical fiber i.

9. The optical link according to claim 1, wherein, for at least one of said fibers i?[[1;N]] in said optical link, said optical core has a minimal refractive index n.sub.1i=n.sub.Cli, and said optical cladding also comprises an inner cladding layer directly surrounding said optical core, with an inner radius R.sub.1i and an outer radius R.sub.2i?R.sub.1i, said inner cladding layer having a constant refractive index n.sub.2i, such that n.sub.2i?n.sub.Cli and n.sub.2i?n.sub.trenchi.

10. The optical link according to claim 1, wherein, for at least one of said fibers i?[[1;N]] in said optical link, said optical core has a minimal refractive index n.sub.1i?n.sub.Cli, and said optical cladding also comprises an inner cladding layer directly surrounding said optical core, with an inner radius R.sub.1i and an outer radius R.sub.2i?R.sub.1i, said inner cladding layer having a constant refractive index n.sub.2i such that n.sub.2i=n.sub.1i and n.sub.2i>n.sub.trenchi.

11. The optical link according to claim 1, wherein, for at least one of said fibers i?[[1;N]] in said optical link, said optical core has a minimal refractive index that equals n.sub.Cli, said optical cladding also comprises an inner cladding layer directly surrounding said optical core, with an inner radius R.sub.1i and an outer radius R.sub.2i?R.sub.1i, said inner cladding layer being an extension of said single graded-index profile of said optical core, and said inner cladding layer has a minimal refractive index n.sub.1i=n.sub.trenchi.

12. An optical system comprising the optical link of claim 1.

Description

5. BRIEF DESCRIPTION OF THE FIGURES

(1) Other features and advantages of embodiments of the invention shall appear from the following description, given by way of an indicative and non-exhaustive examples and from the appended drawings, of which:

(2) FIG. 1 schematically depicts an optical link according to an embodiment of the invention, such an optical link comprising several concatenated few-mode optical fibers;

(3) FIG. 2 schematically depicts a cross section of an optical fiber according to one or more embodiments described herein;

(4) FIG. 3A graphically provides the refractive index profile of an optical fiber according to a first embodiment of the invention;

(5) FIG. 3B graphically provides the refractive index profile of an optical fiber according to a second embodiment of the invention;

(6) FIG. 3C graphically provides the refractive index profile of an optical fiber according to a third embodiment of the invention;

(7) FIG. 4 illustrates how Differential Mode Group Delays decrease as a function of R.sub.1 for few-mode fibers supporting 6 to 16 LP guided modes for graded-index trench-assisted structures in accordance with the invention;

(8) FIG. 5 shows the C criterion set forth in the invention for few-mode fibers supporting 6 to 16 LP guided modes as a function of R.sub.1 for graded-index trench-assisted structures in accordance with the invention;

(9) FIG. 6 graphically depicts the Max|DMGD| as a function of wavelength for some embodiments of the invention;

(10) FIG. 7 illustrates the Max|DMGD| as a function of a for some embodiments of the invention;

(11) FIGS. 8A and 8B illustrate optical systems according to embodiments of the invention.

6. DETAILED DESCRIPTION

(12) The general principle of the invention is to propose a carefully designed trench-assisted graded index few-mode optical fibers link, showing reduced Differential Mode Group Delay and supporting more LP modes over prior art FMFs. More precisely, the purpose of such an optical link is, among others, to compensate for small profile variations that can occur during the manufacturing process of a few-mode fiber by concatenating several FMFs showing different features. Such an optical link allows reaching an improved trade-off over prior art FMFs between reduced Differential Mode Group Delay, reduced bend loss and increased leakage loss. Moreover, designing such DMGD-compensated FMF links is an efficient and robust way to reach low DMGDs.

(13) Light travelling in an optical fiber actually forms hybrid-type modes, which are usually referred to as LP (linear polarization) modes. The LP.sub.Op modes have tow polarization degrees of freedom and are two-fold degenerate, and the LP.sub.mp modes with m?1 are four-fold degenerate. These degeneracies are not counted when designating the number of LP modes propagating in the fiber. Hence, a few-mode optical fiber having two LP modes supports the propagation of all of the LP.sub.01 and LP.sub.11 modes, or a few-mode fiber guiding 6 LP modes supports the propagation of all of the LP.sub.01, LP.sub.11, LP.sub.02, LP.sub.21, LP.sub.12 and LP.sub.31 modes.

(14) Reference will now be made in detail to embodiments of few-mode optical fibers comprised in an optical link according to an embodiment of the invention, examples of which are illustrated in the accompanying drawings. Whenever possible, the same reference numerals will be used throughout the drawings to refer to the same or like parts.

(15) One embodiment of a few-mode optical fiber link according to the invention is schematically depicted in FIG. 1. The optical link 70 is built by concatenating several few-mode fibers (for example such an optical link 70 comprises p spans of optical fibers, with p?2, which are spliced together. FIG. 1 only shows optical fiber 701 and optical fiber 70p, all the other potential optical fibers i?[[1; N]] in the optical link being symbolized by dashed lines).

(16) Few-mode fiber 1 has a length L.sub.1, few-mode fiber i has a length L; and few-mode fiber p has a length L.sub.p. As will be described in more details below, such lengths L.sub.1, . . . L.sub.i, . . . L.sub.p are chosen such that the optical link 70 satisfies a criterion of quality of optical communications defined by the following equation:

(17) $C=10.Math.\frac{\mathrm{Max}\left|{\mathrm{DMGD}}_{\mathrm{link}}\right|}{{\left(R_{1\mathrm{link}}^2.Math.{\mathrm{Dn}}_{1\mathrm{link}}\right)}^3}?18$
where DMGD.sub.link is the Differential Mode Group Delay between two guided modes in optical link 70,
where

(18) $R_{1\mathrm{link}}=\frac{\underset{i=1}{\overset{p}{.Math.}}{R}_{1i}{L}_i}{\underset{i=1}{\overset{p}{.Math.}}{L}_i},$
and where

(19) ${\mathrm{Dn}}_{1\mathrm{link}}=\frac{\underset{i=1}{\overset{p}{.Math.}}{\mathrm{Dn}}_{1i}{L}_i}{\underset{i=1}{\overset{p}{.Math.}}{L}_i}$
with Dn.sub.1i=n.sub.0i?n.sub.Cli is the core-cladding index difference at ?=1550 nm for optical fiber i?[[1; N]]. In addition, at least one of the optical fibers 1 to p in link 70 has an optical core radius R.sub.1i such that R.sub.1i?13.5 ?m, i?[1; p].

(20) Few-mode fibers 1 to p are hence spliced together to form an optical link 70 of length L=L.sub.1+ . . . +L.sub.i+ . . . +L.sub.p, which can be of several tens or several hundreds of kilometers.

(21) Of course, the present disclosure encompasses any number of few-mode fibers concatenated to form an optical link; as a mere example, such a link may comprise only two FMFs, four FMFs, or even several tens of FMFs.

(22) The following disclosure will now focus on the structure and characteristics of optical fibers making up an optical link according to the invention.

(23) 6.1 Few-Mode Optical Fibers

(24) One embodiment of a few-mode optical fiber used in an optical link according to the invention is schematically depicted in cross section in FIG. 2. The optical fiber 10 generally has a glass core 20 surrounded by a glass cladding. The glass core 20 generally has a radius R.sub.1i from about 13.5 ?m to about 20 ?m. The cladding generally has an inner radius R.sub.1 and an outer radius R.sub.4. In the embodiments shown and described herein, the core 20 and the cladding generally comprise silica, specifically silica glass. The cross-section of the optical fiber 10 may be generally circular-symmetric with respect to the center of the core 20. In some embodiments described herein, the radius R.sub.4 (i.e. the radius of the glass portion of the optical fiber 10) is about 62.5 ?m. However, it should be understood that the dimensions of the cladding may be adjusted such that the radius R.sub.4 may be greater than or less than 62.5 ?m. The optical fiber 10 also comprises a coating 60 of inner radius R.sub.4 and of outer radius R.sub.5. Such a coating may comprise several layers, and it may notably be a dual-layer coating, although these different layers are not shown on FIG. 2. It must be noted that R.sub.4 and R.sub.5 are the lower and upper limits of the coating, whatever the number of layers in-between. In some embodiments described herein, the radius R.sub.5 is about 122.5 ?m (but it could be greater or less than 122.5 ?m). In alternative embodiments, other dimensions could be such that R.sub.4=40 ?m or R.sub.4=50 ?m, and R.sub.5=62.5 ?m.

(25) All few-mode fibers in an optical link according to the invention share the common features described above in relation to FIG. 2.

(26) FIG. 3A depicts the refractive index profile n(r) of optical fiber 10 according to a first embodiment of the invention. It describes the relationship between the refractive index value n and the distance r from the center of the optical fiber. The x-axis represents radial position with x=0 representing the center of the core region, and the y-axis represents refractive index, expressed as an index difference Dn unless otherwise stated.

(27) According to embodiments of the invention, it is possible that only one, or several, or all, or even none of the optical fibers making up an optical link according to the invention show the refractive index of FIG. 3A.

(28) In that first embodiment, the optical fiber 10 has an optical core 20 having a refractive index profile n(r) defined as follows:

(29) 0$n\left(r\right)=n_0.Math.\sqrt{1-2.Math.?.Math.{\left(\frac{r}{R_1}\right)}^{?}}r?R_1$
where:
r is a variable representative of the radius of the optical fiber,
R.sub.1 is the optical core radius,
? is the normalized refractive index difference, with

(30) $?=\frac{n_0^2-n_1^2}{2n_0^2}$
n.sub.1 is the minimal refractive index of the optical core,
n.sub.0 is the maximal refractive index of the optical core,
? is a non-dimensional parameter that defines the index profile shape of the optical core.

(31) The alpha refractive index profile of the optical core 20 allows reducing intermodal dispersion of the optical fiber 10.

(32) The optical core 20 is directly surrounded by an optical cladding, which comprises at least a depressed-index ring 40, also called a trench, with inner radius R.sub.2 and outer radius R.sub.3, and an outer cladding layer 50 with inner radius R.sub.3. In some embodiments such an outer cladding layer 50 comprises pure silica glass (SiO.sub.2) and its refractive index n.sub.Cl is hence that of silica glass. This trench 40 has a negative refractive index difference Dn.sub.3=n.sub.trench?n.sub.Cl with respect to the refractive index of the outer cladding, and its position and size are designed so as to improve bend-loss resistance of the fiber.

(33) Preferably, the trench 40 is designed so as to fulfill the following criterion:
55?1000.Math.|(R.sub.3?R.sub.2).Math.Dn.sub.3.Math.(R.sub.1.sup.2.Math.Dn.sub.1)|?150
where Dn.sub.3=n.sub.trench?n.sub.Cl is the trench-cladding index difference at ?=?.sub.C, a central wavelength of an operating band for which the optical fiber is intended (for example ?=1550 nm).
Such a criterion allows reaching a good trade-off between bend losses and leakage losses in the fiber.

(34) The cladding may also optionally include an inner cladding layer 30, with inner radius R.sub.1 and outer radius R.sub.2. The trench 40 may hence be spaced apart from the core 20 by the inner cladding layer 30. Alternatively, the trench 40 may surround and directly contact the core portion 20.

(35) In this first embodiment, the inner cladding 30 has a constant refractive index n.sub.2, such that n.sub.2>n.sub.trench, and which may either show a negative or a positive (shown in dashed lines on FIG. 2A) refractive index difference Dn.sub.2=n.sub.2?n.sub.Cl with respect to the optical fiber outer cladding.

(36) The different portions 30, 40, 50 in the cladding may comprise pure silica glass (SiO.sub.2), silica glass with one or more dopants, which increase the index of refraction (e.g. GeO.sub.2 or any other known dopant), such as when the portion of the cladding is up-doped, or silica glass with a dopant, which decreases the index of refraction, such as fluorine, such as when the portion of the cladding is down-doped (e.g. for the trench 40).

(37) Although not illustrated on FIG. 1, the outer cladding 50 may also comprise other portions or layers of lower or higher refractive indexes, for r>R.sub.3.

(38) In the first embodiment illustrated by FIG. 3A, the minimal refractive index of the core n.sub.1 is equal to the refractive index of the outer cladding n.sub.Cl.

(39) The down-doped trench 40 can provide lower bending loss.

(40) FIG. 3B depicts the refractive index profile n(r) of an optical fiber according to a second embodiment of the invention. Once again, there may be embodiments of the invention where only one, or several, or all, or even none of the optical fibers in said optical link show the profile of FIG. 3B.

(41) Such a profile differs from that of the first embodiment in that the minimal refractive index of the core n.sub.1 is not equal to the refractive index of the outer cladding n.sub.Cl but may either show a negative or a positive (shown in dashed lines on FIG. 2B) refractive index difference with respect to the optical fiber outer cladding. In case the cladding comprises an inner cladding layer 30, the minimal refractive index of the core n.sub.1 is equal to the constant refractive index of the inner cladding n.sub.2, which may either show a negative or a positive (shown in dashed lines on FIG. 2B) refractive index difference Dn.sub.2=n.sub.2?n.sub.Cl with respect to the optical fiber outer cladding.

(42) Like in the first embodiment, the outer cladding 50 may also comprise other portions or layers of lower or higher refractive indexes, for r>R.sub.3.

(43) Like in the first embodiment, the trench 40 is preferably designed so as to fulfill the following criterion:
55?1000.Math.|(R.sub.3?R.sub.2).Math.Dn.sub.3.Math.(R.sub.1.sup.2.Math.Dn.sub.1)?150
where Dn.sub.3=n.sub.trench?n.sub.C1 is the trench-cladding index difference at ?=?.sub.C, a central wavelength of an operating band for which the optical fiber is intended (for example ?=1550 nm).

(44) FIG. 3C depicts the refractive index profile n(r) of an optical fiber according to a third embodiment of the invention. Once again, there may be embodiments of the invention where only one, or several, or all, or even none of the optical fibers in said optical link show the profile of FIG. 3C.

(45) In this third embodiment, the inner cladding layer 30 is an extension of the graded index core 20, such that both the optical core 20 and the inner cladding layer 30 have a refractive index profile n(r) defined as follows:

(46) $n\left(r\right)=n_0.Math.\sqrt{1-2.Math.?.Math.{\left(\frac{r}{R_2}\right)}^{?}}r?R_2$
where:
r is a variable representative of the radius of the optical fiber,
R.sub.2 is the outer radius of the inner cladding layer 30,
? is the normalized refractive index difference, with

(47) $?=\frac{n_0^2-n_1^2}{2n_0^2}$
n.sub.1 is the minimal refractive index of the inner cladding layer (i.e. the refractive index at radius R.sub.2),
n.sub.0 is the maximal refractive index of the optical core,
? is a non-dimensional parameter that defines the index profile shape of both the optical core and the inner cladding layer.

(48) Hence, in this third embodiment, the term single-? graded-index profile has a slightly different meaning as compared to the first two embodiments, since this graded-index profile goes beyond the optical core until the outer edge of the inner cladding layer.

(49) The optical cladding also comprises at least a depressed-index ring 40, with inner radius R.sub.2 and outer radius R.sub.3, and an outer cladding layer 50 with inner radius R.sub.3. In some embodiments such an outer cladding layer 50 comprises pure silica glass (SiO.sub.2) and its refractive index n.sub.Cl is hence that of silica glass. The trench 40 has a negative refractive index difference Dn.sub.3=n.sub.trench?n.sub.Cl with respect to the refractive index of the outer cladding, and its position and size are designed so as to improve bend-loss resistance of the fiber.

(50) Like in the first and second embodiments, the outer cladding 50 may also comprise other portions or layers of lower or higher refractive indexes, for r>R.sub.3.

(51) Like in the first and second embodiments, the trench 40 is preferably designed so as to fulfill the following criterion:
55?1000.Math.|(R.sub.3?R.sub.2).Math.Dn.sub.3.Math.(R.sub.1.sup.2.Math.Dn.sub.1)|?150
where Dn.sub.3=n.sub.trench?n.sub.Cl is the trench-cladding index difference at ?=?.sub.C, a central wavelength of an operating band for which the optical fiber is intended (for example ?=1550 nm).

(52) FIG. 4 illustrates how the maximum of the Differential Mode Group Delays Max|DMGD| between any two LP modes guided in the optical fiber decreases as a function of the core radius R.sub.1 for FMFs guiding 6, 9, 12 and 16 modes according to one of the embodiments of FIGS. 3A-3C. The x-axis depicts the core radius of the fiber R.sub.1, ranging from 12 to 16 ?m. The y-axis depicts the Max|DMGD| expressed as ps/km on a logarithmic scale. Curve 31 corresponds to a FMF guiding 6 LP modes; curve 32 corresponds to a FMF guiding 9 LP modes; curve 33 corresponds to a FMF guiding 12 LP modes, while curve 34 corresponds to a FMF guiding 16 LP modes.

(53) FIG. 5 jointly illustrates how the criterion

(54) $C=10.Math.\frac{\mathrm{Max}\left|\mathrm{DMGDs}\right|}{{\left(R_1^2.Math.{\mathrm{Dn}}_1\right)}^3},$
where DMGD is the Differential Mode Group Delay between two guided modes in said optical fiber and where Dn.sub.1=n.sub.0?n.sub.Cl is the core-cladding index difference at ?=?.sub.C, a central wavelength of an operating band for which the optical fiber is intended, also decreases as a function of the core radius R.sub.1 for FMFs guiding 6, 9, 12 and 16 modes according to one of the embodiments of FIGS. 3A-3C. The x-axis depicts the core radius of the fiber R.sub.1, ranging from 12 to 16 ?m. The y-axis depicts the C criterion ranging from 0 to 30. Curve 41 corresponds to a FMF guiding 6 LP modes; curve 42 corresponds to a FMF guiding 9 LP modes; curve 43 corresponds to a FMF guiding 12 LP modes, while curve 44 corresponds to a FMF guiding 16 LP modes.

(55) As can be observed from both figures, a good trade-off may be obtained by setting the core radius R.sub.1?13.5 ?m. This allows reaching low values for Max|DMGD|, whatever the number of LP guided modes in the fiber. By setting the lower limit of the core radius at 13.5 ?m, it is possible to guide a high number of LP modes in the FMF, and thus reach a good per-fiber capacity, while, thanks to low Max|DMGD| values, bridge long distances.

(56) Once the core radius has been set at a minimum value of 13.5 ?m, it can be deduced from FIG. 5 that an adequate upper limit for the C criterion can be set at a value of 18: C<18. Such an upper limit allowable for FMFs of the invention is illustrated by horizontal straight line 45 on FIG. 4.

(57) As can be observed from FIGS. 4 and 5, for FMFs supporting 6 LP guided modes, the normalized frequency

(58) $V=\frac{2?R_1}{?}\sqrt{n_0^2-n_{\mathrm{Cl}}^2},$
where ? is the operating wavelength) is preferably between 7.8 and 9.8. Max|DMGD| is preferably <25 ps/km, and more preferably <15 ps/km, at ?, here 1550 nm. More generally, such values can be achieved for any central transmission wavelength ?.sub.C of any operating wavelength band for which the optical fiber is intended, such as the C-band, or the L-, S-, or U-band for example. Max|DMGD| is also preferably <50 ps/km and more preferably <30 ps/km from 1530 to 1570 nm. More generally, such values can be achieved for any operating wavelength band [?.sub.C???; ?.sub.C+??] where 2?? is a width of said operating band, preferably ??=20 nm, such as the C-band, or the L-, S-, or U-band for example.

(59) For FMFs supporting 9 LP guided modes, V is preferably between 9.8 and 11.8. Max|DMGD| is preferably <100 ps/km, and more preferably <60 ps/km, at ?, here 1550 nm (and more generally for any central transmission wavelength ?.sub.C of any operating wavelength band for which the optical fiber is intended). Max|DMGD| is also preferably <200 ps/km and more preferably <120 ps/km from 1530 to 1570 nm. More generally, such values can be achieved for any operating wavelength band [?.sub.C???; ?.sub.C+??] where 2?? is a width of said operating band, preferably ??=20 nm, such as the C-band, or the L-, S-, or U-band for example.

(60) For FMFs supporting 12 LP guided modes, V is preferably between 11.8 and 13.8. Max|DMGD| is preferably <150 ps/km and more preferably <120 ps/km, at ?, here 1550 nm. More generally, such values can be achieved for any central transmission wavelength ?.sub.C of any operating wavelength band for which the optical fiber is intended, such as the C-band, or the L-, S-, or U-band for example. Max|DMGD| is also preferably <300 ps/km and more preferably <250 ps/km from 1530 to 1570 nm. More generally, such values can be achieved for any operating wavelength band [?.sub.C???; ?.sub.C+??] where 2?? is a width of said operating band, preferably ??=20 nm, such as the C-band, or the L-, S-, or U-band for example.

(61) For FMFs supporting 16 LP guided modes, V is preferably between 13.8 and 15.9. Max|DMGD| is preferably <300 ps/km and more preferably <250 ps/km, at ?, here 1550 nm. More generally, such values can be achieved for any central transmission wavelength ?.sub.C of any operating wavelength band for which the optical fiber is intended, such as the C-band, or the L-, S-, or U-band for example. Max|DMGD| is also preferably <600 ps/km and more preferably <500 ps/km from 1530 to 1570 nm. More generally, such values can be achieved for any operating wavelength band [?.sub.C???; ?.sub.C+??] where 2?? is a width of said operating band, preferably ??=20 nm, such as the C-band, or the L-, S-, or U-band for example.

(62) Moreover, for FMFs supporting 4 LP guided modes, the normalized frequency V is preferably between 5.7 and 7.8. Max|DMGD| is preferably <20 ps/km, and more preferably <10 ps/km, at 1550 nm. More generally, such values can be achieved for any central transmission wavelength ?.sub.C of any operating wavelength band for which the optical fiber is intended, such as the C-band, or the L-, S-, or U-band for example. Max|DMGD| is also preferably <30 ps/km and more preferably <20 ps/km from 1530 to 1570 nm. More generally, such values can be achieved for any operating wavelength band [?.sub.C???; ?.sub.C+??] where 2?? is a width of said operating band, preferably ??=20 nm, such as the C-band, or the L-, S-, or U-band for example.

(63) All LP guided modes of FMFs according to an embodiment of the invention have effective areas, A.sub.eff<400 ?m.sup.2, preferably <350 ?m.sup.2, and bend losses <100 dB/turn, preferably <50 dB/turn, at 10 mm bend radius at 1550 nm, and all LP leakage modes have leakage losses >0.1 dB/m, preferably >0.5 dB/m, at 1550 nm, so that they are cut-off after few tens of meters of propagation (>19.34 dB (Leakage) loss). As used herein, the effective area of an optical fiber is the area of the optical fiber in which light is propagated and is determined at the specified mode (e.g. LP.sub.01), at a wavelength of 1550 nm, unless otherwise specified.

(64) Table 1 gives the parameters of the index profiles of examples of FMFs according to the embodiment of FIG. 3B, and results on Max|DMGD|, specific core Criterion C and trench criterion T.

(65) TABLE-US-00001 TABLE 1 Ex. 0 Ex. 1 Ex. 2 Ex. 3 Ex. 4 Ex. 5 Ex. 6 Ex. 7 Ex. 8 Ex. 9 Ex. 10 # Guided LP modes 4 6 9 12 16 Alpha 1.9581 1.949 1.951 1.954 1.944 1.945 1.934 1.934 1.926 1.928 1.931 R1 (?m) 14.00 13.50 14.00 15.00 14.00 15.00 14.00 15.00 14.00 15.00 16.00 Dn1 (?10.sup.3 at 6.31 10.69 9.95 8.66 14.4 12.55 19.65 17.13 26.02 22.7 19.06 1550 nm) R2 (?m) 15.88 14.66 15.30 16.56 15.05 16.12 14.91 15.44 14.68 15.98 16.63 Dn2 (?10.sup.3 at 0 0 0 0 0.12 0 ?0.59 0 ?0.89 ?1.06 0.2 1550 nm) R3 (?m) 22.23 20.53 21.42 23.18 19.57 22.56 19.39 19.30 19.08 20.77 20.41 Dn3 (?10.sup.3 at ?4.81 ?4.81 ?4.81 ?4.81 ?5.78 ?4.81 ?5.78 ?3.85 ?5.78 ?5.78 ?4.81 1550 nm) Max|DMGD| 2.9 11.3 8.6 8.0 25.4 21.2 73.5 87.3 152.7 124.1 96.1 (ps/km) Core Criterion 15.3 15.3 11.6 10.8 11.3 9.4 12.9 15.2 11.5 9.3 8.3 Trench 37.8 55.0 57.4 62.1 73.7 87.6 99.6 57.3 129.8 141.5 88.7 Criterion

(66) In table 1, the core criterion is the C parameter such that

(67) $C=10.Math.\frac{\mathrm{Max}\left|\mathrm{DMGDs}\right|}{{\left(R_1^2.Math.{\mathrm{Dn}}_1\right)}^3}.$
The trench criterion T is defined as T=1000.Math.|(R.sub.3?R.sub.2).Math.Dn.sub.3.Math.(R.sub.1.sup.2.Math.Dn.sub.1)|. According to a preferred embodiment 55?T?150. As can be noticed, for Example 0 of a FMF guiding 4 LP modes, the T criterion is not met, since T=37.8, although the C criterion is met with C=15.3<18. However, with such a low number of LP modes (i.e. 4), the trade-off between the bend losses and the leakage losses is much more easily met.

(68) Table 2 gives the characteristics of the LP modes of the Examples of Table 1 supporting 4 LP guided modes, i.e. modes LP.sub.01, LP.sub.11, LP.sub.21 and LP.sub.02.

(69) TABLE-US-00002 TABLE 2 Leakage Bend DMGD vs. Dneff Loss A.sub.eff CD Loss LP01 Ex. 0 (?10.sup.?3) (dB/m) (?m.sup.2) (ps/nm-km) (dB/turn) (ps/km) LP01 4.68 / 158 20.7 0.0 / LP11 3.01 / 213 20.9 0.0 ?2.4 LP21 1.36 / 286 21.2 1.5 0.5 LP02 1.37 / 323 21.3 4.8 ?1.5 LP12 <0 13.7 LP31 <0 2.4

(70) In table 2, as well as in tables 3 to 6 disclosed below, Dneff stands for the effective index difference, CD stands for the chromatic dispersion expressed as ps/nm-km (chromatic dispersion is the sum of the material dispersion, the waveguide dispersion and the inter-modal dispersion), and Bend Losses, expressed as dB/turn, are given at 10 mm bend radius. A.sub.eff expressed as ?m.sup.2 designates the effective area of the LP guided mode. The Differential Mode Group Delay DMGD is measured with respect to the first guided mode LP.sub.01 and expressed as ps/km. LP.sub.12 and LP.sub.31 are leaky modes.

(71) Table 3 gives the characteristics of the LP modes of the Examples of Table 1 supporting 6 LP guided modes, that is to say Examples 1, 2 and 3, at a wavelength ?=1550 nm.

(72) TABLE-US-00003 TABLE 3 Leakage Bend DMGD vs. Dneff Loss A.sub.eff CD Loss LP01 (?10.sup.?3) (dB/m) (?m.sup.2) (ps/nm-km) (dB/turn) (ps/km) Ex. 1 LP01 8.50 / 117 20.0 0.0 / LP11 6.25 / 157 20.3 0.0 ?10.3 LP21 4.01 / 211 20.5 0.0 ?8.5 LP02 4.02 / 238 20.5 0.0 ?8.0 LP12 1.79 / 254 20.7 6.5 ?7.6 LP31 1.77 / 254 20.8 1.3 1.0 LP03 <0 53.4 LP22 <0 33.5 LP41 <0 3.0 Ex. 2 LP01 7.71 / 126 20.1 0.0 / LP11 5.62 / 169 20.4 0.0 ?7.6 LP21 3.54 / 227 20.6 0.0 ?6.5 LP02 3.55 / 256 20.6 0.0 ?7.8 LP12 1.48 / 274 20.9 7.3 ?7.5 LP31 1.46 / 273 20.9 1.6 0.8 LP03 <0 37.8 LP22 <0 21.4 LP41 <0 1.8 Ex. 3 LP01 6.70 / 144 20.3 0.0 / LP11 4.87 / 194 20.5 0.0 ?4.4 LP21 3.06 / 260 20.7 0.0 ?4.3 LP02 3.07 / 294 20.7 0.1 ?7.5 LP12 1.27 / 314 21.1 7.6 ?5.2 LP31 1.25 / 313 21.0 2.0 0.5 LP03 <0 17.4 LP22 <0 9.9 LP41 <0 0.9 LP.sub.03, LP.sub.22 and LP.sub.41 are leaky modes.

(73) Table 4 gives the characteristics of the LP modes of the Examples of Table 1 supporting 9 LP guided modes, that is to say Examples 4 and 5, at a wavelength ?=1550 nm. As can be observed, LP.sub.13, LP.sub.32 and LP.sub.51 are leaky modes.

(74) TABLE-US-00004 TABLE 4 Leakage Bend DMGD vs. Dneff Loss A.sub.eff CD Loss LP01 (?10.sup.?3) (dB/m) (?m.sup.2) (ps/nm-km) (dB/turn) (ps/km) Ex. 4 LP01 11.75 / 104 19.3 0.0 / LP11 9.23 / 140 19.6 0.0 ?22.5 LP21 6.72 / 188 20.0 0.0 ?25.1 LP02 6.74 / 212 19.9 0.0 ?20.1 LP12 4.24 / 227 20.3 0.0 ?24.3 LP31 4.22 / 226 20.3 0.0 ?16.7 LP03 1.76 / 311 20.4 12.0 ?16.1 LP22 1.75 / 284 20.5 5.4 ?17.8 LP41 1.72 / 260 20.6 1.0 0.3 LP13 <0 137.1 LP32 <0 47.4 LP51 <0 1.6 Ex. 5 LP01 10.41 / 119 19.6 0.0 / LP11 8.21 / 161 19.9 0.0 ?18.5 LP21 6.03 / 215 20.2 0.0 ?21.2 LP02 6.04 / 243 20.2 0.0 ?16.6 LP12 3.87 / 260 20.4 0.0 ?21.1 LP31 3.85 / 260 20.4 0.0 ?15.4 LP03 1.71 / 357 20.6 9.8 ?15.6 LP22 1.69 / 326 20.6 2.9 ?17.1 LP41 1.67 / 298 20.7 0.7 ?3.3 LP13 <0 31.2 LP32 <0 11.4 LP51 <0 0.5

(75) Table 5 gives the characteristics of the LP modes of the Examples of Table 1 supporting 12 LP guided modes (examples 6 and 7). LP.sub.04, LP.sub.23, LP.sub.42 and LP.sub.61 are leaky modes.

(76) TABLE-US-00005 TABLE 5 Leakage Bend DMGD vs. Dneff Loss A.sub.eff CD Loss LP01 (?10.sup.?3) (dB/m) (?m.sup.2) (ps/nm-km) (dB/turn) (ps/km) Ex. 6 LP01 16.58 / 89 18.4 0.0 / LP11 13.63 / 119 18.8 0.0 ?53.2 LP21 10.69 / 160 19.1 0.0 ?73.5 LP02 10.72 / 181 19.1 0.0 ?60.9 LP12 7.80 / 193 19.5 0.0 ?64.8 LP31 7.77 / 193 19.5 0.0 ?70.6 LP03 4.90 / 266 19.8 0.0 ?27.5 LP22 4.89 / 242 19.9 0.0 ?38.5 LP41 4.85 / 221 19.9 0.0 ?45.3 LP13 2.00 / 260 19.0 5.8 ?68.4 LP32 1.98 / 283 19.6 2.2 ?31.3 LP51 1.94 / 247 20.2 0.2 ?5.8 LP04 <0 218.5 LP23 <0 149.9 LP42 <0 31.9 LP61 <0 0.6 Ex. 7 LP01 14.63 / 102 18.8 0.0 / LP11 12.05 / 137 19.1 0.0 ?48.2 LP21 9.49 / 184 19.5 0.0 ?68.6 LP02 9.52 / 208 19.5 0.0 ?56.5 LP12 6.97 / 222 19.9 0.0 ?52.1 LP31 6.94 / 222 19.8 0.0 ?66.4 LP03 4.44 / 304 20.1 0.0 18.1 LP22 4.43 / 278 20.2 0.0 ?2.6 LP41 4.40 / 254 20.2 0.0 ?37.0 LP13 1.90 / 298 17.4 77.5 ?69.2 LP32 1.89 / 324 19.0 20.7 ?3.7 LP51 1.86 / 283 20.3 2.8 6.7 LP04 <0 1077.0 LP23 <0 750.2 LP42 <0 145.0 LP61 <0 3.0

(77) Table 6 gives the characteristics of the LP modes of the Examples of Table 1 supporting 16 LP guided modes (examples 8, 9 and 10). LP.sub.14, LP.sub.33, LP.sub.52 and LP.sub.71 are leaky modes

(78) TABLE-US-00006 TABLE 6 Leakage Bend DMGD vs. Dneff Loss A.sub.eff CD Loss LP01 Ex. 8 (?10.sup.?3) (dB/m) (?m.sup.2) (ps/nm-km) (dB/turn) (ps/km) LP01 22.50 / 76 17.2 0.0 / LP11 19.10 / 103 17.6 0.0 ?88.8 LP21 15.72 / 139 18.1 0.0 ?131.5 LP02 15.75 / 157 18.0 0.0 ?112.3 LP12 12.39 / 168 18.5 0.0 ?133.6 LP31 12.35 / 167 18.5 0.0 ?142.9 LP03 9.06 / 230 19.0 0.0 ?83.8 LP22 9.04 / 210 19.0 0.0 ?104.8 LP41 8.99 / 192 19.0 0.0 ?121.8 LP13 5.72 / 225 19.3 0.0 ?34.2 LP32 5.69 / 245 19.5 0.0 ?43.8 LP51 5.64 / 214 19.6 0.0 ?67.9 LP04 2.39 / 301 16.8 6.4 ?106.8 LP23 2.37 / 272 17.5 2.2 ?80.2 LP42 2.34 / 276 19.0 0.4 ?8.2 LP61 2.29 / 234 19.9 0.0 9.8 LP14 <0 271.8 LP33 <0 125.1 LP52 <0 11.8 LP71 <0 0.2 Dneff LL Aeff CD BL DMGD Ex. 9 (?10.sup.?3) (dB/m) (?m.sup.2) (ps/nm-km) (dB/turn) (ps/km) LP01 19.61 / 88 17.8 0.0 0 LP11 16.65 / 119 18.2 0.0 ?73.7 LP21 13.70 / 159 18.5 0.0 ?110.7 LP02 13.73 / 180 18.5 0.0 ?94.6 LP12 10.80 / 192 18.9 0.0 ?117.5 LP31 10.77 / 192 18.9 0.0 ?124.1 LP03 7.89 / 264 19.4 0.0 ?85.1 LP22 7.87 / 241 19.4 0.0 ?102.0 LP41 7.84 / 220 19.4 0.0 ?113.4 LP13 4.98 / 258 19.6 0.0 ?59.7 LP32 4.95 / 281 19.7 0.0 ?64.5 LP51 4.91 / 246 19.8 0.0 ?78.7 LP04 2.07 / 345 18.1 6.5 ?106.0 LP23 2.06 / 312 18.6 3.3 ?91.4 LP42 2.03 / 317 19.5 0.9 ?42.7 LP61 1.99 / 269 20.1 0.1 ?26.8 LP14 <0 123.0 LP33 <0 57.0 LP52 <0 5.4 LP71 <0 0.1 Leakage Bend DMGD vs. Dneff Loss A.sub.eff CD Loss LP01 Ex. 10 (?10.sup.?3) (dB/m) (?m.sup.2) (ps/nm-km) (dB/turn) (ps/km) LP01 16.58 / 102 18.5 0.0 / LP11 14.03 / 138 18.8 0.0 ?51.3 LP21 11.50 / 186 19.1 0.0 ?82.6 LP02 11.52 / 210 19.1 0.0 ?77.7 LP12 9.00 / 224 19.4 0.0 ?90.4 LP31 8.97 / 224 19.4 0.0 ?96.1 LP03 6.50 / 308 19.8 0.0 ?67.2 LP22 6.49 / 281 19.8 0.0 ?75.7 LP41 6.46 / 257 19.8 0.0 ?89.8 LP13 4.00 / 301 20.2 0.0 ?11.5 LP32 3.98 / 327 20.2 0.0 ?29.6 LP51 3.94 / 286 20.2 0.0 ?60.6 LP04 1.50 / 401 16.9 221.3 ?82.0 LP23 1.49 / 364 17.9 73.4 ?48.5 LP42 1.46 / 368 19.6 27.1 0.0 LP61 1.43 / 312 20.4 2.1 ?12.3 LP14 <0 916.3 LP33 <0 522.3 LP52 <0 100.4 LP71 <0 1.9

(79) FIG. 6 illustrates the evolution of Max|DMGD| as a function of wavelength for few-mode fibers supporting from 6 to 16 LP guided modes. More precisely, FIG. 6 shows the Max|DMGD| as a function of wavelength for Ex. 2, 5, 6 & 9 listed in Table 1. Such examples correspond to few-mode fiber according to the second embodiment of the invention, as depicted in FIG. 3B.

(80) The x-axis depicts the wavelength of the light guided by the fiber, ranging from 1530 to 1570 nm. The y-axis depicts the Max|DMGD| between any two LP guided modes, expressed as ps/km and ranging from 0 to 200. Curve 51 corresponds to the FMF guiding 6 LP modes of Example 2; curve 52 corresponds to the FMF guiding 9 LP modes of Example 5; curve 53 corresponds to the FMF guiding 12 LP modes of Example 6, while curve 54 corresponds to the FMF guiding 16 LP modes of Example 9.

(81) As can be seen, the Max|DMGD| remains low in the entire extended C-band from 1530 to 1570 nm. The Max|DMGD| slope in this extended C-band is in absolute value <3 ps/km/nm, preferably <2 ps/km/nm, and more preferably <1 ps/km/nm.

(82) FIG. 7 depicts the evolution of Max|DMGD| for FMFs supporting from 6 to 16 LP guided modes as a function of the ? parameter of the graded-index profile. More precisely, FIG. 7 shows the Max|DMGD| as a function of ? for Ex. 2, 4, 7 & 8 listed in Table 1. Such examples correspond to few-mode fiber according to the second embodiment of the invention, as depicted in FIG. 3B.

(83) The x-axis depicts the value of ?, which is a non-dimensional parameter that defines the index profile shape of the graded-index optical core, with a ranging from 1.91 to 1.99. The y-axis depicts the Max|DMGD| between any two LP guided modes, expressed as ps/km and ranging from 0 to 200. Curve 61 corresponds to the FMF guiding 6 LP modes of Example 2; curve 62 corresponds to the FMF guiding 9 LP modes of Example 4; curve 63 corresponds to the FMF guiding 12 LP modes of Example 7, while curve 64 corresponds to the FMF guiding 16 LP modes of Example 8.

(84) As shown on FIG. 7, there are optimum values for ?, for which these Max|DMGD| have minimum values. ? lower and higher than these optimum ? generally exhibit DMGDs with opposite signs.

(85) By carefully choosing the value of ?, as close as possible to the optimum, it is possible to design a few-mode fiber minimizing the Max|DMGD| value. The few mode optical fibers 10 according to the disclosure have a low loss and a small differential group delay, and are suitable for use in optical transmission systems, particularly those that utilize space-division multiplexing and that are configured for long-haul transmission.

(86) 6.2 Few-Mode Optical Fibers Link

(87) As mentioned above, there are optimum values for ?, for which Max|DMGD| have minimum values, and a lower and higher than these optimum ? generally exhibit DMGDs with opposite signs.

(88) As a consequence, the inventors have reached the conclusion that, if a FMF is off-target in term of a (i.e. if the ?-value of the FMF is either slightly higher or lower than the optimum ? shown on FIG. 7), it is possible to associate it with another FMF showing an appropriate a (i.e. either higher than the optimum ? if the off-target ? is smaller, or smaller than the optimum ? if the off-target ? is higher), by choosing the appropriate lengths for both FMFs, in order to realize a DMGD-compensated link.

(89) The resulting Max|DMGD| of the link can then be very close to the minimum value shown on FIG. 7. This association can, for instance, compensate for process variability that may result in FMFs with slightly off-target Alphas.

(90) Tables 6, 7, 8 & 9 show examples of these associations for FMFs supporting 6, 9, 12 and 16 LP guided modes, respectively.

(91) TABLE-US-00007 TABLE 6 Alpha.sub.Opt Alpha.sub.Off-Target Alpha.sub.Comp Combination of Alphas 1.951 1.956 1.910 with L.sub.Comp/L.sub.Opt = 0.128 DMGD vs. LP01 DMGD vs. LP01 DMGD vs. LP01 DMGD vs. LP01 Ex. 2 (ps/km) (ps/km) (ps/km) (ps/km) LP01 / / / / LP11 ?7.6 1.6 ?81.6 ?7.9 LP21 ?6.5 11.7 ?151.8 ?6.9 LP02 ?7.8 10.1 ?150.3 ?8.1 LP12 ?7.5 17.5 ?206.1 ?7.9 LP31 0.8 27.2 ?209.7 0.3 Max|DMGD| 8.6 27.2 209.7 8.4 (ps/km)

(92) Table 6 shows the values of DMGDs and Max|DMGD|, expressed as ps/km, at a wavelength ?=1550 nm for few-mode fibers supporting 6 LP guided modes according to example 2 (already discussed in Table 1 and Table 2). According to the results shown on FIG. 7, the optimum value of a for such a FMF is ?.sub.opt=1.951. Column 2 of table 6 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for a FMF according to Example 2 showing an optimum value of ?. Column 3 of table 6 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for a FMF according to Example 2, which would show a slightly off-target ?, for example ?.sub.off-target=1.956.

(93) Column 4 of table 6 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for a FMF according to Example 2, which would show a value of ?, for example ?.sub.comp=1.910, which could be used in concatenation with the FMF of column 3, to build a DMGD-compensated optical link.

(94) The last column in table 6 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for an optical link built by concatenating the FMF in column 3 with a length L.sub.opt and the FMF in column 4 with a length L.sub.comp. The total length of the link is L.sub.link=L.sub.opt+L.sub.comp, where L.sub.opt and L.sub.comp are such that L.sub.comp/L.sub.opt=0.128.

(95) Such a ratio is chosen so as to be equal to, or very close to, the ratio of the absolute values of the DMGDs of the modes having the highest DMGDs values for both fibers in the link. In example 2 disclosed in table 6, such a highest mode is the LP.sub.31 mode, for which the ratio of the absolute values of the DMGDs for both fibers in the optical link is |27.2/?209.7|=0.13. In this embodiment, we choose a ratio of the optical fibers lengths close to 0.13, such that L.sub.comp/L.sub.opt=0.128. The lengths L.sub.opt and L.sub.comp are hence chosen so as to minimize Max|DMGD| on the optical link.

(96) As shown in table 6, the values of DMGDs and Max|DMGD| for the DMGD-compensated optical link are hence very close to the minimum values. Such an optical link allows guiding six LP modes, with a very low DGMD, and thus a very good system reach.

(97) Such an optical link meets the core criterion C<18, and is made up of two FMFs, which both individually meet the core criterion C<18.

(98) TABLE-US-00008 TABLE 7 Alpha.sub.Opt Alpha.sub.Off-Target Alpha.sub.Comp Combination of Alphas 1.944 1.938 1.990 with L.sub.Comp/L.sub.Opt = 0.134 DMGD vs. LP01 DMGD vs. LP01 DMGD vs. LP01 DMGD vs. LP01 Ex. 4 (ps/km) (ps/km) (ps/km) (ps/km) LP01 / / / / LP11 ?22.5 ?35.7 74.6 ?22.7 LP21 ?25.1 ?51.4 168.2 ?25.4 LP02 ?20.1 ?46.2 172.4 ?20.4 LP12 ?24.3 ?62.6 259.1 ?24.6 LP31 ?16.7 ?55.7 270.9 ?17.1 LP03 ?16.1 ?63.6 337.2 ?16.2 LP22 ?17.8 ?66.4 342.9 ?18.0 LP41 0.3 ?50.7 377.1 ?0.1 Max|DMGD| 25.4 66.4 302.5 25.4 (ps/km)

(99) Table 7 shows the values of DMGDs and Max|DMGD|, expressed as ps/km, at a wavelength ?=1550 nm for few-mode fibers supporting 9 LP guided modes according to example 4 (already discussed in Table 1 and Table 3). According to the results shown on FIG. 7, the optimum value of ? for such a FMF is ?.sub.opt=1.944. Column 2 of table 7 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for a FMF according to Example 4 showing an optimum value of ?. Column 3 of table 7 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for a FMF according to Example 4, which would show a slightly off-target ?, for example ?.sub.off-target=1.938.

(100) Column 4 of table 7 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for a FMF according to Example 4, which would show a value of ?, for example ?.sub.comp=1.990, which could be used in concatenation with the FMF of column 3, to build a DMGD-compensated optical link.

(101) The last column in table 7 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for an optical link built by concatenating the FMF in column 3 with a length L.sub.opt and the FMF in column 4 with a length L.sub.comp. The total length of the link is L.sub.link=L.sub.opt+L.sub.comp, where L.sub.opt and L.sub.comp are such that L.sub.comp/L.sub.opt=0.134.

(102) Such a ratio is chosen equal to, or very close to, the ratio of the absolute values of the DMGDs of the modes having the highest DMGDs values for both fibers in the link. In example 4 disclosed in table 7, such a highest mode is the LP.sub.41 mode, for which the ratio of the absolute values of the DMGDs for both fibers in the optical link is |?50.7/377.1|=0.13. In this embodiment, we choose a ratio of the optical fibers lengths close to 0.13, such that L.sub.comp/L.sub.opt=0.134. The lengths L.sub.opt and L.sub.comp are hence chosen so as to minimize Max|DMGD| on the optical link.

(103) As shown in table 7, the values of DMGDs and Max|DMGD| for the DMGD-compensated optical link are hence very close to the minimum values. Such an optical link allows guiding nine LP modes, with a very low DGMD, and thus a very good system reach.

(104) Such an optical link meets the core criterion C<18, and is made up of two FMFs, which both individually meet the core criterion C<18.

(105) TABLE-US-00009 TABLE 8 Alpha.sub.Opt Alpha.sub.Off-Target Alpha.sub.Comp Combination of Alphas 1.934 1.942 1.900 with L.sub.Comp/L.sub.Opt = 0.230 DMGD vs. LP01 DMGD vs. LP01 DMGD vs. LP01 DMGD vs. LP01 Ex. 7 (ps/km) (ps/km) (ps/km) (ps/km) LP01 / / / / LP11 ?48.2 ?30.4 ?125.6 ?48.2 LP21 ?68.6 ?33.2 ?222.0 ?68.5 LP02 ?56.5 ?21.3 ?208.7 ?56.3 LP12 ?52.1 0.2 ?278.0 ?51.8 LP31 ?66.4 ?13.7 ?294.6 ?66.2 LP03 18.1 86.1 ?274.9 18.6 LP22 ?2.6 65.9 ?298.1 ?2.1 LP41 ?37.0 32.7 ?338.1 ?36.6 LP13 ?69.2 13.7 ?425.5 ?68.5 LP32 ?3.7 80.3 ?365.5 ?3.0 LP51 6.7 92.9 ?365.0 7.2 Max|DMGD| 87.3 126.1 425.5 87.1 (ps/km)

(106) Table 8 shows the values of DMGDs and Max|DMGD|, expressed as ps/km, at a wavelength ?=1550 nm for few-mode fibers supporting 12 LP guided modes according to example 7 (already discussed in Table 1 and Table 4). According to the results shown on FIG. 7, the optimum value of a for such a FMF is ?.sub.opt=1.934. Column 2 of table 8 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for a FMF according to Example 7 showing an optimum value of ?. Column 3 of table 8 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for a FMF according to Example 7, which would show a slightly off-target ?, for example ?.sub.off-target=1.942.

(107) Column 4 of table 8 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for a FMF according to Example 7, which would show a value of ?, for example ?.sub.comp=1.900, which could be used in concatenation with the FMF of column 3, to build a DMGD-compensated optical link.

(108) The last column in table 8 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for an optical link built by concatenating the FMF in column 3 with a length L.sub.opt and the FMF in column 4 with a length L.sub.comp. The total length of the link is L.sub.link=L.sub.opt+L.sub.comp, where L.sub.opt and L.sub.comp are such that L.sub.comp/L.sub.opt=0.230.

(109) Such a ratio is chosen equal to, or very close to, the ratio of the absolute values of the DMGDs of the modes having the highest DMGDs values for both fibers in the link. In example 7 disclosed in table 8, such a highest mode is the LP.sub.51 mode, for which the ratio of the absolute values of the DMGDs for both fibers in the optical link is |92.9/?365.0|=0.25. In this embodiment, we choose a ratio of the optical fibers lengths close to 0.25, such that L.sub.comp/L.sub.opt=0.230. The lengths L.sub.opt and L.sub.comp are hence chosen so as to minimize Max|DMGD| on the optical link.

(110) As shown in table 8, the values of DMGDs and Max|DMGD| for the DMGD-compensated optical link are very close to the minimum values. Such an optical link allows guiding twelve LP modes, with a very low DGMD, and thus a very good system reach.

(111) Such an optical link meets the core criterion C<18, and is made up of two FMFs, which both individually meet the core criterion C<18.

(112) TABLE-US-00010 TABLE 9 Alpha.sub.Opt Alpha.sub.Off-Target Alpha.sub.Comp Combination of Alphas 1.926 1.916 1.990 with L.sub.Comp/L.sub.Opt = 0.150 DMGD vs. LP01 DMGD vs. LP01 DMGD vs. LP01 DMGD vs. LP01 Ex. 8 (ps/km) (ps/km) (ps/km) (ps/km) LP01 / / / / LP11 ?88.8 ?118.7 94.4 ?90.9 LP21 ?131.5 ?190.7 233.4 ?135.4 LP02 ?112.3 ?171.2 251.7 ?116.0 LP12 ?133.6 ?221.5 411.6 ?139.0 LP31 ?142.9 ?231.2 403.3 ?148.5 LP03 ?83.8 ?199.8 638.7 ?90.5 LP22 ?104.8 ?221.2 619.2 ?111.6 LP41 ?121.8 ?239.1 604.9 ?129.0 LP13 ?34.2 ?176.4 854.4 ?42.0 LP32 ?43.8 ?187.4 852.3 ?51.9 LP51 ?67.9 ?213.5 837.5 ?76.5 LP04 ?106.8 ?270.8 922.4 ?115.2 LP23 ?80.2 ?245.7 957.3 ?88.9 LP42 ?8.2 ?177.3 1049.5 ?17.4 LP61 9.8 ?163.6 1090.3 ?0.1 Max|DMGD| 152.7 270.8 1090.3 148.5 (ps/km)

(113) Table 9 shows the values of DMGDs and Max|DMGD|, expressed as ps/km, at a wavelength ?=1550 nm for few-mode fibers supporting 16 LP guided modes according to example 8 (already discussed in Table 1 and Table 5). According to the results shown on FIG. 7, the optimum value of a for such a FMF is ?.sub.opt=1.926. Column 2 of table 9 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for a FMF according to Example 8 showing an optimum value of ?. Column 3 of table 9 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for a FMF according to Example 8, which would show a slightly off-target ?, for example ?.sub.off-target=1.916.

(114) Column 4 of table 9 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for a FMF according to Example 8, which would show a value of ?, for example ?.sub.comp=1.990, which could be used in concatenation with the FMF of column 3, to build a DMGD-compensated optical link.

(115) The last column in table 9 gives the values of DMGDs and Max|DMGD| at a wavelength ?=1550 nm for an optical link built by concatenating the FMF in column 3 with a length L.sub.opt and the FMF in column 4 with a length L.sub.comp. The total length of the link is L.sub.link=L.sub.opt+L.sub.comp, where L.sub.opt and L.sub.comp are such that L.sub.comp/L.sub.opt=0.150.

(116) Such a ratio is chosen equal to, or very close to, the ratio of the absolute values of the DMGDs of the modes having the highest DMGDs values for both fibers in the link. In example 7 disclosed in table 9, such a highest mode is the LP.sub.61 mode, for which the ratio of the absolute values of the DMGDs for both fibers in the optical link is |?163.6/1090.3|=0.15. In this embodiment, we choose a ratio of the optical fibers lengths that equals 0.15, such that L.sub.comp/L.sub.opt=0.150. The lengths L.sub.opt and L.sub.comp are hence chosen so as to minimize Max|DMGD| on the optical link.

(117) As shown in table 9, the values of DMGDs and Max|DMGD| for the DMGD-compensated optical link are very close to the minimum values. Such an optical link allows guiding sixteen LP modes, with a very low DGMD, and thus a very good system reach.

(118) Such an optical link meets the core criterion C<18, and is made up of two FMFs, which both individually meet the core criterion C<18.

(119) The examples of optical links described above in relation to tables 6, 7, 8 and 9 are hence all built by concatenating two FMFs having the same overall refractive index profile, according to any of the embodiments of FIGS. 3A to 3C, having the same radii R.sub.1, R.sub.2 and R.sub.3 and index differences Dn.sub.1, Dn.sub.2 and Dn.sub.3, but having slightly different values of ?, one with a value of a slightly lower than the optimum ?, and one with a value of a slightly higher than the optimum ?, so that the DMGDs with opposite signs can compensate each other. It is thus possible to build a DMGD-compensated link with robust features.

(120) However, according to another embodiment, it is also possible to concatenate more than two FMFs: as long as at least two of them guide at least one mode with DMGDs showing opposite signs, compensation can occur to build a DMGD-compensated link.

(121) Moreover, according to another embodiment, it is also possible to concatenate FMFs having the same value for ?, but having different values of R.sub.1 or of Dn.sub.1 for example. The same DMGD compensation can occur by choosing the appropriate optical fiber lengths.

(122) It is also possible, according to another embodiment, to associate few-mode fibers showing different refractive index profiles, such as, as a mere example, by combining a few-mode fiber according to Example 1 with a few-mode fiber according to Example 3, or even by combining a FMF according to Example 8 with a FMF according to Example 10.

(123) More generally, it is possible to associate FMFs having any of the refractive index profiles of FIGS. 3A to 3C. It is also possible to associate one or several FMFs having one of the refractive index profiles of FIGS. 3A to 3C, with any other FMF having an optical core with a single ? graded-index profile and an optical cladding comprising a trench.

(124) It is also possible to associate in an optical link several optical fibers, which individually satisfy the criteria R.sub.1i?13.5 ?m and C?18, or which do not individually satisfy such criteria, or for which one, or several of them only, individually satisfy such criteria.

(125) As long as the appropriate lengths of optical fibers are carefully chosen, such that the optical link meets the criterion

(126) $C=10.Math.\frac{\mathrm{Max}\left|{\mathrm{DMGD}}_{\mathrm{link}}\right|}{{\left(R_{1\mathrm{link}}^2.Math.{\mathrm{Dn}}_{1\mathrm{link}}\right)}^3}?18,$
any association of any number of any type of FMFs is possible to build a DMGD-compensated optical link.

(127) FIGS. 8A and 8B illustrate embodiments of an optical system according to the invention. According to the first embodiment in FIG. 8A, such an optical system comprises transceivers 81 and receivers 85 optically connected by an optical fiber link 70 that includes at least two spans of fiber. Transceivers 81 comprise light sources (such as lasers) and generate n LP modes, referenced 1, 2, . . . , n used in the optical system of FIG. 8A. A mode multiplexer 82 multiplexes the n LP modes and is optically connected to optical link 70, which guides the n multiplexed LP modes, towards a mode demultiplexer 83, which is optically connected to the end of optical link 70.

(128) Mode demultiplexer 83 demultiplexes the n multiplexed LP modes, and feeds each LP mode into an amplifier 84. At the output of amplifiers 84, LP modes enter receivers 85.

(129) Such an optical system may comprise M optical links. In an example, M=1; in another example, M=2; in another example M=5; in yet another example, M=10. In case the optical system comprises M optical links, it also comprises M mode multiplexers 82, M mode demultiplexers 83, and M amplifiers 84 for each LP mode guided by the optical system.

(130) The embodiment in FIG. 8B differs from the first embodiment in FIG. 8A in that amplifier 84 amplifies all LP modes guided by the optical link 70; as such, amplifier 84 is optically connected between the output of optical link 70 and the input of mode demultiplexer 83. In this second embodiment, when the optical system comprises M optical links, it also comprises M amplifiers 84; however, there is only one mode multiplexer 82, optically connected between transceivers 81 an optical link 70, and only one mode demultiplexer 83, optically connected between amplifier 84 and receivers 85.

(131) The embodiments of FIGS. 8A and 8B are given as mere examples, and an optical link according to the invention may of course be used in any other kind of optical system.