WO2006006192A1 - Optical fibers with very long polarization correlation length and method for obtaining optical fibers with very long polarization correlation length - Google Patents

Abstract

A method for determining at least a parameter of an almost periodic spin function, α(z), to be applied to an optical fiber during the drawing by means of a spinning process, such that the polarization correlation length Lc of said optical fiber, corresponding to any input state of polarization, is considerably increased. Said method is characterized in selecting the spin function so that I (see formula I) where δ is about 1 km, ∈ is about 0.01 and the vector II (see formula II) is such that III (see formula III) where LB and LF are, respectively, the beat length and the birefringence correlation length of said fiber, and IV (see formula IV). The method is effective for substantially increasing the polarization correlation length of optical fibers drawn while applying a spin function as determined above.

Description

OPTICAL FIBERS WITH VERY LONG POLARIZATION CORRELATION LENGTH AND METHOD FOR OBTAINING OPTICAL FIBERS WITH VERY LONG POLARIZATION

CORRELATION LENGTH

DESCRIPTION

Technical Field

This invention is about optical fibers and, in particular, single mode fibers with very long polarization correlation length. This invention also relates to a method for obtaining optical fibers with very long polarization correlation length.

The invention may apply to all kinds of fibers such as, for example, dispersion unshifted fibers, dis¬ persion shifted fibers, dispersion flattened fibers, dispersion compensating fibers, fibers for ampli¬ fiers (e.g. erbium doped fibers, or fibers for Raman amplification), fiber sensors, highly nonlinear fibers, highly birefringent fibers, polarization maintaining fibers, fiber gratings (e.g. Bragg gratings), microstructured fibers, photonic crystal fibers.

Background Art

It is well known that the so-called single-mode fibers are affected by several kinds of perturbations. These perturbations are caused by asymmetric deformations of the core and/or the cladding, by mechanical stresses at the core-cladding interface, and by mechanical stresses due to the external environment. Because of these perturbations, two different modes can propagate along the fiber with different phase and group velocities; the two modes are non-degenerate and orthogonally polarized. The main consequence of this phenomenon is that the state of polarization of the electromagnetic field does not remain unchanged but varies during signal propagation.

It is useful to introduce the tridimensional Stokes versor, s — (si, S₂, S₃)^τ which represents the polar¬ ization state of the electromagnetic field (the apex ^τ represents hereinafter the transposition, see for example E. Collet, "Polarized light", Marcel Dekker Inc., ISBN 0824787293, 1993). Generally, s is a function of the coordinate z which identifies the position of interest along the fiber. Under a phenomenological point of view, a perturbed single mode fiber can be described as a bire¬ fringent fiber where two polarization eigenstates (corresponding to the non-degenerate modes) prop¬ agate. Each mode is characterized by its propagation constant Jt₁ and fc₂. respectively. Mathematical¬ ly, the birefringence may be represented by means of a real tridimensional vector /3 = (βι, fa, βz)^τ , called "birefringence vector". Its direction identifies the polarization eigenstates of the perturbed fiber in the Stokes space; its norm, "birefringence", is equal to the absolute value of the difference between the two constants of propagation

where the symbol • represents the internal product between vectors. It is usually introduced also the definition of "beat length" L_# as

Ls - J² . (2,

When the perturbations act uniformly along the fiber, the birefringence vector remains constant; this is implicitly assumed in (1) and (2). Consequently, the state of polarization evolves periodically along the fiber, with a period equal to LB- On the other hand, if the state of polarization is parallel to one of the two eigenstates (i.e. to the birefringence vector), the polarization will remain unchanged during the entire propagation. Therefore, the fibers with constant birefringence vector are called "polarization maintaining fibers". On the contrary, when the perturbations do not interact uniformly along the fiber, the birefringence vector varies section by section and the propagating state of polarization is neither periodic, nor constant.

In particular, in the case of non polarization-maintaining fibers, the perturbations act on the fiber in a random way and so the birefringence vector can be represented by a stationary stochastic process which is dependent on the coordinate z. In such case, the beat length is defined as

^{B ~}7wWϊ ^{( )} where (β²(z)} is the mean square value of the birefringence. Moreover, one can define the "birefrin¬ gence correlation length", LF, as the distance at which

(β(z) . β(z + L_F)) ₌ 1

(β(z) - β(z)} e ' ^{( )} where e = 2.71828... is the Neper number. In practice, Lp is inversely proportional to the speed at which the birefringence vector, i.e. the perturbations along the fiber, looses correlation with its initial state. It is assumed that in polarization maintaining fibers, Lp tends to infinity. In the literature, many techniques have been proposed in order to measure and to estimate the beat length LB and the correlation length Lp (see, for example: D. Q. Chowdhury, D. A. Nolan, "Perturba¬ tion model for computing optical fibre birefringence from a two-dimensional refractive-index profile", Optics Letters, vol. 20, no. 19, pp. 1973-1975, 1995; Y. Park et al., "Residual stress in a doubly clad fiber with depressed inner cladding", Journal of Lightwave Technology, vol. 17, no. 10, pp. 1823- 1834, 1999; J. G. Ellison, A. S. Siddiqui, "Automatic matrix-based analysis method for extraction of optical fiber parameters from polarimetric optical time domain reflectometry data", Journal of Light¬ wave Technology, vol. 18, pp. 1226-1232, 2000; A. Galtarossa, L. Palmieri, M. Schiano, T. Tambosso, "Measurement of birefringence correlation length in long single-mode fibers", Optics Letters, vol. 26, pp. 962-964, 2001 ; M. Wegmuller, M. Legre, N. Gisin, "Distributed beatlength measurement in single- mode fibers with optical frequency-domain reflectometry", Journal of Lightwave Technology, vol. 20, pp. 828-835, 2002). In the case of non polarization-maintaining fibers, typical values for LB and L_F are between 1 m and 50 m, and between 0.1 m e 20 m, respectively. When dealing with polarization maintaining fibers, LB is typically shorter than 0.1 m and Lp is about tens of kilometers.

When the birefringence vector is random, even the polarization state evolves in a random and not predictable way. This phenomenon may be quantified by means of the so-called "polarization cor- relation length", Lc, which is described as follows. Consider a statistic set of randomly birefringent fibers (i.e. non polarization-maintaining fibers). In each fiber an electromagnetic field with a fixed initial polarization state s(0) (the same for all fibers) is launched. Due to the random evolution of the birefringence vector, the polarization state does not remain unchanged but evolves during the signal propagation so that the state of polarization at the distance z will be equal to s(z). The polarization correlation length Lc is defined as the length at which

IKs(Lc))II = V(KLc)) - (HLc)) = - , (5) where (s(z)} is the mean (over the statistical ensemble of fibers) of the polarization state of the elec¬ tromagnetic field at position z. If the state of polarization does not remain fixed and if no relationship exists among the polarization states, then §(z) assumes different values in each fiber of the statistical ensemble. Therefore, the norm of the quantity (s(z)) tends to zero. On the contrary, the longer a suit¬ able state of polarization (or the relation among the states of polarization) is preserved , the slower IKs(Z))H tends to zero, and the longer is the polarization correlation length Lc- Actually, the polarization correlation length is a parameter that describes the peculiarity of a fiber to maintain a suitable polarization state and/or a suitable relation among the states of polarization. The longer is Lc, the more evident is this characteristic of the fiber. It is worth noting that the polarization correlation length depends in general on both the initial polarization state and the fiber properties, i.e. on the beat length LB, and on the birefringence correlation length Lp. In the case of non polarization-maintaining fibers (commonly used for telecommunications and sen- sors), the polarization correlation length varies from a few meters up to a few tens of meters, depend¬ ing on the kind of fiber. This means that in this kind of fibers neither the state of polarization of the electromagnetic field, nor a suitable relation among the states of polarization is preserved. Indeed, the state of polarization of the electromagnetic field evolves randomly and loses the correlation with its initial state only after a few meters or a few tens of meters. On the contrary, in a polarization maintaining fiber the polarization correlation length is tens of kilometers long.

As it is well known in the literature, the possibility of preserving suitable polarization states or suitable relations among the states of polarization, i.e. of getting a fiber with long polarization correlation length, is very important for many applications. For example, the efficiency of several nonlinear op¬ tical phenomena (such as, for instance, wavelength conversion, Raman amplification or parametric amplification) is increased when the state of polarization of the electromagnetic fields is maintained constant (Y. R Svirko, N. I. Zheludev, "Polarization of light in nonlinear optics", John Wiley & Sons, ISBN 0471976407, 1998). As a further example, the use of a preserved polarization state is par- ticularly useful for transmission applications, such as, but not only, coherent transmission with het- erodyne/homodyne detection, or polarization multiplexing transmission (see, for example: T. Okoshi, K. Kikuchi, "Coherent optical fiber communications", Kluver Academic Publisher, ISBN 9027726779, 1988) or those cryptogaphic techniques based on quantistic phenomena. Finally, even many sensors based on optical fibers, such as for example gyroscopes, temperature, stresses or pressure sensors, and sensors based on the Faraday effect, accrue a big benefit from the preservation of the state of polarization (or of suitable relations among the states of polarization).

Because of their practical importance, some techniques to able produce optical fibers with long polar¬ ization correlation length, i.e. fibers able to maintain suitable polarization states or suitable relations among the states of polarization, have been proposed. In the following, four known techniques to produce fibers with long polarization correlation length are reported.

A first known method (see, for example: US patent n. 4,354,736; WO patent n. 00/60390) consists in the realization of special preforms that do not have circular symmetric cross-sections, but show one or more plane of symmetry (as for exampe in the case of a fiber with an elliptical core). In this way the produced fiber has a constant birefringence vector, and it is therefore able to maintain a pair of mutually orthogonal states of polarization.

A second known method for producing optical fibers able to preserve suitable states of polarization (see, for example: US patent n. 5,452,394; US patent n. 4,578,097; EP patent n. 0 413 387 A1) consists in introducing in the preform some elements that break the circular symmetry of the preform section. These elements are introduced in the direction parallel to the preform axis and usually have optical, thermal or mechanical properties different from those of the preform. In this way, during the fiber drawing (and possibly simultaneous torsion) mechanical stresses are introduced, and therefore the core is made birefringent.

A third known method for producing optical fibers able to preserve suitable states of polarization (see, for example: A. J. Barlow, J. J. Ramskov-Hansen, D. N. Payne, Birefringence and polarization mode- dispersion in spun single-mode fibers, Applied Optics, vol. 20, pp. 2962-2968, 1981 ; T. Okoshi, Heterodyne and Coherent Optical Fiber Communications: Recent Progress, IEEE Trans. On Mi¬ crowave Th. And Tech., MTT-30, pp. 1138-1149, 1982; T. Okoshi, K. Kikuchi, "Coherent optical fiber communications", Kluver Academic Publisher, ISBN 9027726779, 1988) consists in twisting the al¬ ready completely drawn fiber; in this way it is possible to realize a fiber with two circular or elliptical eigenstates. However, to reach this purpose, the fiber has to be twisted at a speed of tens of turns per meter, that may seriously impair the structural integrity of the fiber itself.

A fourth known method for producing optical fibers able to preserve suitable states of polarization, or suitable relations among the states of polarization of the optical signals (see, for example: A. J. Bar- low, J. J. Ramskov-Hansen, D. N. Payne, Birefringence and polarization mode-dispersion in spun single-mode fibers, Applied Optics, vol. 20, pp. 2962-2968, 1981) consists in spinning the fiber dur¬ ing its drawing. This process, called "spin" or "spinning", forces the fiber birefringence vector to rotate, without introducing any additional stresses to the fiber. Techniques to apply the spin process to fibers are reported, for example, in the patents: WO n. 83/00232; US n. 5,298,047; US n. 5,418,881. Hereinafter, α(z) will represent the "spin function" that is the speed of rotation of the fiber at point z.

The first and the second method (described in the mentioned patents), require to realize special preforms, that are substantially different from those commonly used for producing non polarization- maintaining fibers. Furthermore, the fibers realized according to the first and second method, described in the above mentioned patents, show attenuation per unit of length higher than that of the common non polarization- maintaining fibers; moreover, the capacity of preserving a certain state of polarization of the optical signal strongly depends on the temperature. The fibers produced according to the first, the second and the third method, described in the above mentioned patents, are able to preserve only a pair of states of polarization (i.e., those parallel to the birefringence vector in the Stokes space). Conversely, all the other states evolve periodically with a period equal to LB- AS it is known, for these kind of fibers LB is typically much shorter than 1 m, that means that the evolution of the states of polarization not parallel to the eigenstates is very fast. The two eigenstates in the fibers produced according to the first, the second and the third method, described in the above mentioned patents, have different group velocities. Therefore, unless only one eigenstate is excited at each time, the optical signal experiences a strong modal dispersion and it is transmitted to the end of the fiber with a strong distorsion. The fibers produced according to the fourth method tend to have a long polarization correlation length. Moreover, the polarization eigenstates of the fibers obtained following the fourth method tend to have the same group velocity; thus, the modal dispersion observed in the fibers produced with the above mentioned first, second and third method is strongly reduced. However, according to the literature and to the prior art related to the fourth method, it is not known how to appropriately calibrate the parameters of the spinning process so to maximize the optical fiber performances in order to preserve suitable states of polarization and/or suitable relations among the states of polarization.

These considerations make the above mentioned methods not fully satisfactory.

Disclosure of the Invention

The inventors have found that by properly selecting either the spin function, or one or more of its parameters, it is possible to increase the fiber polarization correlation length up to N times the polar¬ ization correlation length that the same fiber would have if it were not subjected to the spin process. Preferably, N is approximately 5; more preferably, N is approximately 50; even more preferably N is approximately 5000 or even larger.

Moreover, the inventors have found precise mathematical relations, that allow to determine the spin function and its parameters so to guarentee a considerable increase of the polarization correlation length.

The truthfulness of such mathematical relations has been verified by means of numerical simulation techniques, that are well known and exploited in the scientific literature because of their reliability and adherence to the physical reality (see for example: R K. A. Wai, C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence", Journal of Lightwave Technology, vol. 14, pp. 148-157, 1996; A. Galtarossa, L. Palmieri, M- Schiano, T. Tarn- bosso, "Statistical properties of fiber random birefringence", Optics Letters, vol.. 25, pp. 1322-1324, 2000; A. Galtarossa, L. Palmieri, A. Pizzinat, B. S. Marks, C. R. Menyuk, "An analytical formula for the mean differential group delay of randomly birefringent spun fibers", Journal of Lightwave Technology, vol. 21 , pp. 1635-1643, 2003). The inventors underline that a wide scientific literature regarding the so-called "almost periodic func¬ tions" is available (see for instance, C. Corduneanu, "Almost periodic functions", lnterscience Pub¬ lishers, ISBN 0470173955, 1968). Additionally, the inventors put in evidence that the beat length L_B, and the birefringence correlation length Lp can be measured or estimated by means of techniques already known in the literature (see for instance: D. Q. Chowdhury, D. A. Nolan, "Perturbation model for computing optical fibre birefringence from a two-dimensional refractive-index profile", Optics Let¬ ters, vol. 20, no. 19, pp. 1973-1975, 1995; Y. Park et at., "Residual stress in a doubly clad fiber with depressed inner cladding", Journal of Lightwave Technology, vol. 17, no. 10, pp. 1823-1834, 1999; J. G. Ellison, A. S. Siddiqui, "Automatic matrix-based analysis method for extraction of optical fiber pa¬ rameters from polarimetric optical time domain reflectometry data", Journal of Lightwave Technology, vol. 18, pp. 1226-1232, 2000; A. Galtarossa, L. Palmieri, M. Schiano, T. Tambosso, "Measurement of birefringence correlation length in long single-mode fibers", Optics Letters, vol. 26, pp. 962-964, 2001; M. Wegmuller, M. Legre, N. Gisin, "Distributed beatlength measurement in single-mode fibers with optical frequency-domain reflectometry", Journal of Lightwave Technology, vol. 20, pp. 828-835, 2002).

The polarization correlation length of a randomly perturbed optical fiber (i.e. a fiber with random bire¬ fringence, i.e. a non polarization-maintaining fiber) subjected to the spin process, can be studied only after having determined, as a first step, a mathematical model for describing the fiber birefringence, and, as a second step, the equations that describe the mean evolution of the states of polarization. Concerning the mathematical model for the description of the fiber random birefringence, it is con¬ venient to use a well known model introduced in the literature (see: P. K. A. Wai, C. R. Menyuk, "Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence", Journal of Lightwave Technology, vol. 14, pp. 148-157, 1996), according to which, in the tridimensional Stokes space with coordinate axes {ύι, U₂, «₃}, the birefringence vector reads

β(z) (6)

where the symbol b = 2-K/LB has been introduced for convenience, LB is the beat length of the fiber to be modelled, and

A(z) = J" a(t)dt , (7)

Jo a(z) being the generic spin function applied to the optical fiber, and where the function θ{z) is given by the equation dB_

= ση(z) , dz (8) in which the symbol σ = 1/Λ/2LF has been introduced for convenience, Lp is the birefringence correlation length of the fiber to be modeled, and in which η(z) is a white Gaussian noise (see, for example: B. øksendal, "Stochastic Differential Equations", Springer Verlag, ISBN 3540047581 , 2003). The model introduced just now is known in the scientific literature because of its reliability and adherence to the physical reality of optical fibers.

If s(z) denotes the versor representing in the Stokes space, {ύι, «₂, «₃}, the state of polarization of the electromagnetic field present in z, then the polarization evolution along the fiber is described by the equation (see, for example: R. Ulrich, A. Simon, "Polarization optics of twisted single-mode fibers", Applied Optics, vol. 18, pp. 2241-2251, 1979)

f_z = β(z) x s(z) . (9)

The inventors have observed that it is convenient to write equation (9) in a new reference frame, that rotates around the axis «₃ with speed equal to 2α(z) + 2dθ/dz. If v(z) indicates the Stokes versor in the new reference frame, then v(z) is related to s(z) by

v{z) = R[-2A(z)]R[-2θ(z)]s(z) , (10)

where cos φ — sin φ 0

R(φ) = sin φ cos φ 0 (11)

0

Please note that R(φ)R^τ(φ) = I₃, where R^τ{φ) = R(-φ) is the transpose matrix of R(φ), and J₃ is the identity matrix of dimension 3 x 3. As a consequence, after performing the proper mathematical transformation, equation (9) becomes

By definition, the polarization correlation length is the distance z = Lc such that

Il (S(Lc)) Il = V(HLc)) ^■ (HLc)) = - , (13)

where the angled parenthesis, (•}, represent the statistical average. Since the spin function a(z) is independent of η(z), from equation (10) it follows that

(s(z)) = R[2A(z)](R[2θ(z)]υ(z)) , (14)

and consequently,

\\(R[2θ(z)]ϋ(z))\\ . (15)

Thus, in order to determine Lc it is necessary to evaluate the mean of the vector

v\ cos 2Θ — V₂ sin 2Θ w(z) = R[2θ(z)]ϋ(z) = v\ sin 2Θ + V₂ cos 2Θ (16)

V3 Such a mean can be calculated by exploiting the theory of stochastic differential equations (see, for example: B. øksendal, "Stochastic Differential Equations", Springer Verlag, ISBN 3540047581, 2003). In fact, by putting together equation (8) with equation (12), the following stochastic differential equation is obtained:

where Vi is the i-th component of ϋ(z), and the application to equation (17) of the rules of the theory of stochastic differential equations, yields the following ordinar differential equation:

x(z) , (18)

where x(z) = ((vi), (U₂), («3), (ι>i cos20), (v₂ cos20), («3003261), (υi sin20), (υ2sia2θ), (u₃ sin20))^r . If the initial state of polarization is s(0) = (si(0), S₂(O), S₃(O))^T, then the initial condition for equation (18) is

5(0) = ^ (0, 0, 2s₃(0), S₁(O), S₂(O), 0, S₂(O), -S₁(O), 0)^T . (19) By solving equation (18), the evolution of the vector w(z) is determined and consequently, also the polarization correlation length Lc is determined.

From the analysis of equation (18), the inventors understood that it is advantageous to apply a proper almost periodic spin function to an optical fiber, because this allows to obtain a considerable increase of the polarization correlation length of said optical fiber. It can be noted that a spin function a(z) (representing the rotation speed applied to the fiber in the spin process) is called almost periodic when it can be expressed in the following manner:

M

"O²O = ∑ c_k eχ-p(-j2πf_kz) , (20) fc=-M where c__fc = c*_k (the symbol * means complex conjugation), the frequencies f_k are arbitrary real numbers, such that /__& = -f_k, and M is a proper integer number. As a consequence, the periodic functions are a particular sub-case of the almost periodic functions, obtained when f_k = kf₀ with /₀ φ 0 fundamental harmonic. Another subcathegory is represented by the constant spin functions that are obtained when c_k, corresponding to the null frequency f_k = 0, is the only coefficient different from zero.

Because of the convenience of the choice of an almost periodic spin function, the inventors decided to determine precise mathematical conditions that allow to calculate the parameters of such almost periodic spin functions, in order to maximize the polarization correlation length of a fiber.

Any input state of polarization. The inventors have observed that for any input state of polariza- tion, indicated with x(z) = (χi(z), χi{z), #₃(2), 35₄(2:), 3:₅(2:), χe(z), χγ(z), χs{z), 0:9(2:)) the solution to equation (18), it follows

ViMz))² + (W₂(Z))² + (w₃(zψ = (21) = VMz) - Z₈(^)]² + [X5(z) + X7(zψ + X3(z)² , (22)

and consequently, in order to determine the polarization correlation length Lc it is necessary to analyze system (18). The evolution of χ{z) is determined by the Lyapunov characteristic exponents, λj, of the system (18) and by the corresponding eigenvectors, α* (see, for example: W. Hahn, "Stability of motion", Springer Verlag, ISBN 0387038299, 1968). Actually, let λ_mαx be the Lyapunov characteristic exponent with largest real part, and α_mαx be the corresponding eigenvector, then asymptotic evolution of x(z) is

x(z) ~ exp(λ_mαxz)(x(0) ^■ α_mαx)α_mαx , (23)

where x{0) = |(θ, 0, 2s₃(0), sχ(0), S₂(O), 0, S₂(O), -si(O), θ)^r is the initial condition of the system (18) for any input state of polarization.

Upon the application of the well known direct Lyapunov method (see again: W. Hahn, "Stability of motion", Springer Verlag, ISBN 0387038299, 1968), the inventors have observed that, for any almost periodic spin function, the Lyapunov characteristic exponents of (18) have negative real part. Thus, from (23), it follows that the larger the real part of λ_mαx, i.e. the larger Re [λ_mαa;], (and consequently, the closer to zero, because in any case it has to be Re [X_max\ < 0), the more slowly x(z) decays to zero, and the longer is the polarization correlation length Lc-

As it is well known from the scientific literature (see for example the already cited W. Hahn, "Stability of motion", Springer Verlag, ISBN 0387038299, 1968), Re [λ_mαx] can be calculated by means of the following relation:

RePW] = -f LQ = z→ lim+∞ *M zΦ]I . (24)

As a consequence, according to these argumentations, the inventors have come to the conclusion that, to achieve the purpose of producing an optical fiber with polarization correlation length Lc longer than a given value L_min, it is necessary to select at least one of the parameters of the almost periodic spin function α(z) such that

Additionally, the inventors understood that preferably the given value L_min is equal to 1/e times the polarization correlation length, Lc,o, that the same fiber would have, if it were not spun. As a consequence, condition (25) is rewritten in the following way:

where' χ(z; α(z)) is the solution to the system (18), and x(z; 0) is the solution to the same system when α(z) — 0.

The inventors have verified that preferably e is approximately equal to 0.01; more preferably e is approximately equal to 0.001 ; even more preferably e is approximately equal to 0.0001. From a practical point of view, the inventors realized that a considerable increase of the polarization correlation length can be achieved also if condition (26) is replaced by the following

where δ is a suitable finite value approximately equal to 1 km; more preferably, δ is approximately equal to 10 km; even more preferably, δ is approximately equal to 100 km and more. Moreover, preferably e is approximately equal to 0.01; more preferably e is approximately 0.001 ; even more preferably e is approximately equal to 0.0001.

Advantageously, the absolute maximum of the modulus of the spin function a(z) is greater than 2-K/LB- Preferably, LB is longer than 0.05 m; more preferably it is longer than 5 m. In general, it is not possible to derive explicitly the solution of the system (18), and it is not possible to solve explicitly conditions (25), (26) e (27). However, it is always possible to optimize numerically the spin function (or one, or some of its parameters). Indeed, chosen an arbitrary almost periodic spin function, equation (18) can be solved numerically, and the parameters of said spin function can be varied iteratively, until condition (27) is verified. The numerical simulations can be carried out by means of suitable commercial software, such as, for instance, Matlab™, or Mathematica™. By means of such commercial software it is also possible to verify the actual increase of the polarization correlation length, by plotting in a graph the evolution of || (s(z))\\ as a function of z. The solution of equation (18) and conditions (25), (26) and (27) depend not only on the particular spin function, but also on the beat length LB and on the birefringence correlation length L_F. Regarding this point, the inventors observe that in any case it is possible to have an accurate estimate of L_B and LF also before the fiber drawing. As an example, in the case of a production process based on preforms that share substantially the same profile, it is possible to evaluate L_B and LF from the analysis of the preform ellipticity. Additionally, a few fiber kilometers may be drawn from such preform, so to measure the LB and LF values of said fiber by means of known techniques (see, for example: J. G. Ellison, A. S. Siddiqui, "Automatic matrix-based analysis method for extraction of optical fiber parameters from polarimetric optical time domain reflectometry data", Journal of Lightwave Technol¬ ogy, vol. 18, pp. 1226-1232, 2000; A. Galtarossa, L. Palmieri, M. Schiano, T. Tambosso, "Statistical characterization of fiber random birefringence", Optics Letters, vol. 25, pp. 1322-1324, 2000; A. Gal¬ tarossa, L. Palmieri, M. Schiano, T. Tambosso, "Measurement of birefringence correlation length in long single-mode fibers", Optics Letters, vol. 26, pp. 962-964, 2001 ; M. Wegmuller, M. Legre, N. Gisin, "Distributed beatlength measurement in single-mode fibers with optical frequency-domain reflectometry", Journal of Lightwave Technology, vol. 20, pp. 828-835, 2002). The inventors put in evidence that the fiber birefringence parameters (i.e. L_B and Lp) obtained with such measurements, remain the same also for the optical fibers obtained by drawing (with or without the spinning process) the remaining part of the preform. Alternatively, the birefringence parameters can be estimated also by means of suitable models proposed in the scientific literature (see, for example: D. Q. Chowdhury, D. A. Nolan, "Perturbation model for computing optical fibre birefringence from a two-dimensional refractive-index profile", Optics Letters, vol. 20, no. 19, pp. 1973-1975, 1995; Y. Park et al., "Residual stress in a doubly clad fiber with depressed inner cladding", Journal of Lightwave Technology, vol. 17, no. 10, pp. 1823-1834, 1999; D. Q. Chowdhury, D. Wilcox, "Comparison between optic fiber birefrin¬ gence induced by stress anisotropy and geometric deformation", IEEE Journal of Selected Topics in Quantum Electronics, vol. 6, no. 2, pp. 227-232, 2000).

Circular input state of polarization. The inventors observed that when the input state of polar¬ ization is circular, the solution of equation (18) is x(z) — (χi(z), X₂(z), χs(z), 0, 0, 0, 0, 0, θ)^τ, so it comes out that {wι(z)) — {w₂{z)) = 0. This means that when the input state of polarization is circular

<S(z)>|| = \\ (w(_Z)) \\ =

= \(v₃(z))\ = x₃(z) , (28)

and consequently, in order to determine Lc it is sufficient to analyze the system

(29)

_κxz(z)j

The evolution of x(z) is determined by the Lyapunov characteristic exponents, ψ_it of the system (29) and by the corresponding eigenvectors, C₁ (see, for example: W. Hahn, "Stability of motion", Springer Verlag, ISBN 0387038299, 1968). In fact, let ψ_mαx be the Lyapunov characteristic exponent with largest real part and ^_x be the corresponding eigenvector, then the asymptotic evolution of x(z) is

x(z) ~ e-χp(φ_mαxz)(x(0) ^■ c_mαχ)c_mα (30)

where x(0) — (0, 0, l)^τ is the initial condition of the system (29) in the case of circular input state of polarization.

Upon the application of the well known Lyapunov direct method (see again: W. Hahn, "Stability of motion", Springer Verlag, ISBN 0387038299, 1968), the inventors have observed that, for any almost periodic spin function, the Lyapunov characteristic exponents of (29) have negative real part. Thus, from equation (30) it derives that the larger the real part of φ_mαχ, i-e., the larger Re [φ_mαx], (and consequently the closer to zero, since in any case it has to be Re [ψ_mαχ] < 0), the more slowly x(z) decays to zero, and the longer is the polarization correlation length Lc- As it is well known in the scientific literature (see for example the already cited W. Hahn, "Stability of motion", Springer Verlag, ISBN 0387038299, 1968), Re [φ_mαx] can be calculated by means of the following formula:

Re [VW] = z— H >m+oo -« Z ^ . (31) As a consequence, according to these argumentations, the inventors have come to the conclusion that, in the case of a circular input state of polarization, to achieve the purpose of producing an optical fiber with polarization correlation length Lc longer than a given value L_min, it is necessary to select at least one of the parameters of the almost periodic spin function α(z) such that it, H!M1 > ⁱ . ₍₃₂₎ z→+oo z L_min Additionally, the inventors understood that preferably the given value L_min is equal to 1/e times the polarization correlation length Lc,o that the same fiber would have, if it were not subjected to the spin process. As a consequence, condition (32) is rewritten in the following way:

where x(z; α{z)) is the solution of system (29), and x(z- 0) is the solution of the same system when α(z) = 0.

The inventors have verified that preferably e is approximately equal to 0.01 ; more preferably e is approximately equal to 0.001 ; even more preferably e is approximately equal to 0.0001. Under a practical point of view, the inventors realized that a considerable increase of the polarization correlation length can be achieved also if condition (33) is replaced with the following

_z→δ ln ||ic(z; 0) || where δ is a suitable finite value approximately equal to 1 km; more preferably, δ is approximately equal to 10 km; even more preferably, δ is approximately equal to 100 km and more. Moreover, preferably e is approximately equal to 0.01; more preferably e is approximately equal to 0.001; even more preferably e is approximately equal to 0.0001. Advantageously, the absolute maximum of the modulus of the spin function α(z) is greater than 2TT/LB- Preferably, LB is longer than 0.05 m; more preferably it is longer than 5 m. In general, it is not possible to derive explicitly the solution of system (29), and it is not possible to solve explicitly conditions (32), (33) and (34) a part from the case of some types of almost periodic spin profiles described in the following. However, it is always possible to optimize numerically the spin function (or one, or some of its parameters). Indeed, in analogy with the treatment in the case of any input state of polarization, chosen an arbitrary almost periodic spin function, equation (29) can be solved numerically, and the parameters of such spin function can be varied iteratively, until condition (34) is verified. The numerical simulations can be carried out by means of suitable commercial software, such as, for instance, the already cited Matlab™ or Mathematica™. By means of such commercial software, it is also possible to verify the actual increase of the polarization correlation length, by plotting in a graph the evolution of ||(s(^))|| as a function of z.

The solution of equation (29) and conditions (32), (33) and (34) depend not only on the particular spin function, but also on the beat length LB and on the birefringence correlation length Lp- Regarding this point, the inventors state again that in any case it is possible to have an accurate estimate of L_B and Lp also before the fiber drawing, in the way described from line 19 at page 12 to line 9 at page 13.

The inventors observed that for some types of almost periodic spin functions conditions (32), (33) and (34) can be simplified.

A first case in which it is possible to perform said simplification is when the spin function a(z) is constant as a function of z (i.e. when the fiber is rotated with constant speed during the drawing process). Indeed, according to the known theory of ordinary differential equations, when a(z) — cto is constant in z, then it results

where Re [•] indicates the real part of a complex variable, and p_max is the eigenvalue of the matrix

with largest real part. If p_ma_x,o denotes the eigenvalue of the matrix

with largest real part, then it is possible to conclude that in the case under examination, conditions (32), (33) and (34) simplify to

where preferably, e is approximately equal to 0.01 ; more preferably, e is approximately equal to 0.001 ; even more preferably, e is approximately equal to 0.0001. Advantageously, α₀ is greater than 2τr/Lβ. The inventors have also observed that condition (38) simplifies to a higher degree when the ratio LB/ Lp, between the beat length and the birefringence correlation length is sufficiently big. In detail, fixed do = 2τr/p, with p the period of the torsion applied to the fiber by the constant spin function, condition (38) simplifies to

/ A T \ 2 ^{< £} - ⁽³⁹⁾ V P /

Condition (39) is completely equivalent to condition (38) when the latter holds, and in addition the following condition is verified

where preferably, 7 is approximately 0.1; more preferably, 7 is approximately 0.01 ; even more prefer- ably, 7 is approximately 0.001. Moreover, with reference to condition (39), the inventors understood that preferably, e is approximately equal to 0.01 ; more preferably, e is approximately equal to 0.001; even more preferably, e is approximately equal to 0.0001. The demonstration of the equivalence of conditions (39) and (38) is based on the adoption of well known perturbation techniques (see, for example: A. H. Nayfeh, "Perturbation methods", John Wiley & Sons, ISBN 0471399175, 2000). As it is evident, (39) gives an explicit condition on p, consequently on α₀ = 2τr/p, consequently on α(z) = OQ, that guarentees a very long polarization correlation length Lc-

A second case in which it is possible to perform said simplification is when the spin function α(z) is such that the function A(z) = f^" α(t)dt , (41 )

J 0 is periodic with an arbitrary period p (i.e., A(z + p) — A(z)). This takes place when α(z) is periodic with period p (i.e., α(z + p) = α(z)) and moreover

α {z)dz = 0 . (42)

/ Jo'

In this case, according to the Floquet-Lyapunov theorem (see for example: A. V. Yakubovich, V. M. Star- zhinskii, "Linear differential equations with periodic coefficients", Krieger Publishing Company, ISBN 0470969539, 1975), it is possible to write the generic solution of equation (29) as

x(z) = F(z) exp(Kz)x{0) , (43)

where F(z) is a periodic matrix of period p, such that F(O) = Is (with I₃ identity matrix of dimension 3 x 3), and where K is a suitable constant matrix. The same Floquet-Lyapunov theorem allows also to conclude that lim ^ln !!*(*) H ₌ Re βn Mmαx] ₍₄₄₎

2→+oo Z p ' where μ_max is the eigenvalue of the matrix eχp(Kp) with largest modulus. As a consequence, conditions (32), (33) and (34) can be rewritten in the following equivalent way:

pRe [/w,o] ' ■ ^{( }} where p_maχ_β is the quantity already defined at line 23 of page 15 in the present document. Preferably, e is approximately equal to 0.01 ; more preferably, e is approximately equal to 0.001 ; even more preferably, e is approximately equal to 0.0001. Advantageously, p is shorter than LB- The simplified condition (45) can be used only if the matrix exp(Kp) is known. It can be noted that from (43) it follows that x(p) = exp(Kp)x(0), thus the matrix eχp(Kp) can be calculated simply by solving equation (29) with z belonging to the interval [0, p], and for the three different initial conditions (1, 0, 0)^τ, (0, 1, 0)^τ, (0, 0, l)^r, or for some suitable linear combinations of them. The solution of (29), whenever it cannot be calculated analytically, can be determined numerically with the already cited commercial software. However, condition (45) simplifies to a higher degree when the ratio LB/ Lp, between the beat length and the birefringence correlation length is sufficiently high. In detail, the result is

where A(z) has been defined in equation (41). Condition (46) is completely equivalent to condition

(45) when the latter holds and also the following condition is verified

where preferably, 7 is approximately 0.1 ; more preferably, 7 is approximately 0.01 ; even more prefer¬ ably, 7 is approximately 0.001. Moreover, concerning condition (46), the inventors have understood that preferably e is approximately equal to 0.01 ; more preferably e is approximately equal to 0.001 ; even more preferably e is approximately equal to 0.0001. The proof of the equivalence of (46) with (45) is based on the use of the well known Lyapunov-Schmidt reduction technique (see, for exam¬ ple: N. Sidorov et al., "Lyapunov-Schmidt methods in nonlinear analysis and applications", Kluwer Academic Publishers, ISBN 1402009410, 2002). The integrals that appear in (46) and (47) can be accurately determined by means of numerical techniques, or by means of suitable commercial software, such as, for example, the already cited Matlab™ o Mathematica™. Linear input state of polarization. The inventors have observed that when the input state of polarization is linear, the solution of equation (18) is x(z) = (0, 0, 0, χ^(z), 2:₅(2), χ_&{z), X₇(z), χs(z), XQ{Z))^T, that yields (10₃(2)) = 0. This means that when the input state of polarization is linear

H(J(Z))II

y/[χ₄(z) - χ_B(z))* + [X₅(Z) + x₇(z)]* _t (48)

and consequently, in order to determine Lc it is sufficient to analyze the system

-4σ² 2a(z) 0 0 -4σ²

-2a(z) -4σ² -b 4σ² 0 dx 0 b -2σ² 0 0

(49) dz 0 4σ² 0 -4σ² 2a(z)

-4σ² 0 0 -2a(z) -4σ²

0 0 0 0 b

The evolution of x(z) is determined by the Lyapunov characteristic exponents, τ_x, of the system (49) and by the corresponding eigenvectors, t_% (see, for example the already cited, W. Hahn, "Stability of motion", Springer Verlag, ISBN 0387038299, 1968). In fact, indicated with τ_mαx the Lyapunov characteristic exponent with largest real part, and with t_mαx the corresponding eigenvector, then the asymptotic evolution of x(z) is

x(z) ~ exp(r_maa;2)(a;(0) • t_max)t_rι (50)

where S(O) = l/2(si(0), S₂(O), 0, -5₂(O)₁ Si(O), 0)^τ is the initial condition of the system (49) in the case of linear input state of polarization.

Upon the application of the well known Lyapunov direct method, (see, for example: W. Hahn, "Stabil¬ ity of motion", Springer Verlag, ISBN 0387038299, 1968), the inventors have observed that, for any almost periodic spin function the Lyapunov characteristic exponents of the system (49) are negative. Then, from (50) it follows that the larger the real part of τ_mαx, i.e., the larger Re [τ_mαχ], (and conse- quently the closer to zero, since in any case it has to be Re [τ_mαx] < 0), the more slowly x(z) decays to zero, and the longer is the polarization correlation length LQ. It has already been shown previously that Re [τ_mαx\ can be calculated by means of the following relation:

According to these argumentations, the inventors have come to the conclusion that, in the case of a linear input state of polarization, to achieve the purpose of producing an optical fiber with polarization correlation length Lc longer than a given value L_min, it is necessary to select at least one of the parameters of the almost periodic spin function a{z) such that

_{≥MA >} i z→+oo Z L_min

Additionally, the inventors understood that preferably the given value L_min is equal to 1/e times the polarization correlation length Lc,o that the same fiber would have, if it were not spun. As a consequence, condition (52) is rewritten in the following way:

where x(z; a(z)) is the solution of system (49), and x(z;0) is the solution of the same system when a(z) = 0. The inventors have verified that preferably e is approximately equal to 0.01 ; more preferably e is approximately equal to 0.001 ; even more preferably e is approximately equal to 0.0001. Under a practical point of view, the inventors realized that a considerable increase of the polarization correlation length can be achieved also if condition (53) is replaced with the following

z→δ ln ||z(;z; 0) || ' ^V ' where δ is a suitable finite value approximately equal to 1 km; more preferably, δ is approximately 10 km; even more preferably, δ is approximately 100 km and more. Moreover, preferably e is approx¬ imately equal to 0.01; more preferably e is approximately equal to 0.001 ; even more preferably e is approximately equal to 0.0001. Advantageously, the absolute maximum of the modulus of the spin function a(z) is greater than 2-K/LB- Preferably LB is longer than 0.05 m; more preferably it is longer than 5 m.

In general, it is not possible to write explicitly the solution of the system (49), and it is not possible to derive explicitly the conditions (52), (53) and (54) a part from the case of some particular types of almost periodic spin functions described in the following. However, it is always possible to optimize numerically the spin function (or one, or a few of its parameters). Indeed, in analogy with the treatment done in the case of circular input state of polarization, chosen an arbitary almost periodic spin func¬ tion, equation (49) can be solved numerically, and the parameters of such spin function can be varied iteratively, until condition (54) is verified. The numerical simulations can be carried out by means of suitable commercial software, such as, for example, the already cited Matlab™ or Mathematica™. By means of such commercial software, it is also possible to verify the actual increase of the polarization correlation length, by plotting in a graph the evolution of ||(s(z))ll as a function of z.

The solution of equation (49) and conditions (52), (53) and (54) depend not only on the particular spin function, but also on the beat length LB, and on the birefringence correlation length Lp. Regarding this point, the inventors state again that in any case it is possible to have an accurate estimate of LB and Lp also before the fiber drawing, in the way described from line 19 at page 12 to line 9 at page 13.

The inventors observed that for some types of almost periodic spin functions, conditions (52), (53) and (54) can be simplified.

A first case in which it is possible to perform such simplification is when the spin function a(z) is constant as a function of z (i.e., when the fiber is rotated with a constant speed during the drawing process). In fact, according to the known theory of ordinary differential equations, when a(z) — ao is constant in z, then it results lim Ms(^S) — K.Θ [/Ϊ77icιa;J (55)

2-t+∞ Z where κ_max is the eigenvalue of the matrix

with largest real part. If κ_maχfi denotes the eigenvalue of the matrix

with largest real part, then it is possible to conclude that in the case under examination, the conditions (52), (53) and (54) simplify to

Re [KmαzJ

< e , (58)

Re [K-maxβ] where preferably, e is approximately equal to 0.01 ; more preferably, e is approximately equal to 0.001 ; even more preferably, e is approximately equal to 0.0001. Advantageously, α₀ is bigger than 2π/Lβ- The inventors have also observed that condition (58) simplifies to a higher degree when the ratio LB/ Lp, between the beat length and the birefringence correlation length is sufficiently big. In detail, fixed α₀ = 2τr/p, with p the period of the torsion applied to the fiber by the constant spin function, condition (58) simplifies to

\ P J Condition (59) is completely equivalent to condition (58) when the latter holds, and in addition the following condition is verified

where preferably, 7 is approximately 0.1 ; more preferably, 7 is approximately 0.01; even more prefer¬ ably, 7 is approximately 0.001. Moreover, with reference to (59), the inventors understood that prefer- ably, e is approximately equal to 0.01; more preferably, e is approximately equal to 0.001; even more preferably, e is approximately equal to 0.0001. The demonstration of the equivalence of conditions (59) and (58) is based on the adoption of well known perturbation techniques (see, for example: A. H. Nayfeh, "Perturbation methods", John Wiley & Sons, ISBN 0471399175, 2000). As it is evident, (59) gives an explicit condition on p, consequently on αo — 2π/p, consequently on α{z) = αo, that guarantees a very long polarization correlation length Lc-

A second case in which it is possible to perform said simplification is when the spin function α(z) is such that the function A(z) defined in (41) is periodic with an arbitrary period p (i.e., A(z + p) = A{z)). In this case, according to the Floquet-Lyapunov theorem (see for example: A. V. Yakubovich, V. M. Starzhinskii, "Linear differential equations with periodic coefficients", Krieger Publishing Com¬ pany, ISBN 0470969539, 1975), it is possible to write the generic solution of equation (49) as x(z) = G{z) exp{Hz)x{0) , (61) where G(z) is a periodic matrix of period p such that G(O) = I_& (with J₆ identity matrix of dimension 6 x 6), and where H is a suitable constant matrix. The same Floquet-Lyapunov theorem allows also to conclude that _lim In PKz)II _^ Re [In⁽W] ₍₆₂₎ z→+00 Z p ' where ζ_mαx is the eigenvalue of the matrix exp(Hp) with largest modulus. As a consequence, conditions (52), (53) and (54) can be rewritten in the following equivalent way:

Re [In C ^_nmαx i p TRJe r [Kmαxfl 1] < ^e > (⁶³) where κ_mαxβ is the quantity already defined at line 16 of page 20 in the present document. Preferably, e is approximately equal to 0.01 ; more preferably, e is approximately equal to 0.001 ; even more preferably, e is approximately equal to 0.0001. Advantageously p is shorter than LB- The simplified condition (63) can be used only if the matrix exp(Hp) is known. It can be noted that from (61) it follows that x(p) — eχp(Hp)x(0), thus the matrix exp(Hp) can be easily calculated by solv- ing equation (49) with z belonging to the interval [0, p], and for six different linearly independent initial conditions. The solution of (49), whenever it cannot be calculated analytically, can be determined numerically by means of the already cited commercial software.

However, condition (63) simplifies to a higher degree when the ratio LB/ Lp, between the beat length and the birefringence correlation length is sufficiently big. In detail, it comes out that

where A{z) has been defined in equation (41). Condition (64) is completely equivalent to condition

(63) when the latter holds, and additionally the following condition is verified

where preferably, 7 is approximately 0.1 ; more preferably, 7 is approximately 0.01; even more prefer- ably, 7 is approximately 0.001. Moreover, with reference to (64), the inventors have understood that preferably e is approximately equal to 0.01 ; more preferably e is approximately equal to 0.001; even more preferably e is approximately equal to 0.0001. The proof of the equivalence of condition (64) with (63) is based on the use of the well known reduction technique of Lyapunov-Schmidt (see, for ex¬ ample: N. Sidorov et al., "Lyapunov-Schmidt methods in nonlinear analysis and applications", Kluwer Academic Publishers, ISBN 1402009410, 2002). The integrals that appear in (64) and (65) can be ac¬ curately determined by means of numerical techniques, or by means of suitable commercial software such as, for example, the already cited Matlab™ or Mathematica™.

General considerations. As soon as the almost periodic spin function, or one or more of its pa- rameters, have been defined according to at least one of the conditions (25), (26), (27), or according to at least one of the conditions (32), (33), (34), (38), (39), (45), (46), or accorrding to at least one of the conditions (52), (53), (54), (58), (59), (63), (64), then the spin function α(z) must be applied to the fiber.

Those who have practice in the production of optical fibers, know that in order to apply a spin function α(z) to the fiber, it is necessary to rotate the fiber and/or the preform with angular velocity propor¬ tional to said function α(z), during the fiber drawing process. This can be realized by means of well known techniques described in several patents (see, for example: US patent n. 5,298,047; WO patent 5 request n. 99/67180).

Those who have practice in the production of optical fibers, know also that some apparatuses for the application of the spin function are not always able to guarantee the equivalence between the imposed spin function and the spin function actually applied to the optical fiber. However, this discrepancy can be reduced to acceptable levels, so that the invention presented in this document can be referred to io the spin function used in the fiber drawing, also if said function may not correspond exactly to the spin function actually applied to the optical fiber.

The inventors believe it is advantageous that the absolute maximum of the modulus of the spin func¬ tion a(z) is lower than 400 rad/m. Moreover, they find it is preferable that said maximum is greater than 15 1 rad/m, and they find it is more preferable that said maximum in greater than 2π/L_B. Advantageously, LB is longer than 0.05 m, more advantageously it is longer than 5 m.

It has been observed that it can be difficult to transfer to the optical fiber spin functions with too large amplitudes (expressed in turns/meter) when the drawing speed is too large, for example larger than

20 10 m/s, as it happens for telecommunication fibers (non zero-dispersion shifted fibers, non zero dis¬ persion fibers, dispersion shifted fibers). On the contrary, special fibers have not a very fast drawing speed. Some examples of special fibers are: doped fibers for optical amplification, fibers for disper¬ sion compensation, fibers with gratings, fibers for sensors, photonic crystal fibers, microstructured fibers. The fibers of special type typically have shorter lengths with respect to telecommunication

25 fibers, in general shorter than 2 km, preferably shorter than 500 m, more preferably shorter than 100 m. In addition, special fibers are in general characterized by a higher birefringence with respect to the birefringence typically present in telecommunication fibers, because they present strong step indexes between the inner core region and the cladding. Typically, the fibers of special type have a reduced mode diameter and a numerical aperture NA greater than 0.2, where the numerical aperture

₃₀ is defined (in the case of step index fibers) as NA = JnI₀ - n² _cl, where n_co and n_cι are the core and cladding refractive indexes, respectively. Such fibers present a beat length shorter than 4 m. In the case of photonic crystal fibers and microstructured fibers, air holes are voluntarily introduced in the preform; the big step index between the glass and the air is the cause of strong birefringence. According to what exposed above, it can be noted that the most commonly used fibers with long po¬ larization correlation length are fibers with strong birefringence and, consequently, they are inevitably characterized by a higher attenuation and by a strong polarization mode dispersion measured in tens and sometimes even hundreds of ps/km.

Optical fibers to which the spin functions chosen according to what described in the present patent have been applied, can be used in optical telecommunication systems. An example of said systems is shown in figure 1, and it is indicated in the following with the reference number 10. It is made up of a transmitting device 12, a transmission line 18, and a receiver 30. The system 10 can include also one or more devices to insert or drop optical signals from the system 10, such as, for example, the optical add/drop multiplexer indicated with the numbers 14 and 28 in figure 1. These devices can be based on wavelength division multiplexing techniques (WDM, coarse wavelength division multiplexing, CWDM, dense wavelength division multiplexing, DWDM), or on optical time division multiplexing techniques (OTDM), or on techniques with code division multiplexing access (CDMA), or on polarization multiplexing techniques.

In case it is necessary to increase the optical signal power, (as, for example, in the broadcast trans¬ mission, i.e. when a big number of receivers are connected to the same transmitter), one or more optical amplifiers 22 may be used. Sometimes it may be advantageous to control the state of polarization of the signal generated by the transmitter 12. To this purpose, a polarization controller 16 may be inserted. The polarization controller can be realized according to what described in the scientific literature and in the patents. In particular, it can be realized in optical fiber, or by means of optical or electro-optical devices, or with liquid crystals. The polarization controller 16 can, if necessary, vary the state of polarization of the optical signal in a controlled and known way. The transmission line 18 includes one or more spans of optical fiber. At least one of such optical fiber spans has been subjected, during the production, to a spinning process with a spin function a(z) chosen according to what described in the present document. The whole length of the transmission line 18 depends on the type of its task and can vary from a few hundreds of meters in the case of connections in a local network, to a few kilometers in the case of connections in a metropolitan net- work, up to some hundreds of kilometers and also more in the case of terrestrial and/or submarines backbones. Just as an example, some spans of optical fiber are indicated with 20 and 24 in figure 1. The transmitter 12 can be, for example, a narrow bandwidth optical source, such as a DFB laser directly or externally modulated, for instance by a Mach-Zender or by an acousto-optic modulator; alternatively, the transmitter 12 can be a Fabry-Perot laser followed by an optical filter; alternatively it can be a DBR laser; alternatively, it can be a gas or a solid state laser; alternatively, it can be a fiber optic laser; alternatively, it can be a combination of these devices. The optical communication system can support any kind of modulation format, such as, for exam- pie: NRZ (non return to zero), RZ (return to zero), CRZ (chirped return to zero), AM (amplitude modulated), FM (frequency modulated), PM (phase modulated); there are no limitations to the bit rate.

The receiver 30 can be also a demultiplexer or a router, and it has the purpose of converting the optical signal in an electric signal. If a polarization controller 16 has been used in transmission to change cyclically the polarization of the signal to be sent in the line 18, a polarization controller 26 that works in a similar manner can be inserted in front of the receiver. In this way, even if in the line

18 several signals with different polarizations are present at the same time, an operator can receive correctly, if authorized, the signal sent in 18 only if he has the availability of the exact polarizations sequence to be sent to the polarization controller 26. As it is well known to those with practice of optical telecommunication systems, the different elements that make part of the transmission system 10, can be connected one to the other by means of fusion splices, mechanical splices or connectors.

Optical fibers, to which a spin function a(z) chosen according to the criteria described in the present document has been applied, can be inserted in optical cables. Eventually, said optical fibers can be inserted in ribbons made up of more fibers.

Optical fibers, to which spin functions chosen according to what described in the present patent have been applied can be used also as devices of parts of an article. A non exhaustive list of said articles is the following: fiber Bragg gratings, modules for the active or passive compensation of optical fibers chromatic dispersion, optical amplifiers based on optical fibers doped with suitable rare earths (for example: erbium, ytterbium, etc.), optical amplifiers based on the Raman effect, fiber lasers, gyroscopes, pressure sensors, temperature sensors, mechanical stress sensors, electro-magnetic field sensors, jumpers, fiber couplers, etc.. These optical devices can be used alone or in combination with other optical devices or other optical fibers.

Example 1. As a first example, the inventors considered the following almost periodic spin function

where pi = 3 m, p₂ — (3/_Λ/2) m and αo is the parameter to be determined with the aim of increasing considerably the polarization correlation length. By means of numerical simulations, said spin function has been applied for different values of α₀ to an optical fiber with LB — 25 m and L_F = 7 m. In a first case, it has been assumed an elliptical input state of polarization to the optical fiber under examination. According to equation (24) the polarization correlation length Lc,o of the same fiber when not subjected to the spinning process has been calculated. Said polarization correlation length resulted to be equal to Lc,o = 7.5 m. Afterwards, the behavior of the following ratio has been numerically evaluated as a function of the parameter αo,

, , s

Φi(αo) (67)

where x(z; 0) is the solution of the system (29) with a(z) = 0 and x(z; aι(z)) is the solution of the system (29) with a(z) = aχ(z). It can be observed that the ratio Φi(αo) is the quantity that appears in the condition (27), so that imposing Φi(αo) < e corresponds to imposing exactly condition (27). Moreover, it should be recalled that, due to the given definition, Φi(αo) is the ratio between the polar¬ ization correlation length of the fiber not subjected to the spinning process, Lc,o, and the polarization correlation length of the same fiber when subjected to a spinning process, Lc- The behavior of Φi(αo) has been reported in figure 2 as a function of the parameter αo. It can be clearly seen that Φi(α₀) has some well defined and evident local minima for particular values of the parameters αo, in whose correspondence the polarization correlation length Lc is considerably increased. The inventors, in fact, understood that it is very advantageous to choose the parameter αo so that the ratio Φi(αo) has a value close to one of said local minima.

From the graph in figure 2 it can be understood also that, if one wants to realize an optical fiber with polarization correlation length Lc at least 100 times longer than Lc,o, it is advantageous to choose α₀ in an interval approximately comprised between 0.83 turns/m and 1 turns/m. Moreover, it can be observed that this interval is reported here only as an example; indeed, other intervals exist for the αo value, for which Lc is at least 100 times longer than Lc,o,' such intervals are actually those where

Φi(αo) < 1/100.

Alternatively, a different purpose may be to realize an optical fiber with polarization correlation length Lc longer than, or equal to a given value, for example Lc > 1 km. In the example under examination, given the values of LB and L_F, it is known that L_Cβ = 7.5 m, so that, in order to obtain Lc ≥ 1 km it is necessary that Φi(α₀) < 0.0075. As it can be seen from the graph, this can be obtained, for instance, by choosing αo in the interval approximately comprised between 1.38 turns/m and 1.53 turns/m. Successively, the inventors believed it was convenient to verify that a fiber with very long polarization correlation length is able to preserve a suitable state of polarization, or a suitable relation among the states of polarization. To reach this purpose, the evolution of ||(s(z))|| as a function of the distance z has been numerically simulated (please, recall equation (5)), and for some values of αo- The results are shown in figure 3: the curves 1 , 2 and 3 refer respectively to the cases α₀ = 0, a_o = 0.67 turns/m and α₀ = 1.4 turns/m, for which it comes out Lc = Lc,o = 7.5 m, Lc = 114.3 m and Lc — 1249 m, respectively.

It is evident that for α₀ = 1-4 turns/m, corresponding to one of the Φi(αo) minima shown in figure 2, the fiber is able to preserve a state of polarization or a suitable relation among the states of polarization, for distances much longer than the other two cases obtained for a_Q = 0 and α₀ = 0.67 turns/m. This, is actually due to the very long polarization correlation length Lc that in this particular example, is obtained for Ct₀ = 1-4 turns/m.

In a second case, the polarization at the input of the optical fiber under examination has been as- sumed to be circular. According to equation (31), the polarization correlation length Lc,o, that the same fiber would have had if it were not spun, has been calculated. Said polarization correlation length resulted to be equal to Lc,o — 5.4 m.

Afterwards, the behavior of the following ratio has been numerically evaluated as a function of the parameter ao

where x(z; 0) is the solution of the system (29) with a(z) = 0 and x(z; aχ(z)) is the solution of the system (29) with a(z) = aχ{z). It can be observed that the ratio Θ₁(αo) is the quantity that appears in condition (34), so that imposing θi(αo) < e corresponds to imposing exactly condition (34). Moreover, it should be recalled that, due to the given definition, θi(αo) is the ratio between the polarization cor- relation length of the fiber not subjected to the spinning process, Lc,o, and the polarization correlation length of the same fiber when subjected to the spinning process, Lc-

The behavior of Θi(αo) is reported in figure 4 as a function of the parameter α₀- It can be clearly noted that Θi (ao) presents some well defined and evident local minima for particular values of the pa¬ rameters ao, in whose correspondence the polarization correlation length is considerably increased. The inventors, in fact, understood that it is very advantageous to choose the parameter a_Q so that the ratio Θ_I(Ω:O) has a value close to one of said local minima.

From the graph in figure 4 it can be understood also that, if one wants to realize an optical fiber with polarization correlation length Lc at least 100 times longer than Lc,o, it is advantageous to choose ao in an interval approximately comprised between 1.38 turns/m and 1.54 turns/m. Moreover, it can be observed that this interval is reported here only as an example; indeed other intervals exist for the ao value, for which Lc is at least 100 times longer than Lc,o,' such intervals are actually those where θi(αo) < 1/100. Alternatively, a different purpose may be to realize an optical fiber with polarization correlation length Lc longer than, or equal to a given value, for example Lc ≥ 1 km. In the example under examination, given the values of L_B and L_F, it is known that Lc,o = 5.4 m, so, in order to obtain Lc ≥ 1 km it is necessary that Θχ(αo) < 0.0054. As it can be observed in the graph, this can be obtained, for instance, by choosing α₀ in the interval approximately comprised between 1.84 turns/m and 1.94 turns/m. Successively, the inventors believed that it was convenient to verify that a fiber with very long polar¬ ization correlation length Lc is able to preserve a suitable state of polarization or a suitable relation among the states of polarization. To reach this purpose, the evolution of ||(s(z))|| as a function of the distance z has been numerically simulated (please, recall equation (5)), for some values of a₀- The results are shown in figure 5: the curves 1 , 2 and 3 refer respectively to the cases αo = 0, OiQ = 0.67 turns/m and αo = 1.4 turns/m, for which it comes out Lc = Lc,o = 5.4 m, Lc = 65 m and Lc = 720 m, respectively.

It is evident that for α₀ = 1-4 turns/m, corresponding to one of the Θi(αo) minima shown in figure 4, the fiber is able to preserve a state of polarization or a suitable relation among the states of polarization, for distances much longer than the other two cases obtained for αo = 0 and αo = 0.67 turns/m. This is actually due to the very long polarization correlation length Lc that in this particular example, is obtained for αo = 1.4 turns/m.

In a third case, the polarization at the input of the optical fiber under examination is assumed to be linear. According to equation (51), the polarization correlation length Lc,o, that the same fiber would have had if it were not spun, has been calculated. Said polarization correlation length resulted to be equal to Lc,o = 8.7 m.

Successively, in analogy to the first case, the behavior of the following ratio has been numerically evaluated as a function of the parameter αo

^Φ ₁ ^(αo⁾ = z→ ^lim+∞ 'T lnl||aT;(z;i0)f|| _' <⁶⁹> where x(z; 0) is the solution of the system (49) with a(z) = 0 and x(z; aι(z)) is the solution of the system (49) with a(z) = a_\{z). It can be observed that the ratio Φχ(αo) is the quantity that appears in condition (54), so that imposing Φi(α₀) < e corresponds to imposing exactly condition (54). Moreover, it should be recalled that, due to the given definition, Φi(αo) is the ratio between the polarization cor¬ relation length of the fiber not subjected to the spinning process, Lc,o, and the polarization correlation length of the same fiber when subjected to the spinning process, Lc-

The behavior of Φi(αo) >^s reported in figure 6 as a function of the parameter αo. It can be clearly noted that Φi(αo) presents some well defined and evident local minima for particular values of the parameter α₀, in whose correspondence the polarization correlation length Lc is considerably in¬ creased (analogously to what happens in figure 4). The inventors, in fact, understood that also in this case it is very advantageous to choose the parameter αo so that the ratio Φi(αo) has a value close to one of said local minima. From the graph in figure 6 it can be understood also that, if one wants to realize an optical fiber with polarization correlation length Lc at least 100 times longer than Lc,o, it is advantageous to choose αo in an interval, approximately comprised between 0.8 turns/m and 1.05 turns/m. Moreover, it can be observed that this interval is reported here only as an example; indeed other intervals exist for the α₀ value, for which Lc is at least 100 times longer than Lc,o; such intervals are actually those where Φi(αo) < 1/100. Alternatively, a different purpose may be to realize an optical fiber with polarization correlation length Lc longer than or equal to a given value, for example Lc ^~≥ 1 km. In the example under examination, given the values of LB and LF, it is known that Lc,o = 8.7 m, so in order to obtain Lc ≥ 1 km it is necessary that Φi (αo) < 0.0087. As it can be observed in the graph, this can be obtained, for instance, by choosing αo in the interval approximately comprised between 1.37 turns/m and 1.54 turns/m. Successively, the inventors believed it was convenient to verify that, also relatively to a linear input state of polarization, a fiber with a very long polarization correlation length Lc is able to preserve a suitable state of polarization, or a suitable relation among the states of polarization. To reach this purpose, the evolution of ||(s(z))|| as a function of the distance has been numerically simulated (please, recall again equation (5)), and for some values of αo. The results are shown in figure 7: the curves 1 , 2 and 3 refer respectively to the cases α₀ = 0, αo = 0.67 turns/m and αo = 1.4 turns/m, for which it comes out Lc = L_Cβ = 8.7 m, Lc = 144 m and Lc = 1410 m, respectively. It is evident that for αo = 1.4 turns/m, corresponding to one of the Φi(αo) minima shown in figure 6, the fiber is able to preserve a state of polarization or a suitable relation among the states of polarization, for distances much longer than the other two cases obtained for α₀ = 0 and αo = 0.67 turns/m. This is actually due to the very long polarization correlation length Lc that in this particular example, is obtained for α₀ = 1.4 turns/m.

Example 2. As a second example, the inventors considered the following periodic spin function

Ot₂[Z) = α_o sm , (70)

V Ps J where p₃ — 2 m and α₀ is the parameter to be determined with the aim of increasing considerably the polarization correlation length. By means of numerical simulations, said spin function has been applied, for different vales of α₀. to an optical fiber with LB — 20 m and Lp — 5 m. In a first case, the polarization at the input of the optical fiber under examination has been assumed to be elliptical. According to equation (24), the polarization correlation length Lc,o of the same fiber when not subjected to the spinning process has been calculated. Said polarization correlation length resulted to be equal to Lc,o — 6-35 m. Afterwards, the behavior of the following ratio has been numerically evaluated as a function of the parameter αo

Φ₂(αo) = (71)

where x(z; 0) is the solution of the system (29) with a(z) = 0 and x(z; a₂(z)) is the solution of the system (29) with a(z) = ot₂(z). It can be observed again that imposing Φ₂(α₀) < e corresponds to imposing exactly condition (27). Moreover, it should be recalled that, due to the given definition, Φ₂ (CK₀) = Lcfl/Lc, where Lc is the polarization correlation length of the fiber subjected to the spin¬ ning process, whereas Lc,o is the polarization correlation length of the same fiber when not subjected to the spinning process. The behavior of #₂(0:0) has been reported in figure 8 as a function of the parameter α₀. It can be clearly seen that also in this second example, $₂(0₀) has some well defined and evident local minima for particular values of the parameter α₀, in whose correspondence the polarization correlation length Lo is considerably increased. Such minima are in any case different with respect to those of Φi(αo). The inventors confirm that it is advantageous to choose the parameter αo so that the ratio #₂(0₀) has a value very close to one of said local minima. From the graph in figure 8 it can be understood also that, if one wants to realize an optical fiber with polarization correlation length Lc at least 500 times longer than Lc,o, it is advantageous to choose «o in an interval approximately comprised between 2.89 turns/m and 3 turns/m. Moreover, it can be observed that this interval is reported here only as an example; indeed, other intervals exist for the αo value, for which Lc is at least 100 times longer than Lcp, such intervals are actually those where

Alternatively, a different purpose may be to realize an optical fiber with polarization correlation length Lc longer than or equal to a given value, for example Lc ≥ 5.5 km. In the example under exam¬ ination, given the values of LB and LF, it is known that Lc,o — 6.35 m, so that, in order to obtain Lc ≥ 5.5 km it is necessary that Φ₂(«o) < 0.0012. As it can be observed in the graph, this can be obtained, for instance, by choosing αo in the interval approximately comprised between 5.28 turns/m and 5.33 turns/m. Successively, the inventors believed it was convenient to verify that a fiber with very long polarization correlation length is able to preserve a suitable state of polarization, or a suitable relation among the states of polarization. To reach this purpose, the evolution of ||(s(z))|| as a function of the distance z has been numerically simulated (please, recall equation (5)), for some values of α₀. The results are shown in figure 9: the curves 1 , 2 and 3 refer respectively to the cases αo = 0, α₀ = 2.6 turns/m and α₀ = 2.93 turns/m, for which it comes out Lc — Lc,o — 6.35 m, Lc — 58 m and Lc = 4195 m, respectively.

It is evident that for αo = 2.93 turns/m, corresponding to one of the Φ₂(cκo) minima shown in figure 8, the fiber is able to preserve a state of polarization or a suitable relation among the states of polarization, for distances much longer than the other two cases obtained for αo = 0 and αo = 2.6 turns/m. This, is actually due to the very long polarization correlation length Lc that, in this particular example, is obtained for αo = 2.93 turns/m.

In a second case, the polarization at the input of the optical fiber under examination has been as¬ sumed to be circular. According to equation (31), the polarization correlation length Lc,o, that the same fiber would have had if it were not spun, has been calculated. Said polarization correlation length resulted to be equal to Lc,o — 4.4 m.

Successively, the behavior of the following ratio has been numerically evaluated as a function of the parameter α₀

(72) where 5(2; 0) is the solution of the system (29) with a(z) = 0 and x(z; ot₂(z)) is the solution of the system (29) with a(z) = 0₂(2). It can be observed again that imposing ©₂(0:₀) < e corresponds to imposing exactly condition (34). Moreover, it should be recalled that, due to the given definition, ©₂(0:₀) = Lcfi/Lc, where Lc is the polarization correlation length of the fiber subjected to the spin¬ ning process, whereas Lc,o is the polarization correlation length of the same fiber when not subjected to the spinning process.

The behavior of ©₂(0₀) is reported in figure 10 as a function of the parameter α₀. It can be clearly noted that, also in this second example, Θ₂(α₀) presents some well deifned and evident local minima for particular values of the parameter αo, in whose correspondence the polarization correlation length is considerably increased. Anyway, such minima are different from those of Θi(αo). The inventors confirm that it is advantageous to choose the parameter αo so that the ratio 6₂(0:₀) has a value close to one of said local minima. From the graph in figure 10 it can be understood also that, if one wants to realize an optical fiber with polarization correlation length Lc at least 1000 times longer than Lc_1O, it is advantageous to choose αo in an interval approximately comprised between 4.5 turns/m and 4.535 turns/m. Moreover, it can be observed that this interval is reported here only as an example; indeed other intervals exist for the αo value for which L_c is at least 1000 times longer than Lc,o', such intervals are actually those where Θ₂(α₀) < 1/1000. Alternatively, a different purpose may be to realize an optical fiber with polarization correlation length Lc longer than or equal to a given value, for example Lc ≥ 5.5 km. In the example under examination, given the values of L_B and Lp, it is known that Lc,o = 4.4 m, so in order to obtain Lc ≥ 5.5 km it is necessary that ©₂(0:₀) < 0.0008. As it can be observed in the graph, this can be obtained, for instance, by choosing αo in the interval approximately comprised between 5.285 turns/m and 5.32 turns/m. Afterwards, the inventors believed it appropriate to verify that a fiber with very long polarization cor¬ relation length Lc is able to preserve a suitable state of polarization or a suitable relation among the states of polarization. To reach this purpose, the evolution of ||(s(z))|| as a function of the distance z hase been numerically simulated (please, recall equation (5)), and for some values of ceo. The results are shown in figure 11 : the curves 1 , 2 and 3 refer respectively to the cases αo = 0, αo = 2.6 turns/m and αo = 2.93 turns/m, for which it comes out Lc = Lc,o — 4.4 m, Lc = 34 m and Lc = 2500 m, respectively.

It is evident that for αo = 2.93 turns/m, corresponding to one of the ©₂(0₀) minima shown in figure 10, the fiber is able to preserve a state of polarization or a suitable relation among the states of polarization, for distances much longer than the otherr two cases obtained for αo = 0 and αo = 2.6 turns/m. This is really due to the very long polarization correlation length Lc that in this particular example, is obtained for αo = 2.93 turns/m.

In a third case, the polarization at the input of the optical fiber under examination is assumed to be linear. According to equation (51) it has been calculated that Lc,o = 7.1 m. Similarly to the first case of this second example, the behavior of the following ratio has been numerically evaluated as a function of the parameter αo

Φ₂(αo) = iim ;' ;_; y , 73 z→+∞ ln ||a;(2; O) H where x(z; 0) is the solution of the system (49) with a(z) = 0 and x(z; a₂(z)) is the solution of the system (49) with a(z) = α₂(z). It can be observed that Φ₂(αo) < e corresponds to imposing exactly condition (54). Moreover, it should be recalled that, due to the given definition, it comes out that *2(«o) = Lc,o/Lc. The behavior of ^₂(^₀) has been reported in figure 12 as a function of the parameter α₀. It can be clearly noted that *₂(ao) presents some well defined and evident local minima for particular values of the parameters α₀, in whose correspondence the polarization correlation length Lc is considerably increased (analogously to what happens in figure 10). The inventors, in fact, understood that that also in this case it is very advantageous to choose the parameter α₀ so that the ratio Φ₂(QΌ) has a value close to one of said local minima.

From the graph in figure 12 it can be understood also that, if one wants to realize an optical fiber with polarization correlation length Lc at least 500 times longer than Lc,o_> it is advantageous to choose cϋo in an interval approximately comprised between 2.92 turns/m and 2.97 turns/m. Moreover, it can be observed that this interval is reported here only as an example; indeed other intervals exist for the αo values, for which Lc is at least 500 times longer than Lc,o', such intervals are actually those where Φ₂(α₀) < 1/500.

Alternatively, a different purpose may be to realize an optical fiber with polarization correlation length Lc longer than, or equal to a given value, for example Lc ≥ 10 km. In the example under examination, given the values of LB and LF, it is known that Lc,o = 7.1 m, so that, in order to obtain Lc ≥ 10 km it is necessary that Φ₂(αo) < 0.00071. As it can be seen in the graph, this can be obtained, for instance, by choosing αo in the interval approximately comprised between 5.29 turns/m and 5.315 turns/m. Afterwards, the inventors believed it appropriate to simulate numerically the evolution of ||(s(.z))|| as a function of the distance z, for some values of αo. The results are shown in figure 13: the curves 1 , 2 and 3 refer respectively to the cases α₀ = 0, αo = 2.6 turns/m and α₀ = 2.93 turns/m, for which it comes out, Lc = £c,o = 7.1 m, Lc = 70 m and Lc = 5000 m, respectively.

It is evident that for αo = 2.93 rad/m, corresponding to one of the *₂(«o) minima shown in figure 12, the fiber is able to preserve a state of polarization or a suitable relation among the states of polarization, for distances much longer than the other two cases obtained for α₀ = 0 and αo = 2.6 turns/m. This is actually due to the very long polarization correlation length Lc that, in this particular example, is obtained for αo = 2.93 turns/m.

Example 3. As a third and last example the inventors considered a constant spin function

«3(2) = αo , (74) where αo is the only parameter to be determined in order to increase considerably the polarization correlation length. By means of numerical simulation, said spin function has been applied, for different values of αo, to an optical fiber with LB — 18 m and Lp — 5 m.

In a first case, the polarization at the input of the optical fiber under examination has been assumed to elliptical. According to equation (24), it has been calculated that ∑c_to = 5.5 m (obtained for α₀ — 0). Afterwards, the behavior of the following ratio has been numerically evaluated as a function of the parameter αo

^Φ^ = Λ?_∞ ln ||x(z; 0)l| ' ^(?5) where x(z; 0) is the solution of the system (29) with a(z) — 0 and x(z; az(z)) is the solution of the system (29) with a(z) = 0:3(2). Again, it can be observed that the condition Φ₃(α₀) < e corresponds exactly to condition (27). Moreover, it should be recalled that, due to the given definition, it results Φ₃(α₀) = L_c,o/L_c. The behavior of $3(00) has been reported in figure 14 as a function of the parameter αo- It can be clearly observed that in this third example, $₃(0:₀) does not present any local minima, but it decreases monotonically when α₀ increases.

For instance, from the graph in figure 14 it can be understood also that, if one wants to realize an optical fiber with polarization correlation length Lc equal to 30 km, i.e. approximately equal to 5455Lc,o. it is advantageous to choose α₀ approximately equal to 1.6 turns/m (value for which it is φ₃(ao) = Lcfi/Lc c=: 1/5455).

Afterwards, the inventors verified that a fiber with very long polarization correlation length LQ is able to preserve a suitable state of polarization, or a suitable relation among the states of polarization. To reach this purpose, the evolution of ||(s(z))|| as a function of the distance z has been numerically simulated (please, recall equation (5)), and for some values of αo- The results are shown in figure 15: the curves 1 , 2 and 3 refer respectively to the cases αo = 0, αo = 0.1 turns/m and αo = 1 tunrs/m, for which it comes out Lc = Lc,o = 5.5 m, Lc = 110 m and Lc = 10974 m, respectively. It is evident that for αo — 1 turns/m, the fiber is able to preserve a state of polarization, or a suitable relation among the states of polarization, for distances much longer than the other two cases obtained for αo = 0 and α₀ = 0.1 turns/m. This, is actually due to the very long polarization correlation length Lc that, in this particular example, is obtained for α₀ = 1 turns/m.

In second case, the polarization at the input of the optical fiber under examination has been assumed to be circular. According to equation (31), it has been calculated that Lc,o = 3.9 m (obtained for α₀ = 0).

Successively, the behavior of the following ratio has been numerically evaluated as a function of the parameter αo , ( ,₇76«)

where x(z; Q) is the solution of the system (29) with a(z) - 0, and x(z; a_s(z)) is the solution of the system (29) with a(z) — az{z). Again, it can be observed that condition Θ₃(α₀) < e corresponds to condition (34). Moreover, it should be recalled that, due to the given definition, it results ©₃(0₀) = Lc,o/Lc-

10 The behavior of ©₃(0₀) is reported in figure 16 as a function of the parameter α₀. It can be clear¬ ly noted that, in this third example, 63(0:₀) does not present any local minima, but it decreases monotonically when αo increases.

For instance, from the graph in figure 16 it can be understood also that, if one wants to realize an optical fiber with polarization correlation length Lc equal to 20 km, i.e. approximately equal to

15 5128Lc_1O, it is advantageous to choose αo approximative^ equal to 1.9 turns/m (value for which it results Θ₃(α₀) = L_c,ϋ/L_c c- 1/5128).

Afterwards, the inventors verified that a fiber with very long polarization correlation length Lc is able to preserve a suitable state of polarization, or a suitable relation among the states of polarization. To reach this purpose, the evolution of ||(s(«))|| as a function of the distance z has been numerically

20 simulated (please, recall equation (5)), and for some values of αo. The results are shown in figure 17: the curves 1, 2 and 3 refer respectively to the cases αo = 0, αo = 0.1 turns/m and αo = 1 turn/m, for which it comes out Lc = Lc,o = 3.9 m, Lc — 70 m and Lc = 6490 m, respectively. It is evident that for α₀ = 1 turns/m, the fiber is able to preserve a state of polarization or a suitable relation among the states of polarization, for distances much longer than the other cases obtained for

25 α₀ = 0 and αo — 0.1 turns/m. This is really due to the very long polarization correlation length Lc that in this particular example, is obtained for α₀ = 1 turns/m.

In a third case, the polarization at the input of the optical fiber under examination is assumed to be linear. According to equation (51), it has been calculated that Lc,o = 6.3 m (obtained for αo = 0). 30 Therefore, the behavior of the following ratio has been numerically evaluated as a function of the parameter αo

where x(z; 0) is the solution of the system (49) with a(z) = 0 and x(z; as(z)) is the solution of the system (49) with a(z) = a_${z). Again, it should be recalled that the condition Φs(αo) < e corresponds to condition (54), and that it comes out *₃(ao) = Lc,o/Lc-

The behavior of Φ₃(α₀) has been reported in figure 18 as a function of the parameter α₀. It can 5 be clearly noted that also in this second case of the third example Φs(αo) does not have any local minima, but it decreases monotonically when αo increases.

For instance, from the graph in figure 18 it can be understood also that, if one wants to realize an optical fiber with polarization correlation length Lc equal to 100 km it is advantageous to choose αo approximately equal to 3.2 turns/m (value for which it results Φs(α₀) = Lc,o/Lc ^ 6.3 • ICT⁵). Afterwards, the inventors simulated numerically the evolution of ||{s(z))|| as a function of the distance z, for some values of α₀. The results are shown in figure 19: the curves 1 , 2 and 3 refer respectively to the cases α₀ = 0, α₀ = 0.1 turns/m and α₀ = 1 turns/m, for which it comes out Lc — Lc,o — 6.3 m,

Lc = 140 m and Lc = 13000 m, respectively.

It is evident that for αo = 1 turns/m, the fiber is able to preserve a state of polarization, or a suitable relation among the states of polarization, for distances much longer than the other two cases obtained for αo = O and α₀ = 0.1 turns/m. This is actually due to the very long polarization correlation length

Lc that, in this particular example, is obtained for αo = 1 turns/m.

Brief Description of Drawings The advantages provided by the present invention are rendered more evident by the detailed descrip¬ tion, that makes use of suitable figures in which: fig. 1 shows the scheme of a telecommunication scheme based on optical fiber; fig. 2 shows, for an elliptical input state of polarization, the ratio between the polarization correlation length of a fiber not subjected to the spinning process, and the polarization correlation length of the same fiber when subjected to the spinning process according to the almost periodic spin function defined in equation (66), as a function of the amplitude of said spin function; fig. 3 shows the evolution of ||(s(z))|| for an elliptical input state of polarization, when a spin function (66) is applied to the fiber for three different values of a₀, i.e.: αo = 0 (curve 1), α₀ = 0.67 turns/m (curve 2), and αo = 1.4 turns/m (curve 3); fig. 4 shows, for a circular input state of polarization, the ratio between the polarization correlation length of a fiber not subjected to the spinning process, and the polarization correlation length of the same fiber when subjected to the spinning process according to the almost periodic spin function defined in equation (66), as a function of the amplitude of said spin function; fig. 5 shows the evolution of ||(s(z))|| for a circular input state of polarization, when a spin function (66) is applied to the fiber for three different values of αo, i.e.: αo = 0 (curve 1), αo = 0.67 turns/m (curve 2), and αo = 1.4 turns/m (curve 3); fig. 6 shows, for a linear input state of polarization, the ratio between the polarization correlation length of a fiber not subjected to the spinning process, and the polarization correlation length of the same fiber when subjected to the spinning process according to the almost periodic spin function defined in equation (66), as a function of the amplitude of said spin function; fig. 7 shows the evolution of ||(sOO)|| for a linear input state of polarization, when a spin function (66) is applied to the fiber for three different values of ao, i.e.: «₀ = 0 (curve 1), a₀ = 0.67 turns/m (curve 2), and αo = 1.4 turns/m (curve 3); fig. 8 shows, for an elliptical input state of polarization, the ratio between the polarization correlation length of a fiber not subjected to the spinning process, and the polarization correlation length of the same fiber when subjected to the spinning process according to a sinusoidal spin function as a function of the amplitude of said spin function; fig. 9 shows the evolution of || (s(z)) || for an elliptical input state of polarization, when a sinusoidal spin function is applied to the fiber for three different values of ao, i.e.: α₀ = 0 (curve 1), ao — 2.6 turns/m (curve 2), and αo = 2.93 turns/m (curve 3); fig. 10 shows, for a circular input state of polarization, the ratio between the polarization correlation length of a fiber not subjected to the spinning process, and the polarization correlation length of the same fiber when subjected to the spinning process according to a sinusoidal spin function as a function of the amplitude of said spin function; fig. 11 shows the evolution of || (s(z)} || for a circular input state of polarization, when a sinusoidal spin function is applied to the fiber for three different values of a_Q, i.e.: «₀ = 0 (curve 1), αo = 2.6 turns/m (curve 2), and α₀ = 2.93 turns/m (curve 3); fig. 12 shows, for a linear input state of polarization, the ratio between the polarization correlation length of a fiber not subjected to the spinning process, and the polarization correlation length of the same fiber when subjected to the spinning process according to a sinusoidal spin function as a function of the amplitude of said spin function; fig. 13 shows the evolution of IKs(²O)II for a linear input state of polarization, when a sinusoidal spin function is applied to the fiber for three different values of ao, i.e.: ao = 0 (curve 1), αo = 2.6 turns/m (curve 2), and α₀ = 2.93 turns/m (curve 3); fig. 14 shows, for an elliptical input state of polarization, the ratio between the polarization correlation length of a fiber not subjected to the spinning process, and the polarization correlation length of the same fiber when subjected to the spinning process according to a constant spin function, as a function of the amplitude of said spin function; fig. 15 shows the evolution of || (s(z))\\ for an elliptical input state of polarization, when a constant spin function is applied to the fiber for three different values of a₀, i.e.: α₀ = 0 (curve 1), a₀ = 0.1 turns/m (curve 2), and a₀ = 1 turns/m (curve 3); fig. 16 shows, for a circular input state of polarization, the ratio between the polarization correlation length of a fiber not subjected to the spinning process, and the polarization correlation length of the same fiber when subjected to the spinning process according to a constant spin function, as a function of the amplitude of said spin function; fig. 17 shows the evolution of ||(s(z))|| for a circular input state of polarization, when a constant spin function is applied to the fiber for three different values of α₀, i.e.: a₀ = 0 (curve 1), a₀ = 0.1 turns/m

(curve 2), and αo = 1 turns/m (curve 3); fig. 18 shows, for a linear input state of polarization, the ratio between the polarization correlation length of a fiber not subjected to the spinning process, and the polarization correlation length of the same fiber when subjected to the spinning process according to a constant spin function, as a function of the amplitude of said spin function; fig. 19 shows the evolution of ||(s(z))|| for a linear input state of polarization, when a constant spin function is applied to the fiber for three different values of a₀, i.e.: a_Q = 0 (curve 1), α₀ = 0.1 turns/m

(curve 2), and α₀ = 1 turns/m (curve 3);

Modes for Carrying Out the Invention

A. A method for determining at least one parameter of an almost periodic spin function, a(z), to be applied to an optical fiber during the drawing (spinning process), such that the polarization corre- lation length Lc of said optical fiber corresponding to any input state of polarization is considerably increased. Such method is characterized in that the spin function is selected according to

where δ is approximately 1 km, e is approximately 0.01 and the vector χ(z; ζ) = {χ_\{z), X₂{z), χ_${z), Xi(z), x${z), xβ{z), xj{z), χ_&{z), X_d(z)) is such that

— dx2 = ~ c2 ξt xi r{z \) - — ¹ X2[ iz \) - -τ ²—^π xz i(z \) az Lp LB dxs 2τr . .

-T- = -j—X2(z) dz LB

^ = ~XA{Z) + 2ξ x,{z) - ^-xs(z)

dxQ 2π 1 .

^■7k ⁼ LΪ^XBW - LΪ^X'W

dx& τ; 2^X5{z)~~h>^X7{z)+H 2^x&{z) 2π

— = --X₄[Z) - 2ξ X₇[Z) - J-X8{Z) - -jT-X₉[z)

where L_B and Lp are, respectively, the beat length and the birefringence correlation length of said fiber, and x(0; ξ) ≠ 0. From now on, the horizontal bar over a variable (as in x) denotes the symbol of vector.

The inventors observed that preferably, δ is approximately 1 km; more preferably, δ is approximately 10 km; even more preferably, δ is 100 km or more.

The inventors also observed that preferably, e is approximately 0.01 ; more preferably, e is approxi¬ mately 0.001 ; even more preferably, e is approximately 0.0001.

B. An optical fiber made up of at least one section to which an almost periodic spin function, a(z), has been applied. The almost periodic spin function has been selected so that

where δ is approximately 1 km, e is approximately 0.01 and the vector χ(z; ξ) = (χ_\(z), χ₂{z), £₃(2), xi{z), χ${z), XQ{Z), X₇{z), χs{z), χg{z)) is such that

(81 )

where L_B and Lp are, respectively, the beat length and the birefringence correlation length of said fiber, and x(0; ξ) ≠ 0.

The inventors observed that preferably, δ is approximately 1 km; more preferably, «5 is approximately 10 km; even more preferably, δ is approximately 100 km or more. The inventors observed that preferably, e is approximately 0.01; more preferably, e is approximately 0.001; even more preferably, e is approximately 0.0001.

Preferably, L_B is longer than 0.05 m, more preferably, it is longer that 5 m. Advantageously, the modulus of the spin function a(z) has an absolute maximum smaller than 400 rad/m, preferably, such maximum is smaller than 100 rad/m. Preferably, such maximum is greater than 1 rad/m, more preferably, it is greater than 2-π/L_B.

The almost periodic spin functions can be of different types, according to the particular layout. Only as examples, the inventors mention the periodic spin functions, suitable linear combinations of these periodic spin functions, and the constant spin functions. C. A method for determining at least a parameter of an almost periodic spin function, a(z), to be applied to an optical fiber during the drawing (spinning process), such that the polarization correlation length Lc of said optical fiber corresponding to a substantially circular input state of polarization, is considerably increased. Such method is characterized in that the spin function is selected according to ln ||a;(,z; a(z) lim ≤ e . (82) z→δ In |]z(2:; O)H where δ is approximately 1 km, e is approximately 0.01 and the vector χ(z;ξ) = (χι(z), X₂{z), χz{z)) is such that

= -2 ξx_x(z) - —x₂{z) - —X₃(z) (83) • Lp LB

wher LB and L_F are, respectively, the beat length and the birefringence correlation length of said fiber, and (Z₁(O; ξ), X₂(O; ξ), ^₃(O; ξ)) ≠ 0.

The inventors observed that preferably, δ is approximately 1 km; more preferably, δ is approximately 10 km; even more preferably, <5 is 100 km or more. The inventors also observed that preferably, e is approximately 0.01 ; more preferably, e is approxi¬ mately 0.001 ; even more preferably, e is approximately 0.0001.

D. An optical fiber made up of at least one section to which an almost periodic spin function, a(z), has been appied. The almost periodic spin function has been selected so that

lim — — TTT-, — ΓTV— ≤ e , (84) z→δ ln ||£(z; 0)|| where δ is approximately 1 km, e is approximately 0.01 and the vector χ{z; ξ) = (χi(z), χi{z), χ_%{z)) is such that

X3{z) (85)

where LB and Lp are, respectively, the beat length and the birefringence correlation length of said fiber, and On(O; ξ), x₂{0; ξ), X₃(O; ξ)) ≠ 0. 15 The inventors observed that preferably, δ is approximately 1 km; more preferably, δ is approximately 10 km; even more preferably, δ is 100 km or more.

The inventors also observed that preferably, e is approximately 0.01 ; more preferably, e is approxi¬ mately 0.001 ; even more preferably, e is approximately 0.0001. Preferably, LB is longer than 0.05 m, more preferably, it is longer than 5 m. Advantageously, the

20 modulus of the spin function a(z) has an absolute maximum smaller than 400 rad/m, preferably, said maximum is smaller than 100 rad/m. Preferably, said maximum is greater than 1 rad/m, more preferably, it is greater than 2-K/LB-

The almost periodic spin functions can be of different types, according to the particular layout. Only as examples, the authors mention the periodic spin functions, suitable linear combinations of these

25 periodic spin functions, and the constant spin functions.

E. A method for determining at least a parameter of an almost periodic spin function, a(z), to be applied to an optical fiber, such that the polarization correlation length Lc of said optical fiber, corresponding to a substantially linear input state of polarization is considerably increased. Such method is characterized in that the spin function is selected so that

where <5 is approximately 1 km, e is approximately 0.01 and the vector χ(z; ξ) = (2₄(2), χ_$(z), XQ(Z), 5 x₇(z), x_&(z), XQ{Z)) is such that

—j^ = -2 £2₄(2) - — x₅(z) - —x_e(z) + —x₇(z) dxQ 2τr 1

- ^{~ =} Ts^x*^{{z) '} T_F ^x&{z) (87)

dx8 2 _{/ s r}- ^ _{/ x} 2 2τr

— = -—xφ) - 2ξ x₇(z) - — X₈(Z) - -Xg(Z) dx₉ 27T 1

-dz^~ = L^^{X8iz) ~} L^^{X9 {Z)} where LB and Lp are, respectivley, the beat length and the birefringence correlation length of said optical fiber, and (^χ ₄(0; ξ), ^χ _δ(0; ξ), X₆(O; ξ), X₇(O; ξ), ^s(O; O. ^9(⁰J 6) ≠ 6. The inventors observed that preferably, δ is approximately 1 km; more preferably, δ is approximately 10 10 km; even more preferably, δ is 100 km or more.

The inventors also observed that preferably e is approximately 0.01 ; more preferably, e is approxi¬ mately 0.001 ; even more preferably, e is approximately 0.0001. F. An optical fiber made up of at least one section to which an almost periodic spin function, a(z), has been applied. Such almost periodic spin function has been selected so that

where δ is approximately 1 km, e is approximately 0.01 and the vector χ(z; ξ) — {X_A{Z), χ_${z), χe(z), X₇(z), X₈(z), X₉{z)) is such that

dx₈ 2 2 2τr

— = -— a^; ₄(«) - 2 ^₇(Z) - ^^ - ^W) ώg 2π 1

where LB and L^ are, respectively, the beat length and the birefringence correlation length of said fiber, and (x₄(0; £), X₅(O; ξ), Z₆(O; ξ), X₇(O; ξ), X₈(O; ξ), x₉{0; ξ)) φ 0.

Preferably, δ is approximately 1 km; more preferably, δ is approximately 10 km; even more preferably, δ is approximately 100 km or more. Preferably, e is approximately 0.01 ; more preferably, e is approximately 0.001 ; even more preferably, e is approximately 0.0001.

Preferably, LB is longer that 0.05 m, more preferably it is longer than 5 m. Advantageously the absolute maximum of the modulus of the spin function a(z) is smaller than 400 rad/m, preferably, said maximum is smaller than 100 rad/m. Preferably, said maximum is greater than 1 rad/m, more preferably, it is greater than 2TΓ/LB-

The almost periodic spin functions can be of different types, according to the particular layout. Only as examples, the inventors mention the periodic spin functions, suitable linear combinations of these periodic spin functions, and the constant spin functions.

G. An article relative to an optical fiber telecommunication system comprising: comprising: an opti- cal transmission line, at least a transmitter to insert a signal in said line, at least a receiver to receive said signal from said line, characterized in that the optical transmission line comprises at least a section of optical fiber realized according to what described in anyone of the points from A to F. H. An article comprising at least a section of optical fiber realized according to what described in anyone of the points from A to F.

Claims

1) A method for determining at least one parameter of an almost periodic spin function, a(z), to be applied to an optical fiber during the drawing by means of a spinning process, such that the polarization correlation length Lc of said optical fiber corresponding to any input state of polarization, is considerably increased. Said method is characterized in selecting the spin function so that

⁶ ' where δ is about 1 km, e is about 0.01 and the vector x(z; ξ) = (χχ(z), χ_%(z), χz(z), χ₄(z), χs(z), XQ(Z), χγ(z), χs(z), X₉{z)) is such that

where LB and Lp are, respectively, the beat length and the birefringence correlation length of said fiber, and 5(0; ξ) ≠ 0.

2) A method, as described in claim 1), but characterized in that e is about 0.001.

3) A method, as described in claim 1), but characterized in that e is about 0.0001.

4) A method, as described in anyone of the claims from 1) to 3), but characterized in that <5 is about 10 km. 5) A method as described in anyone of the claims from 1) to 3), but characterized in that δ is about 100 km or more.

6) A method for determining at least one parameter of an almost periodic spin function, a(z), to be applied to an optical fiber during the drawing by means of a spinning process, such that the polarization correlation length of said optical fiber Lc, corresponding to a substantially circular input state of polarization, is considerably increased. Said method is characterized in selecting the spin function so that ≤ e .

where δ is about 1 km, e is about 0.01 and the vector x(z; ξ) = (χi{z), 0:₂(2), χs(z)) is such that ^f I ^da;r2 -- „.W, _M"^{) + «} 1 *^»< ,"> s 2π — = -2^s₁(Z) - ^^-(z) - -j— W)

where L_# and L_F are respectively, the beat length and the birefringence correlation length of said fiber, and z(0;ξ) ≠ O. 5 7) A method as described in claim 6), but characterized in that e is about 0.001.

8) A method as described in claim 6), but characterized in that e is about 0.0001.

9) A method as described in anyone of the claims from 6) to 8), but characterized in that δ is about 10 km.

10) A method as described in anyone of the claims from 6) to 8), but characterized in that <5 is about 10 100 km or more.

11) A method for determining at least one parameter of an almost periodic spin function, α(z), to be applied to an optical fiber, so that the polarization correlation length Lc of said optical fiber, corresponding to a substantially linear input state of polarization is considerably increased. Said method is characterized in selecting the spin function so that

^> where δ is about 1 km, e is about 0.01 and the vector

x(z; ξ) = (0:4 (2), 0:5 (2) , x₆(z), x₇(z), x₈ (z), XQ (Z) ) is such that dx^ 2 2

_ ₌ --X₄(Z) + 2 ξ x₅(z) - —x₈(z)

_ dx_{8 =} ^ 2 ₄(Z) _ _{2 ξ x7{z)} _ _ 2 _{X8 {z} .₎ _ ^ 2τr _{X9 {z)} dx_Q 2τr 1

^~d7 ⁼ I^^⁸^) - L-_F ^x^^z)

20 where LB and Lp are, respectively, the beat length and the birefringence correlation length of said fiber, and x(0; ξ) ≠ O. 12) A method as described in claim 11), but characterized in that e is about 0.001.

13) A method as described in claim 11), but characterized in that e is about 0.0001.

14) A method as described in anyone of the claims from 11) to 13), but characterized in that δ is about 10 km.

15) A method as described in anyone of the claims from 11) to 13), but characterized in that δ is about 100 km or more. 16) A method as described in anyone of the claims from 1) to 15), but characterized in that the maximum amplitude of the modulus of the spin function a(z) is more than 2π/Lβ-

17) A method as described in anyone of the claims from 1) to 15), but characterized in that the maximum amplitude of the modulus of the spin function a(z) is less than 400 rad/m.

18) A method as described in anyone of the claims from 1) to 15), but characterized in that the maximum amplitude of the modulus of the spin function a(z) is more than 1 rad/m.

19) A method as described in anyone of the claims from 1) to 18), but characteirzed in that the spin function a(z) is given by the sum of sinusoids with different periods, phases and amplitudes.

20) A method as described in anyone of the claims from 1) to 18), but characterized in that the spin function a(z) is periodic of period p. 21) A method as described in claim 20), but characterized in that the period p is shorter than LB-

22) A method as described in anyone of the claims 20) and 21), but characterized in that the spin function a(z) is a sinusoidal function.

23) A method as described in anyone of the claims 20) and 21), but characterized in that the spin function a(z) is a triangular function. 24) A method as described in anyone of the claims 20) and 21), but characterized in that the spin function a(z) is a trapezoidal function.

25) A method as described in anyone of the claims from 1) to 18), but characterized in that the spin function a(z) is a constant function.

26) A method for making an optical fiber that comprises to heat a preform and to draw from that preform said optical fiber, applying simultaneously a spin described by the almost periodic spin func¬ tion a(z), characterized in that at least one of the parameters of said spin function is determined according to anyone of the claims from 1) to 25).

27) A method as described in claim 26), but characterized in that L_B is longer than 0.1 m.

28) A method as described in claim 26), but characterized in that NA ≥ 0.2. 29) A method as described in claim 28), but characterized in that LB is shorter than 5 m.

30) A method as described in anyone of the claims from 26) to 29), but characterized in that the optical fiber is a photonic crystal fiber.

31) A method as described in anyone of the claims from 26) to 29), but characterized in that the optical fiber is a microstuctured fiber.

32) An optical fiber produced according to anyone of the methods described in the claims from 26) to 31).

33) An optical fiber comprising at least a section, such that said section has been produced accord- ing to anyone of the methods described in the claims from 26) to 31).

34) An optical fiber telecommunication system, comprising: an optical transmission line, at least one transmitter for adding a signal to said transmission line, at least one received for receiving the signal form said line, characterized in that the optical transmission line comprises at least one optical fiber as described in the claims 32) and 33). 35) An article comprising at least one optical fiber as described in the claims 32) and 33).