Joint-characteristic Function of the First- and Second-orderPolarization-mode-dispersion Vectors in Linearly Birefringent Optical Fibers

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  • ABSTRACT

    This paper presents the joint characteristic function of the first- and second-order polarization-modedispersion (PMD) vectors in installed optical fibers that are almost linearly birefringent. The joint characteristic function is a Fourier transform of the joint probability density function of these PMD vectors. We regard the random fiber birefringence components as white Gaussian processes and use a Fokker-Planck method. In the limit of a large transmission distance, our joint characteristic function agrees with the previous joint characteristic function obtained for highly birefringent fibers. However, their differences can be noticeable for practical transmission distances.


  • KEYWORD

    Polarization mode dispersion , Optical fiber , Optical fiber transmission , Optical communication

  • I. INTRODUCTION

    Random fluctuations of fiber birefringences cause polarization- mode dispersion (PMD) effects detrimental to high bit-rate channels. The PMD effects can be described using a PMD vector whose magnitude is equal to the differential group delay between two principal states [1], [2]. When there are many optical channels or the channel bit rate is high, the angular frequency derivative of the PMD vector becomes also important [3]. Often, the conventional PMD vector is called the first-order PMD vector while its angular frequency derivative is called the second-order PMD vector.

    In [4], Foschini and Shepp derived a joint characteristic function for two white Gaussian vector processes in a compact functional form using a sine-cosine Fourier expansion representation. The joint characteristic function is a Fourier transform of the joint probability density function (pdf) of two vector processes. Shortly after [4], it was shown that, when the fiber transmission distance is large, the joint characteristic function can be used for the first- and second-order PMD vectors in highly birefringent fibers [5]. This joint characteristic function has been used extensively to describe various distribution characteristics of PMD vectors [6-8]. The birefringence vector model used in [5] assumes a fixed linear birefringence with much smaller linear and circular random birefringences in equal amounts. Interestingly, the joint characteristic function of [5] agrees with the result of [9] that uses the model of [5] without the fixed linear birefringence component. However, installed optical fibers are almost linearly birefringent [10]. Thus it is worthwhile to justify the joint characteristic function of [5] for installed fibers.

    Regarding the birefringence components as white Gaussian processes, a Fokker-Planck equation [11] was derived in [12] to investigate the pdf behavior of the first-order PMD vector in linearly birefringent fibers. The Fokker-Planck method was used also in [13] to find a more correct joint characteristic function for the birefringence vector model of [9]. The second-order angular frequency derivative of the fiber birefringence vector was included in [13] but was neglected in [5] and [9]. The birefringence vector distribution of [9] and [13], having linear and circular random birefringences in equal amounts, has a spherical symmetry in Poincare space. This symmetry helps to find the joint characteristic functions of [9] and [13]that are dependent only on the magnitude of two PMD vectors and the angle in between in their Fourier domain. However, the spherical symmetry does not hold for installed optical fibers and, to our knowledge, the joint characteristic function for installed optical fibers is still unknown.

    In this paper, we find the joint characteristic function for installed optical fibers. We use the Fokker-Planck method and include the second-order angular frequency derivative of the fiber birefringence vector. Our final result obtained after a complicated procedure has a compact functional form similar to the previous joint characteristic functions [5, 9, 13].

    At first, we will present details of the Fokker-Planck method of [12] along with a new shortcut procedure for finding the pdf of the first-order PMD vector. Then we extend the approach to find a Fokker-Planck equation (in Fourier domain) for the first- and second-order PMD vectors. The asymptotic solution of our Fokker-Planck equation yields the joint characteristic function.

    II. PDF FOR THE FIRST-ORDER PMD VECTOR

    The dynamical PMD equation,

    image

    describes the evolution of the first-order PMD vector, τ =(τ 1, τ 2, τ 3) along the fiber [1]. L is the fiber transmission distance. β is the birefringence vector and βΩ=∂β /∂Ω, where Ω is the angular frequency. For linearly birefringent fibers, we set

    image

    where

    image

    is a zero-mean white Gaussian process having a correlation property,

    image

    denotes the ensemble average and δij is the Kronecker delta. This modeling neglects the spatial size of the correlation length compared with the large parameter L [9], [12], [13]. The proportionality factor a is a function of Ω and we will denote da/dΩ as aΩ. Similarly, dβ/dΩ is simplified to βΩ=a

    image

    etc.

    We note that the vector equation (1) is composed of three Langevin equations,

    image

    where N(=3) is the number of variables. The matrix formed by gij is

    image

    Then we find the Fokker-Planck equation as follows [11]:

    image

    where Di and Dij are drift and diffusion coefficients, respectively, defined as

    image

    From these relations, we have D1=-a2τ1, D2 =-a2τ2, D3=-a2τ 3, and

    image

    P=P(τ, L) is the pdf for τ . Since the input signal has experienced no PMD degradations, the initial condition is a Dirac delta function, P(τ ,0) =δ (τ).

    The Fokker- Planck equation for the first-order PMD vector is given by

    image

    where

    image

    As is shown in Fig. 1, we use a spherical coordinate using the following transform relations [14]:

    image
    image
    image

    The positive τ 3-axis is the polar axis. θ and φ are polar and azimuth angles, respectively. Since our problem is invariant to the rotation about the τ 3-axis, we may set ∂/∂φ=0. Then the differential operator K can be simplified greatly as

    image

    This equation implies that (5) is a diffusion equation along τ 1, τ 2, and θ directions. Therefore, in the asymptotic region, where the transmission distance L is large, P=P(τ , L) becomes smooth and widely spread in τ space.

    We expand P(τ , L) in terms of Legendre polynomials as

    image

    The n-th order Legendre polynomial, Pn(cosθ), is an eigenmode of the a2 term such that

    image

    Since Pn(cosθ) is not the eigenmode of aω2 terms of (10), there are couplings between these modes during the transient region. The initial condition, P(τ , 0)=δ(τ ), belongs to the P0(cosθ) mode and An(τ , 0) for n≠0. As L increases from zero, the magnitude of An(τ , L) (n≠0) increases owing to the coupling with the n=0 mode. Actually, these couplings exist only between even- modes owing to the even symmetry of our problem with respect to the θ coordinate. Note that the a2 term in (10) is dependent only on the angular variable θ while the aω2 terms are not. Thus, in the asymptotic region, where P=P(τ , L) becomes smooth, the a2 term becomes dominant compared with the aω2 terms for n≠0 modes. If aω2 terms are neglected, An(τ , L) decays in proportion to exp{-n(n+1)a2L}. It implies that, in the asymptotic region, we have P(τ , L)=A0(τ , L) ultimately. This can be verified also using a multiple-scale method [15].

    To find A0(τ , L) in the asymptotic region, we set ?/?θ= ?/?φ=0 for the spherical coordinate representations of ?2/? τ 1 2 and ?2/?τ 2 2 terms in K, which gives

    image

    Next, we remove the terms in K that couple A0(τ ) with other An(τ , L) (n≠0) components. This can be achieved by applying (1/ 2)fπ0 d θ sin θon K, which leaves only the P0(cosθ) component in K. The result is

    image

    Thus, in the asymptotic region, P(τ , L) becomes a Gaussian, exp

    image

    with a variance σ2(L)=4aω2L/3. This also implies that the magnitude of the first order PMD vector's pdf converges to a Maxwellian [2].

    We have used the same initial condition, P(τ, 0)=δ(τ), to solve (13). This can be verified by taking a Laplace transform of (5) to find

    image

    where

    image

    is the Laplace transform of

    image

    with respect to L. The initial condition term, δ(τ), also belongs to the P0(cosθ) mode and remains the same in the asymptotic region. Thus the initial condition, P(τ, 0)=δ(τ), can be used for (13).

    III. JOINT CHARACTERISTIC FUNCTION FOR THE FIRST-AND SECOND-ORDER PMD VECTORS

    From (1), we find a differential equation for the secondorder PMD vector, τω=?τ/?ω=(τωl, τω2, τω3), as follows:

    image

    where βωω=?2β /?ω2=aωω1, γ1, 0). We note that (1) and (14) can be regarded as six Langevin equations, ? ? =

    image

    where N=6. xi=τi for i = 1, 2, 3 and xiωi for i = 4, 5, 6. Now gij set forms a 6×6 matrix that will be used to find new drift and diffusion coefficients in the Fokker-Planck equation, (3). Its solution is the joint pdf of τ and τω which will be denoted as P=P(τ, τω, L).

    We will solve the Fokker-Planck equation in the Fourier domain by introducing a joint characteristic function,

    image

    where k=(k1, k2, k3) and kω=(kω1, kω2, kω3). Our definition of the join]t characteristic function is the complex conjugate of the conventional one. The differential equation for the joint characteristic function is given as

    image

    where the differential operator M is given by

    image
    image
    image
    image
    image
    image
    image

    In (18), we have introduced angular momentum operators defined as Lk =-jk ×∇k = (Lk1, Lk2, Lk3), L= -jk ×∇=(Lkω1, Lkω 2, Lkω 3), and Ltot=Lk+L=(Ltot1, Ltot2, Ltot3) [16]. ∇k and ∇ are del operators in k and kω domains, respectively. Ltot=Ltot1+Ltot2 and Ltot-=Ltot1-Ltot2 are raising and lowering operators, respectively, for the total angular momentum operator, Ltot. Since the input signal has experienced no PMD degradations, the initial condition is Q(k, kω, 0) = 1 or P(τ ,τω, 0)=δ(τ)δ(τω).

    As is shown in the Appendix A, we can find E2}=4a2ωL from (16) using the relation, E2}=-∇2kQ(k, kω, L)'k=kω=0 . In a similar way, E2ω} is found as 16a4ωL2 /3+4a2 ωωL(1-a4ω /9a2 ωωa2)+2a 4ω {1-exp (-6a2L )}/ 27a4. As L becomes large, the ratio E2ω}/(E2})2 converges to 1/3 [5]. In this asymptotic region, Q(k, kω, L) becomes localized to the origin k=kω=0.

    Since E2ω} >>E2}, Q(k, k, L) shrinks faster in kω domain than in k domain as L increases.

    The L2 tot operator has eigenvalue Ltot(Ltot +1), where Ltot is a non-negative integer. We may expand Q(k, kω, L) using the eigenmodes of L2tot. The L2tot operator is composed of angular variables only and becomes dominant in the asymptotic region. The other terms in M become less effective. This is because, as is mentioned above, Q(k, kω, L) shrinks to the origin k=kω=0 and the shrink speed is faster in kω domain. Thus eigenmodes with Ltot ≠ 0 become negligible in the asymptotic region and only the eigenmode with Ltot = 0 survives. Note that this procedure is similar to the foregoing first-order PMD vector analysis. The Ltot = 0 condition also implies that the only possible eigenvalue of Ltot3 is zero since -LtotLtot3Ltot. Thus we have Ltot±Q =Ltot3Q = 0 or

    image

    From (24), we find kLQ =kωLk Q = 0. Using these relations, it can be shown that D and F terms in the operator M disappear, i.e., DQ = FQ = 0 . As a result, there are no differential terms for kω in the operator M and kω becomes just a parameter in the asymptotic region. The symmetry in (24) implies that the asymptotic solution should be dependent only on the magnitudes of k and kω vectors and the angle in between. This can be proved simply by rotating the coordinate such that one of k or kω vector becomes a polar axis.

    To find the asymptotic solution satisfying this condition, let's assume that the kω vector is located in the k vector domain as is shown in Fig. 2. Since our problem is invariant under the rotation about the k3-axis, we have set kω2. We rotate the (k1, k2, k3) coordinate about the k2 -axis into the (k1', k2', k3') coordinate, where the k3' -axis coincides with the kω vector's direction.

    The corresponding rotation matrix is

    image

    where θωk is the polar angle of kω before the rotation. With (25), we apply the transform relations

    image

    and

    image

    to M, where Rji is the matrix element of the inverse rotation matrix R(-θ). Then the remaining operators, B, C, and E, are transformed into

    image
    image
    image

    In the transformed (k1', k2', k3') coordinate, the asymptotic solution is not dependent on the azimuth angle φ′. Thus we set ?/?φ'=k1'?/k2'-k2'?/?k1=0 in M. In addition, we apply

    image

    on M to remove θ and φ′ dependences in M. This operation removes the couplings between the -Ltot = 0 mode and the other modes with -Ltot ≠ 0. Then cos2 θωk, sin2 θωk, and cosθωksinθωk, factors become 1/3, 2/3, and 0, respectively. Also, cos2φ', sin2φ', and cosφ'sinφ' factors become 1/2, 1/2, and 0, respectively. Besides, we find ?/?k1'2 = ?/?k2'2.

    Finally, we obtain the following differential equation for the asymptotic solution:

    image

    where we have omitted the prime notation. As before, the initial condition remains the same, Q(k, kω, 0) = 1. We decompose Q(k, kω, L) as

    image

    where q satisfies

    image

    Equation (31) can be solved using the separation-of-variables method as

    image

    where H2m(.) is the Hermite polynomial. Using an expansion formula for a Gaussian function,

    image

    we find

    image

    Thus, for an arbitrary direction of kω, the joint characteristic function becomes

    image

    where σ2 =E12}=E22}=E32}=4a2ωL/3 as in the previous section. k// is the component of k parallel to kω and k21=k2-k2//.

    In the limit of large L, the cosh (kωσ2) term in (35) implies that the effective range of kω decreases as O(L-1) since σ2L. Also, the exponential terms in (35) imply that the effective range of ki(i=1, 2, 3) decreases as O(L-1/2). Thus we may neglect the aωω term in (35) as L increases indefinitely. In this limit, (35) becomes the same as the joint characteristic function of [5] with reevaluated appropriate to the linearly birefringent fiber. This feature also holds when the fiber has both linear and circular birefringences in equal amounts [9], [13]. The aωω term can be meaningful for practical transmission distances. For example, the second-order PMD vector components can be Gaussian-like distributed instead of a soliton shape [2]. This point has been shown in [13] and presented in Appendix B.

    With the

    image

    condition, the first-order PMD vector distribution has an azimuthal symmetry for all L in [12]. However, the azimuthal symmetry is not evident in our analysis of joint-characteristic function where the number of variables is doubled from 3 to 6. In fact, our final asymptotic solution (35) has spherical symmetry. The coordinate orientation does not affect our result. This kind of spherical symmetry for the asymptotic joint-characteristic function is also found in [5], [9], and [13].

    It is also interesting that (35) is quite similar to the result of [13]. The only difference is the change of L into 2L/3 in our case. In [13], the birefringence vector was assumed as

    image

    is also a zero-mean white Gaussian process having a correlation property,

    image

    Note that the number of random birefringence components is larger than our case. Thus, for given a, aω, and aωω values, the joint characteristic function reaches to its asymptotic form faster than our case as the transmission distance increases. In this respect, the scaling factor 2/3 can be regarded as the ratio of the birefringence component numbers.

    IV. CONCLUSION

    We have found a joint characteristic function for the first- and second-order PMD vectors in installed fibers. As the fiber transmission distance increases indefinitely, our joint characteristic function resembles that of [5] obtained for highly birefringent fibers. This agreement may not hold for practical transmission distances owing to the secondorder angular frequency derivative of the fiber birefringence vector included in our analysis. As an example, we have shown that the second-order PMD vector components can be Gaussian-like distributed instead of a soliton shape. If the fiber length is scaled by a factor of 2/3, our result obtained for linearly birefringent fibers has the same functional form with the previous joint characteristic function of [13] obtained for fibers having a spherically symmetric birefringence vector distribution in Poincare space.

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  • [FIG. 1.] Spherical coordinate for τ space.
    Spherical coordinate for τ  space.
  • [FIG. 2.] Rotation of k vector space coordinates about the k2-axis.
    Rotation of k vector space coordinates about the k2-axis.