Abstract

Channel matrix plays a critical role in receiver design and ultimate channel performance. To fully describe the channel matrix of a few-mode fiber (FMF), we choose the generalized high-dimensional Gell-Mann matrices, an equivalent of the 2-dimensional Pauli matrices used for a single-mode fiber (SMF), as the basis for the channel matrix decomposition. The frequency dependence of channel matrix can be studied in terms of the autocorrelation function (ACF), showing how fast channel changes in frequency domain. In this paper, we derive a canonical stochastic differential equation (SDE) for the FMF channel matrix in the regime of strong coupling. With the SDE, we develop an analytical form for the ACF of FMF channel matrix, from which the channel correlation bandwidth is obtained.

© 2013 OSA

1. Introduction

Few-mode fiber (FMF) is being actively explored as a promising transmission medium to surmount the capacity crunch in future fiber optic communication [13]. FMF supports multiple spatial modes co-propagating in one medium simultaneously, enhancing the capacity of information transmission. However, the random mode coupling caused by fiber index imperfections and mechanical perturbations leads to signal dispersion in FMF, called differential modal delay (DMD) [4]. While large DMD and mode coupling can improve the channel diversity, making the system robust against mode-dependent loss (MDL) [5], excessive DMD would increase the computational complexity of digital signal processing (DSP) required for the channel equalization [1]. An in-depth understanding of the statistic properties of FMF channel subject to DMD is essential to assist system design for achieving an optimal system performance.

With respect to single-mode fiber (SMF), there is a large body of studies on the statistics of SMF channel affected by polarization modes dispersion (PMD). The stochastic analysis is conducted in terms of either the PMD vector in the Stokes space [6,7] or the channel matrix in the Jones space [8,9]. Though the Stokes space and the Jones space are isomorphic [10], a PMD vector is more suitable for the analysis of time-domain direct detection systems, whereas a Jones matrix is more useful for the analysis of coherent detection systems. Based on a white Gaussian noise assumption for the birefringence vector with strong coupling, standard tools of stochastic calculus can be used to derive some important statistic properties for the PMD vector and Jones matrix [69].

The basic concepts and stochastic study methods for SMF can be extended to FMF. Since each spatial mode supports two orthogonal polarization modes, FMF with N spatial modes supports totally 2N modes with different group velocities. An extended 2N-dimensional vector is used to represent the propagating modes in FMF, and the linear effect of FMF channel can be described as a 2N x 2N channel matrix in the generalized Jones space. Recently, study has been conducted on the statistics of mode dispersion (MD) vector (a general form of PMD vector in SMF) for FMF [11,12]. A generalized Stokes space has been formulated to allow convenient representation of the mode coupling vector (a general form of birefringence vector in SMF) and the MD vector in FMF. Based on a white Gaussian noise model for the mode coupling vector, the statistic properties of MD vector have been derived in [11,12]. The statistics of group delays in FMF has also been derived in [5,13] by dividing fiber into many cascaded sections modeled by random matrices.

On the other hand, the latest decade has seen the revival of coherent detection, where DSP is performed on the channel matrix of fiber optic link [14,15]. Subsequently, the statistical analysis of FMF channel matrix has become critically important. In particular, the statistical analysis provides us the autocorrelation function (ACF) of FMF channel matrix, which gives an insight into the frequency behavior of FMF channel dispersion and helps to find the optimal receiver design. The channel correlation bandwidth derived from the ACF is also important for the coherent detection system design, as it is often needed for the channel estimation, for instance, for the minimum mean-square error (MMSE) based channel estimation [16].

As an extension of the Pauli matrices used as the basis of the Stokes space in SMF [9], high-dimensional trace orthogonal matrices have been proposed in [11] as the basis of the generalized Stokes space in FMF. In this work, we also resort to the trace orthogonal matrices as the basis for FMF channel matrix. In this report, a canonical stochastic differential equation (SDE) for FMF channel matrix is derived in the regime of strong coupling. Applying the standard tools of stochastic calculus to the SDE, we develop an analytical form for the autocorrelation function (ACF) of FMF channel matrix, from which the channel correlation bandwidth is obtained. A comparison of simulation and analytical results demonstrates the accuracy of our theoretical model.

2. FMF channel matrix and decomposition

The main notations used in the paper are as follows: |V denotes a column vector in the generalized Jones space of dimension 2N, V denotes a column vector in the generalized Stokes space of dimension 4N21, V˜ denotes a column vector of dimension 4N2, a column vector with elements vi is denoted by [vi1;vi;vi+1]; Symbol ‘’ represents the transpose conjugate operation for a matrix, and ‘Τ’ represents the transpose operation; E[ ] denotes the ensemble average, and Tr{ } denotes the trace of a matrix.

The optical field in an N-mode FMF at length z and frequency ω can be expressed as a 2N-dimensional complex vector |ψ(z,ω). The linear channel of FMF can be treated as a 2N input 2N output linear system given by [11,12]

|ψ(z,ω)=U(z,z0,ω)|ψ(z0,ω),
where U is the channel matrix of dimension 2N x 2N, describing the random mode coupling and group delays in optical channel. Without consideration of MDL, the channel matrix U is a unitary matrix with unit determinant.

Similar to the Pauli matrices used as the basis for the 2x2 Jones matrix of SMF [9], 4N2trace orthogonal matrices form the basis for the 2N x 2N channel matrix of FMF. The trace orthogonal matrices and their associated properties have already been developed in [11]. Specifically, to represent the channel matrix U, Λ0=I/Nand 4N21 traceless Hermitian matrices Λi (i = 1 to 4N21) are constructed, satisfying the trace-orthogonal condition [11]

Tr{ΛiΛj}=2δij
where δij is the Kronecker symbol (i,j=0to 4N21). Note that in (2) we use a normalization factor of 2 instead of 2N in [11]. This follows the convention in other works on special unitary groups [17,18]. Hence, the channel matrix U could be expressed as a superposition of the trace orthogonal matrices [11]
U=i=04N21uiΛi=u˜Λ˜
where the vector Λ˜=[Λ0;Λ1;;Λ4N21] is an ensemble of the trace orthogonal matrices, and the complex vector u˜=[u0;u1;;u4N21] is an ensemble of the weights of each trace orthogonal matrix. Since one set of weights ui can exclusively identify a matrix U, the properties of a 2N x 2N channel matrix U is fully represented by a vector u˜ of dimension 4N2. According to the trace-orthogonal condition (2), the weights ui of trace orthogonal matrices Λi could be easily extracted by [11]

ui=12Tr{ΛiU}.

In group theory, the traceless Hermitian matrices Λi(i = 1 to 4N21) form the generators (elements of the Lie algebra) of special unitary group of degree 2N (SU(2N)), satisfying [17]

ΛmΛn=δmnNI+k(ifmnk+dmnk)Λk
ΛnΛm=δmnNI+k(ifmnk+dmnk)Λk
where the indices k, m, n take values from 1 to 4N21, and the coefficients f and d are the structure constants. As provided by [11], a cross product “×” of vectors A and B in the generalized Stokes space can be defined with the structure constants f as
A×B=mnkfmnkAmBnek=B×A,
where Am is the m-th element in vector A, Bn is the n-th element in vector B, and ek is a unit vector with 1 for the k-th element. For the convenience of the mathematical description of our problem, we also introduce a product “” of vectors A and B defined with structure constants d as
AB=mnkdmnkAmBnek=BA.
We call “” circle dot product to distinguish it from the conventional dot product ‘’. Then, the matrices AΛ and BΛ satisfy the relations
(AΛ)(BΛ)(BΛ)(AΛ)=2i(A×B)Λ,
(AΛ)(BΛ)+(BΛ)(AΛ)=2N(AB)I+2(AB)Λ
Adding (9) and (10), we get the product of the matrices AΛ and BΛ

(AΛ)(BΛ)=1N(AB)I+(AB+iA×B)Λ

The choices for the trace-orthogonal matrices are many. The only guide for the choice is the simplicity of calculation. Modified Gell-Mann matrices have been used to build the trace-orthogonal matrices in [11]. In this paper, we select the generalized Gell-Mann matrices, since the structure constants (f and d), extracted from the generalized Gell-Mann matrices, are sufficiently simple for the calculation in appendix B. The generalized Gell-Mann matrices are described in appendix A. The off-diagonal ones are identical to those in [11]. Only the diagonal ones are slightly different from [11]. The generalized Gell-Mann matrices coincide with the Pauli matrices when N = 1.

3. SDE for channel matrix

The evolution of FMF channel matrix U along fiber position z is governed by the mode coupling vector β, a generalization of the birefringence vector in SMF

dU(z,ω)dz=i2[β(z,ω)Λ]U(z,ω)
βΛ is required to be a Hermitian matrix to ensure the unitarity of U. Such that, all the elements in β(z,ω) are real [11]. For a ‘long’ fiber, where the overall fiber length is significantly larger than the correlation length of local modal dispersion, the white Gaussian noise model is valid for the mode coupling vector given by [11]
β(z,ω)=μωn(z),
where the elements of n(z) are statistically uncorrelated white Gaussian processes with zero mean and unit variance, and the parameter μ represents the mode coupling strength.

Substituting the channel matrix U in (12) with its vector representation (3), and using the product defined in (11), we obtain the evolution equation of the vector representation u˜=[u0;u] as

du0=iμω2NudWdu=iμω2N[u0dW+N(udWiu×dW)]
where dW=ndz is the differential of the Brownian motion W(z). Equation (14) can be written in a canonical form of SDE
du˜(z)=Q(u˜(z))dW(z)
where Q(u˜) is the diffusion matrix of dimension 4N2x4N21with entries to be polynomials of the components in u˜

Q=iμω2N[uΤu0I+N(uiu×)]

We note that, normally a SDE is on a real-value space, whereas the dynamic vector u˜ in Eq. (15) is complex. However, we can consider (15) as a short hand for real-value SDEs. Namely, we can separate the real and imaginary part of the dynamic vector u˜, and treat the two parts individually as real elements. As long as the Brownian motion term W(z) is real, the real and complex forms of SDE describe the same physics of stochastic process. For the sake of concision, we use the complex vector u˜ in our discussion in the remainder of this paper.

Derived from (3), (12) and (13), the m-th column q˜m(u˜) in the diffusion matrix Q satisfies

q˜mΛ˜=iμω2Λm(u˜Λ˜).
Denote qx,y as the entry at x-th row and y-th column in the diffusion matrix Q. Using the trace-orthogonality of Λ in (17), qx,y can be extracted as
qx,y=12Tr{Λx1(q˜yΛ˜)}.
Some important statistic properties of the dynamic vector u˜ can be derived from the SDE (15) using standard tools of stochastic calculus. However, those standard tools are usually applied for the Ito form of SDE, whereas the product in (15), involving the Brownian motion, should be interpreted as the Stratonovich product [8,9]. Before (15) can be directly used for stochastic calculus, it should be converted from the Stratonovich form into its equivalent Ito form. In the following context, we will use two different approaches to carry out the conversion.

3.1 Stratonovich-Ito conversion algorithm approach

In SDE theory, there is a standard algorithm to convert SDE (15) from the Stratonovich form to its equivalent Ito form [19], which is given by

du˜=c˜(u˜)dz+Q(u˜)dW
where the diffusion matrix Q(u˜) is the same as that in Eq. (15), and c˜(u˜) is the drift correction vector of dimension 4N2, whose l-th component cl is given by
cl=12n=14N2m=14N21qn,mql,mun1.
Since qn,m can be obtained using (18), the product of the vectors c˜ and Λ˜ can be written as
c˜Λ˜=12n=14N2m=14N2112Tr{Λn1(q˜mΛ˜)}q˜mΛ˜un1,
where q˜mΛ˜ is already known in (17). Noting u˜Λ˜/un=Λn, (21) can be rewritten as

c˜Λ˜=μ2ω28[m=14N211NumΛm+n=14N21m=14N2112Tr{ΛnΛm(u˜Λ˜)}ΛmΛn].

After some length calculation shown in appendix B, we obtain

c˜Λ˜=4N218Nμ2ω2(u˜Λ˜).
As such, the drift correction vector c˜ in (19) becomes
c˜=4N218Nμ2ω2u˜.
The i-th component in c˜ is proportional to the i-th component in u˜.

3.2 Stochastic integral definition approach

We can also develop the correction term c˜ from the first principle of definitions for the Stratonovich and Ito integral.

According to the definition of stochastic integral [20], the integral of canonical SDE (15) can be interpreted as

abQ(u˜(z))dW(z)=limδ0i=0n1Q(λu˜(zi+1)+(1λ)u˜(zi))ΔWi
where zi(i = 0 to n-1) are n partitions in the intervals [a,b] with mesh δ=max(zi+1zi), ΔWi=W(zi+1)W(zi), and λ is the parameter defining different methods of integral. The case of λ = 0 corresponds to the Ito integral, whereas λ = 1/2 corresponds to the Stratonovich integral.

SDE (15) is the canonical form of dynamic Eq. (12) with a vector representation for the channel matrix U. When we substitute the canonical SDE (15) in (25) with (12), the integral of dynamic Eq. (12) is given by

iμω2ab[dW(z)Λ]U(z)=iμω2limδ0i=0n1[ΔWiΛ](λU(zi+1)+(1λ)U(zi))
From (26), the correction term, defined as the conversion between the Ito and Strantonovich integral, is a given by
iμω4[dWΛ]dU
Substituting dU in (12) into (27), and using the rules of dWidWi=dz, dWidWj=0(ij) [21, Chapter III], and ΛΛ=(4N21)I/N, we can finally reach the correction term as
4N218Nμ2ω2Udz
The correction term (28) we get here is consistent with that we developed using the standard Stratonovich-Ito conversion algorithm in last section (24).

It follows that the dynamic equation of U (12) in its Ito form can be expressed as

dU(z,ω)dz=4N218Nμ2ω2U+iμω2[n(z,ω)Λ]U(z,ω)
The dynamic equation of u˜ (14) in its Ito form is

du0=(4N21)μ2ω28Nu0dz+iμω2NudWdu=(4N21)μ2ω28Nudz+iμω2N[u0dW+N(udWiu×dW)]

The canonical form of SDE (30) is (19), with the diffusion matrix Q defined by (16), and the drift vector c˜defined as (24). The standard tools of stochastic calculus can be directly applied to the Ito Eq. (19).

4. Autocorrelation function of channel matrix

The frequency correlation of channel matrix U can be defined as

R(z,ω1,ω2)=U(z,ω1)U(z,ω2)
R is also a unitary matrix, describing the disparity between the channel matrices U at frequencies ω1 and ω2. Its expectation value, E[R], is the ACF of channel matrix, giving a measure of the statistical dependence of U at different frequencies, and indicating how quickly U diverges in frequency domain. The bandwidth of E[R] is called channel correlation bandwidth, representing a frequency span over which U can be approximately considered as a constant matrix from the viewpoint of DSP.

Taking derivative of (31) with respect to length z, the evolution of R along z is

dR(z,ω1,ω2)dz=dU(z,ω1)dzU(z,ω2)+U(z,ω1)dU(z,ω2)dz
Substituting the evolution equation of U (12) into (32), we obtain
dR(z,ω1,ω2)dz=iμ(ω2ω1)2[U(z,ω1)(n(z)Λ)U(z,ω1)]R(z,ω1,ω2)
Term U(z,ω1)(n(z)Λ)U(z,ω1) is a rotated white Gaussian noise, and can be denoted as n'(z)Λ, where n' can be regarded as the white Gaussian noise vector in a new coordinate system. From (33), we can observe that R is determined only by the frequency difference Δω=ω2ω1, such that (33) can be rewritten as
dR(z,Δω)dz=iμΔω2[n'(z)Λ]R(z,Δω)
Equation (34) has the same form as (12), substituting ω with Δω. Following the steps in last section, we can also obtain the Ito equation for (34) in a similar form to (19), given by
dr˜=c˜(r˜)dz+Q(r˜)dW
where r˜ is the vector representation of R on the basis matrices R=r˜Λ˜. The diffusion matrix Q(r˜) is given by
Q(r˜)=iμΔω2N[rΤr0I+N(rir×)]
The drift vector c˜(r˜) is given by

c˜(r˜)=4N218Nμ2Δω2r˜

The Fokker-Planck equation (FPE) is a powerful tool of stochastic calculus, which can be used to derive the distribution of a dynamic process governed by SDE [21]. Applying FPE to the Ito Eq. (35) gives the probability density function (pdf) p of vector r˜

p(r˜,z)z=12n=14N2m=14N22Omnp(r˜,z)rm1rn1k=14N2ckp(r˜,z)rk1
where O=QQΤ. Though the distribution of the vector representation u˜ for the channel matrix experiences an instant spread on unit sphere after fiber input due to the white Gaussian noise model adopted for the mode coupling vector [9], the distribution of the vector representation r˜ for the ACF shall experience a gradual spread on unit sphere as the random coupling is mitigated by the transpose conjugate operation in (31). The solution to the pdf in FPE (38) will be discussed in a separate future submission.

On the other hand, the Dynkin formula gives the expectation of any smooth function f of the dynamic process r˜without the knowledge of the pdf [21]

dE[f(r˜)]dz=E[Gf(r˜)]
where G is the Ito generator given by
G=12n=14N2m=14N2Omn2rm1rn1+k=14N2ckrk1
Since E[ri]is what we need to obtain the ACF, the smooth functions in our case should be f(r˜)=ri. The second order differential operator to ri in (40) equals to 0, and only the first order differential operator left. Such that, the evolution of E[ri] is only determined by the drift term ck. Therefore, the Dynkin formula gives
dE[ri(z)]dz=4N218Nμ2Δω2E[ri(z)].
The correlation matrix R at the input of fiber is an identity matrix I for any frequencies. With the initial condition r0(0)=N and r(0)=0, we obtain
E[r0]=Nexp(4N218Nμ2Δω2z),E[ri]=0(i0)
Consequently, the ACF of channel matrix can be expressed as
E[R]=exp(4N218Nμ2Δω2z)I
From (43), the 3dB channel correlation bandwidth, where the frequency correlation level falls to 0.5 [22], is obtained
BU=28Nln0.5(4N21)μ2z.
Equation (44) shows that the frequency dependence of channel matrix is related to the mode number N, and decreases with the square root of fiber length z.

Taking derivative of (31) with respect to frequencies ω1 and ω2, we obtain

2Rω1ω2=U(ω1)ω1U(ω2)ω2
Setting ω1=ω2 in (45), and noting that the Hermitian MD matrix is defined as
ΩΛ=j2UωU/N
where the subscript ‘ω’ stands for the derivative over ω, we can relate the frequency derivative of R to the MD matrix by
Tr{2Rω1ω2|ω1=ω2}=N4Tr{(j2UωU/N)(j2UUω/N)}=Nτ22
where τ=|Ω|. To derive (47), we have first applied the identity of (46) then of (11). Provided that R is determined only by the frequency difference Δω, 2R/ω1ω2=2R/2Δω. Substituting this into (47), we have
E[τ2]=2NTr{2E[R](Δω)2}|ω=0=(4N21)μ2zN
Equation (48) is consistent with the study reported in [11]. The MD vector Ω(z,ω) of FMF undertakes a 4N21 dimensional isotropic random walk, and its root-mean-square value E[τ2] grows as the square root of the fiber length z [11,13].

From (43) and (48), the mean-square value of MD vector E[τ2] and the ACF of channel matrix E[R] can be related by

Tr{E[R]}=2Nexp(E[τ2]Δω28)

Applying (48) into (44), the correlation bandwidth in relation to the channel matrix can be written as BU=4.7/E[τ2]. On the other hand, the correlation bandwidth in relation to the square modulus of MD vector, as given in [11], is Bτ2=3.2(4N21)/(N2E[τ2]), which is 0.6841/N2 times of BU. We conclude that the MD vector is always slightly more stable in frequency domain compared to the channel matrix. Our conclusions about the ACF of the FMF (43) and (49) coincide with those of the SMF studied in [9] when N = 1. We extend the work of [9] to a more general case of arbitrary number of modes. We note that the MD vector Ω defined in (46) has a factor of N difference than that in [11]. With the MD vector definition in (46), the ACF is independent of the number of modes for the same mean-square value of MD vector E[τ2]. Namely, E[τ2] will be a good indicator of the correlation bandwidth regardless of the number of modes under the condition of strong coupling.

5. Simulation

We conduct the Monte-Carlo simulation to verify our theoretical result obtained in the last section. A FMF is divided into 100 sections, assuming that the length of one section is longer than the correlation length. The local principle modes (PMs) in separate sections are considered independent and uniformly distributed. The propagation of each section is modeled as a 2N x 2N matrix with equivalent statistical properties, given by [5,13]

V(ω)=C1T(ω)C2
where C1 and C2 are independent random unitary matrices corresponding to the random coupling of the PMs at the input and the output of the section, and T is a diagonal matrix representing the impact of group delays on the PMs
T(ω)=diag[ejωt1,ejωt2,...,ejωt2N]
where ti (i = 1 to 2N) are the group delays of the local PMs. The average delay is a constant by setting ti=0. The channel matrix of the whole fiber is obtained by multiplexing the propagation matrices V for each section. The choice of the individual group delays does not have significant impact on the statistical properties of the global fiber as long as the modes are strongly coupled [13]. We just need to ensure that the MD in each section T=2ti2/N satisfies the requirement of the global MD τ=T100.

After the channel matrices U at different frequencies are obtained using the Monte-Carlo simulation, their correlation R could be calculated using (31). We measured the ACF for FMFs with a 10-ps global MD in the bandwidth of 250 GHz. 10000 fiber configurations are used in the simulation to obtain the ensemble averaging. The simulation results and theoretical predictions of Tr{E[R]}/2N for FMFs with 2 to 4 spatial modes are plotted in Fig. 1. The accuracy of our theory result is justified by the excellent match between the simulation and theoretical results.

 

Fig. 1 The comparison of the analytical and simulation result for the ACF of the FMF channel matrix with 2 to 4 spatial modes.

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6. Conclusion

In this paper, we have used two different approaches to derive the SDE for the FMF channel matrix in the regime of strong coupling. The channel matrix is decomposed over the generalized high-dimensional Gell-Mann matrices, an equivalent of two-dimensional Pauli matrices. We also develop the ACF of channel matrix for a strong coupling FMF. The channel correlation bandwidth obtained from the ACF indicates that the frequency dependence of the channel matrix decreases with the square root of fiber length. The validity of our analytical result is confirmed by the Monte-Carlo simulation.

Appendix A

We choose the generalized Gell-Mann matrices as the basis matrices for the decomposition of the channel matrix. The n-dimensional Gell-Mann matrices are constructed by the following algorithm:

The first n-1matrices are diagonal with the form

Λk=2k(k+1)[11k00](k=1ton1).
where the first k diagonal entries are 1, and the k + 1th diagonal entry is –k. The rest entries are all 0. The coefficient 2/(k(k+1)) is added to satisfy the trace-orthogonal condition Tr{ΛiΛj}=2δij.

The rest n2n Gell-Mann matrices are off-diagonal. Half of them are symmetric

Λpq=[11],
where the entries at the p-th row, q-th column and the q-th row, p-th column are 1, and the rest entries are 0. The other half are antisymmetric
Λpq=[ii],
where the entry at the p-th row and q-th column is –i, the entry at the q-th row and p-th column is i, and the rest entries are 0.

Following this construction algorithm, we can easily prove out thatmΛmΛm=2(n21)I/n.

Appendix B

The product ΛnΛm and ΛmΛn in (22) can also be expanded as (5) and (6). Using the trace-orthogonality of Λ, the structure constants fmnk and dmnk can be extracted by

dmnk=14Tr{Λk(ΛmΛn+ΛnΛm)}
fmnk=i4Tr{Λk(ΛmΛnΛnΛm)}.
Switching indices m with n in the above functions, we have dmnk=dnmk, andfmnk=fnmk. Substituting n with m in (53), we havefmmk=0. Then, ΛmΛm=I/N+kdmmkΛk. Since we knowmΛmΛm=(4N21)I/N, we can easily prove out mdmmk=0.

Substituting (5) and (6) into the second term on the right side of (22), we have

n=14N21m=14N2112Tr{ΛnΛm(u˜Λ˜)}ΛmΛn=n=14N21m=14N21[δmnNu0+k=14N21(ifmnk+dmnk)uk][δmnNΛ0+l=14N21(ifmnl+dmnl)Λl].
Using the properties of coefficients fmnk and dmnkwe have discussed above, (54) can be simplified as
n=14N21m=14N2112Tr{ΛnΛm(u˜Λ˜)}ΛmΛn=(4N21)Nu0Λ0+klmn(fmnkfmnl+dmnkdmnl)ulΛk.
For the Gell-Mann matrices described in appendix A, we can easily prove out that mn(fmnkfmnl+dmnkdmnl)=0for the case of kl, and mn(fmnk2+dmnk2)=(4N22)/Nfor the case of k=l. Thus, (22) can be written as

c˜Λ˜=4N218Nμ2ω2[u0Λ0+k=14N21ukΛk].

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10. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000). [CrossRef]   [PubMed]  

11. C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fiber with multiple mode transmission,” Opt. Express 20, 11718–11733 (2012).

12. C. Antonelli, A. Mecozzi, and M. Shtaif, “Modeling of linear and nonlinear coupling in multiple-mode fiber optic transmission with MIMO signal processing,” Signals, Systems and Computers (Asilomar, 2012), 645–649.

13. K.-P. Ho and J. M. Kahn, “Statistics of group delays in multimode fiber with strong mode coupling,” J. Lightwave Technol. 29(21), 3119–3128 (2011). [CrossRef]  

14. A. Li, A. Al Amin, X. Chen, and W. Shieh, “Transmission of 107-Gb/s mode and polarization multiplexed CO-OFDM signal over a two-mode fiber,” Opt. Express 19(9), 8808–8814 (2011). [CrossRef]   [PubMed]  

15. A. Al Amin, A. Li, S. Chen, X. Chen, G. Gao, and W. Shieh, “Dual-LP11 mode 4×4 MIMO-OFDM transmission over a two-mode fiber,” Opt. Express 19(17), 16672–16679 (2011). [CrossRef]   [PubMed]  

16. M. Hsieh and C. Wei, “Channel estimation for OFDM systems based in comb-type pilot arrangement in frequency selective fading channels,” IEEE Trans. Consum. Electron. 44(1), 217–225 (1998). [CrossRef]  

17. S. Weigert, “Baker-Campbell-Hausdorff relation for special unitary groups SU(N),” J. Phys. Math. Gen. 30(24), 8739–8749 (1997). [CrossRef]  

18. M. Gell-Man, “Symmetries of Baryons and Mesons,” Phys. Rev. 125(3), 1067–1084 (1962). [CrossRef]  

19. L. Arnold, Stochastic Differential Equations Theory and Application (Wiley 1974).

20. T. C. Gard, Introduction to Stochastic Differential Equations (Marcel Dekker, INC. 1988).

21. B. Øksendal, Stochastic Differential Equation (Springer-Verlag Berlin Heidelberg 1989).

22. D. C. Cox and R. Leck, “Correlation bandwidth and delay spread multipath propagation statistics for 91-MHz urban mobile radio channels,” IEEE Trans. Commun. 23(11), 1271–1280 (1975). [CrossRef]  

References

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  1. A. Li, A. A. Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” OFC/NFOEC2011, PDPB8 (2011).
  2. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle, “Mode-division multiplexing over 96 km of few-mode fiber using coherent 6x6 MIMO processing,” J. Lightwave Technol.30(4), 521–531 (2012).
    [CrossRef]
  3. N. Bai, E. Ip, Y. K. Huang, E. Mateo, F. Yaman, M. J. Li, S. Bickham, S. Ten, J. Liñares, C. Montero, V. Moreno, X. Prieto, V. Tse, K. Man Chung, A. P. T. Lau, H. Y. Tam, C. Lu, Y. Luo, G. D. Peng, G. Li, and T. Wang, “Mode-division multiplexed transmission with inline few-mode fiber amplifier,” Opt. Express20(3), 2668–2680 (2012).
    [CrossRef] [PubMed]
  4. R. Olshansky, “Mode-coupling effects in graded-index optical fibers,” Appl. Opt.14(4), 935–945 (1975).
    [CrossRef] [PubMed]
  5. K.-P. Ho and J. M. Kahn, “Frequency diversity in mode-division multiplexing systems,” J. Lightwave Technol.29(24), 3719–3726 (2011).
    [CrossRef]
  6. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fiber with randomly varying birefringence,” J. Lightwave Technol.14(2), 148–157 (1996).
    [CrossRef]
  7. G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol.9(11), 1439–1456 (1991).
    [CrossRef]
  8. A. Bononi and A. Vannucci, “Statistics of the Jones matrix of fibers affected by polarization mode dispersion,” Opt. Lett.26(10), 675–677 (2001).
    [CrossRef] [PubMed]
  9. A. Vannucci and A. Bononi, “Statistical characterization of the Jones matrix of long fibers affected by polarization mode dispersion (PMD),” J. Lightwave Technol.20(5), 783–793 (2002).
    [CrossRef]
  10. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A.97(9), 4541–4550 (2000).
    [CrossRef] [PubMed]
  11. C. Antonelli, A. Mecozzi, M. Shtaif, and P. J. Winzer, “Stokes-space analysis of modal dispersion in fiber with multiple mode transmission,” Opt. Express20, 11718–11733 (2012).
  12. C. Antonelli, A. Mecozzi, and M. Shtaif, “Modeling of linear and nonlinear coupling in multiple-mode fiber optic transmission with MIMO signal processing,” Signals, Systems and Computers (Asilomar, 2012), 645–649.
  13. K.-P. Ho and J. M. Kahn, “Statistics of group delays in multimode fiber with strong mode coupling,” J. Lightwave Technol.29(21), 3119–3128 (2011).
    [CrossRef]
  14. A. Li, A. Al Amin, X. Chen, and W. Shieh, “Transmission of 107-Gb/s mode and polarization multiplexed CO-OFDM signal over a two-mode fiber,” Opt. Express19(9), 8808–8814 (2011).
    [CrossRef] [PubMed]
  15. A. Al Amin, A. Li, S. Chen, X. Chen, G. Gao, and W. Shieh, “Dual-LP11 mode 4×4 MIMO-OFDM transmission over a two-mode fiber,” Opt. Express19(17), 16672–16679 (2011).
    [CrossRef] [PubMed]
  16. M. Hsieh and C. Wei, “Channel estimation for OFDM systems based in comb-type pilot arrangement in frequency selective fading channels,” IEEE Trans. Consum. Electron.44(1), 217–225 (1998).
    [CrossRef]
  17. S. Weigert, “Baker-Campbell-Hausdorff relation for special unitary groups SU(N),” J. Phys. Math. Gen.30(24), 8739–8749 (1997).
    [CrossRef]
  18. M. Gell-Man, “Symmetries of Baryons and Mesons,” Phys. Rev.125(3), 1067–1084 (1962).
    [CrossRef]
  19. L. Arnold, Stochastic Differential Equations Theory and Application (Wiley 1974).
  20. T. C. Gard, Introduction to Stochastic Differential Equations (Marcel Dekker, INC. 1988).
  21. B. Øksendal, Stochastic Differential Equation (Springer-Verlag Berlin Heidelberg 1989).
  22. D. C. Cox and R. Leck, “Correlation bandwidth and delay spread multipath propagation statistics for 91-MHz urban mobile radio channels,” IEEE Trans. Commun.23(11), 1271–1280 (1975).
    [CrossRef]

2012 (3)

2011 (4)

2002 (1)

2001 (1)

2000 (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A.97(9), 4541–4550 (2000).
[CrossRef] [PubMed]

1998 (1)

M. Hsieh and C. Wei, “Channel estimation for OFDM systems based in comb-type pilot arrangement in frequency selective fading channels,” IEEE Trans. Consum. Electron.44(1), 217–225 (1998).
[CrossRef]

1997 (1)

S. Weigert, “Baker-Campbell-Hausdorff relation for special unitary groups SU(N),” J. Phys. Math. Gen.30(24), 8739–8749 (1997).
[CrossRef]

1996 (1)

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fiber with randomly varying birefringence,” J. Lightwave Technol.14(2), 148–157 (1996).
[CrossRef]

1991 (1)

G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol.9(11), 1439–1456 (1991).
[CrossRef]

1975 (2)

R. Olshansky, “Mode-coupling effects in graded-index optical fibers,” Appl. Opt.14(4), 935–945 (1975).
[CrossRef] [PubMed]

D. C. Cox and R. Leck, “Correlation bandwidth and delay spread multipath propagation statistics for 91-MHz urban mobile radio channels,” IEEE Trans. Commun.23(11), 1271–1280 (1975).
[CrossRef]

1962 (1)

M. Gell-Man, “Symmetries of Baryons and Mesons,” Phys. Rev.125(3), 1067–1084 (1962).
[CrossRef]

Al Amin, A.

Antonelli, C.

Bai, N.

Bickham, S.

Bolle, C.

Bononi, A.

Burrows, E. C.

Chen, S.

Chen, X.

Cox, D. C.

D. C. Cox and R. Leck, “Correlation bandwidth and delay spread multipath propagation statistics for 91-MHz urban mobile radio channels,” IEEE Trans. Commun.23(11), 1271–1280 (1975).
[CrossRef]

Esmaeelpour, M.

Essiambre, R.

Foschini, G. J.

G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol.9(11), 1439–1456 (1991).
[CrossRef]

Gao, G.

Gell-Man, M.

M. Gell-Man, “Symmetries of Baryons and Mesons,” Phys. Rev.125(3), 1067–1084 (1962).
[CrossRef]

Gnauck, A. H.

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A.97(9), 4541–4550 (2000).
[CrossRef] [PubMed]

Ho, K.-P.

Hsieh, M.

M. Hsieh and C. Wei, “Channel estimation for OFDM systems based in comb-type pilot arrangement in frequency selective fading channels,” IEEE Trans. Consum. Electron.44(1), 217–225 (1998).
[CrossRef]

Huang, Y. K.

Ip, E.

Kahn, J. M.

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A.97(9), 4541–4550 (2000).
[CrossRef] [PubMed]

Lau, A. P. T.

Leck, R.

D. C. Cox and R. Leck, “Correlation bandwidth and delay spread multipath propagation statistics for 91-MHz urban mobile radio channels,” IEEE Trans. Commun.23(11), 1271–1280 (1975).
[CrossRef]

Li, A.

Li, G.

Li, M. J.

Liñares, J.

Lingle, R.

Lu, C.

Luo, Y.

Man Chung, K.

Mateo, E.

McCurdy, A. H.

Mecozzi, A.

Menyuk, C. R.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fiber with randomly varying birefringence,” J. Lightwave Technol.14(2), 148–157 (1996).
[CrossRef]

Montero, C.

Moreno, V.

Mumtaz, S.

Olshansky, R.

Peckham, D. W.

Peng, G. D.

Poole, C. D.

G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol.9(11), 1439–1456 (1991).
[CrossRef]

Prieto, X.

Randel, S.

Ryf, R.

Shieh, W.

Shtaif, M.

Sierra, A.

Tam, H. Y.

Ten, S.

Tse, V.

Vannucci, A.

Wai, P. K. A.

P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation, and diffusion in optical fiber with randomly varying birefringence,” J. Lightwave Technol.14(2), 148–157 (1996).
[CrossRef]

Wang, T.

Wei, C.

M. Hsieh and C. Wei, “Channel estimation for OFDM systems based in comb-type pilot arrangement in frequency selective fading channels,” IEEE Trans. Consum. Electron.44(1), 217–225 (1998).
[CrossRef]

Weigert, S.

S. Weigert, “Baker-Campbell-Hausdorff relation for special unitary groups SU(N),” J. Phys. Math. Gen.30(24), 8739–8749 (1997).
[CrossRef]

Winzer, P. J.

Yaman, F.

Appl. Opt. (1)

IEEE Trans. Commun. (1)

D. C. Cox and R. Leck, “Correlation bandwidth and delay spread multipath propagation statistics for 91-MHz urban mobile radio channels,” IEEE Trans. Commun.23(11), 1271–1280 (1975).
[CrossRef]

IEEE Trans. Consum. Electron. (1)

M. Hsieh and C. Wei, “Channel estimation for OFDM systems based in comb-type pilot arrangement in frequency selective fading channels,” IEEE Trans. Consum. Electron.44(1), 217–225 (1998).
[CrossRef]

J. Lightwave Technol. (6)

J. Phys. Math. Gen. (1)

S. Weigert, “Baker-Campbell-Hausdorff relation for special unitary groups SU(N),” J. Phys. Math. Gen.30(24), 8739–8749 (1997).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. (1)

M. Gell-Man, “Symmetries of Baryons and Mesons,” Phys. Rev.125(3), 1067–1084 (1962).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A.97(9), 4541–4550 (2000).
[CrossRef] [PubMed]

Other (5)

L. Arnold, Stochastic Differential Equations Theory and Application (Wiley 1974).

T. C. Gard, Introduction to Stochastic Differential Equations (Marcel Dekker, INC. 1988).

B. Øksendal, Stochastic Differential Equation (Springer-Verlag Berlin Heidelberg 1989).

C. Antonelli, A. Mecozzi, and M. Shtaif, “Modeling of linear and nonlinear coupling in multiple-mode fiber optic transmission with MIMO signal processing,” Signals, Systems and Computers (Asilomar, 2012), 645–649.

A. Li, A. A. Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” OFC/NFOEC2011, PDPB8 (2011).

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Figures (1)

Fig. 1
Fig. 1

The comparison of the analytical and simulation result for the ACF of the FMF channel matrix with 2 to 4 spatial modes.

Equations (59)

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| ψ(z,ω) =U(z, z 0 ,ω)| ψ( z 0 ,ω) ,
Tr{ Λ i Λ j }=2 δ ij
U= i=0 4 N 2 1 u i Λ i = u ˜ Λ ˜
u i = 1 2 Tr{ Λ i U}.
Λ m Λ n = δ mn N I+ k ( i f mnk + d mnk ) Λ k
Λ n Λ m = δ mn N I+ k (i f mnk + d mnk ) Λ k
A × B = mnk f mnk A m B n e k = B × A ,
A B = mnk d mnk A m B n e k = B A .
( A Λ )( B Λ )( B Λ )( A Λ )=2i( A × B ) Λ ,
( A Λ )( B Λ )+( B Λ )( A Λ )= 2 N ( A B )I+2( A B ) Λ
( A Λ )( B Λ )= 1 N ( A B )I+( A B +i A × B ) Λ
dU(z,ω) dz = i 2 [ β (z,ω) Λ ]U(z,ω)
β (z,ω)=μω n (z),
d u 0 = iμω 2 N u d W d u = iμω 2 N [ u 0 d W + N ( u d W i u ×d W )]
d u ˜ (z)=Q( u ˜ (z))d W (z)
Q= iμω 2 N [ u Τ u 0 I+ N ( u i u ×) ]
q ˜ m Λ ˜ = iμω 2 Λ m ( u ˜ Λ ˜ ).
q x,y = 1 2 Tr{ Λ x1 ( q ˜ y Λ ˜ ) }.
d u ˜ = c ˜ ( u ˜ )dz+Q( u ˜ )d W
c l = 1 2 n=1 4 N 2 m=1 4 N 2 1 q n,m q l,m u n1 .
c ˜ Λ ˜ = 1 2 n=1 4 N 2 m=1 4 N 2 1 1 2 Tr{ Λ n1 ( q ˜ m Λ ˜ )} q ˜ m Λ ˜ u n1 ,
c ˜ Λ ˜ = μ 2 ω 2 8 [ m=1 4 N 2 1 1 N u m Λ m + n=1 4 N 2 1 m=1 4 N 2 1 1 2 Tr{ Λ n Λ m ( u ˜ Λ ˜ )} Λ m Λ n ].
c ˜ Λ ˜ = 4 N 2 1 8N μ 2 ω 2 ( u ˜ Λ ˜ ).
c ˜ = 4 N 2 1 8N μ 2 ω 2 u ˜ .
a b Q( u ˜ (z))d W (z) =lim δ0 i=0 n1 Q(λ u ˜ ( z i+1 )+(1λ) u ˜ ( z i )) Δ W i
iμω 2 a b [d W (z) Λ ]U(z) = iμω 2 lim δ0 i=0 n1 [Δ W i Λ ](λU( z i+1 )+(1λ)U( z i ))
iμω 4 [d W Λ ]dU
4 N 2 1 8N μ 2 ω 2 Udz
dU(z,ω) dz = 4 N 2 1 8N μ 2 ω 2 U+ iμω 2 [ n (z,ω) Λ ]U(z,ω)
d u 0 = (4 N 2 1) μ 2 ω 2 8N u 0 dz+ iμω 2 N u d W d u = (4 N 2 1) μ 2 ω 2 8N u dz+ iμω 2 N [ u 0 d W + N ( u d W i u ×d W )]
R(z, ω 1 , ω 2 )= U (z, ω 1 )U(z, ω 2 )
dR(z, ω 1 , ω 2 ) dz = d U (z, ω 1 ) dz U(z, ω 2 )+ U (z, ω 1 ) dU(z, ω 2 ) dz
dR(z, ω 1 , ω 2 ) dz = iμ( ω 2 ω 1 ) 2 [ U (z, ω 1 )( n (z) Λ )U(z, ω 1 )]R(z, ω 1 , ω 2 )
dR(z,Δω) dz = iμΔω 2 [ n '(z) Λ ]R(z,Δω)
d r ˜ = c ˜ ( r ˜ )dz+Q( r ˜ )d W
Q( r ˜ )= iμΔω 2 N [ r Τ r 0 I+ N ( r i r ×) ]
c ˜ ( r ˜ )= 4 N 2 1 8N μ 2 Δ ω 2 r ˜
p( r ˜ ,z) z = 1 2 n=1 4 N 2 m=1 4 N 2 2 O mn p( r ˜ ,z) r m1 r n1 k=1 4 N 2 c k p( r ˜ ,z) r k1
dE[f( r ˜ )] dz =E[Gf( r ˜ )]
G= 1 2 n=1 4 N 2 m=1 4 N 2 O mn 2 r m1 r n1 + k=1 4 N 2 c k r k1
dE[ r i (z)] dz = 4 N 2 1 8N μ 2 Δ ω 2 E[ r i (z)].
E[ r 0 ]= N exp( 4 N 2 1 8N μ 2 Δ ω 2 z), E[ r i ]=0(i0)
E[R]=exp( 4 N 2 1 8N μ 2 Δ ω 2 z)I
B U =2 8Nln0.5 (4 N 2 1) μ 2 z .
2 R ω 1 ω 2 = U ( ω 1 ) ω 1 U( ω 2 ) ω 2
Ω Λ = j2 U ω U / N
Tr{ 2 R ω 1 ω 2 | ω 1 = ω 2 }= N 4 Tr{ ( j2 U ω U / N )( j2U U ω / N ) }= N τ 2 2
E[ τ 2 ]= 2 N Tr{ 2 E[ R ] (Δω) 2 } | ω=0 = (4 N 2 1) μ 2 z N
Tr{E[R]}=2Nexp( E[ τ 2 ]Δ ω 2 8 )
V(ω)= C 1 T(ω) C 2
T(ω)=diag[ e jω t 1 , e jω t 2 ,..., e jω t 2N ]
Λ k = 2 k(k+1) [ 1 1 k 0 0 ]( k=1 to n1 ).
Λ pq =[ 1 1 ],
Λ pq =[ i i ],
d mnk = 1 4 Tr{ Λ k ( Λ m Λ n + Λ n Λ m )}
f mnk = i 4 Tr{ Λ k ( Λ m Λ n Λ n Λ m )}.
n=1 4 N 2 1 m=1 4 N 2 1 1 2 Tr{ Λ n Λ m ( u ˜ Λ ˜ )} Λ m Λ n = n=1 4 N 2 1 m=1 4 N 2 1 [ δ mn N u 0 + k=1 4 N 2 1 (i f mnk + d mnk ) u k ] [ δ mn N Λ 0 + l=1 4 N 2 1 (i f mnl + d mnl ) Λ l ] .
n=1 4 N 2 1 m=1 4 N 2 1 1 2 Tr{ Λ n Λ m ( u ˜ Λ ˜ )} Λ m Λ n = (4 N 2 1) N u 0 Λ 0 + kl mn ( f mnk f mnl + d mnk d mnl ) u l Λ k .
c ˜ Λ ˜ = 4 N 2 1 8N μ 2 ω 2 [ u 0 Λ 0 + k=1 4 N 2 1 u k Λ k ].

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