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11 :I 5am-I1:30am TuJ3 Jones transfer matrix for polarization mode dispersion fibers Alessandra Orlandini (1) and Luca Vincetti (2) (1) Dipartimento di Ingegneria dell’Informazione, (2) CNIT, Consorzio Nazionale Italian0 delle Telecomunicazioni Universiti degli studi di Parma, Parco Area delle Scienze 181/A, 43100 Parma Tel. +39-521-905750 Fax +39-521-905758 e-mail: orlandiQtlc.unipr.it With the advent of long distance high bit rate optical sys- tems, polarization mode dispersion (PMD) has become an im- portant source of limitation for the system performance. In a first order approximation, PMD, that is described by a dif- ferential group delay (DGD) between two orthogonal states of polarization (PSPs), causes an indesired output pulse broad- ening; the frequency dependence of DGD and PSPs produces other distorting effects, considered as higher order PMD ef- fects. A useful theoretical means of predicting the overall dis- tortion of the transmitted signal is the evaluation of the Jones transfer matrix of the fiber but, unfortunately, the statistics of its coefficients are not avaliable up to now. On the other hand, the statistical behavior of the three-dimensional disper- sion vector, that characterizes the PMD of the fiber in the Stokes space and can be measured, is known up to a second order PMD approximation [6]-[’7]. Consequently, finding the analytical relationship between the PMD vector and the co- efficients of the Jones matrix is mandatory. In literature a lot of analytical models have been developed up to second order PMD [l]-[5]. In particular, the model of Bruyire [l], that seemed to be simple and effective, has been recently cor- rected by Kogelnik et al. in [3] and by Penninckx et al. in [4]. The last one has proposed an approach to the problem different from Bruybre, giving a new expression for the Jones matrix; unfortunately, this model refers to the input principal states of polarization and it cannot be directly reconducted to the fiber output dispersion vector. In the present work, the right methodology of calculating the Jones matrix, start- ing from the knowledge of the PMD vector, is shown. This new method is used to determine the output temporal pulse expression in a second order PMD approximation and it is applied to evaluate the performance of a system affected by PMD. The results obtained with the present model and with the model proposed in [3] are compared to the performance evaluated by numerical simulations, where all order PMD ef- fects are taken into account; our model gives a performance curve that is more accurate in the approximation of all order PMD effects. In the Stokes space the disyersion vector at the ouput of the PMD fiber is described as R(w) = AT& where AT is the dif- ferential group delay and t^ represents the direction of the slow principal state of polarization. Besides, the Jones transfer ma- trix of the fiber (under the assumption of no polarization de- pendent losses) is T(w) = e-(a(y)L+ja(,)L)U(w), where a(w), &U) and L are the attenuation, the mean propagation con- stant and the length of the fiber respectively and U(w) is the unitary matrix To determinz the analytical relationship between the disper- sion vector R(w) and the Jones matrix T(w), we start from deriving with respect to w both members of d,t(w) = T(w)&ni,l (1) where_&,, is the input field, supposed coneant with frequency and E.,,(w) is the output field. Writing E,,*(w) =I E.,,(w) I e-J4(u)20yt(w), where *(U) is the phase and %.,*(U) the out- put state of polarization, we obtain the differential equation where (3) and X = v. The principal states of polarization result to be the eigenvectors of the eigenvalue problem j(Q + XI)8,,t = 0, correspondent to the eigenvalues A+ = &dm. Note that equation (2) refers to the output principal states of po- larization e^oul(w) becauseJhey are directly connected to the output dispersion vector R(w). On the contrary, in [4] they have solved an eigenvalue equation that, although similar to (2), refers to the input principal states of polarization. The output states of polarization and the input states of polar- ization are related by a frequency-dependent rotation matrix that is unknown. Passing from Jones space to Stokes space, it is possible to demonstrate that Rewriting the (3) as first order differentia?&tions - jQU and using (4), we obtain two whey R1 , R2,Rs are the three components of the Stokes vec- tor 0. Solving the (5) is not simple because RI,R~,R~ are complicated functions of w. Neverthless, u$ng a @st o_rder Taylor expansion_for the dispersion vector R(w) = Ro + Rhw, where Ro and 0; are the dispersion vector and its deriva- tive calculated at WO respectively, it is possible finding an analytical solution of the_(5) in a second order PMD ap- proximation. We define Rg = AT'^ + ATOP; where A70 = AT AT’ = ;~G;.I~~~~ and p’ = filpl = glw=uo, so that, if tly slow PSP is i = (-l,O,O) and I; = (0,-l,O), we have R(w) = (-As0 - AT’~,-AToI~~w,O). Furthermore dAr we make the assumptions 24w) = arctan (e) N Iplw and lG(w)l = ~(ATo + Ark)* + (A~0Iplw)2 N ATO + AT’w. Then, using t.he substitution w = u1 +jug and z = u1 - ju:, supposing that AT‘ = 0 and with the condition of output 0-7803-5947-xl00/$10.00@2000 IEEE 218

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11 :I 5am-I1:30am Tu J3

Jones transfer matrix for polarization mode dispersion fibers Alessandra Orlandini (1) and Luca Vincetti (2)

(1) Dipartimento di Ingegneria dell’Informazione, (2) CNIT, Consorzio Nazionale Italian0 delle Telecomunicazioni Universiti degli studi di Parma, Parco Area delle Scienze 181/A, 43100 Parma

Tel. +39-521-905750 Fax +39-521-905758 e-mail: orlandiQtlc.unipr.it

With the advent of long distance high bit rate optical sys- tems, polarization mode dispersion (PMD) has become an im- portant source of limitation for the system performance. In a first order approximation, PMD, that is described by a dif- ferential group delay (DGD) between two orthogonal states of polarization (PSPs), causes an indesired output pulse broad- ening; the frequency dependence of DGD and PSPs produces other distorting effects, considered as higher order PMD ef- fects. A useful theoretical means of predicting the overall dis- tortion of the transmitted signal is the evaluation of the Jones transfer matrix of the fiber but, unfortunately, the statistics of its coefficients are not avaliable up to now. On the other hand, the statistical behavior of the three-dimensional disper- sion vector, that characterizes the PMD of the fiber in the Stokes space and can be measured, is known up to a second order PMD approximation [6]-[’7]. Consequently, finding the analytical relationship between the PMD vector and the co- efficients of the Jones matrix is mandatory. In literature a lot of analytical models have been developed up to second order PMD [l]-[5]. In particular, the model of Bruyire [l], that seemed to be simple and effective, has been recently cor- rected by Kogelnik et al. in [3] and by Penninckx et al. in [4]. The last one has proposed an approach to the problem different from Bruybre, giving a new expression for the Jones matrix; unfortunately, this model refers to the input principal states of polarization and it cannot be directly reconducted to the fiber output dispersion vector. In the present work, the right methodology of calculating the Jones matrix, start- ing from the knowledge of the PMD vector, is shown. This new method is used to determine the output temporal pulse expression in a second order PMD approximation and it is applied to evaluate the performance of a system affected by PMD. The results obtained with the present model and with the model proposed in [3] are compared to the performance evaluated by numerical simulations, where all order PMD ef- fects are taken into account; our model gives a performance curve that is more accurate in the approximation of all order PMD effects. In the Stokes space the disyersion vector at the ouput of the PMD fiber is described as R(w) = AT& where AT is the dif- ferential group delay and t̂ represents the direction of the slow principal state of polarization. Besides, the Jones transfer ma- trix of the fiber (under the assumption of no polarization de- pendent losses) is T ( w ) = e-(a(y)L+ja(,)L)U(w), where a ( w ) , &U) and L are the attenuation, the mean propagation con- stant and the length of the fiber respectively and U(w) is the unitary matrix

To determinz the analytical relationship between the disper- sion vector R(w) and the Jones matrix T ( w ) , we start from

deriving with respect to w both members of

d , t ( w ) = T(w)&ni,l (1)

where_&,, is the input field, supposed coneant with frequency and E.,,(w) is the output field. Writing E,,*(w) =I E.,,(w) I e-J4(u)20yt(w), where * (U) is the phase and %.,*(U) the out- put state of polarization, we obtain the differential equation

where

(3)

and X = v. The principal states of polarization result to be the eigenvectors of the eigenvalue problem j ( Q + XI)8,,t = 0, correspondent to the eigenvalues A+ = & d m . Note that equation (2) refers to the output principal states of po- larization e^oul(w) becauseJhey are directly connected to the output dispersion vector R(w). On the contrary, in [4] they have solved an eigenvalue equation that, although similar to (2), refers to the input principal states of polarization. The output states of polarization and the input states of polar- ization are related by a frequency-dependent rotation matrix that is unknown. Passing from Jones space to Stokes space, it is possible to demonstrate that

Rewriting the (3) as first order differentia?&tions

- jQU and using (4), we obtain two

whey R1 , R2,Rs are the three components of the Stokes vec- tor 0. Solving the (5) is not simple because R I , R ~ , R ~ are complicated functions of w. Neverthless, u$ng a @st o_rder Taylor expansion_ for the dispersion vector R(w) = Ro + Rhw, where Ro and 0; are the dispersion vector and its deriva- tive calculated at W O respectively, it is possible finding an analytical solution of the_(5) in a second order PMD ap- proximation. We define Rg = AT'^ + ATOP; where A70 = AT AT’ = ; ~ G ; . I ~ ~ ~ ~ and p’ = filpl = glw=uo, so that, if t ly slow PSP is i = ( - l , O , O ) and I; = (0,-l,O), we have R(w) = (-As0 - A T ’ ~ , - A T o I ~ ~ w , O ) . Furthermore

dAr

we make the assumptions 2 4 w ) = arctan (e) N Iplw and lG(w)l = ~ ( A T o + A r k ) * + (A~0Iplw)2 N ATO + AT’w. Then, using t.he substitution w = u1 +jug and z = u1 - ju: , supposing that AT‘ = 0 and with the condition of output

0-7803-5947-xl00/$10.00@2000 IEEE 218

lo - ’ PSPs aligned to the input PSPs at w = W O , the solutions of the ( 5 ) become

= [ld sin($w) sin (+) + acos(J$w) cos (qw) + ._ h %3

( 6 ) 3 + jAro cos( $U) sin (:U)] f u ; ( w ) = cos(Fw)sin ( ~ w ) + asin(!$w)cos ( ~ w ) + a 9

al .z m =I c - s 0

1 o-2

(+ jaro sin( Tu) sin (&)I$ where a = d m . The dispersion vector correspon- dent to the (6) , calculated through the (4), results t.o be

The (7) represents a vector of constant magnitude that ro- tates on a circonference in the equatorial plane of the Poincark sphere with angular velocity lfl, The Taylor expansizn of (7) presents the two terms 0, = (-Aro,O,O) and 0; = (0, -1AAr0,O) that are equal to the ones of t,he exact s_olution, but, because of the assumptions made on+(w) and IR(w)l, it also presents terms of higher order, like 0: = (lp1’Aro,O,O). Therefore, the ( 6 ) are a good approximation of the second

order PMD exact solutions on the signal bandwidth if !%$! can be neglected with respect to I@l, that is when 9 << 1. The Taylor expansion of the output _dispersio_n vector deriv- ing from [4] presents the two terms Ro y d Ri equal to the o_nes found with our model but its t e e is different, being 0: = (Ip1’Aro, 0, Ifl AT,”). In fact, if R;,, is the input disper- sion vector and M the Miller matfix of the-fiber evaluated at w = W O , we can write RO = MR;,, and Ob = 131 but we can’t go further with the n-th derivatives. Besides, the model of Penninckx needs a more st.rict. condition %w << 1 for its validity, meaning that it fails not only for high values of lp1 but also for high values of A n . To have a more concrete idea of how the PMD distortion acts on the transmitted signal, it is worth finding the output pulse expression in the temporal domain. Consider an input signal &(t) = Ei,.(t)di,(t), where E;,(t) is the envelope and din is the input state of polarization: defining the vector v’= (1, -j), at the output of the fiber represented by the ( 6 ) the field is

- -- &(U) equal to the Fourier transform of E;,(t), A I , ~ = &[(Id + a)b T Ad*], -42,s = &[(-Id - a)b* f A~obl and b = + j e + . The output signal is a combination of four replicae of the input signal, differentially delayed of &a). Note that, in the (8) the angular velocity lp7 contributes to de- termine the delay of each replica together ATO. If = 0, the two replicae delayed of *+ are found, according to the first order PMD approximation. Our analytical model has been applied in the study of optical system performance. The cumulative probabilities on the sen- sitivity penalty at the receiver, calculated at BER = lo-’’ for a mean differential group delay of 20 ps, have been eval- uated with first order PMD approximation, with our model, the models of Kogelnik and BruyBre and with all order PMD numerical model. The cumulative probabilities are reported in Fig l., where each curve is the result of 15000 realizations.

I I 0.3 0.6 0.9 1.2

Sensitivity Penalty (dB)

Figure 1: Cumulative probabilities on sensitivity penalty at B E R = and mean DGD=20 ps

In the case of all order PMD effects the fiber is simulated as a concatenation of N = 100 PMFs with random PSPs uniformly distributed on the Poincark sphere and with a local DGD 67; uniformly distributed. Using the models up to second order, opportune Jones rotation matrices have been applied t,o U(w): they correspond to the rotation in the SJokes stace of the equat.orial plane into the one defined by 620 and 62b deriving from all order PMD simulations. Our new model gives an approximation of the performance curve with all order PMD that is better of those obtained using the models proposed in [l] and 131; observe also that the model of Bruybre oversti-

extensive analysis of the results obtained applying our model in the evaluation of the system performance will be shown at presentation time.

mates the second order effects according to [31 and [4]. A more

PI

PI

131

141

I51

161

I71

REFERENCES F. Bruyire, “Impact of first and second order PMD in optical digital transmission systems ”, Opt. Fiber Technol., v01.2, pp.

C. Francia et al., “PMD second order effects on pulse propa- gation in single-mode optical fibers”, IEEE Photon. Technol. Lett., vol. 10, no. 12, pp. 1739-1741, Dec. 1998. R. Kogelnik et al., “Jones matrix for second order polarization mode dispersion”, Optics Letters, vol. 25, no. 1, pp. 19-21, Jan. 2000. D. Penninekx and V. Morenas, “Jones matrix of polarization mode dispersion”, Optics Letters, vol. 24, no. 13, pp. 876-877, July 1999. A. Eyal et al., “Representation of second order polarization mode dispersion”, Electronics Letters, vol. 35, no. 19, pp. 1658- 1659, Sept. 1999. G. J. Foschini and C. D. Poole, “Statistical Theory of Polariza- tion Dispersion in Single-Mode Fibers”, J . Lightwave Technol., vol. 9, no. 11, pp. 1439-1456, Nov. 1991. G. 3. Foschini et al.,“The statistics of PMD-induced chromatic fiber dispersion”, J . Lightwave Technol., vol. 17, no. 9, pp. 1560- 1565, Sept. 1999.

269-280,1996.

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