Effect of local PMD and PDL directional correlation on the principal state of polarization vector autocorrelation

The effect of local PMD and PDL directional correlation is considered for the first time in a single mode fiber communication link. It is shown that the autocorrelation between the real and imaginary part of the complex principal state vector is nonzero in general. Experimental results verifying the local correlation between PMD and PDL directional are reported. © 2003 Optical Society of America OCIS codes: (260.2030) Dispersion; (260.5430) Polarization; (060.2330) Fiber optics communications. References and links 1. J.P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97, 4541 (2000). 2. N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119 (1997). 3. Ricardo Feced, Seb J. Savory and Anagnostis Hadjifotiou, “Interaction between polarization mode dispersion and polarization-dependent losses in optical communication links,” J. Opt. Soc. Am. B 20, 424 (2003). 4. Y. Li and A. Yariv, “Solution to the dynamical equation of polarization-mode dispersion and polarizationdependent losses,” J. Opt. Soc. Am. B 17, 1821 (2000). 5. Liang Chen, Saeed Hadjifaradji, David S. Waddy and Xiaoyi Bao, “Principal state vector autocorrelation in a fiber optic system having both polarization-mode dispersion and polarization dependent loss,” ICAPT’2003, SPIE proceeding (in press). 6. M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939 (1999). 7. Liang Chen, Ou Chen, Saeed Hadjifaradji and Xiaoyi Bao, “PMD Measurement method using the equation of motion for a system with PDL and PMD,” ICAPT’2003 SPIE Proceeding (in press). Polarization mode dispersion (PMD) [1] describes the group velocity dependence on the state of polarization, and it has been widely recognized as one of the major factors limiting high bit-rate communication systems. Polarization dependent loss (PDL) is a varying insertion loss arising from the dependence of a component’s transmission coefficient on the state of the polarization. Most passive optical components have some level of anisotropy with a transmission coefficient that is sensitive to the state of polarization. In a complex system with optical fibers, couplers, isolators, filters, multiplexers/demultiplexers, combiner/splitters, variable optical attenuators (VOAs), Erbium doped optical amplifiers (EDFA) and optical add/drop multiplexing switches (OADMS), the key question is: what is the combined effect of PMD and PDL? Gisin and Huttner were the first to consider such a question in their seminal work [2]. They showed that the combination of PMD and PDL will lead to a complex principal state vector. The autocorrelation function for such a complex principal state vector has been reported by several groups [3, 4, 5]. However, those calculations either assume a zero correlation between (C) 2003 OSA 17 November 2003 / Vol. 11, No. 23 / OPTICS EXPRESS 3141 #2917 $15.00 US Received August 26, 2003; Revised November 11, 2003 local PMD and PDL polarization directional [3] or strong correlation [4, 5]. The object of this work is to generalize the principal state vector autocorrelation considering arbitrary local PMD and PDL directional correlations. Experimental measurements are compared with our analytic results. We start our investigation by using the standard waveplate model. Because we are going to take the continuum limit at the end we consider the following simple arrangement: each PMD element β j is followed by a PDL element α j leading to the following transmission Jones matrix, Tj(ω) = exp[−iω β j · σ/2]exp[ α j · σ/2]. The output electric field Eout(ω) is connected with the input electric field Ein(ω) by: Eout(ω) = ANeCDTN(ω) Ein(ω) = ANeiφCD(ω)e−iω βN · σ/2e αN · σ/2 · · ·e−iω β1· σ/2e α1· σ/2 Ein(ω) (1) where AN = exp [ −α0 −∑j=1 α j ] , and α0 represents the polarization independent attenuation; φCD(ω) stands for the total chromatic dispersion (CD) of the system; β j = β jβ̂ j represents j-th PMD waveplate having differential group delay (DGD) β j and the fast axis polarization is expressed by the unit vector β̂ j in the Stokes space; α j = α jα̂ j stands for the j-th PDL waveplate with value expressed in decibel by 20|α j| log10 e and the maximum transmission polarization is denoted by the unit vector α̂ j in the Stokes space; and σ are the standard Pauli matrices. The complex principal state of polarization (PSP) vector WN(ω) is accordingly defined by the following [2]: − i 2 WN (ω) · σ= ∂TN(ω) ∂ω T−1 N (ω) = − i 2 βN · σ + e−iω βN · σ/2e αN · σ/2 ( ∂TN−1 ∂ω T −1 N−1 ) e− αN · σ/2eiω βN · σ/2 = − i 2 βN · σ + e−iω βN · σ/2e αN · σ/2 ( − i 2 WN−1 · σ ) e− αN · σ/2eiω βN · σ/2 (2) which leads to the following recursion relationship: WN (ω) = β̂N { βN +(1− cosβNω) [ coshαN ( β̂N · WN−1 (ω) ) +isinhαN ( β̂N · ( α̂N × WN−1 (ω) )) − (coshαN −1) ( α̂N · β̂N )( α̂N · WN−1 (ω) )]} +α̂N { isinhαN sinβNω ( β̂N · WN−1 (ω) ) − (coshαN −1)cosβNω ( α̂N · WN−1 (ω) )} + WN−1 (ω) { coshαN cosβNω− isinhαN sinβNω ( α̂N · β̂N )} + ( α̂N × β̂N ){ (coshαN −1)sinβNω ( α̂N · WN−1 (ω) )} +isinhαN cosβNω ( α̂N× WN−1 (ω) ) +coshαN sinβNω ( β̂N× WN−1 (ω) ) . (3) Writing the complex PSP vector explicitly in its real and imaginary part WN(ω) = ΩN(ω)+ i ΛN(ω), we would like to find its autocorrelation functions < ΩN(ω) · ΩN(ω′)>, < ΩN(ω) · ΛN(ω′)> and < ΛN(ω) · ΛN(ω′)>. Where 〈· · ·〉 means the average over statistical fluctuations of the PMD and PDL element. However, before we go any further we would like to mention the main object of this work, namely the local correlation <α̂ j · β̂ j> between the PMD unit vector β̂ j and the PDL unit vector α̂ j, could be neither zero [3] nor ±1 [4, 5]. In what follows we will assume a general local correlation between the PMD and PDL polarization directions. Now, it (C) 2003 OSA 17 November 2003 / Vol. 11, No. 23 / OPTICS EXPRESS 3142 #2917 $15.00 US Received August 26, 2003; Revised November 11, 2003 is convenient to recall some mathematical properties for averaging over a random unit vector n̂ (e.g., vectors α̂ j and β̂ j) for given constant vectors A and B: 〈 n̂ · A 〉 ≡ 〈 n̂ · B 〉 〈( n̂ · A )( n̂ · B )〉 = 3 ( A · B ) . (4) Furthermore, considering the local correlation between α̂ j and β̂ j, we also have: 〈( α̂ j · A )( β̂ j · B )〉 = 1 3 〈 α̂ j · β̂ j 〉( A · B ) (5) where we have used the generic expression <α̂ j · β̂ j> to represent the local PMD and PDL directional correlation. To go to the continuum limit, we adopt the strategy first introduced by Karlsson and Brentel [6], namely we are going to assume in the waveplate model that each section has the same DGD, and PDL, i.e., β j ≡ β and α j ≡ α , except treating their directions α̂ j and β̂ j as random unit vectors. The continuum limit is taken in such a way that N → ∞, α → 0, β → 0 while keeping Nα 2 = 〈 η 2 〉 , Nβ2 = 〈 ∆τ 2 〉 constants. Using the above strategy, we can easily get the following: 〈 WN (ω) · WN ( ω′ )〉 = β +q 〈 WN−1 (ω) · WN−1 ( ω′ )〉 = β 1−qN 1−q (6) with q = [1+2cosβ (ω−ω′)]/3, and remembering 〈 W0 (ω) · W0 (ω′) 〉 = 0. Taking the continuum limit as described above, we have: 〈 W (ω) · W (ω′)〉 = lim N→∞ 〈 WN (ω) · WN ( ω′ )〉 = 3 (∆ω) [ 1− e−〈∆τ 2〉(∆ω)2/3 ] (7) where ∆ω = ω−ω′. Similarly we can find another complex correlation function: 〈 WN (ω) · W ∗ N ( ω′ )〉 = β +q 〈 WN−1 (ω) · W ∗ N−1 ( ω′ )〉 = β 1− pN 1− p (8) with p = 3 [ cosh2α (1+ cosβ∆ω)+ cosβ∆ω−2i 〈 α̂ · β̂ 〉 sinh2α sinβ∆ω + 〈( α̂ · β̂ )2〉 (cosh2α −1)(cosβ∆ω−1) ]

Polarization mode dispersion (PMD) [1] describes the group velocity dependence on the state of polarization, and it has been widely recognized as one of the major factors limiting high bit-rate communication systems.Polarization dependent loss (PDL) is a varying insertion loss arising from the dependence of a component's transmission coefficient on the state of the polarization.Most passive optical components have some level of anisotropy with a transmission coefficient that is sensitive to the state of polarization.In a complex system with optical fibers, couplers, isolators, filters, multiplexers/demultiplexers, combiner/splitters, variable optical attenuators (VOAs), Erbium doped optical amplifiers (EDFA) and optical add/drop multiplexing switches (OADMS), the key question is: what is the combined effect of PMD and PDL? Gisin and Huttner were the first to consider such a question in their seminal work [2].They showed that the combination of PMD and PDL will lead to a complex principal state vector.The autocorrelation function for such a complex principal state vector has been reported by several groups [3,4,5].However, those calculations either assume a zero correlation between local PMD and PDL polarization directional [3] or strong correlation [4,5].The object of this work is to generalize the principal state vector autocorrelation considering arbitrary local PMD and PDL directional correlations.Experimental measurements are compared with our analytic results.
We start our investigation by using the standard waveplate model.Because we are going to take the continuum limit at the end we consider the following simple arrangement: each PMD element β j is followed by a PDL element α j leading to the following transmission Jones matrix, The output electric field E out (ω) is connected with the input electric field E in (ω) by: where A N = exp −α 0 − ∑ N j=1 α j , and α 0 represents the polarization independent attenuation; ϕ CD (ω) stands for the total chromatic dispersion (CD) of the system; β j = β j β j represents j-th PMD waveplate having differential group delay (DGD) β j and the fast axis polarization is expressed by the unit vector β j in the Stokes space; α j = α j α j stands for the j-th PDL waveplate with value expressed in decibel by 20|α j | log 10 e and the maximum transmission polarization is denoted by the unit vector α j in the Stokes space; and σ are the standard Pauli matrices.The complex principal state of polarization (PSP) vector W N (ω) is accordingly defined by the following [2]: (2) which leads to the following recursion relationship: Writing the complex PSP vector explicitly in its real and imaginary part W N (ω) = Ω N (ω) + i Λ N (ω), we would like to find its autocorrelation functions Where ••• means the average over statistical fluctuations of the PMD and PDL element.However, before we go any further we would like to mention the main object of this work, namely the local correlation < α j • β j > between the PMD unit vector β j and the PDL unit vector α j , could be neither zero [3] nor ±1 [4,5].In what follows we will assume a general local correlation between the PMD and PDL polarization directions.Now, it is convenient to recall some mathematical properties for averaging over a random unit vector n (e.g., vectors α j and β j ) for given constant vectors A and B: Furthermore, considering the local correlation between α j and β j , we also have: where we have used the generic expression < α j • β j > to represent the local PMD and PDL directional correlation.To go to the continuum limit, we adopt the strategy first introduced by Karlsson and Brentel [6], namely we are going to assume in the waveplate model that each section has the same DGD, and PDL, i.e., β j ≡ β and α j ≡ α, except treating their directions α j and β j as random unit vectors.The continuum limit is taken in such a way that N → ∞, α → 0, β → 0 while keeping Nα 2 = η 2 , Nβ 2 = ∆τ 2 constants.Using the above strategy, we can easily get the following: , and remembering W 0 (ω) • W 0 (ω ) = 0. Taking the continuum limit as described above, we have: where ∆ω = ω − ω .Similarly we can find another complex correlation function: Likewise, taking the continuum limit, we would find: with the abbreviation g=[4<η 2 >−<∆τ 2 >(∆ω) 2 ]/3 and h=4< α • β> ∆τ 2 η 2 (∆ω) /3.Substituting the complex PSP vector by their real and imaginary parts in the above two complex autocorrelations Eq. ( 7) and Eq.(10), we could get the following: These are the main results of this work.They agree with the published result [3] when one takes < α • β >= 0, and it also agrees with the result [5] when one takes < α • β >= 1.It is interesting to note the fact that PMD and PDL polarization directional correlation is nonzero and can be linked with the finite values of the cross autocorrelation between the real and imaginary parts of the PSP vector.We now discuss a few interesting aspect of our results.First, because of the PMD and PDL interaction the parameter ∆τ 2 is not equal to the measured average square DGD, rather it corresponds to the pristine average square DGD (PASDGD).Neither is η 2 proportional to the measured average square PDL in dB unit due to the mutual interaction of PMD and PDL .Again, we call η 2 the pristine average squared PDL (PASPDL).However, PASDGD ∆τ 2 and PASPDL η 2 can be calculated by the measurement of the autocorrelation at ∆ω = 0: Lastly the parameter, α • β , describing the local PMD and PDL directional correlation, can be linked to the derivative of the cross autocorrelation at ∆ω = 0: It is worthwhile to note that complex PSP vector can be directly measured via the equation of motion method [7], thus all the parameters introduced in this paper can be determined.Lastly the parameters PASDGD ∆τ 2 , PASPDL η 2 and < α • β> are only system parameters needed to exactly evaluate the eye diagram for given pulse sequence in a highly mode coupled fiber optic link.We will publish this result in a future work.
In order to check if the local PMD and PDL polarization directional correlation < α • β > is relevant we have performed an experiment on a 52km field fiber with both buried and aerial segments.Two measurements (10 days apart) were performed by launching three states of polarization and scanning (3GHz steps) across optical frequencies from 186.2 to 196.6THz (1526 to 1611nm).The fiber was observed to have an average DGD of 1 ps. Figure 1 shows the experimental autocorrelation functions and their corresponding analytical fits for two different instants: one was taken during the daytime ((a), (b), and (c)), and the other during the nighttime ((d), (e), and (f)).Figure 1(a) plots the autocorrelation between the real parts of the PSP vector, Fig. 1(b) plots the autocorrelation between the imaginary parts of the PSP vector, and Fig. 1(c) plots the autocorrelation between the real and the imaginary parts of the PSP vector.While Figs. 1(d), (e), and (f) show the same autocorrelations for another instant.Note the choice of the fitting parameter < α • β >= 1 and < α • β >= 0.15 shown in the inset of Fig. 1(c); clearly < α • β >= 1 does not fit.Thus, one has to consider finite PMD and PDL directional correlation in field fiber.Overall the agreement between the experiment and the analytic result is satisfactory.However, notice in Fig. 1(c) that the experimental value is not zero at ∆ω = 0, this might be related to several effects like the experimental accuracy as well as the fact that the whole link might be not treated as a continuum highly coupled limit.The fitting parameters chosen for ∆τ 2 = 1.31ps 2 and η 2 = 0.073 (equivalent PDL 2.2dB) are obtained from solving Eq. ( 12), while the value < α • β >= 0.15 was chosen based on eying the overall fit.The positive value of the fitting parameter < α • β >= 0.15 demonstrate that the fast PMD vector lies more closely in the direction of maximum transmission polarization.Now it is interesting to note the same comparison of the fiber at different instants (here is the nighttime data) as plotted in the Figs.1(d), 1(e), and 1(f).However, it is found that < α • β >= −0.32 seems to fit the experiment the best.This means that the fast PMD vector lies more closely in the direction of minimum transmission polarization.The other fitting parameter ∆τ 2 = 1.26ps 2 and η 2 = 0.011 (equivalent PDL 0.84 dB) were obtained similarly by solving Eq. ( 12).This time one notices more obvious oscillatory behaviors in the Fig. 1(e) for the imaginary part of the PSP autocorrelation despite their overall small magnitude.Again, this could be linked to the finite number of optical components in the fiber span, signaling the deviation from our analytic result which is only valid in the continuum limit.It is also interesting to notice the significant variations in the fitting parameter PASPDL η 2 between the two instants of the experimental measurement, while the parameter PASDGD ∆τ 2 seems to be relatively constant between those two instants.This indicates the importance of taking PDL fluctuations into consideration in real field fiber links.
Because of the relatively small PDL values observed in the field fiber link we have constructed a PMD and PDL emulator made of 14 sections of polarization maintaining fiber spliced together (PDL is contributed by the splicing).Figure 2 plots the three experimental autocorrelation functions and their analytical fits.Clearly the overall agreement between theory and experiment are much better than those shown in Fig. 1.Note the fitting parameter PASPDL η 2 = 2 (equivalent PDL 11.3 dB) is relatively large as well the fitting parameter PASDGD ∆τ 2 = 46.4ps 2 is bigger than those in the Fig. 1.From this result one can conclude that the emulator constructed seems to be closer to the continuum limit that is required by the analytical result presented here.Figure 3 illustrates the last interesting consequence of the finite PMD and PDL directional correlation (i.e., < α • β > = 0).Analytically it is clear that all three autocorrelation functions show oscillatory behavior as a function of ∆ω = ω 1 − ω 2 for large PASPDL η 2 .This is illustrated by the case ∆τ 2 = 64.5 ps 2 , η 2 = 10 and < α • β >= 1.
In conclusion, we have derived analytical autocorrelation functions between complex PSP vectors considering the finite local PMD and PDL directional correlation.Analytical results are compared with a field fiber span and an emulator.Both experimental results illustrate the importance of considering the effect of PMD and PDL directional correlation.
We wish to acknowledge the financial support from Canadian funding agency: NSERC and the Canadian networks of excellence: CIPI, AAPN.

Fig. 1 .
Fig. 1.The autocorrelation functions for a buried field fiber of 52km (thick lines) are compared with the analytical fitting curves (thin lines).(a), (b) and (c) are correspond to one instant, and (d), (e) and (f) to another instant.The overall agreement demonstrates the necessity of finite PMD and PDL directional correlation.Note: ω 1 = ∆ω + ω 2 here.

Fig. 2 .Fig. 3 .
Fig.2.The autocorrelation functions (lines with dots) of an emulator are compared with analytical fitting curves(solid lines).The overall agreement is very good.