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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 23 — Nov. 17, 2003
  • pp: 3141–3146
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Effect of local PMD and PDL directional correlation on the principal state of polarization vector autocorrelation

Liang Chen, Saeed Hadjifaradji, David S. Waddy, and Xiaoyi Bao  »View Author Affiliations


Optics Express, Vol. 11, Issue 23, pp. 3141-3146 (2003)
http://dx.doi.org/10.1364/OE.11.003141


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Abstract

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

Polarization mode dispersion (PMD) [1

1. J.P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97, 4541 (2000). [CrossRef] [PubMed]

] 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

2. N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119 (1997). [CrossRef]

]. 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

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). [CrossRef]

, 4

4. Y. Li and A. Yariv, “Solution to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,” J. Opt. Soc. Am. B 17, 1821 (2000). [CrossRef]

, 5

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).

]. However, those calculations either assume a zero correlation between local PMD and PDL polarization directional [3

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). [CrossRef]

] or strong correlation [4

4. Y. Li and A. Yariv, “Solution to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,” J. Opt. Soc. Am. B 17, 1821 (2000). [CrossRef]

, 5

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).

]. 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 β⃗j is followed by a PDL element α⃗j leading to the following transmission Jones matrix, Tj (ω)=exp[-iωβ⃗j ·σ⃗/2]exp[α⃗j ·σ⃗/2]. The output electric field E⃗out (ω) is connected with the input electric field E⃗in (ω) by:

Eout(ω)=ANeiφCD(ω)TN(ω)Ein(ω)
=ANeiφCD(ω)eiωβN·σ2eαN·σ2eiωβ1·σ2eα1·σ2Ein(ω)
(1)

where AN =exp[-α 0-j=1N αj ], and α 0 represents the polarization independent attenuation; φCD (ω) stands for the total chromatic dispersion (CD) of the system; β⃗j =βjβ^j represents j-th PMD waveplate having differential group delay (DGD) βj and the fast axis polarization is expressed by the unit vector β^j in the Stokes space; α⃗j =αjα^j stands for the j-th PDL waveplate with value expressed in decibel by 20|αj |log10 e and the maximum transmission polarization is denoted by the unit vector α^j in the Stokes space; and σ⃗ are the standard Pauli matrices. The complex principal state of polarization (PSP) vector W⃗N (ω) is accordingly defined by the following [2

2. N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119 (1997). [CrossRef]

]:

i2WN(ω)·σ=TN(ω)ωTN1(ω)
=i2βN·σ+eiωβN·σ2eαN·σ2(TN1ωTN11)eαN·σ2eiωβN·σ2
=i2βN·σ+eiωβN·σ2eαN·σ2(i2WN1·σ)eαN·σ2eiωβN·σ2
(2)

which leads to the following recursion relationship:

WN(ω)=β̂N{βN+(1cosβNω)[coshαN(β̂N·WN1(ω))
+isinhαN(β̂N·(α̂N×WN1(ω)))
(coshαN1)(α̂N·β̂N)(α̂N·WN1(ω))]}
+α̂N{isinhαNsinβNω(β̂N·WN1(ω))
(coshαN1)cosβNω(α̂N·ŴN1(ω))}
+ŴN1(ω){coshαNcosβNωisinhαNsinβNω(α̂N·β̂N)}
+(α̂N×β̂N){(coshαN1)sinβNω(α̂N·WN1(ω))}
+isinhαNcosβNω(α̂N×WN1(ω))+coshαNsinβNω(β̂N×WN1(ω)).
(3)

Writing the complex PSP vector explicitly in its real and imaginary part W⃗N (ω)=Ω⃗N(ω)+iΛ⃗N(ω), we would like to find its autocorrelation functions <Ω⃗N(ω)·Ω⃗N(ω′)>, <Ω⃗N(ω)·Λ⃗N(ω′)> and <Λ⃗N(ω)·Λ⃗N(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 <α^j·β^j> between the PMD unit vector β^j and the PDL unit vector α^j, could be neither zero [3

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). [CrossRef]

] nor ±1 [4

4. Y. Li and A. Yariv, “Solution to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,” J. Opt. Soc. Am. B 17, 1821 (2000). [CrossRef]

, 5

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).

]. 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 (e.g., vectors α^j and β^j) for given constant vectors A⃗ and B⃗:

n̂·An̂·B
(n̂·A)(n̂·B)=13(A·B).
(4)

Furthermore, considering the local correlation between α^j and β^j, we also have:

(α̂j·A)(β̂j·B)=13α̂j·β̂j(A·B)
(5)

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

6. M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. 24, 939 (1999). [CrossRef]

], namely we are going to assume in the waveplate model that each section has the same DGD, and PDL, i.e., βj =β and αj =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 2=〈η 2〉, 2=〈Δτ 2〉 constants. Using the above strategy, we can easily get the following:

WN(ω)·WN(ω)=β2+qWN1(ω)·WN1(ω)=β21qN1q
(6)

with q=[1+2cosβ(ω-ω′)]/3, and remembering 〈W⃗ 0(ωW⃗ 0(ω′)〉=0. Taking the continuum limit as described above, we have:

W(ω)·W(ω)=limNWN(ω)·WN(ω)=3(Δω)2[1eΔτ2(Δω)23]
(7)

where Δω=ω-ω′. Similarly we can find another complex correlation function:

WN(ω)·WN*(ω)=β2+qWN1(ω)·WN1*(ω)=β21pN1p
(8)

with

p=13[cosh2α(1+cosβΔω)+cosβΔω2iα̂·β̂sinh2αsinβΔω
+(α̂·β̂)2(cosh2α1)(cosβΔω1)].
(9)

Likewise, taking the continuum limit, we would find:

W(ω)·W*(ω)=limNWN(ω)·WN*(ω)=Δτ2gih[egih1]
(10)

with the abbreviation g=[4<η 2>-<Δτ 2>(Δω)2]/3 and h=4<α̂·β̂>Δτ2η2(Δω)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:

Ω(ω)·Ω(ω)=32(Δω)2[1eΔτ2(Δω)23]+Δτ22(g2+h2)[gegcoshg+hegsinh]
Λ(ω)·Λ(ω)=32(Δω)2[1eΔτ2(Δω)23]+Δτ22(g2+h2)[gegcoshg+hegsinh]
Λ(ω)·Ω(ω)=Δτ22(g2+h2)[hegcoshh+gegsinh]=Ω(ω)·Λ(ω).
(11)

These are the main results of this work. They agree with the published result [3

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). [CrossRef]

] when one takes <α^·β^>=0, and it also agrees with the result [5

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).

] 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 〈Δτ 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 〈Δτ 2〉 and PASPDL 〈η 2〉 can be calculated by the measurement of the autocorrelation at Δω=0:

{Ω(ω)·Ω(ω)=Δτ2[12+38η2(e4η231)]Λ(ω)·Λ(ω)=Δτ2[1238η2(e4η231)].
(12)

Lastly the parameter, 〈α^·β^〉, describing the local PMD and PDL directional correlation, can be linked to the derivative of the cross autocorrelation at Δω=0:

dΛ(ω+Δω)·Ω(ω)dΔω|Δω=0=38α̂·β̂[Δτ2η2]32[e4η23(143η2)1].
(13)

It is worthwhile to note that complex PSP vector can be directly measured via the equation of motion method [7

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).

], thus all the parameters introduced in this paper can be determined. Lastly the parameters PASDGD 〈Δτ 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.

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 1ω+ω 2 here.

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.
Fig. 3. The analytical autocorrelation functions are plotted for large PASPDL with finite PMD and PDL polarization directional correlation. Notice the oscillatory behavior.

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 Δω=ω 1-ω 2 for large PASPDL 〈η 2〉. This is illustrated by the case 〈Δτ 2〉=64.5 ps2, 〈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.

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). [CrossRef] [PubMed]

2.

N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. 142, 119 (1997). [CrossRef]

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). [CrossRef]

4.

Y. Li and A. Yariv, “Solution to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,” J. Opt. Soc. Am. B 17, 1821 (2000). [CrossRef]

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). [CrossRef]

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).

OCIS Codes
(060.2330) Fiber optics and optical communications : Fiber optics communications
(260.2030) Physical optics : Dispersion
(260.5430) Physical optics : Polarization

ToC Category:
Research Papers

History
Original Manuscript: August 26, 2003
Revised Manuscript: November 11, 2003
Published: November 17, 2003

Citation
Liang Chen, Saeed Hadjifaradji, David Waddy, and Xiaoyi Bao, "Effect of local PMD and PDL directional correlation on the principal state of polarization vector autocorrelation," Opt. Express 11, 3141-3146 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-23-3141


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References

  1. J.P. Gordon and H. Kogelnik, �??PMD fundamentals: Polarization mode dispersion in optical fibers,�?? Proc. Nat. Acad. Sci. 97, 4541 (2000). [CrossRef] [PubMed]
  2. N. Gisin and B. Huttner, �??Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,�?? Opt. Commun. 142, 119 (1997). [CrossRef]
  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). [CrossRef]
  4. Y. Li and A. Yariv, �??Solution to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,�?? J. Opt. Soc. Am. B 17, 1821 (2000). [CrossRef]
  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). [CrossRef]
  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).

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