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

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

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]

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]

*β⃗*

_{j}is followed by a PDL element

*α⃗*

_{j}leading to the following transmission Jones matrix,

*T*

_{j}(

*ω*)=exp[-

*iωβ⃗*

_{j}·

*σ⃗*/2]exp[

*α⃗*

_{j}·

*σ⃗*/2]. The output electric field

*E⃗*

_{out}(

*ω*) is connected with the input electric field

*E⃗*

_{in}(

*ω*) by:

*A*

_{N}=exp[-

*α*

_{0}-

*α*

_{j}], and

*α*

_{0}represents the polarization independent attenuation;

*φ*

_{CD}(

*ω*) stands for the total chromatic dispersion (CD) of the system;

*β⃗*

_{j}=

*β*β ^ represents

_{j}_{j}*j*-th PMD waveplate having differential group delay (DGD)

*β*

_{j}and the fast axis polarization is expressed by the unit vector

*in the Stokes space;*β ^

_{j}*α⃗*

_{j}=

*α*α ^ stands for the

_{j}_{j}*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

*in the Stokes space; and*α ^

_{j}*σ⃗*are the standard Pauli matrices. The complex principal state of polarization (PSP) vector

*W⃗*

_{N}(

*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]

*W⃗*

_{N}(

*ω*)=Ω⃗

_{N}(

*ω*)+

*i*Λ⃗

_{N}(

*ω*), we would like to find its autocorrelation functions <Ω⃗

_{N}(

*ω*)·Ω⃗

_{N}(

*ω*′)>, <Ω⃗

_{N}(

*ω*)·Λ⃗

_{N}(

*ω*′)> and <Λ⃗

_{N}(

*ω*)·Λ⃗

_{N}(

*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}*> between the PMD unit vector*β ^

_{j}*and the PDL unit vector*β ^

_{j}*, could be neither zero [3*α ^

_{j}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]

*n̂*(e.g., vectors

*and*α ^

_{j}*) for given constant vectors*β ^

_{j}*A⃗*and

*B⃗*:

*and*α ^

_{j}*, we also have:*β ^

_{j}*·*α ^

_{j}*> to represent the local PMD and PDL directional correlation.*β ^

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

*β*

_{j}=

*β*and

*α*

_{j}=

*α*, except treating their directions

*and*α ^

_{j}*as random unit vectors. The continuum limit is taken in such a way that*β ^

_{j}*N*→∞,

*α*→0,

*β*→0 while keeping

*Nα*

^{2}=〈

*η*

^{2}〉,

*Nβ*

^{2}=〈Δ

*τ*

^{2}〉 constants. Using the above strategy, we can easily get the following:

*q*=[1+2cos

*β*(

*ω*-

*ω*′)]/3, and remembering 〈

*W⃗*

_{0}(

*ω*)·

*W⃗*

_{0}(

*ω*′)〉=0. Taking the continuum limit as described above, we have:

*ω*=

*ω*-

*ω*′. Similarly we can find another complex correlation function:

*g*=[4<

*η*

^{2}>-<Δ

*τ*

^{2}>(Δ

*ω*)

^{2}]/3 and

**20**, 424 (2003). [CrossRef]

*·*α ^

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

*·*α ^

*>=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.*β ^

*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 〈Δ

*τ*

^{2}〉 and PASPDL 〈

*η*

^{2}〉 can be calculated by the measurement of the autocorrelation at Δ

*ω*=0:

*·*α ^

*〉, describing the local PMD and PDL directional correlation, can be linked to the derivative of the cross autocorrelation at Δ*β ^

*ω*=0:

*τ*

^{2}〉, PASPDL 〈

*η*

^{2}〉 and <

*·*α ^

*> 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.*β ^

*η*

^{2}〉=2 (equivalent PDL 11.3 dB) is relatively large as well the fitting parameter PASDGD 〈Δ

*τ*

^{2}〉=46.4 ps

^{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.

*·*α ^

*>≠0). Analytically it is clear that all three autocorrelation functions show oscillatory behavior as a function of Δ*β ^

*ω*=

*ω*

_{1}-

*ω*

_{2}for large PASPDL 〈

*η*

^{2}〉. This is illustrated by the case 〈Δ

*τ*

^{2}〉=64.5 ps

^{2}, 〈

*η*

^{2}〉=10 and <

*·*α ^

*>=1.*β ^

## References and links

1. | J.P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. |

2. | N. Gisin and B. Huttner, “Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,” Opt. Commun. |

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 |

4. | Y. Li and A. Yariv, “Solution to the dynamical equation of polarization-mode dispersion and polarization-dependent losses,” J. Opt. Soc. Am. B |

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. |

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

- J.P. Gordon and H. Kogelnik, �??PMD fundamentals: Polarization mode dispersion in optical fibers,�?? Proc. Nat. Acad. Sci. 97, 4541 (2000). [CrossRef] [PubMed]
- N. Gisin and B. Huttner, �??Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers,�?? Opt. Commun. 142, 119 (1997). [CrossRef]
- 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]
- 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]
- 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).
- M. Karlsson and J. Brentel, �??Autocorrelation function of the polarization-mode dispersion vector,�?? Opt. Lett. 24, 939 (1999). [CrossRef]
- 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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