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Second-Order PMD-Compensation

Component for Second-Order Compensation of Polarization-Mode Dispersion

Jörn Patscher and Ralf Eckhardt
Technical University Hamburg-Harburg, 21071 Hamburg, Germany
_{April 1997}
The Compensation of polarization-mode dispersion in long fibers by using principal states of polarization is limited in bandwidth, when these states vary with frequency. We characterize a compensator, containing high birefringent fibers and polarization controllers, which is adapted to these variation and promises enlarged bandwidth.

This work is also published in Electronics Letters [5] and reproduced here by kind permission of the IEE. A Postscript file is available for downloading and printing. We do also provide more information about our group Optics and Instrumentation. See also our Publication List.

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Introduction

Polarization-mode dispersion (PMD) remains the dominant bitrate-limiting effect in long single mode (SM) fibers, when chromatic dispersion is reduced by state-of-the-art techniques like compensating fibers or chirped gratings. Recently published techniques for PMD compensation [1] are based on splitting the signal after transmission into two parts, which are aligned with either of the two principal states of polarization (PSP) of the fiber. Each part on its own does not suffer from PMD in first order approximation, but has undergone different group delays. The differential group delay (DGD) may be equalized as shown in Fig.1 by launching the fast signal part into the slow eigenstate of a polarization maintaining (PM) high birefringent fiber and vice versa.

This approach works well as long as the PSP of the compensator (i.e. the combination of polarization controller (PC) and PM fiber) matches to the PSP of the SM fiber, but residual DGD appears due to misalignment, if the PSP's vary with the signal wavelength. Taking e.g. the data from [2], the PSP variation in a 147km long cable would restrict the bandwidth, which could be covered simultaneously by this compensation method, to roughly 10-100 GHz. Considering now transcontinental cable lengths, wavelength multiplexing, and future multigigabitrates, PMD compensation will require close approximation of the PSP's of SM fiber and compensator PSP over a larger bandwidth, i.e. the 1st-order approach of treating the compensator PSP as constant must be extended to PSP's, which are linear or higher order function of frequency. Here, we theoretically and experimentally describe a compensator, which provides linear PSP variation.

Principle of Operation

The components of a PMD compensation setup (Fig.1) can be described compactly by polarization dispersion vectors [2], which point to the PSP's on the Poincaré sphere [3] and indicate the DGD by their lengths. Then it can be shown, that the residual DGD after compensation (

t) can be expressed as vectorial sum

$\Delta\tau=...$

(1)

where

_s and

_c are the PSP vectors of the SM fiber and the compensator at their connection (A). Both are generally functions of frequency w. Assuming good compensation at an arbitrarily chosen center frequency w₀, (i.e.

_s(w₀)

_c(w₀)), a taylor series expansion yields the residual DGD at frequency deviations

w=w-w₀ as

$\Delta\tau=...$

(2)

with

_s-(-

_c) and the symbols (') and ('') denoting the 1st and 2nd derivatives with respect to w, taken at w₀. A first-order compensation, as indicated in Fig.1, provides

(w₀)=

and leaves a finite 1st-order term

', which causes the DGD to grow linearly with

w. Further minimization of

t(w) can be achieved by additionally cancelling this 1st-order term (

), as is provided by an extended 2nd-order compensator (Fig.2). Here, a PSP variation rate

'_c is introduced, which is subject of the following analysis.

Fig.1: Single-Stage PMD Compensation The fast PSP of the long SM-fiber at (A) is aligned to the slow axis of the PM-fiber by the polarization controller (PC) and vice versa.

Fig.2: Experimental Setup LASER = Erbium-Ringlaser (1525-1570nm), P=Polarizer, PC1,PC2 = Polarization Controllers, PM#1, #2 = 2.00m and 1.15m length of PM fibers (Fujikura SM6/150, Bow-Tie), t₁=1.58 ps, t₂=0.91ps. PM#1 and PM#2 were temporarily arranged in reversed order. In a real system, the arrangement between (A) and (B) would replace the single PM-fiber in fig.1..

Theoretical Analysis

The PSP and the DGD of the core of the compensator, i.e. the components between (A) and (B) in Fig.2, are to be calculated. Each part of the setup can be characterized by a unitary Jones matrix J_i:

(3)

where i={1,2,PC2,c} indicates the PM fibers PM#1 and PM#2, PC2, and the combination of them, given by J_c=J₂×J_PC2×J₁. The transformation properties of PC1 and PC2 are assumed not to depend significantly on frequency, so, using eqn.3, the elements of J_PC2 can be noted generally as

(4)

where

denotes the angular distance on the Poincaré sphere between the slow eigenstate of the first PM fiber and the State of Polarisation (SOP), to which this eigenstate is transformed by PC2. As PC1 would contribute just a left-multiplication with a constant matrix, it is omitted here. The PM fibers are characterized by

(5)

where t_1,2 are the DGD, which were experimentally verified to have negligible variations within the used bandwidth, and

_1,2 are the mean differential phase delays. An eigenvalue analysis of J_c(w), which is given in detail e.g. in [4], now yields the total DGD (t_c) of the compensator,

(6)

and the slow input PSP (

_a+):

(7)

where

represents a common phase and amplitude. The angels

_a and

_a identify coordinates on the Poincaré sphere, as indicated in Fig.3b, and will be expressed explicitly in eqn.8 and eqn.9. As a result of the symmetrical structure of the setup between (A) and (B), the corresponding expressions for the output PSP (

_b+,

_b and

_b) at (B) can be obtained from eqn.7 by exchanging the terms u_1,2 against u_2,1, which are implicitly contained in u_c' and v_c'. A straightforward calculation yields:

(8)

and

(9)

These Eqn's. describe a circular variation of the compensator PSP on the Poincaré sphere (Fig.3b). The radius r_a,b=|sin2

_a,b|, the DGD (t_c), and also the PSP variation rate (|

'_c/t_c| = t₁×r_a) are fully determined by the same set of three parameters, (t₁, t₂ and

) and thus do not depend on frequency. Modification of the parameters

_i has the same effect as applying a frequency offset

w. The first PC, which has been ignored so far, provides for

(w₀)=

and

'(w₀)=

by transforming the input PSP of the compensator to any arbitrary position and by also controlling the variation direction

'_c(w₀)/t_c, while the variation rate |

'_c| is controlled by PC2.

Fig.3: Poincare representation
a) Sample trajectory of output SOP at (B) while frequency sweeping. Circles (o): measured SOP's in steps of 30 GHz (not all shown). Solid part of the line: Frequency interval 900 GHz.
b) PSP at (B), determined from SOP's. Angular coordinates according to eqn.7. 2r=diameter of PSP trajectory.

Experimental

Some assumptions about component properties were used in Eqn's (4) and (5), so for experimental verification of the Eqn's (8) and (9), a compensator was set up and characterized (Fig.2). The DGD and the PSP were analyzed by sweeping the input wavelength by

36 THz from 1525 to 1570 nm and monitoring the output SOP trajectory (Fig.3a) at various settings of PC1 and PC2. Local rotation axes of the SOP were identified for determination of the DGD (t_c) and the PSP (

_b, Fig.3b). The DGD was found to be independent of frequency within 2% measurement accuracy, while the PSP exhibited the predicted circular variation. The slope of the phase

_b of the circular variation (Fig.4a) agrees closely to eqn.8. The corresponding PSP variation at the input (A), was obtained by exchanging the two HiBi-fibers against each other. The slope of the measured circulation phase

_a meets eqn.8 within 3%. As

(w₀) was arbitrarily set to zero due to technical reasons, no information about

, or

(w₀) could be obtained. Finally (Fig.4b) the measured circulation radii r_a,b were compared to calculated values from eqn.9. The setting of

, which is used in eqn.9, was obtained from measured values of the total DGD (t_c) by reverse evaluation of eqn.6.

Fig.4: Experimental results
a) Phase of the PSP circulation at the output and at input. Solid lines: predicted from eqn.8, dots: measured.
b) Radii r_a,b of PSP circulation vs. total DGD t_c(). Dots: measured at port (B), squares: measured at (A), solid lines: predicted from eqn.9.

Conclusion

As experimentally verified, the above described compensator allows for PMD compensation within a frequency interval, where the PSP trajectory of the communication fiber can be approximated by a single arc on the Poincaré sphere. Compensation of variations of the DGD will require one further compensator stage. However, the effect of PSP variation is expected to dominate the latter [2], so, compared to single-stage (1st order) compensators [1], this setup promises substantial increase of bitrate and/or repeater distance in multigigabit systems.

Acknowledgement

The authors wish to thank R.Ulrich and E.Brinkmeyer for fruitful discussions during this investigation and H. Rosenfeldt for great technical support.

References

[1]	B.W.Hakki: "Polarization Mode Dispersion Compensation by Phase Diversity Detection", IEEE Photonics Technology Letters, 9, pp.121-123, 1997 ;

[2]	C.D.Poole, N.S.Bergano, R.E.Wagner, and H.J.Schulte: "Polarization Dispersion and Principal States in a 147-km Undersea Lightwave Cable", Journal of Lightwave Technology, LT-6, pp.1185-1191, 1988 ;

[3]	A.Simon and R.Ulrich : "Evolution of Polarization along a single-mode fiber", Applied Physics Letters, 31, pp.517-520, 1977 ;

[4]	C.D.Poole and R.E.Wagner: "Phenomenological approach to polarization dispersion in long single-mode fibers", Electronics Letters, 22, pp.1029-1030, 1986 ;

[5]	J.Patscher and R.Eckhardt: "Component for second-order compensation of polarisation-mode dispersion", Electronics Letters, 33, pp.1157-1159, 1997 ;

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