1. Introduction

Polarization-mode dispersion (PMD) phenomena in lightwave systems are the collective result of birefringence in the transmission path. They lead to pulse distortion and system penalties, particularly for high bit-rate transmission. The PMD of a fiber is commonly characterized [1,2] by two specific orthogonal states of polarization called the principal states of polarization (PSP’s) and by the differential group delay (DGD) between them. These entities can be encapsulated in the three-component Stokes vector, $\overrightarrow{\tau }$ =Δτ $\overrightarrow{p}$ , where $\overrightarrow{p}$ is a unit Stokes vector pointing in the direction of the slower PSP, and the magnitude, Δτ, is the DGD. The vector $\overrightarrow{\tau }$ (ω) usually varies with optical angular frequency, ω. Typical DGD values encountered in transmission systems range between 1 ps and 100 ps. Note that the $\overrightarrow{\tau }$ (ω) vector that we use to characterize PMD here is defined in a right-circular Stokes space where right circular polarization corresponds to Stokes parameter S ₃=+1. The vector $\overrightarrow{\tau }$ (ω) is closely related to Poole’s $\overrightarrow{\Omega }$ (ω) vector [1], which is defined for left-circular Stokes space (see Appendix).

Measurement methods that determine PMD vectors with sufficient frequency resolution and signal-to-noise ratio (SNR) include Jones Matrix Eigenanalysis (JME) [3] and the Müller Matrix Method (MMM) [4,5]. Both of these frequency-domain techniques require differentiation with respect to frequency of measured matrix data. Note that PMD vector information is required for the determination of second- and higher-order PMD. When only DGD information is needed, other methods can be employed such as the technique using sinusoidal amplitude modulation described by Williams [6]. Here we discuss a time-domain method for measuring PMD vectors, the Polarization-Dependent Signal Delay Method (PSD Method) [7]. After describing the method, we present experimental results and compare them to data obtained by more traditional methods, and finally discuss some advantages offered by the method.

2. Theory

Figure 1 shows an example of the delay imposed on a pulse as it traverses a system. The green curve in Fig. 1 shows results similar to those that would be obtained when applying a delay measurement to a fiber span having +124 ps/nm of dispersion and no PMD. By taking the derivative of the measured delay with respect to wavelength, the chromatic dispersion can be determined. Any PMD present in the fiber can increase or decrease the delay by as much as half the DGD. The red curves in Fig. 1 show examples of the maximal and minimal delays for a fiber span having a dispersion of +124 ps/nm and a mean PMD of 35 ps. The upper curve is the delay that would be observed when launching a pulse with polarization aligned with $\overrightarrow{\tau }$ (ω) at each wavelength, while the lower red curve would be observed when launching a pulse aligned with -$\overrightarrow{\tau }$ (ω) at each wavelength. The delay measured for a fixed launch polarization will occupy the region bounded by the red curves. The markers in Fig. 1 show the delay measured for the specific launch polarization, Ŝ1, an axis of the Poincaré sphere in Stokes space. It can be seen that at some wavelengths, the delay for this launch is nearly equal to that of the fast PSP, while at other wavelengths, it is nearly equal to that of the slow PSP. Usually a launched polarization is a combination of the two PSP’s, so the delay lies somewhere between the two extrema. In fact, it can be shown that the delay is a power-weighted average of the two extreme delays, where the weights are the fractions of power launched into the two PSP’s. This is one of the arguments leading to Eq. (1) below.

2.1 Moments

The PSD method uses the polarization dependence of the pulse propagation time to determine the PMD vector. To extract PMD vector information from measurements such as those shown by the blue markers in Fig. 1, the PSD method uses a result of Mollenauer, et al. [8], relating $\overrightarrow{\tau }$ , the launch polarization, and the mean signal delay,

Here $\overrightarrow{s}$ and $\overrightarrow{\tau }$ are, respectively, the Stokes vectors of the light and the PMD vector at the fiber input. The mean signal delay, τ _g , is defined by the first moment of the pulse envelope in the time domain [8–10], while τ _o is the polarization-independent delay component. More precisely, τ _g is expressed as the difference between the normalized first moments at the fiber output and input,

where z is the distance of propagation in the fiber. Here W=∫dt $\overrightarrow{E}$ ^† $\overrightarrow{E}$ =∫dωẼ ^† Ẽ is the energy of the signal pulse represented by the complex field vector $\overrightarrow{E}$ (z,t) with Fourier transform Ẽ(z,ω), and W ₁(z)=∫dtt $\overrightarrow{E}$ ^† $\overrightarrow{\tau }$ =j∫dωẼ ^† Ẽ _ω is the first moment. Eq. (1) assumes that τ _o and $\overrightarrow{\tau }$ (ω) do not vary significantly over the bandwidth of the signal. If this approximation is not valid, Eq. (1) can be generalized by expressing the right-hand side in an integral over ω [9,11].

 

Fig. 1. Mean signal delay of a fiber having a dispersion of +124 ps/nm. The green curve shows the delay that would be obtained if the fiber had no PMD. The red curves show the green curve plus and minus half the differential group delay. The markers show the measured delay for a specific launch polarization state.

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Mean signal delay measurements at four appropriately chosen input polarizations allow the determination of τ _o and the vector $\overrightarrow{\tau }$ . For example, consider the case of the four launch polarizations coinciding with the Poincaré sphere axes, Ŝ ₁, -Ŝ ₁, Ŝ ₂, and Ŝ ₃, with corresponding mean signal delay measurements τ_g1, τ_g(-1), τ_g2, and τ_g3. Then τ _o and the vector components of $\overrightarrow{\tau }$ follow from Eq. (1):

where i=1,2,3. Eq. (3) can be generalized to any four non-degenerate input polarization launches $\overrightarrow{s}$ _i (i=1,2,3,4) that span Stokes space. The corresponding signal delay measurements, τ_gi , can be grouped into a four-dimensional (4-D) vector $\overrightarrow{\tau }$ _g =(τ_g1,τ_g2,τ_g3,τ_g4), and similarly, the desired delay components can be grouped into a 4-D vector $\overrightarrow{\tau }$ _4D=(τ₀,τ₁,τ₂,τ₃). The four launches together with Eq. (1) generate the required four linear equations for the four unknowns represented by $\overrightarrow{\tau }$ _4D. Thus, $\overrightarrow{\tau }$ _g =X$\overrightarrow{\tau }$ _4D, where X is a 4×4 matrix containing the components of the launched polarizations,

Here s_ij is the jth component of polarization launch $\overrightarrow{s}$ _i . The equivalent of Eq. (3) is then $\overrightarrow{\tau }$ _4D=X ^-1$\overrightarrow{\tau }$ _g , and therefore τ _o and the other vector components of $\overrightarrow{\tau }$ _4D can be determined by inverting the matrix X at each frequency of interest.

2.2 Sinusiodal modulation

Instead of determining moments from pulse-shape measurements, the signal delays, τ _gi , can also be obtained by observing phase shifts of transmitted sinusoidal signals and benefiting from the precision of sensitive phase-detection techniques. (Note that Williams [6] has used sinusoidal modulation for the determination of the scalar DGD.) For most purposes, it suffices to only measure changes in τ _gi with polarization and optical frequency and to not resolve the ambiguity presented by the use of a signal with periodic modulation. For a sinusoidal signal, with the assumption of frequency-independent PSP’s and DGD, we can use the dot-product rule [11] to determine the power fractions, 1/2(1+$\overrightarrow{p}$ ·$\overrightarrow{s}$ ) and 1/2(1-$\overrightarrow{p}$ ·$\overrightarrow{s}$ ), coupled into the slow and fast PSP’s, respectively. The phase of the detected sinusoidal output intensity, ϕ_out=ω _m τ_ϕ, defines a signal delay τ_ϕ that obeys the relation

where ω _m is the angular modulation frequency. For small ω _m , the signal delays τ_ϕ (defined by sinuosidal phase) and τ _g (defined by momenta) are approximately equal. When the four launch polarizations coincide with the Poincaré sphere axes, Ŝ ₁, -Ŝ ₁, Ŝ, and Ŝ ₃, Eq. (3) is still valid for sinusoidal modulation: ${\tau }_o=\frac{1}{2}\left({\tau }_{\varphi 1}+{\tau }_{\varphi \left(-1\right)}\right)$. Recalling that $p_1^2$+$p_2^2$+$p_3^2$=1, expressions for τΔ (the magnitude of $\overrightarrow{\tau }$ ) and the components, p_i , of the unit vector pointing in the direction of $\overrightarrow{\tau }$ can also be determined and solved exactly.

Like Eq. (3), Eq. (5) can be generalized to any four non-degenerate input polarization launches $\overrightarrow{\tau }$ _i (i=1, a, b, c) that span Stokes space. We orient the input Stokes space so that $\overrightarrow{s}$ ₁=Ŝ ₁. Then, with $\overrightarrow{s}$ _a =(a ₁, a ₂, a ₃), $\overrightarrow{s}$ _b =(b ₁, b ₂, b ₃), and $\overrightarrow{s}$ _c =(c ₁, c ₂, c ₃), we measure the corresponding delays τ_ϕ1, τ_ϕa, τ_ϕb, and τ_ϕc. We first express Ŝ ₁, Ŝ ₂, and Ŝ ₃, in terms of $\overrightarrow{s}$ _a , $\overrightarrow{s}$ _b , and $\overrightarrow{s}$ _c ,

where the coefficients α_i , β_i , and γ_i are obtained from

Note that $\overrightarrow{s}$ _a , $\overrightarrow{s}$ _b , and $\overrightarrow{s}$ _c should not lie in a common plane. If $\overrightarrow{s}$ _a , $\overrightarrow{s}$ _b , and $\overrightarrow{s}$ _c are coplanar, for purposes of analysis, rotate to a different Stokes space with Ŝ ₁’ aligned with one of the other launch polarizations.

Substituting Ŝ ₁=α ₁ $\overrightarrow{s}$ _a +β ₁ $\overrightarrow{s}$ _b +γ ₁ $\overrightarrow{s}$ _c into Eq. (5) gives

leading to a transcendental equation for τ _o :

A first trial for τ₀ can be obtained by linearizing Eq. (9),

After solving for τ₀ and using $p_1^2$+$p_2^2$+$p_3^2$=1, an expression for the magnitude, Δτ, can be found:

The components, p_i , of the PMD vector, $\overrightarrow{\tau }$ , can then be determined from

Although the above procedure will yield any computational accuracy desired, it is often not necessary. For small modulation frequencies, ω _m , we can approximate tan (x)⋍x in Eq. (5) and use the procedure outlined above (i.e. Eq. (3–4)). These linear expressions are valid for sinusoidal modulation to within 6% as long as ω _m Δτ<π/4. For instance, for the peak DGD we observed here, 88 ps, better than 6% accuracy will be obtained for modulation frequencies, f _m =ω _m /2π, less than 1.5 GHz.

3. Experiment

We implemented the PSD Method using sinusoidal modulation and the apparatus shown in Fig. 2. A tunable laser provided the signal light, which was amplitude modulated by a LiNbO₃ modulator typically operating at 1 GHz with an extinction ratio of 1/3. A wavemeter monitored the carrier wavelength. Three of the four launch polarizations were sequentially provided by a portion of a commercial polarization analyzer instrument containing three linear polarizers that could be switched into the beam. The fourth polarization was set by polarization controller 2 (PC2) and obtained by switching all three polarizers out of the signal path. In principle, the fourth polarization could have been provided by a polarizer as well. The phase of the modulation was measured with a network analyzer. The apparatus contained a polarimeter to allow PMD measurements by the MMM to verify the accuracy of the PSD method. The polarimeter also provided a convenient monitor when adjusting PC2.

The launch polarizations were three linear polarization states at 0°(=Ŝ ₁), 60.4°, and 120.6°, and right-circular polarization, Ŝ ₃. Figure 3 shows the wavelength dependence of the signal delays for these four inputs in a fiber span with 35-ps mean DGD. The span included an 8-km length of high-PMD dispersion compensating fiber and a 54-km length of standard fiber to compensate the chromatic dispersion to a residual dispersion of +124 ps/nm at 1542 nm. The fiber’s PMD causes the polarization dependence of the signal delays, and the variations in these relative signal delay curves originate from the change of direction and magnitude of $\overrightarrow{\tau }$ (ω) as well as from the underlying chromatic dispersion. Because the signal delays through the 62-km fiber span are about 300 µs, relative signal delays have been plotted in Fig. 3, by subtracting a reference delay from all measured data. The Ŝ ₁, 60.4°, and 120.6° delay curves shown in Fig. 3 have been corrected for the ≈2 ps additional delay caused by the thickness of the polarizers. The data in Fig. 3, however, have not been corrected for drift caused by changes in the optical length of the fiber during the course of the measurement.

 

Fig. 2. Apparatus used for the Polarization-Dependent Signal Delay Method measurement of PMD. MOD: Lithium Niobate amplitude modulator, A: optical amplifier, PC: polarization controller, Rx: receiver.

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Fig. 3. Measured wavelength dependence of the signal delay of a single-mode fiber span having 35-ps mean DGD for four fixed input polarizations.

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4. Results and Discussion

Figure 4 shows both the input PMD vector $\overrightarrow{\tau }$ (ω) and the relative polarization-independent delay, τ_0R, for the fiber span, obtained from the data of Fig. 3. We used the linear approximation to the tangent, i.e. Eq. (4) and ${\overrightarrow{\tau }}_{4D}=X^{-1}{\overrightarrow{\tau }}_{\varphi }$, where τ̃_ϕ=(τ_ϕ1,τ_ϕ2,τ_ϕ3,τ_ϕ4). Here, τ_0R is the polarization-independent delay through the fiber span at each wavelength compared to the delay at the center wavelength, 1542.0 nm. The modulation frequency was 1.0 GHz, providing ω _m Δτ=0.55 at the maximum DGD. The measurement points were separated by 0.02 nm. We show a scan of only 1 nm so that the magnitude and components of $\overrightarrow{\tau }$ (ω) are discernible. The drift in fiber length was tracked and corrected by performing a delay measurement at the center wavelength every five steps. Also shown in Fig. 4 is an MMM measurement of the input $\overrightarrow{\tau }$ (ω) (see Appendix) for the same fiber span. This measurement, using an MMM step-size of 0.03 nm and 0.01-nm interleave steps, was taken immediately after the PSD measurement. The two techniques agree to within a maximum discrepancy of 2 ps for the DGD and 4 ps for the components, where some of the discrepancy is caused by drift of the fiber’s PMD vector during the measurements. We also computed $\overrightarrow{\tau }$ (ω) from the PSD data using the exact expression in Eq. (5). For this measurement using 1-GHz modulation frequency, the largest deviation of the linear expression from the exact expression was 0.80 ps.

 

Fig. 4. Wavelength dependence of the first-order input PMD vector, $\overrightarrow{\tau }$ (ω), including magnitude (DGD) and components, and the relative polarization independent delay, τ_0R, for the single-mode fiber span with 35-ps mean DGD. Markers show the Polarization-Dependent Signal Delay Method (PSD) results using 1-GHz modulation frequency; curves without markers show the results obtained from the Müller Matrix Method (MMM).

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To further investigate the range of applicability of the linear approximation to the tangent in Eq. (5), we measured signal delays for the same fiber span using a modulation frequency of 3 GHz. Figure 5 shows a comparison of $\overrightarrow{\tau }$ (ω) and τ_0R computed from the 3-GHz data using the exact expression and the linear approximation. For this case, ω _m Δτ=1.66 at the maximum DGD. The largest difference between the two DGD curves was 2.79 ps at 1542.14 nm, corresponding to a 5.3% difference. Note that with a 3-GHz modulation frequency, the modulation sidebands are spread out over 48 pm, so there is averaging of the signal delay over this bandwidth.

The PSD method has a number of advantages over the well-known frequency-domain techniques for PMD vector measurement. Generally, the PSD technique benefits from the accuracy of sensitive phase detection compared to the accuracy of a polarimeter. The wavelength step is created electronically by the modulator and is accurately known. In addition, the accuracy of the PSD method is not dependent on the measured difference between two optical frequencies. Because the PMD vector is determined at the wavelength where the measurements are made, the PSD method could eventually allow continuous monitoring of the PMD at a specific wavelength. It also offers more simplicity as the receiver end requires only electronics and not a polarimeter. One disadvantage of the PSD method is that the drift of the fiber length must be corrected. A second disadvantage is that four polarization launches at each optical frequency are required, instead of two launches as with the JME or MMM. Although the PSD method does not yield the fiber rotation matrix, adding a polarimeter at the output would allow this determination, without sacrificing the other advantages of the PSD method.

 

Fig. 5. PSD measurement using 3-GHz modulation frequency of $\overrightarrow{\tau }$ (ω) and τ_0R for the same single-mode fiber span as in Fig. 4. Red curves show the PSD results for τ_0R and magnitude and components of $\overrightarrow{\tau }$ (ω) computed with the linear approximation; blue curves show the results obtained from the exact expressions. The markers show measured data, while the lines serve to guide the eye.

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5. Conclusion

We have described a time-domain method for determining first-order PMD vectors by measuring polarization-dependent signal delays. The method also provides a way of accurately measuring chromatic dispersion in the presence of PMD, to our knowledge the only method available for this today. Experimental results show that the PMD vectors obtained by the method agree with those obtained by frequency-domain techniques. No frequency differentiation is necessary for first-order PMD vectors, however, the data appear smooth, allowing differentiation of the vector components to determine second-order PMD vectors as in the Müller Matrix Method.