## Abstract

In this paper, we propose an in-service method to simultaneously monitor both nominal and effective values of differential group delay (DGD) in wavelength-division multiplexing (WDM) optical communication systems, in a per channel basis. The method is based on coherent heterodyne detection of the optical signal. We have demonstrated that the technique is capable to recover nominal DGD values from 0 ps to 90 ps while, at same time, to provide the effective DGD parameter, related to the impairment of optical channels. The relationship between the Q factor and effective DGD was also demonstrated, both numerically and experimentally, for distinct nominal values of DGD inserted on the system, by varying the state of polarization (SOP) of the optical signal at the input of the DGD element.

© 2013 OSA

## 1. Introduction

The current demand for high capacity long-haul optical communication systems leads to a continuous increase of the transmission bit rate, as well as to the deployment of reconfigurable optical networks. In this scenario, the polarization mode dispersion (PMD) becomes one of the major impairments to limit optical transmission capacity and performance [1–4]. The PMD induces a differential group delay (DGD), or first-order PMD, between the two principal fiber polarization axes due to its birefringence, as well as the second-order PMD, caused by the random variation of the orientation of the birefringence along the fiber length [5–7]. Those mechanisms distort the signal waveform, increasing the bit error rate (BER) and limiting the receiver performance, due the significant intersymbol interference (ISI) [8].

PMD measurement techniques have attracted a great deal of interest in recent years [1–4] and several methods to monitor the optical fiber link PMD have been reported in the literature [9–23]. Some of those methods, such as the pulse delay method [9], the interferometric method [10,11], the Poincarè arc method [12], as well as the Jones and Mueller Matrix methods [13,14], can be used to evaluate the PMD of the fiber link. However, they require a disruption of data traffic, becoming unattractive, from the network operator point-of-view.

As an alternative, among a variety of in-service monitoring techniques proposed in literature, Anderson’s group have proposed a fully optical method [15] to provide a DGD evaluation between the optical transmitted channels, recently revisited by some of us [16]. The method is based on low-frequency polarization modulation of each optical channel at the transmitter side, thereby requiring a modification on the transmission side of the optical system. Also, the maximum value of measurable DGD is limited by the WDM channel spacing: reducing the channel spacing limits the measured DGD range.

There are also techniques which allow the in-service DGD monitoring through the degree of polarization (DOP) measurement of the received signal [17]. This technique is based on the assumption that the laser signal is completely polarized while the amplifier spontaneous emission (ASE) noise is non-polarized. On this basis, we have recently demonstrated a procedure to simultaneously measure DGD and optical signal-to-noise ratio (OSNR) [18]. The technique is simple and bit-rate independent, but also requires modification on the transmission side of the optical system.

As an alternative, several papers [19–23], have investigated a non-intrusive, in-service, and in-band technique for real-time DGD evaluation. Essentially, the effective value of first-order DGD is calculated by measuring the polarization walk-off between two frequency components of the signal spectrum. By using heterodyne coherent detection, the technique shifts the signal spectrum from the optical domain to the radiofrequency (RF) domain. Next, filtering and power detection are employed to convert the signal power into Stokes vector information, providing an evaluation of the system effective DGD (DGD_{eff}), i.e, the DGD actually experienced by the optical signal.

However, it must be pointed out that the effective value of DGD experienced by a propagating optical signal (DGD_{eff}) can be very different from the nominal DGD value, related to the PMD parameter of the optical fiber. This difference is because the DGD is defined as the delay originated from the different propagation velocities between the two principal states of polarization (PSPs), which is an intrinsic characteristic of the optical fiber itself. On the other hand, the effective DGD is related to the orientation of the input state of polarization (SOP) of the optical pulse relative to the PSP of the fiber. In fact, if the optical pulse is aligned to one of the PSPs, it will not suffer any distortion, meaning that the effective DGD would be zero (although the nominal DGD will still be the same as before). Conversely, if the input SOP is aligned at 45 degrees between the PSPs, the effective DGD will be equal to the nominal DGD. In intermediate cases, effective and nominal DGD are related by: $0\le DG{D}_{eff}\le DG{D}_{nom}$.

As a consequence, the nominal and effective DGD values can be indeed very different and it is important to simultaneously monitor both DGD parameters. Measuring only the effective DGD can yield a very low value and the network operator may be misled to believe that the optical link has a low value of DGD and would never suffer from PMD impairments. Information on the nominal DGD of the fiber link can be very useful to network providers, in order to prevent future link failures, once they are aware to the upper-limited that can be reached by the effective DGD in a given instant. In [21] the authors shown that the nominal mean DGD of the fiber can be obtained through the measurement of instantaneous effective DGD, although requiring a relatively long time to complete the measurement for the statistical distribution. In contrast, in our proposed technique, we are focused on the measurement of nominal instantaneous DGD, which varies both in time as well as wavelength.

In this context, we propose a modification on the non-invasive technique presented in [20,21] to allow simultaneous measurement of both DGD parameters. Section II presents our experimental procedure. In comparison to the original proposition [20,21], which focus only on the effective DGD, significant modifications were made and several components were added. Next, in section III, we present the mathematical background to support our technique, based on fundamental concepts of geometry and linear algebra. Sections IV and V discuss experimental set-up and results, respectively. Section VI concludes the paper.

## 2. Measurement procedure

The non-intrusive DGD monitoring system, based on heterodyne detection, is illustrated in Fig. 1
. The experimental arrangement requires the tapping of a small portion of the transmitted optical signal. In practical implementations, the monitoring port of an optical amplifier can be used for this purpose. In the photodetector (PD), the heterodyne coherent detection shifts the information spectrum from the optical domain to the RF domain, beating the transmitted optical signal to an optical local oscillator, whose polarization is controlled by a polarization state generator (PSG). The resulting spectrum is centered at an intermediate frequency, f_{IF}, given by the difference between the carrier frequency, f_{C,} and the local oscillator frequency, f_{LO}. Next, this signal is amplified by a low noise amplifier (LNA). Also, tuning the local oscillator laser frequency, f_{LO}, allows the monitor scanning of all the WDM channels.

The method proposed here is based on a modification of the measurement system originally discussed in [20,21]. The setup of our novel PMD monitor is depicted in Fig. 1(a). In comparison to previous implementations, see Fig. 1(b), two more measurement arms are added, corresponding to two additional measurement frequencies and associated electronics. The goal is to enable simultaneous per channel calculation of both nominal and effective DGD in the fiber link, which would not be possible without this new arrangement.

Specifically, in the previous method [20,21], illustrated in Fig. 1(b), a 2-way RF power divider is used to equally split the signal power, previously amplified by the low noise amplifier (LNA). Each half of the signal power is sent to narrow bandpass RF filters, centered at frequencies f_{1} and f_{2}, respectively. The resulting narrowband signals are power detected through a Schottky RF envelope detector diode.

In the optical domain, the angular separation φ, between the Stokes vectors regarding these selected frequencies f_{1} and f_{2}, is proportional to the effective DGD acting on the optical channel. In other words, the knowledge of these Stokes vectors allows the measurement of the frequency-dependent polarization walk-off between the slices of the signal spectrum, and, consequently, the measurement of the effective DGD induced by the fiber link. The polarization state generator (PSG) assures an orthonormal basis to recover the state of polarization corresponding to each slice of the signal spectrum. Upon coherent detection, it can be easily shown that the angular separation φ can be extracted from the resulting RF spectrum filtering and power detection at the selected frequencies f_{1} and f_{2} [20,21].

In contrast, our novel experimental arrangement employs a 4-way RF power divider, depicted in Fig. 1(a), which divides the signal power into four equal parts. Each one is filtered by a narrow bandpass filter, centered at one of the equally spaced frequencies f_{1}, f_{2}, f_{3} and f_{4}. An illustration of each narrowband spectrum slice is depicted at Fig. 2
. By measuring the RF power of each narrowband spectrum at four distinct frequencies it is now possible to track four distinct Stokes vectors. Then, one is capable to obtain not only the angle φ between Stokes vectors related to the filtered slices of the signal spectrum, which provides the effective DGD, but also to construct a least-squares Π plane, established by an orthogonal regression, considering the distance from each vector tip to the Π plane and using singular value decomposition (SVD) method [24], as illustrated in Fig. 3
. This is a very useful result because the vector normal to this plane is collinear to the optical fiber principal state of polarization vector. In the next section we will demonstrate that the angle between each Stokes vector projection on the Π plane provides the nominal fiber DGD value. As a consequence, the use of four RF frequencies, in place of the conventional two-way setup, allows simultaneous, real-time monitoring of both effective and nominal DGD values.

## 3. Mathematical background

In order to describe the mathematical background supporting our method, we start by the power detected from each slice of the signal spectrum after the heterodyne detection, given by the expression [25]:

The signal power within the bandwidth centered at frequency${f}_{i}$ will be given by:

According to [2,25], Eq. (2) can be written as:

In this way it is possible to obtain the power of a specific narrowband slice of the signal spectrum, related to the frequency${f}_{i}$, on a specific state of polarization (SOP) through the Eqs. (4) and (5) below,

As demonstrated in Appendix A, based on [8,25], the normalized Stokes vectors of each frequency slice can be recovered by measuring the RF power at frequencies f_{1}, f_{2}, f_{3} and f_{4}, when the local oscillator is set to one of the six states of polarization by the polarization state generator (PSG) of Fig. 1. Considering these six states of polarization of the local oscillator as being the linearly horizontal polarized (LHP), linearly vertical polarized (LVP), linearly polarized at + 45° (L + 45), linearly polarized at −45° (L-45), right-hand circular polarized (RCP), and left-hand circular polarized (LCP), respectively represented by ${\overrightarrow{S}}_{LO}^{(0)}$, ${\overrightarrow{S}}_{LO}^{(1)}$, ${\overrightarrow{S}}_{LO}^{(2)}$, ${\overrightarrow{S}}_{LO}^{(3)}$, ${\overrightarrow{S}}_{LO}^{(4)}$, ${\overrightarrow{S}}_{LO}^{(5)}$ and, their corresponding RF powers as ${P}_{i}^{(0)}$, ${P}_{i}^{(1)}$, ${P}_{i}^{(2)}$, ${P}_{i}^{(3)}$, ${P}_{i}^{(4)}$ and ${P}_{i}^{(5)}$, we show in Appendix A how one can obtain the normalized Stokes vectors parameters for each frequency ${f}_{i}$. The result is given in Eq. (6), as follow:

Therefore, the knowledge of the Stokes vectors, obtained from the power readings of each narrowband spectrum slice centered at one of the equally spaced frequencies f_{1}, f_{2}, f_{3} and f_{4,} allows the determination of a least squares Π plane on the basis of these four distinct vectors, through orthogonal regression and considering the distance from each vector tip to the Π plane [24]. This plane is limited by the circumference drawn by the intersection with the Poincarè sphere, as illustrated in Fig. 3. Also, it is possible to compute the center of the circumference, the principal state of polarization vector ${\overrightarrow{\Omega}}_{PSP}$ and the Stokes vectors projections in the Π plane. The angle φ between two Stokes vectors will provide the effective DGD value while the angle θ between their projections will provide the nominal DGD value experienced by the optical channel. In contrast, the previous method [8,15–19] does not furnish the projection plane, making impossible the calculation of the angle θ and, consequently, the nominal DGD value. The first-order PMD, or differential group delay (DGD), is defined as

_{2}– f

_{1}= f

_{3}– f

_{2}= f

_{4}– f

_{3}and, Δθ is the angle between projections of the correspondent Stokes vectors in plane Π, in radians. As it can be seen in Eq. (7), the separation between the selected center frequencies f

_{1}, f

_{2}, f

_{3}and f

_{4}of the narrowband filters is inversely proportional to the measurable DGD range. For example, if the frequency separation between the RF filters is 10 GHz, it will correspond to a maximum of 50 ps of DGD range. Reducing the frequency separation to 5 GHz leads to a maximum of 100 ps of DGD range.

## 4. Experimental setup

In Fig. 4 , the back-to-back experimental setup is shown. The goal is to evaluate the performance of the proposed method as a PMD monitor. The setup was composed by a 10 Gb/s optical pattern generator, a manual polarization controller (PC), to vary the SOP of the incoming light signal, a DGD module to emulate different DGD values, a 90/10 splitter and an oscilloscope, to register the eye diagram for each tested case.

The PMD monitor itself follows the setup of Fig. 1. The separation between signal and local oscillator frequencies is set to 20 GHz in order to allow detection of the entire RF signal band. In this case, for an intermediate frequency (f_{IF}) of 20 GHz, the RF filters f_{1}, f_{2}, f_{3} and f_{4} were centered at 12.5 GHz, 17.5 GHz, 22.5 GHz, 27.5 GHz, respectively. In this way, the maximum effective DGD which can be measured is around 100 ps.

When the state of polarization at the input of the DGD module is varied through the manual polarization controller (PC), the PMD monitor measures different values of the per-channel effective DGD for each nominal DGD set in the module. It also provides a value of measured nominal DGD. The effective DGD represents the instantaneous DGD to which the channel is subjected and it is directly related to the BER. However, for small values of effective DGD and, consequently, very low BER, the direct measurement of the BER via the pattern generator is not possible. In those cases, the high sampling oscilloscope is used to provide the eye diagram and the electrical SNR, from which a measurement of the Q factor is carried out and the BER is extracted.

## 5. Results

Evaluation results concerning the proposed technique are presented in Figs. 5
and 6
. In Fig. 5, we show plots of Poincarè spheres and the set of experimentally recovered normalized Stokes vectors of some measured cases (SOP states) in the system under test. Figures 5(a)–5(d) correspond to nominal DGD inserted in the system of 15 ps, 25 ps, 45 ps and 90 ps, respectively. Each Poincarè sphere contains the representation of the measured Stokes vectors in dashed lines for the 4 RF frequencies f_{1}, f_{2}, f_{3} and f_{4}, as explained in Figs. 1 and 2. The calculated principal states of polarization vector ${\overrightarrow{\Omega}}_{PSP}$, the least squares plan obtained from the experimental obtained Stokes vectors, and the projection of each Stokes vector in the least squares plan in solid lines, are also shown in this figure. The effective DGD is calculated through the angle between two Stokes vectors of consecutive frequencies, and the nominal DGD is calculated through the angle between projection vectors on the least squares plan, also from two consecutive frequencies.

After obtaining all the required geometrical information, Fig. 6 depicts the DGD measurement evolution for different SOPs at the input of the DGD module. Figures 6(a)–6(d) correspond to experimental results with nominal DGD of 15 ps, 25 ps 70 ps and 90 ps, respectively. In these figures, 50 SOP input states are used, varying from the worst condition (SOP number 1), where the SOP is very close to 45 degrees from the two PSP (slow and fast axis) and the effective DGD (DGD_{eff}) is very close to the nominal DGD (DGD_{nom}), to the SOP almost perfectly aligned to the PSPs (SOP number 50), when the expected effective DGD is close to zero. As it can be observed in Fig. 3, combining two consecutives Stokes vectors, related to consecutive frequencies, one obtains three distinct values of the nominal DGD (DGD_{nom}). Our final result is simply the arithmetic mean between those three values. Next, in Fig. 7
, one has the equivalent Q factor evolution, obtained through the electrical SNR from the oscilloscope eye diagram, for each measured case.

One can observe that for relatively small DGD values, i.e. 15 ps and 25 ps in Figs. 6(a) and 6(b), the nominal DGD measured through our proposed method was very close to the value inserted in the DGD module (nominal DGD), for all input SOPs. For relatively high DGD values, i.e. 70 ps and 90 ps in Figs. 6(c) and 6(d), the measured nominal DGD was quite constant, no matter the input SOP, but with a small offset relative to the nominal value. This can be probably attributed to artifacts in RF power acquisition, related to inaccuracies of the electronic hardware. Overall, these results indicate that the proposed technique is really capable to measure the nominal DGD no matter the input SOP. Also, the proposed method is still capable to measure the DGD_{eff}, as can be seen in Figs. 6(a)–6(d). In fact, all cases the DGD_{eff} displays a variation from its maximal value (SOP number 1), corresponding to the nominal DGD inserted in the system (15 ps, 25 ps, 70 ps and 90 ps), to a minimum value (SOP number 50).

Regarding the Q factor, no significant variation can observed for relatively small values of inserted DGD of 15 ps and 25 ps, as it can be seen in Figs. 6(a) and 6(b). For these DGD values the BER impairment cannot be detected by eye diagram distortion. In other words, the distortion in eye diagrams is quite small. However, for higher values of inserted DGD (70 ps and 90 ps), as in Figs. 6(c) and 6(d), one can observe the expected correlation of Q factor and effective DGD. In fact, the Q factor is inversely related to the system effective DGD. When the input SOP is aligned to the PSP of the birefringent element of DGD module, a small effective DGD means the optical signal is not distorted and the eye aperture is preserved.

In order to understand the relationship between effective DGD and the Q factor, we have performed computer simulations of the experimental setup depicted in Fig. 4. In these simulations, the SOP angle at the input of the DGD module was varied from 0 to 45 degrees, i.e., from SOP perfectly aligned to the PSP of the DGD module to the worst condition where input signal is at 45 degrees to each one of the DGD element PSP (slow and fast axis). In the last case, the optical power is equally split in two parts, propagating at different velocities (slow and fast axis). Thus, for each value of DGD inserted in the system, the Q factor will vary from no impairment due to the DGD parameter (SOP at 0 degrees from PSP) to the maximum possible impairment (SOP at 45 degrees from PSP). In Fig. 7(a) we plot the results of the simulations. In order to relate the SOP angle to the effective DGD, we used the relationship:

where DGD_{nom}is the nominal DGD of the module, DGD

_{eff}is the effective DGD at a specific SOP and φ is the angle between SOP and the PSP. If φ = 0, than DGD

_{eff}= 0, as all the optical power of the signal will travel in one of the DGD module PSP and, thus, no relative delay between modes of polarization will be observed. On the other hand, if φ = 45, than DGD

_{eff}= DGD

_{nom}and the maximum impairment due to relative delay between the two modes of polarization is observed. In this case optical power of the signal will be split equally between the two optical polarization modes. The corresponding Q factor will decrease with the increase of effective DGD, until the maximum value of DGD

_{eff}= DGD

_{nom}. In Fig. 7(a), we present simulation results for the system described in Fig. 4, for values of DGD

_{nom}from 15 ps to 90 ps. As expected, the Q factor varies from 22 to 6 in the case of DGD

_{nom}= 90 ps. These simulated results were compared with the experimental results of Fig. 7(b), where the same qualitative behavior can be observed. Although results are qualitative, we can see that the Q factor varies from 22 to 4, for the case of 90 ps nominal DGD, in excellent agreement with simulated results.

The observed flatness of the Q factor as a function of SOP input of Figs. 6(a) and 6(b) can finally be understood, by means of Figs. 7(a) or 7(b). For small DGD inserted in the system, one can observe that the variation of Q factor is less them 1 dB (refer to the simulated values of Fig. 7(a)), when the inserted DGD is less than 25 ps. On the other hand, when the DGD inserted is 45 ps, the variation begins to be appreciable, exceeding 3 dB.

## 6. Second-order PMD effects and estimation

Second-order PMD may, in principle, distorts the arc described by the SOPs leading to erroneous nominal DGD evaluation [26–28]. In fact, considering the DGD vector is given by [7]:

where the $\tau (\omega )$is the magnitude of the DGD and $\widehat{q}$ is an unit vector pointing to the PSP direction. The second-order PMD is given by the derivative of the DGD vector in Eq. (9) as:As it is well known, the relationship in Eq. (10) shows that the second-order PMD is composed of two terms. The first term causes the polarization dependent chromatic dispersion (PDCD) and the second term represents a frequency dependent rotation of the PSPs (SOPMD), known as depolarization term. The second term leads to a distortion of the arc whose Stokes vector describes around the PSP, as the PSP it self-rotates as a function of the frequency. Consequently, Eq. (7), used to calculate 1st order PMD (DGD), is not valid except in the case where the second-order PMD is negligible.

Fortunately, as stated in literature [26–28], there is a bandwidth range, known as bandwidth of principal states of polarization, or $\Delta {f}_{PSP}$ in which an approximation can be used to represent real optical fibers links. Specifically, it turns out that the mean DGD of an optical fiber, $\overline{\tau}$, and $\Delta {f}_{PSP}$ are related by the simple relationship:

where k is a constant. Several values for this constant have been proposed in literature. In [26], the constant k is found to be 0.8, and in [27], k was found to be 0.26 for dispersion shifted fibers (DSF) and 0.45 for single-mode fibers (SMF). A rigorous analytical study in [28] has demonstrated that $k=2\sqrt{2}/\pi \approx 0.90$.For a SMF, the worst case reported in [26–28] is k = 0.45. Even then, using our total bandwidth of 15 GHz (4 filters spaced by 5 GHz), that is, $\Delta {f}_{PSP}$ = 15 GHz, it turns out that the mean DGD for which the approximation in Eq. (11) holds is $\overline{\tau}=30$ ps. Considering, as it is well known, that a fiber DGD obeys to a Maxwellian distribution, the maximum and the mean DGD values will follow the relationship ${\tau}_{\mathrm{max}}=3\text{\hspace{0.17em}}\overline{\tau}$. Then, one obtains an upper limit of 90 ps, using the k value reported by [27] On the other hand, if we use the more recent result of [28], results in a mean DGD of 60 ps and maximum DGD of 180 ps. This indicates that the our filter spacing is wide enough and the achieved results will not be impaired by SOPMD in a practical application with real fiber links.

Further improvement can be achieved by reducing the frequency spacing of the RF filters used. For example, using 2.5 GHz filters, reduces by 2 the total bandwidth, increasing the maximum DGD value to 180 ps, considering k value of 0.45 proposed in [27]. Using 1 GHz (total of 3 GHz), would augment to 450 ps the maximum DGD, according to Eq. (11), without impairment of second-order PMD. In addition, we stress that the minimum bandwidth of the PSPs even on long cables, as determined in [26] is always over 50 GHz, well above of the total bandwidth of 15 GHz used in our setup.

On the other hand, this technique may provide a way to monitor the second-order PMD. Specifically, if the second-order PMD is strong in the fiber link, it would be possible to evaluate it by, for example, using 3 RF filters of lower frequencies, to calculate a given PSP_{1} (and its unit vector ${\widehat{q}}_{1}$, pointing in the direction of the fast principle axis) associated to DGD_{1} (${\tau}_{1}$), and use 3 RF filters of higher frequencies to calculate a given PSP_{2} (and its unit vector ${\widehat{q}}_{2}$), associated to a DGD_{2} (${\tau}_{2}$). From these data, it is possible to estimate the terms of Eq. (10). In particular, the depolarization term could be written as:

Otherwise, if in-band measurements are not feasible due to a high value of $\Delta {f}_{PSP}$, is it always possible to measure neighbor channels and estimate the SOPMD from these measurements. The same relation in Eq. (12) applies, but with $\Delta {f}_{21}$ now being the optical channel separation, for example, 50 GHz or 100 GHz.

## 7. Conclusion

We presented an in-service method to simultaneously monitor nominal and effective differential group delay (DGD) of optical communication systems in a per channel basis. We have demonstrated the proposed monitor is able to recover nominal DGD values from 0 ps to 90 ps, while at the same time providing the effective DGD, related to the PMD impairment of the optical channel. By varying the state of polarization (SOP) of the optical signal, we also obtained the relationship between Q factor and effective DGD for various DGD elements inserted in the system, both numerically and experimentally. We have also experimentally shown how to obtain in-band Stokes vectors and its geometrical relationship with effective and nominal DGD quantities, thereby demonstrating that the knowledge of the nominal DGD is of interest to fiber network operators, in order to prevent future link outages caused by PMD.

## Appendix A

In order to obtain Eq. (6) from Eq. (5) we proceed as follows. The coefficient $\left({\eta}_{i}\text{\hspace{0.17em}}{P}_{S}\left({f}_{i}\right)\text{\hspace{0.17em}}{P}_{LO}\right)/2$ of Eq. (5) results from the signal heterodyne detection, using two orthogonal states in the local oscillator, provided by a polarization controller (or a polarization state generator). Considering the six principal states of polarization in a Poincarè sphere, given by the linearly horizontal polarized (LHP), linearly vertical polarized (LVP), linearly polarized at +45° (L+45), linearly polarized at −45° (L-45), right-hand circular polarized (RCP), and left-hand circular polarized (LCP), respectively represented by ${\overrightarrow{S}}_{LO}^{(0)}$, ${\overrightarrow{S}}_{LO}^{(1)}$, ${\overrightarrow{S}}_{LO}^{(2)}$, ${\overrightarrow{S}}_{LO}^{(3)}$, ${\overrightarrow{S}}_{LO}^{(4)}$, ${\overrightarrow{S}}_{LO}^{(5)}$. In optical domain, the Stokes vectors ${\overrightarrow{S}}_{LO}^{(0)}$ and ${\overrightarrow{S}}_{LO}^{(1)}$, related to the polarizations linearly horizontal (LHP) and linearly vertical (LVP) are related by ${\overrightarrow{S}}_{LO}^{(1)}=-{\overrightarrow{S}}_{LO}^{(0)}$. In this case,

Adding this two Eqs. (13) and (14), one have

Then, for each frequency${f}_{i}$ we can obtain a proportionality constant $\left({\eta}_{i}\text{\hspace{0.17em}}{P}_{S}\left({f}_{i}\right)\text{\hspace{0.17em}}{P}_{LO}\right)/2$, through the power measured in two anti-parallel states of polarization, ${P}_{i}^{(0)}$ and ${P}_{i}^{(1)}$. In consequence, we can also write the same proportionality constant obtained through the power from others anti-parallel states of polarization as follow:

Setting the polarization controller (or the polarization state generator) in three states of polarization defined as linearly horizontal polarized (LHP), linearly polarized +45° (L+45), and right-hand circular polarized (RCP), respectively represented by ${S}_{LO}^{(0)}$, ${S}_{LO}^{(2)}$, and ${S}_{LO}^{(4)}$, one can obtain the power for each frequency${f}_{i}$, for these three states of polarization as Eqs. (4) and (5):

Thus, using three states of polarization imposed by the polarization controller or by the polarization state generator, one can solve the above linear system and determine the Stokes vector for the spectral bandwidth centered in frequency${f}_{i}$, given by ${\overrightarrow{S}}_{S}({f}_{i})$. The method is made simpler if the states of polarization used in the local oscillator to obtain the Stokes vector ${\overrightarrow{S}}_{S}({f}_{i})$ are the following: linearly horizontal (${S}_{LO}^{(0)}$), linear at +45° (${S}_{LO}^{(2)}$), and right-hand circular (${S}_{LO}^{(4)}$). In this case, one has:

Then, the linear system reduces to:

And each signal Stokes parameters can be written as:

Replacing the proportionality constant $\left({\eta}_{i}\text{\hspace{0.17em}}{P}_{S}\left({f}_{i}\right)\text{\hspace{0.17em}}{P}_{LO}\right)/2$ for each Stokes parameter of Eq. (25) according the relationships obtained in Eq. (16), one can write:

## Acknowledgment

This work was supported by the FUNTTEL/FINEP funding agency of the Brazilian government, under the CPqD GIGA Project.

## References and links

**1. **F. Heismann, “Polarization mode dispersion: Fundamentals and impact on optical communications systems,” in *Proceedings of IEEE 24th European Conference on Optical Communication* (Institute of Electrical and Electronics Engineers, Madrid, 1998), pp. 51–79, http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=729518.

**2. **J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. **97**(9), 4541–4550 (2000), http://www.pnas.org/content/97/9/4541.full.pdf+html(PNAS). [CrossRef] [PubMed]

**3. **C. R. Menyuk and A. Galtarossa, *Polarization mode dispersion: Optical and fiber communications reports* (Springer Science + Business Media, Inc., 2005).

**4. **C. Yu, “Polarization mode dispersion monitoring,” in *Optical performance monitoring: advanced techniques for next-generation photonic networks*, Calvin C. K. Chan ed., (Elsevier Academic Press, 2010), Chap. 4.

**5. **R. M. Jopson, L. E. Nelson, and H. Kogelnik, “Measurement of second-order polarization-mode dispersion vectors in optical fibers,” IEEE Photon. Tech. Lett. **11**(9), 1153–1155 (1999), http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=784234.

**6. **H. Kogelnik, L. E. Nelson, J. P. Gordon, and R. M. Jopson, “Jones matrix for second-order polarization mode dispersion,” Opt. Lett. **25**(1), 19–21 (2000). [CrossRef] [PubMed]

**7. **B. J. Soller, “Second-order PMD in optical components,” in Luna Technologies, May 13rd, 2005, http://lunainc.com/wp-content/uploads/2012/08/2nd-Order-PMD.pdf, last access: 2012, October 29th.

**8. **R. Hui and M. O'Sullivan, *Fiber optic measurement techniques* (Elsevier Academic Press, 2009).

**9. **C. D. Poole and C. R. Giles, “Polarization-dependent pulse compression and broadening due to polarization dispersion in dispersion-shifted fiber,” Opt. Lett. **13**(2), 155–157 (1988). [CrossRef] [PubMed]

**10. **N. Gisin, J.-P. Von der Weid, and J.-P. Pelleaux, “Polarization mode dispersion of short and long single-mode fibers,” J. Lightwave Technol. **9**(7), 821–827 (1991), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=85780. [CrossRef]

**11. **J.-P. Von der Weid, L. Thenenaz, and J.-P. Pelleaux, “Interferometer measurements of chromatic dispersion and polarization mode dispersion in highly birefringent single mode fibers,” Electron. Lett. **23**(4), 151–152 (1987), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4257386. [CrossRef]

**12. **P. Hernday, “Dispersion measurements,” in *Fiber-optic test and measurement*, Denis Derickson ed., (Prentice Hall, 1998).

**13. **B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis,” IEEE Photon. Tech. Lett. **4**(9), 1066–1069 (1992), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=157151.

**14. **B. L. Heffner, “Accurate, automated measurement of differential group delay dispersion and principal state variation using Jones matrix eigenanalysis,” IEEE Photon. Tech. Lett. **5**(7), 814–817 (1993), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=229816.

**15. **M. Sköld, B. E. Olsson, M. Karlsson, and P. A. Andrekson, “Low-cost multiparameter optical performance monitoring based on polarization modulation,” J. Lightwave Technol. **27**(2), 128–138 (2009), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-27-2-128. [CrossRef]

**16. **C. Floridia, G. C. C. P. Simões, E. W. Bezerra, M. M. Feres, and M. A. Romero, “Simplified approach to low-cost multiparameter monitoring based on low frequency polarization modulation,” Microw. Optic. Technol. Lett. **54**(8), 1820–1824 (2012), http://onlinelibrary.wiley.com/doi/10.1002/mop.26956/full*.* [CrossRef]

**17. **G.-W. Lu, M.-H. Cheung, L.-K. Chen, and C.-K. Chan, “Simple PMD-insensitive OSNR monitoring scheme assisted by transmitter-side polarization scrambling,” Opt. Express **14**(1), 58–62 (2006). [CrossRef] [PubMed]

**18. **C. Floridia, G. C. C. P. Simões, M. M. Feres, and M. A. Romero, “Simultaneous optical signal-to-noise ratio and differential group delay monitoring based on degree of polarization measurements in optical communications systems,” Appl. Opt. **51**(17), 3957–3965 (2012). [CrossRef] [PubMed]

**19. **B. Fu and R. Hui, “Fiber chromatic dispersion and polarization-mode dispersion monitoring using coherent detection,” IEEE Photon. Tech. Lett. **17**(7), 1561–1563 (2005), http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1453677.

**20. **R. Hui, R. Saunders, B. Heffner, D. Richards, B. Fu, and P. Adany, “Non-blocking PMD monitoring in live optical systems,” Electron. Lett. **43**(1), 53–54 (2007), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4068491. [CrossRef]

**21. **J. Jiang, S. Sundhararajan, D. Richards, S. Oliva, and R. Hui, “PMD monitoring in traffic-carrying optical systems and its statistical analysis,” Opt. Express **16**(18), 14057–14063 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14057. [CrossRef] [PubMed]

**22. **J. Jiang, D. Richards, S. Oliva, and R. Hui, “PMD and PDL monitoring of traffic-carrying transatlantic fibre-optic system,” Electron. Lett. **45**(2), 123–124 (2009), http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4752656. [CrossRef]

**23. **J. Jiang, D. Richards, S. Oliva, P. Green, and R. Hui, “In-situ monitoring of PMD and PDL in a traffic-carrying transatlantic fiber-optic system,” in *National Fiber Optic Engineers Conference*, OSA Technical Digest (CD) (Optical Society of America, 2009), paper NMB1, http://www.opticsinfobase.org/abstract.cfm?URI=NFOEC-2009-NMB1.

**24. **http://mathforum.org/library/drmath/view/63765.htm, last access: 2010, September 10th.

**25. **I. Roudas, G. A. Piech, M. Mlejnek, Y. Mauro, D. Q. Chowdhury, and M. Vasilyev, “Coherent frequency-selective polarimeter for polarization-mode dispersion monitoring,” J. Lightwave Technol. **22**(4), 953–967 (2004), http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-22-4-953. [CrossRef]

**26. **S. Betti, F. Curti, B. Daino, G. De Marchis, E. Lannone, and F. Matera, “Evolution of the bandwidth of the principal states of polarization in single-mode fibers,” Opt. Lett. **16**(7), 467–469 (1991). [CrossRef] [PubMed]

**27. **O. Aso, “Measurement of a parameter limiting the analysis of the first-order polarization-mode dispersion ef fect,” Opt. Lett. **23**(14), 1102–1104 (1998). [CrossRef] [PubMed]

**28. **M. Karlsson and J. Brentel, “Autocorrelation function of the polarization-mode dispersion vector,” Opt. Lett. **24**(14), 939–941 (1999). [CrossRef] [PubMed]