Abstract:
Methods and systems allow an in situ determination of the magnitude of PMD in an optical network and provide an estimate of the PMD impairment in the transmitted signal even when PMD is time dependent.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]    This application relates to the co-pending application Ser. No. ______ (Attorney Reference No: 10020844), filed on the same day, entitled “Method and Apparatus for a Jones Vector Based Heterodyne Optical Polarimeter” by Szafraniec owned by the assignee of this application and incorporated herein by reference. 
     
    
     
       BACKGROUND OF INVENTION  
         [0002]    Evaluation of transmission quality is an important aspect of fiber optic communications systems. Prior art evaluation of transmission quality is performed by electronic detection where the detected sequence of digital information is compared using a functional relationship to the actual value transmitted along with the information such as parity checks or error correction coding. However, the detection of errors does not provide an indication of the origin or cause of the transmission error. Many factors can produce transmission factors including limited received power, chromatic dispersion effects, poor optical signal-to-noise ratio, polarization mode dispersion (PMD) and nonlinear effects. The issue of PMD is of particular interest as it is expected that PMD will become the major source of error for optical networks transmitting information at data rates greater than 20 Gbits/s. Hence, it is important to measure PMD and determine the impact of PMD or PMD impairment on individual dense wavelength-division multiplexing (DWDM) channels. It is important to distinguish between PMD and PMD impairment. The PMD describes the birefringence of the optical link while the PMD impairment describes the effect of that birefringence on a DWDM channel or frequency band. Even large PMD may not cause PMD impairment if all optical frequencies comprising a frequency band propagate throughout the link in predominantly the same polarization state.  
           [0003]    PMD refers to the temporal pulse distortion that arises from different propagation speeds for light of differing polarization states through an optical medium such as a single mode optical fiber. PMD arises from the birefringence in an optical fiber that increases with fiber length. The larger the birefringence, the larger the PMD and the more rapidly the polarization state changes with wavelength and with fiber length. Hence, a typical method of determining PMD involves analyzing the evolution of the polarization state with wavelength. The PMD induced delay is defined as:  
             τ   =     Δθ     2      πΔ                 v               (   1   )                               
 
           [0004]    where Δθ is the rotation angle on a Poincare sphere and Δv is the optical frequency span that produced Δθ. To determine PMD in an operational network requires that the polarization state analysis be performed over the width of a single channel or frequency band of the DWDM system carrying data. Thus, spectral width is related to the frequency band spacing. The present International Telecommunications Union (ITU) grid is placed at 100 GHz or 0.8 nm with further reduction of frequency band spacing being planned. This requires that the polarization state measurements are performed with high spectral selectivity.  
           [0005]    Westbrook et al., in “Wavelength sensitive polarimeter for multichannel polarization and PMD monitoring,” OFC 2002, pp. 257-259, have disclosed a wavelength selective polarimeter that is based on fiber grating technology. The disadvantage of this approach is that the current grating technology is limited to a resolution of about 0.01 nm. Roudas et al., in “Coherent heterodyne frequency-selective polarimeter for error signal generation in higher-order PMD compensators,” OFC 2002, pp. 299-301, disclosed a heterodyne polarimeter based on Stokes vector measurements that requires sequential switching of the local oscillator (LO) polarization state. The heterodyne polarimeter potentially offers high resolution but the technique disclosed by Roudas et al. resembles that of classical intensity based polarimeters and does not take advantage of the phase information provided by the heterodyne signal. Therefore, sequential switching of the polarization state is required. This may lead to erroneous measurements in systems where the polarization state is time dependent.  
         SUMMARY OF THE INVENTION  
         [0006]    Methods and systems in accordance with the invention provide an in situ determination of the magnitude of PMD in an optical network and provide an estimate of the PMD impairment in the transmitted signal even when PMD is time dependent. Estimates of PMD impairment aid in determining the quality of the data transmitted in the individual DWDM channels or frequency bands while also providing a feedback signal to PMD compensators used to minimize PMD effects. These methods are typically based on the polarization state evolution within a single DWDM frequency band or in an ensemble of frequency bands of a DWDM system. For the purposes of this application, the term “frequency band” is used to denote an arbitrary fraction (proper or improper fraction) of a DWDM channel.  
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0007]    [0007]FIG. 1 shows a Poincare sphere with a coordinate system in accordance with the invention.  
         [0008]    [0008]FIG. 2 shows a polarization state tracing a length of arc on a Poincare sphere in accordance with the invention.  
         [0009]    [0009]FIG. 3 a  shows that the probability of a random polarization state ρ(ζ) having a value ζ on the Poincare sphere, is equal to sin ζ within a multiplicative constant in accordance with the invention.  
         [0010]    [0010]FIG. 3 b  shows an embodiment in accordance with the invention.  
         [0011]    [0011]FIG. 4 shows an embodiment in accordance with the invention.  
         [0012]    [0012]FIG. 5 shows an exemplary single stage PMD compensator. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0013]    In accordance with the invention, a highly selective heterodyne polarimeter is typically used that employs direct phase measurement of the heterodyne beat term to estimate the polarization state. The heterodyne polarimeter used may be a phase sensitive optical heterodyne detector as described in detail in “Method and Apparatus for a Jones Vector Based Heterodyne Optical Polarimeter” referenced above and incorporated by reference. Alternatively, a heterodyne polarimeter as described by, for example, Roudas et al, in “Coherent heterodyne frequency-selective polarimeter for error signal generation in higher-order PMD compensators”, OFC 2002, WQ2, may also be used to determine the Stokes vector with sufficient frequency selectivity in accordance with the invention but requires sequential switching of the polarization state.  
         [0014]    Two parameters, α and ψ which describe the polarization state and are shown in FIG. 1 on Poincare sphere  100  are typically determined by the Jones vector based heterodyne polarimeter. The polarization state is described by a Jones vector:  
             V   =     (           cos                 α                    ψ        sin                 α           )             (   2   )                               
 
         [0015]    The description of the polarization state may be rewritten in terms of a normalized Stokes vector P using the same parameters, α and ψ from the Jones vector based heterodyne optical polarimeter:  
             P   =     (           cos                 2      α               sin                 2      αcosψ               sin                 2      αsinψ           )             (   3   )                               
 
         [0016]    Normalized Stokes vector P may be viewed as a position vector capable of locating any point on a unit radius Poincare sphere. Note that the fourth parameter of the Stokes vector that describes the degree of polarization is omitted. Eq. (3) defines the polarization state shown in FIG. 1 in Cartesian coordinates x, y, and z and is useful for describing a parameter which corresponds to the length of an arc on the unit Poincare sphere such as Poincare sphere  100  in FIG. 1. Note that the normalized Stokes vector may also be obtained, for example, from a heterodyne polarimeter as described in Roudas et al.  
         [0017]    Over a comparatively narrow frequency range, as for example, the frequency band related to the ITU grid of 100 GHz, the polarization state can be viewed as tracing an arc on Poincare sphere  100  that has an axis of rotation defined by the principal states of polarization. This behavior of the polarization start is characteristic for wavelength independent PMD known also as first order PMD. As the PMD increases, the path traced on Poincare sphere  100  may become more complex because the axis of rotation as defined by the principle states of polarization becomes wavelength dependent and moves about Poincare sphere  100 . It is typically a good assumption to take the axis of rotation as nearly stationary over a single ITU grid of 100 GHz as is done here although it is possible to deal with a more complex evolution of the polarization state. This requires subdividing path  110  traced on Poincare sphere  100  into shorter arcs  115 ,  116 , and  117  that each have a nearly stationary axis of rotation defined by the principle states of polarization. In practice, this may be achieved by determining the principle states of polarization from consecutive polarization measurements in accordance with Eq. (6).  
         [0018]    The angle of rotation Δθ provides a measure of PMD and the corresponding PMD induced delay is determined in accordance with the invention from Eq. (1). As noted above, A v in Eq. (1) denotes the range of optical frequencies over which the measurement of the polarization state is performed. In accordance with the invention, it is possible to work with small optical signals because heterodyning offers high dynamic range. Small optical signals typically occur on the tails of the typical optical spectrum describing, for example, non-return to zero or return to zero modulation. Hence, Δ may be measured well below (20 to 40 dB) the peak of the spectrum.  
         [0019]    With reference to FIG. 2, the length of arc  222  depends on the position of the polarization state with respect to axis of rotation  210  which is the birefringence axis. The length of arc  222  provides a measure of the PMD impairment. PMD impairment is the polarization dispersion observed in a particular optical channel having some polarization state. The length of arc  222  may be used as a feedback signal that controls a PMD compensator (see FIG. 5) to provide for birefringence compensation by adjusting the relative optical path length of the fast and slow polarization states.  
         [0020]    The length of arc  222  is not typically the distance between the endpoints P 1  and P N . The distance between P 1  and P N  is given by the fractional length of the great circle that lies between them. Arc  222  is taken to contain the points P 1 , P 2  . . . , P N  where each point P i =(x i , y i , z i ) is described in Cartesian coordinates according to Eq. (3). The angles α and ψ are typically output from the Jones vector based heterodyne optical polarimeter referenced above.  
         [0021]    The length of arc  222  is typically approximated by summing the distances between the individual points P 1 , P 2  . . . , P N  forming arc  222 . Because the radius of Poincare sphere  100  is unity, a suitable selection criteria for choosing the points P 1 , P 2  . . . , P N  is that the distance between the points be a small fraction of 1, for example in the range of 0.01 to 0.1. The distance from point P i  to point P i+1 , where (x i , y i , z i ) and (x i+1 , y i+1 , z i+1 ) are the respective Cartesian coordinates, is approximated using the distance formula:  
           d   i ={square root}{square root over (( x   i   −x   i+1 ) 2 +( y   i   −y   i+1 ) 2 +( z   1   −z   i+1 ) 2 )}  (4)  
         [0022]    The approximate length of arc  222 , L, which is a measure of the PMD impairment, is then given by:  
             L   ≈       ∑   i                     d   i               (   5   )                               
 
         [0023]    To determine the rotation angle Δθ that subtends arc  222  it is necessary to find the axis of rotation determined by the principle polarization states. The vector axis of rotation or the axis of birefringence is orthogonal to the plane defined by any three distinct points that make up arc  222 , for example, points P 1 , P N/2 , P N  which can be viewed as unit vectors from the origin to the respective coordinates on the surface of Poincare sphere  200 . Hence, the vector axis of rotation lying along the principle polarization state can be determined from the cross product:  
           {right arrow over (X)} =(           N/2 −           1 )×(           N −           N/2 )  (6)  
         [0024]    which after normalization becomes (note the hat indicates a unit vector):  
               X   ^     =       X   →            X   →                    (   7   )                               
 
         [0025]    The angle Γ, as shown in FIG. 2 can be typically found from the cross product of           with            i  for polarization states represented by points P 1  . . . P N  that form arc  222  on Poincare sphere  200  with principle polarization state          . Note the angle Γ is fixed for any point on arc  222 . This allows determination of the radius r corresponding to polarization evolution arc  222 :  
           r =sin Γ=|         ×           i|   (8)  
         [0026]    The rotation angle Δθ subtends arc  222  and may be found explicitly by constructing two vectors of length r that extend from           to points P 1  and P N , respectively, and that lie in the plane of arc  222 . The two required vectors are given by            n −(           i ·         )          and            1 −(           i ·         )         . The normalized dot product of the two vectors yields cos Δθ where Δθ is the angle between the two vectors by construction. The rotation angle Δθ is then given by:  
             Δθ   =         cos     -   1            [         cos                 ΔΨ     -       cos   2        Γ           sin   2        Γ       ]       ≈     L   r               (   9   )                               
 
         [0027]    where cos Δψ=           1 ·           N , cos Γ=           i ·          and sin Γ is given by Eq. (8) with PMD then being determined by Eq. (1).  
         [0028]    Another parameter other than the length of the arc that may be used as a feedback signal to a PMD compensator is the degree of polarization (DOP) as described by N. Kikuchi, “Analysis of signal degree of polarization degradation used to control signal for optical polarization mode dispersion compensation,” in Journal of Lightwave Technology, Vol. 19, No. 4, 2001, pp.480-486. The DOP of an optical signal reflects the degree of waveform degradation caused by PMD and therefore the amount of DOP decrease corresponds to the amount of signal pulse distortion caused by PMD.  
         [0029]    If a specific channel is affected more by the PMD, the degree of polarization of the channel is less than that of a channel whose optical frequencies are predominately in a single polarization state which necessitates that DOP≈1. The DOP is closely related to the optical spectrum and the distribution of polarization states over that frequency band. Both the optical spectrum and distribution of polarization states over the spectrum are measured by the phase sensitive heterodyne polarimeter that uses a swept local oscillator and is described in “Method and Apparatus for a Jones Vector based Heterodyne Optical Polarimeter” and is incorporated by reference. The optical spectrum is described by power spectral density function ρ(v) while the distribution of polarization states may, for example, be represented on Poincare sphere  200 . The DOP may be defined by the centroid of arc  222  on the surface of Poincare sphere  200  and is equal to the distance of the centroid from the center of Poincare sphere  200 . If the centroid lies at the center of Poincare sphere  200  then arc  222  is a great circle and the DOP is zero. Similarly, if all frequencies in a spectrum of a single channel have the same polarization state, then, the centroid lies on the surface of Poincare sphere  200  and the DOP=1.  
         [0030]    For clarity, it has been assumed above that the power distribution is uniform. For cases of a non-uniform spectrum the determination of the centroid must include a power spectral density function ρ(v). Arc  222  on Poincare sphere  200  can be parameterized in terms of the frequency v where v=v 0 +yt. Hence, arc  222  can be described parametrically by functions x(v), y(v), z(v) that form the Stokes vector S of Eq. (3). The functions x(v), y(v), z(v) represent the normalized components of the Stokes vector for the 0° linear, 45° linear and the right circular polarized components. The centroid coordinates x 0 , y 0 , z 0  may be determined by calculating the individual Cartesian coordinates:  
               x   0     =       ∫       ρ        (   v   )            x        (   v   )               v           ∫       ρ        (   v   )               v                   (   10   )                 y   0     =       ∫       ρ        (   v   )            y        (   v   )               v           ∫       ρ        (   v   )               v                   (   11   )                 z   0     =       ∫       ρ        (   v   )            z        (   v   )               v           ∫       ρ        (   v   )               v                   (   12   )                               
 
         [0031]    where integration is performed over the spectral width of a DWDM channel. Hence, the DOP is given by:  
           DOP={square root}{square root over (x 0   2 +y 0   2 +z 0   2 )}   (13)  
         [0032]    To obtain accurate determinations of PMD it is desirable to have a long arc to obtain reliable estimates of the angle of rotation Δθ. One embodiment in accordance with the invention controls the polarization of the signal in a given frequency band to ensure that the length of the arc is not near a minimum. The polarization of the signal in the frequency band is typically sequentially switched between states that are 90° with respect to each other in the reference frame of Poincare sphere  200 .  
         [0033]    Typically, multiple DWDM frequency bands transmitted through predominately the same optical network have uncorrelated and random polarization states. Because the polarization states are random, the length of the arcs for polarization evolution vary depending on how near the individual polarization states are to the corresponding principle polarization states. Knowing the expected length of the arc for the random polarization states allows determination of the rotation angle Δθ.  
         [0034]    With reference to FIG. 3 a , the probability of a random polarization state ρ(ζ) having a value ζ, is equal to sin ζ within a multiplicative constant and is equal to the radius of circle  310  on Poincare sphere  300 . Circle  310  is formed by all polarization states having the particular value of which is equal to 2α (see Eqs (2) and (3)). If the principle polarization states define axis  320  as shown in FIG. 3 a , the probability that a random polarization state is a principle polarization state approaches zero while the most probable polarization has ζ=π/2. The length of an arc on Poincare sphere  300  can also be expressed in terms of the angle ζ. The length of an arc L (ζ) on Poincare sphere  300  is proportional to the radius of circle  310 , sin ζ, and is equal to Δθ sin ζ where Δθ is the rotation angle around the axis of birefringence  320 .  
         [0035]    Given the probability density function ρ(ζ) and the function for the length of the arc L(ζ), the expected value of the length of the arc can be determined from:  
               L   _     =       ∫       L        (   ζ   )            ρ        (   ζ   )               ζ           ∫       ρ        (   ζ   )               ζ                   (   14   )                               
 
         [0036]    where {overscore (L)} is the mean or expected value for the length of the arc. By substituting for L(ζ) and ρ(ζ):  
               L   _     =       Δθ          ∫   0   π          sin                 ζsinζ                      ζ               ∫   0   π          sin                 ζ                      ζ                   (   15   )                               
 
         [0037]    that yields:  
               L   _     =     Δθ          π   4     .               (   16   )                               
 
         [0038]    The mean length of the arc may also be calculated as an average of the length of the individual arcs. The length of an individual arc may determined using the method described above for a single DWDM frequency band. Hence an average length {overscore (L)} may be determined  
               L   _     =       1   M            ∑   j   M          L   j                 (   17   )                               
 
         [0039]    where the average is calculated over M measured DWDM frequency bands. Combining Eqs.(16) and (17) gives for Δθ:  
               Δ                 θ     =       4     π                 M              ∑   j                M              L   j     .                 (   18   )                               
 
         [0040]    Eq.(18) approximates the rotation angle Δθ from the average length of the arcs and hence estimates PMD from an average PMD impairment. It is assumed that all the lengths L i  are measured for the same spectral width Δ v as described above. Then the PMD is determined using Eq. (1).  
         [0041]    In accordance with an embodiment of the invention, M measurements may be made on a single frequency band instead of measuring M frequency bands if the frequency of measurements is low enough to ensure that the measurements are uncorrelated. This assumes that some birefringence wander is always present in an optical network which results in polarization state wander. Typical sources for birefringence wander are environmental fluctuations such as temperature.  
         [0042]    [0042]FIG. 3 b  shows an embodiment in accordance with the invention for determining the PMD and PMD impairments along an optical signal path. Note that PMD impairment typically varies from frequency band to frequency band and typically a different PMD impairment will be associated with each frequency band. In step  351  multiple frequency bands of optical signals are transmitted over the optical fiber. The multiple frequency bands may be generated sequentially using a tunable swept laser or by a number of different laser sources. A single frequency band may be created by intensity modulating a laser directly or typically externally by using an intensity modulator. In step  352  the polarization of each optical band is measured over its spectral width at a receiver location that is sufficiently far from the transmitter that birefringence and hence first order PMD is an important effect. From the measured polarization parameters, such as, for example, α and ψ, the associated polarization states are determined in step  353 . In step  354 , a parameter is computed to determine the PMD impairment along the optical fiber path for each measured frequency band. Then the PMD is calculated, using for example, Eqs. (18) and (1).  
         [0043]    [0043]FIG. 4 shows a simplified block diagram for a typical optical digital communication system which is typically affected by PMD in accordance with an embodiment of the invention. Tunable laser source  405  typically operating around 1.55 microns is coupled to modulator  415  which is driven by modulator driver  410 . Note that in typical implementations of an optical digital communication system there is typically more than one tunable laser source. The input pulses typically couple into both the slow and fast polarization modes which results in PMD distortion over longer transmission paths. The PMD impairment depends on the polarization state of laser  405 . Amplifiers  420 ,  425 ,  430  and  435  amplify the signal along optical fiber path  480 . Demultiplexer  440  routes the optical signal on a wavelength basis to receiver  445  that is typically one of many, which in turn relays the signal to a 1.3 micron intra-office link from which the signal proceeds to exchange  455 . Without correction, the optical signal at receiver  445  typically suffers from PMD due to birefringence associated with the optical fiber path. Optical heterodyne polarimeter  475  is optically coupled to optical path  480  to measure the polarization of signals traveling over optical fiber path  480 . Processor  490  is coupled to optical heterodyne polarimeter  475 . Processor  490  typically calculates the PMD impairment and PMD induced delay as discussed above. The PMD induced delay information is used to adjust PMD compensator  510  to remove the first order PMD impairment. Alternatively, the PMD impairment information may be used to assist with electronic-based methods for mitigation of the PMD.  
         [0044]    [0044]FIG. 5 shows an exemplary single stage PMD compensator  510 . Signal  505  enters polarizing beamsplitter  520  from polarization controller  507 . Polarizing beam splitter  520  separates the signal into faster polarized state  530  and slower polarized state  540 . The path length for polarized state  530  is adjustable using moveable corner cube mirror  565 . Polarized state  530  is recombined with polarized state  540  in beam splitter  525  and the combined signal is launched into fiber  506 . Moveable mirror  565  is used to adjust the path length for polarized state  530  so that it is delayed by the PMD induced delay τ determined in accordance with the invention as described above. Hence, PMD compensator  510  serves to remove the first order PMD impairment due to the birefringence of the optical fiber by delaying faster polarized state  530  with respect to slower polarized state  540  by the PMD induced delay T. Operationally, PMD compensator  510  may theoretically be inserted anywhere between modulator  415  and receiver  445  in the optical digital communication system of FIG. 4 when the communications system is sufficiently linear. Therefore, if the total PMD induced delay is τ, polarized state  530  may be predelayed by τ at modulator  415  so that both polarized states  530  and  540  are “in phase” at receiver  445 . Typically, PMD compensator  510  is inserted before receiver  445 .  
         [0045]    Alternatively, instead of using polarizing beam splitters and spatially separating optical waves in orthogonal polarization states one can use a birefringent element in the form of a wave plate or a section of polarization maintaining (PM) optical fiber. The use of multi-stage compensators allows for the compensation of first order and second order PMD. The advantages and disadvantages of typical PMD compensation techniques are, for example, described by H. Sunnerud et al in “A Comparison Between Different PMD Compensation Techniques,”  Journal of Lightwave Technology , Vol. 30, No  3 , March 2002, pp. 368-378.  
         [0046]    While the invention has been described in conjunction with specific embodiments, it is evident to those skilled in the art that many alternatives, modifications, and variations will be apparent in light of the foregoing description. Accordingly, the invention is intended to embrace all other such alternatives, modifications, and variations that fall within the spirit and scope of the appended claims.