# Patent

## Title
Method based on stokes parameters for the adaptive adjustment of PMD compensators in optical fiber communication systems and compensator in accordance with said method

## Abstract

A method for the adaptive adjustment of a PMD compensator in optical fiber communication systems with the compensator comprising a cascade of adjustable optical devices through which passes an optical signal to be compensated and comprising the steps of computing the Stokes parameters S0, S1, S2, S3 in a number Q of different frequencies of the signal output from the compensator, producing control signals for parameters of at least some of said adjustable optical devices so as to make virtually constant said Stokes parameters computed at different frequencies. A compensator comprising a cascade of adjustable optical devices (12-14) through which passes an optical signal to be compensated, an adjustment system which takes the components y1(t) e y2(t) on the two orthogonal polarizations from the signal at the compensator output, and which comprises a controller (15, 16) which on the basis of said components computes the Stokes parameters S0, S1, S2, S3 in a number Q of different frequencies of the signal output by the compensator and which emits control signals for at least some of said adjustable optical devices so as to make virtually constant the Stokes parameters computed at the different frequencies.

## Background
The present invention relates to methods of adaptive adjustment of PMD compensators in optical fiber communication systems. The present invention also relates to a compensator in accordance with said method.
In optical fiber telecommunications equipment the need to compensate the effects of polarization mode dispersion (PMD) which occur when an optical signal travels in an optical fiber based connection is known.
It is known that PMD causes distortion and dispersion of optical signals sent over optical fiber connections making the signals distorted and dispersed. The different time delays among the various signal components in the various polarization states acquire increasing importance with the increase in transmission speeds. In modern optical fiber based transmission systems with ever higher frequencies (10 Gbit/s and more), accurate compensation of PMD effects becomes very important and delicate. This compensation must be dynamic and performed at adequate speed.
The general purpose of the present invention is to remedy the above mentioned shortcomings by making available a method of fast, accurate adaptive adjustment of a PMD compensator and a compensator in accordance with said method.
In view of this purpose it was sought to provide in accordance with the present invention a method for the adaptive adjustment of a PMD compensator in optical fiber communication systems with the compensator comprising a cascade of adjustable optical devices over which passes an optical signal to be compensated comprising the steps of computing the Stokes parameters S0, S1, S2, S3 in a number Q of different frequencies of the signal output from the compensator, producing control signals for parameters of at least some of said adjustable optical devices so as to make virtually constant said Stokes parameters computed at the different frequencies.
In accordance with the present invention it was also sought to realize a PMD compensator in optical fiber communication systems applying the method and comprising a cascade of adjustable optical devices over which passes an optical signal to be compensated and an adjustment system which takes the components y1(t) and y2(t) on the two orthogonal polarizations at the compensator output with the adjustment system comprising a controller which on the basis of said components taken computes the Stokes parameters S0, S1, S2, S3 in a number Q of different frequencies of the signal output from the compensator and which emits control signals for at least some of said adjustable optical devices so as to make virtually constant the Stokes parameters computed at the different frequencies.



To clarify the explanation of the innovative principles of the present invention and its advantages compared with the prior art there is described below with the aid of the annexed drawings a possible embodiment thereof by way of non-limiting example applying said principles. In the drawings—
FIG. 1 shows a block diagram of a PMD compensator with associated control circuit, and
FIG. 2 shows an equivalent model of the PMD compensator.



With reference to the FIGS FIG. 1 shows the structure of a PMD compensator designated as a whole by reference number 10. This structure consists of the cascade of some optical devices which receive the signal from the transmission fiber 11. The first optical device is a polarization controller 12 (PC) which allows modification of the optical signal polarization at its input. There are three polarization maintaining fibers 13 (PMF) separated by two optical rotators 14.
A PMF fiber is a fiber which introduces a predetermined differential unit delay (DGD) between the components of the optical signal on the two principal states of polarization (PSP) termed slow PSP and fast PSP.
In the case of the compensator shown in FIG. 1 the DGD delays at the frequency of the optical carrier introduced by the three PMFs are respectively τc, ατc and (1−α) τc with 0<α<1 and with τc and α which are design parameters.
An optical rotator is a device which can change the polarization of the optical signal upon its input by an angle θi (the figure shows θi for the first rotator and θ2 for the second) on a maximum circle on the Poincarè sphere.
An optical rotator is implemented in practice by means of a properly controlled PC.
In FIG. 1, x1(t) and x2(t) designate the components on the two PSPs of the optical signal at the compensator input whereas similarly y1(t) and y2(t) are the components of the optical signal at the compensator output.
The input-output behavior of each optical device is described here by means of the so called Jones transfer matrix H(ω) which is a 2×2 matrix characterized by frequency dependent components. Designating by W1(ω) e W2(ω) the Fourier transforms of the optical signal components at the device input the Fourier transforms Z1(ω) e Z2(ω) of the optical signal components at the device output are given by:




  
    
      
        
          (
          
            
              
                
                  
                    Z
                    1
                  
                  ⁡
                  
                    (
                    ω
                    )
                  
                
              
            
            
              
                
                  
                    Z
                    2
                  
                  ⁡
                  
                    (
                    ω
                    )
                  
                
              
            
          
          )
        
        =
        
          
            H
            ⁡
            
              (
              ω
              )
            
          
          ⁢
          
            (
            
              
                
                  
                    
                      W
                      1
                    
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                
              
              
                
                  
                    
                      W
                      2
                    
                    ⁡
                    
                      (
                      ω
                      )
                    
                  
                
              
            
            )
          
        
      
    
    
      
        (
        1
        )
      
    
  




Thus the Jones transfer matrix of the PC is:




  
    
      
        (
        
          
            
              
                
                  h
                  1
                
                ⁢
                
              
            
            
              
                h
                2
              
            
          
          
            
              
                
                  -
                  
                    h
                    2
                    *
                  
                
                ⁢
                
              
            
            
              
                h
                1
                *
              
            
          
        
        )
      
    
    
      
        (
        2
        )
      
    
  




where h1 e h2 satisfy the condition |h1|2+|h2|2=1 and are frequency independent.

Denoting by φ1 and φ2 the PC control angles, h1 and h2 are expressed by:

h1=−cos(φ2−φ1)+j sin(φ2−φ1)sin φ1  (3)

h2=−j sin(φ2−φ1)cos φ1

Clearly if the PC is controlled using other angles or voltages, different relationships will correlate these other parameters with h1 and h2. The straightforward changes in the algorithms for adaptive adjustment of the PMD compensator are discussed below.
Similarly, an optical rotator with rotation angle θi is characterized by the following Jones matrix:




  
    
      
        (
        
          
            
              
                
                  cos
                  ⁢
                  
                    
                  
                  ⁢
                  
                    θ
                    i
                  
                
                ⁢
                
              
            
            
              
                sin
                ⁢
                
                  
                
                ⁢
                
                  θ
                  i
                
              
            
          
          
            
              
                
                  
                    -
                    sin
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    θ
                    i
                  
                
                ⁢
                
              
            
            
              
                cos
                ⁢
                
                  
                
                ⁢
                
                  θ
                  i
                
              
            
          
        
        )
      
    
    
      
        (
        4
        )
      
    
  




The Jones transfer matrix of a PMF with DGD τi may be expressed as RDR−1 where D is defined as:




  
    
      
        D
        ⁢
        
          =
          ^
        
        ⁢
        
          (
          
            
              
                
                  e
                  
                    
                      jωτ
                      i
                    
                    /
                    2
                  
                
              
              
                0
              
            
            
              
                0
              
              
                
                  e
                  
                    
                      -
                      
                        jωτ
                        i
                      
                    
                    /
                    2
                  
                
              
            
          
          )
        
      
    
    
      
        (
        5
        )
      
    
  




and R is a unitary rotation matrix accounting for the PSPs' orientation. This matrix R may be taken as the identity matrix I without loss of generality when the PSPs of all the PMFs are aligned.

As shown in FIG. 1, to control the PMD compensator a controller 15 is needed to produce optical device control signals of the compensator computed on the basis of the quantities sent to it by a controller pilot 16 termed controller driver (CD).
The CD feeds the controller with the quantities needed to update the compensator optical device control parameters. As described below, these quantities will be extracted by the CD from the signals at the input and/or output of the compensator.
The controller will operate following the criterion described below and will use one of the two algorithms described below.
To illustrate the PMD compensator adaptive adjustment algorithms let us assume that the controller can directly control the parameters φ1, φ2, θ1 and θ2 which we consolidate in a vector θ defined as:

θ{circumflex over (=)}(φ1, φ2, θ1, θ2)T

If it is not so, in general there will be other parameters to control, for example some voltages, which will be linked to the previous ones in known relationships.
The time instants in which the update of the compensator parameters is realized are designated tn (con n=0, 1, 2 . . . , ), and Tu designates the time interval between two successive updates, thus tn+1=tn+Tu. In addition, θ(tn) designates the value of the compensator parameters after the nth update.
In accordance with the method of the present invention the criterion for adjusting the compensator parameters employs the so-called Stokes parameters. Computation of the Stokes parameters for an optical signal is well known to those skilled in the art and is not further described.
Again in accordance with the method the parameters θ of the compensator are adjusted to make constant the Stokes parameters computed at different frequencies on the compensator output signal. The four Stokes parameters S0, S1, S2 e S3 computed at the frequency fl are designated by:

S0|f=fl{circumflex over (=)}S0,l 

S1|f=fl{circumflex over (=)}S1,l 

S2|f=fl{circumflex over (=)}S2,l 

S3|f=fl{circumflex over (=)}S3,l 

Similarly, the Stokes parameters computed at the frequency fp are designated by S0,p, S1,p, S2,p e S3,p.
Using these Stokes parameters the following unitary vectors are constructed with components given by the three Stokes parameters S1, S2, S3 normalized at the parameter S0. (.)T below designates the transpose while (.)* designates the complex conjugate:




  
    
      (
      
        
          
            S
            
              1
              ,
              l
            
          
          
            S
            
              0
              ,
              l
            
          
        
        ,
        
          
            S
            
              2
              ,
              l
            
          
          
            S
            
              0
              ,
              l
            
          
        
        ,
        
          
            S
            
              3
              ,
              l
            
          
          
            S
            
              0
              ,
              l
            
          
        
        ,
      
      )
    
    T
  
  ⁢
  
    
  
  ⁢
  and
  ⁢
  
    
  
  ⁢
  
    
      (
      
        
          
            S
            
              1
              ,
              p
            
          
          
            S
            
              0
              ,
              p
            
          
        
        ,
        
          
            S
            
              2
              ,
              p
            
          
          
            S
            
              0
              ,
              p
            
          
        
        ,
        
          
            S
            
              3
              ,
              p
            
          
          
            S
            
              0
              ,
              p
            
          
        
        ,
      
      )
    
    T
  




In the absence of PMD these two vectors are parallel. Consequently, if their quadratic Euclidean distance is considered G1p(θ):




  
    
      
        
          
            G
            lp
          
          ⁡
          
            (
            θ
            )
          
        
        =
        
          
            
              (
              
                
                  
                    S
                    
                      1
                      ,
                      l
                    
                  
                  
                    S
                    
                      0
                      ,
                      l
                    
                  
                
                -
                
                  
                    S
                    
                      1
                      ,
                      p
                    
                  
                  
                    S
                    
                      0
                      ,
                      p
                    
                  
                
              
              )
            
            2
          
          +
          
            
              (
              
                
                  
                    S
                    
                      2
                      ,
                      l
                    
                  
                  
                    S
                    
                      0
                      ,
                      l
                    
                  
                
                -
                
                  
                    S
                    
                      2
                      ,
                      p
                    
                  
                  
                    S
                    
                      0
                      ,
                      p
                    
                  
                
              
              )
            
            2
          
          +
          
            
              (
              
                
                  
                    S
                    
                      3
                      ,
                      l
                    
                  
                  
                    S
                    
                      0
                      ,
                      l
                    
                  
                
                -
                
                  
                    S
                    
                      3
                      ,
                      p
                    
                  
                  
                    S
                    
                      0
                      ,
                      p
                    
                  
                
              
              )
            
            2
          
        
      
    
    
      
        (
        6
        )
      
    
  




which is a function of the parameters θ of the PMD compensator it will be zero when the PMD is compensated at the two frequencies considered fl and fp.

Now consider a number Q of frequencies fl, l=1, 2, . . . , Q. Compute the Stokes parameters at these frequencies and construct the corresponding units defined as explained above, i.e. with components given by the three Stokes parameters S1, S2, S3 normalized with respect to the parameter S0. All these units are parallel if and only if the sum of their quadratic Euclidean distances is zero.
Consequently, to adaptively adjust the PMD compensator parameters we define the function G(θ) which is to be minimized as the sum of the quadratic distances G1p(θ) with 1,p=1, 2, . . . , Q, i.e. the sum of the quadratic distances of the pair of vectors at the different frequencies fl and fp, for l,p=1, 2, . . . Q:




  
    
      
        
          G
          ⁡
          
            (
            θ
            )
          
        
        ⁢
        
          =
          ^
        
        ⁢
        
          
            ∑
            
              l
              =
              2
            
            Q
          
          ⁢
          
            
              ∑
              
                p
                =
                1
              
              
                l
                -
                1
              
            
            ⁢
            
              
                G
                lp
              
              ⁡
              
                (
                θ
                )
              
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




The update rule for the compensator parameters to be used in accordance with the present invention are:




  
    
      
        
          
            
              
                
                  
                    
                      
                        φ
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          
                            n
                            +
                            1
                          
                        
                        )
                      
                    
                    =
                    
                      
                        
                          φ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      -
                      
                        γ
                        ⁢
                        
                          
                            ∂
                            
                              G
                              ⁡
                              
                                (
                                θ
                                )
                              
                            
                          
                          
                            ∂
                            
                              φ
                              1
                            
                          
                        
                      
                    
                  
                  
                
                
                  θ
                  =
                  
                    θ
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                    =
                    
                      
                        
                          φ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      -
                      
                        γ
                        ⁢
                        
                          
                            ∑
                            
                              l
                              =
                              2
                            
                            Q
                          
                          ⁢
                          
                            
                              ∑
                              
                                p
                                =
                                1
                              
                              
                                l
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                ∂
                                
                                  
                                    G
                                    lp
                                  
                                  ⁡
                                  
                                    (
                                    θ
                                    )
                                  
                                
                              
                              
                                ∂
                                
                                  φ
                                  1
                                
                              
                            
                          
                        
                      
                    
                  
                  
                
                
                  θ
                  =
                  
                    θ
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                
                  
                    φ
                    2
                  
                  ⁡
                  
                    (
                    
                      t
                      
                        n
                        +
                        1
                      
                    
                    )
                  
                
                =
                
                  
                    
                      
                        φ
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                    -
                    
                      γ
                      ⁢
                      
                        
                          ∂
                          
                            G
                            ⁡
                            
                              (
                              θ
                              )
                            
                          
                        
                        
                          ∂
                          
                            φ
                            2
                          
                        
                      
                    
                  
                  ⁢
                  
                    ❘
                    
                      θ
                      =
                      
                        θ
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                    
                  
                
              
            
          
          
            
              
                
                  
                    =
                    
                      
                        
                          φ
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      -
                      
                        γ
                        ⁢
                        
                          
                            ∑
                            
                              l
                              =
                              2
                            
                            Q
                          
                          ⁢
                          
                            
                              ∑
                              
                                p
                                =
                                1
                              
                              
                                l
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                ∂
                                
                                  
                                    G
                                    lp
                                  
                                  ⁡
                                  
                                    (
                                    θ
                                    )
                                  
                                
                              
                              
                                ∂
                                
                                  φ
                                  2
                                
                              
                            
                          
                        
                      
                    
                  
                  
                
                
                  θ
                  =
                  
                    θ
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                
                  
                    
                      
                        θ
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          
                            n
                            +
                            1
                          
                        
                        )
                      
                    
                    =
                    
                      
                        
                          θ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      -
                      
                        γ
                        ⁢
                        
                          
                            ∂
                            
                              G
                              ⁡
                              
                                (
                                θ
                                )
                              
                            
                          
                          
                            ∂
                            
                              θ
                              1
                            
                          
                        
                      
                    
                  
                  
                
                
                  θ
                  =
                  
                    θ
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                
              
            
          
          
            
              
                
                  
                    =
                    
                      
                        
                          φ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      -
                      
                        γ
                        ⁢
                        
                          
                            ∑
                            
                              l
                              =
                              2
                            
                            Q
                          
                          ⁢
                          
                            
                              ∑
                              
                                p
                                =
                                1
                              
                              
                                l
                                -
                                1
                              
                            
                            ⁢
                            
                              
                                ∂
                                
                                  
                                    G
                                    lp
                                  
                                  ⁡
                                  
                                    (
                                    θ
                                    )
                                  
                                
                              
                              
                                ∂
                                
                                  θ
                                  1
                                
                              
                            
                          
                        
                      
                    
                  
                  
                
                
                  θ
                  =
                  
                    θ
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                
                  
                    θ
                    2
                  
                  ⁡
                  
                    (
                    
                      t
                      
                        n
                        +
                        1
                      
                    
                    )
                  
                
                =
                
                  
                    
                      
                        θ
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                    -
                    
                      γ
                      ⁢
                      
                        
                          ∂
                          
                            G
                            ⁡
                            
                              (
                              θ
                              )
                            
                          
                        
                        
                          ∂
                          
                            θ
                            2
                          
                        
                      
                    
                  
                  ⁢
                  
                    ❘
                    
                      θ
                      =
                      
                        θ
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                    
                  
                
              
            
          
          
            
              
                =
                
                  
                    
                      
                        θ
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                    -
                    
                      γ
                      ⁢
                      
                        
                          ∑
                          
                            l
                            =
                            2
                          
                          Q
                        
                        ⁢
                        
                          
                            ∑
                            
                              p
                              =
                              1
                            
                            
                              l
                              -
                              1
                            
                          
                          ⁢
                          
                            
                              ∂
                              
                                
                                  G
                                  lp
                                
                                ⁡
                                
                                  (
                                  θ
                                  )
                                
                              
                            
                            
                              ∂
                              
                                θ
                                2
                              
                            
                          
                        
                      
                    
                  
                  ⁢
                  
                    ❘
                    
                      θ
                      =
                      
                        θ
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




where γ>0 is a scale factor which controls the amount of the adjustment.

In vector notation this means that the vector of the compensator parameters is updated by adding a new vector with its norm proportionate to the norm of the gradient of G(θ) and with opposite direction, i.e. with all its components having their sign changed. This way, we are sure to move towards a relative minimum of the function G(θ).
All this is equivalent to:




  
    
      
        
          
            
              
                θ
                ⁡
                
                  (
                  
                    t
                    
                      n
                      +
                      1
                    
                  
                  )
                
              
              =
              
                
                  θ
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
                -
                
                  γ
                  ⁢
                  
                    ∇
                    
                      G
                      ⁡
                      
                        (
                        θ
                        )
                      
                    
                  
                
              
            
            
          
          
            θ
            =
            
              θ
              ⁡
              
                (
                
                  t
                  n
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        =
        
          
            
              θ
              ⁡
              
                (
                
                  t
                  n
                
                )
              
            
            -
            
              γ
              ⁢
              
                
                  ∑
                  
                    l
                    =
                    2
                  
                  Q
                
                ⁢
                
                  
                    ∑
                    
                      p
                      =
                      1
                    
                    
                      l
                      -
                      1
                    
                  
                  ⁢
                  
                    ∇
                    
                      
                        G
                        lp
                      
                      ⁡
                      
                        (
                        θ
                        )
                      
                    
                  
                
              
            
          
          ⁢
          
            ❘
            
              θ
              =
              
                θ
                ⁡
                
                  (
                  
                    t
                    n
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




A simplified version of (9) consists of an update by means of a constant norm vector and therefore an update which uses only the information on the direction of ∇G(θ). In this case the update rule becomes.




  
    
      
        
          θ
          ⁡
          
            (
            
              t
              
                n
                +
                1
              
            
            )
          
        
        =
        
          
            
              
                θ
                ⁡
                
                  (
                  
                    t
                    n
                  
                  )
                
              
              -
              
                γsign
                ⁢
                
                  ∇
                  
                    G
                    ⁡
                    
                      (
                      θ
                      )
                    
                  
                
              
            
            ⁢
            
              ❘
              
                θ
                =
                
                  θ
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          =
          
            
              
                θ
                ⁡
                
                  (
                  
                    t
                    n
                  
                  )
                
              
              -
              
                γsign
                ⁢
                
                  
                    ∑
                    
                      l
                      =
                      2
                    
                    Q
                  
                  ⁢
                  
                    
                      ∑
                      
                        p
                        =
                        1
                      
                      
                        l
                        -
                        1
                      
                    
                    ⁢
                    
                      ∇
                      
                        
                          G
                          lp
                        
                        ⁡
                        
                          (
                          θ
                          )
                        
                      
                    
                  
                
              
            
            ⁢
            
              ❘
              
                θ
                =
                
                  θ
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




where sign (z) designates a vector with unitary components and of the same sign as the components or the vector z.

Two methods are now described for computing the gradient of the G(θ) function and obtaining the required control parameters.
First Method
To implement the update rules (8) the partial derivatives of G(θ) for θ=θ(tn) can be computed using the following five-step procedure.

    
    
        Step 1. find the value of G[θ(tn)]=G[φ1(tn), φ2(tn), θ1(tn), θ2(tn)] at iteration n. To do this, in the time interval (tn, tn+Tu/5) the Stokes parameters at the above mentioned Q frequencies are derived and the value of the function G(θ) is computed using equations (6) and (7).
        Step 2. find the partial derivative
    
    






  
    
      
        ∂
        
          G
          ⁡
          
            (
            θ
            )
          
        
      
      
        ∂
        
          φ
          1
        
      
    
    
  
  
    θ
    =
    
      θ
      ⁡
      
        (
        
          t
          n
        
        )
      
    
  




    
    
         at iteration n. To do this, parameter φ1 is set at φ1(tn)+Δ while the other parameters are left unchanged. The corresponding value of G(θ), i.e. G[φ1(tn)+Δ, φ2(tn), θ1(tn), θ2(tn)], is computed as in step 1 but in the time interval (tn+Tu/5, tn+2Tu/5). The estimate of the partial derivative of G(θ) as a function of φ1 is computed as:
    
    






  
    
      
        
          
            
              
                ∂
                
                  G
                  ⁡
                  
                    (
                    θ
                    )
                  
                
              
              
                ∂
                
                  φ
                  1
                
              
            
            
          
          
            θ
            =
            
              θ
              ⁡
              
                (
                
                  t
                  n
                
                )
              
            
          
        
        ≅
        
          
            
              
                
                  
                    G
                    ⁡
                    
                      [
                      
                        
                          
                            
                              φ
                              1
                            
                            ⁡
                            
                              (
                              
                                t
                                n
                              
                              )
                            
                          
                          +
                          Δ
                        
                        ,
                        
                          
                            φ
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                        ,
                        
                          
                            θ
                            1
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                        ,
                        
                          
                            θ
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                      
                      ]
                    
                  
                  -
                
              
            
            
              
                
                  G
                  ⁡
                  
                    [
                    
                      
                        
                          φ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          φ
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          θ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          θ
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                    
                    ]
                  
                
              
            
          
          Δ
        
      
    
    
      
        (
        11
        )
      
    
  




    
    
        Step 3. Find the partial derivative:
    
    






  
    
      
        ∂
        
          G
          ⁡
          
            (
            θ
            )
          
        
      
      
        ∂
        
          φ
          2
        
      
    
    
  
  
    θ
    =
    
      θ
      ⁡
      
        (
        
          t
          n
        
        )
      
    
  




    
    
         at iteration n. To do this the parameter φ2 is set at φ2(tn)+Δ while the other parameters are left changed. The corresponding value of G(θ), i.e. G[φ1(tn), φ2(tn)+Δ, θ1(tn), θ2(tn)], )], is computed as in step 1 but in the time interval (tn2Tu/5, tn+3Tu/5). The estimate of the partial derivative of G(θ) with respect to φ2 is computed as:
    
    






  
    
      
        
          
            
              
                ∂
                
                  G
                  ⁡
                  
                    (
                    θ
                    )
                  
                
              
              
                ∂
                
                  φ
                  2
                
              
            
            
          
          
            θ
            =
            
              θ
              ⁡
              
                (
                
                  t
                  n
                
                )
              
            
          
        
        ≅
        
          
            
              
                
                  
                    G
                    ⁡
                    
                      [
                      
                        
                          
                            φ
                            1
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                        ,
                        
                          
                            
                              φ
                              2
                            
                            ⁡
                            
                              (
                              
                                t
                                n
                              
                              )
                            
                          
                          +
                          Δ
                        
                        ,
                        
                          
                            θ
                            1
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                        ,
                        
                          
                            θ
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                      
                      ]
                    
                  
                  -
                
              
            
            
              
                
                  G
                  ⁡
                  
                    [
                    
                      
                        
                          φ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          φ
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          θ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          θ
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                    
                    ]
                  
                
              
            
          
          Δ
        
      
    
    
      
        (
        12
        )
      
    
  




    
    
        Step 4: Find the partial derivative:
    
    






  
    
      
        ∂
        
          G
          ⁡
          
            (
            θ
            )
          
        
      
      
        ∂
        
          θ
          1
        
      
    
    
  
  
    θ
    =
    
      θ
      ⁡
      
        (
        
          t
          n
        
        )
      
    
  




    
    
         at iteration n. To do this, parameter θ1 is set at θ1(tn)+Δ while the other parameters are left unchanged, the corresponding value of G(θ), i.e. G[φ1(tn), φ2(tn), θ1(tn)+Δ, θ2(tn)], is computed as in Step 1 but in the time interval (tn+3Tu/5, tn+4Tu/5) and the estimate of the partial derivative of G(θ) with respect to θ1 is computed as:
    
    






  
    
      
        
          
            
              
                ∂
                
                  G
                  ⁡
                  
                    (
                    θ
                    )
                  
                
              
              
                ∂
                
                  θ
                  1
                
              
            
            
          
          
            θ
            =
            
              θ
              ⁡
              
                (
                
                  t
                  n
                
                )
              
            
          
        
        ≅
        
          
            
              
                
                  
                    G
                    ⁡
                    
                      [
                      
                        
                          
                            φ
                            1
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                        ,
                        
                          
                            φ
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                        ,
                        
                          
                            
                              θ
                              1
                            
                            ⁡
                            
                              (
                              
                                t
                                n
                              
                              )
                            
                          
                          +
                          Δ
                        
                        ,
                        
                          
                            θ
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                      
                      ]
                    
                  
                  -
                
              
            
            
              
                
                  G
                  ⁡
                  
                    [
                    
                      
                        
                          φ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          φ
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          θ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          θ
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                    
                    ]
                  
                
              
            
          
          Δ
        
      
    
    
      
        (
        13
        )
      
    
  




    
    
        Step 5: Find the partial derivative:
    
    






  
    
      
        ∂
        
          G
          ⁡
          
            (
            θ
            )
          
        
      
      
        ∂
        
          θ
          2
        
      
    
    
  
  
    θ
    =
    
      θ
      ⁡
      
        (
        
          t
          n
        
        )
      
    
  




    
    
         at iteration n. To do this the parameter φ2 is set at φ2(tn)+Δ while the other parameters are left changed. The corresponding value of G(θ), i.e. G[φ1(tn), φ2(tn), θ1(tn), θ2(tn)+Δ], is computed as in step 1 but in the time interval (tn+4Tu/5, tn+Tu). The estimate of the partial derivative of G(θ) with respect to φ2 is computed as:
    
    






  
    
      
        
          
            
              
                ∂
                
                  G
                  ⁡
                  
                    (
                    θ
                    )
                  
                
              
              
                ∂
                
                  θ
                  2
                
              
            
            
          
          
            θ
            =
            
              θ
              ⁡
              
                (
                
                  t
                  n
                
                )
              
            
          
        
        ≅
        
          
            
              
                
                  
                    G
                    ⁡
                    
                      [
                      
                        
                          
                            φ
                            1
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                        ,
                        
                          
                            φ
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                        ,
                        
                          
                            θ
                            1
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                        ,
                        
                          
                            
                              θ
                              2
                            
                            ⁡
                            
                              (
                              
                                t
                                n
                              
                              )
                            
                          
                          +
                          Δ
                        
                      
                      ]
                    
                  
                  -
                
              
            
            
              
                
                  G
                  ⁡
                  
                    [
                    
                      
                        
                          φ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          φ
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          θ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          θ
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                    
                    ]
                  
                
              
            
          
          Δ
        
      
    
    
      
        (
        14
        )
      
    
  




The above parameter update is done only after estimation of the gradient has been completed.
Note that in this case it is not necessary that the relationship between the control parameters of PC and optical rotators and the corresponding Jones matrices be known.
Indeed, the partial derivatives of the function with respect to the compensator control parameters are computed without knowledge of this relationship. Consequently if the control parameters are different from those assumed as an example and are for example some voltage or some other angle, we may similarly compute the partial derivative and update these different control parameters accordingly.
Lastly, it is noted that when this algorithm is used the CD must receive only the optical signal at the compensator output and must supply the controller with the Stokes parameters computed at the Q frequencies fl, l=1, 2, . . . , Q.
Second Method
When an accurate characterization of the PC and of each optical rotator is available the update rules can be expressed as a function of the signals on the two orthogonal polarizations at the compensator input and output.
In this case, for the sake of convenience it is best to avoid normalization of the three Stokes parameters S1, S2 e S3 with respect to S0 and use the function H(θ) defined as:




  
    
      
        
          
            H
            ⁡
            
              (
              θ
              )
            
          
          =
          
            
              ∑
              
                l
                =
                2
              
              Q
            
            ⁢
            
              
                ∑
                
                  p
                  =
                  1
                
                
                  l
                  -
                  1
                
              
              ⁢
              
                
                  H
                  lp
                
                ⁡
                
                  (
                  θ
                  )
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        where
      
    
    
      
        (
        15
        )
      
    
  
  
    
      
        
          
            H
            lp
          
          ⁡
          
            (
            θ
            )
          
        
        =
        
          
            
              (
              
                
                  S
                  
                    1
                    ,
                    l
                  
                
                -
                
                  S
                  
                    1
                    ,
                    p
                  
                
              
              )
            
            2
          
          +
          
            
              (
              
                
                  S
                  
                    2
                    ,
                    1
                  
                
                -
                
                  S
                  
                    2
                    ,
                    p
                  
                
              
              )
            
            2
          
          +
          
            
              (
              
                
                  S
                  
                    3
                    ,
                    1
                  
                
                -
                
                  S
                  
                    3
                    ,
                    p
                  
                
              
              )
            
            2
          
        
      
    
    
      
        (
        16
        )
      
    
  




Consequently we have new update rules similar to those expressed by equation (8) or equivalently (9) with the only change being that the new function H(θ) must substitute the previous G(θ).
Before describing how the gradient of this new function H(θ) is to be computed let us introduce for convenient an equivalent model of the PMD compensator.
Indeed it was found that the PMD compensator shown in FIG. 1 is equivalent to a two-dimensional transversal filter with four tapped delay lines (TDL) combining the signals on the two principal polarization states (PSP). This equivalent model is shown in FIG. 2 where:

c1{circumflex over (=)}cos θ1 cos θ2h1 

c2{circumflex over (=)}−sin θ1 sin θ2h1 

c3{circumflex over (=)}−sin θ1 cos θ2h2*

c4{circumflex over (=)}−cos θ1 sin θ2h2*

c5{circumflex over (=)}cos θ1 cos θ2h2 

c6{circumflex over (=)}−sin θ1 sin θ2h2 

c7{circumflex over (=)}sin θ1 cos θ2h1*

c8{circumflex over (=)}cos θ1 sin θ2h1*  (18)

For the sake of convenience let c(θ)designate the vector whose components are the c1 in (17). It is noted that the tap coefficients ci of the four TDLs are not independent of each other. On the contrary, given four of them the others are completely determined by (17). In the FIG for the sake of clarity it is designated β=1−α.
The gradient of H1p(θ) with respect to θ is to be computed as follows:




  
    
      
        
          ∇
          
            
              H
              lp
            
            ⁡
            
              (
              θ
              )
            
          
        
        =
        
        ⁢
        
          4
          ⁢
          
            (
            
              
                S
                
                  1
                  ,
                  l
                
              
              -
              
                S
                
                  1
                  ,
                  p
                
              
            
            )
          
          ⁢
          Re
          ⁢
          
            {
            
              
                1
                
                  T
                  u
                
              
              ⁢
              
                
                  ∫
                  
                    t
                    n
                  
                  
                    t
                    
                      n
                      +
                      1
                    
                  
                
                ⁢
                
                  [
                  
                    
                      
                        
                          y
                          
                            1
                            ,
                            l
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          a
                          l
                          T
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    -
                    
                      
                        
                          y
                          
                            2
                            ,
                            l
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          b
                          l
                          T
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    -
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
            
              
              ⁢
              
                
                  
                    
                      y
                      
                        1
                        ,
                        p
                      
                      *
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    
                      a
                      p
                      T
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                +
                
                  
                    
                      y
                      
                        2
                        ,
                        p
                      
                      *
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    
                      b
                      p
                      T
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
              ]
            
            ⁢
            
              d
              tJ
            
          
          }
        
        +
      
    
  
  
    
      
        
        ⁢
        
          4
          ⁢
          
            (
            
              
                S
                
                  2
                  ,
                  l
                
              
              -
              
                S
                
                  2
                  ,
                  p
                
              
            
            )
          
          ⁢
          Re
          ⁢
          
            {
            
              
                1
                
                  T
                  u
                
              
              ⁢
              
                
                  ∫
                  
                    t
                    n
                  
                  
                    t
                    
                      n
                      +
                      1
                    
                  
                
                ⁢
                
                  [
                  
                    
                      
                        
                          y
                          
                            2
                            ,
                            l
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          a
                          l
                          T
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    +
                    
                      
                        
                          y
                          
                            1
                            ,
                            l
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          b
                          l
                          T
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    -
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
            
              
              ⁢
              
                
                  
                    
                      y
                      
                        2
                        ,
                        p
                      
                      *
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    
                      a
                      p
                      T
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                -
                
                  
                    
                      y
                      
                        1
                        ,
                        p
                      
                      *
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    
                      b
                      j
                      T
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
              ]
            
            ⁢
            
              d
              tJ
            
          
          }
        
        -
      
    
  
  
    
      
        
        ⁢
        
          4
          ⁢
          
            (
            
              
                S
                
                  3
                  ,
                  l
                
              
              -
              
                S
                
                  3
                  ,
                  p
                
              
            
            )
          
          ⁢
          Im
          ⁢
          
            {
            
              
                1
                
                  T
                  u
                
              
              ⁢
              
                
                  ∫
                  
                    t
                    n
                  
                  
                    t
                    
                      n
                      +
                      1
                    
                  
                
                ⁢
                
                  [
                  
                    
                      
                        
                          y
                          
                            2
                            ,
                            l
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          a
                          l
                          T
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    +
                    
                      
                        
                          y
                          
                            1
                            ,
                            l
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          b
                          l
                          T
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    -
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
            
            ⁢
            
              
                
                  
                    y
                    
                      2
                      ,
                      p
                    
                    *
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  
                    a
                    p
                    T
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              -
              
                
                  
                    y
                    
                      1
                      ,
                      p
                    
                    *
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  
                    b
                    p
                    T
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
            ]
          
          ⁢
          
            d
            tJ
          
        
        }
      
    
  




where:

    
    
        y1,l(t) and y2,l(t) are the signals y1(t) and y2(t) at the compensator output respectively filtered through a narrow band filter centered on the frequency fl (similarly for y1,p(t) and y2,p(t));
        al(t) and bl(t) are the vectors:
    
    






  
    
      
        
          
            a
            l
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          (
          
            
              
                
                  
                    x
                    
                      1
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      1
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        ατ
                        c
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      1
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        c
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      1
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        c
                      
                      -
                      
                        ατ
                        c
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        ατ
                        c
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        c
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        c
                      
                      -
                      
                        ατ
                        c
                      
                    
                    )
                  
                
              
            
          
          )
        
      
    
    
      
        
          
            b
            l
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          (
          
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                    *
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        2
                        ⁢
                        
                          τ
                          c
                        
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                    *
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        c
                      
                      -
                      
                        βτ
                        c
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                    *
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        c
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                    *
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        βτ
                        c
                      
                    
                    )
                  
                
              
            
            
              
                
                  -
                  
                    
                      x
                      
                        1
                        ,
                        l
                      
                      *
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          2
                          ⁢
                          
                            τ
                            c
                          
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  -
                  
                    
                      x
                      
                        1
                        ,
                        l
                      
                      *
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          c
                        
                        -
                        
                          βτ
                          c
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  -
                  
                    
                      x
                      
                        1
                        ,
                        l
                      
                      *
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          c
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  -
                  
                    
                      x
                      
                        1
                        ,
                        l
                      
                      *
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          βτ
                          c
                        
                      
                      )
                    
                  
                
              
            
          
          )
        
      
    
  




    
    
         with x1,l(t) and x2,l(t) which are respectively the signals x1(t) and x2(t) at the compensator input filtered by a narrow band filter centered on the frequency fl (similarly for y1,p(t) and y2,p(t));
        J is the Jacobean matrix of the transformation c=c(θ) defined as
    
    






  
    
      
        J
        ⁢
        
          =
          ^
        
        ⁢
        
          (
          
            
              
                
                  
                    ∂
                    
                      c
                      1
                    
                  
                  
                    ∂
                    
                      φ
                      1
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      1
                    
                  
                  
                    ∂
                    
                      φ
                      2
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      1
                    
                  
                  
                    ∂
                    
                      θ
                      1
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      1
                    
                  
                  
                    ∂
                    
                      θ
                      2
                    
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      c
                      2
                    
                  
                  
                    ∂
                    
                      φ
                      1
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      2
                    
                  
                  
                    ∂
                    
                      φ
                      2
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      2
                    
                  
                  
                    ∂
                    
                      θ
                      1
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      2
                    
                  
                  
                    ∂
                    
                      θ
                      2
                    
                  
                
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                ⋮
              
              
                ⋮
              
            
            
              
                
                  
                    ∂
                    
                      c
                      8
                    
                  
                  
                    ∂
                    
                      φ
                      1
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      8
                    
                  
                  
                    ∂
                    
                      φ
                      2
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      8
                    
                  
                  
                    ∂
                    
                      θ
                      1
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      8
                    
                  
                  
                    ∂
                    
                      θ
                      2
                    
                  
                
              
            
          
          )
        
      
    
    
      
        (
        18
        )
      
    
  




The parameters θ are updated in accordance with the rule




  
    
      
        
          
            
              θ
              ⁡
              
                (
                
                  t
                  
                    n
                    +
                    1
                  
                
                )
              
            
            =
            
              
                θ
                ⁡
                
                  (
                  
                    t
                    n
                  
                  )
                
              
              -
              
                γ
                ⁢
                
                  
                    ∑
                    
                      l
                      =
                      2
                    
                    Q
                  
                  ⁢
                  
                    
                      ∑
                      
                        p
                        =
                        1
                      
                      
                        l
                        -
                        1
                      
                    
                    ⁢
                    
                      ∇
                      
                        
                          H
                          lp
                        
                        ⁡
                        
                          (
                          θ
                          )
                        
                      
                    
                  
                
              
            
          
          
        
        
          θ
          =
          
            θ
            ⁡
            
              (
              
                t
                n
              
              )
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




or in accordance with the following simplified rule based only on the sign:





  
    
      
        
          
            
              θ
              ⁡
              
                (
                
                  t
                  
                    n
                    +
                    1
                  
                
                )
              
            
            =
            
              
                θ
                ⁡
                
                  (
                  
                    t
                    n
                  
                  )
                
              
              ⁢
              γsign
              ⁢
              
                
                  ∑
                  
                    l
                    =
                    2
                  
                  Q
                
                ⁢
                
                  
                    ∑
                    
                      p
                      =
                      1
                    
                    l1
                  
                  ⁢
                  
                    ∇
                    
                      
                        H
                        lp
                      
                      ⁡
                      
                        (
                        θ
                        )
                      
                    
                  
                
              
            
          
          
        
        
          θ
          =
          
            θ
            ⁡
            
              (
              
                t
                n
              
              )
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




When the control parameters are different from those taken as examples we will naturally have different relationships between these control parameters and the coefficients ci.
For example, if the PC is controlled by means of some voltages, given the relationship between these voltages and the coefficients h1 and h2 which appear in (2), by using the equations (17) we will be able to express the coefficients ci as a function of these new control parameters.
Consequently in computing the gradient of the function H(θ), the only change we have to allow for is the expression of the Jacobean matrix J, which has to be changed accordingly.
Lastly it is noted that when this second method is used the CD must receive the optical signals at the input and output of the compensator. The CD must supply the controller not only with the Stokes parameters for the optical signal at the compensator output and computed at the Q frequencies fl, l=1, 2, . . . , Q but also with the signals x1,l(t), x2,l(t), y1,l(t) e y2,l(t) corresponding to the Q frequencies fl, l=1, 2, . . . , Q.
It is now clear that the predetermined purposes have been achieved by making available an effective method for adaptive control of a PMD compensator and a compensator applying this method.
Naturally the above description of an embodiment applying the innovative principles of the present invention is given by way of non-limiting example of said principles within the scope of the exclusive right claimed here.

## Claims

1. Method for the adaptive adjustment of a PMD compensator in optical fiber communication systems with the compensator comprising a cascade of adjustable optical devices over which passes an optical signal to be compensated comprising the steps of:
computing the Stokes parameters S0, S1, S2, S3 in a number Q of different frequencies of the compensator output signal, and
producing control signals for parameters of at least some of said adjustable optical devices so as to make virtually constant said Stokes parameters computed at different frequencies.

2. Method in accordance with claim 1 comprising the steps of computing the Stokes parameters in pairs of frequencies fl and fp with l,p=1, 2, . . , Q, obtaining at the lth and pth frequencies of the Q frequencies the two series of Stokes parameters S0,l, S1,l, S2,l, S3,l and S0,p, S1,p, S2,p, S3,p, computing a vector function of each series of Stokes parameters and producing the control signals in such a manner that said vectors function of the two series of parameters are virtually parallel to each other.

3. Method in accordance with claim 2 in which said vectors are unitary norm vectors with components given by the Stokes parameters S1, S2, S3 normalized to the Stokes parameter S0, i.e.:



  
    
      (
      
        
          
            S
            
              1
              ,
              l
            
          
          
            S
            
              0
              ,
              l
            
          
        
        ,
        
          
            S
            
              2
              ,
              l
            
          
          
            S
            
              0
              ,
              l
            
          
        
        ,
        
          
            S
            
              3
              ,
              l
            
          
          
            S
            
              0
              ,
              l
            
          
        
        ,
      
      )
    
    T
  




  
    
      
        and
        ⁢
        
          

        
        (
        
          
            
              S
              
                1
                ,
                p
              
            
            
              S
              
                0
                ,
                p
              
            
          
          ,
          
            
              S
              
                2
                ,
                p
              
            
            
              S
              
                0
                ,
                p
              
            
          
          ,
          
            
              S
              
                3
                ,
                p
              
            
            
              S
              
                0
                ,
                p
              
            
          
          ,
        
        )
      
      T
    
    .

4. Method in accordance with claim 3 in which is defined the function:



  
    
      G
      ⁡
      
        (
        θ
        )
      
    
    ⁢
    
      =
      ^
    
    ⁢
    
      
        ∑
        
          l
          =
          2
        
        Q
      
      ⁢
      
        
          ∑
          
            p
            =
            1
          
          
            l
            -
            1
          
        
        ⁢
        
          
            G
            lp
          
          ⁡
          
            (
            θ
            )
          
        
      
    
  




  
    
      with
      ⁢
      
        
      
      ⁢
      
        
          G
          lp
        
        ⁡
        
          (
          θ
          )
        
      
    
    =
    
      
        
          (
          
            
              
                S
                
                  1
                  ,
                  l
                
              
              
                S
                
                  0
                  ,
                  l
                
              
            
            -
            
              
                S
                
                  1
                  ,
                  p
                
              
              
                S
                
                  0
                  ,
                  p
                
              
            
          
          )
        
        2
      
      +
      
        
          (
          
            
              
                S
                
                  2
                  ,
                  l
                
              
              
                S
                
                  0
                  ,
                  l
                
              
            
            -
            
              
                S
                
                  2
                  ,
                  p
                
              
              
                S
                
                  0
                  ,
                  p
                
              
            
          
          )
        
        2
      
      +
      
        
          (
          
            
              
                S
                
                  3
                  ,
                  l
                
              
              
                S
                
                  0
                  ,
                  l
                
              
            
            -
            
              
                S
                
                  3
                  ,
                  p
                
              
              
                S
                
                  0
                  ,
                  p
                
              
            
          
          )
        
        2
      
    
  


and the control signals are produced to minimize said function G(θ).

5. Method in accordance with claim 4 in which the optical devices comprise a polarization controller with controllable angles φ1, φ2 and two rotators with controllable rotation angles respectively θ1 and θ2, and to minimize the function G(θ) the updating of φ1, φ2, θ1 and θ2 of the compensator observes the following rules to go from the nth iteration to the n+1th iteration:



  
    
      
        
          
            
              
                
                  φ
                  1
                
                ⁡
                
                  (
                  
                    t
                    
                      n
                      +
                      1
                    
                  
                  )
                
              
              =
              
                
                  
                    φ
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
                -
                
                  γ
                  ⁢
                  
                    
                      ∂
                      
                        G
                        ⁡
                        
                          (
                          θ
                          )
                        
                      
                    
                    
                      ∂
                      
                        φ
                        1
                      
                    
                  
                
              
            
            
          
          
            θ
            =
            
              θ
              ⁡
              
                (
                
                  t
                  n
                
                )
              
            
          
        
      
    
    
      
        
          =
          
            
              
                
                  φ
                  1
                
                ⁡
                
                  (
                  
                    t
                    n
                  
                  )
                
              
              -
              
                γ
                ⁢
                
                  
                    ∑
                    
                      l
                      =
                      2
                    
                    Q
                  
                  ⁢
                  
                    
                      ∑
                      
                        p
                        =
                        1
                      
                      
                        l
                        -
                        1
                      
                    
                    ⁢
                    
                      
                        ∂
                        
                          
                            G
                            lp
                          
                          ⁡
                          
                            (
                            θ
                            )
                          
                        
                      
                      
                        ∂
                        
                          φ
                          1
                        
                      
                    
                  
                
              
            
            ⁢
            
              |
              
                θ
                =
                
                  θ
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
              
            
          
        
      
    
  




  
    
      
        
          
            
              
                
                  φ
                  2
                
                ⁡
                
                  (
                  
                    t
                    
                      n
                      +
                      1
                    
                  
                  )
                
              
              =
              
                
                  
                    φ
                    2
                  
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
                -
                
                  γ
                  ⁢
                  
                    
                      ∂
                      
                        G
                        ⁡
                        
                          (
                          θ
                          )
                        
                      
                    
                    
                      ∂
                      
                        φ
                        2
                      
                    
                  
                
              
            
            
          
          
            θ
            =
            
              θ
              ⁡
              
                (
                
                  t
                  n
                
                )
              
            
          
        
      
    
    
      
        
          =
          
            
              
                
                  φ
                  2
                
                ⁡
                
                  (
                  
                    t
                    n
                  
                  )
                
              
              -
              
                γ
                ⁢
                
                  
                    ∑
                    
                      l
                      =
                      2
                    
                    Q
                  
                  ⁢
                  
                    
                      ∑
                      
                        p
                        =
                        1
                      
                      
                        l
                        -
                        1
                      
                    
                    ⁢
                    
                      
                        ∂
                        
                          
                            G
                            lp
                          
                          ⁡
                          
                            (
                            θ
                            )
                          
                        
                      
                      
                        ∂
                        
                          φ
                          2
                        
                      
                    
                  
                
              
            
            ⁢
            
              |
              
                θ
                =
                
                  θ
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
              
            
          
        
      
    
  




  
    
      
        
          
            
              
                
                  θ
                  1
                
                ⁡
                
                  (
                  
                    t
                    
                      n
                      +
                      1
                    
                  
                  )
                
              
              =
              
                
                  
                    θ
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
                -
                
                  γ
                  ⁢
                  
                    
                      ∂
                      
                        G
                        ⁡
                        
                          (
                          θ
                          )
                        
                      
                    
                    
                      ∂
                      
                        θ
                        1
                      
                    
                  
                
              
            
            
          
          
            θ
            =
            
              θ
              ⁡
              
                (
                
                  t
                  n
                
                )
              
            
          
        
      
    
    
      
        
          =
          
            
              
                
                  θ
                  1
                
                ⁡
                
                  (
                  
                    t
                    n
                  
                  )
                
              
              -
              
                γ
                ⁢
                
                  
                    ∑
                    
                      l
                      =
                      2
                    
                    Q
                  
                  ⁢
                  
                    
                      ∑
                      
                        p
                        =
                        1
                      
                      
                        l
                        -
                        1
                      
                    
                    ⁢
                    
                      
                        ∂
                        
                          
                            G
                            lp
                          
                          ⁡
                          
                            (
                            θ
                            )
                          
                        
                      
                      
                        ∂
                        
                          θ
                          1
                        
                      
                    
                  
                
              
            
            ⁢
            
              |
              
                θ
                =
                
                  θ
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
              
            
          
        
      
    
  




  
    
      
        
          
            
              
                
                  θ
                  2
                
                ⁡
                
                  (
                  
                    t
                    
                      n
                      +
                      1
                    
                  
                  )
                
              
              =
              
                
                  
                    θ
                    2
                  
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
                -
                
                  γ
                  ⁢
                  
                    
                      ∂
                      
                        G
                        ⁡
                        
                          (
                          θ
                          )
                        
                      
                    
                    
                      ∂
                      
                        θ
                        2
                      
                    
                  
                
              
            
            
          
          
            θ
            =
            
              θ
              ⁡
              
                (
                
                  t
                  n
                
                )
              
            
          
        
      
    
    
      
        
          =
          
            
              
                
                  θ
                  2
                
                ⁡
                
                  (
                  
                    t
                    n
                  
                  )
                
              
              -
              
                γ
                ⁢
                
                  
                    ∑
                    
                      l
                      =
                      2
                    
                    Q
                  
                  ⁢
                  
                    
                      ∑
                      
                        p
                        =
                        1
                      
                      
                        l
                        -
                        1
                      
                    
                    ⁢
                    
                      
                        ∂
                        
                          
                            G
                            lp
                          
                          ⁡
                          
                            (
                            θ
                            )
                          
                        
                      
                      
                        ∂
                        
                          θ
                          2
                        
                      
                    
                  
                
              
            
            ⁢
            
              |
              
                θ
                =
                
                  θ
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
              
            
            .

6. Method in accordance with claim 5 in which the partial derivatives of G(θ) for θ=θ(tn) are computed in accordance with the following steps:
Step 1. find the value of G[θ(tn)]=G[φ1(tn), φ2(tn), θ1(tn), θ2(tn)] at iteration n; to do this, in the time interval (tn, tn+Tu/5) the Stokes parameters at the Q frequencies are derived and the value of the function G(θ) is computed
Step 2. find the partial derivative





  
    
      
        ∂
        
          G
          ⁡
          
            (
            θ
            )
          
        
      
      
        ∂
        
          φ
          1
        
      
    
    
  
  
    θ
    =
    
      θ
      ⁡
      
        (
        
          t
          n
        
        )
      
    
  



 at iteration n; to do this, parameter φ1 is set at φ1(tn)+Δ while the other parameters are left unchanged, the corresponding value of G(θ), i.e. G[φ1(tn)+Δ, φ2(tn), θ1(tn), θ2(tn)], is computed as in step 1 but in the time interval (tn+Tu/5, tn+2Tu/5) and the estimate of the partial derivative of G(θ) with respect to φ1 is computed as:





  
    
      
        
          ∂
          
            G
            ⁡
            
              (
              θ
              )
            
          
        
        
          ∂
          
            φ
            1
          
        
      
      
    
    
      θ
      =
      
        θ
        ⁡
        
          (
          
            t
            n
          
          )
        
      
    
  
  ≅
  
    
      
        
          
            
              G
              ⁡
              
                [
                
                  
                    
                      
                        φ
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                    +
                    Δ
                  
                  ,
                  
                    
                      φ
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                  ,
                  
                    
                      θ
                      1
                    
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                  ,
                  
                    
                      θ
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                
                ]
              
            
            -
          
        
      
      
        
          
            G
            ⁡
            
              [
              
                
                  
                    φ
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
                ,
                
                  
                    φ
                    2
                  
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
                ,
                
                  
                    θ
                    1
                  
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
                ,
                
                  
                    θ
                    2
                  
                  ⁡
                  
                    (
                    
                      t
                      n
                    
                    )
                  
                
              
              ]
            
          
        
      
    
    Δ
  



Step 3. Find the partial derivative:





  
    
      
        ∂
        
          G
          ⁡
          
            (
            θ
            )
          
        
      
      
        ∂
        
          φ
          2
        
      
    
    
  
  
    θ
    =
    
      θ
      ⁡
      
        (
        
          t
          n
        
        )
      
    
  



 at iteration n; to do this the parameter φ2 is set at φ2(tn)+Δ while the other parameters are left changed, the corresponding value of G(θ), i.e. G[φ1(tn), φ2(tn)+Δ, θ1(tn), θ2(tn)], is computed as in step 1 but in the time interval (tn+2Tu/5, tn+3Tu/5) and the estimate of the partial derivative of G(θ) with respect to φ2 is computed as:





  
    
      ∂
      
        G
        ⁡
        
          (
          θ
          )
        
      
    
    
      ∂
      
        φ
        2
      
    
  
  ⁢
  
    |
    
      θ
      =
      
        θ
        ⁡
        
          (
          
            t
            n
          
          )
        
      
    
  
  ⁢
  
    ≅
    
      
        
          
            
              
                G
                ⁡
                
                  [
                  
                    
                      
                        φ
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                    ,
                    
                      
                        
                          φ
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      +
                      Δ
                    
                    ,
                    
                      
                        θ
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                    ,
                    
                      
                        θ
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                  
                  ]
                
              
              -
            
          
        
        
          
            
              G
              ⁡
              
                [
                
                  
                    
                      φ
                      1
                    
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                  ,
                  
                    
                      φ
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                  ,
                  
                    
                      θ
                      1
                    
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                  ,
                  
                    
                      θ
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                
                ]
              
            
          
        
      
      Δ
    
  



Step 4. Find the partial derivative:





  
    
      ∂
      
        G
        ⁡
        
          (
          θ
          )
        
      
    
    
      ∂
      
        θ
        1
      
    
  
  ⁢
  
    |
    
      θ
      =
      
        θ
        ⁡
        
          (
          
            t
            n
          
          )
        
      
    
  



 at iteration n; to do this, parameter θ1 is set at θ1(tn)+Δ while the other parameters are left unchanged, the corresponding value of G(θ), i.e. G[φ1(tn) φ2(tn), θ1(tn)+Δ, θ2(tn)], is computed as in Step 1 but in the time interval (tn+3Tu/5, tn+4Tu/5) and the estimate of the partial derivative of G(θ) with respect to θ1 is computed as:





  
    
      ∂
      
        G
        ⁡
        
          (
          θ
          )
        
      
    
    
      ∂
      
        θ
        1
      
    
  
  ⁢
  
    |
    
      θ
      =
      
        θ
        ⁡
        
          (
          
            t
            n
          
          )
        
      
    
  
  ⁢
  
    ≅
    
      
        
          
            
              
                G
                ⁡
                
                  [
                  
                    
                      
                        φ
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                    ,
                    
                      
                        φ
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                    ,
                    
                      
                        
                          θ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      +
                      Δ
                    
                    ,
                    
                      
                        θ
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                  
                  ]
                
              
              -
            
          
        
        
          
            
              G
              ⁡
              
                [
                
                  
                    
                      φ
                      1
                    
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                  ,
                  
                    
                      φ
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                  ,
                  
                    
                      θ
                      1
                    
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                  ,
                  
                    
                      θ
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        n
                      
                      )
                    
                  
                
                ]
              
            
          
        
      
      Δ
    
  



Step 5. Find the partial derivative:





  
    
      ∂
      
        G
        ⁡
        
          (
          θ
          )
        
      
    
    
      ∂
      
        θ
        2
      
    
  
  ⁢
  
    |
    
      θ
      =
      
        θ
        ⁡
        
          (
          
            t
            n
          
          )
        
      
    
  



 at iteration n; to do this the parameter φ2 is set at φ2(tn)+Δ while the other parameters are left unchanged, the corresponding value of G(θ), i.e. G[φ1(tn), φ2(tn), θ1(tn), θ2(tn)+Δ], is computed as in step 1 but in the time interval (tn+4Tu/5, tn+Tu) and the estimate of the partial derivative of G(θ) with respect to φ2 is computed as:





  
    
      ∂
      
        G
        ⁡
        
          (
          θ
          )
        
      
    
    
      ∂
      
        θ
        2
      
    
  
  ⁢
  
    |
    
      θ
      =
      
        θ
        ⁡
        
          (
          
            t
            n
          
          )
        
      
    
  
  ⁢
  
    ≅
    
      
        
          
            
              
                
                  G
                  ⁡
                  
                    [
                    
                      
                        
                          φ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          φ
                          2
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          θ
                          1
                        
                        ⁡
                        
                          (
                          
                            t
                            n
                          
                          )
                        
                      
                      ,
                      
                        
                          
                            θ
                            2
                          
                          ⁡
                          
                            (
                            
                              t
                              n
                            
                            )
                          
                        
                        +
                        Δ
                      
                    
                    ]
                  
                
                -
              
            
          
          
            
              
                G
                ⁡
                
                  [
                  
                    
                      
                        φ
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                    ,
                    
                      
                        φ
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                    ,
                    
                      
                        θ
                        1
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                    ,
                    
                      
                        θ
                        2
                      
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                  
                  ]
                
              
            
          
        
        Δ
      
      .

7. Method in accordance with claim 1 comprising the steps of computing the Stokes parameters in pairs of frequencies fl and fp with l,p=1, 2, . . . , Q, to obtain at the lth and pth frequencies of the Q frequencies the two series of Stokes parameters S1,l, S2,l, S3,l e S1,p, S2,p, S3,p, defining the function:




  
    
      
        
          H
          ⁡
          
            (
            θ
            )
          
        
        =
        
          
            ∑
            
              l
              =
              2
            
            Q
          
          ⁢
          
            
          
          ⁢
          
            
              ∑
              
                p
                =
                1
              
              
                l
                -
                1
              
            
            ⁢
            
              
            
            ⁢
            
              
                H
                lp
              
              ⁡
              
                (
                θ
                )
              
            
          
        
      
    
  
  
    
      
        
          with
          ⁢
          
            
          
          ⁢
          
            
              H
              lp
            
            ⁡
            
              (
              θ
              )
            
          
        
        =
        
          
            
              (
              
                
                  S
                  
                    1
                    ,
                    l
                  
                
                -
                
                  S
                  
                    1
                    ,
                    p
                  
                
              
              )
            
            2
          
          +
          
            
              (
              
                
                  S
                  
                    2
                    ,
                    l
                  
                
                -
                
                  S
                  
                    2
                    ,
                    p
                  
                
              
              )
            
            2
          
          +
          
            
              (
              
                
                  S
                  
                    3
                    ,
                    l
                  
                
                -
                
                  S
                  
                    3
                    ,
                    p
                  
                
              
              )
            
            2
          
        
      
    
  




with
and producing said control signals to minimize said function H(θ).

8. Method in accordance with claim 7 in which the optical devices comprise a polarization controller with controllable angles φ1, φ2 and two rotators with controllable rotation angles respectively θ1 and θ2, and for minimizing the function H(θ) the updating of φ1, φ2, θ1 and θ2, of the compensator follows the following mles for passing from the nth iteration to the n+1th e iteration:




  
    θ
    ⁡
    
      (
      
        t
        
          n
          +
          1
        
      
      )
    
  
  =
  
    
      
        θ
        ⁡
        
          (
          
            t
            n
          
          )
        
      
      -
      
        γ
        ⁢
        
          
            ∑
            
              l
              =
              2
            
            Q
          
          ⁢
          
            
          
          ⁢
          
            
              ∑
              
                p
                =
                1
              
              
                l
                -
                1
              
            
            ⁢
            
              
            
            ⁢
            
              ∇
              
                
                  H
                  lp
                
                ⁡
                
                  (
                  θ
                  )
                
              
            
          
        
      
    
    ⁢
    
      |
      
        θ
        =
        
          θ
          ⁡
          
            (
            
              t
              n
            
            )
          
        
      
    
  




or the following simplified rule:




  
    θ
    ⁡
    
      (
      
        t
        
          n
          +
          1
        
      
      )
    
  
  =
  
    
      θ
      ⁡
      
        (
        
          t
          n
        
        )
      
    
    -
    
      γsign
      ⁡
      
        [
        
          
            
              ∑
              
                l
                =
                2
              
              Q
            
            ⁢
            
              
            
            ⁢
            
              
                ∑
                
                  p
                  =
                  1
                
                
                  l
                  -
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                ∇
                
                  
                    H
                    lp
                  
                  ⁡
                  
                    (
                    θ
                    )
                  
                
              
            
          
          ⁢
          
            |
            
              θ
              =
              
                θ
                ⁡
                
                  (
                  
                    t
                    n
                  
                  )
                
              
            
          
        
        ]
      
    
  




with ∇HLP(θ) equal to the gradient of H1p(θ) with respect to {tilde over (θ)}.

9. Method in accordance with claim 7 in which said parameters are consolidated in a vector θ which is updated in accordance with the rule




  
    
      
        
          θ
          ⁡
          
            (
            
              t
              
                n
                +
                1
              
            
            )
          
        
        =
        
          
            
              θ
              ⁡
              
                (
                
                  t
                  n
                
                )
              
            
            -
            
              γ
              ⁢
              
                
                  ∑
                  
                    l
                    =
                    2
                  
                  Q
                
                ⁢
                
                  
                
                ⁢
                
                  
                    ∑
                    
                      p
                      =
                      1
                    
                    
                      l
                      -
                      1
                    
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    ∇
                    
                      
                        H
                        lp
                      
                      ⁡
                      
                        (
                        θ
                        )
                      
                    
                  
                
              
            
          
          ⁢
          
            |
            
              θ
              =
              
                θ
                ⁡
                
                  (
                  
                    t
                    n
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        19
        )
      
    
  




or the following simplified rule based only on the sign:




  
    
      
        
          θ
          ⁡
          
            (
            
              t
              
                n
                +
                1
              
            
            )
          
        
        =
        
          
            θ
            ⁡
            
              (
              
                t
                n
              
              )
            
          
          -
          
            γsign
            ⁡
            
              [
              
                
                  
                    ∑
                    
                      l
                      =
                      2
                    
                    Q
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      ∑
                      
                        p
                        =
                        1
                      
                      
                        l
                        -
                        1
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      ∇
                      
                        
                          H
                          lp
                        
                        ⁡
                        
                          (
                          θ
                          )
                        
                      
                    
                  
                
                ⁢
                
                  |
                  
                    θ
                    =
                    
                      θ
                      ⁡
                      
                        (
                        
                          t
                          n
                        
                        )
                      
                    
                  
                
              
              ]
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




with ∇HLP(θ) equal to the gradient of Hlp(θ) with respect to {tilde over (θ)}.

10. Method in accordance with claim 9 in which between the controller and an optical rotator and between optical rotators there are fibers which introduce a predetermined differential unit delay maintaining the polarization.

11. Method in accordance with claim 1 in which the PMD compensator is modeled like a two-dimensional transversal filter with four tappered delay lines combining the signals on the two principal states of polarization (PSP).

12. Method in accordance with claim 11 in which the gradient ∇HLP(θ) with respect to θ is computed as:




  
    
      
        
          ∇
          
            
              H
              lp
            
            ⁡
            
              (
              θ
              )
            
          
        
        =
        
        ⁢
        
          4
          ⁢
          
            (
            
              
                S
                
                  1
                  ,
                  l
                
              
              -
              
                S
                
                  1
                  ,
                  p
                
              
            
            )
          
          ⁢
          Re
          ⁢
          
            {
            
              
                1
                
                  T
                  u
                
              
              ⁢
              
                
                  ∫
                  
                    t
                    n
                  
                  
                    t
                    
                      n
                      +
                      1
                    
                  
                
                ⁢
                
                  [
                  
                    
                      
                        
                          y
                          
                            1
                            ,
                            l
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          a
                          l
                          T
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    -
                    
                      
                        
                          y
                          
                            2
                            ,
                            l
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          b
                          l
                          T
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    -
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
            
              
              ⁢
              
                
                  
                    
                      y
                      
                        1
                        ,
                        p
                      
                      *
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    
                      a
                      p
                      T
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                +
                
                  
                    
                      y
                      
                        2
                        ,
                        p
                      
                      *
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    
                      b
                      p
                      T
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
              ]
            
            ⁢
            
              d
              t
            
            ⁢
            
              
            
            ⁢
            J
          
          }
        
        +
      
    
  
  
    
      
        
        ⁢
        
          4
          ⁢
          
            (
            
              
                S
                
                  2
                  ,
                  l
                
              
              -
              
                S
                
                  2
                  ,
                  p
                
              
            
            )
          
          ⁢
          Re
          ⁢
          
            {
            
              
                1
                
                  T
                  u
                
              
              ⁢
              
                
                  ∫
                  
                    t
                    n
                  
                  
                    t
                    
                      n
                      +
                      1
                    
                  
                
                ⁢
                
                  [
                  
                    
                      
                        
                          y
                          
                            2
                            ,
                            l
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          a
                          l
                          T
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    +
                    
                      
                        
                          y
                          
                            1
                            ,
                            l
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          b
                          l
                          T
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    -
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
            
              
              ⁢
              
                
                  
                    
                      y
                      
                        2
                        ,
                        p
                      
                      *
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    
                      a
                      p
                      T
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
                -
                
                  
                    
                      y
                      
                        1
                        ,
                        p
                      
                      *
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                  ⁢
                  
                    
                      b
                      j
                      T
                    
                    ⁡
                    
                      (
                      t
                      )
                    
                  
                
              
              ]
            
            ⁢
            
              d
              t
            
            ⁢
            
              
            
            ⁢
            J
          
          }
        
        -
      
    
  
  
    
      
        
        ⁢
        
          4
          ⁢
          
            (
            
              
                S
                
                  3
                  ,
                  l
                
              
              -
              
                S
                
                  3
                  ,
                  p
                
              
            
            )
          
          ⁢
          
            I
            ⁢
            m
          
          ⁢
          
            {
            
              
                1
                
                  T
                  u
                
              
              ⁢
              
                
                  ∫
                  
                    t
                    n
                  
                  
                    t
                    
                      n
                      +
                      1
                    
                  
                
                ⁢
                
                  [
                  
                    
                      
                        
                          y
                          
                            2
                            ,
                            l
                          
                          *
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          a
                          l
                          T
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    +
                    
                      
                        
                          y
                          
                            1
                            ,
                            l
                          
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                      ⁢
                      
                        
                          b
                          l
                          T
                        
                        ⁡
                        
                          (
                          t
                          )
                        
                      
                    
                    -
                  
                
              
            
          
        
      
    
  
  
    
      
        
          
            
            ⁢
            
              
                
                  
                    y
                    
                      2
                      ,
                      p
                    
                    *
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  
                    a
                    p
                    T
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
              -
              
                
                  
                    y
                    
                      1
                      ,
                      p
                    
                    *
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ⁢
                
                  
                    b
                    p
                    T
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
            ]
          
          ⁢
          
            d
            t
          
          ⁢
          
            
          
          ⁢
          J
        
        }
      
    
  



where y1,l(t), y2,l(t) and y1,p(t), y2,p(t) are respectively the components y1(t) e y2(t) on the two orthogonal polarizations of the compensator output signal filtered respectively through a narrow band filter centered on the frequency fl and fp; and
al(t) e bl(t) are the vectors:





  
    
      
        
          
            a
            l
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          (
          
            
              
                
                  
                    x
                    
                      1
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      1
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          τ
                          c
                        
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      1
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        c
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      1
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        c
                      
                      -
                      
                        ατ
                        c
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    t
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          τ
                          c
                        
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        c
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        c
                      
                      -
                      
                        α
                        ⁢
                        
                          
                        
                        ⁢
                        
                          τ
                          c
                        
                      
                    
                    )
                  
                
              
            
          
          )
        
      
    
    
      
        
          
            b
            l
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          (
          
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                    *
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        2
                        ⁢
                        
                          τ
                          c
                        
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                    *
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        c
                      
                      -
                      
                        β
                        ⁢
                        
                          
                        
                        ⁢
                        
                          τ
                          c
                        
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                    *
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        τ
                        c
                      
                    
                    )
                  
                
              
            
            
              
                
                  
                    x
                    
                      2
                      ,
                      l
                    
                    *
                  
                  ⁡
                  
                    (
                    
                      t
                      -
                      
                        β
                        ⁢
                        
                          
                        
                        ⁢
                        
                          τ
                          c
                        
                      
                    
                    )
                  
                
              
            
            
              
                
                  -
                  
                    
                      x
                      
                        1
                        ,
                        l
                      
                      *
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          2
                          ⁢
                          
                            τ
                            c
                          
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  -
                  
                    
                      x
                      
                        1
                        ,
                        l
                      
                      *
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          c
                        
                        -
                        
                          β
                          ⁢
                          
                            
                          
                          ⁢
                          
                            τ
                            c
                          
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  -
                  
                    
                      x
                      
                        1
                        ,
                        l
                      
                      *
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          τ
                          c
                        
                      
                      )
                    
                  
                
              
            
            
              
                
                  -
                  
                    
                      x
                      
                        1
                        ,
                        l
                      
                      *
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          β
                          ⁢
                          
                            
                          
                          ⁢
                          
                            τ
                            c
                          
                        
                      
                      )
                    
                  
                
              
            
          
          )
        
      
    
  



 with x1,l(t) and x2,l(t) which are respectively signals x1(t) and x2(t) on the two orthogonal polarizations of the compensator input signal filtered with a narrow band filter centered on the frequency fl (similarly al(t) and bl(t) for y1,p(t) and y2,p(t)) with the frequency fl), and
J is the Jacobean matrix of the transformation c=c(θ) defined as





  
    
      
        J
        ⁢
        
          =
          ^
        
        ⁢
        
          (
          
            
              
                
                  
                    ∂
                    
                      c
                      1
                    
                  
                  
                    ∂
                    
                      φ
                      1
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      1
                    
                  
                  
                    ∂
                    
                      φ
                      2
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      1
                    
                  
                  
                    ∂
                    
                      θ
                      1
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      1
                    
                  
                  
                    ∂
                    
                      θ
                      2
                    
                  
                
              
            
            
              
                
                  
                    ∂
                    
                      c
                      2
                    
                  
                  
                    ∂
                    
                      φ
                      1
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      2
                    
                  
                  
                    ∂
                    
                      φ
                      2
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      2
                    
                  
                  
                    ∂
                    
                      θ
                      1
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      2
                    
                  
                  
                    ∂
                    
                      θ
                      2
                    
                  
                
              
            
            
              
                ⋮
              
              
                ⋮
              
              
                ⋮
              
              
                ⋮
              
            
            
              
                
                  
                    ∂
                    
                      c
                      8
                    
                  
                  
                    ∂
                    
                      φ
                      1
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      8
                    
                  
                  
                    ∂
                    
                      φ
                      2
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      8
                    
                  
                  
                    ∂
                    
                      θ
                      1
                    
                  
                
              
              
                
                  
                    ∂
                    
                      c
                      8
                    
                  
                  
                    ∂
                    
                      θ
                      2
                    
                  
                
              
            
          
          )
        
      
    
    
      
        (
        18
        )
      
    
  



 with c1, . . . , c8 which are the tap coefficients of the four tappered delay lines.

13. Method in accordance with claim 1 in which said optical devices comprise a polarization controller with control angles φ1, φ2 and two optical rotators with rotation angles θ1 and θ2 and said parameters comprise said control angles φ1, φ2 and said rotation angles θ1 and θ2.

14. PMD compensator in optical fiber communication systems comprising: a cascade of adjustable optical devices over which passes an optical signal to be compensated, and an adjustment system configured to take components y1(t) and y2(t) on the two orthogonal polarizations from the compensator output signal signal, the adjustment system comprising a controller which is operable to compute, on the basis of said components taken, the Stokes parameters S0, S1, S2, S3 in a number Q of different frequencies of the compensator output signal and to emit control signals for parameters of at least some of said adjustable optical devices so as to make virtually constant the Stokes parameters computed at the different frequencies.

15. Compensator in accordance with claim 14 characterized in that said optical devices comprise a polarization controller with control angles φ1, φ2 and two optical rotators with rotation angles θ1 and θ2 and in which said parameters for which the controller is operable to emit the control sianals consist of said control angles θ1, θ2 and said rotation angles θ1 and θ2.

16. Compensator in accordance with claim 15 characterized in that between the controller and an optical rotator and between optical rotators there are fibers which introduce a predetermined differential unit delay maintaining the polarization.