Abstract:
The invention provides a method for stabilising a spectrometric transducer for an optical spectrum measuring instrument by obtaining an instantaneous central wavelength of a thermally controlled tunable filter of the instrument, calibrating for a selected ambient temperature, determining a heat variance of the filter and controlling the filter to compensate for heat variance

Description:
[0001]    This application is a continuation or U.S. patent application Ser. No. 12/901,238 filed Oct. 8, 2011 which is a continuation of U.S. patent application Ser. No. 12/318,040 filed on Dec. 19, 2008 which is a continuation of U.S. patent application Ser. No. 11/754,781 filed on May 29, 2007 which is a continuation of U.S. patent application Ser. No. 11/038,490 on Jan. 21, 2005 which claims priority from U.S. Provisional Patent Application No. 60/537,919 filed on Jan. 22, 2004; the contents which are incorporated herein by reference. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The present invention relates to a method for the stabilization of spectrometric transducers. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0003]    An embodiment of the present invention will now be described by way of example only with reference to the accompanying drawings, in which: 
           [0004]      FIG. 1  is a schematic diagram of an optical spectrum measurement instrument. 
           [0005]      FIG. 2  is a schematic representation of a portion of the apparatus shown in  FIG. 1 . 
           [0006]      FIG. 3  is a flow diagram of a thermal stabilization method for a spectrometric transducer. 
           [0007]      FIG. 4  is a plot showing the variation of controlling current for a particular set of parameters. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0008]    In  FIG. 1  there is shown an optical spectrum measurement instrument  20  for analyzing optical signal  12 . The optical signal  12  is sampled by sampler  14 , which provides optical spectrum x(λ)  16  to the optical spectrum measurement instrument  20 . The optical spectrum measurement instrument  20  includes spectrometric transducer  30 , including a tunable filter  32  and photodiode  34 , and Digital Signal Processor (DSP)  40 . Optical spectrum x(λ)  16  is separated into its optical components by tunable filter  32 , after which its components are converted from optical to current by photodiodes  34 , resulting in an electrical spectrometric data representation y(t)  36  of optical spectrum x(λ)  16 . The spectrometric data representation y(t)  36  is then usedby the DSP  40  to compute an estimate {tilde over (x)}(λ)  42  of the optical signal  12  optical spectrum, which is then provided to user display  50 . Optionally, the DSP  40  may also compute various parameters relating to the optical signal  12  optical spectrum such as power measurements, OSNR, BER, etc. or may implement signal analysis or reconstruction algorithms. The response of a filter to an input optical signal whose spectrum is x(λ)  16 , i.e. the output current of the photodiode  34 , may be modelled by an integral operator of the form: 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       F 
                       y 
                     
                      
                     
                       [ 
                       
                         
                           ∫ 
                           
                             - 
                             ∞ 
                           
                           
                             + 
                             ∞ 
                           
                         
                          
                         
                           
                             g 
                              
                             
                               ( 
                               
                                 λ 
                                 , 
                                 t 
                               
                               ) 
                             
                           
                           · 
                           
                             x 
                              
                             
                               ( 
                               λ 
                               ) 
                             
                           
                           · 
                           
                              
                             λ 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   1 
                 
               
             
           
         
       
     
         [0000]    where λ is wavelength, t is time, g(λ, t) is the filter response function, and F y [•] is a slightly nonlinear scalar function of a scalar variable. For a monochromatic input signal, whose spectrum is x(λ)=δ(λ−l), this model responds with: 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         F 
                         y 
                       
                        
                       
                         [ 
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             
                               + 
                               ∞ 
                             
                           
                            
                           
                             
                               g 
                                
                               
                                 ( 
                                 
                                   λ 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                             · 
                             
                               δ 
                                
                               
                                 ( 
                                 
                                   λ 
                                   - 
                                   l 
                                 
                                 ) 
                               
                             
                             · 
                             
                                
                               λ 
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         F 
                         y 
                       
                        
                       
                         [ 
                         
                           g 
                            
                           
                             ( 
                             
                               l 
                               , 
                               t 
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   2 
                 
               
             
           
         
       
     
         [0000]    and to the input signal whose spectrum is flat, x(λ)=1, it responds with: 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         F 
                         y 
                       
                        
                       
                         [ 
                         
                           
                             ∫ 
                             
                               - 
                               ∞ 
                             
                             
                               + 
                               ∞ 
                             
                           
                            
                           
                             
                               g 
                                
                               
                                 ( 
                                 
                                   λ 
                                   , 
                                   t 
                                 
                                 ) 
                               
                             
                              
                             
                                 
                             
                             · 
                             
                                
                               λ 
                             
                           
                         
                         ] 
                       
                     
                     ≡ 
                     
                       
                         
                           F 
                           y 
                         
                          
                         
                           [ 
                           
                             
                               g 
                               0 
                             
                              
                             
                               ( 
                               t 
                               ) 
                             
                           
                           ] 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   3 
                 
               
             
           
         
       
     
         [0009]    The response g(l, t), defined by Equation 2, is a gaussoid-like function with a maximum changing monotonically with the wavelength l. The function of time g 0 (t), defined by Equatiou 3, characterizes the amplitude variability of the filter response along the wavelength axis. 
         [0010]    As shown in  Figure 2 , the instantaneous central wavelength of the filter is externally controlled by the current i(t)  38 . In the preferred embodiment, the filter  32  is a thermally controlled filter in which the central wavelength depends on its internal temperature. The current i(t)  38  drives a variable resistance heater incorporated in the filter. Such filters utilize the characteristic that as the temperature on tunable filters varies, such as a particular optical material&#39;s change in its index of refraction n with temperature T, so its center wavelength varies, resulting in thermally-controlled tunable filters. The temperature of the filter depends on the heat supplied from the resistence heater and the heat loss/gain to/from the ambient environment. To compensate for the heat loss/gain, a signal containing information about the ambient temperature is used to compensate the current i(t)  38  supplied to the heater. The current is adjusted by the controller that implements a control algorithm to maintain the filter at the desired center frequency. 
         [0011]    Assume that the function λ=F 0 (u) is the result ofwayelength calibration for a selected ambient temperature T 0  represented by the voltage U 0 . Assume, moreover, that the input signal is generated by a battery of lasers whose central wavelengths are uniformly distributed in the interval [λ min , λ max ]: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       λ 
                       j 
                     
                     = 
                     
                       
                         
                           λ 
                           min 
                         
                         + 
                         
                           
                             
                               ( 
                               
                                 j 
                                 - 
                                 1 
                               
                               ) 
                             
                             
                               J 
                               - 
                               1 
                             
                           
                            
                           
                             ( 
                             
                               
                                 λ 
                                 max 
                               
                               - 
                               
                                 λ 
                                 min 
                               
                             
                             ) 
                           
                            
                           
                               
                           
                            
                           for 
                            
                           
                               
                           
                            
                           j 
                         
                       
                       = 
                       1 
                     
                   
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     J 
                     . 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   4 
                 
               
             
           
         
       
     
         [0012]    Then, the maxima y 0,j   max  of the response of the stabilized filter to such a signal may be easily identified together with the corresponding values of the voltage u(t): u 0,j =F 0   −1 (λ j ) for j=1, . . . , J. The practical purpose of controlling the current i(t)  38  may be now formulated as follows. For an arbitrary ambient temperature T, find i(t) such that the maxima of the stabilized filter response remain as close as possible to the maxima determined for T 0 , i.e. the coordinates of the maxima for the ambient temperature T, the values u 1 , . . . , u j  of the voltage u(t) and the values y 1   max , . . . , y j   max  of the output signal y(t), satisfy the condition: 
         [0000]        w [( u   1   −u   0,1 ) 2 +. . . , +( u   j   −u   0,j ) 2 ]+(1− w )[( y   1   max   −y   0,1   max ) 2 +. . . +( y   j   max   −y   0,j   max ) 2 ]→MIN   Equation 5
 
         [0000]    where w∈[0, 1] is a weighing factor. 
         [0013]    The task of control may be significantly simplified (at the price of the sub-optimality of the solution) by an appropriate parameterization of the heating current i(t). An example of such a parameterization is defined by the formula: 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           0 
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               t 
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             i 
                             1 
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               t 
                             
                             ∈ 
                             
                               [ 
                               
                                 0 
                                 , 
                                 
                                   t 
                                   2 
                                 
                               
                               ] 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   
                                     i 
                                     3 
                                   
                                   - 
                                   
                                     i 
                                     2 
                                   
                                 
                                 
                                   
                                     t 
                                     3 
                                   
                                   - 
                                   
                                     t 
                                     2 
                                   
                                 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   
                                     t 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               i 
                               2 
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               t 
                             
                             ∈ 
                             
                               [ 
                               
                                 
                                   t 
                                   2 
                                 
                                 , 
                                 
                                   t 
                                   3 
                                 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   6 
                 
               
             
           
         
       
     
         [0000]    where i 1 , i 2 , i 3 , t 2 , and t 3  are parameters of the current to be optimized by minimization of the left-hand side of Equation 5. The parameterization of controlled current enables one to use an empirical procedure of optimization that does not require any explicit reference to the mathematical model of the stabilized filter. This procedure is implemented in the controller and is depicted by the flow chart shown in FIG  3 . The sequence of steps composing the procedure is indicated by the sequence of blocks  62  to  70 . 
         [0014]    In block  62 , for the selected ambient temperature T 0 , the values of i 1 , i 2 , i 3 , t 2  and t 3  are chosen so as to produce a relatively uniform distribution of u 0,j =F 0   −1 (λ j ) for j=1, . . . , J. 
         [0015]    Then in block  64 , on the basis of measurements performed for the same ambient temperature, the matrix S y  of the sensitivity of the maxima y 1   max , . . . , y j   max  is computed for a small change in the ambient temperature ΔT and for small changes Δi 1 , Δi 2 , Δi 3 , Δt 2  and Δt 3  in the parameters i 1 , i 2 , i 3 , t 2  and t 3 . 
         [0016]    Similarly, in block  66 , on the basis of measurements performed for the same ambient temperature, the matrix S u  of the sensitivity of the corresponding maxima voltage values u 1 , . . . , u j  is computed for a small change in the ambient temperature ΔT and for small changes Δi 1 , Δi 2 , Δi 3 , Δt 2  and Δt 3  in the parameters i 1 , i 2 , i 3 , t 2  and t 3 . 
         [0017]    In block  68 , the following minimization problem is solved: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       w 
                        
                       
                         
                            
                           
                             
                               S 
                               u 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       T 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         i 
                                         1 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         i 
                                         2 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         i 
                                         3 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         t 
                                         2 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         t 
                                         3 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                            
                         
                         2 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           1 
                           - 
                           w 
                         
                         ) 
                       
                        
                       
                         
                            
                           
                             
                               S 
                               y 
                             
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       T 
                                     
                                   
                                 
                                 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         i 
                                         1 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         i 
                                         2 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         i 
                                         3 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         t 
                                         2 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       Δ 
                                        
                                       
                                           
                                       
                                        
                                       
                                         t 
                                         3 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                            
                         
                         2 
                       
                     
                   
                   → 
                   
                     MIN 
                     . 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   7 
                 
               
             
           
         
       
     
         [0000]    with respect to Δi 1 , Δi 2 , Δi 3 , Δt 2  and Δt 3  for an assumed (sufficiently small) increment ΔT of the ambient temperature. 
         [0018]    Following which, in block  70 , the ambient temperature is changed to the value T=T 0 +ΔT; the new values of i 1 , i 2 , i 3 , t 2  and t 3  are computed using increments Δi 1 , Δi 2 , Δi 3 , Δt 2  and Δt 3  determined in block  68 ; i 1 , i 2 , i 3 , t 2  and t 3  and are empirically corrected in such a way as to satisfy the condition defined by Equation 5. 
         [0019]    Finally, blocks  62  to  70  are repeated itcratively as to cover the whole range of ambient temperatures the stabilized filter is assumed to operate in. 
         [0020]    It should be noted that the whole process of optimization is subject to the constraint concerning the admissible values of current and heating time. 
         [0021]    A further refinement of the thermal stabilization of the stabilized filter is possible during the software pre-processing of the data provided by the stabilized filter using the above described hardware means. The residual instabilities may be characterized during general calibration of the stabilized filter, and the results of this characterization may next be used for correction of the raw data before their pre-processing. 
         [0022]    A closer empirical study of the current i(t) control based on Equation 6 provides the following:
       t 2  and t 3  may be fixed to the values 1 ms and 5 ms, respectively;   abrupt changes of current i(t) should be avoided;   a convex control in the interval [t 2 , t 3 ] would be desirable.       
 
         [0026]    Consequently, a second example of the current i(t) parameterization may been designed. 
         [0000]    It is defined by the formula: 
         [0000]    
       
         
           
             
               
                 
                   
                     i 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           0 
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               t 
                             
                             &lt; 
                             0 
                           
                         
                       
                       
                         
                           
                             
                               
                                 a 
                                 1 
                               
                                
                               
                                 t 
                                 2 
                               
                             
                             + 
                             
                               
                                 b 
                                 1 
                               
                                
                               t 
                             
                             + 
                             
                               c 
                               1 
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               t 
                             
                             ∈ 
                             
                               [ 
                               
                                 0 
                                 , 
                                 1 
                               
                               ] 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 
                                   a 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                               2 
                             
                             + 
                             
                               
                                 b 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   t 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             + 
                             
                               c 
                               2 
                             
                           
                         
                         
                           
                             
                               for 
                                
                               
                                   
                               
                                
                               t 
                             
                             ∈ 
                             
                               [ 
                               
                                 1 
                                 , 
                                 5 
                               
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   8 
                 
               
             
           
         
       
     
         [0000]    where the parameters a 1 , b 1 , c 1  and a 2 , b 2 , c 2  should satisfy the following conditions: 
         [0000]        i (0)= I   0   , i (1)= I   1   , i (3)= I   3 , and  i (5)= I   5    Equation 9
 
         [0027]    The solution of the above algebraic problem has the form: 
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             a 
                             1 
                           
                         
                       
                       
                         
                           
                             b 
                             1 
                           
                         
                       
                       
                         
                           
                             a 
                             2 
                           
                         
                       
                       
                         
                           
                             b 
                             2 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             1.000 
                           
                           
                             
                               - 
                               1.750 
                             
                           
                           
                             1.000 
                           
                           
                             
                               - 
                               0.250 
                             
                           
                         
                         
                           
                             
                               - 
                               2.000 
                             
                           
                           
                             2.750 
                           
                           
                             
                               - 
                               1.000 
                             
                           
                           
                             0.250 
                           
                         
                         
                           
                             0.000 
                           
                           
                             0.125 
                           
                           
                             
                               - 
                               0.250 
                             
                           
                           
                             0.125 
                           
                         
                         
                           
                             0.000 
                           
                           
                             
                               - 
                               0.750 
                             
                           
                           
                             1.000 
                           
                           
                             
                               - 
                               0.250 
                             
                           
                         
                       
                       ] 
                     
                     · 
                     
                       [ 
                       
                         
                           
                             
                               I 
                               0 
                             
                           
                         
                         
                           
                             
                               I 
                               1 
                             
                           
                         
                         
                           
                             
                               I 
                               3 
                             
                           
                         
                         
                           
                             
                               I 
                               5 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   10 
                 
               
             
           
         
       
     
         [0000]    and c 1 =I 0 , c 2 =I 1 . An example of the current i(t) generated according to the above formula is shown in  FIG. 4  for I 0 =100 mA, I 1 =50 mA, I 3 =120 mA, and I 5 =150 mA. 
         [0028]    The components of a spectrometer or the device in which it is used such as an OPM are subject to aging. Consequently, the parameters of the OPM drift in time. In particular, the absolute accuracy of wavelength estimation is deteriorating. Taking into account that the contemporary DWDM transmitters contain highly stable lasers, one may use the time series of wavelength estimates as the basis for compensation of the time drift of the OPM. 
         [0029]    The idea of using time series of wavelength estimates provided by the OPM for time drift compensation of this OPM is based on an assumption that the average deviation of the central wavelength of the laser signal used for this purpose is close to zero, i.e. there is no systematic evolution of this wavelength over time. Consequently, the average deviation of the central wavelength as computed by the OPM should be expected to also be close to zero. If not, this average deviation may be used to model the OPM&#39;s drift and provide a way of stabilizing the OPM by compensating for this time drift. 
         [0030]    A wavelength such as described above may be modelled by means of a stochastic process: 
         [0000]          l ( t ) = {circumflex over (l)}+ δl (t)   Equation 11
 
         [0000]    where t is time, {circumflex over (l)} is the central wavelength value according to the ITU grid, and  δl(t)  is a stochastic process modelling the wavelength deviation from the ITU-grid value. The latter process is assumed to be zero-mean and stationary. Consequently, the time sampling of l(t) at equidistant time points, t 1 , . . . , t N , yields the vector of random variables: 
         [0000]          l =[ l (t 1 )  . . .    l ( t   N ) ] T    Equation 12
 
         [0000]    such that: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                        
                       
                         [ 
                         
                           1 
                           _ 
                         
                         ] 
                       
                     
                     = 
                     
                       
                         
                           
                             [ 
                             
                               
                                 ? 
                               
                                
                               
                                   
                               
                                
                               … 
                                
                               
                                   
                               
                                
                               
                                 ? 
                               
                             
                             ] 
                           
                           T 
                         
                          
                         
                             
                         
                          
                         and 
                          
                         
                             
                         
                          
                         
                           Cov 
                            
                           
                             [ 
                             
                               1 
                               _ 
                             
                             ] 
                           
                         
                       
                       = 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   σ 
                                   2 
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   c 
                                   
                                     1 
                                     , 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                               
                                 ⋱ 
                               
                               
                                 ⋮ 
                               
                             
                             
                               
                                 
                                   c 
                                   
                                     N 
                                     , 
                                     1 
                                   
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   σ 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                         T 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ? 
                     
                      
                     
                       indicates text missing or illegible when filed 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   13 
                 
               
             
           
         
       
     
         [0031]    Under the above assumptions, a result of central wavelength measurement, provided by the OPM drifting in time, may be modelled with: 
         [0000]        {circumflex over ( l )}( t   n ) =   l ( t   n ) +Δ l ( t   n )= {circumflex over (l)}+Δl ( t   n )+ δ l ( t   n )  for n=1, . . . , N   Equation 14
 
         [0000]    where Δl(t n ) is the time drift of wavelength to be estimated on the basis of the realizations {circumflex over (l)}(t n ) of the random variables  {circumflex over (l)}(t n ) . The vector {circumflex over ( I )}=[ {circumflex over (l)}(t             ) . . .  {circumflex over (l)}(t N ) ] T  has the following statistical properties: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                        
                       
                         [ 
                         i 
                         ] 
                       
                     
                     = 
                     
                       
                         [ 
                         
                           i 
                           + 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             
                               i 
                                
                               
                                 ( 
                                 
                                   t 
                                   1 
                                 
                                 ) 
                               
                             
                              
                             
                                 
                             
                              
                             … 
                              
                             
                                 
                             
                              
                             i 
                           
                           + 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             
                               i 
                                
                               
                                 ( 
                                 
                                   t 
                                   N 
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                       T 
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                    
                   
                     
 
                   
                    
                   
                     
                       Cov 
                        
                       
                         [ 
                         
                           i 
                           _ 
                         
                         ] 
                       
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   σ 
                                   2 
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   c 
                                   
                                     1 
                                     , 
                                     N 
                                   
                                 
                               
                             
                             
                               
                                 ⋮ 
                               
                               
                                 ⋱ 
                               
                               
                                 ⋮ 
                               
                             
                             
                               
                                 
                                   c 
                                   
                                     1 
                                     , 
                                     N 
                                   
                                 
                               
                               
                                 … 
                               
                               
                                 
                                   σ 
                                   2 
                                 
                               
                             
                           
                           ] 
                         
                         T 
                       
                       ≡ 
                       Σ 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   15 
                 
               
             
           
         
       
     
         [0032]    Assuming that the solution is to be approximated by a linear combination: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Δ 
                        
                       
                         ? 
                       
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         j 
                       
                        
                       
                           
                       
                        
                       
                         
                           P 
                           j 
                         
                          
                         
                           
                             ϕ 
                             j 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       ? 
                     
                      
                     
                       indicates text missing or illegible when filed 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   16 
                 
               
             
           
         
       
     
         [0000]    of known functions φ j (t), such as, for example, a polynomial, an orthogonal polynomial, a trigonometric polynomial, a b-spline. etc, with unknown coefficients p j  forming the vector p=[p 1  . . . p j ] T . This vector is to be estimated on the oasis of the approximate equations: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           J 
                         
                          
                         
                             
                         
                          
                         
                           
                             p 
                             j 
                           
                            
                           
                             
                               ϕ 
                               j 
                             
                              
                             
                               ( 
                               
                                 t 
                                 n 
                               
                               ) 
                             
                           
                         
                       
                       ≡ 
                       
                         
                           
                             ? 
                           
                            
                           
                             ( 
                             
                               t 
                               n 
                             
                             ) 
                           
                         
                         - 
                         
                           ? 
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           for 
                            
                           
                               
                           
                            
                           n 
                         
                         = 
                         1 
                       
                       , 
                       … 
                        
                       
                           
                       
                       , 
                       N 
                     
                      
                     
                       
 
                     
                      
                     
                       
                         ? 
                       
                        
                       
                         indicates text missing or illegible when filed 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   17 
                 
               
             
           
         
       
     
         [0000]    that may be given a matrix form: 
         [0000]      Φ p ≅Δ {circumflex over (l)}   Equation 18
 
         [0000]    where: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       Φ 
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   ϕ 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   ϕ 
                                   j 
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               ⋮ 
                             
                             
                               ⋱ 
                             
                             
                               ⋮ 
                             
                           
                           
                             
                               
                                 
                                   ϕ 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     N 
                                   
                                   ) 
                                 
                               
                             
                             
                               … 
                             
                             
                               
                                 
                                   ϕ 
                                   j 
                                 
                                  
                                 
                                   ( 
                                   
                                     t 
                                     N 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     and 
                      
                     
                       
 
                     
                      
                     
                         
                     
                      
                     
                       
                         Δ 
                          
                         
                           ? 
                         
                       
                       = 
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     ? 
                                   
                                 
                               
                               
                                 
                                   ⋮ 
                                 
                               
                               
                                 
                                   
                                     ? 
                                   
                                 
                               
                             
                             ] 
                           
                           . 
                           
                             
 
                           
                            
                           
                             ? 
                           
                         
                          
                         
                           indicates text missing or illegible when filed 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   19 
                 
               
             
           
         
       
     
         [0033]    Under an assumption that there is no correlation between consecutive samples provided by the OPM, i.e. all c n,v =0, the LS solution of this equation has the form: 
         [0000]      {circumflex over (p)}=(Φ T Φ) −1 Φ T Δ {circumflex over (l)}   Equation 20
 
         [0000]    and the corresponding covariance matrix is: 
         [0000]      Cov[ {circumflex over (p)} ]=(Φ T Φ) −1 σ 2    Equation 21
 
         [0034]    If the correlation between samples cannot be neglected, then the solution takes on the form: 
         [0000]      {circumflex over ( p )}=(Φ T Σ −1 Φ) −1 Φ T Σ −1 Δ{circumflex over (l)}  Equation 22
 
         [0000]    and the corresponding covariance matrix is: 
         [0000]      Cov[ {circumflex over (p)} ]=(Φ T Σ −1 Φ) −1    Equation 23
 
         [0035]    If the samples may be considered uncorrected, then estimates {circumflex over (p)} 1 , . . . , {circumflex over (p)} K  of the parameters p 1 , . . . , p K , characterizing the drift of K DWDM channels, may be obtained using the LS method, described in the previous section, in an integrated numerical process: 
         [0000]      [{circumflex over (p)} 1  . . . {circumflex over (p)} K ]=(Φ T Φ) −1 Φ T [Δ{circumflex over (l)} 1  . . . Δ{circumflex over (l)} K ]  Equation 24
 
         [0000]    where Δ{circumflex over (l)} k  is the vector of deviations of the results of measurements of the central wavelength in the kth channel from the ITU-grid value of this wavelengthth. In general, the channels may differ in the laser wavelength deviation; the corresponding variances σ 1   2 , . . . , σ K   2  may be estimated 
         [0036]    according to the formula: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         σ 
                         ^ 
                       
                       k 
                       2 
                     
                     = 
                     
                       
                         1 
                         
                           N 
                           - 
                           J 
                         
                       
                        
                       
                         
                            
                           
                             
                               Φ 
                                
                               
                                 
                                   p 
                                   ^ 
                                 
                                 k 
                               
                             
                             - 
                             
                               Δ 
                                
                               
                                 
                                   l 
                                   ^ 
                                 
                                 k 
                               
                             
                           
                            
                         
                         2 
                         2 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   for 
                    
                   
                     
 
                   
                    
                   
                     
                       k 
                       = 
                       1 
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     K 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   25 
                 
               
             
           
         
       
     
         [0037]    Then the uncertainty-based weighing may be applied to obtain the solution: 
         [0000]    
       
         
           
             
               
                 
                   
                     p 
                     ^ 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         K 
                       
                        
                       
                           
                       
                        
                       
                         
                           1 
                           
                             σ 
                             k 
                             2 
                           
                         
                          
                         
                           
                             p 
                             ^ 
                           
                           k 
                         
                       
                     
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         K 
                       
                        
                       
                           
                       
                        
                       
                         1 
                         
                           σ 
                           k 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   26 
                 
               
             
           
         
       
     
         [0000]    characterizing the wavelength-averaged drift of the OPM over time. 
         [0038]    If the maximum deviation of the central wavelength of the laser signal used for the OPM drift compensation is negligible with respect to the assumed maximum error of wavelength measurements performed by the OPM, then the time drift of the OPM may be corrected on the basis of a single result of measurement. 
         [0039]    As well, if the standard deviation of the central wavelength averaged over a time interval ΔT is negligible with respect to the assumed standard deviation of the error of wavelength measurements performed by the OPM, and the OPM drift during the time interval ΔT is negligible, then the time drift of OPM may be corrected on the basis of the average of results of measurement performed during ΔT. 
         [0040]    Although the present invention has been described by way of a particular embodiment and examples thereof, it should be noted that it will be apparent to persons skilled in the art that modifications may be applied to the present particular embodiment without departing from the scope of the present invention.