Abstract:
Disclosed herein is a dual parallel Mach-Zehnder-modulator (DPMZM) device comprising a DPMZM  10  having first and second inner MZMs arranged parallel to each other. The first inner MZM generates an in-phase component E I  of an optical signal in response to a first driving voltage V I , and the second inner MZM generates a quadrature component E Q  of said optical signal in response to a second driving voltage V Q . Further disclosed is a calculation unit  52  configured for receiving an in-phase component y I  and a quadrature component y Q   _  of a desired base-band signal, and for calculating pre-distorted first and second driving voltages V I , V Q . The calculation of the pre-distorted first and second driving voltages V I , V Q  is based on a model of said DPMZM  10  accounting for I-Q cross-talk, and using an algorithm that determines said first and second driving voltages V I , V Q  each as a function of both of said in-phase and quadrature components y I , y Q  of said base-band signal.

Description:
FIELD OF THE INVENTION 
       [0001]    The present invention is in the field of optical data transport technology. More particularly, the present invention relates to a dual parallel Mach-Zehnder-modulator (DPMZM) device for modulating the amplitude and phase of a laser carrier according to data to be transmitted, as well as a method for operating a DPMZM. 
       BACKGROUND OF THE INVENTION 
       [0002]    The structure of a conventional DPMZM  10  is shown in  FIG. 1 . As seen therein, the DPMZM  10  comprises an optical input  12  for inputting an optical carrier signal, and an optical output  14  for outputting a QAM-modulated optical signal. Downstream of the optical input  12 , the DPMZM  10  branches into a first and a second arm  16 ,  18 , respectively, that are rejoined at the optical output  14 , thereby forming what is referred to as an “outer MZM” in the present disclosure. 
         [0003]    Within each of the first and second arms  16 ,  18  of the outer MZM, respective first and second “inner” MZMs  20 ,  22  are provided. The first inner MZM  20  comprises electrodes  24  for applying a first driving voltage V I  for generating an in-phase component E 1  of the optical signal to be transmitted. In other words, the first driving voltage V 1  is intended for modulating the part of the carrier signal propagating along first arm  16  of the outer MZM according to the I-component of a base-band signal, and said driving voltage V I  is hence generally an AC-signal. In addition to AC-driving, a bias voltage is applied to the first inner MZM  20 . While in practical implementations AC-driving and bias would typically be applied using different electrodes, for simplicity a single set of electrodes  24  is shown in  FIG. 1 . 
         [0004]    Likewise, a pair of electrodes  26  is associated with the second inner MZM  22 , for applying a second driving voltage V Q  for generating a quadrature component E Q  of the optical signal and for also applying a suitable bias voltage. Finally, a set of electrodes  28  is provided in the second arm  18  of the outer MZM in order to introduce a desired phase shift of 90° between the in-phase and quadrature components E 1  and E Q  of the modulated signal before these I- and Q-modulated signals are combined at the optical output  14 . 
         [0005]    The amplitudes E 1  and E 2  of the electrical fields of the portions of the carrier entering the first and second arms  16 ,  18  of the outer MZM can be modulated in response to the driving voltages V I , V Q  to give the in-phase and quadrature components E I , E Q  as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       E 
                       I 
                     
                     = 
                     
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               
                                 π 
                                 2 
                               
                               · 
                               
                                 
                                   V 
                                   I 
                                 
                                 
                                   V 
                                   π 
                                 
                               
                             
                             + 
                             
                               ϕ 
                               1 
                             
                           
                           ) 
                         
                       
                        
                       
                         E 
                         1 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         E 
                         Q 
                       
                       = 
                       
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 
                                   π 
                                   2 
                                 
                                 · 
                                 
                                   
                                     V 
                                     Q 
                                   
                                   
                                     V 
                                     π 
                                   
                                 
                               
                               + 
                               
                                 ϕ 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                           E 
                           2 
                         
                       
                     
                     , 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    assuming that the DPMZM device is composed of ideal inner and outer MZMs. As seen herein, the in-phase and quadrature components E I , E Q  depend non-linearly from the corresponding driving voltages V I , V Q . V π  is a device dependent constant and φ 1  and φ 2  are constant phases which can be adjusted by introducing a suitable bias at the electrodes  24 ,  26 , respectively. 
         [0006]    Unfortunately, a DPMZM is far from an ideal device: By its construction principle, it has non-linear input-output characteristics, and due to manufacturing imperfections, it generates cross-talk and amplitude imbalance between in-phase and quadrature components of the output signal. It is seen that the manufacturing imperfections are related to the extinction ratio (ER) of the DPMZM, i.e. the ratio of the maximum and minimum output power at the optical output  14  of the outer MZM over a sweep of the first and second driving voltages V I , V Q . An ideal DPMZM has an infinite ER, but series manufactured MZMs rarely achieve a guaranteed ER greater than 20 dB. With the introduction of new technologies like CMOS Photonics, it can be envisaged that newer and cheaper DPMZMs will become available, but their extinction ratios will likely be even well below 20 dB. 
         [0007]    The non-ideal characteristics of the DPMZM impair the quality of the transmit signal and result into a performance penalty depending on the adopted signal constellation. State-of-the-art 100 G (˜100 GB/s) optical systems employ 4-point quadrature amplitude modulation (4 QAM) which tolerates well the imperfections of currently available DPMZMs. However, 200 G and 400 G systems will likely rely on 16 QAM that is very sensitive to DPMZM limitations. Future systems may also employ bigger QAM constellations or orthogonal frequency division multiplexing (OFDM) which suffer from even larger penalties. It would therefore seem that for these applications, an increased manufacturing effort is unavoidable in order to achieve the required signal quality. 
       SUMMARY OF THE INVENTION 
       [0008]    The problem underlying the invention is to provide a DPMZM device that allows for an improved signal quality with minimum increase in manufacturing effort. 
         [0009]    This problem is solved by a DPMZM device according to claim  1  and by a modulation method according to claim  17 . Preferable further developments are defined in the dependent claims. 
         [0010]    Similar to the prior art DPMZM of  FIG. 1 , the DPMZM device of the invention comprises first and second inner MZMs arranged parallel to each other, the first inner MZM for generating an in-phase component E I  of an optical signal in response to a first driving voltage V I  and the second inner MZM for generating a quadrature component E Q  of said optical signal in response to a second driving voltage V Q . In addition, the DPMZM device of the invention comprises a calculation unit configured for
       receiving an in-phase component y I  and a quadrature component y Q  of a desired base-band signal, and   calculating pre-distorted first and second driving voltages V I , V Q  based on
           a model of said DPMZM device accounting for I-Q cross-talk, and   using an algorithm that determines said first and second driving voltages each as a function of both of said in-phase and quadrature components y I , y Q  of said desired base-band signal.   
               
 
         [0015]    Note that in the present disclosure, the term “DPMZM” refers to the optical component only, while the term “DPMZM device” in addition includes electronic components, such as the calculation unit. For example, the DPMZM may be part of an optical chip, and the calculation unit may be formed by a further electronic chip. However, the present invention is not limited to any specific structure in this regard. For example, the optical and electronic components could also be integrated on a single chip. Further, the “calculation unit” need of course not be a separate entity but could be resembled by an electronic component implementing further functionalities, other than the pre-distortion such as for example electrical functionalities of a transmitter. 
         [0016]    Instead of trying to avoid imperfect optical behavior of the DPMZM at any price, the present invention simply put suggests to take the deficiencies as they are but to instead mitigate the adverse effects of these deficiencies by a new way of calculating first and second driving voltages V I , V Q  accounting for these deficiencies. The invention is based on the observation that a main source of the errors encountered is due to the finite inner extinction ratios of the individual first and second inner MZMs, which turn out to be the major source of cross-talk between the in-phase and quadrature output components E I  and E Q . Accordingly, the present invention suggests to calculate the first and second driving voltages V I , V Q  based on a model of the DPMZM accounting for I-Q cross-talk. 
         [0017]    Further, according to the invention the calculating unit uses an algorithm that determines the first and second driving voltages V I , V Q  each as a function of both of said in-phase and quadrature components y I , y Q  of the desired base-band signal. Namely, since due to the finite inner extinction ratios cross-talk as such cannot be avoided, in the framework of the invention the first and second driving voltages V I , V Q  can be calculated in an anticipatory view of the prospective cross-talk, thereby leading to adapted or “pre-distorted” first and second driving voltages V I , V Q  that will lead to the desired in-phase and quadrature components y I , y Q . The term “pre-distortion” hereby indicates that the driving voltages are “distorted” as compared to the driving voltages occurring in the ideal situation of Eq. (1), and they are “pre”-distorted in the sense that the distortion accounts for the imperfections of the inner MZMs and in particular the I-Q cross-talk in advance. This becomes possible if each one of the first and second driving voltages V I , V Q  are determined as a function of both of said in-phase and quadrature-components y I , y Q  of said base-band signal. 
         [0018]    In a preferred embodiment, the model employed by the calculation unit is based on one or more model parameters corresponding to or at least partially reflecting the finite inner extinction ratios of the inner MZMs. In fact, as will be demonstrated below, the calculation unit may employ a model of the DPMZM in which the finite inner extinction ratios are explicitly employed as model parameters, and it turns out to be possible to greatly increase the signal quality by pre-distortion of the driving voltages V I , V Q  based on this model. However, the model may also be based on model parameters that only reflect, at least in part, the finite inner extinction ratio, for example by being functionally dependent on the inner extinction ratios. 
         [0019]    The one or more model parameters can be determined upon manufacturing and may be stored in a memory of or accessible by the calculation unit. However, in a preferred embodiment, the DPMZM device comprises a parameter calculating unit which is adapted to
       receive a quality indicator fed-back from a receiver receiving the optical signal, and   to modify the one or more model parameters such as to optimize the quality indicator.       
 
         [0022]    This variant is based on the consideration that the better the model captures the imperfections of the DPMZM device, the more successfully can the quality impairments of the output signal be reduced by means of pre-distortion of the first and second driving voltages V I , V Q . Accordingly, an optimized quality indicator at the same time indicates that the one or more model parameters are optimally chosen. In a preferred embodiment, the quality indicator is an estimate of a bit-error-rate at the receiver and the parameter calculating unit is adapted to modify the one or more model parameters, in response to said back-fed bit-error-rate, such as to minimize the same. Alternatively, the quality indicator may be an estimate of the power of a residual carrier and the parameter calculating unit is adapted to modify the one or more model parameters, in response to the back-fed residual carrier power, such as to minimize the same. 
         [0023]    A yet further example of a quality indicator is a deviation, in particular a mean square error between the desired transmit signals and the actual transmit signals, wherein the parameter calculating unit is adapted to modify the one or more model parameters such as to minimize said deviation. 
         [0024]    In a preferred embodiment, the DPMZM device further comprises an AC-coupling for coupling said first and second driving voltages V I , V Q  as calculated by the calculation unit to the respective first and second inner MZMs. Further, first and second biasing units are associated with the first and second inner MZMs, respectively, said first and second biasing units being adapted to apply first and second bias components to said first and second inner MZMs. Herein, the first and second bias components may at least approximately represent corresponding DC-components of the driving voltages V I , V Q  as calculated by the calculation unit. 
         [0025]    Note that generally, depending on the model of the DPMZM device employed by the calculation unit, it is not to be expected that the average values of the calculated pre-distorted first and second driving voltages V I , V Q  arising for all in-phase and quadrature components y I , Y Q  of a desired base-band signal are zero, only because the average values of these components y I , y 2  themselves should be zero. Instead, as will be shown below with reference to an explicit embodiment, it is seen that the first and second driving voltages V I , V Q  arising from this calculation have a DC-component or “offset”. However, when AC-coupling the first and second driving voltages V I , V Q  with the first and second inner MZMs, this DC-component would be lost. According to this embodiment, this lost DC-component is re-introduced by first and second biasing units associated with the first and second inner MZMs, respectively, which are adapted to apply first and second bias components to the first and second inner MZMs making up for the missing DC-components of the AC-coupled V I , V Q . 
         [0026]    Herein, it is referred to first and second bias “components” because the first and second biasing units will in general provide a further bias as would occur in an ordinary bias control scheme, for example for compensating the phase offsets φ 1 , φ 2  in Eq. (1), so that the bias corresponding to the DC-components of the calculated driving voltages V I . V Q  resembles only a component of the total bias applied to the first and second inner MZMs. Further, in practical applications, the bias corresponding to the respective DC-component of V I  and V Q  will not be applied separately, but together with the bias according to a customary control. But in this case too, the biases applied by the first and second biasing units do account for the DC-components of V I  and V Q  which would be lost due to AC coupling. 
         [0027]    In one embodiment, the value of these first and second bias components could simply be obtained from the average of the calculated pre-distorted driving voltages V I , V Q . These computed bias components could then be added to the bias as determined by customary bias control In the alternative, however, the DPMZM device may further comprise a bias component control unit adapted to adjust the first and/or second bias components in response to
       an error indicating signal determined at a receiver receiving the optical signal, or   a quality indicator, in particular a residual carrier in the optical signal, determined at a receiver of the optical signal.       
 
         [0030]    Namely, if a DC-component of the pre-distorted first and second driving voltages V I , V Q  is truncated in the AC-coupling, this will lead to an increased signal error or a reduced quality of the signal at the receiver. By sending an error indicating signal or quality indicator determined at a receiver of the optical signal to the bias component control unit, the first and second bias components can be adjusted in a way that the error represented by the error indicating signal is reduced or the quality as indicated by the quality indicator is increased, which will then reintroduce at least approximately the bias that was lost due to truncation of the DC-component of the first and second driving voltages V I , V Q . 
         [0031]    In a preferred embodiment, the bias component control unit is configured to adjust said first and second bias components using a gradient descent algorithm minimizing the error indicated by said error indicating signal or optimizing said quality indicator as a function of said first and/or second bias components. 
         [0032]    According to the previous description, the parameter calculating unit and the bias component control unit rely on quality indicators or error indicating signals sent back to the DPMZM device from a receiver. Herein, the receiver may be a far-end receiver or a local monitoring receiver associated with the DPMZM device. In particular, the receiver may be a coherent receiver. 
         [0033]    In a preferred embodiment, the aforementioned model of the DPMZM device employed by the calculation unit is based on the assumption that each of the first and second inner MZMs are independently biased to deliver the minimum possible output power when the first and second driving voltages V I , V Q , respectively, are zero. Note that this is generally not the biasing state which will lead to the optimum signal quality and might hence at first sight not be an obvious starting point for the model. However, it turns out that this model allows for a very simple and efficient calculation of the driving voltages V I , V Q  which then turn out to comprise a DC-component reflecting and in fact “repairing” the non-perfect biasing underlying this model. As explained before, these DC-components can then be accounted for by the bias component control unit in practical implementations. 
         [0034]    In a preferred embodiment, the model employed by the calculation unit is represented by a system of two coupled non-linear equations relating the in-phase and quadrature components y I , y Q  of the desired base-band signal to the driving voltages V I , V Q . 
         [0035]    In one embodiment, the system of two coupled linear equations can be represented as follows: 
         [0000]    
       
         
           
             
               y 
               I 
             
             = 
             
               
                 sin 
                  
                 
                   ( 
                   
                     
                       π 
                       2 
                     
                     · 
                     
                       
                         V 
                         I 
                       
                       
                         V 
                         π 
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   γ 
                   Q 
                 
                 · 
                 
                   cos 
                    
                   
                     ( 
                     
                       
                         π 
                         2 
                       
                       · 
                       
                         
                           V 
                           Q 
                         
                         
                           V 
                           π 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 y 
                 
                   Q 
                    
                   
                       
                   
                 
               
               = 
               
                 
                   sin 
                    
                   
                     ( 
                     
                       
                         π 
                         2 
                       
                       · 
                       
                         
                           V 
                           Q 
                         
                         
                           V 
                           π 
                         
                       
                     
                     ) 
                   
                 
                 - 
                 
                   
                     γ 
                     I 
                   
                   · 
                   
                     cos 
                      
                     
                       ( 
                       
                         
                           π 
                           2 
                         
                         · 
                         
                           
                             V 
                             I 
                           
                           
                             V 
                             π 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    wherein V π , γ I  and γ Q  are positive characterizing constants of the DPMZM device. More precisely, the constants γ I , γ Q  represent the square root of the inverse inner power extinction ratios of the first and second inner MZMs. 
         [0036]    Note that the coupling of the two coupled non-linear equations is introduced by the terms proportional to γ Q  and γ I . A smaller inner extinction ratio corresponds to a larger value of γ I  or γ Q  and hence to an increased cross-talk. Conversely, if the inner extinction ratios approach infinity, the constants γ I  and γ Q  converge to zero, which means that the system of non-linear equations decouples and in fact converges towards the situation from Eq. (1). 
         [0037]    Accordingly, the above system of two-coupled non-linear equations resembles a model of the DPMZM device accounting for first and second inner MZMs having finite inner extinction ratios and in fact directly employs two model parameters corresponding to the finite inner extinction ratios of the inner MZMs. 
         [0038]    In a preferred embodiment, the aforementioned algorithm used by the calculation unit resembles an iterative solution of the coupled non-linear equation system. As will be shown in detail below with reference to a specific embodiment, the iterative solution can be provided with low computational effort in real time using look-up tables for only two non-linear functions requiring moderate memory. Herein, the iterative solution includes at least two, preferably at least three iterations. 
         [0039]    In an alternative embodiment, the calculation performed by said calculation unit comprises solving the following equations: 
         [0000]    
       
         
           
             
               V 
               I 
             
             = 
             
               
                 
                   2 
                    
                   
                     V 
                     π 
                   
                 
                 π 
               
                
               a 
                
               
                   
               
                
               
                 sin 
                  
                 
                   ( 
                   
                     
                       P 
                       1 
                     
                      
                     
                       ( 
                       
                         
                           y 
                           I 
                         
                         , 
                         
                           y 
                           Q 
                         
                       
                       ) 
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
               V 
               Q 
             
             = 
             
               
                 
                   2 
                    
                   
                     V 
                     π 
                   
                 
                 π 
               
                
               a 
                
               
                   
               
                
               
                 sin 
                  
                 
                   ( 
                   
                     
                       P 
                       2 
                     
                      
                     
                       ( 
                       
                         
                           y 
                           I 
                         
                         , 
                         
                           y 
                           Q 
                         
                       
                       ) 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    wherein P 1  (y I ,y Q ) and P 2  (y I ,y Q ) are polynomials of y I  and y Q . Herein, P 1  (y I ,y Q ) is preferably a first order polynomial in y I  and a two or higher order polynomial in y Q . Likewise, P 2  (y I ,y Q ) is preferably a first order polynomial in y Q  and a two or higher order polynomial in y I . The coefficients of the polynomials are examples of the aforementioned model parameters which can be determined by a parameter calculating unit in a way described above. 
         [0040]    Preferably, the calculation unit comprises a look-up table for the function a sin(x), and preferably also a look-up table for the function cos(x) or cos(a sin(x)). Using for example a single look-up table for a sin(x), the above polynomial based expression for V I , V Q , can be evaluated. Using in addition a look-up table for cos(x) or cos(a sin(x)), the iterative solution of the above stated system of two coupled non-linear equations can be efficiently solved. 
     
    
     
       SHORT DESCRIPTION OF THE FIGURES 
         [0041]    For the purposes of promoting an understanding of the principles of the invention, reference will now be made to the preferred embodiments illustrated in the drawings and specific language will be used to describe the same. It will nevertheless be understood that no limitation of the scope of the invention is thereby intended, such alterations and further modifications in the illustrated device and method and such further applications of the principles of the invention as illustrated therein being contemplated therein as would normally occur now or in the future to one skilled in the art to which the invention relates. 
           [0042]      FIG. 1  shows a schematic view of a prior art DPMZM, 
           [0043]      FIG. 2  shows a flow diagram of a single iteration of a pre-distortion algorithm according to the invention, 
           [0044]      FIG. 3  shows a flow diagram of two consecutive iterations of a pre-distortion algorithm according to the invention, 
           [0045]      FIG. 4  shows a flow diagram of three consecutive iterations of a pre-distortion algorithm according to the invention, 
           [0046]      FIG. 5  shows a DPMZM device employing a modified version of a generally pre-known bias control, in which error offsets are introduced, 
           [0047]      FIG. 6  shows a DPMZM device employing a novel bias control relying on a feedback channel from a far-end receiver, 
           [0048]      FIGS. 7 to 9  show the performance of the iterative pre-distortion algorithm as indicated in  FIGS. 3 and 4  in simulative investigations, 
           [0049]      FIG. 10  shows a DPMZM device including a parameter calculating unit adapted to modify model parameters such as to optimize a quality indicator received from a far-end receiver, 
           [0050]      FIG. 11  shows a DPMZM device including a parameter calculating unit employing an indirect learning architecture, 
           [0051]      FIG. 12  shows a DPMZM device including a parameter calculating unit, and employing an inverse calculation unit, and 
           [0052]      FIG. 13  shows a general setup of a DPMZM and an inverse system providing pre-distorted complex input to the DPMZM. 
       
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0053]    Turning back to the DPMZM  10  of  FIG. 1 , in the following discussion we assume an infinite or a perfectly compensated outer extinction ratio of the outer MZM and concentrate on the inner MZMs  20 ,  22 . Further, for the moment we assume that each inner MZM  20 ,  22  is biased to deliver the minimum possible output power when the corresponding driving signal V I , V Q  is zero. Note that from an operation point of view, the latter would actually not be an ideal starting point, as this does not correspond to the biasing that eventually leads to the best signal quality. However, the assumption is made purely for mathematical purposes, as it leads to very simple equations, the solutions of which then lead to DC-components in the pre-distorted first and second driving voltages V I , V Q  that account for a more proper biasing. 
         [0054]    With appropriate normalization of the electric field amplitude by the input electric field amplitudes in the respective first and second arms  16 ,  18  of the outer MZMs, the input/output relations of the first and second inner MZMs  20 ,  22  are as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                     I 
                   
                   = 
                   
                     
                       1 
                       
                         1 
                         + 
                         
                           β 
                           I 
                         
                       
                     
                      
                     
                       [ 
                       
                         
                           exp 
                            
                           
                             ( 
                             
                               j 
                               · 
                               
                                 π 
                                 2 
                               
                               · 
                               
                                 ( 
                                 
                                   
                                     
                                       V 
                                       I 
                                     
                                     
                                       V 
                                       π 
                                     
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             β 
                             I 
                           
                           · 
                           
                             exp 
                              
                             
                               ( 
                               
                                 
                                   - 
                                   j 
                                 
                                 · 
                                 
                                   π 
                                   2 
                                 
                                 · 
                                 
                                   ( 
                                   
                                     
                                       
                                         V 
                                         I 
                                       
                                       
                                         V 
                                         π 
                                       
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                    
                   and 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       E 
                       Q 
                     
                     = 
                     
                       
                         1 
                         
                           1 
                           + 
                           
                             β 
                             Q 
                           
                         
                       
                        
                       
                         [ 
                         
                           
                             exp 
                              
                             
                               ( 
                               
                                 j 
                                 · 
                                 
                                   π 
                                   2 
                                 
                                 · 
                                 
                                   ( 
                                   
                                     
                                       
                                         V 
                                         Q 
                                       
                                       
                                         V 
                                         π 
                                       
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               β 
                               Q 
                             
                             · 
                             
                               exp 
                                
                               
                                 ( 
                                 
                                   
                                     - 
                                     j 
                                   
                                   · 
                                   
                                     π 
                                     2 
                                   
                                   · 
                                   
                                     ( 
                                     
                                       
                                         
                                           V 
                                           Q 
                                         
                                         
                                           V 
                                           π 
                                         
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0055]    Herein, E I  and E Q  are the normalized in-phase and quadrature components of the optical signal generated in response to the first and second driving voltages V I , V Q , respectively. V π , β I  and β Q  are positive characteristic constants of the DPMZM  10 . More particularly, the constants β I  and β Q  represent the ratio of the electric field amplitude in the two arms of the first and second inner MZMs  20 ,  22 , respectively. In other words, a value β I =1 would resemble a situation where the power is evenly split between the two arms of the first inner MZM  20 , while any deviation from this ideal behavior would lead to a value β I  different from 1. The symbol “j” resembles the imaginary part of a complex number in the usual manner. 
         [0056]    It is further assumed that the outer MZM is biased, by means of the electrodes  28  provided in the second arm  18  thereof, such as to establish a 90° phase shift between the in-phase and quadrature components E I , E Q  in the combined or total electric field amplitude E, i.e. 
         [0000]        E=+E   I   +j·E   Q .  (4)
 
         [0057]    Introducing the expressions from Eqs. (2) and (3) above and carrying out a number of arithmetic steps, we obtain the following expression for the combined or total electric field amplitude E: 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                     = 
                     
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               π 
                               2 
                             
                             · 
                             
                               
                                 V 
                                 I 
                               
                               
                                 V 
                                 π 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           γ 
                           Q 
                         
                         · 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 π 
                                 2 
                               
                               · 
                               
                                 
                                   V 
                                   Q 
                                 
                                 
                                   V 
                                   π 
                                 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         j 
                         · 
                         
                           [ 
                           
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     π 
                                     2 
                                   
                                   · 
                                   
                                     
                                       V 
                                       Q 
                                     
                                     
                                       V 
                                       π 
                                     
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 γ 
                                 I 
                               
                               · 
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     
                                       π 
                                       2 
                                     
                                     · 
                                     
                                       
                                         V 
                                         I 
                                       
                                       
                                         V 
                                         π 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where we introduced: 
         [0000]    
       
         
           
             
               
                 
                   
                     γ 
                     I 
                   
                   = 
                   
                     
                       1 
                       - 
                       
                         β 
                         I 
                       
                     
                     
                       1 
                       + 
                       
                         β 
                         I 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 and 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     γ 
                     Q 
                   
                   = 
                   
                     
                       
                         1 
                         - 
                         
                           β 
                           Q 
                         
                       
                       
                         1 
                         + 
                         
                           β 
                           Q 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0058]    Herein, γ I  and γ Q  are the square root of the inverse power extinction ratios ER I , ER Q  of the first and second inner MZM  20 ,  22 . Usually the extinction ratios ER 1  and ER Q  of the first and second inner MZMs  20 ,  22  are represented as dB values as follows: 
         [0000]      ER I =−20·log 10 |γ I |  (8)
 
         [0000]      and 
         [0000]      ER Q =−20·log 10 |γ Q |.  (9)
 
         [0059]    From Eq. (5) it can be seen that the DPMZM  10  exhibits an intrinsic sinusoidal non-linearity and, if the inner ERs are finite, additional I-Q-cross-talk. 
         [0060]    In a next step, we describe an algorithm for the computation of the driving voltages V I  and V Q  that produce the desire transmit signal if applied to the electrodes  24  and  26  of the non-ideal DPMZM. Assuming the above model of the imperfect DPMZM with non-vanishing inverse extinction ratios β I , γ Q , from Eq. (5) it is seen that the appropriate driving voltages V I  and V Q  to produce the I- and Q-components γ I , γ Q  of a desired transmit signal amount to a solution of the following non-linear equation system: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       y 
                       I 
                     
                     = 
                     
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               π 
                               2 
                             
                             · 
                             
                               
                                 V 
                                 I 
                               
                               
                                 V 
                                 π 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           γ 
                           Q 
                         
                         · 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 π 
                                 2 
                               
                               · 
                               
                                 
                                   V 
                                   Q 
                                 
                                 
                                   V 
                                   π 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     y 
                     Q 
                   
                   = 
                   
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             π 
                             2 
                           
                           · 
                           
                             
                               V 
                               Q 
                             
                             
                               V 
                               π 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         γ 
                         I 
                       
                       · 
                       
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 π 
                                 2 
                               
                               · 
                               
                                 
                                   V 
                                   I 
                                 
                                 
                                   V 
                                   π 
                                 
                               
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
         [0061]    For solving the above equation system for V I  and V Q , the following iterative pre-distortion algorithm may be employed: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               V 
                               I 
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                             = 
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     V 
                                     π 
                                   
                                 
                                 π 
                               
                                
                               a 
                                
                               
                                   
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       y 
                                       I 
                                     
                                     - 
                                     
                                       
                                         γ 
                                         Q 
                                       
                                       · 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               π 
                                               2 
                                             
                                             · 
                                             
                                               
                                                 V 
                                                 Q 
                                                 
                                                   ( 
                                                   
                                                     n 
                                                     - 
                                                     1 
                                                   
                                                   ) 
                                                 
                                               
                                               
                                                 V 
                                                 π 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               V 
                               Q 
                               
                                 ( 
                                 n 
                                 ) 
                               
                             
                             = 
                             
                               
                                 
                                   2 
                                   · 
                                   
                                     V 
                                     π 
                                   
                                 
                                 π 
                               
                                
                               a 
                                
                               
                                   
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     
                                       y 
                                       Q 
                                     
                                     + 
                                     
                                       
                                         γ 
                                         I 
                                       
                                       · 
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             
                                               π 
                                               2 
                                             
                                             · 
                                             
                                               
                                                 V 
                                                 I 
                                                 
                                                   ( 
                                                   
                                                     n 
                                                     - 
                                                     1 
                                                   
                                                   ) 
                                                 
                                               
                                               
                                                 V 
                                                 π 
                                               
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                      
                     
                         
                     
                      
                     
                       ( 
                       
                         
                           n 
                           = 
                           1 
                         
                         , 
                         2 
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         K 
                       
                       ) 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the positive integer K is a number of iterations, and V I   (n)  and V Q   (n)  are the approximations of the desired driving voltages V I  and V Q  at the n th  iteration. From the above iteration algorithm, the concept of “pre-distortion” becomes particularly apparent. For example, in a perfect DPMZM without cross talk, γ Q  would be zero and V I   (n)  would be dependent on the desired I-component y I  of the transmit signal only. With non-vanishing y Q , a “distortion” is introduced to V I , which distortion is dependent on V Q  and in fact accounts for the cross-talk introduced by non-vanishing parameters γ I , γ Q  in an anticipatory manner. 
         [0062]    The algorithm can be conveniently initialized with 
         [0000]        V   I   (0) =0 
         [0000]        V   Q   (0) =0,  (12)
 
         [0000]    which results into 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       V 
                       I 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           2 
                           · 
                           
                             V 
                             π 
                           
                         
                         π 
                       
                        
                       a 
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               y 
                               I 
                             
                             - 
                             
                               γ 
                               Q 
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       V 
                       Q 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           2 
                           · 
                           
                             V 
                             π 
                           
                         
                         π 
                       
                        
                       a 
                        
                       
                           
                       
                        
                       
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 y 
                                 Q 
                               
                               + 
                               
                                 γ 
                                 I 
                               
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0063]    If desired, the first iteration can be slightly improved with negligible effort by using the following initialization: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       V 
                       I 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           2 
                           · 
                           
                             V 
                             π 
                           
                         
                         π 
                       
                        
                       a 
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               y 
                               I 
                             
                             - 
                             
                               
                                 
                                   γ 
                                   Q 
                                 
                                 · 
                                 E 
                               
                                
                               
                                 { 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     
                                       
                                         π 
                                         2 
                                       
                                       · 
                                       
                                         
                                           y 
                                           Q 
                                         
                                         
                                           V 
                                           π 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 } 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       V 
                       Q 
                       
                         ( 
                         1 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           2 
                           · 
                           
                             V 
                             π 
                           
                         
                         π 
                       
                        
                       a 
                        
                       
                           
                       
                        
                       
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 y 
                                 Q 
                               
                               + 
                               
                                 
                                   
                                     γ 
                                     I 
                                   
                                   · 
                                   E 
                                 
                                  
                                 
                                   { 
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         
                                           π 
                                           2 
                                         
                                         · 
                                         
                                           
                                             y 
                                             I 
                                           
                                           
                                             V 
                                             π 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                   } 
                                 
                               
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0064]    Herein, E{•} denotes a stochastic expectation. 
         [0065]      FIG. 2  illustrates a single iteration of the pre-distortion algorithm. In case of a digital implementation, the non-linear functions can be conveniently approximated by spline, i.e. piecewise linear interpolation. Since the non-linear functions are completely unaffected by the actual inner extinction ratios, the spline coefficients can be pre-computed offline and implemented via constant look-up tables. The parameters γ I  and γ Q , on the other hand, need to be calibrated for each individual DPMZM  10 , either upon manufacturing or upon operation and can possibly be tracked and adjusted over time in a fashion described in more detail below. 
         [0066]    Notably, when cascading multiple pre-distortion stages in the iterative solution according to Eq. (11), the input non-linear functions at each stage can be combined with the output non-linear functions of the previous stage and conveniently implemented in a single operation. Consequently, every iteration requires in fact the computation of only two non-linear functions with one real input and one real output which can be carried out rapidly in real-time under operation of the DPMZM  10 . 
         [0067]    To appreciate this fact, in  FIG. 3  a 2-stage pre-distortion algorithm is displayed, where the constants I init  and Q init  were introduced as the initial values of the iteration. If the initialization of Eq. (12) is adopted, we set 
         [0000]        I   init =1 
         [0000]        Q   init =1.  (15)
 
         [0068]    Alternatively, for the initialization of Eq. (14) we set 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       I 
                       init 
                     
                     = 
                     
                       E 
                        
                       
                         { 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 π 
                                 2 
                               
                               · 
                               
                                 
                                   y 
                                   I 
                                 
                                 
                                   V 
                                   π 
                                 
                               
                             
                             ) 
                           
                         
                         } 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       Q 
                       init 
                     
                     = 
                     
                       E 
                        
                       
                         
                           { 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   π 
                                   2 
                                 
                                 · 
                                 
                                   
                                     y 
                                     Q 
                                   
                                   
                                     V 
                                     π 
                                   
                                 
                               
                               ) 
                             
                           
                           } 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0069]    For further exemplification purposes, in  FIG. 4  an implementation of the 3-stage pre-distortion algorithm is shown. From this, generalizations to more stages will be apparent to the person skilled in the art. 
         [0070]    As mentioned above, the recursion of Eq. (11) was derived under the assumption that each of the inner MZMs  20 ,  22  is independently biased to deliver the minimum possible output power when the corresponding modulating signal V I  and V Q  is zero. However, if the inner extinction ratio is finite, in the absence of pre-distortion, this biasing condition results into imperfect carrier suppression in the optical output of the DPMZM  10 . The consequent residual carrier component is detrimental because it impairs the demodulation algorithms, wastes part of the available optical power, and enhances nonlinearities in the fiber. Therefore, the assumed biasing point in the above mathematical derivation, although leading to simpler pre-distortion equations, is actually neither optimal nor desirable in the absence of pre-distortion. 
         [0071]    In the presence of pre-distortion, the suboptimal bias is not problematic, since the recursion of Eq. (11) is able to suppress the residual carrier and to produce the desired optical signal. In the model considered so far, this is possible because the pre-distortion as derived by the iteration of Eq. (11) will automatically lead to values of V I  and V Q  including a DC offset representing the optimal biasing condition. In the mathematical model presented, the pre-distortion hence injects a DC-offset into the driving voltages and thereby effectively corrects the biasing point. However, in practical implementations, the driving voltages V I , V Q , i.e. the modulating signal, will generally be AC-coupled with the inner MZMs  20 ,  22  and a bias correction provided through the driving voltages V I  and V Q  would not reach the DPMZM  10 . 
         [0072]    In the present invention, this difficulty can be overcome by adopting a proper automatic bias control scheme which suppresses the residual carrier. Such a bias scheme, which in fact reintroduces the DC-correction that has been removed from the modulating signal by the AC-coupling, guarantees that the sum of the bias and driving voltage and, thus, the optical output of the DPMZM  10  remain unchanged. Remarkably, the pre-distortion can still be computed according to the simple recursion (11), i.e. under the assumption that each inner MZM  20 ,  22  is biased for minimum output power, because the resulting offset is filtered out anyway and does not interfere with the bias control. 
         [0073]    Several automatic bias control schemes for a DPMZ have been proposed in prior art, see e.g. P. S. Cho, J. B. Khurgin, and I. Shpantzer, “ Closed - loop bias control of optical quadrature modulator ,” IEEE Photonics Technology Letters, vol. 18, no. 21, pp. 2209-2211, November 2006 and M. Sotoodeh, Y. Beaulieu, J. Harley, and D. L. McGhan, “ Modulator bias and optical power control of optical complex E - field modulators ”, IEEE Journal of Lightwave Technology, vol. 29, no. 15, pp. 2235-2248, August 2011. These schemes control the bias voltages according to a gradient descent algorithm applied to an algorithm-specific error signal. Since they are designed to work on an ideal DPMZM with infinite ERs, they fail to suppress the residual carrier when the inner MZMs have a finite ER. However, it is possible to adapt such standard bias controls by injecting a proper offset into the error signal. The offset can be determined by factory calibration in such a way that the modified scheme suppresses the residual carrier. 
         [0074]      FIG. 5  illustrates a DPMZM device employing a modified version of the standard bias control. As is seen in  FIG. 5 , digital representations of the driving voltages V I , V Q  as computed with the recursion of Eq. (11) are inputted at inputs  30  and DA-converted at DA-converters  32 . The thus obtained analog signals are AC-coupled to the DPMZM  10 . Due to the AC-coupling, a DC-component of the driving voltages V I  and V Q  as obtained from the iteration of Eq. (11) will be lost. 
         [0075]    A part of the optical output signal  14  of the DPMZM  10  is branched off and detected with a photo detector  34 . The detection signal of the photo detector  34  is coupled to a bias error computer  36  which computes an error signal in a way per se known from the above citations. Two I- and Q-error signals  38 ,  40  are outputted from the bias error computer  36 , to which error offsets are added using adders  42 . The error offset can be determined by factory calibration in such a way that the modified scheme suppresses the residual carrier. These additional error offsets for residual carrier suppression are not provided for in known automatic bias control schemes and specifically relate to the operation of the DPMZM device of the present invention accounting for finite inner ERs. 
         [0076]    The error signals  38 ,  40 , together with the added offsets, are introduced to a bias voltage computer  44  which in turn computes a bias  46  for the first inner MZM  20  and a bias  47  for the second MZM  22  which are added to the AC-coupled analog driving voltages V I , V Q  by further adders  42  prior to introducing them to the DPMZM  10 . 
         [0077]    A further DPMZM device according to an embodiment of the invention employs a novel bias control that relies on a feedback channel from a far-end receiver to the transmitter. In the case of coherent transmission, digital demodulation algorithms at the receiver are able to detect the power of the residual carrier generated at the transmitter. The demodulator shall send back the power of the detected residual carrier to the bias control, which uses this information in a gradient descent algorithm to suppress directly the residual carrier. 
         [0078]    A corresponding DPMZM device is shown in  FIG. 6 . The DPMZM device of  FIG. 6  is able to carry out a conventional bias control scheme using the photo detector  34 , the bias error computer  36  and the bias voltage computer  44  as shown in  FIG. 5 , but without the error offset introduction of  FIG. 5 . This standard biasing algorithm, which is based on the standard local error signal defined along the lines of the previously cited references is, however, only provided for as a fallback means in the initial convergence phase. The proper bias control is based on an error signal  51  generated by a residual carrier detector  50  provided in a far-end receiver  48 . This error signal  51  may in fact directly correspond to the power of the residual carrier. The bias error computer  44  may then iteratively adapt the bias voltages  46 ,  47  by minimizing the error indicated by error signal  51 . The bias error computer  36  and the bias voltage computer  44  hence in combination form an example of the bias component control unit referred to in the summary of the invention. Note that this bias component control unit provides I- and Q-bias values which “account for” the DC component of the calculated driving voltage, but which also account for the customary bias control. In other words, the contribution corresponding to the DC-components of V I  and V Q  are not separately determined or applied, but are nevertheless automatically accounted for in the embodiment of  FIG. 6 . 
         [0079]    Both DPMZM devices of  FIGS. 5 and 6  hence allow for implementing a proper bias control scheme, i.e. a bias control scheme that suppresses the residual carrier and is compatible with the proposed iterative pre-distortion as defined in Eq. (11). 
         [0080]    With reference to  FIGS. 7 to 9 , the performance of the proposed pre-distortion algorithm is demonstrated by means of simulative investigations. In the analysis, a 16-QAM transmission at 31 Gsymbols/s in the presence of root Nyquist spectral shaping with a digital pre-distortion (DPD) running at two samples per symbol was considered. For the sake of simplicity, the quantization noise introduced by the DAC was neglected, and the same inner ERs for the first and second DPMZMs  20 ,  22  were assumed. The back-to-back performance (i.e. without transmission fiber in between) was evaluated on the basis of the bit error rate (BER) as a function of the optical signal-to-noise ratio (OSNR) and compared with the performance of a reference system without DPD. In the absence of DPD, the biasing points of the inner MZMs  20 ,  22  were optimized for minimal BER. In the presence of DPD, as discussed above, the biasing point was chosen such that the residual carrier is maximally suppressed. For a fair comparison, the same average optical power with and without DPD was transmitted. In the absence of DPD, the swing of the driving voltages V I , V Q  is smaller and the system benefits from using only a limited region of the MZM characteristics. In the presence of DPD, which enhances the signal peaks, i.e. the maximum values of V I  and V Q , a larger non-linear portion of the MZM characteristics is used, but the nonlinearity and the I-Q-cross-talk are pre-compensated. 
         [0081]      FIGS. 7 and 8  illustrate the cases of an inner ER of 20 dB and 15 dB, respectively. When the DPD is active, the inner MZMs  20 ,  22  are fully driven, i.e. the peak-to-peak swings of the driving voltages V I , V Q  are fixed to 2·V π . The simulation results show that, remarkably, only two iterations of the iterative pre-distortion algorithm of Eq. (11) are sufficient to achieve the ideal performance corresponding to a hypothetical ideal DPMZM. By contrast, the system without pre-distortion suffers from a significant OSNR penalty. 
         [0082]      FIG. 9  illustrates related simulation results for inner ERs as low as 10 dB. In order to avoid clipping in the implementation of the non-linear functions, the power of the modulated signal was backed-off by 1 dB, i.e. the swing of the modulated signal was reduced by 1 dB as compared to the previous two examples. In this case, three iterations of the DPD algorithm were necessary to achieve the ideal performance, whereas the system without DPD exhibits more than 3 dB penalty at a BER of 10 −3 . 
         [0083]    The iterative solution according to Eq. (11) of the system of coupled non-linear equations (10) turns out to be very attractive. One advantage is the low computation cost, which is both due to the underlying mathematical structure and the fact that in practice very few iterations are necessary. Further, the memory needed for look-up tables (if employed) for the non-linear functions is very moderate. However, the present invention is not limited to this particular algorithm, and the present invention instead also considers calculation units employing other algorithms. An alternative very useful calculation is derived from a 2-stage implementation of the iteration algorithm of Eq. (11). From  FIG. 2 , and noting that cos(a sin(x))=√{square root over (1−x 2 )}, the second iteration of the driving voltages can be obtained as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       V 
                       I 
                       
                         ( 
                         2 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           2 
                           · 
                           
                             V 
                             π 
                           
                         
                         π 
                       
                        
                       a 
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               y 
                               I 
                             
                             - 
                             
                               
                                 γ 
                                 Q 
                               
                               · 
                               
                                 
                                   1 
                                   - 
                                   
                                     
                                       ( 
                                       
                                         
                                           y 
                                           Q 
                                         
                                         + 
                                         
                                           
                                             γ 
                                             I 
                                           
                                           · 
                                           
                                             I 
                                             init 
                                           
                                         
                                       
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       V 
                       Q 
                       
                         ( 
                         2 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           2 
                           · 
                           
                             V 
                             π 
                           
                         
                         π 
                       
                        
                       a 
                        
                       
                           
                       
                        
                       
                         
                           sin 
                            
                           
                             ( 
                             
                               
                                 y 
                                 Q 
                               
                               + 
                               
                                 
                                   γ 
                                   I 
                                 
                                 · 
                                 
                                   
                                     1 
                                     - 
                                     
                                       
                                         ( 
                                         
                                           
                                             y 
                                             I 
                                           
                                           - 
                                           
                                             
                                               γ 
                                               Q 
                                             
                                             · 
                                             
                                               Q 
                                               init 
                                             
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0084]    In the above expressions, the square roots can be approximated by a Taylor expansion. If the square roots are expanded to second order terms in the components y I , y Q  of the desired transmit signal, the following alternative solution can be obtained: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       V 
                       I 
                     
                     = 
                     
                       
                         
                           2 
                           · 
                           
                             V 
                             π 
                           
                         
                         π 
                       
                        
                       a 
                        
                       
                           
                       
                        
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               y 
                               I 
                             
                             + 
                             
                               
                                 a 
                                 I 
                               
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                       Q 
                     
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                        
                       a 
                        
                       
                           
                       
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                             ( 
                             
                               
                                 y 
                                 Q 
                               
                               + 
                               
                                 
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                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where the coefficients a I , b I , c I , a Q , b Q , c Q  depend implicitly on γ I  and γ Q . This expression suggests an alternative implementation that requires only two evaluations of a single non-linear function besides the computation of the multivariate polynomials in y I  and y Q . Again, the non-linear functions can be conveniently approximated by spline, i.e. piecewise linear interpolation. In this implementation, only two look-up tables for the function a sin(x), i.e. one for the I- and one for the Q-component, are necessary. 
         [0085]    In practice, the effectiveness of the pre-distortion requires an accurate characterization of the imperfect DPMZM. In particular, the recursive implementation according to Eq. (11) depends directly on γ I  and γ Q  which need to be precisely determined. Also, the polynomial implementation according to Eq. (18) depends on six real coefficients, which themselves are implicitly dependent on γ I  and γ Q  and can be individually tuned. In both cases, the parameters of the DPD can be set during factory calibration. In addition or alternatively, they can be continuously adapted at run time. 
         [0086]      FIG. 10  is an example of a DPMZM device according to a preferred embodiment of the present invention. The DPMZM device comprises, in addition to the DPMZM  10  itself, a calculation unit  52  receiving desired in-phase and quadrature components y I , y Q  for the transmit signal to be generated. The calculation unit  52  is designated as “MZM −1 ” in the figure, because effectively, it represents the inverse operation of the DPMZM  10  to the extent that the underlying model captures the true characteristics and deficiencies of the DPMZM  10 , and to the extent that the algorithm provides an exact solution of the set of coupled equations, depending on the number of iterations and the like. 
         [0087]    The calculation unit  52  outputs the first and second driving voltages V I , V Q , which are converted to analog signals by a DAC  32  and are applied to the first and second inner MZMs  20 ,  22  (not shown in  FIG. 10 ) of the DPMZM  10 . Again, the DC-components of the first and second driving voltages V I , V Q  are lost due to AC-coupling (not shown in  FIG. 10 ) which is compensated by adding corresponding bias components using adders  42  in a manner described with reference to  FIGS. 5 and 6 . Note that for simplicity the bias control units are not shown in  FIGS. 10 to 12 . 
         [0088]    Further shown in  FIG. 10  is a far-end receiver  48  receiving the optical output signal  14  transmitted by the DPMZM  10 . The far-end receiver  48  returns, through a feedback channel  54 , a quality indicator, in the present example an estimated BER, to a parameter calculation unit  56 . The parameter calculation unit  56  adjusts the parameters of the model employed by the calculation unit  52  such as to minimize the estimated BER. The parameters calculated by the parameter calculating unit  52  can for example be the values γ I  and γ Q  characterizing the finite ERs of the first and second inner MZMs  20 ,  22 , or the coefficients a I , b I , c I  and a Q , b Q , c Q  in the polynomial approximation according to Eq. (18), which is why in the embodiment shown in  FIG. 10 , the parameter calculation unit  52  is referred to as “coefficient calculation”. Note, however, that the model parameters employed by the models of the invention and determined by the parameter calculation unit  52  generally do not need to be coefficients but could also be other types of parameters. 
         [0089]      FIG. 11  shows an alternative DPMZM device allowing for adapting the parameters or coefficients by means of what is referred to as an “indirect learning architecture” introduced by C. Fun and E. J. Powers in “ A new Volterra predistorter based on the indirect learning architecture ”, IEEE Transactions on Signal Processing, pp. 223-227, January 1997. 
         [0090]    In  FIG. 11 , the calculation unit receives frequency dependent target values Y I   tgt (f), (short noted as “e(f)”) and in response calculates corresponding driving voltages referred to as “z(f)” for short in  FIG. 11 . 
         [0091]    A local coherent monitoring receiver  58  receives part of the output signal  14  and provides hence the actual in-phase and quadrature components Y I   act (f) and Y Q   act (f). The better the pre-distortion, or, in other words, the closer the model employed by the calculating unit  52  is to the true MZM  10 , the more similar Y 1   act (f), Y Q   act (f) should be to the target values Y I   tgt (f), T Q   tgt , respectively. 
         [0092]    In  FIG. 11 , Y I   tgt  (V Q   act ) and Y I   act  (V Q   act ) are not directly compared. Instead, in the indirect learning architecture of  FIG. 11 , Y I   act  and Y Q   act  are applied to a copy of the calculating unit  52  designated at  60 . The output of this copy  60  are hence driving voltages referred to as “z′(f)” which, if the models underlying the calculation unit  52  and its copy  60  were precisely reflecting the true MZM  10 , should be identical with the applied driving voltages z(f). At a subtracting unit  62 , the difference between z(f) and z′(f) is inputted as a frequency dependent error signal into a parameter calculating unit  64  which adapts the parameter (coefficients) such as to minimize the error. 
         [0093]    Note that the parameter calculation unit  64  allows for introducing artificial offsets y I   off , y Q   off  in y I , y Q  using a further subtractor  62 , meaning that the locations of the QAM states are purposefully shifted in the two-dimensional plane. This can be desirable to better exploit the capabilities of the actual DPMZM. 
         [0094]      FIG. 12  indicates yet an alternative DPMZM device which is similar to that of  FIG. 11 . The main difference is that instead of a copy of the calculation unit  52 , which essentially reflects an inverse model of the DPMZM  10 , this version employs an “inverse” calculation unit  66  which performs the inverse calculation of the calculation unit  52  and can hence be regarded as the direct model of the DPMZM  10 . Note in this regard that of course the calculation units  52 ,  60  and  66  all are based on the same model of the true DPMZM but differ by the corresponding algorithm receiving y I , y Q  as inputs and yielding V I  and V Q  as outputs or vice versa. 
         [0095]    The inverse calculation unit  66  receives the driving voltages (referred to as “z(f)” in  FIG. 12 ) and calculates target signals based thereon, from which actual signals as picked up by the coherent monitoring receiver  58  are subtracted by the subtracting unit  62 . 
         [0096]    The parameter calculation or coefficient adaption as employed in  FIGS. 11 and 12  can be based on a minimization of the mean square error (MSE) between the desired and actual transmit signals. The MSE can be expressed either in the frequency or in the time domain. A general frequency formulation is 
         [0000]      MSE f   =∫w ( f ) E [( Y   I   act ( f )− Y   I   tgt ( f )) 2 +( Y   Q   act ( f )− Y   Q   tgt ( f )) 2   ]df,   (19)
 
         [0000]    where w(f) is the desired weighting function, E[•] denotes stochastic expectation, Y I   tgt (f)+j·Y Q   tgt (f) is the desired transmit signal at frequency f and Y I   act (f)+j·Y Q   act (f) is the actual transmit signal at frequency f, as captured by the monitor receiver  58 . A possible formulation in the time domain is 
         [0000]      MSE t   =E [( y   I   act   −y   I   tgt ) 2 +( y   Q   act   −y   Q   tgt ) 2 ].  (20)
 
         [0097]    To provide the DPD with additional degrees of freedom, it may be advantageous allowing a DC offset on the transmit constellation. In this case 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                   21 
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         [0098]    Note that in  FIGS. 11 and 12 , the calculation units  52  are referred to as “MZM −1 ”, because essentially the calculation unit establishes a model of the actual, imperfect DPMZM  10  but carries out a calculation that inverts its operation. From a more general viewpoint, a DPMZM  10  can be modelled as a memory-free non-linear system with complex input and complex output. To pre-compensate the DPMZM  10 , according to the invention one generally synthesizes the inverse system and then inserts it between the source of the baseband modulating signal and the DPMZM. The desired base-band signals are provided to the inverse system, which then delivers the corresponding pre-distorted complex input to the DPMZM. This generic setup is illustrated in  FIG. 13 . If the inverse system is modelled with sufficient accuracy, the DPMZM  10  returns a close approximation of the desired signal as a response to the pre-distorted input. 
         [0099]    Generally, a memory-free non-linear system can be represented by a look-up table. Accordingly, one could think of synthesizing the inverse DPMZM simply by a look-up table with complex input and complex output. However, such a more straight-forward approach has significant implementation drawbacks. If real and imaginary parts of the complex signals are represented with n bits, respectively, the required look-up table would have 2n input bits and 2n output bits and would require a local storage of (2n)2 2n  bits. This can quickly become a prohibitive size, especially if circuit parallelization is also taken into account, as required to support optical data rates on integrated digital circuits. Instead, employing a model as described above and solving a corresponding set of coupled non-linear equations iteratively or in some other approximated way is a much more efficient way of handling this problem. 
         [0100]    The embodiments described above and the accompanying figures merely serve to illustrate the method according to the present invention, and should not be taken to indicate any limitation of the method. The scope of the patent is solely determined by the following claims. 
       LIST OF REFERENCE SIGNS 
       [0000]    
       
           10  dual parallel Mach-Zehnder-modulator (DPMZM) 
           12  optical input 
           14  optical output 
           16  first arm 
           18  second arm 
           20  first inner MZM 
           22  second inner MZM 
           24  set of electrodes 
           26  set of electrodes 
           28  set of electrodes 
           30  input 
           32  DA-converter 
           34  photo detector 
           36  bias error computer 
           38  error signal 
           40  error signal 
           42  adder 
           44  bias voltage computer 
           46  bias 
           47  bias 
           48  far-end receiver 
           50  residual carrier detector 
           51  error signal 
           52  calculation unit 
           54  feedback channel 
           56  parameter calculation unit 
           58  local coherent monitoring receiver 
           60  copy of the calculating unit  52   
           62  subtracting unit 
           64  parameter calculating unit 
           66  inverse calculation unit 
       
     
       LIST OF ABBREVIATIONS 
     AC Alternating Current 
     ADC Analog-to-Digital Converter 
     BER Bit Error Rate 
     CMOS Complementary Metal-Oxide-Semiconductor 
     DAC Digital-to-Analog Converter 
     DC Direct Current 
     DPD Digital Pre-Distorsion 
     DPMZM Dual Parallel Mach-Zehnder Modulator 
     ER Extinction Ratio 
     MSE Mean Square Error 
       [0132]    I In-phase component 
       MZM Mach-Zehnder Modulator 
     OFDM Orthogonal Frequency Division Multiplexing 
     OSNR Optical Signal-to-Noise Ratio 
       [0133]    Q Quadrature component 
       QAM Quadrature Amplitude Modulation 
       [0000]