Abstract:
A method of nonlinear polarimetry for measuring higher order moments of the E field of an optical signal is provided. The method includes imposing a phase delay on a first polarization of a received optical signal with respect to a second polarization of the optical signal to produce an intermediate optical signal having a time varying polarization. A polarization of the intermediate optical signal is suppressed. The intermediate optical signal is detected with a plurality of photodetectors, with at least one photodetector configured to be responsive to a nonlinear optical process. Spectra of the photodetector outputs are calculated to determine higher order moments of the E field, and the moments are transformed to obtain the polarization measurement.

Description:
CROSS-REFERENCE TO RELATED APPLICATION  
       [0001]     This Application is a Divisional of prior application Ser. No. 10/413,962 filed on Apr. 15, 2003, currently pending, to Paul S. Westbrook. The above-listed Application is commonly assigned with the present invention and is incorporated herein by reference as if reproduced herein in its entirety under Rule 1.53(b). 
     
    
     TECHNICAL FIELD OF THE INVENTION  
       [0002]     This invention relates to a method and apparatus for measuring the polarization of light.  
       BACKGROUND OF THE INVENTION  
       [0003]     High-speed optical fiber communication systems operate by encoding information (data) onto lightwaves that typically propagate along optical fiber paths. Most systems, especially those used for medium to long distance transmission employ single mode fiber. As implied by the name, single mode fibers propagate only one mode of light below cutoff. The single mode typically includes many communications channels. The communications channels are combined into the one transmitted mode, as by wavelength division multiplexing (WDM) or dense wavelength division multiplexing (DWDM).  
         [0004]     While only one mode is transmitted, that mode actually comprises two perpendicular (orthogonal) polarizations. The polarization of these two components varies undesirably as the waves propagate along a fiber transmission path. The distortion of the optical signals caused by the varying polarization is called polarization mode dispersion (PMD). PMD can be corrected through a combination of measurements of the PMD and the control of active corrective optics.  
         [0005]     Polarimeters measure the polarization of light. Polarimeters can generate signals representing a measured degree of polarization that can be useful for diagnostic purposes. The signals can also be advantageously used for polarization correction using feedback techniques to minimize PMD.  
         [0006]     Polarimeters generally employ one or more photodetectors and related electro-optical components to derive basic polarization data. The raw photodetector signal measurements are typically transformed by mathematical techniques into standard polarization parameters. In the prior art, the photodetector outputs are generally averaged, as by some electronic time constant, and then multiplied as part of the signal processing and transformation process. The problem with averaging at detection is that instantaneous temporal information lost through averaging cannot be retrieved later.  
         [0007]     What is needed for more accurate polarization measurements is a polarimeter that instantaneously measures polarimeter photodetector outputs without averaging, multiplies the unaveraged signals early in signal processing, and then averages and transforms the signals into polarimetry parameters.  
       SUMMARY OF THE INVENTION  
       [0008]     An improved method and apparatus for the measurement of the polarization of light uses nonlinear polarimetry. The higher order moments of the E field are measured and then transformed into standard polarimetry parameters yielding the polarization of the light. In a first embodiment, the light to be measured is transmitted through a rotating retarder capable of rotating at a plurality of angles with at least two retardances Δ. The retarder is optically coupled to a fixed analyzer. The light from the analyzer is then detected by linear and nonlinear photodetectors. The spectra from the detectors is calculated and transformed, to obtain the polarization. In a second embodiment, the light to be measured is received by an optical fiber comprising a plurality of fiber birefringences to retard the light. Polarization sensitive gratings along the length of the fiber scatter the light, and photodetectors detect the scattered light. The signals from the photodetectors can then be transformed to obtain the polarization.  
         [0009]     Apparatus in two preferred embodiments can perform the inventive method. In the first embodiment, nonlinear and linear photodetectors are preceded by a rotating retarder, rotating at a plurality of angles with a retardance, and an analyzer, such as a fixed polarizer. In the second preferred apparatus, a plurality of photodetectors are located adjacent to polarization sensitive gratings situated in a birefringent optical waveguide located between each of the polarization sensitive gratings.  
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0010]     The advantages, nature and various additional features of the invention will appear more fully upon consideration of the illustrative embodiments now to be described in detail in connection with the accompanying drawings. In the drawings:  
         [0011]      FIG. 1  shows an apparatus to perform the method of nonlinear polarimetry;  
         [0012]      FIG. 2  shows an alternative apparatus to perform the method of nonlinear polarimetry using optical fiber and polarization sensitive gratings;  
         [0013]      FIG. 3  shows an apparatus as in  FIG. 1 , including a linear and a nonlinear detector;  
         [0014]      FIG. 4  shows a detector arrangement comprising a linear and a nonlinear detector;  
         [0015]      FIG. 5  shows a first preferred embodiment of an apparatus to perform the method of nonlinear polarimetry; and  
         [0016]      FIG. 6  shows a second preferred optical fiber apparatus to perform the method of nonlinear polarimetry. 
     
    
       [0017]     It is to be understood that the drawings are for the purpose of illustrating the concepts of the invention, and except for the graphs, are not to scale.  
       DETAILED DESCRIPTION  
       [0018]     This description is divided into two parts. Part I describes the inventive method for polarimetry and two embodiments for making polarization measurements according to the inventive method. For those skilled in the art, Part II further develops, defines, and introduces the concepts of invariance, state of polarization and degree of polarization, and the foundation equations governing nonlinear polarimetry as best understood by applicants at the time of the invention.  
         [0000]     Part I: Nonlinear Polarimetry  
         [0019]     Standard polarimeters use linear detectors and thus measure terms quadratic in the E field. These can be considered 2 nd  order moments of the E field and are related to the power and the Stokes parameter of the E-field. A detector measuring intensity squared, though would measure 4 th  order moments of the E field. Such higher order moments can have more information about the E-field. Simply put, a higher order moment of some time varying quantity is simply the time average of a higher power of the quantity. The first power is always just the mean. The second power is the standard deviation and so on.  
         [0020]     Nine moments of the E field can be measured with apparatus  10  as shown in  FIG. 1 . Incoming light  11  is retarded by retarder  12  at angle C, with a retardance of Δ. Analyzer A  13  is a polarizer that precedes the nonlinear detector as represented by photodiode  14 . This apparatus can be accomplished using bulk optics or integrated electro-optical techniques.  
         [0021]     Retarder  12  is an optical component that retards one polarization with respect to the orthogonal polarization. In terms of E fields, the retarder gives one of the polarizations a phase with respect to the other orthogonal E component. Examples are ½ λ or ¼ λ retarders.  
         [0022]     A ¼ λ wave retarder causes a  
       π   2       
 
 delay difference:  
       Δ   =         (         1   4     ⁢   λ     λ     )     ⁢   2   ⁢   π     =     π   2           
 
         [0023]     Similarly, a ½ λ wave retarder causes a delay difference of π:  
       Δ   =         (         1   2     ⁢   λ     λ     )     ⁢   2   ⁢   π     =   π         
 
         [0024]     Here, the retarder  12  is a generic retarder. It has an arbitrary angle and arbitrary phase retardance. The angle C sets the two linear states of polarization on which the phase difference Δ is applied.  
         [0025]     Analyzer  13  is a polarizer. It passes the light of polarization A, and suppresses all other polarizations. By rotating analyzer  13 , light of polarization A is a continuous sampling of all 2π polarizations. A detector viewing the light output of a continuously rotating analyzer registers a periodic waveform. The Fourier spectra of that waveform contains a DC component (near 0), and all other components of the spectra.  
         [0026]     A preferred alternative version of this embodiment rotates retarder  12 , with a fixed analyzer  13  to generate the sine and cosine quadrature components of the Fourier spectra of the detector output. These components yield the nine E field higher order components.  
         [0027]     The response of the nonlinear detector is:  
               V   detector     =     I   optical   2                 =       [              E   x          2     +            E   y          2       ]     2               
 
 A linear detector would measure:  
             I   =       ⁢       1   2     [       S   0     +       [         S   1     ⁢   cos   ⁢           ⁢   2   ⁢   C     +       S   2     ⁢   sin   ⁢           ⁢   2   ⁢   C       ]     ⁢   cos   ⁢           ⁢   2   ⁢     (     A   -   C     )       +                       ⁢         [         S   2     ⁢   cos   ⁢           ⁢   2   ⁢   C     +       S   1     ⁢   sin   ⁢           ⁢   2   ⁢   C       ]     ⁢   sin   ⁢           ⁢   2   ⁢     (     A   -   C     )     ⁢   cos   ⁢           ⁢   Δ     +                       ⁢       S   3     ⁢   sin   ⁢           ⁢   2   ⁢     (     A   -   C     )     ⁢   sin   ⁢           ⁢   Δ     )     ]             
 
         [0028]     The nonlinear detector would measure I 2 , and the filter in the DC electronics would determine an averaging time, as in the linear case:  
         V   detector     =         ∫     T   RC       ⁢     ⅆ     tI   2         =     ⋯   ⁢       〈       S   0     ⁢     S   1       〉       T   RC       ⁢   ⋯           
 
         [0029]     All nine components can be measured if one rotates both analyzer A and retarder C in a manner analogous to the linear Stokes case. By performing measurements at the different sum and difference frequencies proportional to nine linearly independent superpositions of          S i S j           , a 9×9 inversion matrix may then be applied to calculate the nine moments.  
         [0030]     For the nonlinear polarimeter one would toggle between:  
         A   =       π   4     +     ɛ   1         ,     Δ   =       π   4     +       ɛ   2     ⁢           ⁢   where   ⁢           ⁢     ɛ   1     ⁢             ⁢             ⁢   and   ⁢           ⁢     ɛ   2     ⁢             ⁢             ⁢   are   ⁢           ⁢   small             
     and     
       A   =         π   4     ⁢           ⁢   and   ⁢           ⁢   Δ     =     π   4           
 
 and rotate C at a fixed rate. Then the nine moments can be extracted from the nine (quadrature) components: 1 (DC), cos2C, sin2C, cos4C, sin4C, cos6C, cos8C, sin8C. However, it can be advantageous to have more oscillating components, since it is less desirable to measure at a frequency that appears in the DC or non-oscillating response as this would be subject to DC noise. 
 
         [0031]     A static measurement of the moments can also be done with apparatus  20  as shown in  FIG. 2 , but with polarization sensitive gratings  22  and fiber birefringences  23  for the retarder. Here nonlinear detectors  24  detect the light scattered by polarization sensitive gratings  22 . Birefringent optical fiber  23  causes the birefringences. In the limit of weak scattering for each grating, the scattered E-field is the same as in the case of the retarder and the analyzer. As before, there are nine detectors and a resultant 9×9 matrix to connect the detector values to the moments.  
         [0032]     Here each grating with its nonlinear detector  24  will generate an output signal which is proportional to a linear transformation of the Stokes parameters. Each detector  24  signal is linearly related to a Stokes tensor component. Therefore with proper grating  22  alignments, the nine detector  24  outputs have a linear relationship with the nine Stokes tensor components. Gratings  22  are each aligned in different directions. Gratings  22  are each aligned azimuthally about the axis of the optical fiber. Both the grating  22  alignments and birefringences are aligned such that the 9×9 calibration matrix is invertible.  
         [0033]     The measured moments have several uses. The degree of polarization (DOP) is most useful with the Stokes vector because it does not depend on the SOP. That is you can bump the fiber, and the DOP will not change. In other words, the DOP is invariant (see definition of invariant later in Part II) under unitary or lossless transformations. This makes it valuable as a monitoring quantity since a fiber bump does not change it, at least not as much as a bump causes a change in S 1  or S 2 . The higher order moments also have invariants. To understand the invariance of          S i S j           , remember that          S 1 S 2 S 3            is a vector and unitary transformations correspond to a rotation on the Stokes sphere R ij . With the higher order moments then, (         S 0 S 1           ,          S 0 S 2           ,          S 0 S 3           ) transforms as a vector. Therefore their magnitudes are fixed and:  
         ∑   1   3     ⁢       〈       S   0     ⁢     S   i       〉     ⁢     〈       S   0     ⁢     S   i       〉           
 
 is invariant. 
 
         [0034]     But, there are more terms, since          S i S j           =T ij  is a tensor  
         ∑     i   =   1     3     ⁢       ∑     j   =   1     3     ⁢       T   ij     ⁢     T   ij             
 
 is also invariant. 
 
 A proof of this is shown as follows (All duplicate indices are summed from 1 to 3): 
 
 Rotation of the Stokes tensor: T ij ′R im R jn =T mn , invariant:  
             T   ij     ⁢     T   ij       =     T   ij   ′       ⁢               
           R   im     ⁢           ⁢     R   jn     ⁢           ⁢     T   kl   ′       ⁢               
               ⁢         R     k   ⁢           ⁢   m       ⁢           ⁢     R     i   ⁢           ⁢   n         =         T   ij   ′     ⁢     T   kl   ′     ⁢           ⁢     R   im     ⁢     R   mk     -   1       ⁢     R   jn     ⁢     R   nl     -   1         =     
     ⁢         R   mk     -   1       ⁢       R   nl     -   1       ⁡     (     these   ⁢           ⁢   are   ⁢           ⁢     3   ⨯   3     ⁢           ⁢   rotation   ⁢             ⁢             ⁢   matrices     )         ⁢     
     =       ɛ   ij     ⁢           ⁢     T   ij   ′     ⁢           ⁢     T   ij   ′                 
 
 Another invariant is:  
           ∑   1   3     ⁢       T   ii     ⁢     T   ii         ,       
 
 and one can also get invariants from the determinants: det(T ij ), where i,j=1,2,3. Therefore a list of some invariants is:  
           ∑     i   ,     j   =   1       3     ⁢     T   ij   2       ,       ∑     j   =   1     3     ⁢     T     0   ⁢   j     2       ,       ∑     j   =   1     3     ⁢     T   jj   2       ,       
 
 and det(T ij ) where i,j=1,2,3. These invariants can all represent useful monitoring quantities. Since higher moments are usually most interesting when combined with the lower moments to give fluctuations of the E-field, it would be useful to build in the same linear measurement done in normal polarimetry. 
 
         [0035]      FIG. 3  shows an apparatus to accomplish this measurement comprising incoming light  31  retarded by retarder  32  at angle C, with retardance Δ. Analyzer A  33  comprises coupler  36 , and nonlinear and linear photon detector  34  and  35 . The response of detectors is V l =k l I for detector  34 , and V l =k l I 2  for detector  35 . By building four more gratings into the device of  FIG. 3 , for a total of 13 gratings, the averages can be subtracted from higher order moments.  
         [0036]     Using such an embodiment, one can measure aV l   2 −V l , where a is such that when the signal is constant, aV l   2 −V n =0, then V n −aV l   2 ≧0, since intensity fluctuations always make          I 2           &gt;         I           2 . DOP=0 gives the extreme case, since the linear detector is constant in this case.  
         [0037]     An important advantage to having both a linear and nonlinear detector is that the nonlinear detector can be “nonlinearized” by subtracting out the linear part. This is illustrated by  FIG. 4 , where the response to light  41  of detector  34  is V l =CI, and the response of detector  35  is V n =aI 2 +bI. Thus:  
         V     nonlinear   quadratic       =       cV   n     -       bV   1     ⁢       aacI   2     .             
 
 This would allow for lower powers to be used with the nonlinear detector. Of course the noise would still be as large as it is for one detector, but one could extend the nonlinear concept previously discussed and measure the linear and nonlinear moments simultaneously. This embodiment of the invention needs nine nonlinear (quadratic) and four linear detectors. The 13 detectors would have a linear relationship to the 13 linear and quadratic moments as related by a 13×13 matrix. Rotating polarizers or static birefringence can be used. 
 
       Examples  
       [0038]      FIG. 5  shows a first preferred embodiment of the nonlinear polarimeter. Here, rotating retarder  52  receives light  31 . Fixed polarizer  53  is optically coupled to rotating retarder  52  and coupler  36 . Coupler  36  splits the light from fixed polarizer  53  to the two photodetectors, linear detector  34  and nonlinear detector  35 . This embodiment can be accomplished in bulk optics or by using integrated electro-optics fabrication techniques.  
         [0039]      FIG. 6  shows a second preferred embodiment of a nonlinear polarimeter to accomplish static measurement of the moments. Here, fiber  23  receives light  31 . The light from polarization sensitive gratings  22  is detected by four linear detectors  34  and nine nonlinear detectors  35 . Each polarization sensitive gratings  22  has a different scattering angle. Birefringent optical fiber  23  causes the birefringences. In the limit of weak scattering for each grating, the scattered E-field is the same as in the case of specific retarder and the analyzer positions. As before, there are nine detectors and a resultant 9×9 matrix to connect the detector values to the moments. This embodiment can be fabricated with optical fibers and fiber components or by integrated electro-optic fabrication techniques. Here the additional four detector outputs yield a 13×13 calibration matrix. The polarization sensitive gratings&#39;  22  scattering angles and the sections of birefringent optical fiber  23  are set such that the 13×13 calibration matrix is invertible.  
         [0040]     Actual fabrication forms and techniques suitable for constructing the inventive apparatus in general, includes, but is not limited to, bulk optical components, optical fibers and optical fiber components, and integrated techniques, including planer waveguides, and other integrated optical components.  
         [0000]     Part II: Theoretical Development of Nonlinear Polarimetry Including the Definition of Invariance  
         [0041]     Invariance: A polarization transformation is said to be invariant when there is a polarization transformation in which the two principle states are delayed by less than the coherence length of the light. This is an invariant transformation. In mathematical terms:  
         [0042]     ∫dtE 1 (t)E 2 (t+τ c )≠0, τ c =correlation time, E 1 , E 2  are principal states, and τ invariant &lt;&lt;τ c . In short: Invariant=unitary with τ&lt;τ c  where τ is the maximum time delay between polarization components. Also the ratio of the two principle states must remain fixed, i.e., the “fiber touch” cannot be before a large PMD element such as a fiber link, since changing the launch polarization into a fiber with PMD will change the ratio of the two principle states and hence alter the output pulse shape and its higher order moments. The “fiber touch” that we wish to avoid being sensitive to through the use of invariants is that directly before the polarization monitor. With standard polarimeters the only invariants are the total power and the DOP.  
         [0043]     State of polarization and Degree of Polarization: It is useful to provide a clear definition of “state of polarization” (or SOP), with respect to an optical signal propagating through a fiber. In general, if the core-cladding index difference in a given optical fiber is sufficiently small, then the transverse dependence of the electric field associated with a particular mode in the fiber may be written as: 
 
 E ( z,t )= {circumflex over (x)}A   x   exp ( iφ   x )+ ŷA   y   exp ( iφ   y ) 
 
 where A x  and A y  define the relative magnitude of each vector component and the phases are defined as follows: 
 
φ x   =βz−ωt+φ   0 , and 
 
φ y   =βz−ωt+φ   0 −δ, 
 
 where β defines the propagation constant, ω defines the angular frequency, φ 0  defines an arbitrary phase value, and δ is the relative phase difference between the two orthogonal components of the electric field. 
 
         [0044]     In accordance with the teachings of the present invention, the state of polarization (SOP) of an optical fiber will be described using the Jones calculus and the Stokes parameters, since these are both complete and commonly used. The Jones vector J that describes the field at any location z or point in time t is given by the following: 
 
 J= ( A   x   exp ( iφ   x ), A   y   exp ( iφ   y ))= exp ( iφ   x )( A   x   ,A   y   exp (− i δ)). 
 
 In practice, the factor exp(iφ x ) is ignored, so that the state of polarization is described by the three main parameters: A x , A y  and δ. The physical interpretation of these three parameters is most commonly based on the polarization ellipse, which describes the path traced out by the tip of the electric field vector in time at a particular location, or in space at a particular time. It should be noted that the Jones vector description is valid only for monochromatic light, or a single frequency component of a signal. 
 
         [0045]     A more complete description of the state of polarization is based on the defined Stokes parameters, since this method also accounts for the degree of polarization (DOP) of a non-monochromatic signal. In terms of the Jones vector parameters, the four Stokes parameters are defined by: 
 
 S   0   =A   x   2   +A   y   2  
 
 S   1   =A   x   2   −A   y   2  
 
 S   2 =2 A   x   A   y  cos δ
 
 S   3 =2 A   x   A   y  sin δ, 
 
 and the degree of polarization (DOP), 0≦DOP≦1, is defined to be:  
       DOP   =             S   1   2     +     S   2   2     +     S   3   2           S   0       .         
 
 A partially polarized signal can be considered to be made up of an unpolarized component and a polarized component. The DOP is used to define that fraction of the signal which is polarized, and this fraction may be described by either the polarization ellipse or Jones vector. It is to be noted that, in strict terms, there are four parameters that fully describe the elliptical signal: (1) the shape of the ellipse; (2) the size of the ellipse; (3) the orientation of the major axis; and (4) the sense of rotation of the ellipse. Thus, four measurements can unambiguously define the signal. These four parameters are often taken to be A x , A y , the magnitude of δ, and the sign of δ. The four Stokes parameters also provide a complete description of fully as well as partially polarized light. The Jones vector may be derived from the Stokes parameters according to: 
 
 A   x =√{square root over ( S   0   +S   1 )}/√{square root over (2)}
 
 A   y =√{square root over ( S   0   −S   1 )}/√{square root over (2)}
 
δ=arctan( S   3   /S   2 ) 
 
 It is to be noted that the last equation above does not unambiguously determine δ. Most numerical implementations of θ=arctan(x) define the resulting angle such that −π/2&lt;θ&lt;π/2. Thus, for S 2 ≧0, the expression δ=arctan(S 3 /S 2 ) should be used, where as for S 2 &lt;0, the expression δ=arctan(S 3 /S 2 )±π should be used. Therefore, with the knowledge of the four Stokes parameters, it is possible to fully determine the properties of the polarized signal. 
 
         [0046]     It has been recognized in accordance with the teachings of the present invention that the full state of polarization (SOP) cannot be determined by merely evaluating the signal passing through a single polarizer. Birefringence alone has also been found to be insufficient. In particular, a polarimeter may be based on a presumption that the optical signal to be analyzed is passed through a compensator (birefringent) plate of relative phase difference Γ with its “fast” axis oriented at an angle C relative to the x axis (with the light propagating along the z direction). Further, it is presumed that the light is subsequently passed through an analyzer with its transmitting axis oriented at an angle A relative to the x axis. Then, it can be shown that the intensity I of the light reaching a detector disposed behind the compensator and analyzer can be represented by: 
 
 I ( A,C,Γ )=½{ S   0   +S   1 [cos(2 C )cos(2[ A−C ])−sin(2 C )sin(2[ A−C ])cos(Γ)]+ S   2 [sin(2 C )cos(2[ A−C ])+cos(2 C )sin(2[ A−C ])cos(Γ)]+ S   3  sin(2[ A−C ])sin(Γ)}. 
 
 In this case, S j  are the Stokes parameters of the light incident on the compensator, such that S 0  is the incident intensity. If the compensator is a quarter-wave plate (Γ=π/2), then the intensity as defined above can be reduced to: 
 
 I ( A,C,π/ 2)=½{ S   0   +[S   1  cos(2 C )+ S   2  sin(2 C )]cos(2[ A−C ])+ S   3  sin(2[ A−C ])}
 
 whereas if the compensator is removed altogether (Γ=0), the equation for the intensity I reduces to: 
 
 I ( A,−, 0)=½−{ S   0   +S   1  cos(2 A )+ S   2  sin(2 A )}. 
 
 This latter relation illustrates conclusively that it is impossible, without introducing birefringence, to determine the value of S 3 , and hence the sense of rotation of the polarization ellipse. 
 
         [0047]     Following from the equations as outlined above, a polarimeter may be formed using a compensator (for example, a quarter-wave plate), a polarizer, and a detector. In particular, the following four measurements, used in conventional polarimeters, unambiguously characterize the Stokes parameters: 
        1) no wave plate; no polarizer→I(−,−,0)=S 0       2) no wave plate; linear polarizer along x axis→I(0,−,0)=½(S 0 +S 1 )     3) no wave plate; linear polarizer at 45°→I(45,−,0)=½(S 0 +S 2 )     4) quarter-wave plate at 0°; linear polarizer at 45°→I(45,0,π/2)=½(S 0 +S 3 ). 
 
 In a conventional polarimeter using this set of equations, the measurements may be performed sequentially with a single compensator, polarizer and detector. Alternatively, the measurements may be performed simultaneously, using multiple components by splitting the incoming beam of light into four paths in a polarization-independent fashion. 
 
 Nonlinear Polarimeters: 
       
 
         [0052]     Standard polarimeters measure the degree of polarization (DOP), or Stokes parameters that represent the polarization, by taking time averaged measurements of the x and y components of the E-field as represented by: 
 
 S   1   =E   x   E   x   *−E   y   E   y *         
 
 But, higher order moments can be measured as well as: 
 
         E x E x *E x E x *          or          E y E y *E y E y *         
 
         [0053]     A nonlinear polarimeter is a device that measures the higher order moments. These measurements can provide extra information about the bit stream or any polarized or partially polarized signal.  
         [0054]     The number of moments that can be measured can be determined in two ways. The E-field representation as mentioned above is one way:  
         [0000]     Define (m, n) where m=#E x &#39;s and n=#E y &#39;s  
         [0000]     This gives 1×(4,0)+1×(0,4)+2(1,3)+2(3,1)+3(2,2)=9 or 
 
         E x E x *E x E x *                   E y E y *E y E y *                   E x E y *E y E y *                   E y E x *E y E y *                   E x E x *E y E x *                   E x E x *E x E y *                   E x E y *E x E y *                   E y E x *E y E x *                   E x E x *E y E y *         
 
         [0055]     Alternatively, the un-averaged Stokes products S i S j  can be constructed. These are the 2 nd  order moments before averaging: 
 
 S   0   S   1 =( E   x   E   x   *+E   y   E   y *)( E   x   E   x   *−E   y   E   y *) 
 
 They are linear superpositions of the four product E field averages. The independent quantities are: S 0 S 1 , S 0 S 2 , S 0 S 3 , S 1 S 1 , S 2 S 2 , S 3 S 3 ,S 1 S 2 , S 2 S 3 , S 3 S 1 . Again there are nine higher order moments. Note that these are not the same as Stokes parameters: 
 
         S 1 S 2           ≠         S 1                     S 2           
 
 Also, S 0 S 0  is not independent, because before averaging DOP=1, therefore, before time averaging, S 0 S 0 =S 1 S 1 +S 2 S 2 +S 3 S 3 .