Abstract:
A polarimeter is proposed that utilizes additional Stokes parameter measurements to determine both an average Stokes vector, as well as any rotation of the state of polarization around the Stokes vector. The optical polarimeter is configured to measure the state of polarization (SOP) under multiple, different conditions that yield both averaged Stokes vector and at least one other secondary (filtered) Stokes vector, the latter thus being determined from a subset of the conditions used to create the average Stokes vector. The secondary Stokes vector created from a filtered input will necessarily exhibit changes over time as a function of polarization transformations (based on filter-dependent changes), while the average Stokes vector will retain a constant value. Thus, a comparison of the average Stokes vector to the changing secondary Stokes vector allows for these polarization-dependent transformations to be recognized.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. Provisional Application Ser. No. 61/798,153, filed Mar. 15, 2013 and herein incorporated by reference. 
    
    
     TECHNICAL FIELD 
     The present invention relates to an optical polarimeter and, more particularly, to a polarimeter that is capable of measuring both an average Stokes vector as well as any rotation of the state of polarization around the Stokes vector. 
     BACKGROUND OF THE INVENTION 
     Polarimeters measure the state of polarization of an input optical signal. Measurement of light polarization and its variation in time is important for many photonic applications, including telecommunications and fiber sensors. The transfer properties of fiber-based optic devices such as isolators, couplers, amplifiers and the like depend on the polarization state in the fiber itself. Thus, to completely characterize these devices, the relationship between the input and output states of polarization (SOP) of the fiber-based system must be known. 
     A conventional method for measuring the SOP of a light beam includes aligning a waveplate and a linear polarizer in the optical path of the beam. The waveplate is rotatable about the optical axis and is typically a quarter-wave plate. An optical sensor, such as a photodetector, is positioned to measure the intensity of light transmitted by the waveplate and polarizer. In operation, the waveplate is sequentially rotated to a minimum of four angular positions about the optical axis relative to the linear polarizer, and the transmitted light intensity is measured at each position by the photodetector. 
     The art has led to the development of in-line fiber polarimeters that utilize a set of four solid-state detectors that absorb only a small, scattered portion of a propagating optical signal, allowing for the remainder of the signal to continue along and impinge each detector in turn. Each detector develops an electrical signal proportional to the polarization-dependent fraction of light that it absorbs from the fiber. The four electrical output signals are then used to determine the four Stokes parameters of light in the fiber via an instrument matrix determined by calibration (at times, referred to as a “calibration matrix”). 
     The four Stokes parameters S 0 , S 1 , S 2  and S 3  (which combine to form the “Stokes vector”) are generally defined as follows: S 0  is the total power, S 1  is the linearly-polarized horizontal component minus the linearly-polarized vertical component, S 2  is the linearly-polarized component at 45° minus the linearly-polarized component at −45°, and S 3  is the right-hand circularly polarized component minus the left-hand circularly polarized component. Thus, a 4×4 calibration matrix C can be developed to define the relationship between a set of four detector output signals D 1 , D 2 , D 3 , D 4  and the four Stokes parameters S 0 , S 1 , S 2 , S 3 : 
                 [           S   0               S   1               S   2               S   3           ]     =       [           C   00           C   01           C   02           C   03               C   10           C   11           C   12           C   13               C   20           C   21           C   22           C   23               C   30           C   31           C   32           C   33           ]     ⁡     [           D   1               D   2               D   3               D   4           ]         ,         
where the Stokes vector S is the sum of the four parameter components.
 
     Classic polarimeters such as those described above are designed to accurately measure the time-averaged state of polarization (SOP) of the light source. When the degree of polarization (DOP) is 100%, the time-averaged polarization will be seen as the instantaneous polarization. For instance, when a signal has a sufficiently narrow linewidth, the time-averaged SOP will be the same as the instantaneous SOP. In most cases, however, optical signals will exhibit a finite bandwidth that extends over a range such that the polarization of the signals varies over their spectral bandwidth. Therefore, at any point in time, the “actual” (instantaneous) SOP may differ from the measured average value. Equivalently, their polarization can vary in time at a rate that is too great for conventional detectors to track. In this case, the DOP will be less than 100%. It has been found that if the original optical signal has a sufficiently large bandwidth, it is possible that the value of the Stokes vector S will vary across the spectrum of the optical signal. In this case, the DOP as measured from the SOP may be less than unity at any given time. 
       FIG. 1  is a Poincare sphere representation of the average Stokes vector S, showing a variation that may exist across the spectrum, where this variation is also referred to as a spectral “string” s of polarization states or, simply, “SOP string”. The set of polarizations forming the SOP string s may also be referred to as the SOP of the signal, in contrast to the “average SOP”. The bold arrow representation of the average Stokes vector S is the frequency/time-averaged vector measured by a standard polarimeter. A measurement of the Stokes vector for an exemplary single frequency component ω is represented as S ω  in  FIG. 1 . It is to be noted that a corresponding string s that varies as a function of time (instead of frequency) may also be depicted for a time domain representation of the Stokes vector, which is derived from a Fourier transformation of the spectral data into temporal data. 
     While a conventional polarimeter can measure the average state of polarization, it cannot measure rotations of the SOP string about the time average SOP. If the polarization state of the propagating optical signal changes as a function of time, it is possible for the averaged Stokes vector S to remain fixed (measured, for example, by using rotating half-waveplates and quarter-waveplates), while the underlying spectral string rotates about this average value.  FIG. 2  is a Poincare sphere representation of this time-dependent characteristic, illustrating the rotation of the exemplary string s through an angle α on the sphere to position s′, while the Stokes vector S remains constant (the shape of the spectral string also remains fixed for the case of  FIG. 2 ). 
     In the case of either a relatively large bandwidth or large time-dependent changes in polarization (or both), information about the rotation of the spectral (and/or temporal) SOP string with respect to the average SOP will not be recognized by a conventional polarimeter, rendering the measurements incomplete. It would be desirable to add a functionality to a polarimeter so as to make it sensitive to the polarization transformations that cause the SOP string to rotate about the average SOP value. 
     Prior art spectral polarimeters have been developed to provide a measurement of the Stokes vector as a function of wavelength for wideband applications. In these arrangements, wavelength filtering (or the use of polarization sensitive elements with sufficient wavelength dispersion) is required to be used so that the polarization properties may be recorded as a function of wavelength. However, such measurements require complex dispersive elements, movable polarization elements, large detector arrays and/or slowly scanning filters, all of which make the resulting device large, slow, costly and complicated. Moreover, such spectral polarimeters are not necessary to provide information with respect to rotations of the SOP about the average SOP. A need remains, therefore, for a more efficient and compact, as well as higher speed, arrangement to provide information regarding the rotation of the SOP string with respect to average SOP. 
     SUMMARY OF THE INVENTION 
     The needs remaining in the prior art are addressed by the present invention, which relates to an optical polarimeter and, more particularly, to a polarimeter that utilizes additional Stokes parameter measurements to determine both an average Stokes vector as well as any rotation of the state of polarization around the average Stokes vector value. 
     In accordance with the present invention, an optical polarimeter is formed that measures the SOP under multiple, different conditions that yield both the conventional average Stokes vector and at least one condition-dependent Stokes vector, allowing for any wavelength-dependent and/or time-dependent variations in the polarization state of a propagating optical signal with respect to the average Stokes vector value to be observed. 
     In one embodiment, a conventional in-line, fiber-based optical polarimeter is enhanced to include a second Stokes measurement arrangement, in one case, a spectral filter used in combination with conventional grating and detector pairs, to create a “condition-dependent” Stokes vector—the “condition” in this case being the filter characteristics. The inclusion of a second Stokes measurement arrangement provides a separate set of measurements for the propagating optical signal that yields a wavelength-dependent view of the Stokes vector (i.e., the spectral “string”). Additionally, if the measurements are made over time, information about changes in the spectral string as a function of time can be calculated and used to define the state of polarization of the propagating signal, as well as a time-dependent function. 
     In another embodiment, a plurality of different filters is utilized, each having a different set of characteristics and thus yielding a different set of measurements. The filters themselves may take any desired form and, in fact, electrical filtering may be used to filter the photodetector output signals prior to calculating the Stokes vectors. Yet another embodiment of the present invention utilizes a single detector/filter pair as the second Stokes measurement arrangement. Thus, broadly speaking, a single measurement may be used to create the second Stokes vector measurement, as long as the single measurement is not parallel to the average Stokes vector measured by the first Stokes measurement arrangement (which would therefore not contain any additional information). 
     While the polarimeter examples presented in this specification describe the utilization of fiber-based gratings to form an in-line polarimeter, it is to be understood that other types of polarimeter arrangements may be used. These arrangements include, but are not limited to, rotating waveplate polarimeters and division-of-amplitude polarimeters. 
     The second Stokes vector measurement arrangement of the inventive polarimeter may be disposed in a separate portion of the optical signal path with respect to the first Stokes vector measurement arrangement (i.e., the conventional polarimeter portion), or disposed at same axial location along the signal as the conventional polarimeter, but spatially separated therefrom (i.e., in a fiber-based polarimeter, the second arrangement is azimuthally separated from the first arrangement). 
     While preferred embodiments comprise an in-line, all-fiber polarimeter, it is possible to form the optical polarimeter of the present invention from discrete elements, if desired. 
     A specific embodiment of the present invention may be defined as an optical polarimeter for identifying polarization transformations in a propagating optical signal, including a first Stokes measurement arrangement for creating an average Stokes vector S defining a state of polarization of an optical signal propagating along an optical signal path and a second Stokes measurement arrangement including at least one filtering element, the second Stokes measurement arrangement configured to create a second Stokes component (which may be a scalar or vector quantity) that is modified by the presence of the at least one filtering element such that the second Stokes component is a subset of the average Stokes vector S, wherein the combination of the average Stokes vector S and the second Stokes vector S′ provides information regarding polarization transformations in the propagating optical signal. 
     Another specific embodiment of the present invention may be defined as a method of measuring polarization transformation of an optical signal comprising the steps of: (1) passing a portion of the optical signal through a first Stokes vector measurement arrangement to measure an average Stokes vector S defining a state of polarization of a propagating optical signal; (2) passing a portion of the optical signal through a second Stokes measurement arrangement including at least one filtering element, the second arrangement providing a second measured Stokes component that is a subset of the average Stokes vector and (3) providing information regarding changes of the averaged Stokes vector S with respect to wavelength and time. 
     Other and further embodiments and aspects of the present invention will become apparent during the course of the following discussion and by reference to the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Referring now to the drawings, 
         FIG. 1  is a Poincare sphere, showing a three-dimensional representation of an averaged Stokes vector, as well as a single Stokes vector measured at a particular frequency, and the associated spectral string, illustrating the movement in the Stokes vector over the associated spectrum; 
         FIG. 2  is a representation similar to that of  FIG. 1 , in this case illustrating the rotation of the spectral string about the (fixed) averaged Stokes vector, where this rotation can occur as a function of time; 
         FIG. 3  is a Poincare sphere representation of the aspects of the present invention associated with creating at least one additional Stokes vector S′, created using a subset of conditions, the representation illustrating the additional information associated with utilizing at least one additional Stokes vector S′; 
         FIG. 4  is an exemplary optical polarimeter formed in accordance with the present invention, using an optical filter to create a second Stokes vector S′; 
         FIG. 5  is an end view of an exemplary embodiment of the present invention, showing a polarimeter having both a first and second Stokes measurement arrangements formed at the same location along the length of the optical fiber circumferentially displaced therefrom; 
         FIG. 6  is a high level diagram depicting an alternative embodiment of the present invention, utilizing two separate filters for forming two different Stokes vectors S′ and S″, for use in analysis with the averaged Stokes vector S; 
         FIG. 7  illustrates another embodiment of the present invention, in this case using only a single grating and associated photodetector as a second Stokes measurement arrangement; 
         FIGS. 8(   a )-( c ) illustrate a set of three different filter characteristics useful in the second Stokes measurement arrangement, in accordance with embodiments of the present invention; 
         FIG. 9  shows yet another embodiment of the present invention, in this case utilizing an electrical filter to form a second Stokes vector S′; 
         FIG. 10  shows a prior art polarization controller arrangement; 
         FIG. 11  is a Poincare sphere representation of the output Stokes vector associated with the prior art arrangement of  FIG. 10 ; 
         FIG. 12  is a diagram of a polarization controller arrangement formed in accordance with the present invention, where the polarimeter performs two separate measurements and the feedback path uses these two measurements to create the control signal for the polarization controller; and 
         FIG. 13  is a Poincare sphere representation of the arrangement of  FIG. 12 . 
     
    
    
     DETAILED DESCRIPTION 
     As mentioned above, the present invention involves measuring the state of polarization (SOP) under multiple, different conditions so as to produce additional Stokes vectors that supplement the conventional average Stokes vector measured by a traditional optical polarimeter.  FIG. 3  illustrates the premise of the present invention in a Poincare sphere representation. As before, the average Stokes vector S and the spectral string s are shown. In accordance with the present invention, a second Stokes vector S′ is measured, perhaps using a filtered subset of wavelengths, the use of a thinner line for S′ being indicative of its subset nature. Since the second measurement is a subset of the averaged Stokes vector S, it will change as the conditions are modified (i.e., polarization transformations are applied to the signal over time, over bandwidth, etc.). This “change” will appear as a rotation of the S′ about S—in the same manner as string s rotates. Therefore, by creating this additional measurement opportunity with S′ (and, perhaps, additional subsets creating additional vectors S″, as shown in  FIG. 3 , S′″, etc.), polarization transformations of the signal that rotate about the average SOP can be detected. The rotation shown by angle α in  FIG. 3 . 
       FIG. 4  illustrates an exemplary optical polarimeter  10  formed in accordance with the present invention that includes a spectral (optical) filter  12  for creating the conditions where a “subset” of the complete average Stokes vector S is created as a second Stokes vector S′ in accordance with the present invention to provide measurements of the spectral string characteristics. The particular embodiment illustrated in  FIG. 4  is an “all-fiber” polarimeter, with an optical signal coupled into a core region  16  of an optical fiber  14 . Optical fiber  14  is selected to exhibit some degree of birefringence to ensure that polarization rotation occurs during the measurement of the average Stokes vector. It is to be understood that the polarimeter of the present invention may alternatively be formed of discrete components. Indeed, this is only one exemplary embodiment of the present invention while others will be described in detail hereinbelow. 
     Referring to  FIG. 4 , optical polarimeter  10  is shown as comprising a first Stokes measurement arrangement  20  that comprises the components found in a conventional fiber-based optical polarimeter. In particular, first Stokes measurement arrangement  20  includes a set of four gratings  22   1 - 22   4  of specific characteristics formed within core region  16  of optical fiber  14  (e.g., each grating is linearly polarized at a different orientation with respect to the optical axis OA). As a propagating optical signal passes through gratings  22   1 - 22   4 , a portion of the beam will be out-coupled and directed into a set of associated photodetector  24   1 - 24   4 , associated with gratings  22  in a one-to-one relationship. The detector output signals D 1 , D 2 , D 3  and D 4  are electrical signals, used to create the average Stokes vector S by employing the 4×4 instrument matrix C in the manner described above and shown below for the sake of completeness: 
                   S   -&gt;     ⁡     (   t   )       =       [             S   0     ⁡     (   t   )                   S   1     ⁡     (   t   )                   S   2     ⁡     (   t   )                   S   3     ⁡     (   t   )             ]     =         [           C   01           C   02           C   03           C   04               C   11           C   12           C   13           C   14               C   21           C   22           C   23           C   24               C   31           C   32           C   33           C   34           ]     ⁡     [             D   1     ⁡     (   t   )                   D   2     ⁡     (   t   )                   D   3     ⁡     (   t   )                   D   4     ⁡     (   t   )             ]       =     C   ⁢           ⁢       D   -&gt;     ⁡     (   t   )               ,   where                     〈       S   -&gt;     ⁡     (   t   )       〉     time     =     C   ⁢       〈       D   -&gt;     ⁡     (   t   )       〉     time         ,   and                   〈       D   1     ⁡     (   t   )       〉     time     =         ∫     -   ∞       +   ∞       ⁢                E   1     ⁡     (   t   )            2     ⁢     ⅆ   t         =       ∫     -   ∞       +   ∞       ⁢                E   1     ⁡     (   ω   )            2     ⁢     ⅆ   ω                 
where E i  is the electrical field at a given detector  24   i . The indicated “time” averages are taken over all time, but can be generalized to a given time interval that is appropriate for the specific devices used as detectors  24 .
 
     In accordance with the present invention, optical polarimeter  10  is formed to further comprise a second Stokes measurement arrangement  30 , used to measure a second Stokes component, in this case a vector S′ that is created based on a subset of the information used to create the average Stokes vector. In this  FIG. 4  embodiment, second Stokes measurement arrangement  30  is formed to include spectral (wavelength) filter  12 , which thus removes some of the wavelength values utilized in the formation of average Stokes vector S and creates a second Stokes vector S′ based upon a subset of wavelength values. 
     Similar to first Stokes measurement arrangement  20 , second Stokes measurement arrangement  30  includes a set of gratings  32   1 - 32   4 , disposed along a portion of core region  16 , which are used to out-couple portions of the propagating optical signal. In accordance with this embodiment of the present invention, the out-coupled signal portions first pass through optical filter  12  before entering a set of photodetectors  34   1 - 34   4 , thus modifying the spectral properties of the signals presented to photodetectors  34 . Optical filter  12 , as will be discussed in more detail below in association with  FIG. 8 , may take a variety of different forms, including a bandpass filter that limits the specific wavelengths that are permitted to pass through and reach detectors  34 . 
     While the embodiment of  FIG. 4  shows Stokes measurement arrangements  20  and  30  as located along separate portions of optical fiber  14  (that is, separated longitudinally along optical axis OA), is it possible that they may be located at essentially the same longitudinal position along fiber  14 , and in this case spatially separated around the perimeter of the fiber (i.e., circumferentially separated). An end view of this circumferentially-separate embodiment is shown in  FIG. 5 . 
     In the case of either longitudinally-separated or circumferentially-separated Stokes measurement arrangements, the presence of optical filter  12  as part of second Stokes measurement arrangement  30  modifies the signals reaching detectors  34  (as compared to the signals reaching detectors  24 ). As a result, the second Stokes vector S′ will be different from, and is actually a subset of, the original average Stokes vector S, where S′ is defined as follows: 
                     S   ′     -&gt;     ⁡     (   t   )       =       [             S   0   ′     ⁡     (   t   )                   S   1   ′     ⁡     (   t   )                   S   2   ′     ⁡     (   t   )                   S   3   ′     ⁡     (   t   )             ]     =         [           C   01           C   02           C   03           C   04               C   11           C   12           C   13           C   14               C   21           C   22           C   23           C   24               C   31           C   32           C   33           C   34           ]     ⁡     [             D   1   ′     ⁡     (   t   )                   D   2   ′     ⁡     (   t   )                   D   3   ′     ⁡     (   t   )                   D   4   ′     ⁡     (   t   )             ]       =     C   ⁢           ⁢         D   ′     -&gt;     ⁡     (   t   )               ,   where                     〈         S   ′     -&gt;     ⁡     (   t   )       〉     time     =     C   ⁢       〈         D   ′     -&gt;     ⁡     (   t   )       〉     time         ,   and                         〈       D   i   ′     ⁡     (   t   )       〉     time     =       ⁢       ∫     -   ∞       +   ∞       ⁢       ⅆ   t     ⁢              ∫     -   ∞       +   ∞       ⁢         ⅆ   ω       2   ⁢   π       ⁢     F   ⁡     (   ω   )       ⁢       E   i     ⁡     (   ω   )       ⁢     ⅇ     ⅈω   ⁢           ⁢   t                2                     =       ⁢       ∫     -   ∞       +   ∞       ⁢         ⅆ   ω       2   ⁢   π       ⁢              F   ⁡     (   ω   )       ⁢       E   i     ⁡     (   ω   )              2                       =       ⁢       ∫     -   ∞       +   ∞       ⁢       ⅆ   t     ⁢              ∫     -   ∞       +   ∞       ⁢       ⅆ     t   ′       ⁢     F   ⁡     (     t   ′     )       ⁢       E   i     ⁡     (     t   -     t   ′       )                2           ,               
where F(ω) is the filter response. As illustrated in  FIG. 3 , as long as S′ is not parallel to S, rotations about S can be measured by measuring changes in S′. In particular, the S′ vector can be broken up into its components that are parallel to and perpendicular to S. The component that is perpendicular to S will give the rotation of the SOP about S.
 
     While the embodiment shown in  FIG. 4  utilizes a set of four detectors in second Stokes measurement arrangement  30 , it is to be understood that additional detectors beyond this set of four may be added to improve the accuracy of the measurement (in general, additional detectors may be added to either arrangement  20  or  30 ). 
     Additional measurements may be also be created by adding a second filter to the configuration, as shown in high level form in the diagram of  FIG. 6 , thus creating a second “subset” of measurements that may be used to observe rotations of the SOP with respect to the average Stokes vector. Upper section I of  FIG. 6  is equivalent to the embodiment of the present invention as shown in  FIG. 4 , and also illustrates the relative positions of average Stokes vector S and second Stokes vector S′ (i.e., the “filtered” version, created from a subset of wavelength values). As discussed above, second Stokes vector S′ will rotate about average Stokes vector S over time, as different wavelengths are impacted by changes in the instantaneous polarization state of a propagating optical signal (this rotation being “lost” in the creation of the conventional average Stokes vector). 
     Lower section II of  FIG. 6  depicts an embodiment of the present invention where a third Stokes measurement arrangement  40  is added to the above-described configuration and includes a second optical filter  42  (having different characteristics than optical filter  12 ). By virtue of using different filtering, a different subset of wavelengths will be involved in the measurements, creating a different Stokes vector S″. The Poincare sphere associated with lower section II illustrates this embodiment, shown the location of both S′ and S″ with respect to average Stokes vector S (where both S′ and S″ are shown as “thinner” vectors, since they are based on a subset of values). Inasmuch as both S′ and S″ were produced by filtering, both will be frequency-dependent. Therefore, the combination of S, S′ and S″ can be used to approximate the complete spectral string s underlying the time-averaged Stokes vector S. 
     While the embodiments of the present invention as described thus far have used a complete set of gratings and detectors in the creation of the second Stokes vector, it is to be understood that fewer grating/detector pairs may also be used and, indeed, an exemplary second Stokes measurement arrangement formed in accordance with the present invention may be used on the utilization of only a single (filtered) measurement (as provided by a single grating/detector pair, for example). 
       FIG. 7  illustrates an exemplary optical polarimeter  100  formed in accordance with this “single measurement” embodiment of the present invention. As shown, optical polarimeter  100  includes a first Stokes measurement arrangement  200  that is similar to arrangement  20  described above. In the configuration of polarimeter  100 , however, a second Stokes measurement arrangement  300  is shown as comprising only a single grating  110 , which is used out-couple a portion of the propagating optical signal and direct it through filter  12  into a photodetector  112 . In this embodiment, photodetector  112  will measure the projection of S′ onto the polarization state associated with grating  110 , this projection denoted as the scalar value S gr . As long as S′ has some component along S gr  (i.e., S′ is not parallel to S gr ), the single-detector embodiment of the present invention will provide some information regarding polarization rotations about S (as long as S′ does not align with the measured polarization). 
     This can be understood since the inverse of the calibration matrix C has row elements that are simply the measurement polarizations, S i , the Stokes vector used in the projection for detector D i . In this case, therefore, the detector value D i  is simply the dot product of a measurement polarization with the input Stokes vector S: 
               D   i     =           ∑   j     ⁢     C     -   1         ⁢     |     i   ⁢           ⁢   j       ⁢     S   j       =       ∑   j     ⁢       S   j             ⁢   i       ⁢     S   j                         D   i   ′     =           ∑   j     ⁢     C     -   1         ⁢     |     i   ⁢           ⁢   j       ⁢     S   j   ′       =       ∑   j     ⁢       S   j             ⁢   i       ⁢       S   j   ′     .                 
It can be shown that rotations about S will create a variation in even one detector output signal D′ i . Therefore, the single detector arrangement of  FIG. 7  is a suitable embodiment to address this situation. For example, consider without loss of generality, the following parameters: S={circumflex over (3)} on the Poincare sphere, with R s  a rotation by angle α around the fixed Stokes vector S, and S′ defined as the Stokes vector associated with the arrangement as shown in  FIG. 7 :
 
                     S   =     [         1           0           0             S   3           ]       ,               S   ′     =     [         1             S   1   ′               S   2   ′               S   3   ′           ]       ,   and                       R   S     =       [         1       0       0       0           0         cos   ⁡     (   α   )             sin   ⁡     (   α   )           0           0         -     sin   ⁡     (   α   )               cos   ⁡     (   α   )           0           0       0       0       0         ]     .           
In the conventional optical polarimeter of the prior art, it was not possible to measure α, the rotation of the spectral string s associated with the Stokes vector S. However, inasmuch as D′ i  is sensitive to this rotation, a measurement can be found in accordance with the teachings of the present invention. That is:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                                         sin 
                                         ⁡ 
                                         
                                           ( 
                                           α 
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         S 
                                         2 
                                         ′ 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           sin 
                                           ⁡ 
                                           
                                             ( 
                                             α 
                                             ) 
                                           
                                         
                                       
                                       ⁢ 
                                       
                                         S 
                                         1 
                                         ′ 
                                       
                                     
                                     + 
                                     
                                       
                                         cos 
                                         ⁡ 
                                         
                                           ( 
                                           α 
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         S 
                                         2 
                                         ′ 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   
                                     S 
                                     3 
                                     ′ 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           1 
                           + 
                           
                             
                               S 
                               1 
                               
                                 
                                     
                                 
                                 ⁢ 
                                 i 
                               
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 α 
                                 ) 
                               
                             
                             ⁢ 
                             
                               S 
                               1 
                               ′ 
                             
                           
                           - 
                           
                             
                               S 
                               2 
                               
                                 
                                     
                                 
                                 ⁢ 
                                 i 
                               
                             
                             ⁢ 
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 α 
                                 ) 
                               
                             
                             ⁢ 
                             
                               S 
                               1 
                               ′ 
                             
                           
                           + 
                           
                             
                               S 
                               3 
                               
                                 
                                     
                                 
                                 ⁢ 
                                 i 
                               
                             
                             ⁢ 
                             
                               S 
                               3 
                               ′ 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     Assume 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     without 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     loss 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     of 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     generality 
                     ⁢ 
                     
                       : 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       S 
                       2 
                       ′ 
                     
                   
                   = 
                   
                     
                       0 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         S 
                         0 
                         
                           
                               
                           
                           ⁢ 
                           i 
                         
                       
                     
                     = 
                     1 
                   
                 
               
             
           
         
       
     
     Thus, D′ i  will depend on rotations about the average Stokes vector S, even if only a single detector is used in the creation of S′, as shown in  FIG. 7 . Moreover, with sufficient calibration of the detectors and polarization optics, this single-detector embodiment can be used to estimate α from a time series of such data. Of course, if S 1   i =S 2   i =0, there will be no variation and a single detector will be insufficient in providing enough data to measure α Obviously, increasing the number of detectors overcomes this problem. 
     It is important to understand that there is no constraint on the shape of the optical filter used to form the additional Stokes vector(s) in the polarimeter of the present invention. For example, the filter may be periodic, quasiperiodic, static, or even vary arbitrarily in value as a function of frequency, ω. In one special case, it may be a periodic filter such as a Fabry-Perot filter with a predetermined Free Spectral Range (FSR) and finesse value, where FSR is defined as the spacing in optical frequency (or wavelength) between two successive reflected or transmitted signals and the finesse is defined as the FSR divided by the FWHM bandwidth. In this case, if the signal is of the order of one FSR, then it can be expected that the detector will see primarily one frequency component and, therefore S′ will be the Stokes vector at that component. In reality, this value will depend on the finesse of the filter but will, in general, not be the same Stokes vector as the frequency (or time) averaged Stokes vector S. 
     An important example of an appropriate use of such a periodic filter is a WDM system where the propagating optical signals are separated by a known channel spacing. In this case, if the FSR is set to the known channel spacing, then F(ω) measures the Stokes vector at a given point within the channel bandwidth. The optical polarimeter in this embodiment therefore operates as an advanced “colorless” (i.e., wavelength-independent) polarimeter with the ability to measure the string rotation angle. 
       FIG. 8  illustrates a variety of other filter forms that may be used in the polarimeter of the present invention.  FIG. 8(   a ) shows a bandpass filter with a step edge falling within the bandwidth of the propagating optical signal.  FIG. 8(   b ) illustrates a periodic filter with a predetermined FSR, and  FIG. 8(   c ) illustrates an arbitrary filter shape. In general, the optical filter used in the polarimeter of the present invention may comprise a thin film etalon, an arrayed waveguide (AWG) splitter, a fiber Bragg grating, a long period grating, a bulk grating, a thin film filter, or the like. 
     In yet another aspect of the present invention, it is also considered beneficial if the filter can be tuned so as to maximize the difference between S and S′. This tuning ensures that the spectrum of the signal overlaps with the filter. Moreover, if it appears that S′ is very close to the average Stokes vector S, the ability to tune the filter to move S′ away from S is useful. The tuning may be accomplished by shifting its wavelength, modifying its bandwidth, or a combination of these parameters. As long as the filter is stable after the tuning is completed, a proper measurement can be made. 
     As previously mentioned, it is possible to replace the use of optical filtering with electrical filtering in the configuration of the second Stokes measurement arrangement. In this case, filtering of the electrical output signals (D) from the detectors is used to create additional Stokes vectors S′, S″, etc., for use providing additional information about the spectral string and rotation (if any) associated with the original average Stokes vector S.  FIG. 9  illustrates an exemplary optical polarimeter  500  of the present invention utilizing electrical filtering to create a second Stokes vector S′. As shown, a first Stokes measurement arrangement  520  of polarimeter  500  remains the same as the “first” Stokes measurement arrangements described above and used to create the average Stokes vector S. Referring to  FIG. 9 , first Stokes measurement arrangement  520  is shown as comprising a set of four gratings  522  that are used in combination with a set of four detectors  524  to measure the average Stokes vector S. 
     In this particular embodiment, a second Stokes measurement arrangement  530  comprises a set of four gratings  532 , with the various scattered signals out-coupled by gratings  532  directly applied as inputs to an associated set of four photodetectors  534 . As long as gratings  532  are oriented in the same manner as gratings  522 , the optical signals received by photodetectors  534  will be essentially the same received by photodetectors  532  and used to create the average Stokes vector S. 
     In accordance with this embodiment of the present invention, a “subset” of the average Stokes vector is created within second Stokes measurement arrangement  530  by utilizing an electrical filter  510 . As shown in  FIG. 9 , electrical filter  510  is positioned at the output of detectors to electrically filter signals D 1 -D 4  and create a set of unique electrical output signals D e1 , D e2 , D e3 , D e4 . Using the same calibration matrix C as before, a unique, filtered Stokes vector S′ will be formed and used to provide the additional string/rotation information as above. Electrical filter  510  (or multiple electrical filters) may be configured in a variety of forms to create the subset of signals used to generate second Stokes vector S′, where filter  510  may comprise a low pass filter, bandpass filter, etc. The filters may be “fixed” in terms of their characteristics, or tunable in the manner as described above to provide the desired separation between the average Stokes vector S and second Stokes vector S′. Indeed, it is contemplated that one embodiment of the present invention may utilize a combination of both optical and electrical filtering to create the additional Stokes vector measurements. 
     It is also contemplated that a polarimeter formed in accordance with the present invention may be utilized to construct an automatic polarization controller with improved performance over prior art automatic polarization controllers.  FIG. 10  shows a prior art polarization control loop arrangement. An input optical signal with varying polarization, denoted OS in . is sent through a polarization controller  600  (e.g., fiber squeezers, or LiNbO 3 ) and then applied as an input to a polarimeter  610  that measures that state of polarization of the output optical signal, OS out . A portion of output optical signal OS out  is tapped off and applied as an input to a feedback element  620 , where feedback element  620  also has as an input a signal S set , which is defined as the polarization “setpoint” of the polarization control loop. Feedback element may be a computer-controlled element. An electrical control signal C is then generated by determining the difference |S out −S set | and applied as an input to polarization controller  600 . Ultimately, when S out =S set , control signal C will have a value of zero, and no further modifications to the polarization controller will be required. 
       FIG. 11  illustrates the output Stokes vector for the arrangement of  FIG. 10  when the loop has stabilized so that S out =S set . As discussed above, if the polarization string S out  rotates about S set  (without any other changes in polarization state), the error signal will not change. That is, the value of S out  will be fixed and these rotations of the polarization string may freely occur even when the loop has stabilized. 
       FIG. 12  shows an improved polarization control loop using the disclosed polarimeter of the present invention. In this embodiment a feedback signal is created that includes the additional measurement S′, which is sensitive to rotations about S. The control loop includes a conventional polarization controller  700 , similar to controller  600  of the prior art arrangement shown in  FIG. 10 , with an input optical signal OS in  with a varying polarization state applied as an input to polarization controller  700 . The output from polarization controller  700  is then passed through a polarimeter  710  formed in accordance with the present invention that performs two separate measurements: S and S′. The additional measurement S′ allows for an error signal to be generated that prevents rotation of the polarization string OS out  about S set . In particular, the control signal C is defined as the quantity |S out −S set |+|S out ′−S set ′I. Thus, when the feedback loop is stabilized not only will the Stokes vector be stabilized, but any rotation about the Stokes vector will also be stabilized. The stabilized output of this feedback loop is shown in  FIG. 13 . It is to be understood that the value of S set  may be fixed, or may be varied as a function of time, depending on the application. 
     Note that this is just an example of a feedback loop. Different error signals, polarization controllers and electronic configurations may be possible. The measurement S′ may not be a full polarization measurement as described above. For instance, if there are only one or perhaps two additional detectors, then the enhanced feedback loop of  FIG. 12  can be of similar speed and complexity to prior art automatic polarization controllers. Feed-forward configurations may also be possible. The intent of this example is to show an enhanced control loop using a multiple measurement polarimeter formed in accordance with the present invention. 
     Indeed, many other alternatives, modifications, and variations of a polarimeter that uses rotations of states of polarization will be apparent to those skilled in the art in light of the above teachings. Of course, those skilled in the art readily recognize that there are numerous applications of the invention beyond those described herein. While the present invention has been described with reference to one or more particular embodiments, those skilled in the art recognize that many changes may be made thereto without departing from the spirit and scope of the present invention. It is therefore to be understood that within the scope of the appended claims and their equivalents, the invention may be practiced otherwise than as specifically described herein.