Patent Publication Number: US-8121480-B2

Title: Methods and apparatus for recovering first and second transmitted optical waves from a polarization multiplexed optical wave

Description:
BACKGROUND 
     There are many ways to transmit data through an optical link of an optical transmission system. One simple way employs on-off keying, where an optical signal is simply turned “on” or “off” to define a binary data stream. Turning the optical signal “on” and “off” can be viewed as a simple form of amplitude modulation. 
     To improve the spectral efficiency of an optical transmission system, increasingly complex modulation formats may be used. Some of the more complex modulation formats may involve phase modulation or combined amplitude and phase modulation. One of the simplest forms of digital phase modulation is binary phase-shift keying (BPSK). A spectrally more efficient form of digital phase modulation is quadrature phase-shift keying (QPSK). In the QPSK modulation format, a signal&#39;s phase can take four discrete states. Both BPSK and QPSK belong to the same family of digital phase modulation formats, and are particular forms of n-ary phase shift keying. Another family of digital phase modulation formats combines amplitude shift keying (ASK) and phase shift keying (PSK). A subclass of these formats is sometimes denoted as quadrature amplitude modulation (QAM). 
     The spectral efficiency of an optical transmission system may be further improved by using two orthogonal polarization states for simultaneous transmission of two optical signals. This technique is known as polarization multiplexing. Polarization multiplexing doubles the efficiency of optical transmission. 
     In a typical polarization multiplexed optical transmission system, an optical transmitter uses a polarization beam combiner (PBC) to combine two optical waves having orthogonal polarization states. For example, the two optical waves could have horizontal and vertical polarization states. The output of the PBC is therefore an optical wave that carries the digital information of the two combined optical waves. The output optical wave is sometimes referred to as a “polarization multiplexed optical wave”. Of note, the polarization state of the polarization multiplexed optical wave is not constant, but instead varies in response to phase and amplitude changes in either or both of the two combined optical waves. 
     A polarization multiplexed optical wave propagates through an optical link to an optical receiver. However, as a result of birefringence in optical fiber, and because of the dependence of birefringence on environmental factors such as temperature and vibration, the polarization states of the optical waves combined in a polarization multiplexed optical wave change as the waves propagate through an optical link. As a result, when the polarization multiplexed optical wave arrives at an optical receiver, the polarization states of the combined optical waves are different from what they were at the optical transmitter. For example, if the two combined optical waves were transmitted in horizontal and vertical polarization states, it is almost a certainty that their polarization states will not be horizontal and vertical when they arrive at an optical receiver (although their orthogonal relationship will be maintained in the absence of polarization dependent loss). Nonetheless, optical receivers are typically configured to separate a polarization multiplexed optical wave into optical waves having horizontal and vertical polarization states. As a result, the originally combined optical waves are not properly recovered, and a transform is needed to recover the originally combined optical waves. Once the transform is found, it can then be applied to the polarization multiplexed optical wave by means of a polarization controller that aligns the polarization states of the combined optical waves with respect to the principal axes of an optical receiver. Alternately, the transform can be applied to digital representations of the optical waves, as extracted from a polarization multiplexed optical wave by an optical receiver. For example, digital representations of extracted optical waves having horizontal and vertical polarization states may be mathematically transformed into digital representations of the originally combined optical waves. 
     As disclosed by Tsukamoto et al. in “Coherent Demodulation of 40-Gbit/s Polarization-Multiplexed QPSK Signals with 16-GHz Spacing after 200-km Transmission”,  Optical Fiber Communication Conference and Exposition and the National Fiber Optic Engineers Conference , Technical Digest (CD) (Optical Society of America, 2005), paper PDP29, and as disclosed by Tseytlin et al. in “Digital, Endless Polarization Control for Polarization Multiplexed Fiber-optic Communications”,  Optical Fiber Communications Conference,  2003, Vol. 1, p. 103 (Mar. 23-28, 2003), the way to find a transform for recovering originally combined optical waves is to undertake an ‘iterative search’ for the transform. Other disclosures in the art also discuss the need to undertake an ‘iterative search’. However, iteratively searching for a transform can be slow and unreliable. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Illustrative embodiments of the invention are illustrated in the drawings, in which: 
         FIG. 1  illustrates an exemplary polarization multiplexed optical transmission system; 
         FIG. 2  illustrates, on a Poincaré sphere, a great circle of polarization states of a polarization multiplexed optical wave transmitted from an optical transmitter; 
         FIG. 3  illustrates, on a Poincaré sphere, a rotated great circle of polarization states, corresponding to the polarization states of the polarization multiplexed optical wave received at an optical receiver; 
         FIG. 4  illustrates a method of recovering first and second transmitted optical waves from a polarization multiplexed optical wave; 
         FIG. 5  illustrates an exemplary optical receiver; 
         FIG. 6  illustrates, on a Poincaré sphere, clusters of measured polarization states for a polarization multiplexed optical wave composed of QPSK-modulated first and second transmitted optical waves; 
         FIG. 7  illustrates a first exemplary embodiment of the polarization multiplexed optical transmission system shown in  FIG. 1 , for implementing the method shown in  FIG. 4 ; and 
         FIG. 8  illustrates a second exemplary embodiment of the polarization multiplexed optical transmission system shown in  FIG. 1 , for implementing the method shown in  FIG. 4 . 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  illustrates an exemplary polarization multiplexed optical transmission system  100 . In general, the system  100  comprises an optical transmitter  102  and an optical receiver  104 , coupled to one another via an optical link  106 . 
     The optical transmitter  102  transmits a polarization multiplexed optical wave  108  over the optical link  106 . As is the case in polarization multiplexed optical systems, the optical transmitter  102  forms the polarization multiplexed optical wave  108  by combining first and second transmitted optical waves  110 ,  112  having orthogonal polarization states. The optical waves  110 ,  112  are combined using a polarization beam combiner (PBC)  114 . 
     For purposes of this description, the first and second transmitted optical waves  110 ,  112  that are combined to form the polarization multiplexed optical wave  108  are respectively designated as: the horizontal wave, h, and the vertical wave, v. These designations are purely arbitrary and are made solely for the purpose of establishing a frame of reference within a polarization multiplexed optical transmission system  100 . 
     Having respectively designated the first and second transmitted optical waves  110 ,  112  as the horizontal wave, h, and the vertical wave, v, the polarization multiplexed optical wave  108  may be described by the following Jones vector, E: 
                     E   =       (           z   h               z   v           )     =     (             a   h     ⁢     exp   ⁡     (     j   ⁢           ⁢     φ   h       )                     a   v     ⁢     exp   ⁡     (     j   ⁢           ⁢     φ   v       )               )         ,           (   1   )               
where a h  and a v  are the respective amplitudes of the first and second transmitted optical waves  110 ,  112 , and φ h  and φ v  are their respective phases. Both elements, z h  and z v , of the Jones vector, E, are complex. By factoring out √{square root over (a h   2 +a v   2 )}exp(jφ h ), the Jones vector, E, may be rewritten as:
 
                     E   =           a   h   2     +     a   v   2         ⁢     exp   ⁡     (     jφ   h     )       ⁢     (           cos   ⁡     (   α   )                   sin   ⁡     (   α   )       ⁢     exp   ⁡     (   jφ   )               )         ,           (   2   )               
where a h /√{square root over (a h   2 +a v   2 )}=cos(α), a v /√{square root over (a h   2 +a v   2 )}=sin(α), and φ=φ v −φ h .
 
     The normalized Jones vector 
                     (           cos   ⁡     (   α   )                   sin   ⁡     (   α   )       ⁢     exp   ⁡     (   φ   )               )     ⁢           ⁢   of           Eq   .           ⁢     (   2   )                 
describes the polarization state of the optical wave that results from combining the optical waves  110 ,  112 . Polarization state can also be described by the Stokes vector. In this regard, the Jones vector of Eq. (1) can be converted to the Stokes vector, S, as follows:
 
     
       
         
           
             
               
                 
                   S 
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               s 
                               0 
                             
                           
                         
                         
                           
                             
                               s 
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                               - 
                               
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                   ( 
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     The elements or parameters, s 0 , s 1 , s 2 , s 3 , of the Stokes vector, S, represent the power, content of 0° linear light, content of 45° linear light, and content of circular light in an optical wave. For the normalized Jones vector 
                     (           cos   ⁡     (   α   )                   sin   ⁡     (   α   )       ⁢     exp   ⁡     (   φ   )               )     ,           Eq   .           ⁢     (   3   )                 
can be rewritten in the following form:
 
     
       
         
           
             
               
                 
                   S 
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               s 
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                       . 
                     
                   
                 
               
               
                 
                   ( 
                   4 
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     When the power component of the Stokes vector is normalized to “1”, as reflected in Eq. (4), the point (s 1 , s 2 , s 3 ) describes a polarization state on the sphere known as the Poincaré sphere (to be described below). 
     A representative Poincaré sphere  200  is shown in a three-dimensional space in  FIG. 2 . The three-dimensional space is a Cartesian space having three axes for plotting the normalized Stokes vector, or polarization state, of an optical wave. The first of the axes is the H/V axis. It describes the content of 0° linear light in an optical wave. The point “H” designates horizontally-polarized light, and the point “V” designates vertically-polarized light. Polarization states “H” and “V” are orthogonal and appear on opposite sides of the Poincaré sphere  200 . The second of the axes is the +45/−45 axis. It describes the content of 45° linear light in an optical wave. The point “+45” designates +45° linearly polarized light, and the point “−45” designates −45° linearly polarized light. Polarization states +45° and −45° are orthogonal and appear on opposite sides of the Poincaré sphere  200 . The third of the axes is the L/R axis. It describes the content of circular light in an optical wave. The point “L” designates left-circular polarized light, and the point “R” designates right-circular polarized light. Polarization states R and L are orthogonal and appear on opposite sides of the Poincaré sphere. 
     The point (s 1 , s 2 , s 3 ), in  FIG. 2 , designates the Cartesian coordinates of a polarization state defined by Eq. (3). The point (s 1 , s 2 , s 3 ) can alternately be described by the angles α and φ. The angles α and φ define the Jones vector of Eq. (2). On the Poincaré sphere, the angle 2α is an angle of rotation from the 0° linear polarization state “H”, along an equator  202  defined by the plane containing the H/V and +45/−45 axes. The angle φ is an angle of rotation about the H/V axis, beginning from the polarization state defined by the angle 2α. 
     If the first and second transmitted optical waves  110 ,  112  are nearly equal in amplitude, and both are predominantly phase modulated, the Jones vector that describes a polarization state of the polarization multiplexed optical wave  108  (see Eq. (2)) takes the simpler form of: 
                   E   =       (         1             exp   ⁡     (   jφ   )             )     .             (   5   )               
Based on Eq. (4), the parametric description of the trajectory traced on the Poincaré sphere  200  by the polarization state of the polarization multiplexed optical wave  108  described by the Jones vector of Eq. (5) is therefore: (0, cos(φ), sin(φ)). This description is applicable to the polarization multiplexed optical wave  108  at the optical transmitter  102 , where the orthogonal polarization states of the first and second transmitted optical waves  110 ,  112 , respectively, coincide with the points H and V shown in  FIG. 2 . Then, the trajectory of the polarization state of the polarization multiplexed optical wave  108  is a great circle  204  that contains the L and R poles of the Poincaré sphere  200 , as well as a set of ±45° linear polarization states. The polarization states of the first and second transmitted optical waves  110 ,  112  are therefore identified by an axis  206  that is normal to the plane containing the great circle  204 .
 
     In the case of digital phase modulation, the polarization states that define the plane of the great circle  204  may take discrete values. For example, if the phase modulation is quadrature phase shift keying (QPSK), the phase difference, φ=φ v −φ h , between the first and second transmitted optical waves  110 ,  112  defines four discrete polarization states (e.g., states Q 1 -Q 4 ) belonging to the great circle  204 . In the more general case, when the first and second transmitted optical waves  110 ,  112  are modulated by an n-ary phase shift keying, Eq. (4) describes n discrete polarization states that belong to the great circle  204 . 
     Typically, the optical link  106  through which the polarization multiplexed optical wave  108  is transmitted will be an optical fiber (e.g., a fiber-optic cable), but in some cases, the optical link  106  could take other forms, including that of free space. Because of the birefringence of optical fiber, and because of the dependence of birefringence on environmental factors such as temperature and vibration, the polarization states taken by the polarization multiplexed optical wave  108  evolve with time and position as the polarization multiplexed optical wave  108  propagates through the optical link  106 . However, the polarization states of the first and second transmitted optical waves  110 ,  112  remain orthogonal, or at least substantially so, i.e., on the Poincaré sphere  200 , the polarization states of the first and second transmitted optical waves  110 ,  112  are always on opposite sides of the sphere. As a result of the polarization state evolution, the position of the great circle  204  that defines the polarization states of the polarization multiplexed optical wave  108  is typically in a different position when the polarization multiplexed optical wave  108  arrives at the optical receiver  104 . For example, the great circle  204  might move to the position  300  shown in  FIG. 3 . Similarly, the axis  302  defining the orthogonal polarization states of the first and second transmitted optical waves, h and v, is in a new position. Yet, the relationships between 1) the great circle  204  and the axis  206  of  FIG. 2 , and 2) the great circle  300  and the axis  302  of  FIG. 3  remain constant, or at least substantially so (i.e., the polarization states of the polarization multiplexed optical wave  108  are on or near a great circle, and the axis normal to the plane containing the great circle identifies the polarization states of the first and second transmitted optical waves  110 ,  112 ). 
     At the optical receiver  104 , a polarization beam splitter (PBS)  116  splits the polarization multiplexed optical wave  108  into first and second received optical waves  118 ,  120  having orthogonal polarization states. If the principal axes of the optical receiver  104  (e.g., the principal axes of the PBS) are aligned with the polarization states of the first and second transmitted optical waves  110 ,  112 , then the first and second received optical waves  118 ,  120  will be the first and second transmitted optical waves  110 ,  112 . However, because of the birefringence of fiber, and because of the dependence of birefringence on the environmental factors described above, such an alignment rarely exists, and the polarization states of the first and second transmitted optical waves  110 ,  112  are typically rotated with respect to the principal axes of the PBS  116 . This is shown in  FIG. 3 , where the principal axes of the PBS  116  are assumed to be aligned with the H/V axis of the Poincaré sphere  200 , and where the orthogonal polarization states of the first and second transmitted optical waves are designated by the points (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ). The points (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) lie on the axis  302  of the Poincaré sphere  200 . As a result, to recover the first and second transmitted optical waves  110 ,  112 , a transform is needed to align the orthogonal polarization states (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) of the first and second transmitted optical waves  110 ,  112  with the principal axes H and V of the receiver  104 . 
     In light of the need to recover the first and second optical waves at the optical receiver  104 ,  FIG. 4  illustrates an exemplary method  400  for recovering first and second transmitted optical waves  110 ,  112  from a polarization multiplexed optical wave  108  (where the first and second transmitted optical waves have orthogonal polarization states). The method  400  comprises measuring an electrical field of the polarization multiplexed optical wave  108  at an optical receiver  104  (at block  402 ). From the measured electrical field, a plurality of polarization states of the polarization multiplexed optical wave  108  is determined (at block  404 ). From the plurality of polarization states, a transform that aligns the orthogonal polarization states of the first and second transmitted optical waves  110 ,  112  with respect to the principal axes of the optical receiver  104  is estimated (at block  406 ). The first and second transmitted optical waves  110 ,  112  are then recovered by applying the transform to one of i) the polarization multiplexed optical wave  108  and ii) the measured electrical field of the polarization multiplexed optical wave  108 . 
     By way of example, the transform may be applied to the polarization multiplexed optical wave  108  by altering the polarization state of the polarization multiplexed optical wave  108 . Alternately, measuring the electrical field of the polarization multiplexed optical wave  108  may comprise determining a Jones vector that describes the polarization multiplexed optical wave  108 . The transform may then be applied to the measured electrical field of the polarization multiplexed optical wave  108  by applying the transform to the Jones vector. 
     Exemplary ways to implement various aspects of the method  400  will now be described. 
       FIG. 5  illustrates an exemplary optical receiver  500  for measuring various parameters of the optical field of the polarization multiplexed optical wave  108 . By way of example, the optical receiver  500  is a polarization diverse heterodyne receiver using a pair of 90° optical hybrids. 
     The optical receiver  500  comprises a PBS  116  that receives a polarization multiplexed optical wave  108  comprised of first and second transmitted optical waves  110 ,  112  having orthogonal polarization states. The PBS  116  splits the polarization multiplexed optical wave  108  into first and second received optical waves, S TE  and S TM , having orthogonal polarization states, where TE stands for Transverse Electric and TM stands for Transverse Magnetic. This notation is frequently used in connection with planar lightwave circuits to denote a component of the electromagnetic wave that is orthogonal to the plane of the optical structure. In a Cartestian system of coordinates, the z axis can be used to denote the direction of propagation, while the x and y axes are parallel and perpendicular to the planar structure. The linear polarization mode (state) aligned with the x-axis is called TE, and the linear polarization mode (state) aligned with the y-axis is called TM. Because of rotation of the polarization states of the first and second transmitted optical waves  110 ,  112  with respect to the principal axes of the PBS  116 , the first and second received optical waves  118 ,  120  will likely not be the same as the first and second transmitted optical waves  110 ,  112 . Typically, the first and second received optical waves, S TE  and S TM , are linear combinations of the first and second transmitted optical waves  110 ,  112 . Therefore, the first and second received optical waves contain the information needed to recover the first and second transmitted optical waves  110 ,  112 . 
     After being split from the polarization multiplexed optical wave  108  by the PBS  116 , the first received optical wave, S TE , is provided to an input of a first 90° optical hybrid  502 , and the second received optical wave, S TM , is provided to an input of a second 90° optical hybrid  504 . The optical receiver  500  also receives (or generates) a local oscillator signal, labeled LO in  FIG. 5 . The local oscillator signal is provided to each of the first and second 90° optical hybrids  502 ,  504  in a polarization state that is aligned with TE or TM optical waves. In some embodiments, the local oscillator signal may be provided to the 90° optical hybrids  502 ,  504  via a 50/50 splitter  506 . 
     Each of the 90° optical hybrids  502 ,  504  further divides the signal (S TE  or S TM ) into two optical waves whose phases differ by π/2. Similarly, the LO signal which is provided to the optical hybrids  502 ,  504  is also further divided into two optical waves whose phases differ by π/2. The resulting four optical waves can be combined in the four different ways shown below. This process leads to the creation of optical signals that, after detection, result in electrical in-phase (I) and quadrature (Q) signals. It will be shown below that having quadrature signals I and Q allows for reconstruction of the amplitude and phase of the received optical waves S TE  or S TM . The operation of the 90° optical hybrids  502 ,  504  is known to those skilled in the field of optical communications. The optical signals that are created within the 90° optical hybrid  502  are:
 
 I   1   =S   TE   +LO  
 
 I   2   =S   TE   −LO  
 
 Q   1   =S   TE   +jLO  
 
 Q   2   =S   TE   −jLO   (6)
 
     The optical signals that are created within the 90° optical hybrid  504  are:
 
 I   3   =S   TM   +LO  
 
 I   4   =S   TM   −LO  
 
 Q   3   =S   TM   +jLO  
 
 Q   4   =S   TM   −jLO   (7)
 
     The above optical waves are used to illuminate four balanced detectors  508 ,  510 ,  512 ,  514 . In some embodiments, each of the balanced detectors  508 ,  510 ,  512 ,  514  may comprise a pair of photodiodes (e.g., the photodiodes  516 ,  518  of detector  508 ), which photodiodes are connected in series between first and second biasing voltages. Each of the photodiodes is illuminated by a respective optical wave and thereby produces an electrical current that is proportional to the optical intensity of the optical wave. Thus, the optical signals I i  produce an electric current that is proportional to |I i | 2 . Similarly, the optical signals Q i  produce an electrical current that is proportional to |Q i | 2 . Within the four balanced detectors, the produced currents are subtracted in pairs to produce the four electrical signals shown below:
 
 R   1   =|I   1 | 2   −|I   2 | 2   =|S   TE   +LO|   2   −|S   TE   −LO|   2 =4 |S   TE   ∥LO |cos(φ S     TE   −φ LO )
 
 R   2   =|Q   1 | 2   −|Q   2 | 2   =|S   TE   +jLO|   2   −|S   TE   −jLO|   2 =4 |S   TE   ∥LO |sin(φ S     TE   −φ LO )
 
 R   3   =|I   3 | 2   −|I   4 | 2   =|S   TM   +LO|   2   −|S   TM   −LO|   2 =4 |S   TM   ∥LO |cos(φ S     TM   −φ LO )
 
 R   4   =|Q   3 | 2   −|Q   4 | 2   =|S   TM   +jLO|   2   −|S   TM   −jLO|   2 =4| S   TM   ∥LO |sin(φ S     TM   −φ LO )  (8)
 
     The electrical signals R 1 , R 2 , R 3  and R 4  can be further amplified. For simplicity, the responsivity of the photodiodes and any further amplification is omitted from this description. Typically, after digitization, the electrical signals R 1 , R 2 , R 3  and R 4  are combined into complex quantities as shown below:
 
 z   x   =R   1   +jR   2 =4 |S   TE   ∥LO |exp( j (φ S     TE   −φ LO ))
 
 z   y   =R   3   +jR   4 =4 |S   TM   ∥LO |exp( j (φ S     TM   −φ LO ))  (9)
 
     Thus, assuming that the power of the local oscillator signal, LO, is known, the polarization diversity receiver  500  allows for reconstruction of the electrical field of an optical wave at its input, with a precision of an unknown LO phase shift φ LO . Since the unknown phase shift can be removed in the demodulation process, it is correct to rewrite the above equations in a simplified form without loss of generality. Therefore,
 
 z   x   ∝a   x exp( jφ   x )
 
 z   y   ∝a   y exp( jφ   y )  (10)
 
     Based on Eqs. (1)-(3), the measured Jones vector 
                   (           z   x               z   y           )           
represents the electrical field of the polarization multiplexed optical wave  108  at the optical receiver  104  (with a precision of an unknown LO phase shift φ LO ). Furthermore, the measured Jones vector
 
                   (           z   x               z   y           )           
uniquely defines the polarization state of the polarization multiplexed optical wave  108 . Therefore, the polarization diversity receiver  500  can be used as a polarimeter. In particular, the polarization diverse optical receiver  500  can be used as a polarimeter that determines the position of the rotated great circle  300  in  FIG. 3  with respect to the principal axes of the optical receiver  500 .
 
     Furthermore, based on the position of the great circle  300  in  FIG. 3 , a transform that aligns the orthogonal polarization states of the first and second transmitted optical waves  110 ,  112  with respect to the principle axes of the PBS  116  may be estimated. As discussed previously, this may be done by: fitting a plane in Stokes space to the plurality of measured polarization states; identifying a normal to the plane, the normal containing the center of the Poincaré sphere  200 ; and then determining a transform that aligns the normal with respect to the principal axes of the optical receiver  500  (which axes are defined by the axes of the PBS  116 ). 
       FIG. 6  illustrates an exemplary fit of a plane to a plurality of measured polarization states. As shown, the plane intersects the Poincaré sphere  200  to define a circle of polarization states. The circle may in some cases be a great circle  300 , but it need not be. Because a plurality of measured polarization states may exhibit some scatter, the plane may not intersect each and every one of the measured polarization states. The scatter exhibited by the polarization states may result from various factors, such as limited bandwidth of the transmitter  102  and receiver  104  ( FIG. 1 ), electrical and optical noise sources, or environmental factors such as temperature and vibration. It is therefore desirable to fit the plane to a large number of polarization states (e.g., tens to thousands of polarization states), thereby minimizing the effects of scatter on the precision of the fit. It is possible, however, to fit a plane to only three polarization states. It is also possible to fit a plane to only two polarization states and the center of the Poincaré sphere  200 . However, such “small sample size” plane fittings can lead to undesirable loss of precision. 
     Ideally, the polarization states of a particular sample set should not belong to the same cluster. That is, the polarization states should be distributed equally amongst a plurality of locations or regions of a circle, such as the great circle  300 . For example,  FIG. 6  illustrates a distribution of polarization states amongst four “clusters”, with each cluster representing polarization states Q 1 -Q 4 , as might be expected in a system where the first and second transmitted optical waves  110 ,  112  of a polarization multiplexed optical wave  108  have been phase modulated in accord with QPSK. 
     Under ideal conditions, the plane that is fit to the plurality of polarization states intersects the center of the Poincaré sphere  200  and defines a great circle on the Poincaré sphere  200 . However, even if the plane does not contain the center of the sphere  200 , and does not define a great circle, the normal  600  ( FIG. 6 ) to the plane is chosen to contain the center of the sphere  200 . Therefore, the normal  600  intersects the sphere  200  in two points located on opposite sides of the sphere  200 , i.e., in orthogonal polarization states. The axis  302  intersects the sphere  200  at two points, (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ), which points correspond to the rotated polarization states of the first and second transmitted optical waves  110 ,  112 . Because each of the points (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) describes a Stokes vector that is normal to the plane (or great circle  300 ) of the measured polarization states, either of the points may be used to estimate a transform that aligns the orthogonal polarization states of the first and second transmitted optical waves  110 ,  112  with respect to the principal axes of the receiver  500 , as defined by the PBS  116 . For the purpose of the below description, the point (x 1 , y 1 , z 1 ) is used to define the normal  600  to the plane of the great circle  300 . The point (x 2 , y 2 , z 2 ) defines a polarization state that is orthogonal to the polarization state defined by point (x 1 , y 1 , z 1 ), therefore, (x 2 , y 2 , z 2 )=(−x 1 , −y 1 , −z 1 ). 
     Based on Eq. (4), the angles α 1  and φ 1  for the point (x 1 , y 1 , z 1 ) may be found using the following relationships:
 
cos(2α 1 )= x   1  
 
sin(2α 1 )cos(φ 1 )= y   1  
 
sin(2α 1 )sin(φ 1 )= z   1   (11)
 
     Thus,
 
φ 1 =arctan( z   1   /y   1 )
 
α 1 =arctan(√{square root over ( y   1   2   +z   1   2 )}/ x   1 )/2  (12)
 
     If the polarization multiplexed optical wave  108  received by the polarization diversity receiver  500  is described by the Jones vector: 
                     E   o     =           (           z   x               z   y           )               (   13   )               
where z x  and z y  are the complex numbers described in Eq. (10), and a birefringence matrix (B) and a rotation matrix (R) are defined as:
 
                     B   =     (           exp   ⁡     (     -       jφ   1     2       )           0           0         exp   ⁡     (       jφ   1     2     )             )       ⁢     
     ⁢       R   =     (           cos   ⁡     (     α   1     )             sin   ⁡     (     α   1     )                 sin   ⁡     (     -     α   1       )             cos   ⁡     (     α   1     )             )       ,             (   14   )               
then the first and second transmitted optical waves  110 ,  112  may be recovered from the first and second received optical waves  118 ,  120  by multiplying the Jones vector (E 0 ) by the birefringence and rotation matrices, as follows:
 
 E   i   =R·B·E   o .  (15)
 
     The defined transformation converts the Stokes vector (x 1 , y 1 , z 1 ) and the corresponding Jones vector 
                   (           cos   ⁡     (     α   1     )                   sin   ⁡     (     α   1     )       ⁢     exp   ⁡     (     φ   1     )               )           
into the Jones vector
 
               (         1           0         )     .         
Thus, it operates to align the polarization states of the first and second transmitted optical waves  110 ,  112  with horizontal and vertical polarization states described by the Jones vectors
 
     
       
         
           
             
               ( 
               
                 
                   
                     1 
                   
                 
                 
                   
                     0 
                   
                 
               
               ) 
             
             ⁢ 
             
                 
             
             ⁢ 
             and 
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 ( 
                 
                   
                     
                       0 
                     
                   
                   
                     
                       1 
                     
                   
                 
                 ) 
               
               ⁢ 
               
                   
               
               . 
             
           
         
       
     
     A transform such as the one defined by Eq. (15), or an equivalent transform, may be used in various ways to recover the first and second transmitted optical waves  110 ,  112  from the polarization multiplexed optical wave  108 . In some embodiments, the transform that is equivalent to that of Eq. (15) may be applied to the polarization multiplexed optical wave  108  by, for example, configuring a polarization controller  700  placed upstream from the receiver  500 , as shown in  FIG. 7 . The exact settings of the polarization controller  700  depend on its construction. For example, the polarization controller  700  may contain a half-waveplate and a quarter-waveplate, the positions of which are selected to accomplish the polarization state transformation described in Eq. (15). As shown in  FIG. 7 , a control system  702  receives the complex signals z x =R 1 +jR 2  and z y =R 3 +jR 4  generated by the receiver  500 . For n samples of the complex signals z x =R 1 +jR 2  and z y =R 3 +jR 4 , the control system  702  determines the polarization state of the polarization multiplexed optical wave  704 . From the determined polarization states, the control system  702  then estimates a transform such as the one defined by Eq. (15), or an equivalent transform, and provides appropriate control signals to the polarization controller  700 , such that the transform is applied to the polarization multiplexed optical wave  108 . In this manner, the polarization state of the polarization multiplexed optical wave  108  can be altered such that the receiver  500  is caused to separate the first and second transmitted optical waves  110 ,  112  from the polarization multiplexed optical wave  704  (i.e., the receiver is aligned with the first and second transmitted optical waves  110 ,  112  described by Eq. (1)). A phase demodulation stage  706  then receives the complex signals z x ∝a h  exp(jφ h +jφ LO ) and z y ∝a v  exp(jφ v +jφ LO ) and demodulates these electrical representations of the first and second transmitted optical waves  110 ,  112  to extract the data  708  that is encoded in the first and second transmitted optical waves  110 ,  112 . Techniques for demodulating complex signals are known to those skilled in the field of optical communications. 
       FIG. 8  illustrates another way to recover the first and second transmitted optical waves  110 ,  112  from the polarization multiplexed optical wave  108 . In  FIG. 8 , the polarization controller  700  is dispensed with, and the polarization multiplexed optical wave  108  is received directly into the receiver  500 . As shown, a control system  800  receives the complex signals z x =R 1 +jR 2  and z y =R 3 +jR 4  generated by the receiver  500 . For n samples of the complex signals z x =R 1 +jR 2  and z y =R 3 +jR 4 , the control system  800  determines the polarization state of the polarization multiplexed optical wave  108 . From the determined polarization states, the control system  800  then estimates a transform such as the one defined by Eq. (15) (or an equivalent transform). The transform is then applied to the complex signals z x =R 1 +jR 2  and z y =R 3 +jR 4  by means of Eq. (15), where 
               E   0     =             (           z   x               z   y           )     ,             
to yield the complex signals z′ x ∝a h  exp(jφ h +jφ LO ) and z′ y ∝a v  exp(jφ v +jφ LO ), thereby transforming the measured complex signals into the complex signals that represent the first and second transmitted optical waves  110 ,  112 .
 
     In  FIG. 8 , the control system  800  is arbitrarily divided into a portion  802  labeled “Poincaré Sphere Analysis” and a portion  804  labeled “Transformation”. The portion  802  labeled “Poincaré Sphere Analysis” determines the polarization states of the polarization multiplexed optical wave  108  and outputs the angles α 1  and φ 1  defined by Eq. (14). The portion  804  labeled “Transformation” then estimates and applies a transform, based on the angles α 1  and φ 1 , to the complex signals z x =R 1 +jR 2  and z y =R 3 +jR 4  by means of Eq. (15), where 
               E   0     =             (           z   x               z   y           )     ,             
to yield the complex signals z′ x ∝a h  exp(jφ h +jφ LO ) and z′ y ∝a v  exp(jφ v +jφ LO ) that define the first and second transmitted optical waves  110 ,  112 .
 
     A phase demodulation stage  806  then receives the complex signals z x ∝a h  exp(jφ h +jφ LO ) and z y ∝a v  exp(jφ v +jφ LO ) and demodulates these electrical representations of the first and second transmitted optical waves  110 ,  112  to extract the data  808  that is encoded in the first and second transmitted optical waves  110 ,  112 . Techniques for demodulating complex signals are known to those skilled in the field of optical communications. 
     In some embodiments, part or all of the control systems  702 ,  800  shown in  FIGS. 7 &amp; 8  may be implemented using hardware, firmware or software. Hardware implementations may variously comprise a programmed circuit such as a field-programmable gate array (FPGA) or microprocessor. Firmware and software implementations may comprise machine-readable instructions stored on one or more tangible computer-readable media (e.g., a random access memory (RAM), read-only memory (ROM), or a fixed or removable magnetic or optical disk). When read and executed by a processor, the machine-readable instructions cause the processor to implement some or all of the functions of the control system  702  or  800 . 
     The methods and apparatus disclosed herein are useful, in one respect, in that they eliminate the need to iterative search for a transform that aligns the orthogonal polarization states of the first and second optical waves with respect to principal axes of a receiver. Instead, the transform is estimated from measured polarization states of a polarization multiplexed optical wave. Because the iterative searching is eliminated, and because the transform is based on measurements of actual polarization states, the methods and apparatus disclosed herein are capable of estimating a transform more reliably, more quickly, and more precisely than iterative search algorithms. 
     Although exemplary embodiments of the methods and apparatus disclosed herein have been described with respect to an n-ary phase shift keying of first and second transmitted optical waves  110 ,  112 , the methods and apparatus disclosed herein are operable to recover any pair of optical waves having orthogonal linear polarization states, so long as at least one of the optical waves has a modulated phase. With more complex modulation schemes, involving modulation of phase and amplitude, multiple planes or three-dimensional objects can be formed within the three-dimensional Stokes space. However, if the orientation of these three-dimensional objects can be uniquely determined in a three-dimensional space, the disclosed methods and apparatus are still applicable. In addition, the methods and apparatus disclosed herein can be adapted for the recovery of only a single optical wave having an arbitrary phase or amplitude modulation, as described in the following paragraphs. 
     In certain cases, it is possible that the first or second transmitted optical wave  110 ,  112  ( FIG. 1 ) is a zero-amplitude optical wave. In these cases, the PBC  114  effectively transmits a single optical wave over the optical link  106 . The single optical wave propagates through the optical link  106  in a single polarization state that evolves along the link  106  and with time. Thus, upon arrival at the optical receiver  104 , the polarization state of the single optical wave is likely to be misaligned with the principal axes of the receiver  104  (as defined by the PBS  116 ). However, using the disclosed method it is possible to align the polarization state of the single optical wave with respect to an axis of the optical receiver  104 . That is, the optical receiver  104  (or  500 ,  FIG. 5 ) can still measure the electrical field of a received optical wave, even though the received optical wave is no longer really a polarization multiplexed optical wave. The control system  702  or  800  ( FIG. 7  or  8 ) can also determine a plurality of polarization states of the received optical wave. However, instead of the measured polarization states being distributed in clusters about a circle (or great circle  300 ), as shown in  FIG. 6 , the measured polarization states will belong to a single cluster, such as the cluster surrounding the point Q 1 . In such a case, there is no need to fit a plane to the polarization states. Nor is there a need to determine a normal to any plane. This is because the measured polarization states directly correspond to the polarization state of the single optical wave that was transmitted by the optical transmitter  102  (subject, that is, to polarization dependent loss). Instead of fitting a plane and determining a normal, the measured polarization states of a single cluster may be defined by a vector (q 1 , q 2 , q 3 ) representing, for example, the average, mean or center of mass of the cluster. Alternately, it is possible to measure only a single polarization state, although this may fail to provide sufficient precision. 
     For sake of this explanation, it is considered that the average of the polarization states distributed about point Q 1  is simply the point Q 1 . Having identified the polarization state of point Q 1 , a transform that aligns this polarization state with a principal axis of the optical receiver  104  (or  500 ) can be estimated, and the single optical wave transmitted by the optical transmitter  102  can be recovered. Thus, using the apparatus shown in  FIG. 7  or  8 , a zero-amplitude optical wave will also be recovered. However, power measurements for the two recovered waves (one being absent) can be used to determine which of the recovered waves is meaningful. 
     Given the description provided in the above paragraph, it should be clear that the method shown in  FIG. 4  can also be adapted for use outside of a polarization multiplexed optical system  100  ( FIG. 1 ). That is, even in a system that does not employ polarization multiplexing, the electrical field of a received optical wave may be measured; a plurality of polarization states of the received optical wave may be measured; a transform may be estimated; and the transform may be used to recover the optical wave that was actually transmitted.