Abstract:
A method and apparatus for determining the optical parameters of a device under test (DUT) is disclosed. A first portion of an optical signal is modulated to generate a first modulated signal. The first modulated signal is applied to the DUT to output a test signal. A second portion of the optical signal is modulated to create a reference signal. The test signal and reference signal are optically combined into a combined signal. An electrical signal generated from the combined signal is processed to determine at least one optical parameter of the DUT. Processing the electrical signal includes demodulating the electrical signal.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is related by subject matter to U.S. application for patent Ser. No. 11/112,457, (Attorney Docket #10040927-1), entitled “Elementary Matrix Based Optical Signal/Network Analyzer,” which was filed on Apr. 22, 2005. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    Determining the optical characteristics of optical components and networks is an important element to the successful design and operation of optical communications networks. Important characteristics of an optical component or network include (but are not limited to) group delay, differential group delay (DGD), power, and polarization dependent loss (PDL). 
         [0003]    The group delay is a measure of the dispersion of an optical signal traveling through a device under test (DUT). The differential group delay is a measure of dispersion with respect to the polarization of the optical signal traveling through the DUT. Power is a measure of the intensity of the optical signal after traveling through the DUT. PDL is a measure of the polarization state dependent attenuation of the optical signal. 
         [0004]    A traditional way of measuring the group delay and the differential group delay is the phase shift method. In the phase shift method, an intensity-modulated input optical signal is transmitted through a device under test (DUT). The optical signal at the output of the DUT is then directly detected. For example, the phase shift in the intensity modulation between the input and output optical signals provides a measure of the group delay. In the case of the differential group delay, multiple measurements need to be performed for different polarization states of the input optical signal. The phase shift method also allows a measurement of power and PDL from the strength of the received signals. 
         [0005]    The strength of the phase shift method is its immunity to optical phase noise and vibration. However, it requires direct detection of the optical signal from the DUT, which yields electrical signals proportional to the intensity of the received optical signal. Thus, the dynamic range of this method is relatively limited. 
         [0006]    A traditional interferometric heterodyne optical system overcomes the limited dynamic range of the phase shift method. In a traditional heterodyne optical system, a signal from an optical local oscillator is combined with an optical signal from the DUT. As a result, the detected optical signal generates electrical signals that are proportional to √{square root over (I s I LO )}, where I LO  denotes the intensity of the optical local oscillator signal, and I s  denotes the intensity of the optical signal transmitted through the DUT. In this manner, even a weak optical signal from the DUT can be amplified by mixing with a strong optical local oscillator signal. 
         [0007]    However, the previous interferometric heterodyne method needed to measure the amplitude and phase of the generated electrical signals to determine a DUT&#39;s group delay and differential group delay characteristics. This method is reliable so long as those quantities of amplitude and phase are measurable. Unfortunately, DUTs with long optical signal paths (e.g. optical fiber spools) produce electrical output signals with high frequencies and difficult-to-measure phases due to the local oscillator sweep and/or the local oscillator optical phase noise. 
       SUMMARY OF THE INVENTION 
       [0008]    A method and apparatus for determining the optical parameters of a device under test (DUT) is disclosed. A first portion of an optical signal is modulated to generate a first modulated signal. The first modulated signal is applied to the DUT to output a test signal. A second portion of the optical signal is modulated to create a reference signal. The test signal and reference signal are optically combined into a combined signal. An electrical signal generated from the combined signal is processed to determine at least one optical parameter of the DUT. Processing the electrical signal includes demodulating the electrical signal. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0009]      FIG. 1  depicts a system for determining optical characteristics of a DUT, according to embodiments of the present invention. 
           [0010]      FIG. 2  is a block diagram view of the processing unit, according to embodiments of the present invention. 
           [0011]      FIG. 3  shows exemplary signals from within the processing unit. 
       
    
    
     DETAILED DESCRIPTION 
       [0012]      FIG. 1  depicts a system  10  for determining optical characteristics of a DUT  30 , in accordance with embodiments of the present invention. The system includes a laser source  12 , an optical splitter  14 , a test branch  16 , a reference branch  18 , a DUT interface  20 , a coupler  22 , an optical sensor  24 , and a processing unit  28 . For description purposes, the system is connected to a DUT  30  although the DUT is not necessarily a part of the system. 
         [0013]    Referring to  FIG. 1 , the laser source  12  generates an optical signal  32 . In one embodiment, the laser source  12  is a tunable, highly coherent laser that may be continuously swept. During DUT characterization, the optical signal  32  is typically swept across a range of wavelengths (frequencies) in order to characterize the DUT over the range of wavelengths (frequencies). The sweep rate of the laser source  12  is represented by γ. 
         [0014]    The laser source  12  is in optical communication with the optical splitter  14 , which splits the optical signal  32  into a test signal  34  that travels along the test branch  16 , and a reference signal  36  that travels along the reference branch  18 . The reference signal  36  may also be referred to as a local oscillator. In one embodiment, the test and reference branches  16  and  18  are two branches of an interferometer. The test and reference signals  34  and  36  are generated when the optical signal  33  is split by the optical splitter  14 . 
         [0015]    The test branch  16  includes a modulator  40 , a polarization controller  41  and the DUT interface  20 . The DUT interface  20  optically connects the DUT  30  to the system  10 . In the configuration of  FIG. 1 , the DUT interface is intended to include any optical system or mechanism that enables the DUT to be optically connected between the modulator  40  and the coupler  22 . The DUT  30  may be any component having optical characteristics that need to be determined, e.g.: a fiber, a filter, a multiplexer, a demultiplexer, a circulator, tissue samples, etc. The DUT may also be an optical network that is made up of multiple optical components. 
         [0016]    The test branch  16  optically connects the optical splitter  14  to the coupler  22  such that the test signal  34  propagates from the optical splitter  14 , through the modulator  40 , the polarization controller  41 , and the DUT  30 , to the coupler  22 . 
         [0017]    The modulator  40  modulates the test signal  34  at a frequency f 2  and generates a modulated test signal  35  having optical sidebands created at the frequencies v c ±nf 2 , where v o  is the optical frequency and n is the sideband number. The modulator  40  can be any phase, intensity, or polarization modulator. A polarization modulator modulation properties are polarization state dependent. 
         [0018]    The polarization controller  41  controls the polarization state of the optical sidebands in the modulated test signal  35 , generating test signal  37 . The functions of the modulator  40  and the polarization controller  41  may be combined into one component. In one embodiment, the functions of the modulator  40  and the polarization controller  41  are performed by a lithium niobate polarization (LiNbO 3 ) phase modulator whose electro-optic coefficients are different for the TE and TM propagation modes. 
         [0019]    At this point, it may be helpful to the reader&#39;s understanding to see a mathematical representation of the electric field of the test signal  37 , prior to the test signal  37  passing through the DUT  30 . For the sake of simplicity, only the first order sidebands of the test signal  37  (n=1) will be considered. However, it should be noted that other sidebands in the test signal  37  may also be used in characterizing the DUT  30 . In Jones vector notation, the electric field that describes these first order sidebands is: 
         [0000]        E   1 =exp( j 2π v   o   t )(exp(− j 2πf 2   t ) E   o +exp( j 2π f   2   t ) E   0 )   (1) 
         [0000]    where the Jones vector 
         [0000]    
       
         
           
             
               E 
               0 
             
             = 
             
               ( 
               
                 
                   
                     
                       cos 
                        
                       
                         ( 
                         α 
                         ) 
                       
                     
                   
                 
                 
                   
                     
                       
                         sin 
                          
                         
                           ( 
                           α 
                           ) 
                         
                       
                        
                       
                         exp 
                          
                         
                           ( 
                           
                             j 
                              
                             
                                 
                             
                              
                             ϕ 
                           
                           ) 
                         
                       
                     
                   
                 
               
               ) 
             
           
         
       
     
         [0000]    and the angles α and φ uniquely describe on the Poincare sphere the polarization state created by the polarization controller  41 . 
         [0020]    The test signal  37  then passes through the DUT  30 , generating the DUT test signal  39 . The positive and negative first order sidebands of the test signal  37  are perturbed in an opposite way by the DUT  30 , resulting in a phase shift between the positive and negative sidebands. This behavior is described in more detail in U.S. application Ser. No. 11/112,457 which is referenced above. The electric field of the DUT test signal  39  can be described using Jones vector notation as 
         [0000]        E   test =exp( j 2π v   o   t )(exp(− j 2π f   2   t )(1−2π f   2   N ) E   0 +exp( j 2π f   2   t )(1+2π f   2   N ) E   0 )   (2) 
         [0000]    where N is an elementary matrix describing the DUT  30  and comprising elementary perturbations as described in the Appendix. The modulation frequency f 2  for the modulator  40  is selected so that the system  10  is able to detect the resulting phase shift between the positive and negative sidebands. The larger the modulation frequency f 2 , the larger the phase shifts 2πf 2 N as seen in equation (2). In one embodiment, the frequency f 2  is between hundreds of MHz to a few GHz. 
         [0021]    The reference branch  18  of the system  10  includes a modulator  42 . The reference branch  18  optically connects the optical splitter  14  to the coupler  22  such that the reference signal  36  can propagate from the optical splitter  14  through the modulator  42  to the coupler  22 . The modulator  42  modulates the reference signal  36  at a frequency f 0  and generates a modulated reference signal  38 . The modulator  42  is a polarization modulator whose modulation properties are polarization state dependent. The modulation frequency f 0  is typically much smaller than f 2 , typically in the tens or hundreds of kHz. This is to avoid interactions between the optical sidebands in test and reference signals once they are recombined at the coupler  22 . In one embodiment, the modulator  42  is a LiNbO 3  polarization modulator. 
         [0022]    In one embodiment, the modulator  42  also depolarizes the reference signal  36 . Depolarized light (sometimes referred to as “pseudo-depolarized light”) refers to a scenario in which, if all the polarization states of the depolarized light were to be mapped onto a Poincare sphere, they would form a trajectory on the surface of the Poincare sphere, and the average of those polarization states would be the center of the Poincare sphere. In mathematical terms, for a sinusoidal modulation and a simple birefringence modulator, the light is depolarized when the 0 th  Bessel function of the phase difference modulation is 0, i.e.: J 0 (a 0 −b 0 )=0, where a 0  and b 0  denote the modulation depths of the orthogonal linear polarization modes (TE and TM). The significance of depolarizing the reference signal  36  shall be discussed in more detail below. 
         [0023]    The electric field of the modulated reference signal  38  can be described in Jones vector notation as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                     ref 
                   
                   = 
                   
                     
                       exp 
                        
                       
                         ( 
                         
                           
                             j 
                              
                             
                                 
                             
                              
                             2 
                              
                             π 
                              
                             
                                 
                             
                              
                             
                               v 
                               o 
                             
                              
                             t 
                           
                           - 
                           
                             j 
                              
                             
                                 
                             
                              
                             2 
                              
                             πγτ 
                              
                             
                                 
                             
                              
                             t 
                           
                         
                         ) 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               exp 
                                
                               
                                 ( 
                                 
                                   
                                     ja 
                                     o 
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           o 
                                         
                                          
                                         t 
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               exp 
                                
                               
                                 ( 
                                 
                                   
                                     jb 
                                     o 
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       
                                         2 
                                          
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                                          
                                         
                                             
                                         
                                          
                                         
                                           f 
                                           o 
                                         
                                          
                                         t 
                                       
                                       ) 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             1 
                           
                         
                         
                           
                             
                               exp 
                                
                               
                                 ( 
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   ξ 
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where a 0  and b 0  are modulation depths, γ is the laser source sweep rate, ξ is the polarization state parameter, and τ is the interferometer free spectral range (FSR) or delay. The quantity γτ represents the frequency shift between the test and reference branches  16  and  18  due to the sweep of the laser source  12 . 
         [0024]    The coupler  22  optically combines the DUT test signal  39  and modulated reference signal  38  into a combined signal  44  and outputs at least a portion of the combined signal  44  to the optical sensor  24 . The coupler  22  is typically a fiber optic coupler, such as an SM coupler. 
         [0025]    The optical sensor  24  is a square-law detector, and includes at least one photodetector (such as a photodiode) that generates electrical signals in response to the received optical signal. Since it is a square-law detector, the optical sensor  24  produces current i D  proportional to the intensity of the combined optical waves: 
         [0000]        i   D   =|E   test   +E   ref | 2    (4) 
         [0026]    The signal i D  is passed to the processing unit  28  for use in characterizing the DUT  30 . In particular, the characteristics that the processing unit  28  can determine include (but are not limited to): the group delay, differential group delay, loss, and PDL of the DUT  30 . 
         [0027]      FIG. 2  is a block diagram view of the processing unit  28 , according to embodiments of the present invention. The processing unit  28  includes a pre-amplifier  50 , a mixer  52 , a low-pass (LP) filter  54 , an amplitude modulation (AM) demodulator  56 , a squaring operator  57 , and a phase sensitive detector (PSD)  58 . 
         [0028]    The pre-amplifier  50  amplifies the signal i D  to the desired level and generates an amplified signal  64 . Next, the amplified signal  64  is mixed at a mixer  52  with a signal at the frequency f 1 , producing a mixed signal  66 . The mixed signal  66  is sent through a low-pass filter  54 , producing a filtered signal  68  that contains the frequency difference Δf=f 2 −f 1 . The frequencies f 2  and f 1  are selected such that the frequency difference Δf=f 2 −f 1  and γτ±Δf are well within the pass-band of the low-pass filter  54 . The filtered signal  68  is then passed through an AM demodulator  56 , producing a demodulated signal  70  that has all high frequency carrier phase information discarded. 
         [0029]    In the past, the AM demodulation was not performed and the amplitude and phase of the filtered signal  68  was measured directly. As described above, direct measurement works for short DUTs. However, the high-frequency signals and phase noise produced by a long DUT made it difficult to measure those quantities directly.  FIG. 3  shows an exemplary filtered signal  68  and a demodulated signal  70 . As can be seen in  FIG. 3 , when the frequency and phase noise of the filtered signal  68  get too high (as is the case with long DUTs), it becomes impossible to measure frequency and/or phase directly. 
         [0030]    The effect of the AM demodulator  56  is to create a demodulated signal  70  that is essentially just the signal envelope of the filtered signal  68 . The desired characterization information about the DUT  30  can still be extracted from the signal envelope of the filtered signal  68 . The signal envelope is at a frequency Δf=f 2 −f 1 . Thus, even when filtered signal  68  is high-frequency or noisy, it can still be analyzed by the processing unit  28  to characterize the DUT  30 . 
         [0031]    Referring back to  FIG. 2 , the signal  70  is squared by a squaring operator  57  to produce a squared signal  72 . After squaring, the desired characterization information in the squared signal  72  is at a frequency 2Δf. The PSD  58  selects a 2Δf component of the squared signal  72  by mixing it with the reference signal at that frequency and low-pass filtering. The frequency 2Δf is used to select the first sidebands of the optical signal; however, other frequencies could be used if other sidebands were to be selected. For example, the second sidebands can be selected by using the frequency 2 f 1  instead f 1  in the mixer  52 . Then, the envelope of the demodulated signal  70  would contain the frequency 2Δf. The phase shift detection would then be performed, after squaring, at the frequency 4Δf. 
         [0032]    The PSD  58  generates the in-phase (I) and phase-quadrature (Q) components of the squared signal  72 . The I component of the demodulated signal  70  is equal to x(t). The Q component of the squared signal  72  is equal to y(t). By following the procedure described above the closed form solutions can be found for all the signals described above. The details of the derivation are omitted as they will be apparent to one of ordinary skill in the art. The equations for x(t) and y(t) are shown below: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       x 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                     = 
                     
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                       + 
                       
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         [0000]    where ξ describes the polarization state parameter of the modulated reference signal  38  and i, q, u and v are elements of the Stokes vector describing the polarization state of the test signal  37 . 
         [0033]    In one embodiment, the reference signal  36  is depolarized by the modulator  42  to create a depolarized modulated reference signal  38 . In general, the depolarization of the reference signal  36  is not necessary. However, it simplifies the process of characterizing the DUT  30  because when the modulated reference signal  38  is depolarized, the 0 th  order Bessel function J 0  of the series expansion of the in-phase signal x(t) and the quadrature signal y(t) is equal to zero, i.e.: J 0 (a 0 −b 0 )=0. 
         [0034]    Thus, when the modulated reference signal  38  is depolarized, the in-phase component of the phase sensitive detection at the frequency 2Δf is reduced to: 
         [0000]        x ( t )= i    (7) 
         [0035]    The quadrature component of the phase sensitive detection is reduced to: 
         [0000]        y ( t )/2 =ip   0   −qp   1   −up   2   +vp   3    (8) 
         [0000]    where p i  are the sought elementary parameters, and i, q, u and v are elements of the Stokes vector describing the polarization state of the test signal  37 . The intensity of the optical wave, i, is determined from the in-phase demodulated signal in equation (7). The remaining Stokes vector parameters are related to the earlier introduced angles α and φ by the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
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                   = 
                   
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                      
                     
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                                  
                                 
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                                      
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                                   ϕ 
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                   9 
                   ) 
                 
               
             
           
         
       
     
         [0036]    Therefore, by making 4 measurements for four different polarization states (i i ,q i ,u i ,v i ), where the subscript i denotes the number of polarization state, the elementary parameters p 0 , p 1 , p 2 , and p 3  can be determined from a set of four equations each having a form of equation (8). The DUT group delay (GD) is directly determined by p 0 : 
         [0000]      GD=p 0    (10) 
         [0037]    The differential group delay (DGD) of the DUT is determined as follows: 
         [0000]        DGD =√{square root over (p 1   2   +p   2   2   +p   3   2 )}  (11) 
         [0038]    The loss and PDL of the DUT can be determined from the in-phase signal x(t). The process for determining loss and PDL are obvious to one of ordinary skill in the art, and therefore will not be discussed in further detail here. 
       Appendix 
       [0039]    The elementary perturbations represented by N are described by the following matrices. 
         [0040]    The group delay matrix can be represented as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       N 
                       0 
                     
                     = 
                     
                       
                         p 
                         0 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 - 
                                 j 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 - 
                                 j 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where p 0  represents the group delay. 
         [0041]    The 0° linear birefringence matrix can be represented as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       N 
                       1 
                     
                     = 
                     
                       
                         p 
                         1 
                       
                        
                       
                         ( 
                         
                           
                             
                               j 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 - 
                                 j 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where p i  represents the 0° component of the differential group delay. 
         [0042]    The 45° linear birefringence matrix can be represented as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       N 
                       2 
                     
                     = 
                     
                       
                         p 
                         2 
                       
                        
                       
                         ( 
                         
                           
                             
                               0 
                             
                             
                               j 
                             
                           
                           
                             
                               j 
                             
                             
                               0 
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where p 2  represents the 45° component of the differential group delay. 
         [0043]    The circular birefringence matrix can be represented as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       N 
                       3 
                     
                     = 
                     
                       
                         p 
                         3 
                       
                        
                       
                         ( 
                         
                           
                             
                               0 
                             
                             
                               
                                 - 
                                 1 
                               
                             
                           
                           
                             
                               1 
                             
                             
                               0 
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where p 3  represents the circular component of the differential group delay. 
         [0044]    The differential absorption matrix can be represented as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       N 
                       4 
                     
                     = 
                     
                       
                         p 
                         4 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 - 
                                 1 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 - 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where p 4  represents absorption per unit frequency (the frequency derivative of the absorption). 
         [0045]    The differential 0° linear dichroism matrix can be represented as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       N 
                       5 
                     
                     = 
                     
                       
                         p 
                         5 
                       
                        
                       
                         ( 
                         
                           
                             
                               1 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 - 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where p 5  represents the 0° component of the polarization dependent loss frequency derivative. 
         [0046]    The differential 45° linear dichroism matrix can be represented as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       N 
                       6 
                     
                     = 
                     
                       
                         p 
                         6 
                       
                        
                       
                         ( 
                         
                           
                             
                               0 
                             
                             
                               1 
                             
                           
                           
                             
                               1 
                             
                             
                               0 
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where p 6  represents the 45° component of the polarization dependent loss frequency derivative. 
         [0047]    The differential circular dichroism matrix can be represented as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       N 
                       7 
                     
                     = 
                     
                       
                         p 
                         7 
                       
                        
                       
                         ( 
                         
                           
                             
                               0 
                             
                             
                               
                                 - 
                                 j 
                               
                             
                           
                           
                             
                               j 
                             
                             
                               0 
                             
                           
                         
                         ) 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where p 7  represents the circular component of the polarization dependent loss frequency derivative. 
         [0048]    The above definitions differ slightly from those presented by Jones. Instead of thin slices of material as presented by Jones, small increments of optical frequency are considered a preferred embodiment of the present invention. Since the net perturbation can be induced by several optical phenomena, it is represented by a sum of the elementary matrices defined above: 
         [0000]        N=N   0   +N   1   +N   2   +N   3   +N   4   +N   5   +N   6   +N   7 .   (20)